text
stringlengths 14
5.77M
| meta
dict | __index_level_0__
int64 0
9.97k
⌀ |
|---|---|---|
Q: How to prevent textview below imageview from getting cut off inside relativelayout? - Android I have an imageview then below it is a textview. These are contained inside a relativelayout. Why is the textview getting cut off?
my_fragment:
<FrameLayout xmlns:android="http://schemas.android.com/apk/res/android"
xmlns:tools="http://schemas.android.com/tools"
android:layout_width="wrap_content"
android:layout_height="wrap_content">
<RelativeLayout
android:layout_width="wrap_content"
android:layout_height="wrap_content"
android:layout_centerInParent="true"
android:layout_gravity="center">
<ImageView
android:layout_width="wrap_content"
android:layout_height="wrap_content"
android:src="@drawable/icon_unique_img"
android:id="@+id/uniqueImg"
android:layout_centerVertical="true"
android:layout_centerHorizontal="true" />
<TextView
android:layout_width="wrap_content"
android:layout_height="wrap_content"
android:text="my multiple line text\ntest"
android:id="@+id/txtTest"
android:textSize="16dp"
android:layout_below="@+id/uniqueImg"
/>
</RelativeLayout>
</FrameLayout>
parent_container.. The fragment is being added by code:
<RelativeLayout
android:id="@+id/content"
android:layout_width="match_parent"
android:layout_height="match_parent"
android:layout_above="@id/myFooter"
android:layout_below="@id/myHeader"
android:gravity="center" >
<android.support.v4.view.ViewPager
android:id="@+id/vpPager"
android:layout_width="match_parent"
android:layout_height="wrap_content">
</android.support.v4.view.ViewPager>
</RelativeLayout>
A: I'd suggest you remove android:layout_centerVertical="true" from the ImageView that is messing up the TextView below and give a padding/margin instead (if you want). Hope this helps!
A: Think this will do.
<FrameLayout xmlns:android="http://schemas.android.com/apk/res/android"
xmlns:tools="http://schemas.android.com/tools"
android:layout_width="match_parent"
android:layout_height="match_parent">
<RelativeLayout
android:layout_width="wrap_content"
android:layout_height="wrap_content"
android:layout_centerInParent="true"
android:layout_gravity="center">
<ImageView
android:layout_width="wrap_content"
android:layout_height="wrap_content"
android:src="@drawable/icon_unique_img"
android:id="@+id/uniqueImg"
android:paddingBottom="20dp"
android:layout_centerVertical="true"
android:layout_centerHorizontal="true" />
<TextView
android:layout_width="wrap_content"
android:layout_height="wrap_content"
android:text="my multiple line text\ntest"
android:id="@+id/txtTest"
android:textSize="16dp"
android:layout_below="@+id/uniqueImg"
/>
</RelativeLayout>
</FrameLayout>
|
{
"redpajama_set_name": "RedPajamaStackExchange"
}
| 3,508
|
module JsonSpec
module Exclusion
extend self
def exclude_keys(ruby)
case ruby
when Hash
ruby.sort.inject({}) do |hash, (key, value)|
hash[key] = exclude_key?(key) ? 1 : exclude_keys(value)
hash
end
when Array
ruby.map{|v| exclude_keys(v) }
else ruby
end
end
def exclude_key?(key)
excluded_keys.include?(key)
end
def excluded_keys
@excluded_keys ||= Set.new(JsonSpec.excluded_keys)
end
end
end
|
{
"redpajama_set_name": "RedPajamaGithub"
}
| 782
|
\section{Introduction
We consider the higher-order KdV-type equation,
\begin{equation}\label{eq:kdv}
\left\{\begin{array}{ll}
\partial_t u + (-1)^{j+1} \partial_x^{2j+1}u + \frac12 \partial_x(u^2)=0, \hspace{1em} &(t,x) \in {\mathbb R} \times {\mathbb T}, \\
u(0,x) = u_0(x), &u_0 \in H^s({\mathbb T}),
\end{array}
\right.
\end{equation}
for $j \in \mathbb{N}$ and $u$ is a real-valued function. Especially, \eqref{eq:kdv} is called KdV and Kawahara equation when $j=1,2$, respectively. These types of equations have conservation laws such as
\begin{align}
M[u]&= \int_{\mathbb{T}} u dx, &\text{(Mean)} \label{eq:conservation laws} \\
E[u] &= \int_{\mathbb{T}} u^2 dx, &\nonumber\\
H[u] &= \int_{\mathbb{T}} \frac{1}{2} \left(\partial_x^j u\right) ^2 - \frac{1}{6} u^3 dx. &\text{(Hamiltonian)} \label{eq:Hamiltonian}
\end{align}
Furthermore, \eqref{eq:kdv} is the Hamiltonian equation with respect to \eqref{eq:Hamiltonian}. In other words, we can rewrite \eqref{eq:kdv} as follows:
\begin{equation*}
u_t = \partial_x \nabla_u H\left(u\left(t\right)\right) = \nabla_{\omega} H\left(u\left(t\right)\right)
\end{equation*}
where $\nabla_u$ is the $L^2$ gradient and $\nabla_{\omega} = \nabla_{\omega_{-\frac{1}{2}}}$ is the symplectic gradient (see \eqref{eq:symplectic form}). These three conservation laws play various roles (in particular, the global behavior) in the study on the partial differential equations. In this paper, we focus on the global well-posedness and the nonsqueezing property of \eqref{eq:kdv} for any $j \le 3$ and $j \le 2$, respectively. Thus, they are importantly used to prove our results as well.\\
\subsection{Global well-posedness}
The local and global well-posedness of \eqref{eq:kdv} were widely studied. For the local well-posedness result, Gorsky and Himonas \cite{Gorsky:2009eg} firstly proved this problem for $s\ge-\frac12$ and Hirayama \cite{Hirayama} improved for $s \ge -\frac{j}{2}$.
Both works are based on the standard Fourier restriction norm method. Hirayama improved bilinear estimate by using the factorization of the resonant function.
The results of the global well-posedness for \eqref{eq:kdv}, when $j=1,2$ were proved by Colliander et al. \cite{CKSTT1} and Kato \cite{Kato}, respectively, via "I-method". In this paper, we extend results of \cite{CKSTT1} and \cite{Kato} for $j\ge3$. The method also basically follows the argument in \cite{CKSTT1} for periodic KdV equation, while some estimates are slightly different. We encountered difficulties in the algebraic factorization of resonant functions. In order to overcome this issue, we use another argument (see Lemma \ref{lem:algebra} below) comparing with Hirayama's proof (Lemma 2.2 in \cite{Hirayama}). Remark that $s =-\frac{j}{2}$ is sharp in the sense that the bilinear estimate in $X^{s,\frac12}$ space fails for $s < -\frac{j}{2}$ (see Theorem 1.4. in \cite{Hirayama}).
The following theorem is one of the main results in this paper:
\begin{theorem}
Let $j \ge 3$ and $s \ge -\frac{j}{2}$. Then \eqref{eq:kdv} is globally well-posed in $H^s({\mathbb T})$
\end{theorem}
\subsection{Nonsqueezing property}\label{subsec:Nonsq prop}
The first contributor of the nonsqueezing property is Gromov \cite{Gromov:1985ww}. He proved the finite dimensional nonsqueezing theorem by using \emph{Darboux width}. Thereafter, Hofer and Zehnder \cite{Hofer:2011vo} developed this to the symplectic capacity. Furthermore, Kuksin \cite{Kuksin:1995ue} introduced an abstract method that the solution map of a given Hamiltonian PDE can be regarded as an approximate symplectic map on the appropriate function space. Concrete examples are presented by Bourgain \cite{Bourgain:1994tr} for the 1D cubic NLS and Colliander et al. \cite{CKSTT3} for the KdV equation. Recently, Roum\'egoux \cite{Roumegoux:2010sn} also proved the nonsqueezing property of the BBM equation and Mendelson \cite{Mendelson:2014vh} proved the nonsqueezing of the Klein-Gordon equation on $\mathbb{T}^3$ via a probabilistic approach. Also, the first author and Kwon \cite{HK2015} obtained the result of nonsqueezing property for the coupled KdV-type system without the Miura transform in the symplectic phase space $H^{-\frac12}({\mathbb T}) \times H^{-\frac12}({\mathbb T})$.
First of all, we introduce the finite dimensional nonsqueezing theorem.
\begin{theorem}[Nonsqueezing property: finite dimensional version \cite{Kuksin:1995ue}]\label{thm:finite dimensional nonsq.}
Let $\mathcal{S}$ be a symplectic map on the $2n$-dimensional phase space. Let $B_R$ and $C_{k,r}$ be a ball of radius $R$ and a cylinder of radius $r$ at $k$-the component, respectively. If
\begin{equation*}
\mathcal{S}\left(B_R\right) \subseteq C_{k,r},
\end{equation*}
then $r \ge R$.
\end{theorem}
Intuitively, Theorem \ref{thm:finite dimensional nonsq.} means that the symplectic map cannot transform any $R$-ball into a hole of $r$-pipe placed in the \emph{basis} direction. To apply Theorem \ref{thm:finite dimensional nonsq.} to \eqref{eq:kdv}, we need a global solution in the phase space with a symplectic form and a symplectic transform. Moreover, we need appropriate truncation of the solution map on the finite dimensional function space.
We firstly find a symplectic form with respect to the given Hamiltonian \eqref{eq:Hamiltonian}. Let $\omega_{-\frac{1}{2}}$ be the symplectic form in $H_0^{-\frac{1}{2}}$ of the form of
\begin{equation}\label{eq:symplectic form}
\omega_{-\frac{1}{2}}\left(u,v\right) := \int_{\mathbb{T}} u \partial_x^{-1} v dx,
\end{equation}
for all $u,v \in H_0^{-\frac{1}{2}}$.
Hence, we can rewrite \eqref{eq:kdv} as follows:
\begin{equation*}
u_t = \nabla_{\omega_{-\frac{1}{2}}} H\left(u\left(t\right)\right)
\end{equation*}
by the following observation
\begin{align*}
\omega_{-\frac{1}{2}}\left(v, \nabla_{\omega_{-\frac{1}{2}}}H\left(u\left(t\right)\right)\right)&:= \left.\frac{d}{d\varepsilon}\right|_{\varepsilon=0} H \left(u+\varepsilon v\right) \\
&= \left. \int \partial_x^j \left(u+ \varepsilon v\right) \cdot \partial_x^j v - \frac{1}{2} \left(u+\varepsilon v\right)^2 v dx \right|_{\varepsilon=0} \\
&= \int \partial_x^j u \cdot \partial_x^j v - \frac{1}{2}u^2v dx \\
&= \int \left[\left(-1\right)^j \partial_x^{2j} u - \frac{1}{2}u^2\right]v dx \\
&= \int \left[\left(-1\right)^{j+1} \partial_x^{2j+1} u + \frac{1}{2}\partial_x\left(u^2\right)\right] \partial_x^{-1} v dx \\
&= \omega_{-\frac{1}{2}} \left(\left(-1\right)^{j+1}\partial_x^{2j+1} u+ \frac{1}{2} \partial_x \left(u^2\right),v\right) \\
&=\omega_{-\frac{1}{2}} \left(v,-\left(-1\right)^{j+1}\partial_x^{2j+1} u- \frac{1}{2} \partial_x \left(u^2\right)\right).
\end{align*}
Since the solution map of \eqref{eq:kdv} is a symplectic transform from $H^{-\frac{1}{2}}_0$ to itself, we can regard the function space and the solution map as the phase space and the symplectic transform, respectively. Remark that the symplectic form does not depend on $j$, so we do not need to consider other symplectic forms or phase spaces for each $j$. With the obtained phase space and the symplectic transform, we state the second main theorem, the nonsqueezing property of \eqref{eq:kdv}.
\begin{theorem}[Nonsqueezing property: analytic version]\label{thm:Nonsqueezing thm}
Let $j \ge 2$, $ 0 < r< R$, $u_* \in H^{-\frac{1}{2}}_0 \left( \mathbb{T}\right)$, $k_0 \in \mathbb{Z}^*(=\mathbb{Z}\setminus \left\{0\right\})$, $z \in \mathbb{C}$ and $T>0$. Then there exists a global $H^{-\frac{1}{2}}_0$-solution $u$ to \eqref{eq:kdv} such that
\begin{equation*}
\left\|u_0 - u_*\right\|_{H^{-\frac{1}{2}}_0} \leq R
\end{equation*}
and
\begin{equation*}
\left|k_0\right|^{-1/2} \left| \mathcal{F}_x\left(S_H\left(T\right)u_0\right) \left(k_0\right) - z\right| >r,
\end{equation*}
where $\mathcal{F}_x$ and $S_H$ are the spatial Fourier transform and the solution map of \eqref{eq:kdv}, respectively.\end{theorem}
The proof of Theorem \ref{thm:Nonsqueezing thm} follows arguments in \cite{Bourgain:1994tr} and \cite{CKSTT3}. In \cite{Bourgain:1994tr}, Bourgain proved the nonsqueezing property of the nonlinear Schr\"odinger equation on $L^2_x\left(\mathbb{T}\right)$ space. After taking the frequency truncation to the original equation, he applied Gromov's nonsqueezing theorem for the finite dimensional Hamiltonian system. From the approximation argument, the result is extended to the infinite dimensional NLS flow. Bourgain used basic (or a sharp) frequency truncation and $X^{s,b}$ space for this argument.
Later, this argument extended by Colliander et al. \cite{CKSTT3} for the KdV flow on its phase space $H^{-1/2}_x\left(\mathbb{T}\right)$ with two more additional ingredients. Firstly, they found a counter example that the sharp truncated flow does not approximate to the original flow. Hence, they used a smooth truncation to resolve this problem. Secondly, they used the Miura transform to close the approximation argument. Indeed, they obtained the approximation result for the KdV equation by using the mKdV approximation result and the bi-continuity of the Miura transform. They proved approximation by truncated flow for mKdV flow and using the bi-continuity of the Miura transform in the some sense, concluded the approximation for the KdV flow.
Like former results, our main tasks are also to find appropriate truncation and prove the approximation argument. We use the sharp truncation like Bourgain's approach. Even if (\ref{eq:kdv}) has the same symplectic phase space and the strength of the nonlinearity as in the KdV equation, much stronger modulation effect than that in the KdV equation facilitates that the finite dimensional system well approximates to the original infinite dimensional system without using the smooth truncation and the Miura transform. We note that with analytic version, the nonsqueezing property tells that the solution flow does not transfer the energy between low and high frequencies on the symplectic manifold, $H^{-\frac{1}{2}}_0$.
From now on, we consider a concrete truncated equation and other objects. Let $P_{\le N}$ be the Fourier projection for the spatial frequency as in \eqref{eq:Fourier multiplier}, we introduce the truncated equation
\begin{equation}\label{eq:truncated equation}
\left\{\begin{array}{ll}
\partial_t u + (-1)^{j+1} \partial_x^{2j+1}u + P_{\le N}\left(\frac12 \partial_x(u^2)\right)=0, \hspace{1em} &(t,x) \in {\mathbb R} \times {\mathbb T}, \\
u(x,0) = u_0(x), &u_0 \in P_{\leq N}H_0^s({\mathbb T}).
\end{array}
\right.
\end{equation}
Denote the nonlinear flow of \eqref{eq:truncated equation} by $S_H^N(t)$. Using \eqref{eq:symplectic form}, we know that \eqref{eq:truncated equation} has the truncated Hamiltonian,
\begin{equation*}
H_N\left(u\left(t\right)\right) := \int_{\mathbb{T}} \frac{1}{2} \left(\partial_x^j u\right) ^2 - P_{\le N}\left(\frac{1}{6} u^3\right) dx.
\end{equation*}
Thus, this flow is the finite dimensional symplectic map, so we can apply Theorem \ref{thm:finite dimensional nonsq.} directly (see Lemma \ref{lem:Nonsqueezing of trun. fow}). Also, the equation (\ref{eq:truncated equation}) is locally and globally well-posed by using the similar argument as in \cite{Hirayama} and Section 3, respectively. In Section 4, we provide the proof of the approximation argument, and hence, we can completely obtain the nonsqueezing property of (\ref{eq:kdv}).
We now restate Theorem \ref{thm:Nonsqueezing thm} geometrically for better understanding. To do this, we may define balls and cylinders. Let $B^{\infty}_r\left(u_*\right)$ be an infinite dimensional ball in ${H^{-1/2}_0}$ of radius $r$ and centered at $u_* \in {H^{-1/2}_0}$ and $C^{\infty}_{k,r}\left(z\right)$ be an infinite dimensional cylinder in ${H^{-1/2}_0}$ of radius $r$ and centered at $z \in \mathbb{C}$:
\begin{align*}
B^{\infty}_r\left(u_*\right) := \left\{u \in {H^{-1/2}_0} : \left\|u-u_*\right\|_{{H^{-1/2}_0}} \leq r\right\}, \\
C^{\infty}_{k,r}\left(z\right) := \left\{u \in {H^{-1/2}_0} : \left|k\right|^{-1/2}\left|\widehat{u}\left(k\right)-z\right| \leq r\right\}.
\end{align*}
The following is the geometric version of Theorem \ref{thm:Nonsqueezing thm} with respect to \eqref{eq:kdv}
\begin{theorem}[Nonsqueezing property: geometric version]
Let $0 < r< R$, $u_* \in H^{-\frac{1}{2}}_0 \left( \mathbb{T}\right)$, $k_0 \in \mathbb{Z}^*$, $z \in \mathbb{C}$ and $T>0$. Then
\begin{equation*}
S_{H} \left(T\right) \left( B^{\infty}_R \left(u_*\right)\right) \not \subseteq C^{\infty}_{k_0,r} \left(z\right),
\end{equation*}
where $S_H$ be the solution map of \eqref{eq:kdv} when $j>1$.
\end{theorem}
\subsection{Notations}
We clear some terminologies for our results. We use the spatial Fourier transform, the inverse Fourier transform and the space-time Fourier transform as follows:
\begin{align*}
\mathcal{F}_x \left(u\right) =\widehat{u}\left(k\right) &= \int _{\mathbb{T}} e^{-ikx}u\left(x\right) dx, \\
u \left(x \right) &= \frac{1}{2\pi}\int_{{\mathbb Z}} e^{ikx}\widehat{u}\left(k\right) dk:=\frac{1}{2 \pi}\sum_{k \in \mathbb{Z}} \widehat{u}\left(k\right) e^{ikx},\\
\mathcal{F}\left(u\right) = \wt{u}\left(\tau,k\right) &= \iint_{\mathbb{T} \times \mathbb{R}} e^{-ikx}e^{-i\tau t} u\left(x,t\right) dxdt.
\end{align*}
We have the spatial Sobolev space
\begin{equation}\label{eq:H^s space}
\left\|u\right\|_{H^s} = \left\|\left<k\right>^s \widehat{u}\right\|_{\ell^2_k} := \frac{1}{\left(2\pi\right)^{1/2}} \left(\sum_{k \in \mathbb{Z}} \left<k\right>^{2s}\left|\widehat{u}\right|^2\right)^{1/2}
\end{equation}
for $s \in \mathbb{R}$, where $\left<k\right> = \left(1+\left|k\right|^2\right)^{1/2}$. For each dyadic number $N$, we define the Fourier multipliers,
\begin{equation}\label{eq:Fourier multiplier}
\begin{split}
\widehat{P_Nu}\left(k\right) := 1_{N \leq \left|k\right| <2N}\left(k\right) \widehat{u}\left(k\right), \\
\widehat{P_{\leq N} u}\left(k \right) := 1_{\left|k\right| \leq N}\left(k\right) \widehat{u}\left(k\right), \\
\widehat{P_{\geq N} u}\left(k \right) := 1_{\left|k\right| \geq N}\left(k\right) \widehat{u}\left(k\right),
\end{split}
\end{equation}
where $1_{\Omega}$ is a characteristic function on $\Omega$. By the mean preserving \eqref{eq:conservation laws} and the Galilean transform, we have the mean zero function space with the same norm as in (\ref{eq:H^s space}) as follows:
\begin{equation*}
H^s_0 = \left\{u \in H^s : \int_{\mathbb{T}} u =0 \right\}.
\end{equation*}
We define the general $X^{s,b}$ norm associated to \eqref{eq:kdv},
\begin{equation*}\label{eq:X^sb norm}
\left\|u\right\|_{X^{s,b}} = \left\|\left<k\right>^s\left<\tau-k^{2j+1}\right>^b \widetilde{u} \right\|_{L^2_{\tau}\ell^2_k}.
\end{equation*}
Using this, we define $Y^s$ and $Z^s$ spaces for the solution and nonlinear term under the norms
\[\norm{f}_{Y^s} = \norm{f}_{X^{s,\frac12}} + \norm{\bra{k}^s\wt{f}}_{\ell_k^2L_{\tau}^1},\]
\[\norm{f}_{Z^s} = \norm{f}_{X^{s,-\frac12}} + \norm{\bra{k}^s\bra{\tau - k^{2j+1}}^{-1}\wt{f}}_{\ell_k^2L_{\tau}^1}.\]
For $x, y \in {\mathbb R}_+$, $x \lesssim y$ denotes $x \le Cy$ for some $C >0$ and $x \sim y$ means $x \lesssim y$ and $y \lesssim x$. Using this, we denote $f = O(g)$ by $f \lesssim g$ for positive real-valued functions $f$ and $g$. Moreover, $x \ll y$ denotes $x \le cy$ for small positive constant $c$. Let $a_1,a_2,a_3 \in {\mathbb R}$ and $b_1,b_2,b_3,b_4 \in {\mathbb R}$. The quantities $a_{max} \ge a_{med} \ge a_{min}$ can be defined to be the maximum, median and minimum values of $a_1,a_2,a_3$, respectively, Also, $b_{max} \ge b_{sub} \ge b_{thd} \ge b_{min}$ can be defined similarly as before.
The paper is organized as follows: In Section \ref{sec:bi-,tri-}, we give algebraic results for the resonant functions, and prove the bi- and trilinear estimates for the global well-posedness and the nonsqueezing property. In Section \ref{sec:global}, we prove the global well-posedness of \eqref{eq:kdv}. In Section \ref{sec:nonsqueezing}, we prove the nonsqueezing property of the solution flow of \eqref{eq:kdv} by showing the approximation argument between the original and truncated flows.
\textbf{Acknowledgement.} The authors would like to thank their advisor Soonsik Kwon for his helpful comments and encouragement through this research problem. The authors are partially supported by NRF (Korea) grant 2015R1D1A1A01058832.
\section{Bi- and Trilinear estimates}\label{sec:bi-,tri-}
In this section, we will prove some algebraic analysis, bi- and trilinear estimates which are useful tools to prove the global well-posedness and nonsqueezing property of \eqref{eq:kdv} in section \ref{sec:global} and \ref{sec:nonsqueezing}, respectively. We first observe some algebraic analysis results.
\begin{lemma}\label{lem:algebra}
Let $j \in {\mathbb N}$.
{\rm{(a)}} If $x,y,z \in {\mathbb R}$ with $x+y+z=0$. Then we have
\begin{equation}\label{eq:algebra1}
P_3(x,y,z) = x^{2j+1} + y^{2j+1} + z^{2j+1} = xyz\cdot Q_3(x,y,z),
\end{equation}
where $|Q_3(x,y,z)| \sim \max(|x|,|y|,|z|)^{2j-2}$.
{\rm{(b)}} If $x,y,z,w \in {\mathbb R}$ with $x+y+z+w=0$. Then we have
\begin{equation}\label{eq:algebra2}
P_4(x,y,z,w) = x^{2j+1} + y^{2j+1} + z^{2j+1} +w^{2j+1}= (x+y)(x+z)(x+w)\cdot Q_4(x,y,z,w),
\end{equation}
where $|Q_4(x,y,z,w)| \sim \max(|x|,|y|,|z|,|w|)^{2j-2}$.
\end{lemma}
\begin{proof}
(a) can be obtained by the similar argument for (b) (or see \cite{Hirayama}). Hence, we only prove the second part of Lemma \ref{lem:algebra}. We may assume that $|x| \ge |y| \ge |z| \ge |w|$ without loss of generality. If at least one of $x+y$, $x+z$ and $x+w$ is zero, we can easily see $x^{2j+1} + y^{2j+1} + z^{2j+1} +w^{2j+1} = 0$ and thus it suffices to show $|Q_4(x,y,z,w)| \sim \max(|x|,|y|,|z|,|w|)^{2j-2}$.
\textbf{Case I.} $|x|\sim|y|\gg|z|$. From $x+y+z+w =0$, we may assume that $x>0$ and $-y >0$. Then, \eqref{eq:algebra2} is equivalent that for $x,y >0$,
\[x^{2j+1} - y^{2j+1} + z^{2j+1} +w^{2j+1}= (x-y)(x+z)(x+w)\cdot Q_4'(x,y,z,w),\]
where $x-y+z+w=0$ and $|Q_4| = |Q_4'|$. By the mean value theorem (MVT), we have
\begin{equation}\label{eq:calculation1}
\frac{x^{2j+1} - y^{2j+1} + z^{2j+1} +w^{2j+1}}{x-y} = (x^{\ast})^{2j} + \frac{z^{2j+1} +w^{2j+1}}{x-y},
\end{equation}
for some $y < x^{\ast} < x$. For the rest term of the right-hand side of \eqref{eq:calculation1}, from the following identity
\begin{align*}
\frac{z^{2j+1} +w^{2j+1}}{x-y}&= - \frac{z^{2j+1} +w^{2j+1}}{z+w} \\
&=- (z^{2j} - z^{2j-1}w + \cdots -zw^{2j-1}+w^{2j}),
\end{align*}
and $|z| \ll |x|$, we have
\[\left| \frac{z^{2j+1} +w^{2j+1}}{x-y} \right| \ll |x|^{2j}.\]
Hence, we conclude that
\[\left|\frac{x^{2j+1} - y^{2j+1} + z^{2j+1} +w^{2j+1}}{x-y} \right| \sim |x|^{2j},\]
which implies $|Q_4'| \sim |x|^{2j-2}$ from $|x+z|,|x+w| \sim |x|$.
\textbf{Case II.} $|x| \sim |z| \gg |w|$. From $x+y+z+w =0$, we may assume that $x>0$ and $-y,-z >0$. Moreover, we have $|x+y|,|x+z| \sim |x|$ and $|x^{2j+1} + y^{2j+1} + z^{2j+1} +w^{2j+1}| \sim |x|^{2j+1}$ , and thus $|Q_4| \sim |x|^{2j-2}$.
\textbf{Case III.} $|x|\sim|w|$. We may assume that $x,w >0$ and $-y,-z>0$. Then, similarly as before, \eqref{eq:algebra2} is equivalent that for $x,y,z,w >0$,
\[x^{2j+1} - y^{2j+1} - z^{2j+1} +w^{2j+1}= (x-y)(x-z)(x+w)\cdot Q_4'(x,y,z,w),\]
where $x-y-z+w=0$ and $|Q_4| = |Q_4'|$. Using the MVT twice, we can obtain that
\begin{align*}
\frac{x^{2j+1} - y^{2j+1} - z^{2j+1} +w^{2j+1}}{(x-y)(x-z)} &= \frac{(x^{\ast})^{2j} - (z^{\ast})^{2j}}{x-z}, \hspace{1.5em} y < x^{\ast} < x, ~ w < z^{\ast} < z\\
&= (x^{\ast\ast})^{2j-1}, \hspace{1.5em} z^{\ast} < x^{\ast\ast} < x^{\ast}.
\end{align*}
Hence, we conclude from $|x+w| \sim |x|$ that $|Q_4'| \sim |x|^{2j-2}$.
\end{proof}
Now, we state the $L^4$-Strichartz estimate which is a useful tool to prove bi- and trilinear estimates.
\begin{lemma}\label{lem:strichartz}
Let $j \in \mathbb{N}$. For any function $u \in {\mathbb T} \times {\mathbb R}$, we have the $L^4$-Strichartz estimate for \eqref{eq:kdv}
\begin{equation}\label{eq:strichartz}
\norm{u}_{L_{t,x}^4} \lesssim \norm{u}_{X^{0,\frac{j+1}{2(2j+1)}}}.
\end{equation}
In particular, we have $\norm{u}_{L_{t,x}^4} \lesssim \norm{u}_{X^{0,\frac13}}$.
\end{lemma}
\begin{proof}
This type estimate was first introduced by Bourgain \cite{Bourgain1993} associated to the Schr\"odinger and the KdV equations. Moreover, one can also find the comment for Lemma \ref{lem:strichartz} in \cite{Bourgain1995}. The proof of this lemma is almost similar as in \cite{Bourgain1993} and hence, we omit the detailed proof. We also refer \cite{Tao1} and \cite{Tao2} for the proof.
\end{proof}
From now on, let us consider the bi- and trilinear estimates which are the main results in this section. We already know the bilinear estimate proved by Hirayama in \cite{Hirayama} as follows:
\begin{proposition}[Hirayama \cite{Hirayama}]
Let $j \in {\mathbb N}$ and $s \ge -j/2$. Then, the following bilinear estimate holds:
\begin{equation}\label{eq:bilinear1}
\norm{{\mathcal{F}}^{-1}[\bra{\tau - k^{2j+1}}^{-1}\wt{\partial_x uv}]}_{Z^s} \lesssim \norm{u}_{Y^s}\norm{v}_{Y^s}.
\end{equation}
\end{proposition}
However, for our analysis, refined estimates of \eqref{eq:bilinear1} are needed. The following lemma will be used to prove the global well-posedness.
\begin{lemma}\label{lem:bilinear2}
Let $j \in {\mathbb N}$ and $s \ge -j/2$. Let $u_i = P_{N_i}u$ and $|k_i|\sim N_i\ge1$, $i=1,2,3$. Then we have
\begin{equation}\label{eq:bilinear2}
\norm{P_{N_3}\partial_x(u_1u_2)}_{X^{s,-\frac12}} \lesssim (N_1N_2)^{-\frac12}N_3^{s+\frac12}N_{max}^{1-j}\norm{u_1}_{X^{0,\frac12}}\norm{u_2}_{X^{0,\frac12}}.
\end{equation}
\end{lemma}
\begin{proof}
Due to the total derivative in the left-hand side of \eqref{eq:bilinear2}, we may assume that $k_3 \neq 0$. Let $\lambda_i = \tau_i -k_i^{2j+1}$, $i=1,2,3$. Then, from the definition of $X^{s,b}$-norm, we can reduce \eqref{eq:bilinear2} by
\begin{equation}\label{eq:equiv}
\begin{aligned}
&\normo{\sum_{\substack{k_1,k_2\in{\mathbb Z}^{\ast}\\k_1+k_2=k_3}}\int_{\tau_1 + \tau_2 = \tau_3}\frac{|k_3|\bra{k_3}^s|k_3|^{-s-\frac12}|k_1k_2|^{\frac12}N_{max}^{j-1}}{\prod_{i=1}^{3}\bra{\lambda_i}^{\frac12}}\wt{u}_1(k_1,\tau_1)\wt{u}_2(k_2,\tau_2)\: d\tau_1}_{\ell_{k_3}^2L_{\tau_3}^2}\\
&\hspace{28em}\lesssim \norm{u_1}_{L_{t,x}^2}\norm{u_2}_{L_{t,x}^2}.
\end{aligned}
\end{equation}
Without loss of generality, we assume $|\lambda_1|\le|\lambda_2|\le|\lambda_3|$. Then, from the identity
\[\lambda_1 + \lambda_2 = \lambda_3 - P_3(k_1,k_2,-k_3),\]
where $P_3$ is defined as in \eqref{eq:algebra1}, and Lemma \ref{lem:algebra} (a), we have $|\lambda_3| \gtrsim |k_1k_2k_3|\max(|k_1|,|k_2|,|k_3|)^{2j-2}$. By using this and duality, the left-hand side of \eqref{eq:equiv} is dominated by
\begin{equation}\label{eq:integral}
\int_{{\mathbb R} \times {\mathbb T}} u_3{\mathcal{F}}^{-1}[\bra{\tau_1-k_1^{2j+1}}^{-\frac12}\wt{u}_1]{\mathcal{F}}^{-1}[\bra{\tau_2-k_2^{2j+1}}^{-\frac12}\wt{u}_2]\:dxdt,
\end{equation}
where $\norm{u_3}_{L_{t,x}^2} \le 1$. We apply the H\"older inequality and Lemma \ref{lem:strichartz} ($X^{0,\frac13} \subset L_{t,x}^4$) to \eqref{eq:integral}, then \eqref{eq:integral} is bounded by
\begin{align*}
&\norm{u_3}_{L_{t,x}^2}\norm{{\mathcal{F}}^{-1}[\bra{\tau_1-k_1^{2j+1}}^{-\frac12}\wt{u}_1]}_{L_{t,x}^4}\norm{{\mathcal{F}}^{-1}[\bra{\tau_2-k_2^{2j+1}}^{-\frac12}\wt{u}_2}_{L_{t,x}^4} \\
\lesssim& \norm{u_3}_{L_{t,x}^2}\norm{u_1}_{X^{0,-\frac16}}\norm{u_2}_{X^{0,-\frac16}}\\
\lesssim& \norm{u_1}_{L_{t,x}^2}\norm{u_2}_{L_{t,x}^2},
\end{align*}
and this completes the proof.
\end{proof}
The following lemma will be used to obtain the nonsqueezing property.
\begin{lemma}
Let $j \in {\mathbb N}$. Let $u_i = P_{N_i}u$ and $|k_i|\sim N_i \ge 1$, $i=1,2,3$. Then we have
\begin{equation}\label{eq:bilinear3}
\norm{P_{N_3}\partial_x(u_1u_2)}_{Z^{-\frac12}} \lesssim N_{max}^{1-j}\norm{u_1}_{Y^{-\frac12}}\norm{u_2}_{Y^{-\frac12}}
\end{equation}
\end{lemma}
\begin{remark}
Thanks to the frequency decay bound $N_{max}^{1-j}$, $j > 1$, one can easily obtain an error bound in the proof of Lemma \ref{prop:approx1}, and this guarantees the approximation of the higher-order KdV flow without the Miura transform.
\end{remark}
\begin{proof}
We also assume $k_3 \neq 0$ due to the same reason in the proof of Lemma \ref{lem:bilinear2}.
We first control the $\ell_k^1L_{\tau}^2$ part. From the definition of function spaces, it suffices to show that
\begin{equation}\label{eq:bi1}
\normo{\sum_{\substack{k_1,k_2 \in {\mathbb Z}^{\ast}\\k_1+k_2=k_3}}\int_{\tau_1 + \tau_2 = \tau_3}\frac{|k_1k_2k_3|^{\frac12}\wt{u}_1(k_1,\tau_1)\wt{u}_2(k_2,\tau_2)}{\bra{\lambda_3}\bra{\lambda_1}^{\frac12}\bra{\lambda_2}^{\frac12}}\: d\tau_1}_{\ell_{k_3}^2L_{\tau_3}^1} \lesssim N_{max}^{1-j}\norm{u_1}_{X^{0,0}}\norm{u_2}_{X^{0,0}},
\end{equation}
where $\lambda_i$ is defined in the proof of Lemma \ref{lem:bilinear2}. From \eqref{eq:algebra1}, we have $\max(|\lambda_1|,|\lambda_2|,|\lambda_3|) \gtrsim |k_1k_2k_3|\max(|k_1|,|k_2|,|k_3|)^{2j-2}$. If $|\lambda_1|=\max(|\lambda_1|,|\lambda_2|,|\lambda_3|)$, \eqref{eq:bi1} is reduced by
\begin{equation}\label{eq:bi2}
\normo{\sum_{\substack{k_1,k_2 \in {\mathbb Z}^{\ast}\\k_1+k_2=k_3}}\int_{\tau_1 + \tau_2 = \tau_3}\bra{\lambda_3}^{-1}\wt{u}_1(k_1,\tau_1)\bra{\lambda_2}^{-\frac12}\wt{u}_2(k_2,\tau_2)\: d\tau_1}_{\ell_{k_3}^2L_{\tau_3}^1} \lesssim \norm{u_1}_{L_{t,x}^2}\norm{u_2}_{L_{t,x}^2}.
\end{equation}
Since $\bra{\lambda_3}^{-\frac23}$ is $L^2$-integrable, by the Cauchy-Schwarz inequality with respect to $\tau_3$, the left-hand side of \eqref{eq:bi2} is bounded by
\begin{equation*}
\normo{\sum_{\substack{k_1,k_2 \in {\mathbb Z}^{\ast}\\k_1+k_2=k_3}}\int_{\tau_1 + \tau_2 = \tau_3}\bra{\lambda_3}^{-\frac13}\wt{u}_1(k_1,\tau_1)\bra{\lambda_2}^{-\frac12}\wt{u}_2(k_2,\tau_2)\: d\tau_1}_{\ell_{k_3}^2L_{\tau_3}^2}.
\end{equation*}
Then, by duality and $X^{0,\frac13} \subset L_{t,x}^4$, we can obtain
\begin{align*}
\int_{{\mathbb R} \times {\mathbb T}} {\mathcal{F}}^{-1}[\bra{\lambda_3}^{-\frac13}\wt{u}_3]&u_1{\mathcal{F}}^{-1}[\bra{\lambda_2}^{-\frac12}\wt{u}_2]\: dxdt \\
&\lesssim \norm{{\mathcal{F}}^{-1}[\bra{\lambda_3}^{-\frac13}\wt{u}_3]}_{L_{t,x}^4}\norm{u_1}_{L_{t,x}^2}\norm{{\mathcal{F}}^{-1}[\bra{\lambda_2}^{-\frac12}\wt{u}_2]}_{L_{t,x}^4} \\
&\lesssim \norm{u_3}_{X^{0,0}}\norm{u_1}_{X^{0,0}}\norm{u_2}_{X^{0,-\frac16}},
\end{align*}
where $\norm{u_3}_{L_{t,x}^2} \le 1$. The last term implies the right-hand side of \eqref{eq:bi2}. By symmetry, we do not need to consider $|\lambda_2| = \max(|\lambda_1|,|\lambda_2|,|\lambda_3|)$. Next, we consider the case when $|\lambda_3| = \max(|\lambda_1|,|\lambda_2|,|\lambda_3|)$. If $\bra{\lambda_1} \gtrsim |P_3(k_1,k_2,-k_3)|^{\frac{1}{100}}$, we can reduce the left-hand side of \eqref{eq:bi1} as
\[\normo{\sum_{\substack{k_1,k_2 \in {\mathbb Z}^{\ast}\\k_1+k_2=k_3}}\int_{\tau_1 + \tau_2 = \tau_3}\bra{\lambda_3}^{-\frac12-\frac{1}{600}}N_{max}^{1-j}\bra{\lambda_1}^{-\frac13}\wt{u}_1(k_1,\tau_1)\bra{\lambda_2}^{-\frac12}\wt{u}_2(k_2,\tau_2)\: d\tau_1}_{\ell_{k_3}^2L_{\tau_3}^1}.\]
Since $\bra{\lambda_3}^{-\frac12-\frac{1}{600}}$ is $L^2$-integrable, using the Cauchy-Schwarz inequality with respect to $\tau_3$, duality, the H\"older inequality and $X^{0,\frac13} \subset L_{t,x}^4$, we have the \eqref{eq:bi1}. Finally, we assume $\bra{\lambda_i} \ll |P_3(k_1,k_2,-k_3)|^{\frac{1}{100}}$ for $i=1,2$. Then the left-hand side of \eqref{eq:bi1} is bounded by
\[\normo{\sum_{\substack{k_1,k_2 \in {\mathbb Z}^{\ast}\\k_1+k_2=k_3}} |P_3|^{-\frac12}N_{max}^{1-j}\int_{\bra{\lambda_1} \ll |P_3|^{\frac{1}{100}}}\int_{\bra{\lambda_2} \ll |P_3|^{\frac{1}{100}}}\wt{u}_1(k_1,\tau_1)\wt{u}_2(k_2,\tau_2)\:d\tau_1d\tau_2}_{\ell_{k_3}^2}.\]
Applying the Cauchy-Schwarz inequality in $\tau_1$ and $\tau_2$ separately, above norm is dominated by
\[\normo{\sum_{\substack{k_1,k_2 \in {\mathbb Z}^{\ast}\\k_1+k_2=k_3}} |P_3|^{-\frac12+\frac{1}{100}}N_{max}^{1-j}F_1(k_1)F_2(k_2)}_{\ell_{k_3}^2},\]
where
\[F_i(k_i) = \norm{\wt{u}_i(\tau_i)}_{L_{\tau_i}^2}.\]
From \eqref{eq:algebra1}, we have $|P_3| \gtrsim |k_i|^{2j}$ for $i=1,2,3$, which implies
\[|P_3|^{-\frac12+\frac{1}{100}} \lesssim |k_3|^{-j+\frac{j}{50}},\]
and this should be $\ell^2$-summable in $k_3$. Hence, we finally have
\[\normo{\sum_{k_1 \in {\mathbb Z}^{\ast}}N_{max}^{1-j}F_1(k_1)F_2(k_3-k_1)}_{\ell_{k_3}^{\infty}},\]
this is bounded from the Cauchy-Schwarz inequality in $k_1$ that
\[N_{max}^{1-j}\norm{F_1}_{\ell_{k_1}^2}\norm{F_2}_{\ell_{k_2}^2} = N_{max}^{1-j}\norm{u_1}_{X^{0,0}}\norm{u_2}_{X^{0,0}}.\]
For the $X^{-\frac12,-\frac12}$ part, it follows directly from Lemma \ref{lem:bilinear2}, when $s=-\frac12$. Hence we complete the proof.
\end{proof}
The following trilinear estimates will be also helpful to prove the global well-posedness.
\begin{lemma}\label{lem:trilinear}
Let $j \in {\mathbb N}$ and $-j/2 \le s < 0$. Let $u_i = P_{N_i}u$ and $|k_i|\sim N_i \ge 1$, $i=1,2,3$. Suppose that $k=k_1+k_2+k_3$, $|k_1|\ge|k_2|\ge|k_3|$ and $P_4(k_1,k_2,k_3,-k) \neq 0$, where $P_4$ is defined as in Lemma \ref{lem:algebra}.
{\rm{(a)}} If $|k|\sim|k_1|$, then
\begin{equation}\label{eq:trilinear1}
\norm{u_1u_2u_3}_{X^{-s,\frac12}} \lesssim N_1^{-s+j}N_2^{\frac12}N_3^{\frac12}\norm{u_1}_{Y^0}\norm{u_2}_{Y^0}\norm{u_3}_{Y^0}
\end{equation}
{\rm{(b)}} If $|k_1|\sim|k_2|$ and $j \ge 2$, then
\begin{equation}\label{eq:trilinear2}
\norm{u_1u_2u_3}_{X^{-s-j,\frac12}} \lesssim N_1^jN_3\norm{u_1}_{Y^0}\norm{u_2}_{Y^0}\norm{u_3}_{Y^0}, \hspace{2em} \mbox{for} \hspace{1em} |k_3| \ge |k|,
\end{equation}
or
\begin{equation}\label{eq:trilinear3}
\norm{u_1u_2u_3}_{X^{-s-j-\frac12,\frac12}} \lesssim N_1^jN_3^{\frac12}\norm{u_1}_{Y^0}\norm{u_2}_{Y^0}\norm{u_3}_{Y^0}, \hspace{2em} \mbox{for} \hspace{1em} |k| \ge |k_3|.
\end{equation}
\end{lemma}
\begin{proof}
From the Plancherel theorem, we have the identity
\[\norm{u_1u_2u_3}_{X^{-s,\frac12}} = \normo{\bra{k}^{-s}\bra{\tau-k^{2j+1}}^{\frac12}\sum_{\substack{k_1,k_2,k_3 \in {\mathbb Z}^{\ast}\\k_1+k_2+k_3=k}}\int_{\tau_1 + \tau_2 + \tau_3 = \tau}\prod_{i=1}^3\wt{u}_i(\tau_i,k_i)\: d\tau_1d\tau_2}_{\ell_k^2L_{\tau}^2}.\]
{\rm{(a)}} We first consider $\bra{\tau - k^{2j+1}} \lesssim \bra{\tau_i - k_i^{2j+1}}$ for some $i=1,2,3$. We may assume that $\bra{\tau - k^{2j+1}} \lesssim \bra{\tau_1 - k_1^{2j+1}}$. Then, \eqref{eq:trilinear1} is restricted by
\begin{equation}\label{eq:tril1}
\norm{u_1u_2u_3}_{X^{-s,0}} \lesssim N_1^{-s+j}N_2^{\frac12}N_3^{\frac12} \norm{u_1}_{X^{0,0}}\norm{u_2}_{Y^0}\norm{u_3}_{Y^0}.
\end{equation}
Since $|k|\sim |k_1|$, from the Young's and the Cauchy-Schwarz inequalities, we have
\begin{align*}
\mbox{LHS of } \eqref{eq:tril1} &\sim N_1^{-s}\normo{\sum_{\substack{k_1,k_2,k_3 \in {\mathbb Z}^{\ast}\\k_1+k_2+k_3=k}}\int_{\tau_1 + \tau_2 + \tau_3 = \tau}\prod_{i=1}^3\wt{u}_i(\tau_i,k_i)\: d\tau_1d\tau_2}_{\ell_k^2L_{\tau}^2}\\
&\lesssim N_1^{-s}\norm{\wt{u}_1}_{\ell_k^2L_{\tau}^2}\norm{\wt{u}_2}_{\ell_k^1L_{\tau}^1}\norm{\wt{u}_3}_{\ell_k^1L_{\tau}^1}\\
&\lesssim N_1^{-s}N_2^{\frac12}N_3^{\frac12}\norm{u_1}_{X^{0,0}}\norm{\wt{u}_2}_{\ell_k^2L_{\tau}^1}\norm{\wt{u}_3}_{\ell_k^2L_{\tau}^1},
\end{align*}
the last term implies the right-hand side of \eqref{eq:trilinear1}.
Now, we consider $\bra{\tau - k^{2j+1}} \gg \bra{\tau_i - k_i^{2j+1}}$ for all $i=1,2,3$. Then, from \eqref{eq:algebra2}, we have
\begin{align*}
|\tau - k^{2j+1}| &\sim |P_4(k_1,k_2,k_3,-k)|\\
&\sim |(k-k_1)(k-k_2)(k-k_3)||k_1|^{2j-2}\\
&\lesssim |k_1|^{2j}|k_2|.
\end{align*}
Hence,
\begin{align*}
\norm{u_1u_2u_3}_{X^{-s,\frac12}} &\lesssim N_1^{-s+j}N_2^{\frac12}\norm{u_1u_2u_3}_{L_{t,x}^2} \\
&\lesssim N_1^{-s+j}N_2^{\frac12}\norm{u_1u_2}_{L_{t,x}^2}\norm{u_3}_{L_{t,x}^{\infty}}.
\end{align*}
From the H\"older and the Sobolev inequalities, \eqref{eq:strichartz} and $Y^s \subset C_t H^s$ we have
\begin{align*}
\norm{u_1u_2}_{L_{t,x}^2}\norm{u_3}_{L_{t,x}^{\infty}} &\lesssim N_3^{\frac12}\norm{u_1}_{L_{t,x}^4}\norm{u_2}_{L_{t,x}^4}\norm{u_3}_{L_t^{\infty}L_x^2} \\
&\lesssim N_3^{\frac12}\norm{u_1}_{X^{0,\frac13}}\norm{u_2}_{X^{0,\frac13}}\norm{u_3}_{Y^0}
\end{align*}
and this implies the right-hand side of \eqref{eq:trilinear1}.
{\rm{(b)}} We consider firstly $\bra{\tau - k^{2j+1}} \lesssim \bra{\tau_1 - k_1^{2j+1}}$ similarly as in (a). Then, \eqref{eq:trilinear2} and \eqref{eq:trilinear3} are also restricted by
\begin{equation}\label{eq:tril3}
\norm{u_1u_2u_3}_{X^{-s-j,0}} \lesssim N_1^jN_3 \norm{u_1}_{X^{0,0}}\norm{u_2}_{Y^0}\norm{u_3}_{Y^0},
\end{equation}
and
\begin{equation}\label{eq:tril4}
\norm{u_1u_2u_3}_{X^{-s-j-\frac12,0}} \lesssim N_1^jN_3^{\frac12} \norm{u_1}_{X^{0,0}}\norm{u_2}_{Y^0}\norm{u_3}_{Y^0}.
\end{equation}
Since $j \ge 2$, both $\bra{k}^{-s-j}$ and $\bra{k}^{-s-j-\frac12}$ are $\ell_k^2$-summable, and from the H\"older and the Young's inequalities, we obtain
\begin{align*}
\mbox{LHS of } \eqref{eq:tril3} &= \normo{\bra{k}^{-s-j}\sum_{\substack{k_1,k_2,k_3 \in {\mathbb Z}^{\ast}\\k_1+k_2+k_3=k}}\int_{\tau_1+\tau_2+\tau_3=\tau}\prod_{i=1}^3\wt{u}_i(\tau_i,k_i)\:d\tau_1d\tau_2}_{\ell_k^2L_{\tau}^2}\\
&\lesssim \normo{\sum_{\substack{k_1,k_2,k_3 \in {\mathbb Z}^{\ast}\\k_1+k_2+k_3=k}}\int_{\tau_1+\tau_2+\tau_3=\tau}\prod_{i=1}^3\wt{u}_i(\tau_i,k_i)\:d\tau_1d\tau_2}_{\ell_k^{\infty}L_{\tau}^2}\\
&\lesssim \norm{\wt{u}_1}_{\ell_k^2L_{\tau}^2}\norm{\wt{u}_2}_{\ell_k^2L_{\tau}^1}\norm{\wt{u}_3}_{\ell_k^1L_{\tau}^1}\\
&\lesssim N_3^{\frac12}\norm{u_1}_{X^{0,0}}\norm{\wt{u}_2}_{Y^0}\norm{\wt{u}_3}_{\ell_k^2L_{\tau}^1},
\end{align*}
and
\begin{align*}
\mbox{LHS of } \eqref{eq:tril4} &= \normo{\bra{k}^{-s-j-\frac12}\sum_{\substack{k_1,k_2,k_3 \in {\mathbb Z}^{\ast}\\k_1+k_2+k_3=k}}\int_{\tau_1+\tau_2+\tau_3=\tau}\prod_{i=1}^3\wt{u}_i(\tau_i,k_i)\:d\tau_1d\tau_2}_{\ell_k^2L_{\tau}^2}\\
&\lesssim \normo{\sum_{\substack{k_1,k_2,k_3 \in {\mathbb Z}^{\ast}\\k_1+k_2+k_3=k}}\int_{\tau_1+\tau_2+\tau_3=\tau}\prod_{i=1}^3\wt{u}_i(\tau_i,k_i)\:d\tau_1d\tau_2}_{\ell_k^{\infty}L_{\tau}^2}\\
&\lesssim \norm{\wt{u}_1}_{\ell_k^2L_{\tau}^2}\norm{\wt{u}_2}_{\ell_k^2L_{\tau}^1}\norm{\wt{u}_3}_{\ell_k^1L_{\tau}^1}\\
&\lesssim N_3^{\frac12}\norm{u_1}_{X^{0,0}}\norm{\wt{u}_2}_{Y^0}\norm{\wt{u}_3}_{\ell_k^2L_{\tau}^1},
\end{align*}
each last term implies the right-hand side of \eqref{eq:trilinear2} and \eqref{eq:trilinear3}, respectively.
Next, we consider $\bra{\tau - k^{2j+1}} \gg \bra{\tau_i - k_i^{2j+1}}$ for all $i=1,2,3$. Then, from \eqref{eq:algebra2}, we have similarly as before that
\begin{align*}
|\tau - k^{2j+1}| &\sim |P_4(k_1,k_2,k_3,-k)|\\
&\sim |(k-k_1)(k-k_2)(k-k_3)||k_1|^{2j-2}\\
&\lesssim |k_1|^{2j}\max(|k_3|,|k|).
\end{align*}
For $|k_3| \ge |k|$, since $\bra{k}^{-s-j}$ is $\ell^2$-summable, we use the similar argument as above to show
\begin{align*}
\norm{u_1u_2u_3}_{X^{-s-j,\frac12}} &\lesssim N_1^{j}N_3^{\frac12}\norm{\wt{u}_1 \ast \wt{u}_2 \ast \wt{u}_3}_{\ell_k^{\infty}L_{\tau}^2} \\
&\lesssim N_1^{j}N_3^{\frac12}\norm{\wt{u}_1}_{\ell_k^2L_{\tau}^2}\norm{\wt{u}_2}_{\ell_k^2L_{\tau}^1}\norm{\wt{u}_3}_{\ell_k^1L_{\tau}^1}\\
&\lesssim N_1^{j}N_3\norm{u_1}_{X^{0,0}}\norm{\wt{u}_2}_{Y^0}\norm{\wt{u}_3}_{\ell_k^2L_{\tau}^1}.
\end{align*}
For $|k| \ge |k_3|$, by the same argument, we have
\begin{align*}
\norm{u_1u_2u_3}_{X^{-s-j-\frac12,\frac12}} &\lesssim N_1^{j}\norm{\wt{u}_1 \ast \wt{u}_2 \ast \wt{u}_3}_{\ell_k^{\infty}L_{\tau}^2} \\
&\lesssim N_1^{j}\norm{\wt{u}_1}_{\ell_k^2L_{\tau}^2}\norm{\wt{u}_2}_{\ell_k^2L_{\tau}^1}\norm{\wt{u}_3}_{\ell_k^1L_{\tau}^1}\\
&\lesssim N_1^{j}N_3^{\frac12}\norm{u_1}_{X^{0,0}}\norm{\wt{u}_2}_{Y^0}\norm{\wt{u}_3}_{\ell_k^2L_{\tau}^1}.
\end{align*}
Thus, we complete the proof.
\end{proof}
\section{Global well-posedness for $j \ge 3$.}\label{sec:global}
In this section, we will prove the global well-posedness of \eqref{eq:kdv} for $-\frac{j}{2} \le s < 0$$\footnote{Due to the $L^2$-conservation laws, it suffices to consider the case when $-\frac{j}{2} \le s < 0$.}$, when $j \ge 3$. We use the method of almost conservation law (so called "I-method") in \cite{CKSTT1}. Before introducing the modified energy, we introduce some definitions.
\begin{definition}
An $n$-multiplier is a function $m : {\mathbb R}^n \to \mathbb{C}$. We say an $n$-multiplier $m$ is symmetric if
\[m(\xi_1, \cdots, \xi_n) = m(\xi_{\sigma(1)},\cdots \xi_{\sigma(n)}) \hspace{2em} \mbox{for all} \hspace{1em} \sigma \in S_n,\]
where $S_n$ is the group of all permutations on $n$ objects, with the symmetrization
\[[m(\xi_1,\cdots,\xi_n)]_{sym}:= \frac{1}{n!}\sum_{\sigma \in S_n}m(\xi_{\sigma(1)},\cdots \xi_{\sigma(n)}).\]
\end{definition}
Even though the domain of $m$ is ${\mathbb R}^n$, we will only be interested in $m$ on the hyperplane $\xi_1 + \cdots + \xi_n = 0$.
\begin{definition}
An $n$-linear functional $\Lambda _n$ acting on functions $u_1,\cdots,u_n$ generated by an $n$-multiplier $m$ is given by
\[\Lambda_n(m:u_1,\cdots,u_n) := \int_{\Gamma_n}m(\xi_1, \cdots, \xi_n) \wh{u}_1(\xi_1) \cdots \wh{u}_n(\xi_n),\]
where $\Gamma_n = \set{(\xi_1 \cdots,\xi_n) \in {\mathbb R}^n : \xi_1 + \cdots + \xi_n =0}$. In particular, when $u_1, \cdots, u_n$ are the same functions, we write $\Lambda_n(m)$.
\end{definition}
Now, let us define an operator $I$ which operates $\wh{Iu} = m(\xi)\wh{u}(\xi)$, and acts like an identity and an integral operator on low and high frequencies, respectively, by choosing a smooth monotone multiplier satisfying
\begin{equation}\label{eq:multiplier}
\begin{split}
m(\xi) = \left\{
\begin{array}{cl}
1, & |\xi| < N \\
N^{-s}|\xi|^s, & |\xi| > 2N ,
\end{array}
\right.
\end{split}
\end{equation}
for fixed $N$ (which will be chosen later).
Let us define the modified energy $E_I^2(t)$ by
\[E_I^2(t) = \norm{Iu}_{L_x^2}^2 = \Lambda_2(m(\xi_1)m(\xi_2)).\]
The last equality follows from the Plancherel theorem and the facts that $u$ is real-valued, $m$ is even. In order to approach our goal, we further define modified energies (so called, \emph{correction terms}) by using the following lemma:
\begin{lemma}
Suppose $u$ be a solution of \eqref{eq:kdv} and $m$ is a symmetric $n$-multiplier. Then
\begin{equation}\label{eq:normal}
\frac{d}{dt}\Lambda_n(m) = \Lambda_n(m\alpha_n) -i\frac{n}{2}\Lambda_{n+1}([m(\xi_1, \cdots, \xi_{n-1}, \xi_n+\xi_{n+1})\set{\xi_n+\xi_{n+1}}]_{sym}),
\end{equation}
where
\[\alpha_n = i(\xi_1^{2j+1} + \cdots + \xi_n^{2j+1}).\]
\end{lemma}
\begin{proof}
See the Proposition 1 in \cite{CKSTT1}.
\end{proof}
We compute the time derivative of $E_I^2(t)$,
\[\frac{d}{dt}E_I^2(t) = \Lambda_2(m(\xi_1)m(\xi_2)\alpha_2) -i\Lambda_{3}([m(\xi_1)m(\xi_2+\xi_3)\set{\xi_2+\xi_3}]_{sym}).\]
The first term vanishes since $\xi_1 + \xi_2 = 0$ implies $\alpha_2 = 0$, and hence we have from the remainder that
\[\frac{d}{dt}E_I^2(t) = \Lambda_{3}(-i[m(\xi_1)m(\xi_2+\xi_3)\set{\xi_2+\xi_3}]_{sym}).\]
Let us denote
\[M_3(\xi_1,\xi_2,\xi_3) = -i[m(\xi_1)m(\xi_2+\xi_3)\set{\xi_2+\xi_3}]_{sym},\]
and define the new modified energy
\[E_I^3(t) = E_I^2(t) + {\Lambda}_3(\sigma_3),\]
where the symmetric 3-multiplier $\sigma_3$ will achieve a cancellation. Using \eqref{eq:normal} again, we have
\[\frac{d}{dt}E_I^3(t) = \Lambda_3(M_3) + \Lambda_3(\sigma_3\alpha_3) + \Lambda_4(-i\frac32[\sigma_3(\xi_1,\xi_2,\xi_3+\xi_4)\set{\xi_3+\xi_4}]_{sym}).\]
Taking
\[\sigma_3 = -\frac{M_3}{\alpha_3}\]
gives a cancellation of the first two terms. With this choice, similarly as before, the time derivative of $E_I^3(t)$ is a 4-linear expression $\Lambda_4(M_4)$, where
\[M_4(\xi_1,\xi_2,\xi_3,\xi_4) = -i\frac32[\sigma_3(\xi_1,\xi_2,\xi_3+\xi_4)\set{\xi_3+\xi_4}]_{sym}.\]
In the same manner, we define the third modified energy by
\[E_I^4(t)= E_I^3(t) + \Lambda_4(\sigma_4)\]
with
\[\sigma_4 = -\frac{M_4}{\alpha_4},\]
and we obtain
\[\frac{d}{dt}E_I^4(t) = {\Lambda}_5(M_5),\]
where
\[M_5(\xi_1,\cdots,\xi_5) = -2i[\sigma_4(\xi_1,\xi_2,\xi_3,\xi_4+\xi_5)\set{\xi_4+\xi_5}]_{sym}.\]
Under this setting, in order to show the global well-posedness for \eqref{eq:kdv}, we need to show that $E_I^2(t)$ is comparable to $E_I^4(t)$ at first, and next $E_I^4(t)$ is almost conserved. Let us start with obtaining some pointwise estimates which play a crucial role to show Proposition \ref{prop:comparable} and \ref{prop:quinti}, later. We define and state some calculus properties. If $m$ is of the form \eqref{eq:multiplier}, then $m^2$ satisfies
\begin{equation}\label{eq:multiplier1}
\begin{split}
&m^2(\xi) \sim m^2(\xi') \quad \mbox{for} \quad |\xi| \sim |\xi'|,\\
&(m^2)'(\xi) = O\left(\frac{m^2(\xi)}{|\xi|}\right),\\
&(m^2)''(\xi) = O\left(\frac{m^2(\xi)}{|\xi|^2}\right),
\end{split}
\end{equation}
for all non-zero $\xi$. With this notion, we can observe two forms of the mean value formula which follow directly from the fundamental theorem of calculus. If $|\eta|,|\lambda| \ll |\xi|$, then,
\begin{equation}\label{eq:mean1}
m^2(\xi+\eta) - m^2(\xi) = O\left(|\eta|\frac{m^2(\xi)}{|\xi|}\right)
\tag{MVT}\end{equation}
and
\begin{equation}\label{eq:mean2}
m^2(\xi + \eta + \lambda) - m^2(\xi + \eta) - m^2(\xi + \lambda) + m^2(\xi) = O\left(|\eta||\lambda|\frac{m^2(\xi)}{|\xi|^2}\right).
\tag{DMVT}\end{equation}
From the following two lemmas, the multiplier $\sigma_3$ can be smoothly extended on ${\mathbb R}^3$ as in \cite{KT}.
\begin{lemma}\label{lem:extension}
Let $m$ is of the form \eqref{eq:multiplier}. Then for each dyadic $\lambda \le \eta$, there is an extension of $\sigma_3$ from the diagonal set
\[\set{(\xi_1,\xi_2,\xi_3) \in \Gamma_3 : |\xi_1|,|\xi_2| \sim \eta, \quad |\xi_3| \sim \lambda}\]
to the full dyadic set
\[\set{(\xi_1,\xi_2,\xi_3) \in {\mathbb R}^3 : |\xi_1|,|\xi_2| \sim \eta, \quad |\xi_3| \sim \lambda}\]
which satisfies the size and regularity conditions
\begin{equation}\label{eq:extension}
|\partial_{\xi_1}^{\beta_1}\partial_{\xi_2}^{\beta_2}\partial_{\xi_3}^{\beta_3}\sigma_3(\xi_1,\xi_2,\xi_3)| \lesssim m^2(\lambda)\eta^{-2j-\beta_1-\beta_2}\lambda^{-\beta_3}.
\end{equation}
The implicit constant does not depend on $\lambda, \eta$, but may depend on $\beta_i's$, $i=1,2,3$.
\end{lemma}
\begin{proof}
We may assume that $|\xi_1| \gtrsim N$, otherwise $\sigma_3 \equiv 0$. Since $\xi_1 + \xi_2 + \xi_3 = 0$, we have from \eqref{eq:algebra1} that $\alpha_3 = i\xi_1\xi_2\xi_3Q_3(\xi_1,\xi_2,\xi_3)$ with the size $|\alpha_3| \sim \lambda \eta^{2j}$ and
\[M_3(\xi_1,\xi_2,\xi_3) = -i[m(\xi_1)m(\xi_2+\xi_3)\set{\xi_2+\xi_3}]_{sym} = i(m^2(\xi_1)\xi_1+m^2(\xi_2)\xi_2+m^2(\xi_3)\xi_3).\]
If $\lambda \sim \eta$, we extend $\sigma_3$ by
\begin{equation}\label{eq:M3-1}
\sigma_3(\xi_1,\xi_2,\xi_3) = C \frac{m^2(\xi_1)\xi_1+m^2(\xi_2)\xi_2+m^2(\xi_3)\xi_3}{\xi_1\xi_2\xi_3Q_3(\xi_1,\xi_2,\xi_3)}
\end{equation}
and if $\lambda \ll \eta$, we extend $\sigma_3$ by
\begin{equation}\label{eq:M3-2}
\sigma_3(\xi_1,\xi_2,\xi_3) = C \frac{m^2(\xi_1)\xi_1-m^2(\xi_1 + \xi_3)\set{\xi_1 + \xi_3}+m^2(\xi_3)\xi_3}{\xi_1\xi_2\xi_3Q_3(\xi_1,\xi_2,\xi_3)}.
\end{equation}
From \eqref{eq:multiplier1} and \eqref{eq:mean1}, we have the desired result.
\end{proof}
From Lemma \ref{lem:extension}, we can easily obtain the pointwise bound for $M_3$. If $|\xi_1|\sim|\xi_2|\sim|\xi_3|$, we have directly
\begin{equation}\label{eq:M3}
|M_3(\xi_1,\xi_2,\xi_3)| \lesssim m^2(\xi_3)|\xi_3|,
\end{equation}
from \eqref{eq:M3-1}, the triangle inequality and \eqref{eq:multiplier1}. Otherwise (i.e., if $|\xi_1| \sim |\xi_2| \gg |\xi_3|$), from \eqref{eq:M3-2}, we also have \eqref{eq:M3} by using \eqref{eq:mean1}.
Next, we give the pointwise estimate for $M_4$ which is the most important thing to show the almost conservation of $E_I^4(t)$.
\begin{lemma}\label{lem:M4}
Let $m$ is of the form \eqref{eq:multiplier}. For $N_i,N_{jk}$ dyadic and $N_1 \ge N_2 \ge N_3 \ge N_4$ where $|\xi_i| \sim N_i$ and $|\xi_j + \xi_k| \sim |N_{jk}|$, we have
\begin{equation}\label{eq:M4}
|M_4(\xi_1,\xi_2,\xi_3,\xi_4)| \lesssim \frac{|\alpha_4|m^2(\min(N_i, N_{jk}))}{(N+N_1)^j(N+N_2)^j(N+N_3)^{2j-1}(N+N_4)}.
\end{equation}
\end{lemma}
\begin{proof}
This proof is almost same as the proof of Proposition 5.3 in \cite{GW}. Moreover, when $N_4 \ll N/2$ and $N/2 \lesssim N_{12} < N_1/4$, we obtain exact \eqref{eq:M4} from the bound of
\[\sigma_3(-\xi_3,-\xi_4,\xi_3+\xi_4)(\xi_3+\xi_4).\]
Indeed, let $N_4 \ll N/2$ ($\Rightarrow N_{13} \sim N_1$) and $N/2 \lesssim N_{12} < N_1/4$. From \eqref{eq:algebra2} and \eqref{eq:extension}, we get
\[|\alpha_4| \sim |\xi_3+\xi_4|N_1^{2j} \gtrsim N/2 \cdot N_1^{2j}\]
and
\[|\sigma_3(-\xi_3,-\xi_4,\xi_3+\xi_4)(\xi_3+\xi_4)| \lesssim |\xi_3+\xi_4|N_3^{-2j} \lesssim N_3^{-2j+1},\]
respectively, and hence
\[\left|\frac{\sigma_3(-\xi_3,-\xi_4,\xi_3+\xi_4)(\xi_3+\xi_4)}{\alpha_4}\right| \lesssim \frac{1}{N_1^{2j}N_3^{2j-1}N}.\]
In the other cases, using \eqref{eq:algebra2}, \eqref{eq:mean1}, \eqref{eq:mean2} and \eqref{eq:extension}, one can obtain better bounds than the right-hand side of \eqref{eq:M4}. See \cite{GW} for the detailed proof.
\end{proof}
From the definition of $M_5$ and Lemma \ref{lem:M4}, we have the following pointwise bound for $M_5$:
\begin{lemma}\label{lem:M5}
Let $m$ is of the form \eqref{eq:multiplier}, and for $N_i,N_{jk}$ dyadic such that $|\xi_i|\sim N_i$ and $|\xi_j + \xi_l| = |\xi_{jl}| \sim N_{jl}$. Suppose $N_1 \ge N_2 \ge N_3 \ge N_4 \ge N_5$.
{\rm{(a)}} If $N_{12} \sim N_3$, then
\begin{equation}\label{eq:M5a}
|M_5(\xi_1,\xi_2,\xi_3,\xi_4,\xi_5)| \lesssim \frac{N_{12}}{(N+N_3)^{2j}(N+N_4)^{2j-1}(N+N_5)}.
\end{equation}
{\rm{(b)}} If $N_3 \sim N_4$, then for $N_5 \ge N_{12}$, we have
\begin{equation}\label{eq:M5b}
|M_5(\xi_1,\xi_2,\xi_3,\xi_4,\xi_5)| \lesssim \frac{N_{12}}{(N+N_3)^{j}(N+N_4)^{j}(N+N_{12})^j(N+N_5)^j},
\end{equation}
and otherwise, we have
\begin{equation}\label{eq:M5c}
|M_5(\xi_1,\xi_2,\xi_3,\xi_4,\xi_5)| \lesssim \frac{N_{12}}{(N+N_3)^{j}(N+N_4)^{j}(N+N_{12})^{j+\frac12}(N+N_5)^{j-\frac12}}.
\end{equation}
\end{lemma}
\begin{proof}
Under the condition, we may assume that $N_1 \sim N_2 \gtrsim N$, since $M_5$ vanishes when $N_1 \ll N$. From the definition of $M_5$ and \eqref{eq:M4}, we have
\[|M_5(\xi_1,\xi_2,\xi_3,\xi_4,\xi_5)| \lesssim |\sigma_4(\xi_3,\xi_4,\xi_5,\xi_1+\xi_2)(\xi_1+\xi_2)|.\]
Using \eqref{eq:M4}, (a) can be easily proven. For (b), from the fact that if $N \ge M \ge 1$,
\[\frac{1}{N^{2j-1}M} \lesssim \frac{1}{N^{2j-1-\alpha}M^{1+\alpha}} \qquad \mbox{for} \quad 0 \le \alpha \le 2j-1\]
holds, and \eqref{eq:M4}, we can also easily prove \eqref{eq:M5b} and \eqref{eq:M5c}.
\end{proof}
Now, going back to the main parts in this section, we first prove that $E_I^2(t)$ is comparable to $E_I^4(t)$.
\begin{proposition}\label{prop:comparable}
Let $-\frac{j}{2} \le s < 0$ and $N \gg 1$. Then,
\begin{equation*}
|E_I^4(t) - E_I^2(t)| \lesssim \norm{Iu(t)}_{L^2}^3 + \norm{Iu(t)}_{L^2}^4.
\end{equation*}
\end{proposition}
\begin{proof}
In view of $E_I^4(t)$, we know that $E_I^4(t) = E_I^3(t) + \Lambda_4(\sigma_4) = E_I^2(t) + \Lambda_3(\sigma_3) + \Lambda_4(\sigma_4)$, so it suffices to show
\begin{equation}\label{eq:comparablepf1}
|{\Lambda}_3(\sigma_3)| \lesssim \norm{Iu(t)}_{L^2}^3,
\end{equation}
and
\begin{equation}\label{eq:comparablepf2}
|{\Lambda}_4(\sigma_4)| \lesssim \norm{Iu(t)}_{L^2}^4.
\end{equation}
Let us now use $k_i$ and $k_{jl}$ as variables instead of $\xi_i$ and $\xi_{jl}$ to prevent confusion from the notations throughout the paper.
We first show \eqref{eq:comparablepf1} and may assume that the $\wh{u}$ is nonnegative. Let us define $v = Iu$. From \eqref{eq:M3-1}, we need to show that
\begin{equation}\label{eq:comparablepf1-1}
\left|{\Lambda}_3\left(\frac{m^2(k_1)k_1+m^2(k_2)k_2+m^2(k_3)k_3}{k_1k_2k_3Q_3(k_1,k_2,k_3)m(k_1)m(k_2)m(k_3)}\right)\right| \lesssim \norm{v}_{L^2}^3.
\end{equation}
We make a Littlewood-Paley decomposition and without loss of generality, assume $N_1 \ge N_2 \ge N_3$ for $|k_i| \sim N_i$ (dyadic). If $N_1 \le \frac{N}{2}$, then ${\Lambda}_3$ vanishes, so we also assume $N_1 \sim N_2 \gtrsim N$. We consider two cases separately: $N_3 \ll N$ and $N_3 \gtrsim N$.
\textbf{Case I.} $N_3 \ll N$. From \eqref{eq:M3}, \eqref{eq:algebra1} and \eqref{eq:multiplier}, \eqref{eq:comparablepf1-1} is reduced to
\[\sum_{N_1 \sim N_2 \ge N_3}\left|{\Lambda}_3\left(\frac{N^{2s}}{N_1^{2(s+j)}} : v_1,v_2,v_3 \right)\right| \lesssim \prod_{j=1}^3\norm{v_j}_{L^2}.\]
Since $s \ge -\frac{j}{2}$, we know that $N_1^{-2(s+j)} \le N_1^{-j}$, so \eqref{eq:comparablepf1-1} is reduced as
\[N^{2s}\sum_{N_1 \sim N_2 \ge N_3}N_1^{-j}\int v_1v_2v_3 \: dx\lesssim \norm{v_1}_{L^2}\norm{v_2}_{L^2}\norm{v_3}_{L^2}.\]
We use the H\"older and the Sobolev inequalities to show
\begin{align*}
N^{2s}\sum_{N_1 \sim N_2 \ge N_3}N_1^{-j}\int v_1v_2v_3 \: dx&\lesssim N^{2s}\sum_{N_1 \sim N_2 \ge N_3}N_1^{-j}\norm{v_1}_{L^4}\norm{v_2}_{L^4}\norm{v_3}_{L^2}\\
&\lesssim N^{2s}\sum_{N_1 \sim N_2 \ge N_3}N_1^{-j}N_1^{\frac12}\norm{v_1}_{L^2}\norm{v_2}_{L^2}\norm{v_3}_{L^2},
\end{align*}
which implies the right-hand side of \eqref{eq:comparablepf1-1}.
\textbf{Case II.} $N_3 \gtrsim N$. From \eqref{eq:M3}, \eqref{eq:algebra1} and \eqref{eq:multiplier}, \eqref{eq:comparablepf1-1} is reduced to
\[\sum_{N_1 \sim N_2 \ge N_3}\left|{\Lambda}_3\left(\frac{N^{s}N_3^s}{N_1^{2(s+j)}} : v_1,v_2,v_3 \right)\right| \lesssim \prod_{j=1}^3\norm{v_j}_{L^2}.\]
Similarly as before, we have from the H\"older and the Sobolev inequalities that
\begin{align*}
N^{s}\sum_{N_1 \sim N_2 \ge N_3\gtrsim N}N_1^{-j}N_3^s\int v_1v_2v_3 \: dx&\lesssim N^{2s}\sum_{N_1 \sim N_2 \ge N_3}N_1^{-j}N_3^s\norm{v_1}_{L^4}\norm{v_2}_{L^4}\norm{v_3}_{L^2}\\
&\lesssim N^{2s}\sum_{N_1 \sim N_2 \ge N_3}N_1^{-j}N_3^sN_1^{\frac12}\norm{v_1}_{L^2}\norm{v_2}_{L^2}\norm{v_3}_{L^2},
\end{align*}
which also implies the right-hand side of \eqref{eq:comparablepf1-1}.
We turn to prove \eqref{eq:comparablepf2}. We make again a Littlewood-Paley decomposition and without loss of generality, assume $N_1 \ge N_2 \ge N_3 \ge N_4$ for $|k_i| \sim N_i$ (dyadic). If $N_1 \le \frac{N}{2}$, then ${\Lambda}_4$ vanishes, so we also assume $N_1 \sim N_2 \gtrsim N$. From \eqref{eq:M4} and \eqref{eq:multiplier}, we need to show, similarly as \eqref{eq:comparablepf1}, that
\begin{equation}\label{eq:comparablepf2-1}
\begin{split}
\sum_{N_1 \sim N_2 \ge N_3\ge N_4}\left|{\Lambda}_4\left(\frac{1}{(N+N_1)^{2j}(N+N_3)^{2j-1}(N+N_4)\prod_{i=1}^{4}m(k_i)}:v_1,v_2,v_3,v_4\right)\right|\\
\lesssim \prod_{i=1}^4\norm{v_i}_{L^2}.
\end{split}
\end{equation}
From \eqref{eq:multiplier}, we know
\[\left|\frac{1}{(N+N_1)^{2j}(N+N_3)^{2j-1}(N+N_4)\prod_{i=1}^{4}m(k_i)}\right| \lesssim \frac{N^{4s}}{N_1^{2(s+j)}\bra{N_3}^{s+2j-1}\bra{N_4}^{1+s}}.\]
From the H\"older and Sobolev inequalities, we have
\begin{align*}
\int v_1v_2v_3v_4\:dx &\lesssim \norm{v_1}_{L^2}\norm{v_2}_{L^2}\norm{v_3}_{L^{\infty}}\norm{v_4}_{L^{\infty}}\\
&\lesssim (N_3N_4)^{\frac12}\norm{v_1}_{L^2}\norm{v_2}_{L^2}\norm{v_3}_{L^2}\norm{v_4}_{L^2}.
\end{align*}
Hence, we finally obtain that
\begin{align*}
\mbox{LHS of } \eqref{eq:comparablepf2-1} &\lesssim \sum_{N_1 \sim N_2 \ge N_3\ge N_4} \frac{N^{4s}(N_3N_4)^{\frac12}}{N_1^{2(s+j)}\bra{N_3}^{s+2j-1}\bra{N_4}^{1+s}}\prod_{i=1}^{4}\norm{v_i}_{L^2}\\
&\lesssim N^{4s}\sum_{N_1~N_2}N_1^{-j}\norm{v_1}_{L^2}\norm{v_2}_{L^2} \sum_{N_2 \ge N_3 \ge N_4}\bra{N_3}^{-\frac32(j-1)}\bra{N_4}^{\frac{j}{2}+1}\norm{v_3}_{L^2}\norm{v_4}_{L^2},
\end{align*}
which implies the right-hand side of \eqref{eq:comparablepf2-1}, and hence we complete the proof of lemma.
\end{proof}
Now, we prove that $E_I^4(t)$ is the almost conserved quantity for $t \in (0,1]$. In order to show this, since
\begin{equation}\label{eq:almost}
|E_I^4(t) - E_I^4(0)| \lesssim \left|\int_0^1 \Lambda_5(M_5) \: dt\right|,
\end{equation}
we shall control the quintilinear form.
\begin{proposition}\label{prop:quinti}
Let $-\frac{j}{2} \le s < 0$ and $N \gg 1$. Then,
\begin{equation*}
\left| \int_{0}^{1} \Lambda_5(M_5)\:dt\right| \lesssim N^{5s}\norm{Iu}_{Y^0}^5.
\end{equation*}
\end{proposition}
\begin{proof}
We may assume that $\wt{u}$ be nonnegative and let us define $v= Iu$. Then, it suffices to show
\begin{equation}\label{eq:quinti1}
\int_0^1 \Lambda_5\left(\frac{M_5(k_1,k_2,k_3,k_4,k_5)}{m(k_1)m(k_2)m(k_3)m(k_4)m(k_5)}\right) \: dt \lesssim N^{5s}\norm{v}_{Y^0}^5.
\end{equation}
We make a Littlewood-Paley decomposition $v_i = P_{N_i}v$ for dyadic numbers $N_i$, $i=1,2,3,4,5$. Let $|k_i| \sim N_i$ and $|k_j+k_l| = |k_{jl}| \sim N_{jl}$, and without loss of generality, we may assume $N_1 \ge N_2 \ge N_3 \ge N_4 \ge N_5$. From Lemma \ref{lem:M5}, we only consider two cases that $N_3 \sim N_{12} \gtrsim N$ and $N_3 \sim N_4 \gtrsim N$. For the $N_3 \sim N_{12} \gtrsim N$ case, from \eqref{eq:multiplier} and \eqref{eq:M5a}, we have
\begin{align*}
\mbox{LHS of } \eqref{eq:quinti1} &\lesssim N^{5s}\int_0^1 \Lambda_5(|k_{12}|\bra{k_1}^{-s}\bra{k_2}^{-s}\bra{k_3}^{-s-2j}\bra{k_4}^{-s-2j+1}\bra{k_5}^{-s-1})\: dt\\
&\lesssim N^{5s}\sum_{N_1\sim N_2 \ge N_3 \ge N_4 \ge N_5}N_1^{-s}N_2^{-s}N_3^{-s-2j}\bra{N_4}^{-s-2j+1}\bra{N_5}^{-s-1}\norm{\partial_x(v_1v_2)v_3v_4v_5}_{L_{t,x}^1}.
\end{align*}
From the Cauchy-Schwarz inequality, we reduce \eqref{eq:quinti1} to the following two estimates:
\begin{equation}\label{eq:quinti2}
N_1^{-s}N_2^{-s}\norm{\partial_x(v_1v_2)}_{X^{s,-\frac12}} \lesssim N_1^{-2s-j}N_3^{s+\frac12}\norm{v_1}_{Y^0}\norm{v_2}_{Y^0},
\end{equation}
and
\begin{equation}\label{eq:quinti3}
N_3^{-s-2j}\bra{N_4}^{-s-2j+1}\bra{N_5}^{-s-1}\norm{v_3v_4v_5}_{X^{-s,\frac12}} \lesssim N_3^{-2s-j}\bra{N_4}^{-s-2j+\frac32}\bra{N_5}^{-s-\frac12}\prod_{j=3}^5\norm{v_j}_{Y^0}.
\end{equation}
Since $N_1,N_2,N_3 \ge N \gg 1$ and $N_3 \sim N_{12}$, we have \eqref{eq:quinti2} and \eqref{eq:quinti3} directly from \eqref{eq:bilinear2} and \eqref{eq:trilinear1}, respectively. Hence, we obtain
\begin{align*}
\mbox{LHS of } \eqref{eq:quinti1} &\lesssim N^{5s}\sum_{N_1\sim N_2 \ge N_3 \ge N_4 \ge N_5}N_1^{-2s-j}N_3^{\frac12-j}\bra{N_4}^{-s-2j+\frac32}\bra{N_5}^{-s-\frac12}\prod_{i=1}^{5}\norm{v_i}_{Y^0}\\
&\lesssim N^{5s}\norm{v}_{Y^0}^3\sum_{N_1}N_1^{-2s-j}\norm{v_1}_{Y^0}\norm{v_2}_{Y^0},
\end{align*}
which shows \eqref{eq:quinti1} for $ -\frac{j}{2} \le s < 0$.
Now, we consider the $N_3\sim N_4\sim N$ case. We further divide this case into $N_{12} \ge N_5$ and $N_5 \ge N_{12}$ cases. In these cases, since we have the upper bound of $M_5$ as \eqref{eq:M5b} and \eqref{eq:M5c} in Lemma \ref{lem:M5}, \eqref{eq:quinti1} is reduced by the same manner as above that
\begin{equation*}
N_1^{-s}N_2^{-s}\norm{\partial_x(v_1v_2)}_{X^{s,-\frac12}} \lesssim N_1^{-2s-j}\bra{N_{12}}^{s+\frac12}\norm{v_1}_{Y^0}\norm{v_2}_{Y^0},
\end{equation*}
and
\begin{equation*}
N_3^{-s-j}N_4^{-s-j}\bra{N_5}^{-s-j}\norm{\prod_{j=3}^{5}v_i}_{X^{-s-j,\frac12}} \lesssim N_3^{-s}N_4^{-s-j}\bra{N_5}^{-s-j+1}\prod_{j=3}^5\norm{v_j}_{Y^0}
\end{equation*}
for $N_{12} \ge N_5$, and
\begin{equation*}
N_3^{-s-j}N_4^{-s-j}\bra{N_5}^{-s-j+\frac12}\norm{\prod_{j=3}^{5}v_i}_{X^{-s-j-\frac12,\frac12}} \lesssim N_3^{-s}N_4^{-s-j}\bra{N_5}^{-s-j+1}\prod_{j=3}^5\norm{v_j}_{Y^0}
\end{equation*}
for otherwise. These estimates can be obtained directly from \eqref{eq:bilinear2}, \eqref{eq:trilinear2} and \eqref{eq:trilinear3}, respectively. Hence we obtain
\begin{align*}
\mbox{LHS of } \eqref{eq:quinti1} &\lesssim N^{5s}\sum_{N_1\sim N_2 \ge N_3 \sim N_4 \ge N_5}N_1^{-2s-j}N_3^{-2s-j}\bra{N_5}^{-s-j+1}\bra{N_{12}}^{s+\frac12}\prod_{i=1}^{5}\norm{v_i}_{Y^0}\\
&\lesssim N^{5s}\norm{v}_{Y^0}\sum_{N_1}N_1^{-2s-j}\norm{v_1}_{Y^0}^2\sum_{N_3}N_3^{-2s-j}\norm{v_3}_{Y^0}^2,
\end{align*}
which shows \eqref{eq:quinti1} for $ -\frac{j}{2} \le s < 0$, and hence we complete the proof.
\end{proof}
Finally, we sketch the proof of the global well-posedness by using the same argument in \cite{CKSTT1}. From the scaling property, \eqref{eq:kdv} with initial data $u_0 \in H_0^s({\mathbb T})$ is invariant under the following scaling:
\begin{equation*}\label{eq:scaling}
u_{\mu}(t,x) = \mu^{-2j}u(\mu^{-2j-1}t,\mu^{-1}x), \quad u_{0,\mu}(x) =\mu^{-2j}u_0(\mu^{-1}x),
\end{equation*}
with
\[\norm{u_{0,\mu}}_{H_0^s({\mathbb T}_{\mu})} = \mu^{-s-2j+\frac12}\norm{u_0}_{H_0^s({\mathbb T})}.\]
Proposition \ref{prop:comparable} and \eqref{eq:almost} with Proposition \ref{prop:quinti} give
\begin{equation}\label{eq:global1}
\sup_{0 \le t \le N^{-5s}}\norm{Iu(t)}_{L^2} \lesssim \norm{Iu(0)}_{L^2}.
\end{equation}
Moreover, a direct calculation also gives
\begin{equation}\label{eq:global2}
\norm{Iu_{\mu}(0,\cdot)}_{L^2} \lesssim \mu^{-s-2j+\frac12}N^{-s}\norm{u_0}_{H_0^s},
\end{equation}
Taking $\mu \ge 1$ satisfying
\[\mu^{-s-2j+\frac12}N^{-s} = \epsilon_0 \ll 1,\]
implies with \eqref{eq:global1} and \eqref{eq:global2} that
\begin{align*}
\sup_{0 \le t \le T}\norm{u(t)}_{H_0^s} &\le \mu^{s+2j-\frac12}\sup_{0 \le t \le \mu^{2j+1}T}\norm{Iu_{\mu}(t)}_{L^2}\\
&\lesssim \mu^{s+2j-\frac12}\norm{Iu_{\mu}(0)}_{L^2} \\
&\lesssim N^{-s}\norm{u_0}_{H_0^s},
\end{align*}
when $\mu^{2j+1}T \le N^{-5s}$. Furthermore, for our global-in-time solution of \eqref{eq:kdv}, we have the uniform time growth bound of $H_0^s$-norm,
\begin{equation}\label{eq:uniform bound}
\sup_{0 \le t \le T} \norm{u(t)}_{H_0^s} \lesssim T^{\frac{2s+4j-1}{10s+16j-7}}\norm{u_0}_{H_0^s},
\end{equation}
for $-j/2 \le s < 0$.
\begin{remark}
In fact, in order to use the scaling argument in the proof of the global well-posedness, we need to consider the $\mu$-periodic function, $\mu \ge 1$. However, all estimates obtained in Section \ref{sec:bi-,tri-} for the global well-posedness do not depend on the $\mu$-scale, even though we prove those estimates under the $\mu$-periodic setting. Hence, we can use the scaling argument without further work. See Appendix \ref{sec:lambda} for the details.
\end{remark}
\section{Nonsqueezing property when $j\ge2$}\label{sec:nonsqueezing}
In this section, we prove the nonsqueezing property of \eqref{eq:kdv} when $j\ge 2$. As mentioned in Section \ref{subsec:Nonsq prop}, we first state the nonsqueezing property of (\ref{eq:truncated equation}) as an application of Theorem \ref{thm:finite dimensional nonsq.}.
\begin{lemma}\label{lem:Nonsqueezing of trun. fow}
Let $N \geq 1$, $ 0 <r <R$, $u_* \in P_{\le N} H^{-\frac{1}{2}}_0\left(\mathbb{T}\right)$, $0< \left|k_0\right| \leq N$, $ z \in \mathbb{C}$ and $T>0$. Let $S_H^N \left(t\right) : P_{\le N} H^{-\frac{1}{2}}_0 \left(\mathbb{T}\right) \to P_{\le N} H^{-\frac{1}{2}}_0 \left(\mathbb{T}\right)$ be the solution map to \eqref{eq:truncated equation}.
Then
\begin{equation*}
S_H^N \left(T\right) \left(B_R^N \left( u_* \right)\right) \not \subseteq C_{k_0,r}^N \left(z\right).
\end{equation*}
\end{lemma}
Our task, in this section, is to prove the closeness between two flows, $S_H\left(t\right)$ and $S_H^N\left(t\right)$. Since there are two differences between two flows, initial data and solution map, we can show the closeness by proving the following propositions, respectively:
\begin{proposition}\label{prop:approx}
Let $T>0$, and $N \gg 1$. Let $u_0, \underline u_0 \in H^{-\frac{1}{2}}_0$ be such that $P_{\le2N} u_0 = P_{\le 2N}\underline u_0$. Then we have
\[\sup_{|t| \le T} \norm{P_{\le N}(S_{H}(t)u_0 - S_{H}(t)\underline u_0)}_{H^{-\frac{1}{2}}_0} \leq C\Big(T,\left\|u_0\right\|_{H^{-\frac{1}{2}}_0}\left\|,\underline u_0\right\|_{H^{-\frac{1}{2}}_0}\Big) N^{-\sigma}\]
for some $\sigma>0$.
\end{proposition}
\begin{proposition}\label{prop:approx2}
Let $T>0$ and $N\gg1$. Let $u_0 \in H^{-\frac{1}{2}}_0$ have Fourier transform supported in the range $\left|k\right| \le N$. Then we have
\[\sup_{|t| \le T} \norm{P_{\le N^{1/2}}(S_{H}(t)u_0 - S_{H}^{N}(t)u_0)}_{H^{-\frac{1}{2}}_0} \leq C\Big(T,\left\|u_0\right\|_{H^{-\frac{1}{2}}_0}\Big)N^{-\sigma}.\]
for some $\sigma >0$.
\end{proposition}
\begin{remark}
Proposition \ref{prop:approx} tells that change in the initial data at frequencies $\ge 2N$ does not significantly affect the solution at frequencies $\le N$ in the $H^{-\frac{1}{2}}_0$.
\end{remark}
Now, we first prove Proposition \ref{prop:approx} by using estimates in Section \ref{sec:bi-,tri-}. We use the same argument in \cite{CKSTT3}. From the local well-posedness theory and the uniform bounds \eqref{eq:uniform bound}, Proposition \ref{prop:approx} can be reduced to the following lemma:
\begin{lemma}\label{prop:approx1}
Let $N' \gg 1$ and $u_0,\underline u_0 \in H_0^{-\frac12}$ satisfying $P_{\le N'}u_0 = P_{\le N'}\underline u_0$. Then if $T'$ is sufficiently small depending on $\norm{u_0}_{H_0^{-\frac12}}$ and $\norm{\underline u_0}_{H_0^{-\frac12}}$, we have
\begin{equation*}
\sup_{|t| \le T'} \norm{P_{\le N' - (N')^{1/2}}(S_{H}(t)u_0 - S_{H}(t)\underline u_0)}_{H^{-\frac{1}{2}}_0} \leq C\Big(\norm{u_0}_{H_0^{-\frac12}}, \norm{\underline u_0}_{H_0^{-\frac12}}\Big)\left(N'\right)^{-\sigma},
\end{equation*}
for some $\sigma > 0$.
\end{lemma}
\begin{remark}
In \cite{CKSTT3}, Colliander et al. explain why proving this proposition is enough to prove Proposition \ref{prop:approx}. See the section 5 in \cite{CKSTT3}.
\end{remark}
\begin{remark}
The proof of Lemma \ref{prop:approx1} is easier and simpler than the proof of the Proposition 5.1 in \cite{CKSTT3}. Since we can obtain the good frequency decay bound from the bilinear estimate \eqref{eq:bilinear3}, no more techniques such as the Miura transform in \cite{CKSTT3} is required for our analysis as mentioned in Section \ref{subsec:Nonsq prop}. Moreover, since the right-hand side of \eqref{eq:bilinear3} has the coefficient depending only on $N_{max}$, it is sufficient to separate $u$ into low and high frequencies different from the argument in \cite{CKSTT3}.
\end{remark}
To simplify our argument, consider
\begin{equation}\label{eq:kdv1}
u_t + (-1)^{j+1} \partial_x^{2j+1}u = F(u,u),
\end{equation}
where $j>1$ and $F(u,v) = \frac12\partial_x(uv)$ with
\[F(u,v)\wh{\:}(k) = -\frac12\sum_{\substack{k_1,k_2 \in {\mathbb Z}^{\ast}\\k_1+k_2=k}} ik \wh{u}(k_1)\wh{v}(k_2).\]
Note that in contrast with analysis in \cite{CKSTT3}, since we control the quadratic form, the resonant term $F_0$ as in \cite{CKSTT3} is not considered.
\begin{proof}[Proof of Lemma \ref{prop:approx1}]
From the local well-posedness theory of \eqref{eq:kdv}, we have the local estimates
\begin{equation}\label{eq:small_1}
\norm{u}_{Y^{-\frac12}} + \norm{\underline u}_{Y^{-\frac12}} \le C,
\end{equation}
by choosing the sufficiently small time $T'$ depending on the $H_0^{-\frac12}$-norms of $u_0$ and $\underline u_0$.
Let $M \in \left[N' - \left(N'\right)^{\frac{1}{2}}, N'\right]$ be an integer. We separate $u$ as
\[u = u_{lo} +u_{hi},\]
where
\[u_{lo} := P_{\le M}u, \hspace{1em} u_{hi}:= (1-P_{\le M})u.\]
From \eqref{eq:small_1}, we have
\begin{equation}\label{eq:split}
\norm{u_{lo}}_{Y^{-\frac12}}, ~ \norm{u_{hi}}_{Y^{-\frac12}} \le C.
\end{equation}
We also split $\underline u$ and obtain the similar result as \eqref{eq:split} for $\underline u$. Applying $P_{\le M}$ to \eqref{eq:kdv1}, $u_{lo}$ obeys the equation
\begin{equation}\label{eq:low freq eq}
(\partial_t + \partial_x^3)u_{lo} = P_{\le M}F(u,u).
\end{equation}
In order to control the right-hand side of \eqref{eq:low freq eq} except for $F(u_{lo},u_{lo})$, define the \emph{error terms} to be any quantity with $Z^{-\frac12}$-norm of $O(\left(N'\right)^{-\sigma})$. From \eqref{eq:bilinear3}, we can easily know that all terms except for $P_{\le M}F(u_{lo},u_{lo})$ are error terms.
Indeed, if the nonlinear term contains $u_{hi}$, then from the bilinear estimate \eqref{eq:bilinear3}, we have $M^{1-j}$ decay bound. Thus, $u_{lo}$ obeys the equation
\begin{equation}\label{eq:error1}
(\partial_t + \partial_x^3)u_{lo} = P_{\le M}F(u_{lo},u_{lo}) + \mbox{error term}.
\end{equation}
By the same manner, the function $\underline u_{lo}$ obeys the equation
\begin{equation}\label{eq:error2}
(\partial_t + \partial_x^3)\underline u_{lo} = P_{\le M}F(\underline u_{lo},\underline u_{lo}) + \mbox{error term}.
\end{equation}
Since $u_{lo} = \underline u_{lo}$, we have from the standard local well-posedness theory that
\[\norm{u_{lo} - \underline u_{lo}}_{Y^{-\frac12}} \lesssim \left(N'\right)^{-\sigma},\]
which implies Lemma \ref{prop:approx1} by $Y^{s} \subset C_{t}H^s$.
\end{proof}
For Proposition \ref{prop:approx2}, we use the similar argument as in the proof of Proposition \ref{prop:approx}. The point of the proof of Proposition \ref{prop:approx} is to show \eqref{eq:error1} and \eqref{eq:error2}, and since $P_{\le N}P_{\le 2N} = P_{\le N}$, it suffices to obtain
\[(\partial_t + \partial_x^3)u_{lo} = P_{\le M}F(u_{lo},u_{lo}) + \mbox{error term}\]
and
\[(\partial_t + \partial_x^3)v_{lo} = P_{\le M}F(v_{lo},v_{lo}) + \mbox{error term}.\]
However, those can be easily obtained by the same argument as in the proof of Lemma \ref{prop:approx1}. We omit the detailed proof of Proposition \ref{prop:approx2}.
As the final stage to show the nonsqueezing property, we combine Lemma \ref{lem:Nonsqueezing of trun. fow}, Proposition \ref{prop:approx} and Proposition \ref{prop:approx2}. First of all, we show the following proposition:
\begin{proposition}\label{prop:main approx}
Let $k_0 \in \mathbb{Z}^*$, $T>0$, $A>0$, and $ 0 < \varepsilon \ll 1$. Then there exists a frequency $N_0 = N_0\left(k_0, T, \varepsilon, A\right) \gg \left|k_0\right|$ such that
\begin{equation*}
\left|k_0\right|^{-1/2} \left|\left(S_{H}\left(T\right)u_0\right)\hat{}\left({k_0}\right) - \left(S^{N}_{H} \left(T\right) u_0\right) \hat{}\left(k_0\right)\right| \ll \varepsilon
\end{equation*}
for all $N \geq N_0$ and all $u_0 \in B_A^N \left(0\right)$.
\end{proposition}
\begin{proof}
Let $\underline u_{0,N} = P_{\leq N} u_0$. By the triangle inequality, we have
\begin{align*}
&\left|k_0\right|^{-1/2} \left|\left(S_{H}\left(T\right)u_0\right)\hat{}\left({k_0}\right) - \left(S^{N}_{H} \left(T\right) u_0\right) \hat{}\left(k_0\right)\right| \\
\le& \left|k_0\right|^{-1/2} \left|\left(S_{H}\left(T\right)u_0\right)\hat{}\left({k_0}\right) - \left(S_{H} \left(T\right) \underline u_{0,N}\right) \hat{}\left(k_0\right)\right| \\
+&\left|k_0\right|^{-1/2} \left|\left(S_{H}\left(T\right)\underline u_{0,N}\right)\hat{}\left({k_0}\right) - \left(S^{N}_{H} \left(T\right) \underline u_{0,N}\right) \hat{}\left(k_0\right)\right|
\end{align*}
for $\left|k_0\right| \ll N$.
From Proposition \ref{prop:approx2} and \ref{prop:approx}, we have
\[\left|k_0\right|^{-1/2} \left|\left(S_{H}\left(T\right)u_0\right)\hat{}\left({k_0}\right) - \left(S_{H} \left(T\right) \underline u_{0,N}\right) \hat{}\left(k_0\right)\right| \lesssim N^{-\sigma}\]
and
\[\left|k_0\right|^{-1/2} \left|\left(S_{H}\left(T\right)\underline u_{0,N}\right)\hat{}\left({k_0}\right) - \left(S^{N}_{H} \left(T\right) \underline u_{0,N}\right) \hat{}\left(k_0\right)\right| \lesssim N^{-\sigma},\]
respectively, for $N> N_0\left(k_0,T,\varepsilon,A\right)$ and $\left|k_0\right| \le N^{1/2}$. Thus, we complete the proof.
\end{proof}
Finally, we prove Theorem \ref{thm:Nonsqueezing thm} by combining with Lemma \ref{lem:Nonsqueezing of trun. fow} and Proposition \ref{prop:main approx}.
\begin{proof}[Proof of Theorem \ref{thm:Nonsqueezing thm}]
Choose $0 < \varepsilon < \frac{R-r}{2}$ and the ball $B^{\infty}_{R}\left(u_*\right) \subset B^{\infty}_{A}\left(0\right)$. We also choose $N > N_0\left(T,\varepsilon,k_0,A\right)$ so large that
\begin{equation*}
\left\|u_*- P_{\leq N} u_*\right\|_{H^{-1/2}_0} \leq \varepsilon.
\end{equation*}
From Lemma \ref{lem:Nonsqueezing of trun. fow}, we can find initial data $u_0 \in P_{\leq N} H^{-\frac{1}{2}}_0\left(\mathbb{T}\right)$ satisfying
\begin{equation*}
\left\|u_0 - u_*\right\|_{H^{-1/2}_0} \leq R-\varepsilon
\end{equation*}
and
\begin{equation*}
\left|k_0\right|^{-\frac{1}{2}} \left|\left(S_{H}^N\left(T\right)u_0\right)^{\wedge{}}\left(k_0\right)-z\right| > r+\varepsilon.
\end{equation*}
Then by the triangle inequality, we have
\begin{equation*}
\left\|u_0 - u_*\right\|_{H^{-1/2}_0} \leq \left\|u_0 - P_{\leq N }u_*\right\|_{H^{-1/2}_0} + \left\|P_{\leq N }u_*- u_*\right\|_{H^{-1/2}_0} \leq R.
\end{equation*}
Moreover, by the triangle inequality and Proposition \ref{prop:main approx}, we have
\begin{equation*}
\begin{split}
&\left|k_0\right|^{-\frac{1}{2}} \left|z-\left(S_{H}\left(T\right)u_0\right)^{\wedge{}}\left(k_0\right)\right| \\
\geq& \left|k_0\right|^{-\frac{1}{2}} \left[ \left|z-\left(S_{H}^N\left(T\right)u_0\right)^{\wedge{}}\left(k_0\right)\right| - \left|\left(S_{H}^N\left(T\right)u_0\right)^{\wedge{}}\left(k_0\right)-\left(S_{H}\left(T\right)u_0\right)^{\wedge{}}\left(k_0\right)\right|\right] \\
>& r+\varepsilon -\varepsilon = r,
\end{split}
\end{equation*}
and this completes the proof.
\end{proof}
|
{
"redpajama_set_name": "RedPajamaArXiv"
}
| 7,523
|
Fastly, Inc. participates in and has certified its compliance with the EU-U.S. Privacy Shield Framework and the Swiss-U.S. Privacy Shield Framework. Fastly is committed to subjecting all personal information received from European Union ("EU") member countries and Switzerland, respectively, in reliance on each Privacy Shield Framework, to the Framework's applicable Principles. To learn more about the Privacy Shield Framework, and to view our certification, visit the U.S. Department of Commerce's Privacy Shield List.
Fastly, Inc. adheres to the Principles with respect to personal information we receive from individuals or companies in the EU in reliance on the Privacy Shield. Fastly provides content delivery and related services (our "Services") that permit subscribers to electronically submit (or cause to be submitted) data to the Services for caching, transmission, and associated processing. As the subscribers (or in some cases their end users) control the data to be cached, transmitted and/or processed, Fastly's network, platform and services may be used as a conduit for information. As the Principles do not impose secondary liability, to the extent Fastly, on behalf of its subscriber, merely transmits, routes, switches or caches data, Fastly may rely on its subscribers to comply with legal requirements underlying the Principles with respect to such processing.
We also receive some data in reliance on other compliance mechanisms, including data processing agreements based on the EU Standard Contractual Clauses.
As a data processor, Fastly caches, transmits, discloses and processes data submitted to our Services, which may include personal information about EU personnel received from its subscribers and their users, at the direction of its subscribers in accordance with our agreements. Fastly may access or process your data to provide the Services, to prevent or address technical or service problems, to follow the instructions of our customer who submitted the data, or in response to contractual or legal requirements. Fastly will not access personal information contained in the data submitted to the Services. Fastly may share personal information with its subsidiaries, contractors, or third parties if Fastly undergoes a business transaction, such as a merger, acquisition by another company or sale of all or a portion of its assets.
Fastly is responsible for the processing of personal information it receives, under each Privacy Shield Framework, and subsequently transfers to a third party acting as an agent on its behalf. Fastly complies with the Privacy Shield Principles for all onward transfers of personal data from the EU and Switzerland, including the onward transfer liability provisions.
Fastly may use from time to time third-party service providers, contractors and subprocessors to assist in providing the Services on our behalf. Fastly maintains contracts with these third parties restricting their access, use and disclosure of personal information in compliance with our Privacy Shield obligations, and Fastly may be liable if we fail to meet those obligations and we are responsible for the event giving rise to the damage.
Fastly may be required in certain circumstances to disclose personal information in response to lawful requests by public authorities, including to meet national security or law enforcement requirements.
EU and Swiss individuals have rights to access personal information about them, and to limit use and disclosure of their personal information, except where the burden or expense of providing access would be disproportionate to the risks to the individual's privacy in question, or where the rights of others would be violated. With our Privacy Shield certification, Fastly has committed to respect those rights. Because Fastly personnel have limited ability to access data our subscribers submit to our services, if you wish to request, access, to limit use, or to limit disclosure, please provide the name of the Fastly subscriber who submitted your data to our services. We will refer your request to that subscriber, and will support them as needed in responding to your request.
With respect to personal Information received or transferred pursuant to the Privacy Shield Framework, Fastly is subject to the regulatory enforcement powers of the U.S. Federal Trade Commission.
Fastly will respond to your inquiry or complaint within 45 days. If you have an unresolved privacy or data use concern that we have not addressed satisfactorily, please contact our designated U.S.-based third party dispute resolution provider (free of charge) at https://feedback-form.truste.com/watchdog/request.
|
{
"redpajama_set_name": "RedPajamaC4"
}
| 5,635
|
<?php
/**
* @file
* Contains \Drupal\migrate_drupal\Plugin\migrate\process\d6\BlockRegion.
*/
namespace Drupal\migrate_drupal\Plugin\migrate\process\d6;
use Drupal\Component\Utility\NestedArray;
use Drupal\migrate\MigrateExecutable;
use Drupal\migrate\ProcessPluginBase;
use Drupal\migrate\Row;
/**
* @MigrateProcessPlugin(
* id = "d6_block_region"
* )
*/
class BlockRegion extends ProcessPluginBase {
/**
* {@inheritdoc}
*
* Set the destination block region, based on the source region and theme as
* well as the current destination default theme.
*/
public function transform($value, MigrateExecutable $migrate_executable, Row $row, $destination_property) {
list($region, $source_theme, $destination_theme) = $value;
// Theme is the same on both source and destination, we will assume they
// have the same regions.
if (strtolower($source_theme) == strtolower($destination_theme)) {
return $region;
}
// If the source and destination theme are different, try to use the
// mappings defined in the configuration.
$region_map = $this->configuration['region_map'];
if (isset($region_map[$region])) {
return $region_map[$region];
}
// Oh well, we tried. Put the block in the main content region.
return 'content';
}
}
|
{
"redpajama_set_name": "RedPajamaGithub"
}
| 1,479
|
\section{Introduction}
Benjamini and Kesten~\cite{BenjKesten:96} consider the following
model: Given a known graph $G=(V,E)$ and an unknown function $f:V \to
\{0,1\}$ (i.e., a binary labeling of the vertices, or ``scenery"), let
$v(t)$ (for $t \in \mathbb N} \newcommand{\R}{\mathbb R$) be the position of a particle performing a
random walk on $G$. They give some conditions under which $f$ can be
reconstructed from $\{f(v(t))\}_{t=1}^\infty$ with probability one,
and show that in some cases it cannot be
reconstructed. Lindenstrauss~\cite{Lindenstrauss:1999} showed that
when $G=\mathbb Z} \newcommand{\CalE}{{\mathcal{E}}$ there exist uncountably many functions $f$ that cannot be
distinguished given $\{f(v(t))\}$. On the other hand, Matzinger and
Rolles \cite{Matzinger:2003} showed that when $G=\mathbb Z} \newcommand{\CalE}{{\mathcal{E}}$ and $f$ is chosen
uniformly, then almost all functions $f$ can be
reconstructed. Furthermore, $f$ can be reconstructed in the interval
$[-n,n]$ with high probability from a polynomial sample.
We focus on the case when the graph $G$ is an undirected cycle of size
$n$. One may think of this graph as having $\mathbb Z} \newcommand{\CalE}{{\mathcal{E}}_n = \mathbb Z} \newcommand{\CalE}{{\mathcal{E}}/n\mathbb Z} \newcommand{\CalE}{{\mathcal{E}}$ as its
vertex set, with $(k,\ell) \in E$ whenever $k-\ell \in \{-1,1\}$;
equivalently, $G$ is the Cayley graph of $\mathbb Z} \newcommand{\CalE}{{\mathcal{E}}_n$ with generating set $\{-1,1\}$. We characterize a random walk on this graph by a
step distribution $\gamma$ on $\mathbb Z} \newcommand{\CalE}{{\mathcal{E}}_n$ such that at each turn with
probability $\gamma(k)$ the particle jumps $k$ steps: $\gamma(k) =
\P{v(t+1)-v(t)=k}$. We choose $v(1)$ uniformly from $\mathbb Z} \newcommand{\CalE}{{\mathcal{E}}_n$.
In this setting we explore the conditions under which $f$ can be
reconstructed from $\{f(v(t))\}_{t=1}^\infty$. Let the functions
$f_1:\mathbb Z} \newcommand{\CalE}{{\mathcal{E}}_n \to \{0,1\}$ and $f_2:\mathbb Z} \newcommand{\CalE}{{\mathcal{E}}_n \to \{0,1\}$ be labelings of the
vertices. Fixing the r.w.\ $\gamma$, we say that $f_1$ and $f_2$ belong
to the same equivalence class if the distribution of
$\{f_1(v(t))\}_{t=1}^\infty$ is equal to $\{f_2(v(t))\}_{t=1}^\infty$,
in which case one cannot distinguish between $f_1$ and $f_2$.
A first observation is that if $f_1$ and $f_2$ differ only by a cyclic
shift, i.e. there exists an $\ell$ such that for all $k$ it holds that
$f_1(k)=f_2(k+\ell)$ (again using addition in $\mathbb Z} \newcommand{\CalE}{{\mathcal{E}}_n$), then $f_1$ and
$f_2$ are in the same equivalence class. Hence the equivalence classes
of functions that cannot be distinguished contain, at the least, all
the cyclic shifts of their members.
The trivial random walk that jumps a single step to the right
w.p.\ $1$ clearly induces minimal equivalence classes; that is,
classes of functions related by cyclic shifts and nothing
more. Another trivial random walk - the one that jumps to any $k \in
\mathbb Z} \newcommand{\CalE}{{\mathcal{E}}_n$ with uniform probability - induces equivalence classes that
contain all the functions with a given number of $1$'s.
For some random walks the classes of indistinguishable functions can
be surprising. Consider the following example, illustrated in
Fig.~\ref{fig:hilary-walk} below. The random walk is on the cycle of
length $n$, for $n$ divisible by $6$ ($n=12$ in the figure). Its step
function is uniform over $\{-2,-1,1,2\}$, so that it jumps either two
steps to the left, one to the left, one to the right or two to the
right, all with probability $1/4$.
Let $f_1(k)$ be $0$ for even $k$ and $1$ otherwise. Let $f_2(k)$ be
$0$ for $k \mod 6 \in \{0,1,2\}$ and $1$ otherwise. It is easy to
see that these two functions are indistinguishable, since
the sequence of observed labels will be a sequence of uniform i.i.d.\
bits in both cases.
\begin{figure}[h]
\subfloat[$f_1$]
{
\label{fig:f_1}
\includegraphics[width=0.3\textwidth]{hilary_walk2.pdf}
}
\hspace{80pt}
\subfloat[$f_2$]
{
\label{fig:f_2}
\includegraphics[width=0.3\textwidth]{hilary_walk1.pdf}
}
\centering
\caption{\label{fig:hilary-walk} The random walk depicted here has a
step distribution that is uniform over $\{-2,-1,1,2\}$. It cannot be
used to distinguish the two very different sceneries $f_1$ and
$f_2$ above.}
\end{figure}
The question that we tackle is the following: What are the equivalence
classes for a given r.w.\ $\gamma$? In particular, which random walks
induce minimal equivalence classes, allowing the reconstruction of $f$
up to a shift?
When a random walk allows the reconstruction of any function (up to
shift) we call it {\bf reconstructive}:
\begin{definition}
Let $\gamma:H \to \R$ be the step distribution of a random walk $v(t)$
on a group $H$, so that $v(1)$ is picked uniformly from $H$ and
$\gamma(k) = \P{v(t+1)-v(t) = k}$. Then $v(t)$ is {\bf reconstructive}
if the distributions of $\{f_1(v(t))\}_{t=1}^\infty$ and
$\{f_2(v(t))\}_{t=1}^\infty$ are identical only if $f_1$ is a shift
of $f_2$.
\end{definition}
We are interested in exploring the conditions under which $v(t)$ is
reconstructive.
Howard~\cite{Howard:96} answers this question for $\gamma$ with support
on $\{-1,0,1\}$. For symmetric walks in which $\gamma(-1) = \gamma(1) \neq
0$ he shows that $f$ can be reconstructed up to a shift and a mirror
image flip (that is, $f_1$ and $f_2$ cannot be distinguished when
$f_1(k)=f_2(-k)$). In all other cases (except the trivial $\gamma(0)=1$)
he shows that the equivalence classes are minimal.
Matzinger and Lember~\cite{matzinger2006reconstruction} introduce the
use of the Fourier transform to the study of this question. They
prove the following theorem\footnote{Matzinger and Lember's notion of
a reconstructive random walk is slightly different, in that they
only require reconstruction up to a shift {\em and flip}, where the
{\em flip} of $f(k)$ is $f(-k)$. Thus Theorem~\ref{thm:distinct-f}
differs in this point from theirs.}:
\begin{theorem}[Matzinger and Lember]
\label{thm:distinct-f}
Let $\gamma$ be the step distribution of a random walk $v(t)$ on $Z_n$. Let
$\hat{\gamma}$ be the Fourier Transform of $\gamma$. Then $v(t)$ is
reconstructive if the Fourier coefficients $\{\hat{\gamma}(x)\}_{x \in
Z_n}$ are distinct.
\end{theorem}
We provide the proof for this theorem in the appendix for the reader's
convenience, extending it (straightforwardly) to random walks on any
abelian group\footnote{Matzinger and Lember state it
for ``periodic sceneries on $\mathbb Z} \newcommand{\CalE}{{\mathcal{E}}$'', which are equivalent to
sceneries on cycles.}. We henceforth use $\hat{\gamma}$ to denote the
Fourier transform of $\gamma$.
This condition is not necessary, as we show in
Theorem~\ref{thm:counterexample}. Our main result is that this
condition {\em is} necessary for random walks on $Z_n$ when $n$ is
prime and larger than 5, and $\gamma$ is rational.
\begin{theorem}
\label{thm:prime-distinct}
Let $\gamma$ be the step distribution of a random walk $v(t)$ on $Z_n$, for
$n$ prime and larger than five, and let $\gamma(k)$ be rational for all
$k$. Then $v(t)$ is reconstructive only if the Fourier
coefficients $\{\hat{\gamma}(x)\}_{x \in \mathbb Z} \newcommand{\CalE}{{\mathcal{E}}_n}$ are distinct.
\end{theorem}
When a step distribution $\gamma$ is rational (i.e., $\gamma(k)$ is rational
for all $k$) then it can be viewed as a uniform distribution over a
multiset $\Gamma$. Using this representation we make the following
definition:
\begin{definition}
The {\em drift} of a random walk on $\mathbb Z} \newcommand{\CalE}{{\mathcal{E}}_n$ with step distribution
uniform over the multiset $\Gamma$ is $D(\Gamma) = \sum_{k \in
\Gamma} k$ (with addition in $\mathbb Z} \newcommand{\CalE}{{\mathcal{E}}_n$).
\end{definition}
We show that a walk with non-zero drift is reconstructive on
prime-length cycles.
\begin{theorem}
\label{cor:drift}
Suppose $n$ is prime and greater than 5, and suppose
$v(t)$ is a random walk over $\mathbb Z} \newcommand{\CalE}{{\mathcal{E}}_n$, with step distribution uniform
over the multiset $\Gamma$. Then if $D(\Gamma) \neq 0$ then $v(t)$
is reconstructive.
\end{theorem}
We show that any fixed, bounded rational random walk on a large enough
prime cycle is either symmetric or reconstructive. Symmetric random
walks are those for which $\gamma(k)=\gamma(-k)$ for all $k$. They are not
reconstructive because they cannot distinguish between $f(k)$ and
$f(-k)$.
Given a step distribution $\gamma : \mathbb{Z} \rightarrow [0,1]$ on
$\mathbb Z} \newcommand{\CalE}{{\mathcal{E}}$, let $\gamma_n$ denote the step distribution on $\mathbb{Z}_n$
induced by $\gamma$ via
$$\gamma_n(k)= \sum_{a = k\, mod\, n} \gamma(a).$$
\begin{theorem}\label{cor:bounded}
Let $\gamma : \mathbb Z} \newcommand{\CalE}{{\mathcal{E}} \rightarrow [0,1]$ be a distribution over $\mathbb Z} \newcommand{\CalE}{{\mathcal{E}}$ with
bounded support, and assume $\gamma(a) \in \mathbb{Q}$ for all
$a$. Then either $\gamma$ is symmetric, or there exists an $N$ such
that for all prime $n > N$, $\gamma_n$ is reconstructive.
\end{theorem}
Finally, we extend the result of Theorem~\ref{thm:prime-distinct} to
random walks on any abelian group of the form $Z_{p_1}^{d_1} \times \cdots \times
Z_{p_m}^{d_m}$ where $p_1, \ldots, p_m$ are primes larger than $5$.
\begin{theorem}\label{thm:general}
Let $\gamma$ be the step distribution of a random walk $v(t)$ on
$\mathbb{Z}_{p_1}^{d_1} \times \cdots \times
\mathbb{Z}_{p_m}^{d_m}$, such that $\gamma(k_1, \ldots, k_m) \in \mathbb Q$
for all $(k_1, \ldots , k_m)$, and suppose that $p_i>5$ is prime for
all $i$. Then $v(t)$ is reconstructive only if the Fourier
coefficients $\{\hat{\gamma}(x)\}_{x \in \mathbb Z} \newcommand{\CalE}{{\mathcal{E}}_n}$ are distinct.
\end{theorem}
This result applies, in particular, to walks on $\mathbb Z} \newcommand{\CalE}{{\mathcal{E}}_n$ where $n$ is
square-free (i.e., is not divisible by a square of a prime) and not
divisible by $2$, $3$ or $5$.
We show that the rationality condition of our main theorem is tight by
presenting an example of a r.w.\ with prime $n$, irrational
probabilities and non-distinct Fourier coefficients where
reconstruction is possible. It remains open, however, whether the
result holds for rational random walks on general cycles of composite
length.
This paper will proceed as follows. In
Section~\ref{sec:preliminaries}, we will review some useful algebraic
facts. In Section~\ref{sec:mainproof}, we will prove
Theorem~\ref{thm:prime-distinct}. In Section~\ref{sec:corollaries} we
will prove Theorem~\ref{cor:drift}, on walks with non-zero drift, and
Theorem~\ref{cor:bounded}, on walks with bounded support. In
Sections~\ref{sec:tightness}, and \ref{sec:extensions} we show the
tightness of the main theorem, and extend it to products of
prime-length cycles, respectively. In Section~\ref{sec:openproblems},
we will present some open problems.
\section{Preliminaries}
\label{sec:preliminaries}
Before presenting the proofs of our theorems, we would like to
refresh the reader's memory of abelian (commutative) groups and their
Fourier transforms, as well as introduce our notation. For the cyclic
group $\mathbb Z} \newcommand{\CalE}{{\mathcal{E}}_n$, the Fourier transform of $\gamma: \mathbb Z} \newcommand{\CalE}{{\mathcal{E}}_n \to \mathbb C$ is defined
by
\begin{align*}
\hat{\gamma}(x) = \sum_{k \in \mathbb Z} \newcommand{\CalE}{{\mathcal{E}}_n}\omega_n^{kx}\gamma(k).
\end{align*}
where $\omega_n = e^{-\frac{2\pi}{n}i}$.
Any finite abelian group can be written as the Cartesian product of
cycles with prime power lengths: if $H$ is abelian then there exist
$\{n_i\}$ such that $H$ is isomorphic to the {\em torus}
$\mathbb Z} \newcommand{\CalE}{{\mathcal{E}}_{n_1}\times \cdots \times \mathbb Z} \newcommand{\CalE}{{\mathcal{E}}_{n_m}$. Thus an element $h \in H$
can be thought of as a vector so that $h_i$ is in $\mathbb Z} \newcommand{\CalE}{{\mathcal{E}}_{n_i}$.
When all the $n_i$'s are the same then $H=\mathbb Z} \newcommand{\CalE}{{\mathcal{E}}_n^d$ is a {\em regular
torus}, and when $n$ is prime then it is a {\em prime regular
torus}. Here a natural dot product exists for $h,k \in H$: $h \cdot
k=\sum_{i=1}^dh_ik_i$, where both the multiplication and summation are
over $\mathbb Z} \newcommand{\CalE}{{\mathcal{E}}_n$. The Fourier transform for $\mathbb Z} \newcommand{\CalE}{{\mathcal{E}}_n^d$ is thus
\begin{align*}
\hat{\gamma}(x) = \sum_{k \in \mathbb Z} \newcommand{\CalE}{{\mathcal{E}}_n^d}\omega_n^{k \cdot x}\gamma(k).
\end{align*}
The representation of abelian groups as tori is not unique. The
canonical representation is $H=\mathbb Z} \newcommand{\CalE}{{\mathcal{E}}_{n_1}^{d_1}\cdots\mathbb Z} \newcommand{\CalE}{{\mathcal{E}}_{n_m}^{d_m}$,
where the $n_i$'s are distinct powers of primes. In this general case
an element $h \in H$ can be thought of as a vector of vectors, so that
$h_i$ is in $\mathbb Z} \newcommand{\CalE}{{\mathcal{E}}_{n_i}^{d_i}$. The Fourier transform is
\begin{align*}
\hat{\gamma}(x) = \sum_{k \in H}\prod_{i=1}^m\omega_{n_i}^{k_i \cdot x_i}\gamma(k).
\end{align*}
The {\em $n$-th cyclotomic polynomial} is the minimal polynomial of $\omega_n$; i.e. the lowest
degree polynomial over $\mathbb Q$ that has $\omega_n$ as a root and a
leading coefficient of one. This polynomial must divide all non-zero
polynomials over $\mathbb Q$ that have $\omega_n$ as a root. When $n$ is prime then this polynomial is $Q_n(t)= \sum_{i=0}^{n-1}
t^i$.
\section{Proof of Theorem~\ref{thm:prime-distinct}}
\label{sec:mainproof}
\subsection{Outline of main proof}
Theorem~\ref{thm:prime-distinct} states the distinctness of the Fourier
coefficients of the step distribution $\gamma$ is a necessary (and by
Theorem~\ref{thm:distinct-f} sufficient) condition for reconstruction,
when $n$ is prime and greater than five, and the probabilities
$\gamma(k)$ are rational. In Theorem~\ref{thm:counterexample} we give a
counterexample that shows that the rationality condition is tight,
giving an irrational random walk on $\mathbb Z} \newcommand{\CalE}{{\mathcal{E}}_7$ with non-distinct Fourier
coefficient, which is reconstructive. In Section~\ref{sec:tightness}, we
show that the condition $n>5$ is also tight.
To prove Theorem~\ref{thm:prime-distinct} we shall construct,
for any random walk on a cycle of prime length larger than 5 with rational step distribution
$\gamma$ such that $\hat{\gamma}(x) = \hat{\gamma}(y)$ for some $x \neq y$, two functions $f_1$ and $f_2$ that differ by more than a
shift. We shall consider two random walks: $v_1(t)$ on $f_1$ and
$v_2(t)$ on $f_2$, with the same step distribution $\gamma$. If a
coupling exists such that the observed sequences $\{f_1(v_1(t))\}$ and
$\{f_2(v_2(t))\}$ are identical then $f_1$ and $f_2$ cannot be
distinguished, since the sequences will have the same marginal
distribution. We will show that such a coupling always exists.
\subsection{Motivating example}
Consider the following example. Let the step distribution $\gamma$ uniformly
choose to move 1, 2, or 4 steps to the left on a cycle of length 7. A
simple calculation will show that $\hat{\gamma}(1) = \hat{\gamma}(2)$. In
what is not a coincidence (as we show below), the support of $\gamma$ is
invariant under multiplication by 2 (over $Z_7$): $2\{1,2,4\} =
\{1,2,4\}$. We will use this fact to construct two functions $f_1$ and
$f_2$ that are indistinguishable, even though they are not related by
a shift.
Let $f_1(k)$ equal 1 for $k \in \{0,1\}$ and $f_2(k)$ equal 1 for $k
\in \{0,2\}$, as shown in Fig.~\ref{fig:seven-walk}. (More generally, $f_1$ and $f_2$ can be any two functions such that $f_1(k) = f_2(2k)$ for all $k$.) Let $v_1(t)$ be a
random walk over $Z_7$ with step distribution $\gamma$. We couple it to a
random walk $v_2(t)$ by setting $v_2(t) = 2 \cdot v_1(t)$, so that
$v_2$ always jumps twice as far as $v_1$. Its step distribution is
thus uniform over $2 \cdot \{1,2,4\}=\{1,2,4\}$, and indeed has the
same step distribution $\gamma$. Finally, because $f_2(2k) = f_1(k)$, we
have $f_2(v_2(t)) = f_1(v_1(t))$ for all $t$.
We have thus constructed two random walks, both with step distribution
$\gamma$. The first walks on $f_1$ and the second walks on $f_2$, and yet
their observations are identically distributed. Hence the two
functions are indistinguishable.
Note also that the sum of the elements in the support of $\gamma$ is
zero: $1+2+4=0 \mod 7$. This is not a coincidence; as we show
in Theorem~\ref{cor:drift} this must be the case for every step
distribution that is not reconstructive.
\begin{figure}[h]
\subfloat[$f_1$]
{
\label{fig:f_1}
\includegraphics[width=0.3\textwidth]{seven-walk1.pdf}
}
\hspace{80pt}
\subfloat[$f_2$]
{
\label{fig:f_2}
\includegraphics[width=0.3\textwidth]{seven-walk2.pdf}
}
\centering
\caption{\label{fig:seven-walk} We couple the random walks on $f_1$
and $f_2$ above by choosing, at each time period, either `a', `b'
or `c' uniformly and having each walk take the step marked by that
letter in its diagram. The result is that $f_1(v_1(t)) =
f_2(v_2(t))$. }
\end{figure}
\subsection{Proof of Theorem~\ref{thm:prime-distinct}}
To prove Theorem~\ref{thm:prime-distinct} we will first prove two
lemmas. We assume here that $\gamma(k)$ is rational for all $k$ and that
$n>5$ is prime.
\begin{lemma}\label{lemma:xS=yS} Suppose that $\hat{\gamma}(x) =
\hat{\gamma}(y)$ for some $x, y \neq 0$. Then $\gamma(k
x^{-1} y) = \gamma(k)$ for all $k$, where all operations are in the field $\mathbb{Z}_n$.
\end{lemma}
In other words, the random walk has the same probability to add $k$ or
$kx^{-1} y$, for all $k$.
\begin{proof}
Letting $\omega$ denote $\omega_n=e^{-\frac{2\pi}{n}i}$ and applying the definition
of $\hat{\gamma}$, we have
$$\sum_{k = 0}^{n-1}\gamma(k) \omega^{kx} = \sum_{k=0}^{n-1}\gamma(k)\omega^{ky}.$$
Since $\omega^n = 1$, $\omega$ is a root of the polynomial
$$P(t) = \sum_{k = 0}^{n-1}\gamma(k) t^{kx\, (mod\, n)} - \sum_{k=0}^{n-1}\gamma(k)t^{ky\, (mod\, n)}.$$
or, by a change of variables
$$P(t) = \sum_{k = 0}^{n-1}\left(\gamma(kx^{-1}) -\gamma(k y^{-1}) \right)t^{k}$$
where the inverses are taken in the field $\mathbb{Z}_n$. (Recall $n$ is prime.)
Since $P$ has $\omega_n$ as a root then $Q_n$ divides $P$. However,
$P$ has degree at most $n-1$, so either $P$ is the zero polynomial
or $P$ is equal to a constant times $Q_n$.
The latter option is impossible, since $P(1) = 0$ and $Q_n(1) =
n$. Thus, $P = 0$, so $\gamma(k x^{-1}) = \gamma(k y^{-1})$ for all
$k$, or equivalently, $\gamma(k x^{-1} y) = \gamma(k)$ for all $k$.
\end{proof}
\begin{lemma}\label{lemma:prime-coupling}
If $\hat{\gamma}(x) = \hat{\gamma}(y)$ for some $x, y
\neq 0$, and if $f_1, f_2:\mathbb{Z}_n \rightarrow \{0,1\}$ are such
that $f_1(k) = f_2(x^{-1}yk)$ for all $k$, then $\gamma$ cannot
distinguish between $f_1$ and $f_2$.
\end{lemma}
\begin{proof}
Let $v_1(t)$ be a $\gamma$-r.w.\ on $f_1$. Let $v_2(t)$ be a random
walk defined by $v_2(t) = x^{-1}y v_1(t)$, so that whenever $v_1$
jumps $k$ then $v_2$ jumps $x^{-1}yk$.
By Lemma~\ref{lemma:xS=yS} we have that $\gamma(x^{-1}yk) = \gamma(k)$, so
the step distribution of $v_2$ is also $\gamma$. Furthermore, at time
$t$, $v_1$ will see $f_1(v_1(t))$ and $v_2$ will see $f_2(v_2(t)) =
f_2(x^{-1}yv_1(t))$. But for all $k$, $f_1(k) = f_2(x^{-1}y \cdot
k)$, and so $f_1(v_1(t))=f_2(v_2(t))$ for all $t$.
\end{proof}
To conclude the proof of Theorem~\ref{thm:prime-distinct}, we must
find, for all prime $n > 5$ and all $x \neq y$, $x, y \neq 0$ functions $f_1$ and
$f_2$ such that $f_1$ is not a shift of $f_2$, and $f_1(k) =
f_2(x^{-1}yk)$. We use another argument for the special case $x=0$ or $y = 0$.
\begin{proof}[Proof of Theorem~\ref{thm:prime-distinct}]
For $x, y \neq 0$, by Lemma~\ref{lemma:prime-coupling} we can choose
any $f_1$ and $f_2$ such that $f_1(k) = f_2(x^{-1}yk)$ and $f_1$ is
not a shift of $f_2$; for example, if $x^{-1}y \neq -1$, then
$$f_1(k) = \begin{cases}
1 & k = 0,1\\
0 & o.w.
\end{cases}$$
and
$$f_2(k) = \begin{cases}
1 & k = 0, x^{-1}y\\
0 & o.w.
\end{cases}$$
satisfy the requirements. For $n=7$ and $x^{-1}y=2$ these two
functions are depicted in Fig.~\ref{fig:seven-walk}.
If $x^{-1}y = -1$, then we can choose any function which is not a
reflection or shift of itself; for example,
$$f_1(k) = \begin{cases}
1& k = 0, 1, 3 \\
0 & o.w.
\end{cases}$$
and $f_2(k)=f_1(-k)$.
This leaves us with the case $x=0$ or $y=0$; wlog, suppose $x = 0$. We
still have $P(t) \equiv 0$ by the same proof as above, which gives us
\begin{eqnarray*}
0 & \equiv& P(t)\\
& = & \sum_{k = 0}^{n-1}\gamma(k) t^{kx\, (mod\, n)} - \sum_{k=0}^{n-1}\gamma(k)t^{ky\, (mod\, n)}\\
&=& 1- \sum_{k=0}^{n-1}\gamma(k)t^{ky\, (mod\, n)}\\
&=& 1- \sum_{k=0}^{n-1}\gamma(ky^{-1})t^{k}\\
\end{eqnarray*}
where the last equality is possible because $y \neq x=0$. But then we
must have $\gamma(0) = 1$, so the random walk does not move after the
first vertex is chosen. Thus, $\gamma$ cannot distinguish between any
two functions with the same number of ones.
\end{proof}
\section{Walks with non-zero drift and walks with bounded support}
\label{sec:corollaries}
In this section, we will use the main lemma prove Theorems~\ref{cor:drift} and \ref{cor:bounded}.
\subsection{Drift and reconstruction}
A step distribution $\gamma$ such that $\gamma(k) \in \mathbb Q$ for all $k$ can be
viewed as a uniform distribution over a multiset $\Gamma$. Given
$\Gamma$, define the {\em drift} of a random walk on $\mathbb Z} \newcommand{\CalE}{{\mathcal{E}}_n$ with step
function $\Gamma$ as $D(\Gamma) = \sum_{k \in \Gamma} k$, with
addition in $\mathbb Z} \newcommand{\CalE}{{\mathcal{E}}_n$.
\begin{theorem*}[\ref{cor:drift}]
Suppose $n$ is prime and greater than 5, and suppose
$v(t)$ is a random walk over $\mathbb Z} \newcommand{\CalE}{{\mathcal{E}}_n$, with step distribution uniform
over the multiset $\Gamma$. Then if $D(\Gamma) \neq 0$ then $v(t)$
is reconstructive.
\end{theorem*}
\begin{proof}
We will show that if reconstruction is not possible then the drift
is zero.
If $f$ cannot be reconstructed then there exist $x\neq y$ with
$\hat{\gamma}(x) = \hat{\gamma}(y)$. Then it may be that $x$ or $y$ is 0;
in this case, as we have shown in the proof of
Theorem~\ref{thm:prime-distinct}, the random walk never moves, so
the drift is zero. Otherwise $x, y \neq 0$, which by
Lemma~\ref{lemma:xS=yS} implies $\gamma(k) = \gamma(x^{-1}yk)$, or
alternatively $x^{-1}y\Gamma = \Gamma$. But then
\begin{eqnarray*}
D(\Gamma) &=& \sum_{k \in \Gamma} k\\
&=& \sum_{k \in \Gamma} x^{-1}yk\\
&=& xy^{-1} D(\Gamma)\\
\end{eqnarray*}
so $D(\Gamma) = 0$.
\end{proof}
\subsection{Random walks with bounded step distribution}
Next, we consider random walks with rational transition probabilities,
and with zero probability for steps larger than $c$, for some constant
$c$ independent of $n$. We show that in this case, either the random
walk is symmetric, or for large enough prime cycles, the walk is
reconstructive. A random walk $\gamma$ is symmetric when $\gamma(k) =
\gamma(-k)$ for all $k$. Symmetric random walks are not reconstructive,
since $\hat{\gamma}(k)=\hat{\gamma}(-k)$; they cannot distinguish any
function $f(k)$ from its flip $f(-k)$.
Given a distribution $\gamma :\mathbb Z} \newcommand{\CalE}{{\mathcal{E}} \rightarrow [0,1]$, let $\gamma_n$ denote
the step distribution on $\mathbb{Z}_n$ induced by $\gamma$ via the
natural embedding
$$\gamma_n(k)= \sum_{a = k\, mod\, n} \gamma(a).$$
\begin{theorem*}[\ref{cor:bounded}]
Let $\gamma : \mathbb Z} \newcommand{\CalE}{{\mathcal{E}} \rightarrow [0,1]$ be a distribution over $\mathbb Z} \newcommand{\CalE}{{\mathcal{E}}$ with
bounded support, and assume $\gamma(a) \in \mathbb{Q}$ for all
$a$. Then either $\gamma$ is symmetric, or there exists an $N$ such
that for all prime $n > N$, $\gamma_n$ is reconstructive.
\end{theorem*}
We will need two lemmas in order to prove this theorem.
As we note above, a rational step distribution $\gamma$ can be thought of
as a uniform distribution over a multiset $\Gamma=(a_1,\ldots,a_m) \in
\mathbb Z} \newcommand{\CalE}{{\mathcal{E}}^m$. We denote by $\Gamma_n=(k_1,\ldots,k_m) \in \mathbb Z} \newcommand{\CalE}{{\mathcal{E}}_n^m$ the natural
embedding of $\Gamma$ into $\mathbb Z} \newcommand{\CalE}{{\mathcal{E}}_n$: $k_i = a_i \mod n$.
\begin{lemma}
Let $\Gamma =(a_1,\ldots,a_m) \in \mathbb Z} \newcommand{\CalE}{{\mathcal{E}}^m$ be a multiset. Let
$\Gamma_n=(k_1,\ldots,k_m)$ be the natural embedding of $\Gamma$
into $\mathbb Z} \newcommand{\CalE}{{\mathcal{E}}_n$. Assume further that $1 \in \Gamma$, i.e., $a_i=1$ for
some $i$.
Then there exists a positive integer $N = N(\Gamma)$ such that for
any $n > N$ it holds that if $v\Gamma_n=\Gamma_n$ then $v \in
\{-1,0,1\}$.
\end{lemma}
\begin{proof}
\label{lemma:Gamma_n}
Let $b = \max_i|a_i|$, so that $\Gamma$ is bounded in $[-b,b]$, and
assume w.l.o.g that $b \in \Gamma$. Let $N=2b^2$, $n>N$ and let
$v\Gamma_n=\Gamma_n$. We will show that $v \in \{-1,0,1\}$.
Because $1 \in \Gamma_n$ and $v\Gamma_n = \Gamma_n$ we have $v \in
\Gamma_n$. Assume by way of contradiction that $v \not \in
\{-1,0,1\}$. Then $v \in [-b, -2] \cup [2, b]$. Since $b \in
\Gamma_n$ we also have $vb \in \Gamma_n$, so $vb \in [-b^2, -2b]
\cup [2b, b^2]$. But this is a contradiction because $n > 2b^2$ so
both of these intervals have empty intersection with $\Gamma_n$.
\end{proof}
\begin{lemma}
\label{lemma:symmetric_reconstructive}
Let $\Gamma_n$ be a multiset characterizing a random walk $\gamma_n$
over $\mathbb Z} \newcommand{\CalE}{{\mathcal{E}}_n$, such that if $v\Gamma_n=\Gamma_n$ then $v \in
\{-1,0,1\}$. Then the random walk is either symmetric or
reconstructive.
\end{lemma}
\begin{proof}
Suppose the random is not reconstructive; i.e. there exist $x \neq
y$ with $\hat{\gamma_n}(x) = \hat{\gamma_n}(y)$. We will show that this
means that the random walk is symmetric.
If $x$ is 0, then by the argument in the proof of
theorem~\ref{thm:prime-distinct} we have that $\Gamma = \{0\}$,
which is symmetric, and we are done. Otherwise denote $v =
x^{-1}y$. Then by Lemma~\ref{lemma:xS=yS} we have $\Gamma_n = v
\Gamma_n$, and by this lemma's hypothesis we have $v \in
\{-1,0,1\}$.
Now, $v$ cannot equal $1$ since $v = x^{-1}y$ and $x \neq y$. If
$v=-1$ then the random walk is symmetric. Finally, if $v=0$ then
again $\Gamma = \{0\}$, and again the random walk is symmetric.
\end{proof}
\begin{proof}[Proof of Theorem~\ref{cor:bounded}]
Let $\Gamma \subset \mathbb Z} \newcommand{\CalE}{{\mathcal{E}}$ correspond to $\gamma$ as above (i.e., $\gamma$ is
uniformly distributed over the multiset $\Gamma$). We will show that
for $n$ large enough if $v\Gamma_n = \Gamma_n$ then $v \in
\{-1,0,1\}$, which, by Lemma~\ref{lemma:symmetric_reconstructive}
will show that the random walk is either symmetric or reconstructive.
For any vector of coefficients $c = (c_1, \ldots, c_m)$, define the
multiset $c(\Gamma)$ as
$$c(\Gamma) = c_1\Gamma + \ldots + c_m\Gamma=\{c_1s_1 + \ldots + c_ms_m
: s_i \in \Gamma\}.$$ Note that if $v\Gamma = \Gamma$, then we
also have that $v c(\Gamma) = c(\Gamma)$.
Suppose that the g.c.d.\ (in $\mathbb Z} \newcommand{\CalE}{{\mathcal{E}}$) of the elements of $\Gamma$ is
one. Then by the Chinese remainder theorem, there exists a $c$ such
that $1 \in c(\Gamma)$. Hence we can apply Lemma~\ref{lemma:Gamma_n}
to $c(\Gamma)$ and infer that for $n$ large enough $v c(\Gamma_n) =
c(\Gamma_n)$ implies $v \in \{-1,0,1\}$. But since $v\Gamma_n =
\Gamma_n$, implies $v c(\Gamma_n) = c(\Gamma_n)$ then we have that
for $n$ large enough $v\Gamma_n = \Gamma_n$ implies $v \in
\{-1,0,1\}$.
Otherwise, let $d$ denote the g.c.d.\ of $\Gamma$. Since the
g.c.d.\ of $d^{-1}\Gamma$ is one we can apply the argument of the
previous case to $d^{-1}\Gamma$ and infer that for $n$ large enough
$vd^{-1}\Gamma_n = d^{-1}\Gamma_n$ implies $v \in \{-1,0,1\}$. But
since again $v\Gamma_n = \Gamma_n$ implies $vd^{-1}\Gamma_n =
d^{-1}\Gamma_n$, then again we have that for $n$ large enough
$v\Gamma_n = \Gamma_n$ implies $v \in \{-1,0,1\}$.
\end{proof}
\section{Tightness of Theorem~\ref{thm:prime-distinct}}
\label{sec:tightness}
Theorem~\ref{thm:prime-distinct} states that distinctness of Fourier coefficients is not just sufficient but necessary for reconstructibility of a random walk $\gamma$ on a cycle of length $n$, if the following conditions hold:
\begin{enumerate}
\item $\gamma(k) \in \mathbb Q$ for all $k$,
\item $n> 5$,
\item $n$ is prime.
\end{enumerate}
To see that condition 2 is tight, note that the simple random walk has
$\hat{rw}(k) = \hat{rw}(-k)$ for all $k$, so for $n > 2$, the simple
random walk does not have distinct Fourier coefficients. However,
Howard~\cite{Howard:96} shows that the simple random walk suffices to
reconstruct any scenery up to a flip, and for $n<5$, reconstruction up
to a flip is the same as reconstruction up to a shift, since all
sceneries are symmetric. So for $n = 3,4,5$, the simple random walk is
reconstrive despite having non-distinct Fourier coefficints. For $n =
2$, consider the random walk that simply stays in place.
The following theorem shows that condition 1 is also tight.
\begin{theorem}\label{thm:counterexample}
There exists a reconstructive random walk $\gamma$ on $\mathbb{Z}_7$ such that $\hat{\gamma}(3) =\hat{\gamma}(-3)$.
\end{theorem}
\begin{proof}
Let $\delta = \frac{\cos\left( 6\pi /7\right) + 0.5}{2
\cos\left(6\pi/7 \right) - 1}$, $\gamma(1) = {\textstyle \frac12} + \delta$ and
$\gamma(2) = {\textstyle \frac12} - \delta$. To show that $\hat{\gamma}(3) = \hat{\gamma}(-3)$ is a simple calculation, so we will only show that $\gamma$ can distinguish between any two sceneries.\\
In fact, any random walk with non-zero support exactly on $\{1,2\}$ can distinguish between any two sceneries when $n = 7$. Any r.w. can determine the number of ones in a scenery, so we must show that $\gamma$ can distinguish between non-equivalent sceneries for each number of ones.
\begin{itemize}
\item There is only one scenery with 0 ones and up to a shift, there is only one scenery with 1 one.
\item For 2 ones, $\gamma$ must distinguish among $(1,1,0,0,0,0,0), (1,0,1,0,0,0,0), (1,0,0,1,0,0,0)$. (The others are equivalent to one of these up to a shift.) The third function is the only one for which there will never be two consecutive ones. The first function is the only one for which there will be two consecutive ones, but the substring $(1,0,1)$ never appears. The third function is the only one for which both substrings $(1,1)$ and $(1,0,1)$ will appear.
\item For 3 ones, $\gamma$ must distinguish among:
(a) $(1,1,1,0,0,0,0)$,
(b) $(1,1,0,1,0,0,0)$,
(c) $(1,1,0,0,1,0,0$),
(d) $(1,1,0,0,0,1,0)$, and
(e) $(1,0,1,0,1,0,0)$.
(c) is the only one that will never have three consecutive ones. (e) is the only that will ever have five consecutive zeros. Among (a), (b), and (d), (a) is the only such that (1,0,1) will never occur, and while (1,1,0,1,0,0) will occur for (b), it won't for (d).
\item For 4,5,6, and 7 ones, repeat the previous arguments with the roles of 0 and 1 reversed.
\end{itemize}
\end{proof}
\section{Extensions}
\label{sec:extensions}
\subsection{Extension to prime regular tori}
\begin{theorem}\label{thm:regular_torus}
Let $\gamma$ be the step distribution of a random walk $v(t)$ on $Z_n^d$, for
$n$ prime and larger than five, and let $\gamma(k)$ be rational for all
$k$. Then $v(t)$ is reconstructive {\em only if} the Fourier
coefficients $\{\hat{\gamma}(x)\}_{x \in \mathbb Z} \newcommand{\CalE}{{\mathcal{E}}_n}$ are distinct.
\end{theorem}
\begin{proof}
Suppose that $\hat{\gamma}(x) = \hat{\gamma}(y)$ for some $x \neq
y$, so that the Fourier coefficients aren't distinct. Again letting
$\omega$ denote $\omega_n=e^{-\frac{2\pi}{n}i}$, we have
\begin{align*}
\sum_{k \in \mathbb Z} \newcommand{\CalE}{{\mathcal{E}}_n^d}\gamma(k) \omega^{k \cdot x} =
\sum_{k \in \mathbb Z} \newcommand{\CalE}{{\mathcal{E}}_n^d}\gamma(k)\omega^{k \cdot y}.
\end{align*}
Note that here $k \cdot y$ is the natural dot product over
$\mathbb Z} \newcommand{\CalE}{{\mathcal{E}}_n^d$. If we join terms with equal powers of $\omega$ then
\begin{align*}
\sum_{\ell \in \mathbb Z} \newcommand{\CalE}{{\mathcal{E}}_n}\left(\sum_{\substack{k\;\mathrm{s.t.} \\ k \cdot x
= \ell}}\gamma(k)\right) \omega^\ell =
\sum_{\ell \in \mathbb Z} \newcommand{\CalE}{{\mathcal{E}}_n}\left(\sum_{\substack{k\;\mathrm{s.t.} \\ k \cdot y
= \ell}}\gamma(k)\right) \omega^\ell =
\end{align*}
and by the same argument as in the proof of Lemma~\ref{lemma:xS=yS} we have that,
for all $\ell \in Z_n$,
\begin{align}
\label{eq:same-dist-torus}
\sum_{\substack{k\;\mathrm{ s.t. }\\ k \cdot x = \ell}}\gamma(k) =
\sum_{\substack{k\;\mathrm{ s.t. }\\ k \cdot y = \ell}}\gamma(k)
\end{align}
I.e., if $k$ is distributed according to $\gamma$ then $k \cdot x$ and
$k \cdot y$ have the same distribution. Hence, if we denote
$u_x(t)=v(t) \cdot x$ and $u_y(t) = v(t) \cdot y$ then we also have
identical distributions, under $\gamma$, of $u_x(t)$ and $u_y(t)$.
Fix $g:\mathbb Z} \newcommand{\CalE}{{\mathcal{E}}_n \to \{0,1\}$, and let $f_x(k) = g(x \cdot k)$, and
$f_y(k) = g(y \cdot k)$. Then $f_x(v(t)) = g(u_x(t))$, $f_y(v(t)) =
g(u_y(t))$, and the distributions of $\{f_x(v(t)\}$ and
$\{f_y(v(t))\}$ are identical, by the same coupling argument used in
Lemma~\ref{lemma:prime-coupling} above. Hence $f_x$ and $f_y$ can't
be distinguished.
It remains to show that there exists a $g$ such that $f_x$ and $f_y$
differ by more than a shift. We consider two cases.
\begin{enumerate}
\item Let $x$ be a multiple of $y$, so that $x=\ell y$ for some $
\ell \in \mathbb Z} \newcommand{\CalE}{{\mathcal{E}}_n$, $\ell \neq 1$. Then the problem is essentially
reduced to the one dimensional case of
Theorem~\ref{thm:prime-distinct}, with the random walk projected
on $y$. We can set
$$g(k) = \begin{cases}
1& k = 0, 1 \\
0 & o.w.
\end{cases}$$
when $\ell \neq -1$ and
$$g(k) = \begin{cases}
1& k = 0, 1, 3 \\
0 & o.w.
\end{cases}$$
when $\ell = -1$.
Assume by way of contradiction that $f_x$ and $f_y$ differ by a
shift, so that there exists a $k_0$ such that $f_x(k)=f_y(k +
k_0)$ for all $k$. Since $$f_x(k) = g(x \cdot k) = g(\ell y \cdot
k)$$ and $$f_y(k+k_0) = g(y \cdot (k+k_0)) = g(y \cdot k+y \cdot
k_0)$$ for all $k$, then $$g(\ell m)=g(m+m_0)$$ for some $m_0 \in
Z_n$ and all $m \in Z_n$. That is, $g(m)$ is a shift of $g(\ell
m)$. It is easy to verify that this is not possible in the case
$\ell \neq -1$ nor in the case $\ell = -1$.
\item Otherwise $x$ is not a multiple of $y$. Hence they are
linearly independent.
We here set
$$g(k) = \begin{cases}
1& k = 0 \\
0 & o.w.
\end{cases}$$ One can then view $f_x$ ($f_y$), as the indicator
function of the set elements of $\mathbb Z} \newcommand{\CalE}{{\mathcal{E}}_n^d$ orthogonal to $x$
($y$). Denote these linear subspaces as $U_x$ and $U_y$.
Assume by way of contradiction that $f_x$ and $f_y$ differ by a
shift, so that there exists a $k_0$ such that $f_x(k)=f_y(k +
k_0)$ for all $k$. Then $U_x = U_y + k_0$. Since $0$ is an element
of both $U_x$ and $U_y$ then it follows that $k_0$ is also an
element of both. But, being a linear space, $U_y$ is closed under
addition, and so $U_x = U_y+k_0 = U_y$. However, since $x$ and $y$
are linearly independent then their orthogonal spaces must be
distinct, and we've reached a contradiction.
\end{enumerate}
\end{proof}
\subsection{Extension to products of prime tori}
Recall that any abelian group $H$ can be decomposed into
$H=\mathbb Z} \newcommand{\CalE}{{\mathcal{E}}_{n_1}^{d_1}\times \cdots \times \mathbb Z} \newcommand{\CalE}{{\mathcal{E}}_{n_m}^{d_m}$, where the $n_i$ are distinct
powers of primes. We call $H$ ``square free'' when all the $n_i$'s are
in fact primes. In this case too (for primes $>5$) we show that the
distinctness of Fourier coefficients is a tight condition for
reconstructibility.
\begin{theorem*}[\ref{thm:general}]
Let $\gamma$ be the step distribution of a random walk $v(t)$ on
$\mathbb{Z}_{p_1}^{d_1} \times \cdots \times
\mathbb{Z}_{p_m}^{d_m}$, such that $\gamma(k_1, \ldots, k_m) \in \mathbb Q$
for all $(k_1, \ldots , k_m)$, and suppose that $p_i>5$ is prime for
all $i$. Then $v(t)$ is reconstructive {\em only if} the Fourier
coefficients $\{\hat{\gamma}(x)\}_{x \in \mathbb Z} \newcommand{\CalE}{{\mathcal{E}}_n}$ are distinct.
\end{theorem*}
We can assume w.l.o.g.\ that the $p_i$ are all distinct. We will prove
two lemmas before proving the theorem.
Recall that $\mathbb Q(\omega_{p_1}, \ldots, \omega_{p_{j-1}})$ is the field
extension of $\mathbb Q$ by $\omega_{p_1}, \ldots, \omega_{p_{j-1}}$.
\begin{lemma}
\label{lemma:min_poly}
For all $j$, the minimal polynomial $Q_j$ of $\omega_{p_j}$ over
$\mathbb Q(\omega_{p_1}, \ldots, \omega_{p_{j-1}})$ is
$$Q_j(t) = \sum_{i =
0}^{p_j - 1}t^i.$$
\end{lemma}
\begin{proof}
Let $n = p_1 \cdots p_j$, let $F_j = \mathbb Q(\omega_{p_1}, \ldots,
\omega_{p_j})$, let $F_{j-1} = \mathbb Q(\omega_{p_1}, \ldots,
\omega_{p_{j-1}})$, and let $\zeta =\omega_{p_1}\cdots \omega_{p_j}$
. $F_j$ contains $\zeta$ so $F_j \supset \mathbb Q(\zeta)$, and since each
$\omega_{p_i}$ is a power of $\zeta$, $F_j \subset \mathbb Q(\zeta)$; thus,
$F_j = \mathbb Q(\zeta)$. The degree of $\zeta$ over $\mathbb Q$ is $(p_1-1)\cdots
(p_j-1)$, since $\zeta$ is a primitive $n$-th root of unity and
$(p_1-1)\cdots (p_j-1)$ is the number of integers coprime to
$n$. Thus, $[F_j:\mathbb Q] = [\mathbb Q(\zeta):\mathbb Q] = (p_1 - 1) \cdots (p_j -
1)$. Similarly, $[F_{j-1}:\mathbb Q] = (p_1-1)\cdots (p_{j-1}-1)$, so
$[F_j:F_{j-1}] = [F_j : \mathbb Q] / [F_{j-1}:\mathbb Q] = p_j - 1$, so $p_j -1$
is the degree of the minimal polynomial of $\omega_{p_j}$ over
$F_{j-1}$. Since this is also the degree of $\omega_{p_j}$ over
$\mathbb Q$, the minimal polynomial of $\omega_{p_j}$ over $F_{j-1}$ is the
same as the minimal polynomial of $\omega_{p_j}$ over $\mathbb Q$, which is
the one given in the lemma.
\end{proof}
\begin{lemma}\label{lemma:general_coupling}
Suppose $\hat{\gamma}(x_1, \ldots x_m) = \hat{\gamma}(y_1, \ldots y_m)$ for
$(x_1, \ldots, x_m), (y_1, \ldots, y_m) \in \mathbb{Z}_{p_1}^{d_1} \times
\cdots \times \mathbb{Z}_{p_m}^{d_m}$. Then
for all $(\ell_1, \ldots , \ell_m) \in \mathbb{Z}_{p_1} \times
\cdots \times \mathbb{Z}_{p_m}$, $$\sum_{\substack{(k_1,
\ldots, k_m) \\ k_i \cdot x_i = \ell_i \, \forall i}} \gamma(k_1,
\ldots, k_m) = \sum_{\substack{(k_1, \ldots, k_m) \\ k_i \cdot y_i =
\ell_i\, \forall i}} \gamma(k_1, \ldots, k_m).$$
\end{lemma}
\begin{proof}
We will prove by induction on $j$ that the following statement holds: for all $(\ell_{j+1}, \ldots , \ell_m) \in \mathbb{Z}_{p_{j+1}} \times \cdots \times \mathbb{Z}_{p_m}$,
$$\sum_{\substack{(k_1, \ldots, k_m)\\k_i \cdot x_i = \ell_i \, \forall i>j}} \gamma(k_1, \ldots, k_m) \omega_{p_1}^{k_1 \cdot x_1} \cdots \omega_{p_{j }}^{k_{j } \cdot x_{j}}
= \sum_{\substack{(k_1, \ldots, k_m) \\ k_i \cdot y_i = \ell_i \, \forall i>j}} \gamma(k_1, \ldots, k_m) \omega_{p_1}^{k_1 \cdot y_1} \cdots \omega_{p_{j }}^{k_{j} \cdot y_{j}}.$$
The base case $j = m$ is just the statement $\hat{\gamma}(x_1, \ldots, x_m) = \hat{\gamma}(y_1, \ldots, y_m)$. Now suppose we know that for all $(\ell_{j+1}, \ldots, \ell_m)$
$$\sum_{\substack{(k_1, \ldots, k_m)\\k_i \cdot x_i = \ell_i \, \forall i>j}} \gamma(k_1, \ldots, k_m) \omega_{p_1}^{k_1 \cdot x_1} \cdots \omega_{p_{j}}^{k_{j} \cdot x_{j}}
= \sum_{\substack{(k_1, \ldots, k_m) \\ k_i \cdot y_i = \ell_i \, \forall i>j}} \gamma(k_1, \ldots, k_m) \omega_{p_1}^{k_1 \cdot y_1} \cdots \omega_{p_{j }}^{k_{j} \cdot y_{j}}.$$
Then $\omega_{p_{j}}$ is a root of the polynomial
$$P(t) = \sum_{\substack{(k_1, \ldots, k_m)\\k_i \cdot x_i = \ell_i \, \forall i>j}} \gamma(k_1, \ldots, k_m) \omega_{p_1}^{k_1 \cdot x_1} \cdots \omega_{p_{j-1}}^{k_{j-1} \cdot x_{j-1}} t^{k_j \cdot x_j}
- \sum_{\substack{(k_1, \ldots, k_m) \\ k_i \cdot y_i = \ell_i \, \forall i>j}} \gamma(k_1, \ldots, k_m) \omega_{p_1}^{k_1 \cdot y_1} \cdots \omega_{p_{j-1 }}^{k_{j-1} \cdot y_{j-1}} t^{k_j \cdot y_j}.$$
But as in the proof of the previous theorem, Lemma~\ref{lemma:min_poly} tells us that $P(t) = Q_j(t)$ or $P(t) \equiv 0$, and so $P(1) = 0 \neq p_j = Q_j(1)$ tells us that $P(t) \equiv 0$. Thus, the coefficient of $t^{\ell_j}$ must be zero for all choices of $\ell_j$, establishing the desired statement for $j-1$. When $j= 0$, this gives us the statement of the lemma.
\end{proof}
\begin{proof} [Proof of Theorem~\ref{thm:general}]
Suppose $\hat{\gamma}(x_1, \ldots, x_m) = \hat{\gamma}(y_1, \ldots, y_m)$ for $(x_1, \ldots, x_m) \neq (y_1, \ldots, y_m)$. Lemma~\ref{lemma:general_coupling} allows us to construct the following coupling. Let $v_1(1)$ be uniform and $v_1(t) - v_1(t-1)$ be drawn according to $\gamma$. Let $v_2(t)$ be drawn according $\gamma$, coupled so that $v_2(t)_i \cdot y_i = v_1(t)_i \cdot x_i$ for all $i$. By Lemma~\ref{lemma:general_coupling}, this induces the correct distribution on $v_2(t)$. Now, choose an index $j$ such that $x_j \neq y_j$, and let $f_1(k) = g(x_i \cdot k_i)$ and $f_2(k) = g(y_i \cdot k_i)$. The rest of the proof follows as in the proof of Theorem~\ref{thm:regular_torus}.
\end{proof}
\section{Open Problems}
\label{sec:openproblems}
This work leaves open many interesting questions; we sketch some here.
\begin{itemize}
\item Here, we give an equivalent condition for reconstructivity when (a) the random walk is rational, and (b) the underlying graph corresponds to a group of the form $\mathbb{Z}_{p_1}^{d_1} \times \cdots \times \mathbb{Z}_{p_k}^{d_k}$ for distinct primes $p_1, \ldots, p_k >5$. We know that this condition is not necessary when the random walk is irrational, or when $p_i = 3$ or 5 for some $i$. But is this condition necessary when the random walk is rational and the graph is a cycle of size, say, 27? What (if any) is an equivalent condition when the random walk is irrational?
\item The techniques used in this paper do not extend directly to non-abelian groups, because the Fourier transform on a non-abelian group is very different from the Fourier transform on an abelian group. Nevertheless, it is possible that a sufficient condition for reconstructivity similar to the one given in Theorem~\ref{thm:distinct-f} exists for non-abelian groups; proving such a condition is an interesting challenge.
\item We focus only on characterizing which random walks are
reconstructive. But there are many open questions about the
equivalence class structure of non-reconstructive walks; for
example:
\begin{itemize}
\item For a given r.w., How many non-minimal equivalence classes are
there, and of what size?
\item If the random walk has bounded range, are the equivalence
classes of bounded size?
\item Are sceneries in the same equivalence class far from each
other (i.e. one cannot be obtained from the other by a small
number of changes)?
\end{itemize}
\item One could also ask more practical questions, such as: how many
steps are necessary to distinguish between two sceneries? This could
be related, for example, to the mixing time of the random walk
and/or its Fourier coefficients. A harder problem is: how many steps
are necessary to reconstruct a scenery? Or: find an efficient
algorithm to reconstruct a scenery. Such problems have been tackled
before; see for example~\cite{Matzinger:2003}.
\item An interesting question is the reconstruction of the step
distribution, rather than the scenery. Which {\em known} function
$f(k)$ would minimize the number of observations needed to
reconstruct the {\em unknown} step distribution $\gamma$ of a random
walk $v(t)$?
\item The entropy of $\{f(v_t)\}$, which depends both on the random
walk and on the scenery, is a natural quantity to explore. Specifically, we
can consider the average entropy per observation as the number of observations
goes to infinity. In general, two sceneries that induce the same entropy for a
given random walk do not have to be the same; for example,
$(1,0,1,0,\ldots )$ and $(1,1,1,\ldots)$ both have entropy going to zero for
the simple random walk. (The first random walk will have one bit of entropy
reflecting the starting bit, but as the number of observations grows, this
becomes negligible.) Consider two functions equivalent under a
given random walk when they induce the same entropy. What do these
equivalence classes look like? Are there random walks for which the
equivalence classes are minimal? What is the entropy of a random
scenery under the simple random walk? Does it decrease if the walk
is biased to one side?
\end{itemize}
\section{Acknowledgments} The authors would like to thank Itai Benjamini, Elchanan Mossel, Yakir Reshef, and Ofer Zeitouni for helpful conversations.
\pagebreak
|
{
"redpajama_set_name": "RedPajamaArXiv"
}
| 8,682
|
## MAIL-ORDER HOMES
###### SEARS HOMES AND OTHER KIT HOUSES
###### Rebecca L. Hunter
###### SHIRE PUBLICATIONS
The Sears "Alhambra," such as this Peotone, Illinois, example, was a Mission Revival design. It had the same floor plan as the Dutch Colonial Revival "Martha Washington," (see p. 28) evidencing standardized design.
##### CONTENTS
Pacific Homes 1923 model #385 was copied by Sears and sold as the "Del Rey."
THE HOUSE THAT CAME ON THE TRAIN
EVOLUTION OF THE MAIL-ORDER HOME CONCEPT
MANUFACTURERS OF MAIL-ORDER HOMES
STYLES AND DESIGNS
THE BUILDER OF MODEST MEANS
MARKETING THE MAIL-ORDER HOME
BOOM AND BUST
THE DEMISE OF THE MAIL-ORDER HOME
PLACES TO VISIT
ADDITIONAL RESOURCES
FURTHER READING
The cover of the 1928 Wardway Homes catalog featured the popular Dutch Colonial "Cambridge" model.
##### THE HOUSE THAT CAME ON THE TRAIN
THE MOST intriguing oral history associated with a home is, perhaps, "This house came on the train." A train pulls into the station and leaves two boxcars on a landing. Inside are all the materials needed to construct an entire house. The framing boards are cut to size at the mill to facilitate rapid and accurate construction. The purchaser has chosen the model, selected door and window styles, specified the desired types of wood, decided on the exterior cladding, and even selected the paint colors. These well-designed, practical homes were made of top-quality materials. This is a mail-order kit home.
These homes were marketed on a large scale by mail-order catalog from 1906–83. Eight major U. S. companies and a host of small companies with only local distribution sold such homes primarily in the United States but also in Canada. Aladdin Company once shipped an entire village of homes to Birmingham, England, to provide housing for workers at the Austin Motor car factory.
The catalogs were updated, usually annually, to allow for the addition of newer styles of homes and the deletion of unpopular models, and to accommodate price changes. Companies also placed advertisements in national magazines and newspapers in major cities as a way to promote their homes.
Over time, various terms were used to describe these homes. In a circa 1907 catalog, Aladdin Company advertised "the original Knocked-Down Houses, with every piece cut to right length, breadth and thickness and planed on all four sides." Sears, Roebuck used the terms "cut and fitted," while Gordon-Van Tine featured the "ready-cut system," and Pacific Homes dubbed their models "Ready-Cut" houses. Harris Brothers described their kit homes as "Cut-to-Fit." Lewis and Sterling used no such descriptive terms, but referred to their product as "Lewis-Built Homes" or homes "Built by the Sterling System."
The homes were copies of the U. S. house styles most popular during the period 1900–80. Despite widespread sales of thousands of mail-order homes, they are rare when compared to the total number of homes built. Aladdin sales of 2,800 homes in 1918 comprised 2.3 percent of the 118,000 U. S. housing starts that year.
The popular 1930 Montgomery Ward "Cranford" model is, architecturally speaking, a Dutch Colonial Revival design with a Tudoresque gable added on the front.
Precut housing thrived until after World War II, when government regulation of the U. S. housing industry, tract housing construction methods, and increased popularity of prefabricated and mobile housing meant that precut housing companies could no longer compete financially.
### WHAT THE BUYER RECEIVED
The mail-order house company provided blueprints and construction materials. Lumber was provided either in bulk or as precut framing boards. The latter were known as "kit" homes. One source provided the buyer with all the materials required to build a house: lumber, roofing, doors and windows, flooring, trim boards, hardware, nails, clapboard or cedar shingles for the exterior siding, and enough paint and varnish to put two coats on everything. Of course, an instruction book was included to ensure that the structure was built properly.
This well-maintained "Cranford" is located in Charleston, West Virginia. (Photograph by Rosemary Thornton)
This "Argyle" built in Lexington, Kentucky, has been authenticated by a Sears, Roebuck mortgage record. (Photograph by Dale Wolicki)
As in homes built by standard methods, framing boards were customarily hard southern yellow pine. Flooring in living and dining rooms was typically oak, with maple and/or fir in the other rooms. Trim boards usually matched the flooring. Masonry such as brick and stucco could be chosen as an exterior finish for any design, but this was obtained locally, and the price was adjusted accordingly.
One-story homes with two or three bedrooms predominated mail-order home catalogs. The Sears, Roebuck "Argyle" model with its Arts and Crafts porch columns was built in many locations such as Bridgeport, Connecticut; Rantoul, Illinois; Kokomo, Indiana; Thornton, Iowa; Somerset, Kentucky; Bladensburg, Maryland; and Detroit, Michigan.
Lewis, another Bay City, Michigan, company, offered this "Rowland" model in the 1950s.
Materials were shipped primarily by rail, but also by boat, depending on the locations of the supplier and purchaser. Most customers ordered from the closest supplier, as the buyer usually paid the freight charges.
Researchers are lucky when a mailorder home identified in 2000 still looks exactly like its 1923 catalog image. Gordon-Van Tine's model #562, an American Foursquare design, was popular in the early 1920s.
This Mid-Century Modern Ranch design from the 1954 Aladdin Company catalog is shown with the optional "scenic window." Acknowledging that ranch-style designs evolved from architecture of the southwestern United States, Aladdin named this model "El Rancho."
The mail-order company's responsibility ended with the arrival of the sealed boxcars at the railway depot. The purchaser had several days to unload the material. In the early years, materials most commonly were hauled to the building site using a horse and wagon; later, motor vehicles were used. The majority of building sites appear to have been close to a railroad, but one purchaser is known to have hauled the materials for his house by horse and wagon from the Gordon-Van Tine plant on the Mississippi River in Davenport, Iowa, sixty miles east across the plains to his farm in Illinois.
Shown here as built in Colonial Heights, Virginia, This Gordon-Van Tine model #562 looks original except for the replacement iron porch rail and columns. Compare this picture with the one opposite. (Photograph by Rosemary Thornton)
### COST OF A MAIL-ORDER HOME
Prices shown in catalogs typically included only the building materials; the cost of the finished house, including the materials, lot, foundation, and construction labor was usually at least double the catalog price. Prices for Sears materials ranged from $146.25 in 1911 for a two-room cottage to $9,990 in 1920 for a ten-room, two-and-a-half-story design.
Many upgrades and extras such as heating, lighting, and plumbing fixtures; storm windows; etched glass doors; stained glass windows; cherry or birch woodwork; and built-in cabinetry could be ordered at additional cost. Aladdin Company, in its 1978–79 catalog, listed the following optional materials: scenic windows, insulating glass, kitchen cabinets, insulation, folding closet doors, aluminum storms and screens, and patio doors. In fact, sales records show that Aladdin Company earned more money on upgrades and extras than they earned selling the basic homes.
Manufacturers claimed that the precut system would save the builder up to 30 percent in labor and materials compared to the cost of standard building methods. Few data are available to substantiate the exact amount of savings, however. In any case, the cost of a mail-order home was significantly less than the cost of obtaining a home by the standard method of hiring an architect, purchasing materials from various sources, and paying a carpenter to measure and cut every board. Companies were able to pass their savings on bulk material purchases along to the homebuyers.
In terms of construction time, a buyer of a mail-order home could expect to be living in the house much sooner than a neighbor who built traditionally. Sears, Roebuck claimed that the "Rodessa" model could be built in 352 hours using precut materials, compared to 583.5 hours using standard methods.
### PROUD OWNERS
Homeowners delight in telling the stories of their mail-order homes. In 1913, George E. Allen decided to put up a Sears, Roebuck home on the farm he bought in 1900 in Raymore, Missouri. He chose model #227, a two-story "box house," or American Foursquare design. Allen described the process of building the house in letters to his wife:
_April: Dug the post holes_
_July: Have the cellar done_
_August: All sided up, window frames in and painted_
_September: All the carpenter work done_
In 2000, the farm, still in the Allen family, was designated as a Missouri Century Farm. The family still retains the original contract for purchase of the home.
Susan Stamberg, a Chevy Chase, Maryland resident, initially was dismayed to discover she was living in a home from Sears, Roebuck. "Sears? In New York, you went to Bergdorf's, B. Altman, Saks. You didn't go to Sears," Stamberg told the _Washington Post_ in a 2002 article by Lisa Rauschert. Since then, she has bragged about her "Maywood" model, praising its design.
In September 2009, the Mickey family threw a birthday party for their one hundred-year-old Sears Roebuck model #118 in Poplar Bluff, Missouri. Model #118, later named the "Clyde," is a two-story Victorianesque hipped roof house with gabled extensions. It was one of the very first models offered in the _Modern Homes_ catalog, appearing from 1908–19. The Poplar Bluff home was built in 1909 by Walter R. Mathis for his son Thomas and wife Bertha. Their daughter, Ruth, was born in the finished house in December of that year. The Mickeys purchased the home in 1963.
The "Maywood," such as this example in Glenshaw, Pennsylvania, was offered by Sears, Roebuck from 1927–39. (Photograph by Dale Wolicki)
### DETRACTORS OF MAIL-ORDER HOUSING
It sounds like such a good idea for consumers to be able to obtain excellent housing at low cost that one could easily imagine universal popularity for the idea. However, the construction industry was among those who were not fans of mail-order homes. Architects, carpenters, lumber companies, millwork plants, roofing concerns, and hardware suppliers lost business every time someone purchased a mail-order home. Trade unions, contractors, and architects took out full-page advertisements in national and local publications, insinuating that mail-order homes were inferior and that the purchasers were fools. In April 1928, the United Brotherhood of Carpenters Local 363 took out an advertisement in the Elgin, Illinois, _Courier News_ asking citizens to demand that union carpenters build their homes. However, these advertisements were few in number compared to those placed by mail-order companies advertising their wares.
### IDENTIFYING A MAIL-ORDER HOME
For most U. S. homes, little information is available that indicates the architect or the contractor. Some communities ask for this information on building permits, but most do not. Information can be sought in contemporary newspaper accounts of new construction, in historical society archives, on fragile old blueprints, and in memories of elderly citizens, but the search is often frustrating, time consuming, and fruitless.
A word of caution to would-be seekers of mail-order homes: attempting simply to match a house with a catalog picture is insufficient, since most mailorder models are not unique designs but were closely copied from other companies or architects. Interior evidence and legal records, however, can determine with certainty whether a given house is actually of mail-order origin.
In a precut building, the easiest proof to find is a part number. Framing boards were numbered to facilitate construction. After construction, some of these numbers usually are still visible on ceiling joists, rafters, wall studs, and stair risers and treads. They are not found on trim boards, windows, doors, banisters, built-in cabinets, or mantels. Numbers are seldom visible on every board, and a bright flashlight is usually necessary to see them.
Sears, Harris, and, after circa 1930, Aladdin, used ink. Sears stamped solid numbers and letters near the ends of the boards, usually a capital letter followed by one or more numerals. Harris stenciled the markings in the middle of the boards, usually two-digit numerals or a letter and one or two numerals. Gordon-Van Tine, Pacific Homes, Montgomery Ward, Aladdin, Lewis, and Sterling wrote the numbers by hand in grease pencil, usually in the middle of the boards.
A post-1930 Aladdin part number. (Photograph by Cindy Catanzaro)
A Harris Brothers part number.
It can be difficult to discriminate between part numbers and random scribbling by carpenters. Part numbers are repeated on more than one board; multiple ceiling joists, for example, are likely to bear the same number. Boards serving different functions bear different numbers, but of a similar format. Carpenter's marks are made in the same blue grease pencil as the handwritten part numbers and include such things as arrows, random lines, messages, and arithmetic. In the case of a precut home, it is not necessary to calculate the size of boards, so computations are good evidence that the building is not a precut kit.
Additional proof may include one or more of the following:
• blueprints with a mail-order model name, number, or company logo
• documents including correspondence, guarantees, instruction manuals, or parts lists
• title search and/or tax records listing a mail-order company as architect, buyer, seller, owner, or payer of real estate taxes
• mortgage records, in the case of companies that offered financing
• records of repossession or resale by the mail-order house company during and after the Great Depression
Sears part numbers were stamped on the sides and ends of boards.
Shipping labels, typically applied to an outside board in a bundle of lumber, also can provide evidence, but not proof, of mail-order origin. If the label is found in a house that is identical to a catalog model, it is strong evidence. Otherwise, it is weak, since mail-order companies also sold lumber and millwork regardless of whether buyers also ordered an entire house. The shipping labels are identical in either case.
In 1894, architect George F. Barber of Knoxville, Tennessee, published his book of plans entitled _Artistic Homes_ , showing his latest designs and their floor plans. Prices for the plans ranged from $10 to $50 depending on size and complexity.
##### EVOLUTION OF THE MAIL-ORDER HOME CONCEPT
RAILROAD DEVELOPMENT in the United States after 1865 was a major factor in the evolution of the mail-order home. With rail access to most parts of the country by 1870, purveyors of goods were no longer limited to local markets. Even large items could be transported in a timely manner at a reasonable cost.
A further boon to mail-order marketers came in the form of postal legislation passed in 1879, establishing a reduced rate of one cent per pound for "mail-order" publications. Beginning in 1896, a postal system of nationwide rural free delivery made it easy for the primarily rural population of the United States to receive these publications.
### GENERAL MERCHANDISE CATALOGS
Perhaps the earliest attempt at national sales was initiated as early as 1870 by E. V. Allen of Augusta, Maine. Competitors soon followed his lead. In 1872, Aaron Montgomery Ward of Chicago opened a business devoted entirely to the selling of goods by mail, and he published a single sheet advertising his wares. By 1874, his catalog had increased to seventy-four pages.
What would become the most successful mail-order company in the country was established in 1886 by Richard Warren Sears, a railway station agent in Minnesota. He obtained at low cost a railroad shipment of watches that was refused by a retail jeweler and resold them at a profit to other agents. Later that year, he founded the R. W. Sears Watch Company in Minneapolis.
In 1887, Sears moved his business to Chicago in order to take better advantage of speedy rail service to the entire country. Business was good, but as time passed, Sears realized that the watches he had been selling were aging and might need repairs. Knowing nothing about watches, he decided to hire a watchmaker to meet the anticipated demand for repairs. In 1887, Sears's want ad for a watchmaker was answered by Alvah Curtis Roebuck of Hammond, Indiana. Sears, Roebuck and Company was born.
Max Le Roy Keith, a prolific Minneapolis, Minnesota, architect, published hundreds of house plans from 1910–25, such as this stucco design from his 1911 _100 Designs_.
### PLAN BOOKS
Also taking advantage of the rural free delivery system from 1885 to 1930 were architects selling house plans by mail. Instead of hiring a local architect, homebuilders could order at much lower cost a plan from architects such as George F. Barber of Knoxville, Tennessee, William Radford of Chicago, or Max Le Roy Keith of Minneapolis-St. Paul, Minnesota. These plan book catalogs are different from mail-order home catalogs in that no building materials were offered, only the blueprints. Materials were obtained locally, and nothing was precut.
William Radford's 1908 _Artistic Bungalows_ illustrated many traditional models and a number of unique and unusual designs, such as #5011.
In the mid-1920s, Lord Lumber of LaGrange, Illinois, offered customers catalogs of models from Home Builders Company of Chicago. The plans were purchased from Home Builders, and Lord Lumber hoped to supply the building materials.
During the 1920s, Home Builders Company of Chicago published annual plan book catalogs, each with more than four hundred house designs. Builders all over the country purchased these plans.
Home Builders designs were sometimes featured in small catalogs bearing the name of a local lumber company hoping to increase its lumber sales by offering house plans.
Builder Fred T. Hodgson published numerous texts about construction technology, many of which included plans that could be purchased from architect William Radford or, in the 1919 edition of _The Steel Square_ , from Sears, Roebuck. Popular periodicals such as _Better Homes and Gardens_ and _Ladies' Home Journal_ also published house plans.
Shortly after 1900, purveyors of building materials, such as Gordon-Van Tine, Harris Brothers, Montgomery Ward, and Sears, Roebuck, sought to compete with plan book companies and to increase their sales of building materials by offering plans of their own. The buyer received plans, bulk lumber, and millwork. Nothing was precut or numbered. Only Aladdin Company, Lewis, and Sterling offered precut homes from day one of their entry into the housing market.
The Home Builders "Ardara" model, as built in Elgin, Illinois.
Aladdin Company's 1980 catalog featured ranch house designs.
##### MANUFACTURERS OF MAIL-ORDER HOMES
EIGHT COMPANIES that marketed their homes across the entire United States provided the bulk of mail-order housing. Most west coast homes came from Pacific Homes, Aladdin, Montgomery Ward, or Gordon-Van Tine. All companies except Pacific Homes supplied states in the east, south, and midwest. Aladdin and a Canadian company, Halliday Homes, supplied most of the mail-order homes in Canada.
### BAY CITY, MICHIGAN: THE FIRST PRECUT HOMES
The first of the large mail-order companies offering precut homes evolved in 1906, when a lawyer in Bay City, Michigan, saw the plans for a precut boat. William Sovereign realized that the precut system also would work for a building, and to test the idea he had a precut boathouse designed. The boathouse proved to be marketable, so William teamed up with his brother Otto, an advertising executive, to form the North American Construction Company. William provided the design and manufacturing expertise, while Otto managed the marketing.
Initially, the Sovereign brothers offered architecturally undistinguished homes of two to eight rooms, but they soon added a number of Arts and Crafts-style residences. They ordered the precut lumber locally from Lewis Manufacturing Company. To manage increased precut home sales, Lewis eventually needed to expand their production facilities and, in 1913, the company asked for partial ownership of the Sovereign brothers' business. The Sovereigns refused this request, left Lewis, took their business to nearby International Mill and Timber, and renamed their company "Aladdin" after the mythical genie who built a castle overnight for his master. In retaliation, Lewis published its own 1913 catalog of precut homes, featuring many designs similar to those it had manufactured for Aladdin. In 1925, Lewis changed the name of their housing division to Liberty Homes and continued to produce precut buildings until 1975.
When, in 1915, International Mill and Timber demanded an interest in Aladdin Company, the Sovereigns took their business away, purchased the Eddy-Sheldon Mill in Bay City and resumed production, soon establishing additional facilities in Portland, Oregon; Hattiesburg, Mississippi; Wilmington, North Carolina; and Toronto, Ontario. Aladdin, which remained under the leadership of the Sovereign family, continued to market precut homes and other buildings until 1983.
Aladdin Company of Bay City, Michigan, the first to sell precut homes on a nationwide basis, offered the "Marsden" Arts and Crafts Bungalow.
International Mill and Timber, having lost Aladdin's business, published its first precut house catalog in 1915 under the name Sterling Homes. By 1916, they boasted of mills and forests in Florida, Michigan, Texas, and Washington, as well as Canada. Sterling Company remained in business until 1974 and provided housing for many of the automobile makers in the Detroit area.
The "Marsden" as built in Springfield, Kentucky. (Photograph by Dale Wolicki)
This Aladdin "Victoria" model from the Aladdin Canadian catalog was built in Cambridge, Ontario. (Photograph by Dale Wolicki)
### GORDON-VAN TINE COMPANY
A sawmill established in 1866 by U. N. Roberts in Davenport, Iowa, was the parent of Gordon-Van Tine Company, which incorporated in 1907 to handle building materials. The company name was derived from the middle names of two major stockholders, Horace Gordon Robinson and Harry Van Tine Scott. Gordon-Van Tine issued its first house plan book in 1909 and introduced its "Ready-Cut" home line in 1916. By 1923, the company had additional mills in Hattiesburg, Mississippi, and Chehalis, Washington. From 1917–31, Gordon-Van Tine produced the homes sold by Montgomery Ward. Gordon-Van Tine continued selling homes through 1947, when businessman Sidney Rose of Cincinnati bought the company, closed it, and sold the assets.
The front cover of Gordon-Van Tine Company's 1909 catalog of house plans shows carpenters hard at work in what is obviously a farm setting, perhaps a subtle appeal to the rural market, which at that time included more than half the population of the United States.
### HARRIS BROTHERS
In 1893, an architectural salvage company was incorporated in Chicago as the Chicago House Wrecking Company. Operated by four Harris brothers, Abraham, Frank, Samuel, and David, it secured contracts for the demolition of exhibitions such as the 1893 World's Columbian Exhibition in Chicago and the 1904 St. Louis World's Fair. In addition, the company sold new and salvaged building materials. By 1908, the company began offering house and barn plans to boost the sales of its millwork and lumber.
In 1913, the company name was changed to Harris Brothers. They published their first catalog of precut homes in 1916 and in 1923 opened a branch warehouse at St. Paul, Minnesota, to serve northern customers. According to memoirs written by Samuel Harris, Jr., a son of Samuel Harris, for twenty-five years the company drop-shipped all lumber sold by Sears, Roebuck. In the 1930s, to attract more repeat business, the company began marketing to contractors rather than to individual homeowners. Harris stopped selling homes around 1932.
Harris Brothers claimed that model #6-A was their most popular home; it has been found from Illinois to Minnesota and from New York to Kentucky. This example is in Brainerd, Minnesota. The polygonal tower may appear to be an unusual architectural feature; however, both Sears and Wards offered models with similar towers.
This Wards "Dresden" model, as built in Pensacola, Florida, is reversed from the original catalog image. This common alteration was done at no addtional charge. (Photograph by Dale Wolicki)
Montgomery Ward's late 1920s catalogs featured many Arts and Crafts designs, such as the "Dresden."
In 1934, the company reincorporated as Iron Street Lumber Company and remained in business until 1960, producing doors and windows for prefabricated house companies such as National Homes and Wausau Homes.
### MONTGOMERY WARD
Montgomery Ward, headquartered in Chicago, introduced its house plan books in 1909, featuring turn-of-the-century designs. Montgomery Ward never owned or operated housing production facilities but subcontracted services from other companies. It appears that their early homes were manufactured in Bay City. Beginning in 1917, Gordon-Van Tine produced Montgomery Ward buildings. Precut homes were introduced in 1918 under the brand name "Wardway Homes." Many of these homes are identical to Gordon-Van Tine models from corresponding years; only the model names and prices differ. Unpaid mortgages and decreased sales brought about the demise of Wardway Homes in 1931.
### PACIFIC HOMES
Incorporated in Los Angeles, California, in 1908 under the name Pacific Portable Construction Company, this company initially offered only sectional buildings. Precut homes were added from 1918–40 under the trade name "Pacific Ready Cut Homes." Pacific Homes primarily marketed their buildings west of the Rocky Mountains, but their publications also boast of sales in Belgium, England, and France, as well as Alaska and Hawaii. In 1929, the company began building surfboards under the brand name "Waikiki Surfboards." Pacific Homes ceased production of houses after 1940 but continued to produce surfboards until the end of World War II.
Models shown in the 1922 Crain Ready-Cut Homes catalog were a mixture of typical midwestern and California styles. Model M-209 combines elements of both.
### SEARS, ROEBUCK AND COMPANY
Sears, Roebuck, the most well known of all American mail-order home companies, began offering house plans in 1908 in hopes of boosting building materials sales. In 1915, they offered as a trial five of the 109 plans in the _Modern Homes_ plan book as precut kits. By 1917, most of their plans were available as precuts.
Between 1908 and 1940, Sears, Roebuck marketed more than 450 different models in their _Modern Homes_ catalogs. In addition, a set of 1941 building plans titled "Sears, Roebuck Modern Homes" has been discovered; some of these were built by developers in Elyria, Ohio; Sidney, New York; and Cranford, North Plainfield, and Bergenfield, New Jersey.
Frank Lloyd Wright designed twenty-four precut American System-Built homes. Twelve models were put into production in Milwaukee, Wisconsin. Apparently, only thirteen such structures were built, including this one in Berwyn, Illinois.
### SMALL LOCAL COMPANIES
Some of the many small precut home companies with only local distribution include Crain (Houston, Texas), Bennett (North Tonawanda, New York), Bradley Portable House Company (Randolph, Massachusetts), and Harbing Company (Springfield, Nebraska).
### CASES OF MISTAKEN IDENTITY
Sears, Roebuck has been the most publicized mail-order home company since the 1986 book _Houses By Mail_. Its homes have been featured on television and in periodicals and newspapers. Four of its catalogs (1912, 1913, 1926, and 1932) are available as reprints. Of all the companies that once produced mail-order homes, only Sears, Roebuck is still in existence as of the date of this publication, although they no longer sell homes.
Perhaps, then, it is no wonder that many structures are mistakenly identified as "Sears" homes. This term has become generic, often used erroneously to refer to mail-order homes from every other company, plan book homes, prefabricated homes, homes with unusual provenance, and homes lived in by families whose name happens to be Sears.
This table details the largest mail-order home companies in the United States:
#### Major U.S. Mail-Order Home Companies
**Company** | **Years** | **Estimated* Number of Mail-Order Homes**
---|---|---
Aladdin | 1906–83 | 100,000
Sears, Roebuck | 1908–40 | 60,000–70,000
Harris Brothers | 1908–31 | NA
Gordon-Van Tine | 1909–47 | 75,000
Montgomery Ward | 1909–31 | 30,000
Lewis | 1913–75 | 75,000
Sterling | 1915–74 | 45,000
Pacific Homes | 1918–40 | 37,000
* Data extrapolated from catalog figures and reports listing total annual sales
The cover of the 1929 Sears catalog features a Tudor Revival design, the "Mitchell."
##### STYLES AND DESIGNS
MAIL-ORDER BUILDINGS covered the gamut of architectural styles popular from 1906–83, ranging from two-room vacation cottages to ten-room, two-and-a-half story homes, from chicken coops to sixty-by-ninety-foot barns.
Since the goal of mail-order home purveyors was to sell the maximum number of each design offered, styles tended to follow the popular trends in architecture. Therefore, mail-order homes rarely possess unique or unusual architectural features; instead, they blend in with the most common designs of each decade. This causes confusion a century later as researchers puzzle out how to identify mail-order homes.
The earliest catalogs were dominated by two-story vernacular designs, plus a handful of Victorianesque homes designed to accommodate large families. This is evident in examining Gordon-Van Tine's 1909 catalog offering twenty-two two-story but only eight one-story homes. By 1915–19, Arts and Crafts and American Foursquare styles dominated the catalogs.
After World War I, several factors affected home sizes, among them high inflation, a shortage of materials, and population shifts. Many people were moving to small city building lots, and home size decreased. This became the era of the two-bedroom, one-story bungalow or cottage.
Revival styles proliferated from the late 1920s into the late 1930s: Colonial, Dutch Colonial, Tudor, Spanish, and Mission designs were the latest fad in real estate.
Designs offered by Pacific Homes of Los Angeles differed from those offered in the midwestern and eastern states. Since their market was primarily on the west coast, Pacific Homes models reflect the prevailing California Bungalow style, with low-pitched roofs, many windows, and large porches.
### FARM BUILDINGS, TOO!
Prior to 1920, the largest segment of the United States population was rural. In consideration of the large farming population, major mail-order home producers—Aladdin, Gordon-Van Tine, Harris Brothers, Sears, and Wards—also offered barns and other farm buildings. Most prevalent were designs for gambrel-roofed barns, followed by gabled, gothic, round, and polygonal barns. Also included were hog and chicken houses, implement houses, granaries, and milk houses.
Montgomery Ward's "Farmland" was targeted to rural areas where large families predominated. The nine-room home featured polygonal bays and pedimented gables.
Harris Brothers' model #5010 is a large Arts and Crafts bungalow with a corner tower. This Louisville, Illinois, example is one of only two discovered so far.
Harris Brothers model #1025, such as this example in Elmhurst, Illinois, enjoyed popularity throughout the eastern and midwestern states.
The Sears "Martha Washington," seen here in Lombard, Illinois, is a Dutch Colonial Revival style.
This unusual round barn, model #215, was pictured in the Gordon-Van Tine 1918 farm catalog as a testimonial from a satisfied customer in Manhattan, Illinois. The barn was still standing as of 2011.
In 1909, Gordon-Van Tine offered twelve agricultural buildings; as the demand grew, from 1918–36, so many were offered that separate _Farm Buildings_ catalogs were published.
Aladdin and Pacific Homes played a comparatively minor role in the production of farm buildings, with only a small number of offerings between 1910–19. Montgomery Ward also offered a few agricultural buildings in their house catalogs from 1910–19. Harris Brothers offered only a few farm buildings in their house catalogs from 1912–18.
_Sears, Roebuck 1919 Book of Barns_ has been issued as a reprint.
Sears, Roebuck joined this market in 1911 and became the major competitor of Gordon-Van Tine, offering specialty farm catalogs from 1918–29. As the agricultural population declined after 1920, so did the number of farm buildings offered. The total number of Sears barn models reached a high of twenty-six in 1919, but declined to thirteen in 1929.
### CONSTRUCTION
Precut house designs were standardized to reduce waste in materials and to facilitate milling of the lumber. For example, the same lengths of joists, wall studs, and rafters would be used in many different models. Window and door size variations were limited, and the same basement stairway appears over and over again. Aside from the convenience of precut framing boards, kit homes were constructed exactly like a similar carpenter-built home.
A kit home variant was the sectional home. Four-foot wide sections, be they wall, door, or window, were completely constructed at the mill and included framing and interior and exterior sheathing. The sections could be bolted together quickly at the building site, providing the buyer a home with little effort. The majority of these were small cottages, advertised to potential buyers as vacation homes, but commercial buildings also were available.
Sears, Roebuck dubbed their ready-made buildings "Simplex Sectional" and suggested they could be erected for summer use, then dismantled and stored under canvas until the following year's vacation. In 1916, in addition to cottages, Sears offered sectional garages, a real estate office, a church, schoolhouses, a refreshment stand, election booths, a boathouse, a cobbler shop, a barn, outhouses, and even a photographic studio.
Gordon-Van Tine sold "Portable Houses" from 1910–16 and claimed to have been the "original purveyors of prefabricated houses." However, at least one Cincinnati company offered portable buildings even earlier than that, from 1850–75.
Harris Brothers operated this forty-acre plant at 35th and Iron Streets in Chicago.
In 1920, Harris Brothers published a specialty catalog featuring their sectional buildings.
Pacific Homes started business as Pacific Portable Homes, offering only sectional structures. Harris Brothers introduced sectional cottages in 1912 under the trade name "Perfection Portable Houses" and later changed the name to "Presto-Up." Montgomery Ward sold a few Ready-Made houses from 1912–16.
After World War II, sectional construction was used for the majority of U. S. homes, which became known as "prefabricated" homes.
### LOOK-ALIKES
Given that mail-order homes were copied from the most popular architectural styles, it is not surprising to learn that mail-order companies also copied each others' designs. This resulted in "look-alikes," "clones," or "twins." Of course, because of legal strictures, they could not be exact copies. However, the differences often are trivial and therefore not easy to spot. For example, the Sears "Mitchell" model cannot be visually differentiated from the Aladdin "University" or the Bennett Company "Brentwood." Only part markings and dimensions can reveal for sure which company built it. The Gordon-Van Tine "Kent" and identical Montgomery Ward "Newport" models are the same basic style as the "Mitchell," but have slightly flared rooflines.
Mail-order companies also copied designs from plan books. For example, of the twenty-five homes offered in the 1909 Montgomery Ward catalog, fifteen are close copies of plans published by William Radford in 1908. Harris Brothers' 1908 catalog contained twenty-six plans, twenty of which appear to be based on designs by Minneapolis architect Max Le Roy Keith. Perhaps the mail-order home companies actually purchased plans from these architects, or perhaps they hired their own architects to make minimal changes in others' designs.
For the most part, the architects of mail-order homes remained anonymous. On blueprints, only initials identified the architects. The companies, not individual architects, held the copyrights of the designs. In 1925, Sears hired D. S. Betcone, the only architect of mail-order homes known by name, from competitor Aladdin, where he had worked for eleven years.
### ALTERATIONS: ORIGINAL AND SUBSEQUENT
A common question is "What were the most popular models?" In the absence of sales records, the answer must be a speculation. Popularity varied according to year and geographic location. Sears even published two catalogs for 1920 containing different models: a Chicago and a Philadelphia version. If we assume that the most popular models were sold over the longest period of time, we can come up with some answers: small, architecturally conservative, and inexpensive homes appeared most frequently.
Regardless of which model they purchased, customers were encouraged to personalize their mail-order homes by adding porches, fireplaces, sunrooms, window boxes, trellises, or built-in cabinetry, and by selecting exterior finishes and colors. Of course, over the years, new siding has been put on, additions have been made, roofs have been raised, porches have been enclosed, and dormers have been added. This increases the challenge of finding mail-order homes today, since many no longer look like the catalog pictures.
The Wards "Newport," Aladdin "University," Gordon-Van Tine "Kent," and Bennett "Brentwood" were close copies of this popular Sears "Mitchell" model.
Sears's plain one-story home, the "Winona," was sold from 1913–40, more years than any other model. First known as model #205 and renamed Winona in 1917, it was updated in 1930 by the addition of a gabled dining room bay.
The Sears "Princeville" as it looked in the catalog.
This Sears "Princeville" model in St. Charles, Illinois, no longer bears any resemblance to its catalog image. It was located by means of a testimonial.
The cover of the 1923 Pacific Homes catalog was designed to appeal to the family of "modest means."
##### THE BUILDER OF MODEST MEANS
IN ITS 1925 CATALOG, Aladdin Company stated: "The purpose of the Aladdin plan is to help builders of modest means." Aladdin accurately targeted its audience: most purchasers were working class families, buying a single home for themselves. Although no nationwide studies have been conducted, trends revealed by in-depth studies in a few communities such as Elgin, Aurora, and Rockford in Illinois and Bay City, Flint, and Pontiac in Michigan support the premise that purchasers were indeed people of "modest means." In the Elgin study, 37 percent of original owners were factory employees and 22 percent worked in construction trades. Only 0.02 percent were professionals.
From 1906–20, many buyers were farmers. But with the population shift to urban areas and the growth of suburbs, commuting office and factory workers needed convenient housing. The scene was ripe for the precut mailorder home. Lower in cost and constructed more quickly than standard housing, mail-order homes sprang up in increasing numbers.
Perhaps the most ingenious scheme to attract buyers for suburban homes was that of Grover C. Elmore and his Poultry Farm Colony. Elmore, a Chicago entrepreneur, platted six subdivisions thirty miles southwest of Chicago in Tinley Park, all with double-length lots. He advertised a tempting package: the buyer would get the house model of their choice, a chicken coop, and a hundred chickens. Once the house was ready, the family would move out of polluted Chicago. The husband, using the new commuter rail service, could continue his employment in the city. Meanwhile, the wife and children would enjoy the fresh air and sunshine of Tinley Park. The wife could care for the chickens and gather their eggs. These she would sell, supposedly earning enough money to pay the mortgage on the house. All twenty-one of the identified mail-order homes in Tinley Park are in Elmore subdivisions, some with the chicken coop still in evidence on the back of the lot.
This Sterling "Miracle" was built in Sylvan Lake, Michigan. (Photograph by Dale Wolicki)
### CONTRACTORS AND BUILDERS
In its 1928 catalog, Harris Brothers offers two ways to build:
_"1. Buy the materials for your Harris Plan-Cut Home at the low freight paid prices shown and build it yourself with the guidance of our easily read and readily understood blue print plans, itemized material list and construction details._
_2. Select your Harris Plan-Cut Home and let us send you all the materials we furnish for its construction. Then secure a reliable builder in your locality, arrange a contract for construction and supervise the erection from start to finish as the work progresses."_
We will never know for sure what percentage of purchasers assembled their precut homes themselves. Most probably hired a contractor or carpenter, unless they had carpentry skills.
Some contractors specialized in putting up precut homes. To encourage this, Aladdin took out advertisements in building trade magazines, offering to refer customers and advising builders that they would select one carpenter in each city to handle their orders. Sears published a letter from Des Plaines, Illinois, contractors Elmer and Oscar Blume, who stated they had built 104 mail-order homes. Russell Byrum, a popular local builder, assembled most of the 127 known mail-order homes in Anderson, Indiana. Byrum put up homes primarily from Sears, but also a number from Aladdin and Gordon-Van Tine.
After discontinuing the Modern Homes catalogs in 1940, Sears developed the "Club Plan of home purchase", working through contractors to produce tract housing. Cranford, New Jersey, was the experimental site for the first development, Sunny Acres. Construction of the 172 homes began in 1940. (Photograph courtesy of Sunny Acres Civic Association)
Sears applied "mass production methods to the marketing of complete homes" such as this example in a development of ninety-four homes in Sidney, New York. The homes in each development were sold prior to construction and built simultaneously. The purchaser was not permitted to make any modifications. (Photograph by Anna Blinn Cole)
Although contractors built most mail-order homes, some were, indeed, built by the homeowners themselves. Stories about people who built and lived in mail-order homes have been transmitted by oral tradition. Children whose parents assembled mail-order homes invariably report that their job was to sort the numbered precut lumber. In Phoenixville, Pennsylvania, owners of a 1933 Sears "Laurel" model were visited by a man whose grandfather built the house. He told them that it had been delivered to the back yard by train on a track that is no longer in use.
House plans for the 1940–42 Sears developments were drawn up by architect Randolph Evans of New York. Each house had four rooms and a bath, and an attic which could later be finished. This is one of thirty homes built on a single block in Elyria, Ohio.
The son of one of the original Harris Brothers, Samuel Harris, Jr., recalled in his memoirs that as an added enticement to buyers who were moving west to build and settle, Harris Brothers allowed them to camp out with their families and livestock on the company premises while the materials for their homes were being loaded. When the train was ready to depart, the buyers rode on the boxcars with their new homes.
### COMPANY TOWNS AND INDUSTRIAL HOUSING
After World War I, a nationwide housing shortage induced employers to build or subsidize industrial housing, hoping to attract and keep good factory workers or miners. Large tracts of company-built housing became known as "company towns."
All the major producers of mail-order housing, except Montgomery Ward, at one time or another offered industrial housing. A typical company town contained a limited number of home styles, giving the community a homogeneous appearance. For example, the 152 Sears homes put up in 1917 in Carlinville, Illinois, for Standard Oil Company coal miners included only eight models, with some minor changes to give variety to the neighborhood.
This Sears model #119 was built in 1913 by Peter Larson on a farm in Eagle Grove, Iowa. In 2008, it was purchased at a farm auction and can be seen on the move to nearby Ware, Iowa. (Photograph by Jane Hartsock)
The 252 Aladdin homes built in Birmingham, England, by the Austin Motor Company as a "Garden Community" to house its factory workers consisted mostly of identical one-story homes with identical two-story homes on the corners of each block.
A few of the many employers known to have used mail-order homes for their company towns were:
• 1915: Dupont, Hopewell, Virginia, by Aladdin
• 1915: American Zinc and Chemical Company, Pittsburgh, Pennsylvania, by Gordon-Van Tine
• 1917: Inland Collieries, Detroit, Michigan, by Lewis
• 1918–21: General Motors, Flint, Michigan, by Sterling
• circa 1923: Southern Pacific Railroad, by Pacific Homes
Bachmann Model Railroad Company copied this Sears American Foursquare style farmhouse. The Sears home was sold from 1908–18. Initially identified as model #102, in 1917 it was renamed the "Hamilton."
To catch the interest of buyers, the cover of the 1925 Lewis catalog used the appellation "Homes of Character."
##### MARKETING THE MAIL-ORDER HOME
THE KEY TO SUCCESS in any business is marketing the products and services offered. Mail-order home companies were no exception. The primary marketing campaign consisted of the catalogs themselves. Since these are the only remaining artifacts from the mail-order companies, it would be easy to assume that these are factual historical documents. Far from it! When perusing the past in catalogs, it is important to remember that these are advertising materials and probably have about as much credibility as today's junk mail. Even in 1916, Gordon-Van Tine warned readers to beware of "tricky phrases" found in competitors' guarantees.
All the catalogs detailed how much money could be saved and emphasized that buying directly from the mill avoided the profits of middlemen (architects and suppliers of lumber, millwork, roofing, paint, and hardware). The precut systems claimed to save the buyer anywhere from 25 to 50 percent of the cost of labor. To guarantee the quality of their lumber, Aladdin Company in 1917 promised to pay the customer one dollar for every knot found in the red cedar siding. Unarguably, precut homes did save the buyer money, and an examination of the homes themselves testifies to the high quality of the materials.
The catalogs also contained appealing but unsubstantiated descriptions of the homes, such as "Sterling Homes afford a thousand comforts and conveniences that ordinary plans cannot include." Pictures of interiors were included especially to entice the woman of the house, who, it was supposed, would be less interested in construction details.
Most mail-order home suppliers sold different types of merchandise in other catalogs, thus, cross-marketing was a useful tool. In its 1929 publication _Honor Bilt Modern Homes_ , Sears reminded homebuyers that "behind every Honor Bilt house stands the guarantee of the World's Largest Store. We sell 35,000 different articles. We want to help you furnish that home." Often, a few house catalog pages are devoted to items available at extra cost. The Aladdin 1925 catalog lists as extras special doors, grade cellar entrance, cellar windows, foundation posts, and green slate surfaced shingles. Aladdin advised readers to "send for the complete merchandise catalog," which included heating, lighting and plumbing fixtures, and cabinetry. All the Sears specialty and general merchandise catalogs suggested obtaining the _Modern Homes_ catalogs—for free, of course.
Hardware was not unique to a single company, except for this doorknob marked with the Aladdin logo, which entailed extra cost and so was not ordered by all purchasers.
Mail-order companies placed advertisements in leading popular magazines, such as _National Geographic_ , _Better Homes and Gardens_ , and _Farm Journal_ , encouraging readers to send for their latest house catalogs.
To continually attract new customers, companies offered special deals. Usually, the buyer paid the freight, but from 1925–29, Aladdin stated "we pay the freight." Chicago-based Harris Brothers, in 1928, also advised "We pay the freight to Illinois, Indiana, Iowa, Wisconsin, Michigan, Ohio and western Pennsylvania." Perhaps these were attempts to take business from Sears, Roebuck and Montgomery Ward, which were enticing buyers by offering mortgage financing.
In an effort to appeal especially to the woman of the family, Gordon-Van Tine catalogs included color plates showing interior views.
This advertisement for Lewis/Liberty homes, torn out of a _Farm Journal_ magazine, was found tucked into a 1935 Gordon-Van Tine catalog, showing that buyers compared products.
### NAMING NAMES
Even the names of the companies themselves were used as marketing tools. Company names such as "Aladdin" and "Sterling" clearly were intended to appeal to customers. Model names also could evoke positive emotional responses. In 1920, Sterling models included the "Winsome," the "Miracle," and the "Charmcote." In reality, these homes with charming names had no more curb appeal than the similar Harris Brothers' models known as #1018, #1025, and #60.
Homes named after states and large towns might attract buyers from those locations. In 1917, Aladdin models included "Albany," "Detroit," "Burbank," "Charleston," and "Tucson." Some Lewis names from 1917 have a more international appeal: "Alameda," "Alpine," "Cadiz," and "El Dorado."
When Montgomery Ward began subcontracting its homes from Gordon-Van Tine, they marketed their own designs as well as renamed Gordon-Van Tine models. Later, as these two companies sought to update their catalogs, they simply renamed the same old models in keeping with current fashion in real estate. The Gordon-Van Tine #613 (1924–28) became the "Redwood" in 1929 and the "Emerson" in 1932. Montgomery Ward sold the same house as the "Dresden" (1924–28) and the "Webster" (1929–31). Thus, it is possible that a single house design may bear a model number plus up to five different names. To add to the confusion, Lewis in 1917 named one of its homes the "Sterling."
And, of course, companies used names from their competitors' catalogs. In fact, aficionados of mail-order homes sometimes are confused by models from different companies with the same name that look nothing like each other.
The 1920 Sterling "Charmcote" is certainly designed to charm with its many windows and inviting porch.
### TESTIMONIALS ADD LUSTRE
Sears and Gordon-Van Tine liberally sprinkled their mail-order home catalog pages with comments and letters from satisfied customers. The 1927 Gordon-Van Tine catalog has a testimonial on nearly every page. Montgomery Ward, especially in later years, included numerous letters of praise in the catalogs. In addition, Sears, Roebuck (1912–14), Gordon-Van Tine (1927), and Montgomery Ward (1931) each printed an entire booklet of nothing but testimonials from all over the country, including many photographs of completed homes. Of course, they never printed the communications from customers with grievances.
Rather than use catalog space to print testimonials, Pacific Homes contented themselves with this 1918 statement: "When you come to our office, we will be glad to show you any number of letters from customers." The Bay City companies also tended to avoid cluttering the catalogs with notes from satisfied buyers. Harris Brothers included testimonials in their catalogs and also advised readers to request a folder of letters from their own state.
Harris's #60 is nearly identical to the "Charmcote" and appears even more charming.
This Gordon-Van Tine testimonial includes a photograph of a finished Model #535 built in Delevan, Wisconsin.
Testimonials are a delight to historical architectural researchers as they provide, if not street addresses, at least model names, city and state locations, and often names of buyers. This information can be used to find the houses. Much research is in fact devoted to locating homes, and since only Aladdin preserved its sales records, testimonials for the other companies provide the only primary information available.
### EXHIBITIONS AND DISPLAYS
Print advertising is all well and good, but what if customers could actually see completed homes before making a selection? With that in mind, two companies built permanent display models, and others featured homes at national and regional fairs. Aladdin built examples of most of its homes in Bay City. Visitors to the company offices were given the addresses so they could view them. Pacific Homes put up a permanent exhibition of sample homes at 1220 South Hill Street, Los Angeles. Harris exhibited sectional buildings at their Chicago plant. Sears had a large showroom of building materials, but no complete homes, at its Chicago headquarters.
The Pacific Homes office and all its model homes, advertised in the 1923 catalog, are no longer standing.
To advertise his company's wares, Aladdin chief executive William Sovereign lived in this modified "Villa" model in Bay City, Michigan, from 1922–68. (Photograph by Dale Wolicki)
Aladdin Company erected its "Ideal" cottage at the 1915 Panama-Pacific International Exposition in San Francisco, California. Sears built two replicas of George Washington's home, Mount Vernon: one for the 1931 Paris World's Fair and the second for the Washington Bicentennial Celebration in Brooklyn, New York. Warner Brothers Picture Corporation chose Sears to build a "dream home" for a Pittsburgh, Pennsylvania, advertising campaign.
### FINANCING
Over the years, mail-order home companies offered mortgage financing in order to take business from competitors whose terms were cash on delivery. The first was Sterling, which, in 1916, offered "Wide-Open Credit:" the buyer could pay half down, with the remainder distributed over two years in monthly payments.
In 1917, Sears initiated mortgage financing and soon discovered that the mortgage business was even more profitable than the housing business. Montgomery Ward entered the mortgage business in 1926 and by 1929 claimed that 95 percent of its homebuyers were taking advantage of this program. In 1938, Lewis/Liberty boosted orders by introducing a mortgage program approved by the Federal Housing Authority.
The large Sears pavilion at the 1910 Illinois State Fair, displaying Sears, Roebuck's latest merchandise, was topped with a full-size example of model #151, the "Avondale." (Postcard image courtesy of Lincoln Library, Springfield, Illinois)
The Montgomery Ward credit application was a model of simplicity, asking only a few questions: Do you own the lot? How much unpaid? What utilities? How much cash do you have? How much can you pay each month? What work can you do yourself? Occupation was requested, but it apparently was not important whether the applicant was actually employed. Mortgage payments could be as low as $20 per month. Loans typically were made for a term of five to fifteen years.
Gordon-Van Tine offered financing on a very limited basis from 1924–31, apparently only in the Davenport, Iowa, area. Pacific Homes offered no financing, but suggested that their financial department would be able to assist homebuilders in negotiating bank loans.
In 1928, Harris Brothers offered "most liberal terms:" no cash in advance with five days' inspection time allowed before payment, a 2 percent discount if one quarter of the total was paid with the order, or a 5 percent discount for advance payment in full.
Financially conservative Aladdin's terms were a down payment with the order and the balance COD. This lasted until the 1960s, when they offered two payment plans: send full cash with the order and deduct 2 percent or pay 10 percent with the order and the balance within six months after shipment.
Mortgages to mail-order companies most typically are found in larger urban areas, especially those where mail-order company sales offices were located. Mortgages seldom were found in small towns, where the preference was for local financing.
### GREAT NUMBERS OF HOMES HAVE BEEN SOLD
Mail-order companies sought to improve their credibility by boasting about total sales. Most of these claims are deliberately vague and easily open to misinterpretation.
For example, during its second year of selling homes, Aladdin advertised that "We have sold these homes to hundreds of business men, professional men and farmers." Of course, this begs the question of how many hundreds. In 1925, Aladdin mentioned that "tens of thousands of Americans live in Aladdin homes." Is this the same as saying ten thousand homes have been sold? Of course not: several people typically occupied each home. But readers of the catalog would remember the number "ten thousand" and suppose that was the number of homes sold.
This Sears sales office model of the "Puritan" was discovered on eBay advertised as a dollhouse. (Photograph by Dale Wolicki)
Books published since 1986 state that Sears sold 100,000 homes. However, sales figures do not support this claim. Sears does bandy about the number 100,000, but a careful reading of various catalogs reveals that Sears does not claim to have built 100,000 homes. In 1934, the catalog said "more than 100,000 homes have been built the Sears way." But what was the "Sears way" exactly?
Sterling Company's 1916 catalog graphically demonstrates the company's nationwide delivery system.
##### BOOM AND BUST
BY THE LATE 1920S, mail-order homes were securely established as an affordable way to build new homes. The evolution of commuter rail lines created suburbs on the edges of major urban centers, and the new commuter class was a ready-made market for precut kit manufacturers.
Sears, Roebuck and Montgomery Ward reported their highest home sales in the year 1929. With no idea of the length and depth of the impending Great Depression, mail-order home companies continued to produce new catalogs after the stock market crash of October 29, 1929, presenting new designs embodying the increasingly popular Tudor and Colonial Revival styles.
Considered as a whole, the eight major mail-order home companies had achieved truly nationwide distribution capabilities. Lewis had mills in Bay City, Michigan; Seattle, Washington; Portland, Oregon; Jacksonville, Florida; and Bristol, Tennessee. In 1927, Gordon-Van Tine listed as locations Davenport, Iowa; Chehallis, Washington; St. Louis, Missouri; and Hattiesburg, Mississippi. On the back cover of its 1928 catalog, Montgomery Ward listed several sales office locations: Chicago; Kansas City, Missouri; St. Paul, Minnesota; Baltimore, Maryland; Oakland, California; and Fort Worth, Texas. By 1929, Sears, Roebuck had established a total of sixty-five Modern Homes sales offices in thirteen states and Washington, D.C.
Companies and contractors sometimes built speculation models of mailorder homes in new housing developments. This strategy certainly would have brought a large number of sales, but for the intervention of the Great Depression. By 1931, the spreading ripples of the stock market crash had devastating effects on housing in the United States. New housing starts were down 52 percent from 1929. Acres of newly platted developments in cities and towns sat empty of houses.
Mail-order home producers certainly were not immune to the disastrous economic effects of the Great Depression. In 1931, Montgomery Ward permanently closed its housing division. By 1932, Harris Brothers offered only a handful of precut homes and farm buildings in its general merchandise catalogs. In spring 1933, Sears, Roebuck closed its Modern Homes Division after suffering a loss of more than $8 million in uncollectible mortgages. Although the Modern Homes Division reopened in autumn 1933 offering fewer designs and no mortgage financing, sales never again reached the former profitability, and the division finally closed for good in 1940.
The damaged cover of Sears, Roebuck's 1934 catalog bears the National Recovery Act logo.
F. K. Wheeler, plant manager of the Sears, Roebuck lumber mill in Cairo, Illinois, from 1925–56, wrote that the mill worked "on Sears orders almost exclusively" until the financial crash of 1929 when Sears "virtually stopped selling houses." The mill then turned to making crates and selling bulk lumber. In 1937, they received a government contract for eighteen prefabricated Civilian Conservation Corps camps. In 1940, Sears, Roebuck sold the plant.
Mail-order home companies employed cost-saving measures such as printing separate price sheets, enabling them to adjust prices without the expense of designing and printing an entire new catalog. The Sears, Roebuck Fall 1933 and Spring 1934 catalogs were reduced to half-size pages in order to save on printing costs.
Desperate to remain in business, mail-order home producers explored new marketing strategies. In an effort to reach buyers in a higher economic class, Sears, Roebuck and Gordon-Van Tine began to offer custom design services. "We will prepare a home plan for you FREE," offered Gordon-Van Tine in 1932. "We will take your ideas from photograph, picture, rough pencil sketch or just a written description and develop a plan." In the 1933 publication _America's New Low Cost Homes_ , Sears, Roebuck offered to build "from any plans you already have, or from plans prepared by your own architect."
In 1932, Gordon-Van Tine introduced its "Master Plan" homes, a series of homes with the same floor plan but slightly altered exterior appearance. This was a cost-saving strategy for the company, decreasing the architect's fees and allowing most of the lumber to be standardized.
The Sears, Roebuck 1932 catalog featured photographs of six custom models, including this handsome Anderson, Indiana, home.
No mail-order home company was able to survive the Great Depression solely by selling houses. All opted for strategies such as securing Civilian Conservation Corps contracts, selling pallets and packing crates, and concentrating on sales of bulk lumber and building materials.
Purchasing plans and bulk materials instead of precut lumber had always been a money-saving option for mail-order homes. For example, the four-room Sears "Dundee" model was offered in 1921 for $1,138 already cut, and for $831 not cut and fitted, a savings of 27 percent.
During the 1930s, as evidenced by the higher percentage of homes lacking part numbers, it appears that a larger number of purchasers chose this cheaper option. As shown in the table below, the decrease in total number of models offered each year graphically depicts the challenges faced by mail-order home companies:
#### Mail-Order Home Models
#### Models Offered
**Company** | **1928–29** | **1938–39**
---|---|---
Sears | 82 | 39
Gordon-Van Tine | 83 | 41
Harris Brothers | 95 | 0
Montgomery Ward | 62 | 0
Aladdin | 30 | 43
Lewis | 20 | 25
Sterling | 35 | 35
TOTALS | 407 | 183
In 1930, an Elgin, Illinois, resident decided to build himself a new home on the vacant lot he owned next door. He built this Sears, Roebuck "Strathmore" model, a substantial two-story Tudor Revival home. By the time he finished construction in 1933, however, he could not afford the home and was forced to sell it before he ever moved in.
### INDIVIDUAL TRAGEDIES
During the Great Depression, thousands of Sears, Roebuck and Montgomery Ward houses were repossessed nationwide even though determined homeowners did all they could to avoid losing their homes. In an Elgin, Illinois, study, the son the original owner of a modest Sears, Roebuck five-room "Rodessa" model reported that his family was able to keep the house only because the bank agreed to accept payments of interest only. His mother went to work at the local five-and-ten store in order to save the house.
The Elgin owners of a three-bedroom Sears, Roebuck "Lynnhaven" model kept their house by renting the upstairs to another family. The parents then slept in the dining room, and their son slept in the breakfast nook. And five of Matthew Warner's six sons were able to keep the Sears, Roebuck homes they built because a wealthy relative was willing to hold the mortgages on all five properties. One of the Warner children remembers trudging down a dusty country road south of Elgin to bring the monthly payment to the relative's farm. Another family ordered a Gordon-Van Tine home. When it arrived, the buyer determined he could use the lumber intended for one house to build two homes by spacing boards farther apart and making a few other changes.
The Minimal Traditional Sears "Carver" in West Chicago, Illinois, has no decorative detail.
No sooner had the United States achieved a measure of recovery from the Great Depression than the country entered World War II, further depressing the market for new homes. Mail-order home companies tried to survive the war by building army barracks and making packing crates.
In 1949, Sears reentered the housing market with twelve prefabricated sectional models using the brand name "Homart" (from Homan and Arthington streets where the main Chicago office was located). The Minimal Traditional homes were so lacking in architectural detail as to appear institutional. The construction details were similar to the sectional cottages offered by Sears, Roebuck, Harris Brothers, and Gordon-Van Tine prior to World War I. Available either with or without a basement, they consisted of four-foot wide preconstructed panels that could be bolted together at the building site, quickly providing basic housing at minimal cost.
Perhaps this idea would have met with more success a few years earlier, when, under subsidies from the U.S. Government, numerous companies offered prefabricated homes to meet the immediate postwar housing needs. However, low sales of Homart models are evidenced by the small number of these homes that have been located to date, as well as by the discontinuation of these models by 1952.
A Sears, Roebuck "Homart" model as shown in the 1951 catalog.
By 1950, the nearly 13 million servicemen and women who had returned from the war with the dream of raising their families in peace desired new homes. They wanted something new and possessed sufficient funds to invest in better housing.
### MID-CENTURY MODERN: THE RANCH HOUSE
With the return of prosperity, architects took a new direction in design: the Western Ranch style. Like other architectural styles, it has roots in the past, representing a revival of the nineteenth-century "rancho" homes built in the Southwest by Spanish-speaking settlers.
The one-story rambling plans characterized by a long, low profile became the most popular postwar house design. To add visual interest, architects used a variety of cladding materials such as cedar siding, brick, and Permastone, a type of simulated masonry. Massive chimneys were located centrally or on the front elevation, and a picture window was almost obligatory.
Mid-Century Modern took the country by storm. Due to increased automobile ownership, suburbs no longer were limited to the extent of commuter train, bus, or streetcar lines. The new suburbs offered large lots to accommodate ranch designs.
By the end of World War II, only three of the original mail-order home companies were still in business—Aladdin, Lewis/Liberty, and Sterling—all headquartered in Bay City, Michigan. In the face of increasing competition from makers of prefabricated housing, the Bay City companies continued to do what they knew best: produce precut homes. The cover of the 1954 Aladdin catalog pointed out that the Readi-Cut homes were not prefabricated.
The cover of Sterling's 1948 catalog foreshadowed the new, soon-to-be dominant Western Ranch designs; however, most of the homes in this catalog were Minimal Traditional designs.
Of course, the designs of mail-order homes followed popular trends. Typically, providers of precut homes were cautious about abandoning the tried and true, best-seller models of the past. Although Aladdin's 1948 catalog featured the "Palm Springs" Western Ranch on the cover, it offered only one additional ranch design; the remainder of the catalog consisted of more traditional homes, many continued from earlier catalogs. As it became clear that the ranch was here to stay, mail-order home catalogs contained increasing numbers of ranch models.
Waunakee, Wisconsin, is the home of this Aladdin "Alamo" model. Note the different heights of roof sections, emphasizing the horizontal line.
The Mid-Century Modern Contemporary Style, with its front-facing gable, asymmetrical roof, and angular lines was uncommon in mail-order catalogs. Lewis/Liberty offered this "Carlton" model in 1962–63.
In the mid-1950s and 1960s, Sears, Roebuck again entered the housing market, offering models under the tradename Homart. No catalogs advertising these have been unearthed as of 2010. Blueprints bear the names Homart and Sears, Roebuck but do not indicate whether the buildings were prefabricated, built from kits, or built by standard construction.
As land gradually became more scarce and costly, ranch designers developed split-level and two-story versions of the new style. An attached garage became a standard feature on larger models. The main mail-order home enthusiasts of the new styles were Aladdin and Lewis/Liberty; Sterling never embraced the ranch home to the same extent, offering only one-story models and continuing to market Minimal Traditional designs through 1974.
This 1954 Minimal Traditional concrete block Sears, Roebuck Homart home in Warrenville, Illinois, complete with blueprints, is so far the only known example of its kind.
The conservative Sterling 1971 "Graystone" was designed "with an eye toward economy and yet embodying that modernistic trend."
A row of three identical 1965 Sears Homart models stands in the 3400 block of North Pittsburg Street in Chicago. These bungalows were designed to fit on a narrow city lot.
Lewis/Liberty's 1962 "Marlboro" is a cross-gabled home that blends "the beauty of the ranch house and the flair and compactness of the split-level."
##### THE DEMISE OF THE MAIL-ORDER HOME
STERLING'S 1916 prediction that "within five years, at the present rate of increase, 90 percent of all frame buildings will be cut to fit in special factories" turned out to be a dream rather than a reality. In fact, by the mid-1920s, only an estimated 3 percent of new housing starts were precut buildings. Nonetheless, the precut method of construction proved to be sufficiently popular to keep eight major companies solvent until 1983.
However, major changes in the way the United States built housing proved to be detrimental to the producers of precut kits. Throughout the 1950s, tract housing became increasingly popular. A developer would buy a large tract of land, subdivide it, and put up a large number of homes of similar size and design. Developers required the use of approved designs and/or builders for their subdivisions, making it difficult for anyone to build a precut home in these locations.
The back cover of Gordon-Van Tine's 1927 catalog shows a bygone era, a busy mill on the Mississippi River in Davenport, Iowa.
Competition from the many makers of prefabricated housing, such as Gunnison, National, Economy, U.S. Steel, Harnischfeger, and Lustron had a negative impact on Lewis/Liberty, Sterling, and Aladdin. As a standard part of their services, prefabricated home manufacturers commonly offered delivery by truck and even sent a crew to erect the house. Mail-order home purchasers usually had to pay freight and build the house themselves or hire a carpenter.
The merged Gunnison/U.S. Steel, a major producer, manufactured prefabricated homes in New Albany, Indiana, from 1935–74. Like their competitors' homes, they consisted of factory-constructed panels, complete with interior and exterior cladding. The Lustron Company, a minor competitor, produced all-steel prefabricated homes in a retrofitted aircraft plant in Columbus, Ohio, from 1948–50. These homes commonly are misidentified as Sears homes.
The owners of this Aladdin "Fairlane" model in West Dundee, Illinois, were able to do the construction work themselves only because a contractor friend was willing to claim he was building the house.
The large steel panels, coated with vitreous enamel, are clearly visible on this prefabricated "Lustron" model in Versailles, Indiana.
Mobile homes represented a new facet of the housing market, and they became especially popular as Americans took to the roads in their automobiles. These entirely preconstructed houses on wheels could be installed on a concrete slab. Mobile home owners were able to rent lots countrywide in the new mobile home parks that popped up in the 1950s, sparing the expense of buying and selling land. Should the owner decide to move, the home simply could be pulled to a new location. Because these homes were less costly to build and easier to install, mobile home production gradually surpassed that of precut homes.
With a declining market due to demands for onsite delivery, the development of tract housing, and the growing popularity of prefabricated and mobile housing, the precut home gradually faded away. Sterling and Lewis closed in the 1970s within a year of each other. Aladdin, the pioneer of the mass-marketed kit home, hung on until 1983: the first company in the business also had the distinction of being the last.
The Aladdin "Villa" in Wakefield, Michigan. (Photograph by Dale Wolicki)
##### PLACES TO VISIT
### ILLINOIS
Aurora: a self-guided tour booklet of Sears homes is available from the Planning Department. Visit www.aurora-il.org for information.
Carlinville: "Standard Addition" nine square blocks of 152 Sears homes erected for miners at the Standard Oil Company mine. Visit www.carlinvillechamber.com/sears.html for information.
Elgin: Two local mail-order home surveys and rare mail-order catalogs from multiple companies are available at the Gail Borden Public Library, 270 N. Grove Avenue. Visit www.gailborden.info and click on "research photographs, Illinois Digital Archives" for information.
Libertyville: Cook Memorial Public Library District has a reference book of materials about the town's mail-order homes, in addition to various other titles. Visit www.cooklib.org for information.
### MICHIGAN
Bay City, Michigan: The Bay County Historical Society has information about area mail-order home manufacturers. Visit www.bchsmuseum.org for information.
Mount Pleasant, Michigan: Clarke Historical Library, University of Central Michigan, holds sales records from the Aladdin Company. Visit clarke.cmich.edu for information.
### OTHER LOCATIONS
Birmingham, England: Austin Village, built in 1917 by Herbert Austin to house munitions workers at Longbridge during World War 1.
Hopewell, Virginia: Aladdin Village, 294 homes built for Dupont Company workers. Visit www.hopewellva.gov for information.
Laramie, Wyoming: University of Wyoming American Heritage Center has a large collection of Montgomery Ward mail-order home catalogs. Visit ahc.uwyo.edu for information.
##### ADDITIONAL RESOURCES
### WEBSITES
www.antiquehomestyle.com/plans/sterling
This page features information about Sterling homes in a website about "antique" homes.
www.cooklib.org/Localhistory/MailOrderHouses/MailOrderHouses.html
This site features mail-order homes in Libertyville, Illinois.
www.gordonvantine.com
This company site features Gordon-Van Tine company history and images of houses.
www.kithouse.org
This site features services and publications by historical architectural researcher Rebecca Hunter; includes architectural salvage items.
www.lhi.org.uk/docs/Austin_Village.pdf
This features an Austin Village Preservation Society booklet.
www.lustron.net and groups.yahoo.com/group/LustronHomes
These sites feature information about all-steel Lustron homes as well as a list of home locations.
www.mailorderhomesofparkridge.blogspot.com
This is a blog about mail-order homes in Park Ridge, Illinois.
www.searsarchives.com/homes
This is the website of Sears, Roebuck Archives, including the national Sears Home Registry.
www.searshomes.org
"Sears Modern Homes" is the website of Rosemary Thornton, author of _The Houses That Sears Built_.
www.wardwayhomes.com
This website is dedicated to providing information about homes marketed by Montgomery Ward from 1909–31.
##### FURTHER READING
Barham, Ruth Mathis. _Daily American Republican Newspaper_. Poplar Bluff, Missouri, November 12, 1992.
Emmet, Boris and John E. Jeuck. _Catalogs & Counters_. University of Chicago Press, 1930.
Fetters, Thomas. _The Lustron Home_. McFarland and Co., 2002.
Hunter, Rebecca. _Elgin, Illinois Sears House Research Project_. Elgin Heritage Commission, 1999.
Hunter, Rebecca. _Beyond Sears: Mail Order Homes in Elgin Illinois from Aladdin, Lewis, Sterling, Harris Brothers, Gordon-Van Tine and Montgomery Ward_. Elgin Heritage Commission, 2004.
Hunter, Rebecca. _Putting Sears Homes on the Map: A Compilation of Testimonials Published in Sears Modern Homes Catalogs from 1908–1940_. Rebecca Hunter, 2004
Johnson, Cynthia E. _House in a Box: Prefabricated Housing in the Jackson Purchase Cultural Landscape Region_. Kentucky Heritage Council, 2006.
McArthur, Shirley Du Fresne. _Frank Lloyd Wright System Built Homes in Milwaukee_. Northpoint Historical Society, 1985.
Schweitzer, R. and M. W. R. Davis. _America's Favorite Homes: Mail Order Catalogues As a Guide to Popular Early 20th Century Houses_. Wayne State University Press, 1990.
Stevenson, K. C. and H. W. Jandl. _Houses By Mail_. National Trust for Historic Preservation, 1986.
Thornton, Rosemary. _The Houses That Sears Built_. Gentle Beam Publications, 2002.
Thornton, Rosemary. _Sears Homes of Illinois_. The History Press, 2010.
Thornton, Rosemary and Dale Patrick Wolicki. _Montgomery Ward's Mail Order Homes_. Gentle Beam Publications, 2010
Wheeler, F. K. _Recollection of 36 Years with a Fascinating Lumber Plant_. Unpublished.
Wolicki, Dale Patrick. _The Historic Architecture of Bay City Michigan_. The Bay County Historical Society, 1998.
### CATALOG REPRINTS
Aladdin Company. _Aladdin "Built in a Day" House Catalog 1917_. Athenaeum and Dover Publications, 1995
Bennett Lumber Co. _Bennett's Small House Catalog 1920_. Dover Publications, 1993.
Gordon-Van Tine. _117 House Designs of the Twenties_. Athenaeum and Dover Publications, 1992.
Pacific Homes: _California's Kit Homes: A Reprint of the 1925 Pacific Ready Cut Homes Catalog_. Gentle Beam Publications, 2004.
Sears, Roebuck. _Sears, Roebuck Home Builder's Catalog 1910 Edition_. Athenaeum and Dover Publications, 1990.
Sears, Roebuck. _Sears Roebuck Catalog of Houses 1926_. Athenaeum and Dover Publications, 1991.
Sears, Roebuck. _Homes in a Box: Modern Homes from Sears, Roebuck and Co_. [1912 catalog] Schiffer Publishing, 1998.
Sears, Roebuck. _Sears Roebuck Homes of Today 1932_. Athenaeum and Dover Publications, 2003.
Sears, Roebuck. _Sears, Roebuck Book of Barns: A Reprint of the 1919 Catalog_. R. L. Hunter Press, 2005.
Sears, Roebuck. _Sears Modern Homes, 1913_. Dover Publications, 2006.
Montgomery Ward. _Wardway Homes, Bungalows and Cottages 1925_. Athenaeum and Dover Publications, 2004.
Published in Great Britain in 2012 by Shire Publications Ltd, Midland House, West Way, Botley, Oxford OX2 0PH, United Kingdom.
44-02 23rd Street, Suite 219, Long Island City, NY 11101, USA.
E-mail: shire@shirebooks.co.uk www.shirebooks.co.uk
© 2012 Rebecca L. Hunter.
All rights reserved. Apart from any fair dealing for the purpose of private study, research, criticism or review, as permitted under the Copyright, Designs and Patents Act, 1988, no part of this publication may be reproduced, stored in a retrieval system, or transmitted in any form or by any means, electronic, electrical, chemical, mechanical, optical, photocopying, recording or otherwise, without the prior written permission of the copyright owner. Enquiries should be addressed to the Publishers.
Every attempt has been made by the Publishers to secure the appropriate permissions for materials reproduced in this book. If there has been any oversight we will be happy to rectify the situation and a written submission should be made to the Publishers.
A CIP catalogue record for this book is available from the British Library.
Shire Library no. 645. Print ISBN-13: 978 0 74781 048 3
PDF e-book ISBN: 978 0 74781 112 1
ePub e-book ISBN: 978 1 78200 103 4
Rebecca L. Hunter has asserted her right under the Copyright, Designs and Patents Act, 1988, to be identified as the author of this book.
Designed by Tony Truscott Designs, Sussex, UK and typeset in Perpetua and Gill Sans.
Printed in China through Worldprint Ltd.
12 13 14 15 16 10 9 8 7 6 5 4 3 2 1
© _Shire Publications. Access to this book is not digitally restricted. In return, we ask you that you use it for personal, non-commercial purposes only. Please don't upload this pdf to a peer-to-peer site, email it to everyone you know, or resell it. Shire Publications reserves all rights to its digital content and no part of these products may be copied, stored in a retrieval system or transmitted in any form by any means, electronic, mechanical, recording or otherwise (except as permitted here), without the written permission of the publisher. Please support our continuing book publishing programme by using this pdf responsibly_.
COVER IMAGE
Catalog image of the Sears "Alhambra."
ACKNOWLEDGEMENTS
Without assistance, encouragement, and support from the following, this book could not have been created: Dale Wolicki and Rosemary Thornton, my colleagues in pursuit of the elusive history of mail-order homes; Arlene May, Sears, Roebuck Archivist, consultant, and provider of materials and information; Marilyn Raschka, second-generation Hartford, Wisconsin, Sears "Mitchell" owner; Mary Rowse, Chevy Chase, Maryland, tireless collector of mail-order homes and their histories; Tom Wahl, Elyria, Ohio, discoverer of a subdivision of 1941 Sears, Roebuck homes; Kelli Christiansen, editor, Shire Books; Cindy Catanzaro, Springfield, Ohio; Jean Lythgoe, Rockford, Illinois, Public Library; Historic Preservation Commission, Blue Island, Illinois; Lincoln Library Sangamon Valley Collection, Springfield, Illinois; Preservation Department, City of Aurora, Illinois.
MAIL-ORDER HOME SURVEY SPONSORS
James Chapa, Downers Grove, Illinois; Heritage Commission, Elgin, Illinois; Arlene Lane and Sonia Schoenfield, Cook Memorial Library, Libertyville, Illinois; Heidi Beazley and Fidencio Marbella, Public Library, Melrose Park, Illinois; City Museum, West Chicago, Illinois; Park District, Worth, Illinois; Joe Carney and Ray Featherstone, Anderson, Indiana; Garden Club, Sylvan Lake, Michigan; and Sunny Acres Civic and Improvement Association, Cranford, New Jersey.
PHOTOGRAPHIC CREDITS
All photographs by Rebecca Hunter unless otherwise specified.
Shire Publications is supporting the Woodland Trust, the UK's leading woodland conservation charity, by funding the dedication of trees.
|
{
"redpajama_set_name": "RedPajamaBook"
}
| 6,487
|
{"url":"https:\/\/stat.ethz.ch\/pipermail\/r-sig-mixed-models\/2010q4\/004667.html","text":"# [R-sig-ME] psychometric function fitting with lmer?\n\nDoran, Harold HDoran at air.org\nFri Oct 29 22:05:31 CEST 2010\n\nThe info below is helpful, more comments below.\n\n> -----Original Message-----\n> From: mike.lwrnc at gmail.com [mailto:mike.lwrnc at gmail.com] On Behalf Of Mike\n> Lawrence\n> Sent: Friday, October 29, 2010 3:41 PM\n> To: Doran, Harold\n> Cc: r-sig-mixed-models at r-project.org\n> Subject: Re: [R-sig-ME] psychometric function fitting with lmer?\n>\n> On Fri, Oct 29, 2010 at 3:29 PM, Doran, Harold <HDoran at air.org> wrote:\n> > \u00a0 First, I don't know how one uses OLS\n> > \u00a0to fit a probit model.\n>\n> I've seen it done. Folks usually collapse responses to\n> means-per-value-on-the-x-axis then either use a computationally\n> intensive search algorithm to minimize the squared error on the\n> proportion scale, or fit a simple linear function on the probit scale\n> (when they encounter means of 1 or 0, they \"tweak\" these values by\n> either dropping that data entirely or adding\/subtracting some\n> arbitrary value).\n\nI think you're right, this seems like an inadvisable way to estimate some model parameters. I'll try and ignore this part of the email and focus on the info below. I will most likely explode if I try and figure out a) how this was done and b) why someone would do it this way when there are well-known ways to estimate model parameters in such cases.\n\n> Regardless, I suspect we both agree that these are inadvisable ways of\n> dealing with this data, but I'm not sure we are on the same page with\n> respect to the underlying paradigm motivating the data analysis.\n> Whereas the paper you provided appears to be discussing data derived\n> from questionnaires with different items, etc, I was thinking (and I\n> apologize for failing to be more clear on this earlier) of data\n> derived from studies of temporal order judgement and other\n> psychophysical discrimination studies. Here's an example that I\n> happened to find while searching google for an article not behind a\n> pay-wall:\n>\n> http:\/\/www.psych.ut.ee\/~jyri\/en\/Murd-Kreegipuu-Allik_Perception2009.pdf\n>\n> In such studies, individuals are provided two stimuli and asked \"which\n> one is more X\", where the stimuli are manipulated to explore a variety\n> of values for the difference of X between them. For example, in\n> temporal order judgements, we ask which of two successive stimuli came\n> first, right or left, then plot proportion of \"right first\" responses,\n> accumulated over many trials, as a function of the amount of time by\n> which the right stimulus led the right stimulus (SOA, or stimulus\n> onset asynchrony, where negative values mean the right stimulus\n> followed the left stimulus).\n\nOK, with you here so far, though this kind of thing is a bit away from my field of study. So, let me simplify for sake of argument, we have binary responses at this point where 1 = respondent answer 'right' and 0 = respondent answer 'left'. You also have some observed characteristics of these individuals, call them x.\n\nNow, you use the terms \"likely\" and \"unlikely\" below is something of a colloquial sense, but we can actually quantify this in a model such as:\n\nPr(1|\\theta, \\beta) = 1\/[1 + exp(beta-theta)]\n\nWhich gives the conditional probability that some individual with theta (being an aptitude of some form) will choose the answer \"right\" also conditional on \\beta, which is a characteristic of the item\/task itself. Now, you state that you have other observed characteristics, such as \"time\". You can further condition on these observed characteristics to get the conditional probabilities directly, which seems to be what you are after. If this is right, then the methods in the paper I linked are directly related to this problem, just based on a different data set. It is a general modeling strategy you can employ with lmer.\n\nWhat I don't understand is what bias or variance are you trying to get at? Bias refers to the property that \\beta - E[\\hat{\\beta}] = 0, which would not hold if the parameter estimate were biased. Maybe I am still a bit unclear on the issue.\n\n> This typically yields a sigmoidal\n> function where people are unlikely to say \"right-first\" when the left\n> stimulus leads by a lot (large negative SOA values) and very likely to\n> say \"right-first\" when the right stimulus leads by a lot (large\n> positive SOA values. The place where this function crosses 50% is\n> termed the point of subjective simultaneity (PSS) and the slope of the\n> function indexes the participants' sensitivity (shallow slopes\n> indicate poor sensitivity, sharp slopes indicate good sensitivity).\n> Researchers are often then interested in how various experimental\n> manipulations affect these two characteristics of performance.\n\nNow, if the slope of the curve matters, and it often does, then lmer cannot be used to estimate such a model because the model we demonstrate (Rasch model) assumes all items\/tasks have a constant slope. But, other models extend the conditional probability above and can do this. I believe you can accomplish this using LTM package.\n\n>\n>\n> > Second,\n> > \u00a0why are you treating the observed data as a parameter estimate? Why don't\n> you\n> > \u00a0actually estimate the model parameters (i.e., the item parameters), which\n> are\n> > \u00a0asymptotically unbiased under certain estimation conditions. You can do\n> this in a number of\n> > \u00a0ways in R, lme4 can do this using lmer as described here:\n> >\n> > \u00a0http:\/\/www.jstatsoft.org\/v20\/i02\n> >\n> > \u00a0Or you can use JML methods for Rasch in the MiscPsycho package or you can\n> use\n> > \u00a0MML methods in the LTM package. What you seem to be doing is treating the\n> eICC\n> > \u00a0as some kind of parameter for the item; but this is not reasonable I don't\n> > \u00a0think.\n> >\n> >> This\n> >> > fitting is typically done within each individual and condition of\n> >> > interest separately, then the resulting parameters are submitted to 2\n> >> > ANOVAs: one for bias, one for variability. I wonder if this analysis\n> >> > might be achieved more efficiently using a single mixed effects model,\n> >> > but I'm having trouble figuring out how to approach coding this.\n> >>\n> >\n> >\n> > \u00a0I'm not sure I can help you here as I am unclear on what you are doing\n> > \u00a0exactly. Maybe if we elaborate a bit on what you are trying to do above, we\n> > \u00a0can do this part next.\n> >\n> >>\n> >> Below\n> >> > is an example of data similar to that collected in this sort of\n> >> > research, where individuals fall into two groups (variable \"group\"),\n> >> > and are tested under two conditions (variable \"cue\") across a set of\n> >> > values from a continuous variable (variable \"soa\"), with each cue*soa\n> >> > combination tested repeatedly within each individual. A model like\n> >> >\n> >> > fit = lmer(\n> >> > \u00a0 \u00a0 formula = response ~ (1|id) + group*cue*soa\n> >> > \u00a0 \u00a0 , family = binomial( link='probit' )\n> >> > \u00a0 \u00a0 , data = a\n> >> > )\n> >> >\n> >> > employs the probit link, but of course yields estimates for the slope\n> >> > and intercept of a linear model on the probit scale, and I'm not sure\n> >> > how (if it's even possible) to convert the conclusions drawn on this\n> >> > scale to conclusions about the bias and variability parameters of\n> >> > interest.\n> >> >\n> >> > Thoughts?\n> >> >\n> >\n> >> > _______________________________________________\n> >> > R-sig-mixed-models at r-project.org mailing list\n> >> > https:\/\/stat.ethz.ch\/mailman\/listinfo\/r-sig-mixed-models\n> >\n> > _______________________________________________\n> > R-sig-mixed-models at r-project.org mailing list\n> > https:\/\/stat.ethz.ch\/mailman\/listinfo\/r-sig-mixed-models\n> >","date":"2021-11-28 23:25:33","metadata":"{\"extraction_info\": {\"found_math\": true, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 0, \"math_alttext\": 0, \"mathml\": 0, \"mathjax_tag\": 0, \"mathjax_inline_tex\": 0, \"mathjax_display_tex\": 0, \"mathjax_asciimath\": 1, \"img_math\": 0, \"codecogs_latex\": 0, \"wp_latex\": 0, \"mimetex.cgi\": 0, \"\/images\/math\/codecogs\": 0, \"mathtex.cgi\": 0, \"katex\": 0, \"math-container\": 0, \"wp-katex-eq\": 0, \"align\": 0, \"equation\": 0, \"x-ck12\": 0, \"texerror\": 0, \"math_score\": 0.7947113513946533, \"perplexity\": 3533.4664118248197}, \"config\": {\"markdown_headings\": true, \"markdown_code\": true, \"boilerplate_config\": {\"ratio_threshold\": 0.18, \"absolute_threshold\": 10, \"end_threshold\": 15, \"enable\": true}, \"remove_buttons\": true, \"remove_image_figures\": true, \"remove_link_clusters\": true, \"table_config\": {\"min_rows\": 2, \"min_cols\": 3, \"format\": \"plain\"}, \"remove_chinese\": true, \"remove_edit_buttons\": true, \"extract_latex\": true}, \"warc_path\": \"s3:\/\/commoncrawl\/crawl-data\/CC-MAIN-2021-49\/segments\/1637964358673.74\/warc\/CC-MAIN-20211128224316-20211129014316-00072.warc.gz\"}"}
| null | null |
Meet the Prosthetics Startup Aiding Amputees in Rural India With Helping Hands
by Rina Diane Caballar
Manufacturing - Oct 9 2019 - 4 min read
Social Hardware's prosthetics are purpose-built using generative design to be affordable for agricultural and construction workers in rural India. Images courtesy of Social Hardware.
"There is a gaping hole in the prosthetic market," says Abhit Kumar, cofounder of Indian prosthetics startup Social Hardware. "Our aim is to fill that hole with prosthetic devices designed specifically for low-income rural amputees from the agricultural or construction industry, as this is where most of the casualties occur."
According to a report from India's Ministry of Statistics and Programme Implementation, 2.21% of the country's population lives with a disability. Of that population, 20% have a disability of movement, and most live in rural areas. The highest rate of amputation occurs in these rural settings, where the main employment is in agriculture and construction. Prosthetics are typically imported from overseas and can cost more than six times the average monthly income of a rural Indian family.
After researching existing solutions, Social Hardware cofounders Kumar and Cameron Norris saw a need for a new approach to upper-limb prosthesis design. The pair met through the online community Reddit, where they contributed to an open-source prosthesis project to help a fellow user with a disability. Kumar came from a biomedical and robotics background; Norris worked in the UK startup world.
icon-arrow-lefticon-arrow-right
The latest Social Hardware prosthetic arm with a farming tool attached. Courtesy of Social Hardware.
Alternate render of the prosthetic arm with a farming tool attached. Courtesy of Social Hardware.
The latest Social Hardware assembly with a prosthetic hand attached. Courtesy of Social Hardware.
Alternate render of the Social Hardware assembly with a prosthetic hand. Courtesy of Social Hardware.
Prosthetic arm with both the hand and farming tool attached to the wrist connector. Courtesy of Social Hardware.
Finding a Path Forward
As a recognized startup under the Startup India initiative and a technical partner of Bengaluru-based nonprofit Association of People with Disability (APD), Social Hardware works toward outfitting amputees in rural India with prosthetic devices and providing rehab opportunities—at no cost to the recipients.
Through partnerships with APD and other disability rehab centers, Social Hardware will provide rural amputees access to a rehab program that includes physiotherapy and training to effectively use an assistive device in their daily lives. The startup wants to keep the retail price under ₹20,000 (about $280) for each prosthetic hand; those who successfully complete the program will be fitted with Social Hardware's prosthetics for free.
"By speaking directly with amputees and those working within the prosthetics and orthotics industries, we learned that most devices are unsuitable for end users in rural India," Norris says. "In particular, durability and hygiene are two major issues not being addressed appropriately."
Working prototypes of Social Hardware's latest wrist connectors. Courtesy of Social Hardware.
3D renders of Social Hardware's latest wrist connectors. Courtesy of Social Hardware.
The flip side of the wrist connectors. Courtesy of Social Hardware.
It's All in the Wrist
Once Social Hardware developed a suitable electric prosthetic arm, the team created the Avocado Wrist Connector, an assistive device used to securely attach agricultural and construction tools to their prosthetics. "We took inspiration from the ruggedness of military hardware design," Norris says. "We looked through old patents to help us understand how different attachments were mounted in military settings."
They also wanted it to be compatible with the existing anchor system of connectors and sockets made by ALIMCO, one of India's largest manufacturers of prosthetic and orthotic devices. "Instead of replacing what was already available in the prosthetics market, we wanted to support and supplement it," Norris says. "We wanted to create an add-on device that fits between the bionic hand and the socket, which can be used to attach different tools such as a small hammer or a trowel or whatever tool the individual may need."
But first, the team had to address the issues with existing prosthetic devices. Externally powered prosthetic limbs don't rely on a user's strength but, instead, have limited battery life and typically aren't sturdy enough for high-intensity work. Body-powered prosthetic limbs are generally much more durable and do not require batteries, but they are heavy and have limited functionality. Kumar and Norris wanted to combine the two: the lightweight, feature-rich approach of an externally powered prosthetic limb with the durability and reliability of a body-powered device. "The end goal is to provide the best of both worlds," Norris says.
To achieve this balance, Social Hardware turned to generative design. With measurements from traditionally fabricated sockets, the team used Autodesk Fusion 360 and a five-stage generative design process to bring down the weight of its Avocado Wrist Connector and speed up the development process. "Generative design enabled us to reduce the weight from 300 grams to 96 grams while maintaining the durability we needed," Kumar says. "These results would have taken months of trial and error to achieve through traditional means."
Generative design also helped Social Hardware tackle the challenges amputees experience with their prosthetic sockets and to improve their appearance. "There are issues with sweat and heat, especially in humid environments or if you're working at a high intensity," Norris says. "We used generative design to create a lightweight, breathable socket with a unique aesthetic. Although we're targeting low-income communities, we want them to feel like they have the latest cutting-edge technology."
Social Hardware used generative design to bring down the Avocado Wrist Connector's weight while maintaining its strength. Courtesy of Social Hardware.
Prosthetics for the Masses
As part of their future plan, Kumar and Norris want to transition from a working prototype to an end product ready for mass production. Preparing for clinical trials before releasing their product as a medically certified prosthesis, they plan to work with international aid agencies and humanitarian organizations to bring their devices to other APEC countries that need them.
Moreover, Social Hardware's founders want to provide the first commercial version of their device for preorder as a prosthesis development kit for STEM (science, technology, engineering, and math) education. The kit will include everything hobbyists, researchers, and students need to build their own prosthetic hands.
"We're providing the kit to raise awareness of India's rural amputee population, encourage others to participate in the development of assistive technologies, and for young graduates to understand the concept of design engineering and fabrication," Kumar says. "It also builds on our foundation: frugal innovation and participatory design."
#Generative Design - #Medical Devices - #Product Design - #Prototyping
Intersect: Continuous Positive Airway Pressure Device Can Help Save Babies' Lives
Airbus Continues to Innovate Bionic Design for Future Sustainable Flights
by Ken Micallef
|
{
"redpajama_set_name": "RedPajamaCommonCrawl"
}
| 1,315
|
Police catch trucker on his mobile phone while driving on the M1 motorway in Derbyshire
The M1 motorway.
An HGV lorry driver has been ordered to pay £253 after he was spotted by police using a handheld mobile phone while driving on the M1 motorway in Derbyshire.
Chesterfield magistrates' court heard on January 24 how Romanian Alexandru Bogdan Ursu, 26, of no fixed abode, was spotted by police on the northbound carriageway of the M1 motorway at Junction 28 in Pinxton using his mobile phone.
Prosecuting solicitor Becky Allsop said: "Officers were travelling on the M1 at 11am on January 23 and they began to overtake an HGV and a police officer described looking into the cab of the lorry and seeing the driver with a phone in his left hand with it to his ear.
"He was clearly speaking into the phone and he was observed doing this for ten to 15 seconds. The officer said his position did not change even when he was looking directly at him."
Mrs Allsop added that the articulated lorry was stopped and Ursu apologised and was taken to a police station.
Ursu pleaded guilty to using a handheld mobile phone while driving on the M1.
He told the court: "I was holding the mobile phone while driving but when I was seen or recorded by the officers I was not aware they were police officers and I thought they were just motorists."
Magistrates fined Ursu £138 and ordered him to pay a £30 victim surcharge and £85 costs.
His driving record was also endorsed with six points.
Newbold dad to host bowls tournament for children's hospital charity
|
{
"redpajama_set_name": "RedPajamaCommonCrawl"
}
| 713
|
\section{Introduction}
One of the major goals of the global neutrino physics programme is to explore fundamental symmetries of nature linked to why we live in a matter-dominated universe.
Charge-parity symmetry violation (CPV) in the neutrino sector is one possibility remaining to be explored further experimentally, and neutrino experiments strive to improve current measurements of CPV in the leptonic sector~\cite{Abe:2019vii}.
CPV is obtained from the simultaneous fit of the $\nu_{\mu}$ disappearance and $\nu_e$ appearance oscillation channels separately for neutrinos and anti-neutrinos.
In the absence of CPV and accounting for matter effects, the rates of $\nu_{\mu}\!\rightarrow\!\nu_e$ and $\overline{\nu}_{\mu}\!\rightarrow\!\overline{\nu}_e$ oscillations should be equal.
To convert the measured rate of interactions to a level of CPV, experiments must accurately know the cross section for the interactions of neutrinos and anti-neutrinos with detector materials, which are most commonly hydrogen, carbon, oxygen, argon and iron.
Therefore, systematic uncertainties on neutrino--nucleus interaction cross sections are a key input to such CPV searches.
These interaction cross sections are dependent on modelling neutrino-nucleon interactions occurring within nuclei.
The nuclear models informed by these cross sections have substantial effects on the measured final-state particle kinematic distributions~\cite{Mosel:2016cwa}.
The long baseline neutrino experiments that are currently searching for CPV are the Tokai to Kamioka experiment~(T2K)~\cite{Abe:2019vii} and the NuMI Off-Axis $\nu_{e}$ Appearance experiment~(NOvA)~\cite{Acero:2019ksn}.
The T2K experiment, which currently reports the strongest constraint on CPV in neutrinos~\cite{Abe:2019vii}, has systematic uncertainties of {7}--9\%
after near-detector constraint on the prediction of the rate of far detector electron-like events, with cross section uncertainties being the largest contribution.
The~future Deep Underground Neutrino Experiment~(DUNE)~\cite{abi2020deep} and Hyper-Kamiokande~\cite{abe2011letter} projects will seek to reach {1}--3\%
on that same rate of far detector electron-like events~\cite{acciarri2016long}, with improved systematic errors providing better precision on the CP violating phase.
The key to reducing these uncertainties is to precisely measure the multiplicity and momentum distribution of final-state particles.
\mbox{However, these distributions} are modified by final state interactions (FSI) of the recoiling secondary particles as they traverse the target nucleus.
The most commonly used neutrino generator Monte Carlos (GENIE~\cite{Andreopoulos:2009rq}, NEUT~\cite{Hayato:2009zz} and NuWro~\cite{GOLAN2012499}), simulate FSI with cascade models that are tuned with external hadron--nucleus scattering measurements. The generator GiBUU~\cite{lalakulich2013neutrino} models FSI by solving the semi-classical Boltzmann--Uehling--Uhlenbeck equation.
However, as shown in Figure~\ref{fig:DataProtonXSec}, proton--nucleus scattering measurements are extremely sparse and in many cases do not exist in the relevant energy region and/or on the relevant nuclei.
Therefore semi-empirical parametrisations are used to extrapolate in momentum and atomic mass~\cite{wellisch1996total}.
The~parametrisations are different between the three generators, and yield order-of-magnitude scale differences in the predicted multiplicity and kinematics of final state protons~\cite{dune2018high}.
The proton final state modelling is a key ingredient for neutrino oscillation measurements because it affects the event selection and neutrino energy reconstruction in charged-current (CC) interactions, which is the channel used to measure oscillation parameters and is therefore central to the search for CPV~\cite{Abe:2013hdq}.
For these reasons, FSI contribute substantially to the total neutrino interaction systematic uncertainty~\cite{Abe:2019vii}.
\begin{figure}[H]
\centering
\includegraphics[width=12cm]{files/Figures/DataProtonCrossSections.pdf}%
\caption{Total reaction cross sections for protons on argon, neon, fluorine, oxygen, carbon and helium-4. {Data}~\cite{Carlson:1996ofz} are compared to a semi-empirical {model}~\cite{wellisch1996total}. Figure from~\cite{SPSC-P-355}.}
\label{fig:DataProtonXSec}%
\end{figure}
Moreover, FSI models are in tension with data.
Recent neutrino scattering measurements have shown that the most-used models of neutrino-nucleus interactions (employed by NEUT and GENIE) differ from nature in both cross section and kinematics of final state particles by as much as 30\%~\cite{McFarland:2018aaa}.
These uncertainties cannot be fully mitigated with near/far detector combinations because they come from theoretical model deficiencies that are not cancelled in the near–far extrapolation~\cite{Coloma:2013rqa}.
The key proton kinetic energy range in which to distinguish interaction models is the region below 0.1~GeV.
Figure~\ref{fig:protonsfromargon} shows the proton multiplicity and kinetic energy distributions for $\nu_{\mu}$ CC interactions on argon calculated by the GENIE, NEUT and NuWro neutrino generators for the DUNE experiment.
These distributions are highly discrepant at low proton kinetic energy as shown in the right hand panel. The generators are not designed to handle the low energy region consistently, due~to~the lack of available data.
This is predominantly below the proton detection threshold in liquid Argon TPCs (0.04~GeV), such as those that will be used by DUNE, and in water Cherenkov detectors (0.5~GeV).
\mbox{The lower} threshold in high pressure gas provides a unique opportunity to distinguish between neutrino interaction models for the same nuclear target.
\begin{figure}[H]
\centering
\includegraphics[width=12cm]{files/Figures/protons_from_argon.jpeg}%
\caption{{Predicted proton} kinetic energy (KE) spectra from GENIE, NEUT and NuWro~\cite{Raaf:2018aaa}. \mbox{Energy spectra} up to 1 GeV are shown on the left, and zoomed in to lower energies on the right. The figure uses the Long Baseline Neutrino Facility (LBNF) simulation for DUNE's beam energy and flux. The LBNF beam has a mean energy of approximately 2.5~GeV~\cite{abi2020deep}. The dashed vertical line indicates the expected proton automated-reconstruction/identification threshold in liquid argon, and the solid vertical line shows the same for gaseous argon at 10 atm~\cite{dune2018high}.}
\label{fig:protonsfromargon}%
\end{figure}
We have built a High Pressure gas Time Projection Chamber (HPTPC) prototype and exposed it to a charged particle beam in the T10 beamline at CERN in August and September 2018 \cite{SPSC-P-355}.
The~momentum profile of the T10 beam can be tuned within the range 0.8--6.5~GeV/c (kinetic energy range 0.3--5.6~GeV).
Figure~\ref{fig:utofNoBend} left, shows the time of flight (ToF) spectrum for the T10 beamline tuned to a momentum of 0.8~GeV/c; this measurement was made with our upstream ToF system (see Section~\ref{hptpcPaper:sec:Methods} for details of the ToF systems).
The kinetic energy of the protons calculated from the upstream ToF measurements in this sample is shown in Figure~\ref{fig:utofNoBend} right.
As shown, the flux of protons with kinetic energy less than 200~MeV is negligible.
The physics objective of the HPTPC beam test was to make measurements of protons on argon at kinetic energies below 200~MeV, i.e., below what was available with the T10 beam.
Furthermore, the readout speed of the charge-coupled device cameras (CCD) employed in the HPTPC prototype motivates a limit on the total particle multiplicity in the TPC active~volume.
To enhance the low energy proton flux, a novel technique was employed:
we placed acrylic moderator blocks directly in the beamline, which spread and slowed the beam particles via multiple Coulomb scattering.
By placing the TPC in an off-axis position with respect to the beam direction, \mbox{we observed} a beam composition with lower-energy protons than would otherwise have been possible in the T10 beamline.
These techniques were designed to increase the ratio of protons to MIPs in the TPC, and to decrease the proton momentum and multiplicity in the active region of the TPC.
The flux and composition of beam particles were measured with two ToF systems, \mbox{placed upstream} and downstream of the TPC.
Measurements of protons and MIPs are presented as a function of the off-axis angle and thickness of the moderator.
This paper provides a detailed description of the time of flight systems employed in the beamline in Section~\ref{hptpcPaper:sec:Methods}, the analysis methodology of the ToF data in Section~\ref{hptpcPaper:sec:Analysis}, presentation of the ToF system results in Section~\ref{hptpcPaper:sec:Results} and additional conclusions in Section~\ref{hptpcPaper:sec:Conclusion}.
\begin{figure}[H]
\begin{minipage}[t]{0.49\textwidth}
\centering
\begin{adjustbox}{max totalsize={\textwidth},center}
\input{files/Figures/utofNoBend.tex}
\end{adjustbox}
\end{minipage}
\hfill
\begin{minipage}[t]{0.49\textwidth}
\centering
\begin{adjustbox}{max totalsize={\textwidth},center}
\input{files/Figures/protonKE.tex}
\end{adjustbox}
\end{minipage}
\caption{\label{fig:utofNoBend}Measurements of the unmoderated and unbent T10 beam over a baseline of 10.8~m for \mbox{a selected} beam momentum of 0.8~GeV/c. Measurements are made in the $\mathit{S3}$ detector. The peak between 50~ns and 60~ns is produced by protons. (\textbf{Left}) Time of flight spectrum. (\textbf{Right}) Measured kinetic energy of protons.}
\end{figure}
\section{Beam Line and Detectors}
\label{hptpcPaper:sec:Methods}
\vspace{-6pt}
\subsection{Beam Test Overview}
The beam test took place in the T10 beam line, in the East Area at the Proton Synchrotron (PS) at CERN.
The T10 beamline at CERN is a secondary beam derived from the PS beam which consists primarily of protons, electrons and charged pions~\cite{T10Report}.
The theoretical beam composition as a function of beam momentum is shown in Figure~\ref{fig:beamComp}.
The primary components of the experimental setup are shown schematically in Figure~\ref{fig:setup}.
\begin{figure}[h]
\centering
\includegraphics[width=0.43\linewidth]{files/Figures/t10Comp.png}
\caption{Calculated intensity of the T10 beam as a function of selected beam momentum, separated by particle type~\cite{T10Report}.}
\label{fig:beamComp}
\end{figure}
\begin{figure}[H]
\includegraphics[width=1.0\linewidth]{files/Figures/hptpc_t10_planview.pdf}
\captionsetup{width=1\linewidth}
\caption{Schematic diagram (plan view) of the High Pressure gas Time Projection Chamber (HPTPC) beam test configuration in the T10 area at CERN.}
\label{fig:setup}
\end{figure}
A beam position monitor (BPM) was situated at the beam entrance into the test area, upstream of all the ToF constituents and the TPC.
The TPC was placed 13~m downstream of the BPM.
\mbox{From initial} GEANT4~\cite{brun1993geant} beam simulations, the optimal TPC position to reduce the momentum of particles reaching the detector, without excessively reducing particle flux, was determined to be between 2$^{ \circ }$ and 3$^{ \circ }$ off the beam axis, but space constraints meant the TPC could not be placed that far away from the nominal beam centre. Therefore, the beam was steered approximately 1$^{ \circ }$ away from its nominal position, and the TPC placed 1.5$^{ \circ }$ away from the nominal beam centre so that the TPC active region subtended an off-axis angular range of 1.4--3.8$^{ \circ }$.
There were four ToF constituents:
\begin{itemize}
\item $\mathit{S1}$, a small-area beam trigger, see Section~\ref{subsec:s1s2Exp};
\item $\mathit{S2}$, a coincidence measurement with $\mathit{S1}$, see Section~\ref{subsec:s1s2Exp};
\item $\mathit{S3}$, a panel of plastic scintillator bars placed directly upstream of the TPC vessel, see Section~\ref{subsec:s3Exp};
\item $\mathit{S4}$, a panel of plastic scintillator bars placed directly downstream of the TPC vessel, see Section~\ref{subsec:s4Exp}.
\end{itemize}
A series of acrylic (polymethyl methacrylate) blocks was placed between the $\mathit{S1}$ and $\mathit{S2}$ counters.
Up to four $10\times10\times10$~cm$^3$ acrylic blocks could be placed contiguously on a tripod stand.
Figure~\ref{fig:modblocks} shows the stand with four blocks installed.
The moderator blocks have the effect of both reducing the energies of incoming particles as well as changing their directions.
This tends to increase the proton-to-MIP ratio at low off-axis angles from the beam, while decreasing the total number of protons and MIPs traversing the TPC.
Data were collected with the T10 beam momentum setting at 0.8~GeV/c, and with each configuration of 0 to 4 moderator blocks.
\begin{figure}[H]
\centering
\includegraphics[width=1.0\linewidth]{files/Figures/S1S2S3S4.png}
\caption{Photos illustrating the time of flight (ToF) constituents. (\textbf{Left}) the downstream part of the setup which shows the $\mathit{S3}$, $\mathit{S4}$ detectors and HPTPC. (\textbf{Right}) $\mathit{S1}$ and $\mathit{S2}$ counters and the stand with four acrylic moderator~blocks.}
\label{fig:modblocks}
\end{figure}
The data acquisition (DAQ) systems of the $\mathit{S3}$ (upstream) and $\mathit{S4}$ (downstream) ToF systems were completely independent.
Synchronization between ToF DAQ systems was performed offline using the reference signal from the PS at the beginning of every spill.
T10 received 1--3 spills from the PS during each supercycle, which has a typical duration of 33~s.
The spill duration is 400~ms.
The minimum separation in time between two spills is 1 s, so the start-of-spill signal frequency is less than or equal to 1~Hz.
As a result of the low frequency of start-of-spill signal, it is possible to use it, along with the DAQ file timestamps, to ensure that all spills are matched in both DAQs.
The trigger condition of the upstream ToF was based on the coincidence between $\mathit{S1}$ and $\mathit{S3}$ constituents.
$\mathit{S2}$ signals were also recorded by the upstream ToF DAQ but were not used in the trigger.
The DAQ of the downstream ToF was run in self-triggering mode with a gate open during the spill.
Coincidence signals between $\mathit{S1}$ and $\mathit{S2}$ counters were also recorded by the downstream ToF DAQ and were used in the particle identification (PID) analysis, described in Section~\ref{hptpcPaper:sec:Results}.
\subsection{Survey and Coordinate System}
\label{sec:coord}
The T10 beamline area was surveyed, and the distances to specific components measured with \mbox{a precision} of 0.5~mm by the CERN Survey, Mechatronics and Measurements (SMM) group.
\mbox{Multiple points} on each of $\mathit{S1}$, $\mathit{S2}$, $\mathit{S3}$, $\mathit{S4}$ and the TPC frame have had their positions measured.
The axes of a right-handed coordinate system are defined as follows: $\hat{x}$ refers to the non-beam horizontal direction, $\hat{y}$ to the vertical direction, and $\hat{z}$ the beam direction, as shown in Figure~\ref{fig:setup}.
\mbox{We show} results in terms of two off-axis angles: $\theta$, which is measured in the $\hat{x}-\hat{z}$ plane with positive angles measured in the $+\hat{x}$ direction, and $\phi$, which is measured in the $\hat{y}-\hat{z}$ plane with positive angles measured in the $+\hat{y}$ direction.
The origin is taken to be at $\mathit{S1}$.
Figure~\ref{fig:angularDistS1} shows the angular extent of objects within the beamline using the coordinate system defined above.
Table~\ref{tab:angS1} shows the calculated angular extent of the various beamline components as measured from $\mathit{S1}$.
Table~\ref{tab:distances} shows the distances between the centres of various objects in the T10 beamline.
These distances were calculated using the data gathered by the survey team.
\begin{figure}[H]
\begin{adjustbox}{max totalsize={0.7\textwidth}{.5\textheight},center}
\input{files/Figures/angDistS1.tex}
\end{adjustbox}
\caption{Angular position of various objects within the T10 beamline. The origin in this view is at the centre of $\mathit{S1}$; the true centre of the steered beam is at +1$^{\circ}$ in $\theta$ and 0$^{\circ}$ in $\phi$.}
\label{fig:angularDistS1}
\end{figure}
\begin{table}[H]
\centering
\caption{Angular extents of objects within the T10 beamline as measured from $\mathit{S1}$.}
\begin{tabular}{ccccc}
\toprule
\textbf{Object} & \textbf{Minimum \boldmath{$\theta$}} & \textbf{Maximum \boldmath{$\theta$}} & \textbf{Minimum \boldmath{$\phi$}} & \textbf{Maximum \boldmath{$\phi$}} \\
\midrule
$\mathit{S2}$ & $-3.96^{\circ} \pm 0.03^{\circ}$ & $0.36^{\circ} \pm 0.03^{\circ}$ & $-2.01^{\circ} \pm 0.03^{\circ}$ & $2.94^{\circ} \pm 0.03^{\circ}$ \\
$\mathit{S3}$ & $-5.923^{\circ} \pm 0.004^{\circ}$ & \hspace{6pt} $3.040^{\circ} \pm 0.004^{\circ}$ & $-3.215^{\circ} \pm 0.004^{\circ}$ & $3.344^{\circ} \pm 0.004^{\circ}$ \\
$\mathit{S4}$ & $-6.083^{\circ} \pm 0.003^{\circ}$ & $-0.401^{\circ} \pm 0.003^{\circ}$ & $-1.426^{\circ} \pm 0.003^{\circ}$ & $1.771^{\circ} \pm 0.003^{\circ}$ \\
TPC upstream face & $-3.59^{\circ} \pm 0.01^{\circ}$ & $-1.44^{\circ} \pm 0.01^{\circ}$ & $-2.66^{\circ} \pm 0.01^{\circ}$ & $2.58^{\circ} \pm 0.01^{\circ}$ \\
TPC downstream face & $-3.778^{\circ} \pm 0.009^{\circ}$ & $-1.806^{\circ} \pm 0.009^{\circ}$ & $-2.440^{\circ} \pm 0.009^{\circ}$ & $2.361^{\circ} \pm 0.009^{\circ}$ \\
\bottomrule
\end{tabular}
\label{tab:angS1}
\end{table}
\begin{table}[H]
\centering
\caption{Distances between objects in the T10 beamline. US and DS refer to the upstream and downstream edges of the TPC, respectively.}
\begin{tabular}{cc}
\toprule
\textbf{Points} & \textbf{Distance between Centres/m}\\
\midrule
Beam monitor -- $\mathit{S1}$ & $0.288 \pm 0.001$ \\
$\mathit{S1}-\mathit{S2}$ & $1.419 \pm 0.001$ \\
$\mathit{S1}-\mathit{S3}$ & $10.756 \pm 0.001$ \\
$\mathit{S3}$ -- TPC US side & $1.323 \pm 0.002$ \\
TPC DS side -- $\mathit{S4}$ & $0.918 \pm 0.002$ \\
$\mathit{S2}-\mathit{S4}$ & $12.651 \pm 0.001$ \\
\bottomrule
\end{tabular}
\label{tab:distances}
\end{table}
\subsection{Upstream Beam Counters (S1 and S2)}
\label{subsec:s1s2Exp}
The beam counters $\mathit{S1}$ and $\mathit{S2}$ are shown in Figure~\ref{fig:S1S2headon}.
The $\mathit{S1}$ counter is a $40\times40\times5$~mm$^3$ plastic scintillator cross which is attached to four 1'' Hamamatsu Photonics R4998 photomultiplier tubes (PMTs) at each end for the light readout.
The time resolution of the counter, as measured by the DAQ system of the upstream ToF, was about 30~ps. This is estimated with the distribution of the average PMT hit times; the quantity $t_{\textrm{ave}}=\frac{1}{4}((t_{\textrm{PMT0}}+t_{\textrm{PMT1}})-(t_{\textrm{PMT2}}+t_{\textrm{PMT3}}))$ has the same spread as the simple average but is conveniently centred at zero.
An example of the $t_{\textrm{ave}}$ distribution for one run of $\mathit{S1}$ data is shown in Figure~\ref{fig:s1Res}. The full width at half maximum (FWHM) of the distribution is 62 ps.
\begin{figure}[H]
\centering
\includegraphics[width=0.65\linewidth]{files/Figures/S1S2FrontOn.png}
\caption{The S1 and S2 beam counters. Together the coincidence of signals in the beam counters were recorded by the data acquisition (DAQ) systems.}
\label{fig:S1S2headon}
\end{figure}
\unskip
\begin{figure}[H]
\begin{adjustbox}{width=0.65\linewidth, center}
\input{files/Figures/s1TimeRes.tex}
\end{adjustbox}
\caption{Example of the timing spread of $\mathit{S1}$ hits. The time is calculated as an average of the hit time as measured in each of the four photomultiplier tubes (PMTs).}
\label{fig:s1Res}
\end{figure}
The $\mathit{S2}$ counter is a scintillator tile of size $120\times120\times5$~mm$^3$, coupled to a 2'' Hamamatsu Photonics R1309 PMT~\cite{Hamamatsu}, via a long light-guide as shown in Figure~\ref{fig:modblocks}.
The $\mathit{S2}$ counter was placed $(1.419 \pm 0.001)~\text{m}$ downstream of $\mathit{S1}$.
The transverse position of $\mathit{S2}$ was adjusted to account for the beam divergence in the moderator blocks.
The analog signals from one of the $\mathit{S1}$ PMTs and the $\mathit{S2}$ PMT were fed into LeCroy 620AL NIM discriminator units with a threshold of 30~mV.
Subsequently, the discriminated signals were fed into a NIM coincidence unit, whose output was recorded by the DAQ systems of the downstream ToF ($\mathit{S4}$) panel.
This information was further used for the time of flight analysis of $\mathit{S4}$.
\subsection{Upstream Time of Flight Instrumentation (S3)}
\label{subsec:s3Exp}
The $\mathit{S3}$ `upstream' ToF constituent was placed $(1.323 \pm 0.001)~\text{m}$ upstream of the upstream side of the HPTPC drift volume in the beamline.
A schematic drawing of the $\mathit{S3}$ ToF panel is shown in Figure~\ref{fig:S3sketch} left.
The detector comprises 22 staggered scintillator bars: 20 bars with dimensions $168 \times 6.0 \times 1.0$~cm$^3$ and 2 bars of $150 \times 6.0 \times 1.0$~cm$^3$ placed on top and bottom~\cite{S3-proceedings}.
The overlap between bars was set to 5~mm, thus the active area of the detector was $2.0214~\text{cm}^{2}$.
\begin{figure}[H]
\centering
\includegraphics[width=0.52\linewidth]{files/Figures/uToF_sketch.pdf}
\hfill
\includegraphics[width=0.47\linewidth]{files/Figures/dstofFront2.png}
\caption{View of the time of flight panels.
(\textbf{Left}) The $\mathit{S3}$ panel~\cite{S3-proceedings} upstream of the TPC. (\textbf{Right}) The $\mathit{S4}$ panel downstream of the TPC.}
\label{fig:S3sketch}
\end{figure}
The bars are made from EJ-200~\cite{SCIONIX} plastic scintillator, which provides a brightness of 10,000~photons/MeV~deposited.
It also has a suitable optical attenuation length of 4~m and fast timing, with a rise time of 0.9~ns and decay time constant of 2.1~ns.
The scintillation emission spectrum of EJ-200 peaks in the violet region of the visible spectrum (435~nm)~\cite{EJ200}.
The bars were wrapped in an aluminium foil (60\% reflectivity) to increase the collected light.
Arrays of eight $6 \times 6$~mm$^2$ area silicon photomultipliers (SiPMs) S13360-6050PE from Hamamatsu Photonics \cite{Hamamatsu} were coupled to each end of the bar to collect scintillation photons.
The photon detection efficiency at the peak sensitivity wavelength (450~nm) is 40\%~\cite{Hamamatsu}.
The anode signals of the SiPMs are read out, summed and shaped by a dedicated circuit as described in Ref.\,\cite{S3-readout}.
$\mathit{S3}$ uses a 64 channel data acquisition system based on the SAMPIC chip.
A SAMPIC chip is \mbox{a waveform} and time to digital converter (WTDC) 16-channel ASIC which provides a raw time with ultrafast analog memory allowing fine timing extraction as well as other parameters of the pulse~\cite{SAMPIC}.
Each channel contains a discriminator that can trigger itself independently or participate in a more complex combined trigger.
Three ASIC modules ($16\times3=48$ channels) were connected to the \mbox{44 channels} of $\mathit{S3}$ and were operated in self-triggering mode.
The trigger conditions are as follows: at least three out of the four $\mathit{S1}$ PMTs must have a signal above a 30~mV threshold.
Additionally, there must be at least one signal in $\mathit{S3}$ above 30~mV.
These~$\mathit{S1}$ and $\mathit{S3}$ signals must be coincident within a gate of 70~ns.
A fourth ASIC was used to acquire data from $\mathit{S1}$, the coincidence signal $\mathit{S1} \cap \mathit{S2}$ and the start-of-spill signal from the PS.
The mean time of light signals detected at both ends of a single bar provides a time reference with a resolution of about 100~ps, while the difference between the time of the light signals gives the position of the interaction along the bar, with a resolution of 1.6~cm.
Examples of reconstructed $\mathit{S3}$ spatial distributions are shown in Figure~\ref{fig:s3XY_pion}.
Figure~\ref{fig:s3XY_pion} left, shows the spatial distribution of hits in $\mathit{S3}$ thought to be produced by MIPs when 4 moderator blocks were in the beamline.
Figure~\ref{fig:s3XY_pion} right, shows the spatial distribution of hits identified in $\mathit{S3}$ as protons when 4 moderator blocks were in the beamline.
The pattern of hits is more diffuse, illustrating the scattering effect of the moderator blocks.
When in this position, the measured horizontal FWHM of the unmoderated beam is 16.8~cm while the vertical FWHM is 11.0~cm.
With 4 moderator blocks in the beamline, the measured horizontal FWHM of the beam is 63.8~cm while the vertical FWHM is 60.0~cm.
\begin{figure}[H]
\begin{minipage}[t]{0.49\textwidth}
\centering
\begin{adjustbox}{max totalsize={\textwidth}{.5\textheight},center}
\input{files/Figures/4_pionXY.tex}
\end{adjustbox}
\end{minipage}
\hfill
\begin{minipage}[t]{0.49\textwidth}
\centering
\begin{adjustbox}{max totalsize={\textwidth}{.5\textheight},center}
\input{files/Figures/4_protonXY.tex}
\end{adjustbox}
\end{minipage}
\caption{ \label{fig:s3XY_pion}Reconstructed positions of hits observed in $\mathit{S3}$. (\textbf{Left}) Minimum ionizing particles with four moderator blocks placed in the beamline. (\textbf{Right}) Protons detected with four moderator blocks placed in the beamline. This figure uses local $\mathit{S3}$ coordinates in which $y,x=0~\text{cm}$ is the bottom right corner of the active area when viewed from $\mathit{S1}$.}
\end{figure}
Figure~\ref{fig:utofTrig} shows the required trigger logic for the detection of a beam particle in the upstream ToF instrumentation.
The signal thresholds and timing cuts used for the coincidences are those detailed in this section.
\begin{figure}[H]
\centering
\includegraphics[width=.8\linewidth]{files/Figures/utofTrig.png}
\caption{Simplified trigger logic diagram for the upstream ToF detection of a beam particle, showing the required coincidences. Left and right refer to the silicon photomultipliers (SiPMs) on the opposite ends of the same bar.}
\label{fig:utofTrig}
\end{figure}
\subsection{Downstream Time of Flight Instrumentation (S4)}
\label{subsec:s4Exp}
The $\mathit{S4}$ `downstream' ToF constituent sat $(0.918 \pm 0.001)~\text{m}$ downstream of the downstream edge of the drift volume of the HPTPC prototype in the beamline.
It consists of 10 bars of Nuvia NuDET plastic scintillator which has a wavelength of maximum emission of 425~nm and a decay time constant of 2.5~ns~\cite{Nuvia}.
Each of these bars measure $10 \times 1 \times 140$~cm$^3$.
Attached to each end of these scintillator bars is a 5" Hamamatsu Photonics R6594 PMT~\cite{Hamamatsu}.
The bars are arranged in two rows of five, such that there is complete coverage for any beam particles incident upon the detector.
The bars are wrapped individually in reflective milar sheets to increase the light yield.
The total active area of the $\mathit{S4}$ panel is $1.40 \times 0.78$~m$^2$.
A diagram of $\mathit{S4}$ along with its dimensions is presented in Figure~\ref{fig:S3sketch} right.
The time resolution of the bars and PMTs is measured to be 0.8~ns using a $^{90}$Sr source placed at measured distances along the bar.
Figure~\ref{fig:s4Res} is the measured time difference for signals coming from the PMTs at either end of a bar caused by the $^{90}$Sr at a given position.
Figure~\ref{fig:s4Res} shows an example of the distribution from which the time resolution was derived.
The corresponding spatial resolution of the bars and PMTs was measured to be 7~cm.
\begin{figure}[H]
\begin{adjustbox}{max totalsize={.6\textwidth}{.5\textheight},center}
\input{files/Figures/sampleS4Res.tex}
\end{adjustbox}
\caption{Difference in signal arrival time for PMTs at each end of a bar as measured using a $^{90}$Sr source placed 64~cm from one end of the bar.}
\label{fig:s4Res}
\end{figure}
The anode signals of all 20 of the PMTs are discriminated using LeCroy 620AL NIM discriminators, at a threshold of 20~mV.
The discriminated signals are then fed into a time-to-digital converter (TDC). A signal in $\mathit{S4}$ is deemed to have occurred if a signal is seen in both PMTs, above the discriminator threshold, on the same bar within 20~ns of each other.
This timing window is determined through testing performed with a $^{90}$Sr source at known positions on the bar.
The $\mathit{S1-S2}$ coincidence signal is digitized by the same TDC. This signal is used to calculate the particle time of flight from $\mathit{S2}$ to $\mathit{S4}$.
\subsection{The HPTPC Prototype}
For the characterisation of the beam using the ToF systems described in this paper, the relevant characteristics of the HPTPC prototype are the location and thickness of the steel vessel walls.
\mbox{The cylindrical} steel vessel has a 142~cm outer diameter; the main body is 60 cm in length and the rounded end caps protrude an additional 37~cm on each end.
With 1~cm thick walls it is rated to 6~bar of absolute pressure.
The vessel wall thickness is equivalent to the range of a proton with a kinetic energy of approximately 80~MeV~\cite{rangeTables}.
For the unmoderated beam, the typical energy loss of a proton which does not stop in the vessel is 50~MeV.
This is determined from the Monte Carlo studies detailed in Section~\ref{sec:mcStudies}.
The angular position of the centre of the TPC is approximately $\theta = -2.5^{\circ}$.
More details of the position and extent of the TPC are given in Tables~\ref{tab:angS1} and~\ref{tab:distances}.
The active TPC is a cylinder, 111~cm in diameter and 48~cm in length; the TPC comprised thin steel mesh electrodes (one cathode with \SI{118}{\centi\meter} diameter and three anodes with \SI{121}{\centi\meter} diameter), \mbox{and 12 copper} rings to create the uniform drift field. The anodes were supported by a hexagonal aluminium stiffener on the side facing away from the camera.
Data taking with the TPC made use of both optical and charge readout.
The vessel, electrodes and drift region of the TPC are shown in Figure~\ref{fig:TPC}.
\begin{figure}[H]
\centering
\includegraphics[width=.8\linewidth]{files/Figures/vesselView.pdf}
\caption{Cross-sectional view of the TPC; the thin mesh electrodes and copper ring drift volume can be seen inside the steel vessel. The walls of the vessel shown are 1~cm thick with a vessel outer diameter of 142~cm. At the point of hitting the vessel, the beam centre was 1~cm below the centre of the vessel vertically, where the distance from the inside of the vessel wall to the drift region was 15~cm.}
\label{fig:TPC}
\end{figure}
Throughout the run, the TPC was filled with either pure argon, or a combination of argon and a small percentage of quencher.
The performance of this TPC is the subject of \mbox{a forthcoming~publication~\cite{Deisting:2020aaa}}.
\section{Analysis}\vspace{-6pt}
\label{hptpcPaper:sec:Analysis}
\subsection{Analysis Goals}
The primary aims of this analysis are to assess the feasibility of using the combination of off-axis positioning and a moderated beam to produce particles with momenta covering the range of momenta of particles produced in GeV-scale neutrino interactions and to characterize the incident flux on the TPC and exiting the TPC, for the TPC data analysis.
The numbers of spills recorded for each number of moderator blocks are shown in Table~\ref{tab:spills}.
More~data were collected for 4 blocks as that was the configuration used for the majority of the beam~test.
\begin{table}[H]
\centering
\caption{Total number of spills recorded for each moderator block configuration included in this paper.}
\begin{tabular}{cc}
\toprule
\textbf{Number of Moderator Blocks} & \textbf{Recorded Spills} \\
\midrule
0 & 257 \\
1 & 254 \\
2 & 267 \\
3 & 220 \\
4 & 3884 \\
\bottomrule
\end{tabular}
\label{tab:spills}
\end{table}
\subsection{Time of Flight Analysis}
\label{timeofflightanalysissubsec}
A charged pion with a momentum of 0.8~GeV/c will have a time of flight from $\mathit{S1}$ to $\mathit{S3}$ (a distance of 10.8~m) of 37~ns, while a proton with the same momentum will have a time of flight of 55~ns.
\mbox{For the} same two particles travelling between $\mathit{S2}$ and $\mathit{S4}$ (a distance of 12.7~m), the charged pion would have a time of flight of 43~ns and the proton would have a time of flight of 65~ns.
Figure~\ref{fig:s1s3PredTimes} left and right, shows the predicted time of flight for various particle species across the $\mathit{S1}-\mathit{S3}$ distance and the $\mathit{S2}-\mathit{S4}$ distance, respectively.
\begin{figure}[H]
\begin{minipage}[t]{0.49\textwidth}
\begin{adjustbox}{max totalsize={\textwidth}{.5\textheight},center}
\input{files/Figures/s1s3Times.tex}
\end{adjustbox}
\end{minipage}
\hfill
\begin{minipage}[t]{0.49\textwidth}
\begin{adjustbox}{max totalsize={\textwidth}{.5\textheight},center}
\input{files/Figures/s2s4Times.tex}
\end{adjustbox}
\end{minipage}
\caption{\label{fig:s1s3PredTimes}Calculated time of flight for a number of different particle species as a function of particle momentum. (\textbf{Left}) ToF between $\mathit{S1}$ and $\mathit{S3}$. (\textbf{Right}) ToF between $\mathit{S2}$ and $\mathit{S4}$.}
\end{figure}
Figure~\ref{fig:s3tof} shows the time of flight spectrum recorded in the $\mathit{S3}$ timing point for varying numbers of moderator blocks.
The quicker peak is formed by minimum ionizing particles, while the peak at higher values of $\mathit{t_{S3}} - \mathit{t_{S1}}$ corresponds to protons.
The proton peaks show a double peak feature, with a smaller delayed peak closely following the main proton peak; this feature appeared after the beam was steered so that the full 2.5$^{ \circ }$ off-axis angle could be achieved and is due to a portion of beam scattering in the steering magnets, leading to the slower peak.
The part of the beam which does not impinge on the steering magnets produces the quicker proton peak in the spectrum.
Figure~\ref{fig:utofNoBend} left and right, shows the proton peak for unsteered beam and the double peak structure is gone.
In the black curve, which shows the 0 block data, a deuteron peak can be seen centred at 95~ns.
The timing ranges for particle species selection are chosen using the analytic expectations shown in Figure~\ref{fig:s1s3PredTimes}.
\begin{figure}[H]
\begin{adjustbox}{max totalsize={.7\textwidth}{.5\textheight},center}
\input{files/Figures/utofS1.tex}
\end{adjustbox}
\caption{$\mathit{S3}$ time of flight spectra for varying numbers of moderator blocks.}
\label{fig:s3tof}
\end{figure}
To calculate the correct time of flight, timing delays caused by cabling and equipment are taken into account.
The same method is used to correct the measurements of the time of flight between $\mathit{S1}$ and $\mathit{S3}$, and $\mathit{S2}$ and $\mathit{S4}$.
The initial recorded time, $t_i$, is either $t_{\mathit{S1}}$ or $t_{\mathit{S2}}$ while the final recorded time, $t_f$ is then $t_{\mathit{S3}}$ or $t_{\mathit{S4}}$, respectively.
Timing offsets are measured in the beamline by assuming that the fastest peak in the $t_{f}-t_{i}$ spectrum for the unmoderated data is produced by charged MIPs with a momentum of 0.8~GeV/c.
\mbox{The required} timing shift is then the shift required to move the fastest peak to its expected position, \mbox{given this} assumption.
This shift is then applied to all measured times of flight.
This correction is peformed separately for both the measurement of $t_{\mathit{S3}}-t_{\mathit{S1}}$ and for the measurement of $t_{\mathit{S4}}-t_{\mathit{S2}}$.
The~required timing shift for the $t_{\mathit{S4}}-t_{\mathit{S2}}$ measurement is 43.7~ns.
For the $t_{\mathit{S3}}-t_{\mathit{S1}}$ measurement, the required timing shift is 65.0~ns.
The mass distribution calculated for the dataset without moderator blocks is shown in Figure~\ref{fig:s3tof_mass}.
The time difference between $\mathit{S3}$ and $\mathit{S1}$ counters corresponding to a single particle ($t_{\mathit{S3}}-t_{\mathit{S1}}$) is converted to the mass of the particle, $m$, using Equation~\eqref{eq:recoMass}, where the equation is in natural units.
\mbox{The particle} momentum, $p$, is assumed to be 0.8~GeV/c.
\begin{equation}
m^2 = p^2 \left(
\left(\frac{t_{\mathit{S3}}-t_{\mathit{S1}}}{x_{\mathit{S3}}-x_{\mathit{S1}}} \right)^2 - 1 \right) \,,
\label{eq:recoMass}
\end{equation}
The proton and pion mass positions in Figure~\ref{fig:s3tof_mass} are indicated by vertical arrows.
One can clearly observe distinct peaks corresponding to protons and deuterons.
The insert in the figure shows \mbox{a zoomed} region corresponding to the MIPs.
\begin{figure}[H]
\centering
\begin{adjustbox}{max totalsize={\linewidth}, center}
\input{files/Figures/mSqAnnot.tex}
\end{adjustbox}
\caption{Reconstructed mass spectrum for the data taken without moderator blocks. The spectrum was calculated using the time difference between $\mathit{S3}$ and $\mathit{S1}$. Vertical arrows show predicted position of particles given a momentum of 0.8~GeV/c. Insert: Zoomed view of MIP region of the same spectrum.}
\label{fig:s3tof_mass}
\end{figure}
For the data collected in $\mathit{S3}$, both timing and signal amplitude cuts were used to select protons and MIPs.
Figure~\ref{fig:TvsA} shows an example of the signal size recorded in one of the SiPMs on one of the scintillator bars against the measured value of $t_{\mathit{S3}} - t_{\mathit{S1}}$.
At the beam energies used, due to their higher mass, the protons typically deposit more energy in the detector, resulting in the observation of greater amplitudes.
Therefore, to reduce the number of background events in the proton sample, a minimum signal amplitude is required.
This cut varies, depending on the SiPM in question and is determined from distributions such as those shown in Figure~\ref{fig:TvsA}.
The cut values vary in the range 0.125~V to 0.3~V.
\begin{figure}[h]
\centering
\includegraphics[width=.9\linewidth]{files/Figures/tvsa.pdf}
\caption{Examples of SiPM signal amplitude plotted against $\mathit{S1}$ to $\mathit{S3}$ time of flight for different numbers of moderator blocks. Clockwise from top left 0, 1, 3 and 2 moderator blocks are shown. \mbox{A1 is} the voltage recorded in the SiPM at the end of the bar. The red horizontal dashed line shows the amplitude cut used for this particular SiPM. Events in the area enclosed by the red dashed lines are selected as protons. Events enclosed by the green dashed lines are selected as MIPs.}
\label{fig:TvsA}
\end{figure}
Particles for which $35.75~\text{ns}<t_{\mathit{S3}}-t_{\mathit{S1}}<37.75~\text{ns}$ are identified as MIPs.
Particles which pass the amplitude cut and for which $53~\text{ns}<t_{\mathit{S3}}-t_{\mathit{S1}}<115~\text{ns}$ are identified as protons.
The upper bound of this timing cut is reduced to 80~ns for the unmoderated sample in order to exclude deuterons.
A correction must be applied to the upstream ToF DAQ ($\mathit{S1}$, $\mathit{S2}$ and $\mathit{S3}$) to account for its large dead time.
The $\mathit{S1} \cap \mathit{S2}$ signal is digitised by both the upstream and downstream ToF DAQ.
The dead time of the downstream ToF DAQ is found to be negligible.
A linear relationship between the number of $\mathit{S1} \cap \mathit{S2}$ signals measured in each DAQ is determined for each moderator block sample.
\mbox{Therefore, events measured} in the upstream ToF DAQ are weighted, such that the number of $\mathit{S1} \cap \mathit{S2}$ signals measured in the upstream and downstream ToF DAQs are approximately equal.
Figure~\ref{fig:s4tof} shows the variation in the time of flight spectrum as recorded by $\mathit{S4}$ with a changing number of moderator blocks.
This spectrum is given by the difference in time between observation of a coincidence in the $\mathit{S1}$ and $\mathit{S2}$ timing points and a signal being recorded in $\mathit{S4}$ (the definition of an $\mathit{S4}$ signal is given above).
\begin{figure}[H]
\centering
\begin{adjustbox}{max totalsize={.83\textwidth}{.5\textheight},center}
\input{files/Figures/tofS4_correct.tex}
\end{adjustbox}
\caption{$\mathit{S4}$ time of flight spectra for varying numbers of moderator blocks. For all configurations, a flat background has been fitted and subtracted from the data. Additionally, the plot has also been corrected for the differing efficiencies of the various bars and for the variation in efficiency as a function of position along the bar, as described in Section~\ref{timeofflightanalysissubsec}.}
\label{fig:s4tof}
\end{figure}
Additionally, the reconstructed mass distribution for particles travelling from $\mathit{S2}$ to $\mathit{S4}$ is shown in Figure~\ref{fig:s4tof_mass}, produced using Equation~\eqref{eq:recoMass}.
Unlike the same distribution produced for particles travelling from $\mathit{S1}$ to $\mathit{S3}$ (see Figure~\ref{fig:s3tof_mass}), no deuteron peak is visible.
This is thought to be due to the attenuation of deuterons within the walls of the TPC.
Additionally, the predicted proton position does not line up with the measured proton position.
This is again thought to be caused by the positioning of the TPC in front of $\mathit{S4}$.
Protons passing through the TPC lose energy, resulting in them having less than the original 0.8~GeV/c beam momentum.
In turn, this leads to protons having a larger reconstructed mass than predicted.
The displacement of the proton mass peak in Figure~\ref{fig:s4tof_mass} is consistent with the expected energy loss in the vessel walls.
This consistency is shown with Monte Carlo studies in Section~\ref{sec:mcStudies}.
These Monte Carlo studies also show that, at the energies used in this study, approximately 40\% of protons which impinge on the vessel stop within it.
\begin{figure}[H]
\centering
\begin{adjustbox}{max totalsize={.83\textwidth}{.5\textheight},center}
\input{files/Figures/s4TofMass.tex}
\end{adjustbox}
\caption{Reconstructed mass spectrum for the data taken without moderator blocks. The spectrum was calculated using the time difference between $\mathit{S4}$ and $\mathit{S2}$. Vertical arrows show predicted position of particles.}
\label{fig:s4tof_mass}
\end{figure}
A correction is made for the variation in particle detection efficiency between the bars and for the variation in this efficiency as a function of the position along each bar.
This correction is performed using the cosmic ray flux.
It is assumed that the flux of cosmic rays passing through each part of $\mathit{S4}$ is equal.
Each $\mathit{S4}$ bar is divided into 7~cm segments for analysis, and the number of cosmic rays passing through each segment is measured by assuming that all signals occurring outside of beam spills are produced by cosmic rays.
The efficiency is then found from this distribution by normalising the bin with the highest number of cosmic ray signals to 1.
This efficiency is highest around the middle of the bars (70~cm) because of the requirement that coincident signals are observed in both PMTs on a given bar in order for a hit to be recorded.
An example of one of these distributions is shown in Figure~\ref{fig:s4PosEff}.
Events are then weighted according to the bar in which they are observed and their measured position along this bar.
The weight applied is the inverse of the value shown on the z-axis of Figure~\ref{fig:s4PosEff}.
Additionally, a further weight is applied to all $\mathit{S4}$ events of 1.25.
This weight is derived from tests performed on the $\mathit{S4}$ bars with a $^{90}$Sr source.
Using this source, it was determined that the maximum measured rate of signals produced by the $^{90}$Sr source was equal to 0.8 of the true rate.
\vspace{-6pt}
\begin{figure}[H]
\begin{adjustbox}{width=.65\textwidth, center}
\input{files/Figures/relativeEff1Blocks.tex}
\end{adjustbox}
\caption{Relative detection efficiency of $\mathit{S4}$ as a function of bar number and position along each bar as measured with cosmic rays. The data from bar 10 was not used in the analysis due to the poor efficiency along the bar.}
\label{fig:s4PosEff}
\end{figure}
Using Figure~\ref{fig:s4tof}, protons and MIPs are selected with timing cuts and a flat background is then subtracted.
The particles in the quicker timing window (those for which $36~\text{ns}<t_{\mathit{S4}}-t_{\mathit{S2}}<51~\text{ns}$) are considered to be minimum ionizing particles while those in the slower timing window (those for which $62~\text{ns}<t_{\mathit{S4}}-t_{\mathit{S2}}<125~\text{ns}$) are considered to be protons.
The background is determined by fitting a sum of signal and background functions to the time of flight spectra.
The signal functions are taken to be Gaussians while the background is taken to be flat.
An example of this is shown in Figure~\ref{fig:fitEx}.
The background rates for each sample are shown in Table~\ref{tab:backgrounds}.
These backgrounds have been subtracted from the totals shown in Section~\ref{sec:s4Flux}.
\begin{figure}[H]
\begin{adjustbox}{max totalsize={.75\textwidth}{.62\textheight},center}
\input{files/Figures/bkgSubPaper.tex}
\end{adjustbox}
\caption{Example of the time of flight spectrum observed in $\mathit{S4}$ with combined signal and background functions fitted (shown in red).}
\label{fig:fitEx}
\end{figure}
The backgrounds follow the same pattern as the total measured $\mathit{S4}$ particle rates (see Section~\ref{sec:s4Flux}).
The background rate initially increases with the addition of the addition of more moderator blocks then decreases for the 3 and 4 moderator block configurations.
The ratio of the rate of signal protons to the background rate falls with the addition of moderator blocks.
This is due to increased scattering from the moderator blocks which causes more particles to strike $\mathit{S4}$ without passing through $\mathit{S2}$.
This leads to an increase in false coincidences which contribute to the background rate.
\begin{table}[H]
\centering
\caption{Background rates for the time of flight spectra measured in $\mathit{S4}$. To convert these to the number of expected background events in a spill, the rate is multiplied by the size of the timing window for either MIPs or protons.}
\begin{tabular}{cc}
\toprule
\textbf{Number of Moderator Blocks} & \textbf{Background / $\text{Events} \times \text{spill}^{-1} \times \text{ns}^{-1}$} \\
\midrule
0 & $0.037 \pm 0.004$ \\
1 & $0.066 \pm 0.005$ \\
2 & $0.165 \pm 0.007$ \\
3 & $0.124 \pm 0.009$ \\
4 & $0.085 \pm 0.002$ \\
\bottomrule
\end{tabular}
\label{tab:backgrounds}
\end{table}
\section{Beam Flux Measurement}
\label{hptpcPaper:sec:Results}
\vspace{-6pt}
\subsection{Flux Measurements with S3}
The ToF systems are at an off-axis angle with respect to the beam axis (see Table~\ref{tab:angS1}), in order to probe the reduced proton momentum spectrum, to cover the region most relevant for neutrino experiments and to measure the flux passing through the TPC.
This is quantified in terms of $\theta$ and $\phi$ (see Section~\ref{sec:coord}).
The proton spectra upstream of the TPC are shown in Figure~\ref{fig:s3proke}.
Figure~\ref{fig:s3proke} left, shows the kinetic energy of particles identified as protons is successfully reduced with increasing numbers of moderator blocks, with the range falling from 210-320~MeV for the unmoderated beam, to 60--110~MeV for 4~acrylic blocks.
Figure~\ref{fig:s3proke} right, shows the kinetic energy spectrum of protons crossing into the TPC.
This~figure indicates that the flux of low energy protons (those with a kinetic energy of less than 80~MeV) reaching the TPC was increased from negligible in the 0, 1 and 2 block cases to ($9.7 \pm 0.1$) per spill for the 4 block case.
Comparing Figure~\ref{fig:s3proke} right, with Figure~\ref{fig:protonsfromargon} shows that, for the four moderator block case, the kinetic energy of protons incident upon the TPC is just above the 50~MeV region where the different neutrino interaction generators become discrepant.
These protons lose further energy within the walls of the HPTPC vessel, resulting in a flux of protons below 50~MeV within the TPC.
\begin{figure}[H]
\centering
\begin{minipage}[t]{0.49\textwidth}
\begin{adjustbox}{max totalsize={\textwidth}, center}
\input{files/Figures/proKeAll.tex}
\end{adjustbox}
\end{minipage}
\hfill
\begin{minipage}[t]{0.49\textwidth}
\begin{adjustbox}{max totalsize={\textwidth}, center}
\input{files/Figures/protonKeTpc.tex}
\end{adjustbox}
\end{minipage}
\caption{\label{fig:s3proke}Proton kinetic energy spectrum as measured in $\mathit{S3}$. (\textbf{Left}) All protons. (\textbf{Right}) The subset of protons passing through the HPTPC drift volume. The errors shown in the legend are the statistical error in particle number per spill.}
\end{figure}
The combination of the use of moderator blocks and positioning the TPC in an off-axis position also caused a change in the multiplicity of protons passing through the TPC.
Figure~\ref{fig:s3proke} right, shows that the addition of 1, 2 and 3 moderator blocks increased the number of protons passing through the TPC from ($19.0 \pm 0.9$) per spill in the unmoderated case to ($108 \pm 3$) per spill in the 3 block case.
The addition\mbox{} of the fourth moderator block effectively removes the flux of protons above 100~MeV, leaving ($21.4 \pm 0.2$) per spill to traverse the TPC active volume.
The distributions vs. off-axis angle of MIPs and protons in $\mathit{S3}$ are shown in Figure~\ref{fig:s1s3mips}.
In both cases, the peak beam intensity falls and broadens in $\theta$ with the increasing number of moderator blocks.
At off-axis angles the number of MIPs and protons is increased as the number of moderator blocks is increased.
The TPC lies within this off-axis region.
The spread of particles for unmoderated data was unexpected; this peak was broadened by the beam steering scattering that led to the double proton peaks seen in Figures~\ref{fig:s3tof} and \ref{fig:s3tof_mass}.
For the unsteered and unmoderated beam, the measured horizontal FWHM is 9.6~cm while the vertical FWHM is 11.0~cm.
This is compared with the measured horizontal FWHM for the unmoderated and steered beam of 16.8~cm.
\begin{figure}[h]
\begin{minipage}{0.48\textwidth}
\begin{adjustbox}{max totalsize={\textwidth}{.35\textheight},center}
\input{files/Figures/thetaS1pi.tex}
\end{adjustbox}
\end{minipage}
\hspace{0.3cm}
\begin{minipage}{0.48\textwidth}
\begin{adjustbox}{max totalsize={\textwidth}{.35\textheight},center}
\input{files/Figures/thetaS1pro.tex}
\end{adjustbox}
\end{minipage}
\caption{\label{fig:s1s3mips}Distribution of hits in $\mathit{S3}$ as a function of the horizontal off-axis angle, measured from $\mathit{S1}$, for varying numbers of moderator blocks. No coincident hit in $\mathit{S2}$ was required. (\textbf{Left}) Minimum ionizing particles. (\textbf{Right}) Protons. The errors shown in the legend are the statistical error in particle number per~spill.}
\end{figure}
Figure~\ref{fig:propiratio_s3_horz} shows the proton--MIP ratio measured in $\mathit{S3}$ as a function of the nominal off-axis angle, horizontally and vertically, respectively, and for various numbers of moderator blocks.
For 0, 1, 2 and 3 moderator blocks the ratio falls to a minimum at approximately $1^{\circ}$ with respect to the beam axis.
This corresponds to the true beam centre for the steered beam.
As the angle moves away from the true beam centre, the ratio rises for these configurations.
The peak of the proton--MIP ratio shifts away from the beam centre progressively as more moderator blocks are added (from approximately $1^{\circ}$ away from beam centre for 0 blocks up to approximately $3^{\circ}$ away from beam centre for 3 blocks).
At most values of $\theta$, the proton--MIP ratio falls with the addition of more moderator blocks.
Thus, reducing the kinetic energy of the protons below 100~MeV came at the cost of reducing the purity of the proton beam.
\begin{figure}[h]
\begin{minipage}[t]{0.48\textwidth}
\begin{adjustbox}{max totalsize={\textwidth}{.35\textheight}, center}
\input{files/Figures/thetaS1ratio.tex}
\end{adjustbox}
\end{minipage}
\hspace{0.3cm}
\begin{minipage}[t]{0.48\textwidth}
\begin{adjustbox}{max totalsize={\textwidth}{.35\textheight}, center}
\input{files/Figures/phiS1ratio.tex}
\end{adjustbox}
\end{minipage}
\caption{\label{fig:propiratio_s3_horz}Proton--MIP ratio in $\mathit{S3}$ for varying numbers of moderator blocks as a function of off-axis angle, as measured from $\mathit{S1}$. (\textbf{Left}) Horizontal angle. (\textbf{Right}) Vertical angle. The TPC spans horizontal angles 1.4--3.6$^{ \circ }$ and vertical angles $-2.6$--$+2.6$$^{ \circ }$.}
\end{figure}
\subsection{Flux Measurements with S4}
\label{sec:s4Flux}
Figure~\ref{fig:thetas4mip} left, shows the flux of particles identified as minimum ionizing particles across $\mathit{S4}$.
For all numbers of moderator blocks, the peak number of minimum ionizing particle events occurs at a value of $\theta$ between $-1^{\circ}$ and $-2^{\circ}$.
Similarly the number of proton events per spill, shown in Figure~\ref{fig:thetas4mip} right, peaks at a value of $\theta$ of approximately $-2^{\circ}$.
The fall in the number of events between $\theta = -1^{\circ}$ and $\theta = 0^{\circ}$ is as a result of the beam impinging on the vessel doors at these angles.
The positioning and shape of the pressure vessel doors means that, for particles travelling at these angles, a greater length of steel is passed through compared to those particles which strike the body of the vessel.
Figure~\ref{fig:thetas4mip} left, also shows that initially, an increasing number of moderator blocks results in an increased total MIP flux through $\mathit{S4}$.
This is because both $\mathit{S2}$ and $\mathit{S4}$ are positioned off-axis, so the unmoderated beam particles do not strike these detectors.
Due to scattering processes in the moderator, a greater number of MIPs are incident upon $\mathit{S2}$ and $\mathit{S4}$, with more scattering occurring with greater numbers of moderator blocks.
However, with the fourth moderator block the flux of MIPs is seen to fall.
Similarly, with the addition of the first two moderator blocks, the proton flux shown in Figure~\ref{fig:thetas4mip} right, initially sees an increase in the total number of events in $\mathit{S4}$.
However, with three and four moderator blocks, the total number of protons observed in $\mathit{S4}$ falls.
The initial proton flux increase is similar to that for the MIP flux, with increased scattering causing more protons to pass through the off-axis $\mathit{S2}$ and $\mathit{S4}$ detectors.
The subsequent decrease is due to the larger loss of energy of the protons in the thicker moderator.
In turn, this leads to attenuation of protons in the pressure vessel resulting in fewer observed events in $\mathit{S4}$.
\begin{figure}[h]
\begin{minipage}[t]{0.48\textwidth}
\begin{adjustbox}{max totalsize={\textwidth}{.35\textheight},center}
\input{files/Figures/piS4Horz_new.tex}
\end{adjustbox}
\end{minipage}
\hspace{0.3cm}
\begin{minipage}[t]{0.48\textwidth}
\begin{adjustbox}{max totalsize={\textwidth}{.35\textheight},center}
\input{files/Figures/proS4Horz_new.tex}
\end{adjustbox}
\end{minipage}
\caption{\label{fig:thetas4mip}Distribution of hits in $\mathit{S4}$ as a function of the number of moderator blocks and the horizontal off-axis angle. (\textbf{Left}) Minimum ionizing particles. (\textbf{Right}) Protons. The errors shown in the legend are the statistical error in particle number per spill.}
\end{figure}
Figure~\ref{fig:propiratio_s4_horz} shows the ratio of protons to MIPs as a function of the number of moderator blocks, $\theta$ and $\phi$.
For all of the different block configurations, the ratio is flat across both $\theta$ and $\phi$.
With the addition of moderator blocks, the ratio reduces from its highest level of 0.5 for the 0 block case, to 0.002 for the 4 block data.
As mentioned previously, this is thought to be due to the attenuation of low energy protons within the walls of the pressure vessel.
\begin{figure}[h]
\begin{minipage}[t]{0.48\textwidth}
\begin{adjustbox}{max totalsize={\textwidth}{.35\textheight}, center}
\input{files/Figures/ratioS4Horz_new.tex}
\end{adjustbox}
\end{minipage}
\hspace{0.3cm}
\begin{minipage}[t]{0.48\textwidth}
\begin{adjustbox}{max totalsize={\textwidth}{.35\textheight}, center}
\input{files/Figures/ratioS4Vert_new.tex}
\end{adjustbox}
\end{minipage}
\caption{\label{fig:propiratio_s4_horz}Proton--MIP ratio in $\mathit{S4}$ for varying numbers of moderator blocks as a function of off-axis angle. (\textbf{Left}) Horizontal off-axis angle. (\textbf{Right}) Vertical off-axis angle.}
\end{figure}
\subsection{Monte Carlo Studies}
\label{sec:mcStudies}
In order to ascertain the flux of protons reaching the active region of the TPC, and verify the corrections described above, a Monte Carlo (MC) simulation study was performed.
The simulation was performed using GEANT4~\cite{brun1993geant}, with geometric volumes approximating the vessel, TPC and time of flight systems.
In order to match upstream conditions as closely as possible, particle momenta were drawn from the $\mathit{S3}$ distributions shown in Figure~\ref{fig:s3proke} left, and simulated with trajectories that resulted in the same position distribution as seen in Figure~\ref{fig:s3XY_pion} right.
The same timing cuts described in Section~\ref{timeofflightanalysissubsec} were applied.
The simulated protons are propagated through the vessel to the $\mathit{S4}$ detector.
The momentum profile of simulated protons reaching the $\mathit{S4}$ panel is shown in Figure~\ref{fig:MCS4}.
A proton detection threshold of $\mathit{S4}$ of 140~MeV/c (10~MeV kinetic energy) is included.
The simulation shows a significant reduction in kinetic energy as most particles have travelled through both steel walls of the TPC vessel.
In~particular, in the 4 moderator block case, very few particles have survived through the second vessel wall to reach~$\mathit{S4}$.
\begin{figure}[H]
\centering
\begin{adjustbox}{max totalsize={\textwidth}{.3\textheight}, center}
\input{files/Figures/proKeS4Mc.tex}
\end{adjustbox}
\caption{Energy profile of simulated protons reaching $\mathit{S4}$, with kinetic energy above the detection threshold of 10~MeV.}
\label{fig:MCS4}
\end{figure}
Comparisons of $\mathit{S2}$ to $\mathit{S4}$ time of flight for data and MC are shown in Figure~\ref{fig:tofMC} for varying numbers of moderator blocks.
Figure~\ref{fig:tofMC} shows that, for all numbers of moderator blocks, the peak positions in the data and MC spectra agree to within 2~ns.
This level of agreement confirms that the simulated energy loss in the vessel and TPC is similar to the energy loss in the data.
\begin{figure}[H]
\centering
\begin{adjustbox}{max totalsize={.35\textheight}, center}
\input{files/Figures/tofS4Sim.tex}
\end{adjustbox}
\caption{Comparison of simulated and measured proton ToF between $\mathit{S2}$ and $\mathit{S4}$. Solid lines correspond to the simulated distributions, while points correspond to data. All distributions are area normalised to 1.}
\label{fig:tofMC}
\end{figure}
Systematic uncertainties on the number of protons measured in $\mathit{S3}$ and $\mathit{S4}$ are estimated for both data and MC, and shown in Table~\ref{tab:systematics}.
The systematic uncertainty on the MC simulation is determined by varying the geometric initial conditions of the simulation, including the position of the $\mathit{S1}$ and $\mathit{S2}$ detectors.
These variations induce changes in the direction and momenta of the propagated protons.
Additionally, a study was performed with 1~cm of additional acrylic in the beamline, as a proxy for the uncertainty on other pieces of light material in the beam facility.
This set of calculated errors represents geometric sources of uncertainty in the MC simulation.
For the data, the uncertainty on the overall efficiency of $\mathit{S3}$ is calculated by taking the $\pm1\sigma$ uncertainty on the fitted linear relationship between $\mathit{S1} \cap \mathit{S2}$ signals in the upstream and downstream ToF DAQs (see Section~\ref{timeofflightanalysissubsec}) and calculating the fractional change this causes in the $\mathit{S3}$ proton count.
The uncertainty on the overall efficiency of $\mathit{S4}$ is calculated from the calibration tests performed on the $\mathit{S4}$ bars with a $^{90}$Sr source, as discussed in Section~\ref{timeofflightanalysissubsec}.
The overall efficiency factor of 0.8 was calculated using data taken with a significantly different readout to that used in the beam test and therefore is subject to variation.
The spread in maximum bar efficiencies measured in these $^{90}$Sr source tests for the various $\mathit{S4}$ bars is used as the systematic uncertainty on the overall $\mathit{S4}$ efficiency.
The $\mathit{S4}$ angular correction systematic uncertainty is assessed by varying the number of horizontal bins in Figure~\ref{fig:s4PosEff} from 20 to 10 and taking the fractional change in the number of measured $\mathit{S4}$ protons.
The uncertainty on the $\mathit{S4}$ background subtraction is determined by taking the $1\sigma$ error on the fitted flat background and determining the resulting change in the number of protons.
This has a larger effect in the 4 block case because of the very small number of protons detected in $\mathit{S4}$ relative to the background.
\begin{table}[H]
\centering
\caption{List of systematic errors and values for data and Monte Carlo (MC) simulation. All values are the percent error on the $\mathit{S4}$ proton count with the exception of the uncertainty on the efficiency of $\mathit{S3}$, which is the percent error on the $\mathit{S3}$ proton count. All uncertainties are treated as uncorrelated. $n_{\mathit{S4},~\text{MC}}$ refers to the number of protons reaching $\mathit{S4}$ in MC simulations.}
\begin{tabular}{cccccc}
\toprule
\multicolumn{6}{c}{\textbf{Monte Carlo}} \\
\midrule
& \multicolumn{5}{c}{Number of moderator blocks} \\
& 0 & 1 & 2 & 3 & 4 \\
\midrule
Systematic uncertainty on $n_{\mathit{S4},~\text{MC}}$ & 9.5\% & 8.0\% & 8.5\% & 17.0\% & 8.0\% \\
\midrule
\multicolumn{6}{c}{\textbf{Data}} \\
\midrule
& \multicolumn{5}{c}{Number of moderator blocks} \\
Source of systematic error & 0 & 1 & 2 & 3 & 4 \\
\midrule
Absolute efficiency of $\mathit{S3}$ & 1.1\% & 11.4\% & 7.0\% & 11.4\% & 4.9\% \\
Absolute efficiency of $\mathit{S4}$ & 11.0\% & 11.0\% & 11.0\% & 11.0\% & 11.0\% \\
$\mathit{S4}$ angular correction & 2.9\% & 1.5\% & 6.7\% & 8.2\% & 4.1\% \\
$\mathit{S4}$ background uncertainty & 0.18\% & 0.16\% & 1.1\% & 1.4\% & 8.1\% \\
\midrule
\textbf{Total} & 11.5\% & 16.0\% & 14.7\% & 18.3\% & 18.9\% \\
\bottomrule
\end{tabular}
\label{tab:systematics}
\end{table}
The ratio of number of protons reaching $\mathit{S4}$ to those reaching $\mathit{S3}$ is shown for both simulation and data in Table~\ref{tab:ratios}, which includes the total statistical and systematic error in each case.
The agreement shown relative to the uncertainty provided by the beam test setup provides strong evidence that efficiency corrections described in Section~\ref{timeofflightanalysissubsec} are justified.
The number of simulated particles that penetrate the active area of the TPC are shown in Figure~\ref{fig:MCTPC} left and right, as a function of momentum and kinetic energy, respectively.
Comparing Figure~\ref{fig:MCTPC} right, with the motivation plot shown in Figure~\ref{fig:protonsfromargon}, it is clear that 4 moderator blocks were required to access the momentum region of interest (below 50 MeV).
The off-axis and moderator technique were therefore successful in the extent to which the proton energy was lowered.
The number of protons reaching the active area of the TPC was per spill ($7.0 \pm 0.1$) for 4 moderator blocks, compared with ($12.6 \pm 0.7$) per spill without moderation.
For 4 moderator blocks, ($5.6 \pm 0.1$) of those protons had energies below 100~MeV.
These values were calibrated with the full comparison between data and~simulation.
\begin{table}[H]
\centering
\caption{Ratio of number of protons reaching $\mathit{S4}$ to number protons reaching $\mathit{S3}$ for different numbers of moderator blocks in MC and data. In each instance, the combined statistical and systematic errors are shown.}
\begin{tabular}{cccc}
\toprule
\textbf{Number of Moderator Blocks} & \textbf{Monte Carlo} & \textbf{Data} & \textbf{Data/MC}\\
\midrule
$0$ & $0.027 \pm 0.003$ & $0.049 \pm 0.007$ & $1.8 \pm 0.3$ \\
$1$ & $0.067 \pm 0.005$ & $0.09 \pm 0.01$ & $1.3 \pm 0.2$ \\
$2$ & $0.084 \pm 0.007$ & $0.10 \pm 0.01$ & $1.2 \pm 0.2$ \\
$3$ & $0.06 \pm 0.01$ & $0.036 \pm 0.007$ & $0.7 \pm 0.2$ \\
$4$ & $0.011 \pm 0.001$ & $0.008 \pm 0.001$ & $0.7 \pm 0.1$ \\
\bottomrule
\end{tabular}
\label{tab:ratios}
\end{table}
\begin{figure}[H]
\begin{minipage}[t]{0.48\textwidth}
\begin{adjustbox}{max totalsize={\textwidth}{.35\textheight}, center}
\input{files/Figures/momTpc.tex}
\end{adjustbox}
\end{minipage}
\hspace{0.3cm}
\begin{minipage}[t]{0.48\textwidth}
\begin{adjustbox}{max totalsize={\textwidth}{.35\textheight}, center}
\input{files/Figures/keTpc.tex}
\end{adjustbox}
\end{minipage}
\caption{(\textbf{Left}) Momentum profile of simulated protons reaching the active region of the TPC. (\textbf{Right}) Energy profile of simulated protons reaching the active region of the TPC. The errors shown in the legend are the statistical error in particle number per spill.}
\label{fig:MCTPC}
\end{figure}
\section{Conclusions}
\label{hptpcPaper:sec:Conclusion}
The prototype high pressure gas time projection chamber was operated in the T10 beamline at CERN in August and September 2018 in order to make measurement of low momentum protons in argon.
The vessel was placed at a position off the centre axis of the beam, and a number of acrylic blocks were placed directly in the beamline in order to produce a flux of low momentum protons through the TPC, ensure a low occupancy of these low energy protons within the TPC and change the ratio of MIPs to protons.
Measurements of the beam flux were made using two time of flight systems placed ($1.323 \pm 0.001$)~m upstream and ($0.918 \pm 0.001$)~m downstream of the TPC vessel.
\mbox{These measurements} were used to determine the absolute and relative rates of protons and MIPs as well as their momenta, at different positions off the beam axis, and for varying numbers of moderator blocks.
When the beam was unsteered, the width was measured to be 9.6~cm.
When the beam was steered approximately $1^{\circ}$ off-axis, the beam width increased to 16.8~cm.
These measurements demonstrated that adding moderator blocks reduced the average kinetic energy of protons reaching the TPC from 0.3~GeV with 0~moderator blocks to 0.1~GeV for 4 moderator blocks, accessing the kinematic region of interest.
This indicates that the off-axis moderator technique provides a suitable method for providing low energy hadron beams for neutrino detector tests.
\mbox{The proton/MIP} ratio increased at low off-axis angles, peaking at 1$^{ \circ }$–2$^{ \circ }$ off axis, depending on how many moderator blocks were used and then fell off at higher angles.
The four moderator block configuration yielded a proton/MIP ratio that was substantially lower than 0–3 blocks and also flat versus off-axis angle, but achieved the desired proton energy spectrum.
With calibration from the upstream and downstream time of flight systems, for data with 4 moderator blocks in the beamline the simulated number of protons with energy below 100~MeV reaching the active TPC region was ($5.6 \pm 0.1$) per spill with an energy range of 0 to 50~MeV/c.
\authorcontributions{{Conceptualization, A.B.-F., S.B., L.C., J.H., A.K. (Asher Kaboth), J.M., R.N., J.N., R.S., N.S., Y.S. and M.W. (Morgan Wascko); data curation, S.J., T.N., E.A., D.B., A.D. (Alexander Deisting), A.D. (Adriana Dias), P.D., J.H., P.H.-B., A.K. (Asher Kaboth), A.K. (Alexander Korzenev), M.M., J.M., R.N., J.N., W.P., H.R.-Y., Y.S., A.T., M.U., S.V., A.W. and M.W. (Morgan Wascko); formal analysis, S.J., T.N., P.D., A.K. (Alexander Korzenev), Y.S. and A.W.; funding acquisition, G.B., A.B.-F., S.B., A.K. (Asher Kaboth), J.M., R.N., J.N., S.R., R.S. and M.W. (Morgan Wascko); investigation, S.J., T.N., A.B.-F., S.B., L.C., P.D., P.H.-B., A.K. (Asher Kaboth), W.M., M.M., J.M., R.N., J.N., R.S., Y.S., J.S., M.U., S.V., A.W., M.W. (Mark Ward) and M.W. (Morgan Wascko); methodology, S.J., T.N., E.A., A.B.-F., C.B., S.B., Z.C.-W., L.C., A.D. (Alexander Deisting), A.D. (Adriana Dias), P.D., J.H., A.K. (Asher Kaboth), A.K. (Alexander Korzenev), P.M., J.M., R.N., W.P., H.R.-Y., R.S., N.S., Y.S., A.T., M.U., S.V., A.W., M.W. (Mark Ward) and M.W. (Morgan Wascko); project administration, G.B., S.B., L.C., A.D. (Alexander Deisting), A.K. (Asher Kaboth), J.M., R.N., J.N. and M.W. (Morgan Wascko); resources, S.B., L.C., A.K. (Asher Kaboth), J.M., R.N., J.N., S.R. and M.W. (Morgan Wascko); Software, S.J., T.N., E.A., D.B., Z.C.-W., L.C., A.D. (Alexander Deisting), A.D. (Adriana Dias), J.H., A.K. (Asher Kaboth), J.M., R.N., W.P., S.V., A.W. and M.W. (Mark Ward); supervision, G.B., A.B.-F., S.B., L.C., A.D. (Alexander Deisting), P.D., J.H., A.K. (Asher Kaboth), J.M., R.N., J.N., R.S., A.W. and M.W. (Morgan Wascko); validation, S.J., T.N., E.A., D.B., Z.C.-W., L.C., A.D. (Alexander Deisting), A.D. (Adriana Dias), A.K. (Asher Kaboth), M.M., J.M., R.N., W.P., H.R.-Y. and A.T.; visualization, S.J., T.N., L.C. and R.N.; writing---original draft, S.J., T.N., A.K. (Asher Kaboth), J.M. and M.W. (Morgan Wascko); writing---review and editing, S.J., T.N., E.A., S.B., D.B., Z.C.-W., L.C., A.D. (Alexander Deisting), A.D. (Adriana Dias), P.D., A.K. (Asher Kaboth), M.M., J.M., R.N., J.N., W.P., H.R.-Y., A.T., A.W. and M.W. (Morgan Wascko).}
All authors have read and agreed to the published version of the manuscript.}
\funding{{This research was funded in part by Science and Technology Facilities Council grant number ST/N003233/.}
}
\acknowledgments{We wish to acknowledge support for summer students from the Ogden Trust and St. Andrews University, and outstanding support during the beam test from Johannes Bernhard of CERN as well as
Rebecca Conybeare, Nicole Cullen, Kate Gould, Veera Mikola, Christopher Thorpe, and Simon Williams.}
\conflictsofinterest{{The authors declare no conflict of interest.}
}
\reftitle{References}
|
{
"redpajama_set_name": "RedPajamaArXiv"
}
| 4,304
|
#
### Copyright
HarperCollins _Publishers_
1 London Bridge Street,
London, SE1 9GF
www.harpercollins.co.uk
Published by HarperCollins _Publishers_ 2017
Copyright © Stuart MacBride 2017
Prelims show map of Oldcastle © Stuart MacBride
Cover layout design © HarperCollins _Publishers_ 2017
Cover design © Blacksheep-uk.com
Cover image © Jim Robertson (www.jimrobertson.co.uk)
The quotation from the public information film _Stay at Home_ appears courtesy of the British Film Institute © Crown Copyright (1975). The quotation from William Blake's _Songs of Innocence_ – The Chimney Sweeper (1789) is taken from the British Library's first edition copy.
Stuart MacBride asserts the moral right to be identified as the author of this work
A catalogue record for this book is available from the British Library
This is a work of fiction. Any references to real people, living or dead, real events, businesses, organisations and localities are intended only to give the fiction a sense of reality and authenticity. All names, characters, places and incidents are either the product of the author's imagination or are used fictitiously, and their resemblance, if any, to real-life counterparts is entirely coincidental. The only exception to this are the characters Cecelia Lynch and Robert Shannon who have given their express permission to be fictionalised in this volume. All behaviour, history, and character traits assigned to their fictional representations have been designed to serve the needs of the narrative and do not necessarily bear any resemblance to the real person.
All rights reserved under International and Pan-American Copyright Conventions. By payment of the required fees, you have been granted the non-exclusive, non-transferable right to access and read the text of this e-book on-screen. No part of this text may be reproduced, transmitted, down-loaded, decompiled, reverse engineered, or stored in or introduced into any information storage and retrieval system, in any form or by any means, whether electronic or mechanical, now known or hereinafter invented, without the express written permission of HarperCollins e-books
Source ISBN: 9780007494682
Ebook Edition © JUNE 2017 ISBN: 9780007494705
Version: 2017-08-15
### Dedication
For Sue.
Contents
Cover
Title Page
Copyright
Dedication
Without Whom
Maps
— Exhibit A —
Chapter 1
— Bodies of the Lesser God —
Chapter 2
Chapter 3
Chapter 4
Chapter 5
Chapter 6
Chapter 7
Chapter 8
Chapter 9
Chapter 10
— Every Day We Live — Is a Day Closer to the Day We Die
Chapter 11
Chapter 12
Chapter 13
Chapter 14
Chapter 15
— Callum —
Chapter 16
— Imhotep —
Chapter 17
Chapter 18
Chapter 19
Chapter 20
Chapter 21
Chapter 22
Chapter 23
Chapter 24
Chapter 25
Chapter 26
Chapter 27
Chapter 28
— The Four-Minute Warning —
Chapter 29
Chapter 30
Chapter 31
Chapter 32
Chapter 33
Chapter 34
Chapter 35
Chapter 36
Chapter 37
— Protect and Survive —
Chapter 38
Chapter 39
Chapter 40
Chapter 41
Chapter 42
Chapter 43
Chapter 44
Chapter 45
Chapter 46
Chapter 47
— These Bones Beneath the Earth —
Chapter 48
Chapter 49
Chapter 50
Chapter 51
Chapter 52
Chapter 53
— Father —
Chapter 54
Chapter 55
Chapter 56
Chapter 57
— Open the Coffins —
Chapter 58
Chapter 59
Chapter 60
Chapter 61
Chapter 62
Chapter 63
Chapter 64
Chapter 65
Chapter 66
Chapter 67
Chapter 68
Chapter 69
Chapter 70
— The Bonemonger's Waltz —
Chapter 71
Chapter 72
Chapter 73
Chapter 74
Chapter 75
Chapter 76
Chapter 77
Chapter 78
Chapter 79
— Dearly Departed, — We are Gathered Here Today
Chapter 80
Chapter 81
If you enjoyed _A Dark so Deadly_ , try the latest novel in the Logan McRae series!
About the Author
By Stuart MacBride
About the Publisher
## Without Whom
As always I've received a lot of help from a lot of people while I was writing this book, so I'd like to take this opportunity to thank: Prof. Sue Black, Dr Roos Eisma, Vivienne McGuire, and Dr Lucina Hackman, all of whom do excellent work at the University of Dundee's Centre for Anatomy and Human Identification; Sergeant Bruce Crawford who answers far more daft questions than anyone should ever have to, as does Professor Dave Barclay; Sarah Hodgson, Jane Johnson, Julia Wisdom, Jaime Frost, Anna Derkacz, Sarah Collett, Charlie Redmayne, Roger Cazalet, Kate Elton, Hannah Gamon, Cait Davies, Damon Greeney, Finn Cotton, the eagle-eyed Anne O'Brien, Marie Goldie, the DC Bishopbriggs Super Squad, and everyone at HarperCollins, for doing such a stonking job; Phil Patterson and the team at Marjacq Scripts, for keeping my cat in shoes all these years; Catherine Pellegrino, and Sandra Sawicka for translational help; and let's not forget Cecelia Lynch, or James, Duncan, Katy, and Liz Shannon who helped raise money for two very worthy causes, and Matt Patterson whose wallet makes several guest appearances. And thank you to Tony Dykes of the British Film Institute for permission to quote _Stay at Home_ within these pages.
Of course, there wouldn't be any books without bookshops, booksellers, and book readers – so thank you all, you're stars.
And saving the best for last – as always – Fiona and Grendel.
## Maps
## — exhibit A —
##
The wall whispers to him with splintered wooden lips. _'They'll worship you. They'll worship you. They'll worship you...'_
Its words fill the gloom, rolling around and around and through him, pulsing and pulling. _'They'll worship you.'_
Why?
Why can't he just die?
_'They'll worship you: you'll be a god.'_
Is this what gods feel like? Thirsty. Aching.
Every muscle in his stomach throbs from the repeated heaving. Every breath tastes of bile.
Bile and dark, gritty wood smoke. Filling the low room with its stained wooden walls.
_'They'll worship you: you'll be a god.'_
He slumps back, making the rusty links of chain rattle and clank against each other. Heavy around his throat. Heavier where it's bolted into the wall. The wall that talks.
_'You'll be a god.'_
He can't even answer it, his mouth is desert dry, tongue like a breezeblock, blood booming in his ears. Boom. Boom. Boom.
So thirsty... But if he drinks the foul brown water in the jug, he'll be sick again.
_'A god.'_
He turns his face to the wall. Finds a silent crack in the wood. And stares through into the other room.
_'They'll worship you. They'll worship you.'_
Through there, it's bright: a mix of light and shadow as someone stands on their tiptoes to slot another pole of fish into the rack. Herrings, splayed open, tied in pairs at the tail, their flattened sides like hands. Praying.
Help me...
He opens his mouth, but it's too dry to make words. Too burned by the bile.
_'They'll worship you.'_
Why can't he just die?
Up above, high above the poles of praying fish, eight fingertips brush a blade of sunlight. They run their tips along its sharp edge as the body they belong to sways in the darkness. Caught in the breeze from the open door. Head down – like the fish – arms dangling. Skin darkened to an ancient oak brown.
_'You'll be a god.'_
Then the person on the other side disappears. Comes back with a wheelbarrow piled up with sawdust and small chunks of wood. Dumps the lot in the middle of the room. Stoops to light it. Stands back as pale tendrils of smoke coil up into the air. Backs away and closes the door.
Now the only light is the faint orange glow of the smouldering wood.
_'You'll be a god.'_
He slides down against the wall. Too tired and thirsty to cry. Too tired to do anything but wait for the end to come.
_'They'll worship you...'_
Why can't he just _die_?
## — bodies of the lesser god —
Then the little girl with the lizard's tail jumped into the air with a _whoosh!_ "I've got it!" she shrieked. "We can make an enormous pie out of all the bits of hair and beard!"
Ichabod scowled at her. "That's a horrid idea," he said, because it was. "No one wants to eat a cake made of hair."
"Ah, but the hair of the Gianticus Moleraticus is _magical_ and tastes of everything you like in the whole world! Gumdrops and sausages, baked beans and chocolate biscuits, custard and ham." She scooped up a big handful of hair and shoved it in Ichabod's mouth. "See?"
But to Ichabod it just tasted of hair. The little girl was clearly insane...
R.M. Travis
_The Amazing Adventures of Ichabod Smith_ (1985)
_And if some motherf*cker gonna call the police?_
_I'm-a grab my nine-mill and I'm-a make him deceased._
Donny '$ick Dawg' McRoberts
'Don't Mess with the $ick Dawg'
© Bob's Speed Trap Records (2016)
##
'POLICE! COME BACK HERE, YOU WEE SOD!'
Only that wasn't really right, was it? Ainsley Dugdale _wasn't_ a wee sod – he was a dirty great big lumping hulk of a sod, hammering his way along Manson Avenue. Ape-long arms and short legs pumping, scarf flittering out behind him, baldy head glinting in the morning sunshine.
Callum gritted his teeth and hammered after him.
Why did no one ever come back when they were told to? Anyone would think people didn't _want_ to get arrested.
Squat grey council houses scrolled past on either side of the street, lichen-flecked pantiles and harled walls. Front gardens awash with weeds. More abandoned sofas and washing machines than gnomes and bird tables.
A couple of kids were out on their bikes, making lazy figure eights on the tarmac. The wee boy had sticky-out ears and a flat monkey nose, a roll-up sticking out the corner of his mouth – leaving coiled trails of smoke behind him. The wee girl was all blonde ringlets and pierced ears, swigging from a tin of extra-strong cider as she freewheeled. Both of them dressed in baggy jeans, trainers, and tracksuit tops. Baseball caps on the right way around, for a change.
Rap music blared out of a mobile phone. _'Cops can't take me, cos I'm strong like an oak tree, / Fast like the grand prix, / I'm-a still fly free..._ '
The wee girl shifted her tinny to the other hand and raised a middle finger in salute as Callum ran past. 'HOY, PIGGY, I SHAGGED YER MUM, YEAH?'
Her wee friend made baboon hoots. 'HOOH! HOOH! HOOH! PIGGY, PIGGY, PIGGY!'
Neither of them looked a day over seven years old.
The delights of darkest Kingsmeath.
Dugdale skittered around the corner at the end of the road. Almost didn't make it – banged against the side of a rusty Renault, righted himself and kept on going, up the hill.
'RUN, PIGGY, RUN!' Little Miss Cider appeared, standing on the pedals to keep up, grinning as she flanked him. 'COME ON, PIGGY, PUT SOME WELLY IN IT!'
Her baboon friend pedalled up on the other side. 'FAT PIGGY, LAZY PIGGY!'
'Bugger off, you little sods...' Callum wheeched through the turn, into another row of grubby houses. Low garden walls guarded small squares of thistle and dandelions, ancient rusty hatchbacks up on bricks, the twisted metal brackets where satellite dishes used to be.
'COME ON, PIGGY!'
The gap was narrowing. Dugdale might have got off to an impressive sprint start, but his long game wasn't anywhere near as good – puffing and panting as he lumbered up Munro Place. Getting slower with every step.
'HOOH! HOOH! HOOH!'
He crested the hill with Callum barely ten feet behind him.
The street fell away towards a grubby line of trees and a grubbier line of houses, but Dugdale didn't stop to admire the view: he kept his head down, picking up a bit of velocity on the descent.
The wee kids freewheeled alongside him, Little Miss Cider swigging from her can. 'RUN, BALDY – PIGGY'S GONNA GET YOU!'
One last burst. Callum accelerated. 'I'M NOT TELLING YOU AGAIN!'
Dugdale snatched a glance over his shoulder – little eyes surrounded by dark circles, a nose that looked as if it'd been broken at least a dozen times, scar bisecting his bottom lip. He swore. Then put on another burst of speed.
'NO YOU DON'T!'
'HOOH! HOOH! HOOH!'
Closer. Eight foot. Seven. Six.
Here we go...
Callum leapt. Arms out – rugby-tackle style.
His shoulder caught Dugdale just above the waist, arms wrapping around the top of the big sod's legs. Holding on tight as they both crashed onto the pavement, rolling over and over. Grunts. More swearing. A tangle of arms and legs. Then something the size of a minibus battered into Callum's face.
Now the world tasted of hot batteries.
Another punch. 'GET OFF ME!'
Callum jabbed out an elbow and connected with something solid.
'HOOH! HOOH! HOOH!'
'FIGHT, PIGGY, FIGHT!'
Then the pavement battered off the back of his head and a fist slammed into his stomach. Fire roared through his torso, accompanied by the sound of a thousand alarm clocks all ringing at once.
He swung a punch and Dugdale's nose went from broken to smashed.
'Gahhhh!' Dugdale reared back, blood spilling down over his top lip. He lashed out blind, eyes closed, and that massive fist came close enough to ruffle the hair above Callum's ear.
Distance. Get some distance.
A big black Mercedes slid past, the sweaty-sweet scent of marijuana coiling out from the back windows, a deep _BMMTSHHH_ , _BMMTSHHH_ , _BMMTSHHH_ of hip-hop bass rattling the air. It stopped in the middle of the road, where they could get a good view of the fight. But did anyone get out to help? Of course they sodding didn't.
'KILL HIM, PIGGY, FINISH HIM!'
'HOOH! HOOH! HOOH!'
Callum scrabbled back against a rusty Volkswagen. Yanked out his handcuffs. 'Ainsley Dugdale, I'm detaining you under Section Fourteen of the Criminal Procedure – Scotland – Act 1995—'
'FIGHT! FIGHT! FIGHT!' The kids pulled up their bikes, blocking the pavement, making an impromptu brawl-pit in the space between the Volkswagen and a garden wall. 'COME ON: KILL HIM!'
'Shut up!' Back to Dugdale. 'Because I suspect you of having committed an offence punishable by imprisonment, namely the—'
'HOOH! HOOH! HOOH!'
'GAAAAH!' Dugdale lunged, but not at Callum. He grabbed the wee girl by the throat and yanked her off her bike.
Her tin of cider hit the deck and bounced, sending out a spurt of frothy urine-coloured liquid. 'Ulk...' Eyes wide, both hands clutching onto Dugdale's forearm, legs pinwheeling and kicking at the air.
Oh sodding hell. And things had been going so well right up till that point.
'No, no, no!' Callum scrambled to his feet. 'That's _enough_. Let the girl go.'
Her wee mate hurled his roll-up. It burst against Dugdale's chest in a little hiss of sparks. 'LET HER GO, YOU DIRTY PAEDO!'
'Come on, Dugdale... Ainsley. You don't want to hurt a kid, right?' Hands out, open, nice and safe. 'You're not that kind of guy, are you?'
'PAEDO! PAEDO! PAEDO!'
Callum hissed the words out the side of his mouth. 'You are _not_ helping.'
Dugdale stuck out his other hand. 'Money!'
'Come on, Ainsley, let the girl go and—'
'GIVE US YOUR MONEY!' He gave the girl a shake, sending her legs swinging wildly as her face turned a darker shade of puce. 'NOW!'
'OK, OK. Just let her breathe.' Callum dug out his battered old wallet. The one with the threadbare lining sticking out. He took the last tenner and crumpled fiver from inside. 'Here.' He placed the cash on the floor.
'Is that _it_?' Dugdale glowered at the two sorry notes. 'ALL OF IT, OR I SNAP HER NECK IN HALF!'
Baboon Boy's chant died. 'Paedo...?'
The kicks were getting weaker: those Nike trainers barely moving.
Her wee friend snivelled. Wiped his top lip on the back of his sleeve. 'Please, mister. Don't hurt my sister...'
'That's all the money I've got, OK? Now let the girl go.'
Dugdale growled, then chucked the little girl at Callum.
He ducked for the fifteen quid as Callum dropped the tatty wallet and caught her wee body before it hit the pavement. And that's when everything slowed down.
The tatty wallet bounced off the paving slabs, spinning away, its torn lining waving like a flag.
'Aaaggggh...' She hauled in a huge whoop of air, both hands wrapped around her throat – as if Dugdale hadn't done a good enough job throttling her and she was having a go herself.
But Dugdale didn't snatch up the money, he kept on going, smashing into Callum and the wee girl, sending them slamming back into the Volkswagen. Rocking it on its springs.
A fist connected with Callum's ribs. Arms and legs tangled. Flashes of sky, then concrete, then rusty metal, then sky again.
Then bang – everything was at full speed again.
Callum yanked the pepper spray from his jacket pocket. The little girl wriggled her way out from between them, trainers digging into his thigh as she went. Callum flipped the cap off the spray and thumbed the button, sending a squirt of burning pepper stink out at Dugdale's face.
Missed.
Dugdale didn't. He rammed his hand into Callum's crotch, grabbed hold, and _squeezed_.
Oh _God..._
But when Callum opened his mouth to scream, all that came out was a strangled wheeze – eyes wide as every single ache and pain in his body disappeared, replaced by the thermonuclear explosion going off in his scrotum. It raced out through his stomach, down his legs, up into his chest – a shockwave ripping out from ground zero as Dugdale twisted his handful like a rusty doorknob.
Oh sodding Jesus...
Dugdale let go, but the nuclear war still raged.
_No..._
Water filled Callum's eyes, making the word go all soft focus, but the pain remained pin-sharp. He lashed out with the pepper spray, swinging it in an arc with the button held down.
Someone bellowed in pain.
Then scuffing feet.
Argh...
The clatter of a very large man tripping over a fallen bicycle.
A dull _thunk_ , like a watermelon bouncing off a coffee table.
Oh that _hurt_...
'BLOODY PAEDO!' Some more _thunks_.
'Come on, leave him!'
_Thunk_ , _thunk_ , _thunk_. 'BLOODY BALDY PAEDO WANKER!'
Ow...
'Willow, come on! Before he gets up!'
The sound of someone spitting.
'Grab the cash, Benny. No, you spaz, get the wallet too!'
Then trainers on concrete, the rattle of bicycles being dragged upright, and the growl of tyres fading away into the distance.
One last cry of, 'PIGGY, PIGGY, PIGGY!'
The sound of that big black Mercedes pulling away now the floor-show was over.
And silence.
Callum cursed and panted and wobbled his way up to his knees, one hand clutching his tattered groin.
Sodding... for... _ooogh_...
Deep breaths.
Nope. Not helping.
He scrubbed a hand across his watery eyes.
Dugdale lay on his front, one hand behind his back the other limp in the gutter. His face looked as if someone had driven over it with a ride-on lawnmower.
Callum dragged himself over and slapped on the cuffs. 'You're nicked.'
Ow...
'Little monsters...' Never mind saying _thank you_ – no, that was too much to hope for these days, he'd only saved her life, not as if it was _that_ big a deal – but did they have to take his sodding wallet?
Dugdale twitched and groaned, eyes still closed, the blood crusting on his battered nose. A swathe of red crossed his face, following the pepper spray's less than delicate path, swollen and angry looking. Like the lump on his head. It was going to be impressive when it finished growing – about the size and colour of a small aubergine. Probably have himself a gargantuan headache when he finally woke up. Maybe concussion too.
Good. Served him right.
Callum pulled out his mobile, staying where he was – standing, hunched over almost double, one hand on his knee, holding him upright as he dialled.
Three rings and then a woman's voice came on the line, sounding small and concerned. _'Hello?'_
'Elaine, it's me.'
_'Callum? Are you OK? You don't sound OK. Is everything OK?'_
He gritted his teeth as an aftershock rippled its way through his groin. 'No. Can you phone the bank? I need you to cancel my debit and credit card. Someone's snatched them.'
A sigh. _'Oh, Callum, not your dad's wallet...'_
'Don't start, please. It'll be bad enough when McAdams gets here, don't need you kicking the party off early.'
Silence.
Yeah, way to go, Callum. Smooth. Nice and understanding.
He took a deep breath. 'Sorry, it's... I'm not having the greatest of days.'
_'I'm not your enemy, Callum. I know it's been difficult for you.'_
Understatement of the year. 'All I get is snide comments, nasty little digs, and crap. It's been three solid weeks of—'
_'It's for the best though, remember? For Peanut's sake?'_
Peanut.
He closed his eyes. Tried to make it sound as if he meant it: 'Yeah.'
_'We need the money, Callum. We need the maternity pay to—'_
'Yeah. Right. I know. It's just...' He wiped a hand over his face. 'Never mind. It'll be fine.'
_'And we_ really _appreciate it, me and Peanut.'_ A pause. _'Speaking of Peanut, you know what he'd totally love? Nutella. And some pickled dill cucumbers. Not gherkins: the cucumbers, from the Polish deli on Castle Hill? Oh, and some onion rolls too.'_
'They stole my _wallet_ , Elaine. I—'
_'I didn't_ ask _to get pregnant, Callum.'_ A strangled noise came down the phone, like a cross between a grunt and a sigh. _'Sorry. I don't... There are times when I need a bit of support coping with all this.'_
Support? _Seriously_?
'How am I not supporting you? I put my hand up, didn't I? I took the blame, even though it was nothing to do with—'
_'I know, I know. I'm sorry. It's...'_ Another sigh. _'Don't worry about the Nutella and stuff, it's only cravings. I'll make do with whatever's knocking about here.'_
He limped over to the garden wall and lowered himself onto it with a wince. Took yet another deep breath. Scrunched a hand over his eyes. 'I'm sorry, Elaine. It's not you, it's... Like I said, I'm having a _terrible_ day.'
_'It'll get better, I promise. I love you, OK?'_
'Yeah, I know it will.' It had to, because it couldn't possibly get any worse.
_'Do you love me and Peanut too?'_
'Course I do.'
A shiny red Mitsubishi Shogun pulled into the kerb, the huge four-by-four's window buzzing down as Callum levered himself up to his feet. His crumpled suit and crumpled body reflected back at him in the glittering showroom paintwork.
'Got to go.' He hung up and slipped the phone back in his pocket.
'Constable Useless.' A thin, lined face frowned through the open car window, its greying Vandyke framed by disappointed jowls. The chin-warmer was little more than stubble, matching the patchy salt-and-pepper hair on that jellybean of a head. 'Do these old eyes deceive me? Did you _catch_ Dugdale?'
Callum wobbled up to his feet, one hand on his ruptured testicles, the other holding onto the Shogun for support. 'Oh: ha, ha.' Another wave of burning glass washed through him, leaving him grimacing. 'He's been unconscious for a couple of minutes. You want to take him straight to the hospital, or risk the Duty Doctor?'
Please say hospital, please say hospital. At least there a nice nurse might have an icepack and a few kind words for his mangled groin.
DS McAdams raised an eyebrow. 'I am shocked, Callum. Didn't he have enough cash? No nice bribe for you?'
'Sod off, Sarge.' He let go of his crotch for a moment, pointing off down the hill. Winced. Then cupped his aching balls again. 'Pair of kids got my wallet. We need to get after them.'
'If I had to guess. The reason you're hunched in pain. You have met _The Claw_!' He held up one hand, the fingers curled into a cruel hook, then squashed an invisible scrotum. 'Dugdale's claw attacks. Crush and squish, the pain is great. Bringing hard men low.'
Callum stared at him. 'They – got – my – wallet!'
The frown became a grin. 'A well-turned haiku. It is a beautiful thing. You ignorant spud.' He actually counted the syllables out on his fingers as he spoke.
'For your information, _Sarge_ , I've never taken a bribe in my life. OK? Not a single sodding penny. No perks, no wee gifts, nothing. So you can all go screw yourselves.' He limped over to the back door and swung it open. 'Now are you going to help me get Dugdale in the car or not?'
'That's the trouble with your generation: no poetry in your souls. No education, no class, and no moral fibre.'
'Thanks for nothing.' He bent down. Winced. Clenched his jaw. Then hauled Dugdale's huge and heavy backside across the pavement and up onto the back seat.
'He better not bleed. On my new upholstery. I just had it cleaned.'
'Tough.' Some wrestling, a bit of forcing, a shove, and Dugdale was more or less in the recovery position. Well, except for his hands being cuffed behind his back. But at least now he _probably_ wouldn't choke on his own tongue. Or vomit.
Mind you, if he spewed his breakfast all over Detective Sergeant McAdams' shiny new four-by-four, at least that would be something. Assuming McAdams didn't make Callum clean it up. Which he would.
Git.
Callum clunked the door shut, hobbled around to the passenger side and lowered himself into the seat. Crumpled forward until his forehead rested against the dashboard. 'Ow...'
'Seatbelt.' The car slid away from the kerb.
Callum closed his eyes. 'Think they turned right onto Grant Street. If you hurry we can still catch them: wee boy in jeans and a blue tracksuit top, wee girl in jeans and a red one. About six or seven years old. Both on bikes.'
'You got mugged by toddlers?' A gravelly laugh rattled out in the car. 'That's pathetic even for you.'
'They're getting away!'
'We're not going chasing after little kiddies, Constable. I have _much_ more important things to do than clean up your disasters.'
'That's it. Stop the car.' Callum straightened up and bared his teeth. 'Come on: let's go. You and me. I battered the crap out of Dugdale, I can do the same for you.'
'Oh don't be such a baby.'
'I'm not kidding: stop – the – car.'
'Really, DC MacGregor? You don't think you're in enough trouble as it is? How's it going to look if you assault a senior officer who's dying of cancer? Think it through.' The car jolted and bumped, then swung around to the left, heading down towards Montrose Road. 'And any time our workplace badinage gets too much for you, feel free to pop into Mother's office with your resignation. Do us all a favour.' He slowed for the junction. 'Until then, try to behave like an _actual_ police officer.'
Callum's hands curled into fists, so tight the knuckles ached. 'I swear to _God_ —'
'Now put your seatbelt on and try not to say anything stupid for the next fifteen minutes. I'll not have you spoiling my remarkably good mood.' He poked the radio and insipid pop music dribbled out of the speakers. 'You see, Constable Useless, sometimes life gives you lemons, and sometimes it gives you vodka. _Today_ is a vodka day.'
The jingly blandness piffled to a halt and a smoke-gravelled woman's voice came through. _'Hmmm, not sure about that one myself. You're listening to_ Midmorning Madness _on Castlewave FM with me, Annette Peterson, and today my extra-special guest is author and broadcaster, Emma Travis-Wilkes.'_
McAdams put a hand over his heart, as if he was about to pledge allegiance. 'Today is a _caviar_ day.'
_'Glad to be here, Annette.'_
'A champagne and strawberries day.'
_'Now, a little bird tells me you're writing a book about your dad, Emma. Of course he created_ Russell the Magic Rabbit _, Ichabod Smith, and_ Imelda's Miraculous Dustbin _, but he's probably best known for the children's classic,_ Open the Coffins _.'_
'A chocolate and nipple clamps—'
'All right! I get it: everything's just sodding _great_.' Callum shifted in his seat, setting his testicles aching again. 'One of us got thwacked in the balls, here.'
_'That's right. He's given joy to so many people, and now that he's... well, Alzheimer's is a cruel mistress. But it's been a real privilege to swim in the pool of his life again.'_
'Pfff...' McAdams curled his top lip. 'Listen to this pretentious twaddle. Just because she's got a famous dad, she gets to plug her book on the radio. What about _my_ book? Where's _my_ interview?'
_'And it's lovely to see these memories light up his face, it's like he's right back there again.'_
'Cliché. And, by the way, unless his face is _actually_ glowing like a lightbulb, that's physical hyperbole, you hack.'
Callum glowered across the car. 'We should never have chipped in for that creative writing class.'
McAdams grinned back at him. 'My heart: creative. My soul, it soars with the words. Divinity: mine.'
_'Wonderful stuff. Now, let's have a bit of_ decent _music, shall we? Here's one of the acts appearing at Tartantula this weekend: Catnip Jane, and "Once Upon a Time in Dundee".'_
A banjo and cello launched into a sinister waltz, over a weird thumping rhythm as McAdams pulled out of the junction, heading left instead of right.
Silly old sod.
Callum sighed. 'You're going the wrong way.' He pointed across the swollen grey river, past the docks and the industrial units, towards the thick granite blade of Castle Hill. 'Division Headquarters is _that_ direction. We need to get Dugdale booked in and seen to.'
'Meh, he'll keep.' That skeletal grin had widened. 'It's a vodka day, remember? We, my useless little friend, have finally got our hands on a murder!'
##
The first drop of rain sparkled against the windscreen, caught in a golden shaft of sunlight as McAdams' huge four-by-four slid past the last few houses on the edge of Kingsmeath. A second drop joined it. Then a third. Then a whole heap of them.
McAdams stuck the wipers on, setting them moaning and groaning their way across the glass, smearing the rain into grubby arcs. He pinned his mobile phone between his shoulder and ear, freeing his hand to change gears. Accelerating up the hill. 'Yeah... Yeah, Dugdale was there... No... Not a word of a lie, Mother: the new boy _actually_ caught him. That's right: his anonymous tip-off paid off.' He cast a glance across the car at Callum. 'I know, I know... Ha! That's what I said.'
Callum folded his arms and pushed back into his seat. Stared out of the window at the dull green fields and their dull-grey sheep. The ache in his groin wasn't a full-on testicular migraine any more, it'd settled to more of a dull throbbing – each pulse marking time with the groaning windscreen wipers. 'Oh you're both _so_ hilarious.'
'What did we say about you keeping your mouth shut?' Back to the phone. 'No, not you, Mother: Constable Useless here... Yeah, yeah. Exactly: an actual murder. How long has it been?'
Probably never see his wallet again.
McAdams put his foot down, overtaking a sputtering Mini. 'You on your way?... Uh-huh... Yeah, I couldn't believe it either. Since when does the great Detective Chief Inspector Poncy Powel hand over a murder investigation to the likes of us?... Exactly.'
More fields. More sheep.
OK, so it was just a scruffy, tatty lump of leather and the lining was falling apart, but it had sentimental value.
Bloody kids.
'Did he?... No!... No!' Laughter. 'And did you?... Sodding hell... Yeah, he'll _love_ that.'
Bloody Dugdale too.
He was just visible in the rear-view mirror, lying there with his mouth hanging open, face crusted with blood and bogies. Well, if Dugdale died in custody there was no way Callum was taking the rap for it. If anything happened it was McAdams' fault.
Accepting blame for Elaine's cock-up was one thing, but McAdams? He could sod right off.
'Uh-huh. We're about... five minutes away? Maybe less?... Still can't believe it: a real murder! How long's it been?... Right. Yup. OK. See you there.' He poked a button on his phone's screen then slid the thing back in his pocket, big smile plastered across his skeletal face.
'Am I allowed to ask where we're going?'
'No.'
Git.
McAdams took one hand off the wheel and pointed through the windscreen. 'We go where life rots. Where man's discarded dreams die. We go... to _The Tip_.' Fingers twitching with each syllable.
A large white sign loomed at the side of the road: 'OLDCASTLE MUNICIPAL RECYCLING AND WASTE PROCESSING FACILITY'. Someone had scrawled 'TWINNED WITH CUMBERNAULD!' across the bottom in green graffiti.
The Shogun slowed for the turning, leaving the well-ordered tarmac for a wide gravel road acned with potholes and lined with whin bushes. Their jagged dark-green spears rattled in the rain.
It was getting heavier, bouncing off the rutted track as McAdams navigated his shiny new car between the water-filled craters and up to a cordon of blue-and-white 'POLICE' tape.
He buzzed down the window and smiled at the lanky drip guarding the line. 'Two cheeseburgers, a Coke, and a chocolate milkshake please.'
A sigh and a sniff. Then Officer Drip wiped her nose on the sleeve of her high-viz jacket, sending water dribbling from the brim of her peaked cap. 'Do you _honestly_ think it's the first time I've heard that today?'
'Cheer up, Constable. A little rain won't kill you.' He nodded at the cordon. 'You got our body?'
'Depends. You on the list?' She dug a clipboard from the depths of her jacket and passed it through the window.
McAdams flipped through the top three sheets, making a low whistling noise. 'There's a _lot_ of people here. All for one dead little body?'
'Oh you'd be surprised.'
He printed two more names on the last sheet in blue biro, then handed the clipboard back. 'There we are, right at the end. Now be a good girl and get out of the way. It's the opening chapters: I need to draw the readers in, establish myself as the protagonist, and get on with solving the murder.'
Constable Drip frowned at their names, then into the car. Her mouth tightened as she stared at the bloodied and unconscious Dugdale lying across the back seat. 'Looks like you've already _got_ a body.'
'Oh, this one's not dead, it's just resting. DC MacGregor decided to try his hand at a little police brutality.'
'MacGregor...?' She peered at the list again, then across the car, top lip curling. 'So it _is_ you.'
Callum stared right back. 'Don't: I'm not in the mood.'
She shook her head, stowed her clipboard away, then unhooked a length of the tape barricade and waved them through.
McAdams grinned across the car at Callum. 'My, my, Constable. You just can't stop making friends, can you?'
No.
'That offer of an arse-kicking is still valid, Sarge.'
'Yes, because people don't hate you enough already.'
The Shogun pitched and yawed through the potholes like a boat. God knew how big the rubbish tip was, but from the wide, lumpy road, it stretched all the way to the horizon. A vast sea of black plastic, gulls wheeling and screaming in the air above – flecks of evil white, caught against the heavy grey sky.
And the _smell_...
Even with the car windows wound up it was something special. The rancid stench of rotting meat and vegetables mingled with the sticky-brown reek of used nappies, all underpinned by the dark peppery odour of black plastic left to broil in the sun.
McAdams slipped the four-by-four in behind a line of police vehicles and grubby Transit vans. Had to be, what, eight cars? Twelve if you counted the unmarked ones. About three-quarters of the dayshift, all out here playing on the tip.
The sarcastic half-arsed-poetry-spouting git was right: this was an awful lot of people for one dead body.
McAdams hauled on the handbrake. 'Right, Constable, make yourself useful for a change and go fetch us a couple of Smurf suits, extra-large. Ainsley and I need to have a little chat.'
A chat?
'He's _unconscious_ , Sarge. He needs a doctor. I told you he—'
'Don't be stupid.' McAdams turned in his seat, staring through into the back. 'Give it up, Ainsley, you're not fooling anyone.'
Dugdale didn't move.
'Don't make me come back there, because if I have to...'
One of Dugdale's eyes cracked open. 'I'm dying. Got a brain haemorrhage, or something.'
'You have to have a _brain_ to have a brain haemorrhage, Ainsley. What you've got is a lump of solid yuck wrapped in ugly. Now, Constable Naïve here is going to sod off like a good little boy and you're going to tell me _all_ about what Big Johnny Simpson's up to now he's walked free.' McAdams made a dismissive little waving gesture in Callum's direction. 'Go on, Constable. Two Smurf suits, at the double. I won't ask again.'
One punch in the face. Just one. Right in the middle of his smug, wrinkly face...
What was the point?
It wouldn't change anything.
So Callum gritted his teeth and stepped out into the stinking mud. Closed the car door. Counted out his own muttered haiku. 'Away boil your head. You patronising arse-bag. I hope you get piles.'
Out here the smell was eye-watering. Like jamming your head in a dead badger.
He turned up his collar and hurried through the slimy mud to the nearest Transit van, sheltering in the lee of its open back doors. From here, Oldcastle lay spread out beneath the heavy grey lid of cloud like a cancer beneath the skin. The vast prow of Castle Rock loomed out from the other side of the valley, wound round with the ancient cobbled streets of Castle Hill; the dark sprawl of Camburn Woods peered out from its shadow; the warehouses, shopping centres, and big glass Victorian train station punctuated Logansferry to the left of that. Spires and minarets stabbed up between the slate roofs on the other side of the river, like some vast beast was trapped under the surface, trying to claw its way out. And on _this_ side: the grubby maze of council houses, high-rise blocks of flats, and derelict terraces of Kingsmeath; the rest of the city, hidden by a line of trees at the edge of the tip.
Quite a view for a rancid mass of black plastic bags and mouldering filth.
He reached into the Transit and helped himself to two large blue Tyvek oversuits, two sets of plastic bootees, a pair of facemasks and matching safety goggles. What every well-dressed Scene of Crime officer was wearing this, and every other, season.
One of them appeared from the other side of the van, the hood of her SOC suit thrown back to reveal a sweaty tangle of dark brown hair. Her thin, pale oval face shone with sweat. She took a swig from a leopard-print Thermos, the words coming out on a waft of coffee breath with a faint side-order of Aberdonian. 'Oh, it's _you_.'
'Don't start, Cecelia, OK? I get enough of that from McAdams, don't need the Scene Examination Branch chipping in.' He tucked the suits under his arm. 'We're here for the body.'
She curled her top lip. 'Which one? Started digging at nine this morning and we've already turned up four of the things. Seven if you count those.' She nodded in the vague direction of a red plastic cool box and helped herself to a wad of paper towels. 'Three left feet, severed just above the ankle.'
'Well... maybe their owners aren't dead? Maybe they're limping about somewhere, wondering where their other shoe's gone?'
'Urgh. I'm melting in here.' Cecelia scrubbed the paper towels across her damp face, turning it matt again. 'Bet they don't have this problem in G Division. Bet if you go digging in a Glasgow tip all you turn up is rubbish. Can't open a bin-bag in Oldcastle without finding a sodding corpse.' A sigh. 'Have you got any idea how much work it is to process crime scenes for _seven_ different murder enquiries, all at the same time?' She ticked them off on her fingers. 'One stabbing, one shotgun blast to the face, one God-knows-what, and I'm pretty sure the body we found over by the recycling centre is Karen Turner. You know: ran that brothel on Shepard Lane? Beaten to death.'
At least that explained why most of Oldcastle Division was in attendance, picking their way through the landfill landscape.
'Wow.' Callum frowned out at the acres and acres of black-plastic bags. Suppose it wasn't _that_ surprising the tip was hoaching with corpses – if you had to dispose of a body, where better than here? Clearly the city's criminal element didn't approve of littering. 'Maybe we should set up a recycling box at the front gate, so people can dump their dead bodies responsibly?'
She puffed out her cheeks. 'We should never have started digging here. Just asking for trouble.'
'So, come on then: which one's ours?'
'Body number three: the God-knows-what. That way.' She pointed her Thermos at the middle distance, off to the right, where a handful of blue-suited figures was wrestling with a white plastic tent. 'And Callum?'
He turned back to her. 'What?'
'I know it wasn't you.'
What wasn't...?
She rolled her eyes. 'There's no point standing there looking glaikit. You didn't cock-up that crime scene, Elaine did.'
Oh.
Heat bloomed in his cheeks. 'No she didn't.'
'Yes she did. Elaine worked for me, so I _know_ it wasn't you. One more strike and they'd have fired her.'
He tucked one of the Tyvek suits under his arm. 'I've got no idea what you're talking about.'
Cecelia shook her head, sending a little trickle of sweat running into the elasticated neck of her suit. 'You're a daft sod, Callum MacGregor.'
True.
'Bye, Cecelia.' He turned and marched back to the Shogun.
McAdams was still in the car, mobile clamped to his ear, so Callum struggled into one of the SOC Smurf suits – zipping it up to the chin, hood up. Stood there in the manky mud, rain pattering off his Smurfy shoulders and head.
Come on, you lanky git. Get off the phone.
A rattley green Fiat Panda lumbered its way up the track towards them, bringing a cloud of blue-grey smoke with it. Dents in the bonnet, dents in the passenger side, a long scrape along the driver's door and front wing. Duct tape holding the wing mirror on.
Great, because having to deal with DS Sodding McAdams wasn't bad enough.
The Panda spluttered to a halt behind McAdams' immaculate Castleview Tractor, and its driver peered out through a fogged-up windscreen as the wipers made angry-donkey noises across the glass.
Mother.
She looked right at him and the smile died on her face.
Oh joy.
He gave her a nod. As if that was going to make any difference.
Mother struggled her way out into the rain.
The sleeves of her black fleece were rolled up to the elbows, exposing two large pale forearms – tattoos standing out like faded newsprint against the doughy flesh. A dolphin. Two swallows holding up a little banner with 'LOVE NEVER DIES' on it. A thistle and a rose wrapped around a dagger. What looked like a tribute to the Bay City Rollers – all mullets and tartan scarfs. She glanced about, sending her mass of tight ginger curls bobbing. Sniffed. 'Where's Andy?' Apparently completely unfazed by the rain.
'DS McAdams is in the car, making some calls.'
Her eyes narrowed. 'Have you been upsetting him?'
'Upsetting _him_? He wasn't the one Dugdale tried to neuter! Come on, Mother, how come every—'
'Ah yes, Andy said you'd had a run-in with The Claw.' A tiny smile. 'And how many times do I have to tell you: you haven't earned the right to call me "Mother". As far as you're concerned it's Boss, Guv, or Detective Inspector. Are we crystal?'
'It wasn't a "run-in", Dugdale resisted arrest. Violently. And _for the record_ ,' Callum pointed at the back seat of the Shogun, where Dugdale was now sitting up, 'I said we should take him to the hospital, but DS McAdams refused.'
The tiny smile grew. 'Nobody likes a clype, Constable.'
A clunk and McAdams emerged from the car. 'Mother...' A frown. 'MacGregor, why are you wearing that SOC suit?'
Callum looked down at his blue Tyvek body. 'You told me to get two Smurf—'
'One for me and one for _Mother_ , you idiot. Why the hell would we want you messing up our crime scene?'
He clenched his fists. Stepped forwards. 'You think I won't—'
'All right, that's enough.' Mother held a hand up. 'Andy, we're going to cut the wee boy some slack on account of The Claw. He can come with us.' The hand came down again, till it pointed at Callum. 'Don't make me regret this.'
'Yes, Boss.'
'Now go find someone to keep an eye on Ainsley here,' she nodded at Dugdale in the back seat, 'and fetch me a Smurf suit. We've got a dead body to gawp at.'
##
Wet bin-bags shifted beneath his feet, popping and crackling, crunching and slithering in the rain. Hard not to imagine the surface opening up and swallowing them whole. Pulling them further and further down to drown in its reeking depths.
God that was cheery.
Mother and McAdams struggled on beside him, clinging on to each other to stay upright on the bin-bag sea. They must have made quite a sight: all three of them, dressed in matching blue outfits that were about as flattering as a dose of dysentery, shuffling their way through the rubbish towards the SOC tent.
It stood, a grimy shade of white, poking out of the bin-bag ocean like an iceberg. Or some vast grubby tooth.
Mother sniffed behind her mask. 'What do we know about our victim?'
'Nothing.' McAdams picked his way past a slimy mass of something. 'DCI Powel was even more inscrutable than usual. Probably got his nose out of joint because he had to hand it over to us.'
'Poor darling. Still, as long as it's a murder and we're investigating it, I'm happy.'
McAdams let go with one hand and placed it against his chest, launching into a wobbly but not unpleasant baritone:
'People dismembered with axes and chainsaws,
Someone's been strangled with wire or some string,
A stabbing, a beating, a fresh torture victim,
These are a few of my favourite things...'
'Oh, _very_ good. I like that.' She struggled on a couple of steps. 'Thought you were on haikus today.'
'Decided to branch out a bit.'
A cordon of yellow-and-black tape encircled the SOC tent, the words 'CRIME SCENE – DO NOT ENTER' rippling and spinning in the wind. Every gust making the plastic tape growl. Water ran down the tent's walls, dripping off the sagging roof.
Mother motioned to Callum and he held up the cordon so she could duck under and slip inside. McAdams stopped right next to him, voice low, just audible through the facemask. 'In the three weeks you've been here, you've done nothing but moan, whinge, and disappoint. But if you compromise my crime scene, I'll make you wish Dugdale still had your balls in his fist. Understand?'
Callum just stared back.
'Good.' He turned and pushed through into the tent.
Count to ten.
Don't let him get to you.
Deep breath.
Callum pulled his shoulders back and followed McAdams inside.
Rain thudded against the tent's roof. The wind moaned through the gaps in the plastic, making the walls shudder. Technically, you could have parked a couple of patrol cars in here and still had room for a police motorbike, but instead it was home to a small diesel generator and four workplace lights on six-foot stands.
The stench was something special – so thick it was almost chewy, trapped by the tent's walls and roof, amplified by the warmth of decomposition, and soured with diesel exhaust fumes.
Four figures in the full Smurf kit were kneeling around a hole dug into the rubbish, right in the middle of the tent.
Mother joined them and clapped her hands, raising her voice over the rain and the generator. 'Come on then, what have you got for me?'
One of the figures straightened up with a groan, both hands pressed into the small of his back. 'Mummy.'
She pursed her lips. 'I don't mind a little informality, young man, but that's going a bit too far.'
'Not you.' He pulled down his facemask, showing off a round sweaty face with tiny pursed lips. Like someone had pumped a cherub up on steroids and pies. 'In the hole: it's a mummy. Your actual, curse-of-the-Pharaohs, from-the-leathery-mists-of-time, mummy.'
'Really?' Mother inched her way to the very edge and peered down.
'Or it might be a daddy. Difficult to tell without unfolding the limbs, and I get the feeling they'll snap off if we do that. Teabag tends to frown on our dismembering corpses before he's had a chance to post mortem them.' He dug out a scrap of cloth and dabbed at his shiny face. 'Gah. Like a sauna in here.'
McAdams stepped up beside Mother. 'Ah...'
Callum crept around to the opposite side of the hole, bin-bags shifting beneath his blue-booteed feet, and leaned out over the edge.
The SOC team had shored up the sides of their excavation with sheets of corrugated iron, which held back the mass of garbage, but did nothing to stop the grey-brown liquid seeping out underneath it.
Their body lay on its side at the bottom of the hole, about eight feet down, where the liquid was deepest. Elbows tight in against its ribs, hands drawn up to its chest, knees hard up against them, feet tucked in to the body. Its neck was bent hard forward, so the face was completely hidden by the hands and knees. So far, so murdery, but it was the skin that gave it away. Instead of being all blotched with mould and falling apart it was creased and leathery. Darkened to a dirty mahogany. The only ear visible had shrivelled up till it resembled a dried apricot, clinging to the side of its bald head.
Callum raised his eyebrows. 'Now there's a sight you don't see every day.'
Mother's fists clenched at her sides. 'That rotten, two-faced, lying _bastard_!'
The oversized sweaty cherub in the SOC suit wiped his glistening forehead. 'At a guess, it's got to be about, what... a thousand years old?'
'I should have known! Thought they'd finally given me a proper murder, but _no_. That was asking too much, wasn't it?' She turned and stomped out of the tent.
McAdams didn't follow her, just shouted over his shoulder instead. 'Where are you going?'
Her voice faded away into the distance. 'To tell DCI Powel exactly where he can stick his thousand-year-old mummy!'
The only sound in the tent was the hammering rain and the growling generator.
'Hmmm...' McAdams squatted down, one hand on the bin-bag next to him. 'The body's naked. Wonder what happened to all the bandages.' He glanced up at the Cherub. 'It's a mummy, it should be all wrapped up.'
'Don't look at me.'
Callum eased himself down to his haunches, holding onto the top of a corrugated sheet. No way he was risking an eight-foot plummet into a paddling pool of rancid bin water. 'They've got a mummy just like it in Elgin Museum. On display, naked in a big bell jar. Some Victorian bloke brought it back from Peru: suppose he unwrapped it so the viewing public could get a good look at a real-life dead body.' A small smile shifted against his facemask. 'We used to go there when I was a wee boy. Me and Alastair would...' Yes. Well. The less said about that the better.
McAdams grunted, then stood. Turned to face the sweaty cherub. 'Don't suppose we've got any clue who dumped it here, do we?'
One of the other Smurfs looked up from the contents of a ruptured refuse sack. 'Nah. Back in the good old days, there'd be envelopes and letters and newspapers all through this stuff – dates and addresses in every bag. Now?' He shook his head. 'Recycling: bane of our lives.'
McAdams wiped his hands together. 'Soon as Dr Twining's seen the remains, get them bagged, tagged, and down the mortuary. And if he gives you any grief about it being a waste of his valuable time, tell him tough. Don't see why we should be the only ones.' A click of the fingers, held high overhead, as if McAdams was summoning a waiter in a sitcom. 'Constable MacGregor: we're leaving. Turns out this is more of a short story than a fully-fledged novel.'
Callum stayed where he was, sniffing the air. 'Can you smell that?'
'I said, "We're _leaving_."'
'No, underneath all the rotting rubbishy smell, there's something else. Wood smoke? Like there's been a fire?'
'Don't look at me.' The Cherub shook his head. 'Fifteen minutes in here and you go nose-blind. Can't smell a thing.'
McAdams' voice boomed from outside the tent: _'CONSTABLE MACGREGOR! HEEL!'_
The Cherub shrugged. 'His master's voice.'
Don't suppose it mattered anyway. What was one extra smell on top of all the others?
Callum stood, wiped his gloves on his legs, and slipped back out into the rain.
Halfway back across the slippery bin-bags, his phone launched into its default ringtone. Sodding hell. He peeled off his right glove and fought the bare hand into his SOC suit. Pulled out his phone. Kept on walking. 'Hello?'
_'Ah, hello. Am I speaking to Detective Constable Callum MacGregor?'_
He checked the number. Nope, no idea who it was. 'Can I help you?'
_'Good, good. This is Alex from Professional Standards, we'd like you to pop in for a wee chat.'_
Oh God.
_'How does tomorrow morning sound? I know it's taken us a while to get round to it, but better late than never, yes?'_
No.
'Tomorrow morning?'
_'Excellent. Let's say... Oh, that's lucky: I can fit you in at seven. First thing in the morning, then you can get on with your day without having to worry about it.'_
Might as well get it over with – like ripping off a sticking plaster, wrenching all the hair out with it. 'Right. Yes. Seven tomorrow morning.'
After all, what was the worst that could happen?
They could fire him. Prosecute him. And send him to prison.
_'Good, good. See you then.'_ Alex from Professional Standards hung up.
It would be fine. It would.
Callum put his phone away. 'Yeah, you just keep telling yourself that.'
He crunched his way through the bin-bags to McAdams' shiny new Mitsubishi Shogun. The lanky git was leaning on the roof of Mother's scabby Fiat Panda, one hand making lazy circles in the air as she peeled herself out of her Smurf outfit. Probably working on new ways to make Callum's life even worse. As if it wasn't bad enough already.
Professional Standards.
Gah...
He yanked open the passenger door and pinged his blue nitrile gloves into the footwell. Tore off his SOC suit and bundled it up.
They didn't have anything on him.
They _couldn't_ – he hadn't done anything.
Yeah, but when did that ever stop anyone?
He scowled at his crumpled suit. What was the point taking it back to the station and sticking it in the bin, it was just going to end up right back here anyway. Callum hurled it away. It spun, unfurling in mid-air like a shed skin, before tumbling to the filthy ground.
And when he turned back to the car, there was Dugdale grinning at him from the back seat.
'Oh... sod off.'
The municipal tip shrank in the rear-view mirror. McAdams shifted behind the wheel, dug a packet of gum from his pocket and crunched down a little white rectangle. 'Right, you know what's coming next, don't you?'
Sitting behind him, Dugdale scowled out of the window. 'I want a lawyer.'
'Not talking to you, Ainsley, I'm talking to our special little friend, Constable Crime Scene here.'
Callum folded his arms. 'If it's more haikus, I'm putting in for a transfer.'
'Don't let me stop you. First call all the museums. See whose mummy's gone.'
He stared across the car. 'Oh you have _got_ to be kidding—'
'One of them's lost a mummy. I'll bet if you beaver away _super_ hard for the next two or three months, you'll find out which one.' He smiled. 'Unless you're too busy resigning, of course? Wouldn't want to get in the way of that.'
'Oh for... Why can't Watt do it?'
'Because, dear Constable Useless,' McAdams turned a smile loose, 'I don't like _you_ even more than I don't like _him_.' The smile widened. 'It'll be good for you: character building.'
Callum turned to face the passenger window. 'I'd like to build your character with a sodding claw-hammer.'
'Did you say something, Constable?'
'I said, "Yes, Sarge."'
'Good boy.'
And a nail gun.
Dugdale was still wearing the same scowl, but he'd swapped his clothes for a white SOC suit, bare toes sticking out of a pair of manky grey flip-flops. And he'd washed the dried blood off his face. That would be a bonus when his duty solicitor finally appeared.
Callum stood on the concrete apron and waved him goodbye as a Police Custody and Security Officer led him away, steering Dugdale down the corridor and into the cell with 'M6' stencilled on the thick blue door.
The cell block rang with the sound of someone screaming what sounded like passages from the Bible. All 'thee' and 'thou' and that.
Raw breezeblock walls painted a tired magnolia, with a blue line all the way around it, straddling the bright-red panic strip. A dozen cells in this block, most of them occupied, going by the A4-sized whiteboards mounted next to each closed door. Three assaults, two indecent exposures, a theft from a locked-fast place, a shoplifter, one breach of bail conditions, an attempted murder, and Dugdale.
_'VERILY, SAYETH THE LORD, FOR YE SHALL FEAR MINE WRATH!'_
The PCSO stepped back out into the corridor and clunked the cell shut. Printed, 'RESISTING ARREST, ASSAULT, ARMED ROBBERY', on the custody board, each word smaller than the last as she ran out of space, finishing with a scrawled '& CONSPIRACY 2 PTCOJ'.
_'AND YE SHALL BE SORE AFRAID IN THE TIME OF DARKNESS! FOR LO, IT IS THE WORD OF THE LORD THAT COMES FOR THEE!'_
'Oh shut up, you fruitcake.' The PCSO stuck her marker-pen back in her top pocket and looked Callum up and down. 'Something we can do for you, Constable?'
_'YEA, FOR HE IS THE DARKNESS AND HE IS THE LIGHT!'_
'Can you give me a shout when his solicitor gets here?'
_'AND ALL SHALL KNOW HIS WRATH! THESE ARE THE END OF DAYS, AND—'_
She clicked down the viewing hatch on M3. Tutted. Then, 'Come on, Phil, I thought we had an agreement.'
A muffled, _'Sorry.'_ came from the other side of the door.
'Should think so too, disturbing all our other guests. Poor Ken's trying to sleep.' She clicked the hatch up again. Turned to Callum. 'They picked him up on Chamber Street, "The End Is Nigh" placard in one hand, his "original sin" in the other.'
Lovely. 'So, Dugdale's solicitor...?'
She shook her head. 'Now Kenneth, on the other hand, tried to smash his mother's head in with a china dog from the mantelpiece. Spaniel, I think it was. She wouldn't let him go to the pictures. He's forty-six.'
'Yeah, but Dugdale...?' Eyebrows: up, winning smile: on.
'I can't.' A sigh. 'Oh, don't look at me like that, it's orders. "DC MacGregor is not to be given access to custodies or their representatives without a superior officer being present."'
'You are _kidding_ me!'
'All contact is to be managed through DS McAdams or DI Malcolmson.'
'I can't talk to anyone without McAdams or Mother holding my hand?'
'Nothing to do with me, it's...' She turned away. 'If you were them, would you want to risk it?'
##
'Yes, I understand that, but I'm asking anyway: do you now, or have you at any time, had a human mummy in your museum?'
The smell of chicken curry Pot Noodle coiled its way across the office, warring against a taint of cheesy feet and yesterday's garlic.
From up here, on the third floor of Division Headquarters, the view should have been a lot better than it was: the back of a billboard streaked with pigeon droppings. Rusting supports featured a dozen small grey feathered bodies, strutting about and adding to the stains.
_'A mummy? What, like an Egyptian one?'_ The young man on the other end of the phone sounded about as bright as a broken lightbulb. _'Nah. No. Don't think so.'_ Think, think, think. _'Maybe?'_
Callum turned his back on the window, one hand massaging his temples, the other gripping the phone tight enough to make the plastic creak. Fighting hard to keep his voice reasonable and level. 'Can you check for me? It's important.'
The room was divided up into six bits, each one sectioned off with a chest-high cubicle wall – their grubby blue fabric stained with dribbled coffee and peppered with memos from the senior brass and cartoons cut from the _Castle News and Post_. Six cubicles for six desks, two of which were laden with dusty cardboard boxes and teetering piles of manila folders.
Almost every horizontal surface was covered in a thin grey fuzz of dust.
The top of Dot's head was just visible above the edge of her cubicle, pale-brown hair swept up in a weird semi-beehive do. Schlurping noises marked the death of yet another freeze-dried soy and noodle product.
A tiny kitchen area sat in the corner behind her, complete with kettle, microwave, and a half-sized fridge that gurgled and buzzed. Throw in a sagging assortment of ceiling tiles, scuffed magnolia walls littered with scribbled-on whiteboards, the kind of carpet that looked as if it'd been fished out of a skip, and you had the perfect place to dump police officers while they waited for their careers to die.
Or were too stubborn to realise that their careers already had.
_'Pffff... Suppose. I'll see what I can do. Hang on, gotta put you on hold.'_ Click, and an elevator muzak version of 'American Idiot' dribbled out of the earpiece.
Callum printed the word 'dick' in little biro letters next to the museum's name. It joined a long, long list.
Dot wheeled her chair back till she could peer around her cubicle. 'Callum, you on the phone?' Her scarlet lipstick was smudged and a shiny dot of gravy glittered on one rounded cheek. For some reason she'd decided it was a good idea to dress up in what looked like a black chef's jacket, only with shiny silver buttons and silvery edging.
He held up the receiver. 'On hold.'
'Don't fancy making a chocolate run, do you? Only the machine on the fifth floor's got Curly Wurlies.'
'Can't: I'm on _hold_.' He waggled the phone again to emphasise the point.
'I'd go myself, but I'm avoiding Detective Superintendent Ness. She found out I scratched her new Nissan Micra with Keith. Please?'
His shoulder slumped. 'Dot—'
'Pretty please? Got the doctor at three, need to keep my morale up.'
A voice growled out from the opposite corner: 'For Christ's sake!' Watt stood, glowering over his cubicle wall at them. He'd swept his dark floppy hair back from his high forehead, securing it there with enough product to stick a hippo to the wall. Sunken eyes. Squint teeth. A sad excuse for a beard that looked as if he'd made it himself out of ginger pubic hair. 'Will the two of you shut up? Some of us are trying to _work_.'
Dot narrowed her eyes at him. 'Oh, I'm sorry Detective _Constable_ Watt, are we disturbing your sulk?'
He stuck out his chin and its wispy covering. 'I am _not_ sulking, _Sergeant_. I'm preparing for a deposition, OK? Now will the pair of you shut up and let me get on with it?'
'All I wanted was a Curly Wurly.'
'Fine! Fine. You know what? Here...' He dug into his pocket and hurled a fiver in Callum's direction. It fluttered and tumbled in mid-air, falling to the manky carpet six feet short. 'Go. Get her some sodding chocolate. Just do it quietly.'
Callum held up the phone again. 'Is this thing invisible? I'm – on – _hold_!'
_'Aye, hello?'_ The Scottish idiot on the other end cut 'American Idiot' dead. ' _Hello?... You still there?'_
Finally. 'Hello. Yes.'
_'Right, I've had a word with Davey: he can't remember a mummy, but he's only been here a year longer than me. Marge's been here for donkeys', but she's on holiday till the twelfth. Gone to Norwich for a BDSM festival. You want me to give her your contact details so she can drop you an email when she gets back?'_
Callum folded forward until his forehead rested against his keyboard. Don't swear. Don't swear. 'That would be great. Thanks.'
_'Yeah, OK.'_ And the line went dead.
He hung up.
Dot's chair squeaked across the room. Squeak. Squeak. Squeak. Until it was right next to him. When he looked up, she smiled. 'So... chocolate?' She fiddled with the wheelchair's push rims, twisting the whole thing left and right. All coy and fluttering eyelashes. The left leg of her jeans was stitched closed and trimmed off, just below where her knee should have been.
Suppose a little help getting some chocolate wasn't too much to ask for.
He closed his eyes for a moment. Then nodded. 'Yeah. Could do with a break anyway.' He pushed back from his desk. 'Curly Wurly, coming right up.'
She nodded at the list sitting next to his phone. 'No luck?'
'You got any idea how many museums there are in Scotland?' He stood, bent over and scooped Watt's hurled fiver from the floor. 'Then there's all the universities and private schools with natural history stuff in display cases. Never mind private collections.'
'You want a hand?'
He blinked. Turned back to her.
At least _one_ person on the team didn't treat him like something they'd stepped in. 'Thanks, Dot.'
'Don't get all emotional about it. I'm only helping so you'll be my chocolate monkey.' She wiggled her fingers above her head, cackling it out: 'Fly free, my pretty!'
Over in the corner, Watt gave a frustrated wee scream.
Callum slumped his way up the stairs. Two years since they stopped doing proper meals in the canteen. Two years and the stairwell _still_ smelled of boiled cabbage.
His phone went off as he reached the fourth-floor landing. Sodding hell.
He dragged it out. 'What?'
There was a pause. Then a high-pitched man's voice squeaked in his ear. _'Mr MacGregor? I'm calling from the Royal Caledonian Building Society's Fraud Prevention Department. I need to ask you a few security questions. OK?'_
Callum glowered at the wall. 'No, it's not OK.'
_'I'm sorry, have I called at a bad time?'_
'Someone's just nicked my wallet, and I've got no idea who you are. I'm not giving you my security details. You want to help? You prove who _you_ are by answering _my_ security questions.'
_'I... I don't think we're allowed to do that.'_
'Tough. What's the third, fifth, and first letters of my mother's maiden name?'
_'Errr... Look: why don't you call_ us _, then? That way you'll know it's not a scam. You'll find the number on the back of your cards.'_
'On the back of my _stolen_ cards? The cards I don't have?'
_'Ah... Right.'_ What sounded like an argument echoed up from the floors below, followed by a door clunking shut. _'Well, maybe you could pop into a branch and they can help you?'_ Was that a note of hope and desperation there at the end? _Please_ go away and become someone else's problem.
'Yeah. Why not.' He hung up and clunked his head against the wall. Breathed in the cabbagey smell. Then opened his eyes and swore. No wallet meant no cards. And the little sods had wheeched off with his last fifteen quid, leaving him with... He rummaged in his pocket and came out with two pounds fifty-six in change, a button, and a Mint Imperial that had gone all hairy with pocket fluff. So Elaine could have a jar of Polish pickles _or_ a jar of Nutella, but not both. And forget the onion rolls.
Because it wasn't as if he could steal the change from Watt's fiver.
Could he?
He puffed out a breath. Of course he sodding couldn't.
Callum lumbered up the stairs to the fifth floor. Pushed open the door. And froze.
DCI Powel was standing right in front of him, mug in one hand, manila folder tucked under his arm, phone in his other hand. A big man with ears to match, silver-grey hair swept forward from his temples to cover the bald bits. Smart suit with matching tie. He narrowed his eyes. 'Hang on a minute, Margaret, there's someone I need to talk to.' He lowered the phone.
Callum backed away, into the stairwell again, but Powel followed him.
'Well, well, well, if it isn't our very own answer to Mr Bean: Detective Constable Callum MacGregor.'
'Guv.'
'I hear you managed to catch Ainsley Dugdale this morning, Constable. He's one of Big Johnny Simpson's goons, isn't he? That's a first for you, isn't it? Big Johnny won't like that.'
Don't rise to it.
'And we all know how much you _love_ Big Johnny Simpson, don't we?' A massive finger rose and poked Callum in the chest. 'Don't think I won't screw you to the wall for that, Constable. I don't put up with dirty cops in my division.'
Callum curled his hands into fists. 'Permission to speak freely, Guv?'
'Not a chance.' He leaned in closer, bringing with him the stench of aftershave and dead cigarettes. 'I don't like you, Constable.'
'You hide it well, Guv.'
Was that a twitch of a smile?
Then Powel backed off, turned and marched away down the stairs. 'Enjoy your meeting with Professional Standards, tomorrow. I'll bring in a cardboard box so you can empty your desk afterwards.'
_Clunk_. The door closed, and Callum was alone again.
'And screw you too, Guv.'
Powel's voice echoed up from the landing below: 'I'm still here, Constable.'
Of course he was.
##
Callum logged off his steam-powered computer, stretched, yawned, slumped in his seat for a moment, then hauled himself to his feet.
The office's fluorescent lighting buzzed overhead, giving everything the warm and welcoming ambience of a horror film. Shame he was the only one there to enjoy it.
One more yawn, a sigh, and a rummage in the bottom drawer of his desk for the paperback-sized Tupperware box he'd stuck in there first thing this morning. He went back in for the dog-eared hardback copy of _The Monsters Who Came for Dinner_. Checked his watch. Just gone two. With any luck the lunchtime rush at the building society would have petered out by now, but if it hadn't at least he'd have something decent to read.
Callum pulled on his jacket and stuffed his sandwiches in one pocket, crisps in the other. Right, time to—
The office door swung open and McAdams loomed into the room.
Sod.
McAdams frowned. 'And where, exactly, do you think _you're_ going, Constable MacGregor?'
So near, and yet so far. 'Lunch, Sarge.'
'Lunch? Off to hide in the park reading... What is that, a kid's book?'
'It's a _classic_.'
'Maybe if you're six years old.' He checked his watch. 'And you don't have time. That mummy needs its home found. Get your arse to work.'
Again with the sodding haikus.
'I've _been_ working.' Callum picked up the list, all eight pages of it, and shoogled it. 'Now, I'm going to waste my contractually mandated lunchtime in the building society, trying to get them to give me some of my own money, so I can buy food for my pregnant girlfriend. That all right with you?'
McAdams snatched the list from his hand and flicked through the sheets. Frowned. 'Constable, why do these museums have the word "dick" written next to them?'
Ah...
'I'm waiting, Constable.'
Right. Yes. Er...
Ah, OK: 'It's not "Dick", Sarge, it's "D.I.C.K." Database Incomplete – Currently Checking. Most of them don't have an electronic register of all the exhibits in storage, so they're getting back to me.'
McAdams raised an eyebrow, making a line of wrinkles climb its way up his forehead. '"Checking" doesn't start with a K, Constable.'
Innocent face. 'Doesn't it, Sarge?'
'But I appreciate the creative effort.' He pointed at the empty desks. 'Where's Captain Sulky and The Wheels?'
'DC Watt's off to a deposition – that schoolteacher they caught rubbing himself against old ladies in the big Waterstones. DS Hodgkin has a doctor's appointment.'
'Hmmm...' McAdams' mouth pulled down at the edges. 'Ah well, I suppose it can't be helped.' He clicked his fingers. 'You, with me. Mother's office. Now.'
What?
They weren't going to fire him, were they? They couldn't. Professional Standards hadn't even questioned him yet. They couldn't fire him till _after_ that, surely?
Or maybe they could.
Callum took one last look around the miserable little office – with all its stains and dusty surfaces – then followed McAdams out into the corridor, across the hall, and in through the door opposite. The one with a small brass plaque on it, marked: 'DETECTIVE INSPECTOR MALCOLMSON ~ DIVISIONAL INVESTIGATIVE SUPPORT TEAM'.
Mother's office was a bit nicer than her team's, but not by much. It was just big enough for a scarred Formica desk, a line of filing cabinets down one wall, a whiteboard on the other surrounded by pictures of cats cut out of an old calendar, and a single chair for visitors.
Mother was behind her desk, sooking on the end of a biro, but a uniformed PC stood in the middle of the room, at attention: black trousers; big black boots; black fleece with her ID number on the epaulettes; black, police-issue bowler under one arm. Her curly black hair was pulled back in a bun, exposing the dark skin at the nape of her neck.
OK... Maybe they weren't going to fire him. Maybe they were going to arrest him instead.
Mother wrinkled her mouth around the pen and stared at Callum. 'Is this it?'
McAdams propped himself up against a filing cabinet. 'Everyone else is out.'
'Suppose he'll have to do.' She turned. 'Constable Franklin, this is Detective Constable Callum MacGregor. Not the brightest spade in the undertaker's, but he's all ours. For our sins.' Another grimace. 'Callum, this is Constable Franklin. She's joining us from E Division. That means you're no longer the new boy. You will show her the ropes. You will be nice to her. And most of all,' Mother poked the desk with the sooked end of her pen, 'you will _not_ lead her astray. Are we crystal?'
Babysitting. Even more joy.
'Yes, Boss.'
'Good.' Mother plucked a sheet of paper from her in-tray and held it out. 'Now, if neither of you have anything better to do—'
Callum stuck up his hand. 'Actually, Boss, I—'
'—and I know for a fact that you _don't_ , you can chase this up.'
Constable Franklin took the piece of paper. 'Ma'am.' The word was forced out, resentment dripping from that one syllable like burning pus.
'Tell me, Constable, do you have a fighting suit?'
'A fighting...?' It must have dawned, because she nodded. 'Yes, ma'am.'
'Good. You're a DC now: change out of that uniform. You look like you're about to arrest someone.'
A twitch, a tightening of the hands into fists. A breath. Then: 'Ma'am.'
Oh yeah, babysitting this one was going to be _bags_ of fun.
'Off you go then.'
Franklin turned on her heel, face all pinched and flushed. Narrowed her dark-brown eyes and bared her teeth at Callum. 'Do we have a problem, _Detective_ Constable?' Voice like a silk-covered razorblade.
Wow. She was just... wow. Completely... like a _model_ or something. Not just pretty, but totally—
'I asked you a question.' She curled her top lip, exposing more perfect teeth. 'What's the matter, never seen a black woman before?'
'I... It... No.' He blinked. Stood up straighter. 'I mean: no. No problem. Welcome on board.' He stuck out his hand for shaking, but she just pushed past and marched from the room, slamming the door behind her.
'Bloody hell...' Callum leaned against the wall.
'I know. Magnificent, isn't she?' McAdams grinned at the closed door, then laid a hand against his chest. 'Skin like warm midnight. Her eyes are moonlit rubies. Her heart: frozen steel.' A sniff. 'See if I hadn't already ticked "threesome" off my bucket list?'
Mother smiled. 'Congratulations. Anyone I know?'
'Nah: Beth got someone from her work. Miranda. Nice lady. Presbyterian, but _very_ open minded.' He frowned at Callum. 'Still here, Constable? Haven't you got an angry detective constable to babysit?'
'Yes, Sarge.'
Sodding hell.
Bright yellow diggers and tipper trucks lumbered about on the massive Camburn Roundabout, rearranging it's grass and earth into swathes of rutted mud. The Vauxhall's windscreen wipers made dying-squid noises as Callum took the first exit. He snuck a glance out the corner of his eye at the simmering lump of resentment sitting in the passenger seat.
She'd ditched the uniform in favour of a black suit with weird puffy shoulders, a white shirt, and thin black tie. As if she was on the way to someone's funeral. 'What the hell are you staring at?'
He snapped his eyes front again. 'Nothing.' Yellow-brick cookie-cutter houses stretched out on either side of the road. Bland, safe, and predictable. 'Actually...' He bit his lip. 'If you don't mind my asking...' Deep breath. 'What did you do?'
She turned and gave him the kind of look that could strip flesh from the bone.
'I mean, you know, to end up working for DI Malcolmson?'
DC Franklin faced front again.
'Only, it's not usually—'
'Do you always talk this much?'
'Just thought, if we're going to be working together, we—'
'Let's get something perfectly clear, Detective Constable MacGregor: I am not your friend. I am not your colleague. I am someone who will be out of here very, _very_ soon.' She shot her cuffs, making them exactly the same length where they stuck out of the sleeves of her shoulder-padded jacket. 'I don't intend to spend the remainder of my career lumbered with a bunch of dropouts, has-beens, and never-weres.'
The houses gave way to greying fields and austere drystane dykes. All hard edges softened by the incessant drizzle.
Franklin pulled out her phone and poked away at the screen. Glowering down at it in silence. Ignoring him.
OK, well no one could say he hadn't tried.
About three miles south of Shortstaine, a pair of dark lines swooped out from the tarmac, dug through the roadside verge and punched a hole through a barbed-wire fence. A patrol car sat twenty yards further down, parked up on the side with its flashers going.
Callum indicated and pulled in behind it. 'There's a couple of high-viz jackets in the boot, if you want to... OK.'
She was already out of the car, stalking her way across the verge and down into the field beyond.
'Fine. Catch your death of cold, see if I care.' He helped himself to one of the fluorescent-yellow monstrosities and followed her. Arms out to keep his balance on the slippery grass slope.
A hatchback sat about a hundred yards into the field, on the other side of the fence, at the end of those curling dark lines. Its front end had made friends with a chunk of rock, leaving the bonnet twisted like a sneer.
Franklin was halfway there already, back straight and rigid. Presumably because the stick rammed up her backside was of the extra-large variety.
Callum picked his way down the hill until he stood beside her.
The hatchback was an old Kia Picanto – the kind that looked like a roller-skate on steroids. Originally blue, it was now a muddy grey, with deep scratches along both sides where the barbed wire had raked it. A 'Police Aware' sticker covered most of the driver's window.
Franklin stared at the car, then pulled out a sheet of paper and stared at that instead. Then back to the car. 'Is this _it_?'
Callum walked over to the back window and peered in through the rain-flecked glass.
Inside, the car was a mess. Not just the usual burger wrappers and sweetie papers, but splashes of paint and crusts of what looked like plaster dust. A tool bag lay in the rear footwell, next to two drums of flooring adhesive and a packet of slate tiles.
A voice behind them: 'HOY!'
Callum turned.
A young bloke in uniform was stomping his way across the field towards them, one hand holding the peaked cap on top of his head. 'YOU! WHAT THE HELL DO YOU THINK YOU'RE DOING? GET AWAY FROM THERE!'
Franklin waited till he was six feet away, before hauling out a standard-issue warrant-card holder. 'Constable. Care to explain why I'm wasting my time with a road traffic collision?'
PC Shouty peered at her warrant card, then pulled a face. 'No offence, but could you not have introduced yourself back at the roadside and saved me a trip down...' The expression on Franklin's face must have finally worked its magic, because he shut his mouth with an audible click. Blushed. 'Sorry?'
Her voice got even colder. 'I'm listening.'
'Yes. Right.' He pointed at the car. 'Someone called it in this morning, no sign of the driver or any passengers.'
She stepped closer, looming. 'And I give a toss, because?'
'The boot! There's a body in the boot and we thought... well, _I_ thought – thinking isn't exactly Tony's forte – but—'
'There's a body in the boot?' Her eyes widened. 'YOU BLOODY IDIOT! Why haven't you cordoned off the scene? Where's the common approach path? Why aren't you logging visitors? And where the buggering _hell_ is the SEB?'
He backed off a couple of paces, hands up. 'Whoa. It's not like that. I mean, it's not fresh or anything, it's just, you know, dead, and we—'
'THERE ARE HUMAN REMAINS IN THAT CAR, YOU MORON! Call the pathologist, _now_!'
'No, it's like... Look.' He sidled around to the boot of the car and popped the hatchback lid. Swung it up with a gloved hand. 'See?'
Callum leaned forward and frowned.
There, nestled in amongst the dustsheets and a bucket full of plasterboard fragments was a human body. It lay on its side, arms folded so the hands were pressed against its chest, knees hard up against the hands, feet hard up against the bottom. Head bent forward sharply, so the face was almost completely hidden by the knees. Skin shrunken and wrinkled, the colour of ancient leather.
He groaned. 'Not another one.'
Franklin bared her teeth. 'Is this supposed to be a _joke_ , Constable?' She poked Callum in the shoulder with a rock-hard finger. 'A bit of a laugh at the new girl's expense?' Gearing up for a good bellow. 'WELL, IS IT?'
And there it was again, that smell. Much stronger here than it had been back at the tip, where it had to fight with the stench of a hundred million rotting bin-bags. The rich, warm, but slightly bitter tang of wood smoke, so strong you could taste it at the back of your throat.
'Constable! Constable MacGregor, I'm talking to—'
'Will you shut up a minute?' He snapped on a pair of blue nitrile gloves. Reached in and prodded the body. Solid, as if it'd been carved from a chunk of oak, then dipped in the peatiest whisky in the world.
When he straightened up, Franklin's eyes were wide, her whole person trembling as if she was about to pop.
Before she could get started, he dragged out his Airwave handset and called Control. 'Aye, Brucie? I need a check on a Kia Picanto.' He rattled off the registration number and colour, then clunked the boot shut in the intervening silence.
Franklin squared her puffed-up shoulders. 'Now you listen to me, _Sunshine_ , I will _not_ be spoken to like that! How dare—'
_'Okeydokey.'_ A thick Dundonian accent crackled out of the Airwave's speaker. _'Yer car's registered to a Glen Carmichael, eighteen Walsh Crescent, Blackwall Hill. Twenty-four years old. Ooh, looks like he lives with his mum. You wanting the postcode?'_
'Has he got prior?'
_'Couple counts of housebreaking-and-robbery when he was twelve. Suspended sentence. And an ex-girlfriend got herself a restraining order when he was fourteen. Sounds like a lovely wee lad.'_
'OK, thanks, Brucie.' Callum put his Airwave away. Grinned at Franklin. 'We turned up a mummy at the tip this morning, just like this one. Probably nicked from a museum. The Kia's owner has form for breaking into places he shouldn't and helping himself to things that aren't his. Are _you_ thinking what I'm thinking?'
'I see.' She shot her cuffs again. 'Well, don't just stand there – let's go pick him up.'
##
'Shhh, you're doing great.'
Is he? Then why does he feel so terrible? Why does he just want to lie down and die?
The water around him is cold, but that's not why he's shivering.
A sponge dips into the dark brown liquid, then runs gently across his chest, clearing away the thin white rime of salt. Dissolving the crystals back into the brine.
The wall whispers over the sound of trickling water. _'They'll worship you: you'll be a god.'_
Then the sponge dips into the water again, presses against his forehead sending rivulets running down his lined face.
_'They'll worship you: you'll be a god.'_
'Are you thirsty?' The voice is kind, worried. 'Do you want something to drink?'
He tries to shake his head, but can only tremble. No. No more of the foul water.
'I know it's bitter, but it's good for you. Full of herbs and minerals. Here...'
_'You'll be a god. You'll be a god. You'll be a god.'_
A metal cup presses against his cracked lips, and he hasn't got the strength to keep his jaw clenched shut. Sour liquid fills his mouth, catches the back of his throat. And he coughs, splutters the water out, feels it dripping from his chin onto his chest.
_'They'll worship you.'_
His body rocks back and forward, sending out little waves across the bath.
Why can't he cry?
Only it's not really a bath, is it? It's a large metal trough, big enough for three people, let alone one living skeleton. All the joints are rusty, dark brown as if the thing is bleeding, rivets standing out like nipples on its cold metal skin.
Why can't he just die?
_'You'll be a god, and they'll worship you.'_
'Shhh...' A warm hand on his forehead. A gentle touch and a soft word. 'It'll all be over soon.'
##
Walsh Crescent curled in on itself like a snail shell. Mostly bungalows, but every now and then a second storey sprouted from a converted attic. Box hedges, gravel driveways, nameplates on the garden walls. Pretensions of grandeur. One even had a pair of three-foot-high lions perched either side of the drive, their whitewashed surfaces cracked and showing the concrete below.
No view to speak of, but a nice enough street.
Sitting in the passenger seat, Franklin scowled out at the suburban enclave.
Callum pulled up outside number 18. Killed the engine. Sat there with his wrists draped over the steering wheel. 'Look, I know arresting idiots for stealing mummies from museums probably isn't what you signed up for, but this is all they let us do.'
She didn't move.
'And trust me, this is a lot more interesting than what we're usually lumbered with. At least there's genuine dead bodies involved. Even if they are a thousand years old.'
Franklin let out a low sigh, then unclipped her seatbelt. 'I'm here because I punched a superintendent in the car park.'
'In the car park?' Callum smiled. 'There's a euphemism I've never heard before. Sounds painful.'
'He deserved it. Next thing you know: no more Edinburgh for you, pack your bags, you've been posted to Oldcastle.' Sounding about as pleased as someone who's just discovered their routine check-up has turned into emergency root-canal surgery.
'Welcome to Mother's Misfit Mob.' He pointed through the windscreen. 'Shall we?'
They climbed out into the drizzle and hurried up the path to number 18. Stood beneath the little portico waiting for someone to come answer the bell.
'So?' Franklin stuck her hands in her pockets.
'So what?'
'What did _you_ do?'
'Oh...' Well, she was going to find out sooner or later. 'I cocked up. Contaminated a crime scene, because I wasn't paying attention. Too busy trying to get a conviction.' A shrug. 'You know Big Johnny Simpson?'
'Never heard of him.'
'Well, he walked on a murder charge. Because of me. And no, I'm not happy about it.' At least _that_ part was true.
'So the team's a dumping ground for the unwanted and the incompetent. That's just great.'
'I wouldn't say—'
The door opened. 'Hello?' A middle-aged woman squinted out at them, hair piled on top of her head, a red pinny smeared with grey stains covering polo-shirt and cords. She wiped her hands on a dishtowel. 'Sorry, I was in the studio. Can I...' Her shoulders dipped as she looked them up and down. 'I'm flattered, but I honestly don't want any copies of _The Watchtower_ , leaflets about the Bible being a guide to modern life, or a discussion on accepting Jesus into my heart. So if you don't mind.' She tried to close the door, but Callum stuck his foot in the way.
'Mrs Carmichael? Police. Is Glen in?'
'It's Ms, and no.' Her nose came up. 'Now, if you'll excuse me, I've got clay on the wheel.'
Franklin held out her warrant card. 'There's been an accident: we just found your son's car in a field south of the city. He's not in it. We're worried for his safety.'
A hand fluttered to her mouth. 'Glen...'
'Now can we come in?'
The kitchen was warm enough, every surface covered with pots and bowls and mugs. Some less wonky than others.
Callum stuck the kettle on to boil, then picked up a blue mug with a white rim. 'These are very good. Did you make them yourself?'
Ms Carmichael sat at the small kitchen table, worrying at her clay-greyed dishtowel. 'Is Glen all right?'
Franklin pulled out a chair and sat down opposite her. 'We don't know. We've been in contact with the hospitals and doctors' surgeries, but nothing so far. He's—'
'Oh God...' Her eyes reddened. 'Glen.'
'Let's not jump to any conclusions.' Callum pointed out through the kitchen wall in the vague direction of Shortstaine. 'We didn't see anything in the car to suggest he's badly hurt. He's probably just lying low and feeling a bit bruised and stupid.' That, or Glen had massive internal injuries and was drowning in his own blood somewhere, but his mum _definitely_ didn't want to hear that. Nothing wrong with leaving people with a little hope.
Franklin sniffed. 'Ms Carmichael, your son had something in the boot of his car that we're concerned about. Something that didn't belong to him.'
She stiffened. 'My poor wee boy could be lying dead in a ditch and you're here accusing him of _stealing_?'
Callum put teabags in mugs. 'I know it sounds a bit insensitive,' he gave Franklin a pointed look, 'but we've got to investigate this kind of stuff. It's important.'
'It's because of those burglaries, isn't it?' She poked the table with a clay-greyed finger. 'He was _twelve_ , OK? Just a kid. His dad, God rest his useless little soul, ran out on us the year before. Glen had a hard time adjusting.' A shrug. 'His therapist said he was just trying to get attention. Pushing me to see if I loved him enough to put up with all his crap.'
The kettle grumbled to a boil, spouting steam into the air.
'It wasn't even money he took. It was stupid things: a standard lamp from next door, a bust of Daley Thompson from the sports centre, all the cutlery from Terry's Bistro on Minerva Road. It wouldn't even have been a thing if the bloody sports centre hadn't insisted on pressing charges.'
Callum fished the steaming teabags out and dumped them in the bin. 'What happened with the girlfriend?'
'Gah...' Ms Carmichael stared at the ceiling for a moment. 'Angela. He wouldn't leave her alone. Always buying her little presents and writing her little notes. Following her home from school.' She looked down when Callum put a mug of tea in front of her. 'I tried to talk some sense into him, but you know what teenage boys are like – all hormones, spots, and erections. Her parents called the police, and he was in trouble again.'
The fridge was mostly full of yoghurt and chardonnay, but there was half a pint of semi-skimmed that looked reasonably fresh, so Callum stuck it in the middle of the table. 'Nothing since?'
She wrapped her hands around her mug. 'It took a while, but he grew up a bit. Got over his dad abandoning us for some leggy tart in the roads department. Started doing well in school again. Went to university and got an MA in business management.'
'Sounds like a bright kid.' Callum passed Franklin a slightly wonky green mug, but kept his eyes on Ms Carmichael. 'Is it OK if we take a look at Glen's room?'
'What?' She blinked at him. 'Oh, yes. Right.' She scraped her chair back and stood. Led them out of the kitchen and down a small corridor to a room at the end with 'SECRET EVIL VILLAIN LAIR' printed on a sign hung on the door beneath a radiation symbol. She opened it and stepped to one side, mug of tea clutched to her chest. 'Of course, by the time he graduated no one was hiring. That's the recession, isn't it?'
The floor was barely visible through the patina of discarded socks, T-shirts, jeans, and pants. Walls covered in bookshelves – science fiction and fantasy paperbacks, mostly. A TV hooked up to a PlayStation. A poster of a young woman in a bikini, riding a motorbike. Never mind leathers, she wasn't even wearing a crash helmet. Some people just didn't take basic safety precautions. A collection of photographs pinned to the wallpaper, above a small computer desk that was heaped with envelopes and bits of paper. And a double bed covered in more clothes.
Every breath in here tasted of stale digestive biscuits and mouldering cheese.
Ms Carmichael shrugged. 'Don't look at me, I told him when he turned sixteen: you're a grown-up now. You tidy your own room, or you live in a pigsty. Your choice.'
Franklin picked her way into the middle of the room. 'Was Glen interested in museums?'
'When he was little we'd go to the art gallery, and we'd laugh at all the statues and their naked willies, but other than that...' A shrug.
'Hmm...' She leaned over the desk and pulled a photo from the wall. Held it out. 'Is this him?' Her finger hovered over the central figure in a group of three. It looked like a selfie: three young men, all with grins and tins of lager. Checked shirts and tan braces.
The one on the left had a full-sized Grizzly Adams beard, two squint teeth dominating his smile, all crowned by brown hair cropped close at the sides and floppy on top. He'd got one of those piercings, where they stuck a big round plug in the lobe to stretch it wide – making a dirty big hole. As if he was a Masai tribesman, instead of a peely-wally wee bloke from Oldcastle with a lumberjack fixation.
The one on the right's arm snaked out of the picture – so he'd be the photographer – a shoulder-to-wrist tattoo of Clangers, Soup Dragon, and the Iron Chicken blurring into a colourful mush where the lens couldn't focus. Long hair pulled back in a ponytail. A variety of studs spread about his nose, eyebrow, lip and ears.
And the one in the middle looked as if he'd inherited his great grandad's haircut and glasses. Though where he'd got the massive soup-strainer moustache from was anyone's guess. He was straightening his bow tie, showing off an oversized steel wristwatch on an oversized leather strap. More piercings.
Ms Carmichael squinted at the photo. 'No, that's his friend Ben. Glen's the one on the left with the ridiculous beard.' She grimaced. 'Why these hipsters all want to look like old men from the thirties is beyond me. But there you go.'
'I see.' Franklin produced her notebook. 'Can you tell me what your son was wearing when he left the house this morning?'
A snort. 'This morning? He's not been back here for six weeks. Him and his friends have been staying at the flat they're doing up.' She sighed, looking around the room with its explosion-in-a-laundry-basket décor. 'Brett's got a degree in environmental science, Ben's got a BA in aquaculture, and none of them can find jobs. Recession.'
Franklin scribbled something in her notebook. 'And where is this flat?'
'They bought it at auction. The man who lived there killed himself in the living room – bank was foreclosing on his mortgage.'
'Yes, but where is it?'
'Hold on, it'll be in here somewhere...' Ms Carmichael rummaged through the piles of paper on the computer desk, before emerging triumphant with what looked like a council tax bill. 'Flat twelve, one thirty-five Customs Street, Castleview, OC twenty-one, six QT.' Then she turned and put a hand on Callum's arm. 'You're sure Glen's... not _hurt_?'
He gave her his best reassuring smile. 'We'll let you know as soon as we hear anything.'
'Why did you have to lie to her?'
Callum shrugged, slowing the car at the junction. 'What did you want me to tell her: we've no idea if your wee boy's dead or not? Think that would've helped?' The bridge over the River Wynd made a graceful cobbled arc above the water, marking the border between Blackwall Hill's twisted knot of housing estates, the Wynd's well-ordered Georgian streets, and Castleview's functional industrialisation. All of it grey and miserable in the drizzle.
He took a left at the next roundabout, down a long drab street – blocks of terraced flats, punctuated by shopping centres with more boarded-up windows than new shops.
'What if Glen Carmichael turns up dead from a punctured lung, or a ruptured spleen?'
'Then he'll still be dead whether she's panicking about him or not. Let her have... Oh, hold on.' He slammed on the brakes and pulled their manky Vauxhall into a space between a delivery truck and a skip.
'What the hell do you think you're—'
'I'll only be a minute.' He scrambled out of the car. 'Honestly, five tops.' Callum clunked the door shut, waited for a bus to grumble past, then hurried across the street and into one of the few shops still open.
The Royal Caledonian Building Society's carpet was going threadbare in the middle, drawing a straight line from the door to the counter. A large middle-aged lady sat behind the bulletproof glass, reading a copy of the _Castle News and Post._ She looked up as he reached the counter and pulled on a smile about as natural as a porn-star's breasts.
'How can I help you?'
Callum thumped his warrant card on the countertop. 'Someone stole my wallet and I need you to give me some money from my account.'
She made a face, as if he'd just slapped a used nappy down in front of her. 'I'll have to speak to the manager...'
The sharp-faced woman pulled on her glasses and peered at her computer screen. 'Well, Mr MacGregor, you'll be glad to hear that we appear to have recovered your cards. Someone tried to use them to redeem a number of items at... let me get this right... at a "Little Mike's Pawnshop"? In Kingsmeath?'
Little sods were probably trying to get hold of samurai swords, crossbows, and ninja throwing stars.
Callum crossed his fingers. 'Did they find my wallet?'
Please, please, please, please.
She poked at the keyboard. Frowned. 'I'm sorry, I don't actually have that information. But the proprietor _has_ destroyed the cards, and as there's been no successful purchase made on the account, there won't be any excess to pay.' The frown turned into an expectant smile, as if she was waiting for congratulations and a round of applause. 'We'll get new cards issued to you in the next couple of days.'
'A couple of days? But I need to buy—'
'I'm sorry, but the cards have to be reissued from head office. I'll flag it as urgent, but it'll still take a couple of days. Now is there anything else I can assist you with, Mr MacGregor?'
'Yes: I need to take some money out of my savings account.'
'Ah. I see...' She made the same face as the woman behind the counter.
Franklin glowered at him as he lowered himself into the driver's seat. 'What happened to "five minutes, tops"?'
'Don't start.' He hauled on his seatbelt and started the car. Pulled away from the kerb. 'Just been bent over a bank manager's desk for the last quarter of an hour, being shafted without lubricant. And do you know what for? Fifty-three pounds and seventy-two _sodding_ pence.' He held up the tiny handful of notes and coins. 'Because _that's_ all the money we've got.'
Down to the end of the road, and right onto the main road.
'It's my sodding money! Why do I have to beg for my own fifty-three quid? How is that fair?'
Franklin shook her head. 'Do you do anything other than moan?'
'I'm not moaning, I'm ranting. It's different.' He took a right, into another street full of tenements, heading towards MacKinnon Quay. 'How much did we pay to bail those thieving gits out? Billions. Whole sodding country up to its earholes in debt, people losing their jobs and their houses, all so they can enjoy their bloody yachts and champagne!'
A smile. 'Do you _honestly_ think the manager of a wee building society sub-branch, in a nasty little shopping centre, in a crappy corner of the mediocre cesspit that is Oldcastle, has a yacht?'
'That's not the point.'
Straight through at the roundabout and onto Customs Street. It skirted the edge of the docks, with their large blue cranes and towering tanks of offshore mud, all secured behind a twelve-foot-high fence topped with razor wire.
Just past the harbour, a row of small cottages – jammed in tight as teeth – lined the left side of the road, but the houses opposite were a lot less quaint. They were built into the side of the hill, about six feet above road level: the kind of buildings councils put up to punish people for being too poor to afford somewhere better to live. Long, brutalist rows of grey flats, four storeys tall.
Oh, they'd made an _attempt_ to tart them up, put brightly coloured cladding on the top floor, arranged big concrete planters out front on the four-foot-wide strip of grass that separated the flats from the vertical drop to the road below. But the cladding was chipped and faded, the planters cracked and full of weeds, the grass a patchwork of yellow and brown – landmined by generations of terriers and Alsatians.
Callum slowed the car. 'Which one is it?'
She checked her notebook. 'Number one thirty-five.' Then tapped a finger against the glass, as if she was counting time for a very small orchestra. 'One fifteen. One sixteen. One seventeen...'
'Glen and his mates had three university degrees between them, and they bought a flat down here? So much for modern education.'
'One twenty-two. One twenty-three...'
'Suppose they spend six months doing it up, who'd be daft enough to buy it when they've finished?'
'One twenty-eight. One twenty-nine...'
'Bunch of idiots.'
'One thirty-two. Thirty-three...' She pointed. 'That'll be it there. Top floor.'
Callum parked outside, next to a dilapidated Transit van with, 'DANNY & MIKE ~ CHILDREN'S ENTERTAINERS' on the side. Someone had daubed the words, 'THEYZ PEEDOFILES!!!' underneath, in what looked like blue Hammerite.
He climbed out into the rain. Locked the car soon as Franklin joined him on the pavement. 'You ready?'
She rolled her eyes. 'It's a wee boy with a degree in business management, Constable MacGregor, not Osiel Cárdenas Guillén.' Franklin climbed the steps up to ground level, disappearing from view.
A sigh, then Callum followed her.
Down at road-level, the cottages opposite acted as a windbreak, but up here the drizzle came down sideways, driven in on frigid gusts. MacKinnon Quay sat off to the left, then the grey water of Kings River, then the green line of Dalrymple Park with its big granite monument on the other side. Castle Hill lost in the low grey mist.
On a good day it was probably quite some view, but this wasn't one of them.
Franklin jabbed a finger at the intercom. Then grimaced and pulled it clear. She sniffed the end of her finger and grimaced again. Wiped it on the rough grey wall.
Callum took out a biro and used it to press the button marked 'SERVICES'. Holding it down until someone inside finally got tired of the noise and let them in. He smiled at her. 'Trick of the trade.'
Inside, the corridor was lit by a single flickering bulb in a flyblown fitting. Concrete floor, walls painted magnolia above waist-height and a grubby green below it. The smell of frying onions mingled with the hospital stink of disinfectant. An open stairwell led up into the gloom.
Yeah, Glen and his mates were definitely kidding themselves if they thought _anyone_ was going to buy their flat.
Franklin led the way upstairs.
And Callum tried not to stare at her backside, he really did, but...
Heat rose up his face, making his ears tingle. Yeah, probably better not to ogle his new teammate's rear end. But it _was_ magnificent.
Across the first-floor landing and up another flight of stairs. And there it was, right in front of him again.
Stop it!
Pregnant girlfriend, remember? Even if she had been off sex for the last five months.
Yeah, but...
No. No staring.
He cleared his throat. Stared at the wall instead.
The second floor was almost identical to the first – two pairs of red doors, some with welcome mats, some with browning spider-plants and dying ferns in pots. Numbers on the doors. Plastic or brass nameplates.
A little old man cracked his door and glowered out at them. 'You from the Council? About time. Tell those bloody hooligans to turn their music down! Can't hear myself think in here.' He slammed the door shut again.
OK.
Callum hurried past, trying _very_ hard not to ogle Franklin's bum as she climbed the last two flights and stepped out onto the third floor.
She reached into her jacket, came out with a pair of purple nitrile gloves. Snapped them on. Frowned. 'Are you all right, Constable? Your face is all red.'
'It... I... Just, you know, the stairs and that.' He cleared his throat and snapped on his own gloves: blue. 'You want to kick the door in, or shall we do it the old-fashioned way?'
'Hmm.' She knocked.
A skylight sat in the middle of the ceiling, right above the void in the stairwell. A scuffling scratchy noise followed two blurred outlines across the cloudy glass. Seagulls?
He shifted his feet, locked his eyes on a spot six inches above her head. 'So who's O'Neil Gillen, when he's at home?'
'Osiel _Guillén_ , not O'Neil Gillen. AKA: El Mata Amigos, the Friend Killer. Mexican drug lord.' Franklin knocked again. 'Hello?'
She squatted down and lifted the letterbox.
Music pulsed out onto the landing, Led Zeppelin hammering on and on about giving someone a whole lotta love.
'Hello?' Another knock.
Callum wrinkled his nose. 'Can you smell that?'
Sort of a cross between rancid sausages and pine air freshener.
'Mr Carmichael? Police. I need you to open this door. Mr Carmichael?' She glanced up at Callum. 'Is it just me, or does this scream "dead body" to you?'
He took a step back. 'Two choices: we dunt it in, or we go get a warrant.'
'Hmmm.' Franklin let go of the flap, cutting off the music. 'Dunt it.'
'My thoughts exactly.' He raised a foot and slammed it into the wood, just below the handle. The whole thing rattled in its frame. One more. Then a third and the door sprang open, battering into the wall. It didn't bounce back.
The smell got a hundred times worse.
The music got a lot louder too – thumping away from somewhere deeper inside the flat.
Oh yeah, there was certainly something rotten in there.
Percussion solo.
Franklin gritted her teeth and stepped into the hallway. 'THIS IS THE POLICE! I WANT EVERYONE IN THE FLAT TO STAY WHERE THEY ARE!'
Gloom filled the hallway.
A sheet of plasterboard slouched against the wall, the bottom edge bowing under its own weight, anchored there by two big ten-litre tubs of magnolia paint.
She crept through the door at the end of the hall.
Callum followed her into a reasonably sized living room. Two windows should have given a view out across the harbour and the river, instead they were completely covered with... Yup, that was hardcore pornography. What little light filtered through it picked out the shape of a platform ladder, a collection of hand tools, and a stack of paint pots. A wallpaper table in the corner bent slightly under the weight of a tool belt, three electric drills, and a small, portable CD player – not quite turned up full volume, but close to it.
Franklin switched the thing off.
Now the only noise was the droning buzz of fat lazy bluebottles making drunken circles in the rancid air. The little dead bodies of their fallen comrades crunched beneath Callum's feet.
'GLEN CARMICHAEL?' She reached into her jacket and came out with an extendable baton. Christ knew where she'd been hiding that. A flick of her wrist clacked it out to full length. 'HELLO?'
Callum pulled out his pepper spray. 'COME ON, GUYS, LET'S NOT PLAY SILLY BUGGERS!'
Two bedrooms led off from the living room, their windows similarly coated in bits of porn mag. One of them looked almost finished – the walls smoothly plastered and painted a neutral beige. The other was stripped back to the bare breezeblocks.
The kitchen was awash with pizza boxes and takeaway containers. A bong, half-full of dirty water, sat on the unit by a sink mounded with dirty dishes. A stack of empty lager tins that was taller than Callum.
Three university graduates and they _still_ lived like teenaged boys.
The smell had been much stronger in the corridor than it was in the rest of the flat.
He stopped in the middle of the living room. 'Where's the bathroom?'
She frowned at him.
'These flats didn't go up in Queen Victoria's time, did they? So where's the bathroom?'
Back into the hall, where that big sheet of plasterboard leaned up against the wall.
He hefted the paint pots out of the way, then grabbed the plasterboard and pulled, dragging it over to the other side.
A flat panel door. That would be the bathroom.
Callum turned the handle and it swung open inwards. He—
Oh dear God, the _smell_...
It crashed out into the hall like an avalanche, the dark-sweet taint of rotting meat riding on a wave of cloying pine.
Behind him, Franklin made little retching noises.
He reached for the light switch and clicked it on.
About a million bluebottles leapt into the air, buzzing and swarming, battering at the bare lightbulb. Setting it swinging.
The room was just big enough for a white bathroom suite, which looked brand new, with a shower above the bath. Dark water filled the tub, the surface flecked with floating mats of white and orange mould. A crust of brown made a tidemark around the rim, tiny crystals that glittered in the swaying light.
There was someone in the bath, lying facedown, skin all blackened and swollen. Crawling with little white things where the body's shoulders protruded from the water.
Franklin stepped up beside him. 'Christ...'
Yeah.
And then some.
##
Callum stuck his notebook back in his pocket, stepping out of the stairwell and into the drizzle. The view hadn't improved, if anything it was worse. Low cloud and mist hid everything on the other side of the river, reduced the MacKinnon Quay to little more than a collection of random shapes.
The whole world rendered in shades of grey.
Getting dark too.
Oh no... He checked his watch: just gone half six. The Polish deli would be closed. No pickled cucumbers, onion rolls, or anything else.
So much for Elaine's cravings.
Yeah, he was going to be popular when he finally got home.
He scuffed along the path then down the stairs to road level, made his way past patrol cars and manky Transit vans. Someone had finger-painted a big willy in the dirt across the back doors of one, complete with hairs.
McAdams' shiny red Shogun took pride of place in front of the Willymobile, engine running, inside lights on. Callum limped over to the thing and slid onto the back seat. Closed the door on the cold dreich evening. 'God, it's perishing out there.'
Sitting in the passenger seat, Mother took a sip of something in a large wax-paper cup. 'Well, well, well, if it isn't Detective Constable MacGregor.'
He sighed. 'What am I supposed to have done _now_?'
Her sidekick turned the blowers down and turned in his seat. 'You kicked in the door. Didn't call for permission. You should know better.'
'You're very welcome, _Sarge_.' Callum cupped his hands over the heater mounted between the seats, trying to get some feeling back in his fingertips. 'If it wasn't for me you'd still be investigating odds and sods – I brought in a murder, OK?'
Mother still hadn't turned around. 'What makes you think it's a murder, Callum? Man falls over in the bath, drowns, happens all the time.'
'And did he accidentally drown in the bath, before or after dragging a big sheet of plasterboard and two tubs of paint in front of the bathroom door?' Callum poked at the heater. 'Can you turn this thing up?'
McAdams fiddled with the dashboard and warmth flowed. 'What about the door-to-doors?'
He produced his notebook. 'Sixty-three flats in the immediate vicinity. Twenty-four of them did nothing but complain about their neighbours, thirty-one wouldn't answer the door or weren't in, and nine want their hats re-tinfoiled. Not one of them had a single thing to say about Glen Carmichael or his mates.' Shrug. 'Well, other than the downstairs neighbour complaining about Led Zeppelin playing on a loop, full blast, for the last two days.'
'Interesting...' Mother tapped her fingers along the wax-paper cup. 'Officially, I should reprimand you for breaking into a crime scene without authorisation, Callum, but our new girl put her hand up to it. Said you were dragged along against your better judgement.'
McAdams snorted. 'I didn't even know you had one.'
'So you, my little man, may have a sweetie.' Mother dug into her pocket and produced a bag of jelly babies. Held them out.
Callum helped himself to a green one. 'Thanks.'
She put the bag away. 'I always love this bit. Forensics are going through the scene, we don't know who the victim is, there's a killer on the loose. Excitement. Adventure. And...' She frowned. 'Can't remember the end of the quote, but you know what I mean.'
McAdams nodded. 'The main plot is unfolding. What we need now is a flashback from the killer's perspective then some sort of investigative montage to show how much research the writer's done.' He clicked his fingers again. 'Constable MacGregor, get yourself and your new best friend DC Franklin back to the lair. I want a murder board ready to go by... I'm in the mood for pizza, so call it an hour and a half. And get a lookout request on the go for Glen Carmichael and his two mates while you're at it. Most people stick to rubber duckies in their bathtub, a dead body requires a bit more explaining.'
Ah. 'Sarge, I was kinda hoping to go home and—'
'Oooh.' Mother made a sooking noise. 'And you were doing so well, Callum. I even gave you a jelly baby.'
'Time to be a team player, Detective Constable.'
His shoulders slumped. 'Yes, Boss.'
Yeah, Elaine was going to kill him.
The wet road hissed beneath the pool car's tyres.
Franklin frowned out of the window. 'I thought Division Headquarters was that way?'
' _Technically_ , yes.' Callum took a right at the roundabout, heading back along the boundary between Castleview and The Wynd. 'Just got a quick errand to run first.'
'Oh for God's sake.' She closed her eyes. 'Is this what it's going to be like, Constable? All moaning and "wee errands"?'
'Five, ten minutes tops. I swear.' After all, the traffic wasn't too bad for a Tuesday. 'Someone stole my wallet this morning. A guy might have it at a shop in Kingsmeath.'
A sigh. A shake of the head. 'Thought you were supposed to be a police officer.'
'I was trying to save a little girl's life: that OK with you?' Up and over the Newton Bridge, and back into Blackwall Hill again, with its modern sprawl of cul-de-sacs and middle-class housing estates.
'By losing your wallet?'
Past the lights, the road opened up into dual carriageway, everyone sticking to the outside lane to avoid Oldcastle City Council's award-winning collection of potholes. 'I didn't lose it, it was stolen.'
'This isn't helping us put a murder board together.'
'We'll be fine.'
'They're only going for pizza, we—'
'I've done _loads_ of murder boards: it'll be fine. Trust me.'
She pursed her lips. 'And why on earth would I do that?'
Fair point.
Montgomery Park drifted by on the right-hand side, a bunch of big white marquees with tartan stripes already sprouting on the grass around the boating lake.
'OK. Full one hundred percent honesty time: the reason everyone hates me, is they think Big Johnny Simpson bribed me to sod-up a crime scene so he'd get off. But I _didn't_. Not a penny. Ever.'
She frowned at him. 'Is that supposed to make me feel better? That you're incompetent instead of corrupt?'
'I'm not incompetent!'
'Could have fooled me.'
'Fine. I was trying to share, but why don't you just sit there in sulky silence. See if I care.' He clicked on the radio. Let it drown out her pouting.
_'... headline the main stage on Saturday, of course, it's Oldcastle's very own Donny "Sick Dawg" McRoberts! Donny, my man, good to have you in.'_
A fake London patois burst out of the speaker, not quite good enough to conceal the Kingsmeath burr underneath. _'Yah, it's_ Sick Dawg _, right? Donny's what me foster mum called us, and you ain't my mum, bro.'_
_'Ha, ha. Right. Yeah, I got you, man. Respect. "Sick Dawg" it is...'_
The massive Blackburgh Roundabout loomed before them. Burgh Library sat on a hill in the middle, all lit up like a 1960s idea of a spaceship – glass and concrete, curving walls and wonky rooflines. The Kingsmeath side of the roundabout was ringed by seven massive tower blocks, eighteen-storey headstones soaring above a scrubby patch of woodland. More _1984_ than _Star Trek_.
_'So, "Sick Dawg", welcome to_ Deathbed Discs _on Castlewave FM, where we find out what tracks you'd take with you to the grave. And you're kicking us off with "Stan" from Eminem's fourth album,_ The Marshall _—'_
_'Yah, I been thinking about it, right? And I'm-a not about that no more.'_
Callum swung the pool car around the outside lane, then took the first turning into Kingsmeath.
It was as if someone had turned down the lights, leaving the buildings in gloom. Rows and rows of council houses. Tenements. Grey faces and grey buildings.
_'You're not?'_
_'Nah, man. I go to my grave I'm not gonna be surrounded by stuff from the oldtimers, you know what I'm sayin'? Nah: I'm-a play my_ own _stuff, bro. You know, from the heart.'_
_'OK...'_
An old couple stood on the pavement, screaming at each other, a wee dog cowering on its lead as they yelled.
_'Well, why don't we just play the song anyway. It'll give us time to completely abandon all the music your publicist_ told _us you wanted to talk about and reprogramme the whole show...'_
Fake rain clattered out of the speakers, followed by Dido singing over a heavy bassline.
Franklin made a little growling noise then jabbed her hand out and turned the radio off. 'Bloody rap music.'
After that she kept her mouth firmly shut all the way through the bleak housing estates, past a dilapidated playing park – the swings and roundabouts reduced to slumped blobs of fire-blackened plastic – past Douglas on the Mound with its scaffolding-shrouded spire and vandalised graveyard...
It wasn't until Callum pulled into a potholed car park that she opened it again. 'Is this it?'
The car park was bordered on three sides by what were probably billed as 'single-storey retail units with excellent potential!' but looked more like something off the news when a riot's just passed through. Three of the eight were boarded up; all were covered in a tattoo of graffiti; all had the kind of metal grilles over the window that were meant to roll up out of the way, but probably spent all their time firmly locked in the down position. A newsagents, a chip shop, a convenience store that looked about as welcoming as a shallow grave, a charity shop, and right at the far end: Little Mike's Pawnshop. The sign above the frontage boasted, 'WE BUY AND SELL ALL MANNER OF THINGS!' 'CASH FOR GOLD!' 'PAYDAY LOANS AT EXCELLENT RATES!!!' 'EST. 1995!'
Callum parked in front of it. 'Won't be long.'
'Oh for... You're here to redeem some manky family _heirloom_?'
'Five minutes. Promise.' He climbed out into the rain. Ducked his head and hurried inside.
The door made an electronic _bleep-blonk_ noise as it swung closed behind him. Shelves lined the walls, packed with other people's things. Free-standing display units turned the shop into a labyrinth. Old video game consoles, a collection of musical instruments, microwaves, hairdryers, boxed cutlery, vases, what looked like a brass urn with 'IN MEMORY OF AGNES MAY ~ BELOVED MOTHER' engraved on it. All of it marinating in the gritty stench of dust and mildew.
Callum picked his way through the maze to the counter, where a wee fat man was bent over a copy of the _Castle News and Post_. His white shirt was just a bit too big for him, the collar and cuffs stained and frayed. A maroon waistcoat with buttons missing and brown stains down the front. Bald head glinting in the shop's dim lighting.
'You Little Mike?'
The man behind the counter looked up, squinted, then pulled on a pair of small round glasses. 'I am indeed, young sir, welcome to my emporium of delight.' He swept a chubby hand from left to right, indicating his second-hand wares. 'How may we assist you this drizzly September evening?'
The door made its _bleep-blonk_ noise again and Franklin appeared, as if by magic. 'Are you not finished yet?'
'Ah, I see.' Little Mike smiled like an indulgent parent. Then he folded his paper and moved it off to one side, revealing the glass countertop. A collection of rings and watches sparkled against dusty purple velvet. 'An engagement ring for the lady, perhaps?'
Franklin stiffened. ' _What?_ '
'Definitely not!' Warmth bloomed in Callum's ears. 'Someone tried to use my credit and debit cards in here today. You destroyed them.'
He sighed. 'A shame. You make such a lovely couple.' A finger poked the glass. 'Are you sure I can't tempt you?'
'Did they leave my wallet behind?'
'Or, how about this?' He grabbed something from beneath the counter and stuck it on his head then went back and fiddled a clip-on bow tie into place. 'See? It's a fez and bow tie. You can dress up like Doctor Who, for parties. Isn't that fun?'
'Have – you – got – my – wallet?'
'No? Ah well.' He covered the glass top with his newspaper again. 'The young lady and gentleman concerned _did_ have a wallet with them. A rather tatty affair, with the lining hanging out.'
Oh thank God. 'That's it! That's the one.'
'I see... Well, perhaps I can help.' He disappeared through a door in the back.
Franklin picked the urn from its shelf. 'Who pawns their mother's ashes?'
'Here we are.' Little Mike was back, holding a shoebox. He set it down on the countertop and pulled out a couple of wallets. 'Real leather, look at that stitching, have you ever seen anything so magnificent?'
'What? No. I don't want another wallet, I want the one those little sods stole from me!'
A pained smile. 'I'm sorry, the young lady and gentleman only handed over the cards, not the wallet. But I can do you a _very_ good deal on a new one if—' His eyes went wide behind the little round glasses and he bustled out into the shop. 'If I may?' He held his hands out in front of Franklin.
She gave him the urn.
'Thank you. Mr May would be most distressed if I allowed his mother to leave the shop without him.' Little Mike polished a speck of dust from the urn with a hanky, then returned it to its shelf. 'Now, is there anything else I can interest you in, while you're here? An electric guitar, perhaps? Or how about the sensual delight that comes with an electric foot spa?'
Callum held out his hand. 'Where are the bits of credit card?'
'Ah, of course. You wish to make sure I haven't indulged in anything illicit. Quite proper.' He pulled out a carrier bag and tipped the contents of his wastepaper basket into it. 'Don't worry: as it's loose items, I don't have to charge you for the bag. Now, if I can't tempt you with my esoteric pre-loved wares, I think I might close up for the night. So, if you don't mind...?' He swept a hand towards the door.
They shuffled through the maze to the exit.
Callum stopped with one hand on the handle. Frowned back into the shop. 'The building society said they were trying to redeem something when you cut up the cards.'
'That is correct, yes.'
'What?'
One of Little Mike's eyebrows made a break for freedom. 'Ah... I'm afraid I can't—'
'If you're about to invoke pawnbroker-client confidentiality, don't bother. What did they try to redeem?'
'Very well.' He shook his head, then turned and led them back through the stacks and display cases to a collection of brightly coloured plastic. 'Items F-twenty-three to F-forty-six.'
There was a sandpit, a collection of squeaky toys that looked as if they belonged in a bath, a Wendy house, a kid's tricycle far too small for either of the little monsters to ride. An off-grey teddy bear with only one ear, scuffed button eyes, and stuffing poking out of his side. There were other bits and pieces, but nothing suitable for anyone over the age of three.
Franklin gave Little Mike one of her finest scowls. 'You _pawn_ wee kids' toys?'
He sighed. 'Some people, this is all they have. If they can't pay their bills, their rent, if they can't buy food for their children, what do they do? You want them to go to loan sharks?'
'They're kids' _toys_.'
'I know. But what can _I_ do, turn them away hungry? Let them get thrown out on the street? So I pawn their children's toys, and I know they'll never come back and redeem them, and I know they're worthless, but I do what I can.' He took off his glasses and polished them on the frayed edge of his shirt. 'This is what real life looks like from down here at the bottom, officers. Foodbanks and pawnshops. Who else is going to help these people?'
Callum frowned down at the collection of plastic tat.
A hand on his arm. 'Come on, we need to get that murder board done.'
He puffed out his cheeks. 'How much to redeem the toys? And I'll need their address.'
##
Callum stuffed the multicoloured rocking-horse-shaped-like-a-fish thing in the boot with all the other toys. Closed the lid. Turned and leaned back against the car.
Little Mike rattled down the grille over his pawnshop's front door. Wrestled a thick padlock into position. Then turned and lumbered away into the evening.
A shaft of sunlight broke through the heavy cloud, the low beam of golden light pulling a rainbow from the drizzle. Making the graffiti-wreathed shopping centre shine.
The car's horn blared.
Right.
Callum peered in through the rear window and there was Franklin peering back at him, reaching over from the passenger seat to lean on the horn again.
Mouthing the words, 'Hurry up!'
Funny how some people could start off looking extremely pretty, only to get less and less attractive the more time you had to spend with them. At this rate, by the end of the week, Detective Constable Franklin was going to resemble the underside of Quasimodo's armpit.
He sighed and climbed in behind the wheel. Cranked the engine. 'We've got plenty of time.'
She checked her watch. 'DS McAdams said an hour and a half, thirty minutes ago. We're, what, twenty minutes from DHQ. That leaves—'
' _Plenty_ of time.' He navigated his way through the potholes and back onto the road. 'Just got a little stop to make on the way.'
'God's sake!'
'It's on the way. Won't take five, ten minutes tops.'
'Gah!' She swivelled in her seat to give him the full-on glower. 'I've just started with this team and I am _not_ going to let you screw it up for me.'
'Seriously?' Left at the junction, onto McGilvray Place with its boarded-up terrace and abandoned building site – just foundations and pipes sticking out of the ground to mark the death throes of the local construction industry. 'What happened to, "I'm not wasting my career with you losers"? Thought you wanted nothing to do with us.'
'Let's get something straight, Constable, I'm out of here first chance I get. But until then, I'm going to do the job. _Properly_. Not whatever half-arsed version of it you think you can get away with.'
'It'll take _five_ minutes.' A right, onto Munro Place, taking the car up the hill. 'Then we'll hit Division HQ and _I'll_ do the murder board, OK? And you can feel free to clype on me anytime you like.' After all, it wasn't as if Mother or McAdams could hate him more than they already did.
He slowed for a moment next to the rusty Volkswagen, where Dugdale had deployed The Claw, then over the crest of the hill and down the other side.
Left at the bottom.
Callum checked the slip of paper with 'LITTLE MIKE'S PAWNSHOP ~ PRE-LOVED GOODS & PERSONAL FINANCE SOLUTIONS' in flowery script along the top and, 'BROWN : 45B MANSON AVE.' scrawled beneath it in biro.
Number 45 was on the outside edge of a set of five identical squashed grey council-issue boxes. Each one semidetached, split down the middle – A on the left, B on the right – ten homes per block. Someone probably thought arranging them into wee groups like that would foster a sense of community pride and spirit. It hadn't. A ruptured sofa sat outside the house next door. The one beyond that had a washing machine as a garden ornament, the porthole door open to show a collection of crumpled lager tins. Knee-high weeds from the front door to the garden wall.
Callum parked out front. Hauled on the handbrake. 'Five minutes. You can use the time to compose your formal complaint about me.'
She just scowled at him.
He slipped out of the car, turned and stuck his head in again. 'One of these days, the wind's going to change.' Then clunked the door shut and marched off before she could say anything back.
The garden gate was rusted solid, so he hopped over it onto a path of cracked paving slabs with grass growing in off-green Mohicans between them.
No doorbell.
He gave the chipped wood three loud hard knocks.
The light was on in the living room, shining through a pair of lace curtains. Shadows moved about inside.
Another three knocks.
And a voice came from the other side of the door. Young, female. _'Go away.'_
'Mrs Brown?'
_'If you're from the bailiffs, you can sod off. I don't have to open the door!'_
'It's not the bailiffs, it's the police.' He held his warrant card up to the spyhole. 'See?'
A groan. Then something thunked against the door at head height. _'He doesn't live here, OK? I kicked him out_ six weeks _ago.'_
Callum put his warrant card away. 'Who doesn't live here?'
Franklin was checking her watch, making a big pantomime of pointing at the thing and then pointing at him.
_'Go away.'_
'I've got some stuff for you, OK?'
_'I'm not in.'_
Why bother?
Callum marched back to the car, popped open the boot and hauled out an armful of kid's plastic toys. Dumped them just over the garden wall and went back for another load. Adding to the pile until the boot was empty.
The last thing was the raggedy teddy bear, with its missing ear and herniated stuffing. Plastic tat was one thing, a well-loved teddy bear was another. No way it was getting dumped in the weed-ridden grass.
He returned to the front door. Knocked. Held Teddy up to the spyhole.
Some muttered conversation inside, then the door opened a crack, the chain glinting in the hall light. A thin face peered out at him, blonde hair pulled back tight. She didn't look old enough to leave school, let alone have two small kids. There was a huge bruise on her cheek, dark and angry against the pale skin. She blinked at the bear. 'Mr Lumpylump?'
She shifted, and there was child number three – a baby cradled in her arms, wrapped in a tatty Power Rangers blanket. Face a rounded pink blob, making snuffling noises.
A small child wailed somewhere behind her, sounding as if someone was removing its fingers with a blowtorch. Child number four.
The woman didn't even flinch. 'Shut up, Pinky.'
'I redeemed the rest of the kid's toys. They're in the garden.'
Her hand reached through the gap between the door and the frame, fingers trembling. 'Can I have him. Please?' She licked her lips.
'Look, all I want is my wallet back, OK? There's no money in it anyway, it's just a tatty old wallet that's falling apart. Like the bear.' He gave Mr Lumpylump a wee shoogle, making him dance. 'It's important to me.'
She blinked up at him. 'I don't have it. I don't have any wallet.'
'You could check, though? Ask your children?'
Behind her, the toddler wailed some more, as whoever it was turned the blowtorch on their toes.
'They're not here.' She reached out until the frame and door dug into her arm. Straining for the manky teddy bear. 'Please...?'
What was he going to do, hold a kid's teddy to ransom?
Callum passed her the bear and she snatched it from him, yanking it back inside the house and slamming the door.
He knocked again. 'Hello?' Rested his forehead against the door. 'Hello?'
Silence. Not even the wailing.
Great.
What was the point of trying to help people? Why did everyone have to be so... so _selfish_. And nasty. And horrible?
One last try.
He pulled an official Police Scotland business card from his pocket wrote, 'IF YOU FIND MY WALLET, PLEASE LET ME KNOW' on the back, and slipped it through the letterbox.
Probably be sod-all use, but what other option did he have?
Callum trudged back along the path. Clambered over the rusted gate.
'Hoy, mister?' A young girl's voice, hard with defiance and a broad Oldcastle accent.
He turned.
The little monster from this morning. The one who'd swigged cider from a can. The one Dugdale had used as a human shield. The rotten wee sod who'd stolen his wallet.
She'd ditched the baseball cap and tracksuit top for a T-shirt with a vampire Womble on it, but not the attitude. 'What you doing here, Piggy?'
He nodded at the pile of plastic things.
Her eyes widened. 'Whoa! You got Pinky's toys back?' Then her internal coolometer must have kicked in, her grin turned into a bored expression and a shrug. 'Yeah, so?'
'Swap you for my wallet.'
'Ain't got no wallet, do I? Chucked it.'
His whole face crumpled. 'Oh for...' What was the point? Of course she chucked it, with the credit cards cut up, why would she hold onto it? Wasn't as if there was any cash in there. His shoulders drooped. 'Sodding hell.'
'Don't know what you're greetin' about. Just a crappy old wallet, innit?'
'It was my father's. Only thing I've got of his.'
'Yeah?' She spat into the weeds. 'Well, _my_ dad broke my arm then ran off with one of mum's friends.'
' _Mine_ disappeared when I was five.'
'I was _four_.' Always had to have the last word, didn't she? A competition for who had the crappiest childhood.
'Well _I_ grew up in a care home. Beat that.'
Aha, she couldn't, could she. At least _she_ had a mother. Though going by the bruised face, her mum's taste in men hadn't improved any.
He narrowed his eyes. 'It's Willow, isn't it?' At least, that was what her wee brother had called her when she was kicking three shades out of Dugdale's head. 'Any idea who's been hitting your mum?'
Willow's back stiffened. 'I ain't no snitch, Piggy.'
'Course not.' He produced another business card, stuck his mobile number on the back, and laid it on top of the wall. 'But if you're worried about her or anything...' A shrug. 'You know.'
The lace curtains twitched open, and there was Willow's mum, standing with a toddler on one hip. She had the tatty old teddy bear clutched to her chest like a bible.
Not the kid's bear, hers. Pawned to pay for food, or rent.
How depressing was that?
Callum climbed in behind the wheel. Frowned. Shook his head. Then started the car.
Franklin stared at him. 'Well?'
'No idea.' He pulled away from the kerb, keeping one eye on the rear-view mirror.
The little girl stood and watched them all the way to the corner, then disappeared from view.
'This was all for your stupid wallet, wasn't it?'
He pulled out his Airwave, poking at the buttons with one hand as they navigated their way back towards the real world. 'Control? Can you do a PNC on a Ms Brown, forty-five B Manson Avenue, Kingsmeath? See if anyone's been bothering her.'
_'Aye, will do. Hang on.'_
'Thanks.' He stuck the handset on the dashboard, took them out past a dilapidated community centre – doors and windows boarded up with damp-swollen chipboard – and onto Montrose Road. Pottering along behind a Fiat Punto barely doing twenty miles an hour.
'For God's sake, at least put the blues-and-twos on.' Franklin reached for the button mounted on the dashboard, marked, '999'.
Callum slapped her hand away. 'Are you off your head?'
'We're going to be late!'
'You press that button and the dashboard camera comes on.' He pointed at the little rectangle of plastic mounted against the windscreen, hidden by the rear-view mirror. 'And the GPS starts recording. And it all gets stored for the courts, or in case there's an accident while you're wheeching through traffic. Lights and sirens are for emergencies _only_ , not because you're in a hurry.'
She curled her hand against her chest, as if he'd stabbed it with a fork and scowled at him. 'Where is it then? This magical wallet?'
A stone settled in his stomach, cold and heavy. 'They threw it away.'
'Waste of sodding time.' She checked her watch again. 'Thirty-six minutes to get back to Division Headquarters and make up a murder board.'
'Will you stop moaning on about—'
_'DC MacGregor from Control, safe to talk?'_
He picked up the handset and pressed the button. 'Go ahead.'
_'Aye, right: your woman's a Miss Irene Brown, twenty-three years old. Done for possession four years ago, got off with a caution... Hmm... Looks like that's the last known address for one Jeremy Barron, Jezza to his mates, AKA: Jerome Barton, James Broughton, and Jimmy Bishop. Bit of a scummer from the look of it. Assault, robbery, assault, aggravated assault, possession with intent, serious assault, two counts of sodding about in public with a knife.'_ A clicking keyboard rattled out of the speaker. _'Looks like she's got a bit of a history with violent scumbags. Poor woman couldn't pick a nice bloke out of an empty room if you Sellotaped a balloon to his forehead.'_
Twenty-three years old, with four kids.
And a dirty big bruise on her face.
No wonder she clung onto her teddy bear like that.
Her daughter, the horrible Willow, had to be at least seven years old, so that meant Miss Irene Brown must have been about sixteen when she'd had her.
What a life: trapped beneath a landslide of pregnancy and violence.
Callum tapped his fingers on the handset's plastic case. 'Do me a favour: put a grade one flag on the house, OK? Just in case this Jerome Barton comes back again.'
_'Pfff, can't promise anything, but I'll see what I can do.'_
'Thanks.' Callum slipped his Airwave back in his jacket. Took a left at the roundabout and onto the Calderwell Bridge.
Halfway across the river, Franklin sighed. 'OK, _now_ can we go do this sodding murder board?'
'And _that_ , is that.' Callum pinned the last photo to the corkboard and stepped back, hands on his hips.
Not a bad job, even if he said so himself.
The murder board took up a whole wall of the Divisional Investigative Support Team office. One whiteboard cut up into sections with that thin magnetic tape stuff, all headings spelled correctly, details on the corkboards to either side of Glen Carmichael and his fellow graduate property developers. Ben Harrington with his massive moustache, Brett Millar and his Clangers tattoo. Photos, potted bios, previous brushes with the law, list of known friends and associates. Schedule for the flat from the auctioneer's website along with PNC details for the previous owner.
He checked his watch. 'Done with five minutes to spare.'
Franklin stayed where she was, perched on the edge of her brand-new desk. 'Is that _it_?' A sniff. 'I always thought a murder board would be more... I don't know. Like on the TV.'
'TV people wouldn't know a murder board from a Christmas list.'
The door banged open and in stormed Watt, floppy fringe plastered to his forehead, mouth scrunched up into a twisted pouting sneer, wee pubey beard bristling as he hurled his soggy jacket into the corner. He graced Callum with a glare, then shifted it over to Franklin. 'Who's this?'
She stiffened her back. Drew herself up to full height.
But the door thumped open again before she could lay into him and Dotty wheeled herself into the office. 'Oh don't be such a _princess_ , John. I said I was sorry.'
Might as well do the introductions.
Callum hooked a thumb at Franklin. 'Watt, Dotty, this is our new recruit: Detective Constable Franklin, from E Division. Punched a superintendent, right in the car park.'
Watt wiped his hands down his face and flicked the drips at Dotty. 'I'm bloody _drenched_!'
'It was an _accident_.'
'No it sodding wasn't! You aimed for that puddle on purpose.'
'Franklin: the soggy tit with the beard is Detective Constable Watt. He clyped on his last team at G Division, so the high heedjins had him transferred to Oldcastle. And _we_ are graced with his presence, because none of the other teams will work with the grumpy little git.'
'I didn't know you were standing there.'
'This is because I wouldn't get you chocolate, isn't it?' Watt grabbed his mug from his desk. 'Get your own damn chocolate!'
'The young lady in the wheelchair is Detective Sergeant Dorothy Hodgkin. _She's_ here because some wee radge fancied a high-speed pursuit in a stolen Beamer. Dotty lost her leg above the knee in the crash. Her wheelchair's called "Keith": don't ask.'
'I _will_.' Dotty bared her teeth at Watt. 'And you know what? I _was_ sorry, but I'm not now. You're a sour-faced, childish, chippy, miserable scumbag, John. No wonder nobody likes you.'
Callum shrugged. 'As you can see, we're all one big happy family.'
'Oh, ha-ha.' Watt turned his scowl back on Callum. 'I bet he's not told you why _he's_ here, Franklin, has he? He—'
'Everyone thinks he took a bribe to cock-up a crime scene. I know.' Franklin folded her arms. 'So is everyone on this team a reject? What about McAdams and Malcolmson?'
Dotty wriggled her way out of her jacket. 'DS McAdams has terminal bowel cancer. They _so_ want to send him off on the sick, but he won't go. And DI Malcolmson is just recovering from a massive heart attack.' Dotty held her arms up, flashing victory signs like Richard Nixon. 'Welcome to the Misfit Mob! Abandon hope, all ye who enter here.' She wheeled herself across the manky carpet tiles to Franklin and stuck out a hand with a fingerless leather glove on it. 'Dorothy. Dot or Dotty to my friends.'
After a wee pause, Franklin shook it. 'Rosalind.'
'Rose for short?'
'No.'
'Oh...' Dotty wheeled herself back to her desk. 'Ah well.'
Callum swept his hand around the room. 'And that's us. All the other departments think we're useless, the bosses give us boring or horrible cases, and this is the first exciting enquiry we've had since, well, _ever_. But if you—'
'Knockity, knock.' The door swung open and in waltzed McAdams, a stack of four pizza boxes balanced in one hand. 'Behold, little ones, Mother and I have returned. Lo, I bring _succour_.' A grin. 'Well, one ham-and-pineapple, one meat feast, a four seasons, and a pepperoni, but it's the thought that counts.' He dumped the boxes on the nearest desk. 'I trust you've all been beavering away, advancing the plot and revealing character through action rather than exposition...' A frown. 'Constable MacGregor, why are you still here? Go home.'
Callum pointed at the whiteboard with all its lines and data. 'But you said—'
'Detective Constable Franklin!' McAdams patted her on the back. 'Excellent job on the murder board. Very thorough.'
Her cheeks darkened slightly. 'But I didn't—'
'Nonsense. Credit where it's due.' He picked a sheet of paper from the nearest desk, crumpled it up and hurled it at Watt.
It bounced off his floppy fringe. 'Hoy!'
'What did I tell you about signing off at the end of a shift? I just checked the logs and _apparently_ you're still on duty from yesterday.'
Watt cleared his throat. 'I was busy.'
'I don't care if you're King Busy, ruler of all the Busy Bee people in Busy Buzzy Bee Land: _sign out_! I'm not authorising any overtime till you get that through your pointy wee head.'
'But. Sarge—'
'No.' McAdams glanced at Callum. 'Thought I told you to go home, Constable. You've got a full day tomorrow: all those museums to phone.'
'Oh you are _kidding_ me! I was the one who—'
'To each man his task, according to his merits. Some more than others.' A wink. ' _You_ , for example, can leave the murder investigation to the professionals.'
Callum bit his bottom lip. Arms trembling. Hands curled into fists.
' _Good night_ , Constable.'
He took a step forward.
McAdams grinned.
And there it was: he _wanted_ a punch on the nose. With Franklin, Watt, and Dotty as witnesses, McAdams could go to Professional Standards and get him suspended at the very least. It wouldn't look very good at his review tomorrow either.
Deep breath. Callum forced his hands to open. 'Fine.' Grabbed his coat. 'But I'm taking one of these with me.' He helped himself to a pizza box, warm against his fingertips, and marched out of the door.
'Elaine? Hello?' Callum balanced the pizza in one hand, propped his bike against the wall, and clunked the front door shut. Slipped out of his soggy jacket and kicked off his wet shoes. Left soggy-sock footprints on the laminate flooring through into the kitchen. 'God what a day. Utterly soaked.'
The sounds of some sort of cookery programme oozed out through the closed living room door.
At least the backpack was waterproof. Callum unloaded it onto the kitchen table, raised his voice so she'd hear him in the lounge. 'DID YOU HEAR? THEY SAY IT'S GOING TO BE THE WETTEST SEPTEMBER ON RECORD.'
No reply.
'ELAINE?'
Nothing.
He stuck the Tupperware box for his sandwiches in the sink. Took today's note and put it up on the fridge with all the others she'd sneaked in with his lunches over the last month – little inspirational quotes, terrible puns, and the occasional dirty joke. Most came with a drawing. Today's was a rotund badger with teeny legs, taking a bite out of a pig, above the legend, 'I LOVE YOU MORE THAN DESMOND THE BADGER LOVES BACON'. Which was nice to know.
Callum flicked through _The Monsters Who Came for Dinner_ , smiling at the old familiar illustrations.
Come on: there'd be plenty of time to read it after dinner.
He emptied his pockets, stripped to his pants, and threw his fighting suit in the washing machine. Set it to tumble dry.
Stuck his head back into the hall. 'YOU WANT TEA?'
Nope. Whatever she was watching, it had her.
Callum stuck the kettle on and the oven too. Wandered through to the lounge.
Some posh English bloke with curly hair and big nostrils filled the TV screen – wandering through a forest somewhere, banging on about how tasty squirrels were if you cooked them in a nice ragout.
Elaine was curled up on the sofa with her back to the door, wearing her comfies, a tartan fleecy blanket pulled over her enormous pregnant bulge. A bowl rested in her lap, containing a mixture of marshmallows and crisps.
It wasn't a big living room: barely enough space to take a three-seater sofa and an armchair; a fake coal fire that groaned and flickered; a coffee table with a collection of wooden ornaments on it; a TV, complete with squirrel-mongering celebrity chef; and four floor-to-ceiling bookshelves stuffed to overflowing with novels.
Their blinds were open, the darkness on the other side turning the window into a mirror – reflecting back one thin pasty body in blue underpants. The lights in the houses opposite twinkled through Callum, making him sparkle like the world's least scary vampire. Then the eight o'clock train to Edinburgh rumbled past, its glowing windows making rectangular spotlights sweep across the back garden. Searching.
He crossed the room and closed the blinds, before anyone on board became overwhelmed with desire at the sight of his ancient Marks and Spencer's lingerie going a bit baggy in the elastic. 'I got pizza for tea. Well, technically I _stole_ pizza, and I know it's not Nutella and pickles, but—'
A grunt rattled its way free and Elaine sat up. 'What? M'wake!' She blinked at the room. Then the TV. Then Callum. Brushed the long brown hair from her eyes. 'What time is it?' Cracked a huge yawn, showing off a proper Scottish set of fillings. 'Why are you in your pants?' The corners of her eyes wrinkled. 'What happened to your _face_?'
'It's just gone eight.'
'You look like someone ran over it with a washing machine.'
'I've got pizza.'
'Gah...' Another yawn. Then she held out her arms. 'I had a horrible dream. You abandoned me and Peanut because we got ugly and you didn't love us any more.'
'You're not ugly.' He hugged her and planted a soft kiss on her forehead. 'You're beautiful. You smell of cheese-and-onion, but other than that, you're safe.'
Callum picked one of Elaine's discarded mushrooms and put it on his own slice, adding to the pepperoni. Sat back on the couch and stuffed in another mouthful, trying not to get any on his tartan T-shirt and joggy bottoms.
'Urgh...' She grimaced at him. 'You eat like a wheelie bin.'
'Yronlygelous.' The words all mushed up as he chewed.
Sitting on the bookshelf, the flat's phone launched into a tinny rendition of the _South Bank Show_ theme tune.
Elaine curled her top lip. 'Sod off.' She pointed at the plate resting on top of her bulge like a makeshift tabletop. 'We're eating!'
'If it's your mum, I'm telling her we're not in.'
'Let it go to voicemail. They—'
'Can't. What if it's important?' He stuck his plate back on the coffee table and hauled himself out of the couch, walked round the back to the bookcases. Sooked his fingers clean and picked up the phone. 'Hello?'
Silence.
'Hello?'
Still nothing.
He checked the caller display: 'NUMBER WITHHELD'.
'OK, I'm—'
_Click._
Elaine turned and looked over the back of the couch. 'Who is it?'
'No idea, they hung up.' He put the phone back in the cradle. 'Probably some auto-dialling PPI tossers.'
Probably.
'Callum, while you're up?'
'Mmm?' He turned away from the phone.
'Any chance you can grab the raspberry jam from the kitchen? I think it'll go great with these anchovies.'
He tried not to shudder, he really did...
## [— every day we live —
is a day closer to the day we die](../Text/contents.html#part003)
Sometimes, the worst thing you can imagine – and I mean the worst thing you can _possibly_ think of – that's just the start. Because things can always get worse, dear reader. And in my experience they usually do...
R.M. Travis
_The Monsters Who Came for Dinner_ (1999)
_Damn right you better fear me, cos I'm about to break free,_
_You better f*ckin' hear me, there won't be no all-clear: see?_
_I'm-a sharp like a shark, ma bite's worse than my bark,_
_I attack from the dark, cos violence is ma trademark,_
_Think that you're tough? You ain't even in the ballpark..._
Donny '$ick Dawg' McRoberts
'Unrequited Love Song Number 3'
© Bob's Speed Trap Records (2015)
##
'... another six arrests in the Holyrood sex-ring investigation. Weather now, and there's more rain on the way, sorry, but it should clear up by the weekend for our very own Tartantula Music Festival in Montgomery Park! Fingers crossed. And if you haven't got your tickets yet, stick around – I've got just the competition for you.'
Callum marched back into the bedroom, scrubbing his hair dry on a pink towel.
_'This is_ The Very Early in the Morning Show _and you're listening to me, Jane Forbes, on Castlewave FM, because you're sexy, intelligent, and looking_ fabulous _today!'_
He grimaced at the naked creature in the mirror, then hauled on a pair of pants and yesterday's suit trousers. Maybe not so fabulous. Especially now the bruises Dugdale gifted him had darkened to a deep lustrous purple, ringed with blues and greens. Lucky he hadn't cracked a rib.
_'Right, we've got_ Sensational Steve's Breakfast Drive-Time Bonanza _coming up in thirty minutes, but that gives us loads of time for yet more stonking tunes!'_
Elaine peered out from under the duvet. 'Tmmsit?'
'Half six. Go back to sleep.' A clean white shirt and red clip-on tie.
'No, m'up. M'up.' She let loose a massive yawn. Sat up and had a scratch, long brown hair all flattened on one side.
_'Let's kick off with a Tartantula festival favourite: Nearly Blind Vera, and their new single "Swarm".'_ What sounded like a full orchestra belted out of the speakers, swelling to a—
Elaine thumped her palm down on the clock radio and swivelled her legs out of bed. Shuffled out of the room in pink bunny slippers, rubbing at the small of her back. 'Pfff...'
He pulled on clean socks and dry shoes, dragged a comb through his hair. Scowled at the purple stains on his forehead and chin. Wasn't exactly the best impression to make at a Professional Standards review, but what choice did he have?
Callum knelt by the side of the bed and dragged out a big file box. Rummaged inside for the maroon half-size ring-binders buried under the flat's insurance schedule, the mortgage documents, and the HP agreement for the telly.
Elaine's voice belted out from the kitchen. 'Did you stay up half the night reading again?'
'Maybe.' He tucked the binders into a small backpack, plucked the copy of _The Monsters Who Came for Dinner_ from the bedside cabinet, and wandered through. 'You don't have to do that.'
'Yes I do.' She lumped a couple of slices of white from the bread bin onto the chopping board and slathered them with spread. 'You want cheese-and-pickle, or egg?'
'Go back to bed, it's fine.'
'Just because I'm stuck here with Peanut, doesn't mean I'm useless.'
Callum stepped behind her and kissed the back of her neck. 'No one thinks you're useless.'
'You'll have to have cheese-and-pickle, we're out of eggs.'
The flat's phone launched into its semi-classical theme tune again.
She froze.
'It's OK, I'll get it.' He marched through to the lounge. Grabbed up the phone. 'Hello?'
Nothing.
Checked the caller display. Same as last night. 'NUMBER WITHHELD'.
'Who is this?'
Silence.
_Click_.
Yeah, that was getting old very quickly.
He turned, and there was Elaine, holding out a little Tupperware box in one hand and a banana in the other. 'Who was it?'
'Automated-dialling PPI nonsense again.'
'There's a mini Mars Bar in there too. You know.' She lowered her eyes. 'To keep your strength up.'
He tucked the box and banana into his backpack. 'It's just a boring wee meeting with Professional Standards, it'll be fine. Promise.' That sounded confident, didn't it? Completely unlike the lie it was. He replaced _The Monsters Who Came for Dinner_ in its bookshelf slot, grabbed a tatty paperback at random: _The Beginner's Guide to Shoplifting_ , and added it to the pack.
'Callum...' She put a hand against his chest.
'What can they do? They've got no evidence – they can't, because I didn't _do_ anything, did I?'
She gave him a little pained smile. 'We love you.'
'I know.' A kiss on the cheek. 'Got to go, don't want to be late for the rubber heelers.'
Callum shifted in his seat.
The waiting room was... disturbingly neutral. Blue carpet, magnolia walls, a row of four soft-ish chairs along one wall, a sideboard-sized filing unit thing on the other – complete with the obligatory pile of well-thumbed, ancient magazines. A water dispenser in the corner. A framed painting of Oldcastle's skyline rendered in all manner of bright and unnatural colours.
He checked his phone – 07:13.
Oldest interview technique in the business – leave your victim to stew for a while. Let them work themselves into a state of nervous exhaustion worrying about what you knew.
Well, tough: they knew sod-all. Because there was sod-all to know.
The only thing up-to-date on the sideboard was a copy of that morning's _Castle News and Post_ , the banner headline: 'BODY FOUND IN CASTLEVIEW FLAT' above a photo of the craphole Glen Carmichael and his mates were doing up. There was an inset pic of three figures standing outside the main entrance while SOC Smurfs shuffled past in the background. McAdams, Franklin, and right in the middle – staring straight at the camera – his own face. Looking tired and fed up. So they were right: the camera didn't lie. All three of them got a namecheck, though they'd managed to spell McAdams' name wrong. Which was nice.
Right underneath the main story, was 'DRUG DEN UPSTAIRS MADE LIFE A LIVING HELL', a 'shocking exclusive with Murder Flat's downstairs neighbour!' continued on page six. There was always someone.
Callum dumped the paper and dipped into his rucksack instead, pulling out _The Beginner's Guide to Shoplifting_. Settled back to read the first short story. A bit heavy on the adverbs, but other than that, it was OK.
He was just starting the second one when the door through to the office opened and a middle-aged man in uniform poked his head out. His hair had abandoned its post, retreating to a defensive position around both ears, a set of jowls lightly blued with stubble. A pair of evil-scientist glasses, all narrow with silver frames. He smiled. 'Ah, Callum. Good, good: in you come. Sorry about the wait.' He held the door open and gestured inside.
'No, it's fine.' Callum stood. Stuffed the book in his backpack. 'Gave me a chance to catch up with my reading.'
'Good, good.' He moved aside, then closed the door behind Callum. 'I know we should have done this weeks ago, but you know what it's like. Busy, busy.'
It was a small-ish office, with a desk on one side and a round table in the middle. Some filing cabinets. A coffee machine. A small digital video camera on a tripod.
'Please, please, take a seat. Coffee? I'm having one anyway...?'
'Thanks. Just milk.'
'Perfect.' He wandered over and started pushing buttons and inserting cartridges. 'So, Callum, I understand you're going to be a father in two weeks' time. How exciting. Most fulfilling thing you can do as a man.'
'Well—'
'There you go. One white coffee.' He sank into the chair next to Callum's. 'I can't abide all this "flat white" nonsense, can you? Oh,' he stuck his hand out, 'Chief Inspector Gilmore, we spoke on the phone yesterday, but you can call me Alex.'
OK...
'Chief Inspector.'
'Ah, almost forgot.' He raised himself half out of his seat and pointed a remote control at the camera. A little red light blinked on. 'There we go. Can't do these things without a proper record, can we? The Boss would have my guts for garters. And I understand your good lady is in the job too?'
Callum closed his mouth, then opened it again. 'Well, yes. I mean, she's on maternity leave, but—'
'Let me see now...' He checked a notepad. 'Ah, here we are: Constable Pirie. Elaine. You know, I had an Aunty Elaine when I was wee. Lovely lady, used to give us Advocaat every Christmas because she thought it wasn't alcoholic. And I see she's been seconded to the Scenes Examination Branch?'
What?
Chief Inspector Gilmore held up a hand. 'Sorry, _your_ Elaine, not my aunt. How's she getting on? Weird cravings, I'll bet. My Pauline used to chew the rubber hose from the spin dryer. That dates me, doesn't it? Amazing our sons didn't come out with two heads. How's the coffee?'
Was the man some sort of idiot? How...
Callum sat back in his seat.
No, of course he wasn't. Didn't matter what crime novels and TV dramas said, you didn't get to be a chief inspector without having a considerable amount of grey matter packed between your earholes. The rambling avuncular act was all about putting people at ease and off their game at the same time.
Well that only worked if you didn't know he was doing it.
Callum took a sip. 'It's great. Thanks.'
'Better than the stuff from the canteen anyway. So, Callum: tell me all about Big Johnny Simpson.'
'Well...' He cleared his throat. 'I want to start by saying I've never taken a bribe in my life. Ever.'
'That's the spirit.' Gilmore raised an eyebrow. 'But...?'
'No, no buts.' He picked his rucksack off the floor and upended the contents onto the table. Three burgundy ring-binders, a Tupperware box, and a banana. He retrieved his lunch and pushed the binders towards Gilmore. 'Bank statements. Well, building society statements, but it's the same thing. Feel free – go through them with a nit comb. And if you want to contact the Royal Caledonian, I'll tell them you've got free rein to look at any account I've got.'
'I see. That _is_ awfully kind of you.' Gilmore stacked them into a neat pile on one side. 'But in the meantime,' a smile pulled his jowls up at the edges, 'why don't you tell me all about Big Johnny Simpson?'
'Urgh.' Callum dumped the rucksack on his desk. Collapsed into his seat. Powered up his computer. Grabbed his desk phone and called the control room.
_'Aye, Aye?'_
'Brucie? Any word on my lookout requests?'
_'Hud oan, I'll check...'_
The office was empty, no sign of Dotty or Watt-the-Moaning-Dick. They'd been at the murder board, though: no mistaking Watt's drunken-spider scrawl.
Didn't look as if they'd made a whole load of progress. The column headed 'OPEN TASKS' had gained a bunch of actions allocated to the pair of them, more on the bottom waiting for someone to take them on. Mostly interviewing friends and family of the three amateur property tycoons. Franklin's name appeared on the list only once: 'ATTEND POST MORTAM ~ 10:30'.
God's sake.
_'You still there? Aye: Benjamin Harrington, Brett Millar, and Glen Carmichael – no sightings. You could get yourself a warrant and see if they've used their bank cards?'_
'Thanks, Brucie.' Callum hung up, then hauled himself out of his chair and over to the board. Wiped the word 'MORTAM' out and wrote 'MORTEM' in the gap. Chief Inspector Gilmore might have been putting on an act, but Watt wasn't. He truly _was_ an idiot.
'And what exactly, my dear Constable Callum, are you up to now?'
Wonderful: Haiku Boy.
Callum corrected the spelling of 'INTERVIEW COLLEEGES'. 'I'm fixing the murder board.'
'You keep away from that, young Callum. That's for grown-ups.' McAdams settled on the edge of Dotty's desk. 'While we're at it: what time do you call this? It's ten o'clock. Shift starts at seven a.m., not whenever you feel like it.'
'You know fine well where I was.'
A grin. 'Ah yes, Professional Standards.' He put one hand on his chest. 'They interview cops, who are dirty and bent, / To punish their sins, till they wail and lament, / Then cast them down low, in the dirt at their feet, / And I do hope they fired you, cos that would be sweet.'
'Yeah, go screw yourself, Sarge.' Callum chucked the whiteboard marker back onto Watt's desk, then sank behind his own. 'What happened with Dugdale, he cop to it?'
'That's no longer your concern, Constable.' McAdams checked his watch again. 'When the lovely DC Franklin gets in, you can give her a lift to the overflow mortuary. You're going there anyway.'
Oh _great_.
He sagged back in his seat. 'I am?'
'Of course you are. As a minor character you've been farmed out onto a subplot: discovering which museums have lost their mummies. Mother's even made you _SIO_. Isn't that fun?'
'Gah...' Callum covered his face with both hands. 'I hate you all.'
'And they're post-morteming your first mummy at half ten this morning. Don't be late.'
'No, don't put me on hold, I just need to know if... Hello? Hello?' A pan-pipes version of 'Green Sleeves' rattled out of the phone's earpiece. Wonderful.
Callum printed the letters 'D.I.C.K.' next to the museum's name. Third one in fifteen minutes.
There had to be, what, a dozen active murder investigations in the division right now? And what was he doing? Sodding stolen mummies.
The office door clunked shut.
Probably bloody Andrew McAdams, back for another gloat. Maybe he'd come up with another _hilarious_ poem. Oh ha, ha, ha.
Dick.
Franklin's face appeared over the top of Callum's cubicle wall. 'Where's everyone else?'
He held the handset away from his head and frowned at it. 'Is it just me? Am I hallucinating and this isn't _really_ an actual phone? Is that why I'm the only one who can see it?'
'Somebody's touchy.'
_'Yes, hello?'_ A little voice replaced the pan pipes. _'We've checked and we've never had a human mummy here. We've got a mummified dog and a stuffed polar bear in storage, if that helps?'_
'No. Thanks. You've been a lot of help.' He hung up and stuck two lines through the museum's name. Sat back and massaged his temples.
Franklin sniffed. 'So?'
'So _what_?'
'So where is everyone?'
He pointed at the murder board. 'Off interviewing Glen Carmichael's mates.'
'Ooh, there's stuff on the board.' She disappeared from view. 'Wait a minute, how come _I'm_ down to do the post mortem?'
Callum stood.
She was in front of the murder board, hands on her hips, frown on her face. 'What, I'm stuck in the mortuary with a decomposing corpse while you're all off interviewing people? Thank you very sodding much!'
He pointed at the list of tasks. 'If you didn't want to do it, why put your name down?'
'I _didn't_. None of this was on the board last night.'
Hmm... 'You didn't mark up the actions with Watt and Dotty?'
'No. We ate the pizzas, then Mother told me to head off and not come back in till quarter past ten, as I'd been here till late.'
Lovely. So even though he'd been here three weeks longer than she had, Franklin got to call DI Malcolmson 'Mother' while he had to call her 'Boss'. _And_ she got a lie-in.
Franklin sniffed again. 'What's wrong with _your_ face?'
'Nothing.' He picked his coat off the back of his chair. 'Get your stuff, we're off to the mortuary.'
The pool car slid along Camburn Road, following the edge of the woods. They made a thick blanket of green: leaves and bushes trembling in the rain. There were people in there, on the paths and tracks that wound their way between the trees – walking dogs, wheeling pushchairs, jogging. A wee girl on a bicycle...
Callum slammed on the brakes.
'Aaargh!' Franklin lurched forward against her seatbelt, both hands slapping onto the dashboard – bracing herself. 'What the bloody hell do you think you're—'
'Just be a minute.' He stuck on the hazard lights and scrambled out into the downpour. Flicked his collar up as he jogged between the puddles and in under the canopy of branches. Wiped the rain from his face. 'Willow.'
Her dirty-blue anorak was frayed at the cuffs and shoulders, hood thrown back, gold ringlets stuck to her shiny face. Pink cheeks and Rudolf nose. 'Sup?'
Raindrops pattered on the leaves above them, like a million tiny drummers. The occasional drip made it through the canopy, splashing into a puddle big enough to drown a toddler.
He cleared his throat. 'Is your mum all right?'
'Been waiting on you for ages, Piggy.'
'Did Jerome come back and hit her again?'
Willow tilted her head on one side. 'You perving on my mum?'
'No.'
'Why? What's wrong with my mum?'
'It's OK, I'll keep your name out of it. No one will know you told me who hit your mother.'
'Get bent, Piggy. I ain't no snitch.' She balanced on the pedals, shoogling the bike from side to side to stay upright. 'You got them toys for Pinky from the wee creepy guy with the pawnshop. Why?'
'Because.' Callum shrugged. 'No one should have to pawn their kids' toys just to stay afloat. No matter how much of a pain in the arse those kids are.'
She almost smiled.
'Willow, your dad – the guy who broke your arm when you were four – what was his name?'
'How come you always asking questions, Piggy?' She pedalled around him in a slow circle. 'Nosey, nosey, nosey: oink, oink, oink.'
'Just interested.'
'Always sticking your nose into other people's stuff and that.'
'Hey, it's OK if you don't know.'
'Course I _know_.' She did another lap. 'You saying I don't know?'
'Lots of people have no idea who their dad is. No shame in that.'
'Yeah, well I know: and I ain't no snitch. But see if he ever comes back? I'll break _his_ arm.'
'Sure you will.' Callum turned in place, facing her as she circled.
'Break his little bitch legs too.'
A seven-year-old girl, with blonde ringlets. And the worst thing was: she probably meant it.
'You don't have to _be_ like him, Willow. You can be so much better than that. Hell: put your mind to it and you can be anything you want.'
'You're a nutjob, Piggy.' She pedalled away a couple of feet, then dug into her pocket and came out with a small blue bag – the kind dog-walkers used to collect moist, soft, stinking presents – and chucked it to him.
Please don't let it be warm, please don't let it be warm...
It wasn't. And what was inside wasn't cold and squidgy either, it was a thin, flat rectangle.
Callum opened the bag, and there it was: one tatty leather wallet, the lining dangling loose from one side like a Labrador's tongue. A smile pulled at his face, but when he looked up Willow was already fading into the distance, pedalling for all she was worth.
He took a deep breath and bellowed it out anyway: 'THANK YOU!'
Then the car horn blared from the roadside behind him. Franklin, being her usual patient charming self.
Right.
He puffed out a breath and slipped the poo-bag in his pocket.
Time to visit the dead.
##
'Thanks. Thanks a lot. And now I'm late.' Franklin sat in the passenger seat, arms crossed, scowling.
'It's only just gone half ten.' Callum swung the pool car around the roundabout and into a shabby industrial estate. Past boarded-up business units with empty car parks and rusty chain-link fencing speckled with ancient carrier bags – their colours bleached and brittle. Through puddles the size of lochans, sending arcs of spray up onto the pavements. Windscreen wipers thumping back-and-forth across the glass. 'It's like going to the pictures: first fifteen minutes is all adverts and trailers.'
'I happen to _like_ the trailers.'
Yeah, she would.
Left, past a garage selling shiny four-by-four flatbed trucks, and down to the end of the road.
A thick line of green bushes – at least twelve foot tall – stretched out from either side of a big automatic gate topped with razor wire. An intercom unit sat in front of the gate, mounted on top of a big concrete bollard. Callum pulled up beside it and wound down his window. Pressed the button.
Its speaker crackled and popped, then hissed something unintelligible at him. So he stuck his thumb on the button again and held it there till the gates squealed and rumbled their way open.
The pool car rocked its way over a speed bump and into the compound.
If the architect was going for warm and welcoming when he designed Oldcastle's overflow mortuary he'd done a sodding rotten job of it. The building looked like something out of a Cold War thriller – a concrete bunker with tiny windows along its length. A Transit van sat outside the loading bay, down the far end, two men in grey overalls manhandling a plain gunmetal coffin onto a gurney.
It wasn't the only vehicle there – a handful of manky pool cars had been abandoned as close to the mortuary's front doors as possible. Because clearly police officers weren't waterproof.
Callum parked on the periphery of the clump. 'There you go: five minutes. They'll still be going on about switching off your mobile phone and getting a drink and a snack from the lobby.'
'You're an idiot.' She climbed out into the rain and slammed the door behind her.
'So people keep telling me.' He locked the car and followed her inside.
They'd decorated since last time, the smell of fresh paint fighting against several plug-in air fresheners and the dirty-bowel-like stench of decay. All the posters were new too – motivational landscapes and quotes about peace and forgiveness. As if that was going to do any good to the poor sods who had to come all the way out here to identify their dead child's body. The wee stainless-steel reception desk hadn't changed, and nor had the big dusty rubber plant in the corner. Its thick waxy leaves like slabs of green liver, aerial roots searching the walls for sustenance.
A little old man lounged behind the desk, tongue poking out of the side of his mouth as he wrestled with the _Castle News and Post_ crossword. The effort must have been quite something, because his wrinkles were even more tortured than normal, his hair a mixture of silver and cigarette-tar yellow.
Callum went over and had a look. Poked the newspaper. 'Three across, "Decapitated".'
The old man glanced up, showing off a pair of dark, glittering eyes. 'It doesn't fit.'
'It does if you spell "Robespierre" properly, Dougal. Three Es, two Rs and an I.'
'Oh.' He made the correction, then put the paper to one side. Grinned at Franklin with a big grey wall of perfectly straight false teeth. 'Well, well, well, when DS McAdams called to say you were coming over he didn't tell me you were such a beauty.'
She bared her teeth back at him, but it wasn't a smile. 'Where's the post mortem?'
'Ah, straight to business.' Dougal winked. 'I like that in a woman.'
'Do you also like a punch in the throat?'
'I wouldn't say no to a little light spanking. But maybe I should just show you through to the cutting room?'
'Maybe you should.'
Dougal stepped out from behind the reception desk and led the way through a pair of double doors and into a long corridor with doors opening off either side. 'We've got a full house this morning. Yesterday must've been buy-one-get-one-free on dead bodies.' The door at the end opened on an aisle between two sets of refrigeration units – big rectangles of stainless steel, each one covered in a grid of metal hatches. Four high, eight wide. Each hatch was about the same size as an oven door, only they didn't contain Christmas dinner.
Well, hopefully not anyway.
One of the hatches lay wide open so the two guys from the loading bay could wrestle a body bag out of the gunmetal-grey coffin and onto a sliding drawer. The contents all bendy and awkward.
Dougal waved as they passed. 'Let's not drop the guests, guys.'
A nod. 'Dougie.'
'Bodies, bodies, and more bodies.' He glanced back over his shoulder at Callum. 'It's the same every time you lot go digging about in the tip. Think you'd have more sense.'
At the end of the block, Franklin stopped. Stood there on the damp grey floor with her mouth hanging open. Staring. 'Holy mother of hell...'
From here, the full size of the room became apparent. A mini warehouse, with row after row after row of refrigerated units in it.
She gave a low whistle. 'How many bodies have you _got_ here?'
'One hundred and twelve.' Dougal stuck out his chest, sounding every inch the proud father. 'But we've got space for three hundred and sixty, including the freezers. A seven-three-seven falls out of the sky at Oldcastle airport? We can take every single passenger, a full bendy bus, _plus_ two football teams as well.'
And what a fun weekend _that_ would be.
Callum followed the pair of them into the visitor's changing room, with its rows of lockers, racks of blue wellington boots, boxes of gloves and other assorted paraphernalia. Slipped off his shoes and stuck them in a locker. Helped himself to a pair of size-nine wellies. 'Who's doing the mummies?'
'The mummy?' Dougal scrunched up his wrinkles, then peered at a clipboard hanging on a hook by the door marked 'DISSECTING ROOM ~ SAFETY EQUIPMENT MUST BE WORN BEYOND THIS POINT'.
'Mummies. Two of them.' Callum pulled a plastic apron from the roll by the door and unfurled it. Slipped it over his head and tied the ties. 'Came in yesterday?'
'Right. Right. Well... OK, you've got Lucy Compton.'
'Never heard of her.' He helped himself to a pair of safety goggles.
'New APT. This is her first week. Young lass, you'll like her.'
Callum stared at him. 'Can we at least _pretend_ we're taking this seriously, Dougal? I want a pathologist, not some wee Anatomical Pathology Technician just out of nappies.'
Franklin yanked an apron from the roll. 'What, she's not good enough just because she's a woman?'
'I don't care if she's a man, a woman, or a transgendered squirrel – she's not a pathologist!' He watched Franklin make a cat's breakfast out of tying on her apron. 'You've ripped the plastic.'
Dougal shrugged. 'Don't look at me. All I know is we've got two pathologists on duty and four bodies to PM today. Four to do tomorrow, and four more the day after that. Assuming no one else dies in the meantime. You want to moan at someone? Talk to Teabag and Hairy Harry.'
'Oh don't you worry, I will.'
Franklin tore off another apron and tried again. Finally got herself sorted out with goggles, wellies, a surgical mask, and gloves. Crossed her arms and shuffled on the tiled floor. 'Well?' Looking about as comfortable as a Seventies TV star in a police interview room.
Callum snapped on a pair of blue nitrile gloves. 'Have you been to a post mortem before?'
Her nostrils flared. 'Why, because I'm a weak and feeble—'
'Fine, _sod_ you then.' He nodded at the door. 'Come on, Dougal, let's not keep Detective Constable Franklin waiting. She's keen to see her dead body being hacked apart.'
Dougal opened the dissecting room door and stood back to let them past.
A dozen cutting tables sat in a row down the middle, the air redolent with eau de mortuary. CCTV cameras hung from the ceiling above each one, their black bulbous eyes ready to capture the most intimate and thorough violation anyone would ever experience.
One table was surrounded by half a dozen people doing their best not to look like plainclothes police officers and failing miserably. They'd donned the same safety gear as Callum and Franklin, a couple of them laughing, two looking serious and boot-faced, two taking notes as a tall thin man in purple scrubs arranged a collection of trainers and shoes on the stainless-steel surface. Someone in green scrubs followed him, taking photos – the flash turning everything monochrome for a moment, before the colour seeped back in.
Down at the far end of the room, a dark body lay beneath a set of industrial extractor fans going full pelt. Not that it made much of an impact on the stench. But then it was difficult to imagine what would. Tip three gallons of Febreze in here and it would still stink of perforated bowels and rotten meat.
Someone in green scrubs was washing the body with a sponge, wringing out dirty grey water into a drain set into the floor.
Franklin took a deep breath and stiffened her shoulders. 'That our victim?'
'Shall we?' Douglas offered her his arm, as if they were off to the ball.
She ignored it and marched off, back straight, wellington boots making week-wonk noises on the stained floor.
The far wall was home to a long line of sinks and taps, with a glass wall above them looking in on a viewing gallery. A wee bloke with a red Henry hoover shuffled about inside looking as if he was in need of a post mortem himself.
Only two other tables were occupied – as far away from Franklin's corpse as possible – and both of them sported a mahogany-coloured body, curled up on its side. One of which was being circled by a small figure wearing pink scrubs. Dark curly hair pinned up in a lopsided bun, purple nitrile gloves, surgical mask.
That would be his brand new APT then.
Ah well.
He wandered over. 'Hi. You Ms Compton?'
She stopped and turned to him. 'No, I'm not, sorry, I'm not Ms Compton, who's Ms Compton?' She'd put her pink scrubs on over a black-and-grey stripy top. Its sleeves were rolled up just far enough to expose an inch of yoghurt-pale skin between them and the purple nitrile gloves. Not Ms Compton pointed at the curled body. 'Sorry, I know it's not my case, but I saw the mummies over here and I thought "that looks interesting", I mean I always loved those films when I was little, you know with Boris Karloff all wrapped up in bandages exacting revenge on the archaeologists who dared to disturb his tomb?' The words were delivered like machine-gun fire, in a cheery unplaceable Scottish accent. 'To be honest, I'm supposed to be consulting on another case about some severed feet, but the heart wants what the heart wants.' She stuck out her hand. 'Ooh, and it's Alice, by the way, Alice McDonald, technically it's _Doctor_ Alice McDonald, but that sounds a bit uppity doesn't it, so just Alice is fine, all gets a bit confusing doesn't it, maybe if everyone in the world wore name badges it'd be easier, what do you think?'
Yeah... this one was a freak.
He shook her hand, warm and slightly sticky through his gloves. 'Detective Constable Callum MacGregor.'
'Right, yes, great, good name, couldn't get much more Scottish, could you, not with a name like that, well, I mean it _could_ be, if your middle name was Angus or Hamish. Is it?'
'You said you're consulting on a case. You're not a pathologist are—'
'Oh no, not a pathologist at all, I'm here doing Behavioural Evidence Analysis, which is what we call profiling now, because if we call it profiling people think it'll be just like the movies where the forensic psychologist says, "Whoever killed all these women and ate their uteruses was a white middle-aged man with one leg shorter than the other and an unnatural affinity with the music of Johnny Cash", because it doesn't work like that and lots of people like Johnny Cash but never kill anyone, though I'm not a fan myself. Do you see?'
No.
'Err...' Wait a minute. Forensic psychologist. Alice. Rambling. He lowered his own surgical mask and the dirty-brown smell of the mortuary swelled in his nostrils. 'Dr McDonald? It's me, Callum. I was on the Birthday Boy investigation, five years ago? You were consulting.' No reaction. 'I was on DCI Weber's team?'
She lowered her own mask and shared a slightly painful smile, as if she'd got something bitter caught between her back teeth. 'Ah, sorry, it's nothing personal, but I tend to just see a big sea of faces when I'm up giving presentations and then there's all the different investigations all over the country and there must have been at _least_ three thousand police officers over the years, probably more, and I would love to be able to remember them all, but I haven't got that kind of brain, and I get a bit nervous when I'm up there, so I'm picturing you all in your underwear if that's—'
'Dr McDonald?' A figure appeared at Callum's shoulder, green plastic apron pulled on over a smart dark-grey suit. Half of his face was hidden behind a surgical mask, but there was no mistaking the voice or sticky-out ears. Detective Chief Inspector Powel. 'They're ready for you.'
Alice the weirdo waved at him. 'Hello, Reece, I was just admiring Callum's mummies, aren't they great, did you ever watch Boris Karloff when you were little?'
He barely inclined his head. 'DC MacGregor. I thought they were supposed to fire you this morning?'
'Nope.' Callum leaned against the cutting table. 'You'll just have to try a little harder next time you fit me up.'
Powel cricked his head to one side, then back again – like a boxer getting ready to fight. Then turned back to the professional nutjob in pink. 'Professor Twining's ready to begin, so if you want to come have a look before we take the feet out of their shoes...?'
'Yes, of course, the feet, duh, sorry got distracted. Do you think we should all have name badges, because I think we should all have name badges...' Her voice faded into the distance, swallowed by the background growl of the extractor fans as Powel led her away.
Callum stuck two fingers up at the DCI's back.
_I thought they were supposed to fire you._
Dick.
And how could she not remember him? _He_ remembered _her_. Mind you, she did stand out a bit, what with her whole 'Day-Pass-From-The-Asylum' shtick.
Still, it was nice she'd been interested in his mummies, because no one else seemed to give a sod.
Callum folded his arms. Searched the room for Franklin and her amazing exploding temper. She was standing in the corner, scribbling away in her notebook as the APT finished washing down the swollen corpse.
So, could be worse. At least he wasn't marinating in the Marmite stench of a decomposing body, like Franklin. No, _his_ remains just smelled of... What?
Callum leaned in and took a sniff, but it was just the usual ever-present stink that permeated the mortuary: bleach, bowels, and decay. Which was odd – when they'd opened the car boot yesterday there'd been a distinct smell of wood smoke. And a hint of it back at the tip, with Mummy Number One too. Unless _this_ was Mummy Number One. Kind of difficult to tell them apart.
He inched closer and tried again.
The scent was still there, lying under everything else. Like the old armchair his grandad used to smoke his pipe in. Puffing away, getting the scent of sandalwood and cherry deep into the leather.
Someone cleared their throat behind him. 'Can I _help_ you?'
He flinched up. Smoothed down his thin plastic apron. 'Just...' Warmth tingled in the tips of his ears, as if he'd been caught snogging the remains instead of just sniffing them. 'Callum MacGregor, I'm Senior Investigating Officer.'
'Oh aye?' She was a large woman, compact and powerful looking. The kind of person that could pick up a fridge and beat you to death with it. Her green scrubs looked fresh out of the packet, but her arms looked fresh out of Barlinnie – covered in DIY tattoos. She leaned on the chunk of machinery she'd been wheeling across the mortuary floor. 'You sure?'
' _Yes_. Are you Ms Compton?'
She flexed her muscles. 'Lucy.'
'OK, Lucy.' He pointed at the body. 'Does this smell of wood smoke to you?'
She pulled down her mask, revealing a mole at the corner of her mouth. Sniffed. 'Oak. And...' Another sniff. 'I'm going to go with beechwood.'
'What about the other one?'
Lucy shifted the machinery over to the other cutting table, bent over the curled body and filled her nostrils. 'Definitely beechwood and oak. This one's a lot stronger.'
That would be the one from the car boot. Maybe lying about in the tip for God knew how long masked Mummy Number One's natural smell?
The APT went back to her trolley and pushed it next to the cutting table. Clunked on some sort of footbrake, then fiddled about with pins and levers until a big C-shaped arm swung out from the main unit. It had a box on either end, each about the size of small microwave.
'Right.' She handed him a heavy blue apron. 'Stick that on and we'll get some X-rays done.'
'X-rays?'
She looked at him as if he was a very thick little boy. 'Well we're not going to actually post mortem them, are we? They're _mummies_. Priceless relics of a long-dead civilisation. Cause of death isn't going to do you a hell of a lot of good, is it? Or are you planning on climbing into your DeLorean and travelling back to ancient Egypt with an arrest warrant?'
Yeah, she had a point.
'Now,' the APT pointed at Mummy Number Two, 'help me get it sitting up and we'll see what we can see...'
##
'I know it's not nice, but you need to eat it. It's good for you.'
The spoon is cold against his cracked lips, its contents hard and gritty.
He'd raise his hands and bat the spoon away, but his arms don't work any more. They don't even float in the water, just sink into its filthy depths to lie against the steel tank. _Nothing_ works.
Can't even hold his own head up.
So the Priest holds it up for him, a warm hand on the back of his neck.
'Don't worry, I'll help.'
The other hand forces his mouth open, then pours the grit inside.
It sits there, in his mouth, like tiny stones. Sticking to his tongue and cheeks. Making him gag and cough. But there's not enough breath left to shift anything.
The walls are louder now, singing at the top of their splintered lungs: _'They'll worship you: you'll be a god. They'll worship you: you'll be a god. They'll worship you: you'll be a god.'_
Their voices send a tremor rattling through him, shaking his teeth, making his ribs ache.
'Shhh...' A hand strokes his forehead. 'Shhh...'
Then a kiss.
'I think it's time, don't you?'
Oh God please let it be time to die. Time for the pain to go away. _Please_.
_'They'll worship you, They'll worship you...'_
'Come on.'
The water falls away and he's being carried, arms and legs swinging in the cool air, rivulets of brackish water falling to the floor. There's almost nothing left of him now. Nothing but skin and bone.
_'They'll worship you: you'll be a god.'_
The singing walls swim and pulse around him, worshipping. And finally he makes the transition into the other room. The one where the fish hang in silent prayer.
Even the walls are quiet in here. Reverential. Waiting for the blessed relief.
Soon he'll be dead and all this will be over.
'Here we go.' Gentle hands lay him on the stone floor.
High up above, a sliver of grey sunlight dances with dust motes. Spiralling and swirling.
There's a pressure on his ankles, but not much more than that.
Then the squeal of wood on wood and his legs raise themselves off the ground, then his hips, his back, and finally his head leaves the earth. He sways gently, ascending to heaven with his arms dangling either side of his ears.
Swaying and rising.
Up and up into the darkness.
Up and up into death's comforting embrace.
He opens his mouth to say thank you, but all that comes out is a cascade of little gritty pellets.
The Priest smiles up at him, a thick rope held in one hand. 'You'll be a god...'
A god of skin and bone.
##
'And one more...' Lucy stepped back and the machinery buzzed again. Then clunked. 'OK, all done.' The muscles in her arm rippled as she pushed the portable X-ray machine's arm out of the way, making the tattoos dance. 'Now all we have to do is download the data, format it, and you'll get your glimpse into the ancient past. Might take a while, though.'
He puffed out his cheeks. 'Thanks.'
A grin. 'Who did you piss off?'
'Hmm?'
'To get lumbered with this. No one asks for a PM on a thousand-year-old mummy unless they're being punished for something.' She flipped off the footbrake. 'So who did you piss off?'
Callum forced a smile. 'Pretty much everyone.'
'Thought so.' Lucy took hold of the handles and shoved, setting the X-ray kit rolling. 'You can wait here, in the smell, or you can come through to the IT lab. It'll be warmer. With seats.'
'Yeah.'
'Wise choice. Oh, and on the way? There's a drinks machine in the APTs' lounge, I'll have a hot chocolate.'
Cheeky sod.
A dull buzzing thrum ran through the lab, mingling with the soft whirr of desktop computers, and the _ping-click-ping-click_ of a small electric heater.
Callum took the last slurp from what the machine claimed was a white tea.
It had lied.
He stuck the empty plastic cup on the desk and shifted in his seat. Closed his book and put it down.
'Any good?'
He looked up. 'Hmm?'
' _The Beginner's Guide to Shoplifting_.' Lucy pointed. 'Any good?'
'It was OK.'
'I had a mate who was great at shoplifting. You name it, she could swipe it: food, booze, electric toasters. Made off with a bass guitar once.'
'Yeah, it's more a collection of short stories than a how-to guide.' He stood and stretched, little knots cracking across his spine. 'Pff...' Sagged. Checked his watch. 'Which way to the toilet?'
'Use the disabled: down the hall, on the left. I'm guessing another fifteen minutes? Servers are running like treacle today.'
'What happened to your mate, the shoplifter, she get caught?'
'Married a Glaswegian and emigrated to Newcastle.'
Callum wandered over to the door. 'Might make some calls too.'
Lucy went back to her computers. 'Wouldn't mind another hot chocolate if you're passing...?'
'See what I can do.'
Sodding hand dryers never worked.
He wiped his hands on his trousers as he made his way down the corridor to the far end. A window looked out over the mortuary car park – the reception area just visible in the middle distance.
Callum pulled out his Airwave and called Control. 'Any news on Ainsley Dugdale?'
_'Give us a minute.'_
The rain hadn't let up any, it still hammered down from a slate-coloured sky trying to batter the earth flat. It sparked back from waterlogged potholes, bounced off the parked cars and...
A big man staggered out of the reception doors, both hands clutched over his face.
Was that blood?
It was.
It poured through his fingers, staining the white shirt above the disposable green apron.
_'Aye, Ainsley Dugdale: released on bail pending trial in six weeks' time.'_
'What? How the hell could they let him out on bail?'
The big man lurched against one of the pool cars and stood there, shoulders hunched in the rain, blood turning his shirt dark pink.
_'How should I know? You want details? Speak to the officer in charge: DS McAdams. Anything else?'_
'Yes. I got a PNC on one Irene Brown yesterday, I need a list of all known associates going back about... seven, eight years?'
_'You're kidding, right?'_
'No.'
A sigh battered out of the speaker. Then silence.
Out front, another man joined the first – picking his way between the puddles with a newspaper, or maybe a file folder held over his head as a makeshift umbrella. He stopped beside Captain Bleedy and patted him on the back.
There, there, poor thing.
Ah well, wasn't Callum's problem. 'You still there?'
_'No, I've jetted off to Barbados for piña coladas and a barbequed lobster.'_
He headed back down the corridor. 'Well?'
_'I've got eight different names on here and according to the flags there's about another three that aren't on the system. One armed robber, two drug dealers, one got done for raping a nurse, one unlawful remover, one attempted murderer, and two aggravated assaulters. Well, they've all got charges for that, but two of your woman's mates didn't branch out into anything else. Or they never got caught.'_
Eleven violent scumbags, and Irene Brown was barely in her twenties.
Little Mike was right: this is what real life looks like from down here at the bottom.
'Can you email me the details? Names, dates, convictions, everything you've got.'
_'Urgh...'_ Another big put-upon sigh. _'Fine. Anything else, Your Majesty, or can I go back to working my fingers to the bone?'_
Yes. Away boil your head.
'No. Thank you.' Callum stuck the handset back in his pocket.
Why did everyone have to be such a prima donna?
He paused at the entrance to the technicians' lounge. There was no point getting another tea from the machine – once poisoned, twice shy – but the hot chocolate couldn't be all that bad if Lucy was having another, so—
The door leading through to the dissecting room burst open and DCI Powel stormed into the corridor with his monkey ears bright red. He turned and jabbed a finger at his own feet. 'IN HERE NOW!'
Franklin stepped in after him, shoulders back, chin up, jaw clenched.
Powel's voice dropped to a growling whisper soon as the door shut behind her. 'I do not care what you got away with in Edinburgh, but you do _not_ assault members of _my_ team without _serious_ repercussions. Do you understand?'
She glared back at him. 'With all due respect, sir—'
'I SAID, DO YOU UNDERSTAND?' Bellowing it out, spittle flashing in the overhead lighting.
'Yes, _sir_.' There it was again: that burning-pus tone.
'Now you will wait here and you will not move a muscle while I go check on the man you assaulted. And while I'm gone I want you to have a good hard think about how screwed you are.' Powel barged past her and back into the dissecting room again, slamming the door on his way.
Franklin bared her teeth. 'I will rip your bloody balls off, you...' She must have realised she wasn't alone, because she clicked her mouth shut and turned her glare on Callum instead. 'What the hell are _you_ looking at?'
He wandered over, hands in his pockets. 'Let me guess: the guy with the busted face...?'
'I don't answer to you.'
'Never said you did.'
She paced the width of the corridor in two steps then turned and did it again. 'What is _wrong_ with you people?'
'Me people?'
'He grabbed my arse! Of _course_ I hit him!'
Ooh...
Callum stuck his head on one side. 'Let me guess – big guy with sideburns and a wonky eye? DS Jimmy Blake. AKA: Blakey the Octopus.'
'And do you know what he called me when I hit him? An "effing darkie bitch"!' She thumped her hand against the wall.
'No.' Callum frowned. 'Blakey said, "effing"?'
'Oh shut up.' She went back to short-form pacing again.
The door thumped open, and there was Powel, suit darkened across the shoulders and legs. He hooked a finger an inch from Franklin's face. 'You. With me. Now.' Then turned and marched off.
She caught the door. Took a deep breath and went after him.
She was a pain in the backside, but still...
Callum followed the pair of them into the dissecting room's foetid air, across the stained floor to the cutting table covered in naked disembodied feet. It was surrounded by plainclothes officers, shuffling about and avoiding eye contact. Fidgeting. Looking shifty and embarrassed about the whole thing.
Blakey was a pillar of indignation off to one side, a wad of green paper towels clutched to his nose, shirt stained a lovely Ribena red all down the front.
Powel stopped right in front of him, and did the pointing-at-his-own-feet thing again. 'Detective _Constable_ Franklin. You will apologise to DS Blake and you will do it now.'
She stood where she was told, muscles writhing along her jawline.
' _Now_ , Constable!'
Franklin took a deep breath, opened her mouth—
But Callum got there first. 'Sorry to interrupt, Guv, but I'm sure this is all just a big misunderstanding.'
Powel didn't even look at him. 'This is none of your business, MacGregor.'
'Let's _imagine_ for a moment that DC Franklin was the victim of a serious sexual assault in the workplace. She'd be well within her rights to defend herself, wouldn't she?'
At that, Powel turned. 'She attacked DS Blake.'
'There's a reason they call him "Blakey the Octopus", Guv.'
Blake took the towels from his face. His left eye pointed about ten degrees off from the right one, but both of them had already gone a dark shade of pink. 'I nebber tudged her!'
Ooh... Yeah. Mental note: _never_ try to cop a feel of Franklin's bum. She'd completely flattened Blake's nose, leaving a squint lump of bloody gristle behind.
Franklin bared her teeth. 'You lying, sexist, _racist_ scumbag!' She took a step forward, hands snapping into fists again.
Callum grabbed her arm. 'We can easily check, Blakey.' He pointed up at the dark CCTV globe mounted above the cutting table. 'Big brother sees all.'
'Ah...' Blake stared up at the camera.
'So maybe what _actually_ happened is you slipped on the wet floor and tried to break the fall with your face?'
Blake pressed the towels against his nose again. Looked down at his feet. 'I slibbed ad fell.'
Callum held his arms out, like he was about to accept applause for a magic trick. 'There you go: told you it was nothing but a silly misunderstanding.'
'What was _that_ supposed to be?' Franklin paced the length of the APTs' lounge, between an off-grey sofa and a coffee table covered in tabloid magazines, one arm jabbing in the general direction of the dissecting room.
Callum poked buttons on the vending machine, setting it whirring and clunking. 'You're welcome.'
'I'm _welcome_? Well thank you _so_ much. He grabbed my arse!'
Oh joy.
A plastic cup clunked down in the dispenser, then the hissing and gurgling started.
'He grabs your backside – that's sexual harassment. If you'd _reported_ him, that would've been it: Blake gets hauled up in front of Professional Standards for being a dirty sex offender. End of.' The machine stopped its noises, so Callum extracted the scalding hot cup and pressed the buttons again. 'But no, you had to lash out. You break his nose – that's assault. Now it's _you_ in front of the rubber heelers and everyone thinks _he's_ the victim.'
Gurgling and hissing.
'That what you want?'
She stopped pacing. 'So you just swan in and save the poor powerless black woman, do you? Because _you_ know best.'
He stared at the ceiling. 'I give up. I really do.'
'The Great White Saviour rides again!'
'For God's sake... Do you _never_ stop? All I've done since you got here is try to be nice. I was just looking out for a member of my team, OK? Even though she's hell-bent on leaping feet-first into the sodding wood chipper!'
She threw herself down into the sofa. Sat there seething. Looked away. Closed her eyes. Then let out a barely audible, 'Thank you.'
He picked up one of the scalding cups and held it out. 'Here.'
Franklin took it. Held it in her lap as her head fell back against the back of the couch. 'Why do so many white guys see a black woman and think she's going to be an easy shag? You think it's hard being a police officer? Try doing it when you're a woman, you're black, you're attractive, and everyone thinks you're "gagging for it".' She ran her spare hand across her forehead. 'If I'm sleeping my way to the top, how come I'm still a sodding constable?'
Callum smiled. 'If it's any consolation, I don't think you're in the least bit attractive.' He took a sip of burning-hot chocolate. 'Oh, I did for two or three seconds when I first met you, but soon as I got to know you? Not a chance.'
She gave a little laugh. 'When I was fourteen my geography teacher tried to feel me up in the stationery cupboard, said he'd always wanted to try a black girl. Wasn't the first to chance his arm and he wasn't the last.'
'Did you break his nose too? Or...' Callum sniffed. Wait a minute. He sat on the other end of the couch. 'That's why you got lumped into Mother's Misfit Mob, isn't it? You said you punched a superintendent – all hands was he?'
Franklin held up her cup in salute. 'Welcome to the world of hyper-sexualisation.'
'Yeah: they can't fire us. They _want_ to, but they can't. So they chuck us all together and drip-feed us crappy nothing cases till we get fed up and quit.'
' _And_ he was married.'
Callum had another sip. Lucy was right – the hot chocolate wasn't anywhere near as bad as the tea. 'How you getting on with our body in the bath?'
'You know what I should've done? I should've agreed to go to that hotel with him and recorded him being a sleazy git. Then sent the tape to his wife. Let's see Superintendent Neil Lambert slither his way out of that one.'
'Nice to know your first thought was revenge instead of blackmail.'
She gave her forehead a little slap. 'Blackmail. Damn – why didn't _I_ think of that?' Franklin sagged in her seat. 'Our body's a Caucasian male, five eleven, difficult to tell how old. Turns out the water in the bath was a _very_ heavy brine solution with leaves and flowers and herbs and spices thrown in. Little bits of bark. His lungs are full of it too, so he was alive when he went in. You think you're a bit wrinkly after half an hour in the bath? Our victim looks like he's about ninety.'
'At least all that salt will have preserved the tissue.'
'Only the outside layers. Didn't stop his stomach bursting in the water. Gah...' She had a little shudder. 'God, I _hate_ post mortems.'
'Shouldn't have transferred to Oldcastle then, we're up to our ears in them.' He punched the buttons for a third hot chocolate. 'What do you reckon to Glen Carmichael and his mates being responsible? Three of them, killing the guy in the bath together – holding his head under till he stops struggling. Or it could be one of them. They fall out, fight, same result.' The machine hissed and gurgled. ' _Or_ maybe they get their hands on a load of dodgy drugs? One of them has a catastrophically bad trip so they try to sober him up in the bath. But he drowns. They panic and do a runner.'
She pursed her lips. 'Makes sense. Three blokes, stuck in that flat for months together, the windows all covered with hardcore pornography, getting drunk and wasted. Can't expect sensible decisions from—'
The door battered open and Lucy staggered to a halt on the carpet tiles. 'Callum?' A bit out of breath, but grinning too. 'You are _not_ going to believe it.' She hooked a thumb over her shoulder, tattoos rippling. 'Digital X-rays have finished processing. You've got to see this!'
OK.
He peeled himself out of the couch and followed her down the corridor to the lab.
Lucy pointed at a large computer monitor sitting on a desk covered in empty Quality Street wrappers. 'Look.'
He hunched over and squinted at the screen. It was filled with human bones, the skin reduced to pale wisps of grey around them. The X-ray had been taken from the side, showing the body curled up with its elbows in, hands against its chest and knees against them, neck bent so far forward that the front of the skull was obscured by the kneecaps. Definitely one of the mummies. 'And?'
'This is _so_ cool.' She clattered her fingers across the keyboard and the image zoomed in on the jumbled monochrome mess where the face met the kneecaps. 'See?'
'No.'
'I thought it was an artefact on the machinery – some of the APTs X-ray all sorts of crap for a laugh and if they damage the equipment it shows up on the digital prints – so I checked the anterior plates.' The tip of her tongue poked out between her teeth as she typed, and the picture changed to a close-up that was little more than a mess of white lines and grey masses. ' _Now_ do you see?'
Callum squinted, forcing the image out of focus, letting it... There. OK, those were the eye sockets, there were the cheekbones, difficult to pick them out from the leg bones, but not impossible. That made those the nasal cavities, and _they_ would be the teeth.
Oh.
He sank into the office chair. Of course: it was obvious now. 'The hard white blobs in the mouth.'
'Yup.' Lucy grinned at him, eyebrows way up her forehead. 'I'm no expert on ancient Egyptian burial rituals, but I'm pretty sure Tutankhamun didn't go to his grave with a mouth full of NHS fillings.'
##
'Are you actually _hiding_ in here?' Franklin stood in the doorway, holding the door to the disabled toilets open with her foot.
Callum gave her a grunt, then went back to the sink. Stuck a folded paper towel against the open neck of the disinfectant bottle and upended the thing till it soaked into the off-grey paper. 'I'm not hiding, I'm busy.' He dabbed the damp tissue against the wallet, blotting away a stain.
'Mother's here.'
'Course she is.' More blotting. The stuff was working even better than Lucy had promised. OK, so his dad's wallet would never look new again, but at least it didn't stink like the inside of whatever bin Willow Brown had fished it out of.
'Callum?'
'It was covered in cigarette ash and something I'm hoping was mayonnaise.'
'She's told Dr Jenkins she wants both mummies post-mortemed ASAP. Our bath body's gone back into the fridge till they're done. He's starting the first one in fifteen minutes.'
'Good for you.' Callum dumped the paper towel in the bin and moistened another with disinfectant.
'Well? Aren't you coming? Thought you were SIO?'
'What's the point?' He opened the wallet and started on the inside surfaces. Wiping the square of clear plastic covering the faded photo that took up the whole left-hand side: a happy family, all four of them grinning away at the camera, those bright summer colours faded to autumn tones of brown, orange, and yellow. Not quite in focus. Posed around a picnic table, blue sky, sea, and white sand just visible behind them.
'So you're _sulking_?'
'Like you can talk.' He dabbed away at the other side. 'They're never going to let me be Senior Investigating Officer for a triple murder. A detective constable running down a serial killer? No chance.'
Franklin shook her head. 'You've been inhaling too much of that disinfectant, there's only two murders, not—'
'The mummies have been smoked. That's probably to dehydrate and preserve them. And what do you do before you smoke something? You salt it to draw out excess moisture. You _brine_ it.' He wiped off a crusty smear of red that looked more like tomato ketchup than blood. 'And what did we find floating in a bathtub full of brine?'
When he looked up, Franklin was standing there with her mouth hanging open.
One last go with the soggy paper towel. 'Exactly.' He wiped the wallet dry with a fresh sheet, then dug into his pocket for the cash he'd begged out of the building society yesterday. Slipped it into the slit where the lining was hanging out. 'Our victim was a work in progress.'
'Sodding hell.'
'And God knows how many more bodies he's got out there.' The plastic window was cool beneath his fingertips, its surface scratched in a few places, enough to blot out small sections of the photo beneath. All four of them, just out of focus, smiling their last recorded smile together. 'Mother's going to have her work cut out holding onto the case, never mind me. They'll fly in an MIT from Strathclyde and we'll be back where we started – low-level drug dealers, loan sharks, and pimps.'
Franklin peered over his shoulder. 'Whose are the ugly kids?'
Cheeky sod.
He pointed. 'That's my brother Alastair and me.' Two little boys with matching haircuts and freckles. 'We were five. Mum and Dad at the back.' Mum with her long pale-blonde hair and heart-shaped face, kind blue eyes. Dad with his dark curly mop-top, dimpled chin, and big broad smile. The whole family was dressed for the beach in shorts and flip-flops. T-shirts with cartoon animals on them. A fox for Alastair, an owl for Callum, a cat for Mum, a dog for Dad. Sunburn for everyone. 'Two weeks on a caravan site just outside Lossiemouth.'
Franklin gave a low whistle. 'You've got an identical twin?'
All those years ago...
## — Callum —
'Da-ad, he's touching me again!'
Dad just sighed and turned the radio up, singing along with Mum. Both of them belting it out at the top of their voices as countryside slipped by the car windows. Green fields beneath a dark-grey hat of clouds.
Sitting in the back seat, Alastair grinned his gap-toothed grin. Then reached across and poked Callum again.
Rotten little bumhead.
'Da-ad!'
Dad didn't look around. 'If you two don't cut it out, I'm going to pull this car over. That what you want? You want me to pull over? Because you know what'll happen if you make me do that.'
Alastair stuck his tongue out. His shaggy bowl-shaped haircut was paler than usual, more freckles on his nose and cheeks. A cartoon fox on his brown T-shirt. Tartan shorts and grass-stained knees. Bare feet all sparkly with sand, just like Callum's.
The song on the radio finished, Mum joining in right to the very end. She put her hand on Dad's leg. 'I love that one.'
The man on the radio sounded like he'd eaten a whole nest of bees. _'An oldie, but a goodie – Jimmy Perez and the Mareel Boys, with their breakthrough hit, "Mothcatcher Blues". For an extra five bonus points, name the year that topped the charts.'_
Mum snorted. 'Easy: 1986. Give us a _hard_ one, Scotty.'
'Da-ad?' Callum leaned forward and tapped him on the shoulder.
_'And don't forget we've got – and I'm_ mega _excited about this – the one, the only, the Krankies! They'll be here in fifteen minutes to tell us all about Monday's heee-larious episode of K.T.V. It'll be fan-dabi-dozi!'_
'What did I tell you, Cal?'
_'I'm Scott Kennedy and you're listening to the_ Golden Oldies Quiztime Special _on Castlewave FM...'_
'No, but I need to go wee-wee. I really do.'
_'Now then, my little Quiztronaughts, who was the lead singer on this hit from two years ago? It's the Bangles, and "Eternal Flame".'_ Some sort of horrible old-people music jingled out of the radio – weird pings and things, with a woman being all soft and soppy over the top.
'We're half an hour from home, so you'll have to tie a knot in it.'
'But, Da-ad, I'm _bursting_.'
Mum shook her head, setting her pale-yellow hair swinging. 'Told you it was a mistake to buy him that tin of Fanta. It goes right through—'
'Don't start.'
'I'm just saying.' She pointed through the windscreen at a lumpy blocky building at the side of the road. 'Look, there's a public toilet. Stop.'
'I'm _not_ stopping.'
'Fine. Well, you keep on driving, David MacGregor, and when Callum wees himself, _you_ can clean it up.'
The lady on the radio sang about easing the pain. Which would've been nice, because right now there was a big balloon of pee swelling up in Callum's insides, sending stabby twinges all through his tummy right down to the end of his willy. ' _Please_ , Dad?'
'All right!' Dad thumped his hand on the steering wheel. 'All right, I'll stop. You happy now?'
'David, please, for _once_ can we not—'
'No. That's perfect. I'm stopping.' The car pulled into the lay-by, bumping and rolling along the holey road, caravan lurching away behind it. 'There.'
Mum didn't sing along with the lady on the radio: she just sat there, in the passenger seat, with her arms folded, staring out of the window.
Dad's voice was stretched and twangy, the way it went before someone got spanked for being naughty: 'Alastair, do _you_ need a wee too?'
'No, Dad.'
'Callum?'
'I'm sorry, Daddy.'
'Get out, Callum.'
He scrabbled with his seatbelt, pulled on his flip-flops, and pushed the door open. Hopped down onto the car park's holey surface.
The toilets were a low grey rectangle, sitting in front of a line of trees. Filth streaked the walls and the guttering sagged in the middle. Someone had sprayed 'TORY SCUM OUT!' across the Ladies. There weren't any outside doors, instead a bit of wall was missing at both ends of the building, open and gaping. A cave, full of shadows and horrible smells.
Up above, the sky was dark as an angry cat.
Mum nudged Dad. 'Don't just sit there – go with him.'
'He's not a baby, Nicola. If he wants to go to the toilet he's damn well big enough to go on his own.'
Callum wiped his damp palms on the legs of his shorts.
Maybe he didn't _need_ to go after all?
Maybe he could hold it in all the way home?
But that great big balloon just above his willy didn't want to hold it in. It wanted to pee it out, all down his leg if he didn't—
The car horn blared, and he jumped. Turned.
Dad scowled at him through the driver's window. Alastair grinned from the backseat.
Swallow. Turn.
You can do this, Callum.
You're a big boy now. Big boys can go to the toilet on their own.
He took a deep breath and crept into the Gents. Into the gloom. Into the manky-vinegar stink of old wee.
White tiles covered the walls, the lines in between them all dirty and yellow. Thick scratch marks ran across the brown floor, like something heavy had been dragged from one of the cubicles. Four of them huddled along the left wall, one with its door all splintered and hanging off. Urinals on the wall opposite. Sinks at the back.
A dripping tap went _plink_ , _plink_ , _plink_.
Callum hurried across to the urinals, unzipped his shorts and stood on his tiptoes.
Nothing happened.
Come on. Come on. Come on.
'Hello, little boy.' The voice was big and heavy, thick and slimy. Like a huge slug. 'You're a _pretty_ little boy, aren't you?'
A thin stream of yellow piddle splashed into the urinal, wobbling up and down because Callum couldn't stop shaking.
'Such a pretty blond little boy.'
The Slug slithered closer, breath all heavy and panting.
'Please, my dad—'
'Shut up. Don't spoil it.' Closer. 'Are you a _good_ little boy?'
Callum stood there, with his shrivelled willy in his hand. 'Please.'
'Mmm, I'll bet you are.' The Slug was so close now his butter-minty breath washed over Callum's face. 'This is going to be our little secret. If you tell anyone, I'll know. And I know where you live and I'll come get you. I'll kill your mummy and daddy and I'll _punish_ you. Understand?'
He nodded. Bit his bottom lip to keep the tears in.
'Good.' A warm slimy tongue licked its way up Callum's cheek, slow and minty and wet. 'Now you're going to be very quiet and come with—
'Nah, _course_ Labour's gonna win next year.' A man stumbled into the toilets, voice echoing back from the tiles. 'Stands to reason, don't it?'
Callum flinched.
The warm sticky breath disappeared and the slimy slug trail on his cheek went cold. Now the only thing left was the _plink_ , _plink_ , _plink_ of the dripping tap and the jaggy sour smell of wee.
He fumbled his willy back into his pants. Zipped up with shaky fingers.
'They better win.' Another man – dressed in the same checked shirt and scruffy jeans as his friend – long hair dangling down round his face, cigarette poking from the corner of his mouth. 'Can you imagine another four years of these bawbags?'
No sign of the Slug.
Callum's breath shuddered out. He sagged for a moment. Then scuffed across to the sinks and washed his hands. Scrubbed a wet hand across the cold patch on his cheek. Dried himself on a greying curl of fabric hanging from the towel machine.
Stepped over to the exit.
And froze.
What if the Slug hadn't gone away? What if he was out there, just waiting for him? Waiting to grab him and take him away and punish him and he'd never see his mummy and daddy ever again and it would be horrible and...
The stabby pain was back. He hurried to the urinals, up on his tiptoes again, making little grunty noises as the wee went down the drain.
Then washed his hands again, cos Mum didn't like widdly hands in the car.
Both the guys in the grungy clothes were laughing at some joke about two nuns and a donkey that made no sense at all. Peeing and peeing and peeing like they'd drunk a whole bathtub full of Fanta. They didn't wash their hands either, just lit up cigarettes and sauntered out the exit with their hands in their pockets.
Callum wiped his sweaty hands on his shorts again.
_Plink_. _Plink_. _Plink_.
The sound of a car faded into the distance.
It would be OK. It would.
Dad would get angry about how long he was taking and come get him.
Then he'd shout at Callum, and maybe spank the back of his legs, but he'd scare the Slug away and everything would be OK again.
It would.
Callum swallowed.
Shifted from foot to foot in his gritty flip-flops.
Come on, Daddy. Come on...
It'd been _ages_ now.
What if they'd got fed up, driven off and left him?
What if they'd forgotten he was here, in the toilets?
What if they never came back?
What if the Slug _did_?
Oh no...
Callum hurried outside.
Dad's car and the caravan were still there.
Thank you, thank you, thank you.
He'd never be naughty, ever again. He'd do everything Mum asked him. He'd tidy his room. He'd even be nice to Alastair the bumhead.
A rumble of thunder, off in the distance, mingled with the traffic noises from the road.
He ran fast as a rabbit to Dad's car and grabbed the door handle. But it just clunked up and down. The door didn't open.
Alastair must've locked it. Well, he was going to get a dead arm soon as Mum and Dad weren't looking. It wasn't _funny_ : locking people out of the car when there was a horrible Slug slithering about trying to steal little boys like something horrid from a fairy tale.
Callum knocked on the window.
Tried the handle again.
Still locked.
Stood on his tiptoes, and peered in through the glass.
The bumhead wasn't on the back seat. Or in the footwell.
'Mum?'
She wasn't in the passenger seat. And Dad wasn't behind the wheel. The car was empty.
'Hello?'
Another boom of thunder, loud enough to make him jump. They'd left him. They'd run away and left him.
How could they _leave_ him?
Callum's bottom lip trembled.
He backed against Dad's car. 'Dad?'
They couldn't have left him. They _couldn't_.
It wasn't his fault he needed a wee...
'Mum?'
And what if the Slug came back? A drop of rain burst against the lumpy tarmac.
What if the Slug was _waiting_ for him?
'Please...'
Another drop. Then another. And another. Thumping down on the car roof like the feet of a tiny monster. Soaking through his hair and his T-shirt.
Maybe...
Maybe they'd all gone for a wee too? But then he'd have seen Dad and Alistair, wouldn't he? In the Gents?
Or maybe they were in the caravan?
The breath rushed out of Callum, replaced by a smile. Yeah, that was it: they were in the caravan making cups of tea.
What an idiot. Of course they were.
Boiling the kettle on the little gas cooker.
He ran to the caravan's door. Twisted the handle and climbed inside. Clunked the door shut behind him.
Only there was no one there.
The smile died.
Callum checked under the table, checked the loo, he even checked the cupboards.
No one.
'Mum?'
A flash of white turned the caravan's insides black-and-white, then the thunder roared, rain clattering against the roof. Callum blinked. Rubbed a hand across his eyes. Stared out through the window at the front of the caravan – where the folding table and the benches that turned into Mum and Dad's bed were.
Someone was out there. A figure in the rain: big and hunched, moving with slow lumbering steps.
The Slug.
Callum ran for the caravan door and hauled the handle up, locking it. Backed away.
Another flash, followed by a deafening crash, like someone had jammed a metal dustbin over his head and battered it with a hammer.
He dropped to his knees and scrambled under the table. Curled up against the wall.
Don't move. Don't make any noise. Quiet and still as a mouse.
Outside, something scratched along the caravan's walls. It started over by the chemical loo, grinding and squealing across the metal, working its way slowly around, behind him, and past to the caravan's door.
Stopped.
Callum stared.
The handle twisted. Not far. Just a teeny weeny bit, till the lock stopped it. Twisted again. Then silence.
Maybe the Slug had given up? Maybe he'd gone away? Maybe he'd—
The whole door shook – banging and clattering in its frame.
'No!' Callum wrapped his arms around his head and bit his bottom lip till he could taste pennies. 'Go away, go away, go away...'
Then the noise faded, leaving nothing behind but the battering drone of rain on the caravan roof.
The Slug had given up.
He had to.
The caravan was _locked_ , he couldn't get in.
A trembly sob rattled its way out of Callum. Safe.
And then that dark slimy voice crept through the caravan wall, as if the Slug's lips were right up against it. _'Your mummy and daddy don't love you any more. They say you're ugly and stupid and useless and they don't want you. So they've given you to me.'_
No. They wouldn't do that. They wouldn't leave him.
They couldn't...
_'You're_ mine _now, little boy. You belong to me.'_ Scratching noises against the wall. _'Now open the door and let me in.'_
A hand on his arm. 'Gah!' Callum flinched.
Franklin frowned at him. 'Are you OK?'
He let out a shuddery breath, looking down at the photo of the four of them in their holiday clothes. 'What?'
She pointed at the photo. 'I said, "You've got an identical twin?"'
He clicked the wallet closed and slipped it into his back pocket. 'A long time ago.'
##
Hairy Harry loomed over the wrinkled body on the cutting table, humming away to himself. A huge breezeblock of a man, with rounded shoulders and a bit of a gut on him. He'd tucked the last six inches of his Victorian-style beard into the top of his apron. A blue-camouflage bandanna covered the top of his head, his long furry ponytail poking out the back of it. Hairy Harry's voice was surprisingly soft and warm for someone who looked as if they ate live badgers. 'Now _that's_ interesting...'
He reached into the open body cavity, coming out with a chunk of shrivelled black, holding it aloft like that baboon did at the start of Disney's _The Lion King_. 'Have you ever seen a liver look like that before, all dried out and wrinkly?'
Lucy shook her head and made another note on her clipboard.
'Fascinating.'
They'd laid the body out on its back, not so much uncurling the limbs as snapping them off at the dry brittle joints. Legs and arms, positioned either side of the smoke-coloured ribs.
Franklin had her own arms folded, voice so low it was barely a whisper. 'At least this one doesn't smell as bad.'
Hairy Harry went back in, coming out with what looked like a dehydrated snake. 'Well, well, well...'
Mother and McAdams stood off to one side, heads together, McAdams poking away at his mobile phone as she talked in hushed tones. Every now and then, she'd look up and stare at Callum. Then go back to conspiring with her poetry-spouting sidekick. Probably trying to figure out what crappy job to punish him with next.
'Amazing, when you think about it.' Hairy Harry stuck his gloved hands on the hips of his purple scrubs. 'The only internal organs still attached are the heart and the lungs, everything else has been taken out, preserved, then put back in again. It's almost impossible to tell cause of death from the soft tissue, because there isn't any – it's all like beef jerky.'
The mummy's ribcage lay on a trolley against the wall, its covering of leathery skin too dried-on to remove like in a normal post mortem.
'No external sign of trauma, other than the discolouration around the throat – which _could_ just be pigmentation from the preservation, but looks more like ante-mortem bruising to me. And then there's this.' He held up a little jar full of tiny discoloured spheres and gave it a shake, making them rattle against the glass. 'You'll need to get it tested, but unless I'm very much mistaken, it's silica gel. The kind of thing that comes in those little sachets they stick in bags, shoes, and handbags to sook up moisture and stop them going mouldy. His mouth was stuffed with it. More in the oesophagus, trachea, and sinus cavities. We'll have to rehydrate the stomach to find out, but I'm willing to bet we'll find some there too.'
Mother wandered back to the table. 'Excuse me, Dr Jenkins, I have to borrow Detective Constable MacGregor here.'
Oh. That didn't sound good. Whatever horror she and McAdams had come up with, it was about to spatter down on Callum's head.
'Please, it's Harrison. And by all means. The young man's a bit of a fidget anyway.'
Everyone's a critic.
She pulled on a smile. 'Thank you.' Then headed for the exit. 'Come on, Constable.'
Here we go.
Callum leaned closer to Franklin. 'Try not to punch anyone else, OK?' And followed Mother out, through the changing room, past the rows and rows of refrigeration units, across the reception area, and out into the rain.
She shrugged her shoulders up around her ears and hurried across the puddled tarmac to her battered Fiat Panda. Hurled herself in behind the wheel and beckoned at him from the safety of the car.
What would it be: door-to-doors in the freezing downpour? Digging into the archives for some obscure file that hadn't been seen for three generations? Talking to small children about road safety? Or maybe she was just going to fire him?
He high-stepped between the water-filled potholes, collar pulled up against the rain, and clambered in the passenger side.
A furry penguin hung from the rear-view mirror, along with a yellow air-freshener that smelled of chemical lemons. Inside, the car was a mess. Mud, grit, gravel, and old magazines in the footwells; plastic bags, a collection of cardboard wine-carriers full of empties, and for some bizarre reason a quarter-size inflatable sheep with sunglasses, littering the back seat. Dust coating the dashboard like a furry blanket. The bottles clinked and rattled as he thumped the door shut.
Ooh, sodding hell: it was like climbing into a very filthy fridge. Cold air nipped at his ears.
Mother stuck her hands in her pockets, her breath fogging in front of her face. 'Callum, Callum, Callum... What am I going to do with you?'
Oh great. She'd dragged him all the way out here for a bollocking. Could they not have done it inside in the warm?
'Thought I told you not to lead our new girl astray? And what do I find? She's running around assaulting detective sergeants on DCI Powel's Major Investigation Team. Care to explain yourself?'
What? 'How is this my—'
'I had Powel on the phone this afternoon, and he wasn't a happy hedgehog. Says after the assault you waded in and interfered with the victim – to wit one DS Jimmy Blake. Got him to change his story and say he slipped and battered his _own_ nose to a wee bloody lump.'
'All I did was point out that the whole thing would be caught on the mortuary's CCTV system.' A shrug. 'For some reason, Blakey wasn't keen on anyone seeing it.'
'Right.' Mother nodded. Then sighed. 'Callum, I'm all in favour of sticking up for the team, I really am...'
'But?'
'But probably better get a copy of the footage. Just in case Powel or Blakey decide to make it disappear. Blackmail only works as long as you've got the negatives.' She grinned, then dug a paper bag out of her fleece pocket. 'Have a jelly baby. Hell, take two.'
He did. An orange and a yellow.
Mother shoogled down a bit in her seat and helped herself to a red one. 'And when you get the footage, pop past my office with it. About time someone tried to introduce Blakey the Octopus's nose to his rectum by first-class fist-express; I'm going to get some popcorn in.'
'Yes, Boss.' He popped the yellow baby into his mouth, chewed on its lemony sweetness.
'I don't know what to make of you, Callum, I honestly don't. One minute you're this vast pain in my backside, and the next you're saving Franklin from herself.'
He ripped the head off the orange baby. 'I didn't take a bribe from Big Johnny Simpson. Talk to Professional Standards – they're looking through every penny I've got. Yes: I cocked-up the crime scene, but I didn't do it on purpose.'
'Hmmm...' She chewed in silence for bit.
A squall of wind rocked the car, rain buckshotting the roof, setting it ringing.
Mother devoured another baby. 'They're going to grab this case off us if they can.'
Of course they were.
'Two victims mummified and a third brining, ready for smoking? That spells "serial killer" in eight-foot-tall flashing neon letters. There'll be a media outcry, public panic, press briefings, idiots hanging about outside Divisional Headquarters doing serious pieces to camera...' A yellow jelly baby lost its life. 'They'll want a superintendent running it.'
Callum wrote his name in the dashboard dust. 'Yes, but a superintendent won't want to get their hands dirty, will they? No, they'll want someone _else_ to do the actual police work, in case it all goes horribly wrong. Plausible deniability.'
'Oh goody, a poisoned chalice. My favourite.' She held the paper bag out again. 'We're fighting for this one, Callum. It'll probably be the last chance Andy gets to put a killer away. I won't let them take that away from him.'
'We should run a dental records match on Glen Carmichael and his two mates. Just in case.' He popped a green jelly baby in, feet first. 'And Powel's got a forensic psychologist down to consult on his severed feet, Dr McDonald. She was the one they brought in to work the Birthday Boy case? We could tap her for some Behavioural Evidence Analysis.'
'What's that when it's at home?'
'They're not allowed to call it "profiling" because of the TV. Might help?'
'Not if it's Glen and his mates who're the killers...' A shrug. 'But what the hell. We'll get DNA and a facial reconstruction on the go too. I'll fight with our esteemed masters about the budget later.' She put the sweeties away. 'Anything else?'
Callum wiped the dust from his fingertip onto his trousers. 'When you dragged me out here, I thought you were going to fire me.'
'Did you?' A shrug. 'I just fancied a jelly baby – they always taste funny in the mortuary. Like death.'
Sharp salty cheese, soft claggy bread, smooth silky butter, and the tangy vinegar crunch of Branston Pickle. Callum sat in the APT lounge and chewed.
Elaine had stuck another little note in with his sandwich. Today it was a lumpy drawing of a flat fish, with a speech balloon above its head: 'YOU'RE MY SOLE MATE!', with the subtitle, 'BARRY THE FISH IS TERRIBLE AT PUNS', and a lipstick kiss.
He smiled at Barry, then tucked him into his jacket pocket – ready to join the others when he got home.
A copy of _Hey You!_ magazine lay on the coffee table, all shiny and shallow. Apparently some plastic-faced, talentless, Z-list nonentities were celebrating the first anniversary of the renewal of their wedding vows! Picture exclusive! Oh my God! How exciting!
No wonder people turned into serial killers.
Still, it was his own fault for finishing _The Beginner's Guide to Shoplifting_ that morning, instead of saving it for lunchtime. Could've had something decent to read instead of this.
He flipped the magazine open to a big photo spread of Mrs Plastic Face and her equally gormless-looking husband of eighteen months. Eighteen months married and they'd already reached the heady milestone of a vow-renewal anniversary.
Someone grunted their way down into the couch on the other side of the coffee table.
Callum took another bite. 'According to this, she's just signed a publishing deal: two million quid for four books.'
'How is that fair?' McAdams sighed. 'A book deal for an idiot who can't write her own name, / The public should know better, but they'll buy it just the same, / The publishers will lap it up, to boost their bottom line, / And if they'll publish crap like that, why won't they publish _mine_?'
Callum flipped the page again. 'Move over Pam Ayres, we have a new Poet Laureate.'
'Shouldn't you be doing something?'
'I am. I'm eating the sandwich my pregnant girlfriend made me for lunch.' He held up a finger. 'And before you start: I've already got the DNA sent off from all three bodies, got Lucy to X-ray their heads for dental chart comparisons, contacted Dundee University's facial reconstruction bods, asked the media department to send out "have you seen these men" posters for Glen Carmichael and his mates, and Dr Alice McDonald has agreed to pay us a visit as soon as she's finished drafting her preliminary report on Powel's severed feet.' Another bite of cheesy pickly goodness. 'So yes, right now I'm eating my lunch and reading about vacuous nonentities who spent more cash on a vow-renewal anniversary celebration than you or I will make in a year.'
'Just because Mother's softening on you, doesn't mean _I_ am, Constable. And for the record: summary narrative is the hallmark of a lazy writer.'
He turned the page. 'Ooh, look here: it says she's bringing out a line of perfumes, that'll be nice, won't it? Silicone Implants → la Botox, a fragrance for women.'
'Fine.' McAdams stood. 'When you've finished your meagre repast, I want those dental records checked. And find out who they bought the flat from. Maybe he's the one in the bath. God knows I'd happily kill the idiot who sold us our house.'
'Sarge?' Franklin poked her head around the door. 'Sorry, but there's a Dr McDonald in the observation room asking to see the team. Says she's consulting?'
'That's me.' Callum popped the last chunk of sandwich in his mouth and sooked his fingers clean. Flipped the magazine shut and stood. 'Feel free to tag along, if you like.'
He sauntered out, past a frowning Franklin, and down the corridor into the observation suite. It was subdivided into booths by a series of half-height partitions, each area looking out over one of the dissecting room's twelve cutting tables. The booths all had their own whiteboard, DVD recorder, collection of uncomfortable plastic chairs, and TV screen.
Dr McDonald was sitting cross-legged on the floor right in front of the TV, still wearing her pink scrubs and stripy top, elbows on her knees, hands on her cheeks – holding her head up. Like a little kid watching cartoons. In front of her, the screen had a top-down plan view of the cutting table, a wrinkled leathery body lying dead centre curled up on its side. Figures flickered and swam around it, moving impossibly quickly, lurching in and out of frame.
She'd swapped her mortuary-issue wellies for a pair of red high-tops, and added a pair of glasses to her ensemble. The fast-forward post mortem reflected in their lenses.
She looked up as Callum walked in. 'I've watched it five times now.'
He waited, but nothing else was forthcoming. No babbling. No non sequiturs.
OK...
She unfolded her legs and stood. 'I'll need to see the crime scene.'
'I can probably swing that.'
McAdams marched into the room, followed by Franklin. Still no sign of Mother.
A big smile and McAdams stuck his hand out for shaking. 'Detective Sergeant McAdams, you must be Dr McDonald.'
She looked at the offered hand as if he'd grown a vast pale hairless spider at the end of his arm.
The awkward silence stretched.
He lowered his hand. Stuck it in his pocket instead. 'This is DC Franklin.'
'Before we start, here's how this works,' McDonald walked to the whiteboard and wrote 'VICTIMOLOGY' on it in red marker, 'I give you a series of educated guesses, based on the information you give me. If I don't know something I'll mark it as an assumption and you have to take anything based on that with a whole carton of salt. Agreed?'
'You're going to profile our serial killer?'
'OFFENDER BEHAVIOURAL INDICATORS' went on the board next.
'No, I'm going to give you educated guesses, remember?'
'CRIME SCENE INDICATORS'
McAdams leaned back against the partition wall. 'Go on then, guess away.'
'PSYCHOLOGICAL GEOGRAPHY / BOUNDARIES'
'From what we know right now, our suspect is _probably_ a goal-orientated killer. It's possible preserving the victims turns them into some kind of fertility totem, but I don't think he kills them for sexual release. He kills them so he can mummify their bodies. That's his goal – it means something to him. _What_ is the bigger question.'
'I'll settle for _who_.'
'Statistically it's going to be a white male, mid-twenties. He'll have access to a facility for smoking meat, and or fish, and experience in using it. You don't jump right into this kind of thing without practice.'
McAdams snapped his fingers at Callum. 'I want a list of every smokehouse in a twenty... make it fifty-mile radius.'
Dick.
Callum made a note anyway. 'What about Glen Carmichael, Brett Millar, and Ben Harrington? Any chance the three of them are killing as a team?'
Dr McDonald looked back at the TV, with its flickering ghosts. 'There's a chance, but it's not very likely. Two of them, maybe – one dominant, one submissive – but three would be _very_ unusual. It's hard enough getting three men to agree on what pizza toppings to order, never mind how to select, kill, and preserve their victims.'
Fair enough.
She leaned in closer to the screen. 'Our offender's an artisan and an artist. This kind of work takes time, care, and skill. He's probably unattached, lives alone where no one can interfere with his work. He'll drive a big car, or a van – he needs to be able to transport the bodies.'
Franklin shook her head. 'We found one of them in the boot of a wee Kia Picanto – small four-door hatchback. You don't need that much space.'
'Not when they're mummified, but while they're still alive? You need more room.'
And Franklin explodes: in three, two, one... But she didn't. She just nodded.
'His post-murder activities are highly ritualised too. Removing the organs and preserving them separately, then stitching them back into the body cavity.' She wrapped the fingers of one hand into her hair, fiddling with the curls as her eyes narrowed and her voice dropped off to a murmur. 'You don't just mummify people for fun, do you, no you don't, you do it because you want them to live on in the afterlife, you deify them...' She let go of her hair and straightened. 'I wouldn't be surprised if there was some sort of religious upbringing.' She pointed at the whiteboard, where 'PSYCHOLOGICAL GEOGRAPHY / BOUNDARIES' was written. 'I need to know where the victims came from before we can work out where he's likely to live.'
Callum nodded. 'We're working on it.'
'Also,' McAdams took a marker from the shelf beneath the board and uncapped it, 'we need to decide what we're calling our boy. Can't have a serial killer novel with an unnamed antagonist.' He printed 'IMHOTEP' right in the middle. 'Before the tabloid newspapers come up with something more _lurid_.'
'Ah...' Dr McDonald bit her top lip. 'It's a nice thought, I mean I know we've got to call him something, but "Imhotep" doesn't actually work, does it, because Imhotep was Egyptian and Egyptian mummies are always preserved lying flat, and the curled body posture our suspect uses to pose his victims is more reminiscent of ancient Peruvian burial techniques, which results from a completely different cultural and religious background.' She shrugged. '"Paddington" would probably be more accurate, you know, strictly speaking, because of the Peru connection, I think we should definitely call him Paddington, it just makes a lot more sense.'
'And one final thing.' McAdams smiled. 'Aren't you going to say it?'
Dr McDonald wrote 'PADDINGTON' on the board. 'Aren't I going to say what?'
'It's a cliché of the genre, but the profiler always says it at the end of the briefing.'
A frown. 'Nope, you've lost me.'
'He _will_ kill again!'
'Of course he will.' McDonald stuck the lid back on the marker pen. 'He's a serial killer, it's what he does.'
## — Imhotep —
"Well, well, well," the God Wolf growled, "if I've not just caught the tastiest little morsel in the whole dark world."
"You can't eat me!" gasped Imelda. "I'm made of bones and stones and glass and groans, and if you eat me you'll get a _terrible_ tummy ache and die!"
The God Wolf smiled at her. "I'll take my chances," he said. And swallowed Imelda whole.
R.M. Travis
_Imelda's Miraculous Dustbin_ (1999)
_Stay away from ma b*tches, they ain't down with no snitches,_
_I got me my riches, givin' punks like you stitches!_
Donny '$ick Dawg' McRoberts
'Livin' Free Or Dyin' Tryin''
© Bob's Speed Trap Records (2014)
##
The God-In-Waiting sways gently in the smoke, head down, hands making delicate figure-eight patterns as it swings. No movement of its own, just that final rattling breath, then peace and stillness. Grace and purity.
It's time.
The racks beneath the God-In-Waiting are full of fish, hanging from their poles like the divinity above. He removes the poles, stacking them on the rack next door to cool. It'll be a good batch of smokies. They always are when a new god comes into being. Must be the air.
Or maybe it's the cleansed body, hanging above them as they smoke? Maybe it's the juices that drip like tears from the body as it takes on its final form? Whatever it is, the result is _excellent_ fish.
Next is the scraper – just a plank of wood fixed on the end of a broomstick – he uses it to push the smouldering embers away, heaping them up against the far wall. Then stands beneath the God-In-Waiting.
It's beautiful...
**_Once Upon A Time_**
The man hanging on the wall has got nothing on but a kind of nappy, wrapped around his waist. His skin is a dark, rich wood, polished so much it glows against the cross. Someone's made him a hat of barbed wire, which must hurt something horrible.
A wobbly voice fills the air, echoing back off the church's stone walls. _'Sanctus, Sanctus, Sanctus, Dominus Deus Sabaoth...'_
It's a pretty sound – even if the words are just made-up – floating above the pews, wrapping around the big wooden man. Maybe it makes him happier if people sing to him? He looks very sad.
Father's over by the altar, talking to the priest man. Both of them dressed in black, like crows – though the priest man's got on a kind of dress. Both are wearing those little white things around their necks. Dog collars. Both pretending to be something they're not.
_'Pleni sunt caeli et terra gloria tua. Hosanna in excelsis...'_
The man's been stuck to the cross with big metal nails, and there's holes in his side. Maybe that was mice? There's mice in Father's house and they eat holes in everything. Scurrying about in the dark. Leaving their little black presents behind.
_'Benedictus qui venit in nomine Domini. Hosanna in excelsis—'_
_'NO! NO! DAMN IT, OLIVER!'_ A man's voice, not pretty and floaty, but hard and grating. _'How many times? It's pronounced, "ex-chel-cease". We're going to stay here and do it again and again until – you – get – it – right!'_
Father looks up at the gallery that runs above the back six rows of pews, where the organ is. Then down at him. 'Justin, thumb out of the mouth, eh champ?' He smiles. 'You're a big boy now.'
Justin's not his _real_ name: it's from Father's favourite album, about a little boy who turns into a rabbit and has to save the world from the king of dead things. And Justin's as good a name as any.
He takes his thumb out of his mouth and wipes it dry on his T-shirt. 'Sorry, Father.'
'That's my boy.' Then Father shakes hands with the priest man and wanders down the apse. Ruffles Justin's hair. 'Come on, slugger, time to go home.' He turns and waves at the priest man as they leave the church.
_'Sanctus, Sanctus, Sanctus, Dominus Deus Sabaoth...'_
Down the steps in the warm sunshine, one hand on Justin's shoulder. Steering him to the car with its little Scottish flag fluttering on the end of the aerial. 'In you get.'
Justin does what he's told.
Gravel scrunches and crunches under the wheels as they leave the church grounds.
'Did you hear the singing, Father? Wasn't that—'
The slap is as hard as it is quick, snapping his head to the side, the sound like a gunshot going off.
'Don't you dare embarrass me like that again. Sucking on your thumb like a _baby_. That what you are? A baby?'
He blinks the tears back. Bites his lip. Lets the burning needles sink into his cheek. Don't cry. Feed off the heat. Don't cry. It'll only make it worse.
'You want to wear nappies and sit in your own filth again? IS THAT WHAT YOU WANT?' Little flecks of spit settle against the dashboard. 'ANSWER ME, DAMN YOU!'
Justin takes a deep breath.
Don't cry.
Feel it burn. Own it.
He stares down at his hands, curled in his lap. 'No, Father. Sorry, Father.'
'Good boy.' And just like that the storm passes, the clouds' shadows slip away and Father smiles at him again. 'Come on, why don't we go get some ice cream? We can bring some back for Mummy, she'll like that, won't she?'
Justin nods, even though it's not true. New Mummy doesn't like anything. She just cries all the time.
'And, slugger?' Father ruffles his hair again, the fingers warm and hard where they dig into his scalp. 'You stay away from church music, it's nothing but lies. See these?' He lets go of Justin and unhooks the white band from around his throat. Shakes it like a dead mouse. 'They call them a "dog collar" for a reason. They choke you. There's a chain that clips onto them, so you can go walkies. Because it's all lies: the churches, the hymns, the bible, the whole God-bothering holier-than-thou, deviant filth-mongering lot of them. Lies and liars.'
Justin doesn't move.
This can go one of two ways, and one of them ends with screaming and bruises and getting locked in the Naughty Cupboard – peeing blood for a week.
Father clicks on the car stereo, and the album picks up where it left off. A hissing of drums, then the man's voice comes over the top, quiet as treacle. _'You have to hide right here, right now, you have to stay so still, / Cos Justin, little rabbit boy, the night-time means you ill, / There's monsters here, there's monsters there, and they're prowling through the gloom, / Stay still and oh so quiet, or these woods will be your tomb...'_
Father squeezes his shoulder. 'Come on, champ, let's go get that ice cream.'
But not this time.
**_Now_**
He stares up at the God-In-Waiting. It's not as beautiful as the wooden man on the cross, not yet, but it will be. It'll be better. So much better than a dead thing carved from a dead tree, hanging in a dead building.
It'll be a god...
He unhooks the rope from its cleat on the wall and takes the strain. Lowers the body, hand over hand, until it rests on the freshly cleared floor. Picks it up off the hot stone slabs – it isn't hard, the cleansed remains weigh almost nothing. The weight of sin is gone, purged, purified.
The God-In-Waiting is pale and soft, but that will change.
Everything will.
##
Callum shuffled the printouts together, then slipped them in a plastic folder. 'Thanks.'
Lucy shrugged. 'No probs.' A grin widened her face. 'Tell you the truth, I thought this was going to be a waste of time, but no way. My very first serial killer!'
He tucked the folder under his arm. 'Don't worry, there'll be plenty more where that came from.'
Franklin was in the hallway, outside the lab, standing with her back to the world – forehead resting against the wall, phone clamped to her ear. 'No... Because I don't... How am I supposed to know, Mark? I'm not a mind reader... No... I didn't... Urgh. Just forget it. Doesn't matter.' A sigh. 'OK, OK. I'll call you when I find out.' She hung up and stayed where she was.
Callum cleared his throat. 'You ready?'
She stiffened. Put her phone away. Turned. 'Do you eavesdrop on all your colleagues?'
He shook his head and marched off down the corridor. 'Don't know why I bother.'
Franklin didn't catch up to him till the car park, weaving her way between the puddles in the rain. She came to a halt in front of Mother's manky Fiat Panda. 'Is her face normally that colour?'
Mother was on the phone, eyes scrunched up, cheeks all flushed. Her mouth kept starting in on sentences, but never seemed to get more than a word or two into them before shutting again. Her other hand dug its fingers into her forehead, as if she was trying to force them through the skin into the bone beneath.
'Yeah...' Callum sidled towards the pool car they'd arrived in. 'Whatever that is, there's going to be repercussions and fallout. Don't know about you, but I want to be long gone before then.'
Franklin pushed past him to the pool car's driver's door. 'Keys.'
'Sorry?'
'Keys. Give me the keys, I'm driving.'
He raised an eyebrow. ' _You_ want to drive?'
'Just give me the damned keys.'
'So I'm going to sit in the passenger seat, and you're going to drive _me_ around? Like I outrank you?'
'I'm not spending the rest of the day being dragged all over Oldcastle so you can run "little errands" like yesterday. Now: keys.'
Fair enough. He dug them out of his pocket and tossed them across to her. Walked around to the other side and climbed in out of the rain as she slipped behind the wheel.
Callum settled back in his seat. Stretched out a little. 'I could get used to this.'
Franklin took them through the rolling sea of ruptured tarmac and out into the industrial estate again. Past the boarded-up units, and onto the main road, heading back along the dual carriageway. The City Stadium loomed above the houses on the left, a lopsided bird's nest of steel, concrete, and glass, lording it over the 1950s-style rows of semidetached two-up-two-downs.
It was nice not having to do all the driving for a change. Just sit back and watch the scenery slip by. Even if it _was_ all grey and rain-streaked.
He dug his leprechaun-sized Mars Bar out of his pocket and took a tiny bite. Sweet, sticky, and chocolatey. 'This your first serial killer?'
'Of course it is.'
'Number four for me.'
Franklin looked at him across the car, one eyebrow raised. 'Four serial killers? Yeah, right.' She took them around the roundabout, the granite blade of Castle Hill just visible between the tall concrete buildings ahead. 'I'm not an idiot, constable. There's absolutely _no way_ you've already worked three serial killer investigations.'
A big flat-fronted building went by on the left, little windows in a big granite façade.
'That's Woodrow Hospital. Four years ago, we got complaints of missing dogs in the area. Didn't really pay all that much attention.' He scooted down in his seat, following the hospital in the wing mirror as it faded into the distance. 'Then someone's granny disappeared. Thought it was dementia to begin with, happens a lot with older people: they get confused and they wander off. Then another one went missing. And another. Took us six little old ladies to realise something was wrong.'
The looming green mass of Camburn Woods poked out above the rooftops, getting bigger.
Callum finished off his micro Mars Bar. 'Who's Mark?'
Franklin's jaw tightened. 'Mark is none of your business.'
'Turns out Pawel Sabachevich's parents moved over here from a little village outside Krakow when he was six years old. They brought his maternal grandmother with them. She wasn't very nice to Pawel. And twenty-three years later he abducted, raped, and strangled eight old ladies, dismembered their remains and fed them into the incinerator at Woodrow Hospital. He worked there as an assistant radiologist.' Callum crumpled up his chocolate wrapper and stuck it in his pocket. 'Nice guy. Well, if you overlook the whole murderous raping scumbag bit.'
The diggers were still at it on the huge flanks of Camburn Round-about – making mountains of mud, while a crane erected a lopsided metal trellis and high-viz figures sank into the mire. 'Then there was Ian Zouroudi.' Another shudder followed in the footsteps of the first. 'Gah... The whole team needed therapy after _that_ one.'
'Just because I'm new and a woman, it doesn't make me an idiot.'
'Never said it did.' Camburn Woods reared up and swallowed the car, the thick branches reaching out over the dual carriageway on either side, leaves dark and dripping. 'From what I heard, it's all the mercury in the ground around here. Too big a dose and it screws with brain development.'
'Mercury.'
'We made most of Britain's mustard gas, right here in Oldcastle, for the First World War. Apparently it took a lot of mercury. And now we're the serial-killer capital of Europe. Pretty high on the list for birth defects too.' He sniffed. 'That was a fun day at antenatal class.'
Ruined buildings lurked in the woods to either side of the road, slowly dissolving into the bushes and ivy.
She frowned across the car at him. ' _Three_ serial killers?'
The world opened up in a blast of grey as the road emerged from the depths of Camburn Woods.
'Straight through the next roundabout and it's the third road on the right.' Two parallel lines of shops and flats followed them along the road, at least a quarter of them boarded up. Bookies and charity shops rubbing shoulders with places to sell your gold or pawn your kids' toys.
Callum pointed through the windscreen. 'That's us at the traffic lights.'
Franklin pulled into the turning lane and they sat there with the indicators clicking, waiting for the filter. 'So who was number three?'
'The Birthday Boy. You must've heard about that one: it was in all the papers. Sicko snatches girls just before they turn thirteen, takes photos as he tortures them to death, then turns the pics into homemade cards and sends them to the girls' parents every year on their birthday.'
She glanced across the car. 'And it's all because of the mercury?'
'Meh, what do I know?'
The lights changed and they pulled across the dual carriageway and into a curving street with a collection of cafés, hardware shops, a Sue Ryder and a British Heart Foundation, a newsagent, and finally the reason they'd come.
Franklin nodded. 'There we go.'
The McKibben Dental Practice had a frosted shop window, presumably so you couldn't see their victims writhing in agony, with posters either side of the main door depicting unfeasibly attractive people grinning away with unfeasibly white teeth. Franklin grabbed the nearest parking space, three doors down. 'I can't believe you've worked three serial killers.'
He clambered out into the rain. 'Go have a rummage in the archives at DHQ, there's stuff in there that'll make your hair curl...' He bit his lip. 'I didn't mean that to be—'
'I know what you meant.' She locked the car. Followed him down the pavement to the dentist's. 'And yes, it _is_ naturally this curly.'
He shrugged. 'I quite like it.'
'Are you remembering what happened to Blakey the Octopus?' Franklin pushed through the door and into a warm reception room with seats around the walls of a little waiting annex off to one side. The faint aniseedy tang of oral disinfectant tainted the air. A rack of magazines was mounted on the wall – all the issues considerably newer and classier than the ones Professional Standards had – surrounded by more posters of halogen-white teeth.
'For your information, Detective Constable Franklin, I have a partner I love and she's pregnant with my child. So don't flatter yourself. I've got no intention of groping your backside or anything else.'
An unfeasibly blonde receptionist showed them her unfeasibly perfect teeth in a broad smile. Her voice was unfeasibly cheery too, but nearly as shrill as a dentist's drill. 'Welcome to the McKibben Dental Practice, how can I help you today?'
Callum flashed his warrant card. 'I phoned earlier. We need to speak to someone about Glen Carmichael's dental records.'
'Boss? We've got a match.' Callum switched his phone to the other hand and tucked the folder under his arm again. Rain dripped off the concrete portico that covered the shopping centre's rear doors, darkening the steps down to the car park. 'Took us three lots of dentists to get there, but according to the Leighton Road Dental Association in Blackwall Hill, our body in the bath is Ben Harrington. He's the one in the photo with the auld mannie haircut, glasses, and walrus moustache.'
Silence from the other end of the phone.
'Boss?'
The shopping centre car park was nearly empty, just a handful of old cars and a Shopper-Hopper bus picking up a load of OAPs with their wheelie trollies and battered umbrellas. Franklin was down there too, marching about in the rain, one hand making violent stabby motions in the air as she dumped a shedload of angry into her mobile.
Mother's voice sounded far away, muffled, as if she was talking to someone else. _'You can shift Harrington from the suspect column to the victim one. No, he's definitely dead.'_ Then she was back. _'He was a Blackwall Hill boy, and seeing as you're in the neighbourhood...?'_
Oh joy. 'Death message?'
'Good lad.'
'Dr McDonald wants access to the crime scene.'
_'Meh. The Smurf Patrol have finished with it, so why not? Make sure she comes up with something useful though.'_
'Do my best.' He hung up.
The Shopper-Hopper gave its diesel roar and pulled into the traffic.
Franklin did another lap, jabbing away like she was trying to stab and bludgeon someone all at the same time.
With any luck she'd get it out of her system and there'd be none left to batter him with.
But just in case...
Callum nipped back into the centre and grabbed a couple of fancy pieces and two takeaway teas from the Costa by the lifts. Hunched his shoulders and hurried through the doors, into the rain.
By the time he reached the pool car, she was behind the wheel again, dripping and glowering.
So much for getting it out of her system.
He slipped into the passenger seat and held out his peace offering. 'Here. Tea, milk no sugar, and... Tada!' One paper bag. 'Got a billionaire's shortbread and a rocky-road brownie. You choose.'
The frown didn't shift. 'What's billionaire's shortbread?'
'Like a millionaire's, but there's bits of broken-up Crunchie in there too.'
She went for the shortbread, chewing with her shoulders dipped as the rain thumped down on the car roof. 'Not that it's any of your business, but Mark is my partner.'
Poor sod. Living with Franklin must be like trying to cross a minefield on a pogo stick every day. Blindfold. While sadists threw burning squirrels at you.
Mark was probably up for a medal. Or beatification.
Callum had a bite of brownie, sickeningly sweet, and washed it down with hot tea.
Franklin cracked a chunk off her shortbread. 'His work's hosting a dinner dance for charity Friday night, and apparently _I'm_ being unreasonable because I can't tell him if I'll be there or not. Doesn't matter that I'm working a mass murder, no, the _important_ thing is making him look good in front of his bosses.'
'Actually, a mass murder is when you kill four or more people in the same location without much of a gap between...' He cleared his throat. 'Sorry.'
'You're all the bloody same, aren't you?'
'Sadly.' A slurp of tea. 'What's he do, this Mark of yours?'
'Investment banking.'
And all sympathy for the guy died right there.
She finished her shortbread. 'It's not my fault I got transferred to Oldcastle, is it? I mean, it's not like I can commute here from Edinburgh. I'd have to get the five-thirty train every morning and I _still_ wouldn't be here in time for a seven o'clock start.'
Callum balanced his tea on the dashboard and pulled out his notebook. Flicked through it. 'Mother wants us to drop in on Ben Harrington's parents and give them the bad news. Here we go: sixteen Brookmyre Crescent, Blackwall Hill. About five minutes away.'
'And I am _not_ giving up my career, just to play house in a flat in Portobello.' She bared her teeth, nearly as white as the dental receptionist's only with bits of chocolate stuck between them.
'I can drive, if you like?'
'Why do men have to be such selfish scumbags?'
A young mother slouched past the car, face slumped in permanent disappointment, pushing a buggy with a screaming toddler in it. Rain trickled from the straggly ends of her lank hair.
Callum had another bite of brownie. Kept his mouth shut.
Franklin sighed. Threw back the last of her tea. Then started the car. 'All I ever wanted to be was a police officer. I'm _not_ resigning. Wouldn't give Superintendent Neil Sodding Sexual-Harassment Lambert the satisfaction.'
OK, at least this was safer ground than interfering in her relationship. 'So go to Professional Standards, make a formal complaint.'
'I did. Why do you think they transferred me?' She took them out of the car park. 'Which way?'
'Left, then right onto McAskill Road.'
'And yes, I shouldn't have hit him. I know.' The scowl deepened. 'Dirty, slimy, sleazy little prick got his complaint in first. Who are they going to believe, a black woman PC, or a white middle-aged male superintendent? Because you can bet it's not the woman.'
A lot of the shops around the centre had 'To Let' signs in the window, one advertising a closing down sale. One had its frontage all boarded up and a notice thanking customers for sodding off to Amazon instead of buying their books in a real bookshop.
'That's McAskill Road: take a right.'
She did. 'It's _never_ the woman.'
The road dipped below a railway bridge, the inside scrawled with graffiti tags. A couple of older men huddled in a recess between the supports, sharing a cigarette and a litre bottle of supermarket blended whisky.
North of the line, Blackwall Hill broke out in coiled housing developments, little cul-de-sacs, and sweeping curved streets.
'Take Caldwell.' Callum pointed at the junction up ahead, past the pedestrian crossing. 'You want to deliver the death message?'
'Why, because I'm a _woman_?'
'On second thoughts, maybe a bit of compassion is in order. I'll do it. You can make the tea.' He held up a hand. 'And before you start, it's got nothing to do with "being a woman". You either deliver the death message and sit with them while they grieve, or you make the tea. One or the other. Turn right here.'
That took them onto a wide road with bungalows on either side, that bowed away to the left following the contours of the hill.
Franklin pursed her lips. 'Fine. You make the tea.'
'You sure?'
'Positive.'
It was a weird world when someone thought making four cups of tea was worse than telling a parent that their only child had been murdered. 'Brookmyre Crescent. That's us right there.'
She slowed for the junction, taking them into a dead-end road that cupped twenty or thirty houses in its coiled embrace. Some semidetached, some standing on their own. Most had been extended up into their attics, a few with converted garages, lots of lock-block driveways, wheelie bins arrayed on the pavements like guardsmen ready for inspection.
'Number sixteen: the one with the dark-blue door and hideous garden ornaments.'
Franklin parked outside it as the rain faded to a misty drizzle.
'Right, the mother's name is Christine, father is Tony. No brothers or sisters.'
She nodded. 'Christine. Tony.' Then undid her seatbelt. 'Let's do this.'
Callum followed her out into the damp afternoon gloom.
Number 16 was on the downhill side: a detached bungalow conversion with a room above the garage and dormer windows on the upper floor. Ivy growing up the wall around the door. A wooden wishing well sat in the middle of a gravel lawn, surrounded by gnomes in various rustic poses, and angry tufts of pampas grass.
Classy.
The gap between the house and next door's leylandii hedge was like a little picture postcard, looking down Blackwall Hill, across the river, and up to Castle Hill on the other side. A shaft of sunlight had made it through the heavy lid of slate-coloured cloud, turning the castle and its granite perch a warm shade of honeyed gold, all rendered in soft-focus by the drizzle.
Probably worth a fortune with a view like that.
Franklin leaned on the bell. 'Bet they're not even in.'
'Look, if you'd rather do the teas than deliver the death message, that's OK.'
'No, you idiot. There's no car in the driveway. Family living somewhere like this? They've got more than one car.'
'Maybe it's in the garage?'
She tried the bell again. 'You've never had a garage, have you? It's not for keeping your car in, it's for storing all the crap you moved out of the last house and haven't taken out of the boxes six years later.'
No answer from inside.
'You might be right.' He checked his watch – 15:40. 'Better give it another ten minutes, though. Just to be safe.'
Franklin hunched her shoulders and turned her back on the drizzle. 'I'm not standing here, in the rain, for ten minutes.'
'So we wait in the car. At least it'll be—' His phone burst into life, belting out its anonymous ringtone. 'Hello?'
_'Is that DC MacGregor, I hope so, this is the number he wrote on his business card and I mean he should know what his own mobile number is shouldn't he, mind you I suppose most people don't do they, after all, they don't phone themselves, so why would they remember it?'_ All done in a single breath.
'Dr McDonald. What can I do for you?' He followed Franklin back down the driveway.
_'Psilocybe semilanceata.'_
OK...
'What's that when it's at home?'
_'Liberty Cap mushrooms, AKA: magic mushrooms, AKA: shroooooms. We're halfway through Benjamin Harrington's post mortem and his stomach's full of them, well, not full-full, but there's quite a lot of them and they've not dissolved all that much because he must've died not long after taking them, which isn't surprising because it's still a_ lot _of mushrooms to take in one go, but there's heaps of herbs and things in there as well, only they're going to take a lot longer to identify than the mushrooms, because magic mushrooms always look like magic mushrooms, don't they?'_
Callum settled into the passenger seat. Clunked the door shut. 'Did he eat enough to kill him?'
_'I don't think you_ can _overdose on magic mushrooms, they've got an emetic effect, so you're more likely to vomit them up if you take too many, well, I suppose you could choke on your own sick, but that's not actually overdosing, is it? Anyway, they're running toxicology on the tissue samples from the two mummies to see if they've got any psilocybin in them, did you know they've got their own mass spectrometer here, it's amazing, I've never seen a mortuary with these kinds of facilities before, but Dr Jenkins says they were spending so much money sending samples away for testing that it made a lot more sense getting—'_
'Doctor!' A bit rude, but at least it stopped her. 'There's a bong in the flat where the body was found – the shrooms might be Ben's. He takes too many, dies, Glen and Brett are too stoned to help so they panic and board him up in the bathroom then do a runner.'
Franklin frowned across the car at him, mouthing the word, 'What?'
_'That's why they're rushing through the tox screen on the mummies, if there's psilocybin in the tissue samples, then we've got a link, and that's exciting, but I'd still like to see the flat if I can, can I?_ '
'Yeah. It's fine, SOCOs have finished with it anyway.'
_'OK, I'll see you there, when's good, is now good?'_
'Erm... No. We've got to tell Ben Harrington's parents that he's dead. And you're in the middle of a post mortem, remember?'
Franklin's phone launched into what sounded like Gilbert and Sullivan's ode about policemen being a poor put-upon bunch of sods. 'Yes?... What, _now_?'
_'Oh... Right, well, if you can give me a call when you've done that, that'll be great and we can get on with the geographical side of things and I suppose it won't hurt to spend a little time dealing with the severed feet case, and did I tell you we post mortemed the other mummy?'_
'Is that the one from the tip, or the car?'
_'The tip, and I think I know why it was thrown away.'_
Franklin started the car again. 'Yeah, we'll be there soon as we can.'
Silence from the phone.
'Dr McDonald?'
_'Sorry, dropped my chocolate biscuit. The mummy from the car was eviscerated and the internal organs preserved separately then stitched back inside. The body in the tip wasn't so lucky. He tried to preserve it whole, and mummification only works if you can dry out the remains faster than the microbes inside can decompose it.'_
The gears made complaining grinding noises as Franklin performed a hurried three-point turn. She stuffed her mobile into a pocket. 'Put your seatbelt on.'
He covered the mouthpiece. 'Where are we going?'
'Someone's just broken into Brett Millar's house.'
'When was this?'
'Now. Right now. Neighbour just called it in.'
_'—abdominal cavity is full of slippery moist organs and they go off incredibly quickly if you don't preserve them, that's why undertakers inject everything with preserving fluid when you die, because otherwise you'd probably burst during the eulogies, and that wouldn't be very nice for the mourners, would it?'_
'Any idea who broke in? Did the neighbour recognise them?'
'How should I know?'
_'So my educated guess is that Paddington is one hundred percent committed to the end result. He's venerating these bodies by mummifying them, but they have to be perfect. This one wasn't, so he disposed of it and started again. That also means he's learning.'_
Franklin put her foot down, sending pantile boxes whizzing past the car windows. 'Where am I going, and how do we put on the blues-and-twos in this thing?'
Callum pulled out his notebook and flipped it open, bracing his knees against the door and the dashboard. 'Walderswell Court. Right at the end, then left.' He reached out and poked a switch, setting the sirens wailing and the lights flashing.
_'DC MacGregor?'_
'Still here, Doc.'
_'Please don't call me "Doc" it always makes me feel like I'm meant to be one of the seven dwarfs and I know I'm not the tallest person in the world, but I like to think I'm a bit bigger than that, and if you think about it—'_
'OK, OK, sorry. Not Doc. You're definitely not one of the seven dwarfs.' After all, Snow White's roll call didn't go: Sleepy, Grumpy, Dopey, Doc, Sneezy, Happy, Bashful, and Bug-Eyed Crazy Weirdo Person.
_'You can call me Alice, if you like, or do you prefer to keep things on a formal footing, sometimes that's better in a work evironment, isn't it, or does it just make me seem all distant and aloof, which would be bad, because I think we should operate as a team and—'_
'No, that would be great. Alice it is.' He grabbed the handle above the door as Franklin threw them around the corner past another long sweeping row of houses. 'Go right at the end and it's second on the left.' Back to the phone. 'Was there anything else, Alice? Only we're wheeching across town trying to get to a break-in before the thieving scumbag legs it.'
_'Oh, right. Sorry. That explains the sirens and things in the background, doesn't it? I'll let you go.'_ She hung up.
'Yup. Three hundred and sixty degrees of weird.' He put his phone away as the pool car screeched around the corner and into an older, less gentrified bit of Blackwall Hill. No more lock-block driveways and formation gnomes. No more attic conversions. Just street after street of identical semidetached bungalows, bristling with satellite dishes.
Franklin waved a hand across the car. 'Kill the siren!'
He clicked the button and she hit the brakes, just before the corner, swinging around onto Walderswell Court at a sensible thirty miles per hour. The police vehicular equivalent of whistling a casual tune to kid on you're not up to something.
The houses here were just a bit smaller than the ones on the road outside, jammed in just a bit tighter too. Number 32 was down the far end, next to a building plot. From the signage fixed to the site fencing, someone was chucking up two blocks of 'LUXURY STARTER FLATS!!!' where a pair of wee bungalows used to be.
Yeah, good luck selling those, stuck on the border between Blackwall Hill and Kingsmeath. You could see the dual carriageway from here... Wonder if that was where Brett and his mates got the idea to do up their flat on Customs Street?
Franklin coasted the last twenty feet, engine idling. 'How can we be first on the scene?'
'You drive like a maniac, what do you expect?' Callum popped open the glove compartment and took out the box of nitrile gloves, pulling two from the slot in the top like rubbery blue tissues. Tossed the box across to Franklin. 'Well, come on then.'
He climbed out into the drizzle and snapped his gloves on. Pulled out his pepper spray.
Across the road, a little old man peered out from between a pair of net curtains. Walking stick in one hand, phone in the other. That would be their informant.
Callum half crouched, half ran across the pavement and up the driveway to Brett Millar's house. No sign of forced entry on the front door. The handle was cold in his fingers... and it wouldn't budge. Locked.
Franklin flattened herself on the other side of the door, extendable baton extended. 'Well?'
'Doesn't look like he got in the front way.'
She nodded at the other side of the road. 'Then how did Nosey Norman see it to call it in?'
Good point.
Callum pointed. 'Round the side.'
A six-foot wooden fence marked the boundary between number 32 and the building site, leaving just enough space for a narrow gravel path and a full-height gate. It was hanging off its hinges.
On the other side, a bush was flattened, as if someone had fallen into it. A smear of blood on the harling, probably left by sticking their hand out to break their fall. Oh yeah, this one was a master criminal. With any luck they'd be in the kitchen making themselves a bacon buttie.
Round the back.
The kitchen door was wide open, the glass in the bottom section smashed into regular safety-sized cubes.
Franklin held up a fist, then stuck one finger up and swept it in the direction of the back door. Clenched her fist again.
Callum stared at her. 'Are you off your head? This isn't the A-Team.'
A sigh, then she slipped in through the broken door, bent almost double.
God help us.
He followed her inside.
The kitchen was ground zero for a whirlwind of tins and smashed mugs, jagged shards of plate covering the lino floor, blood-spatters of tomato ketchup on the tiles above the cooker. A shattered jar of mayonnaise lying spent against the dented fridge.
Callum crunched through a drift of Special K. 'Wow. Someone's behind on their housework.'
Franklin did the ridiculous SWAT team signs again, then crouched her way out into the hall.
He wandered after her.
The hallway was a mess of thrown coats and hurled boots, the plasterboard dented where they'd hit the walls. Franklin did a slow three-sixty, then froze and pointed down the hall. Four doors: three shut, one wide open – bangs and crashes thumping out of it. Then a computer monitor bounced off the hall carpet, the display a spider's web of fractured glass.
She crept down the hall, baton raised and ready.
It would be a druggie, off his proverbials on coke, or crack, or jellies, or smack. Sees the house is empty and bingo – tries his hand at a bit of DIY _Bargain Hunt_...
Or maybe it was someone who knew Brett, Ben, and Glen? Someone who knew they might have a stash lying about. Or, going by the destruction, someone they owed money to.
Callum flicked the safety cap off his pepper spray. 'Shall we dance?'
Franklin raised an eyebrow. Looked at him for a moment, then smiled a nasty smile. 'Foxtrot or tango?'
Good. He smiled back. 'Let's see where the music takes us.'
She barged in through the open door. 'POLICE! ON YOUR KNEES, NOW!'
He thumped through less than a breath behind her, into the heart of a disaster. The wardrobe doors were ripped off their hinges, clothes everywhere; a computer desk smashed almost beyond recognition; single bed overturned, the slats cracked and splintered like broken ribs; a disembowelled games console, spilling its electronic innards across the floor; posters torn from the wall.
And there, in the middle of the hurricane, was a man – long greasy hair dangling down his back, sunken eyes, cheekbones you could carve granite with, wrists like two bones wrapped in pink cling film. Skin so pale every vein popped out like a blue-green worm. A solid ring of love bites around his neck. Filthy hoodie, filthy tracksuit bottoms, bare, filthy feet speckled with blood.
Full-on junky chic.
He had both hands above his head – probably not helping with the rotting-cabbage stink of sweat and that stale spicy base-note of old marijuana – holding a desktop computer covered in stickers, cables and a keyboard dangling from the ports in the back.
Captain Filthy just stood there, staring at them.
Franklin whipped the baton back into first-strike position. 'PUT THE COMPUTER DOWN AND GET ON YOUR KNEES!'
He bared his brown-grey teeth.
'I'M NOT TELLING YOU AGAIN: KNEES, NOW!'
'AAAAAAAAAAAAAAAAAAAAAAAAAAAAAA!' The computer went flying, hurled full force at Franklin's head.
She ducked left, but it still caught her on the shoulder, spinning her one way while it went the other, cables flapping.
Captain Filthy lunged for Callum, arms out, hands like claws.
So he got a face full of pepper spray.
Oh crap...
Might as well have sprayed him with lavender floor polish, because Captain Filthy just kept on coming.
'AAAAAAAAAAAAAAAAAAAAAAAAAAA!'
Sodding hell.
A claw whipped past Callum's face, close enough for every dirt-caked fingernail to stand out in perfect focus. And then Captain Filthy was on top of him, snarling, little flecks of stinking saliva spraying against his face.
'Get off me!' He grabbed the guy's ear and twisted: nothing.
A dirty hand raked down Callum's cheek.
The stench of pepper spray was like a mask, choking off the air, blurring his vision as the tears started.
Captain Filthy wrapped his manky fingers around Callum's tie and pulled, trying to throttle him. Must have got a bit of a surprise when the whole thing just pinged free of its clips and came off in his hand, because he reared back, staring at it. Maybe he was distracted by all the pretty colours?
So Callum took a leaf out of Dugdale's book and grabbed Captain Filthy's crotch, digging his fingers into the tracksuit bottoms and crushing the contents. Twisting them.
Still nothing.
Then Captain Filthy threw the tie away and lunged.
Oh dear Jesus, this was it, he was going to die.
Those brown-grey teeth flashed in front of his face, dirty fingers digging into his cheeks.
Callum heaved upward, and the guy's head lurched past his, bashing into the bedroom floor, enveloping his face with that stinking greasy hair. Then knives and bees exploded through Callum's left ear, jerking him to the side, and warmth trickled down and around the back of his head.
'AAAAAAAAARGH! GET OFF ME!'
And just like that, the weight was gone.
'Sodding... _fuck_!' He sat up, one hand clasping his left ear – the fingers slippery and sticky all at the same time.
Captain Filthy was on his tiptoes, hauled back off balance by Franklin's baton as she worked it in around his arm – pulling him into a hammer lock. Blood made a thin red line down one side of her face. 'CALM DOWN!'
Worth a try, but it didn't work.
His mouth opened in a roar and he jackknifed forward, sending Franklin tumbling over his back and crashing into the overturned bed. Then he was off, one foot catching Callum on the way past and thumping him back against the carpet again.
Sodding hell...
He coughed, rolled over. Struggled to his knees. His left hand: bright red and dripping. 'Franklin!'
A little chunk of bloody gristle – about the size of a Wotsit – lay on the bedroom floor in front of him. Complete with the little dimple where an earring used to go.
'Franklin!'
A rattling clatter and she tumbled off the ruined bedframe. 'Bastard...'
His poor ear.
Callum snatched the bit up, clenching it in his fist. Lurched to his feet as Franklin did the same. Bared his teeth. 'He is _not_ getting away!'
##
Out the back door.
The dirty little scumbag was halfway over the fence, leaving bloody footprints as he scrambled up and over the wooden panelling.
Callum tucked the chunk of his ear in his pocket and charged after Captain Filthy. Leapt up. Swung his legs over and dropped down the other side.
It wasn't another garden, it was a path, running straight downhill. Just the thing for little kids to break their necks skateboarding/cycling/rollerblading/sledging down. And Captain Filthy was well on his way, the dirty soles of his feet flapping as he ran.
Franklin cleared the fence and landed on the path, just ahead of Callum. 'Why didn't you pepper-spray him?'
'I did!'
Downhill.
It didn't take much to get going, arms and legs pumping faster and faster as gravity took hold. How the hell did they get planning permission to put a near vertical path down a steep hill? How was this possibly safe?
He leaned back, still getting faster.
Oh sodding hell, this was going to hurt when the inevitable happened and his feet went out from under him and he went tumbling over and over down the tarmac path battering into the garden fences on either side and why did the inside of his head sound like Dr Alice McDonald now?
'Aaargh!' Franklin passed him on the path, leaning back like he was, arms stretched out on either side as if she was about to take flight.
The path levelled out just for long enough to cross another residential street, houses flashing past on either side, and they were on the path again, running.
Captain Filthy was lengthening the gap.
Cannibalistic little sod was probably used to the thing, especially if he grew up here.
_Bdumph_.
Another residential street. More bungalows. More path. Then a set of bollards.
Oh. No...
No wonder there were bollards: that was Branton Street at the bottom of the path. Not a quiet cul-de-sac full of family homes, but a main road lined with shops.
Which meant all three of them were now hurtling full pelt towards the traffic.
A Transit van whizzed past the gap between buildings at the end of the path.
Callum hauled in a breath. 'DON'T BE AN IDIOT!'
But Captain Filthy wasn't listening. He shot between the bollards, still going strong.
There was a squeal of brakes, then a terrible metallic crunch. A horn wailed, accompanied by a car alarm. More screeching tyres.
Callum shifted his weight forward, leaning into it, gathering up a little extra speed, then snatched at the back of Franklin's jacket and skittered and slid on his feet about a dozen yards.
'Get the hell off me!'
They lurched out between the bollards and onto the pavement, still going fast enough to carry them out onto the road. A tricked-out hatchback slammed on its brakes, slithering sideways, just missing Franklin as they stumbled to a halt six feet out from the pavement.
'Wow...'
A ScotiaBrand Chickens van was parked halfway inside an Audi estate, its grille cracked and steaming in the Audi's crumpled interior. Curls of black were scrawled across the tarmac, ending in a Peugeot facing the wrong way with its rear wheels up on the pavement.
And right in the middle of the road was Captain Filthy. Just standing there. Arms dangling at his sides. Head tilted. Staring at the front end of a number 18 bus, stopped about six inches from his nose.
The bus driver still had both hands wrapped around the wheel, his eyes wide, mouth hanging open, shaking.
Franklin shook off Callum's hand.
The hatchback's driver's door opened and a young man clambered out – all spots and sideburns. His car's spoiler was bigger than he was. 'Hoy, you stupid _bitch_! What the fffff...' He pursed his lips into a perfect little bow as Franklin shoved her warrant card in his face.
Callum pulled out his cuffs, marched over to Captain Filthy, slammed him face-first into the bus and snapped the cuffs on. 'Go on: resist arrest. I _dare_ you.'
Callum washed two paracetamol down with a swig of tepid water from a plastic cup. Shuddered. Slumped.
The treatment area wasn't huge – just big enough for an examination table covered in a white paper strip, a plastic chair, and a short section of work surface with cupboards above and below it. A little sink with advice on how to wash your hands, complete with diagrams!
Oh the sodding joy.
A pair of nasty green plastic curtains separated the treatment area from the waiting area. Well, they _called_ it a waiting area, it was really just a line of seven plastic chairs, up against the corridor wall, underneath a sign saying, 'NON-EMERGENCY TREATMENT ZONE' and one of a mobile handset with a line drawn through it.
The curtains hadn't been closed properly, so it was all on show. Including Franklin, sitting right in the middle as if laying claim to the whole thing. Exerting her dominance by ignoring the 'PLEASE TURN OFF YOUR MOBILE PHONE!' sign.
Callum sniffed. Curled his top lip.
Why did disinfectant have to smell so bad? And why did it have to _sting_ so much.
The whole left side of his face throbbed, bleeding into the sharp stabbing grating pulses from what was left of his ear.
Bloody Captain Filthy.
Franklin stuck the phone against her chest and stood. Wandered over to his intimate cubicle of doom and stepped inside. The blood on her face had gone, instead a wad of white gauze, about the size of a Post-it Note, was taped to her forehead above her right eye. 'What happened to the doctor?'
'You tell me. Little git slathered me in Dettol and iodine, said something about consulting someone, sodded off with the chunk of my ear, and I've not seen him since.' Callum shifted on the examination table, making it creak. 'How's the head?'
She pointed at the phone. 'I've got DS McAdams on, Custody Sergeant took one look at our boy and refused to take him into the cells till the hospital say he's not going to die from the drugs or choking on his own vomit.'
'We should be so lucky.'
'So they took him up to A&E, where they stripped him to do a full medical, and guess what?'
'He's off his nipples on cocaine.'
'Nope: they found a tattoo, from here,' she tapped her shoulder with her other hand, 'to here.' Then did the same to her wrist. 'Some sort of kids' cartoon characters. Clangers? Whatever they are.'
'You don't know who the Clangers are?' Some people had no appreciation of the classics. 'When Peanut gets born, he's being raised on a diet of Bagpuss, the Clangers, and Danger Mouse...' Callum scrunched his face up. Sod. Of course it was. 'The tattoo – it's Brett Millar, isn't it? He was breaking into his own house.'
'Got it in one. And when they pumped his stomach, out came magic mushrooms. _Lots_ of them.'
No wonder the pepper spray didn't work. Could have dumped him in a bath of the stuff and he still wouldn't have felt it.
'There were other things in there too, leaves and flowers, so they've sent it all off for testing.'
'Just like Ben Harrington.' Callum massaged his temple, wincing as it pulled at the scratch marks. 'So much for Glen and Brett ganging up on him.'
'Might still be down to Glen Carmichael. He gets the other two stoned, eggs them on to eat more mushrooms than they can handle, then...' Her forehead creased. 'Still doesn't explain the herbs. Unless they thought they could get high from them too?' Then a couple of blinks and she snatched the phone back to her ear. 'Sorry, Sarge, I was checking on DC MacGregor... Yes, Sarge... I appreciate that, Sarge, but—... Sorry.'
A young man in creased blue scrubs hauled the curtain back and joined them in the treatment area. Making it crowded. He cleared his throat and glared at Franklin. 'No mobile phones.'
She stuck two fingers up to him and wandered out into the corridor again. 'Yes, Sarge... Thank you, Sarge.'
'Honestly, some people think the rules apply to everybody but them, don't they?' He checked a clipboard. 'Now: Callum MacGregor?'
A nod.
'Right, I'm afraid we've got a bit of a problem with your ear.'
Of course they did.
'You see, our cosmetic surgery department were going to try and reattach the bit that was... well, bit off. Unfortunate turn of phrase there, sorry. Only a wee girl's come in with third-degree burns. She tipped a boiling kettle all over herself. She's four.'
Callum slumped back against the wall. 'She going to be OK?'
'We hope so, they're taking her into surgery now. But she's going to need a _lot_ of skin grafts. So...?'
He covered his face with his hands. 'You can't fix my ear.'
'I'm sorry, Mr MacGregor, but the little girl...'
'Yeah. I know. She needs it more than I do.'
'But I _can_ stitch up the wound and give you some antibiotics and painkillers. Have you had a tetanus shot in the last ten years? Oh, and we'll need to take some bloods to test for Hepatitis and HIV.'
And to think, this morning a visit to Professional Standards was the worst thing that could happen.
##
Franklin kept sneaking glances at him across the car.
He frowned back at her. _'What?'_
'Nothing.'
The road in front was a long line of vehicles, slowly crawling along in a stop-start-stop-again line. Traffic going the other way was doing the same. Giving everyone time to enjoy the rain.
'How's the ear?'
'Sore. How's the head?'
She shrugged. 'I dodged the tower unit, it was the keyboard that got me.'
Then they listened to the windscreen wipers for a bit, until Callum reached out and clicked on the radio. 'Should be about time for the news. Unless you mind?'
'No.'
Something bland and unthreatening filled the car, the beat just far enough out of time with the wipers' _week-wonk_ to be annoying.
'I can switch it off if you like?'
'Nah, it's OK.'
She was looking at him again.
'Look, there's obviously something, so—'
'Why didn't they send you home?'
His fingers drifted up to the wodge of bandage covering his poor tattered ear. At least it didn't hurt. Not right now anyway. Amazing what a wee injection of local anaesthetic could do. 'Going to be nearly six before we get there at this rate.'
The song on the radio dribbled to an end. _'Wasn't that spectonkular? You, my friend, are listening to_ Crrrrrrrrrrrrrrazy Colin's Rush-Hour Drive-Time Club _, right here on Castlewave FM, and we're counting down the days to Tartantula! Oh yes indeedy-doody.'_
Franklin curled her top lip. 'Why do they always have to be wankers?'
_'Going to be windup o'clock in fifteen minutes, but first here's Gorgeous Gabby with the Naughty News!'_
'I think they grow them in special septic tanks.'
_'Thanks, Colin. Police Scotland refused to comment on claims that a new serial killer is operating in Oldcastle, following the discovery of three mummified bodies yesterday—'_
'Three?'
Callum let his head fall back against the rest. 'Didn't anyone tell you? Oldcastle Division leaks like a chocolate condom. At least it wasn't anyone on the team – we all _know_ how many mummies we've got.'
_'—appeal for calm. A house fire in Logansferry this morning was probably arson, according to Fire Brigade sources. A mother of four was rushed to hospital suffering from smoke inhalation—'_
'Unless whoever leaked said there were three mummies instead of two, so no one would think the informant was on the team?'
'You're very cynical, Detective Constable Franklin.'
_'—announcing road closures for this week's Tartantula Music Festival. Diversions will be in place from Friday lunchtime, add in all the planned roadworks south of the river on Saturday and we can expect_ significant _delays.'_
In Dante's _Divine Comedy_ , Hell was divided into nine circles, each devoted to punishing a particular group of sinners. But up here, in the land of the living, it was roadworks and rush hour.
_'And speaking of the festival, we managed to track down Oldcastle's very own Leo McVey earlier and asked him about Sunday's grand finale performance of his 1980s concept album,_ Open the Coffins _.'_
Franklin turned up the radio and a dark warm voice gravelled out of the speakers. _'Yeah, it's going to be great. I mean we've got some great acts joining us on stage: Lucy's Drowning, Mister Bones, Halfhead, Closed for Refurbishment, Catnip Jane, Donny Sick Dawg McRoberts, and loads of others, you know? Great.'_
The car crawled forward.
Callum puffed out his cheeks. 'It'll pick up a bit when we get onto the dual carriageway.'
_'And we're not just doing highlights, right? We're doing the whole album, start to finish.'_
Let's face it, the traffic couldn't get any slower.
A line of tail-lights, glowing like the fires of hell, flaring in the falling rain.
_'The public's reaction's been great. It's kinda humbling that they still love it after all these years. And I'm_ loving _strutting about like King of the Jungle again._ _Makes me wish I'd come out of retirement years ago.'_ A laugh, black as treacle.
Franklin smiled. 'I loved _Open the Coffins_. We listened to it non-stop when I was at university. Drove the woman downstairs mad...'
_'And all the money's going to charity, right? Which is great. Everyone's giving up their time and their talent to raise money for Alzheimer's research, cos of Ray, you know?'_
'Haven't heard it for ages.'
_'—a terrible shame. I mean the book's genius, yeah? My kids loved it, my grandkids love it, I_ still _love it._ Open the Coffins _: best children's book ever written, that's what I think.'_
_'Leo McVey there. And you can catch him this Sunday at Tartantula, but tickets are going fast, with all proceeds—'_
She turned and frowned across the car at him. 'You didn't answer the question: why didn't they send you home?'
_'—diagnosed with Alzheimer's last year. Weather now, and it looks like we're stuck with this rain till—'_
He clicked the radio off again. 'What's the point of going home early? Won't bring my ear back.'
Besides, if he went home he'd have to explain what happened to Elaine. My earlobe and that gristly bit above it? Oh, nothing much: they were bitten off by a junkie. But at least my HIV test came back negative. Elaine? Hello, Elaine?
He shrugged. 'At least this way I get the overtime. Need all we can get with Peanut on the way. Do you have any idea how much it costs to raise a kid these days?'
Finally the roundabout onto the main road crawled into view. Buses and eighteen-wheelers sending up huge drifts of spray, drenching the smaller cars.
'Urgh.' Franklin crawled the car forward, bumper inches away from the people carrier they'd been stuck behind since leaving the hospital. 'All it ever does in this sodding town is rain.'
'Sometimes.' He drew a frowny face in the mist that crept up the passenger window. 'Did they get any sense out of Brett Millar?'
'Still off his face on mushrooms.' Franklin tapped her fingers against the steering wheel, syncopating with the windscreen wipers. 'They even tried giving him a shot of Narcan; didn't make any difference though.'
'Yeah, well magic mushrooms aren't opioids, are they? Not surprised it didn't work.'
'At least they tried.'
He drew angry eyebrows on the frowny face. 'So we've got Ben Harrington dead in the bath, Brett Millar's so high he can orbit the International Space Station, and Glen Carmichael is missing... You know what I think? I think the three of them aren't serial killers, they're victims. You saw how emaciated Brett Millar was. He's been starved.'
'When did you ever meet a fat junkie? Maybe he's...' Franklin closed her eyes and swore. 'Benjamin Harrington. We've still got to deliver the death message.'
'Oh for God's sake.' Callum peered through the rain-smeared windscreen.
Traffic was solid northbound, so getting over Calderwell Bridge was going to be a nightmare. He checked his watch. 'No point even trying till rush hour's gone. Stick to the plan: at least it's moving southbound.'
Assuming he hadn't just jinxed it.
'What if they find out from the radio, or some scumbag journalist doorsteps them?'
'They _won't_.' Fingers crossed, anyway. 'We pick up Dr McDonald and we head to the flat in Castleview. By the time she's finished poking around, rush hour will have died down and we can sling past Ben Harrington's parents on the way back to the station.'
Franklin edged them closer to the car in front, brow furrowed, eyes narrowed. 'Of course, the real question is: if Brett Millar's running about all over Blackwall Hill, out of his head on magic mushrooms, where's Glen Carmichael?'
'He's already dead.'
##
'AAAAAAAARGH!' The bucket sails through the musty air and bursts against the wall. Water makes a comet's trail, soaking into the bricks.
Where the hell is he?
He _should_ be right there – chained to the wall, but he's not.
Instead, the chain sits on the ground, coiled like a snake. Venomous and treacherous. Useless. Four screws lie in the dirt, still in their Rawlplug shells, torn from the mouldering brickwork, letting the tie-up ring come free from the wall. It's still fixed to the end of the chain by its padlock. The traitorous useless chain.
'You had ONE JOB!'
He grabs it up and hurls it away into the gloom. It clangs and clatters against the long-dead boiler, hissing its way into a deceitful pile.
'AAAAAAAAAAAAAAAAAAAAARGH!'
All that time. All that energy. All those sacred herbs _wasted_.
Weeks and weeks of work. Gone, just like that.
He grinds his teeth, whole body trembling, blood surging in his ears. _Whoom_. _Whoom_. _Whoom_...
How could he be so stupid?
**_Once Upon A Time_**
There's a jackdaw hanging on the fence behind the house. Like a little black kite, caught on its own strings. Wings outstretched. Beak hanging open. Eyes like marbles that've been rolled too many times on rough concrete and gravel, till they're all white and scratched.
The jackdaw is dead.
Everything dies.
He reaches out and touches its feathers. They're cold and soft.
Sometimes things die because they're old, or ill, and sometimes they die because Father makes them dead. Sometimes they get hung from the fence with wasp-eaten wooden clothes pegs. And sometimes they get buried in the cold dark ground.
Justin stands in the kitchen, sniffling. Outside the sun is going down, making the fields look like they're bleeding.
The fields are bleeding and the house is full of smoke.
And Father howls his anger at the walls. Using it like a stick to beat the smoke with. Only the smoke doesn't break as easily as Justin.
The kitchen door bursts open, bouncing off the wall, making the mugs and plates rattle in their cupboards. Father stabs a finger at him. 'It's those bloody jackdaws again!'
Justin doesn't move.
'Building their sodding nests in the bloody chimney...' His face is dark as the smoke, teeth shining like sharp white stones. 'Get the ladder.'
'I...' Justin licks his lips.
Father's hand is like a claw, fingers digging into Justin's arm, squeezing so hard it sends needles and pins and knives stabbing all the way up into his shoulder.
'Aaaaaagh!'
'You're making your mother _cry_. Can you hear her? Can you?' He shakes Justin, making his teeth clack together. 'CAN YOU HEAR HER?'
Faint, muffled sobs come from downstairs, working their way up through the floorboards like sad little seedlings looking for light. But there's no light up here, only blood and smoke.
'I'm sorry, Father, I'm sorry.'
Another shake. 'Then don't make me tell you again.'
A nod. Teeth biting his bottom lip. Blinking back the tears.
Father lets go and Justin _runs_. He runs out the back door and round the side of the house to the garage. Fights with the slippery doorknob. Stumbles into the darkness, wiping tears from his cheeks.
The ladder is bigger than he is, but he gets it down and hauls it out into the back garden. Sticks it up against the wall, so it reaches way up to the guttering. Shuffles his feet on the damp grass, his breath pink and cloudy in the fading light.
Father steps out onto the path. Looks at the ladder. Then looks at him. 'Well?'
Justin stares at his trainers.
'Up you go.'
'But the jackdaws hate me.'
'Of course they hate you. You're destroying their home and killing their babies.' Father smiles his nastiest smile. 'Why would they _like_ you?'
'They'll peck my eyes out and I'll fall off the roof and I don't want to—' The fist is nearly too fast to see, but it smashes into his cheek like a hammer, snapping his head away, making him stumble and fall across the damp grass. The world sounds like cymbals and drums. Then all the air whoomps out of him as Father's boot smacks into his tummy, lifting him off the ground and spinning him over onto his back. Rats gnaw through him, their little pink tails burning his insides.
He rolls over and curls up into a ball. Cries.
And finally, Father squats down beside him. 'Hey, come on, slugger. Dry your eyes, champ.' Gentle hands wipe the tears away. 'There we go. All better.' He helps Justin to his feet. Brushes the grass and dew from his jumper. 'You good?'
Justin nods. Don't tremble. Don't cry.
'Course you are: big boy like you.' He guides him over to the bottom of the ladder. 'Now up you go, and don't forget to kill the babies, OK? OK.'
He stands at the top of the stairs. Father must've left the basement door open again, and a bare lightbulb casts sharp shadows on the rough brick walls.
Justin's not allowed to step on the stairs. If he puts one foot on the stairs something horrible will happen. Father will make sure of it.
So he doesn't. He just stands there, with his face all swollen on one side and bits of stinky bird droppings and sticky blood on his hands. Looking.
New Mummy is there. She's sitting on the dirt floor with her back to an old radiator – all rusty and lumpy edges. She's got her knees drawn up to her chest, arms wrapped around them, face buried in her yellow hair. Shoulders quivering as she cries.
New Mummy must've been bad, because she's got no clothes on. She has to be naked and cold, because she's been bad.
It's like a lump of coal in the middle of his chest. You have to do what Father says. That's the Number One Rule.
Justin moves his foot and the floorboard groans beneath him.
Mummy flinches, like she's been slapped. Then stares up the stairs, eyes wide and soggy, ringed with red. The gag in her mouth has darkened to a deep blood red where the tears and bogies have soaked into it. And the chain around her neck sparkles like no diamond necklace ever did.
She makes... noises. All muffled by the gag. Strangly, angry, pleading noises.
As if he's the one who can help her. As if he isn't every bit as trapped as she is.
Justin reaches out and swings the door closed.
He's never making _that_ mistake again.
There's only one thing for it.
The God-In-Waiting is gone and he's not coming back. Oh, he'll regret running away soon enough. Come crawling back, pleading to be released from his impurities, but as Mummy always found out, pleading never worked. Once you broke the rules you had to take your punishment, because that's how it works.
He had his chance at being a god and he threw it away.
Of course, there's very little chance of him being able to find his way back. The purification ritual is mind-expanding, but often makes the Gods-In-Waiting confused. As if their brains have been rewired to run on a foreign voltage. The signals from the real world get scrambled until they finally achieve divinity and all becomes clear again.
He won't be able to lead anyone back here.
Probably.
But probably isn't definitely, and that was one of Father's many lessons. You never trust a probably, because 'Probably' can't be trusted like 'Definitely' can. A thing is either dead or it's not, 'probably dead' isn't good enough.
So yes, there's only one thing for it.
He unscrews the cap of the petrol canister and pours it down the basement stairs.
It cascades like a little waterfall, making the air swim and wobble in front of him. The sharp heady scent of pear drops and sweet vinegar.
The living room smells of cloves and smoke, with its open fireplace full of twigs and bones. The jackdaws always win in the end. But they won't be back this time. He covers the mouldering furniture in unleaded, sploshes more on the dusty carpet. Pours what's left across the bathroom floor and into the kitchen. Upends it and gives it a little shake, getting the last dribbles out.
Steps out of the back door and into the rain.
Father would _not_ be happy if he could see the garden. His precious vegetable plot disappeared under a war zone of brambles and nettles, battling it out for the last scraps of nutrients from the dark thick earth. The trees heavy with unclaimed fruit, rotting and wasp-riddled on the branches. The garden shed, where so many nights were spent learning not to be a bad little boy.
He dips a strip of fabric into the empty plastic container, strikes a match and puts it to the hanging end. Still enough petrol in there to soak through the scrap of T-shirt and turn it into a torch. Then the whole thing goes in the kitchen door, tumbling over and over, making the same fluttering roar as baby jackdaw wings when they're caught in their nest.
It bounces off the wall, and lands halfway between the kitchen and the living room. Blue flames rush across the floor in both directions, eating their way into the house. Popping and crackling like a baby jackdaw's bones when you hold one of their fluttering little bodies in your hand and _squeeze_.
Two minutes later the basement goes up, growling out its decades of pain.
He waits until the fire has taken hold, then goes back to the van.
It doesn't matter if the ex-God-In-Waiting leads them back here – there will be nothing left to find. And he will have moved on to somewhere new.
The only thing left to do is find someone else to take their place.
##
'Hold on, I'll put you on speaker.' Dr McDonald stood in the middle of the filthy room, dressed in the full-on Smurf outfit: booties on her feet, gloves on her hands, mask, and safety goggles covering most of her face. She had an oversized smartphone in one hand, held out at head height.
Scenes Examination Branch had left the windows alone – still covered with their layers of hardcore pornography, blocking out the evening light, leaving them to the mercy of the single lightbulb dangling from a wire in the ceiling. They'd left everything else as it was as well – the ladder, the wallpaper table, the power tools, the radio. The stack of empty cans in the kitchen and the half-full bong.
Most of the flies were gone from the floor though, so at least every step didn't scrunch.
And Callum's ear throbbed. So much for local anaesthetic. No one said the sodding stuff would wear off in less than an hour.
McDonald poked at her phone's screen and a semi-posh Scottish accent crackled out of the speaker. _'Can you hear me?'_
Franklin stood in the corner, notebook at the ready. But Callum leaned back against the wall, by the porn-covered window, with Cecelia. All of them done up in blue Tyvek oversuits with matching accessories.
Cecelia made a raspberry noise behind her facemask. 'To be perfectly honest, I find this more than a bit insulting.'
Callum shrugged.
'My team's been over this flat with the proverbial nit-comb. We _did_ our job.' She folded her arms, her suit making crinkling noises with each movement. 'And for your information: Tina's confronting Yashnoor about having an affair on _Enders_ tonight. In _ninety_ minutes.'
He groaned. 'You'll be back in time for _EastEnders_.'
'I better be.'
Dr McDonald did a slow pirouette, showing the phone the flat. 'Say "stop" if you spot anything.'
_'Ooh, is that pornography on all the windows? I wouldn't mind a gander at that.'_
Cecelia shook her head. 'Pervert.'
_'Who said that?'_
'Hold on...' Dr McDonald turned the phone around, so they could see the screen.
A creased face blinked out at them – steel-coloured short back and sides, two prominent grey eyebrows, a matching moustache lurking beneath a puckered golf-ball nose. Little rectangular glasses. _'Greetings, minions of Police Scotland! Fear not, for your salvation is at hand.'_
'Bernard, this is Cecelia Lynch, head of the local Scenes Examination Branch. And standing next to her is DC Callum MacGregor and DC Rosalind Franklin. Everyone, this is Professor Bernard Huntly – he's a physical evidence specialist. A bit of an acquired taste, but he's annoyingly good at what he does so we put up with it.'
_'Quite right too.'_ He gave them all a grin. _'Can you turn me up a bit, Alice?'_
She poked at the controls and Huntly's voice got louder.
_'Well, why don't we cut straight through the meat to the bones beneath, Cecelia my love? Bloods?'_
Cecelia pointed at the doorway by the kitchen. 'Biggest quantity was over there, and even then it was less than a teaspoon. Going by the little dots on the floorboards, it wasn't a gusher, more like a hammered thumb.'
_'Semen?'_
'I'd prefer an ice cream, if it's all the same to you?'
_'Oh, we're feisty, are we?'_
'Nothing in the living room, bathroom, kitchen, or hall. There's three sleeping bags in the bedroom, and they're like large down-filled condoms. Scrape them clean and you could artificially inseminate half of Fife.'
_'So our three property tycoons were enthusiastic onanists. Everyone needs to have a hobby.'_ The little face pulled on a smile. _'That's why I wanted to see the pornography, Cecelia. Is it the sort of thing to encourage Ben, Glen, and Brett's nocturnal manipulations, or was it put there by whoever left a drugged body floating in a bathtub full of brine? Alice?'_
Dr McDonald walked over to the window and held the phone up to the bits of magazines taped there.
_'Hmm... Interesting.'_
'You see something?'
_'No, I just didn't think anyone actually bought dirty magazines any more. You can download all this for free from the internet, what sort of idiot pays for it?'_
'It's all heterosexual, well, I mean it's mostly heterosexual except for the lesbian photoshoots and they're basically only there to appeal to heterosexual men, so what I meant is that there's nothing that would suggest the three of them were involved in a romantic way.'
Callum sniffed – the air still had that mouldering sausage smell. 'Unless they're overcompensating? Big display of testosterone: look how manly and laddish we are, and next thing you know they're all running around sharing sleeping bags and playing with power tools.'
'Does it really _matter_? No one cares if they were gay or not.' Cecelia pulled back the edge of her glove and checked her watch. ' _Enders_ starts in eighty minutes.'
_'Nobody cares_ unless _that was why they were targeted, my dear Mrs Lynch. I'm going to need you to scrape out those sleeping bags after all – see if you can find evidence of sexual activity involving more than one person.'_
Her back bowed. 'Oh... lovely.'
Over in the corner, Franklin stiffened as her phone launched into a strangled rendition of 'Dancing in the Moonlight'. She snapped off one of her gloves and dug into her SOC suit, pulled her mobile out and turned to face the wall. 'Mark, this is _not_ a good time.'
_'Now, to the bathroom! Our friend Imhotep had to fill the bath somehow and I fancy we'll find some DNA on the underside of the taps. Easy to contaminate with biological residue, not so easy to clean.'_
Dr McDonald carried Professor Huntly out through the door and into the hall. 'We're not calling him Imhotep, we're calling him Paddington, because of—'
_'The Peruvian-style mummies, yes, I know. But it's hardly a name with dramatic connotations, is it? "Paddington" isn't someone who abducts people, drugs them, drowns them in a bath of brine, then smokes them to a fine wrinkly jerky. Not unless those marmalade sandwiches of his were laced with psilocybe semilanceata. And he wears a duffle coat! What self-respecting monster does that?'_
'But the team agreed—'
_'Paddington's a stupid name for a serial killer, Alice. At least "Imhotep" has a bit of gravitas about it.'_
'Bernard, you can't just waltz in and rename our killer!'
Cecelia hooked a thumb at the doctor and her phone. 'Are they always like this?'
'I have _no_ idea.'
She slouched into the hall, after them. 'Your boy on the phone's a dick, but he's right. Paddington _is_ a stupid name. Jack the Ripper wouldn't have got where he is today with a name like Paddington.'
There was no room for them in the bathroom, so they stayed in the hall.
'So why are you really here?' Callum leaned back against the wall, where the sheet of plasterboard used to live. 'Mother said SEB had finished with the crime scene ages ago. Nothing left for us to contaminate.'
'Ah...' Cecelia rubbed the tips of her gloved fingers together, making them squeak. 'Don't take this the wrong way, but—'
'Oh you have _got_ to be kidding. You're here because of me?'
'The Powers That Be said you couldn't access the scene unless someone from my team made sure you didn't get up to anything. And I can see you making faces at me behind your mask, so don't.' She shifted her feet. Fiddled with her gloves some more. 'I know you didn't cock that last scene up, but everyone else thinks you're a liability.'
Of course they did.
He let his head fall back until it thunked against the wall. Winced as a thousand bees sank their stingers into his ear. 'Ow...'
'So tell them the truth, Callum.'
'I _can't_.' He lowered his voice, even though Franklin was still muttering angrily into her phone in the living room and there was no one else to hear. 'You said it yourself: one more strike and they'd fire Elaine. We need the money. If we didn't have her maternity pay coming in, with the mortgage, and the credit card debt, and all the stuff we've got to buy for the birth and babyproofing the flat, and everything else... It's two weeks away and we're nowhere _near_ ready.'
Cecelia put a hand on his arm. 'Don't panic. Breathe.'
'We're not taking any chances. And _you're_ not telling anyone.'
She gave his arm a squeeze. 'Still think you're an idiot.'
'Join the queue.'
Dr McDonald emerged from the bathroom, phone still in her hand. 'Mrs Lynch, can you do a complete swab of all the taps in the house? Especially the underside of the handles and knobs.'
_'And the flush on the toilet too.'_
'Sorry.'
Cecelia shook her head. 'Fine.' Then stomped off back to the living room.
Dr McDonald frowned behind her safety goggles. 'The three men here were targeted for a reason, we don't know what it was, but the flat's self-contained, a safe zone for Paddington to work, I mean no one's going to see in when you're on the top floor, are they, of course not, so he can do whatever he wants in here and no one's going to notice as long as he's reasonably quiet about it.'
Professor Huntly's voice boomed out in the narrow corridor. _'I've been thinking about your two other victims: the mummies. If you don't get anything from the DNA, you can try the fingerprints. It'll be cheaper than going for facial reconstruction.'_
'The question is: did someone gain access to the flat and decide Glen, Brett, and Ben would make good victims, or did he target them somewhere else and follow them back here?' She pulled back her hood and wrapped a coil of hair around two fingers on her free hand. Holding the phone out in her other like a Dalek's eye stalk.
Callum peered at the face on the screen. 'And we get their prints how? The mummies' fingers are like prunes.'
_'Ah, my dear Constable...?'_
'MacGregor.'
_'I knew it was something like that. Their fingerprints are like prunes because they've been dehydrated. So how would one get them nice and plump again?'_
Of course: 'Soak them in water.'
_'Dear Lord, no, that would be a disaster. We soak them in_ glycerol _. Should make them lovely and soft too.'_
Dr McDonald twiddled with her hair. 'Of course, the fact that there's three of them makes it all a bit more difficult, I mean one person's easy enough to subdue, but three at the same time, when they're all young and fit, that would take a lot more doing, wouldn't it, you could restrain them individually, but then how do you do that without the other two stepping in?'
'Glycerol.'
_'I'm surprised no one thought of it sooner. It's the obvious solution and a lot less expensive than extracting DNA from the tooth pulp cavity and sending it off for analysis.'_
Well, it was worth a try.
'No...' Twiddle, twiddle, twiddle. 'I think they _knew_ their attacker, they invited him into their flat and he brought the magic mushrooms with him, they sit around drinking lager and self-medicating till they pass out and after that Paddington can restrain them easily.'
''Scuse me.' Cecelia squeezed past and into the bathroom, carrying a large square metal case.
'OK, so we get the hands steeping in glycerol, what then?'
_'Then you run the prints. And you get a toxicologist to look at the tissue samples. A_ decent _one, not some wet-behind-the-ears undergrad on work placement. I can probably give you some names if you like.'_ On the little screen, Professor Huntly fluttered his eyelashes. _'Failing that, I'm available at_ very _reasonable rates. ID the drugs and the herbs and you've got somewhere to start looking – he had to get them from somewhere.'_
'So the question becomes where did Paddington meet them, did they have a favourite pub or club, we need to get someone visiting the local bars and ask if Glen, Ben, and Brett were seen there with someone else, because he's going to have his own favourite haunts, areas where he likes to hunt, and if we can get an ID from the other two victims we might find a common denominator, don't you think?' She pulled down her facemask. 'Do you think we could leave here, because the smell is beginning to make me feel a bit sick.'
_'Wimp.'_
'Goodbye, Bernard.' She hung up and put her phone away. 'My arm was getting sore anyway.'
Callum lowered his facemask. 'You know we can just wait for Brett Millar to come down from his trip and _ask_ him what happened.'
'We can, but what if he doesn't know what happened because he can't remember, or maybe the drugs he's been on have caused permanent brain damage, can you imagine what being force-fed magic mushrooms for days would be like, what it would do to your sense of perception?' Dr McDonald struggled her way out of her gloves. 'We have to work on the assumption that he's not going to be any help, that way if he _does_ remember anything about the man who attacked them it's a bonus.'
Yeah, she had a point.
'OK. Well, what if they didn't meet the guy in a pub? He could work for the bank, if they're financing the refurbishment. Or a local estate agent, if they're looking for a valuation?'
'That's certainly worth exploring.'
And the list of people needing interviewed just ballooned to about three times its previous size. Mother would love that.
He took out his notepad. 'So, come on then, you've seen around the flat: who are we looking for? How do we spot him when we see him, assuming Brett Millar doesn't just wake up all bright-eyed and bushy-tailed tomorrow and give us a name and address?'
More hair twiddling. 'He was able to blend in to Glen Carmichael's social circle, that means he _could_ be a bit hipstery. Think beard, lumberjack shirts, skinny jeans, no socks, ironic tattoos, 1930s haircut, but not necessarily in that order. He's big enough to manipulate the unconscious bodies of three large young men and we're on the top floor, that's a lot of stairs to carry someone down to get them in your van. So he's strong. Capable. Not easily flustered.'
Through in the living room, Franklin was getting louder – the words impossible to make out, but the tone of voice was clear as a scream: _not_ sodding happy.
'Assume he waits till the middle of the night to transport Glen and Brett, he still risks being seen by one of the other residents, or someone on the street. So he's confident too. He's got a story for every eventuality.' Alice tilted her head up and to the side, frowning at the plasterboard ceiling. 'He's had a lot of practice. And I'm not just talking about the mummy in the tip and the one in the car – these aren't his first victims. He's been doing this for a long, long time.'
##
**_Once Upon A Time_**
The tattoo ripples like a flag across Father's back as he digs. Faded blue-grey lines and shapes. A little bird. A skull. A big pointy knife.
His spade bites into the black earth, spits out lumps onto a growing pile.
It's getting deep, the hole.
Deep enough that only Father's top half sticks out of it. Sweat all sparkly on his dirty skin. Not a big man, but powerful, like a bulldog. Not the one on the TV ads selling insurance, though, more like the ones Father's friends make fight in wooden pits in barns in the middle of nowhere.
All bulging muscles and dark blood.
Warm sunlight makes the garden shine, green and yellow and red.
And on the fence hang a dozen jackdaws, their bodies all stiff and dead.
But no one's digging them a hole.
'Come on, champ, out you go.' Father holds the car door open. He's wearing his dog collar again, all white and crisp against his freshly shaved neck.
Justin jumps down onto the sticky black tarmac.
The whole street smells like coal and treacle as the sun batters down like a fist. It sparks off the parked cars, so bright it's painful.
He makes sure not to get any tar on his new shoes. Father has been very clear on what'll happen if he does.
'Now, slugger, you know what to do.'
A nod. Then he bites his bottom lip and looks both ways – up and down the street – before skipping across the road. Like he's a little baby, instead of a grown-up six-year-old.
Normally it would earn him a beating, but not this time. This time it's what Father _wants_ and if today goes well, Father will be happy and if Father's happy Justin's happy. So he skips.
The shops are boring, full of stuff no one could ever want: like pots and pans and carpets and things for cleaning dishwashers. But right at the end, by the bus stop, there's a sweetie shop.
It does other stuff, like boring newspapers and magazines, but the wall behind the counter is the best thing ever – rows and rows of old-fashioned plastic jars full of brightly coloured sweets with funny names like 'DIRTY TATTIES', 'POKEY FINGERS', and 'SOOR PLOOMS'.
The air tastes of excitement.
And perfume. Which is sort of like soap, only stronger and a bit chokey, and Father doesn't like it.
The smell's coming from a lady with yellow hair, standing with her elbows resting on the counter. She smiles down at him with shiny white teeth. 'Hello, little man, how can we help you today then?'
She's pretty. Yellow hair, heart-shaped face, little nose, sticky-out boobies. The kind Father always picks.
Justin blinks up at her. 'Ooh, are you a _angel_?' As if he doesn't know that angels aren't real. They're all madey up by liars, like Father says.
'Well, aren't you the wee charmer?'
'My mummy was pretty like you, but she had to go live with Jesus in the Heaven.' He sticks his bottom lip out and makes it wobble, like he's about to burst into tears.
'Oh, sweetheart!' The lady's face goes all wrinkly between the eyebrows and she hurries around the counter to hug him.
It's lovely and warm and she doesn't _really_ smell soapy and chokey. She smells like sunshine.
'Where's your daddy?'
'I...' Sniff. 'I don't know. He went into a shop, but there was a doggy and I went to look at the doggy, and I can't remember which shop...' Justin works the sniffles into a tiny sob. Nothing too wet and snottery. Father didn't raise him to be a whiny little bitch.
She gives him another hug, soft and warm, then holds him at arm's-length and nods. 'How about we get you a nice sweetie, then we go looking for your daddy? I can shut up the shop for ten minutes. Would you like that?'
He pulls on his 'Brave Wee Boy' face. 'You _are_ a angel.'
'How about... sherbet lemons?' She stands and clatters a handful of yellow pebbly things out into a tiny paper bag, then passes it to him. 'I know you're not supposed to take sweeties from strangers, but trust me: they're good.'
He takes one and puts it in his mouth – all nippy and sour and sweet at the same time. 'Thank you.'
She holds his hand and walks him out of the shop. 'Now, let's see if we can't find your daddy.'
Of course they will. Otherwise the plan won't work.
The pretty lady has a nice voice, like the people on the radio, smiling and swinging his hand in hers as they walk down the narrow street. 'And all the tiny mouses sing, "What use have we of golden rings? / All we want is bits of cheese, and socks to warm our feet and knees, / And pies and biscuits by the tonne, and lemon drops for everyone."'
There aren't any shops down here, but she doesn't seem to mind the smell of the bins.
'But Santa frowned and asked again, "Mice, have you seen the silly hen?" / "Oh, no, Santa we have no want of shoes to fit an elephant, / Or zebra shorts, or lion hats, or spats to fit a pussy cat."'
She does a little skippy step every time something rhymes.
There's one parked car on the road, the boot standing open, a man pacing back and forth beside it, wringing his hands. He's the only other person here.
Justin points and breaks free of the nice lady. Runs across to him. 'Daddy!'
Father spins around, eyes wide, then beams and kneels on the cobblestones, sweeps him into a hug. 'Justin! Oh where have you been? I was worried sick!'
'The nice lady helped me, Daddy.'
He lets Justin go and stands. Holds his hand out to the lady. 'Bless you!'
She goes pink in the cheeks. 'Nah, it was nothing. He's a lovely wee lad.'
'Ever since his mother left us...' A sigh. ' _Bless_ you.'
She shakes his hand. 'My pleasure. It's not every day you—'
The fist is fast and only makes a noise when it slams into the side of her head. Then the nice lady's legs buckle and she slumps. But before she's even halfway down, Father sweeps her up in his arms and bundles her into the boot. Wraps her wrists and ankles in silvery sticky tape. Puts another strip of it over her mouth. Slams his fist into her face twice more. Then closes the boot.
Justin stands perfectly still, hands behind his back. No trembling. No crying. No anything.
Father grins at him. 'Who wants chips for tea?'
##
Brookmyre Crescent hissed in the rain. Drops bounced off the glistening tarmac, gathered in the gutters, spreading out in a tiny lake that lapped around the tyres of a new-ish Toyota. Their pool car sent a mini tidal wave sploshing against its hubcaps.
Callum unfastened his seatbelt as they drifted to a halt outside number 16, with its collection of naff garden ornaments. 'You still want to be the one that tells them their son's dead?'
'Why, you think I'm not up to it?' Franklin hauled on the handbrake. 'Think I'm going to—'
'Fine. Whatever.' He shook his head. Winced as a thousand tiny ants dug their pincers into what was left of his ear. 'You know, sometimes, just occasionally, _maybe_ you could try not treating everything I say as some sort of insult to your gender, ethnicity, professionalism, or dress sense.'
She stared down at herself. 'What's wrong with my dress sense?'
'Try a mirror.' A cheap shot, but hey-ho. He grabbed a high-viz jacket from the back seat and clambered out into the rain, hauling it on as he hurried up the lock-block driveway to the door. Turned up his collar and rang the bell.
Rain drummed on his shoulders, hammered at the pampas grass growing around that hideous wishing well and even more hideous gnomes.
Franklin locked the car and jogged her way through the downpour. 'There's nothing wrong with my suit!'
'Keep telling yourself that.'
A light came on inside the hall, filtering out through the fanlight above the door.
'Why don't you stick your—'
The door swung open and Lurch from the Adams Family blinked down at them. He'd swapped the butler's outfit for a brown cardigan and mustard-coloured corduroys, but the huge hands and pale slab of a face were a dead giveaway. But his voice wasn't a deep ringing bass, it was a sharp-edged tenor, clipped and precise. 'Can I help you?'
Callum produced his warrant card. 'Mr Harrington? Can we come in please?'
The only sound was the sibilant hiss of the rain on the drowning world.
Franklin pulled her card out as well. 'It's about Ben.'
Lurch rolled his eyes, then turned and lumbered back down the hall. 'You'd better come in then. Make sure you wipe your feet.' He led the way into a living room lined with bookshelves. No TV, just a fancy stereo surrounded by stacks of vinyl. Leather armchairs that looked worn and soft.
He took up position in the middle of the room, straightened up to his full height, put his hands behind his back. 'If this is about drugs, I can promise you I don't want to know. I told him he was on his own if he _ever_ did anything so stupid again.'
Franklin put a hand on the nearest armchair. 'Maybe you should sit down, Mr Harrington? I'm afraid we've got some bad news...'
'How's he holding up?' Callum fished the teabags out of the mugs and dumped them in the sink.
'Not well.' Franklin puffed out a breath and settled back against the worktop. Ran a hand across her face. 'Doesn't help they had a massive falling out last time they spoke. And now his son's dead and there's nothing he can do to fix it.'
The kitchen was nearly as big as Callum's whole flat, all marble and oak with a huge fridge freezer and a glass-fronted fridge just for white wine. A set of French doors led out onto a patio with wicker furniture dripping in the rain, and a set of steps leading down into a tidy garden with thick borders besmirched by more sodding gnomes. And beyond the fence: that view. Even in the pouring rain it was spectacular. Oldcastle, laid out beneath the heavy lid of grey, slivers of copper and gold caressing the Victorian cobbled streets of Castle Hill as the last gasp of daylight forced its way through the gloom. A slash of Kings River shining like a sharpened knife.
Much better than looking out on a railway line, a manky cluster of allotments, and some tenements.
How the other half lived.
Callum put the milk back in the oversized fridge. 'I called Mother, she's sorting out a Family Liaison Officer. And, according to McAdams, Brett Millar tried to bite off a nurse's fingers, so they've chucked him into a secure psychiatric ward. Straitjacket, padded walls, and twenty-four-hour surveillance.'
'That's the trouble with druggies, once they get the taste for human flesh...' The smile faded. 'Sorry.'
'Should think so too.' He put the mugs on a tray along with a packet of gingersnaps dug out of the cupboard above the kettle. Nodded at the door. 'Go on then.'
He followed her through into the book-lined lounge.
Mr Harrington was crumpled in one of the armchairs, his huge frame shrunken into itself, massive hands wrapped around his knees. Nose and cheeks red, as if he'd been standing out in the rain.
Callum put the tray on the floor and handed him a mug. 'Milk, two sugars.'
A sniff and a nod.
'Is it OK if we ask you a few questions about Ben?'
The lips curled on that slab of a face. 'My son's name is _Benjamin_.'
'OK. Yes, Benjamin.' He took his own tea and settled on the front edge of the other armchair. 'Benjamin bought a flat with his friends, Brett Millar and Glen Carmichael.'
'Gah.' Ben's dad stared down at the mug. 'The Millar boy was always trouble. I should've expelled him, but his parents were just as bad. It didn't matter that their horrible son was a bad influence on our boy, they made it very clear what would happen if I took the appropriate action. Drugs, on school premises!'
He made himself smaller in his chair, knees coming up against his chest. 'Of _course_ they were the Millar boy's drugs. Benjamin didn't do drugs, we brought him up better than that, and now these jumped-up little _nobodies_ are standing in my office telling me they'll go to the papers and say it was all Benjamin's fault.' Ben's dad grimaced into his tea. 'I should have expelled them: Bret _and_ Benjamin. I should have expelled them _both_. A headmaster has to have principles. He has to be uncompromising. He has to be the rule of law.'
Callum nodded. 'But you didn't.'
'How could I? Christine would have died from the scandal. So I made the whole thing go away, and the Millar boy continued to be a bad influence. It's amazing Benjamin got into university. A BA in aquaculture: it should've been law, or medicine. And does he _use_ his degree? No, he buys a worthless flat in a horrible part of town with his two useless friends and thinks he's going to be the next big property magnate.'
'And did Benjamin mention anyone else? Maybe someone he'd met recently? Someone new to their circle?'
'What, a woman?' Ben's dad shook his head. 'We should be so lucky. Oh, don't get me wrong, he isn't _gay_ or anything like that. He's just too busy being a conceited selfish little child to have a proper relationship.'
Franklin cleared her throat. 'Can we look at Benjamin's room, Mr Harrington?'
He wrapped his arms around his knees. 'It's upstairs, down the hall, at the end.' Then he laid his forehead on them and cried.
'Getting dark out there.' Franklin stood at the window, one hand on the _Star Wars_ curtains, looking out at the rain.
The bedroom was immaculate: no oil slick of socks and pants on the floor; all the books in neat little rows on the bookshelf; a fancy workstation with a big monitor, printer, and ergonomic keyboard, all lined up perfectly square; bed made, with the Pokémon duvet cover tucked in tight like they did in hotels.
'You think he tidies up himself, or does his mum do it for him?' Callum snapped on a pair of blue nitriles and tried the bedside cabinet. Socks. Pants. Hankies. Mickey Mouse watch.
'His father's _lovely_.' Franklin put on a slightly deeper voice, mimicking the clipped accent. '"He's not _gay_ or anything like that." Homophobic dick.'
'Check the wardrobe.'
She opened the doors and squatted down in front of it, rummaging through the neat rows of shoeboxes arranged in the bottom. 'Do you buy that whole "everything was Brett Millar's fault" act?'
'Yeah, well Brett Sodding Millar isn't exactly on my Christmas card list this year.' Callum pulled the drawers all the way out and checked the undersides. Nothing Sellotaped there. But there was a pair of socks in the gap beneath the bottom drawer. Probably fell out and popped down the back. 'Wasting our time here.'
'Probably.'
He pulled the socks out, frowned. There was something hard in the middle, something stuffed inside them. They got turned inside out on the bedspread, covering Pikachu's smiley face. 'Or maybe not.'
'You got something?'
'Flash drive shaped like a Lego man, and a wee ziplock baggie of pills.' He held the bag up. The contents looked like small green jelly beans. 'I'm guessing Benjamin was into Temazepam. Don't know about the flash drive, though.'
She pointed at the tower unit sitting under the workstation. 'Could find out easily enough.'
'And compromise the chain of evidence? No thanks. Whatever's on there, I want it admissible in court.' He pulled an evidence bag from his pocket and slipped the flash drive inside. Scribbled down the time, date, location, case number, and both of their names. Did the same with the pills. 'Probably won't go anywhere, but you never know.'
Callum stuck the bags in his pocket. Frowned at the room: the Pokémon duvet cover, the _Star Wars_ curtains, the shelf covered with little SpongeBob SquarePants figurines mounted above a row of kids' books. The framed _Finding Nemo_ print on the wall. 'It's all a bit... childish, isn't it? Like Ben's mum and dad are infantilising him. Keeping him young so they can control him.'
Franklin rolled her eyes, then stuffed the shoeboxes back in the wardrobe. 'You've never met a hipster before, have you? All this crap is "ironic". Watching _My Little Pony_ and getting cartoon characters tattooed all over your body. Listening to bands no one's ever heard of and wearing glasses you don't need just because the frames are "retro". Beards. Haircuts. Tight trousers.'
Callum slid the drawers back again. 'When I was a kid, people dressed up as goths. Or grunge was still a thing. Just.' He stood. 'Not me, obviously.'
'Too cool, were you?' She rifled through a stack of vinyl records.
'The home wouldn't let us wear make-up, or grow our hair. Not even the girls.' His blue nitrile gloves snapped off, got bundled in with the evidence bags. 'Billy Jackson came home from school one day with a pierced ear. Someone did it for him at break-time with a needle and a strawberry Mivvi. Mr Crimon beat the living hell out of him and made him sleep in the bath for a week. Couldn't stand up straight for ages.'
'I mean, look at these bands: Sui-psychedel-icide, the Burning Yesterday Collective, Gerbils from Saturn, Stalin's Wardrobe... Who _listens_ to stuff like this?'
'He runs a garage in Kingsmeath now. Still got a bit of a hunch on him.'
'Ooh, spoke too soon.' Franklin held up an album with a woodcut illustration of a rabbit and a cat dancing in a graveyard on it: _Open the Coffins_. 'Mind you, Harrington's probably only listening to it _ironically_.'
'Meh, the book was better. And speaking of which,' Callum pointed at the bookshelf, with its collection of textbooks and YA novels, 'how about we give this lot a quick rummage, then head?'
'Might as well.' She plucked a thick book from the middle shelf. 'Urgh. Listen to this: _Adaptive Governance_ , colon, _The Dynamics of Atlantic Fisheries Management_ , brackets, _Global Environmental Accord_ , colon, _Strategies for Sustainability and Institutional Innovation_ , close brackets. Sounds _fun_.' She turned it spine-side up and riffled through the pages. Nothing fell out. 'This home you grew up in: did _you_ have to spend nights in the bath?'
'If you were really bad, they half-filled it with cold water first.' He flicked through something about a teenaged spy. 'I guess some people just love working with kids.'
She dumped the textbook and tried another one. 'You report them?'
'You think no one listens to women?' The next book was about the same teenaged spy. How was an eleven-year-old boy supposed to disarm a nuclear weapon? 'Try being the kind of kid that gets labelled "challenging".'
'Hmmm...' Another textbook.
'R.M. Travis came to our school once. Signed my copy of _Ichabod Smith and the Circus of Doom_ and drew a little picture of a rabbit too... I was so nervous I nearly wet myself.' In the next one, Junior Superspy was foiling a global plot to wipe everyone out with Ebola. 'Course, I was too stupid to keep my mouth shut when I got back to the home. All puffed up and proud and showing the book off to all the other kids. So Mr Crimon confiscated it. Never saw it again.'
'Think we should seize the computer?'
'Can if you want, but the IT lab won't do a thing with it till someone bigger than you or me sets a flamethrower to their backsides.' He'd run out of boy-super-spy novels, so Callum moved onto a series about a boy vampire caught up in the Napoleonic wars. 'Maybe we should start with the flash drive and see how we get on?'
They flicked through every book on the shelf and only managed to turn up a voucher for guitar lessons that had expired three years ago. So much for that.
Franklin stuck the last YA novel back on its shelf. 'That's me.'
'Yeah.' Callum made for the door, then stopped as his mobile launched into song. When he pulled it out, the word 'HOME' sat in the middle of the screen. He gave Franklin a wee grimace and pointed towards the stairs. 'I'll catch you up.' Hit the button. 'Elaine?'
_'Hi... Peanut was wondering what time you'd be getting home.'_
'No idea. Late. Probably. You know what it's like with a murder investigation.'
_'Well don't binge on kebabs and pizza, I made tuna casserole for tea. Just make sure you call me when you're heading home so I can pop it in the oven.'_
'Yes, Boss.'
_'And while you're obeying my every whim, can you pick up some pickles and Nutella on your way home? Doesn't have to be the fancy ones with the white-and-green label,_ any _dill cucumbers will do.'_
'Anything else, Your Imperial Majesty?'
_'Love you.'_
'Me too.' He hung up and headed downstairs.
Through in the living room, Ben's dad was still in his seat – all curled up with his forehead against his knees. A living mummy.
Callum cleared his throat. 'Mr Harrington? Is there someone who can stay with you? Maybe a neighbour, or a friend? It's probably—'
The front door rattled and a voice boomed out in the hall. _'Oh for God's sake, Anthony, what have I told you about leaving your muddy shoes on the carpet? Honestly, it's bad enough I have to clean up after idiots all day without coming home to it too.'_
A small woman appeared in the doorway, peeling off a leather jacket. 'You can come help me with the shopping, it's...' She stopped. Stared at Franklin, then did the same to Callum. 'Anthony? Anthony, what's going on? Who are these people?'
Franklin held out her warrant card. 'I'm afraid we've got some bad news, Mrs Harrington.'
##
'No, Marline, I don't. And I'll tell you why I don't, because I never did nothing with him, OK?' Honestly, Marline was _such_ a bitch. 'If he says I did, he's _completely_ a liar.'
No noise from the other end of the phone – Stupid Central.
Ashlee slumped back on her bed and scowled up at the posters on the ceiling. All four members of Mister Bones, shirtless and smiling perfect smiles on some sunny beach somewhere _way_ nicer than crappy old Oldcastle. The three guys from Four Mechanical Mice in a swimming pool, all glistening and muscles and that. $ick Dawg, posing on a motorbike in leather jacket and jeans, all those tattoos on his naked hairless chest. Sexy and mysterious with a superhero mask and utterly cool-shaped moustache/goatee thing. Even if he did have a load of completely thin bitches in the background, posing in their bikinis and showing off. Skanky cows.
_'He said you did.'_
'What did I tell you? _Completely_ a liar.'
He was too. As _if_ Ashlee would ever touch Marline's sloppy seconds. Wasn't even that good looking. And he was a crap kisser. All fat slimy tongue and weird little grunting noises. Freak.
_'He said you snogged him outside the chipper, Sunday.'_
'Ungh. Who you going to believe, Marline: Peter – who utterly dumped you on your birthday – or your best friend in the world, AKA: _me_?'
More silence.
Taylor from Mister Bones was _definitely_ the hottest guy on her ceiling. He had these lovely teeth and a way of singing into the camera that made you know he was doing it just for you. Of course, she wouldn't say no to Zeb from Four Mechanical Mice either. Not with that lovely long hair.
It was nicer than hers.
Mind you, that wasn't difficult these days – hers was like straw. God she was _so_ disgusting.
Why would Zeb or Taylor want to go out with a fat pig like her?
Didn't matter how little she ate, or how many times she did sit-ups and squats and went jogging and everything. Here she was, practically living on rice crackers, sneaking off to throw up after every one of Mum's disgusting fatty meals of slop, and she _still_ wasn't thin. Not _properly_ thin.
She risked a look down at the lines where her ribs poked out beneath the tank top, the hip bones making twin rails through the boxer shorts, the gap between her thighs. There was _completely_ a roll of fat around her middle. Like a beer belly, or something. And she never even drank beer. How was that fair?
_'I'm sorry, Ashlee. I know you'd never do that to me.'_
A long low _'Brrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrring...'_ rang out from downstairs.
'Ungh.' She sat up. Yup, that was massively a roll of fat. 'Someone's at the door.'
_'He's such a liar, isn't he?'_
'Always was. You were _utterly_ too good for him, Marline.' Not true, but that was what you were supposed to say, wasn't it? Not, 'You were a matching pair of bookend freaks.'
She yanked open her door and stuck her head out into the dark hall. Put her phone against her fat-cow chest and shouted down the stairs. 'Door!'
_'Brrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrring...'_
For God's sake, did she completely have to do everything round here?
'DOOR!'
Her mum's voice came up from the kitchen. _'Well answer it then, I'm busy.'_ Probably making more lardy yuck for tea.
'I'M ON THE PHONE!'
_'So call them back!'_
'Aaaargh!' God, it was like... North Korea, or something. 'Fine. Whatever. Don't bother yourself. I'll just stop what _I'm_ doing, shall I?' Back to the phone as she stomped down the stairs. 'Marline?'
_'Everything OK?'_
'Yeah, if you think my MOTHER BEING AN ENTIRE BITCH is OK.' Nice and loud to make sure she could hear it.
_'You want to get wasted for my birthday next week? I can utterly rob a bottle of voddy from my gran.'_
'Yeah, why not. You only turn fourteen once, right?'
The hall, of course, was _completely_ Arctic Circle, because being an entire bitch means you're too tight-fisted to put a radiator in the hall. Not like it's chucking it down winter outside or anything, is it? Noooo.
Ashlee shuffled her feet into a pair of Mum's furry slippers, then grabbed a raincoat from the rack of hooks by the door and pulled it on. Hiding her disgusting fat body.
_'My step-dad wants to have a party down the bowling alley. Laser Quest, dodgems, and burgers, like I'm, I dunno, six years old or something. He's such a_ complete _spazmoidal—'_
'Yeah, hang on, Marline.'
_'Brrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrring...'_
'OK, OK. Jesus.'
Mum appeared from the kitchen, wiping her hands on a tea towel. Typical: turns up when all the hard work's done to take the credit. The lazy cow flicked hair out of her eyes – maybe a decent haircut would help with that? And a proper dye-job too. Honestly: going out in public with an _inch_ of brown roots showing. Never mind the chunky thighs and revolting saggy boobs, because _apparently_ it's OK to massively turn into a slob when you hit thirty. She draped the tea towel over her shoulder, like she worked in Starbucks or something. 'Who is it, Ashlee?'
'Yeah, because I'm COMPLETELY PSYCHIC! Jesus.' Honestly, how thick could you get?
Ashlee twisted the snib on the Yale lock and pulled the door open till the chain jerked tight. Crammed as much scorn into her welcome as she could: ' _What_?'
She's wonderfully thin. _Magnificently_ thin. Glaring up at him, her whole skull visible through the pale thin skin.
He blinks at her, making his bottom lip tremble. His eyes are all red and puffy, like he's been crying – amazing what you can do with a dab of vinegar on a fingertip. He clears his throat. Puts on his best fake Dundee accent, because that makes it a bit more interesting, doesn't it? Being someone else. Someone who doesn't _burn_ inside. 'I'm... I'm sorry, but I'm trying to find my son.'
'And?'
He pulls a sheet of paper from his pocket, printed out at home. 'MISSING! HAVE YOU SEEN SAMUEL (4)?' Underneath the headline is a photo of a wee boy – dark hair and freckled cheeks, a dimple in his chin. 'Please: his name's Sam. He's only four.'
A chunky woman walks up the hallway behind the angry young girl, a tea towel draped over one shoulder. Blonde. Curvy. Maybe a bit curvier than Father would have liked, but still pretty with it. 'Ashlee, don't be rude to the nice man.' She reached past her and unclipped the chain. 'Is he missing?'
'Some woman picked him up from playschool. They said she had dark hair and glasses?'
'Oh, you poor man.'
The girl, Ashlee, folds her arms and rolls her eyes. 'Oh very classy, Mother, completely get your hormones on.'
'Don't listen to her; please, come in.'
'Honestly, ever since "Uncle Eddy" left you've been _completely_ horny. Disgusting in a woman your age. Old people shouldn't be allowed to have sex, ever.'
The mother's cheeks darken, but she forces a big smile anyway. Brave little soul that she is. 'You said his name's Sam?'
'Sam. Yes.' Justin steps into the house.
Father never did get it. All that lying and play acting, dressing up like a priest to make people trust him, using Justin as bait. Silly, when you think about it. Unnecessary.
You don't need a little boy pretending to be lost if you wanted to meet women, you just need an _imaginary_ boy and a photo printed off the internet.
Much easier.
Justin smiles.
Strange, it's been years since he's used that name, but all this thinking of Father has brought it back. Comfortable and warm as an old jumper, or a pair of cosy socks.
So Justin takes his smile and follows the girl and her mother into a kitchen warm with the vanilla smells of baking. Reaches into his pocket and pulls out the knife.
##
EMERGENCY CALL – 09 Sept at 19 hours 52 minutes and 13 seconds
**OPERATOR:** Police Scotland, what's your emergency?
**CALLER:** Oh God, oh God, oh God. [Sobbing]
**OPERATOR:** Hello, can you tell me what's happened?
**CALLER:** He's killing them, he's... They're screaming!
**OPERATOR:** OK. I need you to give me the address.
**CALLER:** Please come... He's killing them.
**OPERATOR:** Where are they? I need an address. Give me the address.
**CALLER:** [Screaming] HE'S KILLING THEM! YOU NEED TO COME NOW!
**OPERATOR:** I need you to calm down. Listen to me. Listen, we can't come if you don't tell me where you are.
**CALLER:** I'm at home. I was on the phone to Ashlee and she was answering the door and the man came in and he said he was looking for his missing kid—
**OPERATOR:** There's a child missing?
**CALLER:** No, you're not listening! He said he was looking for it, but he... he... [Sobbing] and they let him in and now they're screaming!
**OPERATOR:** OK. Where are they? I need an address. Where does Ashlee live?
**CALLER:** With her mum. Two Twenty-Three Johnson Crescent, in Shortstaine. Please, he's got a knife...
**OPERATOR:** Hold on. [Keyboard noises] Cars are on their way. When did it—
**CALLER:** Hurry! You've got to hurry, they're screaming!
**OPERATOR:** It'll be OK. There's police and an ambulance—
**CALLER:** No, listen. They're on my mobile...
[Crackling]
**YOUNG WOMAN:** [Sound is distorted] [Screaming] GET OFF HER! GET OFF HER! GET OFF HER!
**WOMAN:** [Sobbing] Don't hurt my baby! I'll do anything you [Screams]
**YOUNG WOMAN:** NO!
[Grunting] [Banging] [Sound of glass shattering]
**CALLER:** Please, you have to get there!
**OPERATOR:** They're on their way. Can you tell me your name?
**CALLER:** Marline. Marline McFadden. You have to hurry up!
**WOMAN:** I didn't... I didn't...
**MAN:** [Shushing noise] It's OK. It's OK.
**YOUNG WOMAN:** Mummy?
**MAN:** I'll take good care of you [Too quiet to make out] forever. Won't that be nice? Forever and ever.
**YOUNG WOMAN:** Oh God, she's dead. She's dead. She's dead.
**OPERATOR:** Marline, I want you to record the call for me, will your phone let you do that?
**CALLER:** I... Yeah, completely! I've got, like, this app that'll—
**YOUNG WOMAN:** Get away from me!
**MAN:** They'll worship you. You'll be a god and they'll worship you.
**YOUNG WOMAN:** [Screaming]
##
Mother stood with her back to the room, facing the murder board with its growing lines of actions and outcomes. 'You know what I think? I think they need to let us speak to Brett Millar.'
The blinds were open, letting in the darkness from outside. What little streetlight that managed to make its way around to the back of the billboard tainting the shadows with orange and brown.
Watt curled his lip, upsetting the bum-fluff line of ginger pretending to be a beard. 'I talked to a Professor Bartlett over there, he said, and I quote, "Mr Millar is too volatile to remain un-sedated while in this establishment. I will not put my staff, or other patients, at risk."'
McAdams shook his head. 'A doped-up Millar? What use is that to us all? We seek a killer!'
'Then we'll just have to go round there tomorrow and give this professor the opportunity to change his mind, won't we?' Mother turned to face them. 'Right: home time. You can all go get yourselves a good night's sleep and come back bushy-tailed and bright tomorrow morning. Briefing will be seven o'clock sharp.'
Callum stuck his hand up. 'What about the flash drive?'
'The IT Lab have got it, so we should find out what's on the thing by...' she checked her watch, 'about the dawn of the next Millennium.' Then Mother stared at them all. 'Well, come on then: off you go. Home. Shoo.'
Dotty whizzed her wheelchair around in a tight circle. 'Pub?'
'No.' Watt marched out of the room, pulling his jacket on.
McAdams cupped his hands into a loudhailer: 'And remember to sign out this time!' Then a shake of the head and a sigh. Finally a smile pulled at his stubbly-grey Vandyke. 'Come, fair maiden Dot. Let us go from here to a bar. There to drink much beer.' He took hold of the handles on the back of her chair and steered her out into the corridor, throwing a parting shot over his shoulder, 'Dumbarton Arms. Last one there buys the crisps.'
'Boss?' Callum powered down his computer. 'If it's all the same, I'm going to stay and see what's on the drive.'
'Don't be daft, the IT Lab won't even _start_ looking at it for weeks. Dorothy's right: pub o'clock. It's about time Rosalind here did some team bonding.' Mother stuck her hands in her pockets and sauntered out of the office. 'And it's karaoke night down the Bart, how more team bonding can you get?'
Franklin watched the door close behind her. Then groaned. 'I am _not_ singing karaoke.'
Callum crept his way between the empty tables, balancing two pints of Stella, a half of Guinness, a pint of Old Jock, a gin-and-tonic, and a packet of dry-roasted on a wee tray.
The Dumbarton Arms wasn't exactly bustling at nine o'clock on a Wednesday night, which was probably why they'd turned the PA system up to a near-deafening roar. The only other patrons were an auld mannie and his Labrador, and a pair of students – young men that were more spots than skin. All of them blinking up at the little stage where a shiny-faced Franklin and Dotty were belting out an old Meatloaf standard about shagging in a car.
Doing a decent job of it too.
Callum lowered the tray onto the table and slipped back into the booth. Passed McAdams and Mother their drinks. Then gave Mother her change.
McAdams took a sip of Guinness. Raised his voice over the musical onslaught. 'I'd count that if I was you.'
'Oh, don't be such a misery guts, Andy. Callum has been in the wars and deserves a bit of sympathy.'
'Thank you.' Callum helped himself to one of the Stellas.
She nodded at the stage. 'Rosalind's settling in nicely, isn't she?'
Up there, Franklin was getting to the bit about sleeping on it.
Mother glanced at him. 'She says you probably saved her life, earlier. Could've died, running into traffic like that, but you stopped her.'
A shrug, then a gulp of cold lager. 'We need to find out what's on that flash drive.'
' _Really_ , Constable MacGregor?' McAdams let his mouth hang open. 'I hadn't thought of that, you must be some sort of genius!'
'Andy, what did I tell you?'
McAdams chewed on his face for a moment. Then, 'Sorry, Mother.'
'Better. The problem, Callum, is that there's a backlog of stuff waiting to be processed by the Forensic IT people. A _huge_ backlog. You can probably see it from space.'
A nod from McAdams. 'I took a laptop off a dealer six months ago and they haven't even powered it up yet.'
'Yes, but this is a serial killer investigation. Surely we can bump it up the priority list.'
Mother grimaced. 'Easier said than done. I've got no favours left in the IT Lab. How about you, Andy?'
'Do you think I'd wait six months for a laptop if I had?'
Callum scooted forward in his seat as Franklin and Dotty got to the finale. 'How about we call a press conference and tell the world we can't catch this guy, because Police Scotland won't give us the forensic resources?'
Mother and McAdams shared a look, then burst out laughing.
The last triumphant chord battered through the bar. Everyone clapped. Franklin and Dotty took a bow. Then the two spotty students scrambled for the stage as Oldcastle's finest made their way back to the table, grinning.
Dotty wheeled herself in next to McAdams. 'Phew, I'm roasted, is that mine?' She grabbed the pint of Old Jock and gulped at it.
Mother beamed at her and Franklin. 'That was _lovely_.'
Up on stage, the young men launched into ABBA's 'Dancing Queen'.
'You see, my dear Constable MacGregor, if it was that easy everyone would do it.' McAdams took a deep draught of Guinness, getting himself a little white moustache. 'Police Scotland do not give a toss about being showed up at press conferences. All that'll happen is they'll send some bigwigs up from Tulliallan to take over the case, kick us off it, then kick _us_. Hard. Probably in the genitals.'
Mother patted him on the shoulder. 'He's right.'
'I know it might feel like your career's halfway down the chunty right now, but they'll pelt it with used toilet paper and flush like madmen.'
Oh.
'Well...' Callum had a good long hard look at his pint. 'Can we put pressure on the Chief Superintendent instead? He's not going to want an unsolved serial...' Sodding hell. 'It'll be the same with him, won't it?'
'And the penny finally drops!' McAdams gave him a slow round of applause.
'Not telling you again.' Mother slapped the sarcastic git's hands. 'Callum, Andy and I had to fight like wounded badgers so they'd let us keep this case. _Any_ excuse and it's gone.'
The whole team slumped a bit at that.
Up on the stage, the boys danced and warbled their way towards the end of 'Dancing Queen'.
Dotty gulped down the last of her beer. Stuck the empty back on the table with a diaphragm-rattler of a belch. 'So sod them. We find another way!'
Franklin, Mother, and McAdams just shrugged.
'Come on, are we the Misfit Mob, or aren't we?'
McAdams sniffed. 'Suppose.'
'I can't hear you, soldier!'
He rolled his eyes. 'Ma'am, yes ma'am.'
'Could you be more enthusiastic, Andy? I've done jobbies with more life in them than that.'
'Enthusiasm brings me out in a rash.' McAdams downed his Guinness. 'It probably doesn't matter, anyway. Unless Brett Millar is our killer, whatever's on the flash drive has nothing to do with Imhotep.' He stood. 'Same again?'
'But it _might_.' Franklin was barely halfway down her pint. 'Anything that helps has to be a good thing.'
'Ah, the naive enthusiasm of youth.' He grabbed the empties and lumbered off towards the bar.
'Dancing Queen' finished with a lot of fist bumping and whooping.
Franklin rubbed her hands together. 'Who's up for a bit of _Grease_? Callum? I'll even let you sing the man's part.'
'Yeah... _No_. Can't.' He threw his Stella down his throat. 'I've got a pregnant girlfriend to get back to, and Nutella and pickles to buy.' He pulled on his coat. 'You kids have fun, though.'
'Dotty?'
'Oh hell yes.' And the pair of them wheeched and wheeled themselves off to the stage.
Mother ripped her way into the peanuts. 'Andy's not a bad man, Callum.'
'Does a good impersonation of one.'
There he was, standing at the bar, knocking back a sneaky whisky while the barman pulled the pints.
'They've got him on another round of chemotherapy. Being... colourful is how he copes.'
Great. Callum puffed out a breath. 'I'm sorry he's dying. But now and then, it _might_ be nice if he was "colourful" at someone else for a while, because I'm tired of being everyone's kicking post.'
Callum squelched around the supermarket aisles, wheeling a trolley and dripping on the polished floor. Pickled dill cucumbers: check. Nutella: check. Tesco own-brand high-strength paracetamol – not on the official shopping list, but his ear ached like a visit from The Claw, so: check. Bottle of shiraz – _definitely_ not on the official shopping list, but what the hell: check. Multipack of Wotsits: also check, because what was life if you couldn't push the boat out now and then?
One of the fluorescent lights flickered down the end of the cold-meats-and-ready-meals aisle, making the packaging glisten and buzz like something out of a horror film. Up above, the corrugated metal roof pinged and thrummed in the rain.
Be nice to pick up a curry meal-deal for two, but there was tuna casserole waiting at home. Maybe that could be a Friday treat, and sod the budget.
His phone blurted into life.
'Hello?'
Silence.
Checked the screen: 'NUMBER WITHHELD'.
'Hello?'
Callum squelched on through the flickering light, towards the checkouts. 'Hello? Willow, is this you?' More silence. 'It's OK, Willow, you can talk to me. Is someone threatening your mum?'
And the line went dead.
Might be an idea to pop round there tomorrow and make sure her mum hadn't _accidentally_ developed any more bruises.
But first: cycle home through the bucketing rain. Dry off. Painkillers. Wine. Tuna casserole. More wine. Bed.
A decent end to an incredibly crappy day.
And about sodding time...
The bike's lights flickered back from thick dark puddles. Their reflections swept across the dark canopy of leaves overhead, like tiny spotlights. Caught the drips of rain that worked their way through the canopy and made them shine, before disappearing again.
Camburn Woods lurked in the darkness either side of the path. A huge animal, breathing and rustling in the downpour. Waiting. The council still hadn't fixed the streetlights in here: most were topped with broken plastic globes and covered in spray-painted swearing. But the occasional one still glowed a pale gold, casting small pools of light to be swallowed by the forest.
A jogger puffed and plodded into view. A miserable-faced middle-aged man in lycra, lots of wobble as he exercised his way towards a heart-attack. Didn't even nod as Callum cycled past. Too busy sweating.
Probably wasn't the only one out there, sweating and panting in the woods.
Always a lovely thought.
Callum stood in the saddle, legs pumping as the path climbed up over a narrow railway bridge. Freewheeled down the other side. About fifty feet further on, the old familiar footpath led off to the left. Soon be home and...
He coasted to a stop.
Looked back over his shoulder.
The bike's back light cast a blood-red glow that barely touched the forest gloom.
Could've sworn he'd heard something.
A broken streetlight stood sentinel where the footpath snaked off into the undergrowth, leaving the whole area wrapped in darkness.
Callum pulled the bike around, twisting the handlebars, sending the front lights sweeping across the path, the trees, the bushes. 'Hello?'
Nothing. Just the staccato drip-drip-drip of rain on the canopy floor. The muffled grumble of traffic on the dual carriageway a quarter mile away. The dark-brown bitter-sweet tang of decaying leaves.
No one there.
So why were all the hairs standing up on his arms?
Yeah...
Maybe cutting through the woods wasn't the best of ideas at this time of night.
He turned the bike and _pedalled_. Onto the footpath, branches flashing past – caught for a moment in the front light before disappearing behind him. Heart thumping in his chest like a bear in a cage. On, through the gloom, and then BANG, he was out of Brothers Grimm territory and back in the real world.
Oh the glorious joys of tarmac and concrete.
Callum skidded to a halt on the pavement beneath a _working_ streetlight. Sat there in the rain. Panting. Staring back towards the maw of Camburn Woods.
No sign of anything following him.
And breathe.
Of course there wasn't anything following him.
Stupid.
He ran a hand through his wet hair.
Come on. Home.
The windows on Flanders Road glowed like welcoming beacons. Even if it was mostly rabbit-hutch houses and rabbit-hutch flats. Could see his and Elaine's one from here. Well, the side of it anyway. Top floor, third flat on the left, this side of the street. The light was on in the bathroom. Where he was going to take a long hot shower, thank you very much.
He cycled up the pavement and onto the road, lined with bottom-of-the-range hatchbacks and battered estate cars. Let himself into the communal lobby and chained his bike to the rack beneath the stairs. Picked up three small stacks of mail from the windowsill by the back door, and squelched his way up the concrete stairs.
Urgh.
Socks were like sponges, water oozing out of his lace holes with every step.
Callum took his jacket and rucksack off on the third-floor landing, gave them a shake to get rid of the water. Mrs Gillespie's cats had been at Toby's pot plants again – kicking soil in a fan-shape across the concrete in exchange for a little brown 'present'. No wonder his spider plants looked half dead as they sprawled their way up and around the far corner of the landing.
Well, if he didn't want them piddled and crapped in, he shouldn't leave them outside, should he?
Callum poked Toby's mail through the letterbox, then did the same for Mr and Mrs Robson. And, at long last, unlocked the door to his own sodding flat. Light caught the little brass plaque they'd screwed to the wood above the letterbox: 'CALLUM, ELAINE, AND PEANUT ~ THE MACGREGOR-PIRIE CLAN!' He slumped inside and thumped the door shut behind him. Sagged in place, and dripped on the laminate flooring for a moment.
Puffed out a breath. Worked his way out of his shoes and left soggy footprints all the way to the bathroom.
Raised his voice. 'Elaine?' Dumped his wet jacket in the corner and stripped off his shirt. 'We have _got_ to get ourselves a car. It's like trying to cycle through a swimming pool out there.'
Trousers, socks, and pants in a damp little pile. Then he cranked on the shower and stepped inside as soon as steam curled up from behind the curtain. Ahhhhh... Blissful heat.
Should probably keep his bitten ear out of the water, but the taped-on wadding was already drenched from the rain. So too late now.
A clunk as the bathroom door opened. 'Callum?'
'I know it's not top of our priority list, but a car would make life a lot easier when Peanut comes. Nothing fancy. You remember Billy Jackson? Bet he could get us a wee second-hand hatchback on the cheap.'
The curtain clattered back a couple of inches on its metal hoops and Elaine peered in at him as he soaked up the warmth. 'Where have you...' Her eyes widened. 'What happened to your head?'
'Want to get in with me? Be like old times, all soapy and slippery?'
'Callum, your face is all scratched and you've got a _bandage_ on your ear!'
'Come on, when was the last time we took a shower together?'
'Get out of there, now!' She jabbed a finger towards the bathroom door, mouth curled down at the edges. 'You've got a visitor.'
'Who?'
'DCI Powel.'
Callum screwed his face shut and thumped his forehead off the tiles.
Wonderful.
##
Poncy Powel sat on the sofa – in Callum's spot, thank you very much – in his fancypants suit, top two shirt buttons undone, no tie. A mug in one hand. Look at me. Look how at home I am, slumming it with the common man.
Callum loomed in the middle of the room, with a bathsheet wrapped around his middle. Dripping onto the rug. 'What do _you_ want?'
A sigh, then Powel pinched the bridge of his nose. 'Constable MacGregor... Callum, I'm here to give you a bit of friendly advice.'
'Aye, right.'
'Suit yourself.' He put his mug on the coffee table. Stood. 'But don't say I didn't try.'
Callum didn't move.
Another sigh. 'I don't like you. I don't think you're a good police officer. I don't _trust_ you.'
'If this is more crap about me taking a bribe to get Big Johnny Simpson off a murder charge, you can—'
'I understand you arrested Ainsley Dugdale yesterday.'
He bared his teeth. 'So?'
'I got a tip-off this afternoon from a nasty piece of work who breaks people's legs when they don't pay their loan shark. Dugdale's going round shooting his mouth off about how he's going to end you.'
'Ainsley Dugdale can pucker up and kiss my soapy backside.'
'Just... watch yourself, OK? Elaine here,' Powel pointed at her, 'swears blind that you're not as big a disaster as you look, so I'm doing you a favour. Dugdale is _dangerous_. It's not just the drugs and the protection rackets and the punishment beatings, he's implicated in at least two murders.'
'Fine. Consider me warned.' Callum tightened his grip on the towel. 'Now, feel free to sod off.'
'Callum...' Powel dropped his head back and stared at the ceiling. 'I couldn't give a toss if Dugdale kills you, hacks you to bits and chucks them in the Kings River, but you've got a _pregnant girlfriend_ to look after. You're going to be a father in two weeks. Try to think of someone else for a change.'
Think of someone else?
It wouldn't take much. Just two steps and slam a fist right in the middle of the smug git's face. They weren't on duty: it _probably_ wouldn't count as assaulting a superior officer.
Elaine put a warm hand on his arm. 'Callum, _please_. He's trying to help.'
But it would still count as assault.
Deep breath.
He relaxed his hand. Uncurled the fingers. 'Right.' Cleared his throat. 'Thank you.'
'I'm not your enemy, Callum. And you're not the only one he's threatened.' Powel buttoned his jacket shut. 'Well, I'd better get going.'
Damn right you'd better.
'Thanks for the tea, Elaine.'
She squeezed Callum's arm and he stepped back, let Powel past. Then she smiled at the smug-faced lump of yuck. 'Thank you, Reece. I appreciate you letting us know. Callum will be careful, won't you, Callum?'
What choice did he have? 'Of course I will.'
He stayed where he was as Elaine let Powel out of the flat. Sagged when he heard the front door lock thunk shut.
A home visit from DCI Powel and death threat from Ainsley Dugdale. Lovely.
She reappeared a minute later. 'Are you proud of yourself?'
'Since when were you and Poncy Powel on first-name terms?'
'Since we worked that murder/suicide last January. And he's trying to look out for us, OK? You didn't have to be so aggressive – beating your peely-wally chest like a wee shaved monkey. I'm amazed you didn't just drop your towel and measure dicks with him.'
' _He's_ a dick.'
'You know what, Callum MacGregor? Right now, so are you.'
And the worst bit was, she was right.
'Yeah.'
She closed the blinds, shutting out the dark night. 'But you're _my_ dick. Now go get dried and I'll heat up some tuna casserole.'
The flats on the other side of the railway line were mostly dark now. Lights off, time for bed. Wasn't much brighter in the lounge, where only the red glow of the answering machine fought against the night.
A faint rattling snore sounded in the bedroom, muffled by the wall. God knew how they were going to manage with a new baby in a one-bedroom flat. Wasn't as if they were rolling in cash here, even with Elaine's maternity pay.
But they'd make it work. Wouldn't they?
Course they would.
Callum toasted the faint reflection in the window and took another sip of wine. Dark in here, dark out there.
Powel was such a dick. _Dugdale's going to end you._ Yeah, right.
Unless it was Dugdale in the woods – the noises in the gloom – following him home...
Goose pimples rippled their way up his arms and across the back of his neck.
Yes, but it was cold in the living room with the heating turned off.
He'd beaten Dugdale once, he could do it again. In a fair fight, anyway. Which it wouldn't be. Dugdale wasn't a Queensberry rules kind of guy, he was a jump-out-of-the-bushes-with-a-baseball-bat/knife/illegal-firearm/attack-dog/three-friends-with-crowbars kind of guy. The kind you never heard coming until it was too late.
And what if he came after Elaine and Peanut?
What if all those silent phone calls _weren't_ some firm of PPI-claim tossers? What if it was Dugdale?
Something hard and sharp rolled over in Callum's chest.
First chance he got, it was off to the B&Q in Cowskillin for some heavy-duty locks. Fit them to the flat's front door. Maybe rig up a panic button or something? They probably wouldn't let him put a grade-one flag on his _own_ flat, but Poncy Powel could do it.
Worth a try anyway, seeing as he was suddenly all concerned for their wellbeing.
The Callum in the window shifted from foot to foot. Licked his lips. Blood fizzing at the base of his throat.
Dugdale wasn't taking his family away from him, and that was that.
He couldn't.
Callum drained his glass, picked a book from the bookshelf, and went back to bed.
_Nothing_ could.
## — the four-minute warning —
"I'm not sure about this," said Russell. "My nose is twitching like it does when there are goblins around, and goblins are _never_ a good thing."
"Don't be silly," giggled Martha, wriggling under the fence. "We're rabbits! No horrible old goblin could ever catch us!"
But little did they know that the Goblin Queen had sent her minions to the library for books about traps and snares and how to cook silly rabbits who stray into the deep dark woods...
R.M. Travis
_Russell the Magic Rabbit_ (1992)
_My mother didn't love me, so she gave me away._
_Man I hate that b*tch, every God-damned day._
_If she could see me now, she'd be proud as can be,_
_Standin' at the stage door, with her hand out for my money..._
Donny '$ick Dawg' McRoberts
'Mothers' Day'
© Bob's Speed Trap Records (2014)
##
'... extensive roadworks for the next three weeks, so you're going to want to find an alternative route. Jane?'
_'Thanks, Bob. It's competition time and we're giving away three pairs of tickets to Tartantula, this very weekend, folks. Stay tuned for how to win those. But first, how about some words from our sponsors?'_
'Here.' Elaine held out the Sainted Tupperware Box of Lunch. Her pink furry dressing gown hung open, revealing the huge swell of her bulge. It poked out of the gap where her jammies didn't meet any more, outie bellybutton on full display. 'Tuna casserole buttie, with cheese and hot sauce.'
Callum tucked his shirt into his trousers. 'You _do_ know I'm not the one who's pregnant, don't you?'
'Funny. You're a funny guy.'
_'... that's right, this week only, you can get two ScotiaBrand tasty chickens for just eight pounds. They're fan-chicken-tastic!'_
'How's the ear?'
A fresh wad of cotton covered the throbbing remains, glued to his head by half a dozen sticking plasters. Looked terrible, but at least it stayed on. 'Have you seen my red tie?'
'Cupboard.'
He had a rummage through the box. Frowned.
_'... deal of the century at Mad Mark's Motors! You want a new car? You got it! Nought percent finance? You got it! Easy payment terms? I_ must _be mad, cos you got that too!'_
There was a yellow silk tie in there. A proper one, not a clip-on. He picked it up between two fingers, as if it was likely to hiss and bite him as it uncoiled. 'What's this?'
_'... confused about the new tax rates for business? Don't worry, Davis, Wellman, and Manson – chartered accountants – are here to help...'_
'It's a tie.'
'Yes, I can _see_ it's a tie, what I want to know is: what's it doing in my box?'
_'... Oldcastle's premier pizza parlour just got even better!'_
'Isn't it yours?'
'I have clip-ons. Police officers don't wear real ties unless they want throttled by lunatic members of the public.'
_'... three toppings and get another two absolutely free! That's right: free!'_
'Hmm...' She took it from him, turning it over in her hands. 'I found it in the living room, beside the couch. Thought it was yours.'
Oh great. It really was a snake. 'It's Poncy Powel's, isn't it? He waltzes in here like he owns the place, takes his tie off and leaves it. What a _dick_.'
Elaine gave his arm a little punch. 'You should be nicer about Reece. He was only trying to help.'
'Bet he doesn't have to go to work with a tuna-casserole buttie.'
She rolled the tie into a neat little sausage and slipped it into Callum's jacket pocket. 'You can give it to him when you get there. And say thank you _properly_ for coming round to warn us.'
'Gah...'
'And you mock the tuna-casserole buttie, but trust me: Heston Blumenthal wishes he thought of it first.'
Yeah, right.
Franklin beamed, showing off those perfect teeth of hers. 'You look like crap, by the way.'
'Thanks.' Callum settled into his seat and covered a yawn.
' _And_ you missed a great night. We got everyone up for "We Are the Champions".'
He wrapped himself around his coffee. 'Thought you hated karaoke.'
'That was before I tried it.' She turned as Dotty wheeled herself into the room. 'Dot!'
'Am I late? Have we started yet?'
He checked his watch. 'Still got five minutes.'
'Oh, Callum, you missed _such_ a—'
'"We Are the Champions". Franklin told me all about it.'
Dotty wheeled Keith across to the tea- and coffee-making facilities, pausing to share a fist bump with Franklin on the way. 'Rosalind, my man.'
'We have _got_ to do that again.'
More grinning.
'Where's Whiney Watt the Wanky Waster?'
'Not in yet.'
'Good.' Dotty spooned coffee granules into a mug. 'I'm off to interview Brett Millar this morning. Assuming he's not still doped off his monkey. Anyone want to come? Rosalind?'
'Sounds good.'
Callum groaned. 'Noooo. Don't leave me with...'
The office door opened and in stormed everyone's favourite sour-faced git. Watt glowered out from beneath his greasy floppy fringe. 'What?'
'Speak of the devil.'
'Oh go bugger yourselves.' He hung his coat on the rack by the filing cabinets, straightened his nasty brown tie, then thumped himself down in his seat. Powered up his computer. Had a wee seethe on his own.
Ah, the joys of a happy team.
He was still sulking when Mother sailed in, towing McAdams in her wake. 'Gather round, my lovely ones. Time for assembly.' She perched on the edge of Dotty's desk. 'Andy, would you like to lead morning prayers?'
McAdams pulled some sheets of paper from a folder. 'Listen up, both young and old, / For a tale of woe you must behold, / Attention pay, so you may see, / The path of others' misery.' No applause. 'To wit, other stuff on the O Division books we don't actually care about, but have to pay lip-service to. One: they've still not got an ID for any of DCI Powel's severed feet. Two: someone set fire to an abandoned house, about halfway between Castleview and Auchterowan. Third house in a week, so there's an arsonist on the loose who isn't scared of a bit of graft. Three: someone ram-raided the Poundland in Logansferry, _again_. Guess some people just want to reach for the stars. Four: aggravated assault outside the Paris Casino on Holland Street. A group of young "ladies",' he made the quote marks with his fingers, 'attacked a taxi driver and left him with serious internal injuries, a fractured skull, and no sight in his left eye.'
'Ooh.' Dotty bared her gritted teeth. 'Not good.'
'Also not good is number five: mother and teenaged daughter abducted from their home in Shortstaine. Best friend was on the phone at the time and recorded the whole thing. And last, but not least, they got a DNA match from one of the bodies found at the tip on Monday: turns out Karen Turner _didn't_ run off to Portugal with another man, like her husband claimed. She was too busy being battered to death with a golf club and stuffed into bin-bags.' McAdams put his papers down. 'Now, would anyone like to contribute anything to these ongoing cases? No? Didn't think so. Moving on.' He pointed. 'Detective Constable Watt.'
Watt's eyes narrowed, little gingery beard bristling. 'What?'
'When I say you're supposed to sign out at the end of a shift, what I mean is: you – have – to sign off – at the end – of a shift. Not: "Do whatever the monkey-spanking hell you like." Am I getting through to you? Knock once for yes, twice for no.'
'It wasn't—'
'Don't even bother – I checked. And I don't _care_ how you did things in G Division, in O Division you clock off!' He pointed. 'Don't think I won't pull down your pants and spank you in front of the rest of the class.'
Watt's face went redder than his beard.
'Exactly.' McAdams rubbed his hands together. 'Now, children mine, police divine, / Tell me, what do you propose, / Our mystery to diagnose? DS Hodgkin?'
Dotty nodded. 'Rosalind and I are heading up the infirmary to see if we can get any sense out of Brett Millar.'
'Good: feel free to lean on the medical staff. I want a statement off Millar, A.S.A.F.P. Emphasis on the F. And that leaves naughty DCs Watt and MacGregor to canvas every smokehouse in the district. And not just phone calls: I want boots on the ground and signed statements.'
Callum sagged in his seat.
Why did God hate him? Wasn't sacrificing a quarter of his ear enough?
'After that, you can start on Dr McDonald's list: pubs and nightclubs where Ben Harrington, Glen Carmichael, and Brett Millar might have met Imhotep. Start from whatever's nearest the flat they were renovating and work your way out. Probably the Dockmaster's Yard? Let's see if we can't progress the plot a bit today.'
Watt folded his arms. 'I work better on my _own_. Why don't I take the smokehouses and leave the pubs to MacGregor?'
McAdams smiled and fluttered his eyelashes. 'Because I don't want you to be lonely, Detective Constable, that's why. Now off you trot like a good little boy and let's not have any nonsense about you two not playing nice. OK?' A nod. 'OK.'
Mother clapped her hands. 'And that's your lot. Keep me and Andy updated as you go. Class dismissed.'
As everyone else filed out Watt folded forward, elbows on his knees, head held in his hands.
Thanks for the vote of confidence.
'If it's any consolation, Watt, I'm not too thrilled being lumbered with _you_ either.'
He stayed where he was.
Fine.
Callum reached into his jacket pocket and produced the lemon yellow tie, all soft and slippery. 'Right, I've got to go see DCI Powel. In the meantime, do you want to get a list of smokehouses together? Or are you just going to sit there wallowing in your sulk?'
Nothing.
Oh today was going to be lovely.
He wandered out into the corridor. Down the end to the stairs. Clumped his way up to the fourth floor.
The Major Investigation Team had the whole level to themselves, complete with multiple meeting rooms, a series of open-plan offices with swish computers and new furniture, their own mini canteen... All right for some.
Most of the rooms bustled with uniform and plainclothes officers, making phone calls and writing things on whiteboards, tapping away at keyboards that didn't look as if they'd fallen off the ark.
The second door from the end was ajar, its brass plaque polished to a high sheen: 'DETECTIVE CHIEF INSPECTOR REECE POWEL ~ MIT'.
Callum went to knock, then stopped – knuckles half an inch from the wood – as Powel's voice growled out from inside:
_'No, Anita, I don't... Because I_ don't _, that's why... No, you listen to me for a change: marriage counselling didn't work, the second honeymoon didn't work, salsa classes didn't work. I've had enough. Enough of your sniping and your complaining and your nasty little comments. I've had enough of you_ poisoning _my own children against me.'_
Yeah. Probably shouldn't be eavesdropping on this. Still.
_'You know what? Cry all you like. It's over, Anita... No: it's over, because I deserve better than this. I deserve better than_ you _.'_
He had a quick check the corridor was still empty.
_'Of course I am, why do you think I packed a bag? I'll be round for the rest of my things in a couple of days, and if you even_ think _of touching anything, I'll do you for destruction of private property. Are we clear?... You better believe I will – they'll march you out of there in handcuffs.... I don't care: my lawyer will be in touch... No, you know what, Anita?_ You _go screw yourself. Christ knows I never want to again.'_
Then a clattering thump.
Presumably that would be Mr Telephone Handset being forcibly reconciled with Mrs Base Unit.
OK.
Count to five, and knock.
Silence.
Callum pushed the door open. 'Guv?'
Powel was behind his desk, face a threatening cloud of red and fury, glaring at the desk phone. Hands curled into fists either side of it, as if weighing up the pros and cons of smashing it into tiny little bits.
It was a pretty nice office, with a view out across the rooftops and up the hill towards the castle. The spire of St Jasper's, in the middle distance, jabbing at the low clouds. Big wooden desk, a pot plant fern thing, filing cabinets bereft of dents and scratches, framed certificates and news clippings on the walls, a whiteboard broken up into rows and columns full of neat little letters. A small couch, two chairs, and a coffee table. Very swanky.
'Guv?' Callum held up the tie. 'I think you left this at our house.'
He pulled his face up and round, clenched like his fists. His cheeks darkened even more. 'Constable MacGregor.'
'Elaine thought it was mine.' Callum laid the tie on the desk.
'I see.' He uncoiled a hand and picked the thing up. Slipped it into his pocket. Looked somewhere else. 'And how much of that did you hear?'
Innocent face. 'How much of what, Guv? I just walked up and saw your door was open. Took a chance on you being in.'
'Right. Yes.'
'Can I ask a favour? Not for me, for Elaine.'
Powel took a deep breath. Hissed it out. Then sat back in his seat. 'I'm listening.'
'We've been getting silent phone calls. And someone _might_ have followed me home last night. After what you said about Dugdale, I thought, just in case, if we could put a grade one flag on the flat?' A shrug. 'Probably nothing, but if Dugdale does try anything and I'm not there...'
'Yes. Of course.' Still not making eye contact. 'How is—'
A knock on the door and one of Powel's minions stuck his head in. All short-back-and-sides, baggy eyes, and sunken cheeks. An Aberdonian accent you could stun a sheep with at fifty paces. 'Sorry, Boss, but we've got a nine-nine-niner. Some wee wifie's turned up a heid in a shoppin' baggie.'
Powel stared at him. 'A head, in a _carrier_ _bag_?'
'Aye, hacked off at the neck and dumped in Holburn Forest aff nae far frae een o' the car parks. Div yis want ta gan oot and see it in situ?'
'God almighty...' He curled forward until his forehead rested on his desk organiser, talking into the interlocking biro doodles. 'Get a pool car, I'll be down in a minute. And get the pathologist as well. And the SEB. And a PolSA. And DS Blake. And about a dozen search-trained officers to do a fingertip.'
'Aye, Boss.' DC Teuchter pulled a face at Callum, then ducked back out into the corridor and shut the door behind him.
Powel didn't move. 'It never just rains, does it? No, it has to sodding _bucket_ down.' He looked as if someone had driven over him, then reversed a couple of times to make sure he was never getting up again.
Maybe Elaine was right? Maybe Powel was doing his best?
Callum cleared his throat. 'Are you OK?'
'Never better.' A sigh. Then he sat up in his seat. Made a note on a Post-it and stuck it to his monitor. 'OK: grade one flag on your flat in case Dugdale goes after Elaine. Anything else?'
'Actually...' OK, so it wasn't very ethical to take advantage of the man when he'd just split up from his wife, but nothing ventured: 'Don't suppose you've got any pull with the IT Lab, do you?'
Mother blinked at him. 'You're kidding...'
'I'm not.' Callum grinned. 'Probably a once in a lifetime thing, but if we go right now we might get it done before he changes his mind.'
She stared at him, then the Lego man flash drive in its evidence bag, then back to him again. 'Quick as you like.' Her chair juddered back on squeaky wheels and she was off, out of her office and marching down the corridor, pulling out her phone and fiddling with the screen. 'Andy, it's Mother. Callum's got Powel to—... No, of course I'm not checking up on you. But while I've got you, how's it going?... Oh, OK... No, no we're fine. You stay where you are, that's much more important. Listen to the nice doctors... Yes... Yes I will... OK, bye.'
They pushed through the double doors to the stairwell. Waited for the lift to creak and grind its way down from the top floor. When the doors dinged open, they revealed a filing cabinet and a stack of file boxes abandoned in the middle of the lift. Like it was a cupboard.
Mother squeezed inside anyway. 'Room up top for a small one.'
Urgh... He forced his way in, pressed hard up against the filing cabinet.
She thumbed the button for the sub-basement. 'Does it hurt?' Pointing as the doors groaned shut and the lift juddered into life. 'The ear?'
'Yes.'
'Brett Millar _does_ seem to like biting things, doesn't he?' She dug a little paper bag from her pocket. 'Have a jelly baby, it'll make you feel better.'
It probably wouldn't, but Callum took one anyway.
They stood and chewed in silence, squeezed in like beans in a tin.
Yeah, this was comfortable.
Now Mother's breath smelled of strawberries. 'I think it would be nice if we got a card for Andy. Wish him well with his chemotherapy. Maybe get him a cake or something?'
Because that would make _all_ the difference.
Mother popped another jelly baby, humming a happy tune as she munched.
For God's sake, how long did it take a lift to get to the sub-basement?
Callum shuffled his feet.
Stared at the numbers changing on the display above the doors.
_Ding_.
He was first out, popping into the dull grey corridor like a cork from a bottle.
Mother marched out through the double doors and into the warren of corridors and rooms that lurked deep below O Division Headquarters.
He followed her through the maze to a black door with a plastic plaque with 'FORENSIC IT LAB' on it.
She wiggled her fingers, as if she was limbering up to play the piano. 'Remember: I do the talking.' Then pushed through the door.
Callum followed her into a room crammed floor-to-ceiling with metal shelving racks, each one packed with computers, laptops, and cardboard boxes with cables poking out of them. More boxes, hundreds and hundreds of them, each the size of a paperback book, were stuffed into the racks, six or seven deep. A workbench sat against the wall by the door, with a row of computer monitors mounted above it and more dangling cables.
A thin woman in a once-white lab coat was hunched over a netbook, tapping away at the keyboard with purple-nitrile fingers. Tongue poking out the side of her mouth. Glasses balanced on the end of a long straight nose.
Mother knocked on the wall and she flinched hard enough to make her wheelie office chair trundle back from the bench.
'Aaargh...' Mrs Thin turned and scowled. 'Don't _do_ that!'
'Ruby, this is DC MacGregor. We need you to access whatever's on here.' She held up the evidence bag with the Lego man flash drive in it.
The woman in the lab coat raised both eyebrows, then burst out laughing. 'You're kidding, right? Of _course_ you're kidding. Do you have any idea how many bits of electronica are in the queue ahead of you? Let me give you a clue, Flora, it's _hundreds_.' She spun her chair around and waved at the little paperback-sized boxes. 'I've got nearly a thousand mobile phones in here, not to mention everything else. And every time your lot arrest someone another lump gets added to the pile.'
Mother placed the evidence bag on the countertop. 'I know, but we've had the nod from DCI Powel: he wants this bumped to number one priority.'
Ruby's eyes narrowed. 'Oh yes?'
'Yes. It's all been cleared with Cecelia too, you can give her a call if you like. Check.'
She reached for the phone on the desk, then froze. 'This is on the up?'
'Cross my bypassed heart and hope to die.'
That thin pink tongue slithered out between her lips and Ruby picked up the evidence bag instead. 'OK. I guess I can spare five minutes.'
She scooted her chair down a couple of monitors and clicked an ancient black laptop into life. 'We'll run it on a virtual machine, just in case it's chocka with viruses.' She scribbled an entry into a form, copying down all the info from the bag, then pulled the Lego man's legs off and stuck his exposed connector into the side of the laptop. 'Any idea what we're looking for?'
'Not a sausage.'
The machine whirred and clicked.
Mother lowered her voice. 'I like your hair, by the way. Very nice. Frames your face.'
'I was thinking of going blonde.'
'Oh, no. Auburn suits you much better.'
A window appeared on the screen.
'Here we go.' Ruby fiddled with the mouse. 'Looks like it's password protected, so let's see what Aunty Ruby's box of magical tricks can do...' More fiddling. 'Edward Snowden didn't know the half of it.'
Numbers and dialogue boxes flashed in and out of existence.
Mother perched herself on the only other chair in the room. 'So, are you still seeing Charlie from the Finance Team?'
'Not for ages. He was a bit...' She pulled a face. 'I didn't mind the spanking so much, but the PVC all-in-one suit did terrible things for my dermatitis.'
'Spanking?'
'Well it seemed to make him happy, though God knows how he managed to sit down the next day for work. Men are funny creatures, aren't they?' A quick glance at Callum. 'Sorry, but it's true.'
The screen flickered with more boxes. Numbers. Lines. Boxes. Numbers. Lines.
Then the whole thing cleared, leaving a dialogue box.
Ruby hunched over the keyboard, tongue poking out again, and clattered her fingers across the keys. Sat back and smiled. 'Why people never use proper encryption is beyond me. Have you got a clean drive?' She held out her hand and Callum dropped a plain grey USB stick onto her palm. 'Thank you kindly.' It went into the slot next to Mr Lego.
Lights flickered on the stick.
'Just take a minute.'
Mother smiled. 'Next time I bake, you're getting brownies.'
'Brownies are good.' The machine pinged and she pulled out the USB stick. Handed it to Mother. 'You want a quick squint while you're here?'
'What's on it?'
The mouse clicked. 'Looks like we've got a bunch of video files and some word docs. Let's try... this one.' She clicked on an icon in the shape of a piece of film and a new window filled the screen. Black as it loaded. Then...
Mother's eyes widened. 'Oh.'
Callum hissed out a breath. 'Bloody hell.'
But Ruby just nodded. 'Now _there's_ something you don't see every day.'
##
Oh God, they were _completely_ going to die here. Ashlee sniffed back a drip, mouth a trembly wobbling line, cheeks wet with tears. Eyes darting back and forth, pulling shapes from the darkness. How was this fair? How come it couldn't happen to Marline instead? How couldn't _she_ be the one chained up in here? At least Marline would've deserved it!
Another wave of shivers rattled its way through Ashlee, making the water ripple.
Gah. If it _was_ water. The stuff smelled like piss and vinegar and that manky potpourri Mum brought back from Barcelona on her last holiday with 'Uncle Eddy', before he realised how utterly a slob she was.
A metal tank full of cold piss and vinegar and manky potpourri. Like the world's crappiest hot tub.
Ashlee gulped in a big shuddery breath. 'Mum?' Her voice was tiny, high-pitched like a mouse or something. 'Mummy?'
She craned her neck to the side, pushing it as far as it would go, till the chains dug into her skin. 'Mummy, I don't feel so good...'
But Mum didn't move. She just sat there, with her back against the wooden slats, the chain around her neck tight from there to the wall, because she'd slumped a bit to one side. All naked and pale and bloaty.
The bruising was getting worse. It wrapped all the way over the left of her face, dark and purple in the gloom.
The bandages around her hands and wrists were spotted with red and yellow, arms dangling loose at her sides.
'Mummy?'
Ashlee's head fell back, making a dull ringing noise when it hit the metal tub.
They were completely going to die here. Alone and hungry and thirsty, in some crappy wooden room that _stank_ like a chimney fire.
Quiet little sobs popped and crackled from her mouth.
Why couldn't it be Marline?
'Shhhh...' A voice in the dark.
Ashlee froze, eyes widening till it was like they'd pop free or something. 'Please. Please don't hurt me.'
'I'm not going to hurt anyone.' He stepped closer, settled his backside on the edge of the metal tub. Almost invisible in the gloom, like he was a ghost or something. Hands pale as a dead fish. But his voice was all, you know, warm and cheery – like he was a drama teacher or a kindly relative or something, instead of completely a psycho. 'I bet you're thirsty. And cold and tired and lonely. You _must_ be hungry.'
She shrank back, but the chains wouldn't let her go any further. 'I'll scream.'
'That's OK: I've got a few minutes before I head back to work.' He put his fingers in his ears. 'You go ahead if it makes you happy.'
So she did. Long and loud and hard. Over and over till her throat was sandpaper raw and her head rang from the noise.
Ashlee slumped back in the filthy water, panting.
'There we go. Was that good?'
'Please don't...'
'Here, this'll make you feel better.' He pulled one of those plastic sports-bottle things from his jacket, the kind with a pop-up top so you can drink and cycle at the same time. He held it out. 'It's herbal water. It will cleanse away your sins. You'll be pure and free.' Gave the bottle a shoogle. 'You want to be cleansed, don't you?'
Ashlee stared past him, at Mum with her chains and her bruises. 'I want my mummy.'
'Come on, you need to keep your strength up. You're going to be a god.'
'If you let me go... If you let me go, I _promise_ I won't tell anyone. I swear! Just let me go and—'
'Drink.'
'You can keep my mum! Let me go and she'll be _completely_ grateful. Mum's like utterly dirty and—' The slap snapped Ashlee's head to one side. Leaving her face stinging.
'Shhh...' He stroked her hair with his free hand. 'New Mummy isn't dirty. New Mummy will keep us all safe and warm and loved, with her pretty yellow hair and soft cosy lap. New Mummy loves us.'
'You're off your mental nut. We... Just let us go, yeah? _Please_.'
'Don't you want to be a god?' He held the bottle to her lips.
'GET OFF ME!' She flinched back, till the chains yanked tight. 'You're completely a psycho! You're utterly and complete— Ulk.'
He grabbed her face with his other hand, fingers digging into her cheeks, forcing her jaws open. Jamming the sports bottle in and squeezing. Bitter cold water flooded her mouth, full of bits and yuck and no way in hell was she swallowing that crap.
The chains clanged and rattled as she grabbed at his wrists. Sobbing. The whole front of her face burst into flames, burning from the inside out, as that horrible water burst out of her nostrils.
He grabbed her nose and squeezed it shut. Forced more water into her mouth. 'I know it's nasty, but it _is_ necessary. You _have_ to drink it.' Held on tight. 'You want to be a god, don't you? You want people to worship you?'
Her fingertips scrabbled at his forearms. 'Ghhhhhhhaghhh!'
'Shhhh...' He dropped the bottle and forced her mouth shut again, keeping everything inside. 'It'll all be fine. You just drink up.'
Ashlee bucked and thrashed, sending water slopping over the edge of the tank.
Can't breathe.
Can't breathe.
CAN'T BREATHE.
So she did the only thing she could do: she swallowed the horrible bitter water.
He let go of her face. 'There's a good girl.' Smiled as she coughed and retched. Then scooped the plastic bottle out of the tub. 'One mouthful down, six more to go.'
Oh God...
The walls pulsed and groaned, twisting round through ninety degrees before snapping back into place and twisting again. And again. And again. Like hammering back a quarter bottle of Marline's gran's voddy in one go. No coke or nothing.
Ashlee blinked, screwing her face up tight. Bared her teeth.
Don't be sick, OK? Not again.
The bits were still floating all around her from last time. Because the manky bathtub water wasn't bad enough.
His voice came from a long way away, echoing and slow. 'How are you feeling?'
She could barely turn her head, it weighed _so_ much. 'Gnffffmmmmnt...' Why wouldn't her mouth work?
'Good.' His face swam in and out of focus, pulsing in time with the walls. 'I envy you so much. You're going to be a god. Isn't that great?'
'Plnnnnssss...' Her chest was full of rats. Rats and seagulls. And bees. Scrabbling and flapping and buzzing deep inside her.
'They'll worship you and you'll save them, because you'll be a god.'
Rats, seagulls, and bees.
His voice faded away. 'I have to go to work, but I'll be back to check on you. That'll be nice, won't it?' Faded away until there was nothing left but the noises in her chest. Scrabbling, flapping, and buzzing.
_'They'll worship you: you'll be a god.'_
A hole opened up in the base of Ashlee's skull, and the whole world fell through it...
It's her eighth birthday, and all she wants is a pretty princess cake and the new Nerf Elite Hail-Fire blaster, cos you can shoot _a hundred and forty-four_ darts seventy-five feet. Which is as long as a swimming pool. And it would be utterly cool, cos Marline's got an Alpha Trooper CS-18 and that only holds eighteen darts.
But what Ashlee's got is Mum and Dad screaming at each other downstairs in the living room.
She sits on the edge of her bed, picking at the scabs on her left knee. Peeling away the hard bits of skin, making the shiny pink under it bleed.
Then there's a thump and a crash and the sound of heavy feet on the stairs.
Dad throws her door open. 'You. Downstairs. _Now_.'
'But—'
'NOW!' He's bigger than a bear, teeth all shiny, and eyes like burned things. Hands like crushers as he grabs her arm and pulls her out onto the landing. Drags her down the stairs. Shoves her through into the living room.
A blue and red smear is all that's left of her Princess Merida cake, sagging its way down the wallpaper by the window. The coffee table's cracked and broken. Mum's on the carpet next to it, on her hands and knees. Sobbing.
'Tell her!' Dad spits on the back of mum's head. 'TELL HER WHAT YOU DID!'
'I'm sorry. I'm so sorry.' Her pretty yellow dress is all ripped up the side, showing off blotchy purple-and-yellow skin.
Dad drops down to one knee and pulls Ashlee round, so she's got no choice but to look into those burned-thing eyes. 'Because of your slut mother, I'm going to have to go away. That's _her_ fault. She's making me leave.'
Ashlee doesn't say anything, just stands very still.
'I wanted to take you with me, and you could be my little princess, and we'd have adventures all over the world, and we'd be together forever.' He smiles like broken glass. 'But your mother's spoiled it. And I'm going to have to leave you here. With _her_. And I want you to know, it's _all_ her fault. You could be happy, but she won't let you.'
And then he stands and sweeps out of the house like a storm, crackling lightning and thunder, slamming the front door behind him.
Ashlee bites her bottom lip.
All she wanted was a princess cake and a Nerf Elite Hail-Fire blaster.
Mum cries and moans. 'I'm sorry...'
Then Ashlee blinks away tears, turns, and goes back upstairs to her bedroom.
Slams the door.
Just like her dad.
Marline swallows, shudders, and passes the bottle of voddy. Hisses out a breath. 'Gaghh...'
Ashlee takes a swig. God, it's _horrible_. The stuff makes her teeth itch and her mouth fizz, numbing her tongue before she forces it down. Hot and burning. 'Ghaaaa...' She shakes her head from side to side as her eyeballs pop and crackle.
Then it hits her stomach like burning petrol, spreading its fire through her whole body.
Marline pops one shoulder up till it's pressing against her ear. 'Smooth.'
The car park's empty, all the little people headed home for the night. Now there's nothing left but empty spaces and security lights. All the shutters are down on the shopping centre's windows.
Ashlee swings her feet back and forward, back and forward, holding onto the railing with her other hand. 'What you wanna do?'
'Dunno.' Marline takes another scoof of voddy. Shudders. 'Ghaaa...' Holds the bottle out. 'Ooh: you know Peter, right? Sits behind me in English?'
Oh yes. Peter with his squinty eye and his funny teeth. Peter with 'ASSASSIN'S CREED' printed across his school rucksack in flaky enamel paint.
Ashlee helps herself to another mouthful. 'Yurrrrgh...' The warm numbness is getting bigger. Stuff isn't _brilliantly_ revolting when you get used to it.
'Well, Peter completely wants to go to Dougie's party with me.'
'Pfff...' She closes her eyes and listens to them crackle. 'Yeah, that's how it starts. "Oh, go to the party with me. I love you so much." Next thing you know he's trying to finger you in his dad's shed and if you say no he'll tell everyone you're a frigid bitch.'
'Peter?' Marline's voice wobbles, that one word catching and tearing like damp toilet paper.
Ashlee puts the bottle down and wraps her arm around Marline. Gives her a hug. 'No. I'm sure Peter's not like that.' Even though he probably is. They all are.
'Oh, yes, that's _very_ pretty.' Uncle Eddie folds his arms and looks her up and down. Smiles like a crocodile. 'Very pretty indeed.' He licks his lips. 'Now, why don't you try on the red one again?'
The city's spread out before them like burning jewels in the darkness. It's not even that cold, perched up here on the edge of the old castle wall, legs dangling over the edge. Way below, the dual carriageway is a ribbon of streetlights, taxis, and the odd bus. Wouldn't think it was nearly Halloween.
Peter turns to her, with his wonky eye and his funny teeth. 'Are you sure it'll be OK?'
Ashlee swaps the half bottle of Smirnoff to her other hand, then reaches over and cups her hand around the crotch of his trousers. What's inside is hard like a spanner. 'You chicken?'
'But Marline—'
'Marline's an utter munter.' She parts her lips and leans in. 'And I'm _mint_.'
Off in the distance, an airplane roars into the October sky.
The ground rushes up, closer and faster and she's screaming a broken-bottle scream and—
Ashlee fights her way up through the duvet, till she's sitting up, dripping with sweat. Shivering and shaking. Mouth hanging open so she can haul in deep juddery breaths.
Gah...
Just a nightmare. Nothing to worry... about.
There's someone in her room!
She grabs the bedclothes and pulls them up to her chin, scrambling backwards till she thumps into the headboard.
It's Uncle Eddie. Smiling. Her old manky teddy bear sitting in his lap. Covering things while he zips himself up. 'Sweetheart.' He leans forward. 'It's OK, you were having a bad dream. I wanted to make sure you were all right. You're all right, right?'
Ashlee nods.
'Good. Now, you lie down and go back to sleep and I'll stay here to make sure no monsters get you.'
Too late.
''Snot... 'snot fair...' Marline's back heaves as she spatters out this massive flood of Bacardi, all mushed up with a shared poke of chips. 'Hurrrrrgkkk...'
Ashlee rolls her eyes, both hands full of her best friend's hair. Keeping it out of the way as she chucks away perfectly good rum. 'You're too good for him, Marly. He's completely a wanker.'
'How could he... could he... With _her_! Hurrrrrrrgkkkk...'
Because he was a man and that's what men did.
And he wasn't even all that good at it.
'I'm not eating this _slop_!' Ashlee grabs the plate in front of her and flips it up and off the table, sending it spinning till it clatters against the kitchen floor and shatters into three jagged chunks, spraying disgusting spaghetti bolognese everywhere.
Like she's going to eat that?
Spaghetti bolognese? How much fat and carbs are in that? Millions, that's how much.
One hundred and ten percent revolting.
Mum just sits there, bottom lip trembling. Look at the lardy cow cry: big fat tears rolling down her big fat cheeks.
Ashlee stands. 'No wonder no one loves you.'
Little waves lapped the walls of the tub, sloshing the filthy water around as Ashlee sobbed.
The man with blue eyes lied: she wasn't going to be a god. She couldn't be. She was a _monster_. And no amount of bitter water was going to change that.
Why did she have to be so horrible to Marline?
Why did she have to be so horrible to Mum?
To Peter. To everyone...
Spoiled and vile and horrible.
And now she was alone. In a rusty metal tank, in a manky smoky room, with nothing but the darkness and the cold and the itchy feeling in the pit of her stomach for company.
The seagulls and bees were gone – no more buzzing, no more flapping, leaving her innards full of rats.
Any minute they'd wake up and gnaw their way out of her, turning the dirty water a nasty shade of scarlet.
When did she eat the rats?
_Why_ did she eat them?
Ashlee craned her neck round again.
Mum hadn't moved: still slumped against her chains, arms hanging loose at her sides, bruises ripe and dark.
'Mummy?' Ashlee kept her voice down so the rats wouldn't wake up. 'Mummy? Don't let them kill me...'
But Mum didn't answer, because Mum was probably dead.
Selfish cow.
##
'So?' Watt stared at him from the passenger seat.
'So what?'
The pool car thrummed over the cobbles, lurched across a disused set of railway lines, windscreen wipers making a squealing harmony with the screeching gulls.
'You know very well what: what was on that flash drive?'
Callum shuddered. 'I'd rather not think about it.'
The Logansferry docks probably didn't feature in Oldcastle's tourist brochures. It wasn't quaint and old-fashioned like the Kettle Docks across the river – with its gaily coloured wee boats and fishermen's huts – instead it was a rigid grid of huge grey slab-fronted warehouses and chandlers' yards ringed with chain-link fences and barbed wire. Hordes of camera-toting tourists didn't come here, even if they'd managed to get past the security gates they'd end up squashed beneath a forklift truck, articulated lorry, or shipping container.
Watt folded his arms even tighter. 'Don't tell me then.'
'Imagine the most horrible porn you've ever heard of, double it, and add a collection of dogs and farmyard animals. Gah...' The shudder worked its way from the back of one hand, all the way across his shoulders and down the other side. 'I'm never singing "Old MacDonald Had a Farm" ever again.'
'So absolutely _no_ help on the case.'
'Thanks for making it sound as if that's _my_ fault.' Callum took the next turning, along a narrow strip of cobbles. On the left, a waist-high wall separated the road from the river, nothing on the other side of it until the grey lump of Kingsmeath reared up the hill. Ancient stone buildings lined the right-hand side of the road, squat and solid, with rust-reddened corrugated iron roofs and heavy steel doors. He pulled up outside one, two thirds of the way down. 'This is us.'
It had the standard barn-style sliding door, painted a faded blue with 'MEARNS FINE FISH PRODUCTS LTD' in chipped white paint the width of the building.
Watt undid his seatbelt. 'I'm only going to say this once: you are not going to cock this up, do you understand? I will lead the questioning, you will keep your mouth shut and don't touch anything.'
Callum turned in his seat. 'Tell me, _Constable_ , exactly who the hell do you think you are, ordering me about?'
'I'm the police officer who doesn't take bribes to let murderers go free.' He climbed out of the car and slammed the door shut.
Oh no you don't.
Callum clambered out and slammed his own door. The air was heavy with the oily reek of raw diesel and rotting fish. Rain bounced off the pool car's roof and bonnet. 'You know what, you gingery-pube-bearded sack of wank? I've never taken a bribe in my _life_!' He marched around the car, closing the distance. Balling his hands into fists. 'And I am sick and tired of snide sneery comments from arseholes like you.'
Watt looked him up and down. Then stepped back. 'Big Johnny Simpson walked, because of you.'
'IT WAS A SODDING MISTAKE, OK?' He thrust his arms out. 'Have you never cocked anything up in your life? Are you so _buggering_ perfect?'
The bleep-bleep-bleep of a vehicle reversing cut through the gulls' lament.
Behind Watt, the Kings River was a twisted swathe of pewter, dull grey in the rain. Cars on the other side had their headlights on. And up above the sky was the colour of ash.
'Well?'
A shrug. 'You expect me to believe that you didn't take _any_ money from Big Johnny Simpson?'
Callum turned his back. 'Go screw yourself.'
The cobbles were slippery, so he picked his way across them to the big steel door, grabbed the cold metal handle and hauled it back far enough to let out the deep smoky tang of burning wood. Stepped into the gloom.
'Hello?'
The room was easily big enough to park a couple of buses in. Racks along the walls, pallets of boxes in the middle, what looked like a large walk-in cold room on one side with another one beside it. Puddles dotted the damp concrete floor.
'ANYONE IN?'
He worked his way through the boxes to a small office with a grubby window overlooking the warehouse. A thickset woman was behind the desk, wearing a hairnet, heavy-duty white plastic apron, and a bright red fleece. She waved at him and pointed to the phone in her other hand.
Fair enough.
The door rumbled shut, then Watt appeared at his elbow. 'You're a bit touchy, aren't you?'
Callum kept his eyes on the woman. 'You can talk. All you've done since you got here is moan.'
No reply.
Then Watt puffed out a breath. 'I did the right thing, and I got shafted for it. Half my team were on the take and when I went to Professional Standards do you know what happened? Suddenly _I_ was the bad guy.' He folded his arms, tight. 'How is that fair?'
'Yeah, that's pretty crappy.'
'Oh, crappy doesn't even _begin_ to cover it.'
The woman in the office threw back her head and laughed, setting quite a lot of things wobbling.
Watt shifted his feet. 'And just in case you're thinking this is some sort of bonding moment, it isn't. I'm telling you what happened to make it perfectly clear: I don't like bent cops. I _hate_ them.'
'How many times do I have to say this? Eh? How can I make it any clearer? I – didn't – take – any—'
'Aye?' The office door opened and she peered out at them through a pair of black-rimmed glasses. 'Can I help you?'
Watt pulled his warrant card and thrust it under her nose for a second. 'Detective Constable Watt. I'm here to examine your premises.'
She raised an eyebrow. 'Oh aye?'
'And I'll need a list of your employees.'
'Will you now?'
He checked his watch. 'Soon as you like. We've got seven other smokehouses to visit today, so...?'
She folded her arms, making the fleece bulge. There was a lot of muscle in there – probably all that humping heavy boxes of fish about. 'So what?'
Watt leaned in close. 'So: chop, chop.'
Great.
Because _that_ was how you got the public on your side.
'Actually,' Callum produced his own warrant card, nice and gentle, 'what my colleague _meant_ to say was, we really need your help. Any chance you can show us around and answer a few questions?'
Watt stiffened. 'Thank you, Constable, but _I'm_ dealing with this.'
'Please forgive him. He's been in a bad mood ever since he got back from the doctor. They can't do anything about his frighteningly small penis, and it's upset him a bit.'
'What?' Watt wheeled around, mouth pinched, eyes bugged, face darkening.
The woman in the plastic pinny burst out laughing. Then slapped Callum on the back, hard enough to send him staggering. '"Frighteningly small penis." Aye, he looks the type.'
'Hey!'
'Come on, I'll show you around. You too, Wee Willy Winky.'
Watt hurled himself into the passenger seat and slammed the car door.
A bunch of seagulls had taken it upon themselves to respray the windscreen and bonnet with grey and white spatters. All streaked in the rain.
Callum got in behind the wheel. Started her up and set the wipers going, turning the glass opaque. 'Who's next on the list?'
And explosion in five, four, three, two—
Watt thumped his hand down on the dashboard. 'WHAT THE BLOODY HELL DO YOU THINK YOU'RE PLAYING AT?'
A pull of the squooshers sent two streams of blue foamy liquid into the smears, thinning them. 'Are we having a grump?'
'How _dare_ you tell that woman I have a small penis!'
'Well, you were acting like a massive dick, so I thought I'd even things out a bit.' The windscreen was almost clear, so Callum pulled back onto the road. Going slow as the wipers made gaps in the seagulls' art.
'You can sodding well—'
'Know what? I don't think people won't work with you because you clyped on your old team – I think they won't work with you because you're crap at the job.'
'You completely undermined me back there—'
'She wasn't going to tell you anything, Watt. You spoke to her like she needed scraping off your shoe, how was that going to help? You're rubbish at talking to people.'
'I AM SODDING NOT!'
'Marching in there like the King of Dickland. Look at me, I'm so important!'
'I'm going to report you.' Watt thumped back in his seat. 'Soon as we get back to DHQ, I'm putting in a formal complaint.'
'You can't treat people like that and expect them to _help_ you, you idiot.'
They'd barely gone a hundred yards before the next smokehouse appeared in the row of ancient buildings. 'OLDCASTLE SMOKED SEAFOOD SPECIALITIES ~ FABULOUS FISH AT ITS FINEST' boasted a slick plastic hoarding above a stainless-steel set of double doors.
'I'm a _police_ officer.'
Callum parked outside the front. 'I'm sorry, OK? I'm sorry I told her about your minute genitalia, but if you keep on acting like that people are going to assume it's microscopic anyway. At least this way we got a list of every staff member and their rota for the last month.'
The smokehouse had a separate glazed door for enquiries and what looked like a wee factory shop inside.
He undid his seatbelt. 'Are you coming, or are you staying here to sulk?'
Watt scowled at him. 'God, I _hate_ you.'
'Course you do. That's because you're jealous of my jumbo-sized penis.' Callum climbed out, grinned at the rain-dulled river, then turned and hurried in through the glazed door.
After a beat, Watt slammed his car door and thumped after him.
**S CALLOWAY HADDIES**
**(O LD-FASHIONED, HONEST, & BEST)**
_Unit 4, Harbour Road, Logansferry_
Thick greasy coils of smoke filled the room, curling behind them as they stepped inside.
'And this is where the magic happens.' Mr Smug swept a hand upwards. His white coat almost gleamed in the gloom and so did the white porkpie hat.
Callum stood in the middle of the room and stared straight up.
Rows and rows of fish hung, head-down, above – each one suspended by the tail from stainless-steel poles turned yellow by the smoke. Had to be about a thousand of them in here, vanishing up into the smoky heights.
Three smokehouses in and Watt was still grumping. As if it was Callum's fault he'd been born with the kind of genital appendage that could only be seen with an electron microscope. Wasn't his fault that Watt looked like a tit – one hairnet flattening his floppy fringe, another covering his wispy pube-like beard.
A little oily drip splashed on his shoulder.
Callum turned to Mr Smug. 'Is this area ever left unsupervised?'
'What, the smokehouse?' He curled his top lip. 'Oh no, no, no, no. This is a twenty-four-seven operation: we supply haddies to _Harrods_. The only time we shut down is for five days in January to do a thorough deep clean.'
'How about the nightshift?'
'Three staff on at all times. Like I said, it's a twenty-four—'
'Seven. Yes. Thanks.'
**A BERCROMBIE FISHERIES**
**(T RADITIONAL SMOKED FISH SINCE 1826)**
_14 Ship Lane, Logansferry_
'Oh aye. Aye, aye, aye...' Mrs Lumps hauled an empty plastic box on top of the full one and laid out a layer of split herrings in the bottom with quick fluid movements. Then topped them with a fistful of salt, flung with casual precision. 'We've been smokin' fish here, ooh, since the eighteen hundreds. No me personal like.' She gave Callum a wink that bordered on the obscene, layering up more herring as she went.
'And how many people have access to the smokehouse?'
'Och, just me, Jeemy, and the boy Rodger – that's him in the dungers on the forkie. Big lump that he is.' She waved a handful of salt at a bear of a man in dungarees, driving a forklift truck laden with boxes of ice. 'He's our Siobhan's eldest. Well, it's a family business, ken? Has been since the start.'
'OK, so does—'
'We do a lovely hot smoked salmon with Drambuie, chilli, and lemon zest. Our Hot Toddy Salmon was on the TV, you know. We were a "Food Hero"!'
'Yes, that's great. But could anyone else have access to the smokehouse?'
'Oh I doubt it.' Mrs Lumps gave him a big gap-toothed smile. 'We've got the biggest sodding dog you've ever seen. Take your arms off soon as look at you would our Winston.'
**L ENNOX, BREMNER, & WALLACE**
**(L UXURY SEAFOOD SPECIALISTS)**
_Unit 2–4 Consort Lane, Queen's Quay, Castle Hill_
Mr Baldy stuck his hands in his pockets and rocked on his heels. 'And that's it. The grand tour. " _Y daith fawreddog._ " As they say in Welsh.'
Rain hammered down on the large concrete yard, bouncing off the piles of empty fish boxes and refrigerated containers. Made rusty streaks down the ten-foot-high walls that blocked off the outside world. The warm rich smell of smoke wafted out around them, billowing from the open double doors through to the processing plant.
Callum turned. 'Didn't know you were Welsh.'
'I'm not, no. But I _do_ like leeks. So...' Shrug.
'Right. Great.' Why was nobody normal any more? 'You got a big staff?'
'Sixty-two last count. Most are part time – we went into this job-share scheme thing, couple years ago, and you wouldn't believe how many single mothers we've got working here now. Had to open a crèche.'
Which explained the Portakabin in the far corner, behind the containers, all covered in characters from _Winnie the Pooh_ and _SpongeBob SquarePants_.
Callum shifted back a bit, till he was underneath the roof of the loading bay again. 'How many men?'
'Ooh, now you're asking.' He chewed on the inside of his cheek for a bit. 'Fifteen? No: eighteen. I forgot about Mitch, Spanner, and Dingle.'
'Dingle?'
'Don't ask. And Spanner's not much better. What's the point of employing people if they never turn up? I told, Marge, I told her: we need to fire these idiots, but she's soft as Angel Delight, she is.'
'How many in their mid-twenties?'
Mr Baldy did the cheek-chewing thing again. 'Might help if you told me what this was all about.'
Watt pulled down his beard net. 'Sorry, it's an ongoing investigation so we've got to be a bit discreet.'
Dear Lord, was Wee Willie Wattie _actually_ talking to someone like a human being?
'Ah, right. Got you.' Mr Baldy nodded, as if that explained everything. 'I'd have to check the staff records, but I think I can help.'
##
Callum stuck the keys in the ignition and pulled out his phone. 'Hello?'
McAdams' voice drawled into his ear: _'An update I seek, dear Constable Useless, / On Imhotep – killer both nasty and ruthless, / The smokehouses visited, all must be—'_
'Yeah, I get the point. We've done five of the seven smokehouses in Oldcastle: seen round the premises, spoke to the managers, got lists of staff members: when they've been working and where.'
_'I had another three verses.'_
'Thought you were meant to be busy this morning.'
_'If there's one thing you can say about sitting in the Grim Reaper's Soulless Anteroom of Death, with a drip full of poison seeping into your veins, it's that_ any _distraction is a welcome one. Even talking to a lump of gristle like you. Now: An update I seek, dear Constable—'_
'We've done five, so we've got two more to visit in town, and the one over in Strummuir, but...' He tapped his fingers against the wheel as Watt sagged his way into the passenger side.
_'But?'_
'I don't know.' From the car park outside Lennox, Bremner, and Wallace, there was a rain-greyed view across the river to Castleview. Left a bit and there was McKinnon Quay with its background of grim council flats. Squint a bit and you could almost make out the one where Benjamin Harrington died, facedown in bathtub full of brine. 'All these places: they have to conform to EU directives and health-and-safety and food standards. They get inspected by Environmental Health Officers – and you know what the Cheese Police are like: they spot anything, they shut you down.' He frowned out at the rain. 'No. These are commercial enterprises working six days a week, minimum. Someone would notice if you stuck a body in their smoker for a fortnight. It'd get in the way of the kippers.'
Watt wiped the water from his face and flicked it into the footwell. 'Who is it?'
'McAdams.'
_'What?'_
'Not you: Watt. Thing is, I think our boy's built his own smoker. Or he's got access to one that doesn't operate any more. Somewhere you can smoke a body for weeks without any chance of it being found.'
_'And that's supposed to help, is it?'_
'Don't know. Even if he's built his own smoker, he'll have to get the wood he burns from somewhere. Lucy down the mortuary reckons it's a mix of beechwood and oak. Maybe we should get in touch with whoever it is sells sawdust and woodchips to smokehouses? See how many sales they make to hobby smokers?'
Watt shook his head. 'If he doesn't have to comply with food standards, he doesn't have to buy commercial-grade sawdust. He can just get a big bag of it from the local sawmill, or someone that does firewood.'
'You get that?'
McAdams made a hrumphing noise. _'Or he could just—'_
' _Andrew!'_ A woman's voice in the background: _'What did we say about mobile phones?'_
A scrunching noise, and his voice went all muffled. _'Oh bounteous nymph, I hear thy pleas, / but it's police business, so sod off, please.'_
_'And the "please" on the end's meant to make that all better, is it?'_
_'Yes. Now be a good nurse and see if you can rustle up a cuppa and a biscuit. I'm wasting away here.'_ Then McAdams was back at full volume: _'Where was I? Ah, yes:_ _Imhotep doesn't have to_ buy _sawdust at all. He could just get himself a bunch of logs from the forestry commission and make his own.'_
True.
'Worth a try though.' Callum turned the keys, setting the blowers roaring. 'How did Dotty and Franklin get on?'
_'What am I, your secretary? Finish up with the smokehouses, then the pair of you get back to the shop and start chasing up wood suppliers.'_
'But—'
_'That's what you get for interrupting my poem.'_
And he was gone.
**B UCHAN'S CATCH**
**(T HE TRUE TASTE OF SCOTLAND'S FINEST FISH)**
_Buchan House, Brunel Street,
Shortstaine Business Park_
Mr Suit held out his hand for the printouts. 'Thank you, Janice. Tell Ted I'd like to see him in the boardroom in ten minutes please.'
'Yes, Mr Telford.' She pivoted on her heels and clacked out of the room, no-nonsense bob swinging in time with her footsteps.
He scanned the paperwork then slid it across the desk to Callum. 'We have a strict vetting policy and rigorous health-and-safety training for all our staff. All references are followed up. Random drug tests. Etcetera, etcetera.' He made circular motions with his hand – like the Queen waving out of a carriage window – showing off two signet rings and a gold bracelet with his name engraved on it: 'NORMAN'.
The view from his office window wasn't really grand enough to justify the floor-to-ceiling glass. It looked out on the factory complex, in all its stainless steel glory. A ballet of forklifts and containers, hoppers of salt and preservatives, a row of industrial units with their processions of raw and smoked fish. And beyond that, a set of grey warehouse buildings with the ScotiaBrand Tasty Chickens logo on them: a smiling rooster making a thumbs-up with his wing, a mini pastoral scene in the circle behind him. Steam coiled up from the slaughterhouse.
'If you turn to the back, Detective Constable, you'll find Appendix B lists everyone we've had to let go over the last six months.'
Callum scanned the names, then passed the list to Watt. 'Any absenteeism?'
'Oh no.' Mr Suit shook his head. 'We disapprove of that kind of thing. My workforce is highly motivated and dedicated to the task of delivering the most cost-efficient smoked fish and fish-derived products to market.'
Sounded lovely.
'Quick question for you: where do you get your wood from, for smoking?'
'For our luxury _undyed_ range? I'll have to check with procurement.'
Callum forced a smile. 'Thank you, that'll be a great help.'
He pressed a button on the desk phone. 'Janice? Get me Charlie.'
**G ORDON REID & SONS**
**(M AKE IT A FINE FISH DAY)**
_10 Admiralty Place, MacKinnon Quay, Castleview_
Callum nodded. 'Take your time.'
Seagulls screamed on the roof opposite, fighting over something grey and slimy.
Mr Short-And-Limpy curled his top lip, staring down at the printout.
It wasn't a big yard, certainly not compared to Buchan's Catch. Barely room for a stack of empty fish boxes; a couple of Calor gas bottles; a big yellow plastic container heaped with reeking bones, heads, and guts; and a garden shed with a set of folding chairs, a wee card table and an overflowing ashtray. The door wide open to let the cigarette smoke out.
Rain danced on the shed roof, making it rattle like a drum.
Barely room for the four of them in here, but it was better than standing out there in the rain.
Mr Short-And-Limpy shrugged. 'No idea.' He passed the sheet of paper to Mr Ageing Hippy. 'What do you think, Chris? Recognise any of them?'
'Hmmm...' A frown. Mr Ageing Hippy took the fag out of his gob and shook his head, setting the dreadlocks swinging. 'Nope, sorry.'
'Well, thanks anyway.' Callum took the printout back. Folded it so the picture of Brett Millar, Benjamin Harrington, and Glen Carmichael didn't get creased. And slipped it into his pocket. 'It's OK, you finish your break: we'll see ourselves out.'
Watt followed him down a dank corridor, past a billowing cloud of bitter wood smoke, and out into the rain again. 'You know this is a five-minute walk from the flat they were doing up on Customs Street, don't you?'
'Yup.' He ducked his head and hurried across the road to the pool car. Plipped open the locks and scrambled inside.
Bleak granite buildings loomed on either side of Admiralty Place. Ancient warehouses with boarded-up windows, five-storey terraces with rust-streaked fronts, the slow whirl of evil seagulls.
A _thump_ and Watt was in the passenger side. 'Don't you think it's a bit of a coincidence?'
'Could be. Maybe.' He pulled out his Tupperware box and popped the lid. Today's note was just a message, no picture, no puns: 'I KNOW THINGS HAVE BEEN DIFFICULT, BUT NO MATTER WHAT HAPPENS I LOVE YOU EVEN MORE THAN NUTELLA AND PICKLES.' He slipped it into his jacket before Watt saw it.
'Oh come off it – there's a smokehouse right here, and up there Ben Harrington's lying in a bath full of brine, waiting his turn to be kippered. And... Can you not do that please?'
'I'm starving, OK?' Callum stared down at the sandwich – cut on the diagonal as if he were royalty – a bag of Asda's own-brand salt and vinegar, and a 'fun-sized' Snickers bar. 'It's well after two, and we've not stopped for lunch yet.' He pulled out one triangular half of the dubious sandwich and sniffed it. There was a faint whiff of mushrooms and a sort of savoury cheesiness.
Watt humphed.
'What?'
He looked out of the window. 'Left mine back at DHQ.'
Ha, ha, ha. Tough.
Callum caught sight of himself in the rear-view mirror.
Don't be a dick.
Ah well...
He held the half-sandwich out. 'Here.'
An articulated lorry grumbled past, hauling a pair of shipping containers.
Watt didn't move.
'I'll take it back if you don't want it.'
'Thanks.' Watt accepted the triangle and took a bite. Chewed. Frowned. 'Erm, what _is_ it?'
Callum popped open the crisps, poured them into the Tupperware and stuck it on the dashboard in easy reach of them both. 'The thing is, if they're the ones smoking the bodies, why did they leave him to rot in the bath?'
'I mean, I'm not saying there's anything wrong with it, it's just... unusual.'
'Why wouldn't you just pop up there one night, after dark, and bring him back for smoking?'
'No, but really: what's _in_ this?'
'Why just leave him there?' Callum had a bite of his own half. Wasn't too bad. Maybe Elaine had a point after all? 'Leftover tuna casserole, cheese, and hot sauce.'
'Oh.' More chewing. 'I quite like it. Spicy.' He helped himself to some crisps. Crunched. 'You heard Dr McDonald: Imhotep is a perfectionist. Ben Harrington wasn't properly prepared, so he swells up in the bath as the bacteria get to work. His stomach bursts and he's not good enough to preserve. So Imhotep leaves him where he is.'
'Maybe. Worth checking, anyway.'
Fog gathered on the car windows, turning them opaque.
Callum finished his half. Sooked his fingers clean. 'When I was wee, we used to go on caravan holidays. Mum and dad were mad on them – bundle the family into the car and go live in some field, sleeping in what was basically a large aluminium shed. Eight or nine times a year, every chance they got.'
That got a noncommittal grunt from Watt, as he polished off the last of the crisps.
'Nairn, Banff, Sandend, Findochty – that kind of thing. But the favourite was Lossiemouth. Every year, regular as the swallows, the MacGregors would pack up the Travel Scrabble and migrate to their spiritual home, hauling a caravan.'
'Are you going to eat that Snickers?'
'Yes.' Callum unwrapped it. Sighed. Then cracked it in two. 'Go on then.' Popped his own half in his mouth. 'Every time we went to Lossiemouth we'd charge up and down the beach, go rockpooling, collect seashells. And we always spent at least one day in Elgin. Dad would go see some friends – which was code for the pub – and Mum took Alastair and me to the museum. They've got a Peruvian mummy there.' He frowned. 'All naked and curled up with hands against its chest, and its knees against its hands, and its head bent forward... Used to think it was the most fascinating thing in the world. A real live dead body.'
Watt let out a long contented breath and settled back in his seat. 'I wonder if it's worth checking with the planning department to see if anyone's put in an application to build a new smokehouse?'
'Then one day they had a display all about the guy who donated it to the museum. Turned out that in the area where it came from, the mummies weren't just dead people: they were elevated to the position of gods. That freaked me out. Stood there staring at it for ages.'
'And we should crosscheck all the staff lists, see if anyone's been doing the rounds. I'll get them rattled into a spreadsheet and we can sort them by name, or company, or start and end date. Should be pretty straightforward.'
'It wasn't a dead body, it was a god. I was standing there looking at a genuine, one hundred percent, real live Peruvian _god_. And I couldn't help wondering: what would happen if it woke up?'
**S TRUMMUIR SMOKEHOUSE AND VISITOR CENTRE**
**(F ROM GLEN TO SEA, PRESERVING SCOTLAND'S HERITAGE)**
_19 Chapman Street, Strummuir_
Watt coughed. Grimaced. Leaned towards Callum, keeping his voice low. 'Is it just me, or are you getting fed up breathing in smoke?'
'Yup.'
Their guide waved at them to join him in the smokeroom. 'This is the best bit.' Mr Trendy had to be in his forties, far too old to be dressing up in skin-tight jeans and Converse trainers. _Star Wars_ characters posed on his right arm, various X-Men on his left, all of them tattooed in bright LOOK AT ME!!! colours. A tweed waistcoat and a T-shirt with a badger on it completed the ensemble. At least he'd hidden his stupid auld-mannie haircut under the obligatory food-hygiene white hat.
Callum stepped through the big wooden doors and onto the concrete floor. Heat radiated off the pile of wood in the middle of the room, sending up tiny orange sparks and a constant barrage of pungent wood smoke. 'Yes. Very good.'
Mr Trendy pointed upwards. 'We don't churn out "product" like the industrial big boys, but hey, who wants to eat chemically dyed fish, stuffed full of preservatives and additives? Not me!'
Racks of hanging fish reached up into the smoky gloom above them, then the sun must have come out, because that grey mass turned a brilliant white, silhouetting the herrings and haddocks and God knew what else.
'We don't even use stainless steel – all our poles are beech, _sustainably_ harvested from The Swinney.' He held his arms out, as if he'd just won a marathon. 'Isn't it beautiful?'
'Yes. Very good.'
Mr Trendy led the way back onto the processing floor. 'We think natural materials are _very_ important. And it's not just tradition for tradition's sake: the fish tastes better this way. That's why our smokehouse is built from local larch and granite.'
Two men in jeans and T-shirts were layering and salting fish – in wooden boxes, not plastic, of course – listening to some sort of terrible accordion-and-banjo music on a non-traditional and non-sustainable iPod docking station. A walkway ran around the room, about twelve foot up, and a group of cagouled tourists leaned on the handrail, taking selfies with the action in the background. Thrilling.
'And we do a roaring trade in preserving courses for gourmets, gourmands, and the epicurious. But it's not just smoking: it's cheese making, charcuterie, pickling. We're building some wood-fired ovens for a bakery course, if you're interested? Or I run a foraging class, that's always popular – we don't have many hedgerows, but there's mush-rooms, nuts, berries, sorrel, wild garlic?'
Watt dug the printout of Glen Carmichael, Brett Millar, and Ben Harrington from his pocket and held it out. 'Do you recognise any of these men?'
'Hmm...' Some heavy duty frowning. 'I think, maybe this one? On the end with the ink? Sure I've seen him somewhere.'
'What about your staff?'
'I don't know if they'd recognise them, I can ask though?'
Watt gave him a smile. 'Please.'
Mr Trendy marched off with the printout.
As soon as he was out of earshot, Watt lowered his voice again. 'What do you reckon?'
'Definitely has that hipster thing going on. Dr McDonald said he'd be able to blend in with Glen, Ben, and Brett. He's definitely got access to a smokehouse. _And_ he recognised Brett Millar.'
A nod. 'I think Darth Wolverine just became Suspect Number One.'
'Darth Wolverine?'
'You know, because of the tattoos? _Star Wars_ , _X-Men_?'
'Oh.' Callum shrugged. 'I was calling him Mr Trendy.'
Someone else in jeans and T-shirt lumbered in, pushing a hand truck stacked up with more wooden boxes. Short and compact, with a close-cropped haircut and the kind of faded-blue tattoos on his upper arms and wrists that screamed 'I've been in prison!' Mr Trendy waved him over.
Watt shook his head. 'Darth Wolverine's got more of a ring to it.'
'So: we do some digging.' Callum got out his mobile and called Control. 'I need a PNC check on one Finn Noble, mid-forties, don't have an address.'
_'Give us a minute...'_
Mr Hand Truck took off his glasses and frowned at the photo. Then up at Mr Trendy. Who pointed over at Callum and Watt.
Then a frown. A nod. And Mr Hand Truck was off, tipping his boxes, sending ice and gutted fish splashing across the flagstone floor as he darted back out through the door he'd come in.
Watt thumped Callum on the arm. 'We've got a runner!' He sprinted across the room as the tourists swung their mobile phones round, grinning and filming.
'Sodding hell.' It was nearly impossible to get up any speed on the ice-slicked floor, but Callum did his best, hammering after Watt – past Mr Trendy and his staring minions, skidding around a slew of broken boxes, and thumping through the door.
Down a short corridor.
_BANG_ – out through the door at the far end and into a rainy courtyard with a walk-in fridge off to one side and a stack of wooden pallets in the corner.
Mr Hand Truck made straight for them, arms and legs going, head down, with Watt in hot pursuit.
A leap, and the wee tattooed bloke scrambled up the pile of pallets like a monkey. He didn't pause at the top, just hurled himself over the top of the courtyard wall.
Watt clambered up after him.
Come on, up we go...
Callum leapt, grabbed a handhold of splintered wood and hauled himself up the wobbly pile. Sprawled over the lip and onto the top one. The whole stack rocked when he stood up. Yeah, no way this was safe. He lunged for the wall, feet scrabbling at the whitewashed stone as the whole mound of pallets clattered to the flagstones.
Aaaargh...
One leg up and over. Then the other one and he was lying on top of the wall. A short section of roof sloped down towards the swollen river, the slates slick with rain. No sign of Watt or Mr Hand Truck.
Deep breath.
He launched himself onto the roof, crouching low, arms spread out as his shoes slithered on the damp slates. A wooden deck wrapped around the side of the smokehouse about ten feet below, complete with tables, chairs, and patio umbrellas. A handful of people standing and staring, some with their mobile phones out, filming something just out of sight.
Callum dropped onto his backside and slid the last three feet. Popped over the guttering and dropped. Landed with a grunt in the shattered remains of a wooden table and broken patio umbrella.
Watt was on his back next to the wall, wrestling with Mr Hand Truck – trying to grab his hands and failing. A fist smacked into Watt's cheek with a dull slap, sending his head bouncing off the decking.
Then Mr Hand Truck grabbed the patio umbrella's base: a round lump of cast iron, big as a manhole cover but twice as thick, with a foot of splintered wooden pole sticking out of it. Raised the whole thing above his head like a makeshift hammer, muscles in his arms bunching with the strain.
Watt's eyes went wide. He snatched his arms in front of his face. 'NOOOOO!'
Callum dropped his shoulder and charged, hurling himself into the base as it swung down, knocking it sideways into the wall.
_Clang_.
His hip thumped into Mr Hand Truck, sending him bouncing off the wall too.
A knot of arms and legs.
Some swearing.
Thumps.
Then pain ripped its way up Callum's leg – bursting out from his inner thigh. 'Aaargh!'
He snapped around and there was Mr Hand Truck with his teeth buried in Callum's trousers, about eight inches from his groin. 'GET OFF ME!'
Callum smashed the heel of his hand into the biting scumbag's nose. It made a satisfying crunch, and he reared back, eyes closed, blood exploding from his nostrils.
'Aaargh, you dirty...' He grabbed his inner thigh. Dear _God_ , that stung...
Mr Hand Truck pitched backwards onto the decking, making groaning foamy noises as little bubbles of blood popped from his broken nose.
Callum was shoved into the wall again as Watt wriggled out from underneath.
'You!' Watt hauled out his cuffs, floppy fringe all bent and twisted. 'STAY DOWN!'
But Mr Hand Truck wasn't having any of it. He rolled over and fought his way to his feet then lurched off across the decking, scattering tourists all around him.
There was a gap in the railing – stairs down to the grassy river bank.
He staggered down the steps, leaving a trail of red drops on the wooden boards, Watt limping after him.
Callum hauled himself upright. Struggling on his unbitten leg. Gritting his teeth. Making for the stairs in painful hops.
A jetty poked out into the dark-grey water – no more than a dozen feet long, with a couple of rowing boats tied up on both sides. A week of constant rain and the river was in full bore, breaking over the jetty's uprights, pinning one rowing boat against the wooden posts while the other was stretched downstream pulling its mooring line tight.
Mr Hand Truck stumbled his way onto the jetty, both hands clutched over his nose.
Watt closed the gap. 'GET BACK HERE!'
'Urgh.' Come on, move. Callum limp-hopped down the stairs.
Mr Hand Truck came to a halt at the end of the small jetty, looked back over his shoulder, then jumped into the rowing boat on the upstream side. Which promptly overturned and dumped him in the river. 'Aaaaaaaargh!'
He disappeared under the roiling gunmetal water. Thrashed to the surface, snatching handfuls of air as if he could pull himself up with them. Disappeared underwater again.
Didn't come up.
No, no, no, no...
Callum managed a wobbly run, every other step sending rusty shards of metal digging into his thigh.
Watt paced the width of the planks, staring down into the dark water. 'Sodding hell.'
A huge fountain of spray and Mr Hand Truck burst into view again on the opposite side of the jetty. Coughing, spluttering, and screaming. 'HELB! HELB!' Arms flailing.
'Oh God.' Watt froze, his voice just audible as Callum lumbered closer – talking to himself. 'Don't do anything stupid. Don't do anything stupid... Argh!' Then he whipped off his jacket and dived into the water.
'No!' Callum staggered to a halt on the jetty.
Mr Hand Truck thumped into the other rowing boat. 'I CAN'T SWIMB!' His face dipped beneath the water, then he struggled back into the air. 'HELB!'
And there was Watt – bursting out of the gunmetal water right next to him. He grabbed a handful of Mr Hand Truck's T-shirt, his other hand catching hold of the boat.
The river made bow waves against them both, rising up and curling away in breaking white spray. Shoving.
'HELB! HELB!'
Watt bared his teeth. 'Stop struggling!'
Tattooed arms flailed, whipping up spray, eyes wide, mouth open.
Then _CRACK_ , his elbow landed right in the middle of Watt's face, snapping his head back and knocking him under the surface. When he burst into the air again, blood made a dark pink slick down his chin. 'Gagh...'
'HELB!' Mr Hand Truck snatched at him, clambering up Watt like a ladder, forcing him down beneath the surface again.
Oh no...
Callum teetered on the jetty's edge.
Oh God, he was going to have to jump in, wasn't he? Into the fast-flowing dark-grey water. And hope he came out again alive.
He whipped off his jacket, ripped off his clip-on tie.
'HELGGgggggggllllbb!' The river took a firm grip on Mr Hand Truck's head and torso, yanking him back and around, breaking Watt's hold on his T-shirt and sending him spinning away into the torrent.
Oh Christ.
Too late. Should have leapt in straight away. Shouldn't have stopped to think about it.
Watt spluttered his way back to the air, snatched at the boat with both hands as Mr Hand Truck was swallowed by the Kings River.
Should've just _done_ it.
Someone swore.
He turned and faced a barrage of camera phones, all pointed at the swollen water. Half the people who'd been out on the decking were leaning on the rail. The other half had made their way down the stairs to the river bank. And they were all filming.
Oh that was just great – the whole fiasco, captured for all eternity and uploaded onto YouTube. So everyone could see him standing there, doing nothing.
Why the hell hadn't he jumped? He'd hesitated and now someone was probably dead.
Yes, but Watt wasn't. He _still_ needed rescuing.
Do it.
Jump.
Get your cowardly backside into gear and—
Watt hauled himself up the side of the rowboat and tumbled inside. Lay in the bottom of it, on his back, heaving in great gulps of air. Coughing, both hands clutched over his chest. 'Arrgh...'
There was no sign of Mr Hand Truck. Nothing but that roiling expanse of hungry water, growling away around the curl of the bank. He was gone.
And this was officially a monumental cock-up.
##
Callum wrapped one arm around the ladder and reached out with his other. Beneath his feet, the river surged, rain making dimples in the steel-coloured surface that merged and disappeared. Only to be immediately replaced. The rowboat's mooring line was just out of reach, so he tried again, leaning further out over the rolling water... Got it.
Watt was still flat on his back, soaked through, panting, eyes closed. Blood made a dark smear through his beard – bright red at the corner of his mouth.
So Callum pulled. Hauled. Braced his legs against the ladder's steel rungs and dragged the boat closer to the jetty, fighting the river all the way. Until the boat bumped against the bottom of the ladder.
Watt still hadn't moved.
'Anytime you like!'
A groan, then he sat up. Puffed out a breath. Crawled to the prow. 'Did you get him?'
'Can we talk about this on dry land?'
'Oh no.' Watt's face crumpled.
'Just get out of the sodding boat. This thing's _heavy_!'
A sigh, then he reached for the ladder and worked his way up till he was kneeling in the boat. 'I couldn't hold him.'
'I know.'
'I _tried_.'
'I know.' Callum shifted himself to one side, leaving enough room on the rung for Watt to struggle his feet into place. 'Come on. Up you go.'
Watt climbed, dripping water and blood. 'He wouldn't stop struggling.'
Callum let go of the rope and followed Watt up onto the jetty. Stood with him in the rain as he folded forwards and stood with his hands on his knees, coughing. Patted his back.
Then flinched as, 'YOU BASTARDS!' rang out through the downpour. It was Mr Trendy, marching down the path in his health-and-hygiene white trilby, flanked by the T-shirted guys from the processing floor. He cut a path through the onlookers with their mobile phones. 'YOU _KILLED_ HIM!'
'Just calm down, OK?'
'WHAT THE HELL WERE YOU PLAYING AT?' Face dark and trembling, hands balled into fists. 'WHY THE F—'
'All right, that's enough!' Callum held his hand up. 'Calm – down.'
'What did he do, eh? Answer me that! WHY DID YOU—'
'I'm not going to ask you again.' He hauled out his mobile phone and called Control. 'I need urgent assistance: we've got someone in the water, being swept downstream from Strummuir Smokehouse. Individual is an IC-one male with tattooed arms. Said he can't swim.'
_'Putting you on hold.'_
A tinny rendition of something vaguely classical burbled out of the earpiece, and Callum turned back to Mr Trendy. 'What's his name?'
The chest went out. 'I'm not telling you _anything_. Tod was just minding his own business and you hounded him to his death!' A finger poked Callum in the shoulder, hard enough to force him onto his back foot. 'You should be ashamed of yourself.'
Callum kept his voice level. 'I'm going to ask you to back away, _sir_.'
Another poke. 'We've got a very good team of lawyers and I'm going to make sure they hang you from the nearest tree by your balls.'
Watt straightened up, still breathing hard. Spat out a gobbet of foamy red spit. 'All right, that's enough. You have to step back onto the grass, Mr Noble.'
'Don't you tell me what to do! This is _my_ smokehouse and you're a murdering pair of...' His eyes bugged, then he took a swing – fist whistling through the rainy air towards Watt's face.
Only it never got there, because Watt blocked it then rammed his open palm into the centre of Mr Trendy's chest, sending him sprawling onto his back.
Stood over him. 'And you _stay_ down, or I'm arresting you.'
Callum limped up the stairs and through into the main office, carrying two waxed-paper cups of coffee that steamed and nipped at his fingers. Every step sent jagged teeth biting into his thigh. The chafing didn't help, but that's what happened when you wandered about in wet clothes.
One glass wall looked out over the smokehouse car park, then a small line of shortbread-box houses, and out towards a line of trees. All of it buckling under the weight of the rain.
A couple of patrol cars sat blocking off the car park entrance, blue lights flickering. Keeping all the tourists in.
Watt had perched himself on a big leather couch, shivering, a crinkly silver thermal blanket wrapped around his shoulders – giving him the look of a grumpy baked potato. He held a tea towel full of ice cubes against his cheek and jaw. The skin was already darkening there, the first hints of purple blooming around the edges. Working itself into a decent bruise.
Callum handed him one of the cups. 'Didn't know if you take sugar or not.'
'Thanks.' He took a sip and winced. Reapplied the tea towel. 'Any news?'
'They've got the Diving and Marine Unit coming down from Aberdeen. Should be here just in time for rush hour.'
'Urgh.' Watt sagged back on the couch. 'He's dead, isn't he? I couldn't hold onto him, and now he's dead.'
'Wasn't your fault.'
More blue flickering lights made their way along the road, past the line of houses. It pulled up in front of the car park and one of its siblings reversed to let it, and the car after it, to enter, then closed the gap again.
Callum groaned. The second car was a horribly familiar, brand-new, red Mitsubishi Shogun. McAdams. Lovely. No doubt here to dump an industrial-sized lump of doggerel and sarcasm on their heads.
Watt sniffed. 'How's the leg?'
'Why does every scumbag in Oldcastle try to bite a chunk out of me?'
'Yeah.' Another sip of coffee. Another wince. 'Ow... I called Mother – she's getting the review organised. Professional Standards and some Chief Inspector from divisional.'
'Drew blood and everything. Lucky he didn't sink his teeth in a quarter-inch higher, or I'd be circumcised by now.'
'Never been involved in a death before.' Watt shrugged. 'I've seen dead bodies – you know, at crime scenes – and I've delivered death messages, but it's not the same as being responsible.'
'Don't be daft: you're not responsible. _He_ ran. _He_ jumped in. You tried to save him.'
'Should've tried harder...'
There was a knock on the door and Mr Trendy, AKA: Darth Wolverine, AKA Finn Noble shuffled into his own office. 'Sorry.' He'd ditched the stupid white hat, revealing a stupid haircut – short at the sides, all quiffed up with hair gel on top. He closed the door. 'Erm... about earlier, I just wanted to apologise. It was...' He stared at the toes of his Converse trainers. 'I got a bit caught up in the moment. I would _never_ try to punch someone. And you're a police officer, how stupid would that be?'
Watt scowled at him. 'Very.'
'Yes. Yes.' He twisted his fingers into knots. 'Like I said: I'm very, _very_ sorry.'
Callum took out his notebook. 'Let's start with the man we were chasing: name, address, age, shift patterns. Everything you've got.'
'Ah...' Pink flushed in his ears.
'And I'll need the same for everyone who's worked here over the last two years.'
'Erm, no. Not without a warrant. I can't.'
'What happened to "very, _very_ sorry"?'
'We operate an outreach programme, OK? A lot of the people who work here have done time. We help them get back into work, teach them a skill, and it's all rehabilitation, isn't it? Someone's working here: they're making a decent living wage, doing something productive, developing a bit of self-esteem. Not off mugging OAPs and stealing cars.'
'And what was Tod in prison for, Mr Noble?'
'It's not like we employ sex offenders, or anything like that. It's people who've had difficult lives, who've made mistakes, who need a second chance.'
'What – did – he – do?'
Noble licked his lips, all attention focused on those shoes. 'The gang here _have_ to be able to trust me. If they thought I'd just hand over their personal details, rat on them, without putting up a fight? Nah.' He shook his head. 'I couldn't.'
Watt shifted the tea towel full of ice. 'I can still change my mind about charging you.'
'Yes, I know, but I _can't_. It'd undo all the good we've worked for. _Six years_ , ruined.'
'OK, if that's the way you want to play it.' He stood. 'Finn Noble, I'm detaining—'
The door thumped open and in marched McAdams, back straight, face like a clenched fist. 'Well?'
A uniformed PC followed him into the office and shut the door. Stood behind him with her mouth shut and her arms folded.
Callum pointed. 'At approximately fifteen fifty, Detective Constable Watt and I were being shown round the smokehouse by—'
'Thank you, Constable MacGregor, but I think DC Watt can handle it from here. You're going back to the ranch.' No haiku, no rhymes, no mocking asides.
That couldn't be good.
'DC Watt did everything he could. We've got it recorded on about twenty mobile phones: he risked his life to save—'
'I said _thank you_ , Constable MacGregor. Leave the pool car, DC Watt might need it. You can get a lift back with PC Crawford.'
'He dived in the river and—'
' _Now_ , Constable!'
'Right. OK.' He put his coffee down on the desk. 'Fine.' Then followed Crawford out of the office, down the stairs, through the doors, and into the rain.
She didn't say a single word all the way back through Castleview, across Dundas Bridge, and through the windy streets of Castle Hill. It was like crawling along through rush hour traffic, being driven by a shop mannequin. Only with less personality. Crawford just sat there, with her face set straight ahead, ignoring every attempt at conversation.
Ah well, can't say he hadn't tried.
Some officers were like that, though. Couldn't actually talk to people unless they were arresting them. Eventually, the job would weed them out and they'd go utilise their lack of interpersonal skills elsewhere. Like teaching or local politics.
No loss.
When she pulled up outside the back entrance to Division Headquarters he hopped out and gave her a cheery wave. 'Thanks for the lift, it's been fun!'
It wasn't far to the rear doors, barely six feet, but by the time he'd pushed through into the building his clothes had gone from damp to wet again.
They hadn't exactly tried very hard when they were decorating this part: breezeblock walls painted an institutional beige; scuffed concrete floor with suspicious brown stains that were either dried blood or something worse; signs with 'CUSTODY SUITE →', 'PROCESSING →', '← INTERVIEW ROOMS', and 'CUSTODY SERGEANT →' on them. The delightful scent sensation that was microwaved cabbage, fresh urine, and pine disinfectant. A slightly gritty taste of stale digestive biscuits, free with every breath.
'MacGregor.'
Callum stopped. Turned.
A big man with Seventies sideburns was leaning against the back wall, rolling a packet of Fruit Pastilles back and forth in his fingers. DS Jimmy Blake.
Callum nodded. 'Blakey.'
The skin around both eyes had darkened like aubergines and there was a thick T-shaped chunk of plastic covering his nose with the arms of the T taped to his forehead. Clearly, when Franklin punched you in the face, you stayed punched. 'You got a minute?' At least he didn't sound quite so bunged up.
'How's the nose?'
He narrowed his bruised eyes, the left one focusing somewhere over Callum's shoulder. 'You tell your friend, the darkie bitch, I'm not done with her.'
'Yeah...' Callum bared his teeth and hissed in a breath. 'Maybe not the _best_ idea, Blakey. She could kick your arse from here to Kingsmeath and back again without breaking a sweat. Mine too. Let bygones be bygones, eh?'
Blake just scowled at him from behind his plastic nose guard.
'Oh, and Blakey, I know you're bigger than me, but if you ever call DC Franklin a "darkie bitch" again I'll straighten your wonky eye with my fist. OK?'
Outside a siren kicked off, followed by a roaring engine and the screech of tyres. It faded into the distance. A phone rang somewhere in the custody suite.
'OK.' Callum patted him on the arm. 'Good talk.' He turned and limped down the corridor, towards the stairs.
Blakey's voice echoed off the breezeblock walls. 'DCI Powel wants to see you in his office. Don't keep him waiting.'
Sod.
##
Callum stopped outside Powel's door. Straightened his clip-on tie. Brushed at a dirty patch on his suit trousers. Probably dried blood, from when that cannibalistic little sod tried to chew his leg off... Yeah. Maybe best not to dwell on that, given what happened next.
Besides, there'd be plenty opportunities for blame and recriminations coming right up.
The stain didn't come off, just smeared into the damp fabric.
And yes: given this afternoon's monumental fiasco, an internal review was inevitable. Member of the public dies while being pursued by the police? The newspapers would be stumbling over each other, drunk with righteous-indignation and delight, competing to see who could give Police Scotland the biggest kicking.
But did the review have to happen right away? They couldn't even give him half an hour to put on dry clothes?
Of course not.
Ah well, no point putting it off. Callum pulled his shoulders back and knocked.
A voice from inside: _'Come.'_
Deep breath.
He opened the door and stepped inside.
Powel was behind his desk, face poker still. Mother sat on the couch with her back straight, looking disappointed. A bloke in uniform was next to her, with three pips on his epaulettes – that would be the chief inspector Watt mentioned, here to run the review. And last, but not least, everyone's favourite avuncular, fake-bumbling, non-sequitur-spouting, inquisitor from Professional Standards: Chief Inspector 'Call me Alex' Gilmore.
And they were all staring at Callum. Like a firing squad.
Oh joy.
Powel pointed. 'Shut the door, Detective Constable MacGregor.'
He did. Nodded. 'Boss. Detective Chief Inspector. Chief Inspectors.'
Gilmore pulled on a smile. 'I understand you had a spot of bother out at Strummuir, Callum.'
Understatement of the year.
'It wasn't Watt's fault. It wasn't anyone's fault. The guy ran, we chased him, he jumped in the river. Watt went in after him, but he struggled and the river swept him away.'
'I see.'
'It was filmed on about two dozen mobile phones – we commandeered the lot. Soon as they're checked into evidence, watch the footage and you'll see.'
Gilmore nodded. 'I did. At least five of them uploaded the whole thing: Twitter, Facebook, YouTube... We'll have to call in the Police Investigations and Review Commissioner, but it's just a formality. Nothing to worry about. As far as I'm concerned you both did everything you could.'
'Oh.' A smile crept its way across Callum's face. Thank God for that. 'Great. Watt deserves some sort of commendation, though. He blames himself, but he—'
'Moving on.' Powel produced a blue folder from his in-tray and pulled out a sheet of paper. 'We got a call this morning from a little old lady walking her border terrier in Holburn Forest. They'd barely gone twenty feet when "Captain Muffin" hauled a carrier bag out from beneath a bush. And do you know what was _in_ that carrier bag, Detective Constable?'
Callum kept his mouth shut.
'Human remains. A severed head to be precise: female, sawn off between the fourth and fifth cervical vertebrae. Sound familiar?'
What?
Of course it didn't. Why would...
Hold on: yes it did. 'When I was in your office this morning, someone came in and told you about it. You went off to the scene.'
'And that was the first you knew of it?'
'Yes. Why?'
'I see.' Powel held his hands out in a pantomime shrug: all just a silly misunderstanding. Nothing to worry about. 'Then would you like to explain how YOUR DNA GOT ON THE BLOODY REMAINS?'
'My _what_?'
'Your DNA. On a severed woman's head!'
Mother looked away. 'Did you kill her?'
Callum just stared.
'Did – you – kill – her?'
'No, of course I didn't! Why the hell would I kill—'
'Then did you dump the remains for someone else?'
'How... What...' He threw his arms out. ' _No_! I had nothing to do with _any_ of it.'
'You see, Callum,' Gilmore took off his evil scientist glasses and huffed a breath on them, drawing it out as he polished the lenses on a hanky, 'you have something of a reputation for compromising crime scenes, don't you?'
'I haven't been to Holburn Forest for years, how could I contaminate anything?' He jabbed a finger at Powel. 'This is the lab cocking things up again. They couldn't find an angry squirrel in a bean bag, never mind pick out DNA. Just because they buy a machine doesn't mean they know how to use it.'
Powel pulled out another sheet of paper. 'The DNA's degraded, but your name came straight back from the database. Why? Who was she? Why did you kill her?'
'I DIDN'T KILL ANYONE!' Though that might change in the next thirty seconds. The blood thrummed at the back of his skull, pins and needles filling his throat, hands clenched into tight fists.
Mother stood. 'All right, Callum, that's enough.' She pointed at the spare armchair. 'Sit down.'
He stood there, trembling.
'Sit – down. _Now_ , Constable.'
Callum lowered himself into the seat. 'I didn't kill anyone. I didn't interfere with the crime scene. I didn't _do_ anything. It's a mistake.' Or maybe it wasn't a mistake. Maybe it was deliberate? Of course it sodding was. 'Someone's trying to fit me up.'
Powel went back to his folder and pulled out a photograph. Held it out so everyone could see. 'Who was she?'
'I _told_ you: this is nothing to do with me...'
The woman in the photo had to be mid-twenties, early thirties tops. Her long blonde hair, so pale it was almost white, lay plastered against her head – glistening as if it was wet; dark circles around her unfocused blue eyes; skin like the thinnest bone china, speckled with freckles; puffy blue lips; a heart-shaped face; and a neck that ended three or four inches below her chin in a jagged dark-red line.
But that wasn't what made Callum's breath thicken in his throat, made his chest contract. It was her ears.
'Oh God...'
**_A Long, Long Time Ago_**
'Last one there's a bumhead!' Alastair was off running before he'd even finished speaking. Cheating bumhead. Sprinting across the pebbly beach, flip-flops sticking out of his back pocket, bandy net slung over his shoulder.
Callum ran after him, lumbering a bit because of the fish-and-chips sloshing about in Fanta in his tummy. 'Cheater!'
Sunshine sparkled across the water, hissing in and out against the little round stones.
A big fat crab – easily the size of Callum's palm – scuttered across the bottom of the rock pool, between the raspberry-jelly anemones and the floaty bright-green seaweed. All legs and nippers.
Loads of sand and shells and stuff lined the bottom of the pool, along with dull pebbly things that Dad said were bits of glass the sea had ground and ground till it couldn't cut anyone any more.
Alastair dug his hand into the pool and scooped up a bunch of bits. Then Callum reached in and did the same. The pair of them squatted at the pool's edge with dripping handfuls of shells and sand and grit, grinning at each other.
Mum was going to love this.
The lady in the little hut smiled as they tipped their handfuls onto the table. 'Well, let's see what we've got, shall we?'
The wooden walls were clarted with picture frames and animals made of seashells and bits of driftwood. Lots and lots of shelves covered in things covered in more shells: lamps, lumps of rock, more driftwood. Which was kinda cool and kinda dumb, all at the same time.
A sort of ray-gun thing sat in a wee stand, dribbling clear plastic goo from the barrel, making the air smell like the inside of Dad's car on a hot day.
Alastair pointed at a crab claw, still crunchy with sand. 'That one. Use that one.' His hair was full of sand too, his legs sparkled with it.
It was _everywhere_ : in Callum's flip-flops, gritty between his toes, itchy on the back of his knees where it'd started to dry.
The lady made a 'Hrmmmm'ing noise. 'It's a lovely crab claw, but it might be a _bit_ big. How about...' She moved her fingers through the piles and pushed a shell from each – one with a blueish edge, the other pink, both all ruffled and ridged like crinkle-cut chips, both no bigger than Callum's thumbnail. 'I think they'll be very pretty, don't you?'
'Oh come on, David, stop sulking – they're beautiful!' Mum tucks her hair behind her ears and flashes the tiny shell earrings. 'Aren't they beautiful? I have the bestest little boys in the whole wide world.' She kneels and wraps Alastair and Callum up in a big hug. 'I'm never taking them off. Never ever.'
Dad scowls. 'All I'm saying is, I got you a gold bracelet _and_ a bottle of that Priscilla Presley perfume.'
'Well, I love _all_ my presents.' She stood and spanked Dad on his bottom. 'Now get that barbecue fired up, Chief Chef Caveman, the Mighty Empress Birthday Girl demands a sausage-and-steak sacrifice!'
'Callum?' Mother leaned forward and poked him.
He jerked back in his seat. Blinking at the photograph.
Powel nodded. 'You recognise her, don't you, Constable MacGregor? Who is she? Who did you kill?'
Wasn't easy keeping his voice level, but Callum did his best. 'Is this supposed to be some sort of joke?'
'Oh I can assure you, Constable MacGregor, nobody's laughing.'
'Is this supposed to be _funny_?' He lunged, grabbed the printout from Powel's fingers and thumped back down again. Fumbled his wallet out with thick slippery fingers. Opened it and stared.
There, in the photo – Mum, with her big smile and her freckled cheeks. Her cartoon cat T-shirt. Her pale blonde hair, bleached by a week in the sunshine at Lossiemouth. Her earrings, made by a woman in a shed on a caravan site and paid for by two little boys who'd saved up their pocket money.
And the head in the other photograph – Powel's photograph – its ears were two delicate curls of translucent skin and cartilage, with a seashell earring in each lobe. One blue, one pink, both ridged like crinkle-cut chips.
Powel stood. 'You know her, don't you? Who – did – you – kill?'
But it _couldn't_ be.
The walls pulsed in and out in time with each breath.
It was twenty-six years ago. Mum would be in her fifties by now.
He grabbed the photo tighter, as if that would stop the thump-and-hiss of blood in his ears as the room got hotter and hotter.
Maybe... Maybe it was a cousin, or something? A relative he didn't know he'd had.
His mouth flooded with saliva, but he couldn't swallow – his throat was full of brambles.
Which meant there'd been someone who could've taken him in. He didn't _have_ to grow up in a care home. But they hadn't bothered their backsides to help a wee five-year-old boy abandoned by his whole family...
The whole world shrank to the size of the photograph in his hand.
But she was wearing Mum's _earrings_.
How?
'Are you OK?' Mother held out a mug of tea, steam curling from the surface, rain thrumming on the black skin of her umbrella.
From up here, on the station roof, most of Oldcastle was laid out like a tortured Monopoly board. Do not pass go. Do not collect two hundred pounds. Do not hope or dream.
Low cloud hid the upper reaches of Blackwall Hill and Kingsmeath. Scratched at the crest of Castle Hill. Even the tops of the hospital's twin incinerator towers were gone, the red warning lights at their top reduced to a faint bloody Sauron glow in the gloom.
Callum shifted his bum on the metal support and huffed out a breath – opaque in the cold air as rain dripped off the communications array and onto his shoulders. 'Thanks.' He took the mug. Sipped at the scalding liquid. Glanced up at the big metal and plastic drums above his head. At the Airwave transmitters. At the civil defence warning systems. 'Four minutes' warning. If it was you, would you set the siren off, or let the end come as a surprise?'
'You'll catch your death of cold.'
'Four minutes to panic and scream, knowing you're about to die, or blissful ignorance followed by a flash of light and _poof_. You're nothing but a shadow, burned into a concrete wall.'
She sighed and settled onto the support next to him, sheltering them both with her brolly. 'Professor Twining's finished the post mortem. Do you want the details?'
He raised one shoulder in a half-arsed shrug. 'What can you do in four minutes anyway?'
'Twining says the histology proves the remains have been frozen, apparently he can tell from the way the cell walls are ruptured. But we won't know for how long until they've finished the isotope analysis.'
All these years...
A severed head in a plastic bag.
'Say you're at work, or you're in Asda, or you're arresting some scumbag, or maybe you're breaking up a fight when the sirens go off. How are you supposed to spend your final moments? Not as if you can teleport home and be with your loved ones, is it?'
'I had the head of the laboratory on the phone, desperate to apologise for getting it wrong the first time. Someone mixed up the internal and external samples, so it came back looking like a degraded match.' Mother blew out a long slow breath. 'Anyway, Cecelia says your mum's DNA isn't on file; it was the Nineteen Nineties, they had different rules back then. And everyone thought it was a case of child abandonment. If they'd treated it as a murder, or an abduction... but they didn't.'
'So what do you do, phone them? "Sorry, love, it's the end of the world and we're all going to die." Only _everyone's_ trying to call their husband or their wife or their kids or their parents at the same time, the network goes down and you spend your last four minutes on earth swearing at your phone.'
'We're reopening the investigation. Well, Powel is. He's putting an MIT together right now to look into it.'
'Not exactly how I'd want to go out: lonely, pissed off, and scared.'
'Callum.' Mother put a hand on his leg, warmth seeping through the wet fabric to the skin beneath. 'It's OK to be upset.'
He frowned down into the depths of his mug. 'I'd leave the sirens switched off: let people live their last four minutes in blissful ignorance. No one wants to know they're about to die.' Not going by the look of utter horror on Mr Hand Truck's face as the river dragged him away.
How could anyone sane cope with that?
Here comes Death, and he's shouting your name.
Callum ran a hand through his wet hair. 'What about the guy who went in the water?'
'You've had a horrible shock and I want you to take some time off.'
'Let me guess: they haven't found his body. He'll be trapped against the river bank somewhere, or wedged under something beneath the surface, or on his way out to sea.'
'Callum, I'm serious: go home.'
'Yeah...' A fat raindrop dripped from the umbrella's edge and made ripples in the tea's beige surface. 'You've wanted rid of me from the start. Might as well _carpe_ the _diem_.'
'You're a silly sod, you know that, don't you?'
'So people keep telling me.'
'I'm not getting rid of you, Callum.' Mother let go of his leg and wrapped her arm around his shoulders. 'Not when I'm just starting to like you.'
Rain thumped on the brolly. Dripped from the communications array. Hammered down on the buildings and the people and the streets.
She gave him a little squeeze. 'And I'd leave the sirens off too.'
##
Callum wheeled his bike in through the door, locked it up in the rack beneath the stairs. Stood there for a bit with his eyes closed, dripping, blood hissing in his ears like waves on a pebble beach.
All those years...
Deep breath.
Come on.
He flicked through the pile of letters, fliers, and leaflets on the windowsill – took the ones for the top floor and squelched upstairs.
Didn't matter what the weather forecasters said, it was never going to stop raining. Not until they were all drowned and dead.
Mrs Gillespie's cats had been at Toby's pot plants again, black soil spread out in a fan around a wilting rubber plant. A Mylar balloon bobbed in the breeze, the string tied around Mr and Mrs Robson's door handle. 'HAPPY 20TH ANNIVERSARY!'
Glad _someone_ had something to celebrate.
Callum posted their mail then let himself into the flat.
'Elaine?' He peeled off his wet jacket – sodding thing didn't deserve the term, 'Waterproof' it leaked like a teabag. 'Christ, you wouldn't _believe_ the day I've had.'
He dumped his backpack next to the non-waterproof jacket. 'I need a drink.'
The TV was on in the lounge. Some sort of soap opera, probably. Lowered voices and ominous muttering coming through the closed door.
He unlaced his shoes and squelched through to the kitchen. Hauled open the fridge door and helped himself to a Tesco own-brand continental-style lager. Clicked the metal tag and had a good long swig. Not much else in there. Some open jars of pickles and olives, bit of cheese going blue at the corners, some butter, a wilting lettuce... Leftover tuna casserole. He clunked the door shut. 'I'm ordering takeaway. I know we're meant to be saving money, but sod it.'
He peeled off his soggy socks and slapped them into the washing machine.
Took his beer through to the bedroom.
There were two bags, sitting on the bed. One was his battered old suitcase with its wonky wheel and the strip of green fabric tied around the handle so it'd be easy to find on an airport conveyor belt. The other was an aluminium hard-shelled job big enough to fit a small child. Not a scratch on it, so probably brand new.
Callum's shoulders dipped even further.
Great. There he was, worrying about ordering takeaway and Elaine's busy buying expensive luggage off the internet. As if they were going to let her take something that size into hospital. You could give birth _in_ it, it was that—
'Callum?'
He turned, and there she was, wearing a baggy black T-shirt with some sort of communist-chicken design – stretched tight over the bump – and a pair of baggy grey joggy bottoms.
'New luggage? Seriously? I thought we were trying to save up for Peanut's—'
'I'm sorry.' She bit her top lip and stared down at her bulge. 'I'm really, _really_ sorry.'
He ran a hand over his face. Sighed. 'OK. Look, it's only a suitcase, not the end of the world.'
Someone appeared in the doorway behind her. Someone with sticky-out ears and silvery hair. DCI Powel. Wonderful. Just what the doctor ordered – a pain-in-the-arse.
Callum crossed his arms. 'What's this, more "friendly" advice? Cos I'm not in the mood.'
Elaine cleared her throat. 'I was sorry to hear about your mother.'
'Callum, I know the timing's horrible, but it was going to be horrible whenever it happened.' Powel put a hand on Elaine's shoulder.
'Yeah, well, there's never a good time for your mother's head to be found in a carrier bag, is there?' He reached for a dry pair of jeans. 'Anything else?'
A sniff, and Elaine finally dragged her eyes up to look at him. 'Please don't make this any harder than it already is.'
'I just want to get dry and changed, OK?'
'Reece has left his wife.'
'Good for him. But he's not staying here. Barely enough room for the two of us as it—'
'I packed you a bag.'
Callum froze. 'You packed _me_ a bag?'
'It...' She rubbed a hand across her pregnant belly. 'God, why do you have to be like this?'
'I'm not being like anything! What do you mean, you packed me a bag? Why the hell would I need a—'
'It's not your child, OK?' Her voice was loud and trembling. 'Peanut isn't yours, Callum. He's Reece's. Don't you get it?'
The world shrunk to a tiny silent pinprick.
Then blood crashed against the pebble beach. Nails dragged across the caravan's aluminium hull. Thunder _roared_.
Callum blinked. 'He's what?'
'Don't you get it? I was making do. Reece was never going to leave his wife and you were better than nothing. But that's all changed now.'
'Are you kidding? Are you—'
'I _love_ him, Callum.'
Heat rushed through his body, pins and needles crackled between his shoulder blades, fists curled so tight his knuckles burned. 'This was all lies? Peanut isn't... I was better than _nothing_?'
'For God's sake, Callum, listen to yourself.'
'I was good enough to raise someone else's kid, though, wasn't I? Good enough to lie to!'
Powel squeezed past Elaine, putting himself between them. 'All right, that's enough. I need you to—'
'You manipulative, two-timing, backstabbing, lying—'
'I said that's enough, Constable!'
'I took the blame. For you!' He pointed at Powel's child, growing in her belly like a tumour. 'For _that_. And it wasn't even mine?'
Her voice trembled. 'What was I supposed to do?'
'I FLUSHED MY WHOLE CAREER – FOR YOU!'
Powel's open palm thumped into Callum's chest. 'I'm not telling you again.'
And the thunder roared.
Callum grabbed two handfuls of Powel's shirt, yanked him forwards and off balance, then slammed him into the bedroom wall, hard enough to knock framed photos off their hooks. Did it again, harder. Cracks rippled out through the plasterboard where Powel's head crashed into it with a splintering thump.
Kill him.
Callum let go with one hand and smashed his fist into Powel's face. More cracks in the plasterboard as his head bounced.
Kill him.
Elaine screamed, the sound cutting through the thick air like a bone saw. 'GET OFF HIM!'
Powel's eyes rolled up, mouth drooping open.
Kill him.
Getting heavier in Callum's hands.
Kill him.
Callum let go and Powel slumped to the carpet.
KILL HIM!
He drew back a foot, to kick the bastard's head in and—
'STOP IT!' Elaine grabbed at his arm, tried to pull him away. 'LEAVE HIM ALONE!'
Callum stumbled, turned, fist curling up...
She glared at him, tears rolling down her cheeks, face flushed and distorted. 'GET OUT! YOU'RE NOT WANTED, UNDERSTAND? YOU WERE _NEVER_ WANTED!'
He lowered his hand.
Blood crashed on the stony beach.
She covered her eyes. 'Please. Just... go.'
Callum grabbed his battered old case and marched out, scooping up his wet shoes and soggy jacket on the way. Slammed the door behind him. Stood on the landing, jaw clenched so tight his teeth ached. Dragged in a deep, jagged breath.
'Callum?' The door at the far end of the landing was open, just a crack, still on the chain – Toby's jaundiced face barely visible in the gap. 'Is everything OK? I heard screaming.'
'No. Not it's not, "OK".'
He turned and marched down the stairs, suitcase clatter-thumping its way along behind him, past open doors on the second floor – everyone peering out to see what was going on from a safe distance.
Callum unlocked his bike and hurled it out of the front door into the rain. Did the same with the suitcase. Pulled on his wet shoes and jacket, then stormed out after them.
##
Thursday night in the Bart wasn't much busier than the Wednesday.
Three auld biddies in the corner booth were playing dominos. A couple having an argument over by the pool table. A fat beardy bloke, playing with his mobile phone and glancing at the door every two minutes. Probably wondering how long he should give it before admitting defeat and accepting that he'd been stood up.
Callum fed another four pound coins into the jukebox with his left hand, pressed a few buttons and lurched back to the bar as Radiohead's 'Exit Music from a Film' oozed out from the pub's speakers again. Ignoring the groans from everyone else. Feeling no pain.
He tipped back the whisky in his swollen right fist and clunked the empty glass back on the bar. Winced. OK, maybe some pain. But there was an easy way to fix that: 'Same again.'
Hedgehog Dundee sucked a breath in through his teeth then let it out in a long slow hiss. A round wee man with an oversized goatee, shiny face, and long straggly hair, he looked as if his blood was about sixty-four percent cheese. 'Not that I'm ungrateful for your patronage, Constable MacGregor, but you're rather undermining the happy-go-lucky atmosphere we strive for here at the Dumbarton Arms.'
'Double Grouse, no ice or... or water. Pint of Cham-pi-on.' Had to focus a bit to get the word 'Champion' out, because something had gone wrong with his tongue, but it was the thought that counted.
'And while "Exit Music from a Film" is a well-constructed song, and clearly reminiscent of Gustav Mahler's later work, the fact that you've played it fourteen times in a row is beginning to take its toll on the other patrons' _joie de vivre_.'
'An a... an a packet of piggled... onion.' He wobbled his way up onto a barstool.
'Especially as you preceded this tribute to a somewhat lesser known Radiohead song about suicide, with a dozen playings of REM's "Everybody Hurts".' Hedgehog reached beneath the bar and came out with a folded sheet of paper. Placed it in front of Callum. 'This might be more beneficial to your state of mind than the further consumption of alcohol and depressive songs.'
It was a leaflet for the Samaritans.
Callum drained his pint and thumped the empty glass down right on top of the thing. Squinted one eye shut to keep everything in focus. Put on his best police officer voice: 'Hedgehog, I'm going to give you... you a choice. You can either get me my drinks... drinks an crisps, or... or I can call a friend at Food Standards Scotland...' OK, that sounded a bit slurred, but they were difficult words after five or six pints. And double whiskies. 'I'll... I'll get them to come down here an... an give your kitchen the kind of... _examination_ that'd make... make a proctologist's eyes water.'
A sigh. Then Hedgehog turned and pressed a tumbler up beneath the optic of Famous Grouse.
Damn straight.
_No one_ wanted a visit from the Cheese Police.
The whisky went on a coaster in front of him, followed by a foamy pint of dark brown beer, and a silvery green packet.
He fumbled out a tenner and laid it on the bar with exaggerated care, just to prove he wasn't drunk.
Hedgehog took it, then stared over Callum's shoulder, smiling. 'Oh thank the heavens for that: you came.'
'Callum?' Dotty wheeled herself over to his barstool and looked up at him. 'Are you OK?'
'You want... want a drink? I'm buying.' He thumped his left hand down on the bar. 'Hedgehog – pint of... pint of Old Jock for Dotty. Put it... on my tab. No, no, I _insist_. You want crisps? Course you want crisps. Give her some crisps.'
'How much have you had to drink?'
'I'm celebrating.'
'Oh, Callum...'
He took a swig of beer. 'No, it's great. All of it.' A soft warm smile spread across his face. 'My mum didn't... didn't abandon me, she got murd—' The burp tasted of whisky and prawn cocktail. 'Sorry. She got _murdered_. An... an I don't have to raise... raise someone else's baby!' The Grouse set fire to his chest on the way down, making it swell. 'Cos it wasn't mine. You see?'
Hedgehog leaned on the bar. 'Dear, sweet, Detective Sergeant Hodgkin, I would consider it a personal favour – nay, a veritable _boon_ – if you would escort DC MacGregor to another establishment. Perhaps somewhere he can drink copious amounts of coffee, consume some carbohydrates, and prepare for what is no doubt going to be a most terrible hangover? He can settle up and collect his bike when he's sober.'
The song on the jukebox mourned to an end. Then started right back up again.
'Wasn't my baby, Dotty. It never... never was.'
'Have you got somewhere to stay?'
He spread his arms wide, slopping beer across the bar. 'World... world is my oyster.'
She puffed out her cheeks. Grimaced. 'OK, OK. You can stay in the spare room, I'm sure Louise won't mind. Probably. As long as you're not going to be sick – she hates that. You're not going to be sick, are you?'
He lowered his glass. 'They kicked me out of... of my flat. My flat! I paid... paid for it an every... everything.'
'You have to _promise_ not to be sick everywhere.'
'It was my flat.'
'I'm serious about the not being sick, Callum. Don't do that.'
'Cross my heart.' He blinked at her for a bit. Then held out the packet of pickled onion. 'You want... want some crisps?'
Callum wiped his mouth, sighed. Spat out a bitter thread of bile. Rested his forehead against the cool wooden seat.
The shower's hiss disappeared for a moment as he flushed the toilet for the fourth time.
Urgh...
Then he clambered back into the bath again, holding onto the rails built into the bathroom wall to keep him upright. Rinsed the shampoo out of his hair. Then stood there and steamed for a bit, till the water went tepid, then cold.
Finally, he clambered out and dried himself off on a dark-blue towel. Wrapped it around his middle and crept back along the hall and into Dotty's spare room. Stood there with his back to the door, arms hanging by his sides.
Dotty and Louise had gone to town in here: pink chintz cushions; pink floral bedspread; pink floral pillows, curtains, and pelmet; dried flowers on the pine chest of drawers.
It was like standing inside Barbara Cartland.
He sank down the door, until his towelled bum came to rest on the fuchsia-coloured carpet. Cradled his swollen right hand against his chest. Every time he tried to move his fingers it was like rubbing barbed wire into the joints. That's what he got for punching Powel in the face.
Might hurt now, but it felt _great_ at the time. Standing over him, watching the blood seeping out through his open mouth.
Deserved all he got. And more.
How long had it been going on: Powel and Elaine? She was due in two weeks, so that meant at _least_ nine months. Probably longer. Probably ever since they worked that murder/suicide.
All that time, screwing around behind his back...
He groaned.
Yeah, that explained why Elaine had been off sex since April – she was saving herself for sodding DCI Powel.
And what would've happened if Powel hadn't found the balls to leave his wife?
No way Elaine would've come clean, not when she had Callum right there to pay for everything. To change dirty nappies and stay up half the night feeding Poncy Powel's bloody baby.
To take the blame for cocking up a crime scene, so she could keep her paid maternity leave for a baby that wasn't even his. Manipulating him with little love notes and sandwiches. How _stupid_ could he be?
No wonder she treated him like an idiot – that's exactly what he was. A moron. A halfwit. A mug.
Whose mother's severed head was lying in a refrigerated drawer in the city mortuary.
Yeah... this had been a _great_ day.
##
Didn't really matter any more, did it? Who was going to hear? No one.
So Ashlee stopped screaming. Stopped rattling against her chains. Stopped fighting against the darkness.
Just slumped back and let the cold filthy water seep into her bones.
The surface rippled with every shiver that juddered through her, making her teeth clatter.
Maybe the Man with Blue Eyes would come back and let them go?
He said he'd come back...
Or maybe he'd forgotten about them?
How long had it been since he left? Hours. Hours and hours and hours. And no sign of him. Nothing but the cold and the wet and the sound of her own screams.
Mum still hadn't moved. She was just visible in the pale-orange light that seeped through between a couple of the wooden boards. Slumped over to one side with the chain tight around her neck.
Poor cow.
All those years living with Dad, who was _utterly_ a dick. All the shouting and the manipulation and the checking up on her and not letting her have any friends... And then one day he just walks out and never comes back, because occasionally even utter dicks can do something nice for their families.
All those horrible years with Dad, then some more horrible ones with Uncle Eddy who always wanted to tickle Ashlee and take her shopping for pretty dresses and go with her to the swimming. Breathing heavily as she stripped off in the changing rooms. Sitting in her room watching her sleep. Oh, yeah, _completely_ not a paedo. Then Uncle Eddy finds himself another single mother with a much younger daughter and off _he_ goes too.
Good sodding riddance.
And _then_ , when Mum was finally getting herself back together – going out of the house instead of staying at home watching soaps and eating ice-cream and oven chips – this happened.
Ashlee closed her eyes and sobbed in the dark.
Howling it out.
Because who was going to hear her?
She was going to die here. Alone. In the dark.
## — protect and survive —
**Fallout can kill.** Since it can be carried for great distances by the winds it can eventually settle anywhere, so no place in the United Kingdom is safer than any other. The risk is as great in the countryside as in the towns.
Nobody can tell where the safest place will be.
Stay at home – Public Information Film
© Crown Copyright (1975)
"There's no point crying, little girl," said the Bonemonger with his scissor-sharp smile. "No one will hear you, and nobody cares."
R.M. Travis
_Open the Coffins (and Let Them Go Free)_ (1976)
_You better beware, cos yo parents is nowhere,_
_You hear me? I swear, man, you ain't got a prayer,_
_Ain't no love in the air, it's just pain and despair,_
_You grown up in care, and this place is a nightmare._
Donny '$ick Dawg' McRoberts
'The Arsonist's Diary'
© Bob's Speed Trap Records (2015)
##
'Urrrgh...' The kitchen throbbed like the heart of a monster. _Boom_. _Boom_. _Boom_.
Obviously, Dotty's fridge, freezer, and microwave were plotting Callum's death with their horrible reflective metallic surfaces. Sending burning daggers of sunlight stabbing through his eyeballs and into his brain. Making _everything_ burn.
He held the empty pint glass under the tap and filled it up again. Glugged it down, water dripping onto his old grey T-shirt.
Finished and slumped there with his head drooped, panting.
Oh God.
Never, _ever_ again.
He had to put the glass down to turn the tap off – his right hand had swollen up to the size of a space hopper. All purple and yellow and stiff and full of rusty metal. Fingers twisted and rigid.
Callum used his left to fumble a couple of painkillers from the packet by the kettle and dry swallowed them.
His stomach lurched and gurgled.
'Urgh...' He curled forward, resting his elbows on the work surface, head hanging like a sack of burning tatties.
Stay down. Stay down. Stay down.
Please...
The pills stayed where they were and he gulped down another pint of water.
Shuddered.
The sound of breakfast TV burbled through the house and he slouched through to the living room.
It was a lot less chintzy than the guest bedroom – more 'Scandinavian functionalism' than 'Barbara Cartland's innards'. Flatpack furniture with lots of straight edges, Scottish colourist impressions of wee seaside towns, hills, and but-and-bens in big wooden frames. A display cabinet full of random ornaments and wine glasses.
The only book on display was a celebrity biography of someone he'd never heard of, sitting on a big glass-topped coffee table.
'Well, well, well, look what the cat threw up.' The black-leather couch's only occupant took a sip from a huge mug. Greying hair cropped close to her long face. Sharp black suit, red shirt, a pair of no-nonsense schoolmarm glasses perched on the end of her nose. 'You look like a tramp had sex with a wheelie bin.'
'Louise.' Callum wiped a hand down the front of his wet T-shirt. 'Sorry.'
On the TV, a man in waders was chest-deep in a river somewhere. _'... and that's the only good thing about American signal crayfish: they're very, very tasty. Back to you in the studio.'_
'Still, at least you didn't vomit everywhere.'
The picture jumped to a pair of presenters sitting on a curved red sofa with big animated screens behind them. The woman smiled. _'Thanks, Colin. Now here's Valerie with the weather. Any good news for us, Valerie?'_
'Sorry. I'll get packed up and out of your hair.'
Valerie was a sporty-looking type in a stripy dress. _'I do indeed, Claire, as you can see from the satellite map it's going to be a lovely day for the southeast all the way up to Manchester and Newcastle...'_
'Pfff...' Louise waved at him. 'Just because I'm a solicitor, doesn't mean I'm heartless. Dorothy told me what happened with Elaine.'
'Oh.' He sank onto the end of the couch. 'Yeah.'
_'... best of the sunshine in Wales and Northern Ireland...'_
'She worries about you, Callum. Dorothy thinks you've got a self-destructive streak a mile wide.'
'I do genuinely appreciate the bed for a night. She didn't have to take me in.'
'Stray kittens, puppies, injured birds – you name it, she wants to give it a home.'
_'... stubborn band of rain clinging onto the northeast of Scotland, but other than that we've got all the makings of an Indian Summer...'_
He nodded. Stared down at his swollen knuckles.
'I think it's because she's in pain a lot of the time. She's hurting, so she hates to see others suffering. Well, except for that idiot Detective Constable Watt.'
'I'll get a B-and-B sorted out today. Give you your spare room back.'
'All the muscles and nerves are messed up in her right leg, from the crash, but they won't amputate it. Doesn't matter how much she begs.' A sigh. 'How is that fair?'
_'Thanks, Valerie. Now, how many of you remember this?'_
A music clip played from the TV – an orchestra swooping through a guitar-and-bassline.
'Life never is.'
'No.' A small, sour laugh. 'I don't suppose it is.' Louise stood, took a couple of steps closer and squeezed Callum's shoulder. 'You stay as long as you like.'
Then a man's voice, dark and warm belted out over the top. _'Run, little rabbit boy, you'd better run like hell, / Cos the Bonemonger is coming and he's after you as well...'_
Louise gave Callum's shoulder another squeeze. 'Well, let's call it a week. Don't want you cramping our style.'
_'Slick and sharp and sickle-like he smiles his scissor smile, / and he'll catch you and he'll eat you, though you run for miles and miles...'_
Louise let go. 'Sod it: look at the time.' She marched from the room, voice getting louder as she disappeared into the hall. 'There's a spare key hanging in the kitchen – if you're going out, pick up some milk!'
_Clunk_ , the front door shut and he was alone.
Urgh...
He slumped back on the couch. Rubbed his good hand over his eyes.
Probably wouldn't hurt to go back to bed. Maybe a couple of hours' kip would dull the hangover howling its way through his skull.
_'... lovely to have you with us.'_
_'Lovely to be here, Siobhan. Though watching that, I have to wonder what on earth I was thinking. I know it was the Eighties, but oh dear...'_ A familiar laugh, dark and treacly. _'Can't believe the_ Miami Vice _look used to be trendy.'_
Callum peered out through his fingers and an old man had joined Mr Suit and Mrs Casual Dress on the breakfast sofa. He was wearing faded jeans, scuffed cowboy boots, and a dark-blue shirt that had silvery bits speckling the sleeves, leather buckles on his wrists. On the screen behind him was a _much_ younger version of himself in a pastel-green linen suit with the sleeves rolled up, no T-shirt, showing off a lot of chest. The hair on his head swept up and back: coiffured into a big blonde mane.
The mane was still there, but it was thin and white now.
A caption appeared at the bottom of the picture: 'LEO MCVEY ~ SINGER SONGWRITER'.
_'So Leo, of course_ Open the Coffins _was a massive hit in the eighties. But it almost didn't get made, did it?'_
_'Oh yeah. Man that was a_ hard _sell. You should've seen the record execs' faces when I told them I wanted to do a concept album based on this children's book about a wee boy who gets turned into a rabbit and has to fight the Lord of Bones for his sister's soul. "No way!" they said. "You can't make this, it'll be career suicide!"'_
Callum pulled out his phone, winced one eye shut and brought up his call history.
Oh thank God – no drunk-dialling the flat or Elaine's mobile.
_'And then it's like, sixteen weeks at the top of the album charts. Just goes to show you how much guys in suits know.'_ A wink. Then he leaned over and patted the male presenter's knee. _'No offence, Brian.'_
Next check: text messages...
A big sigh let all the pressure hiss out of his lungs.
No angry texts, or weepy ones, not even a big chunk of solid swearing.
_'And of course, Ray and I became_ really _good friends when I was recording the album; have been ever since. Man, we used to hang out_ all _the time. He'd even come on tour with us. And loads of people would bring their books and he'd be sitting out in the auditorium signing them during the interval, you know? Great guy.'_
There were three _incoming_ texts, though. All from Elaine.
Oh.
_'And a great writer too. That's why the album was so successful – the source material was just_ so _great.'_
Callum's thumb hovered over the first one.
Might be a better idea to just delete them, rather than sit here reading all about how it wasn't her fault and he had no right to judge her...
_'So you can imagine how awful it was when we started losing him, yeah? How painful it's been watching Alzheimer's eat Ray. How this horrible disease is consuming the guy we all loved.'_
The male presenter sat forward and oiled out a sympathetic expression. _'But you decided to do something about it, didn't you?'_
Callum selected the first message:
What the hell is wrong with you?
How could you DO that to Reece?
I'm at the hospital because of
YOU!!!
_'Yeah, so me and the lads got together and we said, "We can't just let this happen to Ray!" So I got onto the organisers of the Tartantula music festival, in Oldcastle, and told them, how about we do some sort of benefit...'_
The second message wasn't much better:
You broke three of his teeth!!!!
I used to know you, Callum, but I
don't now. You can't come back
from this. Know what? NEVER come
back again.
_'... came up with the idea of getting all these modern bands to join in and help us raise money for Ray's care and, you know, to help research into Alzheimer's too. And these young guys have been great, it's going to be a terrific gig.'_
Number three:
I can't believe I ever loved you.
'The feeling's mutual.'
_'And if you can't get up to Oldcastle on Sunday, don't worry: we're gonna record the whole thing and it's gonna be this great live collector's edition CD thing and on downloads or whatever it is you kids do these days.'_ Another laugh.
Callum deleted the lot, then added her mobile number to his phone's barred list.
_'And every penny's going to—'_
He killed the TV with the remote and slouched back to bed.
Dotty's voice cracked through the kitchen like a whip. 'You're up!'
Callum didn't look up, just stayed where he was: slumped at the small table, wrapped around a bitter mug of milky coffee while his stomach growled and gurgled. 'Why did you let me drink so much?'
'Don't look at me. According to Hedgehog you'd had about half a bottle of whisky by the time I got there. Not to mention all the beer. You owe him about seventy quid and an apology.'
Seventy quid?
Oh God...
A groan broke free and Callum sagged even further. 'What happened to my bike?'
'You left it at the Bart.' A warm hand rested on the small of his back. 'I know this sounds terrible, but you know what it's like with a murder enquiry, so...?'
He wouldn't be taking a couple of days off after all. 'Mother wants me back at work.'
'Ah. No.' Deep breath. 'They want you to come in and formally identify your mother's remains today. I know they've got the familial DNA match, but they still need someone to come down the mortuary and confirm that it's her. Sorry.'
Of course they did.
He straightened up. 'Yeah.'
Dotty wheeled Keith back towards the door. 'How's the hand?'
He held it out.
'Ooh. Right.' She made a hissing noise. 'We probably better get you to A-and-E first.'
'Are you _sure_ you don't want to go to A-and-E?' Dotty manoeuvred her wheelchair in front of him, blocking the corridor.
'Let's just get this over with.'
Clearly, whoever built Castle Hill Infirmary hadn't given a toss about the bowels of the building looking like what bowels usually contained. Down here, in the sub-basement, they didn't bother with all that fancy terrazzo flooring, calming paintings, and institution-green walls. The floor was concrete with a scuffed line of black paint down the middle, the walls brick and breezeblock. And instead of a fancy-pants suspended ceiling with moon-surface tiles, bundles of cables and pipes snaked their way through the gloom. Because putting in enough lightbulbs to actually _see_ by was just pandering to people. Much better to leave the corridor looking like something out of a horror movie.
It was warm down here too, the air heavy with the weight of the building above, thrumming and ticking and clicking with distant hospital sounds.
'Callum?'
He blinked. Puffed out a breath. 'I'll get the hand seen to when we've finished. Promise.'
A nod, then Dotty spun her wheelchair around and squeaked off down the corridor. 'We're thinking of getting a carryout tonight. I fancy a Thai, if you want to join us?'
Callum limped after her. 'Dotty?'
'What, you don't like Thai food?'
'Thank you.'
'Nah,' she let go of Keith's rim and waved a hand over her shoulder, 'it's only takeaway, no biggie.'
'And not just for taking me in last night, for _everything_. You were the only one who didn't treat me like a deep-fried jobbie when I joined the Mob.'
'Yeah, well, I need _someone_ to raid the vending machines for chocolate, don't I? Not like Wanky Watt the Wrinkled Willy-Wart is going to do it.'
The corridor took a sharp right, only a single working lightbulb to deflect the darkness. A sign on the wall was barely visible, 'MORTUARY →'.
'Maybe you should give him a chance?'
'What, Watt? I'd rather give him a boot up the backside.'
'He risked his life yesterday for a nasty wee scrote who'd tried to kill him. And it wasn't an automatic, gut-reaction, jump-straight-in thing, either: I could see him thinking about it. He knew it was stupid and dangerous, but he did it anyway.'
'Hmmm...' She followed the little black line. 'He's still a moaning dick, though.'
'Oh, _totally_. But wouldn't you be if you were him? If you had to wake up with that face every morning?'
A pair of double doors loomed up on the right, lit from above by a small fluorescent tube that buzzed and flickered. They'd mounted metal bumper plates on the dark-green wood – presumably to protect it – the surface covered with dents, gouges, and scrapes. Unlike living patients, the dead didn't mind if you used the trolley they lay on as a battering ram.
Dotty let Callum do the honours. He held one side open as she wheeled herself inside, then followed her into that familiar dark-brown reek of human waste and disinfectant. Each breath hung in the air, like a ghost, before dissipating away into the cold.
The mortuary looked about two hundred years overdue for a makeover. Its floor was covered in black tiles, cracked and uneven, the walls with filthy ivory-coloured ones, stained like a smoker's fingers. Three stainless-steel cutting tables sat in front of a wall of refrigerated drawers, the other walls lined with metal work surfaces. Glinting in the harsh overhead light.
A tall man in pale-blue scrubs and white wellies worked a mop along the floor tiles, pushing a mini tidal wave of grey-beige water around. Ponytail dangling behind him, high forehead almost as shiny as the cutting tables.
Only one of them was occupied.
The body was huge and tallow pale. Naked. On its back. Thick bloated arms and legs. Dark wiry pubic hair like a mini forest, nestling at the end of a gaping wound that stretched all the way down from his collar bone. Skin peeled back, front of the ribcage removed, torso hollow and glistening. Beard like a bear, but blotchy and yellowed around the mouth. Thick bands of bruises around his wrists and ankles.
The only other person in the room was leaning back against one of the work surfaces, poking away at his phone. Blakey looked up as the mortuary door clunked shut behind them, scowling around that big plastic guard thing taped over his ruined nose. A nod. 'MacGregor. Hodgkin.' Then back to his text, or game of Angry Birds, or whatever.
Dotty parked herself in the middle of the room. 'Come on then, DS Blake, where's your organ grinder?'
Blakey kept on poking. 'Conference call in the office.'
A half-glazed door sat off beside the sinks, the glass frosted like a public lavatory. The word 'PATHOLOGIST' graced a brass plaque with 'PROFESSOR MERVIN TWINING CBE' printed across a laminated sheet of A4 beneath it. The sound of muttered voices was just audible over the chilly drone of the fridges.
'Callum?' Blakey stopped poking, but kept his eyes on the screen. 'I was sorry to hear about your mother.'
'Thanks.'
Dotty wheeled herself over to the opened body. 'Who's your boyfriend?'
Silence.
'In your own time, Blakey, we've nothing better to do.'
He pressed a couple of buttons then put his phone away. 'Fat Archie Benton. Bunch of public-minded citizens decided they didn't fancy their children sharing a tower block with a convicted kiddy-fiddler, so they invited him round for drinks to talk it over.'
Dotty wheeled Keith on a slow lap of the remains. 'Very nice of them.'
'Don't think Fat Archie would agree. They pinned him down, jammed a funnel in his gob and treated him to all the bleach he could drink.'
A clunk and rattle from the office door and Teabag stepped out, flicking the dark floppy hair out of his eyes. 'Detective Sergeant Hodgkin!' He gave Dotty a smile, making the dimple in his square-jawed chin deeper, light glinting off his little round glasses. He smoothed down the top of his purple scrubs, then leaned down and kissed her on the cheek. 'To what do we owe the honour?'
'Professor Twining.' She reached up and put a hand on Callum's arm. 'We're here about the female remains that came in yesterday?'
'Ah yes, the severed head!' His smile grew. ' _Completely_ fascinating. Let me show you.' He strode across the cracked black tiles to the wall of refrigerated units. Opened a door and rattled out the stainless-steel drawer inside. A small body bag sat on the surface, the kind used for children. He scooped it out and placed it on one of the empty cutting tables. 'Here we go.'
The zip hissed open, then Teabag pulled the plastic sides down, exposing the contents.
The breath solidified in Callum's throat. Spread down into his chest like setting concrete.
After all this time.
They'd closed her eyes. Which somehow made it... _better_. Better than the thought of her lying in that plastic bag, with her eyes open, staring into the darkness, in a mortuary drawer, buried deep beneath Castle Hill Infirmary. Her skin was impossibly pale, the freckles looking as if they were fading away. Someone had washed her hair, or at least cleaned the gunk out of it, leaving it like silk. The hideous stump of her neck: wide and purple and gaping.
Callum swallowed something bitter. Stared.
'Now, the truly interesting bit is this.' Teabag snapped on a pair of blue nitrile gloves and picked her head up, turning it over and brushing the hair away from her ear. 'The remains were covered in some sort of gelatinous residue. Took us ages to figure out what it was.'
Dotty looked away. 'Actually, Professor—'
'If you freeze any sort of meat for long enough, it'll end up with freezer burn. Doesn't matter if you put it in a bag or not, if there's _any_ air in there the meat will eventually dehydrate and oxidise. I'm sure you've seen it yourself lots of times: sausages, joints of pork, steaks, they go all pale and gritty looking?'
Twenty-six years...
'That's why producers put an ice glaze on prawns. But ice _sublimates_ , so over extended periods the water molecules will migrate to the coldest spot, leaving the remaining surface exposed, and you get freezer burn again.'
All that time, while he was being shuffled from care home to care home, there she was. Hidden away in someone's freezer.
' _However_ , if you're smart about it, you can get round that by preserving your severed head in aspic before freezing. That's what the residue was: aspic. Isn't that fascinating?'
Dotty's hand tightened on his arm. 'Callum? Are you OK? You look pale.'
'We had to clean it out of the aural, nasal, and sinus cavities with a syringe. Every available orifice was full of it. That's why the remains are so well preserved.'
All those years...
The room went grey around the edges, all colour focused on the head in Teabag's hands.
'Callum?'
'Something like this takes practice. Skill too – you'd probably have to fill all the cavities one at a time and let them set before doing the next one, or the aspic would just ooze out.'
Locked away in the frozen darkness...
'I suppose if you had a big enough bucket you could do it all at once, but you'd have to make sure you didn't leave any air pockets. Not easy.'
All that time...
Behind him, someone coughed.
'Constable MacGregor?' That was Powel's voice. Wonderful. Because things weren't bad enough. 'Callum. I... I can only imagine how difficult this is.'
The words wouldn't come out, blocked by the knot of barbed wire twisting itself at the base of his throat. Callum swallowed and tried again: 'What happened to her earrings?'
'I'm sorry, but we need you to formally identify the remains.'
'They were in the photograph. Tiny shell earrings. One blue, one pink.'
'Is this your mother?'
'WHAT HAPPENED TO HER BLOODY EARRINGS?'
Silence.
He looked up, and there was Powel, staring at him with a look of utter pity on his bruised face. Left cheek all puffy and purpled at the side of his mouth. A scab on his split lip.
Powel nodded. 'They're in evidence. Don't worry: they're safe, nobody's stolen them.'
At least that was something.
Callum closed his eyes and let out a shuddering breath. 'It's her. It's my mother.'
##
The curtain clattered on its rail and Powel stepped into the little treatment booth. 'We need to talk.'
Callum eased his right hand into his jacket sleeve, taking it slow. When he'd finished, a wodge of fibreglass cast poked out of the end. They'd left his thumb free, but all four of his fingers were imprisoned to the tips – partially curled as if he'd been caught in the act of cupping something. 'You're pressing charges.'
Of course he was.
'You chipped two of my teeth and knocked a crown off.'
'Good.' Callum stood. 'You deserved it.'
Powel stared at him. Then looked away. 'Possibly.'
'And this'll be the perfect opportunity to get rid of me, won't it? Polish off your little vendetta.'
A sigh. 'It's not a "vendetta", Constable MacGregor. I don't know how you managed to fool Professional Standards, but you took a bribe and—'
Callum barked out a laugh. 'No. No I didn't.' He fumbled with his jacket zip, not easy using just a thumb and a wodge of fibreglass. 'She didn't tell you, did she?'
'This has nothing to do with Elaine.'
'You've got _no_ idea what she talked me into.'
That pulled a half-smile onto Powel's unbattered side. 'If this is about International Women's Day, it—'
'I didn't take a bribe to cock up that crime scene, and do you know why? Because _I_ wasn't the one who cocked it up. But Elaine couldn't take the blame, could she? Noooo. Not when she was _pregnant_. With all that baby stuff to pay for? We couldn't afford it without her maternity pay.'
Powel's smile died. 'You're genuinely trying to pin this on her?'
'I threw my career in the septic tank to protect _your_ child. It wasn't even mine!'
'Don't be ridiculous. This isn't—'
'Every bloody day! I turn up for work and get treated like filth, so your bloody baby can have a fancy stroller and a crib and everything else.'
'For God's sake, Callum, _listen_ to yourself. You lost, OK? Lying to get revenge won't change anything.'
He paced to the end wall, then back again, barely three steps. 'And you want to know the _really_ funny part? I can't even go to Professional Standards and tell them what actually happened, because that's how completely Elaine's screwed me. I took the blame for her, I _lied_ to an internal enquiry. FOR HER!' Jabbing his broken hand out in the general direction of Flanders Road.
'Callum, don't—'
'Ask her. Ask Elaine.' Another laugh burst free, tasting of bile and betrayal. 'Mind you, she hasn't told the truth for at least nine months, why would she start now?'
A trolley squeaked past in the corridor outside.
The hospital PA system crackled into life: _'Please keep your personal belongings with you at all times.'_
Someone in the next cubicle wailed out in pain.
Powel pinched the bridge of his nose. 'Let's park this for now. OK?'
'And what about the flat? I suppose you expect me to just hand it over?'
'I didn't come here to fight with you, Callum.'
'Because that's _not_ happening. I've paid the mortgage for three years on that place. _My_ name's on the title deeds.'
'I came to ask about your mother.'
All the air went out of Callum's lungs. He settled on the edge of the treatment table, broken hand clutched against his chest. 'Oh.'
'I've been reviewing the original investigation. Not that there's much left in the files after the Great Clear-Out of Ninety-Five. Why they treated it as a case of child abandonment is beyond me. Who abandons a wee boy, _and_ the family car and caravan?' He sniffed. 'Is there anything you remember from that day? Anything at all? Doesn't matter how trivial or insignificant.'
'I was five years old.' Callum fumbled his wallet out and opened it. Showed Powel the photo of the four of them, grinning for the camera, all T-shirts and sunburn. Took a deep breath. 'We'd just spent two weeks on a caravan site outside Lossiemouth and on the way home I needed a pee...'
Steam coated the café windows, turning them nearly opaque. Little rivulets of water trickled across the back of the words 'THE TARTAN BUNNET' – vinyl lettering stuck to the glass in an optimistic arc. The red-and-white checked plastic tablecloth was scarred with ancient cigarette burns, and sticky to the touch.
A TV, mounted high in the corner, was tuned to some auction/car boot sale competition rubbish, with the sound muted and subtitles on, while a radio by the counter burbled out some cheesy pop tune from the eighties.
An auld mannie sat on his own in the corner, buried in that morning's _Castle News and Post_ : 'BEST FRIEND'S PLEA FOR MISSING ASHLEE' above the photo of a painfully thin teenage girl.
Other than that, they were alone.
Powel poured a third sachet of sugar into his mug and stirred it. 'We'll do everything we can, but I won't lie to you: it's a cold case from over twenty-five years ago and we're stretched thin as it is. They won't let me stick a huge team on this while we've got modern-day killers on the loose.'
Callum sat back with his arms folded. 'So that's that, is it? Nothing happens?'
'No. It's going to take _time_ , that's all I'm saying. DS Blake will—'
'Oh well, that's OK then. If Blakey the Octopus is on the case we'll get an arrest by teatime!'
A sigh. 'Callum, you can't just—'
'This is because I punched you in the gob, isn't it?'
'Don't you _dare_.' Powel jabbed a finger at him. 'I don't do "half-arsed", understand? I'm putting Blake on it because he's worked abduction cases before. I do _not_ cock-up investigations out of spite.'
A saggy woman in a checked apron appeared at their table, a plate in each hand. No smile. Mouth surrounded by the kind of wrinkles that suggested she never did. Shiny forehead daubed with thin grey hair. 'Who's gettin' the sausage?'
As if there was any question.
Powel took a breath, then pointed. 'Sausage for him, booby-trapped for me.'
She clattered the plates down in front of them, then shuffled off.
Powel opened the soft white bap on his plate, revealing a thick smear of melting butter and a fried egg – brown and crispy at the edges, bright-yellow wobbly yolk. 'I know you don't think much of DS Blake, but he's like a pit bull with a small child. Once he sinks his teeth in he won't let go.' Salt and pepper on the egg. 'He'll do a good job.'
Callum slathered the inside of his buttie with tomato sauce. 'He's an idiot.'
'For God's sake...' Powel burst the yolk with a fork, spreading it around. Closed his buttie and took a bite. 'Elaine and I didn't set out to hurt you, Callum. It just happened.'
'What, and you think buying me a cup of tea and a sausage buttie, and saying "sorry" makes it all right?'
'No. It... We'd been working that murder/suicide and it was tough, OK? They drowned the kids in the bath first: two beautiful little four-year-old girls. Then Mummy and Daddy took turns eating a shotgun. Blood and brains everywhere.'
The ketchup bottle was the old-fashioned kind: glass. Nearly full. Heavy in Callum's hand. Just the sort of thing for battering Poncy Powel's head in.
'Elaine was upset and we went out for a drink, and it just happened. We—'
'Don't.'
Powel frowned at him over the top of his buttie, voice soft and concerned: 'Callum, I'm only trying to—'
'Well don't.' Callum thumped the sauce bottle down on the sticky tabletop. Stood. 'You want me to what, forgive you? Say it's all fine. All's fair in love and war?' He grabbed his buttie and tossed it across the table, sending the sausages spilling out to roll off the edge of the table and onto the floor, leaving a smear of blood behind them. 'Lost my appetite.' He marched for the door, grabbed the handle.
Powel's voice cut across the room. 'Word of advice, _Constable_.' He reached out a foot and stood on one of the sausages, grinding it into the lino. 'DI Malcolmson's right: you should take a few days off. You don't look well.'
And whose fault is that?
'Callum?'
He looked up from the bench and there was Franklin, wearing yet another Blues-Brothers-tribute-act suit – complete with white shirt and black tie. She'd wangled herself an official Police Scotland golf umbrella with the Crimestoppers' 0800 number emblazoned all over it. It trembled in the downpour.
She had a quick glance around. 'What are you doing, sitting out here in the rain?'
St Jasper's Cathedral reached up into the stained clouds, austere granite walls topped with sandstone spikes and gargoyles. The sandstone wasn't the genteel, pale, creamy-coloured stuff they used in the Wynd, but a dark dirty red like old blood. And nearly five hundred years of sleet and rain had made it bleed into the grey beneath. A big brown scab of rusty scaffolding covered the circular stained-glass window, the sound of a workman's radio burbling out promos for the music festival in Montgomery Park.
Callum toasted her with his can of Fanta. 'Detective Constable.'
Headstones stretched out all around him, most crumbling, many illegible. All those lives: their owners rotted away and forgotten, feeding the massive oak tree that dominated this part of the graveyard.
'I'm... sorry about your mother.'
He nodded. Stared out across the rows of the dead. 'You want to guess what I've spent the last two and a half hours doing? Trying to disentangle myself from Police Constable Elaine Pirie. Bank accounts, council tax, mortgage, electricity bill, BT, hire purchase on the TV...' He rested his elbows on his knees, the cast on his right hand tucked into his sleeve. 'I'd go home and drink a bottle of Bell's in the bathtub, but I don't _have_ a home.'
Franklin stared up at the ribs of her umbrella, voice a low mutter. 'Why do men always have to be such babies?'
'Great, thanks for the pep-talk.'
'Fine, you sit there wallowing in self-pity.'
'Self-pity? My girlfriend's pregnant with someone else's child, I've been thrown out of my own flat, my career's in the crapper, my mother's severed head is lying in the mortuary, and they've put DS Blake in charge of catching her killer. Blakey the Racist-Sexist-Scumbag-Octopus. The halfwit you punched in the face is now the _only_ person looking into my mother's murder. I think I've got every right to complain!'
A seagull landed between two graves, paddling its big orange feet on the grass, trying to lure a worm to its doom.
Franklin shrugged. 'So just sit there, then. Catch pneumonia. Play the tragic jilted hero. See if anyone gives a single toss about it.'
Rain pattered on the shoulders of Callum's jacket. Soaked into his wet hair. 'Feel free to spread your special brand of sunshine and joy somewhere else.'
'Or you can get off your moaning backside and _do_ something about it.'
##
The smell of boiled cabbage haunted Division Headquarters' stairwells. Callum squelched his way down to the locker room, where it was joined by the ghosts of cheesy feet, eggy farts, and cloying deodorant.
Full-length metal lockers lined the room, each one with a number, nametag, and assortment of dents. Some were plastered in headlines and photos culled from newspapers. Some with celebrity photos from the tabloid magazines. Others austere and bare. Nothing to see here: move along.
More lockers made an island in the middle of the room, surrounded by a knee-high wall of slatted benching. Heavy-duty clothes rails were hung with stabproof vests and high-viz waistcoats.
A young PC sat in the far corner, folded forward with his elbows on his knees, face buried in his hands. Shoulders quivering. Making little sniffing noises.
Callum took out his keys and unlocked locker 322. Opened it. Stared at the wall of photos Sellotaped to the inside of the door.
Most of them were selfies of Elaine: grinning for the camera, making pouty duck-lips, flashing the peace sign, pulling faces, going from flat-stomached to swollen pregnant bulge. All nice and normal. A happy little family in the making.
There was even a printout of the sonogram – looking like a radar image. A fan of streaked grey, surrounded by black, and just off the centre of the image, a small dark kidney-shaped blob. Elaine had drawn a circle around it in red crayon and an arrow labelled 'GOD, IT LOOKS LIKE A PEANUT!!!!' surrounded by hearts.
He reached out with his good hand and ran his fingertips along the sonogram's smooth surface. Then curled them into claws and tore it from the door. Raked the photos down after it, letting them flutter and spill across the tiled floor. Peeled off his sodden jacket and wrung it out.
It wasn't easy with one hand in a cast, but he struggled through.
Water spattered down on Elaine's selfies. Soaked into Peanut's first picture.
He did the same with his shirt, trousers, socks and pants. Stood there in the nip, staring down at the puddled photos. Then swept them all up and dumped them in the nearest bin.
Done. Over. Finished with.
A wooden rack sat outside the door through to the showers. He helped himself to a towel from the pile – greying and frayed around the edges, sandpaper-rough to the touch. Scrubbed himself dry on the way back to his locker. Dumped it on the floor to soak up the wrung-out water.
His spare fighting suit was a little crumpled, but at least it wasn't damp. No idea when he'd last had it on. Been a while since someone was sick all over him. They tended to gloss over that bit in the recruitment posters.
Callum bundled up his soggy suit and shoved it into a bin-bag. Tied a clumsy knot in the top.
Then marched out of the room, leaving the PC to cry in peace.
Callum stomped his way up the stairs. Through the double doors and into the MIT's palatial abode. Marching down the corridor, past the meeting rooms, past the open-plan offices, past the mini-canteen.
Poncy Powel's door was shut – probably off getting someone else's girlfriend pregnant.
Good. The chance of Powel getting another punch in the face was about ninety-nine point nine percent. And there were far too many of his team knocking about for that to happen – they weren't exactly going to stand about tutting while Callum battered their boss into a squishy mess.
Across the corridor from Powel's lair was another door with 'SERGEANTS' OFFICE' engraved into its brass plaque. Callum didn't bother knocking.
Inside, six desks were arranged around the walls, all with laptops and monitor stands, ergonomic keyboards and fancy mice. An electronic whiteboard above each desk, displaying photos and timelines.
A large woman with a pixie haircut had her feet up, swivelling back and forth, a mobile phone pinned between her shoulder and her ear as she picked at her fingernails. 'No... Because I _say_ it isn't, Limpy... Well, guess what: I – don't – care.'
Two desks down, a tall thin bloke was hunched over his keyboard like a praying mantis, squinting at the media player on his screen.
And there, right in the far corner, was DS Jimmy Blake: elbows on his desk, hands propping up his face so he could do some industrial-strength frowning at the stack of paper in front of him.
Callum nicked the office chair from the next desk, wheeled it over, and sat. Dumped his bin-bag on the nice new carpet tiles. 'Blakey.'
He didn't look up. 'Go away.'
Mantis Boy must've set his player going, because a young girl's voice crackled out of his computer's speakers. _'You want to get wasted for my birthday next week? I can utterly rob a bottle of voddy from my gran.'_
A slightly muffled answer, dripping with teenaged indifference: _'Yeah, why not. You only turn fourteen once, right?'_
Callum poked Blakey in the shoulder. 'Have you made any progress yet?'
A long-suffering sigh. Finally he turned and looked up, showing off the big plastic guard covering what was left of his nose. Eyes like a panda that'd been on a three-day bender. 'Do you have _any_ idea how many cases I'm working right now?'
_'My step-dad wants to have a party down the bowling alley. Laser Quest, dodgems, and burgers, like I'm, I dunno, six years old or something. He's such a_ complete _spazmoidal—'_
_'Yeah, hang on, Marline.'_
'Have you at least _looked_ at the file?'
'What am I, Dr Who? When am I supposed to have the time?'
The muted sound of a doorbell rang out from the speakers.
_'OK, OK. Jesus.'_ Clunks and rattles. _'What?'_
Callum poked him in the shoulder again. ' _Make_ the time.'
A man's voice, barely audible: _'I'm sorry, but I'm trying to find my son.'_
Blakey stared at him. 'You want me to just drop everything and rake through the ashes of a cold case from twenty-six years ago? No chance.'
The praying mantis swivelled around in his seat. 'Can the pair of you belt up? I'm trying to listen to this.'
_'—his name's Sam. He's only four.'_
'It's my _mother_ , Blakey. Understand? It was _her_ head in that carrier bag.'
Blakey looked away. 'You sure you want to know what happened?'
Was he deaf or something? 'Of course I sodding do!'
_'Oh, you poor man. Please, come in.'_ A woman's voice this time. _'You said his name's Sam?'_
_'Sam. Yes.'_
'Callum, you know well as me: nine times out of ten, when the wife gets murdered it's the husband that did it. That's how people work.'
'It's not—'
'Right now your dad's probably sodded off to the Costa del Sol, or Australia, or something like that. Living under an assumed name with your brother and his grandkids.'
Because Alastair would be able to get _his_ girlfriend pregnant. Wouldn't need some greasy Detective Chief Inspector to come in and do it for him behind his back.
_'What are you doing?'_ The woman's voice again. High and panicky this time. _'No! Leave her alone!'_
Screams ripped out of the speakers.
_'LEAVE HER ALONE!'_
Callum stood. 'You're an idiot, Blakey. And I'm glad Franklin battered your nose to a pulp. Hope it hurt.'
He marched out to a soundtrack of screaming.
How anyone was supposed to find _anything_ down here was a mystery. Rows and rows and rows of shelving units stretched away into the gloom, each one crammed full of brown cardboard file boxes. The air slightly crunchy with the earthy taste of mildew and dust. They'd followed the same lighting design as Camburn Woods – less than one in three strip lights worked. Most were completely dark, the odd one pinging and buzzing, letting out an intermittent flicker before going out and starting all over again.
Callum hauled another box from the shelf and wiped a finger through the thick fur of dust covering the 'CONTENTS' sticker. Checked the crime number. Nope. Stuck it on the floor, and tried another one. And another. Put them all back where they'd come from and tried the shelf below.
Twenty minutes of raking through years' and years' worth of ruined lives and horror.
Still, at least this lot seemed to be in the right kind of era.
His mobile jangled into life and he pulled it out. Who would've thought you'd get a signal down here, in the basement of DHQ? Wonders never ceased.
He hit the button. 'Hello?'
Silence.
'NUMBER WITHHELD.' glowed in the middle of the screen.
Great.
This again.
'Willow, if someone's hurting your mum you need to—'
_'Callum.'_ Not Willow Brown, DCI Reece Scumbag Powel.
He clenched his face. 'What do _you_ want?'
_'I've just had Elaine on the phone.'_
He pinned the phone between his ear and his shoulder and pulled out the next box. 'Well, I'm sure that makes a difference from having her in the bed _I_ bought.'
Nope.
The next box joined it on the concrete floor.
Nope.
'You still there?' Wouldn't be a loss if he wasn't.
Next box.
_'You cancelled the mortgage payments. You know fine well, her maternity pay—'_
'Get stuffed. The pair of you manoeuvred me out of my own sodding flat, do you _really_ think I'm going to keep paying the mortgage so you can shag in it?'
The last one was tucked away at the back of the shelf. Ah. That looked a lot more promising – no crust of dust. Someone had checked it out recently.
_'You can't just—'_
'I've been paying _your_ bills, Powel. Who do you think bought that crib, or the Winnie-the-Pooh mobile, or all those sodding baby clothes?' Getting louder and louder. 'Who bought the Nutella and pickled cucumbers for the last nine months? COS I DON'T REMEMBER IT BEING YOU!'
His words managed a brief echo, before being swallowed by the ranks of shelves and boxes.
_'Are you finished?'_
'Damn right I am: finished being your idiot. Pay your own bloody mortgage.' He thumped the clean box down on the pile and checked the crime number. Had to be at least a dozen of them printed on the sticker in careful blue biro. And third from bottom was an exact match for the reference he'd dug out of the computer.
_'This isn't productive, Callum. We are where we are and throwing tantrums isn't going to change that.'_
'OK, I'm hanging up on you now. Feel free to take your phone and ram it up—'
_'You need to collect your things from the flat.'_
Right.
His back stiffened. 'And will Elaine be there?'
_'I... don't think that would be wise, do you?'_
Definitely not. 'I want my books back.'
_'She's staying with her mother for a couple of days. Come over any time after eight.'_
'I'll turn up when I feel like it. It's _my_ flat.'
_'We've changed the locks, Callum. After eight. I'll be there to make sure you don't do anything foolish.'_ Powel hung up.
Callum lowered his phone, knuckles white, the plastic creaking as he squeezed it. Then he snatched his arm back, ready to hurl the thing into the darkness... Hissed a breath out through his nose. Turned away.
Then back again, slamming his foot into the nearest file box – sending its contents spraying out across the dusty floor.
_'Something foolish.'_ Yeah, like battering Powel's head in.
His shoulders dipped.
There were files and evidence bags everywhere.
He sighed, squatted down, and cleaned it all up.
Powel might have been an adulterous two-faced slimy scumbag, but he was right about one thing: the case file was virtually useless.
Callum flipped to the end of the file and back again. Which didn't take long as it was only two sheets. They had Mum, Dad and Alastair's names, the date they abandoned Callum, a brief note about social services taking him into care, and scribbled on the back in pencil: the name of both officers who worked the case.
No interview notes, no witness statements, no sightings. Nothing of any practical use whatsoever. Not even the name of the rest area they'd left him in.
Either the Great Clear-Out of Ninety-Five was incredibly efficient, or PC Gibbons and DS Shannon hadn't bothered their backsides doing any investigating at all.
Callum jotted their details into his notebook, stuck the file back in its box, and the box back on its shelf.
He signed himself out of the archives and collected his bin-bag full of wet suit. Draped it over his bad arm, freeing his good hand to pull out his phone. Dialled as he slogged up the cabbage-scented stairs. 'Brucie? It's Callum. Do us a favour and run a check on a couple of oldies for me: PC Gibbons, DS Shannon.'
_'You got shoulder numbers?'_
'Nope. But they worked here twenty-six years ago.'
_'Give us a minute...'_
Callum paused on the landing. Rain battered the window, rattling it in its frame. The flickering blue-and-white lights of a patrol car faded in the distance, siren wailing. A double-decker bus grumbled past, going the other way.
_'Right, you got Police Constable Maggie Gibbons – transferred to Strathclyde in 1999. And Sergeant Robert Shannon. Retired twenty-two years ago.'_
'You got an address on Shannon?'
_'What did your last slave die of?'_
'Come on, Brucie.'
_'You're lucky I'm in a good mood. Won fifty quid on a scratchcard this morning.'_ The sound of a keyboard being tortured clattered through the phone. _'Here you go: Robert Michael Shannon, seventy-one, lives at Canaries Cottage, Leveller Road, Fiddersmuir.'_
'Thanks.' Now all he needed was a car.
Callum eased through the double doors into the corridor. Wandered down to the manky little offices of the Divisional Investigative Support Team, nice and casual. Mother's door was closed and so was McAdams'. No sound of voices coming from within.
The main office was quiet too.
Which was nice.
He eased open the door.
Empty. They'd all be out trying to track down Imhotep.
Good. That meant no awkward questions, forced sympathy, crappy haikus, or complaints about him nicking one of the pool cars.
There was a little whiteboard, over by the kettle and microwave, no bigger than a sheet of A4 – split up into three columns. A magnetic hook sat at the bottom of each one, a printed number plate at the top. And a bit in the middle to write your name and why you were taking the associated car.
Dotty's wheelchair-adapted Vauxhall was checked out, as was the battered Audi, leaving just one set of keys dangling on its magnetic hook: the ancient dirty-brown Ford Mondeo estate. And it was an automatic, not usually a plus point, but perfect if you only had one working hand to drive with.
Callum liberated the keys with a 'Yoink!' then scrawled something unintelligible in the details section. Probably wouldn't fool anyone for long, but it was worth a go.
He would've got away with it too, if it wasn't for that pesky DC Franklin.
She was marching down the corridor, clutching a sheet of paper when he slipped out of the office. Stopped and stared at him. 'Callum.'
He wheeched his hand behind his back, hiding the car keys. Pulled on a smile. 'Thought everyone was out.'
'Had to hang about, waiting for this.' Franklin held up the sheet of paper. 'Warrant forcing Strummuir Smokehouse to hand over all their employees' details. I'm off to serve it.'
'Right.' Sod. That meant she needed the last pool car. 'So they've not recovered the body yet? From the river...'
'Could you look more shifty than you do right now?'
'I don't—'
' _And_ you're meant to be on compassionate leave.'
'What happened to "get off your moaning backside and _do_ something about it"?'
Franklin narrowed her eyes. 'What are you up to?'
'Me? Nothing. Nothing at all. Just changed out of my damp suit.' Callum forced the smile a little wider. Jiggled his bad arm and its decorative bin-bag. 'Tell you what, as I'm not doing anything right now, how about I come with you? You know: keep myself occupied. I'll drive if you like? Not a problem.'
'Hrmmm...' Then a nod. 'OK, get the keys.'
'Way ahead of you.'
##
Rain. Rain. Rain. It pattered on the pool car's roof, rippled the windscreen – shifting everything in and out of focus.
The Strummuir Smokehouse car park was nearly empty. Six o'clock on a Friday night. The staff and visitors would be long gone. All except for the owner of the white BMW, parked in a spot marked 'RESERVED FOR MANAGING DIRECTOR ~ MR FINN NOBLE'.
Callum had a scratch at his thumb, where the skin poked out of the cast.
Clicked on the radio.
A weird dirge-like groan filled the speakers, slow and dark. _'And I burn inside like the stars, / A million thoughts and pains and scars, / Running away from you, Angelica...'_
He turned it down a bit.
Should've brought a book.
How? How was he supposed to do that when they were all back at the flat?
Yeah, well, should have thought about that _before_ he stormed out yesterday, shouldn't he?
Sodding hell...
Callum thumped his head back against the rest, then peered through the window.
What the hell was taking Franklin so... Ah, there she was.
Franklin pushed out through the smokehouse front door and into the rain.
_'See me burn, / See me run and hide, / See me dying, / See me cyanide...'_
She hunched her shoulders and ran for the car. Clattered into the passenger side. 'Gah... Does it _never_ stop raining here?'
'You get the names?'
'I swear to God, it's like Oldcastle's cursed.'
In so many ways.
_'But no, / You can't see me, / You can't breathe, / You can't hear me...'_
She shook the rain from her hair. 'And what are you _listening_ to? Sounds like a funeral for depressed monks.'
'No idea.' He cranked the blowers up full, drowning it out. Turned the car around and headed back along the road, through Strummuir. Driving with one hand in a cast wasn't so bad when you didn't have to bother changing gears. 'So: names?'
'Yup.' She dug out her phone and poked at the screen.
'Planning on telling me at any point?'
'Hold on.' Franklin put her mobile to her ear. 'Mother? Yes, it's Rosalind. I got the employee details from the smokehouse... No, rolled right over soon as I flashed the warrant... Uh-huh.'
A right at the roundabout took them out past the rows of little Scottish houses with their grey-harled walls and slate roofs. Fields of green and grey on either side of the road, streaking past as Callum put his foot down.
'According to Mr Noble, the man Watt tried to save from the river was one Tod Monaghan... Hold on, I'll put you on speaker.' She held it in the middle of the car and Mother's voice fought against the blowers' roar.
_'Monaghan, Monaghan... Right, here we go, Andy's just bringing it up now. Tod Monaghan, thirty-five. AKA: Toby Hutchinson, AKA: Timothy Liddell, AKA: Todzilla. Did six years for attempted murder. Released on licence eleven months ago... da, da, da... Oooooh: form for indecent assault. That's interesting, isn't it? And there was a rape case, but the gentleman he attacked didn't want to go to court.'_
'Sounds lovely, doesn't he?'
_'You know what I think, Rosalind? Violent sex offender, attacks men, works in a smokehouse, does a runner soon as John and Callum show up. I think we might have found our Imhotep. Isn't that...'_
Silence from the phone.
A little graveyard slipped by on the left, its church a crumbling ruin. Woods on the right.
Some auld biddie, walking her dog in the downpour, clambered onto the grass verge as they approached. Stuck two fingers up as they passed.
And finally Mother was back: _'Rosalind? Why did you put me on speakerphone?'_
Franklin glanced across the car.
Callum put a finger to his lips and mouthed, 'I'm not here!'
'I'm... driving. Don't have a hands-free set. Safety first.'
_'Oh, yes. That's a good idea.'_
A right at the junction with the main road took them back towards town.
_'Just in case, I'm going to run everything past Dr McDonald. And we better get a warrant sorted for his home address too.'_ Mother's voice faded, as if she'd turned away from the phone. _'Can you sort that for me, Andy? Top floor left, thirty-nine Bellfield Road, Cowskillin. Thanks.'_ Then she was at full volume again. _'Good work, Rosalind.'_
'Thanks, Mother.' The line went dead and she slipped the phone back in her jacket pocket. Then frowned as Callum took the next left onto a country road. 'I thought Division Headquarters was that way?'
'Ah, yes. It is.' He gave her his best smile. 'Just got a _quick_ stop-off to make on the way. Ten, twenty minutes tops.'
Franklin's head fell back against the rest, eyes screwed shut. 'Not again!'
'Knew I shouldn't let you drive.' Franklin scowled out of the passenger window.
'Hey, you told me to get off my backside and do something. Remember?'
The pool car crested the hill and there was Fiddersmuir, sulking at the bottom of a wide dip in the landscape, bordered on one side by a dense swathe of dark-green woods. An irregular grid of streets sat around an oversized town square with a dirty big monument in the middle. A church and town hall on opposite sides, facing each other down in a competition of who could look the most joyless.
The dour grey buildings got smaller the further away from the square they were, three-storey merchant houses giving way to austere Edwardian terraces, and finally miserable wee cottages. Someone had stuck a small housing estate on the far side, all pantiles and cream harling. Looking about as out of place as a vegan in a slaughterhouse.
Franklin kept her face to the glass. 'I meant on your _own_ time.'
'This is my own time. I'm on compassionate leave, remember?' Callum checked the address and took them down a wide road lined with unhappy buildings, around the horrible monument to Prince Albert, and onto Leveller Road.
'You're impossible.'
Right at the end, just before the limits sign, was a large cottage set behind a long stretch of drystane dyke. A small conservatory sat out front, along with a collection of water butts. Tidy garden, thankfully devoid of gnomes. The words 'CANARIES COTTAGE' sat in bright-yellow letters on a green sign.
'Ten, twenty minutes tops.' He pulled up onto the driveway.
'You said that twenty minutes ago!'
'Well, there you go then.' Callum grabbed a high-viz jacket from the back and climbed out into the rain, holding it over his head like a cape. Hurried up the path.
A little laminated notice hung in the glazed panel by the front door: 'IF YOU'RE DELIVERING PARCELS, TRY THE POLYTUNNEL ROUND THE BACK.'
Fair enough.
Franklin appeared with her Crimestoppers umbrella, following him around the side of the house. 'This is a _complete_ waste of my time. I should be back at DHQ, working on—'
'Oh stop moaning. It'll take McAdams at least an hour to get a warrant sorted. Probably longer on a Friday night. All you'd be doing is twiddling your thumbs, listening to Dotty and Watt snipe at each other.'
The back garden was huge: a vast expanse of neatly mowed lawn, peppered with trees and bushes, meandering paths and flowerbeds. Off to one side, the grey arched shape of a polytunnel sat beside a row of beech trees, the plastic quivering in the downpour.
Callum jogged along the path, feet crunching on bark chips. 'Anyway, shouldn't you be more worried about missing your boyfriend's work's do?'
'Ten minutes, then we're heading back to headquarters if I have to drag you there by the balls. Understand?'
Music oozed out through the plastic, something old-fashioned and familiar, turned up loud.
He opened the door and bundled into the warm moist air, full of the toasted bready scent of soil and compost.
Dear God, this thing was bigger than his whole flat. It stretched on and on and on, full of green. Raised beds ran down both sides, packed with sprawling courgette plants, trailing cucumbers, dreels of tatties, rows of spinach, ranks of fancy lettuces...
Come the Zombie Apocalypse you could feed a family of four for a year in here.
Inside it was obvious why the radio was turned up so loud – the whole tunnel rang with every raindrop that thumped into its plastic skin. Thrumming and vibrating like an outboard motor. Fighting against a soft-edged bouncy song about some woman wanting to be with Callum everywhere. Which was a lovely offer, if a bit presumptuous.
Unless she was singing about the only other person in the polytunnel?
He was halfway down, on his knees, footering about with some sort of bean plant. Blue jeans, trainers, grey T-shirt with '1902' across the back in big letters, a swathe of pink skin showing through the top of his close-cropped grey hair.
Callum reached out and clicked the radio off.
The guy stopped footering and turned, frowning at them through a pair of gunmetal-framed glasses. A well-spoken English accent cut through the rain's drumming din, slightly higher-pitched than expected. 'I was listening to that.' His beard was every bit as grey and short as the hair on his head.
'Robert Shannon?' Callum dug out his warrant card. 'We need to talk to you about a child abandonment case you worked in CID.'
'CID?' He levered himself to his feet, brushed the dirt from his hands on a little paunch belly. 'I haven't worked CID for... ooh, has to be twenty-five years now. And it's Bob, not Robert.'
'I was the child.'
'Ah.' Call Me Bob nodded. Turned. And pointed down to the far end of the polytunnel. 'You'd better take a seat.'
The cast-iron patio furniture was comfortable enough: a small round table and four chairs – each with a green-and-yellow cushion, bordered by some recycled chests of drawers on one side and a row of beetroot on the other. Above their heads, the plastic skin trembled.
Franklin checked her watch. 'Why didn't you say this was what we were here for?'
'Would you have come?'
'You should've told me.'
'Blakey hadn't even _looked_ at the crime file. I checked the archive register. He's meant to be SIO and he can't be arsed to read the file?'
Not that it would have done him a lot of good, given how little was actually in there. But he could have put the effort in.
She puffed out a sigh. 'Don't suppose your ex-DS Shannon's done a runner, do you?'
Callum leaned forward in his seat, staring at the polytunnel door. 'Speak of the Devil...'
Shannon hobbled in from the rain, cardboard file box in one hand, brolly in the other. 'Here we go.' He limped his way down to the table and stuck the box in front of Callum. Shrugged his way out of his jacket. Eased himself down into his seat. 'Sorry it took so long; it was right at the back of the attic.'
Franklin raised an eyebrow. 'Sore leg?'
'Hip replacement. Doesn't usually bother me, but all this rain?' A shrug. He took the lid off the box and dumped it on the empty chair. Smiled at her. 'Have a look in the chest of drawers behind you, should be a bottle of red and some glasses.'
'I'm on duty.'
'I'm not.' Callum dipped into the file box, coming out with an overstuffed Manilla folder held together with elastic bands. They crumbled beneath his fingers.
'Men.' She shook her head and went rummaging. Came back with two large wine glasses and a bottle of Malbec. Folded her arms and sat back.
Inside the folder were a bunch of statements from what looked like Dad's work colleagues and the neighbours. More from people who'd been at the caravan park that week.
Shannon opened the wine and poured out two hefty measures. Held one glass out. 'They were going to destroy everything, so I took it home.'
Callum accepted the glass and took a sip. Soft and jammy. 'The Great Clear-Out of Ninety-Five?'
'No.' He settled back in his seat, swirling the wine round and round the glass. 'We've met before, do you remember? Maggie and I interviewed you about a dozen times after it happened. She had a sort of giraffe glove-puppet for you to talk to?'
'I don't...' A frown. There was something there: a soft kindly face with a beauty spot on one cheek. A splotchy orange-and-white animal that talked with an Irish accent. 'Sort of.'
'We spoke to everyone we could think of, put adverts in the papers, posters everywhere, appeals on the radio. I'm sorry, Callum. If we'd had even _one_ witness, maybe we could have done more. I suppose we'll never know what happened to them.'
Franklin scowled at him. 'We know what happened to Callum's mother.'
'Wow.' A chuckle rippled free, bringing a smile with it. 'That's great news. Where has she—'
'Her severed head turned up in Holburn Forest.'
'Oh.' The smile faded. 'I'm sorry.'
Franklin stood. 'Well, maybe if you hadn't treated it as a _child abandonment_ case, you'd have had better luck!'
'Whoa! I'm going to have to stop you there: Maggie and I ran it as an abduction. We'd been on it for months – fighting hard, but getting nowhere – before the top brass pulled the plug. Shifted us off to other cases. Downgraded this thing to "child abandonment" so it would look better for the crime statistics. Thank you and goodnight.' Shannon toasted them with his glass and drank. Sighed. 'So I kicked up a fuss. Wouldn't believe how much trouble _that_ got me into.'
Franklin lowered herself back into her seat, cheeks flushed. 'I see.'
'That was it, far as my career was concerned. Because I wouldn't shut up, they gave me a "development opportunity", AKA: chucking me out of CID and back into uniform. I was supposed to make DI by the time I was fifty, instead of which I got to spend my last three years juggling staff rosters and patrolling Harvest Lane at chucking-out time.'
Callum put the statements back on the table. 'What about the Slug?'
'Hmm? Oh, we don't get a lot of them in here. I was listening to Radio Four the other day and they were banging on about setting a few chickens or ducks loose to eat the slimy little monsters, but they're even worse for guzzling lettuces than slugs are. So I just hand pick them and throw them in the burn.'
'Not slugs plural, _the_ Slug. The man in the toilets?'
'What man?' Shannon leaned forwards, light glinting on his metal-framed glasses. 'Describe him.'
'Paedophile, about six feet tall, hunched, balding, think he had a limp? Breath stank of butter-mint.'
'And this was when you went missing?' Shannon's voice had gone up again. Excited. Eager. 'All those interviews with Constable Giraffe and you never mentioned him once.'
'He...' Callum's mouth clicked shut. 'I didn't?' He cleared his throat.
It was like tying an anvil to his stomach and throwing it overboard. Being dragged down through the water, breath burning in his throat, pressure squeezing him, sunlight fading as the lake swallowed him whole.
_'If we'd even had one witness, maybe we could have done more.'_
All this time.
'Maybe we could have done more.'
They'd _had_ a witness: Callum. And he hadn't told them about the Slug.
_'This is going to be our little secret. If you tell anyone, I'll know. And I know where you live and I'll come get you. Understand?'_
He hadn't told them, because he was too scared. Too cowardly.
'Could have done more.'
DS Shannon and PC Gibbons could have caught the bastard. Could have saved his family.
It was all his fault.
'Are you OK?' Franklin was only visible from the knees down, standing in front of Callum's chair.
'No.' He kept his head between his knees. 'I didn't say anything. I should have _said_ something.'
'Callum, you—'
'The Slug said he knew where I lived. He would come get me if I told anyone. I was terrified of him.'
'You were only five. A wee boy.'
Callum sat up, let his head fall back and covered his face with his fibreglass cast. 'Arrrrrrrrrrrrgh...'
The rain thrummed on the polytunnel walls.
Stupid, stupid, stupid, stupid—
A hand, warm on his shoulder. A squeeze. Franklin's voice, soft and kind: 'It wasn't your fault.'
Her hand was still there when ex-DS Shannon returned.
'Right, I've put in some calls, but it's going to take a while. Everyone I knew on Nonce Patrol is either retired or dead. But Franky Campbell's going to have a root about in his shed, see if he's still got any of the case files from back in the day.'
Franklin's hand slipped from Callum's shoulder. 'Does everyone steal files from the station?'
'Be glad we did.' Shannon pulled a face and sank back into his chair. 'The Great Clear-Out of Ninety-Five. The archives were packed, no one wanted to pay for a new storage facility, so they binned nearly everything not connected to a major case. A lot of people took stuff home rather than see it hurled in a big skip.' A shrug. 'I suppose, in a lot of ways, Oldcastle Police managed to outsource its storage problem to our attics and sheds.'
She checked her watch. 'And is this Franky Campbell going to be long?'
'Hours. And hours. And hours. He's on a Zimmer frame. Arthritis. _Very_ bad.'
Callum nodded, then stood. 'Thanks, Mr Shannon. I appreciate...' A frown. 'Wait a minute: when I asked if they were chucking the files on my family's disappearance because of the Great Clear-Out of Ninety-Five, you said "no".'
'That's right.' He topped up his glass with the last of the Malbec. 'Happened right after they decided to write it off as child abandonment, even though any idiot could see it wasn't.'
'Why?'
'Officially? Not cluttering up the archive with redundant materials. Unofficially: because they were fiddling the crime figures and didn't want anything hanging around that proved it was an abduction. And super-unofficially?' He pursed his lips.
Callum stared.
Franklin cleared her throat. 'Any time you like.'
'Well, and this is just a rumour, and it didn't come out till years afterwards, but _super_ -unofficially: they knew who did it and there was no way they were ever going to bring that person to trial, because that person was famous and that person was protected.'
'Hrmmm...' She flexed her hand. 'If the next words out of your mouth are "Jimmy Savile", I'm going to slap the face off you.'
'Jimmy...? _No_.' Shannon shook his head. 'And it's all just rumours anyway. Oldcastle's never been a mecca for the rich and famous, has it? Too sodding wet and miserable.'
'So who was it?'
'No idea. It was someone's leaving do, a DCI was drunk and mouthing off. Probably just a wine box of supermarket Cabernet Sauvignon talking.' A shrug. 'You know what some cops are like with a drink in them: the Castle Hill Ripper was actually on the city council, Sensational Steve from the radio has a basement full of dead children, Lord Lucan spent his last three years chained to the wall in a warehouse in Logansferry.'
Yeah. Still.
Callum stood. 'Do me a favour: see if you can track down your drunk DCI. Might be rubbish. Might be worth a go.'
Shannon levered himself upright and shook Callum's hand. 'I'm sorry about your mother. I wish we could've done more.'
'Me too.'
##
_'... spectonkular!'_ Grating honking noises blared out of the car radio. _'You're listening to this super bumper edition of_ Crrrrrrrrrazy Colin's Rush-Hour Drive-Time Club _on Castlewave FM, my friend, and we're here live at...'_ pause for dramatic effect, _'the seventh annual Tartantula music festival! Yay!'_ The sound of a crowd baying in the background – whoops, cheers, and whistles.
Franklin turned in the driver's seat. 'It wasn't your fault.'
_'Are we having a great time, or what?'_
More cheering.
Callum frowned down at the cast on his right hand, little pink fingertips poking out of the end. 'I know. But it doesn't help, does it?'
_'That's right-a-roonie, campers: we're not going to let a little rain spoil our fun. And now, are you ready for your next act?'_
Cheers.
The pool car wheeched down the dual carriageway, heading south, back into town.
_'I can't hear you!'_
Cheers.
On the right, the tight spirals and cul-de-sacs of Blackwall Hill. On the left, the necrotic miserable sprawl of Kingsmeath.
_'One more time!'_
Cheers, going on and on and on.
'Callum—'
'I know, OK? I was only five. But...' He rubbed his good hand over his face. 'Maybe Shannon will come up with something. I mean, the trail's only been cold for _twenty-six_ years. What could possibly go wrong?'
_'Darn tootin' you are. Let's give a great big Oldcastle welcome to Overture for a Riot!'_
And the crowd go wild.
A slow, thumping drumbeat wove its way between the screams.
Franklin shook her head. 'You haven't twigged it yet, have you? Yes: the initial incident is twenty-six years cold, but someone dumped your mother's head in the woods Wednesday evening or Thursday morning. That's current. Something's happened to bring him out of retirement.'
She had a point.
The drums got louder. Faster.
She reached across the car and thumped him on the arm. 'So?'
'So we chase up the SEB. Fingerprints on the bag, any foreign DNA found on...' He cleared his throat. 'On the remains.'
Not his mother's head. _Remains_.
'Correct. Then you put out an appeal for witnesses: anyone in the vicinity of the woods. Dog walkers, courting couples, maybe it's a dogging hotspot? Get Traffic to stick up a couple of those "Were you here on the eighth or ninth?" sandwich board things they leave at the site of an accident.'
A single guitar chord sounded, long and trembling.
'Yeah... Only they won't do it for me: I haven't got a budget to give them. It's Blakey's case and he'll do sod-all for as long as he can get away with it.'
'So go _round_ him. Talk to his boss. Get him slapped down.'
Another chord, building on the first.
Callum stared at her for a moment. Then burst out laughing. 'His boss is the one who got my girlfriend _pregnant_!' He held up his cast. 'Whose face do you think I broke this on? Powel's not going to help.'
_'HELLO, OLDCASTLE!'_
Nothing on the radio but cheering.
Franklin pursed her lips for a moment, then nodded. 'You're buggered, then.'
Might as well get _that_ tattooed across his forehead. Save time.
The Blackburgh Roundabout loomed up ahead, the library in the middle dark and lifeless, while lights blazed in Montgomery Park – just past it, on the right. Marquees and anti-aircraft spotlights, a blimp shaped like a massive tartan spider. Its legs trembled in the rain. Yeah, because _that_ wasn't going to give all the kids within a three-mile radius nightmares for months.
_'Wicker Man, Wicker Man, they're dancing while you burn inside, / Run and hide, Wicker Man, your heart's pumping formaldehyde...'_
Franklin's phone rang and she dug it out, tossed it across the car to Callum, then killed the radio. 'Put it on speaker.'
He did, holding the mobile out and keeping his mouth shut.
_'Rosalind? It's Mother. Where are you? We've been worried.'_
'Coming up to the Calderwell Bridge, just heading back to the shop now. Did you get your warrant to search Tod Monaghan's home?'
_'Change of plan – I need you at Kings Park, east entrance.'_ Pause. _'And you can tell Callum he can come too.'_
Franklin raised an eyebrow. 'Callum? I don't—'
_'They didn't make me a detective inspector just because I'm pretty, Rosalind. Now, bottoms in gear, children. We've got ourselves a body.'_
Franklin hunched her shoulders, rain drumming on her Crimestoppers umbrella, picking her way down the gravel path from the car park.
Callum limped along beside her, bundled up in a high-viz jacket, sticking close to stay dry.
Wet grass glistened in the fading light, big rhododendron bushes lurking in the gloom, leaves just starting to turn on the trees. The fancy sandstone bulk of Dundas House lorded it over the manicured grounds – a massive Brideshead Revisited tribute act, covered with pillars and twiddly carved bits – caught in the glare of a dozen spotlights, making it glow beneath the dark sky.
But that was nothing compared to the light show on the opposite side of the river.
Montgomery Park was lit up like a Ferris wheel. Marquees bright as lightbulbs. That looming spider dirigible. Spotlights raking the low grey cloud, the beams solid in the downpour. The pulsing thump and rumble of drum and bass pulsing out across the water like a giant heartbeat.
'Which way?'
Callum pointed.
A line of blue-and-white 'Police' tape turned and whirled in the rain, blocking off the path a hundred yards further on, where the ground fell away towards the river.
They ducked under it and picked their way down the damp stairs to another gravel path, this one bordered by a knee-high stone wall. Probably there to stop the dog walkers and joggers from tumbling down the six feet of muddy bank and into the Kings River.
An aluminium ladder was tied to a couple of metal cleats sticking out the far side of the wall.
Callum peered over the edge.
Down at the bottom of the ladder, two SEB technicians in blue Tyvek Smurf suits squatted beside the broken-ragdoll figure of a man. He'd lost his T-shirt somewhere along the way and gained a deep gash across his back, but the faded prison tats on his arms and wrists were all the ID needed. It was 'Tod' from Strummuir Smokehouse, skin all pale and blotchy in the fading light, face buried in the mud of the river bank.
'Hrmmm...' Franklin's face puckered. 'That him?'
'Yup.' Callum stuck two fingers in his mouth and battered out a harsh whistle.
One of the techs turned and looked up, face completely hidden by the facemask and safety goggles. 'What?'
'Have you gone through his pockets yet?'
'One smartphone, deceased. One wallet full of papier-mâché receipts, two soggy fivers, a couple of sodden business cards, and an Irn-Bru-flavour condom. One handkerchief. A pound eighty-six in change. And a set of keys.'
Callum turned and grinned at Franklin. 'And you know what keys mean, don't you?'
He pulled out his phone and called Mother.
Bellfield Road stretched away into the distance, a long straight street of three-storey terraced granite. No front gardens, just a slab of pavement in front of the slab-faced buildings. A wee shop on the corner was boarded up, tentacles of black soot reaching out across the grey stone. The corner opposite was an aromatherapist's with bars on the windows.
And three doors down, in a block acned in satellite dishes, was number 39. It was one of the few buildings with an attic conversion – an ugly Dutch-barn-style lump of black slate stitched to the top floor. Dirty windows in dirt-streaked UPVC surrounds.
An intercom unit hung by a couple of brightly coloured wires, but the flat numbers on the panel still had names attached. 'TOD MONAGHAN' was printed in green ink next to 'TOP FLOOR LEFT'.
Callum struggled his way into a blue nitrile glove, not easy with one hand in a cast, and slipped the keys out of their evidence bag. 'We ready?'
Mother and Franklin nodded from beneath the Crimestoppers brolly. Standing off to one side, McAdams grunted, rain thumping on the wide brim of his brown leather hat. His face was paler than usual, the lines deeper across his forehead and chin. As if yesterday's chemotherapy had been carved there with a Stanley knife.
'OK.' Keys one through three didn't work, but number four did.
Inside, the building didn't live up to its grim exterior. Instead it was painted a cheery shade of sandy yellow. Bright hall lights in fake-Tiffany lampshades. Pot plants curling out across the landing windowsills.
Callum led the way up the stairs.
By the time they reached the top floor, McAdams was puffing and wheezing, one hand pressed against his stomach. Face pale and shiny.
Top Floor Left had a green door with a welcome mat out front and a potted lily growing in a stand.
Callum knocked.
Mother rubbed McAdams' back as he hunched over. 'There, there. It's OK. You just catch your breath.'
Another knock.
Still nothing.
So Callum went through the keys again. Number Three unlocked the green door with a _click_.
He eased it open with his gloved hand.
Gloom.
'Boss?'
She shook her head. 'You and Rosalind can go first. Andy and me – we'll wait here for a minute. Rest our old bones.'
He stepped over the threshold.
A small irregular-shaped hallway with a coat rack by the door. Leather jacket. Parka with furry hood. Dog lead hanging like a noose.
'Hello?'
A door led off to the right: bedroom. Shrouded in shadow, but clean, tidy, bed made.
Franklin looked back, over her shoulder. 'He shouldn't be out here.'
'Who shouldn't?'
Next door: a galley kitchen, barely wide enough to turn around in. Mugs hanging from hooks beneath the wall units. Plates, bowls, and glasses lined up in the dishrack on the draining board. Spotless cooker.
'DS McAdams. Why haven't they _forced_ him to go off on the sick? He's clearly not coping. Should be at home, or in hospital.'
'He doesn't _want_ to go home; he wants to make a difference before he dies. What's wrong with that?'
Straight ahead: bathroom. Dancing penguins on the shower curtain. Wooden toilet seat. Splodge-free mirror on the medicine cabinet. The sweet lemony scent of bathroom cleaner.
'He needs help. Look at him. How is that healthy?'
Callum glanced down the hall towards the landing. McAdams was still bent double, Mother rubbing his back and talking in a voice too low to make out.
'What are they supposed to do, suspend him? Even if they say they're doing it for health reasons, it'd be a PR disaster: "Police Scotland sack brave cancer hero!"'
'He's going to _die_ here.'
Probably.
One door left. It swung open on a living room.
Oh...
The rest of the flat might have been immaculate, but the living room? Not so much.
Franklin squeezed past him. 'Bloody hell.'
One window gave a view across the rain-slicked rooftops to the vast steel and concrete bird's nest of the City Stadium. The setting sun turned everything to fire and darkness as it burned its way through the gap between clouds and earth. Painting the living room in warm shades of bronze and amber.
The other window didn't give any view at all – it was completely covered in hardcore pornography. Sheets and sheets of it, Sellotaped to the glass. The couch was pushed back against one wall, clearing a space in the middle of the room for a wooden coffee table covered in plastic sheeting. And right in the middle of _that_ , a mahogany-coloured body, curled up on its side, hands against its chest, knees against them, head bent forward at an impossible angle so the face was hidden.
The air was thick with the cosy enveloping smell of wood smoke.
Callum licked his lips. 'Yeah...'
Franklin puffed out a breath. 'Bloody hell. It _was_ him. Tod Monaghan was Imhotep.'
And now he was dead, washed up on the river bank, facedown in the mud.
It was over.
No one else was going to die.
##
Ashlee's head made a dull ringing noise as it thumped back into the metal tank. She sniffed. Blinked. Stared up into the darkness. 'And I'm sorry, Billy. I'm sorry I made fun of your lisp in Mrs Roslin's class. I'm totally, _totally_ sorry.'
The rats were asleep again, but they'd hollowed her out. Now the only things left were the jabbing pain in her stomach and the throbbing fire in her skull. It washed against the back of her eyes, like waves on a pebble beach. _Hissing_. In and out. And in. And out. And in. And out...
'And I'm sorry, Mr Khan. I'm sorry I used to steal Mars Bars from your shop. I'm sorry, I was stupid, and I'm sorry.'
There weren't many tears left, and the words were getting more and more difficult to make with her scorched-earth mouth and sandpaper throat. A tongue like a strip of cork matting, like they used to have in the kitchen before Dad ripped it out. Before he ripped _everything_ out.
The words hurt, but what else could she do?
Dying.
All alone.
Here in this tub of manky water that she can't drink without being sick.
Dying.
In the dark.
There was nothing left, but to say sorry. Sorry for every horrible thing she'd done in her thirteen long horrible years on this cold horrible earth.
Saying it over and over. Grinding through her life: full of lies and petty hurts and jealousies and spite and spit and cruelty. Again and again. Remembering new horrors with every repetition.
Like binge-watching the worst box-set ever.
'And I'm sorry, Marline. I'm sorry for everything.' She scraped in a deep, gritty breath. 'I'm sorry for not being a better friend. I'm sorry for saying you looked like a fat minger all those times. I'm sorry for stealing your lunch money in primary seven. I'm sorry for breaking your hair-straighteners and blaming it on Sarah MacIver. I'm sorry I told everyone you had herpes in second year when it was just a cold sore. I'm sorry for snogging Peter and saying I didn't. I'm sorry for _shagging_ Peter and saying I didn't.'
A little laugh turned into a sob. 'He was crap, by the way.'
The darkness blurred, and when she blinked, the shapes lurking out there refused to come back into focus. Her mother was little more than a fuzzy blob, slumped against the wall, chain tight around her neck.
'And... and I'm sorry, Mummy. I'm completely utterly sorry. I was so horrible to you and you didn't deserve it and I'm a horrible bitch and I lied all those times and I stole and I cheated and...' Her breath rattled like a half-empty cereal box. 'And I let him _in!_ Oh God...' The metal tub rang with the sound of her slamming her head back against it. Once. Twice. Three times. The chain around her neck rattling and clanking. Smashing her head back harder and harder till it drowned out the burning waves inside her brain. 'I let him in and now you're dead and I'm _sorry_. I'm sorry, Mummy. It's all my fault...'
But her mother couldn't forgive her, because her mother was dead.
So Ashlee waited till the sobbing stopped, and the stabbing pains in her stomach faded to a muffled scream, then went back to the start.
'I'm... I'm sorry, Mrs Buchan, for... for stealing money from your purse when you babysitted me...'
Because what else could she do?
##
Callum's phone _finally_ went to voicemail. He gave the old man in Flat Six a pained smile. 'Sorry about that.'
Heat pervaded the living room, throbbing out of the fake coal fire beneath the fake mantelpiece. A single standard lamp glowed in the corner, fighting against the dark rainy night and losing. Not one picture or photo on the wall. No books on the shelves, just a collection of dusty porcelain cat ornaments.
The old man's shoulders rose beneath the threadbare cardigan. 'Don't matter.'
Curled up in his lap, an evil-looking ginger cat scowled out at Callum: _You're not welcome here, this is_ mine.
'And you don't remember seeing any visitors, or anything like that?' He crossed to the window. Good view of the street from here, even at this time of night. Tod Monaghan's building was directly opposite, but this side didn't have an attic conversion, so Callum had to crane his neck up to see the flat. Even from this close, it was impossible to tell that one window was clarted in hardcore porn. Doubtful the old guy could have seen anything more than shadows on the ceiling from here. Like the ones playing across it now; hunched figures thrown into stark silhouettes by the occasional hard-white burst of a camera flash gun.
Two patrol cars blocked this bit of Bellfield Road, one at each junction, their blue-and-whites spinning solid bars through the rain. A dirty Transit van sat outside the building across the road, a slow progression of SOC techs in Smurf suits making their way in and out again. All caught in the streetlights' sickly yellow glow.
The old boy scrunched his face up and let out a huge sneeze. 'Urgh...' He wiped his nose with his fingers, then rubbed them dry on his cardigan. 'No. No visitors. Keep ourselves to ourselves, don't we, Tannhäuser?'
The cat gave Callum another stare. _We hate you: go away._
'Right, well, if you remember anything: give me a call, OK?' Callum left a Police Scotland business card on the mantelpiece and let himself out. Stood on the drab grey landing and pulled out his phone. Checked his call history. 'HOME ~ INCOMING, TODAY 21:05' was right there at the top of the list.
Lovely.
He pressed the button. Listened to it ring. Then Poncy Bloody Powel's voice sounded in his ear.
_'Callum. We said eight o'clock. It's gone nine.'_
'Oh I'm _sorry_ , is our serial-killer investigation interfering with your evening? Because while you're sat on your backside, on _my_ couch, I'm doing door-to-doors here.'
_'Callum, we agreed.'_
'No, _you_ agreed. I'll get there when I get there.' He hung up on Powel and called Mother instead. 'Nothing doing. A whole building full of people and not one of them saw a single thing. Ever.'
_'Ah well, it was worth a try. Thanks, Callum. Get yourself back to the Mobile Incident Centre and we'll call it a day.'_
'Will do.' He thumped down the bare stone stairs and out through the front door. Into the rain.
The Mobile Incident Centre – AKA: DS McAdams' red Shogun – was parked not fifteen feet away, its paint turned the colour of dried blood by the streetlight, exhaust curling up into the cold damp air.
Callum hurried across the road and scooted into the back. 'Any luck?'
Franklin had the other half of the seat, Crimestoppers brolly wedged between her knees. 'Kingdom of the blind. Never seen so many people who never look out of their windows.'
'Welcome to Oldcastle.' Sitting in the passenger seat, Mother rummaged in a small paper bag, then passed it back, chewing around the words: 'Could be our Tod was just very good at not being noticed?'
Franklin helped herself, then tossed the bag to Callum. 'I know this might sound a little odd, but I was expecting a bit more... I don't know, drama? It's never like this in the movies, is it? There's meant to be a big high-octane showdown when you catch a serial killer.'
McAdams turned in the driver's seat. 'Gwyneth Paltrow's head in a box.'
Mother's eyes bugged, then she hit him on the arm. 'Andy! Callum's mum...'
'Oh, yes. Indeed. Sorry, Constable MacGregor.' As if bringing up severed heads was nothing more than a minor faux pas right now.
Callum glared at him.
Mother hit him again. 'Andy, apologise _properly_.'
Sigh. Then a nod. 'Constable MacGregor: I'll admit that I enjoy winding you up, but I would never _ever_ joke about someone's dead mother. It was thoughtless of me and I'm genuinely sorry if my comment upset you.' At least he sounded sincere.
Callum shrugged. 'Fine.'
'Good. Now, where were we? Ah yes, the story's emotional climax. Let's make it... "Jodie Foster being hunted in a darkened basement" instead.'
Franklin glanced at Callum for a moment, then back to McAdams. 'Don't get me wrong, I'm glad we caught him, but it feels like a bit of an anti-climax.'
'I understand. You wanted more drama.' McAdams shook his head. 'My dearest Franklin. Real life is never so bold. Death is a dull thing.'
Callum plucked a green jelly baby from the bag and sent it to its doom. 'Don't knock "dull". When it comes to serial killers, "exciting" is to be avoided at all costs.' He liberated an orange one and passed the bag to McAdams. Peace offering. 'Do you remember Ian Zouroudi?'
'Gah...' McAdams' shudder must've been contagious, because Mother caught it too.
'Changing the subject,' she retrieved her jelly babies, 'we should go out tonight and celebrate.'
Franklin scooted forward in her seat. 'Karaoke?'
Mother smiled. 'There's none as fervent as the recent convert. But I don't see why not, if...' She pointed through the windscreen. 'Clap hands, here comes Cecelia.'
A figure in full Smurf had stepped out of number 39 and peeled back her hood. Now she stood in the rain with her face to the clouds, little tendrils of steam rising from her damp black hair.
Mother poked McAdams. 'Give her a toot.'
He did, leaning on the horn just long enough to make her start. Stare. Then disappear into the filthy Transit van for a moment. When she returned she was clutching a little red umbrella in one hand and an evidence bag in the other. She wandered over and knocked on McAdams' window.
He buzzed it down. 'Four bargain buckets, three with corn-on-the-cob, one with beans, and a Diet Coke, please.'
'Oh. Ha. Haha. Oh.' Her face barely moved. 'Is this you practising your kerb crawling, Andy, or could you just not live without me?'
Mother leaned across the handbrake, smiling up at her. 'Sorry. Just wanted to know if you'd found anything. You know, significant?'
'What, other than the mummified corpse on the coffee table?'
'Hopefully.'
'Well, we've got a number of small ziplock bags full of mushrooms from the fridge _and_ freezer that look pretty damn magic to me. And when we took off the bath's U-bend the thing was full of dark liquid with herbs and wee bits of bark and stick floating in it. Sound familiar?'
'Very.'
'And we found _these_.' She produced the evidence bag from behind her back with two watches, and an assortment of piercings, plugs, and rings in it. 'They were in a shoebox under the bed.'
McAdams held his hand out, as if his drive-through order had actually arrived. Cecelia passed it over and he peered at the contents. 'Looks like serial-killer trophies to me. This big leathery monstrosity, unless I'm hallucinating again, was Ben Harrington's watch.' McAdams held the evidence bag out in the middle of the car, like Mother's sweeties. 'Anyone recognise anything else?'
Difficult to tell, but then one piercing looked very like another.
Franklin pointed. 'I think the red-and-white flesh tunnel might be Glen Carmichael's.'
Callum raised an eyebrow. 'Flesh tunnel?'
'Don't be filthy. It's what they call those hollow tubes they stretch their earlobes with. Glen Carmichael had a red-and-white one with a skull-and-crossbones in it.'
McAdams held the evidence bag up to the car's inner light. 'And there it is. Ladies and gentlemen, our case officially has an airtight lid. The novel is finished. It's time for the epilogue.'
'Cecelia?' Mother gave her a little wave. 'Tell your little friends that we'll be celebrating tonight at the Dumbarton Arms. There _will_ be karaoke, but the first round's on me.'
She took the evidence bag back. 'Deal.'
The sound of fingers battering away at keyboards filled the little office, everyone hunched over their computer, the printer in the corner making whirring clunks every now and then to break the monotonous clicking.
Paperwork, that was another thing they never mentioned on the recruitment posters. Join Police Scotland: spend half your time filling in forms and the other half being covered in other people's sick.
It wasn't easy, typing one-handed, but Callum finished off his final door-to-door report and sent it to the printer.
He powered down his computer, stood, stretched. 'And with that, our handsome protagonist was first to finish.'
Dotty swore.
Franklin and Watt just kept on typing.
'Still all to play for, Dot: silver and bronze aren't as good as gold, but at least you'd be on the winner's podium. It's...' He cleared his throat and nodded. 'Detective Superintendent.'
The woman standing in the doorway had a full-on Kingsmeath facelift – blonde curly hair scraped back from her forehead and imprisoned in an unruly bun at the back of her head. Her grey eyes narrowed as she tilted her head to one side and stared at Callum. 'And the _reason_ you came to an undignified halt just now, Detective Constable MacGregor?'
He forced a smile. 'Just finished my last door-to-door report, ma'am.'
'And we thought we'd rub that in our colleagues' faces, did we?'
The tips of his ears went very warm. 'Yes, ma'am. Sorry, ma'am.'
'Good. Serves them right for being slowcoaches.' Detective Superintendent Ness held up a hand. 'All right, everyone, hands off keyboards for a moment.'
A bit unnecessary, as the whole team had already stopped typing and turned to see how much of a bollocking Callum was going to get. But they all nodded anyway.
Ness stepped into the middle of the room and gifted them a smile. 'I've just been in with DI Malcolmson and DS McAdams, going over Operation Imhotep, and I wanted to stop by and tell you all what a great job you've done. Now I know the Divisional Investigative Support Team often gets the smelly end of the stick, but you've done yourselves, me, Police Scotland, and more importantly: the victims' families proud.'
Even Watt sat up straight at that one, looking like a cat that'd just been offered a spoonful of caviar.
'This was a swift and efficient investigation and you got the right result.' A little sideways nod. 'Yes, it's a shame Tod Monaghan managed to kill himself, cheating the families of a trial, but the important thing is that he's not going to be hurting anyone else. So well done, all of you.' She even gave them a solo round of applause.
Dotty grinned. 'We'll be happy to do it again, ma'am. You give us another killer to catch, we're there.'
'Yes. Quite. Well, the Chief Superintendent wants to add his congratulations to mine. He's putting on a big press conference tomorrow and he'll be making sure everyone knows about your invaluable contributions to solving this case.'
A pat on the back _and_ credit? There would be pigs fluttering past the office window any minute now.
'In the meantime, DI Malcolmson tells me you're all off to the Dumbarton Arms for a well-deserved drink. First one's on me, OK? OK. Good.' Ness wheeched round on her heel and marched out of the office. 'Just make sure your paperwork's up to date before you go.'
'... and one orange juice and lemonade.' McAdams lowered the glass in front of Callum as if it was full of lukewarm urine. 'Takes all sorts.'
Word must've got out about Detective Superintendent Ness paying for the first round, because the Bart was hoaching with off-duty police officers, SEB technicians, and support staff. All clamouring for their free drink. Laughing, shouting, showing off, enjoying themselves.
Made a nice change.
And at least, with the Bart being packed to the walls, Hedgehog didn't have time to corner Callum and demand payment for last night's binge session. Small mercies.
Mother clinked her wedding ring against the double whisky in her other hand, setting the glass ringing. 'All right, people, as my dear old gran used to say: "HUD YER WHEESHT A MINTIE!"'
The hubbub died down to a muted roar.
'That's better.' She held out her hand and McAdams helped her up till she was standing on her chair, looming over the crowd. 'Now, I just want to say—'
A big voice boomed across the pub. 'Thank you, DI Malcolmson, for getting this lot to quiet down for a change.'
Everyone turned to face the door, where a massive lump of a man stood in the traditional black T-shirt and epaulettes, a peaked cap tucked under his arm and a moustache on his face. He held up a hand. 'Now, I know what you're thinking: the old man's just here to make sure no one's sloping off early, but that's not the case. Not this time, anyway.' A dark rumbling chuckle.
He got a couple of sycophantic laughs from the brown-nosed members of the congregation.
Mother forced a smile. 'Ladies and gentlemen: Chief Superintendent McEwan.'
The head of O Division gave a little bow, as if he was expecting a round of applause. He didn't get one. 'Today is a proud day for Police Scotland. Operation Imhotep's excellent result just goes to show what can be done with teamwork and the right management-support from senior officers. This is the essence of modern policing in these challenging times...'
Callum's phone launched into its bland anonymous ringtone. The two officers nearest him pulled pained faces and backed away a couple of paces as he yanked it out of his pocket and hit the button. Keeping his voice low. 'What?'
_'Callum? Callum it's Bob. Bob Shannon? Hello?'_
'... grasp the nettle and forge a new alliance with the public to ensure not just policing by consent, but by active enthusiasm...'
He turned his back on the Chief Superintendent. 'Bob, hi. Sorry, I can't talk, it's a bit—'
_'I know it's late, but I think I've got an ID for your public-toilet paedophile.'_
'... not just a success for DI Malcolmson's Divisional Investigative Support Team, but for all of O Division...'
_'There are three or four possibles, but our most likely lad is one Gareth Pike. Mate of mine in the Sex Offender Management Unit did him at least a dozen times for trying to interfere with little boys in gents' loos. Montgomery Park, Dalrymple Park, Kings Park, Camburn Heritage Centre, all the classics.'_
'... challenges when I took on the role of Chief Superintendent, but I knew that with the right people behind me, I could make a real difference to this division...'
_'Point is, Pike got fed up getting his collar felt, so he started to play away from home. Rest stops on the A90, the services at Montrose, that kind of thing. So a lay-by on the Aberdeen road would have been right up his street.'_
'... but it's important to remember that the transformation in Oldcastle _isn't_ down to the hard work and inspirational leadership of just one man...'
Callum fumbled out his notebook. 'You got an address?'
_'I can do better than that. Tell me where you are and I'll come get you.'_
'Callum? Are you OK?' Mother stood in the Bart's doorway, hands deep in her pockets, breath curling out into the rainy night.
'Yeah. Fine.' Callum tried a smile. 'Just...' He picked at the lining of his fibreglass cast – hadn't even been on a day and already the thing was getting filthy, all greyed and blotchy. Getting it soaked at regular intervals throughout the day probably wasn't helping. 'Elaine and DCI Cock-Face want me to move all my stuff out. Today.'
'Ah.' She winced. 'I heard Reece had left his wife. Didn't know it was for your Elaine. Sorry.'
'She's not _my_ anything.'
Mother nodded.
Water rushed in the gutters like filthy little rivers, washed across the paving slabs, hissed against the sign above the pub door, turned the streetlights into glowing spots – septic and angry.
A taxi grumbled past, a couple screaming at each other in the back seat.
'Callum, do you want a hand moving? It's not going to be easy with just a bicycle.'
He looked the other way. 'Actually, I thought I might borrow one of the pool cars. You know, without telling anyone.'
'Hmmm... Better take the Mondeo, then. You'll get more in the back of an estate.'
'Thanks, Boss.'
She made a tutting noise, then patted him on the back. 'Have you got anywhere to store your things?'
Ah.
There was that.
Because if Powel and Elaine thought they were keeping all the furniture and kitchen stuff he'd bought for the flat, they could carve that thought on a six-foot granite slab and shove it up their collective backsides. And there was no way it was all going to fit in Dotty's spare room. Never mind all the books.
Mother rolled her eyes. 'Men: you're sweetly pretty things, and we wouldn't be without you for the world, but your little heads just aren't suited to the practicalities of life.' She pulled a jailer's bundle of keys from her fleece pocket and worked a small Yale from one of the many rings. 'Here. My Jack has a lockup in Cowskillin: twenty-three Washington Lane, round the back of the processing plant. I'm sure it's all pornography and empty whisky bottles in there, but try not to make too much of a mess.'
Callum took the key. 'Thanks.'
A small green Toyota hatchback pulled onto the street, headlights shining back from the wet road. It parked four buildings up, outside 'DOUGIE'S "FAMOUS" CHIPPER! ~ PIZZAS KEBABS & BAKED TATTIES TOO'. The Toyota switched off its headlights and sat there with the engine running.
'And if you finish before midnight, come back and have a _proper_ drink. No more of that orange-juice-and-lemonade nonsense. OK?'
'OK.'
'Good.' Another pat. 'Better get going: I'm doing "Somethin' Stupid" with Andy in a minute.' She headed back inside. And thirty seconds later, the car parked outside the chip shop flashed its lights and pulled away from the kerb.
Slid to a halt outside the Dumbarton Arms, right in front of Callum.
The driver's window buzzed down, filling the night with Lennon, McCartney, and Harrison imploring a postal worker not to sod off without checking their bag again.
Callum hunkered down with his hands on his knees and peered into the car. 'Can I help you?'
Ex-DS Bob Shannon smiled out at him, voice raised over the Beatles. 'Detective Constable MacGregor. You and I have an appointment with a child molester.'
##
Shannon paused for a moment at the top of the stairs, face flushed and shining. 'Urgh. The _smell_... Why does everyone have to pee in the stairwell? Do they not have toilets here?' He mopped his forehead with a green-and-yellow scarf.
Callum lurched to a halt next to him, breathing hard. Thirteen floors of climbing through the eye-watering reek of other people's urine and his throat burned. The air even tasted of it: sharp and bitter. 'He better be in after this.'
Faulkner Heights had to be the mankiest of the seven tower blocks that enclosed this side of the Blackburn Roundabout. Oh, it didn't _look_ manky from the dual carriageway, or the library, because the council had painted the sides of the building that faced that way. But they'd left the other two sides as dirt-streaked concrete, with all the windows boarded up on the bottom three floors – about as high as a wee scroat could chuck a rock.
They hadn't bothered painting the inside either. Or dousing it with disinfectant. Though, to be honest, tearing the place down and burying the rubble was probably the only hygienic option.
A pile of bin-bags sat by the lift – beneath the 'OUT OF ORDER' sign – leaking rancid brown liquid across the floor. A dull yellow stain marked the wall in the corner, flowing down across pale crystalline growths to the ground – the burning stink of fossilised piddle mingling with the bin-bag's gritty stench.
Graffiti scrawled across the walls: generations of the abandoned, marking their territory in a slightly more permanent way than by peeing on it.
Shannon bared his teeth. 'About time they pulled this block down and stuck up something nicer. Like a crematorium. Or an abattoir.' He pointed at Callum. 'Before we do this, we need some ground rules.'
Callum wiped his good hand across his damp forehead. 'Go on then.'
'One: I know you want to kill this guy, but you don't. Agreed? No beating the living hell out of him, no breaking his fingers, not so much as a Chinese burn.'
'He—'
'No. That _can't_ happen. We're police officers, or at least I used to be, and that means something. If he's the one who killed your mother, he goes to prison for a very, _very_ long time. He doesn't walk free because you played "Batter the Suspect".'
Callum blew out his cheeks. 'Fine.'
'Two: we're going to Good Cop, Bad Cop it, and you're playing the good cop.'
Seriously? 'Come on, you can't—'
'No. Non-negotiable. If you're playing Good Cop you're less likely to twat him one.'
Callum stared at him. Then away down the dull grey corridor with its graffitied walls. 'Agreed.'
'OK, then.' Shannon leaned on the doorbell, but nothing happened: not so much as a _bing-bong_ from inside. So he drew his fist back and gave the door three loud hard knocks. The kind that let everyone know the police were outside and they were _not_ sodding happy.
Which was probably par for the course down here.
Or, strictly speaking, as they were on the thirteenth floor, _up_ here.
Shannon gave the door another three bangs.
No one came out of the other flats for a gawp. The police hammering on someone's door had clearly lost its novelty a _long_ time ago.
Callum checked his watch. 'Maybe he's out?'
'Doubt it. From what I've heard, Pike's pretty much a shut-in these days.' Shannon rolled his shoulders. 'Think we should kick it in? I've not done that in _donkeys_.' A grin tugged his grey beard out of shape. 'I was listening to Radio Four the other day and this guy from the Met said modern UPVC doors can stand up to a battering ram for over half an hour. This thing? One good boot and it's in.' He rapped his knuckles against the tatty wooden door. Someone had stolen the numbers off it, leaving just the dents in the paint to spell out '13-15'.
'We haven't got a warrant. Thought you wanted to do things by the book?'
'I used to love dunting someone's door in. It was like opening a present on Christmas Day, never knowing what you were going to get. Would it be a selection box, a pair of leather football boots, or a druggie with a shotgun?' A sigh. 'Strange, the things you miss when you retire from the force.'
Callum raised his left fist to the wood. 'One more go.'
But before he landed it, there was a _thunk_ from inside. Followed by a light flickering on behind the spyhole.
Then a gravelly voice, dark as gravy, oozed through the wood. Unmistakable, even after all these years. _'Who is it?'_
Heat crawled up Callum's back, spreading pins-and-needles across his shoulders. He swallowed, then held his warrant card up to the spyhole. 'Mr Pike? We need a word.'
_'Ah. I see. Well, I'm sorry, but it's not convenient right now. Not convenient at all. I'm... indisposed.'_
Shannon raised his voice a bit. 'Let's kick it in. Right now. Boot the door right off its hinges.'
_'No! No, don't do that. I... I_ need _my door. Yes. I need it. Please don't do that.'_
'Then open up, _Gareth_.'
A rattling sigh. _'Today doesn't seem to be turning out as well as I'd hoped.'_ The clattering sound of a chain drawing back was followed by several loud clicks. Then the door swung open.
The man standing in the doorway was stooped over, but his large bald head still came within an inch of the frame. Big. Wide. Just like he'd been in the car park, only a lot thicker around the middle. And instead of the black overcoat he was dressed in a stained silk robe, socks, Crocks, an 'I ♥ OLDCASTLE!' T-shirt spattered with what _looked_ like brown sauce, and a pair of polka-dot boxer...
Gah!
Callum recoiled a step. So did Shannon.
Gareth Pike smiled, showing off perfect white teeth. Raised his bushy white eyebrows. 'I did mention I was indisposed.' The front of his underwear was tented out, pointing straight ahead at ninety degrees. 'Now do you gentlemen still wish to come in?'
Pike waved a hand at the couch. 'Sit. Sit.'
No chance.
Piles of newspaper and ready-meal containers teetered against the walls. The carpet, if there was one, lay buried beneath a thick layer of hair and filth. Water stains on the ceiling and walls. A drift of scrunched-up tissues spread out from the couch in a fan shape: all crunchy and flaky. Their faint bleachy smell mingled with the rancid odour of a man who'd clearly fallen out with soap and deodorant.
The only picture on the wall was Mary, cradling the baby Jesus.
An old-fashioned cathode-ray TV was hooked up to a video player, the screen flickering with a close-up of Simon Cowell's sneering face.
Pike lowered himself into a greasy armchair, legs spread wide, tent on full display. 'Oooh... My knees. Not what they once were.'
Hard to believe that this was the man who'd given him nightmares for years.
Callum crossed his arms, just in case his hands accidentally touched something. 'You don't remember me, do you?'
Pike's face creased for a moment. Then, 'No.'
Shannon went to sit, but obviously thought better of it. 'Twenty-six years ago, Gareth. You were interfering with little boys. Public lavatories, mostly. Parks. Service stations. Lay-bys. Winter, summer, autumn, spring: there you'd be, fiddling away. A nonce for all seasons.'
A big fat shrug sent his chins wobbling. 'A wise man once said, "To thine own self be true." So: yes. I admit it. But I've atoned for my sins. Repaid my debt to society with ten years attending to Her Majesty's pleasure at Peterhead Prison.' His hand reached over the brow of his belly and scratched at the boxer shorts, making the tent-pole wobble. 'Oh, this was back when it was a _specialised_ institution, not the modern monstrosity they have now where they'll take just anyone. No, it was a lovely establishment. Loads of character. And _characters_ , too. I remember one time, I was taking tea with a kindred spirit called Haroun and who should walk in but—'
'Twenty-six years ago.' Callum took a step closer. 'It was a lay-by on the Aberdeen road, not far outside Blackwall Hill. A family of four: mother, father, two wee boys.'
Pike raised his bushy eyebrows again. 'Sounds delicious. Were the boys blond and pretty? I _like_ them blond and pretty.'
'I'll blond and—'
Shannon grabbed his arm. 'Good Cop, remember?'
Callum closed his eyes. Took a deep breath of stench. 'It was April. I was five and you tried to touch me in the toilets.'
'Ooh.' Pike squeezed himself through the polka dots. 'You were one of _mine_? You must have been _very_ pretty when you were younger.'
'Two men came into the toilets and scared you off.'
'How rude of them.'
'My mother and father and my brother were waiting for me in the car outside. With a caravan on the back. Do you remember _them_?'
'Mmmm...'
'I swear to God, if you don't stop touching yourself I'm going to—'
'Come on, Callum.' Shannon shook his head. 'We agreed.'
'DO YOU REMEMBER?'
Pike didn't even flinch. 'I remember all my pretty little blond boys.' He squeezed. And squeezed. 'They're all I have left.'
Shannon pulled Callum back. 'All right. Let's just calm down a minute, OK?'
'Ask him. Ask him if he remembers hacking off my mother's head!'
Pike let go of himself and sank back in his seat. 'Oh, no. I would certainly remember something like that.' He raised his squeezing hand to his nose and sniffed the fingers. 'I don't do "mothers". Or fathers, come to that. Grown-ups in general repulse me.' A finger came round to point at Simon Cowell's face on the TV. 'I wasn't sitting here wasting a perfectly good Viagra on him. I mean, I'm not _gay_ or anything. I just didn't think you'd like to see what I'd _really_ been watching. Hmmm?' He licked his lips, dark and slimy like twin slugs circling his mouth. 'It's amazing what one can purchase over the interwebs if one is... discreet.'
Callum pulled his arm free from Shannon's grip. 'Are you saying you didn't kill her?'
'Well of course not. Why on earth would I _do_ something like that? Can you imagine how messy it would be? No: I'm a lover, not a fighter.' He gave a little shudder. 'But I do remember _you_. You ran off and hid in the caravan, didn't you? You wouldn't come out to play.'
'You said my parents didn't love me any more. That they'd given me to you.'
'Oh we could've had such fun, you and I. Before you got so _old_ and _unappealing_.' Sigh. 'I waited so long in the rain for you. I hope you didn't mind my taking care of myself while I waited. But you wouldn't come out and help.'
Shannon curled his top lip. 'Did you see anything? Anyone else at the toilets?'
'Would you like to see the video I was watching? It's very good. I know everyone's obsessed with digital technology, but I'm not supposed to have a computer. I _do_ , of course, but there's something so deliciously nostalgic about videotape.' His eyes widened. 'The way it flickers and buzzes. Oh, my...'
'Did you see anyone else, or didn't you?'
'Sometimes I like to twist the tracking all out of line, so it's like the old days when a tape's been copied and passed around and copied again and again and again. You don't get that sense of tradition with these modern digital films.'
Callum unfolded his arms and balled his left hand into a fist. 'Answer the sodding question, or I'll break the tracking in your _head_.'
'Oh yes. I saw someone. I saw things you wouldn't believe, right there in that car park, all those years ago.' Pike fluttered his eyelashes. 'But it'll cost you.'
Shannon grabbed Callum's arm again. 'He's just screwing with you. It's what people like him do.'
'Oh, no, no, no. All I'm asking is that you sit down and watch my video with me. Me and the Boy Who Got Away, _watching_.'
'OK, that's it, I'm going to kill him.'
Shannon tightened his grip. Leaned in close and whispered into Callum's ear. 'He puts on the video, you do him for possession of indecent images of children, or whatever they're calling it these days. It's not an illegal search, because he's shown you it voluntarily.'
'Well, my once-pretty-little-blond-haired-boy?'
Grab the greasy fat slug by the throat and squeeze. Dig both thumbs into his windpipe and squeeze till his eyes bulge and his face goes purple. Squeeze till he judders and gurgles and dies.
And never find out what happened.
Callum uncurled his fists. Forced it all back down again. 'Go on then.'
Pike clapped his hands. 'Oh how jolly!' He knelt in front of the video player and ejected Simon Cowell. Replaced him with a tape in a grey cassette. Collapsed back in his seat and grinned. 'You see, I was so angry when those ruffians burst into the toilets and spoiled everything I was going to slash their tyres. Serve them right.' He dug a remote control out from between his seat cushion and the arm of the chair. Pointed it at the TV and pressed play. 'But when I got out, another pretty little blond boy was right _there_. I thought it was you for a moment, but he was wearing a different T-shirt.'
A wall of flickering pink filled the screen.
'So I thought, well, today's going to end delightfully after all. Look what the baby Jesus has sent me.'
Callum turned his face away from the TV. Tried not to listen to the muffled sobs and pleas.
'"But," said I to myself, "where are this _delicious_ little offering's parents?" Here's a lovely boy, on his own, in a car park. Surely Mummy and Daddy can't be far away.'
Shannon had his teeth bared, a look of utter disgust slithering its way across his face.
'And then I saw them. Mummy and Daddy. And... Have you ever seen a wildlife programme when a lion meets the hyenas? Here's this magnificent apex predator, with his beautiful blond mane, king of all he surveys, and there's the hyenas. Dirty, squat, and evil. They're not noble like he is, but there's more of them than there are of him so they can chase him off and steal his prey.'
The Slug pouted, eyes fixed on the screen. 'I always thought _I_ was the lion. The noble apex predator. But standing in the car park – watching him beat that man and woman with an iron bar, bind them with duct tape, and bundle them into the boot of his beautiful white Range Rover – I realised I was never the lion. I was the hyena. And there was only one of me.'
Callum cleared his throat. 'Who was he?'
'So I hid in the shadows and watched him take his prey. I watched him scoop up the pretty little blond boy, lower him gently onto the back seat, ruffle his hair, and strap him in. Then I watched the Lion drive away.'
'Who – was – he?'
A dark, slithery smile. 'Oh, you'd know him if you saw him. He's _famous_.' Pike leaned forward, trembling, the images on screen reflected in his bloodshot eyes. 'Isn't it magnificent?'
'I want a sodding name!'
'Shut up. Don't spoil it. This is the best bit.'
Callum grabbed the TV, wrenched it off its stand, and hurled it to the filthy floor. The cathode ray tube popped and crackled, sparks flickered inside the casing, smoke curled up through the vents in the back.
Shannon just stared at him.
'Gareth Pike, I'm arresting you for violation of Section Fifty-Two-A of the Civic Government Scotland Act, as I believe you to be in possession of indecent images of children. You do not have to say anything, but it may harm your defence if you do not mention something that you later rely on in court.'
Pike grinned. ' _Finally_. I thought you'd never ask.'
##
'Interview recommenced at twenty-three fifteen.' Callum held up the evidence bag. 'I am now showing Mr Pike "Exhibit A". Do you recognise this video cassette, Gareth?'
Sitting on the other side of the interview room table, Pike smiled his slug-like smile. 'Of course I do. It's the video I bought from a lovely man in Doncaster, who, I'm afraid, shall have to remain nameless.' Pike raised a hand. 'Oh, I know, I know, but honour among pederasts, Detective Constable. I'd hate to disappoint the brethren of my... shall we say _distinctive_ congregation.'
Interview Room Two stank of fresh paint, the walls, door, and skirting remarkably clean and blemish free. Even the carpet tiles looked new. Bright-white vertical blinds shut out the night, swaying above the pinging radiator. All very clean and hygienic. Which somehow made the stench rolling off Pike all the more cloying.
The uniformed PC sitting next to Callum had her chair pushed as far back as it would go, putting as much distance as possible between herself and the man opposite. She had the same look of revulsion on her face as Shannon had, back at the flat. As if she'd trodden on something she couldn't wipe off. She wouldn't even rest her notebook on the same table Pike leaned on, holding it in her lap instead as she wrote down everything he said.
Just the three of them, steeping in the smell of fresh paint and stale BO.
Callum stuck the evidence bag in front of him, the dirty cast on his right hand clunking against the clean Formica tabletop. 'And you are aware of the contents of this video cassette, Gareth?'
'Oh indeed I am. Very much so.' Pike leaned forward. 'I've watched it many, many times. My favourite bit has worn almost through, but that just adds to the mystique, don't you think? The mind's eye is _so_ much more powerful than reality.' He looked up, stretching out his chins, staring straight into the camera mounted in the corner. 'It's a pornographic video involving two pretty young blond boys and one _very_ lucky man.'
Callum stared at him. 'Why?'
'Oh, because who doesn't dream of having two—'
'No, Gareth: why did you show us the video? Why didn't you hide it? Why did you choose not to have a solicitor present? Why aren't you sitting there saying, "no comment" to everything?'
He curled his shoulders forward, hunching over the table. 'You've seen my home, Detective Constable, would _you_ want to live there? I miss my lovely cell, with its regular meals and its working radiator. I miss my friends. Out here, if I talk to someone with... similar interests, I'm breaking the law, but inside? Ah, the joy of discussing my passions and past triumphs without being spat at!' He winked at the PC. 'How I long to never see another face contorted in ignorance like yours, young lady.'
She glared back, but kept her mouth shut. Wrote it all down instead.
'You _want_ to go back to prison?'
'Who wouldn't? Out here I get excrement posted through my letterbox; in there I can spend time with kindred souls and live in peace. Unmolested.' He spread his chubby fingers wide. 'Perhaps I could finally work on my novel?'
'So you're confessing to the charges.'
'It's about a little boy whose wicked stepfather beats him every night and locks him in the attic. But the stepfather doesn't know that there's a portal to a magical world up there, hidden in an old wooden chest from the First World War. And the little boy goes on adventures with his best friends – a talking cat and a world-weary teddy bear – to save Wunderwelt from the Darkening armies of King Dunkelheit.'
'Are you confessing to the _charges_ , Mr Pike?'
'Oh most certainly.' A frown. 'I can't decide if the teddy bear should harbour the soul of the boy's dear departed grandfather: killed in the trenches, I think. Maybe mustard gas. Or would that be too dark?'
Callum just stared at him.
'Of course, it's semi-autobiographical. I didn't have a talking cat or a haunted teddy bear, but I _definitely_ had a stepfather. I'll leave out the bits where he shared me with his friends, though. No one likes a tattle-tale, do they?'
Was that supposed to make him sympathetic?
Tough.
Callum held out his good hand for the PC's notebook, then slid it across the table. Passed Pike a pen. 'Sign at the bottom there. And date it.'
'I'd like a south-facing cell, if at all possible?' He scrawled his name across the bottom of the page, followed by today's date. 'Would that be possible?'
'You said you recognised the man who abducted my parents and brother.'
'I suppose it all depends on what's available, but it'd be nice to feel the sun on my bars.'
'You said he was famous.'
'Oh I said many things, Detective Constable. And now,' he poked the notebook with its signed confession, 'I've got what _I_ wanted, so why should I help _you_ with anything? Supply and demand.'
'All I want is a name.'
'I know. And all _I_ want is a south-facing cell. Something on an upper floor so there's a nice view. If you can't supply me with that, then our business here is concluded.' A wink. 'So why don't you scurry off and see what you can do about my cell? Off you go. Scurry, scurry.'
Callum snatched the notebook back and returned it to the PC. 'I can't believe I was afraid of you, all those years. You're pathetic.'
'Oh, indubitably. And now I've got power over you all over again.' The slug smile grew. 'Isn't that _delicious_?'
Shannon leaned back against the wall in the cupboard masquerading as the Downstream Monitoring Suite, mug of tea clutched to his chest. 'Told you: men like him, they like screwing with people.'
Callum closed the door behind himself and slumped into one of the three office chairs lined up in front of the monitors – the screen in the middle had a view of Interview Room Two on it, peering down from the corner at the empty chairs and table. He covered his face with his hands. 'Did you see?'
'Oh, "indubitably". Urgh... I mean, who uses words like "indubitably" these days? Dicks, _that's_ who.' A sigh. 'I should've let you play bad cop.'
'He saw him. He saw this guy, this "Lion", attacking my parents and he didn't do a thing.'
'We had twenty-six flights of stairs for him to fall down.'
'He could've called the police. Taken the number plate down. He could've _done_ something.'
'I know.' Shannon's hand landed on Callum's shoulder and squeezed. 'Look on the bright side: at least now we know the rumour was true. That drunken DI was right, it was someone famous. And they drove a white Range Rover.'
'I was terrified of that lardy sack of crap...'
'Callum, it's OK. I'll get my OAPs to go digging through their notebooks and attics and sheds. We'll find out who he is.' One last squeeze. 'But it'll take a day or two. Meantime, you sod off home and get some sleep. I'll give you a call soon as we know anything.'
Callum thumped the Mondeo's door shut and stood on the pavement, in the drizzle, staring up at the third floor. Twenty to twelve and the lights were still on in the flat.
Home.
Or what used to pass for it.
He pulled his shoulders back and let himself in through the communal front door. Ignored the pile of post on the windowsill at the back. Marched up the stairs.
The cats had been at Toby's pot plants again.
Tough.
Callum took out his key and slid it into the lock of 3F-A. It didn't turn. And the little brass plaque above the letterbox was gone too, replaced by a white plastic rectangle with 'R POWEL & E PIRIE' carved into the surface. They hadn't even waited till his grave was cold...
So he bunched his left hand into a fist and gave the door the same three hard knocks Shannon had given Gareth Bloody Pike. The police are here, and they're not sodding happy.
Took a while, but eventually the door swung open and there was Powel, in jeans and a Rolling Stones T-shirt. Big white trainers that looked as if they'd never seen the outside world. He didn't seem as intimidating out of a suit, more like someone's dad trying to be trendy and 'down with the kids'. And failing.
He scowled out at Callum. 'I've been _waiting_.'
'Where's my stuff?' Pitching it as a challenge, rather than a question.
Powel closed his eyes and shook his head, then turned and marched back into the flat on his ridiculous white trainers. 'Elaine packed everything into boxes.'
'I'll bet she did.' Callum followed him. 'And did she pack the TV and the couch and the bed and the microwave and—'
'Oh for God's sake, Callum, will you grow up?' Powel turned, arms out. 'Yes, OK, I get it: you've been betrayed. We hurt your feelings. Everything's terrible and it's all my fault.'
Callum's left hand curled into a fist. Chest out. Shoulders back.
'Does that make you feel better, Callum? I admit it: it's – all – _my_ – fault.'
Grab him by the throat and squeeze the life out of him.
'But do you think you were so easy to live with? Do you think Elaine didn't struggle every day, with your moods and your obsessions and your _neediness_?'
Kill him.
'We fell in love, OK, Callum? We reached out for someone and we found each other.'
Kill him right _now_.
Powel's arms dropped to his sides. 'She didn't love you, Callum. She was just going through the motions because she didn't want to hurt you. It wasn't a conspiracy, it just happened.' He walked through to the living room. Pointed at the cardboard boxes stacked up by the window. 'I know it doesn't help, but I'm sorry.'
They'd obviously raided the nearest supermarket, because the pile was a mixture of small boxes that used to contain wine, big boxes that used to contain frozen chips, boxes for toilet cleaner, crisps, bin-bags, cauliflower florets, and Stork vegetable fat. Each one sealed with brown parcel tape and marked with black pen: 'CLOTHES', 'CDS & DVDS', 'LEGO', and 'MISC'.
But by far the largest number were marked 'BOOKS'.
'Elaine packed your favourite mugs and cookery things. There's some ornaments in there too, and photos of the two of you. She says, if you don't want them just let her know. Don't throw them away: she'd like to hold onto them for old time's sake.'
There it was, his whole life for the last five years, all neatly packed up in scrounged cardboard boxes.
Callum stared at the floor. 'What about the furniture, the TV, the _crib_? All the stuff I paid for?'
A sigh. 'If I write you a cheque, will that make you happy?'
'Happy?'
There was a lamp, sitting on the empty bookcase at the back of the room. They'd bought that on a weekend away in Anstruther. Back before she'd got pregnant.
He picked it up, turned it over in his good hand.
Powel folded his arms. 'And we'll need to sort something out about the flat. Putting it up for sale isn't going to do much good, not with the market like it is.'
Pale-brown pottery, the colour of a hen's egg. A little scene of boats and dinky wee houses wrapped around it. Heavier than it looked. Seagulls on the blue shade. They'd been happy then.
'As I see it, we've got two options: I refund your mortgage payments and we get the title deeds transferred into my and Elaine's names, or we buy you out at the _current_ market value and you pay off half the mortgage.'
Or maybe they hadn't been happy at all. Maybe _he_ 'd been happy, but Elaine was miserable. Maybe she was already shagging Powel behind his back. The pair of them laughing at how stupid he was.
'Though, if I were you I'd go for the first one. The market being what it is, you'd probably end up losing out on the deal. At least if you take the cash you'll get something out of it.'
Poor stupid gullible little Callum. Buying twee lamps, when everything around him was _lies_.
'What do you say, have we got a deal? Like adults?' Powel stuck out his hand for shaking.
Callum stared at it, then at the lamp.
Bared his teeth.
Slammed the lamp back down, grabbed the nearest box, and marched out of the flat.
##
Callum loaded the last box of books into the back of the Mondeo. Looked up at flat 3F-1.
Standing out here, you'd never guess—
Sodding hell. Callum pulled out his phone. 'What?'
Nothing.
He checked the caller display: 'NUMBER WITHHELD'.
Not this again.
'Look, whoever you are, I'm not in the mood, OK? I've had a crappy day, so you can take your phone and jam it up—'
_'Piggy?'_ A little girl's voice, broken and jagged. Her breathing jerky and trembling, punctuated by damp gurgling sniffs.
'Willow?'
_'He's here! He's... he's come... he's come back.'_
Callum closed the Mondeo's boot. 'Who's come back?'
_'Dad. Dad's come back...'_
The man who'd broken his four-year-old daughter's arm as a farewell present.
Right.
Callum marched around to the front of the car and climbed in behind the wheel. 'Where are you?'
_'He's in there with Mum and Pinky and the baby!'_
'OK. You stay away from him. I'll be there soon as I can.' Callum turned the key and put the Mondeo into drive. Stuck his foot down. The fingers of his good hand reached for the '999' button mounted on the dashboard, and the car's siren wailed into the rain, blue-and-white lights flickering behind the radiator grille – reflected back by the wet road.
Callum fumbled his Airwave handset out, the thing lumpy and awkward in his broken hand, working the buttons with his thumb. 'DC MacGregor to Control, I've got report of a domestic at forty-five B Manson Avenue, Kingsmeath.'
_'Oh aye?'_
'Yes, "Oh aye". There's a grade one flag on that property, I need backup—'
_'I'm going to stop you there, Detective Constable. There's no flag on that house.'_
'I asked for one days ago!'
_'Aye, well I'm looking at the system now, and there isn't.'_
'Oh for God's sake...'
Shops and cars and cones flashed past the Mondeo's windows. Then the industrial span of the Calderwell Bridge.
'Who have you got in the area?'
_'Dawson and Cooper, but they're dealing with an assault.'_
Right at the roundabout, the tyres screeching on the wet tarmac.
'Soon as they're done, get them over to Manson Avenue.'
_'Aye, well, I'll do my best, but—'_
'But you can't promise anything. Yeah, I know.' He let go of the button. 'Thanks for nothing.'
Callum swung the car hard left onto Munro Place, tearing up the hill, over the top and down the other side. Threw the Mondeo around onto Manson Avenue.
The depressing rows of flat-faced houses with their tiny weed-strewn gardens reared up on either side, the road lined with parked cars in various stages of decay.
Thirty-nine. Forty-one. Forty-three. There: forty-five.
He jammed on the brakes and slid the car to a halt, right outside. Clambered into the rain, just in time to see a big black Mercedes disappear into the distance.
'PIGGY?' That same little girl's voice, shouting over the wailing siren, sounding so much younger than the last time they'd met.
He turned and puffed out his cheeks. 'YOU OK?'
Willow peered out at him from behind a parked VW Beetle. 'YOU CAME.'
'SAID I WOULD, DIDN'T I?' Callum reached back into the Mondeo and killed the lights and siren.
A couple of curtains twitched on the other side of the road, but other than that Manson Avenue was silent.
Callum nodded at the house. 'He still in there?'
She shook her head. Bit her top lip. 'I wasn't scared or nothing.' A sniff and a shrug. 'Just called you, you know, for _Pinky's_ sake, like. Cos Benny was worried bout her, yeah?'
'Course he was.' The small drift of plastic toys had disappeared from the front garden, leaving it to the weeds. Callum marched up the path.
The front door lay wide open.
He knocked anyway. 'Hello?'
No answer.
The hallway was cold, the wallpaper stained and peeling in the corner above the door. A selection of brightly coloured kids' coats rainbowed a rack on one side. A cracked mirror on the other, reflecting back a spider's-web kaleidoscope.
'Hello? Anyone in?' Callum turned back to Willow. 'What's your mum's name again?'
A shrug.
Yeah, because when you're that age, 'Mum' and 'Dad' was all the name they needed. Assuming you were lucky enough to still _have_ parents.
A flight of stairs sat beyond the mirror. He rested a hand on the newel post, staring up at the landing. 'Hello? Miss Brown? Are you all right?'
No answer.
A little face appeared around the balustrade: sticky-out ears and flat monkey nose. That would be the brother, Benny. AKA: Baboon Boy. Only this time there was no hooting, just a damp-eyed stare.
'Is your mum up there?'
He wiped his face on his sleeve and shook his head.
OK, search downstairs first.
One door at the end of the hall, one on the right. He tried the handle and it opened on a living room just big enough for a saggy armchair covered in throws, a small TV sitting on a tatty sideboard, a stack of kids' toys, two threadbare beanbags, and a flimsy-looking playpen covered in cartoon characters.
Willow's mum was in the corner, sitting with her back to the wall, knees drawn up to her chest, arms wrapped around them, blonde hair hanging over her face as she rocked. The toddler – Pinky? – was holding on to her, face a big flushed tear-stained knot of gristle and snot.
Callum peered into the playpen.
The baby was lying on its back, sooking one of its feet, surrounded by yet more plastic tat.
OK, so at least everyone was still alive.
He squatted down in front of the woman in the corner. 'Miss Brown? It's PC MacGregor. I was the policeman who brought back Mr Lumpylump? You remember?'
She peered out at him from behind her curtain of hair. Looked away.
Callum tried a smile. 'I came because I was worried about you. Are you OK?'
She rested her forehead on her knees, voice soft and mushy – as if she'd been drinking. 'Go way.'
'Willow and Benny's dad's been here, hasn't he?'
No reply.
'Did he hurt you?'
No reply.
'If he hurt you we can _do_ something about it.'
No reply.
Honestly, it was like interviewing a career criminal. Callum settled on the edge of the lone armchair. 'I know this is hard. It's not easy when someone you love hurts you. Trust me, I know what I'm talking about.' He stared down at the filthy fibreglass cast covering his right hand. 'But, you know what? If they hurt us this much, maybe they never really loved us at all? Maybe they don't _deserve_ to be with us. Maybe they never did.'
Willow's mum raised her head, parting that curtain of blonde hair with one hand. A deep red bruise covered one side of her cheek. Her bottom lip was swollen, split, and raw, blood making a wide trail to her chin. No wonder her words sounded slurred.
Another fledgling bruise was spreading across her other cheek. A necklace of them wrapped tight around her throat. Her eyes sparkled with tears. 'He wouldn't hurt me if I didn't make him so angry...'
'What's his name?'
'I shouldn't have had another kid without him. I should have waited for him to come back and not run around with other men like a whore. I ruined everything.'
'You didn't do anything wrong... it's Isobel, isn't it?'
'Irene.' She picked at the knee of her jeans. 'And I _did_. He told me I did.'
'Yeah, well, you didn't and he's a dick.' Callum pointed at the playpen and its foot-sucking inmate. 'You love your baby, don't you?'
A little nod.
'Well then, there you go.' Callum fumbled his notebook and pen out. 'Now, the charmer who beat you: what's his name?'
Irene Brown blinked at him, then looked away. 'I fell. I tripped and I fell. Because I'm a stupid clumsy bitch.'
'Come on, Irene, you're not—'
'Please, just... just leave me alone.'
Callum poured boiling water into the two mugs. The kitchen wasn't huge, and nothing in it looked as if it'd been bought from new, but it was clean and tidy enough.
Willow stood in the doorway, watching as he mashed the teabags against the mugs' inner walls with a teaspoon. Not saying anything as he fished them out and dumped them in the bin. Or when he got the milk out of the fridge. It wasn't till he took the lid off the semi-skimmed and had a sniff that she broke the silence: 'We're not scummers, OK, Piggy?'
'Never said you were.' Both mugs got a dollop of milk. 'Force of habit – you can't trust a pint of milk in a police station. Never know _who's_ been at it.' He took a wee sip of tea. Hot, but bearable. 'Does your mum ever mention your dad's name? Maybe she's got photographs hidden away somewhere?'
Willow rolled her eyes and stomped into the room. Picked the semi-skimmed off the worktop and stuck it back in the fridge. 'What happened to your hand?'
Callum pulled down his jacket sleeve, till it hid the filthy cast. 'You didn't answer the question: your dad's name. Photos? Anything like that?'
'Nah. Not a snitch, yeah? Mum's not a snitch neither. Benny's not fussed, but she dropped him on his head when he was wee, so you can't believe a word he says. Lives in a fantasy world, don't he?'
'This guy comes in here and beats your mum up, and you think it's more important to not be a clype? Thought you wanted to "break his little bitch legs"? Now you want to let him get away with it?'
Willow stuck her head on one side and shrugged. 'Would've kicked the crap out of him, but you know...'
'Sure you would.'
'Yeah, would've killed him right there, but he had this huge shit-eating darkie with him.'
Callum stared at her. 'You can't say things like that.'
'But see if he was on his own?' She mimed punching someone.
'Willow, I'm serious. You want people to think you're some sort of stupid racist lowlife? Because that's what it makes you sound like. You _think_ it makes you sound tough, but it doesn't.'
She clamped her mouth shut.
'Thought you were better than that.'
'You're not my dad!'
'Yeah. Because he's been _such_ a role model, hasn't he?'
Pink swept up her neck and into her cheeks, setting the tips of her ears glowing. Then she glowered at the kitchen floor for a moment, muscles bunching along her jaw, like she was chewing something. Deep breath. 'He had a huge black guy with him. All gold chains and that. He took Mr Lumpylump, cos Dad told him to.'
'They stole your mother's _teddy bear_?'
'I was hiding in the cupboard under the stairs and I saw him take Mr Lumpylump. Should've broken both their legs. Should've killed them both.' Seven years old, going on Charles Manson.
'So tell me his name.'
'What happened to your hand?'
The filthy cast itched. 'I hit someone. _Hard_.'
A nod. 'And I'm not a snitch.'
Callum stepped into the thin drizzle, closed the front door behind him, and slouched down the front path to the Mondeo: still parked in the middle of the street where he'd abandoned it, still full of all his boxes. Which was something of an achievement for Kingsmeath, even at this time of night.
He plipped the locks and sank in behind the wheel.
'00:35', according to the dashboard clock, and he still had to drive all the way to Cowskillin, unload everything into Mother's husband's lockup _and_ drive back to Dotty and Louise's house before he could call it a night.
'Pffff...' Come on. Keys in the ignition and—
A knock on the driver's window made him flinch hard enough to drop the keys.
Sodding hell.
He turned and there was Baboon Boy with his jug ears and pug nose, standing close enough for his breath to fog the glass.
Callum buzzed the window down. 'Benny?'
Benny did a big pantomime of looking up and down the street, then over his shoulder, before turning back and lowering his voice to a whisper. 'They beat on my mum.'
'I know, Benny. But your mum won't tell me who did it, so there's nothing I can do to help her.'
He wiped his nose on the sleeve of his tracksuit top. 'I saw them. Willow thinks I didn't, but I did. Both of them. Cos I see things.'
Callum slumped back in his seat. 'I'm sure you do, but I'm completely knackered, Benny, so...?'
'My Dad's a rock star.'
'Is he now?'
'He's got a helicopter and a plane and a tiger and loads of bitches.'
'Bitches?' Maybe Willow hadn't been lying about Benny being dropped on his head when he was wee.
'In bikinis and stuff, for the dancing.' He did a sort of Michael Jackson crotch-grab-and-twirl thing, finishing off with a finger pointed at the low clouds. 'Owwwww!'
'Right. That makes perfect sense now.'
Benny lowered his pointing hand and nodded, face serious as an aneurism. 'Yeah. I seen him on the telly. With his bitches.'
'OK, well, thanks for letting me know, Benny. I appreciate it. But don't call women "bitches", OK? That's not nice.'
Another serious nod.
Callum scooped up the keys from the footwell. Paused. Turned back to the strange little boy with his snot-silvered tracksuit. 'You don't know your dad's name, do you, Benny? Do you know what he's called?'
Another pantomime check that no one was listening. 'No one's supposed to know.'
'Yes, but do _you_ know, Benny?'
'Mum used to call him Donald when he'd been naughty. But you're not allowed to tell anyone.'
'Donald. Right. It'll be our secret.' Callum stuck the retrieved keys in the ignition and started up the Mondeo. 'Do me a favour? Look after your mum and sisters... And whatever the baby is.'
'Cos I see things.'
'That's right. And if you see your dad round here again, you give me a call, OK?' Callum handed over a Police Scotland business card.
Benny frowned at it, then put it in his back pocket. 'OK.'
And with any luck, next time, they'd catch the cowardly little sod in the act.
Callum put the car in drive, gave Benny a wave, and pulled away into the night.
The lockup wasn't full of pornography, but it _was_ full of booze. Half a dozen demijohns blooped and gurgled on a reclaimed section of kitchen units, complete with worktop. Crates and crates of wine bottles were stacked up in the corner, and a couple of black plastic bins had tea towels draped over their gaping mouths. Everything had the earthy undead smell of live yeast.
Callum piled his boxes against the wall opposite – clothes and things on the bottom, books on the top. Just in case anything happened and there was a homebrew flood. Sod the clothes: save the books.
He stood staring at the boxes, then opened the one marked 'KIDS' BOOKS'. Pulled out a handful of battered paperbacks, their spines cracked and flaking from years of rereading. The ones he was going to read to Peanut, when he was old enough.
_The House at Pooh Corner_ , _The Lion the Witch and the Wardrobe_ , _Black Beauty_ , _Open the Coffins_ , _Biggles Flies Again_... A lifetime of books – every single volume he'd _ever_ owned since he was little – now just a pile of supermarket-scrounged wine boxes in someone else's lockup.
Callum put the books back in their box.
Stared at them.
Then carried the box back out to the car.
At least now he'd have something to read.
## — these bones beneath the earth —
The little chimney boy blew life into a candle, melting back the darkness. "There we is, my dear," he said to Justin. "You just hops yourself up on the kitchen table and I'll warms you a nice bath." Then he pulled a big brass pot from a cupboard, filled it with water, and put it on to simmer.
Justin jumped onto the table and sat there, his brand-new rabbitty ears picking up all manner of scary noises lurking in the gloom. "Why... Why are you putting carrots and onions in my bath?" he asked, trying to sound brave.
"Because they's dirty, and I wants you to wash them for me."
"And are the potatoes, leeks, and salt-and-pepper dirty too?"
"Why, Justin, anyone would think you doesn't trust me..."
R.M. Travis
_Open the Coffins (and Let Them Go Free)_ (1976)
_Cos them bitches be wide with their legs in the air,_
_But he can't barely stand, he's wrapped up in his warfare,_
_His booze and his dreams, his tattoos and his schemes,_
_He's f*cked up inside, and it's time for some screams here._
Donny '$ick Dawg' McRoberts
'Diary of a Motherfunkin' Legend'
© Bob's Speed Trap Records (2016)
##
McAdams held up a hand, eyes clenched shut, wrinkles deep and thick across his forehead. 'If we could keep it down to a deafening scream, that'd be nice.'
He wasn't the only one who looked as if he'd rented his skull out to a Death Metal band. Dotty was slumped in her wheelchair, one hand massaging her temples while the other clutched a large wax-paper cup of coffee. Franklin was wrapped around a bottle of Lucozade, making little grunting noises every time she moved. And Watt sported a pair of dark glasses and a pained expression.
Mother, on the other hand, sat back in her office chair with her knees spread wide, tucking into a bacon buttie and a big mug of tea. She beamed at them, washed down her latest mouthful. 'I don't know what you're all complaining about: if you can't do the time, don't do the tequila shots.'
Callum had a sip of tea.
Dotty buried her face in her hands. 'Urgh... Whose bright idea was it to have flaming Drambuies?'
Watt raised a finger and pointed it at McAdams. Who just stood there, propped up against the wall. Groaning.
'Now, dear children, our masters will be holding their press conference at one, so we have until then to dot-and-cross. Who wants the "i"s and who wants the "t"s? Don't all rush at once.'
Callum raised his mug. 'I need to go see Blakey about the paedophile I arrested last night.'
'When you've done that, get your little friend, Dr McDonald, to look over Tod Monaghan's details. I want a ribbon wrapped around him with a bow on top. Rosalind, how did you get on with our friends at Strummuir Smokehouse?'
Franklin took a scoof of Lucozade, gave another grunt, then picked up a clump of paper. 'The only one _without_ a criminal record is the woman who cooks chips in the canteen. Everyone else has done time: armed robbery, fraud, assault, murder, possession with intent...' She was slumping lower and lower with every word, her other hand digging into her hair, keeping her head from hitting the desk. 'Urgh...'
'Well, we need to add interviewing everyone and checking alibis to the list. Andy? Stick it on the board. Dorothy, you and—'
A knock on the office door, and a spotty young woman in an ill-fitting fighting suit stuck her head around the door. 'Sorry to interrupt, Guv, but you've not seen DCI Powel on your travels, have you? The Super's looking for him.'
Mother turned. 'How's the head today, Erika?'
She pulled a sheepish grin. 'Vodka Red Bull and crème de menthe do _not_ mix.'
'Thought as much when I saw you doing the Lambada with Sergeant Crilley.'
The rest of Erika's face went as red as her spots. 'Oh...'
'And no: I haven't seen Reece this morning. Anyone else?'
There was a chorus of noncommittal grunts.
Callum had another sip of tea.
'Sorry, Erika, your prince is in another castle. Now,' back to the team, 'where was I?'
The DC slipped out of the room, taking her blush with her, leaving nothing but a vaguely minty smell behind.
Mother frowned for a moment. 'Ah yes: _Dorothy_. You and John set a rocket under the lab: they're supposed to be getting us fingerprints on the first two mummified victims. I want those bodies ID'd: we leave no one behind.'
That elicited groans from both of them.
'Don't whinge. A spot of fresh air will do you the power of good. And John?'
'Grnnnng?'
'You actually signed out at the end of your shift last night! A round of applause for DC John Watt, everyone.' What she got didn't even pass for lacklustre. 'Let's see if you can make it two in a row today.' She polished off the last of her buttie. 'Rosalind, seeing as how you won the "Guess How Many Pickled Eggs DI Morrow Can Fit In His Mouth At Once" competition, you can attend Monaghan's post mortem.'
'Oh God...' Franklin's face went a bit more grey.
'Andy and I will stay here and produce briefing notes for Chief Superintendent McEwan, so he doesn't make a fool of himself while claiming all the credit for catching Imhotep.' Mother sooked the last smear of tomato sauce from her fingers. 'Right: off you go. Play nice and no running in the corridors.'
'Well, you've got no one but yourself to blame.' Callum picked his jacket off the back of his chair and pulled it on – wriggling the filthy fibreglass cast on his broken hand down the right sleeve.
Franklin took another scoof of Lucozade. Shuddered. 'I'm never drinking anything ever, _ever_ again...'
He crossed to the door and stepped out into the corridor, just as Watt and Dotty disappeared into the stairwell, making for the lifts – the pair of them groaning and shuffling like a cut-price episode of _The Walking Dead_. Or, in Dotty's case, _The Wheeling Dead_.
Franklin slumped out of the office and followed him down the corridor. 'It's all right for you: you don't have to sit through a bloody post mortem.'
'You're an ungrateful sod, you know that, don't you?'
'No one else has to watch them hack Tod Monaghan up into little squishy pieces.'
He paused with one hand on the double doors at the end of the corridor. 'Teabag never starts his PMs till ten, so Mother's basically given you a free...' he checked his watch, 'two and three-quarter hours to enjoy your hangover in peace.'
'"Enjoy" isn't the word I'd use.'
'Diddums.' Callum pushed through into the cabbagey reek of the stairwell.
Someone on the floors below was whistling the theme tune to _Britain's Next Big Star_ , only flat as an ironing board.
Franklin grimaced one side of her face shut and held the Lucozade bottle against it as they started upstairs. 'This paedophile you arrested last night. It was your Slug man, wasn't it? Bob Shannon found out who he was.'
'You should go back to the office and lie down. Curl up under one of the desks for a bit.'
'And you went round and... did he admit to killing your mum and dad?'
'Or the disabled toilets on the second floor are a _great_ place for a kip. Well, as long as you don't snore. You don't snore, do you?'
'Callum!'
'No. He didn't admit to killing anyone. Says he saw who did, though.'
Through the double doors and into the Major Investigation Team's domain. A lot of the officers milling about here looked every bit as zombied as Watt and Dotty.
Brainzzz...
'So who was it then?'
Callum headed down past the meeting rooms. 'He won't say.'
Blakey was in the Sergeants' Office, scowling away at his computer, elbows on his desk, fingers in his ears.
The only other occupant was DS Praying Mantis, still sodding about with his audio file – the volume turned up far too loud:
_'I need you to calm down. Listen to me. Listen, we can't come if you don't tell me where you are.'_
Callum grabbed one of the empty seats and wheeled it over to Blakey's desk. Thumped down into it. 'Have you interviewed him yet?'
_'I'm at home. I was on the phone to Ashlee and she was answering the door...'_
No response, so he gave Blakey's shoulder a poke. 'Have – you – interviewed – Gareth Pike – yet?'
'Oh Christ, not you _again_.'
_'... a child missing?'_
Franklin settled on the edge of the desk, on the other side, hemming Blakey in. Looming. 'How's the nose, DS Blake?'
_'... said he was looking for it, but he... he...'_ Sobbing belted out of the speakers.
Blakey turned, glowering out from behind his plastic nose guard. 'WILL YOU TURN THAT BLOODY NOISE DOWN!'
DS Praying Mantis stuck out his bottom lip. 'I'm trying to catch a killer here, is that OK with you?'
Callum poked him again. 'Pike's in the cells right now. He's up before the Sheriff at eleven for having indecent images of kids. Get your finger out.'
'Leave me alone!'
_'... mum. Two Twenty-Three Johnson Crescent, in Shortstaine. Please, he's got a knife...'_
'Blakey, he was there when my parents were abducted. He saw who took them!'
_'... on their way. When did it—'_
He dug his fingers into his ridiculous sideburns. 'I don't have _time_ for this, I've got—'
'I swear to God: if you screw this up, Blakey, I'm going to end you.'
_'No, listen. They're on my mobile...'_
The computer's speakers screamed.
_'GET OFF HER! GET OFF HER! GET OFF HER!'_
Blakey shoved his chair back, yanked a drawer open and grabbed a grey stapler from amongst the pencils, pens, and usual office detritus.
_'Don't hurt my baby! I'll do anything you—'_ More screams.
He spun his chair around and hurled the stapler at DS Praying Mantis.
It clattered into the guy's monitor, bounced and went skittering across the desk, shattering a mug of tea and sending the contents exploding across keyboard, paperwork and the Mantis's shirt. 'WHAT THE HELL?' On his feet, staring down at the big beige stain.
_'... on their way. Can you tell me your name?'_
Blakey's face was the colour of an impending stroke. 'IF YOU CAN'T HEAR, GET SOME BLOODY HEADPHONES!'
'Have you lost your mind?'
'ALL DAY, EVERY DAY!' Blakey lurched to his feet, fists clenched. 'THE SAME _BLOODY_ AUDIO CLIP BLARING LIKE A _BLOODY_ AIR-RAID SIREN!' Tears sparked in his eyes.
'Well _excuse me_ for trying to do my job!'
_'I'll take good care of you...'_
'HOW AM I SUPPOSED TO COPE?' Spittle glowed in the office's strip lights. 'TELL ME?' Bottom lip trembling. 'HOW?'
_'... forever. Won't that be nice? Forever and ever.'_
Blakey's shoulders slumped. 'How am I supposed to cope?'
_'Oh God, she's dead. She's dead. She's dead.'_
He bit his bottom lip, then turned and stormed out of the office, one hand rubbing at his eyes.
DS Mantis just stood there, mouth hanging open.
_'Marline, I want you to record the call for me, will your phone let you do that?'_
Through the office windows, most of the MIT zombies stood or peered over their partition walls. Watching Blakey go.
_'I... Yeah, completely! I've got, like, this app that'll—'_
Franklin blew out a low whistle. 'Wow.'
_'Get away from me!'_
Callum sagged in his seat. So much for getting Blakey to actually _do_ something about Gareth Pike.
_'They'll worship you.'_
Shouldn't have pushed him so hard.
_'You'll be a god and they'll worship you.'_
DS Mantis pulled at his shirt, flapping the soggy fabric. 'Absolutely soaked through.'
Still...
More screaming from the speakers.
'You know what?' The Mantis grabbed a handful of tissues from a box of Kleenex and dabbed at himself. 'I'm getting _really_ tired of Blakey's crap.'
Franklin took another swig of Lucozade. 'So what now?'
Good question.
Shame Callum didn't have an answer. 'We can't _force_ Pike to give up the name.'
'Well... maybe we can trick him into it?'
'Oh, so it's _my_ fault Blakey's wife is shagging around on him, is it?' More dabbing. 'Maybe if he wasn't such a dick the whole time, she wouldn't have to.'
Callum looked up. 'Blakey's wife's cheating on him?'
Odds on it was DCI Poncy Powel.
'I mean, I get it: all this macho posturing and sexist rubbish is his way of overcompensating. "Look at how manly I am; no way _my_ wife's having an affair!" But enough's enough.'
Nothing but crashing and banging from the speakers. Muffled cries. A sob.
_'Marline? Can you hear me, Marline? Have you recorded the call?'_
_'I pressed the button. Please, you have to help them!'_
'Well screw him, I'm making a formal complaint soon as Powel gets in.' DS Mantis dumped his soggy tissues in the bin. 'Bloody shirt was clean on this morning.'
_'It's OK, Marline. We're on our way. We'll be there soon.'_
_'You have to hurry!'_
Callum stood. 'Yeah well, I suppose we'll just have to...' A frown. He wandered over to the tea-stained desk. 'Can you play that last bit again?'
'I've put up with his crap for six months now and I'm not doing it any more. I'm not.'
'The last bit of the audio file: play it again.'
Mantis grabbed another handful of tissues. 'I should've marched over there and knocked his ugly block off!'
OK, fine.
Callum scooted around the desk and wiggled the mouse in its little puddle of tea till the cursor on the screen hovered over the media player. A couple of clicks and the audio jumped back in time again.
The authoritarian voice of the Control Room, slightly muffled. Like the recording of a recording: _'Marline, I want you to record the call for me, will your phone let you do that?'_
A young woman, sniffly and frightened: _'I... Yeah, completely! I've got, like, this app that'll—'_
A different voice, another young woman, screeching it out: _'Get away from me!'_
And then a man, barely audible over the shouts and screams, as if he was standing far away from the phone. But there was no mistaking his calm and reasonable tone, the pride in his words: _'They'll worship you. You'll be a god and they'll worship you.'_
Callum turned the volume up and clicked the mouse again.
_'They'll worship you. You'll be a god and they'll worship you.'_
Franklin stared at him. 'Callum?'
'I think we've got more victims out there.'
##
_'They'll worship you. You'll be a god and they'll worship you.'_
Mother sat back in her seat and grimaced. 'Well... Maybe?'
Callum pointed at the media player sitting in the middle of her computer screen. 'Dr McDonald said Imhotep was venerating his victims. The Peruvians used to transform their dead into gods so they could look after the village. This is what he does: he abducts people and he turns them into gods.'
She looked over her shoulder to where McAdams was slouching against the filing cabinet, stirring something white and fizzy in a glass. 'Andy?'
A shrug. 'Play the recording from the start again. Let's hear our boy's voice.'
Callum did, setting the speakers crackling.
They sat and listened all the way through. Frowning at the screen.
_'They'll worship you. You'll be a god and they'll worship you.'_
'You'll be a god and they'll worship you. Hmm...' McAdams stared off into the distance.
Mother stared at him. 'In your own time, Andy.'
'Thinking.' He bared his teeth. 'I know it's all muffled, but does our boy sound local to you? I think there's a big lump of Dundee in there.' The filing cabinet squeaked as he leaned back against it. 'And how did this Marline manage to record the whole call?'
'She's got an app that runs in the background, buffering everything on a loop. Likes to record her boyfriend so she can listen to him over and over again. Find out if he's cheating on her.'
'Ah, the delights of modern technology.' A nod. 'The lad may be right. Imhotep doth make them gods. Now we must find out.'
Mother sniffed. 'I like it better when poems rhyme.'
'Don't blame me, blame the Japanese, / Their haikus flummox, tease, and please, / Though sometimes they may cause unease.' He downed his white-and-fizzy in a single gulp. 'It could be that our story has a third-act twist up its sleeve and our dead serial killer is reaching out from the mortuary slab. Can our brave team of misfits overcome their differences to save Ashlee and Abby Gossard in time?' McAdams put his glass down. 'Well, assuming they're not already dead, tra-la-la.'
'Fair enough.' She nodded. 'All righty. Callum: you and Rosalind go check out this Ashlee Gossard. But do it quickly – let's not get caught out on the "i"s and "t"s, because we got distracted dotting the "j"s as well.'
Johnson Crescent was a big horseshoe development of tiny two-storey houses, all squished together into long tenement blocks.
Callum parked a few doors down from number 223, beneath the yellowing leaves of a sycamore tree.
Franklin sniffed. 'Least it's stopped raining.'
The sun had even managed to poke its way through the city's blanket of dove-grey clouds.
Wonders would never cease.
This side was still wreathed in darkness, though.
The rattle and clank of construction stretching from the Camburn Roundabout cut through the damp air as they climbed out and locked the Mondeo.
Callum stuck his broken hand in his pocket and wandered down the pavement to number 223.
A line of police tape was tied around the door, but there was no sign of anyone guarding it.
He let himself in with the keys from the case file.
A small hallway with stairs up the right-hand side. A row of coats. Laminate flooring with a long smear of dark red curling away down the hall and disappearing through the door at the far end. More smears on both sides, below knee-height, as if someone being dragged had tried to get purchase on the magnolia walls. Smudged bloody handprints on the architrave of the open living room door.
Franklin peered over his shoulder. 'Should we not be in SOC suits, or something?'
'DS McCready says the SEB have been and gone.' Though it had taken a crowbar to get that information out of the Praying Mantis, never mind the keys, or the case file.
Just to be on the safe side, Callum fought his way into a blue nitrile glove and picked his way down the outside edge of the laminate flooring, keeping as far away from the blood smears as possible.
The room they disappeared into was a kitchen.
'Yeah... That's not good.' Franklin stood in the middle and did a slow three-sixty.
She wasn't wrong. There was blood up the walls, little red dots on the ceiling, shattered jars spilling teabags and coffee granules, sugar and cornflakes. A small table lay on its side against the fridge, one leg snapped clean off and sitting in a sticky-looking puddle of scarlet. Two chairs, twisted to splintered bones.
Franklin curled her top lip. 'Why all the blood? Tod Monaghan didn't do this when he attacked Ben Harrington, Brett Millar, and Glen Carmichael. _Three_ of them, and not so much as a drop anywhere. Why the overkill?'
Good question.
'Maybe Ashlee and her mum wouldn't eat the magic mushrooms?'
'Nah, you heard the nine-nine-nine call, he didn't even try. Soon as he was in the house that was it: screaming.'
Callum eased the broken table out from in front of the fridge. The white plastic door was covered in blobs of black fingerprint powder.
'And he's never attacked women before, has he? All the other mummies are men.'
And _that_ was a good point.
'Well, I don't know, do I? Maybe he thought Ashlee Gossard was a better bet: you've seen her photo, she's either bulimic or anorexic. Less body-fat means less water, means easier to preserve.'
'Assuming Ashlee was the target and not her mum.'
Another good point.
Callum sat back on his haunches. 'The only way we'll know for sure is if we find them.'
'If they're not already dead.'
'Will you stop it with the good points already?'
That got him a frown.
He waved a hand. 'Never mind. How did your boyfriend's work's do go?'
'I'm just saying this doesn't look like Imhotep's handiwork. This isn't his MO.'
'I know.'
She opened and closed a couple of the kitchen drawers. 'Apparently the partners kept asking where I was. And he had a miserable time. And it was all _my_ fault, because _I_ wouldn't drop everything and go simper at his side like a little woman _should_.'
Callum stood. 'Want to check upstairs?'
'He's always banging on about how he supports my career, but every time it clashes with _his_ career suddenly I'm being "selfish".'
There was a pool of blood at the foot of the stairs. A couple of dark footprints on the bottom three steps, then a smear down the wall to the ground again. As if someone had made a run for it, but didn't get very far.
Callum tiptoed between them and up the stairs.
Franklin followed him. 'You know what I think? I think Mark doesn't want me to work at all. He wants a trophy wife who'll settle down and do some volunteer work between squeezing out three kids and baking sodding scones.'
The landing was clear – no blood spatters.
She curled her lip. 'I _hate_ scones.'
'Uh-huh.' Callum pushed open one of the three doors – small bathroom with a built-in shower over the bath. Every porcelain surface was clarted with fingerprint powder.
'And why should _I_ sacrifice everything to have kids?'
'I like cheese scones.' Door number two opened on a double bedroom. Nothing fancy. Blue-and-yellow duvet cover with matching pillows. An array of bottles, jars, make-up, brushes, and associated things on a little vanity unit. Scottie Dog cuddly toy thing. A few framed prints of famous Scottish landscapes.
'You can bet if it was _men_ who had to squeeze three and a half kilos of human being out the end of their penis, they wouldn't be so damned keen on a big family.'
'They're nicer if you toast them. Oh, and lots of butter.'
She stared at him. ' _Children?_ '
'Cheese scones, you muppet.'
Door number three: a single bedroom that looked as if a drunken baboon had been locked in there and told to go wild with the clothes and underwear. It was everywhere. On the floor, on the bed, poking out from _under_ the bed, on the chest of drawers, hanging from the top of the wardrobe. Shirts, T-shirts, jumpers, tops, jeans, leggings, jeggings, socks, stockings, tights, shoes, and flip-flops. Add about two dozen bras and pants and mix liberally.
Callum sniffed. The harsh chemical taint of deodorant and air freshener. 'Someone's trashed the place.'
Franklin rolled her eyes. 'You've never been in a teenage girl's bedroom before, have you?'
A collection of posters were stuck to the ceiling above the bed. Popstars and boy bands, a couple of rap artists. Lots of bare chests, tattoos, and flowing hair.
One particularly oily-looking git was posing on a motorbike, surrounded by unfeasibly breasty women in bikinis. They were all pouting at him, as if he were God's greatest creation, instead of a wee nyaff with a shaved chest, stupid facial hair, and a tattoo of a fox poking out from the waistband of his Calvin Kleins.
Franklin followed his gaze. 'Look at them. How are little girls supposed to develop a healthy body image when they're confronted with the Size-Zero Silicone-Mammary Brigade at every turn?'
Callum settled onto the edge of the bed, between a pink fluffy jumper and a pair of leather shorts. Frowned out at the devastation. 'Long as I live, I'll never understand you lot.'
'Try harder.'
He picked up a green sock with orange penguins on it. 'Monaghan raped a guy in Blackwall Hill, but the victim dropped the charges.'
'So?'
'Not sure.' The sock got tossed onto the floor so Callum could pull out his mobile phone and scroll through the contacts list till he got to 'MCDONALD, DR A'.
She picked up on the first ring. _'Hello? Ash?'_
'Alice, it's DC MacGregor. From the Divisional Investigative Support Team?'
_'Oh... I see.'_ Not doing a very good job of hiding her disappointment. _'Anyway, thanks for sending over the file on Tod Monaghan, I've been through it and compared it to the behavioural evidence analysis we did on the initial victim set, well, I say "initial victim set", but it isn't, is it? I mean we don't know who the first victims were, we just know about Glen, Brett, and Ben, but then we can't factor in the first two without an ID to do a victimology work-up from, does that make sense?'_
Sort of.
'Monaghan might have grabbed someone else. We've got a crime scene in Shortstaine, mother and daughter abducted, blood everywhere.'
Franklin leaned back against the chest of drawers, arms folded. Watching him.
_'To be honest, that doesn't sound likely, I mean we've got all these other victims and they're all men and it's_ very _unusual for a killer like this to cross a gender gap once established and—'_
'The daughter's anorexic and she was on the phone to a friend when it happened. Her friend recorded the whole thing. And right at the end, he tells them, "They'll worship you. You'll be a god and they'll worship you." So I thought...?'
_'Ooooh, now that_ is _interesting. It's conceivable that there's someone else going around Oldcastle abducting people and turning them into gods, but it'd be a huge coincidence, wouldn't it, I mean absolutely massive, so if we work on the assumption that it_ was _actually Tod Monaghan, then we'd need a compelling reason to justify his sudden change in victim-gender selection, because it tends to be pretty consistent with serial offenders, oh, it's different if they don't differentiate to start with, but when they make a definite choice they tend to stick with it.'_
'But we're not likely to have two god-making nutbags on the go, are we?'
_'There would have to be a reason for him to suddenly stop selecting male victims and you shouldn't call people "nutbags", these are human beings just like you and me only wired a bit differently due to their brain chemistry and upbringing. Dehumanising them by calling them "nutbags" doesn't help anyone; it doesn't matter how horrible the things they do are, they're_ still _human beings. We should try to remember that.'_
Which was pretty much the same speech he'd given Willow last night. 'Sorry.'
_'According to the notes: eight years ago, Ted Monaghan goes to a picnic area in Moncuir Wood that's a well-known pickup spot for gay men, only there's an argument, the young man he wants rejects him and Monaghan becomes violent. Leaves. Comes back half an hour later with a hammer and tries to beat the young man to death. At the trial Monaghan insists he wasn't looking for sex, because he isn't gay, and that the young man attacked_ him _. The jury doesn't agree and he serves six years for attempted murder.'_
Franklin waved a hand at Callum. 'At least put it on speakerphone.'
'Sorry.'
He pressed the button and a tiny Dr McDonald voice sounded in the pigsty room. _'Five months after Monaghan gets out of prison he's back in Moncuir Wood, only this time he doesn't go looking for a willing partner, he attacks and rapes a different young man. When questioned, Monaghan claims he isn't gay and that he's the real victim. Again. The young man later drops the charges when his car gets set on fire.'_
'So we know Monaghan's violent.'
_'Well, yes, but when he starts turning people into gods, they're_ always _young men, probably because it's young men that he likes, only he can't admit that, because it contradicts his self-image as a manly man, even though he's been having sexual fantasies about them for as long as he can remember, which is why he hangs out in this bit of the woods where it's easy to find someone to explore his sexuality with, only he can't reconcile his sexual needs with his strict upbringing and ends up venting this cognitive dissonance destructively, until one day he rationalises it into something more positive.'_
How did she manage to keep talking for so long without taking a single breath? How was that physically possible?
_'He decides to take the objects of his sexual confusion and turn them into gods, he's venerating what he can't allow himself to physically realise, so their gender is_ very _important to him and the only way he'd change that pattern is if something serious happened, and I mean something_ revelatory _, because he's been planning and fantasising about this for so long, but now it's all different, and it would send him_ right _out of his comfort zone, so I'd expect to see a_ lot _more violence when things don't go exactly as he's planned and he has to improvise his way out of trouble.'_
Callum frowned. 'He didn't improvise anything. We heard him on the recording: he cons his way into their home – pretending he's looking for his missing son – and then attacks them. Blood all over the kitchen. Drag marks in the hall.'
_'Ooh... Now_ that's _interesting, I mean he's all softly-softly with the young men he attacks, but the women are there to be subdued quickly and violently. Maybe they don't deserve subtlety? Maybe women need to be put down hard and fast? What do we know about Monaghan's childhood?'_
'Nothing that isn't in the file.'
_'I think it's a safe bet he had a very difficult relationship with his parents. Probably an abusive father and a submissive mother. She's beneath contempt. She never loved him properly. Father had the right idea – women are dirty, subhuman things that have to be trained like dogs. Chained up and beaten...'_
Silence.
Franklin checked her watch.
'Alice?'
_'Sorry, thinking. Monaghan knew we were looking for him, it was in all the papers. He's feeling threatened and embattled and he needs more gods to protect him. He's running out of time, so he has to cut corners. You say the daughter's anorexic? Well, why starve a young man when you can just abduct a young woman who's done all the hard work for you?'_
Callum grinned at Franklin. 'That's what I said.'
She rolled her eyes.
_'That means the mother is surplus to requirements and if there's blood everywhere, it's probably hers. It'll still take a few days to purify the daughter to make sure she's worthy of godhood, but I'd be shocked if the mother isn't already dead.'_
Sodding hell.
Callum stood. 'So there's still a chance we can save Ashlee?'
_'Not much of one, but yes.'_
'Thanks.' He hung up and turned to Franklin. 'Any ideas?'
'Monaghan had to transport them out of here somehow: car, or a van.' She hauled out her phone. 'The initial investigation must've done door-to-doors.' Her thumb poked at the screen for a moment, then she held it to her ear and wandered out onto the landing. 'Yes. DC Franklin, I need to speak to DS McCready...'
Two could play at that game.
Callum put a call in to the CCTV team. Listened to it ring. Crossed to the window and pulled back the curtains.
The other side of the street still glowed like a packet of fluorescent Fruit Pastilles. This part of Johnson Crescent formed the bottom curve of a big U-shape, so anyone on the left or the right would have a clear view of anything suspicious. Assuming they didn't come down with the traditional Oldcastle amnesia and—
_'Greetings!'_ A woman's voice, crackling with faux-American cheesy cheer. _'You've reached the_ magnificent _Closed-Circuit Television Department, where dreams really_ do _come true. How may I direct your call?'_
'Voodoo? It's Callum.'
The accent disappeared. _'My God, there's a blast from the past! You've not been on the scrounge for a favour since last Wednesday. I was beginning to worry.'_
'I'm looking for a car, or a van, involved in the abduction of a mother and daughter.'
_'What, straight into it? No foreplay?'_
'Your husband says I'm not allowed to get you all fired up and horny. Aggravates his lumbago. The vehicle would have been in the area this Wednesday evening, between seven and nine p.m.: Johnson Crescent. I need to know where they came from and where they went afterwards.'
_'Got makes and models?'_
'Depends if you believe the door-to-doors or not. Probably best to do it blind so we don't miss anything.'
_'Hmmph, you're not asking much, are you? Let's see what we can see...'_ The sound of fingers dancing across a keyboard. _'There's no CCTV cameras on that street. Nearest I've got is Johnson Park, at the wee shopping centre.'_ More clicking. _'We're having a birthday party for Ian next week: sixtieth. You should come. Bring the lovely Elaine, we've not seen her for ages.'_
'Yeah... Not so lovely. We've split up.'
_'Callum MacGregor! You do_ not _get a young lady pregnant and then—'_
'I didn't. It wasn't mine.'
Clickity, clickity, clickity.
_'Oh, Callum, I'm so sorry. I've got ANPR cameras on Camburn Roundabout, one at the traffic lights just before you hit the woods, and another outside ASDA on the Brechin road.'_
'It's probably a van, but any car big enough to hide two bodies in the boot is worth a punt.'
_'Are you_ sure _the baby isn't yours?'_
'She's been shagging DCI Reece Sodding Powel for about a year. Probably longer.'
_'Then you should definitely come to the party. My daughter's just dumped her idiot husband and she could do with a shoulder to cry on.'_ Clickity click, click, click. _'This is probably going to take a while. I'll have to get back to you.'_
'Thanks, Voodoo.'
_'And I mean it about the party, Callum, you and Becky would be perfect together. She's smart; she's pretty; she's_ always _got her head in a book; and she never, ever—'_
'Bye, Voodoo.' He hung up. Stood there, staring out at the brightly coloured houses.
Maybe Oldcastle's answer to Yente was right?
Maybe her daughter _was_ perfect?
And maybe he deserved to be happy for a sodding change?
It wasn't as if Elaine gave a—
'You OK?'
He turned, and there was Franklin, frowning at him. 'Hmmm?'
'Looked like you were miles away.'
'Any luck?'
'Couple of residents mentioned a small grey van parked up the road. One old lady saw a big blue Transit, but nobody else did. And there were three sightings of a big Red Land Rover driving erratically around the time of the nine-nine-nine call. McCready's got two DCs trying to chase them down.'
At least it was a start.
Callum went back to his phone and called Mother.
##
Franklin made a big show of looking at her watch. Again. 'We're going to be late.'
'No we're not.'
To be fair, the traffic was terrible. Whichever moron on the city council thought it was a good idea to dig up the main road through town on the same weekend as that stupid music festival in Montgomery Park needed a stiff kick in the backside. And then a punch in the balls.
The dual carriageway was down to one lane in each direction, crawling with eighteen wheelers; coaches; buses; cars; all blending in an exhaust-fume symphony of grey that stretched from the Camburn Roundabout as far as the eye could see. Didn't help that the rain was on again.
The line of cars ahead of them snaked through the slalom of orange traffic cones, crawling across the central reservation and onto the opposite lane. Then stopped.
Callum cleared his throat. 'OK: how do we trick Gareth Pike into giving up the name?'
'Lie to him.'
'I mean, it's not like I can threaten to put him in jail with a bunch of sex offenders, is it? That's his idea of a social club.'
'You could offer him something, then take it away? Pretend you've found another source and _they've_ already given you the name, so the best Pike can do is corroborate it if he wants any concessions at all?'
'Might work...'
'He has to want something, everyone does. So what does Pike want?'
'South-facing cell with a nice view. Seriously: like he's reserving a room at the Ritz.'
'Good luck with that.' Franklin pulled a face. 'We're definitely going to be late.'
'Look, there's nothing we can do about the traffic, OK? Put the radio on or something.'
Franklin crossed her arms. 'Put on the blues-and-twos more like.'
He tried not to sigh, but it didn't work. 'We've been over this: you hit the nine-nine-nine button and the GPS starts recording, and the dashboard camera starts recording, and...' He tapped his fingers against the steering wheel.
She turned in her seat. 'What?'
Last night – outside Willow Brown's house. When he'd arrived, lights and sirens blaring, that big black Mercedes was just pulling away. Not exactly the kind of car you'd expect to see swanning about Kingsmeath. No: drive something like that down there and you'd be lucky to get home with all the wheels and doors still on it, never mind the hubcaps. So why was it there, on Manson Avenue? Why would—
'Callum!' Franklin poked him in the arm. 'We're moving.'
He blinked.
The cars up ahead had shuffled forward twenty yards.
A horn blared behind him, followed by a rising chorus of angry beeps.
He slid the Mondeo up to the car in front's rear bumper.
It couldn't have been Willow's _dad's_ Mercedes, could it? He'd gone up in the world, if it was... Mind you, since when was being a wife-and-child-beating scumbag any barrier to success?
With any luck, the dashboard cam had got the Merc's number plate before it disappeared around the corner. Then it wouldn't matter how 'not a snitch' Willow and her mum wanted to be – a quick check on the Police National Computer would spit out the wee sod's name, address, and inside-leg measurement.
And speaking of PNC checks, what the hell had happened to the one he'd requested on all Irene Brown's old boyfriends? Have to chase that up. Honestly, you had to stand over people beating them with a stick to get anything done.
'... sometime today?'
'Hmm?' He looked up and the gap had opened in front of them again. 'Sorry, miles away.'
Franklin thumped back in her seat. 'Knew we should've gone the other way.'
'Everything round Montgomery Park is shut for the music festival, everything _near_ the park is all tailbacks and diversions. Doesn't matter which way you go, you're just as stuck.'
'Gah...' She clicked the radio on and a throbbing bassline and kick-drum beat burst into the car.
A woman's voice, rich and dark, amplified over the top: _'Come on, let me see those hands in the air! Yeah!'_
More drum and bass.
Franklin stared out of the passenger window. 'What happened when you went to pick up your stuff, last night?'
_'Sing it with me: You are the fish in my sea.'_
A crowd roared it out, like a football chant. _'YOU ARE THE FISH IN MY SEA!'_
Callum's shoulders itched. 'You know: the usual. Powel banging on about how we're all adults and they didn't mean for it to happen.'
_'You are the birds in my tree.'_
_'YOU ARE THE BIRDS IN MY TREE!'_
His good hand tightened on the wheel. 'Said Elaine never loved me. She was just going through the motions.'
Franklin nodded. 'He is a bit of a dick, isn't he?'
_'You're the honey in my bee.'_
_'YOU'RE THE HONEY IN MY BEE!'_
She clicked the radio off again.
The traffic crawled forwards.
A cough. Then Franklin puffed out her cheeks and sighed. Picked at a stain on the dashboard. Sighed again. 'OK, so Pike wants to go to prison, yes? What if you threaten to take _that_ away from him?'
'We caught him molesting himself to a video of two little boys being raped. He's going to prison and he knows it.'
'Hrmmm... What about telling him you'll put out a statement about how helpful he's been in exposing whatever ring he's part of? Soon as he gets inside they'll tear him apart.'
The traffic crawled forward another six foot.
'Or, how about— Sod.' She hauled out her phone. 'DC Franklin... Uh-huh... Hold on, I'll put you on speakerphone.' She pressed the button and held the phone up.
Dotty's voice thumped out into the car. _'... not telling you again!'_
Watt, in the background: _'You're not allowed to use a mobile phone while driving. It's_ illegal _.'_
_'Oh, go bugger yourself with a loo brush.'_ A pause. _'Rosalind, we've chased the labs up and guess what: fingerprints.'_
_'You've already been in one horrible car crash, Hodgkin, let's not make it two.'_
_'Genuinely, it was the most disgusting thing I've ever seen. They soaked the hands in glucose, peeled them off the bones, and got one of the APTs to wear them like gloves. Urgh...'_
_'I'd like to go home tonight with_ both _my legs, if that's OK with you?'_
Callum raised his voice. 'Did they get an ID off the prints?'
_'Oh, hello, Callum. Can you do me a favour and tell Watt he's being a big – girl's – blouse?'_
'Do you two always have to do this?'
Watt: _'And you can tell her she's being petty, irresponsible, and childish! Traffic laws are there for a reason.'_
He slid the car forward another length. 'Dotty, give Watt the phone.'
_'I don't—'_
'Just do it, OK? Please?'
Watt's voice came through loud and clear. _'Thank you. It's about time_ someone _—'_
'And you can stop being a dick. Stick us on speakerphone.'
_'That's right, take her side. Everyone always—'_
_'Don't be such a Jessie! I'm the one—'_
'OH FOR GOD'S SAKE, THE PAIR OF YOU! You're not six!'
Silence from the phone.
A gap opened up and Callum slid the car into it, crawling along as the line of traffic snaked back to its own side of the road. A massive road grader growled its way along the central reservation, flanked by soggy-looking men in dripping high-viz jackets and hard hats.
The Mondeo thumped over a hard line in the tarmac, marked 'RAMP'.
Still nothing from the phone.
'Fine: I'm sorry.'
Dotty sniffed. _'That's better.'_
'You _are_ six.'
_'It's not my fault he always—'_
'Did you get an ID off the fingerprints, or not?'
_'Hmmph.'_
Watt sounded a bit smug, as if he somehow thought he'd won something: _'I'll answer that one, shall I, Sergeant? Apparently they have to wear the hand in order to flesh out the fingers, so to speak. We got them to run the prints and they came back with two matches.'_
A coach crawled past on the other side of the dual carriageway, full of pre-teen girls waving home-made placards with things like 'WE ♥ MR BONES!!!' and 'MARRY ME TAYLOR!!!' on them.
Franklin scowled at the phone. 'This isn't _Who Wants to Be a Millionaire_ , Watt. Stop milking it.'
_'Sorry, had to check my notes. The body from the tip was one Roger Barrett. Did five years for armed robbery, got out last January. Hasn't been to see his probation officer in nine months.'_ That smug tone was back. _'And you'll never guess where he worked—'_
Dotty hammered in over the top: _'Strummuir Smokehouse!'_
_'I was_ actually _telling them that!'_
_'Rosalind's right: you were milking it.'_
No wonder Mother went around with a pained expression on her face most of the time. 'So who's victim number two?'
_'The mummy from the car boot was one Richard Duffy. No criminal record, but his prints are on file, cos someone broke into his house on Christmas Eve and stole a thousand quid's worth of electronics and jewellery from under the tree. So his wife got him a last-minute fill-in present: a charcuterie and artisanal curing-and-smoking course at Strummuir Smokehouse. He took it in January. His wife reported him missing in March.'_
Which made sense.
Tod Monaghan was a creature of habit, picking his victims from the people he saw at the smokehouse. Too thick to realise that it left a trail leading straight back to Strummuir.
_'We're on our way to break the news to Duffy's wife now.'_
'OK, thanks.' Callum grinned across the car at Franklin. 'And Watt, Dotty? You did good. You make an excellent team.'
There was a small pause, then Watt's voice rang out loud and clear: _'Don't patronise me.'_ He hung up.
Franklin shook her head. 'You did that on purpose, didn't you?'
'Oh hell yes.'
Franklin clambered out into the rain and hurried in through the overflow mortuary's front doors. Paused for a second to turn and wave at him, then disappeared inside.
Callum sat there with the windscreen wipers moaning. Pulled out his phone and called Control. 'What happened to the PNC checks I asked for _days_ ago?'
Silence.
'Hello?'
A woman's voice stabbed out from the speaker. _'Do you want to try that again, DC MacGregor, using words like "please" and "thank you"?'_
Prima donnas. 'Please can you tell me what happened to the sodding PNC checks I asked for _days_ ago? Thank you.'
_'Good manners don't cost anything, you know. And I don't appreciate your sarcasm, by the way.'_
'Fine. I'm sorry. _Please_ can I have my PNC results.'
_'That's better.'_ The sound of a keyboard being thumped into submission. _'I have eight names and details all emailed to you the day before yesterday.'_
Thursday. The day they accused him of murdering his own mother.
'Yes. Sorry I was... It wasn't a good day. I'll check when I get back to the office.'
_'See: there was no reason to be all sarcastic and demanding, was there?'_
'No. Sorry. Thank you.'
He hung up. Ran his good hand across his face.
Well done, Callum. Way to be a complete and utter dick.
'Urgh...'
Let's face it, with all this crap going on, he needed every friend he could get right now.
And speaking of which: he dialled Shannon.
_'What?'_ The word barked down the line.
'Bob, it's Callum. This a bad time?'
_'Oh, OK. Hold on.'_ It sounded like Marilyn Monroe was singing in the background, boop-boop-de-dooping her way into silence. Then Shannon was back. _'Sorry, it's been bloody government boiler schemes and green-energy review calls all sodding morning. Some people need stabbing in the ear with a trowel. Is it too much to expect to watch a film in peace?'_ A grunt. _'Anyway, what can I do for you?'_
'Any news on that name?'
_'The Old Age Police network needs time, Callum. I know it's important, but these guys are wading through nearly thirty years of junk to get at notebooks and case files. And even then, there's no guarantee.'_
Of course there wasn't.
'Sorry.' He rubbed the fingertips of his broken hand across his brows. Trying to massage some life into them. 'Pike's up before the Sheriff at eleven and he's still not saying anything.'
_'He likes screwing with people.'_
'It's not like the name's any good to him.'
_'He needs Viagra to have a wank, Callum, screwing with people is probably the closest he gets to a natural hard-on. And even then, it's probably just a semi.'_
Now there was an image.
'I don't know what to do, Bob.'
_'We'll sort it out, don't worry. Now, if you'll excuse me, Tony Curtis is about to change out of his dress and into a blazer.'_
Callum slipped the phone back into his pocket and sat staring out at the rain.
Checked his watch.
Half an hour and Pike would be making his guilty plea. Then a short car ride to HMP Oldcastle to wait for Social Work reports and sentencing. Taking his secrets with him.
And there was nothing Callum could do about it.
##
The old station house in Castleview had a weird sour coconutty smell, as if it'd got blootered on Malibu the night before and vomited all over itself. Maybe the Security Monitoring And Analysis Department liked to lube themselves up with suntan lotion of a Saturday morning?
Callum parked his backside on the windowsill and tucked his filthy fibreglass cast into his jacket pocket. Hiding it away. 'What do you think?'
The man in the blue hoodie was far too old to be wearing it, or the big fancy trainers, or the 'HALFHEAD ~ BONES & STONES WORLD TOUR!' T-shirt. He'd scraped what little hair remained on his head back into a sumo-wrestler's pigtail, glasses perched on top of his shiny scalp. 'Hmmm...' He picked at his soul patch – greying like the eyebrows. 'Why not: Winston Smith _likes_ a challenge.'
The Mondeo's digital video drive sat in a little plastic cradle, connected to a silvery tower unit. A few clattering keystrokes and lights on the drive flashed green. A few more and the thing bleeped and whirred.
'Of course, I can't promise anything, yeah? Winston never knows what he's going to see till he sees it.'
The room was strangely empty. Just the desk and the computer, one very expensive-looking office chair, one filing cabinet, one window, and a radiator that pinged and gurgled like a fat man's stomach.
Callum checked his watch. Again. Ten past.
Pike would be on his way back to the cells by now.
'Right, here we go...' Smith's fingers flew across the keys and half a dozen little windows appeared on the computer monitor, each playing a view from the Mondeo's dashboard camera. Second-hand flickering lights and speeding streets. He fiddled with the mouse, closing all the windows but the one with yesterday's date – the dual carriageway roared past at triple speed, cars and traffic cones flashing by, over the bridge, round the roundabout, up into Kingsmeath.
Another click and the video slowed to normal speed.
The car swept around onto Manson Avenue.
Froze.
Wound back a bit.
'There we are, one black Mercedes.' Smith tapped the screen, where the back end of the Merc was just disappearing around the corner, partially obscured by an ancient Fiesta. 'Now, let's see what being a genius gets you these days...' He pecked at the keyboard and the window zoomed in on the car's number plate. The footage ran forwards and backwards a few times and the frown on his face deepened. 'Hmmm...'
'What?'
'Well, it's low light, which doesn't help, it's far away, which also doesn't help, and the camera lens isn't the cleanest either. This is as good as we can get.' A blurred and lumpy grey-and-yellow smear.
'Can't you do some sort of image enhancement thing on it?'
'This isn't science fiction, my friend. Winston is a genius, but he's not a miracle worker. These cameras record the image as a big block of pixels and write them to the hard drive. You can zoom in all you like, but there comes a point where all you're doing is making the pixels bigger. You can't magically wring more resolution out of the system, because it just doesn't exist.'
'Oh.'
Well _that_ was a complete waste of time.
Callum stood. 'Well, thanks anyway.'
'Ah, ha, ha!' Smith held up a finger. 'Winston said he wasn't a miracle worker, but that doesn't mean he's incapable of the odd miraculous act. You see the pattern of blurring we've got here, the lights and the darks? That's been formed by the numbers on the number plate, and the way they combine in a given set of lighting conditions at a certain range. Winston can't make them any less blurry, but he can run a very clever bit of software to blur thousands of different number and letter combinations to see what produce the closest matches.' A wink. 'Told you: Winston Smith likes a challenge.'
He curled over his keyboard, face inches from the screen as he typed. Opening up programs, setting things running, clicking and clicking and clicking. 'You might want to grab a cup of tea, this is going to take a while.'
Callum took a sip of scalding hot tea, wandering the corridors. 'No, I just wanted to see if there was any news.'
Mother's voice crackled out of the earpiece as if she were on the other side of the planet, not the other side of the river. _'Well, Dorothy and John are off looking into Roger Barrett. Probably won't add anything to the stew, but it's better to err on the safe side, isn't it? I want us to have a good squint at Ashlee and Abby Gossard's movements in the run-up to the abduction. Monaghan must've bumped into them somewhere, and far as we know they've never been to Strummuir Smokehouse. Which reminds me: we've still not had anything back from the CCTV team, so if you wouldn't mind giving them a prod, that'd be nice.'_
'I'm there now.'
_'Good. And tell Voodoo that Jack and I would love to come to Ian's party. I'll be bringing my famous spinach and artichoke dip. Jack will bring his infamous peapod burgundy.'_
'Will do.'
The CCTV control room door was open, offering a view into a dim room lined with screens – each one displaying a different view of Oldcastle. Half a dozen support staff sat at the long central desk, working the joysticks that moved the cameras and eating cake.
_'Callum?'_
'Yes, Boss?'
_'I hear someone broke Blakey this morning.'_
That was one way to put it.
'Yes, Boss.'
_'He's been signed off on the sick. Stress.'_
'Oh for...' Callum closed his eyes and thunked his head off the corridor wall. 'So there's _no one_ running the case?'
_'I hear you think this paedophile, Gareth Pike, saw what happened to your parents?'_
'How can there be no one running the case? Blakey was useless, but at least he was _there_!'
_'When you get back to the station, we'll see what we can do, OK?'_
His shoulders sagged. 'Yes, Boss.'
_'Good. Now, in the meantime, go chase up Voodoo. And remember: spinach dip.'_
Callum hung up, put his phone away. Thunked his head off the wall again.
Bloody Blakey. _Thunk_. Bloody useless half—
'I hope you're not putting dents in my lovely headquarters, Callum.'
He picked his head off the wall and turned, faked a smile. 'Voodoo.'
She was dressed in Police Scotland black, complete with shiny boots and epaulettes on the shoulders of her T-shirt. A small woman, with close-cropped grey hair and arms like a marathon runner. A big broad smile that made the wrinkles around her eyes deepen. 'I hope you're here to accept my party invitation?'
'Thought it would be wise to see a photo of your daughter first.'
'Cheeky sod.' Voodoo dug into her pocket and produced her phone, swiping away at the screen. 'I know you didn't come all the way over here for that, so what's the _real_ reason?'
'Mother wants to know how you're getting on with the vehicle check for our Johnson Crescent abductions. Vans and big four-by-fours?'
'Ah yes. Walk this way, young man.' She turned and marched off, still fiddling with her phone. 'We've run through every ANPR camera in a half-mile radius, _and_ the security footage from all public spaces in the area.' At the end of the corridor, Voodoo pushed through the double doors and hammered up the stairs, taking them two at a time. 'Keep up, Callum.'
It wasn't easy. 'Did you... did you get... anything?' Puffing and panting all the way. Tea slopping from side to side in his mug.
She stopped on the landing and held out her phone. 'There you go.'
A young woman smiled out of the screen: long brown hair, big brown eyes, long thin nose, suntan, huge smile, and a _tiny_ blue bikini.
'That's our Becky. Still swithering about coming to the party?'
'Well, if I wasn't breathing heavy already...'
'Good boy. She likes sauvignon blanc.' And Voodoo was off again. 'I've got a long-list of about a hundred vehicles, but we've narrowed it down to three likely targets: a small grey Peugeot Bipper, a rust-brown Bedford Rascal, and a green Fiat Fiorino. There's a _lot_ of four-by-fours, but these are the vans that get my juices flowing.'
A young man emerged into the corridor, did a double-take. 'Chief Inspector.' Then flattened himself against the wall as Voodoo strode past.
'Get the kettle on, Williams. I'm gasping.' She kept on going.
'Yes, Chief Inspector.'
She swung through a door near the end, and into a large office with one whole wall given over to at least two dozen TVs. A coffee table and a couch sat in front of them, along with a phone and a wireless keyboard. Voodoo perched herself on the edge of the couch and fiddled with the keyboard.
Pictures sprang into life across the screens: a curling cobbled street in Castle Hill, the bus station on Dalrymple Street, three views of Harvest Lane's rows of nightclubs, the car park just inside The Swinney, two views of Camburn Woods, then MacKinnon Quay, the school on Preston Row, Montgomery Park with its collection of marquees and big inflatable spider... On and on, peering into other people's lives – like being God, or GCHQ.
'Take a seat, Callum, you're making my office look untidy.'
'Sorry.' He sank into the couch next to her and stuck his mug on the coffee table as she did some more fiddling. 'Don't suppose any of these vehicles were registered to a Tod Monaghan, were they?'
'No.'
'Oh...' Well, it was never going to be _that_ easy, was it? Monaghan would be driving someone else's van. Maybe without them even knowing.
The monitors divided up into three huge pictures, stretched across multiple screens. A green van in one, a manky orange-brown van in another, and a grey van in the third. All small vehicles, nowhere near as large as a Transit, and all waiting to go through a different set of traffic lights.
Their number plates sat in a caption box at the top of each image.
'We've got them going into the vicinity of Johnson Crescent between seven and eight, and coming out again between quarter past eight and twenty to nine. And by the way, you owe me six bags of doughnuts – getting my team to drop everything and slog through all that footage required bribery.' She poked at the keyboard again and the images on the screen pulsed like a slideshow as each of the vehicles were picked up on various CCTV and ANPR cameras across the city. 'We followed them as far through the system as we could, but...' a shrug, 'sadly the powers that be won't let me put cameras on every street in the city. If they did, just imagine what we could achieve!'
'Constant and total surveillance, an Orwellian nightmare, only instead of "Big Brother" you'd be "Little Sister"?'
Voodoo smiled. Sighed. 'Ah well, a girl can dream, can't she?' Then jogged over to her desk and pulled a sheet of paper from a tray. Held it out. 'All three vehicles' registered owners and addresses, plus first and last confirmed locations on camera.'
Callum stood and took the sheet. 'Thanks, Voodoo, you're a star.'
'I am, aren't I?' She frowned at him. 'Callum, do you want a little friendly advice?'
No.
'You've had a bad run of late. Don't let it colour everything that happens to you.' She gave him a small hug. 'And come to my party: Becky's a yoga instructor. _Very_ flexible.'
Winston Smith peered out over the top of his glasses. 'Well, yes, Winston did say you should go away and get a cup of tea, but the key part of that sentence is that you should _go away_.'
Callum leaned back against the windowsill again. 'You're all mouth and no trousers, aren't you?'
He narrowed his eyes. 'Winston is very much all trousers, thank you very much. He told you this was going to be a challenge, and he's not going to give up till it's done, but until then you should leave him alone and let him get on with his work.' He raised a hand from his keyboard and made shooing motions. 'Away with you. Winston will call when, ultimately, he is triumphant.'
Police Scotland _really_ needed a moratorium on hiring freaks and weirdos.
'Fine.'
Callum headed back down to the car, phone clamped between his thumb and the fibreglass cast. 'Mother?'
McAdams' voice oiled its way into Callum's ear. _'She goes to stand firm. At the press conference. The top brass to save.'_
He clattered down the stairs. 'A simple, "she's not here" would've done.'
_'Then: "she's not here," you artless spud. And she won't be back till two or three, so if you've got information: spill it.'_
'Voodoo's done the CCTV analysis for us.' Callum pushed out through the old station doors and into the rear car park – surrounded by an eighteen-foot-high brick wall topped with barbed wire. 'They're nominating three small vans as possible abduction vehicles.'
_'Good. Return your backside to DHQ; you and I are going visiting.'_
Callum scuffed to a halt ten feet from the pool car. 'Erm... Maybe you'd be better off staying there and coordinating things? You know, if Mother's going to be tied up at the press conference? Somebody needs to be in charge?'
Fingers crossed...
_'Nice try, Constable.'_
Don't give up!
'And now I think about it, maybe Dotty or Watt would be better—'
_'Backside. Back here. Now.'_
Sod.
##
Division Headquarters was remarkably quiet for noon on a rainy Saturday. No clatter of boots on the stairs, no shouting in the corridors. No drunken singing echoing up from the cells.
A couple of PCs were having a heated argument by the coffee machine outside the Productions Office, but other than that: dead.
And there was still no sign of McAdams. _'I'll be down in five minutes.'_ His arse.
Callum pushed through into the stairwell and froze, fingertips of his broken hand resting on the bannister.
That was Detective Superintendent Ness's voice, wafting up from the floor below. _'... problems. For God's sake, Reece, I know you're having a rough time at the moment, but that's no excuse for not turning up for work! This is_ completely _unacceptable.'_ The sound of feet pacing on the concrete landing. _'Look, call me when you get this, OK? If we have to rejig your workload till things calm down... well, we'll sort something out. Bye.'_
A loud sigh. Then something muttered too low to hear.
Callum waited till the door below clunked shut before scurrying up the stairs like a rat. Along the corridor and into the Misfit Mob's office.
As if he was going to hang around, getting drawn into a conversation about DCI Bloody Powel. No thank you.
He scooted into his chair and logged back into his computer.
The email about Irene Brown's known associates was sitting in his inbox, between a memo about not leaving half-eaten takeaways in the pool cars and a lookout request for an OAP who used to specialise in jacking security vans.
Looked as if Irene Brown had lousy taste in men. Eight of them: all violent, all with criminal records.
What on earth was wrong with some women? How could they possibly find that attractive? Oh, you're an aggressive scumbag who steals things and deals drugs? That sounds _dreamy_!
Callum stuck the names into the Police National Computer and ran them again, just in case. Attempted murder. Drugs. Assaults. Housebreaking. Armed robbery. Stealing cars. Rape... Irene Brown certainly could pick them.
Going by the mugshots on file, she was into the sullen muscly type. Tattoos an added bonus. Like Bachelor Number Four.
Callum scooted forward in his seat.
Previous for shoplifting, theft, breaking into old ladies' houses and robbing them blind, nicking other people's cars, and that was it. Nothing on his docket for the last five years. Either he'd gone straight, he'd died, or he'd gone somewhere else. But the most interesting thing was his name: Donald Newman.
Benny, Willow's brother, said his dad was called Donald.
Mind you, he also said his dad owned a tiger, a helicopter, and "loads of bitches", so: pinch of salt.
But still. Bit of a coincidence if it wasn't.
And Newman was what, eight years older than Irene Brown? That was wholesome, wasn't it? A twenty-four-year-old wannabe gangster talking her into bed on her sixteenth birthday. Assuming he even waited _that_ long.
Callum scowled at the screen.
It wasn't as if they could do anything about it, after all this time. If she was over sixteen it was legal. And if she wasn't, try proving Newman was in violation of sections 13, 14, or 15 of the Sexual Offences (Scotland) Act 2009, seven years after the fact. Assuming she even wanted to press charges after all this time.
Still might be able to do him for breaking his daughter's arm...
Mind you, even then, how would you _prove_ it beyond reasonable doubt?
Might be worth getting in touch with Social Services, though.
Yeah, unless they took one look at Irene Brown and decided her kids would be better off growing up in care.
Sod that.
Callum slouched down the stairs, out the back door, and into the rain. Hurried across the rear car park and into the dysentery-brown Mondeo. Wiped the water from his face.
Checked his watch.
Detective sergeants were a pain in the backside at the best of times, but McAdams took the Jammie Dodger. Still no sign of him.
And you could bet your last fifty-three pounds and seventy-two pence that if anyone caught Callum hanging about in the office, doing nothing, they'd find him something unpleasant to do. Much better to hide out here, waiting for McAdams to turn up. And at least he could do something productive while he was waiting.
Callum took out his phone and made a call.
_'Scenes Examination Branch.'_
'Cecelia? It's Callum. I'm calling about the Gossard crime scene – two twenty-three Johnson Crescent. The abduction?'
_'Oh I remember_ that _one: blood everywhere.'_
'Did you get any decent fingerprints or DNA?'
_'Still working through the samples, but it's all the victims' so far.'_
Sod.
A frown. What was it the posh-sounding bloke had said? The one on Dr McDonald's phone when they were going through Brett, Ben, and Glen's fixer-up flat... Right.
'Did you try under the taps and door handles? He would've been clarted in blood, he's not going to risk being spotted looking like an abattoir's floor. He'll have washed his hands.'
_'Course we did. I got Brian to do it. Hold on...'_ There was a muffled conversation on the other end that escalated into a muffled argument. Then a sigh. And she was back again: _'I'll head round there soon as I've finished my tea.'_
'Thanks, Cecelia, you're a star.'
Callum hung up, stuck the seat back as far as it would go, pulled _Open the Coffins_ from his jacket pocket and settled down to read.
A knock on the window made him flinch. He blinked at the dashboard clock – 12:15. A whole five minutes' peace and quiet.
McAdams slithered into the passenger seat and clunked the door shut. Sat there, staring across the car with one eyebrow raised. 'Any time you're ready.' He'd brought a slightly bitter aroma with him: like pear drops laced with marzipan and vinegar. An unwell smell.
Callum turned the page. 'Hold on, I'm nearly finished this chapter.'
'Oh for goodness' sake, not _more_ kiddies' books. I'm beginning to think you've got an unnatural bent, Constable MacGregor.' He snatched the novel out of Callum's hands, shut it, and grimaced at the cover. ' _Open the Coffins_ is a ridiculous title for a kids' book.'
'It's a _classic_.'
'You know Travis stole that from William Blake, don't you? Or was it Milton? No, definitely Blake. "And by came an Angel who had a bright key. / And he open'd the coffins and set them all free..."'
Callum grabbed the book back and slipped it into his pocket. 'You said five minutes, _twenty_ minutes ago.'
'And he's obsessed: rabbits as a symbol of male innocence and virility, cats as feminine cunning and treachery.'
He hauled his seat forwards, started up the Mondeo and stuck her into drive. 'The rust-brown Berlingo's closest: Milgarvie and Kirk, plumbing supplies and services, Cowskillin.'
'Justin Nevin gets transformed into a rabbit by the Wicked Witch of the Well as a punishment for his theft. And by the way, she's only stuck down the well because the villagers chucked her in there, but Justin thinks it's OK to steal the apples from the tree growing over the well even though they're her _only_ source of food. Well, unless a child falls down there, I suppose.'
Callum slid them out of the car park, right, and onto Camburn Road. 'That's _why_ she's got the apple tree there – so kids will try to steal the apples and fall in the well.'
'And that's something else he's obsessed with: witches eating children. Goblins eating rabbits. Monsters eating children. People eating rabbits that are actually children. It's a smorgasbord of transspecies consumption, posing as anthropomorphic cannibalism, but it's really about venal desire. Consume the flesh, violate the body, and absorb it into your own.'
They skirted the edge of Camburn Woods, steering clear of the main roads. 'They're kids' books. They're about magic and adventure, _not_ sex.'
'Just because you read a lot, Constable, it doesn't mean you read deeply. Skimming across the surface like a water beetle, no idea of the pike swimming through the murky depths below.'
Past the cemetery on the left, where a yellow JCB was busy digging a six-foot hole.
McAdams turned to watch the graves go by. 'And what about Justin Nevin's sister, Arya? Nevin is Croatian for "innocent", Arya is Hausa for "false". So the main female character is _literally_ called False Innocence.'
Callum took a right, up a street of Victorian houses with railing-guarded front gardens, across the road at the end and into a narrow cobbled alley.
He pulled up outside a small shop with a dusty front window and an eight-foot-high gate wide enough to drive a bus through. 'MILGARVIE & KIRK ~ FAMILY PLUMBING SPECIALISTS' in big white letters across the blue-painted wood. 'You finished?'
'All I'm saying is that you've got a terrible taste in literature, and you should feel ashamed about it.'
'Screw you, Sarge.' Callum climbed out into the rain. Slammed the Mondeo's door. Then hurried into the shop.
'Nah...' The man in the overalls handed the photo of Ashlee and Abby Gossard back across a countertop littered with bits of copper pipe, valves, grommets, and washers. 'Sorry.'
Callum put the photo in his pocket and showed him one of Tod Monaghan instead. 'How about him?'
'Nah.'
'But you _were_ on Johnson Crescent Wednesday night?'
'Fixing the most disgusting blocked U-bend you've ever seen in your life. Three women, sharing, and the amount of hair down the bathtub drain looked like they'd drowned a Womble. I can give you their number if you like? Manky cows...'
Dundas Bridge was jammed with cars and trucks trying to avoid the roadworks on the main route through Oldcastle.
McAdams grimaced out of the passenger window. 'Traffic's _terrible_.'
'And yet you made me drive all the way through it to pick you up, then all the way back again.' Callum tightened his grip on the steering wheel. 'Remember?'
'You know, moaning and whingeing isn't an attractive quality in a sidekick. You should watch that. I might have to trade you in for narrative purposes.' McAdams' face was the colour of damp newsprint, his breathing coming in little shallow pulses. The bags under his eyes had darkened since he'd climbed into the car, back at DHQ. He checked the dashboard clock, then reached out and clicked on the radio. 'Press conference should be starting soon.'
They were still doing their live coverage from the festival, only whoever was on stage right now couldn't sing in tune no matter how loud they tried. It wasn't even proper words they were bellowing, just noises.
Callum tightened his grip on the steering wheel. 'I'm _not_ your sidekick.'
'Well you're obviously not the hero, are you? You're not even the comic relief – you have to actually be funny for that. All you do is moan and whinge. No wonder the readers don't like you.' McAdams held up a finger, the other hand pressing against his stomach. A grimace. Then: ' _And_ Arya gets transmogrified into a cat.'
That sickly pear-drop smell was getting stronger.
OK, so McAdams might be a pain in the backside, but still... Callum cleared his throat. 'Are you feeling all right, Sarge? Only you look terrible.'
'Remember the scene when she catches and eats that church mouse, even though it's got a family of six to care for? That's a metaphor for women being soft and fuzzy on the outside and all cruel violence on the inside. How they _consume_ men for their own selfish ends.'
'Seriously, you look awful.'
A shrug. 'I'm dying.'
'You're not planning on doing it right now, though, are you? You can't believe how much extra paperwork that'd give me.'
McAdams smiled. 'I'll do my best to hang on till we get back to the shop.' He stretched in his seat, grimacing. 'Do you ever think about the end of your life, Callum? How it's all going to just... stop?'
'I'm not kidding: if you're going to drop dead do it to Dotty. Or Watt, he deserves all he gets.'
'The doctors say I won't see my forty-third birthday. Can you imagine what that feels like?'
Callum stared at him: the stubbly grey hair, the bags under the eyes, the wrinkles. 'You're only _forty-two_? God, that must have been one hell of a hard paper round.'
'The Reaper reaps all men, in time, / His hand has come to rest on mine...'
The song on the radio droned to a halt as they reached the other side of the bridge, replaced by cheering as Callum took them straight across the roundabout and into the posher part of Castleview. Where the streets were wide and lined with trees, and no one's corpse was floating in a bathtub full of brine and herbs.
_'Wooohooo! Wasn't that fan-chicken-tastic? That's Mr Bones there, sponsored by ScotiaBrand Tasty Chickens Limited, let's give them another big hand!'_
More cheering from a crowd that was either too polite to mention, or too drunk to care, that the band had been unable to carry a tune in a rucksack.
Callum navigated past a chunk of open ground, full of trees, and round onto a car park outside a small line of shops.
_'And now here's Gorgeous Gabby with all your news and weather. Any chance of some sunshine, Gabby?'_
_'Sorry, Chris, but things_ might _be looking up for Sunday. Here's the news. Police Scotland have announced they're not looking for any other suspects in the Imhotep Mummy Murder case, after a man's body was pulled from the Kings River outside Dundas House yesterday...'_
He pointed through the window at a florist's, wedged between a domestic appliances shop and a Co-op funeral director's. A lumpy green van sat out front, the side door slid back so a young woman in trousers, shirt, and tie could load big floral arrangements inside. The number plate matched the one on Voodoo's printout.
_'... over live to Oldcastle Division Headquarters.'_
The sound of a general hubbub died down, punctuated by the occasional clack-whine of a camera going off. Then the Chief Superintendent's voice boomed out, amplified loud enough to cause a squeal of feedback. _'Ladies and gentlemen, I'm pleased to announce that O Division officers have successfully concluded the recent spate of mummification murders...'_
Callum undid his seatbelt, but McAdams stayed where he was, clutching his stomach and grimacing. 'Do you want me to call an ambulance?'
A shake of the head. Teeth gritted. 'It'll be fine. I just... need to sit here for a minute. Make sure... the press conference goes OK.'
_'... tireless work by the officers under my command, preventing further deaths at the hands of a deeply troubled individual...'_
'Honk the horn if you change your mind. I can stick the flashers on and we'll be at CHI in ten, fifteen minutes tops.'
'I told you, I'm _fine_. Now... sod off, you're talking... all over the Chief Superintendent.'
He climbed out. Closed the door. Stood there for a moment, as McAdams grimaced and rubbed at his stomach.
Yeah, definitely _not_ looking well.
Callum turned and marched over to the horrible green van. 'You Mrs Reid?'
The woman gave a little start and yelp. 'Argh, frightened the life out of me.'
'This your van?'
She stuck the last arrangement into the back and slid the side door shut. 'Course it's not my van. It's Mrs Reid's van. How could _I_ afford a van?'
Up close she was a lot younger than she'd looked from the car.
'Are you the driver of this vehicle, then?'
Her mouth slammed shut, then she turned and hurried out of the rain, huddling under the overhang outside the funeral director's. 'You're the police, aren't you? Sodding hell.' She bit at her fingernails – already jagged and rough. 'If this is about that bitch's Porsche, I _swear_ those dents were there before I parked next to her.'
He pulled out his notebook. 'Were you driving this van on Wednesday evening?'
'No.'
'You sure?'
Another fingernail disappeared. ' _Please_ don't tell Mrs Reid, OK? I'm not meant to take the van home, but I was out on a late delivery and my boyfriend had his parents' house to himself and we...' Colour rushed up her cheeks. 'Please don't tell Mrs Reid. I'm on my last warning as it is and I _need_ this job.'
He clamped a pen between his thumb and the dirty fibreglass cast. 'I'm going to need his name, his address, and his number.'
A grey-haired woman emerged from the florist's next door, wiping her hands on a stripy apron. 'Is something wrong, Andrea?'
She slapped on a smile. 'No, Mrs Reid. This gentleman was just asking if we do funeral wreaths. I was giving him our number.'
'Well don't be long, that wedding's at three and I want those pedestals all set up by two at the latest.' She pursed her lips then nodded at Callum. 'I'm sorry for your loss.' Before disappearing back inside.
Callum produced his photographs again. No joy.
Two vans down, one to go...
##
'Are you sure this is it?'
Callum checked Voodoo's printout again. 'Postcode is right.'
The registered address for Van Number Three, the grey Peugeot Bipper, wasn't so much a house as a fire-blackened hulk slumped at the end of a track nearly halfway between Oldcastle and Auchterowan. Bordered on all sides by fields with a clump of woods in the middle distance. Its garden was a riot of weeds and grass, that looked as if it hadn't seen a lawnmower for at least a decade. No neighbours.
The damp undergrowth seeped cold moisture through Callum's trousers. 'Who's going to buy a second-hand van and register it here?'
McAdams howched, then spat a glob of yellowy-green into the rosebay willowherb. 'Oldest trick in the book: clone someone's number plates, or register your dodgy vehicle to someone else. Doesn't matter if you're speeding, or parking on double yellows, the police go after the registered keeper, not you.'
Callum called control on his Airwave, 'Brucie? Can I get a PNC check on a Paul Terence Jeffries, the Cloisters, by Auchterowan, OC25 8TX.'
_'Hud oan.'_
McAdams sniffed and leaned on the roof of the car. 'At least it's stopped raining.'
Overhead, the sky was a looming mass of grey, darkening from dove to charcoal at the horizon.
_'Right, Paul Terence Jeffries: did a six-stretch in the eighties for raping a mother of two, with three other offences taken into consideration. Couple of speeding tickets, then nothing since the early nineties... Oh, and his house burned down.'_
Callum stood, looking at the soot-stained walls, blown-out windows, and partially collapsed roof. 'You don't say?'
_'I do say. And there's no need to sound so sarcastic; it only happened Wednesday.'_
The same day Ashlee and Abby Gossard were abducted. No way that was a coincidence.
'Thanks, Brucie.' He hung up and put his phone away as McAdams fought through the weeds to the front door.
Well, where the front door should have been – it was just a yawning black chasm now.
McAdams disappeared inside.
Silly sod.
Callum followed him as far as the threshold. Stuck his head in. 'Is that safe?'
A smile. 'I'm dying of bowel cancer. What's the worst that can happen: the walls fall in and spare me six more weeks of chemo and a slow lingering death? I'll take my chances.' He wandered down the hall, stepped into another room and was gone.
Just because _he_ had nothing to lose, it didn't mean Callum had to join him.
It wasn't safe. Half the roof was still up there, sagging and fire-blackened, ready to come crashing down at any moment. Bang! Crash! Squish! No more police officers.
Callum groaned. Sighed. Then stepped over the threshold and into the burned-out house.
It reeked of smoke – the sweet scent of charred wood mingling with the acrid tang of fried plastics and fabric.
Every floorboard he stood on creaked...
Urgh.
McAdams reappeared at the end of the hallway, sauntering across and in through another door, hands in his pockets, whistling.
Three more doors led off the hall, one hanging open, revealing what looked like a corridor, one through into a grubby bathroom, and one leading down into the depths of the earth.
Callum peered into the partial gloom.
Stone steps, littered with bits of charcoal.
No chance.
What if the ground floor collapsed while he was down there?
McAdams appeared at his shoulder. 'What have you found here? A dark stairway to Heaven? Or one down to Hell?'
Deep breath.
He took the first step, then the second, then the third.
The floor above didn't fall on his head. 'You know, you can't just chop a sentence into chunks and call it a haiku.'
'Yes I can.'
'That's not poetry, it's bad punctuation.' Callum eased himself off the last stone step and onto a hard-packed dirt floor. 'There's a weird smell down here. Sort of sweet and tangy? Kind of herbal?'
McAdams limped his way down the stairs. Did a slow catwalk turn. 'Definitely what estate agents would call a "fixer-upper".'
The floor was littered with bits of wood from the floor above. Callum looked up, through the holes in the cellar ceiling and out to the heavy grey clouds. A faint wash of sunlight broke free, infiltrating the dark room, casting a warm golden glow onto the wall opposite.
'Ah.' McAdams sucked in a breath. 'Do you see what I see, Constable MacGregor?'
Three sets of chains were fixed to the stonework, each one with a rusty padlock making a noose on the free end. What looked like a melted plastic water bowl beside one of them. The burned frame of a metal bed, mattress gone, springs mottled by the heat, beneath another. An upended bucket by the third.
A fourth chain lay in the corner, still attached to the ring-and-plate that must have fixed it to the wall at one point.
McAdams pointed. 'Do you want to say it, or will I?'
Monaghan hadn't been keeping Rottweilers down here, these were for people. So Dr McDonald was right: _women are dirty, subhuman things that have to be trained like dogs. Chained up and beaten..._
The Peugeot Bipper was Monaghan's van and this was his lair.
Callum pulled out his phone and made the call.
'Rather them than me.' The Dog Officer sooked his teeth as a handful of Smurfs picked their way through The Cloisters' burned-out remains. He was a big man with a list to the left and hair poking out the neck of his black Police Scotland T-shirt. 'Had a friend got trapped beneath somewhere like this for two days till they could dig him out. Lost his arm in that one, retired to Portugal.'
McAdams sniffed. 'All right for some.'
'Nah: plagued with haemorrhoids. Big as a grapefruit.' He swept a hairy hand up and out, indicating another bit of the huge, overgrown expanse of back garden. 'Come on, Penguin, off you go, you lazy sod.'
A black lab in a wee high-viz waistcoat snuffled away into the damp undergrowth, nose down, tail wagging.
Callum turned, one hand held above his eyes like the bill of a baseball cap. A horrible little Fiat Panda was lurching its way down the track towards them, bringing a swirling cloud of grey-blue smoke with it. 'Mother's here.'
That got a grunt from McAdams. 'A pound will get you five, / That she'll skin us both alive, / For delving in the cellar, / Of this terrible Jeffries fellar, / And risking both our instant deaths, / "Reproach" shall be her shibboleth.'
The dog officer raised an eyebrow. 'Just make that up, did you? Cause rhyming "cellar" with "fellar" isn't exactly Wordsworth, is it?'
'Everyone's a critic.'
Mother's Panda came to a juddering halt behind the small collection of SEB Transit vans, gave one last _vrooooom_ , then the gunshot retort of a backfire, and silence.
'I'm just saying: resorting to doggerel in the middle stanza undermines the poetic integrity of the piece. That's all.' Dog Man stuck a hand against his chest. 'For delving in the depths below, / Of this, our dark and deadly foe.' A nod. 'See? Much better.' Then he cupped his hands around his mouth. 'PENGUIN! WHERE THE SODDING HELL DO YOU THINK YOU'RE GOING?'
The black lab disappeared into a massive knot of brambles – five-foot high and covering at least a third of the garden.
'Sodding dog's a pain in my parliament, if you'll pardon the French.'
Mother climbed out, sleeves rolled up on her fleece, bare white arms semaphoring in the sunlight as she marched over to the nearest Smurf.
McAdams stuck his hands in his pockets. 'Save yourself, young Callum. Run away before she gets here. This will not be pretty.'
The Dog Officer nodded. 'A six-eight-six haiku. Very avant-garde. PENGUIN, I'M NOT TELLING YOU AGAIN! Stupid animal.'
Rustling and crackling came from deep within the brambles, but that was it – no Labrador to be seen.
The Smurf turned and pointed in their direction. And Mother was on her way.
'PENGUIN!' A shake of the head. 'Tell you, we get all the rubbish dogs in O Division. Anything that can't find its own tail: they send it here.' Deep breath. 'PENGUIN! OUT HERE NOW, YOU USELESS WEE SOD!'
Mother rounded the corner of the blackened house, shoulders forward, hands curled into fists.
McAdams stood up straight. 'We who are about to die, salute you.'
'PENGUIN!'
'Andrew Thomas McAdams, what in _God's_ holy name do you think you were doing going down there? Are you _insane_?'
He just shrugged.
'PENGUIN, IF YOU'RE NOT OUT HERE BY THE TIME I COUNT TO FIVE, I'LL SKIN YOU AND WEAR YOU AS A POSING POUCH!'
Callum stepped in. 'If DS McAdams hadn't put his neck on the line, we wouldn't have found Monaghan and Jeffries' dungeon.'
'FOUR!'
She turned her scowl on Callum instead. 'And _you_! You should've kept him out of there, you know he's not well!'
'THREE!'
'At least now we know Monaghan wasn't working alone. There was no way he could prepare his victims at that tiny flat on Bellfield Road – he needed room to starve them before he gutted them and stuck them in the smoker. This is where he did it.'
'TWO!'
Mother thumped the Dog Officer on the arm. 'Would you _please_ stop doing that while I'm giving these idiots a bollocking?'
'Cadaver dog, my fuzzy backside.' He zipped up his Police Scotland fleece, hauled on a pair of leather gloves, then stomped towards the brambles. 'PENGUIN!' He dropped to his hands and knees and crawled into the barbed-wire mass. 'AAARGH! SODDING SPIKY SONS-OF-A-BITCHING BRAMBLE... GAAAAAARGH!'
Callum tried his best reasonable voice. 'Dr McDonald thinks there's still a chance to save Ashlee Gossard. Come on, it can't be a coincidence, can it? Someone spots the fire here and calls nine-nine-nine at six twenty – an hour and a half later, Ashlee and her mum are being attacked and abducted from their home. And a van registered to _this address_ is right there in the vicinity.'
'I'M GOING TO KILL YOU, PENGUIN! YOU HEAR ME?'
Mother stared up at the heavy lid of looming clouds for a moment, then sighed. 'Andy, you _know_ it was stupid going down there. What if the floor collapsed?'
McAdams smiled. 'There are worse things in life than death.'
'PENGUIN! WHERE ARE YOU, YOU STUPID...'
Callum pointed at the burned-out house. 'Look at the timing: Brett Millar turns up at his parents' house in Blackwall Hill, this place is set on fire, Ashlee and Abby Gossard get abducted.' He held his hands out, like he was finishing a magic trick. 'Millar got free and escaped – that's why there's a loose chain in the basement, pulled from the wall. Monaghan and Jeffries can't risk him leading us back here, so they torch the house. Only _now_ they don't have anyone to mummify, so they go out and abduct themselves a pre-starved teenaged girl instead. Jeffries is still out there, and he's going to kill her soon as he thinks she's ready to become a god.'
Mother shook her head. 'Yes, well done. All very logical and exact. Only your Paul Terence Jeffries isn't "out there" or anywhere else: he's dead.'
Oh...
'HELLO?'
'I got John and Dotty to go a-rummaging. The Cloisters belongs to an ecclesiastical trust. Jeffries was some sort of lay preacher, so not only did he live here rent-free – they paid him a wee stipend too. He stopped cashing the cheques, wouldn't answer any letters, they couldn't track him down anywhere, so they went to court and eventually had him declared dead. That was twenty years ago.'
McAdams sniffed. 'So who's been staying here?'
'That's the trouble – they didn't know they still owned it, till Dotty phoned and made them go through the files. Turns out they've been paying council tax on a derelict property for over two decades. Not the most efficient biscuits in the tin.'
'HELLO, CAN YOU HEAR ME?'
'Your hairy friend doesn't shut up, does he?' Mother shrugged her shoulders. 'Anyway: even if Jeffries _was_ still alive, he'd be in his seventies by now. Doesn't sound very abducty, does it?'
'So Monaghan knew Jeffries was dead, knew his house was abandoned, and registered the van to this address, in a dead man's name, because he knew no one would ever check.'
'I'M NOT KIDDING ABOUT HERE!'
McAdams spat into the long grass. 'The Dog Man is stuck. Inside the brambles' clutches. Their thorns bind him tight.'
Mother nodded. 'Exactly. So we need to ask: _how_ did Monaghan know? Does he have some connection with the ecclesiastical trust? How many other empty properties does he know about? Because one of _those_ is where he left Ashlee and Abby Gossard before jumping in Kings River.'
'ARE YOU BUGGERS DEAF OR WHAT? SOMEONE NEEDS TO GET IN HERE, NOW!'
She scowled into the brambles. 'Callum, I hate to ask, but can you go see what he's yelling about?'
Oh, lovely.
Callum stayed where he was. 'Maybe we should get this ecclesiastical trust to go through their financial records and find out what other empty properties they've forgotten about?'
'SOON AS YOU LIKE!'
Callum growled out a breath. Buttoned up his suit jacket. 'ALL RIGHT, ALL RIGHT! God's sake...' The long grass was sodden with weeks of rain, making his trousers stick to his legs, soaking through his shoes and into his socks. Urgh...
Dog Man had flattened a path into the brambles and he followed it. Dropped to his hands and knees and picked his way into the spiky mass of horrible bloody stabby jabby— 'OW!'
This stuff was worse than barbed wire. At least a million times sharper. And the bloody thorns came off and stuck in his good hand and they grabbed at his suit jacket and— 'OW! AAAAARGH! I _HATE_ BRAMBLES!'
He battered at them with his filthy fibreglass cast, but they just bounced right back at him. Only now they were angry. 'AAAAAARGH!'
Another ten feet of horrible jaggy stabbing needle-jabbing horror and he emerged into a little hollow, ringed with the old yellow-grey corpses of long-dead brambles. Officer Hairy the Dog Man was sitting off to one side, but Penguin the useless cadaver dog was right in the middle, surrounded by what looked like burrows. Not tiny ones like you'd get with rabbits, but bigger. Maybe a fox, or a badger?
Penguin was lying down, tail thumping against the earth – dry in here under the canopy of horrible spiky tendrils.
Callum sat back on his thighs, still hunched over to avoid their spiny crown. 'You better have a damn good reason for dragging me in here, ruining my suit, and look at my _bloody_ hands!' Literally – covered with scratches and puncture marks seeping red, peppered with dozens of tiny brown thorns. Even the fingertips of his bad hand were lacerated where they protruded from the cast.
Dog Man pointed at a white rock poking out of the ground near one of Penguin's front paws.
Only it wasn't really white at all, it was a sort of off-ivory colour, the size of a half-deflated football. The rock had holes in it, exposed where it poked out of the dirt: one roundish, one an arrow shape. Wait a minute, were those _teeth_?
It was. It was a skull, lying on its side, one eye socket and half the nasal cavities exposed to the air, the rest buried beneath the ground. And it was definitely human.
Maybe there was a good reason no one had heard from Paul Jeffries for over twenty years.
## — Father —
"If there's one thing I've noticed," purred the Goblin Queen, "it's that the people who pretend to be the bestest, and the nicest, are almost always the worstest and most horrible."
"I'm sorry we ate your cabbages," Russell said. "We were hungry and lost and we didn't know they belonged to anyone!"
"That's very honest of you. You're a good little rabbit." She patted him on the head. "But I'm going to have to eat you all the same."
R.M. Travis
_Russell the Magic Rabbit_ (1992)
_And ma dad beat the f*ck outta me as a kid,_
_Got his bones in a box with a button-down lid,_
_And I'll never forgive all them things that he did,_
_But he ain't doin' them no more, cos the b*stard is dead._
Donny '$ick Dawg' McRoberts
'F*ck U (Daddy Dearest)'
© Bob's Speed Trap Records (2014)
##
**_A Whole Lifetime Ago_**
Paul smiles and nods. Wise and trustworthy as the fat bitch in the cardigan drones on about 'Jesus' and 'love' and 'forgiveness' and all the rest of the _crap_ people like this always drone on about.
The vestry is hot and sticky, even with the windows open.
That's the problem with Catholics, though, isn't it? The lingering heat of Hell is never far from their guilty little consciences.
She's still talking – that glistening mouth with its red liver lips. All pious. Like she's never sucked a dick in her life.
Blah, blah, blah.
On and on and on.
A thick floral stench rolls off her, mingling with the sweaty taint of corpulence.
You know what would be nice?
No, what?
Strangle her. Right here, right now. Wrap your hands around that greasy throat and _squeeze_ till her eyes bulge and all the blood vessels burst and flood with red and keep _squeezing_ and _squeezing_...
'... don't you think, Father?'
Paul blinks. It's not the fat bitch, it's the priest in his long black robes. Dressed like a jackdaw. Desperate to be strung up on the back fence to act as a warning to others.
He clears his throat. 'I do indeed.' He gives them the smile and the nod he's practised so many times in front of the mirror. The one he uses to pretend he's human like the rest of them.
But the stench in here is getting too much to bear.
So he makes a big show of looking at his watch. Rolls his eyes and tuts. 'Sorry, I'm afraid I'm going to have to dash. But perhaps we could pick up where we left off next week?'
The man in the dress nods. 'Oh, we'd like that, wouldn't we, Margery?'
'Oh yes. Yes indeed.' She puts one of her flabby hands on his arm, warm and sticky even through his jacket and black shirt, globbing into his skin. 'Do come again. I'm so glad we could help.' Then she hands him the cheque with his name on it.
He slips it into his jacket. 'And please, pass on my thanks – and the thanks of all those poor neglected children – to your kind and generous congregation.'
The fat bitch stops at the threshold, but Father Crossdresser follows him out into the church.
There's a wee boy singing somewhere: _'Sanctus, Sanctus, Sanctus, Dominus Deus Sabaoth...'_ the sound echoes off the walls like an air-raid siren for the damned.
'Well, Father Jeffries, I must say, I'm deeply impressed by the work you do with these poor deprived youngsters. It's an inspiration, it truly is.'
_'Pleni sunt caeli et terra gloria tua. Hosanna in excelsis...'_
Paul gives him a little modest shrug. 'We all must do what we can.'
_'Benedictus qui venit in nomine Domini. Hosanna in excelsis—'_
_'NO! NO, DAMN IT, OLIVER!'_ The exasperated voice of some poor sod trying to make a kid do what it's told. _'How many times? It's pronounced, "ex-chel-cees". We're going to stay here and do it again and again until – you – get – it – right!'_
Paul turns, and there's his own little burden, sucking his thumb like a baby. Five years old and he's _still_ sucking his thumb. How is he ever going to be a man, acting like that? So Paul raised his voice over the wailing chorister. 'Justin, thumb out of the mouth, eh, champ?' He forces a smile. 'You're a big boy now.'
Going to have to beat that out of him.
He'll never learn otherwise.
Beat it out of him till he learns his bloody lesson.
After all, it never did Paul any harm, did it?
The back garden sways and lurches as he... he... why is he out here again?
Oh, yeah, right.
A swig of beer from one hand, while the other fumbles... fumbles his cock from his trousers.
Used beer splashes into the compost bin, adding its steam to the heap's.
Dark out here. Just the light of the moon filtering through clouds.
Watching everything.
Never mind God an Jesus an... an all the rest of that _crap_. Moon. Moon was... what they should be worshipping. Like the old days with... you know... with virgins an sacrifices an... yeah.
Bitch in the basement... crying. Always crying. Specially afterwards.
Pff...
He's not a _bad_ man. No, he's a _priest_!
A proper one.
Not a shep... shepherd.
No.
He keeps his flock... his flock in line the _old_ way. Proper way. Unner the Moon's eternal... eternal eye.
A wolf.
Paul takes another hit of beer, an... an throws back his head an howls out his devotions.
'That's very generous. Thank you so much.' Paul takes the cheque and smiles, even though it's barely half what he got here a year ago. Doesn't do to burn the goose, even if it _is_ only laying silver eggs right now. 'The Romanian orphanages will make good use of this.'
The skinny bitch in the twinset shows off her dentures – stained with dark red lipstick. 'I'm _so_ glad we could help.' Protestants, just as bad as the Catholics, only without the sense of theatre. Holier-than-thou on the outside, deviant scumbags on the inside. 'Now, we _must_ get a photograph for the church newsletter.'
He sighs and shakes his head. Puts a hand on her revolting shoulder, among those nasty little flakes of dandruff. 'I'm just His humble servant, I don't deserve all that limelight and praise.'
'Oh, but—'
'No. _You_ should take the credit, Mrs Ingram. After all, _you're_ the one who raised all this money. I'm just the one who's lucky enough to spend it on a very good cause.'
'Lies and _liars_. All of them.' Paul takes the last swig from his tin of Special, crumples it in his fist and hurls it into the corner. 'And _morons_!'
The living room sways slightly, the wallpaper twisting like the tattoos on a topless dancer.
'Boy! Justin, or whatever. Gimme another beer!'
After all, can't celebrate without beer, can you?
No.
And whisky.
He takes a swig of Glenmorangie, straight from the bottle. It tingles on the way down like a thousand little watch fires, flickering in the darkness.
'BOY!'
And there he is, the horrible little snivelling boy. Standing there in his stupid shorts and cartoon T-shirt. Eyes all big and shiny, like he's going to burst into tears any moment. Pathetic. He's six, for Christ's sake. Far too old to be acting like that.
He holds up a tin of Special, dew beading on the outside, and Paul snatches it off him. 'About time.'
'Sorry, Father.' That annoying, wobbly little voice.
'You should be happy.' Paul grabs him by the back of the neck and pulls him close. 'I saved you, boy. You know that don't you?'
'Thank you, Father.'
'Damn right.' He pushes the little shite away. 'I saved you from the grey. From the beige. From all the crap they shovel down kids' throats.' He rips the tab from the beer and swigs down a mouthful. 'Your mum didn't love you, Justin. She would've killed you and eaten you. I know. I've seen it before.'
'Thank you, Father.'
Better. Bit of respect for his elders and betters.
Paul swills beer between his teeth.
After all, he's a rich man. He _deserves_ respect.
And soon as he's spent it all, there's always another congregation of pious pricks desperate to throw money at him for all his 'good works'.
He toasts the crucifix on the wall. His generous benefactor.
Oh yes, today is a day to celebrate.
'Boy?'
'Yes, Father?'
'Tell your new mummy to wash herself. She's going to be blessed tonight. And if you're a _very_ good boy, I might even let you watch.'
##
Callum turned his back on the garden, the thumb of his broken hand poked into his right ear, phone clamped to the other. 'Sorry, Cecelia, can you say that again?'
The roar of small petrol motors battered back and forth, screaming, then falling, then screaming, then falling as three Smurfs with brush cutters fought against the thicket of brambles. Chopping it down to the ground. Making good progress too – half of it was already gone, the curled jagged stems carted away, leaving the ground grey and bare.
_'I said, we've finished our second sweep of the Gossard house.'_
'Cool.' He marched away from the noise, following a wheelbarrow-pushing Smurf around the side of the burned-out house. 'Anything?'
_'Won't know for sure till we analyse it, but there was a tiny smear of blood under the knobs on the kitchen taps. Nothing in the sink, or on the knobs – so he's tried to clean up after himself – but there's always traces.'_
Professor Whatshisface might have been a bit of a dick, but he obviously knew what he was doing.
'Let me know, OK?'
Wheelbarrow Smurf tipped his load of bramble clippings into a skip parked outside the house.
_'I heard about Elaine.'_
Great.
Callum ran his fingertips across his forehead. 'Can we not do this just now?'
_'I just think it's a good time to tell everyone the truth about who really cocked up that crime scene.'_
'Who's going to take me seriously? They'll all think I'm lying to get back at her for shagging Powel behind my back.'
_'Don't make me stage an intervention, Callum.'_
'Bye, Cecelia.'
Mother appeared from the house's blackened doorway, SOC suit rolled down and tied off around her waist. Talking into her phone as she wandered down the path. 'Is he? That's great, thanks, Duncan... No, we'll be there soon as we can... OK... I owe you one.' She hung up, turned and waved at him. 'Callum, how's it going?'
He pointed back towards the garden. 'Another couple of hours to finish clearing the scrub then they can get the Ground Penetrating Radar in. They'll find the rest of him.'
'Good. Now do me a favour and give Dotty a call: see if she can find us a connection between Jeffries and Monaghan. She's good at digging things up and I get the feeling our boys knew each other.' Mother held out an arm and he took it, helping her balance as she shimmied her way out of the Smurf costume. 'I miss the white Tyvek suits, don't you?'
'Nah.' He pulled a face. 'I always thought we looked like a bunch of sperm in those. Everything You Always Wanted to Know About Crime Scene Investigation But Were Afraid to Ask.' He pulled out his notebook. Flipped through to the details he'd copied down. 'Assuming the ecclesiastical trust people started proceedings on the day Jeffries died, that's twenty-seven years ago. Monaghan would have been eight.'
'Probably not _best_ friends, then.' She puffed out her cheeks and turned to face back towards the house. 'What does your gut tell you, Callum: are these Paul Jeffries' bones?'
'Bit of a coincidence if they're not. Jeffries goes missing twenty-seven years ago, but he never even leaves the property, just turns to bone in a shallow grave.' Callum sniffed. 'Well, it's that or he killed someone, buried them in his back garden, and did a _very_ thorough disappearing act.'
'True.' She stood there, frowning at the burned bones of a long-dead house. Then turned, 'Anyway, there's no point hanging around here, is there? They'll call us when they need us. In the meantime, I think you and I should go on a little field trip, don't you? Spend some quality time together.'
Why did that sound ominous...?
The Fiat Panda roared and spluttered, coughing like an old man on sixty-a-day as they lurched around the City Stadium Roundabout and into the long queue of traffic heading west on the dual carriageway. Orange cones and brake lights stretched ahead of them, fading into the gloom as the first drops of rain splashed against the windscreen.
Fifteen minutes and they'd barely gone a mile and a half.
Callum sat in the passenger seat, with his hands in his lap. Because otherwise there was a risk of touching something in here. The dashboard had developed a new feature: someone had written 'MUCK!' in the dust. With every lurching gear change, bottles clinked in the back.
Something sticky glistened in the passenger footwell and Callum shifted his leg away from it. 'I'm worried about DS McAdams.'
'Welcome to my world.' She cracked the window an inch, letting the drone of traffic _in_ and a fat black fly _out_. 'Rosalind seems to be integrating well, doesn't she? Dotty speaks very highly of her. Very efficient.'
'I'm serious. He's holding his stomach and wheezing, sweating. I've seen bottles of milk with more colour in them.'
'He's dying, Callum. This round of chemo...' Mother sighed. 'I remember the good old days, before the old ticker started acting up.' She tapped herself on the chest. 'It was Andy and me that arrested Ian Zouroudi. We caught Dani McGiven. We nailed Joanne Frankland, even though everyone thought her brother Stephen did it.'
'Maybe... I don't know. He shouldn't be at work, he should be home resting.'
'Then there was that counterfeiting syndicate, operating out of a charity shop on Dresden Street.'
'Tod Monaghan's dead: McAdams got his last serial killer. We'll find Ashlee Gossard and that'll be that. Everything else is just tidying up. Send him home.'
A smile tugged at Mother's cheeks. 'I remember one time, we'd spent all day chasing down an armed-robbery witness, and Andy—'
Callum's phone burst into song, buried deep in his pocket. 'Sorry.' He pulled it out. 'Hello?'
Elaine's 'angry schoolteacher' voice scraiched in his ear: _'Did you pick up your stuff? Because I don't want you turning up at the flat pretending you forgot something.'_
'Oh for God's sake.'
_'If I get home from Mum's and you've—'_
'Bye, Elaine.' Callum hung up. Checked his call history and barred the number she'd dialled from.
The hairs prickled on the back of Callum's neck, and when he turned his head Mother was staring at him. 'What?'
'Do you want to talk about it?'
He folded his arms. 'No.'
'You're going to have to deal with this sooner or later, Callum. That's how relationships work.'
The industrial estate ground past on the right.
Callum kept his arms folded. Scowling out at the traffic in its depressing slow-motion waltz. Brake, two-three. Inch forward, two-three. Brake, two-three. And repeat. On and on till the end of the world...
Up ahead, a big metal sign announced that all this traffic chaos was going to continue for at least the next six weeks. Because everyone wasn't miserable enough already.
'I'm serious, Callum.' Mother indicated, then drifted right into a turning lane marked out with another stretch of orange cones. Sat there with her turn signal clicking. 'Are you going to spend the rest of your life avoiding her?'
He kept his scowl pointed out of the passenger window. At the cars full of miserable people, stuck in their miserable lives, stuck in miserable traffic, in the miserable rain.
A sigh. 'What's going to happen at work, then? When Elaine comes back after the baby?'
Someone coming the other way flashed their lights, leaving a gap in the traffic and Mother put her foot down, sending the car kangarooing across the opposite lane and down a road lined with dark-green gorse.
'And what about Reece? He's still going to be a Detective Chief Inspector, no matter how much you hate him. He'll _still_ be able to give you orders. And if you're fighting with Elaine, he'll win that argument every time.'
'I get it, OK? I'm screwed. You happy? I – am – screwed.'
'That's not what I meant, Callum. You're going to have to make your peace sooner or later. And the longer it takes, the more it'll hurt.'
'Can you just leave it, please?'
A sigh. 'Well, no one can say I didn't try.'
She indicated and turned left, down another gorse-lined road, bordered by fields. A patch of woods. And just beyond them, the hulking grey lump of Her Majesty's Prison, Oldcastle. The kind of building not even an architect could love: brutally minimalist, with featureless walls and a three-storey glass block built out front that had a wide concrete portico like a Seventies hotel.
Mother pulled into the car park, and the Panda did its roar-and-backfire trick again, before dying into silence.
He unclipped his seatbelt. 'Going to tell me what we're doing here?'
She just grinned at him.
Oh joy...
They'd made a bit of effort with the interview room: framed prints of landscapes screwed to the wall; a fake rubber plant in the corner; flattering mood lighting; one of those automatic wall-mounted air-freshener things they sometimes had in toilets, puffing out the occasional fruity whiff. But even that couldn't disguise the fact that this was an eight-by-ten windowless box in the west wing of a prison.
It didn't do much to shift the underlying sour whiffs of grease and BO either.
Mother sat at the interview table – chipped and scarred, with initials and swearwords scratched into the Formica. She poked away at her phone, playing Candy Crush by the look of things.
Callum stood by the radiator, drying off his damp, bramble-ruined trousers. 'Still don't know what we're doing here.'
'Good things come to those who wait. You just have to have faith and patience, and—'
The interview room door opened and a small balding man in thick-rimmed glasses and an ugly jumper sidled into the room. 'Flora. It's lovely to see you again. I was so shocked to hear about your... _incident_. I trust you're fully recovered now, yes? Good. Excellent. Yes.' A nod for Callum. 'You must be the young man, Flora's told me so much about. Nice to meet you.' He stuck out his hand – surprisingly warm, strong, and dry when Callum shook it.
Mother put her phone away. 'Is he ready, Duncan?'
'Oh yes, yes. Yes indeed.' Duncan clasped his hands together. 'Now, I know you've been here before, but there are a few formalities to take care of. You're not to give the inmate anything, and you're not to take anything _from_ him. That includes messages to, and from, the outside world. You're not to let him use your mobile phones. We disapprove of physical contact. And a staff member will be present at all times. OK?'
She spread her hands wide. 'How can I refuse?'
'Good. Yes. Well, let's get started, shall we?' Duncan stuck his wee baldy head out into the corridor. 'You can bring him in now, Rachael, thanks.'
Gareth Pike had to duck to get into the room, his rounded shoulders brushing the door frame. His pale scalp gleamed like a freshly polished cue ball. He paused and pulled on a slow sticky smile. 'Constable MacGregor, how delicious to see you again. Are you here about my south-facing cell?'
Pike.
A prison officer appeared behind him, closing the door and standing with her back to it, at parade rest, face blank and jaw set.
'Hello, Mr Pike.' Mother pointed at the chair on the other side of the table. 'Or can I call you, Gareth?'
'This one's new.' Pike slithered his way into the other seat, and sat hunched forward with his elbows on the table. Like a bear stuffed into a highchair. 'I'm not sure I like her yet.'
'I can quite understand that, Mr Pike. My name's DI Malcolmson and I'm here to formally apologise on behalf of Police Scotland for Detective Constable MacGregor's behaviour.'
_What?_
Pike's eyebrow climbed an inch. 'Are you now? Well that _is_ interesting, isn't it?'
'I've reviewed the footage of your interview and I have to say that I'm more than a little disappointed. After all, you're not a well man, are you, Mr Pike?'
'No, I'm not. And please,' he reached across the table and took Mother's hand, 'Call me Gareth.'
'You've got type-two diabetes, angina, high blood pressure, gallstones, impotency...' 'These are my burdens to bear.'
Mother turned in her seat and stared at Callum. 'Officer MacGregor, is there something you'd like to say to Gareth?'
'What? No. I'm not—'
'Apologise, Constable.'
Was she insane? There was no way he was—
' _Now_ , Constable.'
Silence.
The prison officer didn't move.
Duncan tugged at the hem of his ugly jumper.
And Pike smiled his big slug smile.
Mother sighed. 'Or perhaps you'd like another spell in front of Professional Standards?'
All that caring-sharing crap in the car had been just that: crap. She hated him as much as everyone else, and now she was rubbing his face in it.
That's what he got for trusting her. That's what he got for trusting anyone.
So what choice did he have? Apologise to this vast slimy sack of sick, or get a formal kicking by the rubber heelers for something he didn't even do. _Again_.
Callum cleared his throat. 'I'm sorry, Mr Pike, if I caused you any offence.' Every word burned like battery acid on his tongue.
Pike's smile got bigger. 'Oh, pish and tush, what's a little banter between friends?' He raised his other hand. 'You are forgiven, Constable.'
Mother nodded. 'You're very kind, Gareth. And to show that we in Police Scotland are big enough to admit our mistakes: I've had a word with the Sheriff.' She pulled a bit of paper from her pocket and held it up. 'Given your medical condition, he's agreed to give you a Community Service Order instead of a custodial sentence. You'll be able to stay in your own flat, and only have to spend a few hours a week helping out at the Kingsmeath Animal Shelter. Cleaning out the cages and things like that. Health permitting, of course.' She tucked her paperwork away again. 'Isn't that nice?'
Pike's smile slipped. Then his mouth hung open, eyes widening. 'Oh. Well... it's a very _generous_ offer, but I sincerely fear that my chances of recidivism are—'
'No, let's not hear another word about it. It's the least we can do.' She prised her hand free and stood. 'I'm sure they'll make your stay here as pleasant as possible until the Sheriff makes it official on Monday.'
'But, no... You can't... I mean, I've _forgiven_ Officer MacGregor. I don't expect any special treatment.'
'I wouldn't hear of it.' She turned and clicked her fingers at Callum. 'Come on, Constable, Gareth's had a long day, he'll be needing his rest.'
'But I'm a pederast! You saw the tape, it's got kids on it and—'
'As you say, Gareth, you're impotent. The Sheriff feels that puts you at a much smaller risk of causing actual harm.'
The prison officer put her hand on Pike's big round shoulder. 'Come on, time to go back to our cell.'
'But I'm dangerous! I...' He stared at Callum. 'Tell them. Tell them how dangerous I am! I tried to abduct you!'
And now it all made sense.
'You _were_ dangerous, but like Mother says, you need a little blue pill to get it up these days. And with _your_ blood pressure?' He sucked a breath in through his teeth. 'You're more of a risk to yourself than other people.'
Pike's mouth flapped up and down for a bit, but nothing came out.
Mother patted him on the arm. 'There, there. I know it's all a bit overwhelming, but the knowledge that you're on your way home is thanks enough.'
'I DON'T WANT TO GO HOME!'
The prison officer tightened her grip. 'Easy now.'
'You don't?' Mother frowned.
'I live in a shit hole: don't send me back there!'
'Oh dear and here's me already made that deal with the Sheriff. If only there was _something_ I could do...' She tapped a finger against her forehead, frowning down at the tabletop. 'Come on, Flora: think.' A pained expression. 'I've no idea... Callum? Can you think of anything?'
And for the first time since arriving at the prison, Callum smiled. 'Oh, I'm sure I can come up with _something_. Won't that be fun, Gareth?'
There was a pause, then Pike bared his teeth. ' _Never_.'
'Ah well. We tried, didn't we, DI Malcolmson?'
'We did indeed, DC MacGregor.'
'That's not fair: YOU SET ME UP!'
'Tell you what,' Mother nodded at Rachael the prison officer, 'this nice lady will take you back to your lovely clean comfy cell, with... what was it: three meals a day and the convivial company of like-minded fellows? And you can have a nice _long_ think about living in that manky little flat on the thirteenth floor, surrounded by people who hate you. Who knows, maybe you'll change your mind?'
They waved as he was pulled from the room.
'YOU SET ME UP!'
##
Watt folded over his desk, curling in around the phone in his hand. 'Yes, I _know_ it's seven o'clock on a Saturday evening, but as I told your colleague, this is very, _very_ important.'
The rest of Mother's Misfit Mob sagged in their chairs, McAdams perched on one of the spare desks, Mother leaning against the wall by the whiteboard – twiddling a marker pen.
Dotty sighed. 'Told you _I_ should've phoned. He's useless.'
'No, I understand that... Yes... Yes... Here's the thing,' Watt thumped a hand down on the desk, 'there's a thirteen-year-old girl who's going to die if you don't get us that list. How does that sound? Does that get your ecclesiastical juices flowing?'
'See: he's blown it.'
He stuck the phone against his chest and birled around in his seat. 'Will you _please_ shut your fat cakehole?' Back to the phone. 'Sorry, what was that?... Yes. Right. OK, soon as you can. Here's my email address John— No, J for Jumper, O for Osprey, H for Hawk, N for November...'
Dotty dropped her voice to a barely audible growl: 'I'll give you a "fat cakehole", you gingery wee twat.'
Mother chucked the pen so it bounced off Dotty's desk partition. 'If you kids can't behave, there'll be no jelly and custard.'
Callum stretched in his seat and stifled a yawn. 'Pfff...'
'... dot _police_ dot UK... Uniform Kilo... Yes, UK. Soon as you can, thank you.' Watt hung up and slumped back. 'God save us from religious bawbags.'
'Anytime you're ready, John.'
He grimaced at Mother. 'They thought it'd be OK to stop work at five, as it's a Saturday. Apparently, in their sparkly little world, you only have to comply with a warrant if you've got nothing better to do. They're going back to it now, but it's going to take hours – nothing's on computer, it's all filing cabinets. Be lucky if we hear back by Monday.'
McAdams buried his face in his hands. 'For God's sake.'
'Rosalind?' Mother wandered over and collected her hurled pen.
'Every traffic car, community warden, uniform, and special constable is keeping an eye out for Monaghan's grey Peugeot Bipper. He must've parked it somewhere, but so far no one's spotted it.'
'Dorothy?'
'He shouldn't have called me fat! It's not my fault I'm in a wheelchair, is it? You try exercising when you're stuck in one of these things.'
'Please, Dorothy, we're all tired. How did you get on?'
'The council's sent through a list of all properties currently registered as empty for council tax purposes. It doesn't include buildings considered uninhabitable.'
'What about some sort of Monaghan–Jeffries connection?'
'Still looking.'
'Callum?'
He pointed at his screen. 'I've been through the planning department's records and there's no sign of anyone applying to build a new smokehouse for over forty years. And the last one was converted into, and I quote, "a stylish four-bedroom family home, with off-road parking, outdoor hot tub, and a well-appointed garden" two years ago.' Callum frowned up at the map of Oldcastle pinned to the wall above the kettle. 'Look, according to Voodoo's CCTV search, Monaghan's grey Peugeot Bipper came from somewhere east of Caven Street, Logansferry, and disappeared some-where north of the Royal Williams Hospital in the Wynd. We could try narrowing our search by getting rid of anything in between?'
Franklin pulled a face. 'Risky – he might've dropped them off en route.'
'True.'
Mother thumped down into the only vacant chair. 'Urgh... Ashlee's going to be dead by the time we get there, isn't she? Assuming we _ever_ find her.'
The only sounds were the humming computers and someone squeaking past in the corridor outside.
In the end, it was McAdams that broke the silence. 'We'll find her. We just have to hope she can hold on till Watt's religious time-wasters get back to us.' He hopped down from the desk. 'In the meantime, we go home. Get some rest. There's sod-all we can do here till we get that list.'
##
'Mmsrrry...' Ashlee's head rolls back against the metal tank. The chains clank and click. The walls _thrummmmmm_ in the darkness.
Every muscle aches. Not just from being sick all the time, but from... don't know. Just aches.
'Mmmy. Mmmmy mmsrrry...'
Her mouth barely moves now, all crusted and covered in scabs. She can taste them on the tip of her dry tongue. Split and bleeding, then scabbing over, then splitting again.
Probably all the salt in the water.
It's gritty between her fingers, makes a tidemark of pale-grey crystals around the tank, makes the water undrinkable, no matter how thirsty she is.
'Mmsrrry...'
And she is sorry. Totally, _totally_ sorry.
Sorry she let him in. Sorry she was such a crappy friend. Sorry she was such a crappy daughter. Sorry she was ever born.
But it'll be over soon.
There was this TV show a couple months ago, all about how stupid people died in the woods and deserts and up mountains and at sea and crap like that. Places no one with even _one quarter_ of a brain would go. The woman with the utterly ugly anorak and frizzy hair was banging on about being adrift in a lifeboat, or something. And how there was all this water, but you couldn't drink it, or you'd die.
Cos of the salt.
It does something to your insides and you die of thirst even though you're drinking water. Even if you throw it up again, it screws with your kidneys and you die.
But you're going to die anyway, aren't you?
Cos _not_ drinking anything screws your kidneys and you die.
Her throat is like the road outside the house on a hot summer's day, all dark and sticky, and every horrible breath is like ripping off a sticking plaster.
Maybe dying's not so bad?
Got to be better than this.
Her head lolls to the side, and there's Mum. Still slumped with the chain around her neck. Still not moving. Cos zombies aren't real, are they.
It's better being dead, isn't it, Mum?
Yes it is. It's a relief, to be honest with you. After all the years of struggling and fighting, trying to make ends meet. Trying to make friends. It's nice just to have a bit of a rest.
That's good.
You should try it too, Ashlee. You'll like being dead. Nothing hurts any more.
I will, Mum.
No more lying in that dirty bathwater.
No more.
Maybe you should have a wee drink? That'll make you die of dehydration quicker, won't it? Like they said on the programme? After all, what's the point of hanging on? No one's going to come and save you.
I know, Mum.
It's just you and me, alone in this stinky room that reeks of smoke and puke. And I'm probably going to start smelling a bit soon too. Sorry about that.
It's OK, Mum, you're dead, you can't help it.
Thanks, Ashlee. You're a good girl. And I love you, I always have. Now, why not have a little drink.
So thirsty...
It's OK, no one's looking.
Ashlee lets her head fall further and further, until the salty water stings her broken lips. It's bitter and horrible and wet and sour and fiery and soothing and—
Her stomach clenches and the water sprays out again, frothing from her mouth, burning out of her nose. A hacking cough rattles her back and forth against the metal tub. The chains clink and jitter. The water slops around her in waves.
And finally the coughing fades and she sags, panting in those sticking-plaster breaths in her hot-tarmac throat.
Then, like someone turning on a kettle, whatever's in the water pops and fizzes through her. Getting louder as it boils. Pushing up into her skull and making the insides float. Up and up. Until all the colours sound the same.
He said she was going to be a god, but then he left and never came back.
She's not going to be a god, she's going to be trapped here for all eternity. In the dark. In this filthy metal tub. With this chain around her neck.
I'm sorry, Ashlee. I'm sorry, sweetheart.
Why can't she just die?
## — open the coffins —
"Isn't my kingdom wonderful?" asked the Bonemonger. "All these graves and mausoleums and charnel pits, just waiting for someone to wake up their slumbering guests."
Justin backed away, until his furry shoulders pressed against the cage. "I won't do it!" he shouted, defiantly. "I won't and you can't make me!"
"But, my dear little Rabbit Boy, if you don't, your Sister Cat will sleep forever beneath my darkening ground."
R.M. Travis
_Open the Coffins (and Let Them Go Free)_ (1976)
_All them bones in the dark, and he's proud as can be,_
_Cos he'll open the coffins and let them go free,_
_And you better believe he's as sharp as a church key,_
_He'll cut you to shreds and swallow you whole, see?_
Donny '$ick Dawg' McRoberts
'Little Rabbit Boy (The Bonemonger's Waltz)'
© Bob's Speed Trap Records (2015)
##
'OK, thanks, John.' Callum hung up and slipped his phone back in his pocket.
Thin morning light seeped through the battleship clouds, making the back garden mud glitter and shine. Would have thought, after all this rain, the smell of smoke would've dissipated, but if anything it was stronger than ever, oozing from the burned-out house like fog.
The brambles were gone, but a little village of blue SOC tents had taken their place, arranged almost at random across the garden, each one with 'PROPERTY OF SPSA SCENES EXAMINATION BRANCH – OLDCASTLE' stencilled in white on the side. The nearest had, 'TENT F' added to the end.
Callum pulled up the flap and ducked inside, his blue Tyvek suit rustling and crinkling.
Inside, a small diesel generator growled away in the corner, hooked up to a handful of high-powered work lights on stands. They glared down on a pit in the middle of the tent, five-foot deep and roughly rectangular. All the soil was piled up to one side, a couple of Smurfs on their knees in the hole, trowelling more into a black bucket.
McAdams stood at the head of it, staring into the depths, the hood of his SOC suit thrown back, arms folded. Face pale and shiny.
'That was Watt.' Callum stopped at the opposite end of the trench. 'The ecclesiastical trust must've pulled an all-nighter, because a list of every property they own just arrived in his inbox. All six-hundred-odd of them. They've got some marked as unoccupied, but given they were clueless about _here_...' A shrug.
McAdams didn't look up. 'Have you ever contemplated your own mortality, Callum?'
'He's doing some spreadsheet thing so he can combine their list with the data we got from the council tax people. At least then we'll know what's officially sitting empty. But again: the trust were paying full whack for an abandoned building, so who knows?'
'Ever stood at the edge of a grave and thought, "This will be mine soon. Maybe not this one, but one just like it"?'
'We're going to need a whole heap of bodies on it – visit every single property the trust owns and see what we can see.'
'"For what is man, but doomed to die? / And here within the earth to slumber, / Till naught but bone remains of him, / The merest breath of gods gone by."'
'That's cheery. Pam Ayres again?'
'Stephen P. Dundas, you ignorant spud.' He squatted down and tossed a little clump of dirt at one of the Smurfs. 'Well?'
'Ow!' The Smurf turned, clutching its head with a purple gloved hand. 'What the hell was that for?'
'Have – you – found – anything – yet?'
'Looks like female remains.' He pointed at twin lines of pale grey, just protruding above the black earth. 'That's a forearm, radius and ulna. Going by the scarring, she broke her wrist at _least_ six times. See how it's a bit zig-zaggy? That's because it wasn't treated, just left to heal on its own.'
McAdams grunted. Nodded. 'Any buttons or zips? Bits of clothing?'
'No. If she was wearing anything when she went into the ground, must've been all natural fibres. It's long gone.'
'Same as all the others then.' McAdams stood. Dusted off his hands as he marched out of the tent. 'Seven female bodies, one male. All the women went in naked, but there are buttons and zips in with the man's remains. The women's bones are covered in scars, but not his. What does that tell you?'
Callum closed the tent flap. 'He was a rush job.'
'Show your working.'
'They're all in deep graves, five-foot down at least. Far as we can tell, he was about two, that's probably why the badgers got at him. The women were prepared for burial – stripped, probably washed. He was just tipped in, fully clothed.'
'I'll buy that.' McAdams unzipped his SOC suit and stood with his arms spread wide, steam rising from his chest. 'I want X-rays of all the skulls – see if we can get a match off dental records. And get them to run stable isotope analysis too. How long have they been buried here, how long were they kept here before that, where did they come from? Then we go through every missing persons' report till we find a match. I'd put money on the male remains being Jeffries, but let's widen the net a bit, just in case.'
'Sarge.'
'Urgh...' He let his arms drop and turned, staring out at the SOC tents. 'Seven women. Can you imagine what it must've been like for them?' McAdams shook his head. 'I don't think they were prepared for burial: those chains in the basement have been there a long, long time. They were shackled down there. Naked in the dark. Beaten, raped, and brutalised for months. Maybe even years.'
A depressing thought, but probably right.
'And then, when he was bored with them, he didn't bother stopping: he kept on going and beat them to death. Then buried them in his back garden, and went out to get a new one. Because that's what women are to him: disposable.'
'Sounds lovely, doesn't he?' Callum kicked at a little knot of brambles, still clinging on to the muddy ground. 'You think maybe Monaghan grew up here? That's why he turned out the way he did?'
'Who knows? Maybe it was always in him? Or maybe you just can't live through something like that and not come out broken.' A frown. 'Suppose it doesn't matter in the end.' McAdams pulled out his phone. 'I'll tell Mother, you go get the car warmed up.'
Major Incident Room Two buzzed with the low murmur of voices. A dozen officers sat at the desks, half in uniform, half plainclothes, all staring towards the front of the room as Watt pointed his wee remote at the projector on the ceiling. The screen behind him filled with a spreadsheet – all bars and colour-coded bits and numbers and addresses.
'This is every property N.E.T.H. own within a fifteen-mile radius of Oldcastle. I've ranked each by distance from where Ashlee and Abby Gossard were abducted, where the grey Peugeot Bipper appears on the CCTV system, and where it disappears off again.'
McAdams leaned over and whispered in Callum's ear – breath sticky and sour. 'Because colouring it up like a rainbow is going to sodding help.'
'As you can see here,' Watt poked a button on the remote and a little red dot shone onto the screen, 'I've cross-referenced the dataset with council tax records. Everything marked with a grey arrow is currently registered as vacant.'
'I bet he was touching himself when he put that together. Never seen anyone so turned on by spreadsheets before.'
Mother stood and held up two bits of paper, stapled together. 'Pair up. Each team of two takes one of these. You visit _every_ property on your list and you look for anywhere that could be used as a smokehouse. That includes big sheds, by the way.' She stuck her list back on the pile and handed everything to the nearest uniform. 'Best estimates are that it takes _at least_ a week to cold smoke a whole human being. The neighbours are going to notice something like that. _Ask_ them.'
The uniform took one of the stapled lists and passed the rest on.
'If you see anything, and I don't care how tenuous or irrelevant it seems – if it makes your spidey-sense tingle, you call it in. Understood?'
Everyone nodded.
'Ashlee Gossard is thirteen years old. There's a chance she's still alive, but it's getting smaller every minute. _You_ can save her.' Mother gave them all a big smile, then pointed at the door. 'Now get out there and make me proud.'
Callum, Dotty, Watt, and McAdams stayed where they were till everyone had filed out.
Soon as the door closed behind the last two-person team, Mother slumped back into her chair and rubbed at her face. 'Urgh... Someone tell me it's all going to be OK.'
McAdams picked up one of the remaining lists. 'How many properties are we looking at?'
'Tell me we're not just clutching at ghosts here.'
'Northeast Ecclesiastical Trust Holdings Limited own six hundred and twenty-four properties all over Scotland: from a block of flats in Kirkcaldy to a B-and-B in Cromarty, via a chip shop in Oban.' Watt wheeched his laser pointer across the screen again. 'I've marked commercial properties with a skull-and-crossbones. Fifty-two of them, in total.'
A grunt from McAdams. 'Assuming Monaghan had access to their list of properties at some point. Assuming he's been smoking them locally and not off up the coast in Buckie or Fraserburgh. Assuming he hasn't just built himself a DIY smoker somewhere deep inside Moncuir Wood, or the Swinney, or Holburn Forest. Assuming. Assuming. Assuming.'
Mother let her arms fall by her sides. 'Thanks for that, Andy. I feel _much_ better now.'
'Glad I could help.'
'Does anyone have anything constructive to say? Any ideas at all? The floor's open.'
Franklin was sitting at the back, arms and legs crossed, face tight and angry. 'Public appeal. Any suspicious behaviour. Have you seen smoke coming from your neighbour's shed on a regular basis?' She bared her teeth. 'Heard any screaming lately?'
Mother frowned at her. 'OK... Well, I've got a media briefing at ten, and that's on the list. But you know what's going to happen: every well-meaning citizen, idiot, and attention-seeking special-little-snowflake will be ringing up within the hour. And they're only giving me ten support staff to man the phones. So that'll be fun.' Back to the rest of the room. 'Anyone else? I will literally consider _anything_ at this point.'
Dotty held up her hand. 'I've been looking for any connection between Monaghan and Jeffries, and if there _is_ one, I can't find it. So how did Monaghan get access to all these Northeast God-Bothering Trust properties? Even _they_ didn't know what they owned till we made them look.'
Watt rolled his eyes. 'He doesn't have to have access to _all_ of them, just a couple. Obviously.'
'Don't you "Obviously" me, you gingery wee—'
'Children! That's enough.' Mother stood. 'We stick with the plan, till someone comes up with something better. Rosalind: you and Callum take a list. Dorothy: you and John clearly need to spend some quality time together—'
'Aw, sodding hell...' Dotty folded her arm over her head.
Watt bared his teeth. 'I'd rather staple my scrotum to a leaky tumble dryer full of angry wasps.'
'—give you a chance to bond. And maybe act like grown-ups for a change. Wouldn't that be nice?' Mother clapped her hands together. 'Off you go.'
'Arrrgh. Fine.' Dotty wheeled herself from the room. 'But if I end up killing him, it's _your_ fault.'
Mother waited until Watt snatched a stapled list from the pile and stomped out after her, before grimacing at the ceiling – both hands curled into claws. 'Arrrrgh...'
'Oh, you love it really. Our book is reinvigorated: we have _fresh leads_ to follow. The readers know a big reveal is coming soon and are relishing every page.' McAdams picked up the last remaining list. 'Want to take your car or mine?'
She sagged. 'I can't. I've got a dozen of DCI Powel's cases to review, teams to organise, updates to hear, rotas to organise, overtime to authorise, budgets to work up, etcetera, etcetera, etcetera...' Mother scrubbed a hand across her face. 'Could Reece have _picked_ a worse time to go AWOL?'
Callum stared down at the tips of his shoes.
Don't get involved.
Don't say anything.
'Pff... Anyway. No point hanging about here feeling sorry for ourselves. We've got a missing teenager to find. If she's even still alive.'
Callum held the picture of Ashlee Gossard out again. 'And you're sure you've not seen her? Or her mother?'
The crooked lady in the twinset shook her head, setting free a little flurry of dandruff from her long grey hair. 'Sorry.'
'That's OK. Thanks anyway.' He waited till she'd gone back inside and shut her door before dragging a red pen through address number four on the list.
Franklin was already behind the wheel by the time he got back to the car. Still wearing her best Everyone-In-The-World-Needs-To-Die face.
Oh joy.
He slid into the passenger seat. 'Come on then: who climbed up the drainpipe and crapped down your chimney?'
'This is a waste of time.'
'Have I done something? Because I don't remember doing anything.'
She wrenched the steering wheel around, executing an angry three-point turn. 'We're just out here chasing our backsides.'
'Only you've been chewing a wasp all morning.'
She scowled across the car. 'You're all the same, aren't you?'
'Ah, here we go. Let me guess: Mark?'
'I don't want to talk about it.'
The terraced street gave way to another one, winding back towards the centre of town. 'Suits me.' He checked the list for address number five. 'Hang a right here, then left at the end of the road onto St John Crescent.'
More identical, depressing, featureless houses. Sulking beneath the rain.
Franklin banged her hand on the steering wheel. 'I mean, look at us. Going round and round, achieving nothing. How's that supposed to help Ashlee?'
'Well, if you've got a better idea, we're all ears.'
'The question we should be asking is: who killed all those women?'
'Fair point.' He nodded. 'My money's on Paul Jeffries. Those bodies have been buried at least twenty, thirty years, right? And Jeffries did time for rape. When he gets out of prison he pretends he's put all that behind him and found God, but it's all just a front. He's still a raping little wankmonger, he's just learned how to keep his victims from going to the police.'
'By keeping them chained up in the basement. Then burying them in the back garden when he's done with them.' Franklin took the turning onto St John. More horrible little houses. 'Which begs one _more_ question.'
'Who killed Paul Jeffries and stuffed him into a shallow grave in his own private cemetery?'
She tapped her fingers against the steering wheel. 'Monaghan?'
'He'd only be a kid at the time. More likely Jeffries was part of some sort of ring and he had a falling out with one of his nasty friends.' Callum pursed his lips. 'Course, if this was a film, it'd be one of his victims' husbands. Jeffries gets careless and leaves a clue behind. Our man tracks him down, tortures and kills him, then buries him in his own back garden. And if it's a _good_ film, he finds and saves his wife just in the nick of time.'
Franklin groaned. 'All hail the great male hero!' A snort. 'Sexist piggery. Why can't it be one of the victims' sisters doing the rescuing? Or maybe one of the women escapes and gets revenge on her abuser. Why does it _always_ have to be a man saving the day?'
'True. We are horrible.' A smile. 'Just look at your Mark.'
And just like that, the muscles in Franklin's jaw were clenching again.
Yeah... probably shouldn't have done that. Seemed like fun at the time, but _he_ was the one who'd have to suffer the ensuing foul mood.
Callum flicked through Watt's list of properties: four down, sixteen to go.
Six teams of two, plus them, plus Dotty and Watt. Eight teams. Twenty buildings each. Hundred and sixty houses and/or businesses. Plus whatever odds and sods McAdams was looking into. 'N.E.T.H. Limited have got a _lot_ of interests in Oldcastle. And over six hundred properties Scotland-wide. Must be worth a fortune. Millions.'
Franklin just chewed on her sulk, glowering through the windscreen.
'Right, opposite the chip shop.'
She thumped the steering wheel again. 'I mean, where does he get off, dumping an ultimatum like that on me?'
'Didn't think there'd be so much money in Religion.' Callum stuck the list back in his pocket. 'Why don't they just liquidate the lot and give the proceeds to the poor?'
'What the hell happened to, "Your career is every bit as important as mine"?'
'Mind you, it's a _trust_ , isn't it? Do you think it's all priests' pensions and bishops' investments?'
'Are you even listening to me?'
A shrug. 'Thought you didn't want to talk about it.'
'All the sodding same.' She took the turning at the chip shop.
'I get it: your boyfriend, Mark, is a dick who thinks whatever _he_ wants is more important than what _you_ want.' The houses were getting bigger with every street. 'And do you know _why_ he thinks that?'
Franklin scowled straight ahead, mouth clamped shut.
'Because he's a dick.' Callum turned in his seat. 'And do you know what? He's a merchant banker – that was your clue, right there. Take a left at the roundabout and it should be about halfway down.'
'He wasn't a dick when I met him.'
'Yeah, well, you know the old saying: some men are born dicks, some have dickishness thrust upon them, and some achieve dickosity all on their own.' He gave her a smile. 'I'm paraphrasing a bit.'
'I am _not_ giving everything up to be a bloody housewife with a bloody pinny, two-point-four bloody children, and a Cocker Bloody Spaniel!'
Callum knocked on the passenger window. 'That's us there on the left, number thirty-two.'
She pulled into the kerb. 'I'm not.'
Number 32 was one of Watt's grey-arrow properties, all the windows sealed with chipboard. The garden looked almost as bad as The Cloisters, only without the constant parade of Smurfs, brambles, and body bags.
Callum undid his seatbelt. 'So ditch him. Tell him you've had enough of his crap.'
She chewed and chewed and chewed.
'Someone I know gave me a very good piece of advice once. You want to hear it?'
'No.'
'You can just sit there, playing the tragic jilted hero.' He climbed out of the car, turned, and stuck his head back in. 'Or you can get off your moaning backside and _do_ something about it.'
'... absolutely sod-all. Well, except for the fact I haven't throttled Detective Sergeant I-Do-Nothing-But-Moan Hodgkin. That's an achievement all in itself.'
Callum winced as Dotty's voice scraiked out in the background: _'Oh, you think you're such a delight, do you? You sour-faced, pube-bearded—'_
_'Go roll yourself under a bus.'_
_'I'll roll my boot right up your—'_
'For God's sake! Do you two never stop?' He swapped the phone to his bad hand, freeing up his left to massage the ache growing between his eyebrows.
Mother deserved a knighthood, she genuinely did.
A manky old Renault parked itself next to the Mondeo, a chubby bearded bloke and a thin blonde rock-chick type getting out and having a stretch in the drizzle, before hurrying off across the lay-by. Making for the burger van that lurked along a bit from the public loos.
Dotty was the first to break the silence, obviously trying to sound light and cheerful. And failing. _'What about you and Rosalind?'_
'Naught for seven. Thought we were on to something with a derelict house in Cowskillin, but nothing doing.'
_'Well... there's plenty of time, isn't there? Ashlee's still alive. We'll find her.'_
_'Don't be naive.'_
_'I'm not being sodding naive!'_
And they were off. Again.
_'Tod Monaghan's been dead for three days – unless he left her with plenty of food and water, she's already died of thirst.'_
_'She could_ still _be alive.'_
'Can we not go two minutes without you pair—'
_'All the other victims got starved and dehydrated before he stuck them in his smoker, so he's not going to leave her a fourteen-inch ham-and-mushroom with extra cheese and a big bottle of Diet Coke in case she gets peckish, is he? Use your head.'_
_'I have had just about enough of your bloody lip,_ Constable _.'_
_'Blow it,_ Sergeant _. Genuinely. Out your arsehole, like a trumpet.'_
What was the point?
One last go. 'I'm asking you both nicely: can you try—'
_'That's it: get out of my car.'_
_'It's not "your" anything, so—'_
_'GET OUT OF MY BLOODY CAR!'_
Silence.
Franklin emerged from the ladies' side of the public loos, wiping her hands on her suit trousers. Face wrinkled and sour.
_'GET OUT!'_
_'Fine. Great. You know what, I will.'_
_'Go on then.'_
_'I am.'_ Some clunking and rattling. _'Here!'_
_'Don't you throw stuff at me! You—'_
_'It's your half of the list, you moron.'_ Then the coffin-lid _thunk_ of the car door slamming.
_'AND GOOD RIDDANCE!'_ The sound of an engine revving, then growling, getting louder as she worked through the gears. _'GAAAAAAARGH!'_
Franklin turned her collar up and marched over, weaving her way between the pothole puddles.
_'That man drives me totally insane! He's impossible.'_
'Dotty—'
_'Everything's an excuse to moan and be sarcastic and nip, nip, nip.'_
Franklin hauled open the driver's door and threw herself in behind the wheel. Shuddered. 'God, that toilet is disgusting.'
_'You know what I should do? I should turn this sodding car around and drive right over the top of him!'_
She pointed at the phone in Callum's hand. 'Anything?'
He pulled on a grimace. 'Don't ask.'
_'You heard what he said to me, Callum, didn't you? You heard.'_
'Look, you can't just abandon him, you're police officers. You need to work—'
_'Should turn right round and squash him like the turdbeetle he is! Leave nothing but a skidmark behind. You see if I don't! He can...'_
Callum held the phone against his chest. 'Dotty's thrown Watt out of the car, and driven off without him.'
'Children.'
A big sigh, then he went back to the phone.
_'... never hated anyone so much in my whole sodding life. Not even the wee shite who cost me my leg. He's_ that _bad!'_
'Yeah, well, maybe he only does it because he's secretly in love with you.'
_'Urrrrrrgh... Think I just threw up in my mouth a bit.'_
'You're welcome. Now turn round and go pick him up. We've got a little girl to find.'
– Detective Constable John Watt –
'I HOPE YOUR SODDING WHEELS FALL OFF!' John steps out into the middle of the road and slams the palm of his left hand into the crook of his right arm, punching his fist up as DS Moron drives off into the rain.
Woman's a bloody disgrace.
No way she made sergeant on her own merit. No: must've been a bribe to stop her suing the force after the crash. Which was probably her own fault anyway.
John hurries back onto the pavement, stands under the awning outside a tat shop and has a squint at his watch. Half eleven.
Could head back to Division Headquarters, put in a formal complaint about his useless DS... But what good will it do him? No way they'll fire her, no matter how crap she is. So the only option is to outshine her. Show them how a _real_ police officer does things.
He pulls the other sheet of paper from his pocket – the other half of their list.
DS Hodgkin is such a moron.
Did she actually think he'd printed all those addresses out at random? That he hadn't done a Bayesian statistical analysis, based on the property's location and the location of the victims, and ranked them in order of likeliness? And, having gone to so much trouble, that he wouldn't keep the best ones for himself?
She could've just ridden his coattails to glory, but no. _Hodgkin_ had to be the thorn in his toilet paper, the nettles in his underwear, the razor blade in his sock, the bleach in his eyedrops.
John scans the list. Definitely going to need a car to get round all these.
That's OK, though. Just grab a taxi back to Camburn Road where he parked this morning, collect the Clio, and head out to save the day.
Goodbye, DS Dorothy Pain-in-the-Arse Hodgkin, hello promotion.
And speak of the devil...
Her crappy Wheelchairmobile drifts down the road again, windscreen wipers going. Stupid fat face peering out through the rain-streaked glass.
John ducks down behind the little blue van parked outside the tat shop.
Off you go, Hodgkin. Keep on driving.
There, that's much better, isn't it?
Bye-bye.
Don't let tomorrow's headlines hit you in the arse on the way past: 'POLICE HERO JOHN WATT SAVES MISSING TEEN ~ FIRST MINISTER PAYS TRIBUTE TO BRAVE DETECTIVE CONSTABLE...'
No: 'PRIME MINISTER AWARDS KNIGHTHOOD TO NEWLY PROMOTED HERO COP!'
Yeah, that's better.
John sticks his list in his pocket and hurries off to the nearest taxi rank.
##
'Here.' Callum thumped back into the passenger seat and held out a warm newspaper parcel.
Franklin took it. 'Mayonnaise?'
'They stuck a couple of sachets in there.' He unwrapped his fish supper, filling the car with the loving scent of hot batter and brown vinegar. 'Only chip shop in Oldcastle where they still wrap everything in newsprint. Well, if they know you.'
Steam paled the car windows, hiding the grey street. Rain danced on the roof.
She crunched on a chunk of batter. 'McAdams was on while you were out. The SEB did another sweep of Paul Jeffries' back garden and guess what they found.'
He popped a chip in his mouth, crisp and brown and salty, sharp with malt vinegar. 'Go on then.'
'The male body, with the clothes – there was another set of female remains buried underneath it. And it definitely wasn't like the others: covered in kerf marks. Whoever she was, Jeffries had a serious go at her with a knife.'
'Urgh... He just gets nicer and nicer, doesn't he?'
'That's men for you.' She crunked the top off her tub of mushy peas. 'How many have we still to go?'
The list was on the dashboard. 'Six: four private houses, a block of flats, and a disused green grocer's. Should be done by about four, maybe five o'clock?'
'If we're lucky.' Franklin dug in a chip and scooped out a splodge of neon green. 'Odds on we're—'
Her phone sat on the dashboard, buzzing as it launched into 'Dancing in the Moonlight'. Again. For the third time in twenty minutes. She just grimaced at it. Then ate another pea-smeared chip. Chewing as the ringtone came to a sudden halt.
Callum broke off a chunk of flaky white haddock and tipped it into his mouth. Almost too hot to eat. 'Can I ask you a personal question?'
'No.'
'You don't seem to like this Mark of yours very much.'
'Maybe Monaghan never had access to Northeast Ecclesiastical Trust Holdings Limited list? Maybe he just knew _that_ property was vacant?'
'Maybe.'
'So maybe he knew Paul Jeffries? Just because Dotty can't find a connection doesn't mean one wasn't there. Jeffries was a lay preacher, right? Maybe that's how Monaghan knew him? He was in the congregation.'
'So, if you don't _like_ Mark, why are you still with him?'
Franklin chewed, frowning straight ahead at the opaque windscreen. 'We should find out where Jeffries preached.'
'Life's too short: take it from me.'
'He must have neighbours of some kind, right? They might know a bit about him.'
Callum peeled the outer layer off a pickled onion with his teeth. 'Can't believe I wasted five years of my life with Elaine.'
'You're not helping.'
'Fine: he's a lay preacher. What do lay preachers do?'
'Depends what flavour he was. But there'll be sermons, raising money for charity, organising trips for wayward youths, rescuing fallen women, visiting members of the congregation if they end up in hospital or their partner dies. So officiating at funerals too, probably.' She shovelled in more chips. 'Don't know if they're allowed to give people the last rites or not.'
'Nah, that'll be a union job. Demarcation and all...' Callum put the pickled onion down. 'Dying and elderly members of the congregation: they think Jeffries is God's representative, right? They want to keep in with God on their deathbed, don't they?'
'Argh! Of _course_ they do.' Franklin sooked her fingers clean then picked her phone off the dashboard. Poked at the screen. Held the thing to her ear. 'Hello? Yes, it's DC Franklin. I need someone to get onto the Land Registry Office—... I don't care if it's Sunday. We need to know if a Paul Terence Jeffries owned any properties. Probably left to him by grateful OAPs right before they died... OK... Right. Tell them it's urgent and call me back soon as you hear... OK, thanks. Bye... Bye.' She thumbed the screen, then slipped the phone away. Grinned across the car. 'We're onto something, I can _feel_ it.'
Callum popped a chip in his gob and grinned back. 'We're going to save Ashlee Gossard.'
John's stomach makes a sound like an angry badger trapped in a bath. Should've stopped to grab a sandwich or something on the way. Too late now. Just have to wait till he gets back to town.
His Clio lurches and bumps along the dirt track, little stones pinging in the wheel arches as he slaloms left and right between the potholes.
The outskirts of Holburn Forest run along one side of the road, beech and sycamore giving way to the dark regimented mass of pine trees, stretching away up the hill. The other side is all gorse and broom, spines and spears, reaching down across plowtered fields full of reeds.
And there we go: Thaw Cottages. Number two on the _good_ list.
There's three of them – two semi, one detached, all grey. They look solid enough, but the semidetached cottages are missing glass in their windows, front gardens bounded by a sagging wall with most of the harling hanging off. Nothing but thistles, dock, and nettles growing within its boundaries.
The house next door isn't much better – both front windows boarded up with rain-darkened chipboard, one chimney pot missing, a row of jackdaws glaring down with beady eyes as he parks the Clio outside.
Another sagging wall, another garden full of weeds.
Must've been quite something, living here. Probably _monumentally_ crap in winter: trapped halfway up a hill, at the end of a long winding track, wolves roaming the woods behind the house.
OK, so maybe not wolves, but no way anyone comes anywhere near the arse-end of nowhere like this with a snowplough.
The view, though. That's something.
The hill runs down, past a tumbled-down church and its crumbling graveyard, then out along the River Wynd, nestling in a valley that opens up as it hits the outskirts of Oldcastle. Can see most of the city from up here, lurking beneath a blue-grey lid of heavy cloud tinged with gold and purples.
John zips up his jacket and reaches back between the seats for the umbrella. Scrambles out and opens it with one fluid movement. Pop.
Those two years of contemporary dance were _not_ wasted.
Rain drums on the umbrella skin.
Might as well check the conjoined cottages first.
No sign of a path beaten through the weeds to the front door, but the road hooked around the back of the buildings. Probably garages and things there.
He hops the broken gate and pushes his way through the soggy horrible nettles, holding his elbows up and out to keep both hands away from the stinging leaves. The front door isn't locked, just tied shut with orange string – the kind farmers wrap around bales of hay. He unties it, pushes on the wasp-stripped wood, and steps inside.
Callum drew a red line through property number fifteen: a two-up-two-down on a housing estate in Blackwall Hill. Checked his watch. 'Five to go.'
Franklin started the engine again, and pulled away from the kerb. 'I know there's no point just sitting about till we hear back, but this is _such_ a waste of time.'
'We've been over this.'
'Where's next?'
'Gordon Crescent, Kingsmeath. Back down to the junction then right at the roundabout.'
The car's windscreen wipers grunted and moaned.
Callum's phone joined in the general noise. 'Hello?'
_'Ah, yes, is Winston Smith addressing Detective Constable MacGregor?'_
Because today hadn't contained enough weirdos. 'He is. And does Winston have something for DC MacGregor?'
_'Indeed he does.'_
Rows of squat little houses slid by the Mondeo's windows, all slumped beneath the rain.
'Would he care to tell DC MacGregor what it is?'
_'Winston told you he would be triumphant, and triumphant he is. His software identified one thousand three hundred and fifteen possible three-character number-plate suffixes that would provide a reasonable match to the car on your footage. He then ran that through the DVLA's dataset via a method he'd rather not discuss right now, and narrowed it down to cars that conformed to the manufacturer and make shown on your footage.'_
The Mondeo climbed the hill, over the railway bridge and down the other side. Slowing as Franklin took them left at the roundabout.
'And will Winston be getting to the point sometime soon?'
_'This cut the number of hits to a mere two hundred and ninety across the UK. He then took those and hammered the ANPR system to see if any had been spotted in the vicinity, and lo his genius was rewarded.'_
From here the entirety of Blackwall Hill stretched away down to the river, the garish goings on in Montgomery Park standing out like a grenade in a kid's sandpit. Especially that massive inflatable spider.
'Callum's dying of old age, here, Winston.'
_'Those last three characters are the letters D.W.G. and form the climax of a personalised number plate, currently appearing on a black Mercedes registered to Bob's Speed Trap Records Limited and insured for the use of one Donald Newman.'_
So Newman hadn't died or gone away, he'd gone legit. Or at least as close as passed for it in the music industry.
'OK. Thanks.'
_'You don't seem to grasp the celebrity status of the gentleman in question, DC MacGregor. Donald Newman's stage name is Donny McRoberts, AKA: Sick Dawg. The rap sensation and creator of such modern top-ten classics as "Rock, Paper, Shotgun" and "I'm-a Spit on Yo Grave, Irene".'_
'You're sure it's him?'
_'Winston does not make mistakes. And he has that very vehicle on camera entering Kingsmeath via the Blackburgh Roundabout not thirty minutes earlier.'_ A sniff. _'Now, he takes it your business here is concluded, Detective Constable. And that you will be providing him with a cost centre to write his time against?'_
'Thanks, Winston. I'll get back to you.' Callum hung up before there were any complaints. Then checked his watch again.
So Donald Newman was Donny 'Sick Dawg' McRoberts.
Maybe Willow's brother had been right about the helicopter, tiger, and 'loads of bitches'.
Well, Donald was in for a very nasty shock as soon as Callum caught up with him.
Which probably wouldn't be any time soon.
No way they could abandon the search for Ashlee Gossard to go rattle Newman's teeth for him. No matter how much he deserved it.
Franklin took a right at the traffic lights and onto the dual carriageway, heading for Kingsmeath.
But he _really_ sodding deserved it.
The house is dry and dusty inside, littered with ancient manky furniture riddled with little holes. And the drifts of tiny black 'Tic Tacs' all over the floor explain why. There's mould on the walls by the empty windows, ancient flock wallpaper curled and stained dysentery-brown.
John picks his way through to a bedroom – complete with rusty bedstead and sagging mattress. A wardrobe full of old lady clothes and more mouse droppings.
The bathroom is clean, but dusty. The kitchen cupboards still have tins in them, but they're bloated and furry with dark brown flakes.
As if someone just walked out years ago and never returned.
The kitchen window is dirt-greyed, almost opaque. John huffs a breath on the glass and clears a patch with the side of his hand, revealing an overgrown garden and a collapsed shed. Looks as if next door's is much the same, only there's a greenhouse full of dead brown stalks in there too.
Past the garden is an old bothy – stone walls, corrugated iron roof – and an ancient wooden barn surrounded by chunks of farm machinery slowly disappearing under thistles and brambles.
OK, so finish up in here, check the other two cottages and then—
'What the _sodding_ hell did you think you were doing?'
John freezes. Licks his lips. Forces a smile and turns. 'Sarge. What are you doing here?'
DS McAdams is standing in the kitchen doorway, arms folded, face creased into a pale glower. 'Oh don't look so surprised: I knew exactly what you were up to, soon as you ditched Dotty.'
Oh crap.
John's mouth clicks shut.
'Have you any idea how much trouble you're in right now?'
'I was just—'
'I know exactly what you were "just", Detective Constable Watt.' His head falls back and he stares at the ceiling. 'Why me?'
'It wasn't my fault, Sarge, DS Hodgkin threw me out of the car.'
'Why do I even bother?'
'You _know_ what she's like: incompetent and chippy.'
'I'm dying of bloody cancer, here. I should be lying on a white sandy beach, drinking margaritas, not standing in a manky wrecked house in the middle of nowhere SHOUTING AT YOU!'
John retreats a couple of steps, pulling on his best righteous-indignation face. 'I was only trying to find Ashlee Gossard before something happened to her.'
McAdams' shoulders droop and he runs a hand across his wrinkled eyes. 'You want to know how I found you? You left the USB stick with your spreadsheet on it in the incident-room computer. I went digging.'
'Dotty threw me out the car! It's not my fault she's hormonal and mental.'
'I always knew you were a devious wee shite, Watt. And you're not the only one who understands Bayesian statistics: I saw what you did to the spreadsheet.' He pulls out a sheet of paper. 'Fiddling the ordering so all the highest-probability properties were last to be printed out. So you could take them for yourself.'
John sticks his chest out. 'I'm trying to save a little girl's _life_ here.'
'YOU COULDN'T GIVE A TOSS ABOUT ASHLEE GOSSARD!' A deep breath and McAdams presses a hand against his stomach. 'God damn it, Watt. If you wanted to save her you would have prioritised those addresses and we'd have sent a team to each one _first_. You don't care if she lives or dies, you just care if you can grab all the credit and glory for yourself.'
The only sound is the rain pattering in the long grass, like a thousand little feet. Running away.
McAdams sighs. 'I'm not surprised. I wish I was, but I'm not. _Disappointed_ , but not surprised.'
Heat rushes up John's face. He looks down at the faded lino at his feet. The worst thing isn't being caught, the worst thing is that McAdams is right. 'Sorry, Sarge.'
'Watt, this isn't Hollywood, or some cheesy detective novel, you can't just go running around on your own expecting to solve the case. You've got to be a team player. You've got to work _with_ your team, not piss them off so much they ditch you and drive off on their own.'
'Yes, Sarge.'
'Dorothy's been worried sick about you.'
'Has she, Sarge?'
'Of course not.' Another huge sigh. 'John, do you actually want to be a police officer? I mean honestly, _genuinely_ want to do the job? Cos, believe me, you get one lousy brief stint on this earth and if you're not totally committed to being a cop then you need to find something else. Something you're passionate about. Something you care enough about to do _well_. Understand?'
It's like someone's tied a great big heavy weight around his bowels, dragging him down on the inside. John nods. Can barely squeeze the words out: 'Sorry, Sarge.'
'So: do you want to be a cop?'
'Ever since I was a little boy.'
A tut and a groan. Then a long slow exhale. 'OK. I'm not going to tell Mother about this. Or anyone else. But you have to _promise_ me you'll try harder.'
'Yes, Sarge.'
'We'll divvy up the most likely addresses and get them out to the other teams. Maybe we can find Ashlee before she... Well, we can only do our best.' He spreads his printout on the wobbly kitchen table. 'Which ones are most statistically significant?'
John points them out, ranking them in order – most to least – and McAdams nods, marking them up.
'Good. If anyone asks, _you_ called _me_ and told me these were our best chance. Everything else never happened.' He pulls his phone out and turns for the door. Then stops. 'Have you finished in here?'
'Only just started.'
'OK. Well give it a proper search before you leave. She's got to be out there somewhere.'
'Yes, Sarge.'
The smile looks pained, but at least it's there. 'Watt, you're a bright kid. You were honest enough to turn your old team in for being corrupt – that takes guts. You've got the makings of a good copper in you. Don't let the petty stuff get in the way.'
'Yes, Sarge.'
'Good boy.' McAdams sticks his hands in his pockets. 'And please, in the name of all that's holy, _sign out at the end of your shift_.' Then he limps away down the corridor and out the front door, disappearing into the rain.
John sinks back against the work surface and groans.
It's like every report card he's ever had: _Must try harder_.
Outside, the dark rumbling roar of a four-by-four sounds, then fades away.
Come on: search next door. Then the detached cottage on the end. Then the bothy and barn. Then onto the next address on the list.
After all, he can always claim credit for putting the list together in the first place.
That's got to be worth something. Right?
And with any luck, even if he isn't the one who actually finds her, someone will get to Ashlee Gossard before it's too late.
##
'Hold on a minute...' Callum nipped down the corridor and into the stairwell. The flats weren't bad, a clean six-storey block in a development of three. Landscaped gardens and a row of private garages. There was even a sculpture out front, though God knew what it was meant to be a sculpture of. Looked like a jellyfish having sex with an Oxo Cube. 'Hello?'
McAdams sounded even more tired than usual, the noise of an engine droning away in the background. _'Where are you?'_
'Thompson Court. It's all flats, so no chance anyone's smoking bodies in—'
_'That's great. Listen, I've had a call from Watt. He's refined the spreadsheet and come up with eight high-probability targets for us to hit.'_
Yeah, right. 'And he's just done this now, has he?'
Franklin appeared from the door of number 5, turned and said something to the householder.
_'Don't be so cynical. The important thing is we've got a real chance of saving Ashlee Gossard.'_
'Oh, no: I get it. He fiddled the list, didn't he? Kept all the likely properties to himself.'
Silence from the other end.
The flat door clunked shut and Franklin headed down the corridor towards him.
'I'm right, aren't I?'
_'It's not as if_ you've _never done anything wrong, is it, Constable?'_
Franklin stopped in front of him, both eyebrows up.
Callum pointed at the phone. 'It's McAdams. I'll be down in a sec.' Then soon as she was out of sight he turned his back on the stairs. 'How many times? I – never – took – a – bribe! Is that clear enough for you?'
A sigh. _'I know.'_
'Do I have to tattoo it in six-inch letters on my forehead for you to... Wait, what?'
_'Cecelia told me weeks ago. Your girlfriend messed up the crime scene, and you took the blame so she'd still get maternity pay.'_
'You _knew_?'
_'Can we get back to the topic at hand? I'm on my way to a disused warehouse in Cowskillin now, I need every team en route to the other seven properties A.S.A.B.F.P. Emphasis on the B.F.'_
'Then why the hell have you been treating me like something you trod in?'
_'Because I like screwing with you, Constable MacGregor. You're the gift that never quits.'_
Down below, the flat's communal front door clunked shut and Franklin appeared through the window, hurrying along the path towards their cholera-coloured Mondeo.
'You're an arsehole, you know that, don't you?'
_'I'm a dying man, Callum, I take my fun where I can find it. Now you and Rosalind get your pert little backsides over to number six Creel Lane. It's Kettle Docks, so you can't be more than two minutes away. And try to keep the sexual tension to a bare minimum for the next few pages, it distracts the readers.'_
'Thanks a _lot_.' Callum thumped down the stairs, scowling out the stairwell window at the rain hammering down on the drab grey houses. 'Did you not _maybe_ just think that things were hard enough with everyone else treating me like crap? Didn't need you piling in.'
_'If it makes you feel any better: I'm sorry.'_ This time the sigh was long and rattling. _'I shall add that to my pile of regrets.'_
He pushed the door open and stood beneath the portico, just out of the rain's reach. Grudging every word: 'Are you OK?'
_'There's so many things I'll never get to do, Callum. I'll never sing in a rock band. I'll never climb Mount Kilimanjaro. I'll never win the Booker Prize. Hell, I'll probably never even be published in my own lifetime...'_
Franklin was staring out of the driver's window at him, pointing at her watch.
_'I spent so much time in the procurement of material things_ , _that I forgot to live. Grab every opportunity you get, Callum. You put them off, thinking they'll always come round again, but they don't. One brief spin and we're gone.'_
Now that was cheery.
'Is this us bonding now?'
_'Maybe.'_ A small laugh. _'Yes, well, now that we're besties, you can tell me what the hell I'm going to do with Watt. He means well. Sometimes. When he's not being a gargantuan bell-end.'_
'You give him a good bollocking?'
_'Thought he was going to cry at one point: "I'm not surprised, just disappointed."'_
'Got to love the classics.'
Franklin leaned on the horn and a loud Breeeeeeeeeeep blared out through the rain.
_'Do me a favour: keep an eye on him, Callum. He's his own worst enemy, but it's not for want of trying. There's a good cop in there somewhere. Help him dig it out.'_
Callum licked his lips. 'Are you sure you're OK?'
_'Anyway, I've got the other teams to phone. You and Rosalind hightail it over to Creel Lane. Do me and Mother proud.'_ And he was gone.
Yeah... no way that didn't sound like a last will and testament.
Collar up, Callum jogged for the car and slid into the passenger seat. Ran his good hand through his hair and flicked the water off into the footwell. 'That was McAdams. He says Watt has, and I quote, "refined the list".'
Franklin groaned and rolled her eyes. 'He rigged it, didn't he? So he'd be the one who found Ashlee.'
'Doesn't matter. He's done the right thing now.' Callum hauled on his seatbelt. 'And you and I have got a new address to search. Kettle Docks, and this time let's have a little mood music to help us on our way.' He reached out and poked the 999 button on the dashboard, setting the Mondeo's siren wailing and its lights flickering.
She grinned at him. 'About sodding time.'
Creel Lane: narrow and cobbled, lined on both sides with ancient, thick-walled buildings. Three and four storeys tall. The windows were small, the walls coated in harling and painted various shades of crumbling beige. The road curled away to the right, following the line of the river. One set of buildings facing the water, the other crammed in between the road and a steep hillside, with another set of houses above that.
All very quaint and picturesque in the sunshine. But in the rain? Claustrophobic and grim.
A couple were cloaked in scaffolding, probably on their way to becoming extremely expensive flats.
Number 6 wasn't. It was on the inland side of the road, a flat-fronted building with an archway in the middle – big enough for a Transit van – sealed with a heavy wooden gate that went all the way up. All the windows boarded up. An official-looking notice scowled out from the wall, 'WARNING! PROPERTY IS UNSTABLE AND DANGEROUS!' in big unfriendly letters.
Franklin killed the siren and parked right outside. 'What do you think?'
'Maybe.' Callum hopped out and pulled on his high-viz jacket – the one with 'POLICE' on the back. Hurried around to the gate as Franklin joined him. He pointed at a big brass-coloured lump of metal securing both sides of the gate together. 'That looks like a brand-new padlock to me.'
She snapped on a pair of gloves and ran a finger along the hasp. 'Don't suppose you've got a crowbar?'
He snapped on his own gloves. 'Kick it in?'
'Kick it in.' She braced herself. 'In three, two, one, go!'
A booming thump rattled out into the rain. But the door didn't budge.
'Right, wait here.' Franklin turned and marched across the road to the nearest scaffolding-clad building. She was back a couple of minutes later clutching a claw hammer. 'Might want to stand back.'
BOOM, BOOM, BOOM. The metal head battered down on the padlock, achieving exactly sod-all. Then she flipped the hammer round and dug the claws in behind the hasp and yanked it down. Wood splintered. Metal groaned.
'Come on, you wee bugger!' Putting her weight into it.
And they were in.
The detached cottage is empty. Well, except for the dust. And the mouse droppings. And the _massive_ wasp byke in the kitchen. Which only leaves the bothy and the barn.
John picks up the brolly and does his elbows-out march through the long grass and nettles in the back garden, clambers over the drystane dyke at the bottom, and brushes himself down on the track.
A bent piece of thin metal pipe is hooked through a hasp on the bothy door, keeping it shut. The wood's pale-blue paint is crackled and flaky. The guttering's missing. And this is a sodding waste of time.
He pulls out the pipe and pushes into the bothy.
Dark in here. And the floorboards look about as trustworthy as an angry toddler.
John creeps inside, testing the way before committing each foot.
It'll be just his luck if the floor completely—
'AAAAARGH!'
Sodding hell!
He drags his phone out and the music gets louder. Flips it open and presses the button. Takes a deep breath. 'Hello?'
_'Watt? It's DS Hodgkin.'_
Maybe it'd be better if the floor did collapse and swallow him. 'Sarge.'
_'I got a call from DS McAdams. He says you came up with a new priority order for the houses.'_
'Yeah, this isn't...' He rubs a hand across his forehead. McAdams was right: _Don't let the petty stuff get in the way._ 'Yes. I'm sorry about earlier.'
See, that didn't hurt, did it?
_'Do you, you know, want to meet up and be a team again?'_
John puffs out a breath. _You've got to work_ with _your team, not piss them off so much they ditch you and drive off on their own._ No matter how much of a pain in the ring they are. 'You sure?'
_'Course I am. Not as if I can do a lot of searching on my own, is it? Not in a wheelchair.'_
'Cool. Where are you?'
_'Shortstaine Business Park. The chandler's yard.'_
'Give me...' Five minutes to finish searching, maybe ten minutes to get there if the traffic isn't too bad. 'Call it twenty minutes tops.'
_'We'll find Ashlee. You and me: heroes.'_
'I know. Be there soon as I can.' He hangs up. Lets his chin fall against his chest. 'Pfff...'
The first step is always the worst, though, isn't it? That horrible feeling the ground's not going to be there when your foot goes down and you're just going to fall and fall and fall...
'Here.' Callum handed Franklin a torch, and clicked on his own.
Daylight barely made it over the threshold, swallowed by gloom and shadows.
She swept the cold white beam of her torch across the floor. More cobbles, uneven and buckled, giving way to cracked paving slabs.
Callum did the same for the walls and roof: bare stones and crumbling mortar. A dangling wire with a broken lightbulb hanging from the end.
A single door off to one side.
She pointed at it, then clenched her fist – pumping it once, then flattening her hand out. Nodded at him for confirmation.
'You look like an idiot, you know that, don't you?' Callum marched over and tried the handle. The door was stiff, but a bit of shoulder made it groan open.
Her voice was a hissing whisper, 'We have no idea what's in there.'
'We know Tod Monaghan's dead. And we know the gate was padlocked from the outside, with no way to open it from in here.' Callum stepped through the door. 'So unless you're worried about ghosties and ghoulies, maybe we could get on with it?'
The torch beam picked out an empty room with decaying plaster walls, the lathe exposed like ribs on a carcass. Two doors and a staircase.
'There's nothing wrong with taking precautions.'
'Don't drag me into your SWAT team fantasies.' He put a foot on the stair. The wood creaked. 'What do you think, safe?'
'We should sweep the ground floor first. Work up level by level.'
'Fine.'
Door number one: a kitchen, complete with rusty range cooker and units buried beneath a duvet of grey dust. Door number two: another empty room with skeletal walls. Another door in the far corner.
Franklin held up a fist again. 'Padlock.'
It glinted in the torchlight.
She squared her shoulders, took a step back, then rammed her boot into the wood beneath the hasp. _BOOM_ – but this time the door sprang open in a flurry of crackling splinters. Dust turned their torch beams into solid things.
A stairway led down into the darkness.
Franklin flattened herself against the wall. 'You ready?'
'Seriously, stop it.' Callum squeezed past and hurried down the stairs and onto a bare earth floor. A basement. Bare stone walls. Little archways set into them, lined with brick, the colour of blood in the torchlight.
Franklin crept down after him. 'Anything?'
'Nope. Just an empty...' Something glittered in one of the alcoves.
'What?'
'Shhh...' He edged across the dirt floor, playing the torch across the brick.
It was a chain, hanging from a metal ring screwed into the alcove wall at chest height. There was another one in the next alcove, and one in the alcove after that. Four in total, all hanging empty, all fixed to the wall.
Callum cleared his throat. 'Maybe we should check upstairs. _Now_.'
They scrambled back up to the ground floor, through the kitchen and up the creaking wooden steps.
Four doors leading off the landing.
He jammed his torch under his bad arm and dug out his phone. Scrolled through to McAdams' number. Waited for him to pick up. 'Hello?'
Franklin opened one of the doors and stepped inside.
_'Callum, my new and bestest friend, / Tell me how to make amends, / For all the cruel things I've done, / Like kicking you right up the bum?'_
'Shut up and listen. Six Creel Lane, it's got a basement kitted out like the one at The Cloisters. Chains fixed to the walls.'
_'What about Ashlee, is she there? Have you found her?'_
'Still checking.'
Franklin's torch beam cast long sweeping shadows out into the room.
_'Well get off the bloody phone and check!'_
'We _are_. And we need an SEB team over here – tell them to get a shift on.'
Franklin emerged from the other room, shaking her head. Marched over to one of the other doors and disappeared again.
'Got to go.' He hung up, got his torch and phone sorted. 'Franklin?'
No sign of her.
Callum tried door number three – a big empty room with fancy cornices and a big ceiling rose. Probably the kind of place you hung a chandelier if you were the kind of person who owned a chandelier.
Back out into the other room.
' _Franklin_?'
Nope.
He pushed through door number two. 'Where the hell are...'
This wasn't a room, it was a cavern, three storeys tall, dug into the hillside. Or maybe it was natural and they'd just built the house over the front, sealing it in? either way, it was massive.
He'd emerged onto a landing about six foot square, with no handrail. Stone steps descended to the floor twenty, maybe thirty feet below – dim and grey at the very edge of the torch's reach.
There was another torch down there, though, sweeping across a set of wooden structures – like self-contained rooms, or exhibition stalls. The smell of wood smoke, warm and sweet, mingling with the pungent taint of old fish.
Franklin's torch swung up towards him, voice echoing back off the stone walls. 'CALLUM!' Callum... Callum... Callum...
'ARE YOU OK?' OK... OK... OK...
'I'VE FOUND SOMETHING!' Something... Something... Something...
He picked his way down the stairs, hugging the wall, torch pointed at the steps beneath his feet.
'WILL YOU HURRY UP?' Up... Up... Up...
No.
He walked out onto hard-packed earth, stained almost black.
She was standing in front of a big wooden box, about the size of a large shed, made from rough planks of wood. It stood right next to a big drying rack thing, a good eighteen-foot tall, criss-crossed with notched poles, like the ones they used at Strummuir Smokehouse to be all olde worlde and sustainable.
'They've probably been smoking fish here for generations.' Callum shone his torch across the box, setting another padlock shining. Fish, and other things.
Franklin pulled the claw hammer from her jacket and wrenched the hasp from the door. The padlock clattered into the dirt.
He nudged the door and it swung inwards.
John wades through the grass to the barn door. Gah... Trousers are absolutely sodden now. But he had to take a short cut from the bothy, didn't he? Couldn't go the long way round, by the road, no, that would be too sensible.
Rain batters against his brolly, rolling in, up the valley, in thick grey curls.
Does it _never_ stop raining here?'
The barn has one of those old-fashioned pin-and-bar catches. He clicks it up and pushes through into gloom and the cloyingly familiar scent of wood smoke and dead fish. Nasty and sticky, like every smokehouse he's visited with DC MacGregor.
A smile creeps across his face. Maybe?
The barn's walls are stone on the outside, but wood on the inside, the space divided up into three. An area at the front where old wooden fish boxes are piled. An area on the right set up so poles can be hung at various heights over an open fire – there's still a pile of ashes on the floor where the last burn died out. And last, but not least, an area on the left, sealed away behind a door. No lock, just a metal pin poked through a hasp to stop it opening.
'PRIME MINISTER AWARDS KNIGHTHOOD TO NEWLY PROMOTED HERO COP!'
John takes a deep breath, pulls out the pin, and steps inside.
##
Callum edged inside. The darkness was a solid thing, pushing against his chest and throat, thick and syrupy in his lungs. The torch's beam sliced through it, but the mass healed again soon as the blade was gone.
A metal tank sat off to one side, about the size of a big bathtub, its sides streaked with pale-brown rust.
He edged over, Franklin just behind him, and shone his torch into the tank.
The smell in here isn't just wood smoke and fish, it's tainted with a bitter-scented sourness and something that's half sweet, half horror film.
The gloom seeps out from the walls, leaving just a pale spot of light from the open door.
Should've brought a torch with him.
Too late for that now.
John steps forward. 'Hello?'
Another step.
Then another, scuffing his feet along the dirt floor.
His foot hits something soft and he freezes until his eyes adjust a bit.
'Oh crap...'
It's a woman, sitting on the floor slumped to one side, the chain around her throat stretched tight between there and the wall. No point feeling for a pulse, not with her eyes half-open like that, but he does it anyway. The skin's cold and clammy beneath his fingertips.
At least that explains where the other smell's coming from.
His stomach does a little lurch to one side and he huffs out a breath.
No being sick. This is a crime scene. They'd never let him live it down.
The rest of the room is finally seeping out of the gloom. Rough wooden walls. Another chain, dangling empty from a metal ring. And a metal tank.
He stands. Marches over, back straight.
'Pff...'
It's hard to tell if the body in the tank is male or female. A skeleton, coated in a thin layer of pale skin – crusted with salt just above the filthy waterline. The chain around its neck is looped around a metal pole at this end of the tank, with just enough slack to stop the head disappearing beneath the surface and drowning. Lank, greasy hair, long enough to sink under the surface.
'Ashlee?'
John drops to his knees and reaches for her neck, two fingers touching the point just beneath her jaw where—
Her eyes snap open.
'Aaaaaargh!' He flinches back and goes sprawling on his arse. Sits there, breathing hard. Then laughs. Scrambles forward again. 'Ashlee, my name's John, I'm a police officer. We're going to get you out of here, OK? You're safe now.' Another laugh. 'You scared the living crap out of me, by the way.'
She just stares at him, making little hissing noises from her cracked and bloody lips.
'It's OK. It's OK.' He pulls out his phone. 'I'm going to—'
There's a sound, like ripping fabric. White and cold as the ground rushes up and—
DS Hodgkin hurls a scrunched-up ball of paper at him. _'You're such an arsehole, Watt!'_ She—
The sun's warm on the back of his neck, bees and wasps buzzing through the beer garden as Big Malky gets another round in, all grins and winks, no idea that he and his team are getting a visit from Professional Standards soon as—
Everyone files out of the grubby office. Mother sighs and pats him on the shoulder. 'Maybe you should try being a bit nicer to people, John? Might stop them—'
His father kneels on the sandy beach, holding out a curly shell as big as his fist. 'If you put it up to your ear, you can hear the sea.' He smiles—
Waves crash against the walls and floor.
Why is he lying down? Why isn't—
Mary kisses him, her body pressed hard against his as their song plays on the—
His phone's still in his hand, screen turned towards him, waiting for an input.
Mary. Skin like moonlight. Soft and warm beneath his fingers. That smell of strawberries and sandalwood. A smile like sunshine on a cold winter's—
The light on his phone's screen goes out.
Darkness.
'What do you think?'
Callum ran his torch around the room again. Then back to the tank. Dirty-grey crystals twinkled in the light, all the way down the sides. The bottom of the tank lumpy with them and what looked like bits of twig. 'Difficult to tell for sure, without the SEB, but that looks a hell of a lot like what was in the bathtub at Customs Street. Well, if you left it there for a couple of months till all the liquid evaporated.'
Franklin did a slow three-sixty. 'Chains on the walls, tank full of brine...' A sigh. 'How many more of these torture chambers do you think he has?'
'Can't be many. Too risky. What if Northeast Ecclesiastical decides to sell the property, or turn it into flats?'
Callum stared down into the tank. 'Maybe that's why he abandoned it? He saw all the building work across the road and cut his losses. Found somewhere safer.'
'Doesn't help Ashlee Gossard, though, does it?'
'No. But we've still got the rest of Watt's list to search.' Callum pulled out his phone. Frowned at the screen. 'No signal.'
Franklin turned and marched out of the makeshift wooden tomb. 'Then we'd better get our backsides in gear.'
##
Justin drops the spanner and it clatters off the floor, bounces, spins, then lies on its side like a wounded bird.
He hunkers down and stares at the fallen police officer. He's like a wounded bird too: blood trickling out of his nose and ear.
Hadn't meant to hit him that hard. But hey ho. Eggs and omelettes.
He's still got his mobile phone in his hand, so Justin picks it up. Presses the power button. The screen is a photo of a woman with pale skin, smiling like she's the happiest person that ever lived. Pretty enough.
Justin holds her over the tank of sacred water... and lets go. She splooshes into the liquid, the screen flickering and fizzing as it sinks. That's the trouble with modern electronics – nothing's built to last.
He grabs two handfuls of the poor lad's jacket and drags him over to the wall, where New Mummy is. Or at least, where she _was_. She's gone now, leaving nothing but a shell behind.
Poor New Mummy.
Justin wipes his hands, then kneels and brushes the hair out of her eyes. Blonde and pretty, just like Father wants. Wanted. Whatever.
He shuffles a foot or two to the side, then lies down with his head in her lap. Cold and soft. Just starting to smell. Shame. It would've been nice to just rest here. Sleep with her hand resting on his chest. The two of them joined together in fear, waiting for Father to wake up and the nightmare to start all over again.
He clears his throat. 'Can you hear me, Ashlee?'
A faint hiss sounds in the gloom.
'You're almost there, sweetheart. Soon you'll be a god.'
Silence.
Justin curls his knees up. Wraps his arm around New Mummy's legs. 'Once upon a time, there was a little boy and he had a happy life full of ice cream and adventures. And he had a brother and a mummy and daddy who loved him very, very much. Then one day everything changed...'
**_Once Upon A Time_**
'Here you go, Champ.' Father hands him a burger, all wrapped up in greaseproof paper, with 'WIMPY' written all over it.
Seagulls wheel and scream overhead as Justin takes a big bite.
The sun smiles down on them like a happy god.
Father sits on the bench next to him and wraps an arm around his shoulders. 'We're going to visit Mrs Mason after lunch, that'll be nice, won't it?'
She smells of wee and cats, and shouts cos she can't hear anything, and never gets out of bed, but Justin nods anyway.
'Soon as she signs the will, we're in the money. New car, maybe even a holiday somewhere nice?' He lets go of Justin's shoulders and ruffles his hair instead. 'And you're not going to do anything to cock it up for me, _are_ you?'
The burger turns to gravel in his mouth.
New Mummy shudders and sniffs, holding in the sobs because she's a brave little soldier. Her naked back and shoulders quiver, one arm clutched to her front. Justin creeps down the stairs, pausing with every step to stare up at the cellar ceiling, ears stretched like a bat's for any noise from upstairs.
But the only sound is Father snoring.
Justin gets the blanket from the corner and carries it over to New Mummy.
She flinches as he wraps it around her shoulders, then she blinks up at him, biting her lip and nodding. Her eyes are red as a sunset, tears all over her cheeks and snot dripping from the end of her nose.
The bruises had almost healed from last time.
Justin takes the corner of the blanket and dabs at her face, drying it. 'Shhh...'
It's meant to be calming, but it's a warning too: don't wake him up.
She stares down at the twisted lumpy bits between her elbow and her hand, skin all purple and red and blue and yellow. Like a rainbow, only more horrible.
Justin climbs up onto the bed next to her, curls up on his side, with his head in her lap, and he cries too.
The pair of them sniffling away in the basement.
Because what else can they do?
'Ahhh...' Father licks his lips and smiles, rolling the whisky around in his glass, making it sparkle. Then he picks up a tin of beer and swigs it dry. Crushes the empty in his hand.
It's not the usual cheap beer from the bottom shelf of the supermarket, but stuff in a white tin with a red stripe. And the whisky's all fancy too: with a cork instead of a screwcap.
Father is happy.
He raises his glass. 'Here's to the highlife, Slugger. Think we deserve it, don't we?'
'Definitely. We deserve it.' A big grin and a nod. Because if Father's happy, Justin's happy. And nobody has to get hurt...
The only thing spoiling it is New Mummy. She's not even hiding it, just sobbing and crying and bawling. Like she doesn't care. Like she _wants_ Father to go back down there.
His face turns into an angry-dog snarl and he stamps his foot on the floor. 'I'M NOT TELLING YOU AGAIN!'
The screaming doesn't stop, but it goes all muffled, like she's stuffed something in her mouth to kill the noises.
Father holds the crushed empty out to Justin. 'How about another beer, Kiddo?'
Justin takes it and runs into the kitchen. Yanks open the fridge. Grabs another fancy striped beer and runs back to the living room. 'It's nice and cold.'
Father cracks the ring-pull and drinks. 'Think it's about time you got a new mummy, don't you, Champ?' He takes a deep breath. 'ONE THAT KNOWS HOW GOOD SHE'S GOT IT!'
Another drink and he nods. 'A nice new mummy.' He smiles at Justin. 'There's this little blonde piece, works in a garage outside Ellon. Very sweet. You'd like that, wouldn't you?'
Justin tries not to move. Stares at his shoes. 'Father? Can't we keep this one? I... I like her, she's nice to me. The others were all angry all the time.'
Father stares at him. Stares and stares and Justin's going to pee himself and then the shouting and hitting and kicking and—
'Why not.' Father reaches out and ruffles his hair. 'Just for you. We're celebrating, right?' He raises his tin. 'Here's to Mrs Mason, and the stroke that carried her off. Thanks for the house and all the savings. May you burn in hell, you stinking corpulent bitch.'
Father grunts. His trousers are all muddy and dirty, the ribby bits in his shoes clogged up with earth. He's taken off his shirt, showing off the faded blue tattoos and the little white curly hairs that grow through them. Black soil under his fingernails. Grey dust on his arms.
He's lined his empty tins up on the coffee table in front of him, like little soldiers waiting for orders.
Father throws back the last mouthful of beer and hurls the empty at them, sending his soldiers running for cover. Scowls as they clatter and click. 'Boy: beer!'
Justin grabs another tin from the fridge and holds it out.
He snatches it. Scowls at it. Scowls at everything.
Downstairs, New Mummy is screaming again.
Because she knows what's going to happen.
She knows why Father has been digging that big hole out in the garden.
So she screams and sobs and moans.
Father cracks into his new soldier and throws back a mouthful of beer. The words start out squeezed between his teeth and end up making the whole world tremble: 'Does that _bitch_ never SHUT UP?'
Now the scowl comes round to rest on Justin.
Father's eyes are narrow and pink, one squinted up tighter than the other as he wobbles in his chair. ' _You_.'
Justin backs up a step.
His voice goes all high and whiney: '"Oh, _please_ can we keep her? I _promise_ she'll be good. She's so _nice_ to me..."' He attacks the soldier again. 'I kept her because of you, AND SHE NEVER SHUTS UP!' Father stamps on the floor. 'ALL THE BLOODY TIME! SHUT UP! STOP CRYING!'
But New Mummy keeps on sobbing.
Father bares his teeth. 'I should never have _indulged_ you. I'm too kind, that's my problem. Too _soft_.'
This is how it starts. The first rumble of thunder that brings the storm.
'Well, I'm done being soft. YOU HEAR ME?' He swigs at his beer. 'I should never have rescued you. I should've left you with your stupid mother and your stupid father.' The smile is cold and cruel. 'That's right, you're not even my _real_ son. Did you really think something as ugly and stupid as you could come from my cock? You're just some stupid kid I kept, because I thought it'd be a laugh. You're a joke, Justin.' Father sits forward and laughs in his face. 'That's not even your real name. You don't _deserve_ a real name!'
Father drinks from his beer, then hurls the nearly full can at Justin.
'YOU'RE NOTHING BUT AN EMBARRASSMENT! A SNIVELLING, WORTHLESS, USELESS LITTLE BABY!'
Justin doesn't move as the beer soaks into his jumper.
'No wonder no one ever loved you.'
He does not move and he does not cry. Because crying only ever makes it worse. Doesn't matter how much the words hurt, the beating will hurt even more.
Father curls his lip, then spits on the carpet. 'Get out of my sight: you make me sick.'
Justin gets as far as the kitchen, before Father's voice bellows out from the living room again. 'AND GET ME ANOTHER BEER!'
Another beer.
He opens the fridge and does as he's told.
Father's spade leans against the wall, leaving little blobs of dirt on the floor.
It doesn't matter how much the words hurt. It _doesn't_.
It doesn't.
He's seven years old now, a big boy.
And Father's _wrong_. New Mummy loves him. She said so. She loves him, even if Father doesn't. Because he's a good boy.
'WHAT THE HELL IS TAKING YOU SO LONG?'
A good boy.
Justin goes to the kitchen drawer and pulls out the biggest sharpest knife that'll fit in his hand.
Then walks back into the living room.
The lightbulb flickers, making the basement shadows jump and dance as Justin creeps down the stairs.
He bites his bottom lip. Wipes his eyes on his sticky sleeve.
New Mummy is curled up on the floor by the bed, arms wrapped around her tummy, sobbing.
She's not the smiley pretty lady they picked up at the sweetie shop any more. The one who gave Justin sherbet lemons and sang a song about Santa and the Christmas Mice. The smiley pretty lady who laughed and skipped and smelled of sunshine.
Father's seen to that.
Her nose is twisted and bent, flakey with blood. Both eyes all swelled up and purple. Missing teeth like broken windows when she opens her mouth to wail out another scream. All those bruises. All that pain.
He stops in the middle of the basement. 'Mummy?'
Justin's hands are wet and sticky, his jumper hot where it clings to his arms.
She shrinks back against the wall. 'Please...' The word is all soft and mushy, because her lips are puffy and split.
'It's all right, Mummy. It's all right.' He spreads his sticky red hands so she won't be scared. 'Shh...'
Every finger on her left hand is pointing in a different direction, the joints all swollen and horrid. 'Please...'
He kneels in front of her, reaches out and strokes her hair.
She flinches back.
'It's all right. He can't hurt us now. He _can't_.' Justin's fingers leave dark smears on her yellow hair.
She squints at him with her puffy eyes. At his face, at his dirty hands, at all the blood on his jumper. 'What did... you... do?'
'He won't hurt anyone.'
Her battered eyes flick to the ceiling. Then widen. Then she stares at him. 'Let me go. Please. Please let me go unlock me let me go unlock me unlock me unlock me let me go!'
'I—'
'Let me go, let me go _now_!'
Justin nods. Then digs in his pocket and pulls out the little leather pouch he's not allowed to even look at, never mind touch. 'I've got Father's keys.'
'LET ME GO!'
'I'm doing it.' But his fingers are all red and slippy and the keys fall to the ground and he has to pick them up.
'Unlock me, unlock me, unlock me!'
He flicks through the keys, till he gets to a big brass Yale one. Slips it into the lock and twists. _Click_.
Justin grins at her. 'We can go away and we can be free and he'll never hit us again.'
New Mummy slumps forward, shrugging off the slithery chain. Crawling away from the wall she's been fixed to for months and months. 'Oh God...'
'Come on, Mummy. You can do it.' He helps her to her knees, then up onto her feet. Only one of them doesn't work properly because there's a big lump on her right ankle and her foot's all dangly.
She hisses and groans every time she tries to stand on it.
So Justin takes as much of her weight as he can. A big brave boy as she hobbles and hops and cries and swears her way up the stairs. Slow and painful. Till they're standing in the hall.
Then New Mummy stops, her good hand against the wall, holding herself up, swollen eyes fixed on the open living-room door.
One of Father's legs pokes out from behind the door, trousers matted with dirt and blood. Not moving.
Justin reaches up to take her hand. 'I told you, he won't hurt us ever again.'
'Oh my God...'
'We can be _free_.'
'Where's the phone? There has to be a phone. _Where's the bloody phone?_ '
He points at the living room and she hobbles forwards. Peers around the door.
'We can be free and we can live happily ever after, like in the stories!'
New Mummy limps inside.
The phone is on a little wooden table beside the television. All big and black and forbidden. She stumbles over and grabs the handset from its cradle. Works a shaky finger into the dial.
'We can get a nice house at the seaside and go for walks and eat ice-cream and get a dog! Can we get a dog, Mummy? Can we get a great big—'
The slap sends him crashing against the wallpaper. He leaves a dark red smear of Father's blood behind. Stands there, bottom lip trembling. 'Mummy?'
'I'M NOT YOUR MOTHER!' Little bits of spit land on Justin's cheeks.
'Mummy?'
'You helped him. YOU HELPED HIM KIDNAP ME!'
'But I was scared and—'
'You could've gone for help anytime, you could've called someone, YOU HELPED HIM!'
Justin shrinks back. 'But... But we're supposed to be _together_.'
'GET AWAY FROM ME!'
Justin bites his bottom lip. Blinks back the tears.
_Doesn't matter how much the words hurt, the beating will hurt even more._ Remember?
Only how could any beating hurt as much as this?
She goes back to the phone, sending the dial clicking around. Nine... Nine...
'Mummy?' He reaches for her. 'Mummy, I—'
'I SAID, GET AWAY FROM ME!' She shoves him away with her good hand, hard enough to send him tumbling across the bloodstained carpet.
The knife is right there. Right at his slippy-sticky red fingertips.
Justin picks it up.
The sun peeks over the hills, turning the sky to blood.
The birds are singing, making sure everyone knows they're awake and ready to do whatever it is birds do.
Sweat drips off the end of Justin's nose as he heaves another shovel of soil into the hole.
It took a long time, dragging New Mummy out to the garden and into the hole Father dug for her. Then shovelling in some earth. Then hauling Father out and dumping him in there too. Then more dirt, till the hole is full up to the top again.
Probably should've dug another hole for Father. New Mummy wouldn't like him sleeping on top of her for ever and ever. And maybe if she'd _loved_ Justin, he'd have dug a new one for Father and she could've been all alone in the ground. But she didn't. So he hadn't.
If she'd loved him, they could've had a house by the seaside and a doggie and ice-cream and everything would be nice and happy and they'd sing songs and walk on the beach...
But she didn't.
He wipes his soggy face on his dirty jumper.
Father's lawn is all scuffed and flattened, with nasty red scrapes from here to the kitchen door. He'd have hated that.
And now Justin is all alone.
So in the end, nobody gets what they wanted.
He leaves the spade and trudges back into the house. Locks the kitchen door behind him. Tomorrow he'll have to decide what to do, but for now he's going to curl up in the Naughty Cupboard and sleep and sleep and sleep.
It's been a busy day.
##
Callum jumped back into the car. 'Nothing doing.'
Franklin scored address number seventeen off the list. 'Three more to go.'
'Two thirty-six Banks Road. Next right, then on to the roundabout and left.' It was the same in every direction: bland grey houses for bland grey people living bland grey lives. Callum let out a sigh. Checked the list again. 'Fancy some music, or something?'
'Yeah, OK.'
They both reached for the knob at the same time: their fingers touching. Then flinching back as if they'd been burned.
'Sorry.'
'No, it was me.' Franklin's cheeks darkened.
Callum cleared his throat. Buzzed his window down a crack. Definitely getting hot in here. 'Do you want me to...?'
'I don't mind.'
'Cos we don't have to, if you...?'
'Yes. It's OK.' She kept her eyes fixed on the road.
'Right.' He reached out and clicked the radio on, getting a raucous banjo-and-bagpipe rendition of Pink Floyd's 'Wish You Were Here' in return.
More grey houses went by.
The rain rained.
Franklin made a noise.
'Did you say something?'
'No. I was just... humming along.'
'Right. Yes.'
And then Callum's phone went off.
Oh thank God.
He pulled it out. 'Hello?'
It was Mother: _'_ Please _tell me you've got good news.'_
'Sorry. SEB are hammering six Creel Lane now, but going by the brining tank, Monaghan hadn't been there for months. Maybe years.'
_'Damn it.'_ A clicking noise, like someone drumming their nails on a desk. _'Ashlee Gossard's going to be dead by the time we find her, isn't she?'_
Of course she was. She was probably dead already. 'There's still houses to search.'
_'Gah...'_ A sigh.
'You OK?'
The street gave way to tiny detached houses with steep slate roofs, like a model village for gnomes. A miserable couple wheeled a pushchair through the rain. A bus sat at a bus stop: its driver had an OAP in a headlock, struggling with her in the gutter as the passengers looked on, cheering.
_'Anyway, there's some good news: Gareth Pike has had a chance to think about the error of his ways, and he's decided to identify the man he saw abducting your family. Isn't that public spirited of him?'_
A fire burst into life, right in the middle of Callum's chest. 'Who was it?'
_'He won't say till I promise he's definitely going to prison.'_
'So talk to the Sheriff again! Tell him Pike—'
_'Callum, Callum, Callum... Pike's a paedophile, you caught him with a horrible video and got a confession. There's no way he was ever_ not _going to prison. Do you really think we'd let him walk free?'_
'But you had a thing from the Sheriff, at the prison, I saw—'
_'No, that was just a parking ticket. Should probably get round to paying that...'_
More Noddy Toy-Town houses, then a community centre.
'So...?'
_'None of the other teams have found anything, by the way. Andy's running round like a mad thing – which is_ definitely _not good for him – Dotty's sulking, and God knows where John's got to. Honestly, some days it's like trying to get an angry ginger tom into a pair of Lycra cycling shorts.'_
'What about Gareth Pike?'
Franklin took a left at the roundabout, heading up towards the railway bridge. 'Where now?'
'Make a right, after the postbox.' Back to the phone. 'Boss?'
_'I'm sorry, Callum, but Gareth Pike will have to wait till we've done all we can for Ashlee Gossard. And don't moan and whinge: you know as well as I do.'_
He curled forward until the seatbelt cut across his chest. 'We need to get it out of him tonight. Soon as he finds out he's got what he wants – that he's going to prison anyway – he'll keep his mouth shut just to spite me. This is _fun_ for him.'
_'We'll get him, Callum. I promise. Now you get out there and you do your best. There's a scared little girl hidden away somewhere, dying. Find her.'_
The little old lady frowned out the back door as Franklin disappeared into the shed at the bottom of the garden. 'Are you sure she's all right in there? Unsupervised?'
'It's just procedure, ma'am.' Callum stayed where he was, huddled inside the porch, out of the rain. 'And you're sure you haven't seen either of these women?' He held up the photos of Ashlee and her mum again.
'Only, you know what these _coloured_ people are like. It's always them in those London riots, isn't it? And shooting people.'
It took a lot of effort, but Callum managed a smile. 'I can assure you, Detective Constable Franklin isn't like that. And the majority of rioters were white, by the way.'
'What if she steals my lawnmower?' A sniff. 'And they're so _touchy_ these days, aren't they?'
'You can't say anything or it's a "hate crime".'
'She's _not_ going to steal your lawnmower.'
'When I was a wee girl they were called "nig-nogs" and no one ever complained. If you ask me, that Enoch Powell had the right idea.'
He stared at her. 'Yes. Well. These are more civilised times, aren't they? We don't just _accept_ casual racism. And we _don't_ call people "nig-nogs"!'
Callum fingered the tin of pepper spray in his jacket pocket.
We do not live in a police state. We do not live in a police state...
'Should send them all back where they came from.'
He pointed. 'She's from _Glasgow_.'
A nod. 'There you are then.' As if that ended the argument.
Franklin emerged from the shed and it didn't look as if she'd stuck the old cow's lawnmower under her jacket to make a clean getaway. Instead, she shook her head, brushing cobwebs from her jacket as she marched up the path to the back door. 'Thank you for your cooperation.'
Mrs Enoch Powell smiled at her. 'Not a problem, dearie. I'm only too glad to help.' She followed them through the kitchen, down the hall, and out the front door. Keeping both eyes fixed on Franklin. 'Mind how you go now.'
Callum sank into the passenger seat and clicked on his seatbelt. 'And that's us.' He drew a red line through their final address.
'So what's next?' Franklin pulled away from the kerb, heading back towards the town centre.
The little old racist stood on her front step, watching as they drove away. Probably expecting Franklin to pull a handbrake turn and steal everyone's lawn ornaments.
'Chase up the Land Registry Office?'
'Worth a go.' He took out his phone and called control. 'Brucie? Callum. I need you to light a fire under the Land Registry Office. Tell them there's a little girl's life on the line here.'
_'Your usual slave's got a day off, has he?'_
'Don't be a dick, Brucie. You know it'll sound better coming from you. More official.'
_'Aye, right.'_ A sigh. _'I'll give them a shoogle.'_
'Thanks, Brucie, you're a star.' He hung up. Tapped the phone against his chin. 'There's nothing else we can do for Ashlee right now, is there?'
Franklin shrugged. 'Not till the Registry gets back to us.'
'Exactly.' He called Mother as the housing estate gave way to a short line of shops. 'We're nought for twenty-one. Anyone else?'
_'I should be so lucky.'_
A voice in the background sounded like McAdams: _'Watt, I'm not kidding about here: call me back soon as you get this!'_
'Trouble in paradise?'
_'None of the other teams found anything. Not so much as a smoked sausage.'_
McAdams got louder: _'I trusted you, you wee shite. I thought we had an understanding!'_
The windscreen wipers squeaked and squonked their way back and forth. The gutters were overflowing at the bottom of the hill, making a loch that stretched all the way across the road and about twenty foot long.
'We're heading back to the shop now.'
Mother groaned. _'I think we've blown this one, Callum.'_
_'Where the hell are you?'_
'It was always going to be a long shot.'
_'I know, I know. We—'_
_'When I get my hands on you, Watt, I swear on my oncologist's_ grave _I'm going to—'_
_'Andy! For goodness' sake: enough.'_
Franklin slowed for the water feature, sending arcs of dirty grey splashing up and out.
There was silence from the phone, then a sniff from McAdams. _'_ Fine _. Call me back, Watt.'_
_'And don't look at me like that, you know it's not good for your blood pressure.'_
Callum cleared his throat. 'Boss? You know you said we had to do everything we could for Ashlee Gossard...?'
Her voice was flat as an ironing board. _'You want to talk to Gareth Pike.'_
'Only, we've been through our list, we've got a request in with the Land Registry Office, and there's nothing else we can actually _do_ right now.'
_'Callum, I've got half a dozen of DCI Powel's cases sitting here on my desk, just_ waiting _for someone to—'_
'Boss, please. I need to know.'
A sigh. _'All right. But if something comes up—'_
'Not a problem. You shout and we'll come running.'
And with any luck, by then they'd be one step closer to catching his mother's killer.
The interview room was every bit as depressing as last time. Callum sat at the table, left leg twitching and jumping away to itself, waiting.
Franklin checked her watch. 'What's taking so long?'
'He'll be dragging it out as long as he can. He's lost, and he knows it. Keeping us waiting is the only way he can exert power.'
'Hrmmm...' She paced to the fake rubber plant and back again. 'Even if he gives you a name, there's no guarantees. A lot can happen in twenty-six years.'
'Will you sit down? You're making me itchy.'
'And what if he's just messing with your head, did you think of that? Maybe he didn't see anything at all, and this is just him playing games.'
'I'm not kidding, sit your backside down and...' Callum sat up straight as the door swung open and the little man in the ugly jumper came back in.
'Now, I know you were here yesterday, Detective Constable MacGregor, but this bit is like the safety announcement on aeroplanes: we have to do it.' Duncan took a deep breath. 'You're not to give the inmate anything, and you're not to take anything _from_ him. That includes messages to, and from, the outside world. You're not to let him use your mobile phones. We disapprove of physical contact. And a staff member will be present at all times. OK?'
'OK.'
'OK.' A smile. 'Now, please make sure your seats and tray-tables are in the upright position.' He poked his head back out into the corridor. 'All right, Rachael, bring him in.'
And there was Gareth Pike again, ducking to get in through the door, lowering himself into the seat opposite like a shaved bear. Sitting with his shoulders forward and his back hunched. Lights reflected in his bald head. His mouth turned down at the edges, as if he'd just swallowed something nasty. 'Before we begin this exercise in completely unfair manipulation, I want it made clear that I am only providing this information under the most terrible duress.'
Callum reached into his pocket and produced a sheet of folded paper. Laid it on the table between them. 'You're looking well, Gareth. Have you been polishing your head?'
'Furthermore, I must protest in the strongest terms about being kept waiting for so long. I'm not a well man and the stress is harmful to my conditions.'
'You have a name for me.'
The mouth turned down even further. 'I want assurances that I will not be given a _community service order_.' A small shudder set his jowls wobbling. 'Like some sort of track-suited youth caught shoplifting from Lidl. I will be placed in a suitable residential facility designed to cater for people with my proclivities.'
Callum tapped the piece of paper. 'Things have changed a bit, Gareth. We've found another witness. You're nothing but corroboration now.'
'And I want a south-facing cell.'
'Nope.' He put the sheet back in his pocket and stood. 'Have fun cleaning out those cages. I bet the dogs and cats make one hell of a mess.'
Pike glared up at him.
'Last chance.'
He bared his teeth. 'You're enjoying this, aren't you? After all this time, being the one with the power. No more the scared little boy, cowering in his daddy's caravan, sobbing like a baby and wetting himself.'
'Bye, Gareth.' Callum turned to Franklin. 'Shall we?'
'All right! All right.' Pike balled his chubby hands into two enormous fists. 'I recognised the man who took your parents and brother. He was... I suppose in some circles he probably still is, famous. He's certainly in all the papers right now.'
Callum gave a big theatrical sigh. 'Come on then, Gareth: tell me who you saw and I guarantee you'll go to prison. No point kicking a man when he's down. Even a piss-poor excuse for one, like you.'
'I told you he was a lion, didn't I? That big blond mane of hair, the strut and swagger. A man used to being worshipped and adored.'
Franklin curled her lip. 'Stop milking it.'
Pink flooded Pike's cheeks. 'His name's Leo McVey.'
'No.' She stared. 'Wait, the rock star? _The_ Leo McVey? Leo McVey abducted his parents?'
Pike's eyes widened. 'I know, isn't it delicious?'
##
_'Wow...'_ Mother made a hissing noise. _'Leo McVey?_ The _Leo McVey?'_
Callum tightened his grip on the phone. 'That's what Pike said. Said he attacked them with a length of metal pipe and forced them into the boot of his Range Rover.'
_'Wow...'_
'Can you stop saying that? It's not like this is a claim to fame, here.'
Franklin took them over the Dundas Bridge, windscreen wipers on full pelt. The cars coming the other way populated by hunched men and women, their faces soured by rain.
_'I'm sorry, it's just: Leo McVey. I had_ all _his albums.'_
Right at the roundabout, following the river, picking up a bit of speed for a change.
'Of course, we've only got Pike's word for it.'
_'So what's your plan, Callum? The Sheriff won't give you a warrant on the word of one paedophile, and after twenty-six years...'_
'We're going to go see him.'
_'Leo McVey?'_
'One good thing about this music festival: we know where he'll be right now.' Getting ready to ponce about on stage with all his new showbiz mates. 'Franklin and me are on our way there now.'
_'I see...'_ Her voice sagged a bit. _'Callum, this really isn't a good idea. You're too closely connected, you're upset, you're—'_
'Pike _saw_ him.'
Silence.
A golf course drifted by on the right, trapped between the road and where the River Wynd emptied into Kings River. The fairway was more or less a lake now, punctuated with bunkers and the occasional flag.
'Boss?'
A sigh, then: _'Put me on speaker.'_
He did and her voice crackled out into the car.
_'Rosalind? I'm relying on you to keep this under control. You don't leave Callum alone with Leo McVey. You don't let him say or do anything stupid. And_ most _importantly, you don't get me hauled up in front of the PIRC! Agreed?'_
Franklin nodded. 'We'll tread lightly.'
_'Make sure you do. If there's one thing the seventies taught us, it's: celebrities sue. Even when they're guilty.'_
'Yes, Boss.'
The man at the Portakabin door curled his top lip and stared down at Callum. 'You're kiddin', yeah?' He had to be at least six-five, with a crewcut, black bomber jacket, black jeans and Doc Martens. 'HATE' tattooed on one massive set of knuckles, and 'MUM' tattooed between the other.
A line of metal barriers sealed off this chunk of Montgomery Park from the rest of it, covered walkways keeping the important people's feet out of the shoe-sucking mud and their trendy haircuts out of the rain. Two lines of yurts and tepees were broken up by fancy-looking portable loos and outside-broadcast vans. A marquee with plastic windows was laid out as a fancypants dining room: tablecloths, waiters in black tie, and a real-life chandelier.
But _this_ side of the barricade, the park was a litter-strewn swamp, full of muddy people in ponchos, bouncing up and down to whatever band was currently on stage and belting folk-rock out through the PA system.
And above them all, that _massive_ inflatable tartan tarantula waved its legs in the rain.
None of which seemed to register on the big lump in the bomber jacket. 'You're not gettin' in. Now hop it.'
Callum checked his warrant card, then held it out to Franklin. 'Does this look like it came free in a box of Rice Krispies?'
She folded her arms, eyeing King Kong up and down. 'Are you interfering with a murder investigation, sir?'
He stuck out his chest. 'You're not on the list: you're _not_ comin' in.'
'I'll tell you what's on the—'
'Hello, can I help you?' A trendy-looking specimen with sideburns and a quiff sidled up, clipboard under one arm, three or four lanyards dangling around his neck. Call-centre headpiece cramping his haircut.
King Kong jerked a thumb at Callum. 'This one here thinks he can waltz in, just cos he's a cop.'
'I see. Right. Thanks, Charles, I'll take it from here.' Mr Clipboard clasped it to his chest. 'Now, how can I help?'
'We're here to see Leo McVey.'
'Ah... I'm afraid that's impossible. You see Mr McVey's on the main stage in just a little under thirty minutes. Twenty-seven minutes thirty-nine seconds, to be precise. And he's getting ready in the green room.'
'This won't take long.'
'Yes...' The plastic smile got a bit more stretched. 'Only he's the closing headline act of the whole festival, and we'd rather like him to be at his best when he walks out there to entertain twenty-six _thousand_ people. Not to mention everyone listening at home, and anyone who buys the CD or DVD. So you see...?' A shrug.
'I didn't get your name, sir.'
Franklin reached out and took hold of one of the lanyards. 'Ryan Keen.'
'I see. Yes. Actually, it's _very much_ not a good time and—'
'Have you ever seen the inside of a police cell, Mr Keen?'
He licked his lips. 'Ah...'
Mr Not So Keen stopped outside the door to an oversized yurt – like a cake made of brightly coloured canvas, topped with a big pointy hat. 'Now, _please_ tell me you're A: not going to upset him and B: not going to make him late.'
Franklin put her hand on Keen's shoulder and eased him to one side. 'Thank you for your cooperation.' Then she opened the door and disappeared inside.
Keen fidgeted with his clipboard. 'I'm going to get fired...'
Callum followed her into the cake.
Inside, the sweet-sweaty scent of incense mingled with orange and apple. Oriental rugs overlapped across the floor, and a row of fairy lights twinkled their way around the outside of the large open space. Leather sofas were artfully arranged, with standard lamps casting little golden pools of illumination in the luxuriant gloom.
A woman in full-on French maid costume stood just inside, with a tray of bubbling champagne flutes.
Clearly, people took a lot better care of musicians than they did police officers.
Franklin flashed her warrant card. 'Where's Leo McVey?'
That got them a fixed death grin. 'Mr McVey's in the Absinthe zone.' She pointed at a small tunnel through the yurt wall. 'He's communing.'
'Good for him.' Callum marched over and through the tunnel, coming out in a separate domed expanse. Only this one was lined in pale green, with beanbags instead of leather sofas.
About two dozen people were gathered around a coffee table covered in mugs and glasses – some on the floor, some on the beanbags, others leaning back against the yurt walls. All of them beaming at an old man, as if he were the second coming.
Leo McVey looked just like he had on _Breakfast News_ , Friday morning: tasselled jeans, cowboy boots, dark-blue shirt, leather buckles on his wrists. He leaned forwards. 'So there we are: Mick, David, Noddy, Lemmy, Ozzy, Alice, and me in the hot tub, and the only one with any clothes on is Mick.' He winked at them. 'That's Jagger, not Hucknall. And I have to admit I'd done _quite_ a lot of acid at this point, so when Mick says—'
'Leo McVey, Police.' Callum held up his warrant card. 'We need to talk.'
McVey's smile grew. 'Not quite, but you're close, officer...?'
'MacGregor. Now, let's—'
'Hey, cop!' One of the acolytes stood, right shoulder forward, the other drooped, leather jacket hanging open to show off a shaved chest and a fox tattoo sticking out of the waistband of his underpants where _they_ stuck out of the waistband of his baggy jeans. A golden dollar sign dangled around his neck on a shiny chain. Elaborate moustache and goatee decorating his chin. Both hands circled gangsta signs, pumping himself up with every word: 'You better stop, cos this man rocks, / And you pair of cocks better hit the bookshops, / And learn some respect, cos you incorrect, / I checked, and he ain't no goddamned suspect.'
A _very_ large black man in a blue tracksuit nodded. An American accent so thick you'd need a chainsaw to cut it. 'Word.'
Franklin put one hand in her pocket – the one she usually produced her collapsible baton from, like a very violent magic trick. 'I'm going to need you to sit down, sir.'
Gangsta Boy gave her a good hard leer. 'Damn, bitch, you be _fine_!'
'Word.'
She narrowed her eyes. 'What did you just call me?'
He limp-swaggered closer. 'Bitch, you _know_ I like my women like I like my coffee: strong, sweet, and black.'
'Heh, heh, heh. Word.'
'I'll give you strong, you pasty—'
'Ladies, gentlemen: chill, yeah?' McVey stood, both hands out as if he was about to bless them all. 'Donny, it's OK. I got nothing to hide from these nice police officers. I've not packed a stash since the noughties.'
But Donny just stood there, with his hairless chest puffed out. 'You sure, Leo, cos I can open up a can of righteous deliverance on these sons-a-bitches. Say the word and they _gone_.'
Another nod from the massive sidekick. 'Word.'
'You're very kind, but I'll be fine.' McVey smiled. 'Now, officers, any chance we can get this over with? I'm on in twenty minutes, and my bladder's not as young as it used to be.' He threw his apostles a peace sign. 'Chill here, guys. When I get back we're going to rock this city's socks off!'
They gave him a round of applause and some whoops.
Then Keen took McVey by the elbow and led him out into the main yurt, nearly bent double under the weight of his own obsequiousness. 'I'm sorry about this, Mr McVey. They assure me it won't take more than a couple of minutes. We'll get you out on that stage bang on time, don't you worry. You're going to be magnificent.'
'It's cool.' McVey wandered over to the hospitality table and helped himself to a bottle of beer from the fridge. Cracked it open. Then pointed at a door, hidden away in an alcove. 'Shall we?'
Callum followed him onto a section of decking set out with deckchairs and tables, beneath an awning covered in sponsors' logos.
McVey took a swig and settled his elbows on the railing. 'Amazing, isn't it?' Toasting the view with his bottle.
From here, the park sloped away towards the main stage – a big tessellated hemisphere surrounded by lights and speakers, flanked by a pair of screens three storeys high. Some sort of folk band were on the stage, leaping about and trying to get the crowd to join in.
There were thousands and thousands and _thousands_ of them. All jammed together, waving flags, waving their mobile phones, waving their arms, apparently not minding the fact it'd been bucketing with rain for about a week and they were up to their welly-tops in sticky black mud.
Callum looked over his shoulder.
Franklin lurked nearby, her notebook out and pen at the ready. No sign of Mr Keen or his clipboard.
'So, who was Captain Bare Chest, back there?'
'Donny? He's great, isn't he?' McVey laughed and shook his head. 'Looks like he's barely into his twenties: turned thirty last year. Suppose healthy living and Botox will do that for a man. Well, that and a face-lift, a nose job, and three hours a day with a personal trainer.' Another swig of beer. 'But you didn't come here to talk about Donny Sick Dawg McRoberts.'
AKA: Donald Newman. Willow's dad. The man driving the black Mercedes. The 'man' who broke a little girl's arm. The _man_ who beat his ex-girlfriend and stole her sodding teddy bear.
The man in serious need of a stiff bloody kicking.
But not quite yet.
Callum reached into his jacket and pulled out his father's wallet. Flipped it open. 'Twenty-six years ago, there was a family of four, just back from a fortnight in Lossiemouth.'
'And?'
'There's a lay-by on the Aberdeen road, just outside Blackwall hill.'
'Still listening, still not understanding.'
'They were attacked, Mr McVey. Mother, father, and a five-year-old boy were abducted. Never seen again.'
He shook his head. Took another swig of beer. 'Life can be pretty horrible, can't it?'
'Where were you on Wednesday evening, Thursday morning?'
'This week, or twenty-six years ago?' A shrug. 'Because if it was this week, I was in Brussels with the band. We've not toured in about fifteen years, got the feeling we'd be a bit rusty, so off we lurched to the continent to get our swagger back.' He counted them off on his fingers: 'Hamburg, Friday. Berlin, Saturday. Dusseldorf, Sunday. Amsterdam, Monday. Rotterdam, Tuesday. Brussels, Wednesday. Cologne, Thursday. And back to Dear Old Blighty on Friday, cos I was on _Breakfast News_.'
'And you can prove this, can you?'
McVey laughed. Shook his head. 'Take a look online. There's got to be ten thousand photos of us all over the whatchamacallit: social media. Honestly, nobody actually watches a gig these days – they just stand there filming it on their mobile phones. In my day it was autographs, now everyone wants a selfie.'
'And what about the sixth of April, twenty-six years ago?'
This time he drained his beer. 'Spent most of that decade off my tits on various mind-expanding chemicals. How am I supposed to remember one day?'
'Because you were seen, Mr McVey. At the lay-by. You were seen assaulting the husband and wife with an iron bar. You were seen gagging them and tying them up with duct tape. You were seen loading them into the back of your Range Rover. You were seen abducting the young boy.'
'Doesn't sound like me.'
'You were _seen_.'
'Nah.' McVey pitched his empty bottle, overhand, into a recycling bin eight feet away. 'Ten points.' He stuck his hands in his pockets. 'Because I wasn't there. I didn't attack anyone. And this conversation is like Glam Rock: ridiculous and over.'
Callum stepped right in front of him. 'It's over when I say it is.'
'See, if you actually had anything on me, we'd be doing this down the station, wouldn't we? You're grasping handfuls of cloud and praying they're a parachute.' He raised his voice. 'Mr Keen, ready when you are!'
Shout and the creep will appear. Keen slithered onto the decking, holding his clipboard like a life vest. 'Mr McVey. That's great. I've got the car waiting; we're all set to wheech you over to the main stage.' He checked his watch. 'Don't want to keep your fans waiting...'
McVey turned and patted Callum on the shoulder. 'It's been fun. But if you want to do it again, better get a warrant.' He sauntered off, pausing only to wink at Franklin on the way past. 'Later, beautiful.'
'Mr McVey.' Was she blushing? She was. Unbelievable.
'Call me Leo.' He circled back, took her hand and kissed it. Then turned a wink on Callum too. 'And I never owned a Range Rover. A lot of my circle did, but I was always a Jag man. You got me confused with someone else.'
'Who?'
'Please, Mr McVey, we need to get you over to the main stage, so if we could just...?' Keen pointed at the door.
McVey put one hand against his chest, as if he was about to pledge allegiance to something, or stifling a burp. '"Isn't my kingdom wonderful?" asked the Bonemonger. "All these graves and mausoleums and charnel pits, just waiting for someone to wake up their slumbering guests."' He performed a little bow, then followed Keen out the door and away.
Franklin took a deep breath. Flexed her kissed hand like the fingers were brand new. 'I'm not sure how I feel about that.'
'Just because he's got an alibi for dumping my mother's head in Holburn Forest, doesn't mean he didn't abduct them twenty-six years ago.'
'I mean, eighteen-year-old me wants to never wash this hand again. Feminist grown-up police-officer me wants to rinse it in boiling bleach.'
'And what was that bit at the end supposed to mean?'
'Maybe Pike was lying all this time? He knew he was going to prison anyway, so he's just causing as much trouble as possible. Maybe he didn't see anything at all?'
'He said "the Bonemonger", so it's from _Open the Coffins_. But...' Callum chewed on the inside of his cheek. 'I don't know.'
'Still don't know what to do about my hand.'
'Wash it. _Definitely_ wash it.' Callum pushed through the door and back into the yurt.
McVey was gone, but most of his acolytes were still there – probably waiting their turn in the car to the main stage.
And right there, in the middle, was Donald Newman, AKA: Donny McRoberts, holding court. 'Oh yeah, me and Leo: we go back years, innit? See when I was growing up in a home? He visited me, like _every_ week.' Preening himself. 'That's how come I got him to duet on my very first album. Man's a star, right? Been like a dad to me.'
Leo McVey might have waltzed right out of here smiling, but this sack of vomit wouldn't.
Callum pulled out his warrant card again. 'Donald Newman.'
Newman's face pulled itself to one side, like he was about to spit bile. 'I don't know you, pig, but you better not call me that again. The name's Sick Dawg, yeah? Show some reeeeeee-spect.'
His sidekick folded his thick fat arms. 'Word.'
'You visited a Miss Irene Brown two nights ago, didn't you, Donald? In the black Mercedes your record company pays for.'
'Free country, innit?'
'You're a big man, surrounded by your mates. Think they'd still be your mates if they knew you beat up your ex on Friday night?'
Newman sneered. 'You can blow your lies out your arsehole, Piggy. Ain't nobody here buyin' what you sellin'.'
'You broke your daughter's arm three years ago, didn't you? She was four years old. What was that, a flying visit to abuse her between gigs?'
The sneer became a snarl and Newman lunged forwards, chest out, shoulders back. 'You wanna piece of this _action_? / Man I'm-a put you in _traction!_ / Get my satisfaction from a violent reaction! / My fist and yo face gonna have _interaction_ , / I'm-a beat you down _dead_ , you don't swear a retraction!'
His sidekick stepped up beside him. 'Word.'
Franklin looked the guy up and down: the tracksuit, the heavy gold jewellery, the backwards baseball cap, the sunglasses. 'Nice. Play up to the thick black gang-banger stereotype, why don't you?'
That got her a laugh that set most of him wobbling. 'Hey, bitch, don't you be judging _me,_ / Big Bobby B's got a master's degree, / I came top of ma class at M.I.T., / And they taught me for free, on a scholarship, see?'
Newman gave him a fist bump. 'Word.' He jerked his head towards the door. 'Better scram, pig, before I snap you in two.' Then pumped his chest at Franklin. 'Bitch, you can stay. I'm-a make an exception for your _fine_ ass. Show you what a real man can do to it.' He took a handful of her backside, just to make sure she understood.
Ooh... Not a good idea.
Callum cleared his throat. 'Maybe you shouldn't—'
One second, Newman was standing there, posing, the next he was facedown on the Persian rugs, with his groping hand twisted up behind his back. 'AAAAAAARGH!'
Franklin leaned in and bared her teeth. 'Donald Newman, I'm detaining you under Section Fourteen of the Criminal Justice Scotland Act because I believe you've committed a crime punishable by imprisonment: namely making death threats to one police officer and the sexual assault of another officer.'
'GETOFFMEGETOFFMEGETOFFME!'
'You do not have to say anything, but if you do not mention something you later rely on in court it may harm your defence.'
His sidekick took a step forwards, but Callum got in the way. 'I don't think we got your full name, sir.'
Big Bobby B licked his lips. 'I'm cool.'
'BOBBY, HELP ME! GET THIS BITCH OFF ME!'
'You were there on Friday night, weren't you? When Mr Newman allegedly assaulted Miss Irene Brown. Witnesses saw you enter the premises.'
'Ah... Yeah. About that. I didn't have nothin' to do with beatin' on no woman.'
'BOBBY!'
'Are you _sure_?'
He nodded, setting off a Mexican wave of fat. 'Totally! That was all Donny. I was like—'
'YOU DIRTY TWO-FACED FAT BASTARD! YOU SUPPOSED TO BE MY NIGGA!'
'Hey, suck my balls, Holmes.' He turned and spread his hands. 'I was like, "Man you gotta _stop_ hittin' that poor girl!" and he was like, "No way, this bitch gotta learn her some respect."'
The circle of acolytes backed off a couple of paces. Staring down at Newman as if he was a stain someone had trodden into the carpet.
'BOBBY!'
'Did he now? And what happened to the teddy bear he took from Miss Brown?'
'BOBBY, DON'T YOU DARE!'
'He got his stash hidden inside that poor girl's bear right now. Inna dressing room.'
'I'M-A KILL YOU, BOBBY! YOU HEAR ME? YOU DEAD, BITCH!'
Franklin twisted his arm a little further till the screaming faded to a tiny high-pitched whimper. She produced her handcuffs. 'Threats to kill, sexual assault, actual bodily harm, theft, and possession of a controlled substance. Not your day, is it, Donald?'
The yurt door swung open and Keen oiled his way in. 'Everyone, can I have act-one bands together, please? The car's...' His mouth fell open as he stared at Franklin and Newman. 'What... No... This...'
Callum patted him on the back. 'Looks like you'll have to get someone to fill in for Sick Dawg. He has to go take his medicine.'
##
_Open the Coffins_ belted out of the festival's PA system – Leo McVey growling out the opening song about how boring life was in the village, and how sweet the apples growing over the well looked.
'I can't believe you said that.' Franklin thumped the car door closed and curled her lip across the roof at Callum. '"Sick Dawg has to go take his medicine." Genuinely?'
_'Arya, it's just not fair; We're starving and nobody cares...'_
'Oh come on, that was a classic action-film one-liner.'
'It was cheesy rubbish.'
_'The bones protrude beneath our skin, / Oh, Arya, it's sickening...'_
'Exactly.' He slipped into the passenger seat, turned and smiled. 'So, Donald. Any other offences you'd like us to take into consideration?' He held up the evidence bag with Mr Lumpylump in it. The threadbare bear looked a lot fatter than it had back at Irene Brown's house. Clearly the amphetamines-and-cocaine diet wasn't working. 'Or do you expect us to believe that this is it?'
_'Above the well, so plump and sweet, the apples grow, / And it's not fair that we're both starving, here below...'_
A scowl from the backseat. Newman was hunched forward against his seatbelt, both hands cuffed behind him.
'Nothing to say for yourself?'
'You recognise me, Piggy? You know who I am?'
'Ooh, and there it is: "Don't you know who I am."' Callum grinned. 'Trust me, Donald, if you have to ask that question, you're not going to like the answer.'
'Kiss my ass, Piggy.' He sat back. 'And I ain't sayin' another goddamn word till I get me my lawyer.'
'Probably just as well. You'd only say something stupid and make it worse for yourself.'
_'The witch awakes inside the well, she's getting thinner, / She hears the children up above and dreams of dinner...'_
Franklin started the car, easing them out of the 'FESTIVAL STAFF AND PERFORMERS ONLY!' car park.
'You don't know me, _Piggy_. You don't get to _judge_ me.'
'No, but we _do_ get to arrest you, and that's almost as much fun.' He tapped Franklin on the shoulder. 'Take a left here: cut through Blackwall Hill, miss out most of the roadworks.'
'Hey, you think it's easy? All them people worshippin' you, and kissin' yo ass, and you gotta get up there and, like, _perform_ , man. Don't matter how crap you feel, you gotta make that goddamn stage come alive.'
'You broke a little girl's arm.'
'I grew up in a care home, Piggy. I got pain you ain't even heard of.'
The junction took them out on a road lined with shops.
'She was four years old.'
'I got beat every day I was growin' up, that leaves scars on yo _soul_ , yeah? You wanna see some emotional scars?' He shoogled in his seat, struggling against the seatbelt. Then used his cuffed hands to raise the hem of his leather jacket. A patch of buckled skin, veined through with pale shiny bits, sat in the small of his back, about the size of a dinner plate. 'Bitch ran the place didn't like the way I washed the dishes, so she throws a pot of boiling tatties at me. I was seven.'
'Oh you grew up in care. Boo-hoo. We _all_ grew up in care.'
Franklin shook her head. 'I didn't. My mum was a doctor and my dad worked for BBC Scotland.'
'All right for some.'
'Yo: bitch.' Newman was round the right way again. 'You dislocated my shoulder. Like it rough, do you? Like a bit of angry between your legs? That make you nice and moist?'
She glanced at him in the rear-view mirror, voice like a razorblade. 'Do you want me to stop the car? Because I will.'
Donald Newman licked his lips. Then shrank back in his seat. 'Nah, I'm good.'
'Yeah, I thought not.'
'Officer Franklin!' A grin spread across Callum's face. 'Stop flirting with the prisoner. You're—' His phone burst into song in his pocket and he pulled it out. 'Hello?'
_'Callum.'_ Mother. Silence.
OK...
At the end of the street, Franklin took the main road West. Montgomery Park shrank and disappeared behind them, until only the huge inflatable spider crawled above the rooftops.
'Boss? Are you still there?'
_'I need to ask where you were Friday night between nine p.m. and three a.m.'_
'What? Why?'
_'Callum, please. Just answer the question.'_
'Hold on.' He dug out his notebook and flicked through it to the right day. 'Nine-ish we were doing door-to-doors on Bellfield Road – we'd just searched Tod Monaghan's flat and found the mummified body? Then we went back to the station and did paperwork. Then we went to the Bart for a celebratory drink. Then I got a call from an informant and arrested Gareth Pike in Kingsmeath.'
_'What time?'_
'Stopped interviewing him about half eleven? Then I went round to the flat and collected my stuff. Then quarter past midnight I got a call and went to a domestic on Manson Avenue. Franklin and I have just made an arrest on that one, it—'
_'And was DCI Reece Powel at your flat when you collected your belongings?'_
Callum frowned out of the window. Blackwall Hill sloped down towards the river in a patchwork of houses and small parks. All of it grey and miserable in the rain. 'What's all this about?'
_'Was Reece there?'_
'Well, yes, of course he was. Don't think they'd trust me there on my own, do you? I might have taken a crap on the carpet and spray-painted a few home truths on the walls. Now, what the hell is going on?'
_'Callum, DCI Powel was found in Camburn Woods an hour ago. Someone tried to kill him; he's in surgery now.'_
Oh.
The breath curdled in Callum's lungs. 'And you think, what, that _I_ did it?'
_'I need you to come back to headquarters, Constable MacGregor. I need you to come back right now.'_
Chief Inspector Gilmore sat back in his seat and peered over the top of his evil-scientist glasses. 'I see. Yes.'
Sitting in the other chair, Mother just shook her head.
Some idiot had turned the interview room radiators up full, making sweat prickle between Callum's shoulder blades. He glanced up at the camera's dead black eye. Then down again. 'Of course it wasn't me! Why would I do something like...' he pointed at the photograph sitting on the scarred Formica tabletop, ' _that_.'
DCI Powel lay sprawled on his back, in some undergrowth. Bushes behind him, the roots of a large tree to his left. His face was a mess of scarlet and purple: lumpen, swollen, and misshapen. More bruises on his arms, hands, and wrists where they poked out of his T-shirt. The same Rolling Stones one he'd been wearing that night in the flat, only now the graphic was smeared with blood. His never-been-worn-white trainers, filthy and scuffed.
Gilmore took off his glasses, huffed on them, then polished them with a hanky. 'Do you need me to list the reasons, Callum?'
Mother put another photo on the table, next to the first. A head-and-shoulders portrait of Powel, lying on a hospital trolley. Up close, the damage was even worse. It looked as if someone had driven over his head. Repeatedly.
'It wasn't me!'
'Your girlfriend was cheating on you with Reece Powel, he got her pregnant, you were paying for everything because you thought the child was yours. He told you about the affair the same day you learned that your mother had been murdered. You assaulted him that night and broke your hand...' Gilmore's eyes drifted down to the filthy fibreglass cast on Callum's right hand. 'Your DNA was found on his T-shirt.'
'Of course it was! He was staying in _my_ flat. Sleeping in _my_ bed. Sitting on _my_ couch. Of course he's covered in my DNA!'
Gilmore popped his glasses back on again. 'Then there's the question of your... crime-scene indiscretion. I hear rumours that you now claim it wasn't you who messed up the evidence, it was Elaine. You took the blame so she wouldn't be blamed and fired. You destroyed your career to protect her maternity pay, so you could afford a baby that wasn't even yours.' He shrugged. 'Perhaps you can see why you come top of our list?'
Callum gritted his teeth. 'It _wasn't_ me.'
'He was found an hour ago, by two young girls out playing in the woods. Can you imagine how horrible that must have been for them? And even worse for DCI Powel – the SEB think he'd been lying there for at least a day and a half. Outside. In the rain.'
'I didn't do it. Oh, don't get me wrong, I _wanted_ to do it. I _fantasised_ about doing it, but I – didn't – do – it.'
Mother folded her arms. 'I was just getting to like you as well.'
'How many times do I have to say this?'
Gilmore leaned forwards. 'You were there, at the flat, the night DCI Powel was attacked. You had _very_ good reason to attack him. You'd already attacked him the night before.'
'IT WASN'T ME!' Callum wiped a hand across his sticky forehead. 'I'm not the only one who...' A frown. 'Dugdale! Ainsley Dugdale – it must've been him.'
'Ainsley Dugdale?'
Mother leaned in. 'Big, bald, bad-tempered. Runs loan-sharking and protection rackets for Big Johnny Simpson.'
'He'd threatened Powel. I know, because Powel came round to warn us about it on...' Warmth flushed Callum's face. 'Powel was already in the flat when I got home from work, Wednesday evening, wasn't he? He'd been with _her_. Pretended he'd only dropped by with a warning: Dugdale was shooting his mouth off about getting revenge on the pair of us.'
Gilmore made a note. 'On you and Elaine.'
'No. On me and _Powel_.' Idiot. 'Ask Elaine. Ask her, she'll tell you. Dugdale threatened Powel. That's who attacked him, not me.'
Mother stared at him. 'Elaine returned from Dundee at six o'clock today. She'd been staying with her parents for a couple of days, hadn't she? Keeping out of the way so you could collect your things. Wanting to avoid another fight.'
'I wasn't the one having an affair, OK?'
'She let herself into the flat and called nine-nine-nine to report that something horrible must have happened. Furniture overturned. Ornaments smashed. Blood on the floor. And DCI Powel was missing.'
'Then it must've been Dugdale!'
'She says you've been acting strange for weeks. You're prone to violent outbursts. She's frightened for her safety.'
'That's right: take her word for it. Elaine couldn't tell the truth if you paid her thirty pieces of silver.' He thumped back in his seat. Folded his arms. 'After I collected my stuff from _my_ flat, I got a call to a domestic assault in Kingsmeath. I rushed straight over there. If I'd attacked Powel, I'd be covered in blood, wouldn't I? Ask the householder: Irene Brown. Ask her and her children if I looked like I'd just beaten someone half to death.'
'Callum, you have to see how bad this—'
' _Ask_ them. And I'm not saying another word without a Federation rep and a lawyer.'
Wee Angie Northfield grimaced, then popped a roll-up in her mouth and set a lighter to the end. Sooked in a lungful, setting the tip glowing bright orange. Then let it out in a long hard sigh. 'You shouldn't have agreed to the first interview without representation, MacGregor. That was stupid.'
Rain played a staccato drum roll on the smoking shelter, running down the curved roof in rippled sheets. Splashing on the paving slabs.
Streetlights rocked in the wind, their thin yellow glow swallowed by the downpour, leaving Peel Place washed out and anaemic in the darkness. The war memorial on the other side of the street was a statue of three First World War soldiers, bayonets fixed, kilts billowing out as they charged. Someone had taken pity on them and provided each with a traffic cone hat to keep their heads dry.
Callum scowled out at the overflowing gutters. 'I didn't touch Powel, OK? Well, yes, I punched him _once_ on Thursday night, but I didn't attack him on Friday. And I _didn't_ dump him in the woods.'
'Worst-case scenario: the Procurator Fiscal thinks there's enough to charge you with attempted murder and you're off to the cells till it comes to court. Could be months.'
'It wasn't me!'
'Best case: they decide you _might_ be telling the truth and go after Ainsley Dugdale for it. Either way you're looking at an immediate suspension pending investigation. Probably without pay.'
He let his head fall back till it boinged off the Perspex wall. 'Oh joy.' Then he dug a hand into a pocket and produced his wallet. Checked the contents: a fiver, two used bus tickets, and buy-one-get-one-free voucher for Big Bernie's Pizza Palace on Wallace Lane. 'So I've got to live on five quid and some pocket smush till they clear me?'
'Yeah well, that's how it's going to—'
'Constable MacGregor!' McAdams' voice boomed out from the main doors, cheery as a drunken accountant.
Even better. Now, on top of everything else, here came some sarcastic gloating wrapped up in half-arsed poetry.
Callum bounced his head into the Perspex again.
McAdams limped down the stairs, leaning heavily on the balustrade, and hobbled over to the smoking shelter. Grinned as he stepped inside. He looked as if someone had taken a skeleton and dressed it in an inexpensive suit: cheekbones prominent and sharp, eyes sunken and dark. 'I just heard the news.' Heat radiated off him in sour waves.
'You look like crap.'
'Thank you. I'm dying, in case you didn't hear?' The smile got bigger and more cadaverous. 'Angie, my darling. Can you get our wee boy off, or is he now doooooomed?'
She shrugged, cigarette cupped in her hand. 'Fifty–fifty.'
'Then I have just the thing that may help.' He thumped Callum on the back. 'You, dear Constable Useless, can drive.'
'I can't go anywhere till they decide what's happening. Charge or release.'
'Oh, they've already _done_ that. I've just come from Mother's office with the happy news: you're suspended without pay, pending an investigation.'
Wee Angie Northfield nodded. 'Told you.'
'Bloody hell.' Callum boinged his head off the smoking shelter's wall again.
'Ah yes, but I have a _plan_.' He flashed his death's head smile again. 'To the Misfit-Mob-Mobile!'
##
Rainswept buildings slid past the car windows, turned an unhealthy yellow-grey by the streetlights. Callum took a right, over Dundas Bridge. 'This would go a lot quicker if you told me where we were going.'
The Kings River stretched out on either side, swollen, dark, and angry.
Nothing from McAdams.
Off in the distance, Montgomery Park was lit up like a Wurlitzer. Spotlights raked the low clouds, making the huge inflatable spider glow every time they touched it. Colours flickered and burst out from the giant screens – too far away to make out any detail, just a changing smear of brightness that glittered back from the river.
'Sergeant McAdams!'
'Mmmph?' His head jerked up, eyes blinking. 'What?'
'I said, where – are – we – going?'
'Oh, right.' A long puffed-out breath and a little shake. His face gleamed in the dashboard light, greasy and unwell. 'We, my dear Callum, are going hunting for one Detective Constable John Pain-In-The-Backside Watt. Mother's worried.'
'You _do_ remember I've been suspended?'
'Pff...' He waved a hand. 'Suspended is as suspended does. Besides, we're not undertaking an official investigation here, we're just out looking for a colleague, so I can kick his arse halfway up his back for him.' The sentence ended with a hacking cough that rocked McAdams back and forward in the passenger seat, leaving him panting and slumped.
'Are you sure you're OK to do this?'
'He's not answering his mobile or his landline, so we'll try his flat first. With any luck he's accidentally handcuffed himself to the bed in his favourite gimp suit.'
'Only you _seriously_ look like you should be in hospital.'
'And if he's not there, we widen the search. I've been through his spreadsheet and we've got all properties accounted for. The only ones not searched by other teams are the two he did on his own: the old Patterson-Smith Warehouse in Wardmill, and Thaw Cottages out by Holburn Forest. We'll try those too.'
'I've seen post mortems on healthier looking people than you.'
'I've already checked with all the hospitals and both mortuaries.'
Well, no one could say Callum hadn't tried. 'Where's his flat, then?'
'Take a right at the roundabout.'
The old lady from number 5 lowered the keys into McAdams' palm. 'You sure I can't make you a nice hot cup of tea, dear? Only you look like you need one. It's no trouble.'
'I'd love to, but we're on duty.' He stood on the landing and waved at her until she went back into her flat and closed the door. Then McAdams slumped. Wiped the sweat from his forehead with a trembling hand. Sighed. And passed the keys to Callum. 'You can go first, I just need to catch my breath for a bit.'
OK.
Callum knocked on the door to number 6. Waited.
The only sound was McAdams wheezing.
So he took the spare keys and let himself into Watt's flat. Clicked on the lights. 'Hello? John?'
The hallway was small, but spotless: a wonky rhomboid with four doors leading off it. Kitchen, bedroom, bathroom, and living room. All neat, all clean, all tidy. Strange, would've put money on Watt being a Pot-Noodle bachelor, with posters of wrestlers on the walls and an impressive collection of used pizza boxes. Instead, it was like something out of a decorating magazine.
A row of sympathy cards were lined up on the mantelpiece in the living room, beneath a cheesy posed photo of Watt and a woman so pale she was almost see-through.
Callum picked up one of the cards: 'WE WERE SO SORRY TO HEAR ABOUT MARY. OUR THOUGHTS AND PRAYERS ARE WITH YOU ALWAYS, BILL AND AGGIE.'
He put it down again. As if holding it any longer would make it grubby.
The answering machine flashed a red light in the corner. Callum pressed the button.
_'Have you had a bank loan or credit card in the last six years? Unsure if you're due PPI compensation? Well—'_
He hit delete.
Marched back out to the landing. 'He's not here.'
Callum's torch beam wandered across the large breezeblock wall, catching the faded lettering: 'PATTERSON-SMITH ~ QUALITY FURNITURE YOU CAN DEPEND ON'. And now there was nothing left but dust and the dirty gritty scent of mildew and stagnant water.
McAdams limped out of a door through to the old office, brushing cobwebs from his suit jacket. 'There are spiders in there big as Yorkshire Terriers. I kid you not.'
'Sod-all out here either.' He did a slow turn on the spot. 'Have you tried getting Voodoo to put a lookout request on Watt's car? He jumped out of Dotty's – no way he slogged all the way out here on foot.'
'She's looking.'
'Oh. OK.' So much for that. 'Holburn Forest, then.'
'Holburn Forest.'
'Of course, what I don't get, is why he has to be such a dick the whole time.' McAdams held onto the grab handle above the passenger door as the Mondeo rocked and growled from pothole to pothole along the track, sending up arcs of water.
'Hello, Pot? I have Kettle on line one for you.'
The car's headlights caught the broom and whin crowding the road, sending jagged shadows racing ahead of them.
'That's different. I'm _dying_ , I'm allowed to be a little—'
'Dicklike?'
'I was going to say, colourfully eccentric.' He shifted in his seat as a grating noise sounded somewhere under the car. 'You try being eaten alive by tumours, young Callum. See how altruistic you are then.'
'No thanks.'
The sky was a solid blob of orangey-grey, but a thick black stripe loomed on the horizon. That would be Holburn Forest, lurking in the darkness. Still no sign of any cottages.
'And don't get me started on the chemotherapy...' A thick, rattling sigh. 'You know, I wish I hadn't. Started on it, I mean. I could be dead by now, instead of lurching about like a broken clothes horse.' He nodded. 'But Beth won't let go, so I've got to hold on too.'
The headlights pulled tree trunks from the gloom as they reached the forest's edge. The track disappeared into it, but another track sat at right angles, skirting the boundary.
McAdams pointed. 'Left here, it's at the end of the road.'
Callum took the turning, thumping through another set of waterlogged potholes. Off in the distance, the city lights glittered through the rain. A blanket of stars, draped across the landscape. 'Aaaargh!' The Mondeo lurched like a rollercoaster, setting free another grinding scrape from beneath their feet. 'Be lucky to have any bottom left on the car, after this.'
'You want a bit of advice, Callum?'
'Not you as well.' Why did everyone think he needed their sodding opinion?
There – up ahead – a line of three cottages, sitting between the track and the forest. Grass shone in the rowans, one of the chimneys looked on the verge of collapse. The gardens were full of weeds.
'Live your life like the future's never going to happen. Because before you know it: _plop_. It isn't.' He shook his head. 'Spent my whole life doing the right thing – being responsible, working hard – when I should've been out there _enjoying_ life. Thought there would always be time for that later. Now look at me...' McAdams sighed. 'I'm down to my last few chapters, Callum. I don't think I'm going to make it to the end of the book...'
Three cottages: two semidetached, one standing on its own. Callum parked outside it. Killed the engine and sat there as the hot metal pinged and ticked. The rain got louder, battering off the car roof. 'We should probably check the graveyards.'
McAdams unclipped his seatbelt. 'You'd think, if someone buried him, they'd invite us to the funeral so we could dance on his grave.'
'There were sympathy cards in his flat. You ever hear him talk about someone called Mary?'
'To be honest, I don't know anything about his home life. He's always too... bleccccch to spend that much time with.'
'If she's died recently, he could be visiting her grave. Or out getting hammered somewhere.'
A nod. 'Definitely worth a try.' Then McAdams levered himself out into the rain. 'Are you coming, then?'
Callum grabbed his high-viz from the back and hauled it on. Hunched his shoulders as he followed McAdams up the path to the front door. 'He's definitely been here.' Pointing at a line of trampled grass and weeds leading around the side of the building.
'Of course he was. I _saw_ him here, remember? Honestly, nobody pays any sodding attention.' McAdams pushed through into the house. 'He'd finished searching the cottages, I think. Or just about.'
The hallway smelled of long-dead mice, tainted with the sharp musky odour of fresh rodent urine, and the thick cloying tickle of dust. Other than them and Watt, it looked as if no one had been in here for years.
Callum slid his torch beam through the open door and into a living room. 'This is a complete waste of time. Why would he still be here?'
'Well, I don't know, do I?' McAdams limped past, playing his torch across the peeling wallpaper. 'It was all I could think of to do. He's not at home, he's not at the station, Dotty's not seen him since she turfed him out of her car. He's got to be _somewhere_.'
The bedroom floor sagged towards the corner, where a hole as big as an armchair was rotted through the floorboards. A pair of dark shiny eyes glittered in the torchlight, then disappeared.
'If you're that worried, send out a lookout request. Get the media department issuing statements and posters. Mobilise the nightshift.'
'It's probably nothing. You know what Watt's like – law unto himself, that one. Thinks he's too good to check in with anyone or clock off at the end of the day.' McAdams disappeared down the corridor. 'Nothing in the bathroom.'
The other bedroom was empty too. And the kitchen.
Callum's torch picked out manky worktops and kitchen units, little trails of footprints scrawled through the dust and mouse droppings. It sparkled back from the window above the sink. He opened the back door and ran it around the garden. 'There's outbuildings. A bothy and a big shed-barn thing.'
'I don't understand him, Callum, I honestly don't. You? You're a simple soul – a bran-flakes-and-marmalade kind of guy. But Watt?'
'Screw you. I'm plenty complicated.' He stepped back out into the rain.
'I thought I'd finally got through to him. "Don't be a dick," I said. "You need to work as a team," I said. "Oh yes," he says, "I promise I'll be a good boy from now on!"' McAdams spat into the wet grass. 'Dick.'
The bothy was a squat blocky thing with a rusty corrugated roof. Something had been at the mortar, eating it away, exposing the rocks that made up the walls. It smelled even more of mouse than the house had.
McAdams followed him from room to room. 'See when I get my hands on Watt? I'm going to throttle the life out of him. You're going to have to alibi me. Kid-on he was already dead when we found him.'
An ancient kitchen with a lumpy range that was like a solid heap of decaying metal, floral wallpaper smeared with mildew. The ceiling had collapsed, exposing the roof beams, leaving chunks of plaster all over the floor.
'No, I won't throttle him. I'll tie him to the back of the car and make him run all the way back to Divisional Headquarters. Maybe drag him for a couple of miles too. That'll teach him to do what he's bloody well told.'
A line of swallow or house-martin nests lined the join between ceiling and wall in the next room. Stacks of old tiles and the rotting remains of kitchen units – probably dumped here when they did up one of the cottages decades ago.
McAdams wheezed. Leaned against the horrible wallpaper. Let his head hang.
Callum checked the last room – about the same size as the kitchen, only without the charm. Someone had drawn crude pornographic figures on the walls in crumbling chalk. And there was no way half of it was physically possible.
But no sign of Watt.
Back in the hall, McAdams hadn't moved.
'Right: soon as we've checked the barn, I'm dropping you off at the hospital.'
'I don't want to go to the hospital.'
'Tough. You think Mother's upset about Watt going AWOL? How do you think she'll feel if I let you snuff it out here?'
'I'm _not_ going to the sodding hospital!'
'Keep telling yourself that.' Callum marched back out into the rain. Another trampled path led through the grass and weeds to the barn. So Watt had searched it too.
He took the path anyway, right up to the barn door. Flicked the catch open, pushed the door, and stepped inside.
Stopped.
Everything stank of wood smoke.
'McAdams?' Deep breath. 'MCADAMS! IN HERE!'
The room was split into two bits – one set out with a wooden frame above a pile of ash and burnt logs. The other was a little room, built of rough-hewn wood. Just like the smokehouse they'd found at Creel Lane.
A pile of old wooden fish boxes sat in front of it, still fresh enough to ooze the acidic tang of old seafood.
'MCADAMS!' Callum struggled his good hand into a blue nitrile glove and crept over to the sealed-off section. The door was slightly ajar. He eased it all the way with his foot. Then lurched back a couple of steps, covering his mouth and nose with his fibreglass cast.
The billowing, unmistakable, _greasy_ stench of death collapsed out of the room.
Behind him the barn door thumped.
'Callum?'
'Over here.' He took a deep breath and stepped over the threshold.
A rusty metal tank gleamed in the torchlight. There must've been water in it, because it sent reflections sparkling across the wooden roof. He swung the beam right...
OK, that explained the smell.
A woman's body was slumped on the floor by the wall, held upright by the chain around her neck. Callum set the beam on her dark, swollen face. Abby Gossard. Definitely dead.
'Bastard...'
McAdams appeared in the doorway. 'I found Watt's car parked out back. What's... Oh Christ, is that smell what I think it is?'
'We've found Ashlee's mother.' He ran his torch across the floor around the tank, then back towards Abby Gossard... There was another body, lying against the wall, part hidden by a tarpaulin.
Please don't be Ashlee. Please don't be Ashlee.
Callum inched closer, picked up the edge of the tarp and folded it back.
It wasn't Ashlee.
If anything it was worse.
Detective Constable Watt lay on his side, one knee drawn up, head lying on his arm. Something black had dried in a thin line from his nose to his cheek. Another line down the side of his neck from his ear. His skin was so pale it fluoresced in the torch beam.
'Watt, you silly sod.'
McAdams cleared his throat. 'Is he...?'
Callum knelt beside Watt, laid the torch on the dirt floor, and pressed two fingers in under his jaw.
'Well?'
A tiny quiver pressed against Callum's fingertips. Then another. Faint, but definitely there.
'Call an ambulance! Call it _now_!'
##
Mother barged through the double doors, scattering a couple of paramedics in her wake. 'How is he?'
'Watt's in surgery.' Callum hitched a thumb over his shoulder at McAdams – slumped in a plastic waiting chair with his elbows on his knees and head in his hands. 'This one, on the other hand, probably should be.'
McAdams didn't even move at that.
She puffed out a huge breath. 'He's going to live, though, right? Watt's going to make it?'
'We don't know yet. Someone tried to cave his head in with an adjustable spanner. Nearly succeeded, too.'
'Gah...' She sank into the chair next to McAdams, put a hand between his shoulders and rubbed. 'Are you OK, Andy?'
'No.'
'Callum, get a doctor. Tell them—'
'Oh don't be so melodramatic.' McAdams creaked himself up till his back was straight again. His eyes were red and puffy, shiny in the overhead light. 'I was there, Mother.' He stared down at his hands. 'I was there at Thaw Cottages and I left him.'
'Andy, it's not your fault, it—'
'I gave him a bollocking, I gave him a pep talk, and then I got back in my car and I drove away.'
She cupped his neck with a hand. 'You couldn't have known.'
'If I'd stayed and searched the buildings with him, it might never have happened.'
'Shh...' Mother leaned in and kissed McAdams on the forehead.
Callum pulled up another plastic chair and sank into it. 'I've called the SEB, the Procurator Fiscal, and Hairy Harry. SEB got there before the ambulance, everyone else is on their way.'
McAdams scrubbed a hand across his sunken eyes. 'Sorry.'
'Boss, we've got a problem: whoever tried to kill Watt, it definitely _wasn't_ Tod Monaghan. Not unless he's stitched himself back together after the post mortem and broken out of the mortuary. He was working with someone.'
She stared at the ceiling tiles. 'That's all we need.'
'And whoever it is still has Ashlee Gossard.'
'Even better.'
'So, what if Paul Jeffries didn't die twenty odd years ago?' Callum scooted his chair closer. 'What if the male remains, in the shallow grave, were another victim? Not someone to sexually abuse, but someone to take the blame. What if Jeffries faked his own death and he's still out there?'
'Callum, Callum, Callum.' McAdams shook his head. 'Do you have any idea how many shades of stupid that is?' He held up a hand before Mother could do more than open her mouth. 'And I mean that with the greatest respect. One: if you're faking your own death like that, you need someone to actually _find_ the body, otherwise what's the point? Two: Jeffries would be in his seventies by now, remember? And three: he was a sex offender with a thing for abducting and raping women. Why on Satan's shiny earth would he suddenly change to abducting young men to starve, brine, smoke, and turn into mummies? Think it through.'
Yeah. It was a bit of a stretch.
Callum shrugged as heat bloomed in his cheeks and ears. 'Just playing Devil's advocate.'
'Of course you were.' McAdams sighed, then stood. 'The brass aren't going to like this, not after the triumphant press conference and all the drinks. We're going to need a scapegoat, a statement, and another public appeal.'
'Urgh.' Now it was Mother's turn to curl up into a ball in her seat. 'They're going to blame me for this, aren't they?'
'Tell them it was my fault.' McAdams patted her on the shoulder. 'It _was_ , after all. I should never have left Watt on his own.'
Callum checked his watch. 'If we hurry, we've still got time to get something on the Ten O'clock News. We can probably make the first editions too.'
She looked up at him. 'What are we going to appeal _for_? We have no idea who Monaghan's partner was. We have no idea what they look like. We have exactly sod-all clue what we're doing.'
McAdams pulled his bony shoulders back. 'We'll think of something. We always do.' A sniff. 'Well, usually, anyway.'
Mother rolled her eyes. 'It's official: we're doomed...'
Callum picked at the dirty lining of his cast. 'There's got to be _something_ we can do. What about Brett Millar? He's still off his face on prescription meds right over there, isn't he?' He nodded at the windows lining one side of the corridor. Across a darkened courtyard, the merciless Victorian bulk of the secure psychiatric ward was just visible through the rain. 'We get a warrant and we force them to pump him full of something to bring him back down to earth. Then we sweat him till he tells us what the hell happened in that flat!'
'Andy?'
'They wouldn't give us a warrant yesterday, or the day before.'
Mother gave them a pained smile and a nod. 'Let's give it one more go, then.'
'Good.' Callum pulled out his phone. 'I'll call the—'
'Actually...' She looked away. 'Maybe you should leave it to us.'
'But—'
'Callum, you can't go ordering warrants if you're suspended from duty. I'm amazed you managed to get the SEB, pathologist, and PF to go visit Thaw Cottages. The Sheriff's not going to be that understanding.'
'Oh.' His shoulders slumped.
'I appreciate the thought, though.' She stood, rubbed at the small of her back. 'Right, Andy, let's get cracking. And see if you can drum up a uniform to stand guard at John's hospital bed. I don't want anything else happening to him.'
'Yes, Mother.'
She smiled at Callum. 'Go home. Read a book. We'll let you know if anything happens.'
He stood there, grinding his teeth as they walked away.
Mother and McAdams disappeared through the doors and that was it: abandoned again.
Lovely.
All because of DCI Reece Sodding Powel.
And Ainsley Dugdale, of course.
Callum ran his good hand across his face, straightened up and marched out. Through the double doors, down the corridor past waiting rooms and treatment cubicles and shuffling old people...
A large woman in a flowery blouse and pencil skirt emerged from a door up ahead. Stethoscope around her neck, sleeves rolled up to the elbows. She stepped into the middle of the corridor, hands on her hips, then marched away from him, shoving through the doors. 'Mr McAdams?'
Callum hurried after her, through into the reception area: lined with posters about healthy eating and venereal diseases; packed with miserable-looking people in various stages of despair, waiting their turn.
'Mr McAdams?' She was still going, stout little legs pumping, trainers squeaking on the pale grey floor.
Callum caught up to her as she pushed out of the main doors and into the night. She stopped there, under the hospital canopy, hands on her hips again. Staring out into the rain. 'MR MCADAMS!'
But there was no sign of him, or Mother.
'Oh, for God's sake. Some people just...' She clicked her mouth shut. Looked Callum up and down. 'Can I help you?'
'I work with DS McAdams.'
'Gah...' She rolled her eyes. 'Then can you do me a favour and ask him... no: _tell_ him, _force_ him to call my office and set up an appointment. He's not impressing anyone with this he-man routine.'
'OK. And you are?'
She dug into the pocket on her blouse and produced an NHS Oldcastle card. 'Dr Fitzpatrick. And I'm not kidding about: he needs to call me and make a frigging appointment. Cancer isn't something that just goes away on its own.'
Callum frowned down at the card. 'Is this about his chemotherapy?'
'It would be, if he'd _actually_ turn up to his clinical appointments.' She pinched the bridge of her nose. 'Please, just talk to him, OK? He won't answer any of my calls, or texts. We need to get him in and we need to get his treatment started. He's going to die otherwise. And I'm not talking in some obscure theoretical sense: he – will – literally – die.'
But...
'I mean it: he'll die.'
'Yes. Of course. I'll talk to him.'
'Thank you.' She spun around on a squeaky heel and pulled out her mobile phone, poking at the screen as she thumped back into the hospital. 'Angie: I need a CT scan for Mrs Stoltzman...'
Either she was helping herself to the contents of the medicine cupboard, or something very wrong was going on.
Callum hurried out into the car park, but the thing was huge, stretching all the way from here to the maternity hospital, broken into various chunks along the way. No way of telling where Mother would have parked her manky Fiat Panda.
So he ducked into a bus shelter and scrolled through the contacts on his phone. Clicked on the one marked 'DS CRAP POETRY'.
It rang for a bit, then McAdams was on the line, his voice muffled – almost drowned out by the grumbling engine. _'Are you missing me already? That's sweet.'_
'I just bumped into a friend of yours.'
_'Did you now?'_
'A Dr Fitzpatrick.'
The only noise from the other end was the engine.
'You still there?'
_'No.'_
'She says you've not been going to your chemotherapy sessions.'
_'Well, that is interesting. Hold on, I'll have to text that information to you... Yes.'_
'What the hell are you going on about?'
_'No, don't worry about it. Not a problem... OK, bye.'_ He hung up.
Callum stared at his phone. Dr Fitzpatrick wasn't the only one raiding the medicine cupboard.
He dialled McAdams again, but it went straight through to voicemail: _'You've reached Detective Sergeant Andrew McAdams. I'm sorry I can't take your call at the moment, but—'_
Callum killed the call.
Five seconds later his phone buzzed and dinged an incoming text.
What U playing at? Can't talk
in front of Mother!!!
Shes gt enough 2 worry abt
already!
He leaned back against the bus shelter and poked out a reply:
The doctor said you've not been
going to chemo. You told us you
were.
_Ding_ , buzzzzz.
I'll call U when we get bk to
DHQ. Dnt B a dick about this!
Oh, so Callum was being a dick, was he?
You're the dick! Why aren't you
taking your chemo treatment? Do
you WANT to die?
_Ding_ , buzzzzz.
Yes.
Oh.
The Mondeo was where Callum had left it, double-parked across a pair of matching four-by-fours. He unlocked the door and slid in behind the wheel.
So McAdams was skipping his chemo and didn't want Mother to know about it. Fair enough. It was his life. What was left of it.
Callum rested his forearms on top of the steering wheel. Rain bounced off the car's bonnet, clattered on its roof, made a small river in the gutter, overflowed into a spreading loch in the car park.
Suspended.
Of no use.
Abandoned.
Well, sod the lot of them. Ditch the car back at Divisional Headquarters and hit the Bart instead. Pick up his bike from Hedgehog. Maybe have a pint or three. Or four.
After all, it wasn't as if he had anything better to do.
Not now they'd chucked him off the case.
Something upbeat and poppy was playing as he walked into the room. The Dumbarton Arms was virtually empty, just a pair of youngsters trying to climb inside each other in one of the booths at the back. All hands and tongues.
Callum shook the rain from his hair and stepped up to the bar. 'Hedgehog.'
The barman froze in the middle of loading a case of Bacardi Breezers into the fridge. 'Oh dear.'
'Is that any way to greet a valued customer?'
He straightened up and turned. Tucked his hingin-mince hair behind his ears and faked a smile. 'Detective Constable MacGregor, how delightful of you to grace us with your patronage once more. I trust things are on a more cheerful footing with your good self?'
'Pint of Trade Winds and a packet of pickled onion, thanks.'
'Only I don't think I could realistically survive another night of suicidal music playing on an endless loop.' He pointed at the couple in the corner. 'And it would ruin our young lovers' _esprit d'amour_. Please?'
Callum hauled himself up onto a barstool. 'Thanks for taking care of my bike.'
'Does this mean we'll be permitted to enjoy a Radiohead-slash-REM-free night?'
'Promise.'
All the breath flopped out of Hedgehog. Followed by a smile. 'In that case, dear Detective Constable, this first libation, and its accompanying comestibles, comes to you courtesy of the Dumbarton Arms in fulsome gratitude.' He pulled the pint and set the glass down in front of Callum. 'And while we're talking of fulsome gratitude, will we be settling our bar tab from Thursday at some point today?'
'Ah...' He dug out his fiver and his small handful of change. Set the lot down between them.
'Five pounds, eighty-three pence. And a button?'
'They suspended me without pay, Hedgehog. You're looking at all the cash I've got in the world right now.'
A long, gravelly sigh. Then Hedgehog pushed the small pile of cash back towards Callum. 'In that case, shall we come to an accord? I propose the Dumbarton Arms retain possession of your bicycle until sufficient funds are at your disposal to make the appropriate remuneration. How does that sound?'
Irene Brown pawned her teddy bear and kids' toys to pay for food and rent. And here _he_ was pawning his bike to pay for an alcoholic bender.
Proud moment, Callum. _Really_ proud.
'Thanks, Hedgehog.' He took a sip of his pint, then swore as his mobile dinged and buzzed.
Incoming text message.
No prizes for guessing who that would be from: bloody Haiku Boy.
Callum pulled the phone out and stuck it on the bar top, next to his crisps. But it wasn't McAdams, it was ex-sergeant Bob Shannon.
Callum, I've got some good news:
the OAPs finally came up with a
name!
Where are you? I'll come get you.
He popped open his bag of pickled onion and crunched through a couple. Then wiped his fingers clean on his trousers and replied:
Too late. Gareth Pike gave me the
ID this afternoon. I've already
been round – denies everything.
There was barely time for another mouthful of beer before his mobile burst into life. Shannon again. 'Bob.'
_'Callum. You went round to see him? What happened? Did he cop to it?'_
'Did he hell.' Another couple of crisps vanished in a hail of crunching. 'Said he wasn't there, he didn't attack my parents, he didn't abduct my brother. And if I want to talk to him again I'd better have a warrant with me.'
_'Pfff... What about the night your mother's head was dumped in the woods?'_
'Oh he's got an alibi for that. Hundreds of them: all over Twitter and Facebook.'
_'What?'_
'He was on tour with his band. There's loads of pictures online of him up on stage in Brussels.'
One of the heavy petters surfaced for air, then shuffled his way to the bar, grinning. 'Yeah, can I get a Glenmorangie and a bottle of the strawberry-and-lime cider?' He turned and waved at the bearded friend he'd left behind at the table.
'Bob, you still there?'
_'Sorry. Did you say he was on stage in_ Brussels _? Brussels, like the one in Belgium?'_
Callum took another scoof of beer. 'Have you been hitting the Malbec again, Bob?'
_'No. Just surprised, I suppose. Didn't think he'd be able to perform any more. Given his condition.'_
'You're kidding, right? He's been all over the radio, TV, and papers for about a week, banging on about his grand-finale career-comeback gig tonight. In Montgomery Park?'
_'Erm... Callum, who are you talking about?'_
'The man who abducted my family: Leo McVey.'
_'Yeah. No. That's not the name I've got.'_
##
Shannon pointed through the windshield. 'That's us there.'
The house sat on its own, on the outskirts of Auchterowan, just visible through the trees bordering the property. Not quite a Georgian mansion, more a bungalow with delusions of grandeur. All the lights were on, its windows glowing, casting a warm golden glow out into the front garden. A winding gravel drive, threading its way between rhododendron bushes to the front door and double garage.
At least it had stopped raining. Up above, a patch of stars glared down at them from a hole in the clouds. About as welcoming as a mortuary drawer.
Callum puffed out his cheeks. 'Are you _sure_ about this?'
'It was just a rumour, remember? I'm making no promises.'
'OK. Right. Only... you know.'
A small hatchback sat in front of the garage, its boot gaping open like a hungry mouth. And as they sat there, parked on the road with the engine idling, a woman in a flowery pinny and yellow rubber gloves lurched out of the garage with a big bag and fed the hatchback with it. Wiped her forehead on the back of her arm, then went back inside.
Shannon nodded. 'We don't have to do this if you don't want to.'
'No, I do. But...' He cleared his throat. Took a deep breath. 'But it's R.M. Travis. The man's a hero to millions of kids, he _can't_ be a serial killer.'
'I was listening to Radio Four the other day: someone's started a petition to get him on the New Year's Honours list. Can you believe that?' The car's wheels crunched on the gravel driveway. 'They'll be a bit embarrassed if Pike's telling the truth. Arise, Sir Murdering Tosspot!'
Callum shifted in his seat. 'Maybe Pike's lying?'
A modern extension poked out from one side of the building – a low long box fronted with floor-to-ceiling glass. It was a private library, lined with bookshelves, crammed with books, lit by artfully placed downlighters and standard lamps.
God... To own that many _books_.
'I wouldn't be surprised if he was.' Shannon pulled up outside the front door. 'That's the thing about people like Gareth Pike: lying's like breathing to them. It's a way of life. I don't think Pike would know the truth if it clambered up his bumhole and took up clog dancing.'
Callum undid his seatbelt. Grimaced. 'OK. We can do this...'
'Of course, maybe Pike gave you Leo McVey's name because he was protecting Travis? Maybe they go _way_ back?'
'You think R.M. Travis is a _paedophile_?'
'Wouldn't be the first kiddy fiddler in line for a knighthood.'
Something deep inside his stomach lurched and gurgled. 'Yeah, but...' He hissed out a breath. 'I loved his books, growing up. The home had a complete set. If it turns out he... you know?'
Shannon patted him on the shoulder. 'Look at it this way, we—'
There was a knock on the driver's window.
The woman in the pinny stared in at them, face creased and worried. Brown hair, greying at the roots. Bags under her eyes, going slightly jowly around the chin. Pale. As if she hadn't seen the sun for years.
Shannon opened his door and she stepped back.
'Can I help you?' Voice brittle and sharp.
Callum climbed out. 'We're looking for R.M. Travis.'
She grunted. Then shook her head. 'I'm sorry, but it's late and my father's not up to visitors. Please, he appreciates you reading his books, but he's not well.' She pointed down the drive towards the road. 'Now, if you'll excuse me, I've got things to do.'
He produced his warrant card. 'DC MacGregor.'
If anything, she went even paler. 'What's this about?'
'Can I ask your name, Mrs...?'
'Travis-Wilkes.' She laughed, brittle and high-pitched. Wrung her rubber-gloved hands together. 'And it's _Ms_. Emma. Divorced, single, writer: seeks tall strong man for walks on the beach, cheese toasties, and vigorous lovemaking.' She licked her lips. Cleared her throat. 'Sorry.' Pulled on a smile. 'We get so many fans coming here. Well, not as many as we used to.' A shrug. 'It gets a bit much at times.'
'You live here?'
'As carer, nurse, archivist, biographer, and general dogsbody. You've _no idea_ how much work's involved in looking after someone else's literary legacy. I haven't written a single word of my own for years.'
'We need to ask your father a few questions.'
The laugh sounded forced. 'Good luck. He's not having one of his better days.' She brushed her fringe aside with the bright yellow gloves. 'Half the time he thinks I'm my mother. The other half he hasn't got a clue _who_ I am.' Emma pointed back towards the garage. 'He switched off all the freezers last week. No idea why. You wouldn't _believe_ the smell.'
'We'll try not to take too much of his time.'
'Wouldn't be quite so bad if he didn't have enough rubbish in there to survive a nuclear winter.' Sounding more bitter with every word. 'I swear on the Bonemonger's grave, he hoards leftovers like some people hoard money. We've got pot roasts in there going back to Margaret Bloody Thatcher's...' Emma cleared her throat. Straightened her pinny. 'Sorry. I shouldn't be talking about him like that. It's been a long day.' She sighed. Pulled on another smile. 'Anyway, shall we?'
She snapped off her gloves, stuffed them in the pocket of her pinny, and unlocked the front door. Ushered them into a wide hallway covered in framed book covers – most of their titles barely recognisable in foreign languages: 'ÖFFNEN SIE DIE SÄRGE', 'ZACZAROWANY KRÓLIK RUSSELL', 'LES MONSTRES QUI SONT VENUS DINER', 'EL CUBO DE BASURA MILAGROSO DE IMELDA'...
'I need you to understand, his grip on the real world is... tenuous.' Down to the end of the corridor and left. More book covers. 'He goes off on these rambling discussions where he's playing all the parts. Arguing with himself. I used to just let him get on with it, then I sat down and listened to what he was _actually_ saying.' She looked over her shoulder as they passed a big kitchen with gleaming work surfaces. 'I was trying to get him to talk about his childhood, but he was going on about how the Goblin Queen was rebuilding her army in the depths of the forest.'
A door at the end of the corridor opened on the library. The comforting smell of books mingled with the chemical floral whiff of air freshener.
'It was like he was writing another Russell the Magic Rabbit book. Only instead of battering it out on his old Underwood, he was living it all in his head.'
The bookshelves weren't just around the outside of the room, they made islands in the middle of the space too, dividing it up into discrete areas. Some with armchairs, others with little tables.
How rich would you have to be to own something like that? It was like Heaven, Nirvana, and Jannat ul-Khuld all rolled into one. A Valhalla for bibliophiles.
'So I started recording him. Sometimes it's about Russell, sometimes it's Imelda, sometimes it's Justin and Arya. Sometimes new stuff, sometimes just retreads of the books already published. The only person he never talks about is Ichabod Smith, for some reason.'
Over in the far corner, at a desk piled high with books and papers, was the bent-over figure of a man. White hair circled a shiny bald patch speckled with liver spots. He was curled around a piece of paper, one arm shielding it – like a small child trying not to let the cheat at the next desk copy off him – scribbling away.
'Sometimes I wonder if it's because, writing the books, the characters and worlds he created are more real to him than what happens out here with the rest of us. They say people with dementia find it easier to remember stuff from fifty years ago than this morning. He remembers them instead.' She took a deep breath. 'Dad?'
The man at the desk kept on scribbling.
Emma lowered her voice. 'Just don't be surprised if it all goes a bit surreal. Sometimes he's him, sometimes he's one of his creations. He was the Bonemonger for nearly a week once. That was... _disturbing_.'
Callum took out his phone. 'Thanks.'
She walked over and patted her father on the shoulder. 'Dad?'
He flinched. Turned, blinking up at her. 'Sophie?'
'No, Dad, it's Emma. Remember? Emma?' She turned and held out a hand. 'These nice police officers want to have a chat with you. OK?'
'I'm busy. Tell them to come back tomorrow.' He went back to scribbling.
She pulled a face at Callum, mouthing, 'Told you.' Then back to her father. 'Come on, Dad, it'll only take a minute. Do you need anything? A cup of tea? Some juice?'
'Sophie, I can't find my hat.'
'OK, I'll leave you to it then.' She kissed him on top of the bald spot. 'I've got manky freezers to clean out.' A small shudder, then she was gone.
Callum pulled out the chair on the other side of the table and sank into it. 'Mr Travis.'
No response, just more scribbling.
He produced his wallet and opened it. Laid it on the table between them, turned so the MacGregor family photo was the right way up for Travis. 'Do you recognise these people?'
'There's no point wriggling, little rabbit boy. You're for the pot whether you like it or not.'
'Look at the photograph, Mr Travis.'
'Don't eat me. Please don't eat me.'
'A mother, a father, and two little boys.'
'Don't be silly, I'm _hungry_. And you're a tasty little morsel, all plump and delicious.'
'Mr Travis!'
'But you can't eat me. I'm... I'm full of venomous poison. If you eat me, you'll swell up like a massive balloon and pop!'
Shannon sighed. 'He's not even in the room.'
'You'll pop and there'll be bits of ogre all over the walls and ceiling. And... and your heart will turn _black_ and the crows will take it away to the bonegarden.'
'TRAVIS!' Callum slammed his hand down on the tabletop.
The old man flinched, head snapping up like a trap. Eyes wide. 'I don't know you. Why are you here?'
'Do you recognise them? This family, right here. Twenty-six years ago.'
'Where's Sophie? Where's my wife?' He blinked. Frowned at Shannon. Then back to Callum. 'I don't want to die.'
For God's sake.
Now that he was sitting upright, the sheet of paper he'd been scribbling on was visible. He'd drawn a long-eared rabbit in the middle of it, then surrounded it with the same words over and over: 'KILL AND EAT. DESTROY AND CONSUME. SIN AND INNOCENCE.'
McAdams was right – Travis was obsessed.
One more go. 'Do you recognise this family?'
'They're happy little rabbits. Hopping in the sun. Look at their white tails flashing.'
'Yeah.' Shannon sucked air in through his clenched teeth. 'I'd suggest throwing him down the stairs a few times, but I don't think it'd help.'
'They run and they frolic beneath the happy summer skies. Mummy Rabbit, Daddy Rabbit, and little Justin.'
Callum closed his wallet again. Stuck it back in his pocket. 'You're getting your books mixed up. Justin was a little boy, before he got cursed by the witch. He never had a rabbit mum and dad – they were people.'
'Into the pot you go, little rabbits.'
'Sorry.' Shannon put his hand on Callum's shoulder. 'It was a longshot anyway.'
'So I can't touch Leo McVey, and R.M. Travis is...' A sigh. 'What's the point? Even if it _was_ him we couldn't put him on trial. And no one's sending him to prison like this.'
'Please don't eat us, we promise we'll be good!'
Callum pulled himself to his feet. 'Thank you for your time, Mr Travis. I loved your books when I was growing up. But I don't think I'll be able to read one ever again.'
'What's the point of catching a rabbit if you don't eat it?'
'Come on.' Shannon steered him away from the table. 'We'll go crack open a nice bottle of wine and bitch about the good old days.'
Travis scowled at them. 'Who are you? What did you do with my hat?'
Never meet your heroes.
They made their way back through the house.
'It was just a rumour anyway.' Shannon gave Callum's shoulder a squeeze. 'Gareth Pike was actually there and he _saw_ Leo McVey. So that's what we focus on now.'
'I can't go near McVey without a warrant. And he'll lawyer up, soon as I do.'
Past the fancy kitchen and all the framed foreign-edition covers.
'So we go digging. My OAP network isn't the spryest, but never underestimate the power of bloody-minded old codgers with a lot of free time on their hands. We'll dig into McVey's past till we find something. Then use it to squeeze him till he squeaks.'
They stepped out through the front door and into a thin drizzle.
'Thanks, Bob. I appreciate it.'
'We'll get him. I promise.' Shannon unlocked the car.
'Yeah.' Callum sagged. Then turned. 'Suppose I'd better tell the daughter we're away.'
The back of the hatchback was still gaping open, most of the boot packed with black plastic bags. Not the thin domestic ones – proper thick rubble sacks. Because who wanted their car full of manky rotting meltwater?
He stepped into the garage and the cloying smell of turned meat.
Pff... Place was bigger than his whole flat.
Well, what _used_ to be his flat.
Shelves and storage on one wall, a dusty green Range Rover dominating half the space. White paint on the concrete floor. A rack full of golf bags and clubs near a door that had to lead back into the house. A large collection of tinned beans, hot dogs, peas, macaroni cheese, jars of pickled beetroot, onions, mustard... Callum picked a tin of peaches off the shelf and blew a blizzard of grey off the faded label. Best before June 2001. Yeah. Maybe not.
A row of chest freezers and a couple of uprights sat along the back wall. Some lying open, others still sealed shut.
Emma Travis-Wilkes was bent over one of them, delving into its stinky depths. Divorced – and available – bum wiggling as she reached. A pair of rubble sacks sat on the ground beside her, one full and tied, the other still waiting for more rancid gifts.
He held up the tin of peaches. 'I see what you mean about hoarding food.'
'Gah!' She straightened up with a start. A freezer bag splotched to the concrete at her feet, sending a spatter of brown liquid out as it split. She turned, stepped in front of it. 'You startled me.'
'Sorry. Just wanted to say we're going.'
'Right. OK. Well... thanks for letting me know.' She eased the freezer lid shut behind her. 'Sorry about the smell. How was he?'
'Confused.' Callum took a couple of steps closer. 'Did he ever mention anything about a family of four? It would have been about twenty-six years ago. They were in a lay-by just outside the city.'
'No nothing like that.'
'What about Leo McVey. They used to be close, didn't they?'
'I... yes. They hung out all the time. Got stoned and drunk, trashed hotel rooms on every continent. Cocaine, groupies, and rock-and-roll. No wonder Mum left him.'
Another couple of steps. 'Did he ever say something about McVey that made you suspicious, or worried about him? Something that didn't sound right?'
'No. Thick as thieves, that pair. Uncle Leo could've crapped in the fish tank and Dad would have sworn it was the cat. And you don't want to _hear_ the stories of when they went to Las Vegas.' Emma snapped off her rubber gloves and closed the gap. Stuck her hand out. 'I'm sorry you had a wasted trip.'
'Me too.' He handed her the peaches. Pointed at the rubble sack. 'I can dump that in the car on my way past. I'm going that way anyway.'
'No, you don't have to—'
'You wouldn't believe how strange it was, questioning your father. I was such a huge fan when I was a kid.' Callum marched over and grabbed the sealed rubble sack. Heavier than it looked. And the underside was pale and gritty with frost.
Eh?
Why would _defrosted_ leftovers be cold enough to—
Something solid slammed into the back of his head and the world erupted like a million fire alarms had just gone off at once. Yellow and black spheres popped and crackled across the garage. The freezers. The rubble sacks. He reached out to steady himself and the something solid slammed down again.
Then Callum's knees buckled and the concrete floor welcomed him with open arms.
##
The tin of peaches gave a dull _bang_ as it hit the concrete and buckled. Bright red covered the sell-by date, leaching colour back into the ancient label. It rolled around a lopsided circle and came to rest against Callum's chest.
Garage floor should've been cold. All that concrete. But it wasn't.
Warm and cosy.
Soft and comforting.
Emma's boots appeared, right in front of his face. Then she squatted down. Stroked his forehead. 'I'm so very sorry.' Her bottom lip trembled, eyes sparkling as the tears welled up. 'But I can't.'
The boots faded into the distance.
He blinked.
She was unloading the golf bags from the rack by the door – dumping them on the ground. In the gap behind where they'd been was a tall thin metal locker, fixed to the garage wall.
Warm and comfy, lying sprawled on the cosy concrete floor.
She pulled a bundle of keys from her pocket, sorted through them and unlocked the door. Pulled out a shotgun. 'It wasn't meant to happen like this.'
A _clack_ sounded as she broke the shotgun open. 'But that's life, isn't it? One minute everything's fine and the next you're standing in front of the freezer, looking down at a human head. And it's looking _back_ at you. And everything you've ever known about everything is a lie.' Emma wiped her eyes on her sleeve, then rummaged in a large leather satchel. Slid two red shells into the shotgun's breech.
Flipped it closed again.
_Clack_.
A dull throb bloomed at the back of Callum's skull. Spreading out in jagged waves.
'But you just have to cope, don't you? What else _can_ you do?' She nodded to herself. Then walked out through the open garage door.
Shannon's voice came from outside, ramping up from normal person to police officer in the space of six words: _'What the hell are you... No! Emma, don't do anything stupid. We can talk about this. Put the gun—'_
BOOOOOM...
The squeal of tearing metal, the patter of shattered glass hitting the gravel drive.
_'CALLUM! CALLUM, I'M—'_
BOOOOOM...
Silence.
She stepped back into the garage. Sobbed. Stood there bent almost double beneath the weight of it. 'I... I didn't want this. I _didn't_.'
_Clack_.
Emma broke the shotgun open again and the two spent cartridges spun into the air, twirling as they fell, leaving thin trails of smoke behind. Pinging as they bounced off the concrete.
This was it.
He was going to die here.
She dipped into the cartridge bag for another couple of shells. Slid them home.
_Clack_.
She bit her bottom lip. Sniffed.
At least it would be quick.
But she didn't shoot him. She took a deep shuddering breath and marched through the door that led into the house instead.
Callum forced a hand under himself. Pushed...
Nope.
The throbbing in his head got louder, sharper.
Maybe it was all for the best?
What else did he have going for him?
Crappy childhood. Failed relationship. Ruined career.
The only mark he'd leave on life would be right here on the garage floor. Eight pints of blood. And twenty minutes with a mop and a bucket of bleach would soon get rid of that.
A muffled boom sounded from somewhere deep within the house. Followed by another one.
So get up. Get your useless, _lazy_ , good-for-nothing backside off the ground and _do_ something. What if Shannon was still alive? What if he was lying out there, bleeding to death, because Callum was too busy wallowing in self-pity to get off his arse and help him?
'Grrrrah...' He pushed himself over onto his front, then back and up till he was on his knees.
The garage whirled and roared all around him, like being drunk on the waltzers, making his stomach churn.
Be sick later, get up _now_.
Emma reappeared through the door, her face flushed and shiny, tears glistening on her cheeks. Arms wrapped around the shotgun, like a teddy bear. 'Why does nothing ever turn out the way _I_ want? Tell me that. WHY HAS IT ALWAYS GOT TO BE EVERYONE ELSE?' She rubbed at her eyes again. Then curled her shoulders forward and sobbed.
Get up, get up, get up, get up.
Do it now, while she's distracted.
But his legs just didn't want to.
She's going to shoot you. She's going to stick that shotgun in your face and pull the trigger. They'll have to scrape your brains off the floor with a shovel.
ON YOUR FEET!
Emma shook her head as the sobbing subsided. She blew out a shuddering breath. Wiped her eyes. Sniffed. Turned back to the gun cupboard and dipped into the shell bag again. 'I'm sorry.'
Callum grabbed the chest freezer and pulled himself up to his feet.
_Clack_.
He held a hand out as she turned back towards him. 'Emma, you don't have to do this.'
'Why?' She stepped closer, raised the gun. 'Why did you have to come tonight? Why couldn't you wait till tomorrow?'
'Emma, the man you shot: Bob. There might still be time to save him.'
'It would all have been _gone_ by then.'
'Emma, please, this can still end up OK, I promise.'
Closer.
The twin barrels of the shotgun were huge and dark.
She wiped her eyes again. 'It would all have been gone.'
Goodbye cruel world.
Callum nodded. 'I understand. I'm sorry, Emma, it...' His eyes went wide, staring over her shoulder. 'Bob!' Smiling as he stumbled forwards a couple of inches.
She turned.
And Callum lunged.
Shoved the shotgun to one side and hammered his fibreglass cast into her face _hard_.
She went over backwards, left arm pinwheeling as she fell, gun still held in her other hand.
It went off as her head cracked into the floor, a deafening roar that ripped a huge hole in the Range Rover's driver's door, shattering the window, tearing out through the roof.
He half jumped, half fell on top of her, pinning her gun arm to the ground. Smashing his cast into her face again. And one more time for luck.
Reared back as the shotgun clattered free of her limp fingers.
Grabbed the dented tin of bloody peaches and raised it high above his head, ready to crash it down...
A little bubble of scarlet popped from her squint nose. She coughed and more spattered out of her mouth, leaving her teeth stained dark pink.
Callum let the peaches fall back to the garage floor.
She shook beneath him, eyes screwed shut, tears dribbling down the side of her face. 'I'm sorry. I'm sorry...'
Wiped a hand across his mouth. 'Emma Travis-Wilkes, you're under arrest.'
'I'm so sorry...'
He dragged out his handcuffs. '"There's no point crying, little girl," said the Bonemonger with his scissor-sharp smile. "No one will hear you, and nobody cares."'
Callum lurched out of the garage and onto the driveway. The drizzle was like a soft kiss, cooling and fresh against his face.
Shannon's car was lopsided – the wing peppered with holes, the tyre flat, door hanging open with more holes punched through it.
'Bob?'
Gravel crunched beneath his feet as he staggered over to the door.
Shannon lay on his back across the front seats, one leg in the passenger footwell, the other dangling out of the car.
'BOB!'
A large dark stain covered the front of his yellow T-shirt, spreading across the word 'NORWICH'.
Sodding hell.
Callum clambered around to the driver's side, hauled open the door and felt for a pulse. Swore. Dragged out his phone and called 999. 'OFFICER DOWN, I REPEAT, OFFICER DOWN!'
##
'And we're all done.' The doctor dropped his needle into the kidney dish. 'Excellent stitches, even if I say so myself.' A small round Teletubby of a man with a comb-over and Stalin moustache.
Callum just grunted.
'Now I'm going to write you a prescription for painkillers, but keep off the booze with them, OK? And just in case: get someone to stay with you tonight. Don't want you dying in your sleep, right? Right.' He taped a patch of gauze to the back of Callum's head. 'While you're here, we might as well change the dressing on your ear. Nurse, let's have some disinfectant...'
There was more, but it was just noise.
The treatment cubicle curtains opened and there was Franklin. She'd ditched the black suit and tie for a red lumberjack shirt, blue jeans, and trainers. Flashed her warrant card. 'Will he live?'
Dr Teletubby stepped back to admire his handiwork. 'Well, he's damn lucky she didn't fracture his skull, but other than that? Probably.'
Callum slid down from the table and picked up his jacket. The sleeves and back were stained brown and black. Some of it his blood, some of it Shannon's. 'What about Bob?'
Franklin grimaced. Shook her head. 'They did everything they could.'
Great.
Dr Teletubby pointed through to the reception area. 'Go park yourself for ten minutes while I sort out your pills and get you discharged.'
Callum followed Franklin back to the rows of plastic seating and the buzzing vending machines.
She pointed. 'You want a cup of tea or something?'
'Any word on Watt?'
'I'll get you a tea.'
Soon as she was gone, Callum folded forward till his chest rested against his knees. Wrapped his arms around his head. Pressing them into the gauze taped to the back. Squeezing. Making the stitches scream a sharp bitter song.
Shannon was _his_ fault.
He should never have accepted that lift out to R.M. Travis's house. Should have made his own way there. Should've insisted.
Bloody hell.
Why did everything have to turn to shite? Why did it _always_ have to—
'Callum? Are you OK?' Franklin settled in the chair beside his, the warmth of her body seeping through his dirty shirt.
'No.'
'Do you want me to get the doctor back?'
He blew out a breath. 'No. Thanks. But no.' Another sigh and he sat back up. 'Sorry.'
'Here: drink your tea.'
The plastic cup was scalding hot. And the contents tasted every bit as bitter and foul as he was inside.
Franklin put her hand between his shoulder blades and gave them a little rub. 'You didn't shoot him, Emma Travis-Wilkes did. She killed her dad, she killed Shannon, and she would've killed you too. It wasn't your fault.'
A nod. More horrible tea.
She gave a little laugh. Shook her head. 'This has been a _great_ first week at work.'
He turned in his seat and smiled. 'Welcome to the Misfit Mob.'
'Oh, Callum, you're a poor sod, aren't you?'
'I'm a bloody disaster area.'
'No you're not. You're a good man, and Elaine was an _idiot_ for throwing that away. Didn't know a good thing when she had it.'
Franklin's hand was warm against his back. Her thigh warm against his. Her smile warm and soft. Her lips...
He leaned in, breathing in the scent of her: lemons and jasmine and rosemary. Closed his eyes.
'Argh! Jesus!' Franklin's chair scraped back and she was on her feet. Backing away, staring down at him with her face curdled, as if he'd just coughed up a hairball into her lap.
Heat flooded Callum's face, prickled across the back of his neck, set his ears on fire. 'I'm sorry, I—'
'What is _wrong_ with you? This is—'
'I thought there was a thing and—'
'I've got a fiancé!'
'I'm sorry! I didn't... Arrgh.' He curled up in his seat again. 'God's sake.'
'Just because I'm a black woman, doesn't mean I'm going to jump into bed with every pasty-faced horny bastard who knocks! You're just like all the rest of them!'
Idiot. Bloody stupid halfwit _idiot_.
He gritted his teeth and clenched his eyes shut.
Everything. Every _single_ thing he touched.
Franklin's phone burst into song. Then silenced. 'What?'
There'd been a thing, hadn't there? Between them?
Her voice was hard and brittle. 'Oh, he's here all right.'
Idiot Callum. Such a bloody idiot.
'Yes... OK. Right... No, I'll tell him... OK, bye.'
He took a deep breath. Stood. 'I'm sorry. I didn't mean to make you feel uncomfortable or preyed-on or anything like that. I thought...'
She just glared back at him, arms folded, knuckles pale where she was gripping her phone.
'I don't know what I thought. Maybe it was Emma Travis-Wilkes trying to cave my skull in? Maybe that rattled something loose? Whatever. I wasn't thinking right and I'm sorry. I'm one hundred percent genuinely sorry.'
'I thought you were different, Callum.'
'Yeah, well apparently I'm a moron like every other man on the planet.' He struggled into his filthy jacket. Couldn't look her in the eye. 'It's OK. I'll get a taxi. Thanks for coming. You don't have to stay.'
A nurse squeaked up, clipboard in one hand, small white paper bag in the other. 'Callum MacGregor? I've got some painkillers for you.' She held out the clipboard. 'Just sign there where the X is...' A frown. 'Is everything all right?'
'No. I just made a complete and utter dick of myself.' He scrawled his name in the appropriate box, accepted the paper bag. 'Thanks.' Then turned and limped out through the doors.
Franklin followed him.
He kept going, out from beneath the canopy and into the rain. Stopped. Turned, both arms held out. 'I'm sorry, OK? My life's turned to shit and I'm _sorry_. That wasn't about you, it was about me being _fucking_ useless.' Cold and damp seeped through his filthy jacket. He dropped his arms. 'Just go. Please. I've embarrassed myself enough for one night. I don't need an audience.'
She held up her phone. 'That was DS McAdams. The Duty Doctor's given Emma Travis-Wilkes a clean bill of health. They're putting her in Interview Four now.'
He turned and walked out into the night. 'Good for them.'
'McAdams says you can watch from the viewing suite, if you like?'
Interview Room Four looked bigger on the screen. Not so cramped.
_'Are you sure you want to do this, Emma? These are very serious charges.'_ Mother's mop of bright ginger hair sat in the bottom right-hand corner of the TV, next to McAdams' hunched back in the bottom left.
Emma Travis-Wilkes shifted in her SOC paper oversuit – sperm-white rather than Smurf-blue – and glanced at the man sitting beside her: dark-blue three-piece suit, grey hair pulled back in a ponytail, little round glasses, pointy sideburns. Mr Slick gave her a tiny nod.
Callum sniffed. 'Here come the lies.'
He shifted his cheap plastic chair closer to the monitor. A line of little microphones, perched on the end of bendy metal sticks, poked out at him, each one dark and dead. Waiting for someone to flip the switch.
Franklin stood with her arms folded, leaning back against the door. About as far away as she could possibly get.
Yeah, well, couldn't exactly blame her.
Emma Travis-Wilkes took a deep breath. The left side of her face had swollen and darkened. The bruises taking hold from where Callum introduced her to his fibreglass cast. A sticking plaster made a pale stripe across the bridge of her nose. _'Having discussed the situation with my solicitor, I would like to make the following confession.'_
Mr Slick patted her on the arm, voice almost too low to make out. _'It's OK: in your own time.'_
_'I killed them.'_
There was a pause, then Mother leaned forward. _'Who did you kill, Emma? For the tape.'_
She stared back. _'All of them.'_
Franklin gave a little whistle. 'I genuinely didn't think it'd be that easy. Expensive lawyer like that? Thought he'd make her "no comment" for at least an hour or so. It's a bit of a let-down, to be honest.'
Callum didn't move. 'She's lying.'
_'All of who, Emma?'_
_'I killed the police officer in his car – I shot him once in the stomach and once in the chest. Then I went through into the library and shot my father. Once in the...'_ She cleared her throat. _'Once in the chest and once in the head.'_ Emma stared at the ceiling. Gave a sharp, shuddering breath. _'Then I went into the garage and tried to kill the other police officer, but he was too quick and overpowered me. I'm glad he did. I... wasn't myself. I needed to be stopped.'_
'Hmph.' Franklin didn't sound impressed. 'That's the most half-arsed attempt to plead insanity I've ever seen.'
_'I see, I see...'_ Mother patted McAdams on the shoulder. _'Let's try the photos, Andy.'_
He dipped into a folder and came out with a handful of A4 printouts. Laid them across the table. _'I am now showing Ms Travis-Wilkes exhibits nineteen to twenty-seven.'_
Difficult to see what they were of, from up here – the interview room's CCTV system wasn't high-res enough to show more than a row of grey and pink blurs.
_'Do you recognise any of these, Emma?'_
She licked her lips, then looked away. _'I killed them too.'_
_'Who were they?'_
_'I don't remember.'_
_'How_ many _are they?'_
_'I don't remember.'_
_'You don't remember how many people you killed, cut up, and kept bits of in your father's freezer?'_
_'No. I don't remember. It was a long time ago.'_
Mother picked up one of the pics. _'That's a human hand, right there, Emma. A human hand, severed at the wrist and put into a freezer bag. There's even a date on the label: April fourth, 2015. That's not so long ago, is it?'_
_'I don't remember.'_
_'Here's a bottom jaw, complete with lip, tongue, teeth, and part of the throat. November 2006. Who was he?'_
Travis-Wilkes swallowed. _'I don't... Please.'_
_'Here's a severed head. We found it last week, behind some bushes in Holburn Forest. Do you recognise_ her _?'_
Travis-Wilkes stared.
**_Strange How Much Can Change In Just One Week_**
'Oh, for God's sake, Dad, it's everywhere!' Emma jabs a finger at the kitchen worktops.
Tomato sauce. _Everywhere_.
He's splattered it all over the toaster and the wall, making bloodstains down the units, scarlet puddles on the floor.
'I _told_ you I'd make you something.'
Dad doesn't even answer, just sits there at the breakfast bar, eating his cheese and ketchup sandwich. Chewing as he squints at that morning's paper.
'Could you not have waited _five_ minutes? Look at this mess.'
He picks a pen from his pocket and circles something on the front page.
'Are you even listening to me?'
He circles something else, head on one side as if he's a cat considering whether or not to pounce.
'I said, look at this mess!' She marches over and snatches the paper off him.
'BODY FOUND IN CASTLEVIEW FLAT' sits above a photo of an ugly, soulless block of flats. There's another picture set into it – three people standing outside the building. A pretty black woman, a grey skeleton in a grey suit, and a man with a bruised face. He's the one Dad's drawn a circle around. Another around the name 'DC CALLUM MACGREGOR (31)'.
Her father looks up at her. Then over at the blood-smeared kitchen units. Frowns. Stands. And stalks from the room.
Emma hurls the paper down. 'I'LL JUST STAY HERE AND CLEAR UP AFTER YOU, SHALL I?' God's sake, he just gets _worse_. 'DON'T YOU WALK AWAY FROM ME!'
She storms after him, through into the double garage with its fallout-shelter's worth of antique tinned goods, jars, and all those sodding freezers.
Dad's got one of them open, leaning in to rummage through the contents. Picking things from the deep-frozen depths and dumping them on the concrete floor. Tupperware boxes, freezer bags, carrier bags, lumpy tinfoil parcels thick with frost. They clatter and skitter away.
'DAD!'
Nothing.
'I swear to God, one of these days I'm going to get the shotgun out and blow your _bloody_ head off. And then I'll shoot myself. Who'll look after your literary legacy then?'
He digs and rummages.
'I'm not your skivvy, Dad, I'm your _daughter_.'
Then he straightens up. Closes the chest freezer. Places something on the lid.
'LISTEN TO ME!'
He blinks at her. 'Sophie?' Then frowns.
'No: Emma. EMMA! MY NAME IS EMMA!'
'She was never as beautiful as you, Sophie.' A grin. 'But oh, how she _screamed_.'
Emma takes a step back. 'OK...'
'I can't find my hat.' He turns and walks back into the house, leaving her to clean up after him. Yet again.
'God's sake.' She gathers up the nearest frozen chunks. Marches over to the freezer and stops. Stares at what he's dumped on top of it as her mouth goes dry as a library shelf. 'Dad?'
Oh Jesus... It isn't, is it? It _can't_ be.
But it is.
The Range Rover growls into the forest car park, headlights raking the surrounding trees, turning their bark monochrome. Emma parks as far away from the entrance as possible. Sits there, trembling.
Her breath hammers in her lungs, sharp and shallow, blood thundering in her ears.
She licks her lips and glances into the footwell.
The carrier bag is rippled with a layer of white frost, just visible in the dashboard lights.
Get rid of it. Get rid of it. Get rid of it!
She stumbles out into the cold night air and scrambles around to the passenger side. Grabs the carrier bag like an unexploded grenade.
The plastic burns her skin, right the way down to the bone.
Get rid of it!
So she hurls it, as far and as hard as she can.
It disappears into the gloom... then cracks and thumps mark its progress down through a tree or a bush, finishing with a rattle and a thud.
Oh God.
She shuffles backwards, till the car's warm bonnet stops her going any further.
A head. A human head.
There was a human head in the freezer, with the leftovers and never-weres.
Oh God.
Emma runs a hand over her face.
Calm down. Calm down and breathe.
It's gone now. That's the important thing.
She scrambles back into the car, turns around and drives the hell out of there.
The forest ripples past the Range Rover's windows, caught for a moment in the headlights before vanishing forever.
It's gone. And there's nothing more to...
Her eyes widen and she slams on the brakes. Swears. And swears. And swears.
What if it's not the only thing in the freezers? What if there's other bits? What if the rest of the woman is in there?
Oh God.
Don't panic. It'll be OK.
Oh dear God.
Just... Just go back to the house and look.
And if there's more bits?
Then they go in... bin-bags! Take them to the tip. NOT ALL AT ONCE! Do it gradually. A couple of bags at a time. To different dumps.
No one ever has to know about this.
Emma stares at herself in the rear-view mirror.
Because if they find out...
'Emma? I asked if you recognised her.'
'I... don't remember.'
Mr Slick reached out and turned the photograph face down, then pushed it back across the table. A public-school voice with a superior drawl. _'I believe my client has made her feelings quite clear on this, Detective Inspector Malcolmson. Let's move on, while we're all young, there's a good girl.'_
No one moved.
Franklin bared her teeth. 'Sexist tosser!'
'He did it on purpose to rattle her.'
It didn't seem to work, because Mother just let out a big long sigh. _'Emma, Emma, Emma. I don't believe you killed all those people. Oh, you killed Bob Shannon and your father, and you tried to kill DC MacGregor, but the rest of them? No.'_
_'I killed them all.'_
_'You see, I think your_ father _killed them. I think he's been killing people for a long, long time, and when you found out about it, you did everything you could to protect his legacy. You think, if you take the blame, no one has to find out he wasn't the man they idolised.'_
_'I'm not protecting anyone.'_
_'How many people did he kill, Emma?'_
_'He didn't kill anyone! Why won't you listen to me?'_
_'Was it three? Four? Seven? A dozen?'_
_'NO ONE EVER LISTENS!'_ She battered her fists down on the tabletop. _'_ I _did it. Not_ him _: me.'_
_'Detective Inspector Malcolmson: my client has cooperated with your enquiry and given you a full confession. Now – move – on.'_
Mother tilted her head to one side. _'You loved your father, didn't you, Emma?'_
_'Of course I do. My father is... My father_ was _a saint. He had nothing to do with any of those... remains.'_ She sat up straight. _'I'm the only murderer in our family. Me. Myself. I. Singular and accountable. I did it, because I'm sick and I need help.'_
Silence.
Then Callum leaned forwards and clicked the button beneath the nearest microphone. 'Ask her about Leo McVey. Him and her dad used to get off their faces on drugs and wreck things: hotel rooms, marriages, anything they could get their hands on.'
Mother flinched a little, then put a finger to her ear. Maybe she'd forgotten she'd put the earpiece in? _'Tell me about Leo McVey.'_
Emma's mouth snapped shut.
_'He and your father were very close, weren't they?'_
_'I don't...'_ She glanced at her solicitor. Then back at Mother. _'No comment.'_
_'The drinks, the drugs, the bad behaviour?'_
_'No comment.'_
Callum hit the button again. 'Gareth Pike saw him attacking my parents. He never said McVey was alone in the car. McVey's always had Jaguars, but I'll bet you a hundred quid, R.M. Travis is a Range Rover man.'
Mother spread her hands on the tabletop. _'Your father's got a green Range Rover in the garage, doesn't he? You shot a dirty big hole in it.'_
_'So?'_
_'He always drove Range Rovers, didn't he?'_
_'He liked to buy British. What's wrong with that?'_
_'Oh nothing, nothing.'_ She leaned in, lowering her voice to a pantomime whisper, _'Only we've got a witness who saw Leo McVey attack and abduct a family, twenty-six years ago: mother, father, and little boy. He bundled them into a white Range Rover. Only he's never owned one in his life.'_
Emma fidgeted with the sleeve of her oversuit.
_'Now, wouldn't it be funny if I got on to the DVLA and asked them to search for all the vehicles registered to your father, and up popped a white Range Rover from_ exactly _that time?'_
_'I...'_
_'Remember the severed head in that photo? It belonged to the woman McVey abducted with your father's car. It's been frozen all this time. That's funny too, isn't it?'_
_'No comment.'_ But she didn't sound quite so sure this time.
_'And, of course, our witness said Leo McVey wasn't the only one in the car. He had someone else with him. That would be your father, wouldn't it?'_
Emma blinked. _'It was—'_
_'Actually,'_ Mr Slick put up a hand, _'I think this is a perfect place for a pause. My client and I need to confer.'_
Mother shrugged. _'Call it, Andy.'_
_'Interview suspended at twenty-one minutes past midnight.'_ He stood. _'We'll be outside when you're ready to talk.'_
And the screen went blank.
Franklin groaned. 'Just as they were getting somewhere.'
'I'm genuinely sorry, you know.' Callum swivelled his seat round to face her. 'If I could take it back, I would.'
'Whatever idiot thought it was a good idea to let slimebags have a lawyer present during questioning, needs a good hard kick in the balls.'
'I should never have tried to kiss you.'
'"No comment" this and "I don't remember" that.'
'I'm serious. Can we just go back to how it was before?'
Franklin rolled her eyes. Took a deep breath. 'All right, all right. Just... stop apologising. It's like watching a puppy grovel for scraps.'
'You're probably a crap kisser anyway.'
She scowled. 'I happen to be a _great_ kisser, thank you very much.'
'I promise I'll never try to find out if that's true.'
'Deal.'
'I don't know, maybe a pizza or something? There's a takeaway on Harvest Lane that's open till three. They'll deliver.'
Franklin dug back into her packet of Wotsits. 'Suppose it's better than nothing.'
'Or, if you like curry, there's...'
The screen flickered back into life. _'Interview resumed at six minutes to one. Present: Detective Inspector Flora Malcolmson, DS Andrew McAdams, Miss Emma Travis-Wilkes, and Mr Reginald Flynn.'_ McAdams sat back in his chair.
Mother nodded. _'Well, Emma?'_
She licked her lips. Stared down at the tabletop. _'My father wasn't in the white Range Rover with Leo McVey, it was me. I borrowed the car to go get more vodka and Uncle Leo came with me. He was already pretty wasted – we'd been drinking and snorting cocaine all day – so when I saw the family in that lay-by and decided to abduct them, he had no idea what was going on. I drove them off to the middle of the countryside and I killed them. I don't remember where, so there's no point asking. I dismembered the bodies and kept some parts in the freezer.'_
_'I see.'_ Mother tapped her fingers against the tabletop. _'And how old were you at this point?'_
_'Twenty-one.'_
_'Twenty-one? My, my, my. And did you and Leo McVey kill anyone else?'_
_'Uncle Leo didn't kill anyone. He'd passed out from the drink._ I _did it. All on my own. When he woke up that evening, I told him I'd let the family go. He doesn't know what I did.'_
_'Right. You see, the trouble is: I don't believe you.'_
_'It's the truth. And I have nothing further to say.'_
Mr Slick nodded. _'You've had a full confession from my client. She won't be answering any further questions.'_
The custody area was nearly deserted, but the sounds of singing and swearing echoed through from the cell blocks. An after-midnight serenade, fuelled by cheap lager and low IQ.
Callum shuffled in and up to the desk – an oversized pulpit decorated with computer monitors and public safety posters. A stick insect in an ill-fitting wig sat behind it, leaning over a copy of the _Castle News and Post_ , propping up his long thin face on one fist.
He didn't look up.
So Callum banged a hand down on the countertop. 'Shop.'
A flinch. Then he raised his head and pulled on a pair of oversized glasses. 'Ah, Constable MacGregor. We've been expecting you.'
Silence.
'Can we make this quick, Sarge, only it's been a _really_ long day.'
'Of course we can. It would be my most delightful pleasure.' He rescued a mouse from beneath a stack of paperwork and wriggled it. Clicking away. 'Ah, here we are. Two things. One: when you're processing suspects, try not to cock up the DNA portion of the proceedings. It makes the rest of our lives a lot easier. And Two: your custody from this afternoon wants a word and he wants it with you.'
'Tell him to go screw himself, because I'm going home.'
A jagged smile. 'That's hardly the caring and compassionate face of Police Scotland we're tasked with presenting, is it, Constable?'
'Someone tried to bash my brains in today. A friend was shot and killed. I've been battered and bruised and humiliated. And I don't give a _toss_ what some spoiled rap-star wank-badger wants. I'm – going – home.'
'He's up before the Sheriff at ten tomorrow morning. Make sure you see him before then.'
'I'd _love_ to! Only I've been suspended without pay, so Mr Newman can go crap in his hat.'
Sergeant Stick Insect's eyes widened, magnified out of all proportion by his big glasses. 'Constable MacGregor, language!'
'Bye, Sarge.' Then he turned and marched out into the corridor, through the double doors, and away into the rain.
## — the Bonemonger's waltz —
The old lady puffed on her long willow pipe. "Once upon a time there was a little boy whose soul was dark as the blackest cat. Whose eyes were green as jealousy. And whose skin was pale as the dead."
"What was his name?" asked Justin, eagerly.
"Why child, he didn't have a name back then, just an unpronounceable howl of pain and hatred. And he'd come down from the hills at night and steal skeletons from the villagers' graves. Then he'd take them out to the deepest darkest depths of the woods and dance with their bones till dawn."
R.M. Travis
_Open the Coffins (and Let Them Go Free)_ (1976)
_Ain't nothing so sad as a man in his prime,_
_Got dirt on his knees cos it's grovellin' time,_
_Shoot that poor f*ck in the back of the head,_
_Cos trust me, that b*stard is better off dead._
Donny '$ick Dawg' McRoberts
'Walter Peck, the Bugf*cker'
© Bob's Speed Trap Records (2016)
##
Mother glanced up at the clock on the office wall. 'Well, we'll just have to start without him.'
Which meant it was just Callum, Dotty, Franklin, and Mother for Monday morning prayers. No McAdams.
She clapped her hands together. 'First order of business: the doctors say John's going to be in the high-dependency ward for at least a couple of days, but it looks like he'll be fine. Ish. There might be some brain damage, but they won't know till he wakes up.'
Dotty stuck up her hand. 'I've got a card for everyone to sign.' She held that up as well – something with a teddy bear on crutches surrounded by love hearts. 'GET WELL SOON!!!' in big letters. 'And I've started a whip-round too. Maybe get him something nice so he knows we're all thinking of him.'
'Second: Bob Shannon and Raymond Montgomery Travis are down for post mortem today, starting at ten, if anyone wants to volunteer? Anyone? No?' A shrug. 'Fine, Rosalind, you can do those.'
Franklin sagged. 'Not _again_.'
'Emma Travis-Wilkes has confessed to the murder of Robert Michael Shannon, the murder of her father, and the attempted murder of DC Callum MacGregor. She's _also_ confessed to the abduction and murder of Callum's parents and brother, and an unspecified number of other victims. Which I'm a hundred percent sure is a lie, but can't get her to admit it. She's up before the Sheriff at twelve – expect "remanded without bail" and "sentencing to follow psychiatric reports". But I'll eat my own fleece if she gets less than thirty years.'
Callum folded his arms. 'What about the freezers?'
'SEB are calling in the forensic anthropologists from Dundee Uni. Could take weeks to work out what's human and what's not. How long it takes to identify who the bits _came_ from in the first place is anyone's guess. Maybe years.' Mother perched her backside on Watt's empty desk. 'Which brings us onto a sticky subject.' She pointed. 'Callum is officially suspended without pay, pending an investigation into the aggravated assault on Detective Chief Inspector Reece Powel. If anyone asks, and I mean _anyone_ – don't care if it's your best mate, your mum, or the blessed Chief Superintendent himself – you tell them Callum's here as a witness on the Travis-Wilkes case, and that's all. He is _not_ working on any other cases. Understood?'
Franklin and Dotty nodded.
Callum frowned out at them. 'Just so you know, I didn't touch Poncy Powel. If anyone battered him, it was Ainsley Dugdale. Not me.'
'Just here as a witness, remember?'
He stared up at the manky ceiling tiles. 'Yes, Boss.'
'We're going back to the beginning. Tod Monaghan worked at Strummuir Smokehouse, so did one of his victims, and another victim did a course there. I know we've already interviewed every employee, but we need to do it again. Who did Monaghan hang out with? Did he mention any friends from outside work? Who was he killing with? Ask, ask, and ask again.' She clapped her hands together. 'Right now, Ashlee Gossard has the tiniest, slimmest, most infinitesimal chance of still being alive. Get out there and _find_ her.'
Cecelia was hunched over her desk, poking away at her computer. Most of her long brown hair was tucked back, out of the way, but a strand of it – about as thick as a finger – disappeared into the corner of her mouth. Making little sooking noises as she chewed.
Callum knocked on the doorframe. 'Did your mother never tell you about that?'
'Gah!' Bolt upright, soggy hair swinging loose again. 'Don't sneak up on people, it's rude.'
'I got hauled in by the rubber heelers yesterday. Chief Inspector Gilmore told me he'd heard rumours I was covering for Elaine at that crime scene. Care to explain why?'
She pulled a face. 'Ah. Yes. No. No idea.'
'You're a terrible person, you know that don't you?'
'Oh definitely.' She sat back, swivelling her chair from side to side. 'You here about the freezers, or the lab results from the Gossard house? Because if it's the lab results, that's very much _not_ my fault. I took the samples, I labelled them _properly_ , so whoever arsed it up did it at the laboratory.'
'Officially, I'm just here as a witness, but unofficially: freezers.'
She picked a sheet of paper from the top of her printer. 'We won't know for certain until the anatomy gurus get here from Dundee, but I'd say you're looking at between six and twelve individuals spread among the eight freezers. That's assuming everything _else_ is what the labels say it is. And that's not...' She pointed at his jacket as singing erupted from somewhere deep in his pocket. 'Are you going to answer that?'
'Sorry.' He pulled out his phone. 'Hello?'
A woman's voice that sounded as if it could crush walnuts just by shouting at them. _'Constable MacGregor? Sergeant Price: custody suite. I have a note here that you're to see a Mr Donald Newman this morning before he goes before the Sheriff?'_
'I can't. I've been suspended. I'm not on active duty.' He glanced at Cecelia. 'I'm not even in the building.'
_'Nice try. I'll expect you here before ten.'_
Oh for God's sake. 'But—'
_'Don't make me come looking for you.'_ Then she hung up.
Wonderful.
Cecelia passed over the sheet from her printer. 'Take a look: going by the freezer-bag labels our human remains date from thirty-one years ago through to about twenty ten. The handwriting gets a bit shaky at the end. A lot of the stuff's been frozen, thawed, and frozen again, going by the state of it. Do you know they had bolognese sauce in there going back to the seventies?' She shuddered. 'And I thought my mum was bad for hoarding leftovers. The _seventies_!'
'Lovely.' He scanned the list of items recovered. Maybe one of the hands belonged to his father, or his mother? Or maybe the eyes? Or one of the hearts? 'Any sign of... children's remains. Like a five-year-old boy?'
Alastair.
The bumhead.
In his cartoon-fox T-shirt.
Cecelia shook her head. 'But then I'm only going by the bits that are instantly recognisable. Who knows with the other stuff?'
'Thanks.'
'Now go away, I'm working.'
He backed out of the room.
'And stop sneaking up on people!'
Callum wandered back along the corridor and through into the stairwell, frowning down at the list. A dozen victims over a space of thirty-one years. Not the most prolific killer Oldcastle had ever seen – not even in the same league as someone like Jeff Ashdale – but still...
And the experts from Dundee might ID even more. So—
He jerked to a halt.
Sod.
Detective Superintendent Ness was standing right in front of him, jaw set, shoulders back. 'DC MacGregor.'
'Super.'
Her face barely moved as she spoke. 'I thought I suspended you yesterday.'
'Yes, Super. It... I'm here as a witness? The Travis-Wilkes' case is—'
'Tell me, Constable, why is there a _very_ expensive lawyer sitting in my custody suite threatening to sue Police Scotland if you don't speak to his client?'
Oh sodding hell. So Donny '$ick Dawg' Newman had called in reinforcements.
'I can't speak to him, Super: you suspended me. It's not my—'
'Hello?' Franklin's voice echoed through the stairwell. 'Callum?' She came thundering down the stairs, two at a time. Nodded at Ness. 'Super.' Then back to him. 'Sorry, thanks for waiting.'
'I...?'
She turned her perfect white smile on the superintendent. 'We've been going over his statement from last night. I had to nip off to the toilet. Emergency situation. Anyway,' she took hold of Callum's arm, 'let's get you that lift home.' Another nod. 'Super.'
'Oh no you don't.' Ness held up a hand. 'Constable MacGregor isn't going anywhere until he's seen this Mr Newman and his solicitor. And I want you there for corroboration.'
Franklin's smile slipped. 'Super?'
'You will make sure nothing happens, are we clear, Constable? If I've got to spend the next six months tied up in court, you're going to find yourself doing every crummy crappy horrible job I can find.' A big bright smile. 'Off you go then.'
'Yes, ma'am.' Franklin led him away downstairs. Out through the doors at the bottom. Slumped back against the bare breezeblock wall. 'Oh God... Why did you have to drag me into this?'
'How is this my fault?'
'I don't know, do I? Maybe you antagonised Newman with your "everyone grew up in care" speech? Or maybe you just rub _everyone_ up the wrong way.' She straightened up and poked him in the chest. 'I stood up for you with Ness, and you better not mess this up for me, Callum. You go in there and you be nice to this dick and you say "three bags full" if you have to. Understand?'
Lovely.
'Thanks. Thanks a heap.' He pushed through the double doors and into the custody area.
Peace reigned inside, just the ping and click of the central heating to spoil the silence. Last night's stick insect had been replaced by someone who would've looked more at home on the rugby pitch, or a boxing ring. That would be Sergeant Price. She had her brown hair pulled back in a ponytail, a pair of half-moon glasses perched on the end of her nose as she passed a form across the custody desk to Mr Slick – Emma Travis-Wilkes' lawyer.
So Mr Slick had got himself _another_ celebrity client. That was quite the portfolio he was building: a murdering liar, and a misogynist scumbag druggie. All he needed was a kiddy-fiddling TV presenter from the 1970s to complete the set.
Slick took a fountain pen from his inside pocket and signed. 'Thank you, Sergeant.'
She pointed off to a row of easy-to-hose-down plastic seats. 'Make yourself at home and I'll give you a shout, soon as Ms Travis-Wilkes is ready.'
The room's only other occupant was a chunky middle-aged man in a rumpled suit. Head a combination of Yorkshire Terrier meets pickled egg. Bags under his eyes, stubble on his chins, fingers stained turmeric-yellow from too many cigarettes. He was slouched across a couple of seats, as if he'd forgotten to bring his bones with him.
Mr Slick walked over to the plastic seating, pulled a handkerchief from his pocket, and dusted a chair as far away from Captain Scruffy as possible, before lowering his tailored backside into it. Sat there with his briefcase on his lap. Still as a garden gnome.
Callum slumped up to the counter. 'I came, OK?'
Sergeant Price flashed him a smile. 'Constable MacGregor, is it? Good. Right. Mr Newman's solicitor has laid out some ground rules for your meeting: no recording, no harassing his client, and everything said is to be considered off the record.'
'He can make all the rules he wants: I don't _want_ to meet his client. His client can go—'
'Poop in his hat. Yes, I do read the night logs.' She pointed at a plain veneer door. 'You can use the solicitors' briefing room. Fifteen minutes.' Sergeant Price stared at him with all the warmth of an iceberg. 'Do we have a problem, Constable?' And there was that walnut-crushing tone again.
'No, Sarge.'
A deep breath. A sigh. Then Callum wandered over to the room and pulled open the door. Not even bothering to look at Mr Slick. 'Come on then, let's get this over with.'
The consultation room had three chairs, a frosted window high up by the ceiling, and a Formica table covered in other people's initials and swearwords. Callum collapsed into the chair beneath the window. Franklin was next, leaning back against the wall, arms folded.
And two minutes later, Donald '$ick Dawg' Newman swaggered in. They'd let him keep his own jeans and trainers, but confiscated his belt, shoelaces, and leather jacket. Someone had lent him a scruffy Oldcastle Police polo shirt to cover his naked chest. He slouched into the chair opposite Callum, one hand tucked into the waistband of his trousers, stubble blueing the gaps around his high-maintenance facial hair. 'Sup, Bruv?'
But it wasn't Mr Slick who shambled in after him and closed the door, it was Captain Scruffy. He grunted his way down into the last free chair. Rummaged through his pockets and came out with a packet of nicotine gum. Popped a couple. Chewing through a broad Glaswegian accent. 'Right, gentlemen, youse is all aware of the ground rules, and that? No sneaky recording the conversation. Everything said in here is, like, _mega_ off the record, man.'
This was the _very_ expensive lawyer Superintendent Ness was so worried about?
Took all sorts.
Callum stared across the table. Not saying anything.
Newman grinned at him. 'You look like crap, Bruv.'
He clenched his good hand into a fist. 'I'm not your "Bruv".'
'Chill, man, we just talkin', is all.' He spread his hands on the tabletop. 'You been to see Irene, yeah? And the kids?'
'You assaulted her. You broke Willow's arm. She was _four_.'
'Yeah, well, I been through some tough times in my life. Grew up in care. Got into trouble. And it was the drugs, yeah? They made me do things I'm not proud of, Bruv.' A wee shrug. 'And you caught me, like, bang to rights, innit? Gonna plead guilty and throw myself onna mercy of the courts. Do my time. Get my life in order.'
Captain Scruffy gave a little snort. 'Trust us, Donny, no way you're goin' down. When I get through with the Sheriff, they're gonnae give you a medal for being an upstanding citizen.'
'I'm gonna make amends, Bruv. Gonna make it up to Irene: give that bitch a huge cheque, like an apology for what I did and all that child support I never paid. Bitch can get herself a nice house in Blackwall Hill or something.'
Callum picked at a set of carved initials in the Formica. 'You actually think that'll make it all better?'
'Yeah, I was a dick. Like I said: drugs.'
'And don't call her a bitch.'
There was a pause and Newman tilted his head to the side, one eyebrow raised. 'You still ain't got it, has you, Bruv?'
Outside, in the custody suite, someone coughed like they were trying to expel a lung.
Franklin shifted against the wall.
Captain Scruffy chewed.
Newman sat forward. 'Got into a fight when I was twenty – bit of a barney over who owed who for a load of skunk went missing – sons of bitches broke my nose and cheek and jaw. Had to get plastic surgery to fix it, Bruv. And I thinks to myself, while I is here, might as well get me some more handsome, yeah?'
'There a point to this?'
'You really don't recognise me, do you? _Bruv_.'
'I told you to stop calling me that, I'm not your...'
_Oh yeah, me and Leo: we go back years, innit? See when I was growing up in a home? He visited me, like every week... Man's a star, right? Been like a dad to me._
It couldn't be, could it?
No.
There would be a family resemblance, or something, wouldn't there?
_Looks like he's barely into his twenties: turned thirty last year. Suppose healthy living and Botox will do that for a man. Well, that and a face-lift, a nose job, and three hours a day with a personal trainer._
Newman lifted the waistband of his borrowed polo shirt, showing off his shaved washboard stomach with its tattoo of a cartoon fox. Not identical to the one on his T-shirt all those years ago, but close enough.
Sodding hell...
Callum licked his lips. 'Alastair?'
'There we go.'
'I thought you were dead...'
'Nah, Bruv, I's a superstar and that. Right, Mr McQueen?'
'Oh aye. That you are indeed, Donny.'
Callum stared. 'But... what happened?'
'Tried to get in touch a few years back. You know? Googled you, like, a million times. Thought maybe you were a midfielder for Celtic, but he looks nothing like us, yeah?' The fake London patois was slipping. 'So I hired myself a private detective. And now here we are.'
'No: what happened to _you_? After that day. When you all got abducted.'
Newman... Alastair folded his arms. Looked away. 'Too soon, Bruv. Too soon.'
'Look, you were there. You're a witness. You saw what Leo McVey did – we can put the bastard behind bars!'
Alastair sucked a breath through his teeth. 'Man's been like a dad to me.'
'Only because he helped R.M. Bloody Travis murder our _real_ dad. And our mum!'
'Nah, he's—'
'He _helped_ , Alastair. They'd be alive today and I wouldn't have grown up in sodding care homes. Neither would you. We would've been a family!'
Alastair slumped back in the chair again. Frowned at him. 'You want to do Leo for killing Mum and Dad?'
'Of course I sodding do.'
The only sound in the small room was Captain Scruffy's wheezy breath.
And then Alastair shrugged. 'I'm-a think about it, yeah?' The patois was back. 'You asking me to snitch on the man brought me up. That's cold, Holmes. I'm-a speak to my legal representative now.'
Captain Scruffy took the nicotine gum from his mouth and stuck it to the underside of the table. 'And that concludes our business here this morning.' He nodded at Callum. 'We'll be in touch, pal.'
##
Callum marched past the custody desk and pushed through the door into the custody suite.
Franklin was right behind him. 'Your brother's alive; why have you got a face like a badger's backside? Isn't this a good thing?'
'I'll just be a minute.' Grinding it out between his teeth. 'Five tops.'
She groaned as he shoved through to the female cells: a twin line of blue-painted metal doors. 'Please tell me you're not doing what I think you're doing!'
'Of course I sodding am.' He pointed at the hatch mounted on the door of each cell. 'Help me look. She's in here somewhere.'
'Callum, you're a witness. Worse: you're a _victim_. She tried to kill you. You can't just rock up and have a cosy wee chat with Emma Travis-Wilkes, the Procurator Fiscal will do her nut. Wilkes's lawyer will have a field day!'
'Five minutes tops.'
'Oh for God's sake...'
He slid down the hatches and peered into each cell, working his way along one side until there she was. Emma Travis-Wilkes.
She was sitting on the blue plastic mattress with her back against the wall, legs folded into the lotus position, forearms resting on her knees. A calm smile on her face, even with all the bruises.
Nice to see someone was enjoying themselves.
Callum thumped on the door. 'Hello, Emma. Remember me?'
'How's your head?' She turned the smile up a bit. 'I'm sorry I had to hit you.'
'This lawyer of yours, Flynn. He's not from around here, is he? His suit's worth more than I make in a year.'
'My publishers organised everything. I phoned my publicist, said, "I've just killed my father," and the next thing you know: ta-daaaa.' She unhooked her legs and stood. 'I wanted to thank you for arresting me. For... stopping me.'
'Oh, don't worry, that was a _real_ pleasure.'
Emma padded across the concrete floor on her bare feet. Placed a hand against the door. 'Do you know what it's like to live in the shadow of a famous parent?... No. Of course you don't. Sorry.' She shrugged. 'Believe me, you were lucky. Living with someone like my father, looking after him and his bloody legacy. Did you know, just last week they were talking about putting him on a stamp? On a _stamp_. How is anyone supposed to compete with that?'
She closed her eyes and took a big breath. 'God, I'm finally _free_. I can write again. My _own_ words. Ooooooh... No more adult nappies, no more spoon-feeding him on bad days, no more watching him crumble like a sandcastle as the tides of time rush in.'
'But you didn't kill my mum and dad, did you? It was him.'
'What does it matter? He's dead, isn't he? He can't do...' A frown. 'His legacy is more important than a few dead bodies. Oh, not to _you_ – I get that – but to the world. If he was some sort of monster, parents wouldn't buy his books, would they? Generations of children would miss out on _Russell the Magic Rabbit_. No one would ever read _Open the Coffins_ again.'
'So you're taking the blame to protect his literary estate?'
'I'll be free.'
'How much did they offer you?'
'I can write again!'
'Because I'm going to burn your phoney confession to ash. Then I'm going to do the same with your dad's reputation. And then I'm going after Leo McVey.'
Her eyes narrowed. 'You can't do that.'
Callum inched closer to the hatch and dropped his voice to a harsh whisper. 'I've got an eyewitness. The boy your dad and Leo McVey abducted? My brother. He's still alive. And we're lighting the fires.'
'Hmm...' Emma blinked at him a couple of times, then smiled. 'My lawyer's very, very good. He says if I plead guilty they'll sentence me to treatment at a secure psychiatric facility, where I can get the help I so desperately need. And after two, maybe three years, when they finally declare me cured and fit to return to society, I'll find a _very_ supportive friend in my publisher.' The smile was as sharp as it was cold. 'Because that's the kind of caring socially responsible international corporation they are.'
'Let me guess – it's the same company that publish your father's books.'
'They've been extremely kind to me.'
'Oh grow up. They didn't parachute a fancy-pants solicitor in to look after _your_ best interests, they sent him here to look after _theirs_. How much is your dad's literary estate worth every year: a million? Two? Probably more, now he's dead.'
Franklin grabbed Callum's arm. 'Come on, that's enough.'
'And your publishers don't want anyone to find out he was a serial-killing tosspot because it'll spoil their sales figures.'
A sharp woman's voice boomed out behind them. 'Can I help you?'
Callum turned and there was Sergeant Price, all puffed up, shoulders back. Looking every inch the prop forward.
Franklin stepped in front of her. 'We're looking for Emma Travis-Wilkes.'
'In this station we check with the custody sergeant _before_ we talk to suspects.'
'Right. OK. Thanks. Good pointer.' Franklin took hold of Callum's arm again. 'We'll just be on our way.'
'That's probably a good idea.'
She dragged Callum down the corridor and out into the custody suite. Then through the doors and into the bare-breezeblock corridor. Shoved him up against the wall. 'Are you _insane_?'
'Her solicitor's manipulating her to—'
'If Wilkes tells him about your visit, he – will – have – you – fired. Is that what you want?' Franklin let go. 'Because if it is, you're on your own.'
Callum stared at her. The flared nostrils. The wide eyes. The bared teeth.
'Fine.' He pushed past her, out through the double doors and into the rain. Turned. 'Emma Travis didn't kill my mum and dad, OK? She didn't abduct Alastair. It was her dad and Leo Bloody McVey.'
'For God's sake!' Franklin jabbed a hand back towards the cell block. 'Even if Donny "Sick Dawg" Newman is telling the truth, even if he _is_ your brother, he was five when it happened. Five years old. Do you have any idea how easy it'll be to rip his testimony into tiny frilly little pieces?'
'That's not—'
'And Wilkes _admitted_ it. She killed her own father! She killed Bob Shannon. She nearly killed you! Who's the jury going to believe?'
He closed the gap. 'You didn't see her. She shot Bob and she was in pieces. Sobbing. Horrified. Does that sound like someone who's murdered and dismembered a dozen people? Who's been killing since she was _sixteen_ years old?' He marched away a couple of paces, then back again. 'Because that's when the first chunks of human being went into a freezer-bag at Casa Del Travis: thirty-one years ago. And _apparently_ she's been getting away with it ever since. But shooting a retired copper in a Norwich City T-shirt makes her break down in tears? You believe that?'
Franklin stared up into the downpour for a moment. 'It doesn't matter, OK? It'll all come out at the trial. Newman can—'
'There won't even _be_ a trial! She pleads guilty this morning, and that's it. No jury. No witnesses. No trial. Emma Travis-Wilkes goes off to a rubber room for two years while her father and Leo McVey GET AWAY WITH BUTCHERING MY PARENTS!'
The rain hissed against the bland featureless back of Division Headquarters. Bounced off the patchwork tarmac of the rear podium car park. Drummed on the roofs of the parked patrol cars, pool cars, and assorted private vehicles. Soaked through Callum's hair and trickled down the back of his neck. Leached into his jacket.
He screwed his eyes shut. Bit his lip. Took a deep breath. 'Sorry. I don't... This is all a bit... It's a shock, OK?'
Franklin's hand was warm on his arm. Her voice: soft. 'Maybe you should take a couple of days off?'
'Yeah. Maybe.' After all, it wasn't as if he was already suspended or anything. He turned. 'They killed my mum and dad, they took my brother, and they're going to get away with it.'
'Raymond Montgomery Travis is dead.'
'That's not the point. He shouldn't get to stay a "beloved children's author" – he's a serial-killing dick-monkey. People should be spitting on his grave.' Callum wiped his hands across his face and flicked the water out into the soggy morning. 'They're going to put the bastard on a _stamp_. How is that...' Wait a minute.
'Callum?'
Parked cars circled the gap behind the buildings, the space in between broken up into individual bays. A familiar red Mitsubishi Shogun sat in the far corner, and there was someone slumped in the driver's seat.
'Callum, are you all right?'
He jogged across the car park, splashing through the puddles.
McAdams was a crumpled heap, head thrown back, hands loose in his lap, mouth hanging open. Skin pale as mist. Not moving.
Oh Christ. He was dead, wasn't he.
Callum tried the door. It clunked open.
A sour smell oozed out of the car, layered with the scents of wood smoke and menthol.
'McAdams? Sarge?' He reached in and shook McAdams' shoulder. 'Hello?'
'Nnnghmppph...' McAdams blinked. Shuddered. Then let loose a deep rattling cough. 'Whrm I?'
'You're going to hospital.'
'No. No hospital.' Another cough and he sagged back in his seat. 'I'll be fine.'
Franklin clunked open the passenger door and slid into the seat. 'You look like you already died.'
That got her a smile. 'I love you too, Rosalind.'
'Callum's right, you need to go to hospital.'
He didn't move. 'It's very sweet of you both, but I'm not going to the sodding hospital. Are we clear on that? No – hospital. _Je ne vais pas aller → l'hôpital_.'
'But you're—'
'Dying. I know. And I'm _not_ going to do it in a starchy bed surrounded by strangers and machines that go _ping_.'
Stubborn old git.
Callum sighed. 'Fine, no hospital.'
'Good. Now, where are we going? You're both obviously headed out somewhere. Have we got a lead on Ashlee Gossard?'
'I'm taking Callum home, before he gets himself properly fired.'
McAdams raised an eyebrow. 'You're taking him home? Are you two...?'
'No, we are not.'
'Well, that's probably just as well, you have about as much on-screen sexual chemistry as a loaf of wholemeal bread.' He clicked on his seatbelt. 'Well, climb in, Constable MacGregor. I'll give you a lift. No point getting a nice clean pool car all wet. You too, Rosalind: you can catch me up on the morning's shenanigans.'
She pointed over her shoulder, towards the Divisional Headquarters. 'Maybe I'd be better off—'
'In you get, Constable.'
'Yes, Sarge.' She got in the front.
Callum slid into the back. 'You sure you're OK to drive?'
McAdams grinned. 'Let's find out.' He cranked the engine, setting it roaring, then slid them down the ramp and onto Peel Place.
Some civic-minded soul had removed the traffic cone hats from the war memorial opposite, and given the three bronze figures Oldcastle Warriors scarves instead. The blue fabric hanging limp and dark in the rain.
'Come on then, Rosalind: shenanigans?'
'Callum's brother Alastair's alive, only now he's calling himself Donald Newman, AKA: Donny McRoberts, AKA: Sick Dawg.'
McAdams slammed on the brakes and the Shogun slithered to a halt on the damp tarmac. 'Really? Congratulations, Callum! That's...' He turned and raised an eyebrow. 'Didn't you pair arrest him yesterday for beating up his ex, sexually assaulting a police officer, making threats to kill, and possession of Class A drugs?'
Heat bloomed in Callum's cheeks.
'What a jolly family reunion that must have been. Still, at least you'll know where he is for the next six to eight years.' A wink, then McAdams faced forward again, driving them past the front of Division Headquarters.
A crowd of media people jostled by the main entrance, sheltering beneath umbrellas, doing pieces to camera and taking photos. Behind them were a group of protestors, waving placards with things like 'BRING BACK HANGING!', 'FATHER-KILLING BITCH!, and 'YOU MURDERED THE MAGIC!!!' More people drifted in off the street. By lunchtime there'd probably be a full-on lynch mob.
McAdams pointed. 'It's been on the radio all morning. Tributes to R.M. Travis, from all his celebrity chums. Someone's started fundraising for a statue.' A sniff. 'Morons.'
Callum poked McAdams in the shoulder – all bones. 'Your oncologist wants you to make an appointment.' After all, why _shouldn't_ he share the misery?
'Not this again.'
'You need to start your chemotherapy!'
DHQ faded in the rear-view mirror, swallowed by the rain.
'Have you ever tried it? No, didn't think so.' McAdams took a left, past a squat grey church and its peeling 'THOU SHALT HAVE NO OTHER GODS BEFORE HIM!' posters. 'So don't tell me what I've got to do. I'm not having another round of bloody chemo, and that's final.'
'Your oncologist...' Callum sat back. Frowned. 'Your oncologist told me you've not even _started_ this course of therapy.'
'It's my life, and it's my death too.'
'But you were on the phone – you called from the hospital, wanting an update on the smokehouse searches. I heard your nurse in the background.'
McAdams flashed a smile over his shoulder. 'Good, wasn't it? I recorded four or five of them last time round. Now all I have to do is hit play and people don't bang on about me not going to chemo. It stops Mother worrying.'
Not just a stubborn old git, he was devious too.
'But you're—'
'Do you want to know what I do instead? When everyone thinks I'm strapped into my deathchair in the hospital getting poison pumped into my veins? I go and park outside the castle, or across the water by the golf course, or just a lay-by somewhere up on Blackwall Hill. I sit in my car and look out at the city. And I wonder if anyone's going to remember me when I'm gone...'
A row of little shops went by, windows all dark, waiting for the morning to begin.
McAdams coughed again. Grimaced. Swallowed. 'No one ever remembers the police officers, do they? Oh, if we cock something up it's all over the papers: public enquiries, questions in parliament. Heads must roll!' Right at the roundabout then left, drifting by the closed nightclubs, bars, and takeaways on Harvest Lane. 'And if we actually _catch_ the bad guy, do we get the credit? Do they bang on in the media about our thousands of man hours and dedication and genius? Do they hell. It's all about the killer, isn't it? How many people they murdered. What they did to the bodies afterwards. All the gory sensationalised details.' He shook his head. 'No one ever gives a toss about us.'
Franklin shifted in her seat. 'That's not true.'
'Name a serial killer.'
'Andrei Chikatilo.'
'I'll see Andrei Chikatilo and raise you Dennis Nilsen, Peter Manuel, John Wayne Gacy, Ted Bundy, and Harold Shipman. Name the police officer who caught any of them.'
McAdams pulled up at the traffic lights. Rain battered the Shogun's roof. Outside, an old lady lumbered through the downpour, dragging a tiny terrier along on the end of its leash. The lights turned green and he turned, past a strip joint with 'WE ARE HIRING!' in the window.
'No?' He sucked air through his teeth. 'How about this then: Jeffrey Dahmer. He's _properly_ famous. Never mind who caught him, name one of his victims. Just one.'
Right, onto the main road.
'See, you can't. All people care about is the killer. The rest of us don't matter at all.'
Kings River lay just beyond the docks, swollen and dark, breakwater curling against the supports of Dundas Bridge.
'Oh, one exception: if the victim's famous. People care about them then. JFK, John Lennon, _they_ get remembered. The rest of us are just footnotes in a true-crime book.' He slowed for a small coughing fit. Then shuddered. 'So Emma Travis-Wilkes will probably be famous for generations. A serial killer who murdered her bestselling-children's-author father.'
Callum poked him again. 'She isn't a serial killer, it's all lies.'
'Do you _really_ still believe the popular press and prurient public care about the truth, Constable? How sweet. And where do you stand on the topics of Santa Claus and the Easter Bunny?'
The Shogun growled its way over the bridge.
'You are such a dick.'
'Oh, no doubt.' McAdams gave a small, sour laugh. 'You know what I did this morning? I went up and I sat in the barn behind Thaw Cottages. I shouldn't have left Watt there...'
Franklin turned to him. 'Mother says he's out of surgery and they think he'll be OK.'
'Do they? Oh that's good.' He nodded. 'That's something.'
'Of course he might have brain damage, but— Sod.' She pulled out her ringing phone. 'DC Franklin... Right... No, no put them through... Yes. Thanks.' Franklin held the phone against her chest. 'It's the Land Registry Office.' Back to the call. 'Hello?... Right... Yes, hang on a second.' She produced her notebook, pinning it against the dashboard. 'OK... 14 Lehman Road, Blackwall Hill... Yes.' Wrinkles appeared between her eyebrows. 'How many others?... Can you text them to me?... Yes. Thanks... No, that's great. Bye.'
She hung up. 'According to the Land Registry Office, Paul Terrance Jeffries only owns one property – in Blackwall Hill. He inherited it from a Mrs Georgina Mason. But _before_ that he was left another four houses from various old biddies. And guess who he sold them all to?'
Callum had a stab at that one: 'Northeast Ecclesiastical Trust Holdings Limited.'
'Remember when you asked if N.E.T.H. was some sort of pensions and investment pot for the clergy?'
'But he kept Mrs Mason's house.'
Franklin reached across the car and patted McAdams on the shoulder. 'Change of plan, Sarge. We're off to Blackwall Hill.' Then she was poking at her phone, holding it to her ear as it rang. 'Mother?... Rosalind. I think we know where Imhotep's keeping Ashlee Gossard...'
##
McAdams threw the Shogun around the roundabout, siren wailing over the squealing tyres. Lights flashing, reflecting back from the wet tarmac.
Callum braced his fibreglass cast against the door, leaning into it as the car fishtailed out the other side. Holding onto his phone as tight as possible. 'No, I don't know what the nature of the emergency is, I—'
_'Well how can you possibly need an ambulance if you don't know what the emergency is?'_
'All right, here: one young woman, suffering from extreme dehydration, shock, and starvation. How's that for starters?'
The tyres screeched again.
Franklin clutched the grab handle above her door, pretty much shouting into her mobile phone as the traffic parted before them and the streets raced past. 'NO, _LEHMAN_ ROAD. LEMUR, ECHO, HOTEL, MIKE, ALPHA, NOVEMBER... YES. I NEED A FIREARMS TEAM THERE SOON AS YOU CAN...'
_'An ambulance should be with you in about twenty minutes.'_
'No. Not twenty minutes, _now_!'
'... WELL WHAT USE IS THAT?... NO, THAT'S NOT GOOD ENOUGH. THERE'S A YOUNG GIRL'S LIFE AT STAKE – TELL THEM TO GET THEIR GUNS LOADED AND THEIR FINGERS OUT!'
_'Tell you what, I'll get every available ambulance to open its doors and turf out whoever they've got in the back, shall I?'_
'Just get one there soon as you can.'
Assuming Ashlee Gossard was still alive.
He hung up. 'Ambulance is on its way.'
'YES... THANKS. OK... BYE.' Franklin turned in her seat. 'We're not getting a firearms team for at least half an hour.'
McAdams stuck his foot down, nipping between a bus and a Transit van, coming within an inch of losing the paint off his left wing – earning himself a blare of horns from both vehicles and a lot of rude gestures too. 'Well that's no sodding use, is it?'
'That's what I told them. Apparently they've got to get all the way back to DHQ to get tooled-up before they head out again.'
The Shogun roared through a puddle that stretched all the way across the road, sending a wall of spray slamming into a bus shelter full of people. 'What's the point of having firearms teams if they don't carry firearms on them? Might as well deploy a crack team of Morris dancers.'
Callum's phone went off again. 'Dotty?'
_'We're about ten minutes away. Maybe less if this sodding school bus will get out of the bloody way!'_ The sound of a blaring horn cut through the engine noise. _'MOVE IT, PRINCESS!'_
'You got Mother with you?'
Some hissing fumbling noises, and Mother was on the line. _'Callum? Tell Andy he's not to wait for us, understand? You kick that door in and you save Ashlee. If there's any flack, that's on me. Priority one is saving that little girl's life.'_
'Yes, Boss.'
_'For God's sake, Dorothy, look out for that lorry! Arrrrrrgh...'_ More blaring horns. _'I'll have to call you back, Callum. If we survive. Oh God, I can't look...'_
And they were gone.
McAdams wrenched the wheel left and they drifted sideways around the corner, under the railway bridge, and out the other side. 'Hahahaha!'
'Mother says we don't wait, we go straight in.'
'Excellent!' McAdams grinned over his shoulder. 'Dig into the boot, young Callum. Should be a couple of stabproofs and a bit of MOE gear back there. I _might_ have forgotten to sign them back in after we searched Tod Monaghan's _pied-→-terre_ of mummified delights.'
Callum fiddled with the back of his seat until the other side folded down, letting him drag stuff out of the boot. 'Heads up.' He passed a stabproof vest through the gap between the front seats.
Franklin took it, pulling at the Velcro fasteners.
He did the same with his one: opening it up like a tabard before sticking it over his head and fighting the front part in under the seatbelt as McAdams slid them around another corner.
Callum tossed a pair of thick leather gloves and a couple of elbow protectors to Franklin, then reached back in for the Method Of Entry gear, AKA: one hooley bar. A cross between a crowbar, an ice axe, and a claw hammer. Long enough to be a pain in the backside to extract through the gap where the seat was folded down.
A _crack_ went off like a gunshot and the Shogun's passenger-side wing mirror went flying off.
'Whoops!' A hard right.
Franklin reached out and killed the siren. 'Almost there.'
'Listen up, children. A rescue plan we must have. To save young Ashlee.' McAdams slowed to a more sedate fifty miles an hour, semidetached houses streaking past the windows. 'Callum: you take the hooley bar and pop the front door. Rosalind: you're on pepper spray and truncheon. We don't have time to fanny about here, so we go in hard and fast. Anyone _not_ Ashlee Gossard is to be considered dodgy as hell and completely arrestable. Any questions?'
Franklin pulled out her pepper spray. 'What if he's got a dog? Or a gun?'
'Then we probably get bitten and shot. Try not to, though.'
'And that's our plan, is it? Try not to get bitten or shot?'
He jerked the wheel left, taking them uphill onto Lehman Road. 'Do you have a better plan?'
'Just saying.' She put one hand over the clip of her seatbelt.
Lehman Road was a bit more exclusive than the previous streets, and a bit more rundown too. A cul-de-sac lined with big detached houses – large front gardens secured behind waist-high brick walls topped with six-foot iron railings. Weeds growing out of the cracks in the pavement. Old drooping trees, their leaves already yellowing. A couple of cars crusted with sap and dirt that looked as if they hadn't moved in decades.
Mrs Georgina Mason must've been worth a fair bit before she snuffed it.
McAdams pointed. 'Count it down.'
Franklin nodded. 'Number six... number eight... number ten... twelve... Go!'
The Shogun put on a burst of speed, the front end swinging to the right – up across the pavement and onto the weed-strewn blockwork driveway. Screeched to a halt right in front of the house.
Bang – Franklin was out.
Callum did the same, hooley bar clutched in his good hand, and sprinted up the stairs to number fourteen's front door. Only a few wisps of paint still clung to the wood. He swung the pointy end of the bar at the Yale lock, sank the tip in just in front of it and wrenched the whole bar forwards. A pop and crack, then the lock sprang free of the wood. He twisted the hooley bar round forty-five degrees and swung again, burying the wedge into the gap between the door and its frame. Shoved.
_BOOOM..._
The door flew open and Franklin rushed past, extendable baton clacking open in one hand, pepper spray in the other. Callum charged in after her.
She stuck her head in through an open door. 'Clear!'
He did the other side – an ancient music room, coated in a thick duvet of dust, the chairs sagging and mouse-eaten. 'Clear!'
By the time he'd got out again she was doing another room. 'Clear!'
A wide set of stairs snaked up towards a landing, chairlift rusting away on one side.
The kitchen cabinet doors hung squint in their frames. A collapsed chair lurked by the back door. But a clear line of tracks snaked through the dirt to the sink and back – the dust around the draining board almost non-existent. 'Clear!'
McAdams appeared in the hallway, shaking the rain from his shoulders. He took a quick look around.
'Clear!' Franklin stepped out of the bathroom and made for the stairs, taking them two at a time.
But McAdams marched right past her, to a small door part-hidden under the stairs. Wrenched it open...
A mop, a broom, and a collection of cleaning things collapsed out in a huge billow of grey dust. 'Gah...' He backed away, coughing, one hand waving at the impromptu smokescreen. 'It's an old house: there's got to be a basement somewhere. Find it!'
Callum went back to the kitchen. What looked like a utility room led off from one side, behind the rusted remains of a big round-cornered fridge. Twin-tub washing machine, more sagging cupboards, a collection of rotting wellington boots slumped by a Belfast sink. And a door.
He grabbed the handle and twisted.
Locked.
Well, the hooley bar would soon see to that.
Callum smashed the wedge into the doorjamb and shoved, setting the wood cracking and splintering. Then the door sprang open, bounced back off the wall as he took the first step into darkness.
Should've brought a torch...
Instead, he made do with his mobile phone, holding it out in his good hand, the hooley bar tucked under his other arm. 'Ashlee?'
The wooden steps creaked beneath him as he crept down into the depths.
His phone lit up the wall beside him – brickwork streaked with white where the salt had leached out of the mortar. The air tasted of raw mushrooms, smelled of vinegar and mouse droppings.
'Hello?'
His screen cast a pale-grey glow that barely reached a foot from his hand. Picking out strange rounded shapes all around him. He reached out and brushed a sheet, probably draped over a piece of furniture. His fingertips sent up a little cloud of dust that danced and twirled like midges in the thin light.
The basement was big, had to be about the same length and breadth as the house above. And it was full of unidentifiable _stuff_.
'Callum?' McAdams creaked his way down the stairs. 'Anything?'
'Too dark to tell.'
'Luckily...' A muffled _click_ and a beam of light swept across the room. Shining through the sheets and pulling the shapes of dining chairs and bicycles from within. McAdams played his torch around the weeping brick walls, then down to the floor at their feet.
A clear path was scuffed through the dirt, heading around a stack of tea chests and disappearing behind a supporting wall.
Callum followed it. 'Ashlee? Can you hear me?' Around the edge of the wall. 'Ashlee, it's the police. We're going to get you out of here.'
Assuming the path through the dust wasn't just a well-trodden route for rats.
It took a right, behind another supporting wall...
He froze.
A wooden door. With a brand-new hasp and padlock. A perfectly clean quarter-circle on the floor where it'd been opened outward.
'Well?' McAdams shoved him forward. 'Don't just stand there!'
Callum jammed the hooley bar's claw in under the hasp and shoved his full weight against it.
A groan, a squeal, then a crack as the whole thing ripped free of the wood and clattered to the floor.
He pulled the door open. Wood smoke enveloped him, slipped down into his lungs. Warm and inviting.
Orange light flickered low to the ground inside. Pale and indistinct, but definitely there.
Callum's footsteps echoed up and away, reverberating back from the walls.
The screen on his phone cast just enough of a glow to pick out the brickwork. It was a room, about six foot by twelve. Flagstone floor. It wasn't that warm, even with the fire smouldering away in the middle. He swung his phone up, but all that did was make the smoke glow.
And then a harsh white beam burst into life beside him, turning the smoke into a solid thing as McAdams stepped inside. 'Can't see a bloody thing.'
Something patted against Callum's shoulders. Like tiny raindrops in the dark.
He looked up.
Another drop hit his cheek and he wiped it away. Oily. Greasy between his fingertips. He stuck his phone in his pocket. 'Give me the torch.'
'No chance. Get your own—'
'Give me the bloody torch!' He snatched it out of McAdams' hand and kicked at the fire, scattering the glowing embers. Stood in the middle of the room pointing the beam straight up.
Drips pattered against his face.
Whether it was the door being open, or him kicking the embers out of the way, didn't matter. But something changed and the smoke swirled around his torch, thinning enough for the beam to reach up into the heights.
Rows and rows of filleted fish – tied together at the tail and hooked over the wooden rods that ran from one side of the room to the other – stretched up above him. And above _them_ , a shadow.
And then the smoke cleared.
It was a person, or what was left of them, their skeletal remains hanging head-down, arms dangling free.
'Jesus...'
They'd finally found Ashlee Gossard.
##
McAdams hacked and wheezed on the top step, face buried in an oxygen mask as Dotty's Vauxhall screeched up at the kerb. Mother clambered out into the rain and staggered over to Callum, breathing hard.
He shifted, making room beneath the twisted warty tree, just behind the ambulance. 'Boss.'
'Is Ashlee...?'
'She's so dehydrated they can barely find veins to get fluid into her.'
The ambulance's back doors hung open, both paramedics hunched over the emaciated figure on the trolley. Fighting to get wires and needles and drips fitted. 'Ashlee? Can you hear me, Ashlee?'
'Yes, but will she _live_?'
'Don't know. Maybe. It's possible...' He puffed out a breath.
Franklin marched out through the front door, paused to pat McAdams on the shoulder, then joined them under the tree. 'The only room not covered in eight foot of dust is the DIY smokehouse in the basement. No one's lived here for decades.'
Mother rubbed her hands across her face. Turned her back on the struggling paramedics. 'Good work, both of you.'
'No.' Callum shook his head. 'It was Franklin who got the Land Registry to search for properties belonging to Paul Jeffries, otherwise we wouldn't be standing here. I just went along for the ride.'
Franklin's cheeks went a shade darker. She shrugged. 'Team effort.'
There was a clunk and the Vauxhall's roofbox hinged open, the mechanics inside whirring and bleeping as a black metal arm brought Keith out from his storage bay and lowered him down beside Dotty's open door. Then the arm retracted back out of sight again. As if there was some vast metal spider lurking in the roofbox.
'Well: we've got Ashlee Gossard, that's the important thing. And she's alive.' Mother glanced back at the ambulance. 'Just.'
Dotty popped Keith open, then levered her legs out and swung herself into the seat. Wheeled her way over to them, squeezing under the shelter of the tree. 'Is she alive?'
Callum pointed at the ambulance. 'We've just done that bit.'
'Oh...'
'Excuse me.' Mother walked up the path and settled onto the top step beside McAdams. Put an arm around his shoulders. Talking in a voice too low to hear.
Franklin crossed her arms and leaned in close to Dotty. 'Did you know the silly sod's been ducking his chemo sessions?'
'Didn't you?' She shrugged. Then grinned and slapped Callum on the bum. 'I hear _you_ found your twin brother, and he's alive!'
'He's also an egotistical narcissistic drug-taking misogynist dick who can't decide if he'll help me catch one of the guys who killed our parents. Needs to consult with his lawyer first.' A sigh tore its way free. 'I don't know, Dotty, I genuinely don't. All these years...'
Dotty gave his leg a wee squeeze. 'Give it time.' She rubbed her hands together. 'So: are we any closer to catching Imhotep? Well, I suppose it's more like Imhotep Part Two, " _Son_ of Imhotep!", isn't it?'
'No.'
Dotty pulled a face. 'Didn't think so.'
One of the paramedics hopped down from the ambulance and closed the doors. 'Ashlee's _very_ weak, and I'd be shocked if her internal organs haven't started shutting down, but we've managed to get a little fluid into her. Maybe...?'
Callum handed him a Police Scotland business card. 'If anything happens.'
'We'll do our best.' Then he climbed into the driver's seat, set the lights going and the siren wailing. Pulled away from the house. Getting faster towards the end of the street. Flooring it on the way out.
Rain pattered on the leaves above them. Gurgled in the guttering.
Dotty stared off into the distance.
Franklin fidgeted.
Callum cleared his throat. 'OK, the question we should be asking is: why didn't Paul Jeffries sell this house to Northeast Ecclesiastical Trust Holdings? He turned the rest of them over, presumably for a tidy chunk of cash, why not this one?'
'Maybe he didn't need the money?' Dotty shrugged. 'Or maybe he was planning on living here?'
'He's got that place out in the middle of nowhere, no neighbours to see what he's up to. Why move into town and risk getting caught?'
Franklin pulled out her phone and poked at the screen. 'According to the Land Registry, Mrs Georgina Mason left the property to him thirty-five years ago. Maybe _that's_ when he was killed? He couldn't sell it, because he was propping up a shallow grave. And eight years later, the trust _finally_ notice he's not cashing the cheques any more, and another seven to have him declared dead.'
Dotty smiled. 'Ahoy, hoy – the Smurfs are here.'
A battered Transit van grumbled its way down the road, three faces peering out through the smeared windscreen. They parked in the spot vacated by the ambulance and Cecelia wound down the driver's window. 'This the right address?'
Callum hooked a thumb over his shoulder. 'Top to bottom, we need to know who else has been in there.'
She climbed out into the rain. 'We'll do what we can, but the labs...?' Her two colleagues went round the back and unlocked the rear doors. 'I swear on Jools Holland's grave, I have _never_ had this many lab-result cock-ups in my life. We've got to send about half of them back for retesting.'
One of her minions reappeared, already kitted out in his blue oversuit, and handed another one to Cecelia. 'You want us to start with the basement and work our way up?'
'Sounds like a plan to me.' She pulled the fresh suit out of its plastic wrapper and grabbed Callum to help her stay upright as she wriggled into the thing. 'You think them mixing up the internal and external samples from your mother's head was bad? That isn't even the foothills of Cock-Up Mountain.'
Callum stayed where he was until she'd got herself sorted. 'Well don't let them cock _this_ one up. Whoever Monaghan was working with, they're going to abduct someone else. Soon. We need an ID.'
The minion reappeared. Handed her a facemask and some safety goggles. 'Hi ho?'
'Hi ho.'
He turned and marched off, picking up his mate along the way, the pair of them whistling the tune from _Snow White_ as they disappeared with their kit into the house.
Cecelia gave Callum a pained smile. 'We'll do everything we can.'
Why did everyone keep saying that?
'Here.' Callum put the mug of tea down in front of McAdams, then settled into the seat beside Franklin.
Condensation ran down the inside of the Tartan Bunnet's window, mirroring the rain outside, the steamy air redolent with the round brown scent of frying bacon as the owner worked her sinister magic on a half pack of smoked streaky. All the other tables lay empty, their tops wiped to a sticky gloss, waiting for the next unwary diner to wander in. Like red-and-white checked carnivorous plants.
Dotty hunched forward in her wheelchair, working her way through a huge pile of chips with grim determination and lots of tomato sauce.
A grunt, then McAdams wiped himself a drippy porthole in the steamed-up window and peered out at the street. 'She doesn't look happy.'
From here, Mother was just a dark blob with an orangey bit on the top. Pale arms jabbing and poking as she spoke on the phone.
Franklin added two sugars to her coffee. 'What about going back to the psychiatric ward and _forcing_ them to give us Brett Millar?'
Dotty sighed. 'Here we go.'
'He's the only eyewitness we've got and we're not allowed to interview him?' She waved her spoon at them. 'Who does this Professor Bartlett think he is? We're trying to catch a serial killer and he's playing doctors and nurses! It's—'
'Impossible.' McAdams turned away from the window. 'We need a court order to get Millar's treatment suspended, and no sheriff worth his silly white wig will give us one. And _believe_ me, we've tried.'
'They're keeping him doped up so he won't attack the staff or patients, right? Well, we lock him in a cell and he can shout and scream all he wants. Eventually the drugs will wear off and he'll tell us who Monaghan was working with. We need to—'
'Detective Constable Franklin,' McAdams reached across the table and took one of her hands, 'with the deepest and most sincere respect, in the words of Mother's dear old nan: hud yer wheesht. It's not happening. What we _need_ is another plan.'
Dotty pushed her plate across the table. 'Have a chip, Rosalind. It helps with the feelings of frustration, helplessness, and existential doom.'
There was a pause, then Franklin helped herself to two.
'And Callum, sulking. Tell us pray, what news from court? Your brother, sent down?'
'Yeah, they say, "Bite me, Sergeant McAdams."' Callum took a sip of his own tea, hot and sweet. 'Pleaded guilty. They released him on bail, pending sentencing.'
'He'll be on the first private jet to the Bahamas, if he's got any sense. You'll not see him again.'
Franklin scowled around another stolen chip. 'Don't, OK?'
McAdams fluttered his eyelashes. ' _Moi?_ '
'Yes, you. Try a bit of compassion for once in your life. Can you even imagine how difficult this must be to deal with?'
Dear Lord, was Franklin actually sticking up for him?
'Ah, my dear Rosalind, you're probably right. It's force of habit. Winding up DC MacGregor is one of the few pleasures I have left, in these my twilight chapters.' He held out his hand. 'I'm sorry, Callum. To lose a brother to prison, having only just found him, must be—'
The café's front door banged open and Mother stomped in, shoulders down, fists clenched, cheeks and nose red, eyes narrowed, hair smeared flat by the rain. 'Useless, half-arsed, idiotic, pain-in-the-backside, moronic, _turdwardens_!' She threw herself into the last remaining seat at the table, setting the rubber feet squeaking. Sat there and glowered at her latte.
McAdams grinned. 'Good news?'
'AAAAAAAAAAAAAAAAAAAAARRRRRRRRRRRRRRR-RRRRRGH!' The scream echoed back from the walls, setting the cutlery ringing, then faded away into nothing.
The proprietor didn't even look up from her frying pan.
Mother slumped. Grabbed a handful of napkins and dried her face. 'I'm sorry, children, but I am _more_ than a little upset.'
Callum pushed her latte across the sticky tablecloth, till it was in front of her. 'Is it the Media, Detective Superintendent Ness, or the high heedjins from Tulliallan causing problems?'
'You remember the swabs Cecelia took from under the taps at Ashlee Gossard's house and the flat Ben, Brett, and Glen were doing up? Well our oh-so-wonderful labs came back with the results.' Mother ladled sugar into her latte, thumping each spoonful in, as if she was punishing it. 'Would you like to guess who our prime suspect now is?'
'Lord Lucan.' McAdams still had that grin plastered across his skeletal features. 'No, wait: Anne Widdecombe. Oh, I know: J.R. Hartley! It was, wasn't it?'
She thumped him on the shoulder. 'Don't be facetious.' More sugar was thrown to its death. 'According to the labs, Imhotep is sitting right here. At this very table.'
'It's Callum, isn't it? I always thought his shifty little eyes were hiding something. You can tell by the way they're all piggy and—'
She hit him again. 'It's _you_ , you spanner.'
'Oooooooh.' Both of McAdams' eyebrows made a break for the top of his head. 'Now there's a twist we didn't see coming: the trusted old police officer is actually... dan-dan-daaaaa! A psycho killer! OK, so it's a trope of genre, but who doesn't love the classics?'
Mother stared at him. 'You're just all-the-time hilarious, aren't you?'
'I like to think I have a certain homespun charm, yes.'
'Gah...' She lumped in more sugar. 'If I _ever_ get my hands on the idiot who awarded our forensic-lab services to the lowest bidder, I'll throttle them with their own innards.' She took a sip of her latte and grimaced. Pushed it away and helped herself to a couple of Dotty's chips instead. 'They're rerunning the tests again.'
'I was definitely in the flat after we discovered Ben Harrington's body in the bath. But I don't think I've ever been to the Gossard house, have I?'
Callum joined the free-for-all on Dotty's chips. Stuffing one in his mouth and chewing through the words, 'I'll bet it was the same idiot who cocked up the samples on my mother's head.'
'Oh, I can top that.' McAdams licked a smear of tomato sauce from his fingertip. 'Did you know that they IDed a strangulation victim as Wee Davey Roberts, last week? Didn't seem to matter that Wee Davey is, last time I checked, a fifty-four-year-old man with an artificial leg, and the victim was a twenty-one-year-old woman with all her own limbs. They thought they'd got a DNA match and that was it.'
Mother snorted. 'I heard one of their lab techs has come up as a positive match in _eighteen_ murder cases. Keeps picking his nose when he's running the samples and forgets to change his gloves.'
'Well _I_ heard they had our very own Constable MacGregor down for trying to batter DCI Reece Powel to death. And— Ow!'
Callum kicked him under the table again, for luck. 'Serves you right.'
'It's a disaster.' Mother flopped in her seat, head back, arms dangling. 'All right, we go back to the plan: interview everyone at Strummuir Smokehouse till they squeak. Someone has to know _something_. Don't they?'
Everyone looked away.
Rain clattered against the window.
'Oh for goodness' sake.' McAdams thumped Mother on the back. 'Look at you all sitting there with your faces like fizz. We should be _celebrating_!'
Nobody set off a party popper.
He shrugged. 'All right, so whoever Monaghan was working with has slipped free to strike again. And yes: the labs couldn't identify a breezeblock in a box of cornflakes. But we just saved Ashlee Gossard's life! She's alive because of us, and that's worth celebrating.'
Still no party poppers.
'Isn't it?'
Dotty drained her tea and banged the mug down on the sticky checked tablecloth. Fixed them all with a hard gaze. 'I'll get some more chips.'
Callum looked down at the Mondeo keys nestling in his palm. 'Are you sure?'
The road outside the Tartan Bunnet was packed with cars, parked with a studied disregard for the double-yellow lines on both sides. That was the trouble with police officers – no respect for the law. If they couldn't get a parking spot at Division Headquarters, why not abandon their cars on the surrounding streets?
At the end of Doyle Lane, the ugly Victorian red-brick bulk of DHQ loomed over the surrounding sandstone buildings, like an angry drunk challenging them to start something.
Mother nodded. 'Just don't crash it. Or run anyone over. Nothing that's going to cause me a backside full of paperwork.'
'Thanks, Boss.'
'I'm serious – don't make me have to explain to Professional Standards why I let a suspended DC borrow a pool car.'
He pocketed the Mondeo's keys. 'If you need any help chasing anything down, unofficially, off the books, give me a call, OK?'
She inched back a little, staying in the shelter of the café doorway as the wind shifted, keeping out of the rain. 'How did Andy seem to you? When you were searching the house, was he OK?'
Ah...
Callum licked his lips. 'He gets out of breath all the time. He looks like death. And he's developed a kind of sour funky smell. A bit like a cat that needs a bath?'
'I'm worried about him.'
'He blames himself for what happened to Watt.'
'He's skipping his chemotherapy treatments. He thinks I don't know, but I didn't climb out of a packet of Wotsits yesterday.' Mother picked at the front of her fleece, pulling off little bobbly bits. 'He turns every-thing into a big joke, but he's dying, Callum. He's dying and he's scared and there's nothing I can do about it.'
The door behind her swung open and Franklin squeezed into the doorway. 'You ready?'
Callum nodded. 'If you are.'
Mother reached out and took hold of his arm for a moment. 'Don't tell him I know. Please.'
Her cheek was soft and warm against Callum's lips. 'Our little secret.'
The kiss left her blushing. She mumbled something, turned, and headed back inside.
Franklin watched the café door close, then raised an eyebrow at him. 'So now you're trying to snog Mother? Just can't keep it in your pants, can you?'
'I can wait if you like?' Granite houses slid past the Mondeo's windows, stonework darkened to charcoal by the rain. 'Not a problem. I'm not doing anything anyway.'
Franklin poked away at her mobile phone, not looking up. 'I'll get a lift back with Dotty. We'll be hours.'
'I've got a book to read, I'll be fine. Be glad of the peace, to be honest. You know, after everything.'
'Callum, we've got to interview everyone that works there all over again. Do you have any idea how long that'll take? And you'll be what, pining away in the car, waiting for me? Like a lovesick Labrador?'
Straight through at the roundabout. 'Did it maybe occur to you that you're not as irresistibly desirable as you think?'
'Says the man who thought we "had a thing".'
'I'm just trying to be nice, OK?' Up ahead, the traffic had slowed to a crawl, backed up behind a council lorry laying out yet more orange sodding cones for yet more sodding roadworks. He took a left at the next junction. Cutting through Castleview proper. 'OK, you want the truth? I don't want to go back to Dotty's and sit in the dark, brooding about Alastair, and Leo McVey, and Elaine-and-Powel...' The big granite houses gave way to brick tenements. 'I don't know what to _do_ any more. I've not been on my own since I started seeing Elaine.' Her name was bitter on his tongue. 'I've not had a family since I was five years old. Everything's changed. It's all... It's like someone's cut the anchor free and all these big chunks of me are drifting away.'
Franklin looked up from her phone. 'Oh, Callum...' She reached across the car and squeezed his leg. 'Man up and grow a pair.' Then went back to texting.
'That's the last time I open up to you.'
'Good. Do us both a favour.'
He took a right at the lights. A couple of small tower blocks poked up from the surrounding houses. Gathered in a square.
Franklin put her phone down. Frowned out at the scenery. 'I thought Strummuir was that way?'
'It is. We're just taking a tiny detour. Ten, fifteen minutes tops.'
'Oh God, not _this_ again. Why did I let you drive?'
He headed straight for the tower blocks. 'I'm suspended, remember? I needed someone with me who can still arrest people.'
'You never change, do you?'
The rows of brick tenements gave way to a semidetached council estate, centred on the quartet of tower blocks. And right in the middle of the blocks: some yellowing grass, a little play park, and a shopping centre that looked as if the apocalypse had come early and stayed for tea.
'Ainsley Tyler Dugdale, forty-one, last known whereabouts: the Silver Lady strip club on Calder Road. Home address: fifteen B, Bowmore Avenue, Kingsmeath. Divisional have been looking for him since yesterday.'
'So what are we doing here?'
Callum pointed through the windscreen. 'That.'
A small, old-fashioned-looking pub sat at the corner of the shopping centre. Whitewashed walls and a neon 'T' in the window. Its name was painted in a wide strip of hoarding that ran the length of the building, 'THE PEAR TREE'.
'Dugdale's favourite boozer. And if we're lucky, the devious little sack of crap himself.'
'We're supposed to be finding Monaghan's partner!'
Callum parked outside. 'Dugdale battered a police officer and left him to die in the woods. Doesn't matter if Poncy Powel deserved a kicking or not, he's still one of our own.' And besides, if they did Dugdale for the assault, Professional Standards would sod off and bother someone else for a change.
She sat there, face clenched. 'Five minutes.'
'Fifteen, tops.' Callum climbed out into the rain, locked the car when Franklin joined him, then hurried across the car park and in through the Pear Tree's front door.
Warmth wrapped its arms around him. The smell of beer, peanuts, and Far Eastern spices. It was as old-fashioned on the inside as it was on the out: bare wooden floor; little round tables; chairs, benches, and stools upholstered in red vinyl; hunting prints and landscapes on the walls; and above a crackling fireplace, an oil painting of a tree with a single golden pear nestled within its dark leaves.
About a dozen customers, most of them in their sixties, were gathered around the fire, playing dominos, eating curry, and drinking half-pints.
Callum wandered over to the bar.
The large lady behind it gave him a dimple-cheeked smile. 'What can I get you, love?'
'Looking for a friend of mine: Ainsley Dugdale. He been in?'
Her eyes flicked left for a tiny beat, towards a wooden door with 'GENTS' on it. 'Dugdale?' A frown. 'Dugdale, Dugdale... No, doesn't ring a bell, sorry.'
'Big guy, bald, boxer's nose. This is his regular.'
'Can't say I've ever seen him in here. Maybe you're thinking of another pub? Try the Hare and Goblin on Wisdom Road.' The smile got a bit more strained.
'Right.' A nod. 'I'll just nip to the bogs before I go.'
And the smile disappeared altogether. 'Toilets are for customers only.'
'Fine. I'll take a can of coke. To go.'
'You're barred.'
'Nice try.' He turned and waved a hand at Franklin. 'Shall we?' He marched over and shoved the door to the gents open. Stepped inside.
Black and white tiles on the floor, the grout yellowed and greying. More white tiles on the wall, chipped and broken by the line of three sinks on the left. The sour sharp piddley smell reaching out from the urinals on the far wall. A pair of cubicles on the right.
Humming came from one of them: an old Donna Summer disco tune, from the sound of it.
Then some rattling and a grunt or two.
The sound of a zip being done up.
A toilet flushing.
And the door opened.
Dugdale had kitted himself out in jeans and a black hoodie – hood up, earbuds in, the white cables disappearing into the pocket at the front. His nose, chin and cheeks were a mass of purple and blue bruises, fading away to green and yellow. Willow Brown had obviously given him a serious kicking while he was lying unconscious on the pavement with a face full of pepper spray. But he still hummed along as he swaggered across to the sinks, throwing in a little hop-skip in time to the music.
Franklin nudged Callum. 'Well?'
'Let him wash his hands first. Be more hygienic.'
Another skip-hop-skip and Dugdale turned on the taps, swaying his hips and nodding his head.
Callum let him get as far as lathering up, before stepping right behind him. Reached out and tapped him on the shoulder.
A tiny high-pitched squeal broke through the humming and Dugdale spun around, eyes wide, mouth open. Then he saw Callum. Swore. And lunged.
Both soapy hands smashed into Callum's chest, sending him careering back, crashing into a cubicle door. Arms flailing, feet skittering on the tiled floor, trying to stay upright.
Dugdale was off – barrelling into Franklin.
She bounced off the toilet wall and went sprawling while Dugdale disappeared back into the pub.
Callum hammered after him, jumping over her as she struggled to her hands and knees. Out.
The OAPs were on their feet. One of them grabbed a bottle of Beck's by the neck and smashed its bottom against the fireplace – turning it into a glass dagger.
Dugdale battered out through the front door and Callum followed. Ducking as knives, forks, and dominos were hurled in his direction. Into the rain.
'COME BACK HERE!'
But Dugdale was off, arms and legs pumping, head down.
Well, he was out of luck this time. Callum yanked out the car keys and plipped the Mondeo's locks. Jumped in behind the wheel. Cranked the engine and whacked her into reverse, setting the tyres screeching on the wet tarmac. 'Come on, come on, come on...'
Dugdale was fast, but not fast enough.
Any second now...
He jinked to the right, skidding onto a section of grass, leaping down the bank and onto a path.
Callum hauled the wheel hard over, hauled on the handbrake, and the car spun on its axis, facing the right way as it lurched over the edge and thumped down the grass and onto the path. Slithering and fishtailing as the tyres fought for purchase.
Dugdale risked a glance over his shoulder and his eyes widened again. Head back down.
Closer.
Closer.
Callum tightened his grip on the steering wheel. 'You're _mine_ , sunshine!'
There was no way Dugdale could have heard, but he wheeched to the left – leaping over the waist-high chain-link fence that bordered the playground. Dodging the empty swings. Making for the other side.
Callum slammed on the Mondeo's brakes and the back end slid, caught the chain-link and yanked the whole car to a sudden stop hard enough to set off the airbags. The white balloon punched into Callum's face, forcing his head back, filling the car with the eye-scratching reek of spent fireworks. Leaving the air tasting of rotten eggs.
He coughed and spluttered his way out of the car. Stood in the rain and watched the tiny figure of Dugdale disappear into the distance – vanishing between two houses. And gone.
'SODDING HELL!'
He limped around to the back of the car.
Most of the rear wing was gone, torn off and dangling on the end of a metal fencepost. What was left was gouged and tattered.
Yeah, Mother was going to kill him.
##
Franklin climbed out of the car, stared at the passenger-side wing then ducked her head back inside. 'You're right, Mother's going to kill you.'
Behind her, Strummuir Smokehouse slumped in the rain, a long curl of white snaking up towards the heavy grey clouds.
Scraps of white dangled from the middle of the steering wheel and the dashboard above the glove compartment. Callum gripped one and yanked it free. Dropped it into the footwell. 'Because _today_ wasn't bad enough, was it? No. Course it _sodding_ wasn't.'
'Maybe you could get a garage to fix her up before Mother finds out? Weld on a new panel. Fit replacement airbags...?'
He let his head fall back against the rest. 'If anything happens, if you find out who Monaghan's killing partner was, let me know, OK?'
'We'll do our best.' Then Franklin turned and headed in through the smokehouse doors, leaving Callum alone with the rain.
There was a figure in the office above the main entrance, partially silhouetted in the floor-to-ceiling windows, on the phone. Stupid 1930s haircut, both arms covered in tattoos – _Star Wars_ down one side, _X-Men_ down the other. Skinny jeans. Finn Noble, the smokehouse manager. What was it Watt had named him, Darth Wolverine? He raised his other hand and waved at Callum. Then turned and disappeared back into the room.
Idiot.
Oh, he looked all trendy now, but give it five years when the fashion had moved on to something less lumberjacky. What was he going to do with all those tattoos then?
Callum shook his head, turned the wheel, and steered the crippled Mondeo back towards town. A strange ticking clunk came from the back end now, the engine sounding a lot louder and more gravelly than it had.
He clicked on the radio to drown the noise out, getting a bland poppy number for his troubles.
_'Ooh baby, you know I need you; And I want you; And I'll be true...'_
Maybe Franklin was right – get the car to a mechanic and hope they could hide the damage before anyone else found out. Assuming no one had caught the accident on their mobile phone and uploaded it to YouTube already.
_'Together, we can be free, / We can make love, / Have a baby...'_
'Good luck with that.'
There was Billy Jackson's garage in Kingsmeath. He might do it for cheap with bits from the scrap yard. As long as he matched the colour, who'd know? And it wasn't as if Billy didn't owe him a shed-load of favours.
_'Oh girl, you and me, / Living life, / Raising a family...'_
Blah, blah, blah.
Sodding Dugdale.
The song chuntered on as the Mondeo clicked, rattled, and growled its way across town.
'Face it, Callum, it's not your day today.' A little laugh broke free. 'Day? It's not my week. Month. Year. Hell, it's not my sodding _life_.'
One final close-harmony dose of blandness and the song died.
_'There we go, Mr Bones and "Babylove" from their most excellent live set at Tartantula last weekend. Stick with us, we've got loads more where that came from on the_ Lunchtime Sea of Sound _with me, Chris Pilot! But first it's quarter past one and here's Gabrielle with the news and weather.'_
_'Thanks, Chris. Tributes continue to pour in for R.M. Travis as news of his death spreads around the world—'_
'Dirty murdering bastard.' Callum stuck two fingers up at the radio.
_'—lead singer of ninety's rock band Wolfrabbit.'_
A man's voice: _'Yeah, it's a total nightmare. I mean, he meant everything to us when we were growing up. I know people chuck about the word "genius" like it means nothing nowadays, but he was a genuine hundred percent genius. There's no other word for him.'_
'How about "serial-killing dick-hat"?'
Technically four words, but it was the thought that counted.
A woman: _'It's completely devastating. How could anyone take R.M. Travis from the world? It's insane. I can't believe it.'_
Maybe the garage could fix the radio too, so it wouldn't pour crap out into the car?
Another man: _'R.M. Travis was a not inconsiderable landmark on the British literary landscape. They worship him all over the world, he's practically a religion.'_
And the newsreader was back. _'Staying close to home, Donny McRoberts, more commonly known as "Sick Dawg" appeared in court this morning on charges of possession of class A drugs, making death threats to police officers, and sexual assault. His lawyers issued a statement outside the Sheriff Court.'_
Callum turned it up.
Captain Scruffy's broad Glasgow burr filled the car: _'My client deeply regrets that the pressures of work have led him down the path of substance abuse and is_ determined _to get clean.'_ Though the accent was still there, he'd dropped all the Weegieisms. _'He wants to be a positive role model for his millions of fans, and understands he has a lot of work to do to regain their trust.'_
Not to mention all the violence against women and breaking his daughter's arm.
Callum slumped in the seat.
All these years wanting a family, wishing he still had a brother... Why did Alastair have to grow up such an arsehole?
_'We're asking the court to take into account Donald's very difficult childhood. And I'm afraid I can't say anything else about that at this time, but we will be making a further statement when we can.'_
_'McRoberts was released on bail to his record company who say he will be honouring all tour dates on his schedule. Local news now and police are appealing for witnesses—'_
Difficult childhood.
Suppose that was a bit of an understatement. Growing up in care was bad enough, but God knew what kind of horrors Alastair would have seen the day they were abducted. Did Leo McVey and R.M. Travis make him watch while they killed and dismembered Mum and Dad? Did they _do_ things to him?
The plastic steering wheel creaked in Callum's good hand. Knuckles swollen and pale.
How long did they keep him for, before dumping him on social services as Donald Newman? Then McVey, visiting him in the care home, year after year...
Poor little sod.
Maybe it wasn't surprising Alastair had turned out the way he had?
_'—missing four-year-old was last seen outside the Templer's Vale Shopping Centre in Logansferry. If anyone has any information—'_
Callum's phone went off and he dragged it out. Pinned it between his shoulder and ear so he could turn the radio down. 'Hello?'
Mother, sounding as if she'd just been run over. _'The doctors have been on the phone.'_
That couldn't be good. 'Ashlee's dead, isn't she?'
_'It's John, Callum. A blood clot broke free and... he's had a stroke. They're trying to see how much of him they can save...'_
Oh no.
'Is there anything we can do?'
_'I'm just letting the team know.'_
'Right. Yes. Look, if there was ever a good place to have a stroke it's in the Intensive Care Unit of a big hospital, isn't it? He's going to be OK.'
There was a sniff and a shuddering breath. Then a cough. _'Of course he is. You take care of yourself, Callum. I'll give you a call if I hear anything else.'_
'Thanks.' He waited till she'd hung up to put his phone away.
_'—finally looks like we're going to see an end to this weather. There's high pressure moving in from the Atlantic and that means clear and dry conditions from Wednesday onwards. Best of the sunshine will be on Saturday and Sunday, so bail out your barbecues and get ready for a good weekend.'_
_'Thanks, Gabrielle. Now, who fancies a bit of the Bay City Rollers?'_
'No chance.' Callum switched the radio off again.
'Oooh.' The little man in the greasy grey overalls sucked a breath in through his teeth. Wiped his hands on a rag. 'You've totally buggered that one, haven't you?' He ducked down again and peered into the rear wheel arch, making the hunch between his shoulders stick out even further, showing off the bald patch at the back of his head. 'What did you hit, an elephant?'
'How much, Billy?'
Two cars sat on ramps over matching inspection pits. Shelves and drawers lined the walls, along with a couple of risqué calendars, a portrait of the Queen, a stack of alloy wheels, and a welding kit. A small office off in the corner. The garage's roll-up door was open, letting in the never-ending hiss of rain. It didn't dent the overwhelming smell of old motor oil and diesel though.
Billy stood and sucked his teeth again. 'Your rear wing needs replacing, and the suspension's wrecked, and you're gonna need a new tyre, and the airbags are gone, and the exhaust's loose, and—'
'Bare minimum, on the cheap: how much?'
'Then I've got to order the parts in from Ford, and you know what—'
'No.' Callum held up his hands. 'No dealership parts. We need to salvage everything from the nearest scrap yard. On the cheap, remember?'
'Pffff...' He puffed out his cheeks. 'Depends what I can get my hands on.'
Callum stared at him. 'How long have we known each other, Billy?'
'Oh come on, I'm trying to make a living here!'
'Did I, or did I not save your backside when you set fire to Mr Crimon's car?'
'I've got an ex-wife, two kids, and a cat to support!'
'He caught you, remember? With your jeans all clarted in petrol.'
'That was twenty-two years ago. Just because we grew up in a home, doesn't mean—'
'He was going to kill you, Billy. Literally. Crimon was going to hold you under the bathwater till you _drowned_.'
'It's not—'
'Who hit him with that crowbar? Because it wasn't the Tooth Fairy.'
'Gaaaaagh...' Billy stared at the roof. Slumped. Rubbed at his hunch. 'All right, all right. Much cheapness.'
'Thanks, and much quickness too, I need it back before anyone notices what's happened.'
'I'll make some phone calls.' He produced a battered mobile phone and wandered off, poking away at the screen.
And all Callum had to do now was figure out how to pay for it. Because, somehow, it was doubtful Billy would accept – quick check in the pockets – three pounds, twenty-one pence, and a button for fixing the battered Mondeo.
Couldn't even offer his bike in part exchange, not with the Dumbarton Arms still holding on to it as collateral. And the way his luck was running, buying three scratchcards and hoping for a windfall wasn't going to work either.
So Callum settled back against the workbench to wait. A couple of red-top tabloids sat next to a mug of coffee with 'WORLD'S WORST HUSBAND!' on the side. The front page of both was dominated by a photo of Emma Travis-Wilkes and her father at some sort of black-tie event. The pair of them smiling for the camera – him clutching a chunk of Perspex with a sponsor's logo on it and something trapped inside. 'BOOKED FOR HER FATHER'S MURDER' was one headline. 'THE MOST HATED WOMAN IN BRITAIN?' on the other.
Going by the crowds outside Division Headquarters that morning, she certainly had to be in the running.
There was a small story sharing the front page with Britain's most hated woman: a sidebar with a photo of Alastair with his shaved chest, baseball cap on backwards, and the tattooed cartoon fox poking out the waistband of his pants.
'MY LIVING HELL WITH RAP STAR DRUG FIEND'
Looked as if Irene Brown had sold her story to the papers.
Continued on page four.
Good for her. With any luck she'd got a whole heap of cash for it.
He flipped through to page four.
They'd given her a two-page spread with more photos of Alastair, AKA: Donald Newman (31), AKA: Donny '$ickDawg' McRoberts rapping away on stage. But right across the top was a big picture of Irene Brown (23), sitting in her living room, surrounded by her adoring children. Willow (7) and Benny (6) were striking rapper poses, arms crossed in ridiculous fashion with hands throwing gang signs out the ends. Pouting like ducks. Their little sister, Pinky (4), was dressed up in a long white dress with her hair done up in side buns like Princess Leia, sucking her thumb and clutching a lightsabre – not your standard Disney princess, but it still counted. The baby, Elsa (5 months) sprawled in Mum's lap, all pink arms and legs.
Irene hadn't put any make-up on for the photographer, letting the split lip and bruised cheeks shine through instead.
The article was in full-on tabloid sensationalist mode. The sex: rough. The drinking: constant. The drugs: hard. The violence: all the time. Living in grinding poverty while '$ick Dawg' was off drinking champagne, travelling first class, and not paying a penny in child support. And right at the end, they'd asked her why she'd finally found the courage to confront the aggressive drug addict who'd fathered three of her kids.
Callum raised an eyebrow.
Apparently it was all down to _him_. There it was, in print:
"Detective Constable Callum MacGregor was the first person to be kind to me in years. I'd lost faith in the police, because no one ever cares about people like me," said Irene, holding back the tears. "But he did. And I owe it to him to stand up and tell the truth about what Donny did to us."
Dear Lord...
Something went pop deep inside his chest, spreading warmth across his lungs and down his spine.
Wow.
He closed the paper. Smiled. He'd _actually_ made a difference.
Then frowned. Opened it back up and stared at the photo of the Family Brown again.
Donald Newman was really Alastair MacGregor, so that meant Pinky, Benny, and Willow were Callum's nephew and nieces. He had a family.
A grin spread across his face.
He actually had a family again.
OK, so Willow was a little monster, Benny was _different_ , and Pinky was...
Pinky – dressed up as Princess Leia.
Darth Wolverine – standing there with all his tattoos.
The happy warm feeling seeped away.
Callum dug out his phone and called Franklin. 'Yeah, hi. Just wanted to know how you were getting on.'
_'Believe it or not, we're coping without you. But only just.'_ It shouldn't have been possible, but the sound of Franklin rolling her eyes came down the line loud and clear.
'Sarcasm. Lovely.'
_'Go watch a film, or read a book or something. Some of us have work to do.'_
He wandered over to the roll-up door, standing on the threshold, just out of reach of the rain. Looking out on a manky grey alley in manky grey Kingsmeath. 'Listen, I was thinking about Darth Wolverine. Has anyone—'
_'Darth what?'_
'Watt came up with it: it's a tattoo thing. Finn Noble, runs Strummuir Smokehouse.'
_'Is this going to take long? Only I've got eight more people to interview before the building shuts at five.'_
'He's probably the trendiest hipster there, right?'
_'Callum, can we not—'_
'He's had access to the smokehouse all along. He's the one who decides which ex-cons get to work there. He runs the courses – smoking and charcuterie and all that – so he knows the names and addresses of everyone who attends. He's in charge. He can come and go as he pleases.'
_'You think he's Imhotep?'_
'You didn't see him when Monaghan went into the river: shouting the odds, swearing. He even took a swing at Watt. Bit extreme for someone who just employs the guy, isn't it?'
_'Well...'_
'And Noble said they never hire anyone who's a sex offender. Tod Monaghan had form for indecent assault, and he raped that bloke who wouldn't press charges. How is that not a sex offender?'
_'If the victim wouldn't press charges, maybe Darth Windolene didn't know about it? They probably don't put soft intelligence on file when they send people for work placement.'_
'But they'd put the indecent assault in, wouldn't they?'
_'Hrmmm...'_
Outside, a couple staggered past – the pair of them in ripped jeans and baggy T-shirts. Soaked to the bone, laughing, and sharing a half-bottle of vodka.
'Think about it. He teaches people how to smoke things. He's more hipstery than Ben Harrington, Brett Millar, and Glen Carmichael put together. He'd fit right in, just like Dr McDonald said Imhotep would.'
_'Callum—'_
'He teaches a foraging class too. That means mushrooms. How much do you want to bet he can get his hands on all the psilocybin he wants?'
The laughing couple dissolved into the distance, consumed by the downpour.
Callum turned his back on the rain. 'Come on, Finn Noble's _got_ to be worth a closer look. You said the only person without a criminal record there was the woman who did the chips. What was Noble in for?'
Nothing from the other end.
'You still there?'
_'I'm looking at my notes.'_
'Just ask him where he was when Watt was attacked. See if he's—'
_'Thank you, Constable. I might just be a lowly woman, but I do actually know how a police investigation works.'_
Callum closed his eyes. 'OK, OK, I'm sorry.'
_'Should think so too.'_ A sigh. _'He's gone out for lunch, but Dotty interviewed him this morning. I'll check with her, see what he said.'_
'Great. And call me back?'
_'You're a pain in the backside, you know that, don't you?'_
'Yup.'
'Right, you're in luck.' Billy stuck his head out of the little office. 'Frazer McFee and Son have a diarrhoea-brown Mondeo estate in stock – engine's completely seized, but everything else is salvageable. And they'll let you have the bits you need for three hundred.'
'Pounds?'
'No, Jelly Tots. Of course pounds. Cash, so no VAT, and... What?'
Callum fiddled with a spanner. 'I'm a bit, strapped.'
'Oh for God's sake.' He disappeared back into the office. 'They'll give us a ten percent discount if we dismantle the thing ourselves. And _you'll_ owe _me_ , understand!'
Callum followed him into a gloryhole of paperwork, files, and random bits of machinery. 'How long?'
Billy filled a little kettle from a little sink and stuck it on to boil. 'If I abandon everything else? Lunchtime tomorrow.'
That would be doable, wouldn't it? He'd just have to keep his head down till then and hope Mother didn't ask for her Mondeo back. 'Thanks, Billy, you're a star.'
'Just don't tell anyone.' Billy shook his head. Sighed. 'I'm a fool to myself.' Then delved into the filing cabinet. 'You want a Pot Noodle? Got chicken-and-mushroom, or Bombay Bad Boy.'
Easy. 'Chicken. Why would anyone—' His phone went off and Callum swore. Pulled it out and checked the screen: McAdams. 'Sorry, got to take this.' He walked back out into the workshop while Billy peeled the foil lids off the pots. 'Is Watt OK?'
A small pause. _'He had a stroke. How would he be OK?'_
'When Mother called she sounded... I don't know. Anyway, if it's not Watt, what do you want?'
_'I'm dying.'_
This again.
'I know.' Callum settled back against the workbench and flicked through the other paper. Sex scandal. Sex scandal. 'MY DRUG BINGE HORROR' by some nonentity from a reality TV show.
_'We need to talk.'_
Murder. Sex scandal. '"I'M PROUD OF MY CELLULITE!" SAYS CURVY CORRIE BABE'.
'So talk.' Cellulite-Pride Week seemed to be nothing more than an excuse to print pictures of celebrities in their bikinis. Germaine Greer would be so proud.
A cough rattled down the phone, followed by another one. And another – hacking on and on.
The next page was an editorial about what a genius R.M. Bloody Travis was, and how everyone would miss his magical imagination.
McAdams' coughing gave way to wheezing gasps.
Callum scrunched the page up in his fist, spat on it, then lobbed it at the bin. Missed.
A wheezy voice sounded in his ear again. _'Callum? You still there?'_
'No.'
_'Where are you?'_
'Kingsmeath, visiting an old friend.'
Another cough rattled out of the earpiece. Followed by some panting. _'Urgh... My house. And bring some milk – full fat. No point wasting life on that semi-skimmed rubbish.'_
'McAdams, I'm not traipsing all the way across—'
_'We – need – to – talk. In person. Now.'_
Wonderful.
He didn't bother hiding the sigh. 'I'll see what I can do.'
Callum hung up and went back to the office, where the smell of rehydrating soya product filled the gaps between the oily diesel fug. 'Have you got a car I can borrow? Something's come up. Sorry.'
Billy pulled two forks from his desk drawer and stuck one in each pot. 'I need the truck to go get your Mondeo bits, but I've got something that might help. It's not fancy but it'll get you there.'
'Anything's good.'
That got him a very disturbing grin.
'What?'
'Nothing.' The grin got bigger.
Yeah... why wasn't that reassuring?
##
BILLY JACKSON MOTOR SERVICES ~ MOTS WHILE YOU WAIT didn't stretch to a courtesy car. Instead, Callum wobbled along the side streets, both knees clamped together, buttocks clenched, holding on to the scooter's handlebars as if they were the only thing keeping him from a humiliating and messy death. The crash helmet rattled about on his head, about a size and a half too big, but it was the only one that would go over the gauze padding covering his ear and the back of his head.
This was clearly Billy's way of getting his own back on Callum for abandoning him to dismantle the scrapyard Mondeo on his own. Not to mention the matter of paying for the parts and repairs with an IOU. But still...
Spray made twin arcs either side of the front wheel, there was nothing to keep the rain off, it was freezing cold, and the engine sounded like an angry wasp attacking a PA system.
Sodding DS Sodding McAdams.
Why couldn't he just discuss whatever it was on the phone like a normal person?
Because that would be too easy, that's why. Because then Callum wouldn't have to drive a horrible little scooter through the pouring sodding rain.
A four-by-four passed by, going in the opposite direction, sending up a wall of water that crashed across Callum's arms and chest.
'Aaaaargh!'
That did it – cancer or not, McAdams had to die.
'What took you so long?'
Callum stood on the doorstep, arms outstretched, legs apart, dripping. Plastic bag dangling from his good hand. 'I'm going to kill you.'
'You look like you swam here.'
'I swear to God, if you don't get out of the way and let me into the dry, I'm _seriously_ going to murder you right here.'
A slow smile spread across McAdams' skeletal features. Then he stepped back and gave Callum a low bow, sweeping one hand out to indicate the corridor.
'Gagh...' Callum stepped over the threshold and squelched his way along the parquet flooring and into the tiled dining-kitchen. All slate and black granite, beech units, a big fridge and another one right next to it just for wine.
Must be nice to marry someone with a trust fund.
'Perhaps, dear Constable MacSoggy, / You should change out of your clothes so damp? / You look just like a half-drowned moggy, / You sopping squishy squelchy scamp.' Then there was a wobble. A grimace. And McAdams lowered himself into one of the dining chairs. 'Help yourself to tea, coffee, or a nice glass of wine.' He waved a bony hand at the fridges. 'The Sancerre is particularly good. Far too expensive, but it's not like I can take it with me.'
Callum dumped the plastic bag on the draining board, wriggled out of his sodden jacket and wrung it out in the sink. Hung it over the back of a chair. Kicked off his shoes and poured their contents down the drain. 'This better be important.'
'I'll take a glass, if you don't mind? I'd get it, but my legs don't seem to be cooperating right now.'
His socks splatched and squished against the slate tiles all the way to the kettle. He clicked it on. 'Mother wants you to start your chemotherapy.'
'The glasses are in the cupboard on your right.'
'I'm serious. She's worried about you.' He pulled a white wine glass from the cupboard and stuck it on the countertop. Had a rummage till he found the mugs and stuck one next to it. Opened the tin marked 'TEA'.
Curled his lip.
The tin was full of bits. 'Gah... Don't you have any _proper_ tea?'
McAdams smiled. 'That _is_ proper tea. Beth gets it from a little shop in Aberdeen. One spoon for you, one for the pot.' He pointed at the wine fridge. 'Now: there should be an open bottle of Sancerre at the front.'
Callum pulled the bottle out. Unscrewed the cap. 'Stop being a dick and go to your bloody chemo.' He filled the glass and squelched over to the table. Stuck it in front of McAdams.
'Oh, Callum.' The smile softened at the edges. 'It's too late for that. All the coughing? My lovely cancer has metastasised. I'm riddled with it. Like an old building with rats. Eating the wiring, making holes in the skirting boards, and covering everything in crap.'
The kettle bubbled and growled.
'How long?'
'A week. A fortnight. A month. Does it matter?' McAdams took a sip of wine, eyes closed, then sighed. 'You sure you don't want a glass? It's lovely.'
'You should be in hospital.'
'I'm lucky. The drugs keep most of the pain at bay, for now. And I've still got all my marbles.' A wink. 'For now.'
'Matter of opinion.' A click and the bubbling subsided. Callum dumped two spoons of grey-black bits into the pot. Drowned them with boiling water. 'And for your information: proper tea comes in a teabag. It doesn't look like something you scraped out of the vacuum cleaner.'
'Philistine.' He produced a little notepad and flipped it open to a page covered in cramped handwriting. 'I'm making my bequests while I still can. Dotty's getting a case of Bowmore, because she loves her whisky. Mother's getting a cruise: the Norwegian fjords, because she likes pickled herring and deserves a decent holiday. Watt...' A frown. 'I wasn't sure what to get him. We don't even know if he's going to live now. Maybe his own electric wheelchair, if he pulls through? Or I could send him on holiday too, so he can recuperate?' McAdams took another sip of wine. 'Rosalind gets a diamond necklace. Nothing too flashy, but something dangly that will nestle between those magnificent breasts of hers. Because, let's face it, who wouldn't want to do that?'
Callum stirred the tea. 'What about your wife?'
'Oh, Beth's off to Edinburgh for the week. Apparently I've been a bit more _colourful_ than usual and it's getting on her nerves. Or did you mean, "what does she inherit"?' He took another sip of Sancerre. 'She gets the house and the car and the bank account and the timeshare in Tenerife. Which is peanuts compared to what her dad left her, but there you go.' McAdams put his glass down. 'Which brings us to _you_ , Callum.'
'I'd settle for a towel and a go in your tumble dryer.' He filled his mug from the pot, then pulled a four-pint carton of milk from the plastic bag on the draining board. Sploshed in enough to turn his tea beige.
'It was surprisingly difficult to find something appropriate to leave you. You're a simple soul, yes, but you've got dark depths, don't you? A compelling backstory for crime fiction: family abducted, growing up in care, unlucky in love, rumours of corruption. A mediocre officer in a troubling world, who spends his life trying to get justice for his mother, father, and brother.'
'Hmph.' The fridge was packed with jars and bottles and Tupperware containers. A whole shelf dedicated to kippers. He stuck the milk into the door pocket. Frowned at the shelves. 'You've got enough kippers here to feed an army.'
'And that's why you're so hard to buy for.'
'I didn't, by the way: spend my life trying to get justice.' He unclipped his tie and draped it over the taps. 'Everyone said my family abandoned me: dumped me in the lay-by and sodded off. All my life I thought I'd done something wrong.'
'But I think I finally got you the perfect gift.'
'I became a police officer, because I grew up in the care of people who shouldn't have been allowed within two hundred yards of children. I joined the job, because I wanted to put scumbags that prey on the weak behind bars.'
McAdams nodded. 'Indulge a dying old man and look in the drawer by the toaster.'
Callum did. It was a brown paper parcel, about the size of a ream of paper, only twice as thick. It flopped like a ream of paper when he picked it up too. He held it out. 'This?'
'That.'
Someone had printed the words 'A DARK SO DEADLY' across the front, in black Sharpie. 'I'm guessing it's not a holiday.'
'It's my book.'
Oh joy.
Dotty got a case of whisky, Franklin got a diamond necklace, Mother and Watt both got fancy trips, and what did _he_ get?
'Oh, don't look like that. Everyone else got material stuff, but you: you're a reader. There's so few of you about these days, Callum. So I give you my book. My life's work, distilled into one hundred and ninety-four thousand, five hundred and twenty-eight words. Single-sided double-spaced on A-four.'
Callum put it down. 'You got me to drive all the way across town, on a scooter, in the pouring rain, because you wanted to give me a copy of your book? You said it was urgent!'
'I also needed milk. And how was I to know you'd be on a scooter? What happened to the car Mother lent you?'
Ah...
'Nothing. Thanks for the book.' He cleared his throat. 'Now, any chance of that towel?'
Rain.
It pounded the garden on the other side of the window, battering the bushes into submission, hurtling down from a charcoal sky, turning everything grey. The floral scent of fabric softener filled the utility room, mixing with the tickly smell of warm dust.
Callum tightened his borrowed towelling dressing gown and leaned back against the _whomp-whomp-whomp_ of the tumble dryer. Warmth stroked the back of his legs. 'He what?'
Franklin sounded as if she was sucking on a wasp. _'You heard me: Finn Noble's dead.'_
'How the hell did that happen?' Callum pinned the phone between his ear and his shoulder, freeing up his good hand for a sip of tea. 'Was it an—'
_'We sent a patrol car round his house: No answer. So the uniform peers in through the windows, and there's Finn Noble: hanging from a noose in the hall. He'd tied one end to the balustrade and jumped.'_
'He _killed_ himself?'
_'There's more. He left a suicide note.'_
The tumble dryer bleeped then fell silent.
Callum opened the door and hauled out his suit – all hot and crackling with static electricity. 'It's him, isn't it? Imhotep. He knew we were closing in and he chickened out before we could nab him.' The towelling robe went on the worktop and Callum pulled on his pants – all warm and clingy. Grabbed the edge of the washing machine as the world did a quick swirl when he stood up again. 'Whoa...' Blink. Sniff. 'You still there?'
_'We've sent a copy of the note to Dr McDonald, but it's basically claiming credit for the killings, justifying his actions, and complaining that we spoiled everything by capturing his gods before they could save the world.'_
'AKA: nutjob.' Callum hauled on his trousers, fingers flinching at the hot zip and buttons. Struggled his fibreglass cast down the sleeve of his shirt. 'I can be there in...' Callum checked his watch. Then froze. How _exactly_ was he going to turn up on a scooter and not have everyone asking questions about the missing Mondeo? Borrow McAdams' car? 'Maybe fifteen minutes?'
_'Don't be daft, you're suspended.'_
'Yes, but—'
_'No buts.'_
'Fronting up Finn Noble was my idea! How can you not... Hello?' Silence. 'Franklin? Hello?'
She'd hung up on him.
Perfect.
Thanks. Thanks very sodding much.
He buttoned his shirt. Pulled on hot socks. Checked his shoes – perched on top of the boiler. Twenty minutes up there, stuffed with crumpled-up bits of the _Daily Record_ , and they were _still_ sodden through. Callum changed the damp newsprint for dry bits. Grabbed his jacket then headed back through the door to the kitchen.
The contents of his pockets were where he'd left them, sitting on the kitchen table. He loaded up again. Paused. Sniffed the air. 'Why can I smell smoke?'
McAdams stood in front of the hob, stretching a sheet of clingfilm over a frying pan. 'That's the trouble with most people: no idea how to treat a kipper. It's already cooked, you don't need to grill or fry it – that just dries the flesh out – you stick it in a deep-sided frying pan, or roasting tin, and you pour hot water over it. Seal it for a couple of minutes and you're good to go. Jugged kippers.'
'Finn Noble's hung himself.'
'Has he?' McAdams clattered two plates onto the worktop. 'Let me guess—'
'Left a suicide note, admitting everything.'
'Thank God for that.' He sagged a little, one hand propping himself up. 'Get a couple knives and forks. You like kippers, don't you?'
'Franklin said everyone at Strummuir had a criminal record. What did Finn Noble do?'
'Of course you like kippers. Who _doesn't_ like a proper kipper?' He tapped at the clingfilmed surface, making the condensation form into little round droplets. 'It's the drugs, they mess with your palate. Kippers are one of the few things that still taste right to me. That and the wine. And whisky, of course.'
'McAdams: what did Finn Noble do?'
'Hmmm? Oh, him. Yes. He was arrested for indecent exposure twice – decided it was a good idea to get his willy out on Stone Terrace, outside the youth hostel. Possession of a Class A drug on three occasions. And a handful of burglaries. Nothing major, except for the willy waving.'
'What was the Class A?'
'You know, I think these are just about done.' McAdams peeled back the clingfilm and used tongs and a spatula to manoeuvre one out of the water and onto a plate. Held it out to Callum. 'Why don't you guess what controlled substance Finn Noble was caught with.'
'Magic mushrooms.'
'Give that detective constable a kipper.' He went back to the pan for the other one. 'Try a knob of butter on it, melts right into the smoky flesh.'
'Woops.' Callum sat at the dining table a bit harder than he'd meant to, making the seat creak. 'Mother sent you home, didn't she?'
A shrug. McAdams lowered himself into a chair. Groaned. Then peeled the skin off the top of his kipper. 'She seems to think I'm taking too much on. Well, what am I supposed to do? Hang around here like Banquo's ghost? Eat your kipper.'
The skin was thin and papery, the flesh beneath it plump and moist. Smokey and full of horrible little bones. He worked them to the front of his mouth and picked them out. Wiped them off on the edge of his plate. Looked up to find McAdams smiling at him. 'What?'
'It's like a metaphor for life, isn't it? The flavour is magnificent, but every mouthful comes with a cost. And in the end, all you're left with is a pile of skin and bones.' McAdams reached for the butter dish, dug a chunk off with his knife and dolloped it onto his fish. 'You remember our discussion in the car? You me and Rosalind talking about how no one ever remembers the police officer or the victims, they only remember the serial killer?'
'People remember Gandhi, he wasn't a serial killer.' Callum picked out another mouthful of bones and laid them with their comrades. Like little pale thin soldiers. Took a swig of tea to wash down the saltiness.
'Gandhi doesn't count. He's remembered because he made a _difference_. What difference are you and I going to make?'
More little soldiers. All lined up on parade.
McAdams put his knife and fork down. Topped up Callum's mug from the teapot. 'Normal people don't change history. Normal people die and get forgotten.'
How many bones were there in a kipper. A thousand? Two? A million?
They caught the light and... sort of _glowed_. Little bone soldiers.
'Callum?' A sip of wine. 'Do you believe in God? Or gods? Or anything at all?'
Yeah, McAdams needed to cut back on the wine. His voice was getting a bit wobbly and boomy. Like the grown-ups in a Snoopy cartoon. Whah, whah, whah...
Somewhere, off in the distance, a phone rang. And rang. And rang.
'Don't worry, the answering machine will get it.'
All those glowing bone soldiers.
_Bleeeeeeeeep._
_'Andy? It's Cecelia. I'm sorry, I don't know what the hell they're playing at, but the labs have buggered it up again. I'm putting in a formal complaint.'_
Callum blinked. The soldiers left bright orange streaks on the inside of his eyelids.
'Are you all right?'
He shook his head and the world lurched round by thirty degrees, then slowly drifted back again. Urgh... 'Think I'm coming down with something.' Not helped by driving a scooter all the way across town in the sodding rain.
Or had he said that already?
_'The idiots have taken a liking to you: those samples from the Gossard house and the Carmichael flat have come back as a match_ again _. And now they've got you driving the abandoned Kia Picanto. You know, the one we found Richard Duffy's body in the boot of?'_
So thirsty.
Must be the kipper.
Have another gulp of tea.
_'I've told them: I'm going to make you do these ruddy tests over and over till you get them_ right _. Honestly, it's like trying to teach a lawnmower about particle physics.'_
'Callum? You don't look too well.'
His hands made whooshing sounds when he moved them.
'Can't feel my tongue...'
_'Anyway, just wanted to say sorry. I know it's not_ our _cock-up, but still. Give me a call when you get this, OK? Bye.'_
'That's a shame.' McAdams stood, wobbling like one of those inflatable men they stuck outside car dealerships.
_'Oh, and speaking of Callum, if you see him, tell him the Dundee lot have done a preliminary sweep through the contents of Travis's freezers. They can't be a hundred percent, but they say it_ looks _like it's all adults. No children's remains, if that helps? OK, that's definitely it this time. Bye.'_
_Bleeeeeeeeep._
Callum reached for his mug, but the thing wasn't where it was meant to be and his fingertips caught the handle, tipping it over. Pale brown and beige spread out across the tabletop. 'Sorry...'
'It's all right. It's only tea.' McAdams patted him on the shoulder. 'Well, not _only_ tea, but mostly.' He wobbled over to the worktop and came back with a dishtowel. Laid it out on the beige puddle. The fabric darkened. 'You should've had the Sancerre instead.'
'Mmmph?'
'I only wanted to make a difference. Not a huge one, just a little one. Before I die.' McAdams mopped up the spilled tea. 'It would've worked too, but you...' A sigh. 'Well, I suppose you were only doing your job.'
'Nnnnmph.' The walls lurched in, then out again. Boom. Boom. In time with his pulse. Like waves. In. And out. In. And out.
'But couldn't you have left me with _one_?'
In. And out.
'Just one god for me. To fix all this? To make the world whole again?'
Callum grabbed hold of the tabletop. Dug his fingers into the slippery surface as the grain, buried deep in the wood, twisted and flowed.
'One by one I had to give them up. I didn't want to, but what choice did I have?' His hand was hot on the back of Callum's neck. 'I thought I'd have enough time with Ashlee, she was coming along so well. She'd purified. Made amends. She was ready to transcend.'
The tea. There was something in the tea.
'But now she's gone, back to the venal flesh and dust and darkness. You want to hear a secret? I think surrendering her was probably the hardest thing I've ever done.' A tiny laugh. 'Such a terrible waste. Don't get me wrong: in a weird way, I'm glad she's still alive, but I would have made her _immortal_.'
Callum scraped his chair back. 'Nnngh...'
'I know, I know. Come on.' He hooked his hands under Callum's armpits and pulled him to his feet. 'It's all right. In the end I couldn't help Ashlee, but I can still help you, if you like?'
'Nnnn.'
'Shhh... Here we go. It's going to be fine. Watch your feet on the steps, it's a long way down to the basement.'
##
'There we are, that's better, isn't it?'
The basement floor was warm and soft, like R.M. Travis's garage. Dusty soft warm concrete. The brick walls pulsed – in and out, in and out. Shifting in time with every single one of Callum's breaths. 'Gnnnmmph...'
'Shh... It's OK. You're going to be fine. Trust me. It's all going to be fine.'
Callum's arms were lead, dull and heavy, his fingers like wet balloons.
'You just sit up here. That's right.'
McAdams hauled him into a sitting position.
'Here we go.'
The chain was cool against Callum's neck. The padlock's click reverberated through his skull. Click. Click. Click. Click.
'All safe and secure.'
'Nnngh...'
A sigh. 'I feel bad about Finn, but look at it from my perspective: if I hadn't killed him and written that suicide note, we'd still be looking for Imhotep, wouldn't we? This way it's all finished nice and neat. We've come to the end of the book.' His hand was warm and dry, stroking Callum's cheek. 'You made an excellent antagonist and I _mean_ that. Sincerely. And now you're going to be a god. They'll worship you and you'll be a god. How's that for an epilogue?'
In and out. In and out.
He gave Callum a sad little smile. 'It's all been a bit of a disaster, hasn't it? I didn't mean for the first body to end up in the tip, but Beth came home early and the body was ruined and I panicked, and, and, and...' He rolled his eyes and pulled a face. 'I know, I know: exposition. Anyway, you wouldn't _believe_ how much finessing it took to make sure we got the investigation when they found it. Couldn't have anyone else sniffing around my gods, could I? Of course not.' He leaned forward and kissed Callum on the forehead. 'Don't worry, though: no one will ever find you. You won't end up in Professor Twining's mortuary, like some sort of natural history exhibit. You're going to be a god. You're going to be immortal.'
'Gnnnnnnnnnnngh...'
'I'm going to get you some more tea. You wait here and—' He froze. Staring up at the ceiling as the doorbell gave out two death-knell rings. 'Tsk. Some people just have no sense of timing, do they? Never mind, they'll go away and we can—'
The bell tolled again.
'Oh for goodness' sake. Sorry about this. I'll be right back.' A smile. 'Promise.'
Callum closed his eyes and the walls stopped moving. The sound of McAdams' feet scuffed away then thumped up the stairs. Then the basement door clunked shut. Then silence.
'Urghnngh...'
Die.
Going to.
Going to die.
_They'll worship you, you'll be a god._
Muffled voices came from up above. McAdams and someone else. Someone familiar.
Mother.
'Mmmmnnnfffrrr...!' He took a deep breath, but all that would come out was the same mushy grumble.
The tea. Should never have drunk the tea.
It took a lot, but he hauled his lead arm up and worked wet balloon fingers between his lips. Past his graveyard teeth. Across his slimy tongue. And into his throat. They felt big as—
'Hurrrrkkk...' Pale beige liquid splashed down his wrist, hot and sticky.
Barely a mouthful.
Again.
He dug his fingers in deeper.
'Hurrrrrrrrkkk...' This time his whole stomach heaved, curling his shoulders, hunching his back as tea and kippers splashed out onto the basement floor. 'Hurrrrkkk...' Chewed ribbony chunks of Pot Noodle. 'Hurrrrrrrrrrkkk...' More. Every heave like a kick in the stomach, leaving him hanging on the end of his chain. Spitting bile onto the concrete.
Callum slumped back against the wall.
Blinking.
The bricks started to pulse again.
In and out. In and out.
Mushrooms. It had to be mushrooms. And probably something else as well. Something to make him drowsy. Something to make him weak.
Probably still got enough of the crap in his system to keep a whole squat full of druggies tripping for a week. But at least he'd got rid of the rest.
His whole mouth tasted like an old wheelie bin smelled, though.
He spat again.
Ground his eyes closed.
Pulled his head forwards, then smacked it back into the wall.
A dull ringing flooded through his skull.
Again.
Harder this time.
The gauze padding softened the blow a bit.
Again.
Fire ripped across his scalp, followed by a thousand razorblades. Slicing through the stitches back there. Wrenching his eyes open.
Oh, dear _Jesus_ , that hurt.
But it worked.
He shook his aching head and the walls stayed where they were.
Coughed.
Spat out another string of bitter yellow bile.
Basement. He was in the basement.
The concrete floor rippled beneath him. Hissing.
McAdams had made a wall out of dusty cardboard boxes, blocking this part off from the rest of the room, creating an alcove no more than twelve foot by eight. Three sets of chains were fixed to the brickwork, all of them shiny and new. One for Callum, one dangling above a bare mattress on the floor, and one going around the neck of the woman slumped opposite.
Brown hair with grey at the roots. Naked, except for a grey cardigan that was threadbare at the elbows and cuffs. Knees covered with scratches and bruises.
She didn't say anything. Didn't move. Just slumped there.
Callum cleared his throat. Fumbled his slippery sick-covered fingers to the chain around his neck. Then worked his way back along its length to where it fastened to a ring, bolted to the wall. Gave it a pull.
Nope.
Key.
Get the key.
And how the hell was he meant to do that, chained to the basement wall?
Upstairs, the muffled voices went quiet, as if someone had turned off the radio. Then a solid _thump_ rattled the floor above.
Oh God.
He wiped his fingers on his once-clean, once-dry, shirt.
There was a creak, followed by the sound of something being dragged down the stairs. Thunk. Thunk. Thunk.
McAdams appeared around the boxes, shuffling backwards, bent double, pulling Mother by the armpits. She was limp, on her back, one shoe dangling off as he hauled her over to the mattress.
He struggled her into place, then sat back on his heels, wiping his forehead. 'I'm sorry, Mother, but you couldn't have picked a worse time.'
The chain clattered and rattled as he wrapped it around her neck, fastening it with another heavy padlock and slipping the keys into his pocket.
'But it's OK. It'll all be OK. You'll see.' He leaned forwards and kissed her on the forehead. 'It'll be better than OK.'
Callum slumped against the wall. Groaned.
McAdams turned. 'I know this isn't what we planned, Callum, but we'll make it work. And _now_ we've got a New Mummy to look after us and love us and keep us safe. Isn't that great?' He nodded. 'But, you need more tea, don't you? Right.'
Another groan.
Wrinkles deepened between his eyebrows. 'Have you been sick?'
Callum's head lolled to the right, making the walls pulsate again.
'Oh dear. Never mind. It's OK. It's not your fault.' McAdams stood and walked over. Hunkered down right beside him. Took out a handkerchief and wiped Callum's mouth. 'It can be a bit harsh at first, but the tea's good for you. It purifies your body and your mind. It'll make you ready for divinity.' He brushed the damp hair from Callum's forehead. 'They'll worship you, you'll be a god. You'll be a— Gllllk!'
Callum's good hand slammed into McAdams' crotch. Took a firm grip and _squeezed_ , twisting the contents of McAdams' pants like he was opening a jar of pickled onions.
There was a wet, strangled, scream, then McAdams collapsed to the floor and coiled into a little trembling ball, sobbing while Callum squeezed and twisted.
The words were lumpen and saggy, but Callum forced them out anyway. 'Say hello to... _The Claw_!'
##
Mother was breathing, but there was already a lump growing on the back of her head.
Callum unlocked the padlock from the chain around her neck. 'Boss? Can you hear me?'
He gave her a small slap on the cheek, but it didn't help.
She was still breathing though, that was the important thing.
The lady in the grey cardigan flinched when Callum touched her. So definitely still alive.
'It's all right, I'm a police officer. See?' He pulled out his warrant card. 'Can you tell me your name?'
Her eyes were wide as a rabbit's, waiting for the oncoming car.
'Hello?'
Nope.
Callum unlocked her too, but she scuttered away, squeezing herself into the corner of the room, arms wrapped around her knees. Staring out at him from beneath her greying fringe, with those glittering rabbit eyes. Probably off her face on McAdams' magic mushroom tea.
And that just left the man himself. Still curled up on the basement floor, wrapped around his tortured groin.
Good.
Callum hauled both of McAdams' wrists behind his back and slapped on the cuffs. 'Andrew McAdams, I'm detaining you under Section Fourteen of the—'
'Don't you want to be a god?' Face flushed and shiny. Tears and snot shining on his cheeks. 'Why? What's wrong with you?'
'You're going to prison, _Andy_. It's over.'
He stared. 'It's not too late, I can fix this. I'll get you some more tea. You can be a god, Callum. You can watch over us all. You can fix _everything_ that's wrong!'
'You're a psycho nutbag, and you're going to spend every last miserable day you've got left being someone's bitch in a six-by-six cell.'
And at that, McAdams threw his head back and laughed. Not a fake laugh either, a full-on belly laugh. 'I'm dying of cancer, you idiot. I've got weeks to live. You think they'll lock me up? I won't even stand trial and you know it.'
'I'm detaining you under Section Fourteen of the Criminal Justice, Scotland, Act—'
'You still don't get it, do you? All you've done is stopped me saving everybody's lives. We could have fixed everything. And you ruined it!'
'—because I believe you to have committed an offence punishable by imprisonment—'
'But you know what? It doesn't matter if I die, now. No one ever remembers the cops or the victims, they only remember the serial killer. I get to live _forever_.'
Callum licked his lips. 'You get to die in prison.'
'They'll write books about me – _Sunday Times_ bestsellers. They'll make films about me. Maybe even a TV series. They'll talk about me in hushed whispers when they tuck their children into bed.' A grin. 'Andrew McAdams. Imhotep. Immortal.'
The walls throbbed in time with Callum's heartbeat.
Kill him.
It was the only way.
Arrest him and he'd barely see the inside of a courtroom.
Kill him.
Make it look like suicide – just like McAdams had done with Finn Noble – so everyone would think he just couldn't cope with the cancer. Then get rid of anything incriminating in the house, so no one would ever find out that he murdered all those people. Don't let him take _anything_ from this.
Don't let him win.
McAdams grinned up at him, eyes like polished buttons. 'They'll worship me, I'll be a god!'
'No.' Callum tightened his good hand into a fist. 'No, you won't.'
##
'Ow! Get off me.' Mother slapped the paramedic's hands away. Tried to stand. Got pushed back onto the wheelie-stretcher thing that took up nearly one whole side of the ambulance.
'You've got concussion. Do you _want_ to die? Doesn't bother me: I get paid either way.' The paramedic held out the disinfectant and the wad of gauze again. 'Now, are you going to sit still, or not?'
The ambulance's blue-and-whites spun in the downpour. Throw in the three patrol cars, all with their lights going, and the crime scene was transformed into a very damp, very miserable disco.
Callum climbed up into the back, joining them. 'That's Cecelia and her Smurfs of Doom just arrived.'
Mother nodded. 'Tell them I want— Ow!'
'Well sit still, then. And you're going to need stitches.'
'I don't _need_ stitches.'
'God's sake.' The paramedic handed his gauze and disinfectant to Callum. 'Try to talk some sense into her.' Then he hopped down from the back and walked off under the eaves of the house. Took out a packet of cigarettes and lit up.
Callum peered at the back of Mother's head. 'Yeah... you're going to need stitches.'
'I can't believe Andy hit me. He _hit_ me!' She looked away. 'How's Beth?'
'They've taken her straight to A-and-E. No idea how long he's had her chained up in the basement, but she's gone full-blown psychotic episode.'
'His own _wife_. I was at their silver wedding anniversary.' Mother let her head fall back, then winced and straightened up again. 'Ow!'
'Look on the bright side: it could've been a lot worse. Remember what he did to John?'
'Urgh...' She wiped a hand across her face. 'Andy was my friend, Callum. I've known him for years. I don't even know who he is, now.'
'We're all still alive. He can't hurt anyone else. We've escaped our chains.'
She looked at him, then smiled. 'No more sodding haikus.'
'Deal.' He put a hand on her shoulder and she took it. Gave it a squeeze. Nodded.
'You did good, Callum. You did really, really good.'
'Thanks, Boss.'
'I think, in the circumstances, you can call me "Mother".'
The scent of deep-fried sausages filled the Downstream Monitoring Suite, underpinned by the sharp-sweet tang of cheap tomato sauce. Dr McDonald ripped another bite from her buttie and leaned closer to the screen, chewing as she stared.
Mother folded her arms and slumped back in her seat.
Callum licked tomato sauce and melted butter off the back of his hand. 'Sure you don't want one?'
'Surprisingly enough, I'm not hungry.'
On the screen, Interview Two was full – McAdams and his solicitor on one side of the table, facing the camera; Detective Superintendent Ness and a short fat DS on the other.
_'No comment.'_
Ness sighed. _'Andrew, you know how this works. You've been here often enough.'_
_'No comment.'_
McAdams' solicitor looked as if she was auditioning for a Tim Burton movie. She tucked a strand of jet-black hair behind her ear. _'I believe my client has made himself very clear on this point, Detective Superintendent. He's quite happy to answer any questions, but only to DI Malcolmson.'_
McAdams stared directly into the camera. Smiled. _'And Callum, of course. Only fair to include him. He deserves that much.'_
Dr McDonald took another bite. 'He doesn't even look bothered, does he? It's like he's there to pick up a Chinese takeaway.'
_'You know we can't do that, Andrew. You assaulted them both. It's a conflict of interest.'_
'It's a shame he's dying of cancer, isn't it?' Dr McDonald got even closer, till her nose was only inches from the screen. 'I mean it'd take years to unravel what's going on inside his head, and that would be _very_ interesting wouldn't it and I'd love to have a go, but I don't think they'd let me, do you think they'd let me, or does that sound a bit creepy because of the serial killing and dying thing?'
Mother stared at her. 'Yes.'
_'My client is prepared to make a full confession, but only to DI Malcolmson.'_
_'And have the whole thing ruled inadmissible in court? We'll pass, thanks.'_
McAdams stuck a hand against his chest, as if he was about to pledge allegiance _'Then, "no comment" is the only thing, / That I will mumble, say, or sing, / "No comment" now, "no comment" then, / "No comment" time and time again.'_ Then the smile slipped, as a coughing fit sent him rocking back and forward in his seat, bent over, head nearly touching the tabletop. Leaving him panting and wheezing.
The solicitor patted him on the back. _'I_ insist _you get my client medical attention, right now!'_
Mother groaned. 'I know this is going to sound harsh, but I don't think I like him very much any more.'
## [— dearly departed, —
we are gathered here today](../Text/contents.html#part012)
And when the Bonemonger raised his arms, the earth gave a great rumble as one by one the graves collapsed. Then, from the dark depths below, each and every coffin rose to the above, steaming in the cold morning air.
"Please, don't do it! I've changed my mind!" screamed Justin. "Stop!"
"It's too late," laughed the Bonemonger, "see what we have done!"
He clapped his bony hands and the lids flew off, revealing the dead in all their mouldy finery. They yawned and stretched and sat up in their satin-lined boxes. Then climbed out into the last morning there would ever be. For he'd opened the coffins and let them go free.
R.M. Travis
_Open the Coffins (and Let Them Go Free)_ (1976)
_You better believe I'm-a keepin' it real,_
_Cos there ain't no reprieve when The Man makes you kneel,_
_And I know you all grieve, but I'm gonna appeal,_
_Got some tricks up ma sleeve an' my will's made of steel._
Donny '$ick Dawg' McRoberts
'The Day Them F*ckers Done Fitted Me Up'
© Bob's Speed Trap Records (2017)
##
'I'm sorry for your loss.' Callum shook McAdams' wife's hand.
Beth had put on a little weight in the last week and a bit, but it'd probably be a long time until she lost that chained-up-in-the-basement hollow-eyed look. She stared back at him. 'He was a bastard, and I'm glad he's dead.'
'Yeah.' Callum nodded. 'Me too.' Then moved on to join what was left of the Misfit Mob. Everyone dressed in various shades of black.
Dotty looked up at the bright blue sky. 'Well, at least he got a nice day for it.'
Franklin shuddered. 'I hate crematoriums. Always give me the creeps.'
'Mother? You OK?'
'Hmmm?' She turned, blinking. 'Sorry, miles away. It was a lovely service wasn't it?'
'I thought your eulogy was very good.'
She patted Callum on the arm. 'I know it sounds odd, but I miss him.'
'He tried to kill me. He very nearly killed Watt. He murdered _at least_ six people. And he was going to keep you chained up in his dungeon with his drugged wife. Probably thought he could take turns.'
'When you put it like that...' Mother puffed out her cheeks. Clapped her hands together. 'Right, we'd better get back to work, I suppose.'
Rain drummed against the office window, a counterpoint to the buzzing rattle coming from the radiator.
Callum sat back in his seat, both feet propped up on his desk. Finished the page he was reading and moved on to the next one.
McAdams' book was... different. Not bad, exactly, but a bit long-winded and self-indulgent. Not to mention self-important.
Franklin backed into the room, laden down with box files. 'Working hard are we?'
'Yes.' He nodded towards the pile of paper, sitting on its opened-out brown paper wrapping. 'McAdams believed in the old adage: "write what you know". It's all about Imhotep's quest to save the world, told from the perspective of a little boy called "Justin". Abduction, physical abuse, trauma, desperate need for affection he tries to fulfil with the women his "Father" keeps chained up in the basement.'
She dumped the files on her desk. 'You coming tonight?'
'We don't need to worry about catching Paul Jeffries' killer – it was McAdams. Stabbed him for killing the last in a long line of New Mummies. Then Justin goes into care. More trauma and abuse. But he finally gets himself a nice foster family, changes his name, and everything starts to go right for him. Good exam results. University. Career in the police force. Sense of belonging and self-worth.'
'I mean, it was weird enough going to his funeral, but a wake too?'
'Then he's diagnosed with cancer and it all comes flooding back. That's when the killing starts.' A smile. 'And he would have hated that much summary narrative.'
'I've never been to a celebration wake before. Don't get me wrong, I'm all for celebrating someone's life when they've died, but actually celebrating the _fact_ they've died?'
Callum pointed at the remaining stack of paper. 'Haven't got to the end yet, but five quid says there's nothing in there about getting his balls crushed by The Claw, then being arrested, charged, remanded without bail, or an epilogue where he gets stabbed twenty-three times in the throat by the very disturbed young man from the cell next door.'
'Thought it was nine times?'
'Artistic licence.' Callum marked his spot on the page. 'And, apparently, Tod Monaghan was nothing to do with it.'
Her eyes went wide. 'You're _kidding_ me.'
'Nope. According to chapter forty-two, McAdams just took advantage of the fact Monaghan was dodgy, dead, and in no position to complain about mummified human remains being planted in his living room. Turns out "Justin" sacrificed a god to make it look like Imhotep had drowned himself in Kings River, so we'd stop looking.'
'What about Finn Noble?'
'Haven't got to that bit yet.'
'Wouldn't it be lovely if every murdering wee scumbag wrote their confession out as a novel?' She picked up a sheet of paper and read aloud: '"There's a jackdaw hanging on the fence behind the house. Like a little black kite, caught on its own strings." That's cheery.' The page went back on its pile. 'So, are you coming tonight, or not?'
Callum shrugged. 'Don't see why not. I might even have a bash at karaoke.'
'Just seems... _weird_ , doesn't it? Mind you, I suppose the signs were there – no sane person spouts haikus and bits of doggerel all the time.'
'Hmm.'
'And that thing he did, where he pretended he was doing a literary critique of his own life? Not right in the head.'
'Looks as if McAdams was right, though: he didn't make it to the end of the book after all.'
'Still, I suppose...' She frowned, then pointed at Callum's warbling phone. 'Are you going to get that?'
Might as well. He picked it up. 'Hello?'
A small pause. _'Piggy?'_
Not _again_. 'Willow, we've talked about this. It's "Uncle Callum", not "Piggy".'
Silence.
Franklin raised her eyebrows, then made a letter T out of her two index fingers and pointed at the kettle.
He gave her a thumbs up – not easy with his right hand still in the cast, but she seemed to get the idea. 'Willow, are you there?'
_'Piggy, I need... Mum needs you to come over.'_
'I could probably pop past later on, after work? We could maybe get a pizza or something?'
_'No, Piggy: got to be_ now _, yeah? It's urgent, like.'_
He sat up. 'What happened: is it your dad? Is he hitting her? I can get a police car there in five minutes.'
_'No! Nah, no. It's not Dad. That prick comes here, me and Benny gonna kick his little bitch arse for him. Gonna kick it good.'_ But the bravado didn't sound as brash as it usually did. It sounded brittle. Shaky.
'Are you OK?'
_'Just do it, yeah?'_ A muffled voice in the background was too low to make out. _'Gotta go.'_ And she hung up.
Callum tapped his pen against the desk.
Franklin waved at him. 'We're out of teabags. You want coffee instead?'
He stood. 'Thanks, but I've got to go out for a bit.' Then struggled into his jacket. 'Don't fancy a hurl, do you?'
'Can you see the pile of work I've got on my desk? Every single sodding case McAdams ever worked on, and I have to review the lot.'
'Come on, it'll only take twenty minutes. Thirty tops.'
'Another one of your "wee errands", is it?'
'The woman my brother beat up wants to see me. Apparently it's urgent.'
Franklin smiled. 'Maybe she wants her teddy bear back?'
That was a point. And it wasn't as if they needed it now.
'Why not?'
'So I was thinking, maybe curry?'
Callum shrugged, taking the Mondeo around the Calderwell Roundabout and into Kingsmeath. 'Bit unusual for a wake, but yeah. Sounds good. Get a bit of ballast in us before we hit the Bart.'
Sunlight glittered back from the satellite dishes and double glazing that graced the houses lining the road. Not a cloud in the sky. Even the river was settling back to its normal sedate sludgy grey.
Franklin nodded. 'I know Dotty's getting in some sausage rolls and wee scotch eggs, but I always end up feeling cheated if I don't get a proper dinner.'
'Can't go wrong with fish and chips.'
'True.'
And it'd be quicker than going out for a curry, so there'd be plenty of time to finish reading McAdams' book before heading out to celebrate his death. Would be good to get it finished. Move on to something else. Something a bit less... deranged and _murdery_.
Wonder if anyone would publish it?
Might be an idea to see where they stood, legally, with that one. Suppose the rights would belong to McAdams' wife.
Left, up Munro Place. Over the top of the hill and down the other side.
Maybe they could publish it and all the proceeds could go to the victims' families? Or would that just be fulfilling McAdams' little fantasies of immortality and success?
Callum slowed for the turning onto Manson Avenue. What if they—
Franklin hit him on the arm.
'Ow! What was that for?'
'You haven't been listening to a word I've said, have you?'
'Curry then karaoke.'
She rolled her eyes. 'That was five minutes ago. I was talking about Mark wanting to get back together again. Maybe take a trip to Thailand, or New Zealand.'
'Thought he was "an entitled dick who didn't respect you as a human being"? And I'm quoting there.'
'I don't know why I bother telling you anything.'
'You were the one who said it.' Callum pulled up outside number 45. The block looked pretty much as horrible as it had the last two times, only now the weeds crowding the front garden were bigger. About time Irene cashed her dirty big cheque from Alastair and traded up for somewhere better.
'I've always wanted to go to New Zealand.'
'Mark's a dick. You know it. I know it.' Callum climbed out of the car, then reached into the back for the teddy bear lying on the passenger shelf. ' _Everyone_ knows it.'
She caught up to him halfway along the path. 'The most exotic holiday we ever had was a three-day trip to Belgium. And that was only because he had to go to a conference.'
'So, what: you're going to take him back, shag him, laugh at his horrible jokes, all because he's going to take you to New Zealand? You do _know_ what they call women who only sleep with men to get things, don't you?' Callum stuck his thumb on the doorbell, setting it ringing deep inside the house. 'Because it isn't nice.'
'Oh go bite your own backside.'
'At best, you'd be a gold digger, and at worst—'
The door opened, and there was Benny, in his blue tracksuit and backwards baseball cap. His eyes were small and pink, ringed with red. Cheeks shiny. A glittering trickle of snot on his top lip catching the sunlight. He held up his arm. 'Uncle Callum!'
'Benny, are you OK?'
He looked up at them, bottom lip wobbling, then turned and ran back inside.
Franklin raised an eyebrow.
'He's a bit... odd.' Callum stepped over the threshold. Raised his voice. 'Miss Brown? Irene? Hello?'
'All kids are odd. They don't stop being odd till they hit their twenties, and even then it's—' Franklin's phone launched into 'Dancing in the Moonlight'. She sighed. Dug it out. 'Mark? Is it important, because I'm in the middle of something... I'm not being "like" anything... No, I'm...' She rolled her eyes at Callum. 'Yes. But you remember what I do for a living, don't you?'
Callum pointed into the house and pulled a face.
She nodded and turned back to the path. 'There's no need to get all defensive about it, Mark. If we're going to give this another chance, you need to respect my boundaries... Yes... I didn't _say_ that.'
He left her to it.
Down the hall. 'Irene? You there?' He held up the bear. 'I've got Mr Lumpylump.' Into the living room.
And there she was. Irene Brown. On her knees in front of the room's saggy armchair, one hand covering her mouth, blood dripping between her fingers. The other hand scrabbled at the carpet for what looked like teeth.
What the hell was—
The door battered into him, hard enough to send Callum thumping sideways against the wall, then tumbling to the floor. Sending a plastic tidal wave of kid's toys clattering across the carpet.
Then a boot connected with his ribs, hurling him into the wall again. Slamming all the air from his lungs and setting fire to his whole side. Another boot caught him on the forehead, snapping his head back. Drowning out all sound with a deafening booming ringing noise.
Then a hand grabbed the front of his shirt and hauled him up. 'Think you're clever, don't you? Think you got away.' Ainsley Dugdale grinned at him, the bruises on his butcher's-slab face fading away to a dirty smear. He folded his other hand into a fist. 'See when I take money for a job? I do the job. And you _don't_ get away.'
The world snapped around ninety degrees and everything tasted of hot copper wire.
Callum raised an arm, but the fist battered his face again making something inside his cheek go _snap_. The carpet rushed up and crashed into his chest. 'Ungggghh...'
'Think you're the only one can do detecting?' The boot cracked into his ribs again. 'I open my newspaper and what do I see? This bitch whining on about how her man don't treat her right, even though she's a total whore.' Another kick flipped Callum over onto his back.
Every breath was like running a cheese grater over his lungs.
He coughed and warm red spattered back down onto his cheeks.
'So I thinks to myself, "Oho, this bitch is friends with Detective Constable Callum MacGregor, is she? Let's see if we can't arrange a wee surprise party for him.' This time the boot stamped down on his chest. Once. Twice. Then one cracked his head off the skirting board. 'You enjoying your party? Want to blow out your candles?'
The room lurched, then settled down into a slow, dark throb.
Strange, the carpet looked all cheap and threadbare from a distance, but lying here it was soft and warm and comfortable. Like the floor of R.M. Travis's garage. Or McAdams' basement.
Didn't even hurt all that much any more.
All his arms and legs. Like rubbery lead. Heavy and warm.
Could just go to sleep, right here on the floor.
Dugdale's face was all teeth, shiny and brown, shown off in a grin as he pulled out a lighter and scritched his thumb across the wheel. Sparks. More sparks. Then a big yellow flame. 'What do you think: start with your eyes or your ears? Yeah, let's burn a—'
'Get away from Uncle Callum!' Benny jumped on Dugdale. Eyes wide, tears on his cheeks, snot glistening from his nose. Teeth bared. 'GRRRRRRRRR!'
'What the... Get off me you little freak! You're in for—'
He sank his teeth into Dugdale's arm, shaking his head like a terrier. Blood trickling down his chin.
'AAAAAAAAAAAAAAAAAAAAARGH!'
'Benny!' Willow ran for him, grabbed her brother's arm and pulled. But Benny just bit deeper. So she let go and ran away.
Not so big and tough after all.
'AAAAAAAAAARGH!' A chunk of flesh came free, big as a chicken nugget, the hole it left behind: red and purple, fringed with yellow. Dugdale stared at it for a moment, mouth hanging open. Then he backhanded Benny, sending the little boy flying into the back of the armchair. 'YOU LITTLE SHIT!'
Benny bounced, eyes wide, blood all over his mouth and chin. _Grinning_. Snapping his teeth together. Then lunging again. Sinking his teeth into Dugdale's leg, right at the bulge of his calf.
'AAAAAARGH! KILL YOU!' Dugdale grabbed him by the throat, ripped Benny off his leg and held him up, like a mascot. Then slammed him into the wall hard enough to crack the plasterboard. 'YOU'RE DEAD, YOU LITTLE FREAK!'
'LEAVE MY BROTHER ALONE!' Willow was back, clutching a dirty big kitchen knife, the blade gleaming as she swung it at him.
It never made contact. Dugdale punched her, hard in the face, sending her spinning away to smash into the playpen. Tiny drops of blood glittered in the air, marking her path. The knife thunked blade-down into the carpet.
Dugdale bared his teeth. 'The whole bloody lot of you are going to pay for this. Understand?' He buried another boot into Callum's stomach. 'You're all going to...' His eyes bulged and he looked over his shoulder.
Irene Brown stood right behind him, hunched forwards. She staggered back a couple of steps and raised her hands to cover her mouth. As if trying to hide the horrified expression on her face.
And Dugdale turned to face her. Unsteady on his legs. Shaking.
The handle of the kitchen knife stuck out of his back, just above his belt. Blood darkened his trousers. 'No...'
Callum blinked.
The world pulsed in and out. Warm and inviting as a length of basement chain.
Difficult to keep his eyes open.
When did his eyelids get so heavy?
Dugdale collapsed to his knees, both hands fluttering at the small of his back, fingertips brushing the knife's handle. 'Oh God, oh God, oh God...'
Then the living room door burst open and Franklin charged in, flicking her extendable baton out with a hard sharp _clack_. 'NOBODY MOVE!'
Better late than never.
Callum let the warmth and darkness fold over him like a duvet.
##
'You ready?'
'Yeah.' Callum winced and levered himself out of the big blue hospital armchair. The little private room was far too sodding hot, even with the window cracked as far open as it would go. A handful of get-well-soon cards were pinned up on the corkboard above the telly, the remains of a clichéd bag of grapes going raisiny on the bedside table.
He stood there for a moment. Straightened up. Wobbled.
Mother raised an eyebrow. 'Are you sure? Because I can call a doctor.'
'Four days stuck in here with daytime TV and lukewarm Lucozade is quite enough, thank you.'
'Callum—'
'I'm _fine_. What's a few cracked ribs, fractured cheekbone, and bruised spleen between friends?' He struggled his way into his jacket. 'And there wasn't any blood in my pee this morning. How cool is that?'
McAdams' book was back in its brown paper wrapper. Callum slipped it into his backpack and fastened the catches.
'Did you finish Andy's book?'
'Yup. Want to know what happens in the end?'
She shook her head. 'Certainly not, we don't approve of spoilers.'
'OK. Well, in that case—'
A knock at the door and Franklin stuck her head in. 'Is he still malingering?'
'I am _not_ malingering.'
Mother crossed to the bed and picked up his backpack. Paused to peer out of the window. 'Have you seen the crowd down there? That's an awful lot of people for a Wednesday afternoon.'
Callum grinned. 'Maybe they're here to celebrate me getting out of hospital?'
'Dream on.' Franklin pointed at the door. 'Can we go now? Some of us have work to get back to.'
He limped out into the corridor, following her and Mother out of the ward and along to the lifts. Wheezing like a leaky balloon.
Mother pressed the button and the doors slid open. 'Dotty sends her love, by the way. She'd be here, but they're testing her for her disabled parking badge. You'd think missing half a leg would make her a shoo in, but who are we to question Oldcastle City Council's mysterious ways?'
They shuffled inside.
The doors clunked shut.
Down they went.
Franklin stared up at the numbers. 'Dugdale's regained consciousness, by the way. They've had to remove a big chunk of his bowel, liver, upper and lower intestines, but other than that he's dandy. He'll poop in a bag for the rest of his life, but swings and roundabouts.'
'They pressing charges?'
'What, against Irene Brown? No. She was saving the life of her children and a police officer. They've nominated her for a Queen's Commendation for Bravery. Dugdale, on the other hand, is facing two charges of aggravated assault, two of assaulting children, and the PF's doing him for the attempted murder of DCI Powel too. Turned out Dugdale was shooting his mouth off all over Oldcastle about it.'
Mother sighed. 'Poor wee soul never was the shiniest spoon in the drawer.'
'Hmph.' At least that meant Callum was off the hook. 'Do me a favour: when he attacked me, Dugdale was going on about how he'd been paid to do a job and I wasn't going to get away. Get someone to find out what the hell he was talking about.'
'I can try. But don't get your hopes up. He's not a shiny spoon, but he's sharp.'
The lift gave a shudder, a grinding noise, a ping. Then the door slid open on the hospital's reception area.
Franklin pointed. 'We're parked over in the red zone. Want me to drop you at Dotty's?'
'Any chance we can go via the library?'
'And how long's _that_ going to take?'
'Fifteen minutes. Twenty tops.' Callum limped through the reception area towards the long line of glass doors, when someone stepped in from the overcast afternoon. Alastair.
The murmurs of that huge crowd outside buzzed through the open door, then clunked silent as it closed again.
Alastair / Donny / Donald / $ick Dawg swaggering over with a dirty big grin on his face and a hand out for shaking. 'Bruv!' He'd ditched the shaved-chest-and-leather-jacket look for a simple white shirt and blue jeans. A pair of expensive-looking sunglasses perched on the top of his head. Facial hair neatly trimmed.
'Erm...'
Alastair grabbed his hand. Shook it. Stepped back. Beamed. Then wrapped Callum up into a hug. Lifting him off the ground. Making broken nails burst across his ribs. 'Ha-ha!'
'Arrrgh!'
'Oh, right, yeah. Sorry, Bruv.' A squeeze on the shoulder instead. 'Callum, this is Courtney. Courtney, this is my bruv.'
A woman in a sharp suit appeared – thin, long hair swept back, young and perky looking as she looked Callum up and down, talking in a low Estuary accent. 'I don't like the bruises, makes him look like a boxer or something, but I suppose I can work with it.'
'Courtney's my publicist.'
'Right...'
'OK.' She clapped her hands. 'When we go out there, Sick Dawg's going to make a statement to the media. Then I need you to talk for no more than two minutes: we want to get this on the evening news, so keep it snappy but accessible. How great it is to finally meet your brother, how you're looking forward to getting to know each other again, you've always been a fan of his music, etcetera.'
'But—'
'Great. Don't forget: no more than two minutes. I'll give you a copy of the itinerary once we've done the broadsheet interviews.'
'Itinerary?'
' _The One Show_ , _Breakfast News_ , _Lorraine_. I'm waiting for a call back from _The Graham Norton Show_ , but fingers crossed.'
Callum stared at her. Then at Alastair. Then at Mother. 'But—'
'Excellent.' Courtney checked her watch. 'I've got a table booked at La Poule Française for eight tonight. We can go over any questions you've got then.'
'But—'
'I know, right?' Alastair wrapped an arm around his shoulder. 'Bit of a shock, innit, first time in the bubble? But trust me, Bruv, you is gonna be a natural, right?' He turned a smile on Franklin and Mother. 'Can you give me and my bruv a couple of minutes, yeah? You know, to talk, like?'
Franklin bristled, but Mother put a hand on her arm. Nodded. 'Of course. We'll be over there, raiding the chocolate machine.'
'Cool.' Alastair snapped his fingers then pointed at it. 'Courtney, treat these fine ladies to whatever they like.'
Last of the big spenders.
And as soon as they were gone, Alastair thumped down into one of the waiting room seats, legs splayed out as if they were barely connected, and grinned at Callum. 'Man, we is gonna be on the _front pages_ tomorrow. Can't buy this kinda publicity.'
Callum half sat, half collapsed into the seat opposite. Licked his lips. 'Umm...'
'You always this quiet, Bruv?'
' _Breakfast News_?'
'Yeah, better get used to it. My agent called: we got a bidding war going on for our autobiography, like _big time_. And don't worry: they gonna get some ghost writer guy to do the words for us, we just gotta cash the royalty cheques and sign some books. Already got offers coming in for the film rights.'
'How's that going to work if you're in prison?'
'Nah, Bruv. Got sentenced to two weeks at a five-star rehab retreat. Perks of being a celebrity with a very expensive lawyer.'
Typical.
Callum cleared his throat. 'I want to go after Leo McVey. For what he did to Mum and Dad. For what he did to you.'
Silence.
A nurse squeaked by with a clipboard and a frown.
Someone in the distance coughed.
The machine whirred and clunked as Mother, Franklin, and Courtney threw caution to the wind.
Alastair frowned. 'I remember.' The patois had disappeared, leaving the hard Kingsmeath burr behind. 'Not everything. But some bits. I remember the cage Travis kept me in. I remember the dinner parties, where he'd wheel me out.'
'Dinner parties?'
'Man, you think he kept all those bits of people in the freezer because he was _lonely_? You've read his books.'
_And that's something else he's obsessed with: witches eating children. Goblins eating rabbits. Monsters eating children. People eating rabbits that are actually children. It's a smorgasbord of transspecies consumption, posing as anthropomorphic cannibalism, but it's really about venal desire. Consume the flesh, violate the body, and absorb it into your own._
'Oh Jesus...'
'In the end, it was Uncle Leo got me out of there. Yeah, there was stuff went on before that, but he's the one changed my name. Got me into a care home. Hid me away so Travis couldn't find me.'
'He abducted you in the first place! He helped Travis kill Mum and Dad.'
Alastair picked at the stitching on his jeans. 'Yeah.'
'If we're doing an autobiography, it'll all come out anyway.'
'Yeah.' A small smile. 'Not easy being a superstar. Wearing all these masks, so nobody sees the real you.'
Callum sat forward in his seat. 'Are you going to help me?'
Silence.
Alastair cleared his throat. Kept digging at the stitching. 'I need to tell you something. Ainsley Dugdale: I paid him.'
'You _paid_ him?'
A shrug. 'Yeah, but it was just meant to be a thing, you know? I tell Dugdale to hang around Irene's gaff, I phone in an anonymous tip so you know where he is, then when you turn up and he batters you, I wade in and save the day. Willow and Benny see me, and they're all like, "Whoa, our dad's a hero!" You're like, "My bruv saved my life!"'
Callum stared at him. 'The big black Mercedes.'
'Only I got too stoned and then you got crushed in the balls and I was going to jump in, I really was, but you pepper sprayed him and he fell over and Willow kicked the crap out of him and I was laughing so hard I nearly peed myself.'
Lovely.
At least it explained the whole weird coincidence thing.
'And then, when I heard about that Powel guy, I thought, "No way this scumbag's getting away with shagging my brother's woman. No way." So I got Dugdale to pay _him_ a visit too.'
Callum folded forward and covered his face with his hands. 'Do you have any idea how much trouble that caused me?'
'Yeah, sorry, Bruv. It's the thought that counts, though. Right?'
No, it wasn't.
The silence stretched.
Alastair sniffed. 'You need to know: if you go sniffing around Leo McVey with your warrant card and your police mates, he's going to lawyer up like _that_.' Alastair snapped his fingers. 'Might have to do it... a bit dirty, yeah? You ready for that?'
'You're not talking about fitting him up, are you?'
'No, I'm talking about _justice_ for all them people they killed. I'm talking about _revenge_ for Mum and Dad. I'm talking about pliers and a blowtorch in an abandoned building. I'm talking a shallow grave out in Moncuir Wood where no one's ever going to find him.'
'You want to...' Callum snuck a quick glance around: the nearest person was the receptionist, on her phone. Lowered his voice to a hard whisper: 'You want to murder Leo McVey? No. Not happening. No chance.'
'Hmmm.' A nod. 'OK, so we can't kill him. Courtney said you probably wouldn't go for that.'
'You discussed murdering Leo McVey with your _publicist_? Are you insane?'
'Cos she came up with a better idea: Plan B. We get ourselves a film crew and we go after Leo McVey and that scumbag R.M. Travis on our very own reality TV show. Investigate the background, dig up other victims, all that stuff you cops do, yeah? Play our cards right, we get two seasons out of it.'
'You _are_ insane.'
'Yeah, like a fox.' Alastair stood. 'Right, we got us a press conference to rock. You ready?' He held out his hand.
Callum swallowed. Levered himself out of his seat.
Followed his brother out into an overcast afternoon and a barrage of flash photography.
Oh God...
### If you enjoyed _A Dark so Deadly_ , try the latest novel in the Logan McRae series!
You can click here to buy your copy
### About the Author
Stuart MacBride is the _Sunday Times_ No.1 bestselling author of the Logan McRae and Ash Henderson novels. His work has won several prizes and in 2015 he was awarded an honorary doctorate by Dundee University.
Stuart lives in the north-east of Scotland with his wife Fiona, cats Grendel, Onion and Beetroot, and other assorted animals.
For more information visit StuartMacBride.com
Facebook.com/stuartmacbridebooks
@stuartmacbride
### By Stuart MacBride
The Logan McRae Novels
_Cold Granite_
_Dying Light_
_Broken Skin_
_Flesh House_
_Blind Eye_
_Dark Blood_
_Shatter the Bones_
_Close to the Bone_
_22 Dead Little Bodies_
_The Missing and the Dead_
_In the Cold Dark Ground_
The Ash Henderson Novels
_Birthdays for the Dead_
_A Song for the Dying_
_A Dark so Deadly_
Other Works
_Sawbones (a novella)
12 Days of Winter (short stories)
Partners in Crime (Two Logan and Steel short stories)
The 45% Hangover (a Logan and Steel novella)
The Completely Wholesome Adventures of Skeleton Bob (a picture book)_
Writing as Stuart B. MacBride
_Halfhead_
### About the Publisher
**Australia**
HarperCollins Publishers (Australia) Pty. Ltd.
Level 13, 201 Elizabeth Street
Sydney, NSW 2000, Australia
<http://www.harpercollins.com.au>
**Canada**
HarperCollins Canada
2 Bloor Street East – 20th Floor
Toronto, ON, M4W, 1A8, Canada
<http://www.harpercollins.ca>
**New Zealand**
HarperCollins Publishers (New Zealand) Limited
P.O. Box 1
Auckland, New Zealand
<http://www.harpercollins.co.nz>
**United Kingdom**
HarperCollins Publishers Ltd.
1 London Bridge Street
London, SE1 9GF
<http://www.harpercollins.co.uk>
**United States**
HarperCollins Publishers Inc.
195 Broadway
New York, NY 10007
<http://www.harpercollins.com>
|
{
"redpajama_set_name": "RedPajamaBook"
}
| 6,055
|
Developing Open Source Tools for Differential Privacy
OpenDP is a community effort to build trustworthy, open-source software tools for statistical analysis of sensitive private data. These tools, which we call OpenDP, will offer the rigorous protections of differential privacy for the individuals who may be represented in confidential data and statistically valid methods of analysis for researchers who study the data.
Do you have data to share or an application that can benefit from differential privacy? We can help.
Learn how to use OpenDP Tools
Whatever your background, there are ways in which you can contribute to the OpenDP effort.
The OpenDP team is continually growing! Currently we are accepting applications for our Scientific Staff and for the OpenDP Fellows Program. Please contact us if you are interested in joining us, even if we do not have a suitable job posting at the moment.
Please join our mailing list to get connected with the rest of the OpenDP Community, to stay abreast of our latest developments, and to find opportunities to contribute.
Introducing OpenDP Library v0.4
Dear OpenDP Community,
We're pleased to release version 0.4 of the OpenDP Library. Highlights include several new measurements and transformations, improved floating point handling, and general code cleanup. The full list of changes can be...
Read more about Introducing OpenDP Library v0.4
OpenDP Library Release
In anticipation for the Community Meeting starting Wednesday, September 22, we're pleased to release the initial public version of the...
Read more about OpenDP Library Release
Announcing the Launch of Our New Website
We are excited to announce the launch of the newly designed OpenDP website. We hope you find the website content informative and a valuable resource for differential privacy. Most importantly, we look forward to having you become a part of the OpenDP Community.
... Read more about Announcing the Launch of Our New Website
hp curatior.io feed
|
{
"redpajama_set_name": "RedPajamaCommonCrawl"
}
| 6,876
|
Q: why I get __init__.py problems when using selenium webdrivers chrome? im working on a project and have this as code:
import selenium
from selenium import webdriver
driver = webdriver.Chrome()
print(driver.current_url)
but i get an error and i dont know how to fix it, well mutle tracebacks to error i think, here is the error:
Traceback (most recent call last):
File "C:\Users\ytty\PycharmProjects\test hack\main.py", line 2, in <module>
from selenium import webdriver
File "C:\Users\ytty\PycharmProjects\test hack\venv\lib\site-packages\selenium\webdriver\__init__.py", line 18, in <module>
from .firefox.webdriver import WebDriver as Firefox # noqa
File "C:\Users\ytty\PycharmProjects\test hack\venv\lib\site-packages\selenium\webdriver\firefox\webdriver.py", line 36, in <module>
from .service import Service
File "C:\Users\ytty\PycharmProjects\test hack\venv\lib\site-packages\selenium\webdriver\firefox\service.py", line 21, in <module>
class Service(service.Service):
AttributeError: module 'selenium.webdriver.common.service' has no attribute 'Service'
Process finished with exit code 1
can anyone help?
A: I do it this way:
from selenium import webdriver
from webdriver_manager.chrome import ChromeDriverManager
options = webdriver.ChromeOptions()
options.add_experimental_option("excludeSwitches", ["enable-logging"])
driver = webdriver.Chrome(options=options, executable_path=ChromeDriverManager().install())
This way it will automatically download the optimal ChromeDriver version on your cache and use it every time you need. Also the experimental part helps to avoid some error that might appear due to driver bugs
|
{
"redpajama_set_name": "RedPajamaStackExchange"
}
| 971
|
\section{Introduction}
The Cabibbo-Kobayashi-Maskawa (CKM) matrix~\cite{Kobayashi:1973fv} gives the strength of the cross-generational weak couplings between the up and down type quarks, and is currently the only known source of charge-parity violation in the Standard Model (SM), which is required for understanding the observed matter--anti-matter asymmetry in the universe~\cite{Sakharov:1967dj}.
Measurements of the CKM matrix aim to overconstrain the matrix, testing the unitarity assumption.
Non-unitarity would indicate the existence of an additional, as yet unknown, coupling from Beyond the Standard Model (BSM) physics.
The CKM matrix element \ensuremath{V_{ts}} determines the relative strength of the $t$ quark's weak decay to the $s$ quark compared to other down-type quarks.
The magnitude of \ensuremath{|\vts|}\ determined through fits based on the unitarity of the CKM matrix is $39.78^{+0.82}_{-0.60} \times 10^{-3}$, and the indirect measurement through box diagram oscillations and rare decays involving loops is $38.8 \pm 1.1 \times 10^{-3}$~\cite{PDG}.
However, a recent reanalysis of Tevatron and 8~TeV LHC data has shown that after relaxing the unitarity constraints, \ensuremath{|\vts|}\ can be as large as 0.1~\cite{Clerbaux:2018vup}.
Additionally, a recent CMS analysis of 13~TeV data with the single top channel has given the constraint $\ensuremath{|\vts|} + |V_{td}| < 0.057$ at the 95\%~CL under the SM assumption of CKM unitarity, and $\ensuremath{|\vts|} + |V_{td}|=0.06 \pm 0.06$ after relaxing the unitarity constraints and allowing BSM contributions to the top width~\cite{Sirunyan:2020xoq}.
Further measurements are therefore required to constrain \ensuremath{|\vts|}\ in the most general scenario, and in particular, the decay of $t \to sW$ has not yet been observed.
A direct measurement of $\ensuremath{|\vts|}^2 = \frac{\mathcal{B} (t \to sW)}{\mathcal{B}(t \to qW)}$ at the LHC using the properties of strange hadrons to tag $s$ jets was proposed in~\cite{Ali_2010}.
That proposal made a generator level analysis to argue that the measurement would be feasible at the LHC, showing that under the assumption of perfect non-\ttbar background rejection and perfect top and hadron reconstruction, 10~fb$^{-1}$~is sufficient to observe the $t \to sW$ decay.
Now that the LHC has collected more than an order of magnitude more luminosity than the proposal considered, we extend that study.
In particular, we perform a full reconstruction analysis using the \textsc{Delphes}\ fast detector simulation package, to make a more realistic estimate of the data luminosity required to observe the decay, and analyze the difficulties that would arise in such a measurement.
With our more realistic simulation setup, we investigate the prospects of measuring \ensuremath{|\vts|}\ in several scenarios, including the future High Luminosity LHC (HL-LHC)~\cite{BejarAlonso:2020kmn}.
\section{Simulation and Event Selection}
We used \textsc{MG5\_aMC@NLO}\ 2.4.2 to generate \ttbar events in the dilepton decay channel with up to 2 additional jets at next to leading order~\cite{Alwall_2014}.
We use the next-to-next-to-leading order top pair production cross-section $\sigma(pp \to t\bar{t})= 831.76$~pb for a collision energy of 13~TeV, which was calculated using the {\sc Top}++ program~\cite{Czakon:2011xx}.
We generated about 7 million signal \ttbar events where one of the $t$ quarks is forced to decay to a $s$ quark and about 7 million background \ttbar events are generated where both $t$ quarks decay to $b$ quarks.
Drell-Yan events with 2 additional partons are the dominant non-\ttbar backgrounds for dilepton \ttbar events.
We generated 20 million Drell-Yan plus two parton events for each of the 4 jet flavour categories: $bb$, $cc$, $ss$ and $qq~(q = u,d,g)$ and use the leading order cross section reported by \textsc{MG5\_aMC@NLO}\ for the categories : $bb = 42.9$~pb, $cc = 4.31$~pb, $ss = 4.37$~pb and $qq = 23.8$~pb. In addition to the non-\ttbar backgrounds, we generated 10 million \ttbar plus one $s$ quark and two $s$ quarks respectively with the leading order cross section of 9.41~pb and 1.57~pb.
\textsc{Pythia8}~8.212 was used to simulate parton showering and hadronization~\cite{Sjostrand_2015,Sjostrand_2006} with the FxFx merging scheme~\cite{Frederix:2012ps}.
We modified the \ensuremath{K^0_S}\ decay in \textsc{Pythia8}\ to allow the \ensuremath{K^0_S}\ to decay inside a fiducial volume of a cylinder centered at the proton collision point with a radius of 860~mm and a length of 4400~mm. This is equivalent to the region of the CMS tracking detector where the pions from the decay may still pass through three silicon detectors, and would allow us to reconstruct \ensuremath{K^0_S}\ using reconstructed tracks when it decays to a charged pion pair.
We used \textsc{Delphes}~3.4.2 to simulate the response of a CMS-like detector with particle flow (PF) outputs~\cite{de_Favereau_2014}.
For jet clustering, we use the anti-$k_{t}$ algorithm with jet radius $R = 0.4$ using FastJet 3.3.2~\cite{Cacciari_2012}.
We used the default CMS card included in \textsc{Delphes}\ but updated it to match the CMS setup used of Run 2.
The jet radius was decreased from 0.5 to 0.4.
The $\Delta R$ cone used to calculate lepton isolation was reduced from 0.5 to 0.3 for electrons and 0.5 to 0.4 for muons.
The track transverse momentum resolution formula was updated using the function given in~\cite{CERN-PH-EP-2014-070}.
The b-tagging efficiency was updated to closely match the Run 2 response of CMS~\cite{CMS-PAS-BTV-15-001} and a $b$-tagging based on track counting module was added.
Smearing of the track impact parameter in the transverse plane was also added to emulate a more realistic \ensuremath{K^0_S}\ reconstruction using the associated module and parameters provided with \textsc{Delphes}, in order to replicate the performance of the CMS tracker~\cite{CMS-TRK-11-001}.
In this study, we use dilepton events in order to remove the additional jet activity from the $W$ decay, which additionally suppresses the background contribution from other processes, especially the multi-jet QCD background.
The \ttbar event selection criteria are based on the CMS measurement of the top pair cross-section with the dilepton channel~\cite{CMS-TOP-17-014}.
First, we select events with two isolated leptons, each of which has $p_{T} >$ 25(20)~GeV for a leading (sub-leading) lepton and pseudorapidity $|\eta| <$ 2.4 and an invariant mass of a lepton pair is more than 20~GeV.
Then, we veto $Z$ boson production by excluding events in the dilepton invariant mass range of $|M_{Z} - M_{ll}| <$ 15~GeV, where $M_{Z} = 91.1876$ GeV~\cite{PDG}.
We require events to have missing energy $\cancel{E}_{T} >$ 40 GeV and at least two reconstructed jets with $p_{T} >$ 30 GeV and $|\eta| <$ 2.4.
We define primary jets as reconstructed jets which are matched to a generator level quark $q$ from the $t \to qW$ decay by finding the highest $p_T$ jet within $\Delta R < 0.4$ of the quark.
97\% of top pair production events have one primary jet, while 74\% of events have two primary jets, and therefore fully match the dilepton decay topology after reconstruction.
These primary jets will be used to train two Boosted Decision Trees (BDT).
The Toolkit for Multivariate Data Analysis in ROOT (TMVA) is used to train the BDT using the adaptive boosting algorithm~\cite{Hoecker_2007}.
The first BDT is trained to select the two primary jets out of all the reconstructed jets in the events.
The second BDT is trained to discriminate between $s$-quark initiated jets from other jets.
Once the first BDT selects the two primary jets, the second BDT is applied on these primary jets to look for $t \to sW$ decay.
This process is described in further detail below.
Both BDTs are trained using the signal $t\bar{t} \to sWbW$ and background $t\bar{t} \to bWbW$ samples.
Additional jets in \ttbar events create ambiguities in the selection of the primary jets, so we employ the first BDT to improve the efficiency of the primary jet selection.
We constructed a BDT model with the inputs of two jet and two lepton four vectors, the missing transverse momentum, and the $\Delta R$ between the two jets.
Each jet pair in the event is evaluated by the BDT.
We use the signal \ttbar sample to train this BDT, defining the signal to be the jet pair made from the two primary jets and the background is when one or more of the jets are not the primary jet.
The output of the first BDT is shown in Figure~\ref{fig:BDT1A}.
For each event, the jet pair with the highest BDT output is selected as the \ttbar primary jet candidates. Figure~\ref{fig:BDT1C} shows the top jet pair selection output for the signal, \ttbar, and Drell-Yan backgrounds.
\begin{figure}[tb!]
\centering
\begin{subfigure}[b]{0.5\textwidth}
\centering
\includegraphics[width=\textwidth]{plots/JetDiscrimination/20211111/TOPBDT_MVA_BDT_S.pdf}
\caption{BDT output of primary jet selection}
\label{fig:BDT1A}
\end{subfigure}
\hfill
\begin{subfigure}[b]{0.5\textwidth}
\centering
\includegraphics[width=\textwidth]{plots/Result/Background_study/20211111/Redraw_Background_BDTD__step4_nJetNo_nBJetNo_topRecoNo_def_TOP_BDT.pdf}
\caption{Highest BDT output of primary jet selection as density}
\label{fig:BDT1B}
\end{subfigure}
\hfill
\begin{subfigure}[b]{0.5\textwidth}
\centering
\includegraphics[width=\textwidth]{plots/Result/Background_study/20211111/Stack_Redraw_Background_BDTD__step4_nJetNo_nBJetNo_topRecoNo_def_TOP_BDT.pdf}
\caption{Highest BDT output of primary jet selection as expected number of event on the Run2 integrated luminosity}
\label{fig:BDT1C}
\end{subfigure}
\caption{\ref{fig:BDT1A}) BDT output distribution of signal (Blue) and background (Red) on the BDT for the primary jet pair selection. \ref{fig:BDT1B}, \ref{fig:BDT1C}) BDT output distribution of jet pair selection for signal and background events normalized (\ref{fig:BDT1B}) and scaled to an integrated luminosity of 137.6 fb$^{-1}$ ~(\ref{fig:BDT1C})}
\label{fig:top BDT three graphs}
\end{figure}
Next, in order to distinguish $s$ jets from the background, predominantly $b$ jets from the dominant $t\bar{t} \to bWbW$, we reconstruct \ensuremath{K^0_S}\ candidates inside the primary jet candidates.
In $s$ jets, \ensuremath{K^0_S}\ can be produced directly from the initiating $s$ quark, whereas in $b$ jets they will be produced after a cascade of decays of the $b$ hadron or from the quarks produced in the parton shower.
This means that \ensuremath{K^0_S}\ should be harder (relative to the jet energy) and more collimated in the case of $s$ quark initiated jets.
We reconstruct \ensuremath{K^0_S}\ using its decay into oppositely charged pion pairs, and due to the long lifetime of the \ensuremath{K^0_S}, we require the tracks to come from a displaced vertex within the tracker volume.
Using the charged hadron objects from the \textsc{Delphes}\ particle flow reconstruction, we consider all oppositely charged hadron pairs.
Since general purpose detectors like CMS do not distinguish between pions from other charged hadrons, we assume all charged hadrons to be pions.
We require the charged hadron pair to have $p_{T} >$ 0.95~GeV and $|\eta| <$ 2.4 and the significance of the transverse impact parameter of each track to be greater than two, to ensure the tracks are not from the primary vertex.
Then we select reconstructed \ensuremath{K^0_S}\ candidates with an invariant mass of $|M_{\ensuremath{K^0_S}} - M_{\pi\pi}| < 0.1$~GeV, where $M_{\ensuremath{K^0_S}} = 497.611$~MeV~\cite{PDG}.
We check that the reconstructed \ensuremath{K^0_S}\ candidate is from a primary jet candidate by requiring the angle between the candidate momentum and the jet axis to satisfy $\Delta{}R < 0.4$.
If there are more than one reconstructed \ensuremath{K^0_S}\ candidates, we select the \ensuremath{K^0_S}\ with the highest $p_{T}$.
\begin{figure}[tb!]
\centering
\begin{subfigure}[b]{0.5\textwidth}
\centering
\includegraphics[width=\textwidth]{plots/JetDiscrimination/20211111/JETBDT_MVA_BDTD_S.pdf}
\caption{BDT output of $s$ jet tagging}
\label{fig:BDT2A}
\end{subfigure}
\hfill
\begin{subfigure}[b]{0.5\textwidth}
\centering
\includegraphics[width=\textwidth]{plots/Result/Background_study/20211111/Redraw_Background_BDTD__step4_nJetNo_nBJetNo_topRecoNo_def_JET_BDT.pdf}
\caption{BDT output of $s$ jet tagging on signal and background events as density}
\label{fig:BDT2B}
\end{subfigure}
\hfill
\begin{subfigure}[b]{0.5\textwidth}
\centering
\includegraphics[width=\textwidth]{plots/Result/Background_study/20211111/Stack_Redraw_Background_BDTD__step4_nJetNo_nBJetNo_topRecoNo_def_JET_BDT.pdf}
\caption{BDT output of $s$ jet tagging on signal and background events of the Run2 integrated luminosity}
\label{fig:BDT2C}
\end{subfigure}
\caption{\ref{fig:BDT2A}) BDT output of $s$ jet tagging on signal (Blue) and background (Red). \ref{fig:BDT2B}, \ref{fig:BDT2C}) Highest BDT output of $s$ jet tagging on the primary jets normalized (\ref{fig:BDT2B}) and scaled to an integrated luminosity of 137.6 fb$^{-1}$ (\ref{fig:BDT2C})}
\label{fig:jet BDT three graphs}
\end{figure}
After matching the \ensuremath{K^0_S}\ candidates to the primary jet candidates, we use both hadron and jet information to discriminate $s$ jets from all other jets.
First, from the jet information, we use jet's $p_{T}$, mass, its minor and major axes and their substructure-related quantities such as charged jet multiplicity, charged daughter's $p_{T}$ fraction in a jet, leptonic constituent's $p_{T}$ fraction in a jet, and $p_{T}D$, also called the jet energy sharing, defined as $\frac{\sqrt{\sum{p_{T,i}^2}}}{\sum{p_{T,i}}}$.
To simulate b-tagging, we used a simple track counting method, which tags a jet as a $b$ if more than two tracks are found with a high impact parameter.
From the \ensuremath{K^0_S}\ kinematics, we use the hadron's $p_{T}$ fraction relative to the jet $x = p_T^{had} / p_T^{jet}$, the $\Delta$R between the jet axis and the hadron momentum, the Distance of Closest Approach (DCA) between the two tracks, the cosine of the 2D pointing angle between momentum vector and position vector of hadron and decay length calculated by assuming the \ensuremath{K^0_S}\ vertex is the midpoint of DCA between the two charged tracks. In addition, the following information from both the charged pion daughters are included: the $p_{T}$ fraction compared to the \ensuremath{K^0_S}, the significance of the transverse impact parameter, and of the longitudinal impact parameter.
For training the second BDT, the signal is defined as jets matched to an s-quark from the $t \to sW$ decay with a matching \ensuremath{K^0_S}\ whose momentum fraction, x $>$ 0.15 and the background is all other jets with \ensuremath{K^0_S}\ with x $>$ 0.15.
The result from the s-jet discriminating BDT training is shown in Figure~\ref{fig:BDT2A} and Figure~\ref{fig:BDT2C} shows the s-jet discriminating BDT for the primary jets on the signal and background processes.
For the final $s$ jet selection, we first reject primary jets that are $b$ tagged in each event.
If both the primary jets are $b$ tagged, the event is vetoed from the analysis.
Of the remaining primary jets in each event, we select the jet with the highest $s$ tagging BDT output.
Figure~\ref{fig:BDT2C} shows the final $s$ tagging BDT output for signal and background events.
\section{Results}
The expected number of signal \ttbar and background events is given by
\begin{equation}
\label{eq:Nexp:1}
N_{sig} = \sigma(t\bar{t}) \times \mathcal{L} \times \mathcal{B}(t\bar{t} \to ql^{+}\bar{\nu}\bar{q}l^{-}\nu) \times 2|V_{ts}|^{2}|V_{tb}|^{2} \times \epsilon_{sig}
\end{equation}
\begin{equation}
\label{eq:Nexp:2}
N_{bkg} = \sigma(t\bar{t}) \times \mathcal{L} \times \mathcal{B}(t\bar{t} \to ql^{+}\bar{\nu}\bar{q}l^{-}\nu) \times |V_{tb}|^{4} \times \epsilon_{bkg} + N_{DY}
\end{equation}
where $\sigma(\ttbar)$ is the cross-section, $\mathcal{L}$ is a integrated luminosity, $\mathcal{B}(\ttbar \to ql^{+}\bar{\nu}\bar{q}l^{-}\nu)$ is the branching ratio of dileptonic decay mode, $\epsilon_{sig}$ ($\epsilon_{bkg}$) is the selection efficiency after the selections described in the previous section for the signal $sWbW$ (background $bWbW$) \ttbar sample, and $N_{DY}$ is the expected number of Drell-Yan background events.
\begin{figure}[th!]
\centering
\includegraphics[width=.5\textwidth,trim=0 0 0 0,clip]{plots/Result/RooStats/20211111/mu1_twoside/BDT_All_Lumi_137_61fb_pvalue_Background_BDTD__step4_nJetNo_nBJetNo_topRecoNo_def_mu1_CL0.95.pdf}
\caption{The gaussian fit of the BDT output distribution. The background distribution is shown in blue and the signal in red. The background includes Drell-Yan events with 2 jets as well as $t\bar{t} \to bWbW$ and the signal is $t\bar{t} \to sWbW$
}
\label{fig:gaussianfit}
\end{figure}
\begin{figure*}[th!]
\includegraphics[width=.5\textwidth,trim=0 0 0 0,clip]{plots/Result/RooStats/20211111/LocalP0.pdf}
\includegraphics[width=.5\textwidth,trim=0 0 0 0,clip]{plots/Result/RooStats/20211111/BrazilianBand.pdf}
\caption{Local p$_{0}$ value (Left) and 95\% Confidence Level (CL) upper limit on signal strength $\mu$ (Right) for each integrated luminosity. In the left plot, p$_{0}$ is p value for background-only PDF and the red dashed line is on $\mu = 1$. Black dots in the right plot are $\mu$ corresponding to CL of 95\% at 137 (Run2), 300 (expected for Run3), 600, 1200, 2000 and 3000 (expected for HL-HLC) fb$^{-1}$. }
\label{fig:Limit}
\end{figure*}
\begin{figure*}[th!]
\includegraphics[width=.5\textwidth,trim=0 0 0 0,clip]{plots/Result/RooStats/20211111/mu1_twoside/All_Lumi_137_61fb_pvalue_Background_BDTD__step4_nJetNo_nBJetNo_topRecoNo_def_mu1_CL0.95.pdf}
\includegraphics[width=.5\textwidth,trim=0 0 0 0,clip]{plots/Result/RooStats/20211111/mu1_twoside/All_Lumi_300_0fb_pvalue_Background_BDTD__step4_nJetNo_nBJetNo_topRecoNo_def_mu1_CL0.95.pdf}
\includegraphics[width=.5\textwidth,trim=0 0 0 0,clip]{plots/Result/RooStats/20211111/mu1_twoside/All_Lumi_1200_0fb_pvalue_Background_BDTD__step4_nJetNo_nBJetNo_topRecoNo_def_mu1_CL0.95.pdf}
\includegraphics[width=.5\textwidth,trim=0 0 0 0,clip]{plots/Result/RooStats/20211111/mu1_twoside/All_Lumi_3000_0fb_pvalue_Background_BDTD__step4_nJetNo_nBJetNo_topRecoNo_def_mu1_CL0.95.pdf}
\caption{$p$-value at the Run2 luminosity (top left), the expected luminosity from Run3 (top right), 1200 fb$^{-1}$ (bottom left) and 3000~fb$^{-1}$\ the design luminosity of HL-LHC (bottom right). The dotted line is the median of the expected CL$_{s+b}$, the green (yellow) band is $\pm 1\sigma$ ($\pm 2\sigma$) range.}
\label{fig:CLsb}
\end{figure*}
To study the feasibility of a direct measurement of $\ensuremath{|\vts|}$, we perform an analysis to find the expected significance to reject the hypothesis $H_0$ of no $t \to sW$ decays, the expected upper limit on $\ensuremath{|\vts|}$ and also an expected confidence interval $\ensuremath{|\vts|}$.
We use \textsc{RooFit}~\cite{Verkerke_2003} and \textsc{RooStats}~\cite{Moneta_2010} to perform the statistical analysis using the \textsc{RooStats} asymptotic calculator based on the asymptotic properties of likelihood function~\cite{Cowan_2011}.
We fit the $s$ jet tagging BDT distributions from Figure~\ref{fig:BDT2C} to obtain approximations of the probability density functions (PDFs) for further study.
The PDFs for the signal and background are separately modeled by a sum of four Gaussian distributions and the fitted PDFs are shown in Figure~\ref{fig:gaussianfit}.
The BDT distribution in this figure has a different definition of signal and background from the BDT distribution of jet discriminator.
The BDT distribution shown in the previous section defines the signal (background) as jets matched to generator level s (b) quarks, while the distributions here define the signal as the jet with the highest BDT output in an event from $t\bar{t} \to sWbW$ signal events and the background as the jet with the highest output from $t\bar{t} \to bWbW$ and Drell-Yan events.
The total model is thus
\begin{equation}
\label{eq:PDF}
P_{s+b} = \mu \times N_{sig} \times P_{sig} + \nu \times N_{bkg} \times P_{bkg}
\end{equation}
where $N_{sig}$ is the number of $\ttbar \to sWbW$ events expected by the SM, corrected for the selection efficiency and $\mu$ is the signal strength relative to the SM $\frac{|V_{ts}^{obs}|^2}{|V_{ts}^{CKM}|^2}$, where $|V_{ts}^{CKM}| = 39.78 \times 10^{-3}$~\cite{PDG}, and $\nu$ is a nuisance parameter to control the background level relative to the expected background.
From the model PDF, we generate an Asimov dataset which we use as the observed dataset for the following studies.
Figure~\ref{fig:Limit} and Figure~\ref{fig:CLsb} show the results of one-sided and two-sided scanning using the gaussian fitting for several integrated luminosities. The left plot in Figure~\ref{fig:Limit} shows a median expected local p$_{0}$ under assumption of $H_{0}$ ($|V_{ts}|^2=0$) versus the true signal strength $\mu$. For $\mu = 1$, corresponding to the current SM expectation, the significance of rejecting $H_0$ is expected to be more than 5$\sigma$ when the integrated luminosity is 2000 fb$^{-1}$ and greater than 6$\sigma$ for 3000~fb$^{-1}$, HL-LHC designed integrated luminosity.
The right plot in Figure~\ref{fig:Limit} shows the median expected upper limit on $\mu$ under the assumption of $H_0$ at the 95\% confidence level (CL) for each luminosity as the dashed line and the $\pm 1(2) \sigma$ range as the green (yellow) band. Figure~\ref{fig:CLsb} shows the expected two-sided p-value distribution for the signal strength $\mu$ under the assumption of $\mu=1$.
Table~\ref{tab:upperlimit} summarizes several results shown above: the expected significance to exclude $\ensuremath{|\vts|}=0$, the median 95 \% CL upper limit if $\mu=0$ and the 95\% CL for each luminosity.
Under the assumption of $\mu=1$, the expected significance from the hypothesis test calculation shows a 1.36$\sigma$ significance to reject the background-only hypothesis for the integrated luminosity of the Run2.
The value becomes 2.00$\sigma$ when the integrated luminosity is 300~fb$^{-1}$, as is expected to be collected during Run 3 of the LHC, and 6.34$\sigma$ for the full HL-LHC integrated luminosity of 3000~fb$^{-1}$.
Conversely, if the decay is suppressed, and $\mu=0$, then Run 3 of the LHC will be able to exclude $\mu=1$ at the 95\% CL level.
\begin{table*}[tb]
\centering
\begin{tabular}{|c|c|c|c|}
\hline
Integrated luminosity (fb$^{-1}$) & \vtop{\hbox{\strut Expected significance ($\sigma$)} \hbox{\strut for $\ensuremath{|\vts|}$ = 0 exclusion}} & \vtop{\hbox{\strut Expected 95\% CL}\hbox{\strut median upper limit ($\mu$)}} & \vtop{\hbox{\strut Expected 95\% CL$_{s+b}$} \hbox{\strut median interval ($\mu$)}} \\
\hline
137.6 & 1.36 & $<$ 1.22 & [0.000, 2.30] \\
300 & 2.00 & $<$ 0.822 & [0.0210, 1.98] \\
600 & 2.83 & $<$ 0.582 & [0.307, 1.70] \\
1200 & 4.01 & $<$ 0.411 & [0.509, 1.49] \\
2000 & 5.17 & $<$ 0.319 & [0.619, 1.38] \\
3000 & 6.34 & $<$ 0.262 & [0.689, 1.31] \\
\hline
\end{tabular}
\caption{\label{tab:upperlimit} The expected significance to exclude $|V_{ts}|$ = 0, the expected 95\% CL median upper limit under the assumption of $|V_{ts}|=0$ and the expected 95\% CL$_{s+b}$ median confidence interval on $\mu = \frac{|V_{ts}^{obs}|^2}{|V_{ts}^{CKM}|^2}$ for several integrated luminosities. }
\end{table*}
\section{Conclusion}
We have studied the feasibility of a direct measurement of $|V_{ts}|$ from the dileptonic \ttbar production process, using hadronization by \textsc{Pythia8}\ and \textsc{Delphes}\ detector simulation to produce a more realistic expectation of the results of the LHC experiments.
With an integrated luminosity of 3000~fb$^{-1}$, which is expected to be achieved at the HL-LHC period, $\ensuremath{|\vts|}$ = 0 can be excluded above the 6$\sigma$ significance level.
\section{Acknowledgments}
This article was supported by the computing resources of the GDSC at the Korea Institute of Science and Technology Information.
W.J. is supported by the National Research Foundation of Korea (NRF) grant funded by the Ministry of Science and ICT (MSIT) (2018R1C1B6005826).
J.L. is supported by the NRF grant funded by the MSIT (2019R1C1C1009200).
I.W. is supported by the Brain Pool Program through the NRF funded by the MSIT (2017H1D3A1A01052807).
I.P. is supported by the Basic Science Research Program through the NRF funded by the Ministry of Education (2018R1A6A1A06024977).
|
{
"redpajama_set_name": "RedPajamaArXiv"
}
| 2,184
|
Q: RESTful merge and split of resource I have a REST service that serves Invoice functionality, for example I can create, update and delete an invoice via the REST service.
However I now need to have functionality to split an invoice into several new invoices, i.e. the method needs to create x invoices based on one invoice, and then delete the original invoice in one transaction.
I also need a merge method that does the opposite, i.e. multiple invoices merged into one.
How do I create such architect the RESTful way?
A: We had a similar issue where we wanted to merge 2 resources. After a long debate we decided that a consumer would POST a merge request resource which contained the 2 resources to be merged. Then after looking at it we decided that there was no need for both the complete source and target resource and instead a POST of just the unique identifier was sufficient. We didn't make the merge request with just the 2 ID's come in the body. It was easier to represent in a URL so that's what we did. The response from the POSTing of a merge request is the merged resource.
I guess I'm not a REST guru enough to tell you this was a good strategy or not, but for us it was a simple solution so we went with it.
A: When you say that want to implement this functionality in "one transaction" I assume you have already determined that you should combine generation of the new invoices and deletion of the old one, into one API call; which is the correct approach. With web services you do want to reduce chatter and there is probably some business logic on how this functionality will generate the new invoices and delete the old one. So I am assuming when you ask how to architect this in a RESTful way you are wondering which HTTP verb to use (i.e. GET, POST, PUT or DELETE) for this new API method. Usually these verbs map to CRUD type operations in the following manner:
*
*Create -> POST
*Read -> GET
*Update -> PUT
*Delete -> DELETE
So which verb to use when your function both creates and deletes records. A general rule with REST API's is if there is not a clear mapping to CRUD then use POST if there is a change to the server state and a GET if you are just returning information that does not change the server state. So in this case I would go with a POST.
If you are looking for additional guidance on architecting this please be more specific in what you are looking for and I will try to help.
A: I would do something like,
POST /InvoiceSplitter?sourceInvoiceId=99
and
POST /InvoiceMerger?sourceInvoiceIds=101,87,23,45
|
{
"redpajama_set_name": "RedPajamaStackExchange"
}
| 4,619
|
package net.maizegenetics.dna.snp.io;
import ch.systemsx.cisd.hdf5.HDF5Factory;
import ch.systemsx.cisd.hdf5.IHDF5Reader;
import net.maizegenetics.dna.map.PositionList;
import net.maizegenetics.dna.map.PositionListBuilder;
import net.maizegenetics.dna.snp.GenotypeTable;
import net.maizegenetics.dna.snp.GenotypeTableBuilder;
import net.maizegenetics.taxa.TaxaList;
import net.maizegenetics.taxa.TaxaListBuilder;
import net.maizegenetics.taxa.Taxon;
import org.apache.log4j.Logger;
import java.util.ArrayList;
import java.util.List;
import java.util.TreeSet;
/**
* Provides Builder and methods to build HDF5 files. The methods for merging files work, but they are far from
* optimal. It appears that they are taking twice the memory that they should. Ed has looked for the memory leak
* with a profiler but cannot find it yet.
* Todo: TAS-70 Details the issues with apparent memory leak.
*
* @author Ed Buckler
*/
public class BuilderFromGenotypeHDF5 {
private static final Logger myLogger=Logger.getLogger(BuilderFromGenotypeHDF5.class);
private final String infile;
private BuilderFromGenotypeHDF5(String infile) {
this.infile=infile;
}
// public static BuilderFromGenotypeHDF5 getBuilder(String infile) {
// return new BuilderFromGenotypeHDF5(infile);
// }
//
// //TODO provide options on caching to use, read only some sites, etc.
// //TODO update to the newest version
// //TODO subset??
// public GenotypeTable build() {
// IHDF5Reader reader=HDF5Factory.openForReading(infile);
// if(HDF5Utils.isTASSEL4HDF5Format(reader)) {
// reader.close();
// throw new UnsupportedOperationException("TASSEL4 HDF5 file, please migrate to TASSEL5 using MigrateHDF5FromT4T5");
// }
// TaxaList tL=new TaxaListBuilder().buildFromHDF5Genotypes(reader);
// //TODO move to migration code
//// int numTaxa =HDF5Utils.getHDF5GenotypeTaxaNumber(reader);
//// if(numTaxa!=tL.numberOfTaxa()) {
//// reader.close();
//// IHDF5Writer writer=HDF5Factory.open(infile);
//// HDF5Utils.writeHDF5GenotypesNumTaxa(writer,tL.numberOfTaxa());
//// writer.close();
//// reader=HDF5Factory.openForReading(infile);
//// }
// PositionList pL=PositionListBuilder.getInstance(reader);
// GenotypeCallTable geno=GenotypeCallTableBuilder.buildHDF5(reader);
// return GenotypeTableBuilder.getInstance(geno,pL, tL);
// }
/**
* This merge multiple alignment together into one ByteNucleotideHDF5 File.
* This is designed for putting multiple chromosomes together into one whole genome file.
* @param infiles array of input alignment names
* @param newMerge name of ByteNucleotideHDF5
*/
public static void mergeToMutableHDF5(String[] infiles, String newMerge) {
if ((infiles == null) || (infiles.length == 0)) {
return ;
}
System.out.println("Opening Existing Position List");
PositionListBuilder palBuild=new PositionListBuilder();
//this could be made more efficient by doing the position and taxa list at the same time.
System.out.println("Combining Position List");
for (String infile : infiles) {
System.out.println("..."+infile);
IHDF5Reader reader=HDF5Factory.openForReading(infile);
PositionList pl=PositionListBuilder.getInstance(reader);
palBuild.addAll(pl);
reader.close();
System.gc();
}
System.out.println("Sorting Position List");
PositionList pal=palBuild.build(); //In memory position list
palBuild=null;
System.out.println("Creating Position List Look Up");
int[][] oldSiteToNewSite=new int[infiles.length][];
int misses=0;
int hits=0;
for (int i=0; i<infiles.length; i++) {
System.out.println("..."+infiles[i]);
IHDF5Reader reader=HDF5Factory.openForReading(infiles[i]);
PositionList aPL=PositionListBuilder.getInstance(reader);
oldSiteToNewSite[i]=new int[aPL.numberOfSites()];
for (int j=0; j<aPL.size(); j++) {
oldSiteToNewSite[i][j]=pal.indexOf(aPL.get(j));
if(oldSiteToNewSite[i][j]<0) {
misses++;
System.out.println(oldSiteToNewSite[i][j]);
System.out.println(aPL.get(j).toString());
System.out.println(pal.get(-oldSiteToNewSite[i][j]).toString());
System.out.printf("misses:%d hits:%d %n", misses, hits);
} else {
hits++;
}
}
reader.close();
// aPL=null;
System.gc();
}
System.out.println(misses);
int numberOfSites=pal.numberOfSites();
//Get taxa List
//This could be combined with above.
List<TaxaList> inTL=new ArrayList<>();
TreeSet<Taxon> taxa = new TreeSet<>();
for (String infile : infiles) {
IHDF5Reader reader=HDF5Factory.openForReading(infile);
TaxaList aTL=new TaxaListBuilder().buildFromHDF5Genotypes(reader);
reader.close();
taxa.addAll(aTL);
inTL.add(aTL);
}
TaxaList newTaxaList=new TaxaListBuilder().addAll(taxa).build();
int[][] oldTaxaToNewTaxa=new int[inTL.size()][];
for (int i=0; i<inTL.size(); i++) {
TaxaList aTL=inTL.get(i);
oldTaxaToNewTaxa[i]=new int[aTL.numberOfTaxa()];
for (int j=0; j<aTL.size(); j++) {
oldTaxaToNewTaxa[i][j]=newTaxaList.indexOf(aTL.get(j));
if(oldTaxaToNewTaxa[i][j]<0) {
System.out.println(oldTaxaToNewTaxa[i][j]);
System.out.println(aTL.get(j).toString());
}
}
}
GenotypeTableBuilder ab=GenotypeTableBuilder.getTaxaIncremental(pal,newMerge);
System.gc();
// System.out.println(Arrays.deepToString(oldTaxaToNewTaxa));
//Transfer the genotypes
System.out.println("Opening alignments");
List<GenotypeTable> sourceA=new ArrayList<>();
for (String infile : infiles) {
sourceA.add(GenotypeTableBuilder.getInstance(infile));
}
for (Taxon aT : newTaxaList) {
System.out.println("Write taxon:"+aT.getName());
byte[] geno=new byte[numberOfSites];
for (int i=0; i<sourceA.size(); i++) {
int taxonIndex=sourceA.get(i).taxa().indexOf(aT);
byte[] r=sourceA.get(i).genotypeAllSites(taxonIndex);
for (int j=0; j<oldSiteToNewSite[i].length; j++) {
geno[oldSiteToNewSite[i][j]]=r[j];
}
}
ab.addTaxon(aT,geno);
}
ab.build();
}
}
|
{
"redpajama_set_name": "RedPajamaGithub"
}
| 7,626
|
\section{Introduction}
Actuators and sensors in control systems are often nonlinear. While plants are generally nonlinear as well, they can usually be linearized around an operating point if the control system is well designed. The nonlinear instrumentation, i.e., the actuators and sensors, however, cannot. This is because external random inputs to the system may force them to operate far from their designed operating point, activating nonlinearities in them.
Quasilinear Control (QLC) is a set of methods that can be used to analyze and design control systems with nonlinear actuators and sensors \cite{Ching2010}. It leverages the method of stochastic linearization, which replaces each nonlinearity by an equivalent gain and a bias, based on statistical properties of the stochastic inputs. Consider a nonlinear function $f\left(x\right)=x^2$, as shown in Fig. \ref{fig:The-Approach-of}. The traditional (Jacobian) approach in linearizing such a function is to find the derivative of the function at a suitable operating point, replace the nonlinearity by a linear approximation and shift the origin at that operating point. The method of stochastic linearization, on the other hand, is based on minimizing the expected value of the mean squared error between the nonlinear function and its stochastically linearized approximation, taking into account the probability distribution of the input. This approach is graphically depicted in Fig. \ref{fig:The-Approach-of}, in which the input $x$ has been assumed to follow a Normal distribution with mean 3 and standard deviation 2. The dashed lines represent two Jacobian linearizations performed at (1,1) and (4,16) and the dotted line represents the stochastically linearized approximation. If Jacobian linearization is performed at (1,1), but the operating point shifts to, say, (4,16), the linearization becomes highly inaccurate. However, since stochastic linearization considers the expected value of the derivatives around $x=3$, it performs better at (4,16).
\begin{figure}[t]
\begin{centering}
\includegraphics[width=0.9\columnwidth]{./Figures/sl.eps}
\par\end{centering}
\caption{The Approach of Stochastic Linearization\label{fig:The-Approach-of}}
\end{figure}
The theory of stochastic linearization and quasilinear control has been developed only for nonlinear actuators and sensors having a single input. In practical applications, however, more than just one factor or input affect the performance of a nonlinear actuator, and hence the operation of the control system. As a case in point, the authors are involved in a US Department of Energy Project called ENERGIZE, whose goal is to develop robust and resilient real-time control systems with uncertain distributed energy resources. In such a renewable energy system, that involves aggregation of several distributed energy resources that can both produce and consume electric power, it is often necessary to compute the optimal power set point for these aggregated devices to ensure a robust and resilient operation. It is then desirable to estimate the power limits of such resources. Since the power limits would depend on the number and type of devices being aggregated, both of which are random phenomena, depending on when users decide to turn the devices on or off, the saturation authority of the actuator can be considered to be a stochastic process in such a case.
In this paper, the theory of stochastic linearization is extended to a nonlinear function of multiple variables, such that the inputs form a wide sense stationary (WSS) multivariate Gaussian random vector. The outline is as follows: Section \ref{rsv} provides a brief review of single variable QLC. Section \ref{msl} introduces expressions for stochastically linearizing a generic multivariate nonlinearity. In Section \ref{abs}, the bivariate saturation nonlinearity is introduced and expressions for its equivalent gains and bias derived, using the result of the previous Section. The result is used in Section \ref{qlccl} to find the stochastic linearization of a general feedback control system, in which the actuator is a bivariate saturation, with the randomness in the bounds modeled as a second input to the actuator, taking the reference and disturbance signals to be WSS Gaussian random processes with specified means and standard deviations. Section \ref{cSQL} explores special features of multi-variable QLC compared to single-variable QLC, specifically the effect of the second actuator input and correlation between the actuator inputs. Section \ref{asl} investigates the accuracy of stochastic linearization by a Monte Carlo simulation with different possible input and system parameters. In Section \ref{pe}, a practical example of an optimal controller design has been provided. Section \ref{conc} concludes the paper. In the appendix, series expansions of the integrals in Section \ref{qlccl} have been derived, along with their region of convergence, and an algorithm presented for their calculation.
\section{Review of Single Variable Stochastic Linearization}\label{rsv}
This section presents a brief review of single variable stochastic linearization. For details, please refer to \cite{Ching2010}.
\subsection{Open Loop System}\label{ols}
Consider a single input single output (SISO) system shown in Fig. \ref{fig:p2} driven by a wide-sense stationary Gaussian stochastic input $u(t)$, such that it is governed by the input-output relationship:
\[v(t)=f(u(t))\]
Stochastic linearization replaces the above nonlinearity by a linear approximation $Nu_0(t)+M$, such that the functional:
\[
\epsilon\left(N,M\right)=E\left\{ \left[f\left(u\left(t\right)\right)-Nu_0\left(t\right)-M\right]^{2}\right\}
\] is minimized \cite{Kabamba2015}. Here $N$ is called the \textit{quasilinear gain}, $M$ the \textit{quasilinear bias} and $u_0(t)$ is the zero-mean part of $u(t)$. It can be shown that the values of $N$ and $M$ are:
\begin{equation}\label{neqs}
N=E[f'(u)]
\end{equation}
\begin{equation}\label{meqs}
M=E[f(u)]
\end{equation}
\begin{figure}
\centering
\includegraphics[width=1\linewidth]{./Figures/ssl.eps}
\caption{The process of Single Variable Stochastic Linearization: Here $N$ is the quasilinear gain defined in \eqref{neqs} and $M$ the quasilinear bias defined in \eqref{meqs} and $u_0(t)$ is the zero-mean part of $u(t)$.}
\label{fig:p2}
\end{figure}
\subsection{Closed Loop System}\label{clssv}
Consider a general feedback control system shown in Fig. \ref{fig:Block-Diagram-ofs}. It consists of a plant $P\left(s\right)$ whose output is desired to be controlled using a controller $C\left(s\right)$ and a nonlinear actuator described by the function $f\left(\cdot\right)$. The signal $r(t)$ is the reference signal to be tracked and $d(t)$ is the disturbance. $r(t)$ is generated by passing white noise $w_r(t)$ through a coloring filter $F_{\Omega_r}(s)$, scaling it by $\sigma_r$, and adding a bias $\mu_r$. Similarly, $d(t)$ is generated by passing white noise $w_d(t)$ through a coloring filter $F_{\Omega_d}(s)$, scaling it by $\sigma_d$, and adding a bias $\mu_d$. The $H_2$-norm of the coloring filters are considered to be unity, which allows the reference and disturbance signals to have means $\mu_r$ and $\mu_d$ respectively, along with corresponding standard deviations $\sigma_r$ and $\sigma_d$. The block diagram shows the corresponding state space representations. The nonlinearity, $f(\cdot)$ can be replaced by a stochastically linearized block $Nu_0(t)+M$, where $u_0(t)$ is the zero-mean part of $u(t)$ and $N$ and $M$ are as in \eqref{neqs} and \eqref{meqs} respectively. To calculate $N$ and $M$, we need to consider the entire system, along with the statistics of the input signals.
This is the general idea behind stochastically linearizing a nonlinear control system with a single input to the nonlinearity. Since it is a special case of multivariate stochastic linearization, which will be described in detail in Section \ref{qlccl}, we will not review its details here.
\begin{figure*}[t]
\centering
\begin{tikzpicture}[>=latex,scale=0.8,every node/.style={transform shape}, every text node part/.style={align=center}]
\node (wr) at (0,0){$w_r$};
\node[rectangle,draw] (rf) [below=0.5 of wr,label={left:$F_{\Omega_r}\left(s\right)$}]{$\dot{x}_r=A_{rf}x_r+B_{rf}w_r$\\$r_f=C_{rf}x_r+D_{rf}w_r$};
\node[rectangle,draw] (sr) [below=0.5 of rf]{$\sigma_r$};
\node[circle,draw] (cr) [below=0.5 of sr]{+};
\node (mur) [left=0.5 of cr]{$\mu_r$};
\node[circle,draw] (sum) [below right=0.8 of cr]{};
\node[rectangle,draw] (C) [right=0.8 of sum,label={below:$C\left(s\right)$}]{$\dot{x}_C=A_{C}x_C+B_{C}e$\\$u=C_{C}x_C+D_Ce$};
\node[rectangle,draw] (sat) [right= of C]{$f\left(\cdot\right)$};
\node[circle,draw] (dsum) [right=1.5 of sat]{+};
\node[circle,draw] (cd) [above=0.5 of dsum]{+};
\node (mud) [right=0.5 of cd]{$\mu_{d}$};
\node[rectangle,draw] (sd) [above=0.5 of cd]{$\sigma_{d}$};
\node[rectangle,draw] (df) [above=0.5 of sd,label={right:$F_{\Omega_d}\left(s\right)$}]{$\dot{x}_{d}=A_{df}x_{d}+B_{df}w_{d}$\\${d}_f=C_{df}x_{d}+D_{df}w_{d}$};
\node (wd) [above=0.5 of df]{$w_{d}$};
\node[rectangle,draw] (P) [right= of dsum,label={below:$P\left(s\right)$}]{$\dot{x}_P=A_{P}x_P+B_{P}z$\\$y=C_{P}x_P+D_Pz$};
\node[coordinate] (y) [right= of P]{};
\draw[->] (P)--node[above]{$y\left(t\right)$}(y)--($(y)+(0,-2)$)-|node[pos=0.95,left]{-}(sum);
\draw[->] (wr)--(rf);
\draw[->] (rf)--(sr);
\draw[->] (sr)--(cr);
\draw[->] (mur)--(cr);
\draw[->] (cr)|-node[pos=0.4,below]{$r\left(t\right)$}node[very near end,below]{+}(sum);
\draw[->] (wd)--(df);
\draw[->] (df)--(sd);
\draw[->] (sd)--(cd);
\draw[->] (mud)--(cd);
\draw[->] (cd)--node[left]{$d\left(t\right)$}(dsum);
\draw[->] (sum)--node[above]{$e\left(t\right)$}(C);
\draw[->] (C)--node[above]{$u\left(t\right)$}(sat);
\draw[->] (sat)--node[above,pos=0.35]{$v\left(t\right)$}(dsum);
\draw[->] (dsum)--node[above]{$z\left(t\right)$}(P);
\end{tikzpicture}
\caption{Block Diagram of Control System\label{fig:Block-Diagram-ofs}}
\label{figsv}
\end{figure*}
\section{Multivariate Stochastic Linearization}\label{msl}
Consider a multi-input single output (MISO) system driven by $n$ wide-sense stationary Gaussian inputs $u_1(t),u_2(t),\dots,u_n(t)$, forming a Gaussian random vector $\mathbf{u}(t)$, and modeled by a multivariate nonlinearity $v(t)=f(\mathbf{u}(t))$, where $v(t)$ is the output. The problem is to find a linear approximation to this nonlinearity. Various objective functions have been suggested in the literature for minimizing the error introduced by a possible linear approximation \cite{Iwan1972,Smith1966}, but it has been found that, in general, the mean squared error between the nonlinearity and its linear approximation gives results as good as, if not better, than others \cite{Spanos}. Hence, it is used in the following Theorem to derive the linear approximation.
\begin{theorem}
Let $u_{1}\left(t\right)$,$u_{2}\left(t\right)$,$\ldots$,$u_{n}\left(t\right)$
be $n$ WSS jointly Gaussian processes with expected values $\mu_{1}\left(t\right),\mu_{2}\left(t\right),\ldots,\mu_{n}\left(t\right)$
respectively, $\mathbf{u}\left(t\right)=\left[\begin{array}{cccc}
u_{1}\left(t\right) & u_{2}\left(t\right) & \ldots & u_{n}\left(t\right)\end{array}\right]^{T}$, $\boldsymbol{\mu}\left(t\right)=~\left[\begin{array}{cccc}
\mu_{1}\left(t\right) & \mu_{2}\left(t\right) & \ldots & \mu_{n}\left(t\right)\end{array}\right]^{T}$ and $\mathbf{u_{0}}\left(t\right)=~\mathbf{u}\left(t\right)-\boldsymbol{\mu}\left(t\right)$.
Then, for any piecewise differentiable function $f\left(\mathbf{u}\right):\mathbb{R}^{n}\rightarrow\mathbb{R}$,
the functional
\[
\epsilon\left(\mathbf{N},M\right)=E\left\{ \left[f\left(\mathbf{u}\left(t\right)\right)-\mathbf{N^{T}}\mathbf{u_{0}}\left(t\right)-M\right]^{2}\right\}
\]
is minimized by:
\begin{equation}\label{neq}
\mathbf{N}=E\left[\mathbf{\boldsymbol{\nabla}}f\left(\mathbf{u}\left(t\right)\right)\right]
\end{equation}
\begin{equation}\label{meqd}
M=E\left[f\left(\mathbf{u}\left(t\right)\right)\right]
\end{equation}
where $\mathbf{N}=\left[\begin{array}{cccc}
N_{1} & N_{2} & \ldots & N_{n}\end{array}\right]^{T}$ is a constant vector.
\end{theorem}
\begin{proof}
To minimize $\epsilon\left(\mathbf{N},M\right)$, we set $\frac{\partial\epsilon\left(\mathbf{N},M\right)}{\partial\mathbf{N}}=\mathbf{0}$
and $\frac{\partial\epsilon\left(\mathbf{N},M\right)}{\partial M}=0$. The term $\frac{\partial\epsilon\left(\mathbf{N,M}\right)}{\partial\mathbf{N}}$
is calculated as follows. Let $\mathbf{u_{0}}\left(t\right)=\left[\begin{array}{cccc}
u_{01}\left(t\right) & u_{02}\left(t\right) & \ldots & u_{0n}\left(t\right)\end{array}\right]^{T}$. Then for any $k\in\mathbb{N},\,1\leq k\leq n$,
\[\frac{\partial\epsilon\left(\mathbf{N},M\right)}{\partial N_{k}}\]
{\small \begin{eqnarray*}
& = & E\left\{ \frac{\partial}{\partial N_{k}}\left[f\left(\mathbf{u}\left(t\right)\right)-\mathbf{N^{T}}\mathbf{u_{0}}\left(t\right)-M\right]^{2}\right\} \\
& = & E\left\{ 2\left[f\left(\mathbf{u}\left(t\right)\right)-\mathbf{N^{T}u_{0}}\left(t\right)-M\right]\left[-\frac{\partial}{\partial N_{k}}\left(\mathbf{N^{T}u_{0}}\left(t\right)\right)\right]\right\} \\
& = & E\left\{ 2\left[f\left(\mathbf{u}\left(t\right)\right)-\mathbf{N^{T}u_{0}}\left(t\right)-M\right]\left[-\frac{\partial}{\partial N_{k}}\sum_{i=1}^{n}N_{i}u_{0i}\left(t\right)\right]\right\} \\
& = & E\left\{ 2\left[f\left(\mathbf{u}\left(t\right)\right)-\mathbf{N^{T}u_{0}}\left(t\right)-M\right]\left[-u_{0k}\left(t\right)\right]\right\}
\end{eqnarray*}}
\[\therefore\frac{\partial\epsilon\left(\mathbf{N},M\right)}{\partial N_{k}}\]
{
\begin{eqnarray*}
& = & -2E\left[u_{0k}\left(t\right)f\left(\mathbf{u}\left(t\right)\right)\right]+2E\left[u_{0k}\left(t\right)\mathbf{N^{T}u_{0}}\left(t\right)\right]\\
& & +2E\left[u_{0k}\left(t\right)M\right]\\
& = & -2E\left[u_{0k}\left(t\right)f\left(\mathbf{u}\left(t\right)\right)\right]+2E\left[u_{0k}\left(t\right)\sum_{i=1}^{n}N_{i}u_{0i}\left(t\right)\right]\\
& & +2E\left[u_{0k}\left(t\right)\right]M\\
& = & -2E\left[u_{0k}\left(t\right)f\left(\mathbf{u}\left(t\right)\right)\right]+2\sum_{i=1}^{n}N_{i}E\left[u_{0k}\left(t\right)u_{0i}\left(t\right)\right]
\end{eqnarray*}
}
as clearly, $u_{0k}\left(t\right)=u_{k}\left(t\right)-\mu_{k}\left(t\right)$
are zero-mean processes.
\[\therefore\frac{\partial\epsilon\left(\mathbf{N},M\right)}{\partial\mathbf{N}}\]
\begin{eqnarray*}
= \left[\begin{array}{c}
\frac{\partial\epsilon\left(\mathbf{N},M\right)}{\partial N_{1}}\\
\frac{\partial\epsilon\left(\mathbf{N},M\right)}{\partial N_{2}}\\
\vdots\\
\frac{\partial\epsilon\left(\mathbf{N},M\right)}{\partial N_{n}}
\end{array}\right]& = & -2E\left(\left[\begin{array}{c}
u_{01}\left(t\right)f\left(\mathbf{u}\left(t\right)\right)\\
u_{02}\left(t\right)f\left(\mathbf{u}\left(t\right)\right)\\
\vdots\\
u_{0n}\left(t\right)f\left(\mathbf{u}\left(t\right)\right)
\end{array}\right]\right)\\
& & +2\textrm{cov}\left[\mathbf{u_0}\left(t\right)\right]\left[\begin{array}{c}
N_{1}\\
N_{2}\\
\vdots\\
N_{n}
\end{array}\right]
\end{eqnarray*}
where
\[\textrm{cov}\left[\mathbf{u_0}\left(t\right)\right]=E\left[\mathbf{u_{0}}\left(t\right)\mathbf{u_{0}^{T}}\left(t\right)\right]\]
{ \scriptsize
\[
=\left[\begin{array}{cccc}
\sigma_{u_{01}}^{2} & E\left[u_{01}\left(t\right)u_{02}\left(t\right)\right] & \cdots & E\left[u_{01}\left(t\right)u_{0n}\left(t\right)\right]\\
E\left[u_{02}\left(t\right)u_{01}\left(t\right)\right] & \sigma_{u_{02}}^{2} & \cdots & E\left[u_{02}\left(t\right)u_{0n}\left(t\right)\right]\\
\vdots & \vdots & \ddots & \vdots\\
E\left[u_{0n}\left(t\right)u_{01}\left(t\right)\right] & E\left[u_{0n}\left(t\right)u_{02}\left(t\right)\right] & \cdots & \sigma_{u_{0n}}^{2}
\end{array}\right]
\]
}
is the covariance matrix of $\mathbf{u_{0}}\left(t\right)$.
\[\therefore\frac{\partial\epsilon\left(\mathbf{N},M\right)}{\partial\mathbf{N}} = -2E\left[f\left(\mathbf{u}\left(t\right)\right)\mathbf{u_{0}}\left(t\right)\right]+2\textrm{cov}\left[\mathbf{u_{0}}\left(t\right)\right]\mathbf{N}\]
The term $E\left[f\left(\mathbf{u_{0}}\left(t\right)\right)\mathbf{u_{0}}\left(t\right)\right]$
can be expanded using the following result from \cite{Kazakov}:
\[
E\left[g\left(\boldsymbol{\eta}\right)\boldsymbol{\eta}\right]=E\left[\boldsymbol{\eta\eta^{T}}\right]E\left[\boldsymbol{\nabla}g\left(\boldsymbol{\eta}\right)\right]
\] where $\boldsymbol{\eta}$ is any $n\times1$ jointly Gaussian vector and
\[\boldsymbol{\nabla}=\left[\begin{array}{cccc}
\frac{\partial}{\partial\eta_{1}} & \frac{\partial}{\partial\eta_{2}} & \cdots & \frac{\partial}{\partial\eta_{n}}\end{array}\right]^{\mathbf{T}}\] is the gradient operator.
Hence,
\[\therefore\frac{\partial\epsilon\left(\mathbf{N},M\right)}{\partial\mathbf{N}}\]
\begin{eqnarray*}
& = & -2E\left[\mathbf{u_{0}}\left(t\right)\mathbf{u_{0}^{T}}\left(t\right)\right]E\left[\boldsymbol{\nabla}f\left(\mathbf{u}\left(t\right)\right)\right]+2\textrm{cov}\left[\mathbf{u_{0}}\left(t\right)\right]\mathbf{N}\\
& = & -2\textrm{cov}\left[\mathbf{u_{0}}\left(t\right)\right]E\left[\boldsymbol{\nabla}f\left(\mathbf{u}\left(t\right)\right)\right]+2\textrm{cov}\left[\mathbf{u_{0}}\left(t\right)\right]\mathbf{N}\\
& = & 2\textrm{cov}\left[\mathbf{u_{0}}\left(t\right)\right]\left\{ \mathbf{N}-E\left[\boldsymbol{\nabla}f\left(\mathbf{u}\left(t\right)\right)\right]\right\}
\end{eqnarray*}
$\epsilon\left(\mathbf{N},M\right)$ is minimized when $\frac{\partial\epsilon\left(\mathbf{N},M\right)}{\partial\mathbf{N}}=\mathbf{0}$,
i.e. when $\mathbf{N}=~E\left[\boldsymbol{\nabla}f\left(\mathbf{u}\left(t\right)\right)\right]$,
since $\textrm{cov}\left[\mathbf{u_{0}}\left(t\right)\right]$ is
positive definite.
The term $\frac{\partial\epsilon\left(\mathbf{N},M\right)}{\partial M}$ is calculated as follows.
{\small\begin{eqnarray*}
\frac{\partial\epsilon\left(\mathbf{N},M\right)}{\partial M} & = & E\left\{ \frac{\partial}{\partial M}\left[f\left(\mathbf{u}\left(t\right)\right)-\mathbf{N^{T}}\mathbf{u_{0}}\left(t\right)-M\right]^{2}\right\} \\
& = & E\left\{ 2\left[-f\left(\mathbf{u}\left(t\right)\right)+\mathbf{N^{T}u_{0}}\left(t\right)+M\right]\right\} \\
& = & -2E\left[f\left(\mathbf{u}\left(t\right)\right)\right]+2E\left[\mathbf{N^{T}u_{0}}\left(t\right)\right]+2E\left[M\right]\\
& = & -2E\left[f\left(\mathbf{u}\left(t\right)\right)\right]+2E\left[\sum_{i=1}^{n}N_{i}u_{0i}\left(t\right)\right]+2M\\
& = & -2E\left[f\left(\mathbf{u}\left(t\right)\right)\right]+2\sum_{i=1}^{n}N_{i}E\left[u_{0i}\left(t\right)\right]+2M\\
& = & -2E\left[f\left(\mathbf{u}\left(t\right)\right)\right]+2M\\
& = & 2\left\{ M-E\left[f\left(\mathbf{u}\left(t\right)\right)\right]\right\}
\end{eqnarray*}}
$\epsilon\left(\mathbf{N},M\right)$ is minimized when $\frac{\partial\epsilon\left(\mathbf{N},M\right)}{\partial M}=0$,
i.e. when $M=~E\left[f\left(\mathbf{u}\left(t\right)\right)\right]$.
This completes the proof.
\end{proof}
Equations \eqref{neq} and \eqref{meqd} are similar to \eqref{neqs} and \eqref{meqs} respectively, with some differences. There are now $n$ quasilinear \textit{gains} forming a vector $\mathbf{N}$. Also, the single input $u(t)$ in \eqref{neqs} and \eqref{meqs} is replaced by a multiple-input vector $\mathbf{u}(t)$, and the derivative in \eqref{neqs} replaced by a gradient in \eqref{neq}. The process is illustrated in Fig. \ref{fig:p4}.
\begin{figure}
\centering
\includegraphics[width=1\linewidth]{./Figures/msl}
\caption{The process of Multi-variable Stochastic Linearization. Here $N_1,N_2,\dots,N_n$ are the quasilinear gains forming the vector $\mathbf{N}$ defined in \eqref{neq} and $M$ is the quasilinear bias defined in \eqref{meq}. $u_{01},u_{02},\dots,u_{0n}$ are the zero-mean parts of $u_1,u_2,\dots,u_n$ respectively.}
\label{fig:p4}
\end{figure}
\section{Application to Bivariate Saturation}\label{abs}
In this section, the method of multivariate stochastic linearization is applied to the bivariate saturation nonlinearity, which is illustrated in Fig. \ref{bsn} and defined as follows:
\[\textrm{sat}_{\alpha,\beta}\left[u_{1}\left(t\right),u_{2}\left(t\right)\right]
\]
{\scriptsize
\begin{equation}\label{bsat}
=\begin{cases}
\begin{cases}
\beta+u_{2}\left(t\right), & u_{1}\left(t\right)>\beta+u_{2}\left(t\right)\\
u_{1}\left(t\right), & \alpha-u_{2}\left(t\right)\leq u_{1}\left(t\right)\leq\beta+u_{2}\left(t\right)\\
\alpha-u_{2}\left(t\right), & u_{1}\left(t\right)<\alpha-u_{2}\left(t\right)
\end{cases} & ,u_{2}\left(t\right)\geq\textrm{max}\left(-\beta,\alpha\right)\\
0 & ,u_{2}\left(t\right)<\textrm{max}\left(-\beta,\alpha\right)
\end{cases}
\end{equation}
}
We call $u_1(t)$ the primary input and $u_2(t)$ the secondary input. On substituting the nonlinear function \eqref{bsat} for $f(\cdot)$ in \eqref{neq} and \eqref{meqd}, the values of $N_1$ and $N_2$ can be found out to be:
\begin{figure}[b]
\centering
{\footnotesize{}\begin{tikzpicture}[every node/.style={transform shape}, >=latex]
\draw [->,thick](-2.5,0)--(2,0) node[anchor=west] {$u_1\left(t\right)$};
\draw [->,thick](0,-2)--(0,1.5) node[anchor=south] {$\textrm{sat}_{\alpha}\left[u_1\left(t\right),u_2\left(t\right)\right]$};
\draw[thick] (-2pt,1) node[anchor=east]{$\beta$}--(2pt,1);
\draw[thick] (-2pt,-1.5) node[anchor=east]{$\alpha$}--(2pt,-1.5);
\draw[thick] (-1.5,-2pt) node[anchor=south]{$\alpha-u_2\left(t\right)$}--(-1.5,2pt);
\draw[thick] (1,2pt) node[anchor=north]{$\beta+u_2\left(t\right)$}--(1,-2pt);
\draw [color=gray](-2.5,-1.5)--(-1.5,-1.5);
\draw (-1.5,-1.5)--(1,1);
\draw [color=gray](1,1)--(2,1);
\draw[color=blue, samples=40] plot[domain=1:2] (\x,1+rand*0.1) node[anchor=west] {$u_2 \left(t\right)$};
\draw[color=blue, samples=40] plot[domain=-2.5:-1.5] (\x,-1.5-rand*0.1) (-2.5,-1.5)node[anchor=east] {$-u_2 \left(t\right)$};
\end{tikzpicture}}{\footnotesize \par}
{\footnotesize{}\caption{Bivariate Saturation Nonlinearity}\label{bsn}
}{\footnotesize \par}
\end{figure}
\begin{gather*}
\mathbf{N}=\left[\begin{array}{cc}
N_{1} & N_{2}\end{array}\right]^{T} = E\left\{ \boldsymbol{\nabla}\textrm{sat}_{\alpha,\beta}\left[u_{1}\left(t\right),u_{2}\left(t\right)\right]\right\} \\
= E\left\{ \left[\begin{array}{c}
\frac{\partial}{\partial u_{1}}\textrm{sat}_{\alpha,\beta}\left[u_{1}\left(t\right),u_{2}\left(t\right)\right]\\
\frac{\partial}{\partial u_{2}}\textrm{sat}_{\alpha,\beta}\left[u_{1}\left(t\right),u_{2}\left(t\right)\right]
\end{array}\right]\right\} \\
{\tiny =\begin{cases}
E\left[\begin{array}{c}
\begin{cases}
0, & u_{1}\left(t\right)>\beta+u_{2}\left(t\right)\\
1, & \alpha-u_{2}\left(t\right)\leq u_{1}\left(t\right)\leq\beta+u_{2}\left(t\right)\\
0, & u_{1}\left(t\right)<\alpha-u_{2}\left(t\right)
\end{cases}\\
\begin{cases}
1, & u_{1}\left(t\right)>\beta+u_{2}\left(t\right)\\
0, & \alpha-u_{2}\left(t\right)\leq u_{1}\left(t\right)\leq\beta+u_{2}\left(t\right)\\
-1 & u_{1}\left(t\right)<\alpha-u_{2}\left(t\right)
\end{cases}
\end{array}\right] & ,u_{2}\left(t\right)\geq\textrm{max}\left(-\beta,\alpha\right)\\
0 & ,u_{2}\left(t\right)<\textrm{max}\left(-\beta,\alpha\right)
\end{cases}}
\end{gather*}
or re-written as in \eqref{n1e1} and \eqref{n2e1}.
\begin{figure*}[t]
\centering
\begin{tikzpicture}[>=latex,scale=0.9,every node/.style={transform shape}, every text node part/.style={align=center}]
\node (wr) at (0,0){$w_r$};
\node[rectangle,draw] (rf) [below=0.5 of wr,label={left:$F_{\Omega_r}\left(s\right)$}]{$\dot{x}_r=A_{rf}x_r+B_{rf}w_r$\\$r_f=C_{rf}x_r+D_{rf}w_r$};
\node[rectangle,draw] (sr) [below=0.5 of rf]{$\sigma_r$};
\node[circle,draw] (cr) [below=0.5 of sr]{+};
\node (mur) [left=0.5 of cr]{$\mu_r$};
\node[circle,draw] (sum) [below right=0.8 of cr]{};
\node[rectangle,draw] (C) [right=0.8 of sum,label={below:$C\left(s\right)$}]{$\dot{x}_C=A_{C}x_C+B_{C}e$\\$u=C_{C}x_C+D_Ce$};
\node[rectangle,draw] (sat) [right= of C]{$\textrm{sat}_{\alpha}^{\beta}\left(u,{\beta}_n\right)$};
\node[circle,draw] (dsum) [right=3 of sat]{+};
\node[circle,draw] (cb) [above=0.5 of sat]{+};
\node (mub) [left=0.5 of cb]{$\mu_{\beta}$};
\node[rectangle,draw] (sb) [above=0.5 of cb]{$\sigma_{\beta}$};
\node[rectangle,draw] (bf) [above=0.5 of sb,label={left:$F_{\Omega_{\beta}}\left(s\right)$}]{$\dot{x}_{\beta}=A_{{\beta}f}x_{\beta}+B_{{\beta}f}w_{\beta}$\\${\beta}_f=C_{\beta f}x_{\beta}+D_{\beta f}w_{\beta}$};
\node (wb) [above=0.5 of bf]{$w_{\beta}$};
\node[circle,draw] (cd) [above=0.5 of dsum]{+};
\node (mud) [right=0.5 of cd]{$\mu_{d}$};
\node[rectangle,draw] (sd) [above=0.5 of cd]{$\sigma_{d}$};
\node[rectangle,draw] (df) [above=0.5 of sd,label={right:$F_{\Omega_d}\left(s\right)$}]{$\dot{x}_{d}=A_{df}x_{d}+B_{df}w_{d}$\\${d}_f=C_{df}x_{d}+D_{df}w_{d}$};
\node (wd) [above=0.5 of df]{$w_{d}$};
\node[rectangle,draw] (P) [right= of dsum,label={below:$P\left(s\right)$}]{$\dot{x}_P=A_{P}x_P+B_{P}z$\\$y=C_{P}x_P+D_Pz$};
\node[coordinate] (y) [right= of P]{};
\draw[->] (P)--node[above]{$y\left(t\right)$}(y)--($(y)+(0,-2)$)-|node[pos=0.95,left]{-}(sum);
\draw[->] (wr)--(rf);
\draw[->] (rf)--(sr);
\draw[->] (sr)--(cr);
\draw[->] (mur)--(cr);
\draw[->] (cr)|-node[pos=0.4,below]{$r\left(t\right)$}node[very near end,below]{+}(sum);
\draw[->] (wb)--(bf);
\draw[->] (bf)--(sb);
\draw[->] (sb)--(cb);
\draw[->] (mub)--(cb);
\draw[->] (cb)--node[right]{$\beta_n\left(t\right)$}(sat);
\draw[->] (wd)--(df);
\draw[->] (df)--(sd);
\draw[->] (sd)--(cd);
\draw[->] (mud)--(cd);
\draw[->] (cd)--node[left]{$d\left(t\right)$}(dsum);
\draw[->] (sum)--node[above]{$e\left(t\right)$}(C);
\draw[->] (C)--node[above]{$u\left(t\right)$}(sat);
\draw[->] (sat)--node[above]{$v\left(t\right)$}(dsum);
\draw[->] (dsum)--node[above]{$z\left(t\right)$}(P);
\end{tikzpicture}
\caption{Block Diagram of Control System\label{fig:Block-Diagram-of}}
\end{figure*}
\begin{equation}\label{n1e1}
N_{1}=\int_{\max\left(-\beta,\alpha\right)}^{\infty}\int_{\alpha-u_{2}}^{\beta+u_{2}}\left(1\right)\mathcal{N}\left(\mu_{1},\mu_{2},\sigma_{1},\sigma_{2},\rho\right)du_{1}du_{2}
\end{equation}
\begin{equation}\label{n2e1}
\begin{gathered}
N_{2}=\int_{\max\left(-\beta,\alpha\right)}^{\infty}\int_{-\infty}^{\alpha-u_{2}}\left(-1\right)\mathcal{N}\left(\mu_{1},\mu_{2},\sigma_{1},\sigma_{2},\rho\right)du_{1}du_{2}\\
+\int_{\max\left(-\beta,\alpha\right)}^{\infty}\int_{\beta+u_{2}}^{\infty}\left(1\right)\mathcal{N}\left(\mu_{1},\mu_{2},\sigma_{1},\sigma_{2},\rho\right)du_{1}du_{2}
\end{gathered}
\end{equation}
where $\mu_1$ and $\mu_2$ are the means of the inputs $u_1$ and $u_2$ respectively, $\sigma_1$ and $\sigma_2$ being their corresponding standard deviations, $\rho$ is the correlation between $u_1$ and $u_2$, and:
{\begin{equation}
\begin{gathered}\label{npdf}
\mathcal{N}\left(\mu_{1},\mu_{2},\sigma_{1},\sigma_{2},\rho\right)=\frac{1}{2\pi\sigma_{1}\sigma_{2}\sqrt{1-\rho^{2}}}e^{-\frac{{u}_{1}^{*2}+{u}_{2}^{*2}-2\rho u_{1}^{*}u_{2}^{*}}{2\left(1-\rho^{2}\right)}}
\end{gathered}
\end{equation}}
is the bivariate Gaussian PDF in which
\[u_{1}^*=\frac{u_{1}-\mu_{1}}{\sigma_{1}}\]
\[u_{2}^*=\frac{u_{2}-\mu_{2}}{\sigma_{2}}\]
The value of $M$ can be found from \eqref{meqd} and can be written as:
\begin{equation}\label{me1}
\small
\begin{gathered}
M=\int_{\max\left(-\beta,\alpha\right)}^{\infty}\int_{-\infty}^{\alpha-u_{2}}\left(\alpha-u_{2}\right)\mathcal{N}\left(\mu_{1},\mu_{2},\sigma_{1},\sigma_{2},\rho\right)du_{1}du_{2}\\
+\int_{\max\left(-\beta,\alpha\right)}^{\infty}\int_{\alpha-u_{2}}^{\beta+u_{2}}u_{1}\mathcal{N}\left(\mu_{1},\mu_{2},\sigma_{1},\sigma_{2},\rho\right)du_{1}du_{2}\\
+\int_{\max\left(-\beta,\alpha\right)}^{\infty}\int_{\beta+u_{2}}^{\infty}\left(\beta+u_{2}\right)\mathcal{N}\left(\mu_{1},\mu_{2},\sigma_{1},\sigma_{2},\rho\right)du_{1}du_{2}
\end{gathered}
\end{equation}
The integrals in the RHS of \eqref{n1e1}, \eqref{n2e1} and \eqref{me1} do not have closed form expressions, but can be computed numerically, for example, using vectorized adaptive quadrature, which is used by \texttt{integral2} in MATLAB \cite{shampine}. It has been practically observed that numerical computation is much slower using the integrals as they are. They are much faster to compute, by more than 2 orders of magnitude, if they are transformed as follows and resulting the inner integral reduced to a closed form expression. Applying the transformations:
\[u'_{1}=\frac{\frac{u_{1}-\mu_{1}}{\sigma_{1}}-\rho\frac{u_{2}-\mu_{2}}{\sigma_{2}}}{\sqrt{1-\rho^{2}}}\] and
\[u'_{2}=\frac{u_{2}-\mu_{2}}{\sigma_{2}}\]
we get:
\[N_1=\int_{u'_{2min}}^{\infty}\int_{u'_{1min}}^{u'_{1max}}\left(1\right)\frac{1}{2\pi}e^{-\left(u_{1}^{'2}+u{}_{2}^{'2}\right)}d'u_{1}d'u_{2}\]
where:
\begin{equation}\label{up1min}
u'_{1\textrm{min}} = \frac{\alpha-\left(\mu_{1}+\mu_{2}\right)+u'_{2}\left(\rho\sigma_{1}+\sigma_{2}\right)}{\sigma_{1}\sqrt{1-\rho^{2}}}
\end{equation}
\begin{equation}\label{up1max}
u'_{1\textrm{max}}=\frac{\beta-\left(\mu_{1}-\mu_{2}\right)+u'_{2}\left(\sigma_{2}-\rho\sigma_{1}\right)}{\sigma_{1}\sqrt{1-\rho^{2}}}
\end{equation}
\begin{equation}\label{u2pmin}
u'_{2\textrm{min}}=\frac{\max\left(\alpha,-\beta\right)-\mu_{2}}{\sigma_{2}}
\end{equation}
On simplifying the inner integral, we get:
\begin{equation}\label{n1e2}
N_1=\int_{u'_{2\textrm{min}}}^{\infty}\frac{\sqrt{2}e^{-\frac{{u'_{2}}^{2}}{2}}}{4\sqrt{\pi}}\left[\mathrm{erf}\left(\gamma_1\right)+\mathrm{erf}\left(\gamma_2\right)\right]du'_{2}
\end{equation}
where:
\begin{equation}\label{g1}
\gamma_{1}=\frac{\sqrt{2}\left(\mu_{1}-\alpha+\mu_{2}+\sigma_{2}u'_{2}+\rho\sigma_{1}u'_{2}\right)}{2\sigma_{1}\sqrt{1-\rho^{2}}}
\end{equation}
\begin{equation}\label{g2}
\gamma_{2}=\frac{\sqrt{2}\,\left(\mathrm{\beta}-\mu_{1}+\mu_{2}+\sigma_{2}\,u'_{2}-\rho\,\sigma_{1}\,u'_{2}\right)}{2\sigma_{1}\sqrt{1-\rho^{2}}}
\end{equation}
Similarly,
\begin{equation*}
\begin{gathered}
N_{2}=\int_{u'_{2\textrm{min}}}^{\infty}\int_{-\infty}^{u'_{1\textrm{min}}}\left(-1\right)\frac{1}{2\pi}e^{-\left(u_{1}^{'2}+u{}_{2}^{'2}\right)}du'_{1}du'_{2}
\\+\int_{u'_{2\textrm{min}}}^{\infty}\int_{u'_{1\textrm{max}}}^{\infty}\left(1\right)\frac{1}{2\pi}e^{-\left(u_{1}^{'2}+u{}_{2}^{'2}\right)}du'_{1}du'_{2}
\end{gathered}
\end{equation*}
On simplifying,
\begin{equation}\label{n2e2}
N_{2}=\int_{u'_{2\textrm{min}}}^{\infty}\frac{\sqrt{2}e^{-\frac{u_{2}^{2}}{2}}}{4\,\sqrt{\pi}}\left[\mathrm{erf}\left(\gamma_{1}\right)-\mathrm{erf}\left(\gamma_{2}\right)\right]du'_{2}
\end{equation}
Finally,
\begin{equation*}
\begin{gathered}
M=\int_{u'_{2\textrm{min}}}^{\infty}\left[\int_{-\infty}^{u'_{1\textrm{min}}}\left(\alpha-\mu_{2}-u'_{2}\sigma_{2}\right)\frac{1}{2\pi}e^{-\left(u_{1}^{'2}+u{}_{2}^{'2}\right)}du'_{1}\right.\\+\int_{u'_{1\textrm{min}}}^{u'_{1\textrm{max}}}\left(\mu_{1}+\sigma_{1}\sqrt{1-\rho^{2}}u'_{1}+\rho\sigma_{1}u'_{2}\right)\frac{1}{2\pi}e^{-\left(u_{1}^{'2}+u{}_{2}^{'2}\right)}du'_{1}\\\left.+\int_{u'_{1\textrm{max}}}^{\infty}\left(\beta+\mu_{2}+u'_{2}\sigma_{2}\right)\frac{1}{2\pi}e^{-\left(u_{1}^{'2}+u{}_{2}^{'2}\right)}du'_{1}\right]du'_{2}
\end{gathered}
\end{equation*}
On simplifying,
\begin{equation}\label{me2}
\begin{gathered}
M=\int_{u'_{2 \textrm{min}}}^{\infty}\frac{\sqrt{2}\,{\mathrm{e}}^{-\frac{{u'_{2}}^{2}}{2}}\,\left(\mathrm{erf}\left(\gamma_{1}\right)-1\right)\,\left(\mu_{2}-\alpha+\sigma_{2}\,u'_{2}\right)}{4\,\sqrt{\pi}}du'_{2}\\
+\int_{u'_{2 \textrm{min}}}^{\infty}\frac{\sigma_{1}\sqrt{1-\rho^{2}}}{2\pi}e^{-\frac{{u'}_{2}^{2}}{2}}\left(e^{-\gamma_{1}^{2}}-e^{-\gamma_{2}^{2}}\right)du'_{2}\\
+\int_{u'_{2\textrm{min}}}^{\infty}\frac{\sqrt{2}}{4\sqrt{\pi}}\left(\mu_{1}+\rho\sigma_{1}u'_{2}\right){\mathrm{e}}^{-\frac{{u'_{2}}^{2}}{2}}\left[\mathrm{erf}\left(\gamma_{1}\right)+\mathrm{erf}\left(\gamma_{2}\right)\right]du'_{2}\\
-\int_{u'_{2 \textrm{min}}}^{\infty}\frac{\sqrt{2}\,{\mathrm{e}}^{-\frac{{u'_{2}}^{2}}{2}}\,\left(\mathrm{erf}\left(\gamma_{2}\right)-1\right)\,\left(\beta+\mu_{2}+\sigma_{2}\,u'_{2}\right)}{4\,\sqrt{\pi}}du'_{2}
\end{gathered}
\end{equation}
\section{Quasilinear Control of Closed Loop Control System}\label{qlccl}
\subsection{Description of the Closed Loop System}
Consider the control system shown in Fig. \ref{fig:Block-Diagram-of}. Similar to the system in Fig. \ref{figsv}, it has a plant $P\left(s\right)$ and a controller $C\left(s\right)$. However, this time, the actuator is a bivariate saturation $\textrm{sat}_{\alpha,\beta}\left[u_1\left(t\right),u_2\left(t\right)\right]$, with random bounds modeled as a second input $u_2\left(t\right)$. Similar to the generation of the reference $r\left(t\right)$ and the disturbance $d\left(t\right)$ described in subsection \ref{clssv}, the second actuator input $u_2(t)$ is generated by passing white noise $w_{\beta}(t)$ through a coloring filter $F_{\Omega_{\beta}}(s)$ with $H_2$-norm equal to 1, scaling it by $\sigma_2$, and adding a bias $\mu_2$. This ensures that he reference $r\left(t\right)$, the disturbance $d\left(t\right)$ and the second actuator input $u_2(t)$ are WSS Gaussian random processes with means $\mu_{r}$, $\mu_{d}$, $\mu_2$, and standard deviations $\sigma_{r},\sigma_{d}$ and $\sigma_2$ respectively. The coloring filters band-limit the white noises $w_r$, $w_d$ and $w_{\beta}$ to a desired bandwidth, which is desirable to be close to the system bandwidth. The block diagram shows the corresponding state-space representations.
\begin{figure*}[t]
\centering
\begin{tikzpicture}[>=latex,scale=0.9,every node/.style={transform shape}, every text node part/.style={align=center}]
\node (wr) at (0,0){$w_r$};
\node[rectangle,draw] (rf) [below=0.5 of wr]{$\dot{x}_r=A_{rf}x_r+B_{rf}w_r$\\$r_f=C_{rf}x_r+D_{rf}w_r$};
\node[rectangle,draw] (sr) [below=0.5 of rf]{$\sigma_r$};
\node[circle,draw] (cr) [below=0.5 of sr]{+};
\node (mur) [left=0.5 of cr]{$\mu_r$};
\node[circle,draw] (sum) [below right=0.8 of cr]{};
\node[rectangle,draw] (C) [right=0.8 of sum]{$\dot{x}_C=A_{C}x_C+B_{C}e$\\$u_1=C_{C}x_C+D_Ce$};
\node[rectangle,draw,fill=yellow] (N1) [right= of C]{$N_1$};
\node[circle,draw,fill=yellow] (satc)[right=0.5 of N1]{+};
\node[circle,draw] (dsum) [right=3 of satc]{+};
\node[rectangle,draw,fill=yellow] (N2) [above=0.5 of satc]{$N_2$};
\node (m) [below=0.5 of satc]{$m=M-N_1\mu_1-N_2\mu_{2}$};
\node[circle,draw] (cb) [above=0.5 of N2]{+};
\node (mub) [left=0.5 of cb]{$\mu_{2}$};
\node[rectangle,draw] (sb) [above=0.5 of cb]{$\sigma_{2}$};
\node[rectangle,draw] (bf) [left=0.5 of sb]{$\dot{x}_{\beta}=A_{{\beta}f}x_{\beta}+B_{{\beta}f}w_{\beta}$\\${\beta}_f=C_{\beta f}x_{\beta}+D_{\beta f}w_{\beta}$};
\node (wb) [above=0.5 of bf]{$w_{\beta}$};
\node[circle,draw] (cd) [above=0.5 of dsum]{+};
\node (mud) [right=0.5 of cd]{$\mu_{d}$};
\node[rectangle,draw] (sd) [above=0.5 of cd]{$\sigma_{d}$};
\node[rectangle,draw] (df) [above=0.5 of sd]{$\dot{x}_{d}=A_{df}x_{d}+B_{df}w_{d}$\\${d}_f=C_{df}x_{d}+D_{df}w_{d}$};
\node (wd) [above=0.5 of df]{$w_{d}$};
\node[rectangle,draw] (P) [right= of dsum]{$\dot{x}_P=A_{P}x_P+B_{P}z$\\$y=C_{P}x_P+D_Pz$};
\node[coordinate] (y) [right= of P]{};
\draw[->] (P)--node[above]{$\hat{y}\left(t\right)$}(y)--($(y)+(0,-2)$)-|node[pos=0.95,left]{-}(sum);
\draw[->] (wr)--(rf);
\draw[->] (rf)--(sr);
\draw[->] (sr)--(cr);
\draw[->] (mur)--(cr);
\draw[->] (cr)|-node[pos=0.4,below]{$r\left(t\right)$}node[very near end,below]{+}(sum);
\draw[->] (wb)--(bf);
\draw[->] (bf)--(sb);
\draw[->] (sb)--(cb);
\draw[->] (mub)--(cb);
\draw[->] (cb)--node[right]{$u_2\left(t\right)$}(N2);
\draw[->] (wd)--(df);
\draw[->] (df)--(sd);
\draw[->] (sd)--(cd);
\draw[->] (mud)--(cd);
\draw[->] (cd)--node[left]{$d\left(t\right)$}(dsum);
\draw[->] (sum)--node[above]{$\hat{e}\left(t\right)$}(C);
\draw[->] (C)--node[above]{$\hat{u}_1\left(t\right)$}(N1);
\draw[->] (N1)--(satc);
\draw[->] (N2)--(satc);
\draw[->] (m)--(satc);
\draw[->] (satc)--node[above]{$\hat{v}\left(t\right)$}(dsum);
\draw[->] (dsum)--node[above]{$\hat{z}\left(t\right)$}(P);
\end{tikzpicture}
\caption{Stochastically linearized version of Fig. \ref{fig:Block-Diagram-of}}
\label{sl}
\end{figure*}
Applying stochastic linearization to this system, we get the system of Fig. \ref{sl}. To find the values of the quasilinear gains $N_1$, $N_2$ and the quasilinear bias $M$, it is required to compute the means and standard deviations of $u_1(t)$ and $u_2(t)$. This process can be simplified by separating the input signals into their zero-mean random parts and a constant mean part, considering two different systems and adding the results, as investigated in the following subsection.
\subsection{Decomposition into two sub-systems}
Consider the following stochastic state space equation as a general representation of the system in Fig.~\ref{sl}:
\begin{equation}\label{sde}
d\mathbf{x}=\mathbf{Ax}dt+\sum_{i=1}^{n}\mathbf{b_{i}}dw_{i}
\end{equation}
where $\mathbf{x}=\left[\begin{array}{cccc}
x_{1} & x_{2} & \ldots & x_{n}\end{array}\right]^T$ are the states of the system and $w_i$ are the stochastic inputs. It is known that, provided the system is asymptotically stable, the covariance matrix associated with $\mathbf{x}$, $\mathbf{\Sigma}$, is the solution of the following Lyapunov equation \cite{Brockett}:
\begin{equation}\label{lyapeq}
\mathbf{A}\mathbf{\Sigma}+\mathbf{\Sigma}\mathbf{A^T}+\sum_{i=1}^{n}\mathbf{b_{i}}\mathbf{b_{i}}^{\mathbf{T}}=\mathbf{0}
\end{equation}
where:
{\footnotesize\begin{align}\label{sigma}
\Sigma & =\left[\begin{array}{cccc}
E\left[x_{1}^{2}\left(t\right)\right] & E\left[x_{1}\left(t\right)x_{2}\left(t\right)\right] & \cdots & E\left[x_{1}\left(t\right)x_{n}\left(t\right)\right]\\
E\left[x_{2}\left(t\right)x_{1}\left(t\right)\right] & E\left[x_{2}^{2}\left(t\right)\right] & \ddots & E\left[x_{2}\left(t\right)x_{n}\left(t\right)\right]\\
\vdots & \ddots & \ddots & \vdots\\
E\left[x_{n}\left(t\right)x_{1}\left(t\right)\right] & E\left[x_{n}\left(t\right)x_{2}\left(t\right)\right] & \cdots & E\left[x_{n}^{2}\left(t\right)\right]
\end{array}\right]
\end{align}}is the covariance matrix. For the following Theorem, we denote a system described by \eqref{sde} by $S_{\mathbf{\Sigma}}$ where $\mathbf{\Sigma}$ is its corresponding covariance matrix from \eqref{lyapeq}.
\begin{theorem}
Consider a system $S_{\mathbf{\Sigma}}$, having a covariance matrix $\mathbf{\Sigma}$. Also consider two other systems, $S_{\mathbf{\Sigma}_1}$ and $S_{\mathbf{\Sigma}_2}$, having covariance matrices $\mathbf{\Sigma}_1$ and $\mathbf{\Sigma}_2$ respectively. One of them, $S_{\mathbf{\Sigma}_1}$, is driven by the zero-mean parts of the inputs to system $S_{\mathbf{\Sigma}}$. The other system, $S_{\mathbf{\Sigma}_2}$, is driven by constant inputs having values equal to the means of the inputs driving $S_{\mathbf{\Sigma}}$. Then $\mathbf{\Sigma}=\mathbf{\Sigma}_1+\mathbf{\Sigma}_2$.
\end{theorem}
\begin{proof}
The correlation coefficient between any two states is defined by:
\[
\rho_{ij}=\frac{\textrm{cov}\left(x_{i},x_{j}\right)}{\sigma_{i}\sigma_{j}}=\frac{E\left[x_{i}x_{j}\right]-\mu_{i}\mu_{j}}{\sigma_{i}\sigma_{j}}
\]
where $\mu_i=E\left[x_i\right]$ and $\sigma_{i}=\sqrt{E\left(x_{i}^{2}\right)-\left\{ E\left(x_{i}\right)\right\} ^{2}}$. Then,
\[
E\left[x_{i}x_{j}\right]=\rho_{ij}\sigma_{i}\sigma_{j}+\mu_{i}\mu_{j}
\]
and hence from \eqref{sigma}:
{\scriptsize
\[
\mathbf{\Sigma} =\left[\begin{array}{cccc}
\sigma_{1}^{2}+\mu_{1}^{2} & \rho_{12}\sigma_{1}\sigma_{2}+\mu_{1}\mu_{2} & \cdots & \rho_{1n}\sigma_{1}\sigma_{n}+\mu_{1}\mu_{n}\\
\rho_{21}\sigma_{2}\sigma_{1}+\mu_{2}\mu_{1} & \sigma_{2}^{2}+\mu_{2}^{2} & \ddots & \rho_{2n}\sigma_{2}\sigma_{n}+\mu_{2}\mu_{n}\\
\vdots & \ddots & \ddots & \vdots\\
\rho_{n1}\sigma_{n}\sigma_{1}+\mu_{n}\mu_{1} & \rho_{n2}\sigma_{n}\sigma_{2}+\mu_{n}\mu_{2} & \cdots & \sigma_{n}^{2}+\mu_{n}^{2}
\end{array}\right]\]}
\[
=\left[\begin{array}{cccc}
\sigma_{1}^{2} & \rho_{12}\sigma_{1}\sigma_{2} & \cdots & \rho_{1n}\sigma_{1}\sigma_{n}\\
\rho_{21}\sigma_{2}\sigma_{1} & \sigma_{2}^{2} & \ddots & \rho_{2n}\sigma_{2}\sigma_{n}\\
\vdots & \ddots & \ddots & \vdots\\
\rho_{n1}\sigma_{n}\sigma_{1} & \rho_{n2}\sigma_{n}\sigma_{2} & \cdots & \sigma_{n}^{2}
\end{array}\right]\]
\begin{equation}\label{sigma1}
+\left[\begin{array}{cccc}
\mu_{1}^{2} & \mu_{1}\mu_{2} & \cdots & \mu_{1}\mu_{n}\\
\mu_{2}\mu_{1} & \mu_{2}^{2} & \ddots & \mu_{2}\mu_{n}\\
\vdots & \ddots & \ddots & \vdots\\
\mu_{n}\mu_{1} & \mu_{n}\mu_{2} & \cdots & \mu_{n}^{2}
\end{array}\right]
\end{equation}
When the inputs are zero-mean, so are all the states, and hence, $\mu_i=0$ for $i=1,2,...,n$. Then $\mathbf{\Sigma}=\mathbf{\Sigma}_1$. When the inputs are constant and are equal to the means of the inputs to $S_{\mathbf{\Sigma}}$, there is also no variability in the states, and hence, $\sigma_i=0$ for $i=1,2,...,n$ and $\mathbf{\Sigma}=\mathbf{\Sigma}_2$. Therefore, in general, from \eqref{sigma1},
\[\mathbf{\Sigma}=\mathbf{\Sigma}_1+\mathbf{\Sigma}_2\]
\end{proof}
Hence, for the purpose of analysis, the system can be decomposed into two systems - one having all inputs as zero-mean (this is useful for computing the standard deviations of required signals) and another one - with constant inputs representing the averages of the inputs (this is useful for computing the expected values of the required signals).
\subsection{Calculation of mean and standard deviation of actuator inputs}
\begin{figure*}[t]
\begin{equation}\label{e1}
\mathbf{A}=\left(\begin{array}{ccccc}
A_{rf} & 0 & 0 & 0 & 0\\
0 & A_{\beta f} & 0 & 0 & 0\\
0 & 0 & A_{df} & 0 & 0\\
\frac{B_{c}\,C_{rf}\,\sigma_{r}}{D_{c}D_{p}N_{1}+1} & -\frac{B_{c}\,C_{\beta f}\,D_{p}\,N_{2}\,\sigma_{\beta}}{D_{c}D_{p}N_{1}+1} & -\frac{B_{c}\,C_{df}\,D_{p}\,\sigma_{d}}{D_{c}D_{p}N_{1}+1} & \frac{A_{c}+A_{c}\,D_{c}\,D_{p}\,N_{1}-B_{c}\,C_{c}\,D_{p}\,N_{1}}{D_{c}D_{p}N_{1}+1} & -\frac{B_{c}\,C_{p}}{D_{c}D_{p}N_{1}+1}\\
\frac{B_{p}\,C_{rf}\,D_{c}\,N_{1}\,\sigma_{r}}{D_{c}D_{p}N_{1}+1} & \frac{B_{p}\,C_{\beta f}\,N_{2}\,\sigma_{\beta}}{D_{c}D_{p}N_{1}+1} & \frac{B_{p}\,C_{df}\,\sigma_{d}}{D_{c}D_{p}N_{1}+1} & \frac{B_{p}\,C_{c}\,N_{1}}{D_{c}D_{p}N_{1}+1} & \frac{A_{p}+A_{p}\,D_{c}\,D_{p}\,N_{1}-B_{p}\,C_{p}\,D_{c}\,N_{1}}{D_{c}D_{p}N_{1}+1}\\
\frac{C_{rf}\,D_{c}\,\sigma_{r}}{D_{c}D_{p}N_{1}+1} & -\frac{C_{\beta f}\,D_{c}\,D_{p}\,N_{2}\,\sigma_{\beta}}{D_{c}D_{p}N_{1}+1} & -\frac{C_{df}\,D_{c}\,D_{p}\,\sigma_{d}}{D_{c}D_{p}N_{1}+1} & \frac{C_{c}}{D_{c}D_{p}N_{1}+1} & -\frac{C_{p}\,D_{c}}{D_{c}D_{p}N_{1}+1}\\
0 & C_{\beta f}\,\sigma_{\beta} & 0 & 0 & 0
\end{array}\right)
\end{equation}
\begin{equation}\label{e2}
\mathbf{B}=\left(\begin{array}{ccc}
B_{rf} & 0 & 0\\
0 & 0 & B_{bf}\\
0 & B_{df} & 0\\
\frac{B_{c}\,D_{rf}\,\sigma_{r}}{D_{c}D_{p}N_{1}+1} & -\frac{B_{c}\,D_{df}\,D_{p}\,\sigma_{d}}{D_{c}D_{p}N_{1}+1} & -\frac{B_{c}\,D_{bf}\,D_{p}\,N_{2}\,\sigma_{\beta}}{D_{c}D_{p}N_{1}+1}\\
\frac{B_{p}\,D_{c}\,D_{rf}\,N_{1}\,\sigma_{r}}{D_{c}D_{p}N_{1}+1} & \frac{B_{p}\,D_{df}\,\sigma_{d}}{D_{c}D_{p}N_{1}+1} & \frac{B_{p}\,D_{bf}\,N_{2}\,\sigma_{\beta}}{D_{c}D_{p}N_{1}+1}\\
\frac{D_{c}\,D_{rf}\,\sigma_{r}}{D_{c}D_{p}N_{1}+1} & -\frac{D_{c}\,D_{df}\,D_{p}\,\sigma_{d}}{D_{c}D_{p}N_{1}+1} & -\frac{D_{bf}\,D_{c}\,D_{p}\,N_{2}\,\sigma_{\beta}}{D_{c}D_{p}N_{1}+1}\\
0 & 0 & D_{bf}\,\sigma_{\beta}
\end{array}\right)
\end{equation}
\end{figure*}
The values of $\mathbf{A}$ and $\mathbf{B}$ for the system of Fig. \ref{sl} are shown in \eqref{e1} and \eqref{e2}. The corresponding state vector is:
\[\mathbf{x}(t)=\left[\begin{array}{ccccc}
x_{rf} & x_{\beta f} & x_{df} & x_{C} & x_{P}\end{array}\right]^T\]
with the states as depicted in Fig. \ref{sl}. To calculate the values of $N_1$, $N_2$ and $M$, the values of mean and standard deviation of the first actuator input, $\hat{u}_1\left(t\right)$, are required, along with correlation between the two actuator inputs. This is obtained by solving the Lyapunov equation \eqref{lyapeq} with $\sum_{i=1}^{n}\mathbf{b_{i}}\mathbf{b_{i}}^{\mathbf{T}}=~\mathbf{BB^T}$. The following formula can be used for this purpose \cite{Jarlebring}:
\begin{equation}\label{lyap}
\left(\mathbf{I_{n}}\otimes\mathbf{A}+\mathbf{A^{T}}\otimes\mathbf{I_{n}}\right)\textrm{vec}\left(\mathbf{\Sigma}\right)=-\textrm{vec}\left(\mathbf{B}\right)
\end{equation}
where $\mathbf{\Sigma}=E\left[\mathbf{x}(t)\mathbf{x^T}(t)\right]$, $\mathbf{I_{n}}$ is the $n\times n$ identity matrix, and $\textrm{vec}\left(\cdot\right)$ is the vectorization operator.
First, consider only the zero-mean parts of the signals. In that case, if $\hat{u}_1(t)=\mathbf{C_{1}}\mathbf{x}(t)$, then:
\begin{equation}\label{Csu}
\mathbf{C_{1}}=\left[\begin{array}{c}
\frac{C_{rf}D_{c}\sigma_{r}}{D_{c}D_{p}N_{1}+1}\\
\frac{C_{bf}D_{c}D_{p}N_{2}\sigma_{\beta}}{D_{c}D_{p}N_{1}+1}\\
\frac{C_{df}D_{c}D_{p}\sigma_{d}}{D_{c}D_{p}N_{1}+1}\\
\frac{C_{c}}{D_{c}D_{p}N_{1}+1}\\
\frac{C_{p}D_{c}}{D_{c}D_{p}N_{1}+1}
\end{array}\right]^{T}
\end{equation}
Hence,
\begin{equation}\label{s1}
\sigma_{\hat{1}}^2=\mathbf{C_{1}}\mathbf{\Sigma}\mathbf{C_{1}}^T
\end{equation}
where $\sigma_{\hat{1}}$ is the standard deviation of $\hat{u}_1$ and $\mathbf{\Sigma}$ is as defined in \eqref{sigma}.
Taking only the means of the signals as constant inputs to the system, the mean of the actuator input can be found to be:
\begin{equation}\label{muu}
\mu_{\hat{1}}=-\frac{\left(m+\mu_{d}+N_{2}\mu_{2}-\frac{1}{P_{dc}\mu_{r}}\right)}{N_{1}+\frac{1}{C_{dc}P_{dc}}}
\end{equation}
where $C_{dc}$ and $P_{dc}$ are the DC gains of $C(s)$ and $P(s)$ respectively, and
\begin{equation}\label{meq}
m=M-N_1\mu_{\hat{1}}-N_2\mu_{2}
\end{equation}
\subsection{Correlation between Actuator Inputs}
To find the correlation coefficient between the actuator inputs $\hat{u}_1(t)$ and $u_2(t)$, it is noted that if $u_2(t)=\mathbf{C_{2}x}(t)$, then:
\begin{equation}\label{Cbeta}
\mathbf{C_{2}}=\left[\begin{array}{c}
0\\
C_{bf}\sigma_{2}\\
0\\
0\\
0
\end{array}\right]^{T}\end{equation}
Hence, the correlation coefficient:
\begin{equation}\label{rho}
\rho=\frac{\mathbf{C_{1}\Sigma C_{2}^T}}{\sigma_{\hat{1}}\sigma_{2}}
\end{equation}
\subsection{Solution of Equations}
The values of $N_1$, $N_2$ and $M$ can be found by solving the system of equations \eqref{u2pmin}-\eqref{me2}, \eqref{e1}-\eqref{rho}. From \eqref{n1e2}, \eqref{n2e2} and \eqref{me2}, it can be seen that $N_1$, $N_2$ and $M$ are functions of $\mu_{\hat{1}}$, $\sigma_{\hat{1}}$, and $\rho$, i.e.,
\begin{equation}\label{n1nm}
N_{1}=\mathcal{F}_{N_{1}}\left(\mu_{\hat{1}},\sigma_{\hat{1}},\rho\right)
\end{equation}
\begin{equation}\label{n2nm}
N_{2}=\mathcal{F}_{N_{2}}\left(\mu_{\hat{1}},\sigma_{\hat{1}},\rho\right)
\end{equation}
\begin{equation}\label{mnm}
M=\mathcal{F}_{M}\left(\mu_{\hat{1}},\sigma_{\hat{1}},\rho\right)
\end{equation}
Using \eqref{muu}, \eqref{meq} and \eqref{mnm}, the following equation can be written,
\begin{equation}\label{meq1}
\frac{1}{P_{dc}\mu_{r}}-\frac{\mu_{\hat{1}}}{C_{dc}P_{dc}}-\mu_{d}=\mathcal{F}_{M}\left(\mu_{\hat{1}},\sigma_{\hat{1}},\rho\right)
\end{equation}
Equation \eqref{muu} implies that $\mu_{\hat{1}}$ is a function of $N_1$, $N_2$ and $M$, i.e.,
\begin{equation}\label{mu1nm}
\mu_{\hat{1}}=\mathcal{F}_{\mu_{\hat{1}}}\left(N_{1},N_{2},M\right)
\end{equation}
Furthermore, \eqref{s1} and \eqref{rho} imply that $\sigma_{\hat{1}}$ and $\rho$ are functions of $N_1$ and $N_2$, i.e.,
\begin{equation}\label{s1nm}
\sigma_1=\mathcal{F}_{\sigma_{\hat{1}}}\left(N_{1},N_{2}\right)
\end{equation}
\begin{equation}\label{rnm}
\rho=\mathcal{F}_{\rho}\left(N_{1},N_{2}\right)
\end{equation}
Using \eqref{mu1nm}-\eqref{rnm}, the system of equations \eqref{u2pmin}-\eqref{me2}, \eqref{e1}-\eqref{rho} can be reduced to an equivalent system consisting of \eqref{n1nm}, \eqref{n2nm} and \eqref{meq1}, which together with \eqref{mu1nm}-\eqref{rnm}, can be written as a system of 3 equations with 3 unknowns, $N_1$, $N_2$ and $M$:
\begin{equation*}
N_{1}=\mathcal{F}_{N_{1}}\left(\mathcal{F}_{\mu_{\hat{1}}}\left(N_{1},N_{2},M\right),\mathcal{F}_{\sigma_{\hat{1}}}\left(N_{1},N_{2}\right),\mathcal{F}_{\rho}\left(N_{1},N_{2}\right)\right)
\end{equation*}
\begin{equation*}
N_{2}=\mathcal{F}_{N_{2}}\left(\mathcal{F}_{\mu_{\hat{1}}}\left(N_{1},N_{2},M\right),\mathcal{F}_{\sigma_{\hat{1}}}\left(N_{1},N_{2}\right),\mathcal{F}_{\rho}\left(N_{1},N_{2}\right)\right)
\end{equation*}
\begin{equation*}
\begin{gathered}
\frac{1}{P_{dc}\mu_{r}}-\frac{\mathcal{F}_{\mu_{\hat{1}}}\left(N_{1},N_{2},M\right)}{C_{dc}P_{dc}}-\mu_{d}\\
=\mathcal{F}_{M}\left(\mathcal{F}_{\mu_{\hat{1}}}\left(N_{1},N_{2},M\right),\mathcal{F}_{\sigma_{\hat{1}}}\left(N_{1},N_{2}\right),\mathcal{F}_{\rho}\left(N_{1},N_{2}\right)\right)
\end{gathered}
\end{equation*}
A sufficient condition of the existence of solutions for the system of equations \eqref{n1nm}, \eqref{n2nm} and \eqref{meq1} is discussed in the following Theorem.
\begin{theorem}\label{exist}
Let $\mathcal{N}_{1}$, $\mathcal{N}_{2}$ and $\mathcal{M}$ denote the ranges of $\mathcal{F}_{N_{1}}$, $\mathcal{F}_{N_{2}}$ and $\mathcal{F}_{M}$ respectively in \eqref{n1nm}-\eqref{mnm}, and let $\overline{\mathcal{N}}_{1}$, $\overline{\mathcal{N}}_{2}$ and $\overline{\mathcal{M}}$ denote their closures.
Assume that the following hold:
\begin{enumerate}
\item All the eigenvalues of $\mathbf{A}$ in \eqref{e1} are in the open left half plane $\forall N_1\in{\overline{\mathcal{N}}_{1}}$, $N_2\in{\overline{\mathcal{N}}_{2}}$.
\item $\overline{\mathcal{N}}_{1}$, $\overline{\mathcal{N}}_{2}$ and $\mathcal{\overline{M}}$ are compact and convex sets.
\item If $C_{dc}=\infty$, then $\frac{1}{P_{dc}\mu_{r}}-\mu_{d}\in\mathcal{M}$. If $P_{dc}=\infty$ but $C_{dc}\neq\infty$, then $-\mu_{d}\in\mathcal{M}$.
\item If $C_{dc}=\infty$ or $P_{dc}=\infty$ or both, then:
\[\left|\frac{\partial}{\partial\mu_1}\left[\mathcal{F}_M\left(\mu_{\hat{1}},\sigma_{\hat{1}},\rho\right)\right]\right|\geq d\]
for a fixed constant $d>0$.
\end{enumerate}
Then, the system of equations \eqref{n1nm}, \eqref{n2nm} and \eqref{meq1} has a solution in $\overline{\mathcal{N}}_{1}$, $\overline{\mathcal{N}}_{2}$ and $\overline{\mathcal{M}}$.
\end{theorem}
\begin{proof}
We consider two cases. First assume that $C_{dc}\neq\infty$ and $P_{dc}\neq\infty$. The first assumption implies that for any value of $N_1\in{\overline{\mathcal{N}}_{1}}$ and $N_2\in{\overline{\mathcal{N}}_{2}}$, there is a unique positive definite solution of \eqref{lyap}. Hence, $\sigma_{\hat{1}}$ and $\rho$ exist, from \eqref{s1} and \eqref{rho} respectively, and are continuous functions of $N_1$ and $N_2$. Also, from \eqref{muu} and \eqref{meq}, $\mu_{\hat{1}}$ is a continuous function of $N_1$, $N_2$ and $M$. Therefore, the both the sides of \eqref{n1nm}, \eqref{n2nm} and \eqref{meq1} form continuous functions of $N_1$, $N_2$ and $M$. Since the second assumption holds, by Brouwer's fixed point theorem \cite{smart1980}, \eqref{n1nm}, \eqref{n2nm} and \eqref{meq1} have a solution and the result follows.
For the second case, assume either $C_{dc}=\infty$ or $P_{dc}=~\infty$ or both. Then the LHS of \eqref{meq1} reduces to a constant, independent of $N_1$, $N_2$ and $M$. Since the range of $\mathcal{F}_{M}$ is $\mathcal{M}$, the third assumption ensures a necessary condition for \eqref{meq1} to have a solution. Let $\mathbf{p_1}=\left[\begin{array}{cc}
\sigma_{\hat{1}} & \rho\end{array}\right]^{T}$ and $\mathbf{p_2}=\mu_{\hat{1}}$. Since $\mu_{\hat{1}}$, $\sigma_{\hat{1}}$ and $\rho$ are continuous functions of $N_1$, $N_2$ and/or $M$, so are $\mathbf{p_1}$ and $\mathbf{p_2}$. In addition, since $\mathcal{F}_M$ is continuous and also continuously differentiable with respect to $\mu_{\hat{1}}$, $f\left(\mathbf{p_{1}},\mathbf{p_{2}}\right)=\mathcal{F}_M\left(\mu_{\hat{1}},\sigma_{\hat{1}},\rho\right)$ is a continuous mapping from $\mathbb{R}^{2}\rightarrow\mathbb{R}$ and is continuously differentiable in $\mathbf{x_2}$. By the fifth assumption and Theorem 1 in \cite{zhang2006}, for any value of $N_1\in{\overline{\mathcal{N}}_{1}}$ and $N_2\in{\overline{\mathcal{N}}_{2}}$, and hence for any $\sigma_{\hat{1}}$ and $\rho$ (which are continuous functions of $N_1$ and $N_2$), there exists a unique $\mu_{\hat{1}}=\mathbf{x_{2}}=g\left(\mathbf{x_{1}}\right)=h\left(\sigma_{\hat{1}},\rho\right)$ where $g$ and $h$ are continuous. Hence, the resulting RHS of \eqref{meq1} is continuous. By assumption 2 and Brouwer's fixed point theorem, the result follows.
\end{proof}
There is no closed form solution for this system of equations, but it can be numerically arrived at by using an algorithm like Trust-Region Dogleg \cite{powell1968}, which is used in, for example, MATLAB's \texttt{fsolve}.
\section{Comparison with single variable QLC}\label{cSQL}
The essential difference between single variable QLC and multi-variable QLC is the presence of the second actuator input $u_2(t)$. This leads to effects peculiar only to multi-variable QLC. Two of them are described in the following subsections.
\subsection{Effect of secondary input on primary input saturation}
A particular point of interest is the fact that whether the addition of noise to the actuator bounds leads to an increased saturation in the primary actuator input or not. The following Theorem investigates that.
\begin{theorem}\label{tp}
The probability that the primary input $U_1$ is not saturated in a bivariate saturation nonlinearity with jointly Gaussian inputs $U_1$ and $U_2$ is quantified by the input quasilinear gain, $N_1$, i.e.,
\[P\left(\alpha-U_{2}<U_{1}<\beta+U_{2}\right)=N_{1}\]
\end{theorem}
\begin{proof}
By definition of the joint probability distribution function,
\[
\begin{gathered}
P\left(\alpha-U_{2}<U_{1}<\beta+U_{2}\right)\\
=\int_{-\infty}^{\infty}\int_{\alpha-u_{2}}^{\beta+u_{2}}p_{U_{1},U_{2}}\left(u_{1},u_{2}\right)du_{1}du_{2}
\end{gathered}\]
where $p_{U_{1},U_{2}}\left(u_{1},u_{2}\right)$ is the joint probability distribution function (PDF) of $U_1$ and $U_2$. Since $U_1$ and $U_2$ are jointly Gaussian,
\[p_{U_{1},U_{2}}\left(u_{1},u_{2}\right)=\mathcal{N}\left(\mu_{1},\mu_{2},\sigma_{1},\sigma_{2},\rho\right)\]
which is the bivariate Gaussian PDF defined in \eqref{npdf}. By the definition of the bivariate saturation \eqref{bsat},
\[U_1=0 \textrm{ for } U_2<\max\left(-\beta,\alpha\right)\]
Hence, from \eqref{n1e1},
\[
\begin{gathered}
P\left(\alpha-U_{2}<U_{1}<\beta+U_{2}\right)\\
=\int_{\max\left(-\beta,\alpha\right)}^{\infty}\int_{\alpha-u_{2}}^{\beta+u_{2}}\mathcal{N}\left(\mu_{1},\mu_{2},\sigma_{1},\sigma_{2},\rho\right)du_{1}du_{2}\\
=N_1
\end{gathered}\]
\end{proof}
From Theorem \ref{tp}, it can be seen that the probability that the input is \textit{not} saturated is quantified by $N_1$, which in turn, depends on parameters $\mu_2$, $\sigma_2$ and $\rho$, which are exclusive to the multi-variable case, apart from $\mu_{\hat{1}}$ and $\sigma_{\hat{1}}$, which were present even in the single-variable case. One plot, showing the dependence of $N_1$ on actuator noise $\sigma_2$ and the actuator authority $\beta=-\alpha$, is shown in Fig. \ref{fig:N1sigma2}. It can be seen that as the actuator bounds become more variable, the probability that the primary actuator input is saturated approaches 0.5, as proved in the following Theorem. Intuitively, this is because sometimes the actuator bounds increase, allowing for lesser saturation, and sometimes they decrease, resulting in more saturation.
\begin{theorem}
With other parameters fixed, $N_1 \rightarrow 0.5$ as $\sigma_2 \rightarrow \infty$.
\end{theorem}
\begin{proof}
With other parameters fixed, $\sigma_2 \rightarrow \infty \Rightarrow \gamma_1 \rightarrow \infty$ and $\gamma_2 \rightarrow \infty$, as per \eqref{g1} and \eqref{g2}. Thus, $\textrm{erf}\left(\gamma_{1}\right)+\textrm{erf}\left(\gamma_{2}\right)\rightarrow2$. Also, $u'_{2\min} \rightarrow 0$ from \eqref{u2pmin}. Hence,
\[\lim_{\sigma_{2}\rightarrow\infty}N_{1}=\int_{0}^{\infty}\frac{\sqrt{2}}{4\sqrt{\pi}}e^{-\frac{{u'}_{2}^{2}}{2}}\left(2\right)du'_{2}=0.5\]
\end{proof}
\begin{figure}[t]
\centering
\includegraphics[width=1\columnwidth]{./Figures/N1sigma_2beta.eps}
\caption{Plot of $N_1$ as a function of $\sigma_2$ for varying $\beta=-\alpha$}
\label{fig:N1sigma2}
\end{figure}
\subsection{Effect of Correlation between Actuator Inputs}
The primary actuator input $\hat{u}_1(t)$ in Fig. \ref{sl} is correlated to the secondary actuator input $u_2(t)$ due to the feedback provided by the closed loop system. To find the effect of correlation between the actuator inputs, the value of the correlation coefficient $\rho$ was plotted for varying levels of asymmetry and standard deviation of the secondary actuator input. The results are shown in Fig. \ref{fig:e4p1} and Fig. \ref{fig:e5p1}.
\begin{figure}[t]
\centering
\includegraphics[width=1\columnwidth]{./Figures/E4P1.eps}
\caption{Plot of $\rho$ as a function of asymmetry}
\label{fig:e4p1}
\end{figure}
\begin{figure}[t]
\centering
\includegraphics[width=1\columnwidth]{./Figures/E5P1.eps}
\caption{Plot of $\rho$ as a function of asymmetry and noise variability}
\label{fig:e5p1}
\end{figure}
From the figures it can be seen that as the actuator becomes more asymmetric (i.e., $\beta$, the upper limit of saturation, increases while the lower limit, $\alpha$ is fixed) or when the bounds become more variable (i.e. $\sigma_{2}$ increases), the correlation coefficient increases. However, when the actuator nonlinearity is symmetric ($\alpha=-\beta$), the correlation coefficient is constant for any value of $\sigma_{2}$. This is due to the way $N_2$ and the bivariate saturation function is defined: $u_2(t)$ adds to the upper limit, but subtracts from the lower limit.
\section{Accuracy of Stochastic Linearization}\label{asl}
\begin{figure}[t]
\centering
\includegraphics[width=1\columnwidth]{./Figures/E29P2}
\caption{Histogram plot of error accuracy}
\label{fig:accAO}
\end{figure}
\begin{figure}[t]
\centering
\includegraphics[width=1\columnwidth]{./Figures/E29P1}
\caption{Box plot of output accuracy}
\label{fig:histAO}
\end{figure}
To investigate the accuracy of the method of stochastic linearization, with focus on the effect of the variability of the actuator noise, a Monte Carlo experiment was performed with the following parameters:
\[C(s)=K\sim U\left[0.01,50\right]\]
Systems with 2 types of plants were considered. For half of the systems considered,
\[P(s)=\frac{1}{Ts+1},\textrm{ such that }T\sim U\left[0.01,10\right]\]
For the other half,
\[P(s) =\frac{\omega_{n}^{2}}{s^{2}+2\xi\omega_{n}s+\xi^{2}}\]
such that $\omega_{n}\sim U\left[0.01,10\right], \xi\sim U\left[0.05,2\right]$.
Also, to ensure a fair comparison, the statistical properties of the reference and disturbance signals were assumed to be constant:
\[\mu_{r}=0,\sigma_{r}=1,\mu_{d}=0,\sigma_{d}=1\]
To find the effect of variability in the secondary actuator input $u_2(t)$, the following was assumed about its statistics:
\[\mu_{2}=0, \sigma_{2}\in\left[0,1.25,2.5,3.75,5\right]\]
Finally, the actuator authorities were selected as:
\[\alpha\sim\left[-15,0\right],\beta\sim U\left[0,15\right]\]
3000 systems were considered for simulation, out of which 535 ($\sim 18\%$) unstable systems and those with phase margin $>$ 20 degrees were rejected since they were not practical. All the coloring filters were taken to be of 3rd order Butterworth type, with transfer function \eqref{key}, and cut-off frequency $\Omega=1.43$ kHz, which was found by a separate Monte Carlo experiment to be mean bandwidth of the closed loop systems considered.
\begin{equation}\label{key}
\small F_{\Omega_{d}}\left(s\right)=\sqrt{\frac{3}{\Omega_{d}}}\left(\frac{\Omega_{d}^{3}}{s^{3}+2\Omega_{d}s^{2}+2\Omega_{d}^{2}s+\Omega_{d}^{3}}\right), \Omega_d=1.43 \textrm{ kHz}
\end{equation}
The results for accuracy are shown in Figures \ref{fig:accAO} and \ref{fig:histAO}. Fig. \ref{fig:accAO} shows the histogram of the difference in the square root of second moment of the error $e(t)$ in the nonlinear system and the stochastically linearized system, normalized by the square root of second moment of the nonlinear error. It can be seen that stochastic linearization is fairly accurate for most of the systems. Fig. \ref{fig:histAO} shows a box plot of the difference of the square root of second moment of the output between the nonlinear and the stochastically linearized systems, normalized by the square root of second moment of the nonlinear actuator output. It can be seen that as the actuator bounds become more variable, the relative error increases.
\section{Practical Example - Optimal Controller Design}\label{pe}
In this section, a practical example of designing an optimal proportional controller to reduce the standard deviation of the tracking error is presented.
\begin{figure}[t]
\centering
\includegraphics[width=1\columnwidth]{./Figures/E20cP1}
\caption{Plot of objective function for system with $P(s) = \frac{10}{s(s+10)}$, $C(s)=~K$, $\alpha=-2$, $\beta=1$, $\mu_1=0$, $\sigma_1=1$, $\mu_2=0$, $\sigma_2=1$ and filter bandwidth 48 rad/s.}
\label{fig:objF}
\end{figure}
Consider Fig. \ref{fig:Block-Diagram-of}, with $C(s)=D_C=K$ and $P(s) = \frac{10}{s(s+10)}$. The actuator bounds are chosen to be $\alpha=-2$, $\beta=1$, and the input parameters, $\mu_2=0$, $\sigma_2=1$, $\mu_r=0$, $\sigma_r=1$, $\mu_d=0$ and $\sigma_d=1$. The filter bandwidth is chosen to be 48 rad/s, which is close to the system bandwidth. Let the objective function be the sum of the second moment of the tracking error $e(t)$ and that of the primary actuator input $u_1(t)$. It has a well-defined minimum, as seen in Figure \ref{fig:objF}. With this, the following optimization problem can be formulated:
\begin{equation}\label{opt}
\begin{array}{c}
\textrm{min}\;\mu_{\hat{e}}^2+\sigma_{\hat{e}}^2+\gamma\left(\mu_{\hat{1}}^2+\sigma_{\hat{1}}^2\right)\\
\begin{array}{c}
\textrm{subject to}\end{array}\eqref{u2pmin}-\eqref{me2}, \eqref{e1}-\eqref{rho}
\end{array}
\end{equation}
where $\gamma>0$ is a penalty factor, $\sigma_{\hat{1}}$ and $\mu_{\hat{1}}$ are as in \eqref{s1} and \eqref{muu} respectively, and, similarly,
\[\mu_{\hat{e}}=\frac{1}{C_{dc}}\frac{\frac{1}{P_{dc}}\mu_{r}-m-\mu_{d}-N_{2}\mu_{2}}{N_{1}+\frac{1}{C_{dc}}\frac{1}{P_{dc}}}\]
\[\sigma_{\hat{e}}^{2}=\mathbf{C}_{\mathbf{\hat{e}}}\mathbf{\Sigma}\mathbf{C}_{\mathbf{\hat{e}}}^{\mathbf{T}}\] where:
\[\mathbf{C_{\hat{e}}}=\left(\begin{array}{c}
\frac{C_{rf}\sigma_{r}}{D_{c}D_{p}N_{1}+1}\\
-\frac{C_{bf}D_{p}N_{2}\sigma_{2}}{D_{c}D_{p}N_{1}+1}\\
-\frac{C_{df}D_{p}\sigma_{d}}{D_{c}D_{p}N_{1}+1}\\
-\frac{C_{c}D_{p}N_{1}}{D_{c}D_{p}N_{1}+1}\\
-\frac{C_{p}}{D_{c}D_{p}N_{1}+1}
\end{array}\right)^T\]
and $\mathbf{\Sigma}$ is the solution of \eqref{lyap}.
\begin{figure}[t]
\centering
\includegraphics[trim={0 0 0 35},width=1.1\columnwidth]{./Figures/E20P1}
\caption{Time series plot for baseline controller.}
\label{fig:opt1}
\end{figure}
\begin{figure}[t]
\centering
\includegraphics[trim={0 0 0 35},width=1.1\columnwidth]{./Figures/E20P2}
\caption{Time series plot for optimal controller}
\label{fig:opt2}
\end{figure}
The optimization was performed using \texttt{fmincon} of MATLAB\textsuperscript{\textregistered}, with an initial value of $C(s)=100$ and $\gamma=1$. The optimal controller was found to be $C(s)=0.24$. The cost reduced from 1244.5 to 1.2.
The system was simulated in MATLAB/Simulink\textsuperscript{\textregistered} with an initial value of $K=100$, the results of which are shown in Fig. \ref{fig:opt1}. The upper subplot displays the tracking error in the nonlinear system, $e(t)$, and that in the stochastically linearized system, $\hat{e}(t)$. The lower subplot shows the actuator output from the nonlinear system, $v(t)$, and that from the stochastically linearized system, $\hat{v}(t)$, bounded by the actuator limits: $\alpha-u_2(t)$ and $\beta+u_2(t)$. The nonlinear actuator output can be clearly seen to be saturated by the bounds. Fig. \ref{fig:opt2} shows the same system after optimization. It can be seen that the standard deviation of the nonlinear error reduced from 1.04 to 1.02 and its mean reduced from 0.4 to 0.1. Since the optimization also reduced the primary actuator input, it is no longer saturated, and hence, $e(t)$ and $\hat{e}(t)$ coincide.
\section{Conclusion}\label{conc}
In this paper, the theory of stochastic linearization is extended to functions of multiple variables. A general control system with bivariate saturated actuator is considered for analysis, and expressions for equivalent gains and bias have been derived. The accuracy of stochastic linearization was investigated by a Monte Carlo simulation, and was found to be fairly good. Finally, a practical example of optimal control design is presented to show that the method can be used to design optimal controllers.
|
{
"redpajama_set_name": "RedPajamaArXiv"
}
| 439
|
California Comprehensive Compliance Program and Compliance Declaration
NOTICE: Beckman Coulter, Inc. is providing this information pursuant to the requirements of California Health and Safety Codes §§119400-119402 (the "Statute"), which requires certain medical device companies doing business in California to make available their Compliance Program and annual written Declaration of compliance with the Compliance Program.
The Description and Declaration below covers Beckman Coulter, Inc. ("Beckman").
Beckman is committed to doing business in accordance with the highest standards of business conduct and ethics, and we consider our reputation for ethical and compliant behavior to be a valuable corporate asset. Accordingly, Beckman has established a Compliance Program ("CP") in accordance with California Health and Safety Code, Division 104, Part 15, Chapter 8, Sections 119400?119402 (the "Statute"), based on our good faith understanding of the requirements of the Statute as it may apply to a medical device manufacturer, and the Compliance Program Guidance for Pharmaceutical Manufacturers ("OIG Guidance") published by the Office of Inspector General, U.S. Department of Health and Human Services.
Our Compliance Program, applied in conjunction with our MedTech Healthcare Professional Engagement Policy ("MedTech Policy"), is tailored to Beckman's size, organizational structure, available resources, and the nature of our business as a medical device manufacturer. The medical device industry has established, and Beckman's Compliance Program is designed in accordance with, the AdvaMed Code of Ethics on Interactions with Health Care Professionals (the "AdvaMed Code"). The AdvaMed Code is a voluntary ethical code substantially similar to the Pharmaceutical Research and Manufacturers of America Code on Interactions with Healthcare Professionals (the "PhRMA Code") but was developed with a primary focus on the unique interactions between medical device manufacturers and healthcare professionals and is thus more uniquely suited to Beckman's compliance needs.
This is a description of Beckman's Compliance Program. It reflects the plan and policies we have implemented, but Beckman's Compliance network is a dynamic and continuously developing entity requiring coordination between multiple policies and programs as well as the commitment of our management team and associates. While adoption and implementation of appropriate policies cannot guarantee complete elimination of improper conduct, it is Beckman's expectation that all associates will comply with our Compliance Program and all applicable laws, rules, and regulations. Beckman's passion for continuous improvement in all areas of our business dictates that we regularly review and enhance our Compliance Program to adapt to our evolving compliance needs.
Description of Compliance Program
Leadership and Structure
Our Compliance Program has been developed by Beckman's Compliance leadership and is managed by its Chief Compliance Officer. Company Management works with the Compliance team to ensure Beckman effectively prevents, detects, and reacts to instances of non-compliance.
Beckman has adopted written policies and procedures such as our MedTech Policy to implement our high ethical standards and meet our healthcare compliance obligations under applicable laws, regulations, and industry codes of conduct. These policies apply to all Beckman associates and compliance with those policies is a condition of employment.
Beckman, in accordance with California law, has established an annual limit on promotional spending of USD $2,500 per individual California medical and health professional, which includes persons licensed by California to prescribe prescription products, medical students, and members of drug formulary committees. This limit covers promotional materials, or items or activities that Beckman may give or otherwise provide to an individual medical or healthcare professional in California. This limit represents a spending cap and not an average or target, and in most instances the amount spent per physician will be substantially less than this annual limit.
The annual limit excludes non-promotional spending such as: drug samples or demonstration units; financial support for continuing medical education forums or health education scholarships if provided in accordance with Beckman policy; fair market value compensation for bona fide professional services, and any meals or other expenses related to the provision of such services; training programs necessary for the safe and effective use of Beckman products; items of nominal value with a retail value of less than $10.
We provide training on ethics as well as Beckman's Compliance Program and MedTech Policy to all sales, marketing and research and development associates, both as a part of a regular training program and ad hoc as needed.
Compliance Communications
Our Company is committed to maintaining a work environment where all individuals encourage and embrace open discussion across both geographical and operational boundaries. Employees are encouraged to seek guidance in resolving ethics and compliance questions. They are required to report suspected or actual unethical or non-compliant conduct, violations of law or regulation, and violations of internal company policies. Beckman associates are encouraged to use our reporting hotline which allows anonymous reporting of suspected or actual noncompliance without retaliation.
Our Company performs compliance gap analyses, process reviews and data analysis to identify and address potential issues and opportunities to enhance its processes and practices. These are collaborative efforts between the Compliance Department and the business which has generated the development of new and revised policies, new training and communications initiatives and corrective action.
Investigating and Responding to Potential Violations
Our Compliance Department monitors company functions and activities for potential violations of law, regulation, industry codes, the Code of Conduct and company policies. Potential violations are promptly and thoroughly investigated and appropriate corrective action taken.
In determining the appropriate response to an established violation, our Company considers various options for corrective action. These options include remediation to evaluate current practices and close any gaps in policies, practices or training that may have led or contributed to the violation, as well as the imposition of disciplinary action up to and including termination.
A copy of this description of Beckman's Compliance Program and Declaration may be obtained by calling (714) 993-5321 and speaking with a Customer Service representative.
Posted Date: September 2016
Please select... Analytical Ultracentrifuges Cell Counters/Bioprocess Monitors Cell Sorters Centrifuges (Benchtop, High-Speed, Ultra) Clinical Flow Cytometry Analyzer Flow Cytometry Reagents Research Flow Cytometry Analyzer Genomic Reagents Automation/Liquid Handling Particle Characterization Analyzer Particle Counters (Air, Liquid, TOC) Liquid Handling Tips and Labware Tubes, Bottles, and Liners Rotors
Please select... Information Request Purchasing Quote
Please select... Afghanistan Albania Algeria Andorra Angola Antigua and Barbuda Argentina Armenia Australia Austria Azerbaijan Bahrain Bangladesh Barbados Belarus Belgium Belize Benin Bhutan Bolivia Bosnia and Herzegovina Botswana Brazil Brunei Darussalam Bulgaria Burkina Faso Burma Burundi Cambodia Cameroon Canada Cape Verde Central African Republic Chad Chile China Colombia Comoros Costa Rica Cote d-Ivoire Croatia Cuba Cyprus Czech Republic DR Congo Denmark Djibouti Dominica Dominican Republic East Timor Ecuador Egypt El Salvador Equatorial Guinea Eritrea Estonia Ethiopia Federated States of Micronesia Fiji Finland France Gabon Georgia Germany Ghana Greece Grenada Guatemala Guinea-Bissau Guinea Guyana Haiti Honduras Hong Kong, China Hungary Iceland India Indonesia Iran Iraq Ireland Israel Italy Jamaica Japan Jordan Kazakhstan Kenya Kiribati Kuwait Kyrgyzstan Laos Latvia Lebanon Lesotho Liberia Libya Liechtenstein Lithuania Luxembourg Macedonia Madagascar Malawi Malaysia Maldives Mali Malta Marshall Islands Mauritania Mauritius Mexico Moldova Monaco Mongolia Morocco Mozambique Namibia Nauru Nepal Netherlands New Zealand Nicaragua Niger Nigeria North Korea Norway Oman Pakistan Palau Panama Papua New Guinea Paraguay Peru Philippines Poland Portugal Puerto Rico Qatar Republic of the Congo Romania Russia Rwanda Saint Kitts and Nevis Saint Lucia Saint Vincent and the Grenadines Samoa San Marino Sao Tome and Principe Saudi Arabia Senegal Seychelles Sierra Leone Singapore Slovakia Slovenia Solomon Islands Somalia South Africa South Korea Spain Sri Lanka Sudan Suriname Swaziland Sweden Switzerland Syria Taiwan, China Tajikistan Tanzania Thailand The Bahamas The Gambia Togo Tonga Trinidad and Tobago Tunisia Turkey Turkmenistan Tuvalu Uganda Ukraine United Arab Emirates United Kingdom United States Unlisted Uruguay Uzbekistan Vanuatu Venezuela Vietnam Western Sahara Yemen Zambia Zimbabwe
Please select... Alabama Alaska Arizona Arkansas California Colorado Connecticut Delaware District of Columbia Florida Georgia Hawaii Idaho Illinois Indiana Iowa Kansas Kentucky Louisiana Maine Maryland Massachusetts Michigan Minnesota Mississippi Missouri Montana Nebraska Nevada New Hampshire New Jersey New Mexico New York North Carolina North Dakota Ohio Oklahoma Oregon Pennsylvania Rhode Island South Carolina South Dakota Tennessee Texas Utah Vermont Virginia Washington West Virginia Wisconsin Wyoming
Number of the ID (ex. passport)
ID issued authority
Date of issue for ID
Please keep me informed about Beckman Coulter webinars, products, goods, and services, including products, goods, or services from our related companies.
I Consent I DO NOT Consent
By submitting this form I confirm that I have reviewed and agree with the Privacy Policy and Terms of Use. I also understand my privacy choices as they pertain to my personal data as provided in the Privacy Policy under "Your Privacy Choices".
|
{
"redpajama_set_name": "RedPajamaCommonCrawl"
}
| 5,660
|
\section{Introduction}
\begin{figure*}[tp!]
\setlength{\abovecaptionskip}{-0.2cm}
\setlength{\belowcaptionskip}{-0.4cm}
\begin{center}
\includegraphics[width=0.85\linewidth]{figure/intro.jpg}
\end{center}
\caption[]{(a) Illustration of different domains corresponding to different capture conditions. (b) Our BargainNet utilizes background domain code to guide the foreground domain translation, resulting in consistent foreground and background.}
\label{fig:intro}
\end{figure*}
Image composition synthesizes the composite by combining the foreground from one image with the background from another image. One issue of image composition is the appearance differences between foreground and background caused by distinct capture conditions (\emph{e.g.}, weather, season, time of day). Therefore, making the generated composite realistic could be a challenging task. Image harmonization~\cite{tsai2017deep,xiaodong2019improving,DoveNet2020}, which aims to adjust the foreground to make it compatible with the background, is essential to address this problem.
Traditional harmonization methods~\cite{lalonde2007using,xue2012understanding,multi-scale} improve the quality of synthesized composite mainly by transferring hand-crafted appearance statistics between foreground and background regions, but they could not handle the large appearance gap between foreground and background regions. Recently, more deep learning based harmonization approaches have also been proposed. In \cite{tsai2017deep}, they presented the first end-to-end network for image harmonization. In \cite{xiaodong2019improving}, the spatial-separated attention blocks were proposed to learn the foreground and background features separately. Later in \cite{DoveNet2020}, they proposed an adversarial network with a domain verification discriminator to pull close the domains of foreground and background regions. Nonetheless, previous deep learning based methods neglected the crucial guidance role that background plays in the harmonization task. Therefore, they did not realize the shortcut to addressing image harmonization by posing it as background-guided domain translation.
According to DoveNet~\cite{DoveNet2020}, we can treat different capture conditions as different domains. As illustrated in Fig.~\ref{fig:intro}(a), there could be innumerable possible domains for natural images. Even for the same scene, when the season, weather, time of the day, or photo equipment settings vary, the domain changes.
For a real image, its foreground and background are captured in the same condition and thus belong to the same domain. But for a composite image, its foreground and background may belong to two different domains. In this case, image harmonization could be regarded as transferring the foreground domain to the background domain, making it a special case of domain translation. Domain translation has been extensively explored in \cite{pixelGAN,cycleGAN,starGan,lee2020drit++,choi2020stargan,f2gan,matchinggan}, and most domain translation methods require explicitly predefined domain labels, which are unavailable in our task. More recently, methods without domain labels have also been proposed as exemplar-guided domain translation~\cite{anokhin2020high,wang2019example}, in which an exemplar image provides the domain guidance.
In this paper, we take a further step beyond exemplar-guided domain translation and detail the problem to local region guidance, \emph{i.e.}, background-guided domain translation. As demonstrated in Fig.~\ref{fig:intro}(b), the background and foreground of a composite image belong to different domains. With the guidance of extracted background domain code, which encodes the domain information of background, the composite foreground could be translated to the same domain as background, leading to a harmonious output.
As we propose to address image harmonization problem from a new perspective, one of our main contributions is the proposed \textbf{Ba}ckg\textbf{r}ound-\textbf{g}uided dom\textbf{ain} translation \textbf{Net}work, which is called BargainNet for short. Since partial convolution \cite{Liu2018} only concentrates on the feature aggregation of a partial region, we leverage partial convolution in our domain code extractor to extract the background domain information, which can avoid the information leakage between foreground and background. The obtained background domain code defines the target domain and guides the foreground domain translation. There are various ways of utilizing the target domain code to guide domain translation. For simplicity, we spatially replicate the background domain code to the same size as input image and concatenate them along the channel dimension. The concatenated input, together with the foreground mask, is fed into an attention-enhanced U-net generator~\cite{DoveNet2020} to produce the harmonized result. At the same time, we propose two well-tailored triplet losses to ensure that the domain code extractor can indeed extract domain information instead of domain-irrelevant information (\emph{e.g.}, semantic layout). The proposed triplet losses pull close the domain codes of background, real foreground, and the harmonized foreground, while pushing the domain code of composite foreground apart from them. To verify the effectiveness of our proposed BargainNet, we conduct comprehensive experiments on the image harmonization dataset iHarmony4~\cite{DoveNet2020}.
The contributions of our method are four-fold. 1) To the best of our knowledge, we are the first to formulate the image harmonization task as background-guided domain translation, which provides a new perspective for image harmonization; 2) We propose a novel image harmonization network, \emph{i.e.}, BargainNet, equipped with domain code extractor and well-tailored triplet losses; 3) Our method can extract meaningful domain code, which has other potential usages like inharmony level prediction; 4) Our method achieves the competitive performance on the benchmark dataset.
\vspace{-0.2cm}
\section{Related Work}
\textbf{Image Harmonization: }Image harmonization aims to make the composite foreground compatible with the background. To adjust the foreground appearance, traditional methods mainly leveraged low-level appearance statistics~\cite{reinhard2001color,colorharmonization,poisson,multi-scale}. Later in \cite{lalonde2007using,xue2012understanding,zhu2015learning}, image realism was gradually explored to make the composite image more realistic.
Recently, harmonization methods that synthesize paintings from photo-realistic images have been explored in \cite{luan2018deep,shaham2019singan}. However, they are more like style transfer and different from the photo-realistic harmonization in our task. More related to our work, in \cite{tsai2017deep,xiaodong2019improving,DoveNet2020}, they directly learn a mapping from composite images to real images, with the assistance of auxiliary semantic parsing branch~\cite{tsai2017deep}, inserted attention models~\cite{xiaodong2019improving}, or domain verification discriminator~\cite{DoveNet2020}. Different from these existing methods, our proposed method provides a new perspective by treating image harmonization as a background-guided domain translation.
\textbf{Domain Translation: }The task of domain translation aims to learn the mapping from a source domain to a target domain (\emph{e.g.}, from day to night). Recent works could be divided into two main streams: methods that require domain labels ~\cite{pixelGAN,cycleGAN,huang2018munit,lee2020drit++,choi2020stargan,f2gan,matchinggan} and methods without any predefined domain labels~\cite{anokhin2020high,wang2019example,ma2018exemplar}.
In image harmonization, domains correspond to different capture conditions. Therefore, domain labels are hard to define and hard to solicit from users. So our work is more related to the latter, which is also known as example-guided domain translation. Given an exemplar image as guidance, the input image is translated into the same domain as the given exemplar image. In this paper, we take a further step and pose image harmonization as background-guided domain translation, which utilizes background region instead of an exemplar image as guidance.
\vspace{-0.2cm}
\section{Our Method}
\begin{figure}[tp!]
\setlength{\abovecaptionskip}{-0.1cm}
\begin{center}
\includegraphics[width=\linewidth]{figure/network.jpg}
\end{center}
\caption{The network architecture of our BargainNet, which consists of attention enhanced U-Net generator $G$ and domain code extractor $E$. We employ two types of triplet losses based on four types of domain codes (see Section~\ref{sec:bargainnet}). The test phase is highlighted with red flow lines for clarity.}
\label{fig:flowchart}
\vspace{-0.4cm}
\end{figure}
In image harmonization task, we utilize training pairs of composite image $\tilde I\in \mathbb{R}^{H\times W\times 3}$ and real image $I \in \mathbb{R}^{H \times W \times 3}$, in which $H$ (\emph{resp.}, $W$) is image height (\emph{resp.}, width). The background of $I$ (real background) is the same as the background of $\tilde I$ (composite background). So in the remainder of this paper, we only mention background without distinguishing between real background and composite background. The foreground of $I$ (real foreground) is the harmonization target of the foreground of $\tilde I$ (composite foreground). The binary mask $M\in \mathbb{R}^{H \times W \times 1}$ indicates the foreground region to be harmonized, and therefore the background mask is $\bar{M}=1-M$.
Given a composite image $\tilde I$, the goal of image harmonization task is to use a generator to reconstruct $I$ with a harmonized output $\hat I$, in which the foreground of $\hat I$ (harmonized foreground) should be close to the real foreground.
Next, we first introduce our domain code extractor in Section~\ref{sec:code_extractor}, and then introduce our whole network BargainNet in Section~\ref{sec:bargainnet}.
\vspace{-0.3cm}
\subsection{Domain Code Extractor}\label{sec:code_extractor}
To extract the domain code for a region with an irregular shape, our domain code extractor $E$ is composed of contiguously stacked partial convolutional layers \cite{Liu2018}, which are designed for special image generation with irregular masks.
The output of the domain code extractor only depends on the aggregated features within the masked region, which prevents information leakage from the unmasked region.
For the technical details of partial convolution, please refer to \cite{Liu2018}.
In our task, we use domain code extractor to extract the domain codes of the foreground/background regions of composite image $\tilde I$, real image $I$, and output image $\hat I$.
For example, given a composite image $\tilde I$ and its background mask $\bar M$, $E$ could extract the background domain code of $\tilde I$. To enforce the domain code to contain domain information instead of other domain-irrelevant information (\emph{e.g.}, semantic layout), we use background domain code to guide the foreground domain translation and design well-tailored triplet losses to regulate the domain code, which will be introduced next.
\vspace{-0.2cm}
\subsection{Background-guided Domain Translation Network} \label{sec:bargainnet}
Our proposed \textbf{Ba}ckg\textbf{r}ound-\textbf{g}uided dom\textbf{ain} translation \textbf{Net}work (BargainNet) has two modules: domain code extractor $E$ and generator $G$. We adopt attention-enhanced U-net proposed in \cite{DoveNet2020} as $G$ and omit the details here.
As demonstrated in Fig. \ref{fig:flowchart}, given a composite image $\tilde I$ and its background mask $\bar M$, the domain code extractor takes $\tilde I$ and $\bar M$ as input and outputs the background domain code $z_b$. The extracted background domain code is used as the target domain code for foreground domain translation, which means that the foreground will be translated to the background domain with its domain-irrelevant information (\emph{e.g.}, semantic layout) well-preserved. Besides, the background should remain unchanged if we translate it to the background domain. So for ease of implementation, we simply translate both foreground and background to the background domain.
Inspired by domain translation methods \cite{zhu2017toward,starGan}, we spatially replicate the $L$-dimensional domain code $z_b$ to an $H \times W \times L$ domain code map $Z_b$ and concatenate it with the $H \times W \times 3$ composite image. Besides, based on our experimental observation (see Section~\ref{sec:ablate} and Supplementary), it is still necessary to use foreground mask to indicate the foreground region to be harmonized as in ~\cite{tsai2017deep,xiaodong2019improving,DoveNet2020}, probably because the foreground mask emphasizes foreground translation and enables the foreground to borrow information from the background.
Thus, we further concatenate the input with the $H \times W \times 1$ foreground mask $M$, leading to the final
$H \times W \times (L+4)$ input. After passing the input through the generator $G$, we enforce the harmonized output $\hat I=G(\tilde I, M, Z_b)$ to be close to the ground-truth real image $I$ by using the reconstruction loss $\mathcal{L}_{rec} = \|\hat I-I\|_1$.
We assume that $z_b$ only contains the domain information of background. Because if $z_b$ contains the domain-irrelevant information (\emph{e.g.}, semantic layout) of background, it may corrupt the semantic layout of foreground, which violates the reconstruction loss. To further reinforce our assumption on domain code, we use triplet losses to pull close the domain codes which are expected to be similar and push apart those which are expected to be divergent. Analogous to extracting background domain code $z_b$, we also use $E$ to extract the domain codes of real foreground, composite foreground, and harmonized foreground, denoted as $z_f$, $\tilde{z}_f$, and $\hat{z}_f$ respectively. For ease of description, we define an image triplet as a composite image, its ground-truth real image, and its harmonized output. Given an image triplet, we can obtain $\tilde z_f$, $z_b$, $z_f$ and $\hat z_f$.
First, after harmonization, the foreground is translated from composite foreground domain to background domain. Hence, the domain code of harmonized foreground ($\hat z_f$) should be close to that of background ($z_b$), but far away from that of composite foreground ($\tilde z_f$).
In other words, we aim to pull close $\hat z_f$ and $z_b$ while pushing apart $\hat z_f$ and $\tilde z_f$, which can be achieved by the following triplet loss:
\begin{eqnarray} \label{eqn:triplet_fb}
\mathcal{L}_{\hat fb}\!\!\!\!\!\!\!\!&&= \mathcal{L}(\hat z_f,z_b,\tilde z_f)\\
&&=\max (d(\hat z_f, z_b)-d(\hat z_f, \tilde z_f)+m, 0),\nonumber
\end{eqnarray}
in which $d(\cdot,\cdot)$ is Euclidean distance and $m$ is a margin.
\begin{table*}[tb]
\centering
\begin{tabular}{|l|c|c|c|c|c|c|c|c|c|c|}
\hline
\multicolumn{1}{|c|}{Sub-dataset} & \multicolumn{2}{c|}{HCOCO} & \multicolumn{2}{c|}{HAdobe5k} & \multicolumn{2}{c|}{HFlickr} & \multicolumn{2}{c|}{Hday2night} & \multicolumn{2}{c|}{All}\\ \hline
\multicolumn{1}{|c|}{Evaluation metric} & MSE$\downarrow$ & PSNR$\uparrow$ & MSE$\downarrow$ & PSNR$\uparrow$ & MSE$\downarrow$ & PSNR$\uparrow$ & MSE$\downarrow$ & PSNR$\uparrow$ & MSE$\downarrow$ & PSNR$\uparrow$ \\ \hline
\multicolumn{1}{|c|}{Input composite} & 69.37 & 33.94 & 345.54 & 28.16 & 264.35 & 28.32 & 109.65 & 34.01 & 172.47 & 31.63 \\ \hline
\multicolumn{1}{|c|}{Lalonde and Efros\cite{lalonde2007using}} & 110.10 & 31.14 & 158.90 & 29.66 & 329.87 & 26.43 & 199.93 & 29.80 & 150.53 & 30.16 \\ \hline
\multicolumn{1}{|c|}{Xue \emph{et al.}\cite{xue2012understanding}} & 77.04 & 33.32 & 274.15 & 28.79 & 249.54 & 28.32 & 190.51 & 31.24 & 155.87 & 31.40 \\ \hline
\multicolumn{1}{|c|}{Zhu \emph{et al.}\cite{zhu2015learning}} & 79.82 & 33.04 & 414.31 & 27.26 & 315.42 & 27.52 & 136.71 & 32.32 & 204.77 & 30.72 \\ \hline
\multicolumn{1}{|c|}{DIH~\cite{tsai2017deep}} & 51.85 & 34.69 & 92.65 & 32.28 & 163.38 & 29.55 & 82.34 & 34.62 & 76.77 & 33.41 \\ \hline
\multicolumn{1}{|c|}{DoveNet~\cite{DoveNet2020}} & 36.72 & 35.83 & 52.32 & 34.34 & 133.14 & 30.21 & 54.05 & 35.18 & 52.36 & 34.75 \\ \hline
\multicolumn{1}{|c|}{S$^2$AM~\cite{xiaodong2019improving}} & 33.07 & 36.09 & 48.22 & \bf35.34 & 124.53 & 31.00 & \bf48.78 & 35.60 & 48.00 & 35.29 \\ \hline
\multicolumn{1}{|c|}{Ours} & \bf24.84 & \bf37.03 & \bf39.94 & \bf35.34 & \bf97.32 & \bf31.34 & 50.98 & \bf35.67 & \bf37.82& \bf35.88\\ \hline
\end{tabular}
\caption{Quantitative comparison between our proposed BargainNet and other baseline methods. The best results are denoted in boldface.}
\label{tab:baselines}
\vspace{-0.3cm}
\end{table*}
Next, we consider the relationship among three foregrounds in an image triplet.
The domain code of real foreground ($z_f$) should be close to that of harmonized foreground ($\hat z_f$), but far away from that of composite foreground ($\tilde z_f$). This goal can be achieved by the following triplet loss:
\begin{eqnarray} \label{eqn:triplet_ff}
\mathcal{L}_{f\hat f}\!\!\!\!\!\!\!\!&&=\mathcal{L}(z_f,\hat z_f,\tilde z_f)\\
&&=\max (d(z_f, \hat z_f)-d(z_f, \tilde z_f)+m, 0).\nonumber
\end{eqnarray}
In fact, there could be many reasonable combinations of triplet losses to regulate the domain code. However, based on our experimental observation, a combination of (\ref{eqn:triplet_fb}) and (\ref{eqn:triplet_ff}) has already met all our expectations (see Section~\ref{sec:domain_code}).
So far, the overall loss function for our method is
\begin{eqnarray} \label{eqn:total_loss}
\mathcal{L}\!\!\!\!\!\!\!\!&&=\mathcal{L}_{rec}+\lambda \mathcal{L}_{tri} =\mathcal{L}_{rec}+\lambda(\mathcal{L}_{f\hat f}+\mathcal{L}_{\hat f b}),
\end{eqnarray}
where $\lambda$ is a trade-off parameter.
\vspace{-0.2cm}
\section{Experiments}
\begin{table*}
\centering
\begin{tabular}{|c|c|c|c|c|c|c|c|c|}
\hline
& & $d_{b,f}<d_{b,\tilde f}$ & $d_{b,\hat f}<d_{b,\tilde f}$ & $d_{f,\hat f}<d_{f,\tilde f}$ & $d_{\hat f, f}<d_{\hat f,\tilde f}$ & $d_{\hat f, b}<d_{\hat f,\tilde f}$ & $d_{f, b}<d_{f,\tilde f}$ & All\\
\hline
\multirow{2}*{DoveNet\cite{DoveNet2020}} &Train & 47.08\% & 49.24\% & 72.22\% & 71.47\% & 12.01\% & 11.75\% & 5.93\% \\
\cline{2-9}
~ & Test & 51.34\% &51.58\% &62.34\% &54.65\% &13.68\% &15.64\% & 5.09\% \\
\hline
\multirow{2}*{Ours} &Train & 88.63\% & 97.87\% & 93.65\% & 91.92\% & 96.38\% & 87.98\% & 80.70\% \\
\cline{2-9}
~ & Test & 90.28\% &97.39\% &91.87\% &89.28\% &96.26\% &89.09\% & 81.36\% \\
\hline
\end{tabular}
\caption{The ratio of training/testing image triplets which satisfy the specified requirements of DoveNet and our method. Note that $d_{x,y}$ is short for $d(z_x, z_y)$. For example, $d_{b,f}$ denotes the Euclidean distance between the background domain code $z_b$ and the domain code of real foreground $z_f$.}
\label{tab:domain_code}
\vspace{-0.4cm}
\end{table*}
\vspace{-0.1cm}
\subsection{Dataset and Implementation Details}
We evaluate our method and baselines on the benchmark dataset iHarmony4~\cite{DoveNet2020}, which contains 73146 pairs of synthesized composite images and the ground-truth real images (65742 pairs for training and 7404 pairs for testing). iHarmony4 consists of four sub-datasets: HCOCO, HAdobe5k, HFlickr, and Hday2night. The details of four sub-datasets can be found in the Supplementary.
The extracted domain code is a 16-dimension vector. We set the margin $m$ in Eqn. (\ref{eqn:triplet_fb})(\ref{eqn:triplet_ff}) as 1 and the trade-off parameter $\lambda$ in Eqn. (\ref{eqn:total_loss}) as 0.01.
In our experiments, the input images are resized to $256\times 256$ during both training and testing phases.
Following \cite{tsai2017deep,DoveNet2020}, we use Mean-Squared Errors (MSE) and Peak Signal-to-Noise Ratio (PSNR) as the main evaluation metrics, which are also calculated on $256\times 256$ images. More details can be found in Supplementary.
\vspace{-0.2cm}
\subsection{Comparison with Existing Methods}
Both traditional methods~\cite{lalonde2007using,xue2012understanding} and deep learning based methods~\cite{zhu2015learning,tsai2017deep,xiaodong2019improving,DoveNet2020} are included for quantitative comparisons. Following \cite{tsai2017deep,DoveNet2020}, we train the model on the merged training sets of four sub-datasets in iHarmony4. The trained model is evaluated on each test set and the merged test set as well. Table \ref{tab:baselines} shows the quantitative results of different harmonization methods. The S$^2$AM~\cite{xiaodong2019improving} model is realized with recently released code and the other results of previous baselines are directly copied from \cite{DoveNet2020}. From Table \ref{tab:baselines}, we can observe that our method not only significantly exceeds traditional methods on all sub-datasets, but also outperforms deep learning based approaches on the whole test set.
Besides, following~\cite{DoveNet2020}, we also investigate the the MSE and foreground MSE (fMSE) on the test images in different foreground ratio ranges (\emph{e.g.}, $5\%\sim 15\%$) in the Supplementary.
\vspace{-0.3cm}
\subsection{Ablation Studies}\label{sec:ablate}
We analyze the impact of hyper-parameters (\emph{i.e.}, the margin $m$ in Eqn. (\ref{eqn:triplet_fb})(\ref{eqn:triplet_ff}), $\lambda$ in Eqn. (\ref{eqn:total_loss}) and the domain code dimension $L$) in our method. We also investigate the impact of each type of network input and ablate each type of triplet loss to prove the necessity of mask, background domain code, and two triplet losses. Due to space limitation, we leave the detailed experimental results to Supplementary.
\vspace{-0.3cm}
\subsection{Domain Code Analyses}\label{sec:domain_code}
Recall that we employ two triplet losses Eqn. (\ref{eqn:triplet_fb})(\ref{eqn:triplet_ff}) to regulate the domain code. To verify that the expected requirements are satisfied on the training set and generalizable to the test set, we conduct domain code analyses on both training set and test set. Since DoveNet employs a domain verification discriminator to extract foreground and background domain representations, DoveNet is also included for comparison. As defined in Section~\ref{sec:bargainnet}, an image triplet contains a composite image, its ground-truth real image, and its harmonized output.
We calculate the ratio of training/testing image triplets which satisfy $d(\hat z_f, z_b)<d(\hat z_f, \tilde z_f)$ (\emph{resp.}, $d(z_f, \hat z_f)<d(z_f, \tilde z_f)$) corresponding to Eqn. (\ref{eqn:triplet_fb}) (\emph{resp.}, Eqn. (\ref{eqn:triplet_ff})) for both DoveNet and our method. For brevity, we use $d_{x,y}$ to denote $d(z_x, z_y)$, as shown in Table~\ref{tab:domain_code}.
More generally, in an image triplet, the background, the real foreground, and the harmonized foreground belong to the same domain, while the composite foreground belongs to another domain. Considering that the distance between cross-domain regions should be larger than the distance between same-domain regions, we could construct $6$ groups of (anchor, positive, negative) in the form of triplet loss, leading to $6$ requirements: $d_{b,f}<d_{b,\tilde f}$, $d_{b,\hat f}<d_{b,\tilde f}$, $d_{f,\hat f}<d_{f,\tilde f}$, $d_{\hat f, f}<d_{\hat f,\tilde f}$, $d_{\hat f, b}<d_{\hat f,\tilde f}$, and $d_{f, b}<d_{f,\tilde f}$.
The verification results of each individual requirement and all requirements for DoveNet and our method are summarized in Table~\ref{tab:domain_code}. We can observe the high ratio of training/testing image triplets that satisfy each individual requirement for our method. Moreover, most training/testing image triplets satisfy all six requirements at the same time, which implies that compared with DoveNet, our domain code extractor can indeed extract the domain code which contains domain information as expected.
\vspace{-0.5cm}
\begin{figure*}[ht]
\setlength{\abovecaptionskip}{0.1cm}
\setlength{\belowcaptionskip}{-0.4cm}
\centering
\includegraphics[width=0.90\linewidth]{figure/examples.jpg}
\caption[]{Example results of baselines and our method on four sub-datasets. From top to bottom, we show one example from HAdobe5k, HCOCO, Hday2night, and HFlickr sub-dataset respectively. From left to right, we show the input composite image, the ground-truth real image, and the results of DIH~\cite{tsai2017deep}, DoveNet~\cite{DoveNet2020}, S$^2$AM~\cite{xiaodong2019improving}, our special case BargainNet (w/o $\mathcal{L}_{tri}$) and our proposed BargainNet respectively. The foregrounds are highlighted with red border lines for clarity.}
\label{fig:samples}
\end{figure*}
\vspace{0.1cm}
\subsection{Qualitative Analyses}\label{sec:qualitative}
Given an input composite image from the test set, the harmonized outputs generated by DIH~\cite{tsai2017deep}, DoveNet~\cite{DoveNet2020}, S$^2$AM~\cite{xiaodong2019improving}, BargainNet (w/o $\mathcal{L}_{tri}$) and BargainNet are shown in Fig. \ref{fig:samples}. BargainNet (w/o $\mathcal{L}_{tri}$) is a special case without triplet losses. Compared with other baselines, BargainNet could generate more favorable results with consistent foreground and background, which are visually closer to the ground-truth real images. Besides, by comparing BargainNet with BargainNet (w/o $\mathcal{L}_{tri}$), we can observe that the generated outputs of BargainNet are more harmonious after using triplet losses, which provides an intuitive demonstration that triplet losses contribute to more effective domain code extraction.
In the real-world applications, given a real composite image, there is no ground-truth as the synthesized composite, so it is infeasible to evaluate the model performance quantitatively using MSE or PSNR. Following \cite{tsai2017deep,xiaodong2019improving,DoveNet2020}, we conduct user study on 99 real composite images~\cite{tsai2017deep}, in which we compare our BargainNet with all the other deep learning based methods. The details of user study and harmonization results can be found in the Supplementary.
\vspace{-0.2cm}
\subsection{Background Harmonization and Inharmony Level Prediction}\label{extension}
By inverting the mask fed into the generator and the domain code extractor in the testing stage, our BargainNet could be easily applied to background harmonization, which means adjusting the background to make it compatible with the foreground. We show our background harmonization results and compare with other deep learning based methods in Supplementary.
Besides, one byproduct of our method is predicting the inharmony level of a composite image, which reflects how inharmonious this composite image is.
In particular, based on the extracted domain codes of the foreground region and background region, we can assess the inharmony level by calculating the Euclidean distance between two domain codes. The detailed inharmony level analyses are also left to Supplementary due to space limitation.
\vspace{-0.2cm}
\section{Conclusion}
In this work, we have proposed to formulate image harmonization as background-guided domain translation, which provides a new perspective for image harmonization. We have also presented BargainNet, a novel network that leverages the background domain code for foreground harmonization. Experimental results have shown that our method performs favorably on both the synthesized dataset iHarmony4 and real composite images.
\vspace{-0.3cm}
\section{Acknowledgement}
The work is supported by the National Key R\&D Program of China (2018AAA0100704) and is partially sponsored by National Natural Science Foundation of China (Grant No.61902247) and Shanghai Sailing Program (19YF1424400).
\begin{small}
\vspace{-0.2cm}
\bibliographystyle{IEEEbib}
\section{Dataset Statistics}\label{sec:dataset}
The iHarmony4 dataset contributed by \cite{DoveNet2020} is composed of pairs of synthesized composite images and the ground-truth real images. iHarmony4 consists of 4 sub-datasets: HCOCO, HAdobe5k, HFlickr, and Hday2night.
\textbf{HCOCO} sub-dataset is synthesized based on the merged training and test splits of Microsoft COCO \cite{lin2014microsoft}, containing 38545 training and 4283 test pairs of composite and real images. In HCOCO, the composite images are synthesized from real images and the foreground of composite image is adjusted by transferring the color from another foreground object of the same class in COCO using color mapping functions.
\textbf{HAdobe5k} sub-dataset is generated based on MIT-Adobe FiveK dataset \cite{bychkovsky2011learning}, containing 19437 training and 2160 test pairs of composite and real images. The composite image is generated by exchanging the manually segmented foreground between the real image and its five different renditions.
\textbf{HFlickr} sub-dataset is synthesized based on the crawled images from Flickr, containing 7449 training and 828 test pairs of composite and
real images. The composite images are synthesized similarly to HCOCO, except that the reference foreground is selected from ADE20K~\cite{ade20k} using the dominant category labels generated by pre-trained scene parsing model.
\textbf{Hday2night} sub-dataset is generated based on day2night \cite{zhou2016evaluating}, containing 311 training and 133 test pairs of composite and real images. The composite images are synthesized similarly to HAdobe5k, where the foreground is exchanged between images captured in different conditions.
\section{Implementation Details}\label{sec:implementation}
Our network is trained on ubuntu 16.04 LTS operation system, with 64GB memory, Intel Core i7-8700K CPU, and two GeForce GTX 1080 Ti GPUs. The network is implemented using Pytorch 1.4.0 and the weight is initialized with values drawn from the normal distribution $\mathcal{N}({mean}=0.0, {std}^2=0.02)$.
The domain code extractor $E$ is comprised of five partial convolutional layers with kernel size 3 and stride 2, an adaptive average pooling layer, and a convolutional layer with kernel size 1 and stride 1. Each of the partial convolutional layers is followed by ReLU and batch normalization except the last one.
\section{Comparison with Existing Methods}\label{sec:fmse}
Following DoveNet~\cite{DoveNet2020}, we also report the MSE and foreground MSE (fMSE) on the test images in different foreground ratio ranges (\emph{e.g.}, $5\%\sim 15\%$). The foreground ratio means the area of the foreground over the area of the whole image. Foreground MSE (fMSE) is MSE calculated only in the foreground region.
As shown in Table~\ref{tab:ablate_ratio}, our method outperforms all the baselines in each foreground ratio range, which demonstrates the robustness of our proposed method.
\section{Ablation Studies}\label{sec:ablation_detail}
\subsection{Hyper-parameter Analyses}\label{sec:parameter}
We investigate the impact of three hyper-parameters: the margin $m$ in Eqn. (1)(2), $\lambda$ in Eqn. (3), and the domain code dimension $L$. In Fig.~\ref{fig:hyperparameter}, we plot the performance by varying each hyper-parameter while keeping the other hyper-parameters fixed. It can be seen that our method is robust with $m$ (\emph{resp.}, $\lambda$) in a reasonable range $[2^{-2},2^{2}]$ (\emph{resp.}, $[10^{-4},10^{-1}]$). With the domain code dimension increasing to 16, the performance improves obviously. When $L$ is larger than 16, the performance increases marginally, but more training resources are in demand. So 16-dimensional domain code is a cost-effective choice.
\begin{figure}[tp!]
\begin{center}
\includegraphics[width=1\linewidth]{figure/para.jpg}
\end{center}
\caption{Impact of hyper-parameters, including the margin $m$ in Eqn. (1)(2), $\lambda$ in Eqn. (3), and the domain code dimension $L$. The gray dotted line indicates the default value of each hyper-parameter.}
\label{fig:hyperparameter}
\end{figure}
\subsection{Input Design Choices}\label{sec:ablate}
As described in Section 3.2 in the main text, we concatenate the composite image, foreground mask, and background domain code map as input for our generator $G$. Now we investigate the impact of each type of input and report the results in Table~\ref{tab:ablate_loss}.
When we only use composite image and foreground mask as input (row 1), it is exactly the same as the attention-enhanced U-net introduced in \cite{DoveNet2020}. After adding the background domain code to the input (row 2), the performance is significantly boosted, which demonstrates that background domain code can provide useful guidance for foreground harmonization. We further apply our proposed two triplet losses to regulate the domain code (row 3), which brings in extra performance gain. This is because that the triplet losses impose reasonable constraints for better domain code extraction.
Besides, when we replace the background domain code with the domain code extracted from the whole composite image (row 8), the harmonization performance is degraded by a large margin (row 8 \emph{v.s.} row 3). This is because in composite image, the foreground and background are captured in different conditions and belong to two different domains. Domain code extracted from the whole image will mislead the harmonization with undesirable foreground domain information.
In addition, we also investigate the case in which we only feed the generator with composite image and the background domain code map while removing the foreground mask from input. No matter using triplet losses (row 6) or not (row 7), the performance is significantly degraded after removing the foreground mask (row 6 \emph{v.s.} row 3, row 7 \emph{v.s.} row 2), probably because the foreground mask emphasizes foreground translation and enables the foreground to borrow information from background.
\subsection{Loss Design Choices}
Besides, we also ablate each type of triplet loss (row 4 and row 5) in Table~\ref{tab:ablate_loss}. The results demonstrate that each type of triplet loss is helpful, and two types of triplet losses can collaborate with each other to achieve further improvement.
\begin{table*}[tb]
\setlength{\abovecaptionskip}{0.1cm}
\centering
\begin{tabular}{|l|c|c|c|c|c|c|c|c|}
\hline
\multicolumn{1}{|c|}{Foreground ratios} & \multicolumn{2}{c|}{$0\%\sim 5\%$} & \multicolumn{2}{c|}{$5\%\sim 15\%$} & \multicolumn{2}{c|}{$15\%\sim 100\%$} & \multicolumn{2}{c|}{$0\%\sim 100\%$}\\ \hline
\multicolumn{1}{|c|}{Evaluation metric} & MSE$\downarrow$ & fMSE$\downarrow$ & MSE$\downarrow$ & fMSE$\downarrow$ & MSE$\downarrow$ & fMSE$\downarrow$ & MSE$\downarrow$ & fMSE$\downarrow$ \\ \hline
\multicolumn{1}{|c|}{Input composite} & 28.51 & 1208.86 & 119.19 & 1323.23 & 577.58 & 1887.05 & 172.47 & 1387.30 \\ \hline
\multicolumn{1}{|c|}{Lalonde and Efros\cite{lalonde2007using}} & 41.52 & 1481.59 & 120.62 & 1309.79 & 444.65 & 1467.98 & 150.53 & 1433.21 \\ \hline
\multicolumn{1}{|c|}{Xue \emph{et al.}\cite{xue2012understanding}} & 31.24 & 1325.96 & 132.12 & 1459.28 & 479.53 & 1555.69 & 155.87 & 1411.40 \\ \hline
\multicolumn{1}{|c|}{Zhu \emph{et al.}\cite{zhu2015learning}} & 33.30 & 1297.65 & 145.14 & 1577.70 & 682.69 & 2251.76 & 204.77 & 1580.17 \\ \hline
\multicolumn{1}{|c|}{DIH~\cite{tsai2017deep}} & 18.92 & 799.17 & 64.23 & 725.86 & 228.86 & 768.89 & 76.77 & 773.18 \\ \hline
\multicolumn{1}{|c|}{DoveNet~\cite{DoveNet2020}} & 14.03 & 591.88 & 44.90 & 504.42 & 152.07 & 505.82 & 52.36 & 549.96 \\ \hline
\multicolumn{1}{|c|}{S$^2$AM~\cite{xiaodong2019improving}} & 13.51 & 509.41 & 41.79 & 454.21 & 137.12 & 449.81 & 48.00 & 481.79 \\ \hline
\multicolumn{1}{|c|}{Ours} & \bf10.55 & \bf450.33 & \bf32.13 & \bf359.49 & \bf109.23 & \bf353.84 & \bf37.82 & \bf 405.23 \\ \hline
\end{tabular}
\caption{MSE and foreground MSE (fMSE) of different methods in each foreground ratio range based on the whole test set. The best results are denoted in boldface.}
\label{tab:ablate_ratio}
\end{table*}
\begin{table}[tb]
\centering
\begin{tabular}{c|cc|cc|cc}
\toprule
\# & mask & $z_b$ & $\mathcal{L}_{f\hat f}$ & $\mathcal{L}_{\hat fb}$ & MSE $\downarrow$ & PSNR $\uparrow$ \\
\hline \hline
1 & \checkmark & & & & 60.79 & 34.15 \\
2 &\checkmark& \checkmark& & & 43.70 & 35.43 \\
3 &\checkmark &\checkmark&\checkmark &\checkmark & 37.82 & 35.88 \\
4 &\checkmark&\checkmark&\checkmark & & 41.03 & 35.47 \\
5 &\checkmark& \checkmark& &\checkmark & 41.71 & 35.50 \\
6 & & \checkmark &\checkmark &\checkmark& 115.48 & 31.94\\
7 & & \checkmark& & & 120.49 &31.89 \\
8 &\checkmark & $\circ$ &\checkmark &\checkmark &43.18 &35.50 \\
\bottomrule
\end{tabular}
\caption{Ablation studies on input format and triplet losses. ``mask" means foreground mask, $z_b$ denotes the background domain code, and $\circ$ means that we replace the background domain code with the domain code extracted from the whole composite image. Two triplet losses are $\mathcal{L}_{f\hat f}$ and $\mathcal{L}_{\hat fb}$. }
\label{tab:ablate_loss}
\end{table}
\begin{figure*}[t]
\centering
\includegraphics[width=0.9\linewidth]{figure/bg_harm.jpg}
\caption[]{Example results of background harmonization. From left to right, we show the input composite image and the background harmonization results of DIH~\cite{tsai2017deep}, DoveNet~\cite{DoveNet2020}, S$^2$AM~\cite{xiaodong2019improving}, and our proposed BargainNet. For clarity, we highlight the unchanged foreground with red border lines.}
\label{fig:bg_harm}
\end{figure*}
\begin{table}
\centering
\begin{tabular}{|c|c|}
\hline
Method & B-T score$\uparrow$\\\hline
Input composite & 0.357 \\ \hline
DIH~\cite{tsai2017deep} & 0.813 \\ \hline
DoveNet \cite{DoveNet2020} & 0.897 \\ \hline
S$^2$AM~\cite{xiaodong2019improving} & 1.140 \\ \hline
Ours & \bf 1.266 \\ \hline
\end{tabular}
\caption{B-T scores of deep learning based methods on $99$ real composite images provided in \cite{tsai2017deep}. }
\label{tab:BT_score}
\end{table}
\section{Results on Real Composite Images}\label{sec:user_study}
In reality, there is no ground-truth image for a given real composite image, whose foreground is cut from one image and pasted on another background image. In such a scenario, the foreground is not in the same location and its color distribution is vastly different from the background. Since it is infeasible to evaluate model performance quantitatively, following \cite{tsai2017deep,xiaodong2019improving,DoveNet2020}, we conduct user study on 99 real composite images released by \cite{tsai2017deep}.
The perceptual evaluations in previous works~\cite{tsai2017deep,xiaodong2019improving,DoveNet2020} have shown that deep learning based methods are generally better than traditional methods, so we only compare our proposed BargainNet with deep learning based methods DIH~\cite{tsai2017deep}, DoveNet~\cite{DoveNet2020}, and S$^2$AM~\cite{xiaodong2019improving}.
Similarly, given each composite image and its four harmonized outputs from four different methods, we can construct image pairs $(I_i, I_j)$ by randomly selecting from these five images $\{I_i|_{i=1}^5\}$. Hence, we can construct a large number of image pairs based on $99$ real composite images.
Each user involved in this subjective evaluation could see an image pair each time to decide which one looks more realistic. Considering the user bias, 22 users participate in the study in total, contributing 10835 pairwise results. With all pairwise results, we employ the Bradley-Terry (B-T) model \cite{bradley1952rank,lai2016comparative} to obtain the global ranking of all methods and the results are reported in Table~\ref{tab:BT_score}. Our proposed BargainNet shows an advantage over other deep-based methods with the highest B-T score, which demonstrates that by explicitly using the background domain code as guidance, our method could generate more favorable results in real-world applications.
In Fig.~\ref{fig:real_res}, we present some results of real composite images used in our user study. We compare the real composite images with harmonization results generated by our proposed method and other deep learning based methods, including DIH~\cite{tsai2017deep}, DoveNet~\cite{DoveNet2020}, and S$^2$AM~\cite{xiaodong2019improving}. Based on Fig.~\ref{fig:real_res}, we can see that our proposed method could generally produce satisfactory harmonized images compared to other deep learning based methods.
\begin{figure*}[tp!]
\begin{center}
\includegraphics[width=0.9\linewidth]{figure/inharm.jpg}
\end{center}
\caption{Examples of composite images with different inharmony levels. From top to bottom, we show the network input and the harmonized output of our BargainNet respectively. From left to right, we show the five composite images and the ground-truth real image. The number below each image is its inharmony score.}
\label{fig:inharmonious}
\end{figure*}
\section{Generalization to Background Harmonization}\label{sec:bg_harmonization}
Interestingly, our method could also be used for background harmonization, in which the background is harmonized according to the foreground while the foreground remains unchanged. In particular, we can feed the composite image $\tilde I$, the background mask $\bar M$, and the composite foreground domain code $\tilde z_f$ into our generator $G$. In this way, the background region could be harmonized to the same domain as composite foreground, making the whole image harmonious as well. We show some background harmonization results of different deep learning based methods in Fig.~\ref{fig:bg_harm}. We can observe that our BargainNet is more capable of generating harmonious output. This observation is consistent with the observation in foreground harmonization, which demonstrates the remarkable generalizability and robustness of our method.
\begin{figure*}[tp!]
\begin{center}
\includegraphics[width=1\linewidth]{figure/real_deep.jpg}
\end{center}
\caption{The harmonized results on real composite images, including three deep learning based methods and our proposed BargainNet.}
\label{fig:real_res}
\end{figure*}
\section{Inharmony Level Prediction}\label{sec:inharmony_level}
Based on the extracted domain codes of foreground and background, we can predict the inharmony level of a composite image, reflecting to which extent the foreground is incompatible with the background.
We conduct experiments on HAdobe5k sub-dataset, because each real image in MIT-Adobe FiveK dataset \cite{bychkovsky2011learning} has another five edited renditions of different styles. Given a real image, we can paste the foregrounds of five edited renditions on the background of real image, leading to five composite images with the same background yet different foregrounds in HAdobe5k. Therefore, when feeding the five composite images into $G$, the generated outputs are expected to be harmonized to the same ground-truth real image. Recall that in our BargainNet, we propose to use domain code extractor to extract the domain codes $\tilde z_f$ and $z_b$ for foreground and background respectively. So we can calculate the Euclidean distance $d(z_b, \tilde z_{f})$ as the inharmony score of a composite image, which reflects how inharmonious a composite image is. For the composite images with high inharmony scores, the foreground and the background are obviously inconsistent. After harmonization, the composite foreground is adjusted to be compatible with the background. Therefore, the inharmony score should become lower.
In Fig.~\ref{fig:inharmonious}, we show one ground-truth real image with its five composite images from HAdobe5k sub-dataset, and report two inharmony scores of each image before and after harmonization. In the top row, we can observe that composite images whose foreground and background are obviously inconsistent have higher scores, while the ground-truth real image with consistent foreground and background has the lowest inharmony score. In the bottom row, after harmonization, as the foreground is translated to the same domain as background, the inharmony score of the harmonized output decreases dramatically. Interestingly, even for the ground-truth real image, harmonization using our method can further lower its inharmony score, probably because our network could make the foreground domain closer to the background.
Inharmony level provides an intuitive perspective for inharmony assessment, which is an enticing byproduct of our method and useful for harmonization related tasks. For example, given abundant composite images, we can first predict their inharmony levels and only harmonize those with high inharmony levels for computational efficiency.
\bibliographystyle{IEEEbib}
|
{
"redpajama_set_name": "RedPajamaArXiv"
}
| 4,484
|
layout: post
title: Being Online
---
I started this post with an idea. I got as far as the title, then somewhere between that text box and the text area below it, I completely lost what I was going to write. I just stared at my fingers sitting diligently on the keyboard, waiting for me to give them commands.
These thoughts are mine, but what about when I cannot summon the thoughts. I thought I had several easy-to-hammer-out blog posts ready to go. I thought I would at least last for a few days before I completely blocked myself from being able to type. Apparently, twenty-four hours is enough.
Today is supposed to be a [Meats](http://marketing.linkedin.com/blog/the-blogging-food-groups-a-well-balanced-diet-of-content-infographic/) day for writing. I am not entirely sure what that is supposed to mean, but I think I have plenty of ideas for what to put together. So today's meat post is going to be about what future meat posts I am going to write. These include:
- Cross posts with my Laivani project, wherein I learn about donor management systems and modern WISC coding patterns as I go along.
- A more comprehensive documentation of Knockout's utility functions.
- Other posts about important things.
This experiment is going to be a disaster; I can tell already.
|
{
"redpajama_set_name": "RedPajamaGithub"
}
| 3,060
|
Dies ist eine Liste der Universitäten in Polen.
Es gibt in Polen insgesamt 18 Universitäten sowie zahlreiche technische Universitäten, Wirtschaftsuniversitäten, medizinische Universitäten, landwirtschaftliche und pädagogische Universitäten, eine Musikuniversität, eine theologische Universität als auch zahlreiche den Universitäten gleichgestellte Hochschulen mit Promotionsrecht, wie Akademien und andere Hochschulen.
Universitäten
Universität Białystok
Kazimierz-Wielki-Universität Bydgoszcz
Universität Breslau
Universität Danzig
Universität Ermland-Masuren in Olsztyn
Schlesische Universität Kattowitz
Jan-Kochanowski-Universität Kielce
Jagiellonen-Universität in Krakau
Universität Łódź
Katholische Universität Lublin
Maria-Curie-Skłodowska-Universität in Lublin
Universität Opole
Adam-Mickiewicz-Universität Posen
Universität Rzeszów
Universität Stettin
Nikolaus-Kopernikus-Universität Toruń
Universität Warschau
Kardinal-Stefan-Wyszyński-Universität Warschau
Universität Zielona Góra
Universität für Sozial- und Geisteswissenschaften
Technische Universitäten
Technische Universität Białystok
Technische Universität Breslau
Technische Universität Częstochowa
Technische Universität Danzig
Technische Universität Kielce
Technische Universität Koszalin
AGH Wissenschaftlich-Technische Universität in Krakau
Technische Universität Krakau
Technische Universität Lublin
Technische Universität Łódź
Technische Universität Opole
Technische Universität Posen
Technische Universität Radom
Technische Universität Rzeszów
Schlesische Technische Universität in Gliwice
Westpommersche Technische Universität Stettin
Technische Universität Warschau
Wirtschaftsuniversitäten
Wirtschaftsuniversität Breslau
Wirtschaftsuniversität Krakau
Wirtschaftsuniversität Posen
Warsaw School of Economics
Medizinische Universitäten
Medizinische Universität Białystok
Medizinische Universität Breslau
Collegium Medicum Bydgoszcz der Nikolaus-Kopernikus-Universität Toruń
Medizinische Universität Danzig
Schlesische Medizinische Universität Katowice
Collegium Medicum der Jagiellonen-Universität in Krakau
Medizinische Universität Lublin
Medizinische Universität Łódź
Medizinische Karol-Marcinkowski-Universität Posen
Pommersche Medizinische Universität Stettin
Warschauer Medizinische Universität
Landwirtschaftliche Universitäten
Naturwissenschaftliche Universität Breslau
Technisch-Naturwissenschaftliche Universität Bydgoszcz
Landwirtschaftliche Universität Krakau
Naturwissenschaftliche Universität Lublin
Naturwissenschaftliche Universität Posen
Landwirtschaftliche Universität Warschau
Pädagogische Universitäten
Humanistisch-Naturwissenschaftliche Jan-Kochanowski-Universität Kielce
Jan-Długosz-Universität in Częstochowa
Pädagogische Universität Krakau
Natur- und Geisteswissenschaftliche Universität Siedlce
Musikuniversitäten
Frédéric-Chopin-Universität für Musik Warschau
Kunstuniversitäten
Kunstuniversität Posen
Seefahrt-Hochschulen
Seefahrt-Akademie Gdynia
Seefahrt-Akademie Szczecin
Theologische Hochschulen
Päpstliche Universität Johannes Paul II.
Christlich-Theologische Akademie Warschau
Päpstliche Theologische Fakultät Breslau
Päpstliche Theologische Fakultät Warschau
Ignatianum-Akademie Krakau
Siehe auch
Liste der Sporthochschulen in Polen
Liste von Hochschulen für Bildende Kunst in Polen
Liste der Militärakademien in Polen
Liste der Pädagogischen Hochschulen in Polen
Weblinks
Polnisches Ministerium für Wissenschaft und Hochschulwesen
Büro für Anerkennung akademischer Abschlüsse und für internationalen Austausch – vollständige Liste aller staatlichen Hochschuleinrichtungen in Polen ()
!
Polen
|
{
"redpajama_set_name": "RedPajamaWikipedia"
}
| 6,388
|
Professor Dan Levene is Professor of Semitics and the History of Religion at the University of Southampton.
As a Semitist who specialises in Aramaic dialects and Hebrew my interests are focused on primary sources of a textual nature and specifically on editing manuscripts from late antiquity and the middle ages.
In recent years I have been focusing also on Ethiopia. This is a country that adopted Christianity in the 4th century, has a very rich and ancient literary history and manuscript tradition written in Ge'ez and Amharic.
The biggest focus of my work over the years has been on literature that expresses the belief in the power of the word to heal and harm. The fact that Ethiopia's early traditions still have such an overwhelming presence within living traditions has offered me the chance of extending my research to working with communities where such beliefs are part of the fabric of life.
Looking at the literature of incantation of Jewish, Christian and Gnostic peoples from late antiquity I have become interested in the work of its producers - the scribes. Not only were they the transmitters of this, and other forms of, knowledge, they were the vehicle through whom normal people had access to the supernatural forces that they believed were the cause of misfortune and, more importantly, the key to healing.
More recently I have ventured into an investigation of the incantation culture of Ethiopia. The producers of this literature, the Ethiopian scribes, still produce these amulets in Ge'ez (Old Ethiopic) much in the same way as they did in the medieval period from which the oldest texts of this type survive.
The aim of VMBA is to provide an environment that will allow collaborative work on material that is otherwise difficult to access or unavailable. The material within this archive consists, at this stage, of parts of three collections: The Moussaieff Collection, The Dehays Collection and The Barakat Collection.
Curse or Blessing: What's in the Magic Bowl?', The Ian Karten Lecture 2002, Parkes Pamphlet No. 2.
Levene, D., & Ford, J. N. (2012). "For A?ata-de-?abuh daughter of Imma" Two Aramaic incantation bowls in the Vorderasiatisches museum, Berlin (VA 2414 and VA 2426). Journal of Semitic Studies, 57(1), 53-67.
Levene, D., & Bohak, G. (2012). Divorcing Lilith: from the Babylonian incantation bowls to the Cairo Genizah. Journal of Jewish Studies, 63(2), 197-217.
Levene, D. (2009). Rare magic inscription on human skull. Biblical Archaeology Review, 35(2), 46-50.
Levene, D. (2006). Calvariae Magicae: the Berlin, Philadelphia and Moussaieff skulls. Orientalia, 75(4), 359-379.
Levene, D., & Bhayro, S. (2005). "Bring to the Gates… upon a good smell and upon good fragrances": an Aramaic incantation bowl for success in business. Archiv für Orientforschung, 51, 242-246.
Levene, D. (2003). Heal O' Israel: a pair of duplicate magic bowls from the Pergamon Museum in Berlin. Journal of Jewish Studies, 54(1), 104-121.
Levene, D., & Rothenberg, B. (2001). Early evidence of steelmaking in the Judaic sources. The Jewish Quarterly Review, 92(1-2), 105-127.
Rothenberg, B., & Levene, D. (2001). Forging folklore: about iron in early Jewish folklore. Institute for Archaeo-Metallurgical Studies, 21, 9-10.
Levene, D., & Rothenberg, B. (2000). 'yb' b'š' - a fundamental aspect of the nature of metal. Aramaic Studies, 2, 75-87.
Levene, D. (1999). "... and by the name of Jesus ..." An unpublished magic bowl in Jewish Aramaic. Jewish Studies Quarterly, 6(4), 283-308.
Levene, D. (1998). Expedition to Atika. Institute for Archaeo-Metallurgical Studies, 20, 10-13.
Levene, D., & Geller, M. J. (1998). Magical texts from the Geniza (with a new duplicate). Journal of Jewish Studies, 49(2), 334-340.
Levene, D., Bhayro, S., Ford, J. N., Saar, O-P., Morgenstern, M., & Vilozny, N. (2018). A catalogue of Babylonian incantation bowls in the Vorderasiatisches Museum (Berlin): Descriptive List and Edition of Selected Texts. Brill.
Levene, D. (2013). Jewish aramaic curse texts from late-antique Mesopotamia. (Magical and Religious Literature of Late Antiquity; No. 2). Leiden, NL: Brill.
Levene, D., & Rothenberg, B. (2007). A metallurgical gemara: metals in the Judaic sources. (Metal in History; Vol. 4). London, GB: University College London.
Levene, D. (2003). A corpus of magic bowls: incantation texts in Jewish Aramaic from late antiquity. (Kegan Paul Library of Jewish Studies). New York, US: Kegan Paul International; Columbia University Press.
Levene, D. (2019). Magic Scrolls. In J. Gnisci (Ed.), Treasures of Ethiopia and Eritrea in the Bodleian Library, Oxford Oxford: Oxford.
Levene, D., & Ponting, M. (2015). "Recycling economies, when efficient, are by their nature invisible." A first century recycling economy. In M. J. Geller (Ed.), The Archaeology and Material Culture of The Babylonian Talmud (pp. 39-65). (Studies in Judaica; No. 16). Brill.
Levene, D., & Ponting, M. (Accepted/In press). "Recycling economies, when efficient, are by their nature invisible". A first century Jewish recycling economy. In M. Geller, & S. Shaked (Eds.), Talmudic Archaeology Boston, US: Brill.
Levene, D. (Accepted/In press). Babylonian incantation bowls. In Encyclopedia of Ancient History Chichester, GB: Wiley-Blackwell.
Levene, D. (2010). Amuletic skulls. In F. Vukosavović (Ed.), Angels and Demons, Jewish Magic Through the Ages Jerusalem, IL: Bible Lands Museum.
Levene, D. (2010). Magic bowls and incantations. In J. J. Collins, & D. C. Harlow (Eds.), Dictionary of Early Judaism (pp. 910-912). Grand Rapids, US: Eerdmans.
Levene, D. (2003). A happy thought of the magician: the magical get. In R. Deutsch (Ed.), Shlomo: Studies in Epigraphy, Iconography, History and Archaeology in Honor of Shlomo Moussaieff (pp. 175-184). Tel-Aviv Jaffa, IL: Archaeological Center Publications.
Levene, D. (2003). Zauberschale mit inwendiger beschriftung und zeichnung. In W. Seipel (Ed.), Der Turmbau zu Babel: Ursprung und Vielfalt von Sprache und Schrift ; eine Ausstellung des Kunsthistorischen Museums Wien für die europäische Kulturhauptstadt Graz 2003 (pp. 174-178). Wien, AT: KHM.
Levene, D., & Rothenberg, B. (2001). Tin and tin-lead alloys in Hebrew and Jewish aramaic. In A. Rapoprt-Albert, & G. Greenberg (Eds.), Biblical Hebrew, Biblical Texts: Essays in Memory of Michael P. Weitzman (pp. 100-112). Sheffield, GB: Sheffield Academic Press.
Levene, D. (2002). Curse or blessing, what's in the magic bowl? (Parkes Institute Pamphlet; Vol. 2). Southampton, GB: The Parkes Institute.
Text analyses and critical editing of literature in Aramaic dialects and Hebrew of late antiquity and the medieval period, Jewish magic, the history of technology as represented in Jewish literature from antiquity to the middle ages.
|
{
"redpajama_set_name": "RedPajamaC4"
}
| 7,818
|
\section{Introduction}
Since the early days of supercomputing\footnote{The first Cray-1 (SN1) was delivered to Los Alamos National Laboratory without SECDED memory. Errors were so frequent that SN2 was scrapped and SN3 was delivered to NCAR with SECDED.}, large-scale computing platforms have been engineered to handle unreliability. In contrast, algorithms and applications for large-scale systems have generally assumed a fairly simplistic failure model: The computer is a reliable digital machine, with consistent execution times and infrequent failures. If failure occurs, recovery can be handled by checkpoint-restart (CPR): occasionally storing a snapshot of application state and restarting from that saved state.
Over the past decade, the high performance computing community has become increasingly concerned that preserving the reliable, digital machine model will become too costly or infeasible~\cite{Miskov-Zivanov:2007:SER:1266366.1266680,Karnik:2004:CSE:1032295.1032595}. With the push toward exascale computing, this concern has become even greater~\cite{InterAgencyResilienceWorkshop2012}, and we must explore other models and improve algorithms.
In this paper we discuss possibilities for developing new algorithms that are resilient to hard and soft failures. However, in order to reason about such algorithms, we first need programming models that enable more sophisticated recovery strategies than CPR.
\section{Four Resilience-Enabling Programming Models}
Algorithm-based fault tolerance has certainly been studied, going back many decades~\cite{huang1984algorithm}, and many algorithms have been developed~\cite{Reis:2005:SSI:1048922.1048991,Langou:2007:RPI:1350656.1350657,Wilfredo:2000:SFT:886626,Yang:2007:FTP:1299042.1299048}, but none of these algorithms have made it into broad practical use because we have no standard programming model support. In order to develop effective resilient algorithms and applications, we first need programming models that permit us to reason about failure and implement recovery. Here we present four programming models that we think have strong promise of being useful, ordering them from easiest to hardest to implement in a production system. Even though these models are not widely deployed today, using them as abstractions for developing new algorithms will provide motivation and guidance for development of both new algorithms and the underlying system software and hardware.
\flushleft\fbox{\parbox{3.4in}{Sandia National Laboratories is a multi-program laboratory managed and operated by Sandia Corporation, a wholly owned subsidiary of Lockheed Martin Corporation, for the U.S. Department of Energy's National Nuclear Security Administration (NNSA) under contract DE-AC04-94AL85000.}}
\subsection{Skeptical Programming (SkP)}
Almost all algorithm developers assume that their software will execute reliably or fail obviously by halting. However, if they understand that silent data corruption is possible, they can develop very simple and inexpensive validation tests based on their understanding of the mathematical properties of their algorithms. For example, many algorithms have global properties of orthogonality, or conservation principles, that are implicitly assumed to be true during the execution of an algorithm. If these properties were checked occasionally during execution, the cost can be very low and many silent data corruption events could be detected. Recovery may be as simple as aborting, or may involve rolling back to a previous valid state, or even continuing execution if the error will be damped by subsequent computations.
\subsection{Relaxed Bulk-synchronous Programming (RBSP)}
One of the first impacts of reduced reliability is performance variability. As low-level system failure rates increase, error detection and correction happen more frequently in the hardware and system software layers. These events preserve the reliable digital machine model, but introduce variability in execution time. Many scalable applications are designed under the implicit assumption that equal work implies equal execution time, so that if we balance the work of a parallel application, we should scale well on a parallel computer, even though we synchronize across processors during execution. Performance variability, when coupled with frequent collective operations, leads to severe limitations in scalability, especially as we go to a million or more processes.
With the introduction of MPI-3~\cite{MPI-3.0}, we now have asynchronous neighborhood and global collectives, enabling a ``relaxed'' bulk-synchronous programming model (RBSP). Given RBSP capabilities, we are now able to develop new algorithms that can potentially hide latency.
\subsection{Local Failure, Local Recovery (LFLR)}\label{subsect:LFLR}
For parallel applications based on MPI, the current approach to dealing with the loss of a single process is to kill all remaining processes and restart the application. As we now regularly run on hundreds of thousands to more than a million processors, this approach is not feasible. Instead a local failure should permit a local recovery\footnote{By local recovery we do not mean that no communication is done in the detection or recovery phases. Instead we mean that recovery is logically local from an application developer's perspective. All processes that have valid state are not involved in recovery, except to the extent that they assist in restoring state on the failed process, or take over its workload.}.
One type of local-failure-local-recovery (LFLR) model permits the user to store specific data {\it persistently} for each MPI process and allows a recovery function to be registered, such that, if a process fails, a new process is started and assigned to the rank of the failed process, and the user's recovery function is called, giving access to the persistent data of the old process, as well as the neighbors' persistent data. Using LFLR, we can develop new algorithms for many types of problems.
\subsection{Selective Reliability Programming (SRP)}
The fourth programming model we discuss is Selective Reliability Programming (SRP), which gives the programmer the ability to declare specific data and compute regions to be more reliable that the ``bulk'' reliability of the underlying system (or we can switch the default to be reliable and then selectively be less reliable). By distinguishing between what needs to be highly reliable or not, we can develop new algorithms that store most data and do most computations with low reliability while retaining the robustness of a fully reliable approach.
Although the costs of high reliability will impact the practicality of some approaches, the details of how reliability is implemented is not fundamentally important to reasoning about new algorithms. In some cases, even very expensive approaches such as triple modular redundancy (TMR) can still be much faster than a fully unreliable approach.
\section{Toward Resilient Algorithms}
Resilient algorithms have long been a subject of research. The above four programming models enable further research and drive co-development of the algorithms and computing system features that are required to realize resilient applications. In this section we discuss some of the many possible algorithms that can be developed under the above programming models in order to provide resilience on future systems.
\subsection{Detecting and Responding to Silent Data Corruption}
Skeptical programming can be used to detect silent data corruption such as bit flips, and then determine if the resulting error is ``harmless'' or not. One example of this kind of algorithm is an implementation of GMRES~\cite{Elliott2013} that detects and, optionally, corrects single bit flips very inexpensively as part of the Arnoldi process. Many existing ABFT algorithms can be implemented in using a skeptical algorithm programming approach, since the meta data used to recover state can also be used to detect anomalous behavior.
\subsection{Latency-tolerant Algorithms}
One of the most important and effective algorithm research and development strategies we can explore now is latency tolerance. Many of our scalable algorithms and applications depend on collective operations that, when implemented in a straightforward manner, lead to synchronous global collectives. On emerging high end platforms, these collectives have become severe performance limiters due to poor scaling of collectives. The advent of asynchronous collectives gives us new opportunities. The basic challenge we face as algorithm developers in this situation is finding useful work to do while a collective is completing. Recent work in pipelining algorithms, for example the p$(l)$-GMRES algorithm~\cite{ghysels2012hiding}, shows that latency hiding by unrolling iterations in a Krylov solver can help restore scalability. Similar approaches for many algorithms can lead to relatively minor design changes that result in better tolerance of latency and performance variability.
The impact of successfully redesigning algorithms to be latency tolerant is that performance variability on existing systems can be hidden. But even more importantly, if we can tolerate performance variability due to error detection and correction at the system software and hardware levels, system designers can detect and correct more errors without impacting application scalability, permitting us to extend the viability of the reliable digital machine model.
\subsection{Locally Restarted PDE Computations}
Given the programming features described in Section~\ref{subsect:LFLR}, we have the potential to develop a broad collection of algorithm with local recovery properties. Examples for differential equations include:
\begin{itemize}
\item Explicit methods: As shown in~\cite{GokhaleWong2011}, an explicit time-stepping algorithm can be easily implemented to recover locally, given the LFLR features.
\item Implicit methods: This case is more interesting. The challenge is to restore a local state that is equivalent up to the truncation error of the PDE. Several interesting approaches seem promising.
\item Redundant storage of coarse model: In order to recover state from a lost process, we could explore storing a coarse model representation on neighboring processes that could be used to boot-strap state recovery upon failure.
\end{itemize}
\subsection{Reliable Outer Iterations}
Many algorithms can be cast in an outer-inner formulation. For example, a fault-tolerant GMRES variant, as described in~\cite{2012arXiv1206.1390B}, uses reliable computation and storage in the outer iteration and an ``unreliable'' GMRES in the inner iteration. The result is that most computation and data are in low-reliability mode, leading to presumably cheaper computations. Because the outer iterations are reliable, the solution returned by the inner solve (if it comes back at all) can be analyzed and used or discarded. Even if the inner solve answer is not correct, it can still be used with some effect.
\section{Conclusions}
Resilience is a critical requirement for future high-end computing. In order to effectively develop resilient algorithms and applications, we need robust and usable resilient computing models. In this paper we have identified four specific models that allow us to reason about and develop new algorithms. SkP requires nothing more than a change in attitude on the part of the programmer, from trusting that a machine is reliable digital device to being aware that an incorrect computation may occur. RBSP is already possible with the introduction of MPI 3.0. LFLR requires more support from the underlying system layers, and requires some kind of support from programming languages and libraries. The ULFM library~\cite{Bland2013,ULFM} already provides one approach to supporting LFLR. SRP is the most challenging model, but also firmly addresses one of the biggest challenges we face: silent errors.
Much of the focus of extreme-scale computing is on massive concurrency, which is appropriate. However, without resilient computing models we face a very real risk of application failure rates that are too high to realize the benefits of future systems. In addition, any progress we make in resilient algorithms and applications permits us to utilize lower cost systems in general, systems with lower quality interconnect networks, higher bit failure rates and increased node loss. Resilient algorithms and applications will enable effective extreme scale computing and reduce system costs at other levels.
\bibliographystyle{IEEEtran}
|
{
"redpajama_set_name": "RedPajamaArXiv"
}
| 970
|
{"url":"https:\/\/physics.stackexchange.com\/questions\/228492\/redshift-of-distant-galaxies-why-not-a-doppler-effect","text":"Redshift of distant galaxies: why not a doppler effect?\n\nHow can I explain to my 17 year old pupils that the observed redshift of distant galaxies cannot be interpreted as a doppler effect and inescapably leads to the conclusion that space itself is expanding?\n\nI understand that this redshift is well explained in general relativity (GR) by assuming that space itself is expanding. As a consequence, distant galaxies recede from us and the wavelength of the light is \"streched\". Expansion, redshift and the Hubble law are explained coherently in GR, as well as many other phenomena (e.g. the cosmic microwave background), and the GR predictions about redshift agree with observations.\n\nI understand that the redshift of distant galaxies cannot be explained as a doppler effect of their motion through space. Why exactly is a pupil's doppler interpretation wrong?\n\nMy first answer: \"Blueshifted galaxies (e.g. Andromeda) are only seen in our local neighborhood, not far away. All distant galaxies show a redshift. At larger distances (as measured e.g. with Cephe\u00efds) the redshift is larger. For a doppler interpretation of the redshift distant galaxies we must necessarily assume that we are in a special place, to the discomfort of Copernicus. In this view, space cannot be homogeneous and isotropic.\" Is this answer correct?\n\nMy second answer: \"A doppler effect only occurs at the moment the light is emitted, whereas the cosmological redshift in GR grows while the light is traveling to us.\" My problem with this answer (if it is correct): what observational evidence do we have for a gradual (GR) increase of the redshift, disproving the possibility of an \"instantaneous doppler shift at the moment of emission\"?\n\nMy third answer: \"For galaxies at $z>1$ you can only have $v<c$ if you use the doppler formula from special relativity (SR): $v=\\frac{(z+1)^2-1}{(z+1)^2+1}\\cdot c$\". My problem with this answer: what's wrong with using the doppler formula from SR as long as someone views the universe as static, in a steady state? With just the right amount of dark energy to balance the gravitational contraction, if you wish?\n\nMy fourth answer: \"Recent observations of distant SN Ia show a duration-redshift relation that can only be explained with time dilation [see Davis and Lineweaver, 2004, \"Expanding Confusion etc.\"]\" My problem with this answer: does time dilation prove we have expanding space, in disagreement with a doppler effect?\n\nMy fifth answer would involve the magnitude-redshift relation for distant SN Ia [Davis and Lineweaver], but that's too complicated for my pupils.\n\n\u2022 There are always different levels of explanation. While technically not correctly characterized as a doppler redshift, I would not say that explaining it to 17 year olds that way without involving general relativity is exceptionally poor teaching. Even the \"special place\" Copernican interpretation is not required, unless you want to hold on to a \"rigid space\" model. I would say that at this level letting go of a rigid space is a good idea and using language like \"expanding space makes it look like the galaxies are retreating ever faster and it causes a doppler effect-like redshift\" is OK. \u2013\u00a0CuriousOne Jan 9 '16 at 22:03\n\nIn Stephen Weinberg's well-known book \"The First Three Minutes,\" he talks solely about Doppler shifts. A good review and analysis of the \"expanding space\" vs. Doppler shift question was provided by Bunn and Hogg, \"The kinematic origin of the cosmological red shift,\" Am. J. Phys. 77 (8), 2009, pp. 688-694. They make convincing arguments that the red shift is best understood as a series of Doppler shifts in overlapping space-time regions small enough so that Minkowski (flat) space-time geometry is an excellent approximation. Regardless, they say in the Conclusion: \"There is no \u201cfact of the matter\u201d about the interpretation of the cosmological redshift: what one concludes depends on one\u2019s coordinate system or method of calculation.\" Their argument for the Doppler shift says it is more \"natural\" because it is consistent with a number of well-established facts about the general relativistic theory of gravity.\n\n\u2022 While I don't work in this area, I think that the reason many authors discourage people from thinking about cosmological redshifts as \"Doppler\" is that thinking in terms of \"motion away from us\" makes several basic errors attractive (think \"we're in the middle\" and \"the universe is expanding into pre-existing space\"). Bunn's treatment is sensible but arises from a sophisticated understanding of what \"velocity\" is involved. \u2013\u00a0dmckee Jan 10 '16 at 2:59\n\u2022 Yes I also think that cosmological redshift is stressed under this name to emphasise space expansion. However it is not observationally different from a Doppler shift. \u2013\u00a0Alchimista Nov 18 '17 at 10:36\n\nThere are two parts to understanding this:\n\n1. Space is expanding - we know this to be true because as distance increases, so does recessional velocity (Hubble's law). So the more space in-between, the faster it recedes. Therefore the space itself is responsible for the recessions, and thus must be expanding. There is simply so much evidence for Hubble's law, that the probability that every galaxy (except very close galaxies) just happens to be receding from us is minuscule.","date":"2019-08-23 15:21:29","metadata":"{\"extraction_info\": {\"found_math\": true, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 0, \"math_alttext\": 0, \"mathml\": 0, \"mathjax_tag\": 0, \"mathjax_inline_tex\": 1, \"mathjax_display_tex\": 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\": 0.7526536583900452, \"perplexity\": 800.0731171446291}, \"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-2019-35\/segments\/1566027318894.83\/warc\/CC-MAIN-20190823150804-20190823172804-00340.warc.gz\"}"}
| null | null |
Acquired at an area Cheshire public sale condominium, numerous own albums of WWI photos taken through their past gentleman proprietor, an officer within the Royal box Artillery (RFA) referred to as Harold Cooper Bebington. He was once additionally a member of 1 of the early novice photographic societies at his domestic city, as a result explaining their very good caliber and his curiosity in taking them, even though almost certainly opposed to King's laws! They relate to his education as an officer cadet at the beginning from 1916 in England, operating with horses and box gun limbers, his commissioning, and next hospitalisation in England after wounding at the Western entrance. Afterwards he was once published to the then new anti-aircraft artillery and back in 1918 to the Front.
Additionally, there are interval images of friends and family in uniform as they too went off to struggle, exhibiting diverse regiments which make an excellent uniform examine to counterpoint his story.
They express the "Home entrance" points of military lifestyles protecting officer education and remedy and sport. The photographer himself is a vintage instance of the WWI British officer who observed provider and is strange of these who replied the decision and as such is useful of remembering by means of having his tale told.
John Rushton was once within the Royal military from the age of 16 to nineteen. For numerous years he labored in factories. As a tender grownup, he studied at collage collage and the London college of Economics. His half time paintings integrated barrel organ making a song, time and movement learn and sporting a sandwich board.
The fundamental principles and implications of each state's approach of presidency offer an authoritative and aim foundation to lead and decide the activities of the state's selection makers, together with courts. Christopher Taucar offers a close historical past of the British system's improvement from kingdom energy being exercised by means of centralized royal courts to its present-day detailed legislative, judicial, and government our bodies with varied powers.
Studying within the nice struggle 1917-1919 seems at lifestyles in a major commercial and agricultural city within the south of britain. The publication charts the adjustments that happened in traditional people's lives, a few brought on by the battle, a few of their very own doing. at the floor, studying used to be a peaceful city that obtained on with its company: beer, biscuits, metalwork, seeds and armaments, yet its poverty impacted on business kin resulting in moves.
Fingers and the nation is a historical past of Britain's at the beginning glossy armaments corporation, the Armstrong Whitworth corporation, from its origins in 1854 to 1914. It makes a speciality of the function of Sir William G. Armstrong, an engineer and entrepreneur who reworked his modest mechanical engineering company right into a gigantic business company which invented, built, synthetic and offered heavy weapons and warships through the international.
|
{
"redpajama_set_name": "RedPajamaC4"
}
| 9,554
|
import {
Component,
ElementRef,
Input,
OnChanges,
OnDestroy,
OnInit,
SimpleChange,
ViewEncapsulation
} from '@angular/core';
import { VgAPI } from '../../../core/services/vg-api';
import { Subscription } from 'rxjs';
@Component({
selector: 'vg-scrub-bar-cue-points',
encapsulation: ViewEncapsulation.None,
template: `
<div class="cue-point-container">
<span *ngFor="let cp of cuePoints" [style.width]="cp.$$style?.width" [style.left]="cp.$$style?.left"
class="cue-point"></span>
</div>
`,
styles: [ `
vg-scrub-bar-cue-points {
display: flex;
width: 100%;
height: 5px;
pointer-events: none;
position: absolute;
}
vg-scrub-bar-cue-points .cue-point-container .cue-point {
position: absolute;
height: 5px;
background-color: rgba(255, 204, 0, 0.7);
}
vg-controls vg-scrub-bar-cue-points {
position: absolute;
top: calc(50% - 3px);
}
` ]
})
export class VgScrubBarCuePoints implements OnInit, OnChanges, OnDestroy {
@Input() vgCuePoints: TextTrackCueList;
@Input() vgFor: string;
elem: HTMLElement;
target: any;
onLoadedMetadataCalled: boolean = false;
cuePoints: Array<any> = [];
subscriptions: Subscription[] = [];
totalCues = 0;
constructor(ref: ElementRef, public API: VgAPI) {
this.elem = ref.nativeElement;
}
ngOnInit() {
if (this.API.isPlayerReady) {
this.onPlayerReady();
}
else {
this.subscriptions.push(this.API.playerReadyEvent.subscribe(() => this.onPlayerReady()));
}
}
onPlayerReady() {
this.target = this.API.getMediaById(this.vgFor);
let onTimeUpdate = this.target.subscriptions.loadedMetadata;
this.subscriptions.push(onTimeUpdate.subscribe(this.onLoadedMetadata.bind(this)));
if (this.onLoadedMetadataCalled) {
this.onLoadedMetadata();
}
}
onLoadedMetadata() {
if (this.vgCuePoints) {
// We need to transform the TextTrackCueList to Array or it doesn't work on IE11/Edge.
// See: https://github.com/videogular/videogular2/issues/369
this.cuePoints = [];
for (let i = 0, l = this.vgCuePoints.length; i < l; i++) {
let end = (this.vgCuePoints[ i ].endTime >= 0) ? this.vgCuePoints[ i ].endTime : this.vgCuePoints[ i ].startTime + 1;
let cuePointDuration = (end - this.vgCuePoints[ i ].startTime) * 1000;
let position = '0';
let percentWidth = '0';
if (typeof cuePointDuration === 'number' && this.target.time.total) {
percentWidth = ((cuePointDuration * 100) / this.target.time.total) + "%";
position = (this.vgCuePoints[ i ].startTime * 100 / (Math.round(this.target.time.total / 1000))) + "%";
}
(<any>this.vgCuePoints[ i ]).$$style = {
width: percentWidth,
left: position
};
this.cuePoints.push(this.vgCuePoints[ i ]);
}
}
}
updateCuePoints() {
if (!this.target) {
this.onLoadedMetadataCalled = true;
return;
}
this.onLoadedMetadata();
}
ngOnChanges(changes: { [propName: string]: SimpleChange }) {
if (changes[ 'vgCuePoints' ].currentValue) {
this.updateCuePoints();
}
}
ngDoCheck() {
if (this.vgCuePoints) {
const changes = this.totalCues !== this.vgCuePoints.length;
if (changes) {
this.totalCues = this.vgCuePoints.length;
this.updateCuePoints();
}
}
}
ngOnDestroy() {
this.subscriptions.forEach(s => s.unsubscribe());
}
}
|
{
"redpajama_set_name": "RedPajamaGithub"
}
| 5,669
|
Table of Contents
Preface
Title Page
Copyright Page
Dedication
THE CLAIMING OF SLEEPING BEAUTY
THE JOURNEY AND THE PUNISHMENT AT THE INN
BEAUTY
THE CASTLE AND THE GREAT HALL
THE PRINCE'S BED CHAMBER
PRINCE ALEXI
PRINCE ALEXI AND FELIX
THE SLAVES' HALL
THE TRAINING HALL
THE HALL OF PUNISHMENTS
DUTIES IN THE PRINCES CHAMBER
SERVING MAID
THE BRIDLE PATH
THE QUEEN'S CHAMBER
LADY JULIANA IN THE QUEEN'S CHAMBER
WITH PRINCE ALEXI
PRINCE ALEXI TELLS OF HIS CAPTURE AND ENSLAVEMENT
PRINCE ALEXI'S EDUCATION CONTINUES
THE VILLAGE
**THE EROTIC NOVELS** OF ANNE RICE WRITING AS **A. N. ROQUELAURE**
_**The Claiming of Sleeping Beauty**_
•
_**Beauty's Punishment**_
•
_**Beauty's Release**_
Since 1983, A. N. Roquelaure has envisioned (for the uninhibited reader) a hypnotic and seductive adult fairy tale in the Sleeping Beauty novels. Now, the author of this exquisite erotic trilogy reveals her true identity—beckoning the reader into a sensuous world of forbidden dreams and dark-edged desires ... a world in which traditional ideas of submission and dominance and gender preference are thrown to the winds ... a world made irresistibly inviting by the adventurous spirit and imagination of the unrivaled Anne Rice.
_an_
_erotic novel of_
_tenderness and cruelty_
_for the enjoyment_
_of men and_
_women_
_PLUME_
_Published by the Penguin Group_
_Penguin Putnam Inc., 375 Hudson Street, New York, New York 10014, U.S.A._
_Penguin Books Ltd, 27 Wrights Lane, London W8 5TZ, England_
_Penguin Books Australia Ltd, Ringwood, Victoria, Australia_
_Penguin Books Canada Ltd, 10 Alcorn Avenue,_
_Toronto, Ontario, Canada M4V 3B2_
_Penguin Books (N.Z.) Ltd, 182-190 Wairau Road,_
_Auckland 10, New Zealand_
_Penguin Books Ltd, Registered Offices: Harmondsworth, Middlesex, England_
_Published by Plume, a member of Penguin Putnam Inc._
_Previously published in a Dutton edition._
_First Plume Printing, November, 1990_
_First Plume Printing, This Edition, May, 1999_
_Copyright © A. N. Roquelaure, 1983_
_All rights reserved_
**REGISTERED TRADEMARK—MARCA REGISTRADA**
_Roquelaure, A. N._
_The claiming of Sleeping Beauty_
_I. Title._
PS3568.0696C'.54 82-14715
eISBN : 978-1-440-67392-4
__
__
_Without limiting the rights under copyright reserved above, no part of this_
_publication may be reproduced, stored in or introduced into a retrieval_
_system, or transmitted, in any form, or by any means (electronic, mechanical,_
_photocopying, recording, or otherwise), without the prior written permission_
_of both the copyright owner and the above publisher of this book._
_PUBLISHER'S NOTE_
_This is a work of fiction. Names, characters, places, and incidents either are the_
_product of the author's imagination or are used fictitiously, and any resemblance_
_to actual persons, living or dead, events, or locales Is entirely coincidental._
BOOKS ARE AVAILABLE AT QUANTITY DISCOUNTS WHEN USED TO PROMOTE PRODUCTS OR
SERVICES. FOR INFORMATION PLEASE WRITE TO PREMIUM MARKETING DIVISION, PENGUIN
PUTNAM INC., 375 HUDSON STREET, NEW YORK, NEW YORK 10014
<http://us.penguingroup.com>
_For_
_S. T. Roquelaure_
_with love_
PREFACE
I've always loved the fairy tale Sleeping Beauty, and found something erotic at its core. The Prince awakens Beauty with a kiss. And I thought, all right, what if he brought a kind of liberation, an induction into a world of bizarre yet irresistible delights? It has to be remembered that within the frame of a sadomasochistic fantasy like the Beauty trilogy, the readers are invited to identify with and enjoy the predicament of the slaves. The books aren't about literal cruelty; they're about surrender, the fun of imagining you have no choice but to enjoy sex. Beauty's slavery is delicious, sensuous, abandoned, and ultimately liberating. This is all part of the framework. And it seemed to work exquisitely with the old fairy tale. And of course the fairy tale removes us from everyday life; it removes us from the intrusion of garish headlines, literal violence, and all the ugliness of crime. We go into a gilded dream here, luscious and engulfing, in which we're free to imagine all sorts of things—a fairy-tale world indeed.
As Anne Rice, I'm known for certain kinds of novels; the Roquelaure books retain the name Roquelaure (even with my name added) to indicate that this is something "different." If Anne Rice is one kind of savory dish, well this is another entirely. And some might find it far too spicy for their taste. I don't like the idea of confusing or disappointing readers, so the pen name helps with that. Of course, there are many people who have read all my work, including the Roquelaure novels, and they see me as a multifaceted writer. But the Roquelaure material is erotica, without reservation, and it needs that pen name on the label, so to speak. The pen name says: Anne Rice is doing something very different here.
I felt I needed the anonymity of the pen name to write freely, to pursue an authentic erotica without being inhibited or self-conscious. And it worked wonders to imagine myself "cloaked" by the name Roquelaure, which is a kind of French cloak—named after the Frenchman who popularized it. My father was still living then and I didn't want him to know about the books either. In fact, there were lots of friends and relatives whom I didn't want to worry about as I developed the writing. There was quite a bit of exposure involved in writing such graphic sexual fantasies. It was frightening now and then, and it was thrilling. Eventually, I told my father about the books, asking him not to read them, and I did put my name on them. I adjusted completely to people knowing I'd written them. But only after I'd finished with the trilogy—as I recall.
A pen name enables you not only to cloak what you are doing from friends and family; it gives you a new freedom to do something you would not do as yourself. I have thought of writing some new erotica, and I must confess I imagined using a new pen name for it. I don't know whether I'll pursue it, but I do find the freedom of the pen name attractive.
When the Sleeping Beauty Trilogy books were first published, they were underground books. They had the backing of a major mainstream publisher, yes, but the publication, though dignified and beautiful, was relatively quiet. But different readers embraced the books almost at once. They clearly appealed to young people, and older married people, to gays and straights. And they've sold steadily ever since they first appeared. Women come up to me at signings with babies in strollers and giggle and laugh and say, "We love your dirty books." People of all ages, actually, present the books to be signed.
Why do I think these particular books have been popular? Two reasons. First, I think it is because they involve no harsh, garish violence at all. They involve game playing, really. No one is burned or cut or hurt. Certainly no one is killed. Indeed the whole sadomasochistic predicament is presented as a glorified game played out in luxurious rooms and with very attractive people, and involving very attractive slaves. There are endless motifs offered for dominance and submission, for surrender and love. It's like a theme park of dominance and submission, a place to go to enjoy the fantasy of being overpowered by a beautiful man or woman and delightfully compelled to surrender and feel keening pleasure, without the slightest serious harm. I think it's authentic to the way many who share this kind of fantasy really feel. I think what makes it work for people is the combination of the very graphic and unsparing sexual details mixed with the elegant fairy-tale world.
Unfortunately a lot of hackwork pornography is written by those who don't share the fantasy, and they slip into hideous violence and ugliness, thinking the market wants all that, when the market never really did. Second, this is shamelessly erotic. It pulls no punches at being what it is. It's excessive and it is erotica. Before these books, a lot of women read what were called "women's romances" where they had to mark the few "hot pages" in the book. I said, well, look, try this. Maybe this is what you really want, and you don't have to mark the hot pages because every page is hot. Every page is about sexual fulfillment. Every page is meant to give you pleasure. There are no boring parts. Yet it's very "romantic." And well, I think this worked.
Lots of people enjoy imagining themselves passive, in the hands of a beautiful lover, male or female, who will force them to enjoy themselves. It's a common idea, and it cuts across gender and class. Men love these sorts of fantasies as much as women. And these books offer all kinds of gender combinations; women dominating men and women; men dominating men and women. The books offer ornate and seductive variations on the themes; and all of it is interwoven in stories with real characters, and again, the emphasis is on a lush, sensuous realm in which all this happens. There are very detailed descriptions of physical interaction and response; but the fairy-tale spell is sustained.
I also went all the way with exploring the mind-set of sadomasochism as I saw it, letting the fantasy characters talk in depth about what they felt and what they enjoyed and what thrilled them as they were humiliated and overwhelmed. I suspect that for some readers, this kind of deep exploration of the mentality of the participants was entirely new.
Is this why they appealed to so many, because people want this very combination of elements? Perhaps.
I certainly never found the combination of elements I wanted in anyone else's erotica. So I offered what I could not find; a light touch; elegance; preciseness; a dreamlike kingdom; a dream in which people explore their need to be passive and to "pretend" that someone gorgeous and irresistible is "making" them do it.
Psychiatrists have written volumes on the nature of the sadomasochistic fantasy, but when I wrote the trilogy I didn't know of any fiction that really enabled you to slide in it and "play" the way I wanted to play. So I wrote the books I couldn't find.
I never thought a book as eccentric as _Interview with the Vampire_ would have mass appeal. I only knew that I wanted to "be with the vampire" in the story, tell it from his point of view. I wanted to be inside his head and heart and reveal his voice and his pain. Now as it turned out, other people were exploring this same kind of thing—the backstory of the villain, the monster, or the comic book hero and heroine who'd always been described from a distance or in brittle form. People wanted to explore all kinds of super characters and hear their intimate musings. And I began to see more and more of this—movies made in which Superman could bear his soul, and Lois Lane could really talk about what it meant to love him. The demand for such romantic fantasies grew and grew. But did I have any idea that would happen? No. I wrote what I wanted to read. Well, the same thing is true with the Beauty books.
I didn't know whether that many other people had the fantasies. After all, we didn't talk much about them. Only a small elite knew about the mysterious _Story of O_. But I knew I had these fantasies, and I wanted to share them, and I felt an overwhelming desire to do them "right." I didn't want to compromise, water them down, or shrink from the most humiliating detail. I wanted to really delve into intense sensuous pleasure but put a gilded frame around a safe place for the reader from which he or she could go and come with ease.
Of course these books have from time to time been banned. I never expected a library to stock the Beauty trilogy. I know that many libraries respond to community standards, and I just never thought about it much at all. I did notice and I couldn't help notice that the books sold well and steadily, and that at every signing I gave, people brought them to be signed. Recently, I've signed as many copies of the Beauty books as I have of any other book I've written. So I don't worry too much about being banned. I've always shocked people. Years ago, I published a novel about the eighteenth-century castrati opera singers, titled _Cry to Heaven_. Someone brought a copy back to a bookstore in Stockton, California, and demanded his money back. "This is pornography," he said. There are always some people objecting to what I do. I'm grateful the Beauty books have been embraced and sustained over the years.
As a feminist, I'm very much supportive of equal rights for women in all walks of life. And that includes for me the right of every woman to write out her sexual fantasies and to read books filled with sexual fantasies that she enjoys. Men have always enjoyed all kinds of pornography. How can it be wrong for women to have the same right? We're sexual beings! And fantasy is where we can do the things we can't do in ordinary life. A woman has a right to imagine herself carried away by a handsome prince, and to choose for herself as she writes, the color of his hair and eyes, and imagine his silky voice. She has a right to make him as tall as she wants and as strong as he wants. Why not? Men have always allowed themselves such fantasies.
Famous madams have told us for decades that powerful men love to be dominated and come to them for role playing that allows the male client to be passive. In fact, some madams have said that men who enjoy playing the passive role are often men who are very powerful in real life. Well, women today are more powerful than ever. They're Supreme Court judges, senators, doctors, lawyers, entrepreneurs, executives, soldiers, cops. They can excel in all walks of life. And why shouldn't they be able to go home from the courtroom, the university, or the office and kick back and "pretend" they're being swept away to the Queen's sadomasochistic kingdom where all the fairy-tale court will watch them being ravaged by the handsome Prince?
The literary world today is wide open for all kinds of creative endeavors. We are in a new golden age in which fantasy, science fiction, speculative fiction, historical drama, horror, gothic, and supernatural romance are all mainstream. Well, the same holds true now obviously for erotica. People in general are "out of the closet" as enjoyers of erotic books. The novel _50 Shades of Grey_ has proved this. And I am discovering that the Beauty books in spite of all their playful excess—are for the first time going mainstream.
But I wouldn't continue Beauty's story. I felt that ended just the way I wanted. But I might write some more. I don't think I did all I could do in these books, within the fantasy itself, in admitting how much the slaves enjoyed it—how they loved it. I'd deepen that aspect, and still keep the tension, if I did them today.
People are much more comfortable today admitting and talking about what they enjoy in fiction and film. Much more. People are "out of the closet" about sexuality, period. The whole world knows women are sensual human beings as well as men. It's no secret anymore that women want to read sexy fiction just as men do, and there's a new frankness about the varieties of fantasies one might enjoy. So many clichés have been broken and abandoned. And this is a wonderful thing.
—ANNE RICE
JUNE 2012
**THE CLAIMING OF SLEEPING BEAUTY**
**T** HE PRINCE had all his young life known the story of Sleeping Beauty, cursed to sleep for a hundred years, with her parents, the King and Queen, and all of the Court, after pricking her finger on a spindle.
But he did not believe it until he was inside the castle.
Even the bodies of those other Princes caught in the thorns of the rose vines that covered the walls had not made him believe it. They had come believing it, true enough, but he must see for himself inside the castle.
Careless with grief for the death of his father, and too powerful under his mother's rule for his own good, he cut these awesome vines at their roots, and immediately prevented them from ensnaring him. It was not his desire to die so much as to conquer.
And picking his way through the bones of those who had failed to solve the mystery, he stepped alone into the great banquet hall.
The sun was high in the sky and those vines had fallen away, so the light fell in dusty shafts from the lofty windows.
And all along the banquet table, the Prince saw the men and women of the old Court, sleeping under layers of dust, their ruddy and slack faces spun over with spider webs.
He gasped to see the servants dozing against the walls, their clothing rotted to tatters.
But it was true, this old tale. And, fearless as before, he went in search of the Sleeping Beauty who must be at the core of it.
In the topmost bedchamber of the house he found her. He had stepped over sleeping chambermaids and valets, and, breathing the dust and damp of the place, he finally stood in the door of her sanctuary.
Her flaxen hair lay long and straight over the deep green velvet of her bed, and her dress in loose folds revealed the rounded breasts and limbs of a young woman.
He opened the shuttered windows. The sunlight flooded down on her. And approaching her, he gave a soft gasp as he touched her cheek, and her teeth through her parted lips, and then her tender rounded eyelids.
Her face was perfect to him, and her embroidered gown had fallen deep into the crease between her legs so that he could see the shape of her sex beneath it.
He drew out his sword, with which he had cut back all the vines outside, and gently slipping the blade between her breasts, let it rip easily through the old fabric.
Her dress was laid open to the hem, and he folded it back and looked at her. Her nipples were a rosy pink as were her lips, and the hair between her legs was darkly yellow and curlier than the long straight hair of her head which covered her arms almost down to her hips on either side of her.
He cut the sleeves away, lifting her ever so gently to free the cloth, and the weight of her hair seemed to pull her head down over his arms, and her mouth opened just a little bit wider.
He put his sword to one side. He removed his heavy armor. And then he lifted her again, his left arm under her shoulders, his right hand between her legs, his thumb on top of her pubis.
She made no sound; but if a person could moan silently, then she made such a moan with her whole attitude. Her head fell towards him, and he felt the hot moisture against his right hand, and laying her down again, he cupped both of her breasts, and sucked gently on one and then the other.
They were plump and firm, these breasts. She'd been fifteen when the curse struck her. And he bit at her nipples, moving the breasts almost roughly so as to feel their weight, and then lightly he slapped them back and forth, delighting in this.
His desire had been hard and almost painful to him when he had come into the room, and now it was urging him almost mercilessly.
He mounted her, parting her legs, giving the white inner flesh of her thighs a soft, deep pinch, and, clasping her right breast in his left hand, he thrust his sex into her.
He was holding her up as he did this, to gather her mouth to him, and as he broke through her innocence, he opened her mouth with his tongue and pinched her breast sharply.
He sucked on her lips, he drew the life out of her into himself, and feeling his seed explode within her, heard her cry out.
And then her blue eyes opened.
"Beauty!" he whispered to her.
She closed her eyes, her golden eyebrows brought together in a little frown and the sun gleaming on her broad white forehead.
He lifted her chin, kissed her throat, and drawing his organ out of her tight sex, heard her moan beneath him.
She was stunned. He lifted her until she sat naked, one knee crooked on the ruin of her velvet gown on the bed which was as flat and hard as a table.
"I've awakened you, my dear," he said to her. "For a hundred years you've slept and so have all those who loved you. Listen. Listen! You'll hear this castle come alive as no one before you has ever heard it."
Already a shriek had come from the passage outside. The serving girl was standing there with her hands to her lips.
And the Prince went to the door to speak to her.
"Go to your master, the King. Tell him the Prince has come who was foretold to remove the curse on this household. Tell him I shall be closeted now with his daughter."
He shut the door, bolting it, and turned to look at Beauty.
Beauty was covering her breasts with her hands, and her long straight golden hair, heavy and full of a great silky density, flared down to the bed around her.
She bowed her head so that the hair covered her.
But she looked at the Prince and her eyes struck him as devoid of fear or cunning. She was like those tender animals of the wood just before he slew them in the hunt: eyes wide, expressionless.
Her bosom heaved with anxious breath. And now he laughed, drawing near, and lifting her hair back from her right shoulder. She looked up at him steadily, her cheeks suffused with a raw blush, and again he kissed her.
He opened her mouth with his lips, and taking her hands in his left hand he laid them down on her naked lap so that he might lift her breasts now and better examine them.
"Innocent beauty," he whispered.
He knew what she was seeing as she looked at him. He was only three years older than she had been. Eighteen, newly a man, but afraid of nothing and no one. He was tall, black haired; he had a lean build which made him agile. He liked to think of himself as a sword—light, straight, and very deft, and utterly dangerous.
And he had left behind him many who would concur with this.
He had not so much pride in himself now as immense satisfaction. He had gotten to the core of the accursed castle.
There were knocks at the door, cries.
He didn't bother to answer them. He laid Beauty down again.
"I'm your Prince," he said, "and that is how you will address me, and that is why you will obey me."
He parted her legs again. He saw the blood of her innocence on the cloth and this made him laugh softly to himself as again he gently entered her.
She gave a soft series of moans that were like kisses to his ear.
"Answer me properly," he whispered.
"My Prince," she said.
"Ah," he sighed, "that is lovely."
When he opened the door, the room was almost dark. He told the servants he would have his supper now, and he would receive the King immediately.
Beauty he ordered to dine with him, and to remain with him, and he told her firmly that she was to wear no clothing.
"It's my wish to have you naked and always ready for me," he said.
He might have told her she was incomparably lovely, with only her golden hair to clothe her, and the blushes on her cheeks to cover her, and her hands trying so vainly to shield her sex and her breasts, but he didn't say this aloud.
Rather he took her little wrists and held them behind her back as the table was brought in, and then he ordered her to sit opposite.
The table was not so wide that he couldn't reach her easily, touch her, caress her breasts if he liked. And reaching out he lifted her chin so that he could inspect her by the light of the servants' candles.
The table was laid with roast pork and fowl, fruit in big glistening silver bowls, and immediately the King stood in the door, dressed in his heavy ceremonial robes, a gold crown atop his head as he bowed to the Prince and waited for the command to enter.
"Your Kingdom has been neglected for a hundred years," said the Prince as he lifted his wine goblet. "Your vassals have many of them fled to other lords; good land lies fallow. But you have your wealth, your Court, your soldiers. So much lies ahead of you."
"I am in your debt, Prince," the King answered. "But will you tell me your name, the name of your family?"
"My mother, Queen Eleanor, lives on the other side of the forest," said the Prince. "In your time, it was my great-grandfather's kingdom; he was King Heinrick, your powerful ally."
The Prince saw the King's immediate surprise and then his look of confusion. The Prince understood it perfectly. And when a blush came to the King's face, the Prince said:
"And in those times you served your time in my great-grandfather's castle, did you not, and perhaps your queen also?"
The King pressed his lips together in resignation and slowly nodded. "You are the son of a powerful monarch," he whispered. And the Prince could see that the King would not raise his eyes to see his naked daughter, Beauty.
"I will take Beauty to serve," said the Prince. "She is mine now." He took out his long silver knife and, cutting the hot, succulent pork, he laid several pieces on his own plate. The servants all about him vied with one another to place other dishes near him.
Beauty sat with her hands over her breasts again; her cheeks were moist with tears, and she was trembling slightly.
"As you wish," said the King. "I am in your debt."
"You have your life and your Kingdom now," said the Prince. "And I have your daughter. I will spend the night here. And tomorrow set out to make her my Princess across the mountains."
He had placed some fruit on his plate, and other hot morsels of cooked food, and now he snapped his fingers gently and in a whisper told Beauty to come around the table to him.
He could see her shame before the servants.
But he brushed her hand away from her sex.
"Never cover yourself like that again," he said. He spoke these words almost tenderly, as he lifted her hair back from her face.
"Yes, my Prince," she whispered. She had a lovely little voice. "But it's so difficult."
"Of course it is," he smiled. "But for me you'll do it."
And now he took her and placed her on his lap, cradling her in his left arm. "Kiss me," he said, and feeling her warm mouth on his again, he felt his desire rising too soon for his taste, but he decided he could savor this slight torment.
"You may go," he said to the King. "Tell your servants to have my horse ready in the morning. I won't need a horse for Beauty. My soldiers you've found, no doubt, at your gates," and the Prince laughed. "They were afraid to come in with me. Tell them to be ready at dawn, and then you can say goodbye to your daughter, Beauty."
The King glanced up very quickly to accept the Prince's commands and with unfailing courtesy he backed out of the doorway.
The Prince turned his full attention to Beauty.
Lifting a napkin he wiped at her tears. She kept her hands obediently on her thighs, exposing her sex, and he observed that she did not try to hide her stiff little pink nipples with her arms and he approved of this.
"Now don't be frightened," he said to her softly, feeding a little on her trembling mouth again, and then slapping her breasts so they shivered lightly. "I could be old and ugly."
"Ah, but then I could feel sorry for you," she said in a sweet, tremulous voice.
He laughed. "I'm going to punish you for that," he said to her tenderly. "But now and then just a little very ladylike impertinence is amusing."
She blushed darkly, biting her lip.
"Are you hungry, beautiful one?" he asked.
He could see she was afraid to answer.
"When I ask you will say, 'Only if it pleases you, my Prince,' and I shall know the answer is yes. Or, 'Not unless it should please you, my Prince,' and I shall know the answer is no. Do you understand me?"
"Yes, my Prince," she answered. "I'm hungry only if it pleases you."
"Very good, very good," he said to her with genuine feeling. He lifted a small cluster of glistening purple grapes and fed them to her one by one, taking the seeds out of her mouth and casting them aside.
And he watched with obvious pleasure as she drank deeply from the wine cup he held to her lips. Then he wiped her mouth and kissed her.
Her eyes were glistening. But she had stopped crying. He felt the smooth flesh of her back, and her breasts again.
"Superb," he whispered. "And were you terribly spoilt before and given everything that you wished?"
She was confused, blushing again, and then full of shame she nodded.
"Yes, my Prince, I think perhaps ..."
"Don't be afraid to answer me with many words," he coaxed, "as long as they are respectful. And never speak unless I speak to you first, and in all these things, be careful to note what pleases me. You were very spoilt, given everything, but were you willful?"
"No, my Prince, I don't think I was that," she said. "I tried to be a joy to my parents."
"And you'll be a joy to me, my dear," he said lovingly.
Still holding her firmly in his left arm, he turned to his supper.
He ate heartily, pork, roast fowl, some fruit, and several cups of wine. Then he told the servants to take it all away and leave them.
New sheets and coverlets had been laid on the bed; there were fresh down pillows, and roses in a vase nearby, and several candelabra.
"Now," he said as he rose and set her before him. "We must get to bed as we have a long journey before us tomorrow. And I have still to punish you for your earlier impertinence."
Immediately the tears stood in her eyes; she looked up at him imploring. She almost reached to cover her breasts and her sex, and then remembering herself she made her hands into two little helpless fists at her sides.
"I won't punish you very much," he said gently, lifting her chin. "It was just a little offense, and your first after all. But Beauty, to confess the truth, I shall love punishing you."
She was biting her lip, and he could see she wanted to speak, and the effort to control her tongue and her hands was almost too much for her.
"All right, lovely one, what do you want to say?" he asked.
"Please, my Prince," she begged. "I'm so afraid of you."
"You'll find me more reasonable than you expect," he said.
He removed his long cloak, tossing it over a chair, and bolted the door. Then he snuffed all but a few candles.
He would sleep in his clothes as he did most nights, in the forest, or in the country inns, or in the houses of those humble peasants at which he sometimes stopped, and that was no great inconvenience to him.
And as he drew near her now, he thought he must be merciful and make her punishment quick. And seating himself on the side of the bed, he reached out for her, and pulling her wrists into his left hand he brought her naked body down over his lap so that her legs dangled over the floor helplessly.
"Very, very lovely," he said, his right hand moving languidly over her rounded buttocks, forcing them ever so slightly apart.
Beauty was crying aloud, but muffling her cries into the bed, her hands held out in front of her by his long left arm.
And now with his right hand he spanked her buttocks hard and heard her cries grow louder. It wasn't really much of a slap.
But it left a red mark on her. And he spanked her hard again, and he felt her writhing against him, the heat and moisture of her sex against his leg, and again he spanked her.
"I think you are sobbing more from the humiliation than the pain," he scolded her in a soft voice.
She was struggling not to make her cries too loud.
He flattened out his right hand, and feeling the heat of her reddened buttocks drew it up and delivered another series of hard, loud slaps, smiling as he watched her struggle.
He could have spanked her much harder, for his own pleasure, and without really hurting her. But he thought the better of it. He had so many nights ahead of him for these delights.
He lifted her up now so that she was standing in front of him.
"Toss your hair back," he commanded. Her tear-stained face was unspeakably beautiful, her lips trembling, her blue eyes gleaming with the dampness of the tears. She obeyed immediately.
"I don't think you were so very spoilt," he said. "I find you very obedient and eager to please, and this makes me very happy."
He could see her relief.
"Clasp your hands behind your neck," he said, "under your hair. That's it. Very good." He lifted her chin again. "And you have a lovely modest habit of looking down. But now I want you to look directly at me."
She obeyed shyly, miserably. It seemed she felt her nakedness and her helplessness more fully now as she looked at him. Her lashes were matted and dark, and her blue eyes larger than he had thought.
"Do you find me handsome?" he asked her. "Ah, but before you answer, I should like to know the truth from you, not what you think I should like to hear, or what would be best for you to say, you understand me?"
"Yes, my Prince," she whispered. She seemed calmer.
He reached out, massaged her right breast lightly, and then stroked her downy underarms, feeling the little curve of the muscle there beneath the tiny wisp of golden hair, and then he stroked that full, moist hair between her legs so that she sighed and trembled.
"Now," he said, "answer my question, and describe what you see. Describe me as if you had only just met me and were confiding in your chambermaid."
Again she bit her lip, which he dearly loved, and then, her voice a little diminished by uncertainty, she said:
"You are very handsome, my Prince, no one could deny that. And for one ... for one ..."
"Go on," he said. He drew her just a little closer so that her sex was against his knee, and putting his right arm about her, he cradled her breast in his left hand and let his lips touch her cheek.
"And for one so young to be so commanding," she said, "it's not what one might expect."
"And tell me how does that show itself in me, other than my actions?"
"Your manner, my Prince," she said, her voice gaining a little strength. "The look of your eyes, such dark eyes ... your face. There are none of the doubts of youth in it."
He smiled and kissed her ear. He wondered why the wet little cleft between her legs was so very hot. His fingers could not keep from touching it. Twice already he'd had her today, and he would have her again, but he was thinking he should go about it more slowly.
"Would you like it if I were older?" he whispered.
"I had thought," she said, "that it would be easier. To be commanded by one so very young," she said, "is to feel one's helplessness."
It seemed the tears had welled up and were spilling out of her eyes, so he pushed her gently back so he might see them.
"My darling, I have awakened you from a century's sleep, and restored you father's Kingdom. You're mine. And you won't find me such a hard master. Only a very thorough master. When you think night and day and every moment only of pleasing me, things will be very easy for you."
And as she struggled not to look away, he could see again the relief in her face, and that she was in complete awe of him.
"Now," he said, pushing his left fingers between her legs, and drawing her close again so that she let out a little gasp before she could stop herself, "I want more of you than I've had before. Do you know what I mean, my Sleeping Beauty?"
She shook her head; for this moment she was in terror.
He lifted her up onto the bed and laid her down.
The candles threw a warm, almost rosy light over her. Her hair fell down on either side of the bed, and she seemed on the verge of crying out, her hands struggling to keep still at her sides.
"My darling, you have a dignity about you that shields you from me, much like your lovely golden hair shrouds you and shields you. Now I want you to surrender to me. You'll see, and you'll be very surprised that you wept when I first suggested it."
The Prince bent over her. He parted her legs. He could see the battle she fought not to cover herself or turn away from him. He stroked her thighs. Then with his finger and thumb, he reached into the silky damp hair itself and felt those tender little lips and forced them very wide open.
Beauty gave a terrible shudder. With his left hand he covered her mouth, and behind his hand she cried softly. It seemed easier for her with him covering her mouth and that was all right for now, he thought. She shall be taught everything in time.
And with his right fingers, he found that tiny nodule of flesh between her tender nether lips and he worked it back and forth until she raised her hips, arching her back, in spite of herself. Her little face under his hand was the picture of distress. He smiled to himself.
But even as he smiled, he felt the hot fluid between her legs for the first time, the real fluid which had not come before with her innocent blood. "That's it, that's it, my darling," he said. "And you mustn't resist your Lord and master, hmmmm?"
Now he opened his clothing and took out his hard, eager sex, and mounting her he let it rest against her thigh as he continued to stroke her and work her.
She was twisting from one side to the other, her hands gathering up the soft sheets at her sides into knots, and it seemed her whole body grew pink, and the nipples of her breasts looked as hard as if they were tiny stones. He could not resist them.
He bit at them with his teeth, playfully, not hurting her. He licked them with his tongue, and then he licked her sex, too, and as she struggled, and blushed and moaned beneath him, he mounted her, slowly.
Again she arched her back. Her breasts were suffused with red. And as he drove his organ into her, he felt her shudder violently with unwilling pleasure.
An awful cry was muffled by the hand over her mouth; she was shuddering so violently it seemed she all but lifted him on top of her.
And then she lay still, moist, pink, with her eyes closed, breathing deeply as the tears flowed silently.
"That was lovely, my darling," he said. "Open your eyes."
She did it timidly.
But then she lay looking up at him.
"This has been so hard for you," he whispered. "You could not even imagine these things happening to you. And you are red with shame, and shaking with fear, and you believe perhaps it's one of the dreams you dreamed in your hundred years. But it's real, Beauty," he said. "And it is only the beginning! You think I've made you my Princess. But I've only started. The day will come when you can see nothing but me as if I were the sun and the moon, when I mean all to you, food, drink, the air you breathe. Then you will truly be mine, and these first lessons ... and pleasures ..." he smiled, "will seem like nothing."
He bent over her. She lay so very still, gazing up at him.
"Now kiss me," he commanded. "And I mean, really ... kiss me."
**THE JOURNEY AND THE PUNISHMENT AT THE INN**
**T** HE NEXT morning all the Court was gathered in the Great Hall to see the Prince off, and all of the Court, including the grateful King and Queen, stood with their eyes down, bowing from the waist as the Prince came down the steps with the naked Beauty walking behind him. He had commanded her to clasp her hands on the back of her neck beneath her hair, and to walk just a little to his right so that he might see her in the corner of his eye. And she obeyed, her bare feet making not the slightest sound on the worn stone steps as she followed him.
"Dear Prince," said the Queen, when he reached the great front door and saw that his soldiers stood mounted on the drawbridge, "we are in your eternal debt, but she is our only daughter."
The Prince turned to look at her. She was yet beautiful, though more than twice Beauty's age, and he wondered if she too had served his great-grandfather.
"How can you question me?" the Prince asked patiently. "I have restored your Kingdom, and you know full well if you remember anything of the ways of my land, that Beauty will be much enhanced by her service there."
Then the telltale blush came to the Queen as it had to the King before, and she bowed her head in acceptance.
"But surely you will allow Beauty some clothing," she whispered, "at least until she reaches the border of your Kingdom."
"All those towns between here and my Kingdom have owed their allegiance to us for a century. And in each I will proclaim your restoration and new dominion. Can you ask for more than that? The spring is warm already; Beauty shall suffer no ill effects from serving me immediately."
"Forgive us, your Highness," the King hastened to say. "But is it the same in this age? Beauty's servitude will not be forever?"
"It is the same now as it was always. Beauty will be returned in time. And she shall be greatly enhanced in wisdom and beauty. Now, tell her to obey as your parents commanded you to obey when you were sent to us."
"The Prince speaks the truth, Beauty," the King said in a low voice, still unwilling to look at his daughter. "Obey him. Obey the Queen. And though you find your servitude surprising and difficult at times, be confident you will return, as he says, greatly changed for the better."
The Prince smiled.
The horses were restless on the drawbridge. The Prince's charger, a black stallion, was particularly hard to restrain, so the Prince, bidding them all farewell again, turned and picked up Beauty.
He heaved her easily over his right shoulder, clasping her ankles to his waist, and heard her cry out softly as she fell over his back. He could see her long hair sweep the ground just before he mounted the stallion.
All the soldiers fell into place behind him.
He rode into the forest.
The sun spilled down in glorious rays through the heavy green leaves, the sky now brilliant and blue overhead only to vanish in a shifting green-tinted light as the Prince rode on at the head of his soldiers, humming to himself, and now and then singing.
Beauty's lithe, warm body swayed slightly over his shoulder. He could feel her trembling, and he understood her agitation. Her naked buttocks were still red from the spanking he had given her, and he could well imagine the succulent vision she was to the men who rode after him.
As he walked his horse through a dense glade where the fallen leaves were thick and red and brown beneath him, the Prince tied the rein on his saddle, and with his left hand felt the soft hairy little pelt between Beauty's legs, and leaned his face against her warm hip, kissing it gently.
After a while, he pulled her down into his lap, turning her as before so she rested against his left arm, and he kissed her red face and brushed the long golden strands of her hair away from it, and then he suckled her breasts almost idly as though taking little drinks from them.
"Put your head on my shoulder," he said. And she inclined to him obediently at once.
But when he went to sling her over his shoulder again, she gave a little desperate whimper. He did not allow this to stop him. And having her firmly in place, her ankles clasped to his hip, he scolded her lovingly, and gave her several hard spanks with his left hand until he heard her crying.
"You must never protest," he repeated. "Not with sound, not with gesture. Only your tears may show your Prince what you feel, and never think that he does not wish to know what you feel. Now, respectfully, answer me."
"Yes, my Prince," Beauty whimpered softly.
He thrilled at the sound of it.
When they came to the small town in the middle of the forest, there was great excitement, as everyone had already heard of the enchantment being broken.
And as the Prince rode into the crooked little street with its high half-timbered houses blocking out the sky, people ran to the narrow windows and doorways. They crowded into the cobblestone alleyways.
Behind him, the Prince could hear his men in hushed voices telling the townspeople who he was, that it was their Lord who had broken the enchantment. The girl he carried with him was the Sleeping Beauty.
Beauty was sobbing softly, her body struggling with these sobs, but the Prince held her firmly.
Finally with a great crowd following him, he arrived at the Inn, and his horse, with loud clops, entered the courtyard.
His page quickly helped him down.
"We'll stop only for food and drink," said the Prince. "We can go miles before sundown."
He stood Beauty on her feet and watched with admiration as her hair fell down around her. And he turned her around twice, pleased to see she kept her hands clasped behind her neck and her eyes down as he looked at her.
He kissed her devotedly.
"Do you see how they all look at you?" he said. "Do you feel how they admire your beauty? They are adoring you," he said. And opening her lips again, he sucked another kiss out of her, his hand squeezing her sore buttocks.
It seemed her lips clung to his as if she were afraid to let him go, and then he kissed her eyelids.
"Now everyone is going to want to have a look at Beauty," the Prince said to the Captain of his Guard. "Bind her hands over her head by a rope from the sign over the Inn gate, and let the people have their fill of her. But no one is to touch her. They can look all they like, but you stand guard and see that no one touches her. I'll have your food sent out to you."
"Yes, my Lord," said the Captain of the Guard.
But as the Prince gently gave Beauty over to him, she leaned forward, her lips out to the Prince, and he received her kiss gratefully. "You're very sweet, my darling," he said. "Now be modest and very very good. I should be very disappointed if all this adulation made my Beauty vain." He kissed her again, and let the Captain have her.
Then going inside and ordering his meat and ale, the Prince watched through the diamond-paned windows.
The Captain of the Guard did not dare touch Beauty, except to put the rope about her wrists. He led her by this to the open gate of the courtyard, and throwing the rope up over the iron rod that held the sign of the Inn, he quickly secured her hands above her head, so that she was almost on tiptoe.
Then he motioned for the people to move back, and he stood against the wall with his arms folded as they pressed to look at her.
There were buxom women with stained aprons, and coarse men in breeches and heavy leather shoes, and the young well-to-do men of the town in their velvet cloaks with their hands on their hips as they eyed Beauty from a distance, unwilling to elbow in the crowd. And several young women, their elaborate white headdresses freshly done up, who had come out lifting their hems fastidiously as they looked at her.
At first everyone was whispering, but now people began to speak more freely.
Beauty had turned her face into her arm and let her hair shield her face, but then a soldier came out from the Prince and said:
"His Majesty said to turn her and lift her chin so they might have a better look at her."
An approving murmur went up from the crowd. "Very very lovely," said one of the young men.
"And this is what so many died for," said an old Cobbler.
The Captain of the Guard lifted Beauty's chin, and holding the rope above her, said gently:
"You must turn around, Princess."
"O, please, Captain," she whispered.
"Don't make a sound, Princess, I beg you. Our Lord is very strict," he said. "And it's his wish that everyone admire you."
Beauty, her cheeks flaming, obeyed, turning so the crowd could see her reddened buttocks and then again to show her breasts and her sex as the Captain kept his finger under her chin lightly.
It seemed she breathed deeply as though trying to remain very calm. The young men were calling her beautiful and saying her breasts were magnificent.
"But such buttocks," whispered an old woman nearby. "You can see that she's been spanked. I doubt the poor Princess did anything much to deserve it."
"Not much," said a young man near her. "Except have the most beautiful and pertly shaped buttocks imaginable."
Beauty was trembling.
Finally the Prince himself came out, ready to leave, and seeing the crowd as attentive as before, he himself took the rope down, and holding it like a short leash above Beauty's head, he turned her. He seemed amused by the crowd's grateful nods, and thanks, and bows to him; and very gracious in his generosity.
"Lift your chin, Beauty, I shouldn't have to lift it," he reproved her with a little deliberate frown of disappointment.
Beauty obeyed, her face so red that her eyebrows and eyelashes gleamed golden in the sun, and the Prince kissed her.
"Come here, old man," the Prince said to the old Cobbler. "Have you ever seen such loveliness?"
"No, your Majesty," said the old man. His sleeves were rolled to the elbows, and his legs were slightly bowed. His hair was gray but his green eyes gleamed with a special almost wistful pleasure. "She is truly a magnificent Princess, your Majesty, worth all the deaths of those who tried to claim her."
"Yes, I suppose so, and worth all the bravery of the Prince who _did_ claim her," smiled the Prince.
Everyone laughed politely. But they couldn't conceal their awe of him. They were staring at his armor, at his sword, and above all at his young face and dark black hair that fell to his shoulders.
The Prince drew the Cobbler closer. "Here," he said, "I give you permission if you like just to feel her treasures."
The old man smiled at the Prince gratefully and almost innocently. He reached out, and hesitating a moment, felt Beauty's breasts. Beauty shivered, and tried obviously to repress a little cry.
The old man touched her sex.
Then the Prince drew up her little leash so she was standing on tiptoe; her body stiffened and seemed to grow more tense and at the same time more lovely, breasts and buttocks high, her calf muscles lifted, her chin and throat a perfect line down to her swaying bosom.
"That's all. You must all go now," said the Prince.
Obediently they backed away, but they continued to watch, as the Prince mounted his horse, and instructing Beauty to clasp her hands behind her neck, he ordered her to walk before him.
Beauty led the way out of the Inn yard, the Prince walking his horse behind her.
The people made way for her. They couldn't take their eyes off her lovely vulnerable body, and they squeezed against the narrow walls of the town to follow the spectacle to the edge of the forest.
When they had left the town behind, the Prince told Beauty to come to him. He gathered her up and seated her before him again, and kissed her again, and scolded her:
"You found that so hard," he crooned. "Why were you so proud? Did you think yourself too good to be shown to the people?"
"I'm sorry, my Prince," she whispered.
"Don't you see, if you think only of pleasing me, and pleasing those to whom I show you, it will be simple for you." He kissed her ear, holding her tight to his chest. "You should have been proud of your breasts and your shapely hips. You should have asked yourself, 'Am I pleasing my Prince? Do the people find me pleasing?' "
"Yes, my Prince," Beauty said meekly.
"You are mine, Beauty," the Prince said a little more sternly. "And there is no command that you must shrink from obeying ever. If I tell you to please the lowliest vassal in the field, you will strain to obey me perfectly. He is your Lord then because I have said so. All those to whom I offer you are your Lords."
"Yes, my Prince," she said, but she was in great distress. He stroked her breasts, pinching them firmly now and then, and kissed her until he could feel her body struggling against him, and feel her nipples growing hard. It seemed she wanted to speak.
"What is it, Beauty?"
"Pleasing you, my Prince, pleasing you ..." she whispered, as though her thoughts had spread into a delirium.
"Yes, pleasing me, that is your life now. How many of those in the world know such clarity, such simplicity? You please me and I shall always tell you exactly how to please me,"
"Yes, my Prince," she sighed. But she was crying again.
"I will treasure you all the more for it. The girl I found in the castle room was nothing to me such as you are now, my devoted Princess."
But the Prince was not entirely satisfied with the way in which he was instructing Beauty. He told her when they reached another town at nightfall that he was going to strip a little more dignity away from her to make it easier for her.
And while the townspeople pressed their faces to the leaded glass windows of the Inn, the Prince had Beauty wait on his table.
On her hands and knees she hurried across the rough boards of the Inn floor to fetch his plate from the kitchen. And though she was allowed to walk back with it, she was again on all fours to fetch his flagon. The soldiers devoured their supper, throwing silent glances at her by the light of the fire.
She wiped the table for the Prince and when a morsel of food spilled from his plate to the floor, he commanded Beauty to eat it. With tears spilling from her eyes, Beauty obeyed, and then he gathered her, still on her knees, into his arms and rewarded her with dozens of wet and loving kisses. Obediently she put her arms around his neck.
But this little morsel spilling had given him an idea. He ordered her to quickly fetch a plate from the kitchen again, and then told her to lay it on the floor at his feet.
He put food for her there from his plate, and told her to lift her heavy hair behind her shoulders and eat it only with her mouth.
"You are my kitten," he laughed gaily. "And I would forbid you all those tears if they weren't so beautiful. Do you want to please me?"
"Yes, my Prince," she said.
With his foot he pushed her plate several paces away and told her to turn her buttocks to him as she continued her meal. He admired it, realizing the red marks from her spanking had almost healed. With the toe of his leather boot, he nudged at the silken hair he could see between her legs, felt the moist plump lips beneath the hair, and sighed, thinking her so very beautiful.
When she had finished her meal, with her lips she pushed the plate back to his chair as he ordered her to do it, and then he wiped her lips himself and fed her some wine from his cup.
He watched her long beautiful throat as she swallowed, and kissed her eyelids.
"Now listen to me, I want you to learn from this," he said. "Everyone here can see you, all your charms, you're aware of it. But I want you to be very aware of it. Behind you, the townspeople at the windows are admiring you as they did when I brought you through the town. This should make you proud of yourself, not vain, but proud, proud that you have pleased me, and caught their admiration."
"Yes, my Prince," she said when he paused.
"Now think, you are very naked and very helpless, and you are mine completely."
"Yes, my Prince," she cried softly.
"That is your life now, and you are to think of nothing else, and regret nothing else. I want that dignity peeled away from you as if it were so many skins of the onion. I don't mean that you should ever be graceless. I mean that you should surrender to me."
"Yes, my Prince," she said.
The Prince looked up at the Innkeeper who stood at the kitchen door with his wife and his daughter. They came to attention at once. But the Prince looked only at the daughter. She was a young woman, very pretty in her own way, though nothing compared to Beauty. She had black hair and round cheeks, and a very tiny waist, and she dressed as many peasant women did, in a low-cut ruffled shirtwaist, and a short broad skirt that revealed her smart little ankles. She had an innocent face. She was watching Beauty in wonder, her big brown eyes moving anxiously to the Prince and then shyly back to Beauty who knelt at the Prince's feet in the firelight.
"Now, as I told you," the Prince said softly to Beauty, "all here admire you, and they enjoy you, the sight of you, your plump little rear, your lovely legs, those breasts which I cannot stop myself from kissing. But there is no one here, not the lowliest, who is not better than you, my Princess, if I command you to serve him."
Beauty was frightened. She nodded quickly as she answered "Yes, my Prince," and then very impulsively she bent and kissed the Prince's boot, but then she appeared terrified.
"No, that is very good, my darling," the Prince, stroking her neck, reassured her. "That is very good. If I allow you one gesture to speak your heart unbidden it is that one. You may always show me respect of your own accord in that manner."
Again Beauty pressed her lips to the leather. But she was trembling.
"These townspeople hunger for you, hunger for more of your loveliness," the Prince continued. "And I think they deserve a little taste of it that will delight them."
Beauty kissed the Prince's boot again, and let her lips rest there.
"O, don't think I should really let them have their fill of your charms. O, no," the Prince said thoughtfully.
"But I should use this opportunity, both to reward their devoted attention and teach you that punishment will come whenever I desire to give it. You need not be disobedient to merit it. I will punish when it pleases me. Sometimes that will be the only reason for it."
Beauty couldn't keep herself from whimpering.
The Prince smiled and beckoned to the Innkeeper's daughter. But she was so frightened of him that she didn't come forward until her father pushed her.
"My dear," said the Prince gently. "In the kitchen, have you a flat wooden instrument, for shoveling the hot pans into the oven?"
There was a faint movement throughout the room as the soldiers glanced at one another. The people outside were pressing closer to the windows. The young girl nodded and quickly returned with a wooden paddle, very flat and smooth from years of use, with a good handle.
"Excellent," said the Prince.
But Beauty was crying helplessly.
The Prince quickly ordered the Innkeeper's daughter to seat herself on the edge of the high hearth which was the height of a chair, and told Beauty, on her hands and knees, to go to her.
"My dear," he said to the Innkeeper's daughter, "these good people deserve a little spectacle. Their life is hard and barren. My men deserve it as well. And my Princess can well use the chastisement."
Beauty knelt crying before the girl who, seeing what she was to do, was fascinated.
"Up over her lap, Beauty," said the Prince, "hands behind your neck, and lift your lovely hair out of the way. At once!" he said, almost sharply.
Pricked by his voice, Beauty almost scurried to obey, and all those around her saw her tear-stained face.
"Keep your chin up like that, yes, lovely. Now, my dear," said the Prince looking at the girl who held Beauty over her lap and the wooden paddle in her other hand. "I want to see if you can wield that as hard as a man might wield it. Do you think you can do that?"
He could not keep from smiling at the girl's delight and desire to please. She nodded murmuring a respectful reply, and when he gave her the command, she brought the paddle down hard on Beauty's naked buttocks. Beauty couldn't keep still. She struggled to keep quiet, but she couldn't keep still, and finally even the whimpers and moans escaped her.
The tavern girl spanked her harder and harder, and the Prince enjoyed this, savoring it far more than the spanking he had given Beauty himself.
It was because he could see it much better, see Beauty's breasts heaving, and the tears spilling down her face, and her little buttocks straining, as if, without moving, Beauty might somehow escape or deflect the girl's hard blows.
Finally, when the buttocks were very red but not welted, he told the girl to stop.
He could see his soldiers enthralled and all the townspeople as well, and then he snapped his fingers and told Beauty to come to him.
"Now eat your suppers, all of you, talk amongst yourselves, do as you like," he said quickly.
For a moment no one obeyed him. Then the soldiers turned to one another, and those outside, seeing that Beauty was retired down to kneeling at the Prince's feet, her hair veiling her red face, her raw and stinging buttocks pressed to her ankles, were murmuring and talking at the windows.
The Prince gave Beauty another drink of wine. He was not sure he was entirely satisfied with her. He was thinking of many things.
He called the Innkeeper's daughter to him and told her she had been very good, gave her a gold coin, and took the paddle from her.
Finally it was time to go up. And driving Beauty before him, he gave her a few gentle but brisk spanks to hurry her up the stairs to the bedchamber.
**BEAUTY**
**B** EAUTY STOOD at the foot of the bed, her hands clasped to her neck, her buttocks throbbing with a warm pain that felt so much better now than the spanking she had lately received that it was almost pleasure.
She had for the moment stopped crying. She had only just pulled down the covers for the Prince, with her teeth, her hands clasped behind her back, and then with her teeth taken his boots to the edge of the room.
And now she waited for further commands, trying to watch him, though her eyes were cast down, without his realizing it.
He had bolted the door, and he was sitting on the side of the bed.
And his black hair, loose and curling at his shoulders gleamed in the light of the tallow candle. His face was very beautiful to her, perhaps because in spite of the size of the features, they were all rather delicately molded. She did not know for certain.
Even his hands enthralled her. The fingers were so long, so white, so delicate.
She was terribly relieved to be alone with him. The moments below in the Inn had been such an agony to her, and even though he had brought the wooden paddle with him and might spank her much harder with it than that dreadful girl, she was so glad to be alone with him that she could not be afraid of it. She was afraid, however, that she hadn't pleased him.
She searched her mind for faults. She had obeyed all his commands, and he understood how difficult it was for her. He knew completely what it meant for her to be stripped naked and revealed to everyone, to be helpless and made public and that this surrender of which he spoke could come in acts and gestures long before it could come from her mind. But no matter how hard she tried to excuse herself, she could not help but wondering if she could have tried harder.
Did he want her to cry out more when she was spanked? She was uncertain. Just thinking of that girl spanking her in front of everyone made her cry again, and she knew that the Prince would see her tears, and he might wonder why now, when she'd been told to stand still at the foot of the bed, she was crying.
But the Prince seemed deep in thought.
This is my life, she told herself, trying to calm herself. He has awakened me and claimed me. My parents are restored, their Kingdom is theirs again, and more significantly, life is theirs again, and I belong to him. She felt a great relaxation when she thought these things and a stirring in herself that seemed to make her sore and throbbing buttocks feel suddenly warmer. The pain made her so shamefully aware of that part of her body! But then as she squeezed her eyes against these soft and slow tears, she looked down at her swelling breasts and the tiny hard nipples and felt that same awareness of herself there too, just as if he'd slapped her breasts which he hadn't done in a great while, and she felt softly bewildered.
My life, she struggled to understand. And she remembered that in the afternoon in the warm forest when she had been walking before his horse, she had felt her own long hair on her buttocks, brushing them as she walked ahead of him, and she had wondered if she looked beautiful to him, and she had wished that he would pick her up then, and kiss her and caress her. Of course she had not dared to look back. She couldn't imagine what he would have done had she been so foolish as to do that, but the sun had thrown their shadows ahead of them and she had seen the shadow of his profile, and felt such a pleasure that she was ashamed of it, and her legs had felt weak and there had been the oddest feeling in her, something she had never known in her earlier life, though perhaps in her dreams.
She was awakened now, at the foot of his bed, by his low but firm command.
"Come here, my darling." He motioned for her to kneel before him.
"This shirt is to be opened down the front, and you will learn to do so with your lips and teeth, and I will be patient with you," he said.
She had thought it would be the paddle. And, very relieved, she went almost too quickly to obey, pulling the thick tie that closed the shirt at his throat. His flesh felt warm and smooth to her. Men's flesh. So different, she thought. And she quickly pulled loose the second tie and the third. She had a struggle with the fourth which was at his waist, but he didn't move, and then when she was finished, she bowed her head, her hands as before on the back of her neck and waited.
"Open my breeches," he said to her.
Her cheeks flamed; she could feel it. But again she didn't hesitate. She pulled the fabric forward over the hook until the hook slipped out and let it go. And now she could see his sex, bulging there, painfully twisted. She wanted suddenly to kiss it, but she didn't dare and was shocked at her impulse.
He had lifted it free. It was hard. She thought of it between her legs, filling her, rough and too big for her virginal opening, and of that terrible pleasure which had suffused her and wasted her the night before, and she knew she was blushing furiously.
"Now go to the stand in the corner," he said, "and bring back the basin with water in it."
She almost scurried across the floor. Several times in the Inn he had told her to move fast, and though she had hated it at first, she now did it instinctively. She brought the basin in both hands and set it down. There was a cloth in the water.
"Wring out the cloth tightly," he said, "and bathe me quickly."
She did as she was told at once, staring in amazement at his sex, its length, its hardness, and the tip of it with its tiny opening. She had been so sore from it yesterday, yet that pleasure had paralyzed her. Never had she guessed at such a secret.
"Now, do you know what I want of you?" the Prince said gently. His hand lovingly stroked her cheek, lifting her hair back. She ached to look at him. She wished so much he would command her to look into his eyes. It terrified her, but after the first instant it was so wondrous to her, his expression, that handsome and almost delicate face, and those black eyes that seemed to accept no compromise.
"No, my Prince, but whatever it is ..." she started.
"Yes, darling ... you are being very good. I want you to take it in your mouth, stroke it with your tongue and your lips."
She was shocked. She had never thought of this. She thought suddenly, cruelly of who she had been, a Princess, and she thought of all her young life before she had fallen asleep, and she almost gave a little whimper. But this was her Prince who was commanding her, not some dreadful person she was being given to as a wife who might have demanded this of her. She closed her eyes and took it into her mouth, feeling its huge size, its hardness.
It nudged at the back of her throat, and she pushed up and down on it as the Prince guided her.
The taste of it was almost delicious; and it seemed a salty liquid in tiny droplets came out into her mouth, and then she stopped because he had said it was enough.
She opened her eyes.
"Very good, Beauty, very good," said the Prince.
And she could tell he was in pain with his need suddenly. It made her feel proud, and there was in her, even in her helplessness, a sense of power.
But he had risen and was guiding her to her feet. And she realized as she straightened her legs that that debilitating pleasure had caught hold of her. She felt for a moment that she couldn't stand, but to disobey him was unthinkable. Quickly she stood straight, hands behind her neck, and she struggled to keep her hips from going into some slight humiliating movement. Could he see it? She bit her lip again and felt its soreness.
"You've done marvelously well today, you've learned so very much," he said tenderly. His voice could be so soft and yet so firm at the same time. It made her feel almost drowsy; that pleasure was melting inside of her.
But then she saw that he was reaching for the paddle behind him. She let out a little gasp before she could stop herself, and she felt his hand on her arm, taking her hands away from the back of her neck, and turning her around. She wanted to cry out, "What have I done?"
But his voice came low, crooning in her ear.
"And I've learned a very important lesson myself, that pain softens you, makes it easier for you. You are infinitely more malleable from the spanking given you in the Inn than you were before it."
She wanted to shake her head, but she didn't dare. The thought of all those who had seen her spanked tormented her. She had been turned so those at the windows could see her buttocks and between her legs, and the soldiers could see her face, and it had been excruciating. Well, it would only be her Prince now. If only she could tell him, for him anything, but those others were such punishment ...
She knew this was wrong. It was not what he wanted her to think, what he was trying to teach her. But now she couldn't think.
He was at her side. He held her chin in his left hand, and he had told her to fold her arms behind her back which was difficult for her. It was worse than clasping her hands behind her neck. This position arched her body, forced her breasts out, and made her breasts and face feel painfully naked. She moaned slightly as he lifted her hair and folded the great mane of it over her right shoulder, away from him.
It covered her arm, but he pushed it away from her nipples and pinched both of them hard between his finger and thumb, lifting her breasts and letting them fall naturally as he did so.
Her face was positively smarting. But she knew what was to come would be worse.
"Spread your legs ever so slightly. You must be firmly planted on the ground," he said, "so that you can withstand the blows of the paddle."
She wanted to cry out, and through her tightly pressed lips her sobs sounded very loud to her.
"Beauty, Beauty," he crooned. "Do you want to please me?"
"Yes, my Prince," she cried, her lip trembling uncontrollably.
"Then why are you crying so when you haven't even felt the paddle yet? And your buttocks are only a little sore. Why, the Innkeeper's daughter had little strength."
She cried almost bitterly, as if to say in her soft wordless way that it was all true but it was so difficult.
He held her chin firmly now, bracing her whole body. And then she felt the first crack of the paddle.
It was an explosion of stinging pain on the hot surface of her flesh, and the second spank came much more swiftly than she had thought possible and then there was the third and the fourth, and in spite of herself she was crying aloud.
He stopped and gently kissed her all over her face. "Beauty, Beauty," he said. "Now, I give you permission to speak ... tell what it is you would have me know..."
"I want to please you, my Prince," she struggled, "but it hurts so, and I've tried so hard to please you."
"But, my darling, you please me by bearing this pain. I explained to you earlier that punishment would not always be for a transgression. Sometimes it would be for my pleasure only."
"Yes, my Prince," she cried.
"I shall tell you a little secret about the pain. You are as a tight bowstring. And the pain loosens you, makes you soft as I want you to be. It is worth a thousand little orders and scoldings, and you must not think of resisting it. Do you know what I am saying? You must give yourself over to it. With each crack of the paddle you must think of the next and the next and that it is your Prince doing it to you, giving you this pain."
"Yes, my Prince," she said softly.
He lifted her chin again without further ado and spanked her hard again and again on the buttocks. She felt her buttocks growing hotter and hotter with pain, and the cracks of the paddle sounded loud and somehow shattering to her, as if the sound itself were as dreadful as the pain. She could not understand it.
When he stopped again, she was breathless and almost frantic in her tears, as if the torrent of blows had so humiliated her it was far worse than even a greater pain would have been.
But the Prince folded her in his arms. And feeling his rough clothing against her, and his hard naked chest, and the strength of his shoulders, she felt such a soothing pleasure that her sobs grew soft and open mouthed and languid against him.
His rough breeches were against her sex, and she found herself pressing against him only to have him guide her gently back as if silently reproving her.
"Kiss me," he said, and such a shock of pleasure went through her at the closing of his open mouth over hers that she was almost unable to stand, letting her weight fall against him.
He turned her toward the bed.
"That's enough for tonight," he said softly. "We have a hard journey tomorrow."
And he told her to lie down.
It occurred to her suddenly that he was not going to take her. She heard him moving to the door, and this pleasure between her legs became suddenly an agony. But all she could do was cry softly into the pillow. She tried to keep her sex from touching the sheets because she feared that if it did she could not resist some undulating movement. And she felt sure he was watching her. Of course he'd meant her to feel pleasure. But without his permission?
She lay rigid, afraid, crying.
A moment later she heard voices behind her.
"Bathe her and put a soothing ointment on her buttocks," the Prince was saying, "and you may talk to the Princess if you like, and she to you. You are to treat her with the utmost respect," said the Prince and then she heard his steps dying away.
She lay too afraid to look behind her. The door was closed again. She heard steps. She heard the cloth in the basin of water.
"It's me, dearest Princess," said a woman's voice, and she realized it was a young woman, a woman her own age, and could only be the Innkeeper's daughter.
She buried her face in the pillow. "This is unbearable," she thought, and suddenly with all her heart she hated the Prince, but she was far too humiliated to think of it. She felt the girl's weight on the bed beside her, and just the rough cloth of her apron brushing against Beauty's buttocks caused the sore and stinging flesh to ache more keenly.
She felt as if her buttocks must be enormous, though she knew they were not, or giving off some terrible light with their redness. The girl would feel their heat; this girl, of all girls, who had tried so hard to please the Prince by spanking her far harder than the Prince had realized.
The wet cloth stroked her shoulders, her arms, her neck. It stroked her back and then her thighs and legs and feet, the girl carefully avoiding her sex and the soreness.
But then after the girl had wrung out the cloth, she touched the buttocks lightly.
"O, I know it hurts, dearest Princess," she confided. "I'm so sorry, but what could I do when the Prince commanded me?" The rag was rough on the soreness, and Beauty realized this time that the Prince had left her with a score of welts. She moaned, and though she loathed this girl with a violent feeling she'd never had for anyone else in her brief life, the cloth nevertheless felt good to her.
The moist cloth was cooling her; it was like the gentle massaging of an itch. And Beauty grew quiet as the girl continued to bathe her in a gentle circular motion.
"Dearest Princess," the girl said, "I know how you suffer but he is so very handsome, and he will have his way, there's nothing to be done about it. Please talk to me, please tell me that you don't despise me."
"I don't despise you," Beauty said in a small spiritless voice. "How could I blame you or despise you?"
"I had to do it. And what a spectacle it was. Princess, I must tell you something. You may be angry with me, but maybe it will be a consolation to you."
Beauty closed her eyes and pressed her cheek into the pillow. She did not want to hear it. But she liked the girl's voice, its respect and gentleness. The girl did not mean to hurt her. She could feel that awe in the girl, that humility Beauty had known in all her servants all her life. It was no different, not even with this one who had held her over her knee in a tavern and spanked her in the presence of crude men and villagers. Beauty pictured her as she remembered her from the kitchen door: her dark curly hair in ringlets about her little round face, and those big eyes full of apprehension. How fierce the Prince must have seemed to her! Why she must have been terrified that at any moment, the Prince would order her stripped and humiliated! Beauty smiled to herself, thinking of it. She felt a tenderness for the girl, and for her gentle hands which were now bathing the hot, aching flesh so carefully.
"All right," Beauty said, "what is it you want to tell me?"
"Only that you were so lovely, dearest Princess, that you have such beauty. Even as you were there, why, how many who seem beautiful could have kept their beauty in such a trial, and you were so beautiful, Princess." Over and over she said this word, beautiful, clearly reaching for other words, better words she did not know. "You were so ... so graceful, Princess," she said. "You bore it so well, with such obedience to his Highness, the Prince."
Beauty said nothing. She was thinking of it again, of how it must have seemed to the girl. But it gave Beauty such a frightful sense of herself that she stopped thinking of it. This girl had seen her so closely, had seen the redness of her flesh as it was punished, and had felt her writhing uncontrollably.
Beauty would have cried again, but she didn't want to.
For the first time, through a film of ointment, she felt the girl's naked fingers on her. They massaged the welts.
"Oooh!" the Princess gasped.
"I'm sorry," said the girl. "I am trying so to be gentle."
"No, you must go on. Rub it in well," sighed Beauty, "it feels good, actually. Maybe it's that moment when you take your fingers away." How try to explain it, her buttocks flooded with this pain, itching with it, the welts little hard pebblelike bits of pain, and those fingers pinching them and then releasing them.
"Everyone adores you, Princess," the girl whispered. "Everyone has seen your beauty, with nothing to disguise it or hide your defects, and you have no defects. And they are swooning over you, Princess."
"Is that really so? Or do you say it to console me?" asked Beauty.
"O, it is so," said the girl. "O, you should have heard the rich women out in the Inn yard tonight, all of them pretending they weren't envious, but all of them knew that stripped they couldn't hold a candle to you, Princess. And of course the Prince was so beautiful, so handsome and so ..."
"Ah, yes," sighed Beauty.
The girl had coated the buttocks now and was putting even more ointment into the flesh. And she worked some of it into Beauty's thighs, her fingers stopping just before the hair between Beauty's legs, and again, with fierce annoyance and shame, Beauty felt that pleasure coming back. And with this girl!
"O, if the Prince were to know it," she thought suddenly. She couldn't imagine him being pleased, and it suddenly occurred to her that he might punish her any time she felt this pleasure without his giving it to her. She tried to put it out of her mind. She wished she knew where he was now.
"Tomorrow," the girl said, "when you go on to the Prince's castle, the road all along the way will be lined with those who want to see you. Word is spreading all through the Kingdom ..."
Beauty gave a little start at these words. "Are you sure of it?" she said fearfully. It was too much to think of suddenly. She remembered that peaceful moment in the afternoon forest. She had been alone ahead of the Prince and had some how managed to forget the soldiers following him. And suddenly to think of people all along the road waiting to see her! She remembered the crowded village streets, those inevitable moments when her naked thighs or breasts even had been brushed by an arm or the fabric of a skirt—she felt her breath halt.
"But he wants this of me," she thought. "Not just that he see me but that all see me."
"It gives the people such pleasure to see you," he had said tonight as they entered this little town. He had prodded her on up ahead of him, and she had been crying so fiercely as she saw all about her those shoes and boots from which she dared not look up.
"But you are so lovely, Princess, and they will be telling their grandchildren about it," said the tavern girl. "They cannot wait to feast their eyes upon you, and you will not disappoint them, no matter what they have heard. Imagine that, never disappointing anyone ..." The girl's voice trailed off as though she were in thought. "O, I wish I could follow you to see it."
"But you don't understand," Beauty whispered, unable suddenly to contain herself. "You don't realize ..."
"Yes, I do," said the girl. "Of course I do ... I've seen the Princesses when they come through in their magnificent gowns covered with jewels, and I know how it must feel to be opened to the world as if you were a flower, all of their eyes like fingers prying at you, but you are so ... so splendid finally, Princess, and so rare. And you are his Princess, and he has claimed you and all know you are in his power and must do as he commands you. It is no shame to you, Princess. How could it be, with such a great Prince to command you? O, do you think that there aren't women who would give up everything to take your place, if only they had your beauty?"
Beauty was startled by this. She thought about it. Women giving up everything, taking her place. It had not occurred to her. She remembered that moment in the forest.
But then she remembered being spanked in the Inn, and all of those others watching. She remembered sobbing helplessly, and hating her buttocks propped up in the air, and her legs open, and that paddle coming down again and again. Finally the pain was the least of it.
She thought of the crowds on the road. She tried to picture it. It would happen to her tomorrow.
She would feel this drenching humiliation, this pain, but all those people would be there to witness her humiliation, to amplify it.
The door had opened.
The Prince had come into the room. And the little tavern girl jumped up and was bowing to him.
"Your Highness," the girl said breathlessly.
"You've done your work very well," said the Prince.
"It was a great honor, your Highness," said the girl.
The Prince came to the bed, and clasping Beauty's right wrist, he drew her up out of the bed and stood her beside it. Obediently, Beauty looked down, and not knowing what to do with her hands, quickly brought them to the back of the neck.
She could almost feel the Prince's satisfaction.
"Excellent, my darling," he said. "Isn't she lovely, your Princess?" he said to the tavern girl.
"O, yes, your Highness."
"Did you talk to her and console her as you were bathing her?"
"O, yes, your Highness, I told her how much everyone admired her and how much they wanted to ..."
"Yes, to see her," the Prince said.
There was a pause. Beauty wondered if they were both looking at her, and suddenly she felt herself naked in the sight of both of them. It seemed one or the other she could bear, but both of them staring at her breasts and sex was too much for her.
But the Prince embraced her as if seeing that she needed embracing, and gently squeezing her sore flesh, sent another soft shock of shameful pleasure through her. She knew her face was red again. She had always blushed so easily. And were there other ways in which he could tell what his hands did to her? She would cry again if she could not conceal this mounting pleasure.
"Down on your knees, my darling," said the Prince with a little snap of his fingers.
In a shock Beauty obeyed, seeing the rough floor-boards before her. She could see the Prince's black boots, and then the crude leather shoes of the serving girl.
"Now, approach your servant and kiss her shoes. Show her how grateful you are for her devotion to you."
Beauty didn't stop to think of it. But she felt her tears come again as she obeyed, depositing each kiss on the worn leather of the girl's shoes as gracefully as she could. Above she heard the girl's murmured thanks to the Prince.
"Your Highness," the girl said, "it is I who want to kiss my Princess, I beg you."
The Prince must have nodded, because the girl fell to her knees, and, stroking Beauty's hair, kissed her upturned face with great reverence.
"Now, you see there the posts of the foot of the bed," the Prince said to the girl. Beauty of course knew that the bed had high posts which held a coffered ceiling over it.
"Tie your mistress to those posts with her hands and legs quite wide apart so that as I lie down I can look up at her," said the Prince. "Tie her with these satin bands so her skin won't be injured, but tie her very firmly for she must sleep in this position and her weight must not pull her loose."
Beauty was stunned.
She was in a delirium as she was lifted to stand at the foot of the bed. She obeyed pliantly as the girl told her to spread her legs. She felt the satin go tight around her right ankle and then it firmly bound her left ankle, and then the girl, standing before her on the bed, bound the Princess's hands high on either side of her.
She was spread-eagled, looking down at the bed, and with terror, she realized that the Prince must see how she suffered; he must see the shame of the dampness between her legs, those fluids she could neither check or conceal, and, turning her face into her arm, she whimpered softly.
But the worst of it was that he did not mean to take her. He had tied her here out of reach of himself so that as he slept she must look down on him.
Now the girl was dismissed, secretly depositing a little kiss on Beauty's thigh before she left. And Beauty, crying softly, realized she was alone with the Prince. She did not dare to look at him.
"My beautiful obedient one," he sighed.
And to her horror she felt, as he drew near, the hard handle of that dreadful wooden paddle nudging her moist and secret place, so cruelly exposed by her open legs.
She struggled to pretend this was not happening. But she could feel that revealing fluid, and she knew the Prince knew of her tormenting pleasure.
"I have taught you much, and I am so very pleased with you," he said, "and so now you know a new suffering, a new sacrifice for your Lord and master. I could soothe the burning craving between your legs but I shall let you suffer it and know the meaning of it, and that only your Prince can give you that relief you long for."
She couldn't control her moan, even though she muffled it against her arm. She feared that any moment she might move her hips in helpless, humiliating entreaty.
He had snuffed the candles.
The room was dark.
Beneath her feet she felt the mattress give with his weight.
She leaned her head against her arm and felt secure in the satin bonds as she let herself hang there. But this torment, this torment ... and there was nothing she could do to alleviate it.
She prayed the swelling between her legs would die away, as the throbbing in her buttocks was cooling and dying away. And then falling to sleep, she thought calmly, dreamily almost, of the crowds awaiting her on the roads to the Prince's castle.
**THE CASTLE AND THE GREAT HALL**
**B** EAUTY WAS breathless and flushed as they left the Inn; but it was not so much on account of the crowds that lined the village streets, nor those she would see ahead following the ribbon of road as it ran through the wheat fields.
The Prince had sent couriers ahead, and as Beauty's hair was dressed with white flowers, he told her they would reach his castle by afternoon if they were to hurry.
"We shall be in my Kingdom," he announced proudly, "as soon as we are on the other side of the mountains."
Beauty could not quite anatomize the feeling this aroused in her.
But the Prince, as if sensing her strange confusion, kissed her full on the mouth before mounting his horse, and said in a soft voice so that only those around them could hear:
"When you enter my Kingdom, you shall be mine more completely than ever. You will be mine beyond reprieve, and it will be easier for you to forget all that went before that time, and devote your life to me only."
And now they left the village, the Prince walking his magnificent horse just behind Beauty as she made her way quickly over the warm cobblestones.
The sun was hotter than before, and the crowds were very great, the farmers having all come to the road, and people were pointing and staring, and standing on tiptoe all the better to see, as Beauty felt the soft gravel under her feet and now and then tufts of silken grass or wild-flower.
She walked with her head up as the Prince commanded her, but her eyes were half closed, and she felt the cool air soothing her naked limbs, and she could not stop thinking of the Prince's castle.
Now and then a low voice from the crowd would make her suddenly and painfully aware of her nakedness, and even once or twice a hand shot out to touch her thigh before the Prince behind her cracked his whip immediately.
Finally they entered the dark wooded pass that led through the mountains, and there were only occasional clusters of peasants here and there peeping out from the thick-limbed oaks, and a mist lay upon the ground, and Beauty felt herself drowsy and soft even as she walked. Her breasts felt heavy and soft to her, and her nakedness felt oddly natural.
But her heart became a tiny hammer when the sunlight streamed ahead to reveal an ever-widening green valley.
A great cry rose from the soldiers behind her, and she realized that indeed the Prince was home, and up ahead, across the sloping green, she saw upon a great precipice overhanging the valley the Prince's castle.
It was far greater in size than Beauty's home, a wilderness of dark towers. It might enclose a whole world, it seemed, and its open gates yawned like a mouth before the drawbridge.
Now from everywhere the subjects of the Prince, mere specks in the distance growing ever and ever larger, ran toward the road that wound down and then up again before them.
Riders came over the drawbridge and rode toward them with a blast of trumpets, their banners streaming behind them.
The air was warmer here, as if this place were protected from the sea breeze. It was nothing as dark as the narrow villages and forests through which they had passed. And Beauty could see everywhere the peasants dressed in lighter and brighter colors.
But they were drawing ever nearer to the castle, and in the distance Beauty could see not the peasants whose admiration she had received all along the road, but a great crowd of magnificently dressed Lords and Ladies.
She must have uttered a little cry and bowed her head, because the Prince came up alongside of her. She felt his arm gather her close to the horse, and he whispered :
"Now, Beauty, you know what I expect of you."
But they had already reached the steep approach to the bridge, and Beauty could see it was just as she feared, men and women of her own rank and all clad in white velvet trimmed in gold, or gay and festive colors. She dared not look, and felt the blush in her cheeks again and for the first time was tempted to throw herself on the mercy of the Prince and beg him to conceal her.
It was one thing to be shown to the rustics who praised her and would make a legend of her, but she could already hear the babble of haughty speech and laughter. This was unendurable to her.
But when the Prince dismounted, he ordered her down on her hands and knees and told her softly that this was how she must enter his castle.
She was petrified, her face burning, but she fell quickly to obey, glimpsing the Prince's boots to her left as she struggled to keep up with him in crossing the drawbridge.
Through a great dim corridor she was led, not daring to raise her eyes, though she could see rich gowns and shining boots all around her. Lords and Ladies were bowing to the Prince on either side of her. There were whispers of greeting, and kisses being thrown, and she was naked, moving on her hands and knees as if she were only some poor animal.
But they had reached the mouth of the Great Hall, a room far more vast and shadowy than any in her own castle. An immense fire roared on the hearth, though the sun streamed warm through high narrow windows. It seemed the Lords and Ladies pressed past her, flowing silently along the walls and towards the long wooden tables. Plate and goblets were already set. The air was heavy with the aroma of the supper.
And then Beauty saw the Queen.
She sat at the very end upon a raised dais. Her veiled head was encircled with a gold crown, and the deep sleeves of her green gown were trimmed in pearls and gold embroidery.
Beauty was led forward by a quick snap of the Prince's fingers. The Queen had risen, and now she embraced her son as he stood before the dais.
"Tribute, Mother, from the land over the Mountains, and the loveliest we have received in a long time if my memory serves me. My first love slave, and I am very proud to have claimed her."
"And well you should be," said the Queen in a voice that sounded both young and cold. Beauty dared not look up at her. But it was the Prince's voice which frightened her most. "My first love slave." She remembered his puzzling commiserations with her parents, the mention of their service in this same land, and she felt her pulse quicken.
"Exquisite, absolutely exquisite," said the Queen, "but all the Court must have a look at her. Lord Gregory," she said, and made an airy gesture.
A great murmur rose from the Court gathered around. And Beauty saw a tall gray-haired man approach, though she could not see him clearly. He wore soft leather sock boots, turned down at the knees to reveal a lining of the finest miniver.
"Display the girl ..."
"But Mother," the Prince protested.
"Nonsense, all the common people have seen her. We shall see her," said the Queen.
"And should she be gagged, your Highness?" asked this strange tall man with the fur-lined boots.
"No, that is not necessary. Though punish her surely if she speaks or cries out."
"And the hair, she is shielded by all this hair," said the man, but he was now lifting Beauty and immediately had her wrists clasped over her head. As she stood, she felt herself hopelessly revealed and could not prevent crying. She dreaded a reproof from the Prince, and she could see the Queen all the better though she did not want to see her. Black hair showed beneath the Queen's sheer veil, hanging in ripples over her shoulders, and her eyes were black as the Prince's eyes.
"Leave her hair as it is," said the Prince almost jealously.
"O, he will defend me!" Beauty thought. But then she heard the Prince himself give the order. "Mount her on the table for all to see."
The table was rectangular and stood in the center of the room. It reminded Beauty of an altar. She was forced to kneel on it facing the thrones where the Prince had taken his place beside his mother.
And quickly the gray-haired man placed a large block of smooth wood beneath her belly. She could rest her weight on it and she did, as he forced her knees wide apart and then stretched out her legs so her knees didn't touch the table at all, her ankles bound by leather to the edges. Now her wrists were treated the same. She kept her face hidden as best as she could, weeping.
"You will be silent," said the man icily to her, "or I shall see that you cannot be anything else. Do not misunderstand the Queen's leniency. She does not gag you only because it amuses the Court to see your mouth as it is, and to see you struggle with your own willfulness."
And now, to Beauty's shame, he raised her chin and placed beneath it a long thick wooden chin rest. She could not lower her head, though she lowered her eyes. And she saw all the room about her.
She saw the Lords and Ladies rising from the banquet tables. She saw the immense fire. And then she saw this man, too, with his thin angular face, and gray eyes that were not as cold as his voice, but for the moment seemed even to evince tenderness.
A long shudder went through her as she contemplated herself—spread out, yet mounted so that all could inspect even her face if they chose, and she tried to conceal her sobs by pressing her lips together. Even her hair was no covering, for it fell evenly on either side of her face and cloaked no part of her.
"Young one, little one," said the gray-haired man under his breath. "You're so frightened and it's useless." There seemed a little warmth in his voice. "What is fear, after all? It is indecision. You seek some way to resist, escape. There is none. Do not tense your limbs. It's wasted."
Beauty bit her lip and felt the tears sliding down her face, but she was soothed by his speaking to her. He smoothed back the hair from her forehead. His hand was light and cold as if he were testing for a fever.
"Now be still. Everyone is coming to see you."
Beauty's eyes glazed over, but she could still see the distant thrones where the Prince and his mother were talking to one another quite naturally. But she realized all the Court had risen and was moving towards the dais. The Lords and Ladies were bowing to the Queen and the Prince, before turning and coming towards her.
Beauty squirmed. It seemed the air itself touched her naked buttocks and the hair between her legs, and she struggled to lower her face demurely but the firm wooden chin rest would not yield and all she could do was drop her eyes again.
The first Ladies and Lords were very near and she could hear the rustle of their clothes and see the flash of their gold bracelets.
These ornaments caught the light of the fire and the distant torches, and the dim image of the Prince and the Queen appeared to flicker.
She let out a moan.
"Hush, my darling dear," said the gray-eyed man. And suddenly it was a great comfort that he was so near to her.
"Now look up and to your left," he said now, and she could see his lips spread into a smile. "You see?"
For one instant Beauty beheld what was surely an impossibility, but before she could look again, or clear the tears from her eyes, a great Lady came between her and this distant vision, and with a shock, she felt the Lady's hands upon her.
She felt the cool fingers gathering her heavy breasts, and twisting them almost painfully. She trembled, trying desperately not to cry out. For others had gathered around her, and behind her she felt a pair of very slow and calm hands parting her legs even more. And now someone touched her face, and another hand pinched the calf of her leg almost cruelly.
It seemed her body was all concentrated then in its shameful and secret places. There was a throbbing in the tips of her breasts, and those hands felt cold as if she herself were burning, and now she felt fingers examining her buttocks and prodding even at that tiny and most concealed of openings.
She couldn't help but moan, but she kept her lips tightly shut, and the tears fell down her cheeks.
And for one instant she thought of nothing but what she had glimpsed an instant ago before the procession of Lords and Ladies had intercepted her vision.
High up along the wall of the Great Hall, on a broad stone ledge, she had glimpsed a row of naked women.
It had not seemed possible, but she had seen it. They were all of them young like herself, and they stood with their hands clasped behind their necks as the Prince had taught her to do, and their eyes were down, and she could see the glow of the fire on the curl of pubic hair between each pair of legs, and the swelling, pink nipples of their bosoms.
She could not believe it. She did not want it to be so, and yet if it were so ... well ... again only confusion. Was she all the more terrified, or was she glad that she was not the only one enduring this unspeakable humiliation?
But she could not even think of this, shocking as it was, for the hands were all over her. She had uttered a sharp cry to feel them touching her very sex, and smoothing the hair there, and then to her horror, as her face burned and she shut her eyes tight, she felt a pair of long fingers gliding into her sex and widening it.
It was still sore from the Prince's thrusts, and though the fingers were gentle, she felt that soreness again.
But the most excruciating part was being opened like this and hearing their soft voices now as they talked of her.
"Innocent, very innocent," said one, and another that she had very lean thighs and that her skin was resilient.
That seemed to produce laughter again—that light tinkling laughter, as if all of this were but the greatest amusement, and Beauty realized suddenly that she was straining with all her might to close her legs, but it was quite impossible.
Those fingers were gone, and now someone patted her sex, and pinched shut the hidden little lips, and Beauty squirmed again, only to hear the laughter coming now from the man beside her:
"Little Princess," he said gently in her ear, leaning over so she could feel his velvet cape against her naked arm, "you cannot hide your charms from anyone."
She moaned as if she were trying to appeal to him, but his finger touched her lips.
"Now if I have to seal your lips, the Prince will be very angry. You must resign yourself. You must accept. It is the hardest lesson, compared to which the pain is really nothing."
And Beauty could feel him raising his arm so that she knew the hand that touched her breast now was his. He had imprisoned her nipple and was pressing it rhythmically.
At the same time, someone stroked her thighs and her sex, and to her shame she felt, even in the midst of this degradation, that disgraceful pleasure.
"That's it, that's it," he comforted her. "You must not resist, but rather take possession of your charms, that is, let your mind inhabit your body.
"You are naked, helpless, and all will enjoy you and what can you do? By the way, I should tell you that your squirming only makes you more exquisite. It is very lovely except that it is so rebellious. Now look again, did you see what I pointed out to you?"
Beauty made a soft sound of assent, and fearfully raised her eyes again. It was as she had seen before, the row of young women with their eyes down and their bodies as vulnerable in display as her own.
But what was it she felt? Why must she be subjected to so many confusing feelings? She had thought herself the only one so displayed and humiliated, a great prize for the Prince whom she could no longer see. And was she not displayed here in the very center of the hall?
But then who were these prisoners? Would she only be one of them? Was this the meaning of the odd conversation that had passed between the Prince and her father and mother? No, they could not have served like this. She felt an odd mingling of torrential jealousy and comfort.
It was a ritual, this treatment. Others had suffered it before. It was fixed and she was all the more helpless. She felt herself soften as she thought of it.
But her Lord, the gray-eyed one, was speaking: "Now, for your second lesson. You have seen the Princesses who are tributes here. Now look to your right and you shall see the Princes."
Beauty looked to the other side of the hall as best she could through the shifting figures about her, and there, on another high ledge, in the ghastly shadow-light of the fire, stood a row of naked young men, all of them in the same position.
Their heads were bowed, their hands behind their necks, and they were all of them very handsome to look at, as beautiful each in his own way as the young women of the other side, but their great difference lay in their sex, for their organs were erect and hard to a one, and Beauty could not take her eyes off this sight, for they appeared to her even more vulnerable and subservient.
She knew she had made a little noise again, because she felt the Lord's finger on her lips, and she sensed almost from the air itself that she was now being left by the Lords and Ladies.
Only one pair of hands remained and these she felt touching the tenderest flesh around her anus. She was so frightened by this—for almost no one else had touched her there—that involuntarily she struggled again, only to have the gray-eyed Lord stroke her face again gently.
There was a great commotion in the room. Beauty could just catch the aroma of cooking food, and dishes being brought in, and now she saw that most of the Lords and Ladies were seated at the tables, and there was much talking and lifting of cups, and somewhere a group of musicians had begun to play a low rhythmic music. It was full of horns and tambourines and the strumming of thick strings, and Beauty saw that the long file of naked men and women on either side was moving.
"But what are they?" she wanted to ask. "To what purpose?" But now she saw the first of them appear amid the crowd, carrying silver pitchers with which they filled the goblets at the table, always bowing when they passed the Queen and the Prince, and she watched them, forgetting herself for the moment, with great absorption.
The young men had softly curly hair, cut at the shoulders and neatly combed so that it framed their lean faces. And never did they raise their eyes, though some seemed to move in obvious discomfort from the hardness of their penises. How she could tell this discomfort, she was not sure; it was their manner, a manner of bearing tension and desire, with no expression for it.
And as she saw the first of the long-haired girls bending over the table with her pitcher, she wondered if she too felt this same softly agonizing pleasure. Beauty felt it now just looking at these slaves, and she felt a quiet relief that for a moment she herself was unobserved.
Or so she thought.
Because she could sense a restlessness in the room. Some were rising and walking about, perhaps even dancing to the music. She could not be sure. And others had gone to gather near the Queen, their goblets in hand, regaling the Prince it seemed with stories.
The Prince.
She caught a clear glimpse of him and he smiled at her. How regal he looked, his black hair glossy and full, his long, shining white boots stretched out on the blue carpet before him. He was nodding and smiling to those who addressed him, but now and then his eyes moved to Beauty.
But there was so much to see, and now she felt someone was very near her, and touching her again, and she realized that a line of dancers was just forming to one side of her.
There was a reckless air to things. Much wine was being poured. There were great eruptions of laughter.
And then, quite suddenly, she saw far to her left a young naked boy drop his pitcher of wine, and the red liquid run out on the floor as others hastened to clean it.
At once the Lord at Beauty's side clapped his hands, and Beauty saw three exquisitely dressed Pages, no older than the naked boys themselves, rush forward and seize the boy and hold him up quickly by his ankles.
This brought a loud round of applause from those Lords and Ladies nearest the boy, and at once a paddle was produced, a very beautiful piece of gold enameling and white tracery, and the offender was smartly spanked while all looked on with the greatest fascination.
Beauty felt a fluttering in her heart. If she were to be humiliated like that, punished so immediately and ignominiously for clumsiness, she didn't know how she could bear it. To be displayed was one thing; here she had some grace.
But she could not endure the thought of being held by her ankles as the boy was. She could see only his back, and the paddle flashing down again and again on his reddening buttocks. He held his hands obediently on the back of his neck, and as he was let down on his hands and knees, the young Page with the paddle drove him quickly with a series of loud blows towards the Queen, where the young culprit, his buttocks very red, bowed his head and kissed the Queen's slipper.
The Queen had been in fast conversation with the Prince. She was a mature woman, very full blown but it was from her, obviously, that the Prince had gotten his beauty. She turned, almost indifferently, her eyes darting back to the Prince, and motioning for the young slave to rise a little, she brushed back his hair affectionately.
But then in the same indifferent manner, never withdrawing herself from the Prince, she made a motion to the Page, with a quick frown, that the boy was again to be punished.
The Lords and Ladies nearest applauded with mock scolding gestures, and then obviously enjoyed it very much as the Page put his foot on the second step of the dais before the throne, and hoisted the disobedient slave up over his knee and again, in full view of everyone, soundly spanked him.
A long row of dancers obscured the view for a moment, but again and again Beauty caught glimpses of the unfortunate boy, and she could see that as the paddle came down, he was having a more and more difficult time bearing it. He struggled just a little in spite of himself, and it was also quite obvious that the Page who delivered the paddling was very much enjoying it. His young face was flushed, and he was biting his lip slightly, and he drove the paddle down unnecessarily hard it seemed, and Beauty felt she hated him.
She could hear the Lord beside her laughing. There was a little loose crowd about her now, men and women drinking, talking idly. The dancers moved in a long chain, performing their fluid and graceful movements.
"So you see you aren't the only helpless little creature in this world," said the gray-eyed Lord, "and does it soothe you to see the Tribute that belongs to your Sovereigns ? You are the first Tribute for the Prince and I think that you shall have to set a fierce example. The young slave you saw, Prince Alexi, is very much a favorite of the Queen or he wouldn't be dealt with so lightly."
Beauty saw that the paddling had stopped. Once again, the slave was on his hands and knees and kissing the feet of the Queen as the Page waited in attendance.
Now the slave's buttocks were very red. "Prince Alexi," Beauty thought. It was a lovely name, and he too was of royal blood and high birth. Why, of course, all of them were. It was a delightful thought. What if they had not been, and she were the only Princess?
She stared at his buttocks. There were obvious welts on them and little patches that seemed much redder than the rest, and as the young slave Prince kissed the Queen's feet, Beauty could see also his scrotum between his legs, dark, hairy and mysterious.
It struck her how dreadfully vulnerable he seemed, being a boy, in ways she had never considered.
But he had been released or forgiven. He rose to his feet, and brushed his auburn curly hair out of his eyes and back from his cheek, and she saw his face stained with tears, and reddened too; yet he had about him a marvelous dignity.
He took the pitcher handed him without complaint and gracefully he moved among the standing guests, filling their goblets.
He was only a few paces from Beauty, and drawing ever closer. And she could hear how the men and women teased him.
"Another paddling and you are so wretchedly clumsy," said a very tall blond-haired Lady in a long green gown, with diamonds on her fingers, and she pinched his red cheek, as, with his eyes down, he smiled.
His penis was hard and erect as before, rising up thick and motionless from a nest of dark curly hair between his legs. Beauty could not stop herself from looking at it.
As he came nearer, she held her breath.
"Come here, Prince Alexi," said the Lord with the gray eyes. He snapped his fingers. And then taking a white handkerchief, he had the boy moisten it with the wine.
The boy was so near now Beauty might have touched him. And the Lord took the moistened handkerchief and pressed it to Beauty's lips. It felt good and cool and tantalizing.
But she could not help but look up at the obedient boy Prince who stood waiting, and she saw him looking at her.
And though his face was still slightly pink, and there were tears on his cheeks, he smiled at her.
**THE PRINCE'S BED CHAMBER**
**B** EAUTY AWOKE to new terror.
It was getting dusk; the Feast was over. The Lords and Ladies who remained were very loud and swept up in the fever of the afternoon, but she was being unbound and she did not know what would now happen to her.
Several other slaves had been soundly spanked during the course of the banquet, and it seemed finally that no offense was required, merely the decision of a Lord or Lady. The request was then granted by the Queen—and the unlucky one was thrust up over the Page's knee, his head bowed, his feet dangling off the ground, and down came the golden paddle.
Twice it had been young women.
And one of them had broken into silent sobs. But there was in her manner something that made Beauty a little suspicious. After she was spanked, she scurried all too fast to the Queen's feet, and Beauty hoped she would be spanked again until her sobs were real, and all her scurrying was real, and she found herself vaguely delighted when the Queen ordered it.
Now, as Beauty was awakened, she thought dreamily of all this, and felt sharp fear, and also some sense of drama.
Would she be sent away to some place with all these slaves? Or would the Prince take her?
She was stunned with confusion, when she realized the Prince had risen and given an order to the gray-eyed Lord to bring Beauty after him.
She was untied; she was very stiff. But the Lord now had one of those gold paddles in his hand which he tested loudly upon his palm, and giving her no time to stretch her aching muscles, he ordered her down on her knees and forward.
When she hesitated, his command came very sharp again, but he did not strike her.
She rushed to catch up with the Prince who had just reached the stairway.
And soon she was following him up and down a long corridor.
"Beauty," he stood back. "Open the doors!"
And kneeling up, she quickly opened them and forced them apart and then followed the Prince into a bed chamber.
The fire was already a great blaze on the hearth and the windows were curtained, and the bed had been turned back, and Beauty was quivering with excitement.
"My Prince, shall I begin her training at once?" asked the gray-eyed Lord.
"No, my Lord, I shall attend to it myself the first few days, possibly longer," said the Prince, "though you may of course, whenever the occasion arises, instruct her, teach her manners, the general rules that pertain to all the slaves, and so forth. She does not drop her eyes as she should, as you can see; she is so very inquisitive." And at this he smiled, though Beauty at once looked down, much as she wanted to see it.
She knelt obediently, glad her hair concealed her. And then she checked herself in this thought. She was not learning much if that was what she wanted.
She wondered if Prince Alexi had been ashamed of his nakedness. He had had large brown eyes and such a beautiful mouth, but he was much too lean to be cherubic. She wondered where he was now, and was he being punished more for his clumsiness?
"Very well, your Highness," said the Lord, "but I think you realize that firmness in the beginning is a mercy to the slave, especially when the slave is such a proud and spoilt Princess."
Beauty blushed to hear this.
The Prince gave a low, gentle laugh.
"My Beauty is very like an unstamped coin," said the Prince, "and I wish to draw in the full character. I shall take delight in training her. I wonder if you yourself are as attentive to her faults as I am."
"Your Highness?" the Lord seemed to stiffen slightly.
"You were not yourself so very strict with her in the Great Hall that you prevented her from feasting her eyes on young Prince Alexi. I rather think she enjoyed his punishment as much as her masters and mistresses."
Beauty flushed hotly. She had never dreamed that the Prince had observed her in this.
"Your Highness, she was only learning what will be expected of her, or so I thought..." the Lord answered very humbly. "It was I who drew her attention to the other slaves so she might profit from their obedient example."
"Ah, well," said the Prince wearily and agreeably, "perhaps I am only too enamored of her. After all, she wasn't sent to me as a Tribute, I won her and claimed her myself, and I am too jealous, it seems. Perhaps I seek for some reason to punish her. You're dismissed. Come for her in the morning, if you will, and we shall see."
The Lord, obviously worried that he had failed, left the room quickly.
Beauty was now alone with the Prince, and the Prince was sitting quietly by the fire looking at her. She was in a great state of agitation; she knew she was blushing as always, and that her breasts were heaving slightly. She rushed forward quite suddenly and pressed her lips to the Prince's boot, and it seemed to move as if it welcomed her kiss, rising slightly as over and over again she kissed it.
She was moaning. O, if only he'd give her permission to speak, and when she thought of her fascination with the punished Prince, she blushed all the more.
But her Prince had risen. He took her wrist and lifted her and drawing her hands behind her back so that he held them firmly, he spanked both her breasts hard until she cried out, feeling the heavy flesh sway and the sting of his hands on her nipples.
"Am I angry with you? Or am I not?" he asked softly.
She groaned, imploring him. And he placed her over his knee as she had seen the young Prince over the Page's knee, and with his bare hand he gave her a smart torrent of blows that had her crying aloud in an instant.
"To whom do you belong?" he demanded in a low, but angry voice.
"To you, my Prince, completely!" she cried out. It was dreadful, and then, suddenly unable to control herself she said, "Please, please, my Prince, not in anger, no ..."
But instantly his left hand clamped over her mouth, and she felt another terrible torrent of hot spanks until her flesh was stinging and she couldn't control her crying.
She could feel the Prince's fingers against her lips. But he would hardly be satisfied with this. He had her on her feet now and by her wrists he led her to a corner of the room between the blazing fire and the curtained window. There was a high stool there made of carved wood, and on this he sat while he stood her beside him. She was crying softly, but she dared not beg again, no matter what happened. He was angry, fiercely angry, and though she could endure any pain for his pleasure, this was unbearable for her. She must please him, must make him loving again, and then any pain at all would not be too much.
He turned her and she stood facing him as he sat inspecting her. She dared not look him in the face, and then he drew back his cloak, and laying his hand on the golden buckle of his belt said, "Unfasten this."
At once she went to obey with her teeth without being told that was how she might do it. She hoped and prayed he would be pleased. She pulled on the leather, her breath soft and fast, and then pulled the strap back so that the belt came loose.
"Now pull it off," said the Prince, "and give it to me."
She obeyed at once, even though she knew what would follow. It was a thick, wide leather belt. Maybe it would be no worse than a paddle.
Now he told her to raise her hands and her eyes, and she saw above a metal hook just over her head hanging from a chain on the ceiling.
"You see here we are not without provisions for disobedient little slaves," he said in his usual gentle voice. "Now clasp that hook, though it will put you on tiptoe, and you will not dream of letting go of it, do you understand me?"
"Yes, my Prince," she cried softly.
She had hold of it, and it seemed to stretch her out, and the Prince moved back the stool on which he sat and appeared to make himself comfortable. He had ample room in which to swing the strap which he had made into a loop, and he was silent for a moment.
Beauty cursed herself for ever admiring young Prince Alexi. Yet she was ashamed that his very name had formed in her mind, and when she felt the first hard smack of the belt on her thighs, she let out a frightened little cry but was glad of it.
She deserved this, and she would never again make such a terrible mistake, no matter how beautiful or enticing were the slaves, and her boldness to look at them had been unforgivable.
The wide heavy leather belt struck her with a loud, frightening sound, and the flesh of her thighs, more tender perhaps than her buttocks, even sore as they were, seemed to ignite under it. Her mouth was open, she could not keep herself quiet, and suddenly the Prince ordered her to lift her knees and march in place.
"Quickly, quickly, yes, in rhythm!" he said angrily, and Beauty, astonished, struggled to obey, marching fast, her breasts moving with the effort, her heart pounding.
"Higher, faster," the Prince commanded.
She marched as he commanded, her feet slapping the stone floor, her knees coming up very high, her breasts a terrible aching weight as they swayed, and again came the belt smacking her and stinging her.
The Prince seemed in a fury.
The blows came faster and faster, as fast as she was moving her legs, and very soon, Beauty was writhing and struggling to get away from them. She was crying aloud unable to stop herself but the worst of it, the worst of it, was his anger. If only this were for his delight, if only he were pleased with her. She was crying and burying her face in her arm and the balls of her feet were burning, and her thighs felt swollen and blotched with pain as now again he took out his temper upon her buttocks.
The smacks came so quickly, she had no sense of how many there were, only that it was a great deal more than he'd ever given her before, and it seemed he only grew more agitated, his left hand now thrusting her chin up and closing her mouth so she couldn't cry, all the while he commanded her to march faster and lift her legs higher.
"You belong to me!" he said without ever stopping the loud spanking belt. "And you will learn to please me in all things, and you will never please me with your eyes upon the male slaves of my mother. Is this clear to you? Do you understand?"
"Yes, my Prince," she struggled to say.
But he seemed at wits ends to punish her. And stopping her suddenly by lifting her around the middle, he brought her up over the stool which he had just left, so that dangling from the hook which she held for dear life, she was now thrust over it, the wooden seat of the stool pressing into her naked sex, her legs out helplessly behind her.
And then he sent his worst rain of blows on her, hard snapping spanks that made her calves quiver and sting as her thighs had done before. But no matter how he busied himself with her legs, he always returned to her buttocks, punishing them the hardest so that Beauty was choking with sobs, and felt this as endless.
Quite suddenly he stopped.
"Let go the hook," he commanded, and then he scooped her up over his shoulder and taking her across the room, he flung her down on the bed.
She fell back on the pillow, and immediately beneath her sore and swollen buttocks and thighs felt a prickling and a roughness. She had only to cast her head slightly to the side to see the jewels glittering on the coverlet. And she knew how they would torture her as soon as he had mounted her.
But she wanted him so badly. And when she saw him rise up over her, she felt not the hot throbbing pain in her body but a flood of juices between her legs and a new moan coming out of her as she opened herself to him.
She couldn't keep from lifting her hips, praying it didn't displease him.
He knelt over her, removing his erect cock from his breeches, and then he brought her up on her knees and impaled her upon it.
She cried out. Her head fell back. It was a great hard driving thing inside her sore and quivering orifice. But she felt it bathed with her juices, and as the Prince forced it in deeper and brought her down upon it, it seemed a spit that rubbed against some mysterious core in her, sending the ecstasy washing through her so she was giving great guttural moans in spite of herself. The Prince's thrusts came faster and faster and then he too gave a soft cry, and held her close to him, her breasts aching and pressed to his chest, his lips on the back of her neck, his body softening slowly.
"Beauty, Beauty," he whispered. "You have conquered me as surely as I have conquered you. Don't ever arouse my jealousy again. I don't know what I would do if you did it!"
"My Prince," she moaned and kissed him on the mouth, and when she saw the distress in his face, she covered it with kisses.
"I'm your slave, my Prince," she said.
But he would only moan and press his face into her neck, and seemed bereft.
"I love you," she implored him, and then he laid her down on the bed, and drawing up beside her, took his wine from the bedside stand and, gazing at the fire, seemed for a long time to be thinking.
**PRINCE ALEXI**
**B** EAUTY DREAMED a dream of boredom. She roamed the castle in which she had lived all her life, with nothing to do, and now and then paused in a deep window seat to watch the tiny figures of the peasants in the fields below gathering the fresh mown grass into haystacks. The sky was cloudless and she disliked the look of it, its sameness and vastness.
It seemed she could not find anything to do that hadn't been done a thousand times before, and then suddenly there came to her ears a sound she could not identify.
She followed the sound, and through a doorway saw an old woman, bent and ugly, plying a strange contraption. It was a great turning wheel with a thread that was winding itself upon a spindle.
"What is it?" Beauty asked with great interest. "Come see for yourself," said the old woman, who had the most remarkable voice, because it was young and strong and so unlike her visage.
It seemed Beauty had only just touched this marvelous machine with its whirring wheel when she fell down in a great swoon, and all about her heard the world weeping.
"... sleep, sleep for a hundred years!"
And she wanted to cry out, "Unbearable, worse than death," for it seemed some great deepening of the ennui she had struggled against ever since she could remember, the wandering from room to room ...
But she awoke.
She was not at home.
She was lying in the bed of her Prince, and she felt the prickling of the jeweled coverlet beneath her.
The room was full of the leaping shadows of the fire, and she saw the gleam of the carved posts of the bed, and the drapery fallen about her in rich colors. She felt herself animated and flushed with desire, and she rose up, so eager was she to lose the weight and texture of her dream, and she realized that the Prince was not beside her.
But there he was, by the fire, his elbow against the stone above it which bore a great crest with crossed swords. He wore his brilliant red velvet cloak still and his high turned down leather boots with their pointed toes, and his face was sharpened with brooding.
The pulse between her legs quickened. She stirred, and gave some faint little sigh so that he awoke from his thoughts and approached her. She could not see his expression in the darkness.
"All right, there is but one answer," he said to her. "You shall become accustomed to all the sights of the castle, and I shall become accustomed to seeing you accustomed to them."
He pulled the bell rope by the bed. And lifting Beauty he sat her at the end of the bed so that her legs were curled under her.
A Page entered, as innocent as the boy who had so diligently punished Prince Alexi, and like all the Pages he was extremely tall with powerful arms. Beauty was certain they had all been chosen for these endowments. She had no doubt he could have held her by the ankles had he been ordered, but his face was smooth without the slightest meanness.
"Where is Prince Alexi?" The Prince demanded. He appeared angry and resolute, and he paced back and forth as he spoke.
"O, he is in frightful trouble tonight, your Highness. The Queen is much concerned with his clumsiness. You know he must be her example to others. She has had him tied in the garden, most uncomfortably."
"Yes, well, I shall make him even more uncomfortable. Obtain my mother's permission and bring him to me and bring Squire Felix with him."
Beauty heard all this in quiet amazement. She tried to make her face as smooth as the Page's face. But she was more than alarmed. She was going to see Prince Alexi again, and she could not imagine concealing her feelings from her Prince. If only she could distract him from this.
But when she made a little whispering sound, he ordered her at once to be quiet, and to sit where she was, and to cast her eyes down.
Her hair fell around her, tickling her naked arms and her thighs, and almost with pleasure she realized there was no escaping this.
Squire Felix appeared almost immediately, and as she had suspected he was the Page who had so vigorously spanked Prince Alexi earlier. He had the gold paddle fixed to his belt so it dangled at his side as he bowed to the Prince.
"All of those who serve here are picked for their gifts," Beauty thought, looking at him, for he too was fair, and his blond hair made an excellent frame for his youthful face, though it was somewhat plainer than those of the captive Princes.
"And Prince Alexi?" the Prince demanded. His color was high, his eyes had an almost evil glitter, and Beauty became frightened again.
"We're preparing him, your Highness," said Squire Felix.
"And why should this take so long? How long has he served in this house that he should be so lacking in respect?"
At once Prince Alexi was brought in.
Beauty tried not to admire him. He was naked as before, of course, she'd expected no less, and in the light of the fire she could see that his face was flushed, and his auburn hair hung loose in his eyes which were cast down as if he dared not lift them to the Prince. They were of about the same age, surely, and about the same height, but here stood the darker Prince Alexi quite helpless and humble before the Prince who was striding back and forth before the fire, his face cold and merciless and slighdy agitated. Prince Alexi's organ was rigid. He held his hands behind his neck.
"So you were not ready for me!" the Prince whispered. He drew closer, inspecting Prince Alexi. He looked at the stiffened organ and then with his hand he gave it a rough slap, so that Prince Alexi flinched in spite of himself. "Perhaps you need a little training in being ... always ... ready," whispered the Prince. His words came slowly and with a deliberate courtesy.
He lifted Prince Alexi's chin and looked into his eyes. Beauty caught herself staring at them both without the slightest shyness.
"My apologies, your Highness," Prince Alexi said, and his voice was low in timbre, calm, without rebellion or shame.
The Prince's lips spread slowly in a smile. Prince Alexi's eyes were larger, and they possessed the same calm as the voice. It seemed to Beauty they might even drain away the Prince's anger but this was impossible.
The Prince stroked Prince Alexi's organ and gave it another playful slap, and then another.
The submissive Prince looked down again and there was nothing in him but the grace and dignity Beauty had witnessed before.
"I must behave like this," she thought. "I must have this manner, this strength, to bear it all with the same dignity." Yet she marveled. The captive Prince must at all times show his desire, his fascination, while she could conceal this craving between her legs, and she could not stop herself from wincing as she saw the Prince pinch the tiny hardened nipples on Prince Alexi's chest, and then lift Prince Alexi's chin again to inspect his face.
Beyond them, Squire Felix watched all with obvious pleasure. He had folded his arms, his legs rather wide apart as he stood, and his eyes moved hungrily over Prince Alexi's body.
"How long have you been in the service of my mother?" the Prince demanded.
"Two years, your Highness," said the humble Prince softly. Beauty was quite astonished. Two years! It seemed to her all of her life before had not been so long, but she was more rapt with the sound of his voice than with the words. The voice made him seem more palpable and visible.
His body was a little thicker than that of the Prince, and the dark brown hair between his legs was beautiful. She could see the scrotum, no more than a shadow.
"You were sent here in Tribute by your father."
"As your mother demanded, your Highness."
"And to serve how many years?"
"As long as it pleases your Highness, and my mistress, the Queen," Prince Alexi answered.
"And you are what? Nineteen? And a model among the other Tributes?"
Prince Alexi blushed.
The Prince turned him towards Beauty with a rough blow on the shoulder, and steered him towards the bed.
Beauty drew herself up, feeling her face flushed and warm.
"And the favorite of my mother?" the Prince demanded.
"Not tonight, your Highness," Prince Alexi said with the barest trace of a smile.
The Prince acknowledged this with a soft laugh. "No, you have not comported yourself very well today, have you?"
"I can only beg forgiveness, your Highness," said Prince Alexi.
"You can do more than that," said the Prince into his ear as he pushed him nearer to Beauty. "You can suffer for it. And you can give my Beauty a lesson in willingness and perfect submission."
Now the Prince turned his gaze on Beauty, scrutinizing her mercilessly. She looked down, terrified of displeasing him.
"Look at Prince Alexi," he told her, and when she raised her eyes, she saw the beautiful captive Prince only a few inches from her. His disheveled hair partially veiled his face, and his skin appeared deliciously smooth to her. She was trembling.
Just as she feared he would, the Prince lifted Prince Alexi's chin again, and when Prince Alexi looked at her with his large brown eyes, he smiled very slowly and serenely at her for an instant the Prince could not have witnessed. Beauty drank her fill of him with her eyes because she had no choice and hoped the Prince would see no more than her distress.
"Kiss my new slave and welcome her to this house. Kiss her lips and her breasts," said the Prince. And he lifted Prince Alexi's hands from the back of his neck so they went silently and obediently to his sides.
Beauty gasped. Prince Alexi was smiling at her again, secretly as his shadow fell over her, and she felt his lips close over hers and the shock of his kiss pass through her. She could feel that misery between her legs formed into a tight knot, and when his lips touched her left breast and then her right, she bit into her lower lip so hard she might have drawn blood. Prince Alexi's hair stroked her cheek and her breasts as he carried out the command and then he stood back with that same beguiling equanimity.
Beauty put her hands to her face before she could stop herself.
But immediately the Prince took them away.
"Look well, Beauty. Study this example of the obedient slave. Become accustomed to him so that you do not see him but rather the example he sets for you," the Prince said. And roughly he turned Prince Alexi about so that Beauty could see the red marks on his buttocks.
Prince Alexi had received far worse punishment than Beauty. He was bruised and there were many white and pink welts on his thighs and on his calves. The Prince inspected all this almost indifferently.
"You will not look away again," the Prince said to Beauty, "do you understand me?"
"Yes, my Prince," Beauty said at once, only too eager to show her obedience, and in the very midst of her painful distress, an odd feeling of resignation came over her. She must look at Prince Alexi's exquisitely muscled young body; she must look at his taut and beautifully molded buttocks. If only she could hide her fascination, feign only submission.
But the Prince was no longer looking at her. He had taken both Prince Alexi's wrists in his left hand, and had taken from Squire Felix not the golden paddle, but rather a long flat leather-sheathed stick which appeared heavy and with which he struck Alexi several loud blows on his calves rapidly.
He pulled his captive to the center of the room. He placed his foot on the wrung of the stool as he had done earlier, and pushed Prince Alexi over his knee just as he had done Beauty. Prince Alexi's back was to Beauty and she could see not only his buttocks but also the scrotum between his legs, and she saw the flat leather stick land its blows in red crisscrossed marks over Prince Alexi. Prince Alexi did not struggle. He made hardly a sound. His feet were planted on the floor, and nothing in his form suggested any attempt to escape the aim of the stick as Beauty might.
Yet even as she watched, amazed, wondering at his control and his endurance, she could see the signs of strain in him. He moved ever so slightly, his buttocks rising and falling, his legs quivering and then she could hear the slightest sound from him, a whispered moan which he was obviously concealing behind his closed lips. The Prince flailed at him, the skin growing a darker red with each broad stripe from the stick, and then, when his desire seemed to have reached a crest, he ordered Prince Alexi down on his hands and knees before him.
Beauty could see Prince Alexi's face. It was stained with tears, but the composure had not broken. He knelt before the Prince, waiting.
The Prince lifted his pointed boot and thrust it under Prince Alexi, touching the tip of Prince Alexi's penis.
Then he took Prince Alexi by the hair and lifted his head.
"Open it," he said softly.
Immediately Prince Alexi moved to put his lips to the seam in the Prince's breeches. With a skill that amazed Beauty he unsnapped the hooks that concealed the Prince's bulging sex, and revealed it. The organ was enlarged and hardened, and Prince Alexi freed it from the cloth now and tenderly kissed it. But he was in great pain still and when the Prince thrust the organ into Prince Alexi's mouth he was not prepared for it. He fell backwards a little on his knees and had to reach for the Prince, caressingly, to stop himself from falling. But immediately he sucked the Prince's organ, and he did it with great back and forth motions that amazed Beauty, his eyes closed, his hands hovering at his sides ready for the Prince's command.
The Prince stopped him rather quickly. It was clear he did not want his passion brought to a pinnacle. Nothing so simple would happen.
"Go to the chest in the corner," he said to Prince Alexi, "and bring me the ring that is in it."
Prince Alexi went on his hands and knees to obey. But obviously the Prince wasn't satisfied. He snapped his finger, and Squire Felix at once drove Prince Alexi with his paddle. He drove him to the chest and continued to torment him with the paddle while he opened the chest and with his teeth removed a large leather ring and brought it back to the Prince.
Only then did the Prince send Squire Felix back to his corner, and Prince Alexi was out of breath and trembling.
"Put it on," said the Prince.
Prince Alexi was holding the leather ring not by the leather itself but by some small piece of gold attached to it. And still holding it this way in his teeth, he slipped the ring over the Prince's penis, but he did not release it.
"You serve me, you go where I go," said the Prince, and now he proceeded to walk slowly about the room, his hands on his hips as he looked down on the Prince struggling on his knees, his teeth to the leather ring, to follow him.
It was as if Prince Alexi were kissing the Prince or tethered to him. He scrambled backwards, his hands out so as not to touch the Prince disrespectfully.
The Prince with ordinary strides that took no cognizance of the difficulty of his slave approached the bed, and then turning, made his way back to the fire, his slave struggling before him.
Suddenly he turned his body hard to the left to face Beauty, and when he did Prince Alexi had to take hold of him for balance. Prince Alexi clung to him for a moment and when he did, he pressed his forehead against the Prince's thigh, and the Prince rather idly stroked his hair. It seemed almost affectionate.
"You so dislike the ignominious position, don't you?" he whispered. But before Prince Alexi might have answered, he struck him a hard blow on the face that sent him backwards and away from him. Then he pushed Prince Alexi down on all fours.
"Back and forth across the room," he said with a snap of his fingers to Squire Felix.
As always, the Squire was only too happy to oblige. Beauty hated him! He drove Prince Alexi across the floor to the far wall and then back again to the door.
"Faster!" the Prince said sharply.
Prince Alexi moved as swiftly as he could. Beauty could not bear to hear the anger in the Prince's tone, and she raised her hands to cover her mouth. But the Prince wanted more speed. The paddle came down again and again on Prince Alexi's buttocks and the command came again and again until he was scurrying to obey the commands, and she could see his terrible misery that he had lost all grace and dignity. Now she understood the Prince's little taunt. Prince Alexi's calm and grace had obviously been his consolation.
But had he really lost them? Or was he merely calmly giving this too to the Prince? She couldn't tell. She winced with each spank of the paddle, and each time Prince Alexi turned to go back across the room, she caught a full glimpse of his tormented buttocks.
Quite suddenly, however, Squire Felix stopped. "I've drawn blood, your Highness," he said.
Prince Alexi knelt with his head down, panting.
The Prince looked at him and then he nodded.
He snapped his fingers for Prince Alexi to rise, and again he lifted his chin and looked into his tear-stained face.
"So you are reprieved for the night by virtue of that all too delicate skin of yours," he said.
He turned him towards Beauty again. Prince Alexi's hands were on the back of his neck, and his face, flushed and wet, was indescribably beautiful to her. It was full of unspoken emotion, and as he was led closer to her, she could feel her heart pounding. "If he kisses me again, I shall die," she thought. "I shall never hide my feelings from the Prince."
And if it is the rule that I can be spanked until he draws blood.... She had no real idea what that might mean, except a great deal more pain than she had already felt. But even that would be preferable to the Prince discovering how fascinated she remained with Prince Alexi. "Why does he do this," she thought desperately.
But the Prince thrust Prince Alexi forward.
"Put your face in her lap," he said, "and your arms about her."
Beauty gasped and sat up, but Prince Alexi obeyed immediately. Beauty looked down to see his auburn hair covering her sex as she felt his lips against her thighs, and his arms enclose her. His body was hot and pulsing; she could feel the beating of his heart, and without meaning to, she reached out to clasp his hips with her hands.
The Prince kicked Prince Alexi's legs wide apart and taking Beauty's head roughly in his left hand so that he might kiss her, he drove his organ into Prince Alexi's anus.
Prince Alexi moaned at the roughness and swiftness of the thrusts. Beauty felt the pressure against her as Prince Alexi was driven ever more quickly by it. The Prince had let her go, and she was crying. She held tight to Prince Alexi, and then the Prince gave his final thrust with a moan, his hands pressed to Prince Alexi's back, and he stood still letting his pleasure course through him.
Beauty tried to keep herself quiet.
Prince Alexi let her go, but not without a secret little kiss between her legs right on the crest of her pubic hair, and just as he was being drawn away, again, his dark eyes narrowed in a secret smile for her.
"Mount him in the passage," said the Prince to the Squire. "And see no one satisfies him. Keep him in torment. Every quarter of the hour remind him of his duty to his Prince, but do not satisfy him."
Prince Alexi was taken away.
Beauty sat staring at the open door.
But it was not over. The Prince reached out and taking her by the hair, told her to follow him.
"On your hands and knees, my dear. That is always the way you will move through the castle," he said, "unless told otherwise."
She hurried along, following him out and to the edge of the stairway.
Halfway down was a broad landing from which one might see directly into the Great Hall.
And on the landing was a stone statue that frightened Beauty. It was a pagan god of some sort with an erect phallus.
It was onto this phallus that Prince Alexi was now thrust, his legs bound apart on the pedestal of the statue. His head was laid back on the statue's shoulder. He gave another moan as the phallus impaled him and then he lay still as Squire Felix bound his hands behind his back.
The statue's right arm was upraised, the stone fingers of the hand forming a circle as if they had once clasped a knife or some other instrument. And now the Squire carefully positioned Prince Alexi's head on the shoulder of the statue beneath that hand. And through the clasped hand, he placed a leather phallus, anchoring it so that it fit into Prince Alexi's mouth.
It seemed now that the statue raped him both through his anus and through his mouth, and he was bound to it. And his organ, as stiff as before, lay thrust forward as the phallus of the statue was inside of him.
"Now you are perhaps a little more used to your Prince Alexi," said the Prince softly.
"But this is too terrible," Beauty thought, "that he must spend the night in this misery." Prince Alexi's back was painfully arched, his legs bound wide apart, and the moonlight from the window behind him made a long line down his throat, his smooth chest and his flat belly.
The Prince tugged gently on Beauty's hair which he held wrapped around his right hand and leading her back to bed, he laid her down and told her to sleep, as he would soon be doing beside her.
**PRINCE ALEXI AND FELIX**
**I** T WAS almost dawn. The Prince lay deep asleep. And Beauty who had been waiting for his heavy sleeping breaths, slipped out of the bed, and on all fours, out of stealth, not obedience, crept into the corridor. She had lain for a long time looking at the door, seeing that it had never been really shut, and she might make her small escape without noise if she only had the courage.
She crept to the top of the steps.
The light fell full on Prince Alexi, and she could see that his organ was rigid as before, and Squire Felix was talking to him, softly. She could not hear what the Squire said, but she was furious to see him awake. She had hoped that he too would now be sleeping.
And as she watched, quite unknown to Squire Felix, she saw him come round in front of Prince Alexi, and torment the organ again with a volley of slaps that sounded very loud in the empty stairwell. The captive Prince gave a little moan, and Beauty could see his chest heave with his breath.
Squire Felix walked back and forth restlessly. Then he looked at the Prince, and it seemed he turned his head from left to right as though listening. Beauty held her breath. She was terrified she might be discovered.
Squire Felix drew near to Prince Alexi and putting his arms around his hips, he covered Prince Alexi's organ with his mouth and began sucking it.
Beauty was beside herself with frustration and anger. This was just what she had meant to do. She had imagined herself braving all dangers to do it. And now she was forced to watch as Squire Felix tormented the poor Prince. But to her surprise, Squire Felix was not merely tantalizing Prince Alexi. Squire Felix seemed quite in earnest. He was ravaging the organ with a regular rhythm and Beauty knew from the moans that Prince Alexi couldn't conceal he was now reaching the climax of his passion.
His taut, cruelly bound body shuddered with one protracted groan after another, and then he lay still as Squire Felix drew back and moved into the shadows.
It seemed he spoke to Prince Alexi then. Beauty leaned her head against the stone balustrade.
After a little while, Squire Felix told Prince Alexi to wake, and he gave the organ those tormenting slaps again and when it seemed reluctant, Squire Felix seemed fearful and became threatening. But Prince Alexi was deep asleep in his painful tethers, and Beauty was very pleased to see this.
She turned and silently made her way back to the bedroom door when she realized that someone was near her.
She was so frightened that she almost screamed, a mistake which would surely have destroyed her. But she covered her mouth, and lifting her eyes, she saw in the distant shadows the figure of Lord Gregory watching her. This was the gray-haired Lord who had wanted so to discipline her properly, who had called her spoilt.
Yet he did not move. He stood still watching her.
And when she had stopped trembling, she rushed as quickly as she could back to the Prince's bed, and slipped under the coverlet beside him.
He had never awakened.
She lay in the dark waiting for Lord Gregory to come but he did not, and she soon realized he would not dream of waking the Prince, and then she was half dozing.
She was thinking of Prince Alexi in a thousand ways, of the redness of his sore flesh after the paddle, of his beautiful brown eyes, and his strong, somewhat compact body. She was thinking of his glossy hair against her, the secret kiss he gave her thighs, and how, after this terrible humiliation, he had given her that smile which was so serene and affectionate.
The torment between her legs was no worse than before, and no better. She dared not touch it with her fingers, lest she be discovered, and it was too shameful to think of such things, and she was sure the Prince would never allow it.
**THE SLAVES' HALL**
**I** T WAS late afternoon when Beauty awoke. She realized that the Prince and Lord Gregory were in an argument. Immediately, she was afraid, but as she lay still she perceived that Lord Gregory had obviously not told the Prince what he had seen. Surely her punishment would be terrible if he had. Rather Lord Gregory was arguing only that Beauty must be taken to the Slaves' Hall and properly groomed.
"Your Highness, you are enamored of her, of course," Lord Gregory said, "but you remember, surely, your own censure of other Lords, especially your cousin, Lord Stefan, on account of his excessive love for his slave ..."
"It is not excessive love," the Prince answered sharply, but then he stopped as if Lord Gregory had hit upon the truth. "Maybe you should take her to the Slaves' Hall," he murmured, "though only for the day."
As soon as Lord Gregory had taken her out of the room, he unfastened the paddle attached to his belt and gave her several cruel spanks as she hurried on her hands and knees before him.
"Keep your eyes down and your head down," he said coldly, "and lift your knees gracefully. Your back is to be a straight line at all times, and you are not to look to either side, is this clear to you?"
"Yes, my Lord," Beauty answered timidly. She could see a great expanse of stone before her, and though the paddle smacks had not been very hard, she found she resented them enormously. They had not come from the Prince. And it was just coming to her that now she was in Lord Gregory's power. Perhaps she'd fancied he couldn't strike her, wouldn't be allowed to, but that was obviously not the case, and she realized he might tell the Prince she had disobeyed when she had not, and she might not be allowed to speak for herself.
"Move faster," he told her. "You are always to take a rapid pace showing your eagerness to please your Lords and Ladies," he said, and again there came one of those sharp degrading little spanks that seemed suddenly quite worse than harder ones.
They had come to a narrow doorway and Beauty perceived that a long curving ramp lay before her. It was quite clever as she could not have gone down a staircase on her hands and knees, but this she could follow, and she did with Lord Gregory's pointed leather boots right beside her.
Several times he availed himself of the paddle, so that by the time they reached the door of a vast room on the lower floor, her buttocks were burning a little.
But what concerned her much more was that there were people here.
She had seen no one in the passage above. And she felt torturously shy as she realized that there were many people in this hall moving about and talking to one another.
Now she was told to sit up and back on her heels, with her hands clasped to the back of her neck.
"This will always be your position when you are told to rest," Lord Gregory said, "and keep your eyes down."
Yet even as she obeyed this command, she could see what the room was. There were deep shelves cut into the walls all along three sides of it, and on these shelves, on pallets, slept the many slaves, both male and female.
But she could not see Prince Alexi.
She did see a beautiful black-haired girl with very plump little buttocks who appeared quite deep sleep, and a blond-haired young man who appeared to be strapped on his back, though she could not tell, and others, all of whom were in a drowsy state, if not dozing.
And before her were many tables in a row, and among them pots of steaming water from which came a delicious fragrance.
"This is where you will be bathed and groomed always," said Lord Gregory in that same cold voice, "and when the Prince has had quite enough of sleeping with you as though you were his love, you shall sleep here too, and at any time when the Prince has no specific orders for you. Your groom is named Leon. He will care for you in all details, and to him you shall show the same respect and obedience you show to everyone."
Beauty saw before her the slender figure of a young man, directly beside Lord Gregory. And as he drew nearer, Lord Gregory snapped his fingers and told her to show her respect.
At once Beauty kissed his boots.
"To the lowest scullery maid you owe this respect," Lord Gregory said, "and should I ever detect the slightest haughtiness in you, I snail punish you severely. I am not as ... shall we say, impressed with you as is your Prince."
"Yes, my Lord," Beauty answered respectfully, but she was angry. She felt she had shown no haughtiness.
But Leon's voice calmed her immediately. "Come, my dear," he said, gesturing with a pat of his hand against his thigh for her to follow him, and it seemed Lord Gregory disappeared as Leon led her into a brick-lined alcove where a large wooden tub stood steaming. The scent of the herbs was very strong.
Leon gestured for her to rise up again, and taking her hands, he lifted them over her head and told her to kneel in the tub.
She climbed into it at once and felt the delicious warm water come almost up to her sex. Leon wrapped her hair in a circle on the back of her head and fixed it with several pins. She could see him clearly now. He was older than the Page boys, but just as fair, and his hazel eyes were very appealing in their gentleness. He told her to keep her hands behind her neck and that he was going to give her a thorough cleansing and that she must enjoy it.
"Are you very tired?" he asked her.
"Not so tired, my ..."
"My Lord will do," he said with a smile. "Even the lowliest stable boy is your Lord, Beauty," he said, "and you must always answer respectfully."
"Yes, my Lord," she whispered.
He was already bathing her, and the warm water washing down her did feel very good to her. He lathered her neck and her arms.
"Have you just awakened?"
"Yes, my Lord," she said.
"I see, but you must be tired from your long journey. The first few days slaves are always overexcited. They don't feel their exhaustion, and then after that they begin to sleep for many hours. You'll feel it soon, and there will be an aching in your arms and legs, too. I don't mean from your punishment. I mean only from your fatigue. When that happens I'll massage you and soothe you."
His voice was so gentle that Beauty warmed to him at once. His sleeves were rolled to the elbows and there was golden hair on his arms, and his fingers were very sure as he washed her ears and her face, careful not to get the soap in her eyes.
"And you have been punished very severely, haven't you?"
Beauty blushed.
He laughed softly.
"Very good, my dear, you are learning already. Never answer such a question as that. It could be taken as a complaint if you did. Any time you are asked if you have been punished too much or suffered too much, or anything of that sort, be clever enough to blush."
But even as he spoke almost affectionately, he began washing her breasts just as calmly as he had washed the rest of her, and Beauty's blushes became more painful. She could feel her nipples harden, and she was certain though she could see nothing but the soapy water before her, that he was noticing this, as his hands slowed slightly, and then he pushed at her inner thigh gently. "Spread your legs, dearest," he said.
She obeyed, kneeling with her legs farther apart, and then farther as he pushed her. He had become still, and now drying his hand on the towel at his waist, he touched her sex and she felt herself shudder.
Her sex was moist and swollen with her desire, and to her horror, his hand touched a small hard knot in which much of her craving was accumulated. She drew back involuntarily.
"Ah." He withdrew his fingers, and turning called to Lord Gregory.
"A very lovely flower, this," he said. "Have you observed?"
Beauty was crimson. Her eyes overflowed with tears. It took all her control not to drop her hands to cover her sex as she felt Leon part her legs even wider now and gently touch the moisture there.
Lord Gregory gave a soft laugh.
"Yes, a truly remarkable Princess," he said. "I should have watched her more carefully."
Beauty gave a little muffled sob of shame and yet the driving desire between her legs would not stop, and her face was stinging as Lord Gregory spoke to her.
"Most of our little Princesses are too frightened in the first few days to show such willingness to serve, Beauty," he said in the same cold voice. "They must be awakened and educated. But I see you are very passionate and much enamored of your new masters and all they wish to teach you."
Beauty struggled against her tears. This was more humiliating surely than anything that had happened to her.
And now Lord Gregory was taking her chin as the Prince had taken Prince Alexi's chin and forcing her to look at him.
"Beauty, this is a great virtue in you. You have no cause to be ashamed. It only means that you must learn yet another form of discipline. You are awakened to the desires of your master as you should be, but you must learn to control that desire just as you see the male slaves control it."
"Yes, my Lord," Beauty whispered.
Leon withdrew and a moment later he returned with a small white tray on which were laid several little objects Beauty could not see.
But to her terror, Lord Gregory parted her legs and affixed to that little hard kernel of tormented flesh a plaster of sorts that covered it and adhered to it. He shaped it quickly with his fingers as if he did not wish to have Beauty enjoy this.
And Beauty was all the more relieved, for had she felt the ultimate pleasure, had she commenced to shudder and to blush with the final release from this torment, she would have been absolutely mortified.
But now the little plaster gave her an added torment. What could it mean?
It seemed Lord Gregory read her thoughts.
"That will prevent you from all too easily satisfying your newfound and undisciplined desire, Beauty. It will not alleviate it. It will merely prevent, shall we say, _accidental_ release, until you have gained the proper control of yourself. I had not thought to commence this detailed instruction so soon, but I shall tell you now that you are never allowed to experience full pleasure save at the whims of your master or mistress. Never, never, must you be caught touching your private parts with your own hands, nor trying more secretly to alleviate your obvious ... misery."
"Well-chosen words," Beauty thought, "for all his coldness to me."
But he was immediately gone, and once more Leon was bathing her.
"Don't be so frightened and so ashamed," he said. "You don't realize what a great advantage it is. To be taught to feel such pleasure is very difficult, and far more humiliating. And your passion gives a bloom to you that cannot be achieved otherwise."
Beauty cried softly. The little plaster between her legs made her all the more conscious of her feelings there. Yet Leon's hands and voice were soothing her.
Finally he told her she must lie down in the bath and he must wash her long beautiful hair for her. She let the warm water close over her and thought for a moment that she was covered by it and that felt extremely good to her.
As soon as she had been rinsed and dried, Beauty was put down on one of the beds nearby, and arranged on her face so that Leon could rub an aromatic oil into her skin.
It felt delicious to her.
"Now, surely," he said as he was massaging her shoulders, "there must be questions that you should like to ask me. You may do that if you like. It is not good for you to be confused about things unnecessarily. There is enough for you to fear without fears that are imaginary."
"I may ... talk to you then?" Beauty asked.
"Yes," he said. "I'm your groom. In a way, I belong to you. Each slave, no matter how he or she ranks or pleases or displeases, has a groom, and that groom is devoted to that slave, to that slave's needs and wishes, as well as preparing the slave for the master. Now, of course, there will be times when I shall have to punish you, not because I take pleasure in it, though I can't imagine punishing a more beautiful slave than you, but because your master may order it. He may order you punished for disobedience, or merely readied for him with some blows. But I will be doing it only because I have to ..."
"But do you ... do you take pleasure in it?" Beauty asked timidly.
"It is difficult to resist beauty such as yours," he said, rubbing the oil into the backs of her arms and into the crevices of her elbows. "But I should much rather groom you and care for you." He put down the oil and gave her hair another brisk rub with the towel, adjusting the pillow under her face.
It felt so good to be lying here, with his hands working on her.
"But as I was saying before, you may ask me questions when I give you leave. Remember, when I give you leave, and I have just given it."
"I don't know what to ask," she whispered. "There is so much to ask ..."
"Well, surely you must know already that all punishments here are for the pleasure of your masters and mistresses ..."
"Yes."
"And that nothing shall ever be done which truly harms you. You will never be burned, nor cut, nor _injured,"_ he said.
"Ah, that is a great relief," Beauty said, but in truth she had understood these limits without being told. "But the other slaves," she asked. "Are they here for various reasons?"
"Sent as Tributes mostly," Leon answered. "Our Queen is very powerful and commands many allies. And of course, all Tributes are well fed, well guarded, well treated just as you are well treated."
"And ... what happens to them?" Beauty asked tentatively. "I mean, they are all young and ..."
"They're returned to their Kingdoms when the Queen so wishes, and obviously very much better off for their service here. They're not so vain any longer, they have great self-control, and often a different view of the world, one which enables them to achieve great understanding."
Beauty could scarcely guess what this meant. Leon massaged the oil into her sore calves and the tender flesh behind her knees. She felt drowsy. The sensation was growing ever more delicious, and she resisted it slightly, unwilling to let that craving between her legs torment her. Leon's fingers were strong, almost a little too strong, and they moved to her thighs which the Prince had reddened with his strap as much as her calves and buttocks. She shifted slightly against the soft, firm bedding. Her thoughts slowly cleared.
"Then I might be sent home," she asked, but it had no meaning for her.
"Yes, but you must never mention it, and certainly never ask for it. You are the property of your Prince. You are his slave entirely."
"Yes ..." she whispered.
"And to beg to be released would be a terrible thing," Leon continued. "However in time you will be sent home. There are different agreements for different slaves. Do you see that Princess there?"
In a great hollow in the wall, on a shelf-like bed, lay a dark-haired girl whom Beauty had noticed. She had olive skin, richer in tone than that of Prince Alexi who was also dark, and her hair was so long it lay in rippling strands over her buttocks. She slept with her face to the room, her mouth slightly open on the flat pillow.
"Now, she is Princess Eugenia," said Leon, "and she must be returned in two years by agreement. Her time is almost up and she is broken-hearted. She wants to remain on the condition that her continued slavery will save two slaves from having to come here. Her Kingdom might agree to these terms to keep back two other Princesses."
"You mean she wants to stay?"
"O yes," Leon said. "She is mad for Lord William, the Queen's eldest cousin, and can't bear the thought of being sent home. But there are others who are ever rebellious."
"Who are they?" Beauty asked, but before he could answer, she added quickly, though trying to sound indifferent, "Is Prince Alexi one of those who is rebellious?"
She could feel Leon's hand moving towards her buttocks, and now suddenly all those welts and sore places were brought to life as his fingers touched them. The oil burned slightly as Leon added droplets of it generously, and then those strong fingers commenced to work the flesh, with no regard for its redness. Beauty winced, but even this pain had its pleasure. She felt her buttocks shaped by his hands, lifted, separated, and then smoothed again. She blushed to think that it was Leon doing this who had been talking to her in such a civilized manner, and when his voice went on, she felt a new variant of agitation. "There is no end to it," she thought, "the ways of being humiliated."
"Prince Alexi is the Queen's favorite," said Leon. "The Queen cannot bear to be separated from him for very long, and though he is a model of good behavior and devotion, he is, in his own way, relentlessly rebellious."
"But how can that be?" Beauty asked.
"Ah, you must put your mind on the pleasing of your Lords and Ladies," Leon said, "but I shall say this: Prince Alexi appears to have surrendered his will as a fine slave must, but there is a core in Prince Alexi that no one touches."
Beauty was enthralled by this answer. She thought of Prince Alexi on his hands and knees, his strong back and the curve of his buttocks as he had been driven back and forth across the Prince's bedroom; she thought of the beauty of his face. "A core in him no one touches," she mused.
But Leon had turned her over now, and when she saw him bending down, so close to her, she felt bashful and closed her eyes. He was rubbing the oil into her belly and into her legs, and she pressed her legs together and tried to turn to the side.
"You'll become very accustomed to my ministrations, Princess," he said. "You will think nothing of being groomed in time." And firmly he pressed her shoulders to the pallet. His swift fingers smoothed the oil into her throat and into her arms.
Beauty opened her eyes cautiously to see him intent on his work. His pale eyes moved over her without passion but with an obvious absorption.
"Do you ... derive pleasure from it?" she whispered, and was shocked to hear herself speak these words.
He emptied some oil into the palm of his left hand, and putting the bottle down beside him, he rubbed the oil into her breasts, lifting them and squeezing them as he had done her buttocks. She closed her eyes again, biting her lip. She felt him roughly massaging her nipples. She almost let out a little cry.
"Be still, my dear," he said matter-of-factly. "Your nipples are tender and must be slightly toughened. You've been subjected to very little sport so far from your love-stricken master."
Beauty was frightened by this. Her nipples felt painfully hard to her; she knew her face had colored darkly. It seemed all the feeling in her breasts swelled and pumped towards those tiny hard nipples.
Mercifully, Leon let go of her breasts with a hard squeeze. But then he parted her legs and rubbed the oil into her inner thighs, and this was even worse for her. She could feel her sex throbbing. She wondered if it gave off heat that he could feel with his hands.
She hoped he would be quick.
Yet even as she lay, red faced and trembling, he pushed her legs farther apart, and to her horror, parted the lips of her sex with his fingers as though inspecting her.
"O, please ..." she whispered, turning her head from side to side, her eyes stinging.
"Now Beauty," he scolded gently, "you must never never plead for anything from anyone, not even from your loyal and devoted groom. I must inspect you to see if you are sore, and as I thought, you are. Your Prince has been rather ... devoted."
Beauty bit her lip and closed her eyes as he widened the orifice and now oiled it. She felt as if she were being pulled apart, and even under the plaster that tiny knot of feeling throbbed above the opening Leon's fingers had broadened. "If he touches it, I shall die," she thought, but he was quite careful not to do that, though she felt his fingers entering her, and massaging the lips of her vagina.
"Poor darling slave," he whispered to her with feeling. "Now sit up. If I were to have my way, you would rest. But Lord Gregory wants you to see the rest of the Training Hall and the Hall of Punishments. Let me finish your hair quickly."
He began to brush Beauty's hair and arrange it in coils on the back of her head as she sat, still trembling, her knees drawn up, and her head bowed.
**THE TRAINING HALL**
**B** EAUTY WASN'T certain that she hated Lord Gregory. Perhaps there was something comforting in his air of command. What would it be like to be here without someone who directed her so completely? But he appeared obsessed with his duties.
As soon as he took her out of Leon's hands, he gave her two gratuitous blows with the paddle before ordering her to her knees and to follow him. She was to keep close to the heel of his right boot, and she was to observe all that was around her.
"But you must never look at the faces of your masters and mistresses, you must never try to meet their eyes, and there is not to be a sound out of you," he directed, "save your answers to me."
"Yes, Lord Gregory," she whispered. The stone floor beneath her was swept very clean and polished, but it hurt her knees nevertheless because it was stone. Yet she followed him at once past the other beds on which slaves were being groomed, and the baths in which two young men were being bathed, just as she had been bathed, their eyes flashing over her with mild curiosity as she risked a glance at each of them.
"All handsome," she mused.
But when a stunningly beautiful young woman was driven across her path, she felt a hot flush of jealousy. This was a girl with a mane of silvery hair much fuller and curlier than Beauty's, and as she was on her knees, her huge and magnificent breasts hung down showing their large pink nipples to great advantage. The Page who drove her with the paddle seemed very engaged with her, laughing at her little cries, and forcing her to move faster with the force of his blows as well as the mocking and cheerful commands he gave her.
Lord Gregory paused as if he, too, enjoyed the sight of this girl as she was brought up, and into the bath, her legs forced apart as Beauty's had been. Beauty could not help but notice her breasts again, and how large were the pink nipples. The girl's hips were ample for her size, and to Beauty's amazement, she was not really crying as she was lowered into the water. Her moans were more complaints as the paddle still smacked her.
Lord Gregory made some approving sound. "Lovely," he said so that Beauty could hear him. "And three months ago she was as wild and untamed as a nymph from the forest. The transformation is quite exquisite."
Lord Gregory turned sharply to his left and when Beauty did not at once realize it, he gave her a sound spank and then another.
"Now, Beauty," Lord Gregory said, as they passed through a doorway into a long room, "do you wonder how others are trained to show the passion you exhibit with such abandon?"
Beauty knew her cheeks were crimson. She could not bring herself to answer.
The room was dimly lit by a nearby fire, but its doors were open to the garden. And here Beauty saw that many captives were positioned on tables as she had been in the Great Hall, each with a Page in attendance. And all the Pages worked diligently taking no note of cries or commotion at any other table.
Several young men knelt with their hands strapped behind them. They were paddled steadily while at the same time their penises were also being given pleasure. Here a Page stroked the engorged penis as he worked the paddle. Here two Pages attended the same Prince mercilessly.
Beauty could understand what was happening even when Lord Gregory did not explain it to her. She saw the confusion and misery of the young Princes, their faces caught between struggle and surrender. The Prince nearest her was on all fours, his penis tormented slowly. As soon as the paddling commenced, he went soft. So the paddling ceased, and the hands attended to him again, hardening him.
Along the walls were other Princes, spread-eagled, their ankles and wrists bound to the bricks, their organs being taught obedience with touching and kisses and suckling.
ling. "O, it is worse for them, much worse," Beauty thought, but her eyes and her mind were too filled with their exquisite endowments. She looked at the rounded buttocks of those made to kneel; she loved their polished chests, the lean muscularity of their limbs, and above all, perhaps, the nobility of suffering in their handsome faces. She thought of Prince Alexi again and she wanted to shower him with kisses. She wanted to kiss his eyelids and the nipples of his chest; she wanted to suckle his organ.
Now she saw a young Prince brought over on his hands and knees to suck the penis of another. And as he performed the act with great enthusiasm, he in turn was paddled by the Page who seemed, as all others, to take delight in inflicting torment. The Prince's eyes were closed, he drew on the powerful sex of the other with long caresses of his lips, his own buttocks flinching with each blow, and as the poor Prince whom he suckled seemed on the edge of culminating passion, the suckler was pulled back by the Page who took his obedient slave to yet another erect penis.
"Here, as you can see, the young slave Princes are taught their manners," Lord Gregory said, "to be ever in readiness for their masters and mistresses. A hard lesson to learn and one which you are, in general, spared. It is not that readiness isn't required of you; it is that you are spared having to make such a display of it."
He led her on closer to the female slaves who were being worked in a different manner. Here Beauty saw a lovely red-haired Princess with her legs held apart by two Pages who with their hands massaged that little nodule between her legs. Her hips rose and fell; it was clear she could not control her own movement. She begged to be allowed peace, and just as her face flushed and it seemed she could not control herself, she was let alone, legs held apart so that she groaned miserably.
Another very lovely girl was being spanked and stroked at the same time by a Page who used his left hand between her legs to work her.
And to Beauty's horror several were mounted on phalluses against the wall on which they worked themselves with wild contortions while the Pages in attendance wielded merciless paddles.
"You see each slave receives simple instructions. She is to work herself on the phallus until she achieves satisfaction. Only then will the paddling cease, no matter how sore she is. She soon learns to think of the paddle and pleasure as one in the same, and she soon learns to achieve her pleasure in spite of the paddle. Or on command, I should say. Of course she shall seldom be allowed such satisfaction by her masters and mistresses."
Beauty gazed at the row of struggling bodies. The girls' hands were bound over their heads, their feet below. They had little room for moving on the leather phalluses. They twisted, trying to undulate as best they could, the inevitable tears spilling down their faces. Beauty felt pity for them, yet she so craved the phallus. She knew with deep shame it would not have taken her long to please the Page paddling her. As she watched the nearest Princess, a girl with red ringlets, she saw her finally achieve her goal, her face blood red, her whole body gone to violent quivering. The Page spanked her all the harder. She went limp finally as though too weary to feel shame, and the Page gave her a gentle approving pat and left her.
Everywhere Beauty looked she saw some form of training.
Here a young girl with hands clasped above her head was being taught to kneel still while her private parts were stroked and not to put her hands down to cover herself. Another was being forced to feed her breasts to the Page who suckled them, holding them for him while yet another examined her. Lessons in control, lessons in pain and pleasure.
The voices of the Pages were some of them stern, some of them tender, the dull whacking of the paddle everywhere. And there were the inevitable spread-eagled girls being now and then tormented to awaken them and teach them what they could feel if they did not know it.
"But for our little Beauty such lessons are not necessary," Lord Gregory said. "She is too accomplished as it is. And perhaps she should see the Hall of Punishments, how those disobedient slaves are chastised using the very pleasure they have learned to feel here."
**THE HALL OF PUNISHMENTS**
**A** T THE door of the new hall, Lord Gregory signaled one of the busy Pages.
"Bring Princess Lizetta here," he said raising his voice slightly. "Sit back on your heels, Beauty, with your hands behind your neck and observe all that's presented for your benefit."
The unfortunate Princess Lizetta was apparently just being brought in, and Beauty saw at once that she was gagged but rather simply so. A small cylinder covered with leather and shaped like a dog's bone was forced into her mouth and back so far between her teeth that it was rather like a bit, and apparently she could not have dislodged it with her tongue if she had wanted to.
She was crying angrily and kicking, as the Page who held her hands behind her back gestured for yet another Page to take her about the waist and carry her to Lord Gregory.
She was placed on her knees right before Beauty, her black hair falling down in front of her face, her dark breasts heaving.
"Petulance, my Lord," said the Page rather wearily. "She was to be quarry in the Hunt in the Maze when she refused to give her Lords and Ladies good sport. The usual nonsense."
Princess Lizetta tossed her black hair over her shoulder and let out a little contemptuous growl from behind the gag, which astonished Beauty.
"Ah, and impudence as well," said Lord Gregory. He reached down and lifted her chin. Her dark eyes evinced nothing but anger as she looked up at him and she turned her head so sharply that she was soon free of him.
The Page gave her several hard spanks but she showed no contrition. Her little buttocks looked hard in fact.
"Double her, for punishment," said Lord Gregory. "I think a real punishment is in order."
Princess Lizetta gave several high-pitched groans. They seemed both anger and protest. She seemed not to have bargained for this, and as she was carried ahead of Beauty and Lord Gregory into the Hall of Punishments, the Pages quickly affixed leather cuffs to her wrists and ankles, each cuff with a heavy metal hook imbedded in it.
Now she was raised, struggling, to a great low beam that spanned the room, her wrists hung from a hook above her head and then her legs brought straight up in front of her so that her ankles were fixed to the same hook. She was, in fact, bent double. Her head was then forced between her calves, so that Beauty could see her face clearly. And a leather strap was bound around her, securely pressing her upturned legs against her torso.
But the most cruel and frightening aspect of it for Beauty was the exposure of the Princess's secret parts, for she was hung so that anyone could see her full sex with its pink lips and its dark hair even to the tiny brown orifice between her buttocks. And all this just below her scarlet face. Beauty could imagine no worse exposure and she looked down timidly, glancing up again and again to the girl whose suspended body moved slightly as with a current in the air, the leather links at her wrists and ankles creaking.
But she was not alone. Beauty realized that only yards away, other doubled bodies hung from the same beam just as helplessly.
Princess Lizetta's face remained colored with rage, but she had quieted somewhat and now she turned and tried to conceal her expression against her leg, but the Page nearby adjusted her face forward.
Quickly Beauty looked at the others.
Not very far to the right a young man was mounted in the very same fashion. He appeared very young, no more than sixteen at best, and he was blond with curly hair, and his pubic hair was slightly reddish. His organ was erect, its tip glossy, and there exposed to all the world was his scrotum and again the tiny opening of his anus.
There were more of them, another young Princess and another Prince, but these first two engaged Beauty completely.
The blond Prince was moaning painfully. His eyes were dry, but he appeared to struggle, to shift as he hung from the black leather manacles, and he caused his body to turn a little to the left as he did so.
A young man, meantime, looking somewhat more impressive than the Pages, and differently costumed in dark blue velvet, came down the line of doubled and manacled slaves and appeared to inspect each face and each configuration of mercilessly exposed organs.
He smoothed back the hair from the young Prince's forehead. The young Prince moaned. It seemed he tried to thrust himself forward, and this man in blue velvet stroked the Prince's penis causing the Prince to moan all the louder and more with the sound of one imploring.
Beauty bent her head but she continued to watch the man in velvet as he approached the Princess Lizetta.
"Stubborn one, most difficult," he said to Lord Gregory.
"A day and night of punishment will subdue her," Lord Gregory answered. And Beauty was shocked to think of being exposed for so long, and so uncomfortably. She knew at once she would do anything to prevent this punishment, yet she had the terrible fear that despite all her efforts it might befall her. She at once imagined herself hung in this position, and she let out a little gasp, though she pressed her lips together to stop it.
But to her amazement, the man in velvet had begun to stroke Princess Lizetta's sex with a small instrument that was, as so much here, covered in smooth black leather. This was a three-pronged rod that somewhat resembled a hand, and as soon as he teased the helpless Princess, she began to struggle in her bonds.
Beauty understood at once what was happening. The Princess's pink sex, terrifying to Beauty as it hung so unprotected, appeared to swell, to ripen. Beauty could see tiny droplets of moisture appear on it.
And even as she watched, she felt her own sex ripening in this same manner. She felt the hard plaster that had been placed over the kernel of feeling there, and it seemed to do nothing to prevent the increased throbbing.
As soon as the helpless Princess had been so awakened, the man in velvet left her with an approving smile, and continued his movement back down the row of slaves, stopping again to tease and torment the young blond-haired Prince who without pride or dignity pleaded from behind his leather bone gag.
The victim beside him, another Princess, was even more abandoned in her wordless entreaties for satisfaction. Her sex was small, thick lipped, a mouth amid a thicket of brown curls, and she twisted her entire body struggling to gain some greater contact from the Lord in velvet who left her now to tease and torment yet another.
Lord Gregory snapped his fingers.
Beauty went down on her hands and knees again and followed him.
"Need I tell you that you are well-suited for that sort of punishment, Princess?" he asked.
"No, my Lord," Beauty whispered. She wondered if it was within his power to punish her this way for nothing. She longed for the Prince, and for the time when he alone had power over her. She could think of nothing but the Prince, and why had she ever displeased him by looking at Prince Alexi? Yet she had only to think of Prince Alexi and she was pitched into helpless misery. But if she could be in the Prince's arms, she would think of no one but him. She craved his tender punishment.
"Yes, my dear, you would speak?" Lord Gregory asked, but there was something ruthless in his tone.
"Only tell me how to obey, my Lord, how to please, how to avoid this discipline."
"To begin with, my precious," he said angrily, "stop admiring the male slaves so very much, staring at them at every opportunity. Don't revel so much in all I show you to frighten you!"
Beauty gasped.
"And never, never again think of Prince Alexi."
Beauty shook her head. "I will do as you say, my Lord," she said anxiously.
"And remember, the Queen is none too pleased with her son's passion for you. A thousand slaves have surrounded him ever since he was a young boy, and in none of them has he found an object of devotion such as you. The Queen does not like it."
"O, but what can I do?" Beauty cried softly.
"You can show perfect obedience to all your superiors, and do nothing to make yourself seem rebellious or unusual."
"Yes, my Lord," Beauty said.
"You know that I saw you watching Prince Alexi last night," he said, his voice now a menacing whisper.
Beauty winced. She bit her lip and tried not to cry.
"I could tell this to the Queen at any moment."
"Yes, my Lord," she gasped.
"But you are very young and lovely. And for such an offense as that you would face the most terrible punishment; you would be sent out of the castle to the village, and that would be more than you could bear ..."
Beauty trembled. "The village"—what could this mean? But Lord Gregory continued:
"And no slave of the Queen or the Crown Prince should ever be condemned to such disgraceful punishment, and no favorite slave ever has." He took a deep breath as if to cool his anger. "And when you are properly trained, you shall be a splendid slave. And there is no reason finally why the Prince should not enjoy you, why everyone here should not enjoy you. I am here, therefore, to make something of you, not to see you destroyed."
"You are most kind and gracious, my Lord," Beauty whispered, but the words, _the village,_ made their indelible impression. If only she might ask ...
But a young Lady had come into the room, passing through the door in a great rush, her long yellow hair in thick braids, her dress a rich burgundy color trimmed in ermine. Before Beauty remembered to look down, she caught a full glimpse of the Lady with her ruddy cheeks and large brown eyes which swept the Hall of Punishments now as if searching for someone.
"O, Lord Gregory, how nice to see you," she said, and as Lord Gregory bowed, she curtsied gracefully. Beauty was stunned by her loveliness, and then overcome with her own shame and vulnerability. She stared at the Lady's pretty silver slippers and the rings on the fingers of her right hand which gathered her skirts easily.
"And how may I serve you, Lady Juliana?" asked Lord Gregory. Beauty felt desolate. She was thankful the Lady never looked at her, and then again she felt abysmal. She was nothing to this woman who was dressed and a Lady and free to do all that she pleased, while Beauty was an abject naked slave who could do nothing but kneel before her.
"Ah, but there she is, that wicked Lizetta," said the Lady, and the cheerfulness went out of her face as her lips quivered slightly. There were two little dots of color in her cheeks as she drew near the doubled Princess. "And she has been so spoilt and bad today."
"Well, she is being severely punished for it, my Lady," said Lord Gregory. "Thirty-six hours here should greatly improve her disposition."
The Lady took several delicate steps forward and peered at Princess Lizetta's exposed sex. And to Beauty's amazement, Princess Lizetta did not try to hide her face but stared into the Lady's eyes imploringly. She gave several imploring groans as clearly supplicating as the earlier moans of the Prince beside her. And as she writhed on her hook, her body rocked slightly forward.
"You're a bad girl, you are," whispered the Lady as though reproving a small child. "And you disappointed me. I had prepared the Hunt for the amusement of the Queen and chosen you specially."
Princess Lizetta's groans grew more insistent. She seemed now without hope or pride or anger. Her face was knotted and pink, and her gag looked most painful, her huge eyes flashing as they implored the Lady.
"Lord Gregory," the Lady said, "you must think of something special." Then to Beauty's horror, the lady reached out delicately and fastidiously and pinched Princess Lizetta's pubic lips hard so that they exuded moisture. Then she pinched the right lip and the left, and the girl winced with pain and misery.
Lord Gregory had meantime snapped his fingers for the Lord with the iron clawlike hand, and whispered something Beauty could not hear. "It will strengthen her punishment."
And now the Lord appeared with a little pot and a brush and as the Lady stepped back, he took the brush and bathed Princess Lizetta's naked organ in a heavy syrup. A few droplets fell to the floor, and the Princess again made known her misery. She sobbed softly behind her gag, but the Lady only smiled rather innocently and shook her head. "It will attract any flies we have about," Lord Gregory said, "and if we have none it shall produce its inevitable itching as it dries. It is quite uncomfortable."
The Lady did not seem satisfied. Her pretty and innocent face was smooth however and she sighed. "I suppose it will do for now, but I wish she were bound with her legs apart to a stake in the garden. Then let the flies and the little insects of the air find her honeyed mouth. She deserves it."
She turned to express her thanks to Lord Gregory, and again Beauty was struck by her bright ruddy face. Her braids were done with tiny pearls and thin strands of blue ribbon.
But Beauty, almost lost in her contemplation of all this, was suddenly shocked to realize the Lady was looking at her.
"Oooooh, yes, it is the Prince's lovely one," she said, and now she advanced, and Beauty felt the Lady's hand lifting her face. "And how sweet she is, how truly beautiful."
Beauty shut her eyes, trying to restrain her heaving breaths. She did not believe she could endure the imperious touch of this young Lady. And yet there was nothing she could do.
"O, I should so like to have her take Princess Lizetta's place, it would be a treat for everyone," said the Lady.
"But that is impossible, my Lady," said Lord Gregory. "The Prince is most possessive of her. I cannot allow her to participate in such a spectacle."
"But surely we'll see more of her. Will she be run on the Bridle Path?"
"I feel certain, in time," said Lord Gregory. "There is no accounting for the whim of the Prince. But here, you may examine her if you wish. There is no rule prohibiting it."
He lifted Beauty by her wrists and forced her hips forward with the handle of the paddle. "Open your eyes and keep them down," he whispered. Beauty could not bear to see this lovely Lady's hands as they moved towards her. But Lady Juliana touched her breasts, and then her smooth stomach.
"Why she is radiant and so full of tender spirit."
Lord Gregory laughed softly. "Yes, she is, and you are so discerning to value it."
"They turn out all the better for that," said Lady Juliana with quiet wonder. She pinched Beauty's cheek as she had Princess Lizetta's secret lips. "O, what I would give for a quiet hour alone with her in my chambers."
"In time, in time," said Lord Gregory.
"Yes, and I bet she fights the paddle so, with her tender spirit."
"Only with her spirit," said Lord Gregory. "She is obedient."
"I can see that. Well, my girl, I must leave you. Be confident that you are exquisite. I wish I had you over my knee. I'd paddle you until sundown. You'd play a lot of little games running from me in the garden, you would." And then she kissed Beauty warmly on the mouth, and left as quickly as she had come, in a flurry of burgundy velvet and flying braids.
Just before Beauty took the sleeping potion from Leon she begged to know the meaning of what she had heard. "What is the Bridle Path?" she asked in a whisper, "and the village, my Lord, what does it mean to be sent there!"
"Never speak of the village," Leon cautioned her calmly. "That punishment is for incorrigibles and you are the slave of the Crown Prince himself. As for the Bridle Path, my lovely, you shall know soon enough."
He laid her down in her bed, strapping her ankles and wrists far away from her so that not even in sleep could she touch herself. "Dream," he said to her, "for tonight the Prince will want you."
**DUTIES** **IN THE PRINCES** **CHAMBER**
**T** HE PRINCE was finishing his supper when Beauty was brought to him. The castle hummed with life, torches flaring in the long, high, vaulted corridors. And the Prince sat in a library of sorts, eating alone at a narrow table. Several ministers were about with papers for him to sign, and there was the sound of their soft leather sock boots on the floor, and the crackling of the scrolls of parchment.
Beauty knelt by his chair, listening to the scratch of his pen, and when she was sure he would not see, she looked up at him.
He appeared radiant to her. He wore a blue velvet surcoat trimmed in silver and emblazoned with his coat of arms above a heavy silk girdle. The sides of the surcoat were loosely laced and through them Beauty could see his white shirt, and she admired as well the firm muscles of his legs in the long tight fustian breeches.
He took a few more bites of his meat as a plate was set down on the stones for Beauty. And quickly she lapped up the wine he poured in a bowl for her, and ate the meat as delicately as she could without using her fingers. It seemed he was watching her. He gave her bits of cheese and more fruit, and she heard him give some little sound of satisfaction. She cleaned her plate with her tongue.
She would have done anything to show him how pleased she was to be with him again, and quite suddenly she remembered she had not kissed his boots, and she made up for this immediately. The smell of the clean, polished leather was delicious to her. She felt his hand on the back of her neck, and when she looked up, he fed her a handful of grapes one by one, lifting each one a little higher so that she had to rise off her heels to get it.
He tossed the last grape in the air. She darted up to catch it in her mouth and succeeded. Then overcome with shyness she bowed her head. Was he pleased? After all she'd witnessed during the day, he seemed her savior. She could have wept for happiness now that she was with him.
Lord Gregory had wanted her to dine with the slaves. He had shown her the hall. There were two long rows of Princes and Princesses, all on their knees, hands clasped behind their backs, eating with their quick little mouths from plates on a low table before them. They were bent over so that, as she passed, she saw the row of sore buttocks and felt herself shocked by the sight of so many of them. They were all alike, and yet each body was different. The Princes showed less of themselves if their legs were together, as the scrotum couldn't be seen; but the girls could do nothing to hide their pubic lips. It had alarmed her.
But the Prince had wanted her immediately in his chamber. And now she was with him. Leon had removed the small sealing wax from her secret core of pleasure, and she felt the first stirring of desire. She did not care about the servants moving about, or the last minister waiting nearby with his petition. She kissed the Prince's boots again.
"It's very late," the Prince said. "You've had a long rest, and I see you are much improved for it."
Beauty waited.
"Look at me," he said.
And when she did, she was shocked by the beauty and ferocity of his black eyes. She felt her breath catch in her throat.
"Come," he said, rising and dismissing the minister. "Time for lessons."
He walked fast towards his bed chamber and she followed on her hands and knees, rushing before him as he waited for her to open the door, then going in behind him.
"If only she could sleep here, live here," she thought. And yet she was afraid as she saw him turn with his hands on his hips. She remembered the whipping last night with the strap and she shuddered.
Beside him was a high pedestal table, and he reached into a cloth-covered casket there, and took out what seemed a handful of little brass bells. "Come here, my spoilt dear," he said softly. "Tell me, have you ever attended a Prince in his chamber, dressed him, groomed him?" he asked.
"No, my Prince," Beauty said, and she hurried to his feet.
"Kneel up," he said. She obeyed, hands behind her neck and then she saw the little brass bells he held and that each was fixed to a little spring clamp.
Before she could protest, he applied one very carefully to the nipple of her right breast. It was not tight enough to hurt; nevertheless it bit down on her nipple, pinching her, and causing the nipple to harden. She watched as he clamped the other to her left breast, and then without meaning to, she took a deep breath that made the bells ring ever so faintly. They were heavy. They pulled on her. And she flushed, desperately wishing to shake them loose. They weighted her breasts, made her painfully conscious of them.
But he was telling her to stand up and spread her legs. And as she obeyed she saw another pair of brass bells taken from the casket. They were as large as walnuts. And, whimpering slightly, she felt his hands between her legs as he clamped these bells to her pubic lips quickly.
It seemed she felt parts of herself of which she had been unconscious. The bells touched her thighs. They tugged on the lips and cut into the flesh tightly.
"O, come now, it isn't so dreadful, my little maid," he whispered, and he rewarded her with a kiss.
"If it pleases you, my Prince ..." she stammered.
"Ah, that is lovely," he said. "And now to work, my beautiful one. And I want to see you work fast, yet gracefully. I want to see you do all things correctly, yet with some artfulness. In my closet on a hook you will see my red velvet scapular and gold girdle. Bring these things to me quickly and lay them out on the bed. You are going to dress me."
Beauty rushed to obey.
She had the clothing down from its hooks and hastened to bring it back, moving on her knees, the clothing in her arms. She laid it out on the foot of the bed, and turned waiting.
"Now undress me," said the Prince. "And you must learn to use your hands only when you cannot accomplish something otherwise."
Obediently Beauty took the leather lacings of his surcoat in her teeth, pulled loose the knot and saw them open. The Prince pulled the coat over his head and gave it to her. And now as he seated himself on a stool by the fire, she went to work unfastening his many buttons. It seemed she met with one obstacle after another. She was conscious of his body, its perfume and warmth, and his strange preoccupation. Soon she had the shirt off with his help, and then she must remove his long breeches.
Now and then he would aid her, but most tasks she performed herself, taking the upper lip of his velvet-lined boots carefully in her teeth as she pulled at the heels with her hands until they slipped off easily.
It seemed a long time that she labored, learning every detail of his wear. And now she must dress him.
She placed the white silk undershirt on him with both hands as he slipped his arms into it. And though she laid the placket of buttonholes in place with her hands, she drew each button through with her mouth so that he was very pleased and commended her.
She grew tired; her breasts ached from the heavy brass bells, and she felt the weight of the others between her legs, and that maddening stroking of her thighs and the jingling sound which never quite died away. But when she was finished, and he had just pulled on his new boots to help her, he gathered her in his arms and kissed her.
"As time passes, you will learn to work faster. It will be nothing for you to dress or undress me, to perform any small task I ask of you. I shall have you sleep in my chambers, and attend to everything."
"My Prince," she whispered, and she pressed her breasts against him, aching for him. She kissed his boots quickly, and all she had seen during the day came back to haunt and tantalize her: Princess Lizetta's cruel punishment, the Princes being trained, and then the one she had not seen, but never forgotten, Prince Alexi—all of this came together in her mind, stoking her passion and at the same time frightening her. O, if she could only sleep in the Prince's quarters now. Yet when she thought of all those male slaves she had seen in the Hall ...
But the Prince, as if he sensed her mind was not as attentive to him as it should have been, began kissing her roughly.
Then he ordered her to go down on her hands and knees with her forehead pressed to the floor so that he might see her buttocks turned to him. She obeyed, the cruel little bells reminding her of all the naked parts of her.
"My Prince," she whispered to herself. She felt some change in her heart which she did not fully understand. Yet she was afraid as always.
He ordered her to rise, and again he gathered her into his arms, and this time he said:
"Kiss me as you desire to kiss me."
And overjoyed she kissed the cold smoothness of his forehead, kissed the dark locks of his hair, his eyelids and his long eyelashes. She kissed his cheeks, and then his open mouth. And his tongue passed into her mouth and she weakened all over so that he had to support her.
"My Prince, my Prince," she murmured knowing that she disobeyed. "I am so afraid of all of it."
"But why, beautiful one? Isn't it clear to you now? Isn't it simple?"
"O, but how long will I serve? Will this be all of my life now?"
"Listen to me." He became grave but not angry. He held her by the shoulders, and then he looked at her swollen breasts. The little brass bells shivered as she breathed. She felt his hands between her legs, and then his fingers inside of her, stroking her in an upward motion that caused her to twist her body with the pleasure of it.
"This is all you are to think about, this is all you are to be," he said. "In some former life, you were many things, a lovely face, a lovely voice, an obedient daughter. You've shed that skin as if it were a cloak of dreams, and now you think of these portions of yourself only." He stroked her pubic lips, he widened her vagina. And then he squeezed her breasts almost cruelly. "This is you now, all of you. And your lovely face, only because it is the lovely face of a naked and helpless slave."
Then, as if he could not resist, he embraced her and carried her to the bed. "In a little while, I must take wine with the Court, and you will serve me there, demonstrating your obedience to everyone. But that can wait ..."
"O, yes, my Prince, if it pleases you," she breathed the words so low he might not have heard. She was lying on the jeweled coverlet, and though her buttocks and legs were not as raw as they had been the night before, she felt the painful prickling of the jewels.
The Prince knelt over her straddling her, and then opened her mouth with his fingers, and showing her his hard penis, drove it into her mouth with a quick downward motion. She sucked on it, drew on it. Yet all she need do was lie back helpless for he made the strong thrusts himself, into her, and she closed her eyes, smelling the delicious fragrance of his pubic hair, and tasting the saltiness of his skin, the penis nudging the back of her throat again and again as it all but bruised her lips.
She was moaning in time with its movements, and when suddenly he drew himself out, she gasped, her hands up to embrace him. But he had lain down on her full length, parted her legs, and pulled away the brass bells. Her pubic lips ached as he did so.
He drove into her. She felt herself explode with pleasure, her back arched so rigidly that she lifted his weight with her. Her body was drenched in pleasure. She thrust with her hips in almost a snapping motion, and when he came at last, he gave her cruel thrusts until he lay exhausted.
It seemed she slept; she dreamed. And then she heard him say to someone standing there:
"Take her away, wash her, adorn her. And send her to me in the upstairs parlor."
**SERVING MAID**
**B** EAUTY COULD not believe her bad luck when, entering the upstairs parlor, she saw the lovely Lady Juliana was playing chess with the Prince, and that other beautiful Ladies were seated about at various chessboards, and that there were several Lords as well, including an old man with white hair that flowed down over his shoulders.
Why did it have to be this Lady Juliana, so full of airy gestures and sunshine, her thick braids done tonight with crimson ribbon, her breasts beautifully molded by her velvet gown, and her laughter already filling the air as the Prince whispered to her some little witticism.
Beauty did not know what she felt. Was it jealousy? Was it merely the usual humiliation?
And Beauty had been adorned so cruelly by Leon, it was better to be naked.
First Leon had scrubbed away all the Prince's fluids, then he had braided only a thick lock of Beauty's hair on either side, pinning back these braids so that most of her hair still hung free. Then he had put little jeweled clamps on her nipples, but these were connected to each other by two strands of fine gold chain like a necklace.
The clamps hurt and the chains moved as the bells had with Beauty's every breath. But she had been quite horrified to discover this was not all.
Leon's quick, graceful fingers had probed her navel, then smoothed into it a paste in which he set a glittering brooch, a fine jewel surrounded by pearls. Beauty had gasped. She felt as if someone were pressing her there, trying to enter her, as if her navel had become a vagina. And the feeling continued. She could feel it now.
Then her ears must be hung with heavy jewels on tight gold clamps that stroked her neck when she moved, and her pubic lips of course could not be spared but must wear the same adornment. There were snake bracelets for her upper arms, and jeweled cuffs for her wrists, the effect to make her feel all the more exposed. Adorned and yet exposed. It was mystifying. About her neck finally a choker of jewels, and then on her left cheek a little jewel in paste like a beauty mark.
It caused her such annoyance. She wanted to wipe it away and could imagine it glittering. It seemed she could even see it out of the corner of her eye. But then she had been quite frightened when Leon tipped her head back, and put a delicate little gold ring on the side of her nostril. Its prongs pierced her though not deeply, only enough to hold it in place, but she almost cried because she wanted so to wipe it away like the jewel, indeed, to pull all these adornments loose, though Leon was complimenting her.
"Ah, when they give me something truly beautiful to work with, then I can show my skill," he sighed. He gave her hair a brisk brushing and then said she was ready.
Now she entered this vast shadowy parlor on her hands and knees and hurried to the Prince's side, kissing his boots immediately.
The Prince did not look up from his chessboard, and to Beauty's scalding shame, it was the Lady Juliana who greeted her:
"Ah, but if it isn't the darling one, and how lovely she looks. Kneel up, my precious," she said in that gay, carefree voice, tossing one of her braids back over her shoulder. She laid her hand on Beauty's throat, examining the jewel necklace. It seemed her fingers caused a tingling through Beauty's flesh, but she did not even try to steal a glance at the young woman's face.
"Why am I not sitting there as she is sitting, exquisitely dressed and free and proud," Beauty thought. "What has become of me, that I must kneel here before her and be handled as something less than human? I am a Princess!" And then she thought of all the other Princes and Princesses and felt foolish. "Do they think these thoughts?" This woman, more than any other, tormented her.
But Lady Juliana was not satisfied. "Stand up my dear so that I can have a look at you and don't make me tell you to put your hands behind your neck and spread your legs."
Beauty heard laughter from behind her and someone remarking to someone else that yes, the Prince's slave was well named. And realizing suddenly that there were no other slaves in this room, Beauty felt all the more bereft.
She shut her eyes as she had before when Lady Juliana had inspected her. And she felt the Lady's hands on her thighs and then pinching her buttocks. "O, why can she not leave me alone, doesn't she know what I suffer?" Beauty thought, and through her narrowed eyelids she looked down to see the Lady beaming at her.
"And what does her Highness think of her?" Lady Juliana asked with genuine curiosity, glancing at the Prince who was still deep in contemplation.
"She does not approve," the Prince murmured. "She accuses me of passion."
Beauty tried to remain composed, standing as she was in attendance. She heard laughter and conversation about her. She heard the rumbling of the old man's voice, and a woman say that the Prince's girl should serve the wine, should she not, so they might all see her?
"And haven't they seen me," Beauty thought. Could it be worse than the Great Hall, and what if she spilled the wine?
"Beauty, go to the sideboard and take the pitcher. Serve carefully and well, and come back to me," said the Prince, again without looking at her.
Beauty moved through the shadows to find the gold pitcher on the sideboard. She could smell the fruity aroma of the wine, and she turned, feeling awkward and graceless, and approached the first table. "A common serving girl, slave," she thought, more keenly than she had thought anything when she had been displayed.
With trembling hands she poured the wine slowly into goblet after goblet, and through her glazed vision saw smiles and heard whispered compliments. Now and then some haughty man or woman was quite indifferent to her. She was shocked once by a pinch on her rear and gasped to a general round of laughter.
As she bent over the tables, she felt the nakedness of her belly, saw the chains shimmering as they connected her pinched nipples. Each common gesture made her feel more hopeless.
She backed away from the last table, from a man who sat back with his elbow on the arm of his chair and smiled at her.
And then she filled Lady Juliana's goblet and saw those bright round eyes looking up at her.
"Lovely, lovely, O, I do wish you weren't so possessive of her," said Lady Juliana. "Put the pitcher down, my dear, and come here to me."
Beauty obeyed and returned to the Lady's chair. When she saw the Lady snap her fingers and point to the floor, Beauty blushed. She fell to her knees, and then in a strange impulsive moment, she kissed the Lady's slippers.
It seemed to happen very slowly. She found herself bending down towards the silver slippers and then she touched them with her lips fervently.
"Ah, what a darling," said the Lady Juliana. "Give me only an hour with her."
And Beauty felt the woman's hand on the back of her neck, caressing her, stroking her, and then gathering her hair back and smoothing it tenderly. Tears came to Beauty's eyes. "I am nothing," she thought. And there was that awareness again of some change in her, some quiet despair, except that her heart was racing.
"I would not even have her here," said the Prince under his breath, "save my mother commands it, that she be treated like any other slave, that she be enjoyed by others. Given my own will, I would chain her to my bedpost. I would beat her. I would watch every tear, every change of color."
Beauty felt her heart in her throat like a little fist knocking there faster and faster. "I would make her my wife, even ..."
"Ah, but you are in the grip of madness."
"Yes," said the Prince, "she has done that to me. Are others blind?"
"No, of course not," said Juliana, "she is lovely. But each seeks his own love, you know that. Would you have everyone else equally mad for her?"
"No," he shook his head. And without looking away from the chessboard, he reached out to caress Beauty's breasts, lifting them, squeezing them, so that she winced.
But suddenly everyone was rising.
Chairs slid back on the stones; the assemblage stood bowing.
Beauty turned.
The Queen had come into the room. Beauty glimpsed her long green gown, the girdle of gold embroidery about her hips and that sheer white veil that hung down her back to her hem, only thinly concealing her black hair.
Beauty went down low on her hands and knees not knowing what she must do. Her forehead touched the stones and she held her breath. Yet she could see the Queen approaching. The Queen stood right before her.
"Be seated everyone," said the Queen, "and return to your games. But you, my son, how do you fare with this new passion?"
The Prince was obviously at a loss for an answer.
"Pick her up, display her," said the Queen.
And Beauty realized she was being lifted by her wrists. She rose up quickly, her arms being twisted behind her, her back forced into a painful arc, and suddenly she was standing on her toes moaning. The clamps seemed to tear at her nipples, the jewels between her legs to pull her open. Behind the jewel in her navel, she felt her heart beat, and she felt it too in the lobes of her clamped ears and in her eyelids.
She was looking at the floor but all she could see was that shimmering chain and some great indistinct form that was the Queen standing over her.
Then suddenly the Queen's hand struck Beauty's breasts so hard that Beauty cried out, and at once felt the Page's fingers over her mouth tightly.
She moaned in panic. She felt her tears come, the Page's fingers biting into her cheek. And without meaning to, she struggled.
"There, there, Beauty," whispered the Prince. "You do not show my mother your best disposition."
Beauty tried to calm herself, but the Page forced her forward more harshly.
"She is not so bad," said the Queen, and Beauty could feel the iron in her voice, her cruelty. No matter what the Prince did to her, she did not sense in him such pure cruelty.
"She is only afraid of me," said the Queen. "And I wish you were more afraid of me, my son."
"Mother, be gentle with her, please, I beg you," said the Prince. "Allow me to keep her in my chambers, and to train her myself. Don't send her back to the Hall of Slaves tonight."
Beauty tried to smother her own crying. It seemed the Page's hand over her mouth only made it more difficult for her.
"My son, when she has proven her humility, we shall see," said the Queen. "Tomorrow night, the Bridle Path."
"O, but Mother, it is so soon."
"Such rigor will be good for her; it will make her malleable," said the Queen.
And turning with a broad gesture that loosened the train of her gown and made it fall behind her, the Queen left the parlor.
The Page released Beauty.
. And the Prince at once took her wrists in his hand and urged her out into the corridor, Lady Juliana coming beside him.
The Queen was gone, and the Prince moved Beauty angrily along ahead of him, Beauty's sobs echoing under the dark vaulted ceilings.
"O, dear, poor exquisite dear," said the Lady Juliana.
At last they reached the Prince's apartments, and to Beauty's misery, the Lady Juliana came in as if this were nothing to enter the Prince's chamber.
"Have they no propriety and restraint among themselves," Beauty thought, "or are they degraded with each other as we are degraded?"
But she soon realized it was only the Prince's study, and Pages were about. And the door remained open.
The Lady Juliana took Beauty now from the Prince, her soft cool hands urging Beauty down on her knees before her chair.
Then from somewhere in the folds of her gown, the Lady produced a long narrow silver-handled brush and she commenced to brush Beauty's hair lovingly. "This will soothe you, my poor precious one," she said. "Don't be so frightened."
Beauty broke into fresh sobs. She hated this lovely Lady. She wanted to destroy her. She felt such savage thoughts, and yet she wanted at the same moment to cling to her, to sob against her breast. She thought of friends she'd had at her father's Court, her Ladies in waiting, and how many times they had been easily affectionate with one another, and she wanted to abandon herself to the same affection. The brushing of her hair produced a tingling all through her scalp and through the flesh of her arms as well. And when the Lady's left hand covered her breasts and gently patted them, she felt herself defenseless. Her mouth went slack and she turned towards the Lady Juliana and laid her forehead against her knee, defeated.
"Poor, darling one," said the Lady. "But the Bridle Path is not so dreadful. You will be grateful afterwards that you were used rigorously in the beginning, for it will all the sooner soften you."
"Familiar sentiments," Beauty thought.
"Perhaps," the Lady Juliana went on with the rhythmic stroking of the brush, "I shall ride beside you."
What could this mean?
And then the Prince said:
"Take her back to the Hall now."
Without explanations, without farewells, without tenderness!
Beauty turned and rushed to him on her hands and knees and gave his boots fervent kisses. Again and again she kissed them both, hoping for what she did not know, one real embrace from him perhaps, something to allay her fears of the Bridle Path.
The Prince received her kisses for a long time, and then he lifted her and turned her to Lady Juliana who clasped Beauty's hands behind her back.
"Be obedient, beautiful one," she said.
"Yes, you ride beside her," said the Prince. "But you must make a good show of it."
"Of course, I should very much enjoy making a good show of it," said the Lady Juliana, "and it is best for you both. She is a slave, and all slaves desire a firm mistress and master. If they cannot be free, then they do not like for there to be ambivalence. I shall be most firm with her, but always loving."
"Take her back to the Hall," said the Prince. "My mother will not allow me to keep her here."
**THE BRIDLE PATH**
**A** S SOON as Beauty opened her eyes from sleep, she could feel a new excitement in the castle.
Torches everywhere brilliantly illuminated the Slaves' Hall, and all about her Princes and Princesses were receiving elaborate preparation. The hair of the Princesses was being combed and studded with flowers. The Princes were being polished with oil, their stiff curls combed just as carefully as those of the young women.
But Beauty was hastily taken from bed by Leon who seemed uncommonly excited.
"It's Festival Night, Beauty," he said, "and I've allowed you to sleep a long time. We must hurry."
"Festival Night," she whispered.
But she was already being placed on the table for grooming.
At once he parted her hair and started to braid it. She felt the air on her neck and hated it, and she realized he had started the braids very high on her head so she would look even more girlish than Lady Juliana. A long black leather thong was braided into the hair on both sides, and knotted around the ends with a little brass bell affixed to it. When Leon dropped the braids they were heavy against Beauty's breasts and her neck was exposed as well as all of her face.
"Charming, charming," Leon mused with his usual air of satisfaction. "But now your boots."
And slipping her into a pair of high black leather boots he told her to stand in them while he bent to lace them tightly to her knees and then smooth the leather around her ankles until it was cleaving like a glove there.
Not until Beauty lifted her foot did she realize each boot was fitted at toe and heel with a horseshoe. And the tops were hard and strong so that nothing could hurt her toes.
"But what is happening, what is the Bridle Path?" she asked in a great fluster.
"Shhhhh ..." Leon said, pinching and prodding her breasts to give them as he said, "some color."
He then glossed Beauty's eyelids and eyelashes with oil and smoothed a little rouge into her lips and into her nipples. Beauty drew back instinctively but his touch was sure and quick and he took no notice of her.
But what bothered her most was that her body felt cool and vulnerable. She could feel the sheathing of leather against her calves, and all the rest of her felt worse than naked. It was more terrible than any of the smaller adornments.
"What is going to take place?" she asked again, but Leon had thrust her over the end of the table and now oiled her buttocks vigorously. "Well healed," he said. "The Prince must have guessed last night you would run tonight and he spared you."
Beauty felt his strong fingers plying her flesh and a dread came over her. So they would spank her, but they always did. Only it would be in the presence of many others?
Every humiliating spank she had received before the eyes of others had cost her dearly, though she knew now she would suffer any amount of paddling for the Prince, but she had not really been given a hard, thorough spanking for the pleasure of others since the Inn on the road where the Innkeeper's daughter had spanked her for the soldiers and the common people at the windows.
"But it must come," she thought. And a vision of the Court watching it as some ritual caused her to feel an undeniable curiosity that soon enough gave way to panic. "My Lord, please tell me ..."
Amid the crowd about her, she saw other girls with braided hair and boots. So she was not alone. And there were Princes being fitted with boots also.
Through it all there moved a handful of young Princes on their hands and knees polishing boots as quickly as they could, their own buttocks raw, their necks encircled by a little cord of leather to which was attached a sign that Beauty could not read.
But now as Leon brought her up standing again and gave some finishing touches to her lips and eyelashes, one of these Princes was now buffing her boots though he was weeping. His buttocks were as red as it could have been. And she saw the sign about his neck said, "I am in Disgrace," in small letters.
A Page approached and gave the Prince a sound crack with a belt to hurry him on to another.
But Beauty had no time to think of it. Leon had affixed the accursed little brass bells to her nipples.
She shuddered almost instinctively but they were firmly attached, and he told her to fold her arms behind her back tightly.
"Now forward, only you are to bend your knees slightly and to march, lifting each knee high," he told her.
She started, awkwardly, reluctant to obey, but then she saw all about her other Princesses marching in an almost sprightly manner, their breasts bouncing gracefully as they moved into the corridor.
She hurried, the heavy boots difficult for her to lift with any decorum, but soon she had fallen into a rhythm and Leon was walking beside her.
"Now, darling," he said, "the first time is always difficult. Festival Night is frightening. I had thought some easier duty would be yours this first time, but the Queen has ordered you especially for the Bridle Path, and the Lady Juliana will drive you."
"Ah, but what ..."
"Shhhh, or I shall have to gag you and that will very much displease the Queen as well as make your mouth quite ugly."
All the girls were now in a long room, and through narrow windows on one wall, Beauty could see the garden.
Torches flared in the dark trees, throwing an uneven glare on the leafy boughs above them. The line of girls formed right beside these windows, and Beauty was now able to see more of what lay beyond them.
There was a great roar as of many people conversing, laughing. And then to her shock Beauty saw slaves all through the garden positioned in various ways for their torment.
On high stakes here and there were strapped Princes and Princesses painfully contorted, their ankles bound to the stakes, their shoulders bent over the tops of them. They seemed no more than ornaments, the torchlight causing their twisted limbs to glow, the hair of the Princesses falling free in the air behind them. Surely they could see only the sky above, though all could see their miserable contortions.
And everywhere beneath were the Lords and Ladies, the light falling upon a long embroidered cloak here, a pointed hat there with a veil trailing airily from it. There were hundreds in the garden, these tables placed far back into the trees, as far as Beauty could see in all directions.
Beautifully adorned slaves moved about, pitchers in hand, little gold chains fixed to their breasts, the Princes adorned with gold rings on their erect organs. They hurried to fill the goblets, pass the platters of food, and as in the great hall there was music.
The line of girls before Beauty grew restless. Beauty could hear one girl crying as her groom tried to comfort her, but most of the others were obedient. Here and there a groom rubbed more oil into plump buttocks or whispered in a Princess's ear, and Beauty's sense of apprehension deepened.
She did not want to look into the yard; it frightened her too much, but she could not prevent herself. And each time she saw some new horror. A great wall to the left, was adorned with spread-eagled slaves, and on a huge serving cart she saw slaves fixed to the giant wheels, turned upside down over and over as the cart was moved forward.
"But what will happen to us?" Beauty whispered. The girl in line before her who could not be quieted was now hanging by her ankle from the hand of a strong Page who punished her swiftly. Beauty gasped to see her spanked, her braids falling to the floor beneath her.
"Shh, it is best for her," Leon said, "it will exhaust her fear and drain her slightly. And she will be all the more free on the Bridle Path."
"But tell me ..."
"You must be still. You will see the others first and understand, and as we draw nearer to your turn I shall instruct you. Remember this is a special night of high festivity, but the Queen will be watching. And the Prince will be furious if you fail him."
Beauty's eyes returned to the garden. The great cart of steaming food had moved along, and for the first time she saw the distant fountain. Here too were bound slaves, their arms linked as they stood knee deep in the water, surrounding the central pillar, its sparkling flow pouring down over them. Their bodies glistened under the water.
The groom beside the girl in front of Beauty laughed softly and said that someone was miserable to be missing Festival Night but it was her own fault.
"Surely," Leon agreed when the groom glanced back to him. "They are talking about Princess Lizetta," he told Beauty, "who is still in the Hall of Punishments, and cursing no doubt to miss the excitement."
To miss the excitement! But in spite of her fear, Beauty nodded at this as if it were perfectly natural. A calm descended upon her in which she heard her own heart and felt her body as if there were limitless time in which to know it. She felt the sheathing of the leather boots, the click of her horseshoes on the stones, the air on her neck, her belly. And she thought, "Yes, this is what I am, so I should not wish to miss it either. Yet I rebel in my soul; why do I rebel?"
"O, I despise that miserable Lord Gerhardt, why must he drive me?" asked the girl before her in a low voice. The groom said something that made her laugh. "But he's so slow," she said, "savoring every moment. And I like to run!" The groom laughed at her. She went on, "and what do I get out of it?—the most miserable spanking. I could take the spanking if I could only cut loose and run ..."
"You want everything!" said the groom.
"And what do you want? Don't tell me you don't like it when I'm covered with welts and almost blistered!"
The groom laughed. He had a cheerful face, and was small of build, keeping his hands clasped behind his back, though his chestnut hair fell down over his eyes slightly.
"My dear, I love everything about you," he said. "And so does Lord Gerhardt. Now say something to comfort Leon's little pet, she's so frightened."
The girl turned and Beauty saw her pert face, eyes slanting at the ends somewhat like the eyes of the Queen, but they were smaller, with no cruelty. She smiled with full little red lips. "Don't be frightened, Beauty," she said, "but you have no need for comfort from me. You have the Prince. I have only Lord Gerhardt."
A great current of laughter passed through the garden. The musicians were playing loudly, with much strumming of their lutes and tapping of the tambourines, and then Beauty quite distinctly heard the thunder of hooves approaching. A rider shot past the windows, his cape flying out behind him, his horse bridled in silver and gold which made a streak of light as he rushed forward.
"O, at last, at last," said the girl in front of Beauty. Other riders were coming, and they were making a line all along the wall that almost blocked Beauty's view of the garden. She could not bear to look up at them, but she did and saw they were splendid Ladies and Lords, and each held the reins of the horse in his or her left hand, and in the right a long rectangular black paddle.
"Now, into the room," said Lord Gregory, and the slaves who had waited in a long line were ushered into the next chamber where they stood directly facing the arched door to the garden. Beauty could see now that a young Prince was first in line, and she saw that mounted Lord, his horse pawing the dirt before the archway.
Leon moved Beauty a little to the side. "Now you can see better," he said.
And she saw the Prince clasp his hands behind his neck and step forward.
A trumpet sounded, catching Beauty off guard so she gasped. And a cry rose from the crowd behind the archway. The young slave was forced out and at once greeted by the black leather paddle of the Lord on horse-back.
Immediately the slave commenced to run.
The mounted Lord rode right beside him, and the sound of the paddle came loud and distinct as the murmur of the crowd seemed to rise and mingle itself with faint ripples of laughter.
Beauty was aghast as she saw the two figures disappear down the path together. "I cannot do it, I cannot," she thought. "I cannot be made to run. I will fall. I will fall to the ground and cover myself. To be tied, to be bound in front of so many was dreadful enough, but this is impossible ..."
But another rider was already in place, and a young Princess was forced suddenly forward. The paddle found its mark, the Princess let out a little cry and was immediately running desperately fast along the Bridle Path, the rider after her, spanking her fiercely.
Before Beauty could take her eyes off them, another slave was on the way, and her eyes blurred as she saw far ahead a dim line of torches outlining the path that seemed to go on and on through the trees, past an endless vista of feasting Lords and Ladies.
"Now, Beauty, you see what is required, and don't cry. If you're crying it will be harder. You must put your mind on running fast, keeping your hands on your neck. Here, place them there now. And you must lift your knees high, and try not to squirm to escape the paddle. It will catch you no matter what you do, but I warn you, no matter how many times I tell you that, you will find yourself trying to run away from it. That is the trick, but remain graceful."
Another slave was running, and then another.
And the young girl who had cried earlier was upended again, dangling, as she was spanked.
"Dreadful of her," said the Princess in front of Beauty. "She'll be spanked hard enough in a moment."
Suddenly there were only three slaves before Beauty and the archway.
"O, but I can't ..." she cried to Leon.
"Nonsense, my darling, follow the path. It will unwind slowly before you, you will see its turns well in advance, and stop only if you see the slave before you stopped. Now and then the line is stopped, for as the slaves come before the Queen, they must stop for praise or condemnation. She is on a great pavilion to your right, but don't glance at her when you step out or the paddle will catch you off guard."
"O, please, I shall faint, I can't, I can't ..."
"Beauty, Beauty," said the pretty Princess in front of her, "just follow my example."
And Beauty realized with horror there was no one left but this girl.
But then that one who had just been spanked was placed before her, and ushered out to the waiting paddle. The girl was frantic, sobbing, but she kept her hands on her neck, and soon she was running beside her laughing rider, a tall young Lord who lifted his arm way back as he spanked her.
Suddenly another rider appeared, the elderly Lord Gerhardt, and as Beauty watched in terror, the pretty Princess ran out to receive the first blows and run with graceful lifts of her knees beside him. But for all her complaints, the Lord's horse seemed to move terribly fast and the paddle was loud and merciless.
Beauty was forced to the threshold of the garden. For the first time she stared at the true immensity of the Court, the dozens upon dozens of tables that sprawled out on the green and appeared in great numbers in the forest beyond it. Everywhere were servants and naked slaves. It was perhaps three times the size she had judged from the windows.
She felt tiny, insignificant, for all her terror. Lost and without a name or a soul suddenly. "What am I now," she might have thought, but she could not think. And as if in nightmare, she saw all the faces of those at the nearest tables, Lords and Ladies twisted to see the Bridle Path, and far to her right loomed the pavilion of the Queen, canopied and festooned with flowers.
She was gasping for breath, and when she looked up and saw the splendid mounted figure of Lady Juliana, her eyes filled with tears of gratitude that it was she, though she knew Lady Juliana would spank her perhaps all the harder to do her duty.
The lovely Lady's braids were done with the same silver that threaded her shapely gown. She seemed made for the sidesaddle in which she sat and the handle of her paddle was laced to her wrist. She was smiling.
There was no time to see more, to think more. Beauty was running forward, feeling the crunch of the Bridle Path under her horseshoes, hearing the stomp of hooves beside her.
And though she thought it was not possible to endure such degradation, she felt the first cracking blow to her naked buttocks. It was so forceful it almost knocked her off balance. The stinging pain spread out from it like a warm fire and Beauty realized she was rushing forward.
The stomp of hooves deafened her. And the paddle caught her again and again, almost lifting her and forcing her forward. She realized she was crying out loud through her clenched teeth, her tears making a blur of the torches that clearly defined the path before her. And she was running, running fast towards the enclosing trees, though she could not escape the paddle.
It was as Leon had warned her; it caught her over and over and each time there was some hideous surprise because she was trying to outrun it. She could smell the horse, and when she widened her eyes and gasped for breath, she saw everywhere on either side of her those torchlit and abundantly decked supper tables. Lords and Ladies drank, supped, laughed, turned to glance at her perhaps, she did not know, she was sobbing and running frantically from the blows, which came harder and harder.
"O, please, please, Lady Juliana," she wanted to cry out, but she did not dare to ask for mercy. The path had turned and she was following it only to see more and more Nobles banqueting and dimly before her the figure of the other rider and slave who had greatly outdistanced her.
Her throat was burning as much as her sore flesh.
"Faster, Beauty, faster, and lift your legs higher," Lady Juliana sang out over the wind. "Ah, yes, better, my darling." And there came another shock of pain, and another. The paddle found her thighs with a hard uplifting slap, and then seemed to scoop up her buttocks.
Beauty gave an open-mouthed cry because she could not stop it, and soon she heard her own wordless pleas as clearly as the horse's hooves pounding the cinders.
Her throat constricted, even the soles of her feet burned, but nothing hurt as much as the quick, strong paddling.
Lady Juliana seemed possessed of some evil genius, catching Beauty from one angle and then another, lifting her up again with the blows, smacking her hard and then three or four times in rapid succession.
The path had made another turn, and far ahead Beauty saw the walls of the castle. They were returning now. They would soon reach the Queen's canopied pavilion.
Beauty felt as if all the breath had gone out of her, yet mercifully Lady Juliana slowed her pace as did the riders before her. Beauty ran more slowly, knees high, and felt a great relaxation course through her. She could hear her own choking sobs, and feel the tears slipping down her face, and yet a puzzling sensation was passing over her.
She felt suddenly calmed in some way. She did not comprehend it. She felt no rebellion suddenly, though the obligation to rebel prodded her. Perhaps she was only exhausted. But she knew only that she was a naked slave of the Court and anything might be done to her. Hundreds of Lords and Ladies watched her with amusement. It was nothing to them, as she was only one of many, and it had all been done a thousand times, and would be done again, and she must do her best or take her place tethered to that beam in the Hall of Punishments suffering for the amusement of no one.
"Lift your knees, my precious darling," Lady Juliana said to her as they were moving slowly now. "And O, if you could only see how exquisite you are, you have done splendidly."
Beauty tossed her head. She felt the heavy braids fall against her back, and suddenly when the paddle struck her she felt herself move so languidly with it. It was as if this strange relaxation were softening her all over. Is that what they had meant when they said that the pain would soften her? Yet she feared this relaxation, this despair ... was it despair? She did not know. She had no dignity in this moment. She saw herself as Lady Juliana must surely have seen her, and it seemed she almost preened as she imagined this, tossing her head again, pushing her breasts out proudly.
"That's it, lovely, lovely," Lady Juliana called out. The other rider had disappeared.
The horse picked up its pace; the paddle struck Beauty violently again and drove her through the clustered tables as the crowd grew thicker, the castle coming closer, and suddenly they had stopped before the pavilion.
Lady Juliana turned her mount to the side, and with small prodding spanks brought Beauty beside her to attention.
Beauty did not look up but she could see the long garlands of flowers, the dim white vision of the canopy ballooning gently in the breeze, and a host of figures seated behind the festooned railing of the pavilion.
Her body seemed consumed with fire. She could not catch her breath, and then she could hear the conversation above, the Queen's pure icy voice and others laughing. Her throat was raw, her buttocks pulsing with pain, and now Lady Juliana whispered:
"She's pleased with you, Beauty, now kiss my boot quickly and drop down on your knees and kiss the grass before the pavilion. Do it with spirit, my girl."
Beauty obeyed without hesitation, and as if it were water washing through her, she felt again that calm, that sense of what was it? Release? Resignation?
"Nothing can save me," she thought. All the sounds about her mingled in a din. Her buttocks seemed to glow with pain, and she imagined a great light emanating from them.
And then she was back on her feet, and another hard blow sent her crying into the dark cellar chamber of the castle.
Slaves everywhere were thrown over barrels, their sore bodies being washed quickly with cool water. Beauty felt its flow over her abraded flesh, and then the soft toweling.
At once, Leon had her on her feet. "You've pleased the Queen marvelously. Your form was magnificent. You were born for the Bridle Path."
"But the Prince ..." Beauty whispered. And she felt dizzy, and mistakenly envisioned Prince Alexi.
"Not tonight for you, lovely one, he is quite busy with a thousand amusements. And you must be placed where you can serve and rest, as the exertion of the Bridle Path is quite enough in one night for a novice."
He unfastened her braids and brushed out her hair in ripples. She was breathing deeply and evenly now and bent her forehead against his chest.
"Was I truly graceful?"
"Pricelessly beautiful," he whispered, "and Lady Juliana is thoroughly in love with you."
But now he ordered her down on her knees and told her to follow him.
She was suddenly out in the night again, on the warm grass with the noisy crowd all about her. She saw the table legs, the gathered gowns, hands moving in the shadows. There was a shriek of laughtet nearby and then she saw before her a long banquet table covered with sweets, fruit and pastries. Two Princes attended it and decorative pillars stood at both ends to which slave girls were affixed, their hands above their heads, their feet chained slightly apart at the bottom.
One of these was removed as Beauty approached and she was quickly fastened in the girl's place, standing firmly, her head and swollen buttocks pressed back against the pillar.
She could see the whole feast around her, even with her lids lowered, and she felt herself quite firmly bound in place, unable to move, and it did not matter. The worst was over.
Even when a passing Lord stopped to smile at her and pinch her nipples, she did not care. She was amazed to see the little brass bells had been taken away. She was so weary she hadn't noticed.
Leon was still nearby, at her ear, and she was about to murmur some question as to how long she would be here, when quite distinctly in front of her she saw Prince Alexi.
He was as beautiful as she had remembered, his auburn brown hair curling against the hollows of his handsome face, his soft brown eyes fixed on her. His lips spread easily in a smile though he drew up to the table and gave his pitcher to be filled to one of those in attendance.
Beauty stared furtively out of the corner of her eye. She saw his thick hard sex and the lush hair around it. The vision of the Page, Felix, sucking it filled her with sudden passion.
She must have moaned or stirred because Prince Alexi, glancing up at the distant pavilion before he bent over the table to gather some sweet, suddenly kissed her on the ear, brushing Leon aside as if he were nothing.
"You behave yourself, you wicked Prince," said Leon, but it was not playful.
"I shall see you tomorrow night, my dearest," Prince Alexi whispered with a smile. "And don't be frightened of the Queen for I shall be with you."
Beauty's mouth quivered on the verge of a cry, but he was gone, and now Leon had drawn up to her ear again, cupping his hand as he whispered:
"You're to see the Queen tomorrow night for a few hours in her Quarters."
"O, no, no ..." Beauty wailed, tossing her head from side to side.
"Don't be foolish. This is very good. You could not wish for better," and as he spoke, he slipped his hand between her legs and pinched her lips gently.
She felt herself grow warm there.
"I was on the pavilion while you were running. The Queen was impressed in spite of herself," he went on, "and the Prince said you had always shown such form and spirit. And again, he pleaded for you, and that the Queen should not censure his passion. He agreed then not to see you tonight but to have a dozen or so new Princesses paraded before him ..."
"Don't tell me any more!" Beauty cried softly.
"No, but don't you see, the Queen was enthralled with you and he knew it. She watched you closely as you ran, impatient for you to come to the pavilion. And it was she who said perhaps she should taste your charms herself to see if you were not as spoilt and vain as she had supposed. She will have you in her Quarters tomorrow night after supper."
Beauty cried softly, too spiritless to answer.
"But, Beauty, this is a great privilege. There are slaves here who serve years without ever being noticed by the Queen. You shall have your full opportunity to enchant her. And you shall, my dear, you shall, you cannot fail to do so. And the Prince has been clever for once. He has not worn his heart for all to see it."
"But what will she do to me!" Beauty whimpered. "And Prince Alexi, will he see all of it? O, what will she do?"
"O, she shall only make a plaything of you, of course. And you shall try to please her."
**THE QUEEN'S CHAMBER**
**H** ALF THE night was gone before the Queen came.
Beauty had dozed, then awakened again and again, to find herself still chained in the ornate bedchamber as if in a nightmare. She was bound to the wall, her ankles cuffed in leather, her wrists up over her head, her buttocks pushed against the cold stone behind her.
At first the stone had felt good. Now and then she twisted to let the air touch the soreness. Of course the abraded flesh was much healed from last night's ordeal on the Bridle Path, but she still suffered, and she knew tonight she was surely destined for more torment.
Not the least of it, however, was her own passion. What had the Prince awakened in her that after one night of no satisfaction, she should feel so wanton? It was the stirring between her legs that first brought her out of sleep in the Slaves' Hall, and now and then she felt it as she stood waiting.
The room itself lay in shadow and unbroken stillness. Dozens of thick candles burned in their heavy gilded holders, the wax spilling in rivulets through the traceries of gold. The bed with its tapestried draperies appeared a gaping cavern.
Beauty closed her eyes. She opened them again. And when she was again on the verge of dream, she heard the heavy double doors thrown open and suddenly saw the tall, slender figure of the Queen materialized before her.
The Queen moved to the center of the carpet. Her blue velvet gown cleaved to her girdled hips before flaring gently to cover her black pointed slippers. She gazed at Beauty with narrow, black eyes tipped up at the ends to give her a cruel expression, and then she smiled, her white cheeks dimpling though an instant before they had seemed as hard as white porcelain.
Beauty had lowered her eyes at once. Petrified, she watched covertly as the Queen moved away from her and seated herself at an ornate dressing table, her back to a high mirror.
With an off-handed gesture she dismissed the Ladies who stood at the door. A figure remained there, and Beauty, afraid to look, was certain it was Prince Alexi.
So her tormentor had come, Beauty thought. Her heart pounded in her ears, becoming a roar rather than a pulse, and she felt the bonds holding her helpless so that she could not have defended herself against anyone or anything. Her breasts felt heavy, and the moisture between her legs greatly agitated her. Would the Queen discover it and use it to further punish her?
Yet mingled with her fear was some sense of her helplessness which had come over her the night before and never left her. She knew how she must appear, she was afraid, but she could do nothing and she was accepting it.
Maybe this was a new strength, this acceptance. And she needed all her strength, for she was alone with this woman who had no love for her. Without words, she evoked a memory of the Prince's love, of Lady Juliana's affectionate touch and warm words of praise, even of Leon's caressing hands.
But this was the Queen, the great powerful Queen who ruled all and who felt nothing but coldness and fascination for her.
She shivered against her will. The throbbing between her legs seemed to slacken and then to grow slightly more intense. Surely the Queen was staring at her. And the Queen could make her suffer. And there would be no Prince to witness it, no Court, no one.
Only Prince Alexi.
She saw him now, moving out of the shadows, a naked form exquisitely proportioned, the dark golden skin making him seem a polished statue.
"Wine," said the Queen. And he was moving to pour it for her.
He knelt at her side and he placed the two-handled cup in her hands, and as she drank, Beauty looked up and saw Prince Alexi smiling directly at her.
She was so startled, she almost made a little gasp. His large brown eyes were full of the same gentle affection he'd shown her last night when he passed her at the banquet table. Then he made his mouth into a silent kiss before Beauty looked away in consternation.
Could he feel affection for her, real affection, even desire, as she felt desire for him when she first saw him?
O, how she ached suddenly to touch him, to feel just once for an instant that silken skin, that hard chest, those dark, rose-colored nipples. How exquisite they were on that flat chest, those little nodules that seemed so unmasculine, giving him a touch of feminine vulnerability. How had the Queen punished them, she wondered? Were they ever clamped and adorned as her breasts had been?
They were piquant, those little nipples.
But the throbbing between her legs warned her, and it took an act of will for her not to move her hips.
"Undress me," the Queen said.
And from beneath her half-mast lids, Beauty watched as Prince Alexi obeyed the command skillfully and deftly.
How clumsy she had been two nights ago and how patient the Prince had been with her.
He used his hands but seldom. His first duty was with his teeth to unsnap the hooks of the Queen's dress and this he did, quickly gathering it as it fell down around her.
Beauty was astonished to see the Queen's full white breasts naked under a thin chemise of lace. And then Prince Alexi removed her ornate mantle of white silk to show the Queen's black hair hanging loose in ripples over her shoulders.
He took the garments away.
Then he came back to remove with his teeth the Queen's slippers. He kissed her naked feet before he took the shoes out of sight, and then he brought back to the Queen a sheer nightgown trimmed in white lace, the fabric a lustrous cream color. It was very full and pressed into a thousand pleats.
And as the Queen rose, Prince Alexi pulled down the chemise that she wore, and rising to his full height put the nightgown over the Queen's shoulders. She slipped her arms into the deep pleated bag sleeves, and the garment fell about her like a bell.
And then with his back to Beauty, Prince Alexi on his knees again tied a dozen little bows of white ribbon to close the front of the gown to its hem above the Queen's naked insteps.
As he bent over for the last of these, the Queen's hands played idly with his auburn hair, and Beauty found herself staring at his reddened buttocks where he had obviously been recently punished. His thighs, his tight, hard calves, all of this enflamed her.
"Pull back the curtains of the bed," the Queen said. "And bring her to me."
Beauty's pulse deafened her. It seemed there was a pressure in her ears, in her throat. Yet she heard the tapestries being drawn back. She saw the Queen recline on the coverlet amid a nest of silk pillows. The Queen looked younger now that her hair was free, and her face was without a trace of age as she stared at Beauty. Those eyes were as placid as if they had been painted in her face with enamel.
Then with a shock of unwelcome pleasure, Beauty saw Prince Alexi before her. He obliterated the vision of the menacing Queen. He bent to untie her ankles and she felt his fingers deliberately caress her. When he rose in front of her again, his hands up to free her wrists, she smelled the perfume of his hair and skin, and there seemed something utterly lush about him. For all his hardness, the squareness of his build, he seemed some great spicy delicacy to her, and she found herself staring right into his eyes. He smiled and let his lips touch her forehead. And they stayed secretly pressed to her forehead until her wrists were entirely free and he was holding them.
Then he pushed her gently down on her knees and gestured to the bed.
"No, simply bring her," said the Queen.
And Prince Alexi lifted Beauty and threw her over his shoulder as easily as a Page might have done, or the Prince himself when he took her from her father's castle.
His flesh felt hot beneath her, and thrown over his back as she was, she boldly kissed his sore buttocks.
Then she was laid down on the bed and realized she was beside the Queen, looking up into her eyes, as the Queen, who rested on her elbow, looked down at her.
Beauty's breath left her in rapid gasps. The Queen seemed quite enormous to her. And now she perceived a great resemblance to the Prince, only as always the Queen seemed infinitely colder. Yet there was about her red mouth something which might have once been called sweetness. She had thick eyelashes, a firm chin, and as she smiled dimples showed in her cheeks. Her face was heart shaped.
Flustered, Beauty closed her eyes, biting her lip so hard she might have cut it.
"Look at me," said the Queen. "I want to see your eyes, naturally. I want no modesty from you now, do you understand me?"
"Yes, your Highness," Beauty answered.
She wondered if the Queen might hear her heart beat. The bed was soft beneath her, the pillows soft, and she found herself staring at the Queen's great breasts, the dark circle of a nipple beneath the gown, before she looked at the Queen's eyes again obediently.
A shock passed through her, collecting in a knot in her belly.
The Queen merely studied her in great absorption. Her teeth showed perfectly white between her lips, and those eyes, slanted, long, were black to the core and revealed nothing.
"Sit there, Alexi," the Queen said without looking away.
And Beauty saw him take his position at the foot of the bed, with his arms folded on his chest, and his back to the bedpost.
"Little plaything," the Queen said under her breath to Beauty. "And now I understand perhaps why Lady Juliana is so enraptured over you."
She ran her hand over Beauty's face, her cheeks, her eyelids. She pinched Beauty's mouth. She smoothed back her hair, and then she slapped Beauty's breasts to the right and to the left and again.
Beauty's mouth quivered but she made no sound. She kept her hands still at her sides. The Queen was like a light that threatened to blind her.
If she thought about it, lying here so near the Queen, she would be overcome with panic.
The Queen's hand moved over her belly and her thighs. It pinched the flesh of her thighs and then the backs of her legs at the calves. And in spite of herself Beauty felt a tingling everywhere she was touched as if the hand itself had some dreadful power. She felt hatred for the Queen suddenly, more violently than she had felt it for Lady Juliana.
But then the Queen commenced to examine, slowly, Beauty's nipples. The fingers of the Queen's right hand turned each nipple this way and that, testing the soft circle of skin around it. Beauty's breath became uneven, and she felt the moisture between her legs as though a grape had been squeezed there.
It seemed the Queen was monstrously bigger than she, and as strong as a man, or was it only that to struggle against the Queen was unthinkable? Beauty tried to regain some calm, to think of her feeling of release on the Bridle Path, but it eluded her. It had been fragile all along. Now it was nothing.
"Look at me," the Queen commanded gently again, and Beauty realized as she looked up that she was crying.
"Spread your legs," the Queen ordered.
At once Beauty obeyed. "Now she will see," Beauty thought. "It will be as bad as when Lord Gregory saw. And Prince Alexi will see."
The Queen laughed. "I said spread your legs," she said, and gave Beauty's thighs fierce stinging slaps. Beauty spread her legs much wider and felt graceless as she did so. When her knees were pressed down to the coverlet on either side, she thought she could not endure the ignominy of it. She stared at the coffered ceiling of the bed above her and realized that the Queen was opening her sex as Leon had done. Beauty bit down on her cries. And Prince Alexi witnessed all of it. She remembered his kisses, his smiles. The lights of the room shimmered, and she felt her own shuddering as the Queen's fingers felt the moisture in this secret, exposed spot, playing with Beauty's pubic lips, smoothing the pubic hair, and finally catching a lock of it to pull and tease idly.
It seemed the Queen took both her thumbs and wrenched Beauty open. Beauty tried to keep her hips still. She wanted to rise to escape, like some miserable Princess in the Training Hall who could not endure being so examined. Yet she did not protest; her whimpers were faint and uncertain.
The Queen commanded her to turn over.
Blessed concealment, that she could hide her face in the pillows.
But those cool, commanding hands were playing with her buttocks now, opening them, touching her anus. "O, please," she thought desperately, and she knew that her shoulders shook with her silent crying. "O, this is dreadful, dreadful!"
With the Prince, finally, she had known what was wanted.
On the Bridle Path, finally, she had been told what was wanted. But what did this wicked Queen want of her, that she suffer, that she cringe, that she offer herself or merely endure? And the woman despised her!
The Queen massaged her flesh, prodding it, testing it as if for thickness, softness, resilience. She tested Beauty's thighs in the same manner, and then pushed Beauty's knees so far apart and high on the bed that Beauty's hips rose and she felt she was squatting, sprawled apart, over the coverlet, her sex protruding, hanging down, her buttocks surely split so that she resembled a ripe fruit.
The Queen's hand was under her sex as if weighing it, feeling the roundness and heaviness of the lips, pinching them.
"Arch your back," said the Queen, "and lift your buttocks, little cat, little cat in heat."
Beauty obeyed, her eyes flooded with tears of shame. She was trembling violently as she took a deep breath, and against her will felt the Queen's fingers commanding her passion, squeezing the flame so it burned hotter. Surely Beauty's pubic lips were swelling, their juices flowing, no matter how bitterly she struggled against it!
She did not want to _give_ anything to this wicked woman, this witch of a Queen. To the Prince she would yield; to Lord Gregory, to nameless and faceless Lords and Ladies who showered her with compliments, but to this woman who despised her ... !
But the Queen had sat back on the bed beside Beauty, and hastily she gathered up Beauty as if she were a floppy doll and threw her over her lap, her face away from Prince Alexi, her buttocks surely still exposed to his scrutiny.
Beauty gave an open-mouthed moan, her breasts rubbed against the coverlet, her sex throbbing against the Queen's thigh. It was as if she were some toy in the Queen's hands.
Yes, it was exactly like being a toy, only she was alive, she breathed, she suffered. She could imagine how she appeared to Prince Alexi.
The Queen lifted her hair. She ran a finger down Beauty's back to the tip of her spine.
"All the rituals," the Queen said in a low voice, "the Bridle Path, the stakes in the garden, the wheels, and then the Hunts in the Maze, and all the other clever games devised for my pleasure, but do I ever know a slave until I have this intimacy with the slave, the intimacy of the slave over my lap ready for punishment? Tell me, Alexi. Shall I spank her with my hand only to sustain this intimacy? Feel her stinging flesh, its warmth, as I watch it change color? Shall I use the silver-back mirror, or one of a dozen paddles that are all excellent for the purpose? What do you prefer, Alexi, when you are over my lap? What is it you hope for even as you are crying?"
"You may hurt your hand if you spank her that way," came Prince Alexi's calm answer. "May I get you the silver mirror?"
"Ah, but you do not answer my question," the Queen said. "And do get me the mirror. I shall not spank her with it. Rather I shall see her face with it as I spank her."
In a blur, Beauty saw Prince Alexi move to the dressing table. And then before her, propped against a silk pillow, was the mirror, tilted so she could see the Queen's smooth white face in it distinctly. The dark eyes terrified her. The Queen's smile terrified her.
"But I shall show her nothing," Beauty thought desperately, shutting her eyes, the tears squeezed out down her cheeks.
"Surely, there is something superior about the open hand," the Queen was saying, her left hand on Beauty's neck, massaging it. She slipped it down under Beauty's breasts, and pushing them closer to one another, touched both nipples with her long fingers. "Have I not spanked you with my hand as hard as any man, Alexi?"
"To be sure, your Highness," he answered softly. He was behind Beauty again. Perhaps he had taken his place against the bedpost.
"Now clasp your hands in the small of your back and keep them there," said the Queen. And she closed her hand over Beauty's buttocks just as she had closed her other hand over Beauty's breasts. "And acknowledge my commands to you, Princess."
"Yes, your Highness," Beauty struggled to respond, but to her further shame her voice broke into sobs and she shivered trying to restrain them.
"And be quieter than that," said the Queen sharply.
The Queen commenced to spank her. One great hard slap after another fell on her buttocks, and if a paddle had ever been worse she could not remember it. She tried to be still, to be quiet, to show nothing, nothing, as she repeated that word over and over in her mind, but she could feel herself writhing.
It was as Leon had said with the Bridle Path; you always struggle as if you could escape the paddle, squirm away from it. And she heard herself crying out suddenly in gasps as the slaps stung her. The Queen's hand seemed immense and hard and heavier than the paddle. It shaped itself to her as it spanked her, and she realized she was frantic, full of tears, and cries, and all of this for the Queen to see in her cursed mirror. Yet she could not stop it.
And the Queen's other hand pinched her breasts, stretched her nipples one at a time, letting them go, and stretching them again, as the spanks went on and on until Beauty was sobbing.
Anything would have been better. Rushing through the hall at the end of Lord Gregory's paddle, the Bridle Path, even the Bridle Path, was better for there was some escape in the movement, and here there was nothing but the pain, her enflamed buttocks laid bare for the Queen who now sought out new spots, spanking on the left buttock and then the right, and then covering Beauty's thighs with smacks while Beauty's buttocks seemed to swell and throb unbearably.
"The Queen must tire. The Queen must stop," Beauty thought, but she had thought this only moments before and it went on, so that Beauty's hips were rising and falling, and she found herself squirming to the side only to be rewarded with sounder blows, more rapid blows, as if the Queen were growing ever more violent. It was as when the Prince had beaten her with the strap. It was becoming more frenzied.
Now the Queen worked on the very bottom of her buttocks, that portion which Lady Juliana had so deliberately lifted on her paddle, and she spanked hard and long on either side before moving up again and to the side, and then to Beauty's thighs and back again.
Beauty clenched her teeth to stifle her cries. She opened her eyes in frantic silent pleas seeing only the Queen's hard profile in the mirror. The Queen's eyes were narrowed, her mouth twisted, and then suddenly she gazed through the mirror at Beauty though she never ceased punishing her.
Beauty's hands broke their firm clasp and struggled to cover her buttocks, but the Queen at once moved them aside.
"You dare!" she whispered, and Beauty clasped them tight again, sobbing into the coverlet as the spanking continued.
Then the Queen's hand lay on the burning flesh without motion.
It seemed the fingers were still cold, yet they burned. And Beauty could not control her racing breath or her tears, and she would not open her eyes again.
"You shall tender me your apology for that little slip of decorum," said the Queen.
"I... I ... " Beauty stammered.
" 'I am sorry, my Queen.' "
"I am sorry, my Queen." Beauty whispered frantically.
" 'I deserve only your punishment for it, my Queen.' "
"I deserve only your punishment for it, my Queen."
"Yes," the Queen whispered. "And you shall have it. But all and all ..." The Queen sighed. "Was she not good, Prince Alexi?"
"Very well behaved, your Highness, I should think, but I await your judgment."
The Queen laughed.
She pulled Beauty up roughly.
"Turn around and sit in my lap," she said.
Beauty was astonished. She at once obeyed and realized she was facing Prince Alexi. But he did not matter to her in these moments. Shaken, sore, she sat shivering on the Queen's thighs, the silk of the Queen's gown cool under her burning buttocks, the Queen's left arm cradling her.
The Queen's right hand examined her nipples, and Beauty looked down through her tears to see those white fingers again pulling the nipples.
"I had not thought to find you so obedient," said the Queen, pressing Beauty to her ample breasts, Beauty's hip against the Queen's smooth stomach. Beauty felt tiny as well as helpless, as if she were nothing in this woman's arms, nothing but something small, a child perhaps, no, not even a child.
The Queen's voice grew caressing.
"You are sweet, sweet as Lady Juliana told me you were," she said softly in Beauty's ear.
Beauty bit her lip.
"Your Highness ..." she whispered, but she did not know what to say.
"My son has trained you well, and you show great perception."
The Queen's hand plunged down between Beauty's legs and felt the sex which had never grown cold or dry during all of the worst of the spanking, and Beauty shut her eyes.
"Ah, now why are you so afraid of my hand when it touches you gently?"
And the Queen bent and kissed Beauty's tears, tasting them on Beauty's cheeks and on her eyelids. "Sugar and salt," she said.
Beauty broke into a fresh shower of sobs. The hand between her legs massaged the most moist portion of her, and she knew that her face was flushed, and the pain and the pleasure mingled. She felt overpowered.
Her head fell back against the Queen's shoulder, and her mouth went slack, and she realized the Queen was kissing her throat, and she murmured some strange words that were not words to the Queen, some plea.
"Poor little slave," said the Queen, "poor little obedient slave. I wanted to send you home to get rid of you, to rid my son of his passion for you, my son who is now as enchanted as you were before, under the spell of the one whom he released from the spell, as if all life were a series of enchantments. But you are as perfect in temperament as he said you were, as perfect as more trained slaves, and yet you are fresher, sweeter."
Beauty gasped as the pleasure between her legs washed through her, mounting and mounting. She felt her swollen breasts might burst, and her buttocks, as always, throbbed so that she felt every inch of the abraded flesh relentlessly.
"Now, come, did I spank you so very hard, tell me?"
She took Beauty by the chin and turned her so that Beauty looked into her eyes. They were huge and black and fathomless. The lashes curled upwards, and there seemed a great casing of glass over the eyes, so deep they were, so brilliant.
"Well, answer me," said the Queen with her red lips, and she placed her finger in Beauty's mouth and tugged on her lower lip. "Answer me."
"It was ... hard ... hard, my Queen ..." Beauty said meekly.
"Well, yes, perhaps for such fresh little buttocks. But you make Prince Alexi smile with your innocence."
Beauty turned as if bidden to do so but when she gazed at Prince Alexi she did not see him smiling. Rather he was merely looking at her with the strangest expression. It was both remote and loving. And then he looked to the Queen without haste or fear and let his lips lengthen in a smile as she seemed to wish of him.
But the Queen had tipped back Beauty's head again. She kissed Beauty. The Queen's rippling hair fell down around her, full of perfume, and for the first time, Beauty felt the velvety white skin of the Queen's face, and she realized the Queen's breasts were pressed against her.
Beauty's hips moved forward, she started to gasp, but just before it became too much for her, this shock penetrating to her wet, throbbing sex, the Queen suddenly pushed her down and drew back smiling.
She held Beauty's thighs. Beauty's legs were open. And the hungry little sex wanted for all the world for the legs to be crushed closed against it.
The pleasure subsided slightly, back into that great never ending rhythm of craving.
Beauty moaned, her brows knit in a frown, and the Queen suddenly pushed her off, slapping Beauty's face so hard that Beauty cried out before she could stop herself.
"My Queen, she is so young and tender," said Prince Alexi.
"Don't try my patience," the Queen answered.
Beauty lay facedown on the bed crying.
"Rather ring for Felix and have him bring Lady Juliana. I know how young and tender is my little slave, and how much she has to learn, and that she must be punished for her small disobedience. But that is not what concerns me. I should see more of her, more of her spirit, her efforts to please, and ... well, I have promised Lady Juliana."
It did not make any difference how hard Beauty cried, they would proceed, and Prince Alexi could not stop them. Beauty heard Felix come, she heard the Queen walking about the room, and finally when Beauty's tears were now a steady silent flow, the Queen said, "Get down from the bed, and prepare yourself to greet Lady Juliana."
**LADY JULIANA IN THE QUEEN'S CHAMBER**
**L** ADY JULIANA came into the room exactly as she had come into the Hall of Punishments, her steps light and springing, her round face full of prettiness and animation. She wore a rose pink gown, and there were pink roses threaded through her long thick braids with pink ribbon.
She seemed too full of light and gaiety for the vast shadowy chamber with the torches throwing huge uneven shadows on the high arched ceiling. The Queen sat in the corner on a great chair that resembled a throne, her foot on a plump green velvet cushion. Her arms rested on the chair, and she smiled faintly when Lady Juliana bowed to her. Prince Alexi, sitting on his heels at the Queen's feet, very politely kissed the pretty Lady's slippers.
Beauty knelt in the center of the flowered carpet, still much shaken and tear-stained, and as soon as Lady Juliana approached her she kissed her slippers as Alexi had done, only perhaps a little more fervently.
Beauty was surprised at her response to Lady Juliana. She had been appalled to hear her name, and yet she almost welcomed her. She felt some connection with her. Lady Juliana had, after all, showered Beauty with affectionate attention. She felt almost as if Lady Juliana were on her side, though she had little doubt that she would now be punished by her. Lady Juliana's paddle had been too diligent on the Bridle Path for Beauty to have any doubt of that. Yet she felt almost as if this were a girlhood friend of great confidence and strength, coming to embrace her.
Lady Juliana was beaming at her.
"Ah, Beauty, sweet Beauty, is the Queen pleased?" And as she stroked Beauty's hair and pushed her back to sit on her heels, Lady Juliana glanced at the Queen politely.
"She is all that you said she would be," answered the Queen. "But I wish to see more of her to judge properly. Use your imagination, lovely one. Do as you please, for me."
At once Lady Juliana motioned to the Page. He opened the door to admit yet another young man who carried a great flower basket filled with pink roses.
Lady Juliana took the basket over her arm, and the two Pages retired to the shadows. They stood as still as guards, and Beauty wondered that their presence meant so little to her. For all she cared, there might have been a row of them there. It did not matter.
"Look up, precious, with those beautiful blue eyes of yours," said Lady Juliana, "and see what I have prepared to amuse the Queen, and further demonstrate your lovliness." She lifted a rose which had a rather short stem, no more than eight inches. "No thorns, my pet, and this I show you so you fear only what you should fear, and not carelessness or blunders."
Beauty could see the basket was heaped with such carefully prepared flowers.
The Queen gave a cheerful laugh and shifted in her chair. "Wine, Alexi," she said, "sweet wine, this room is rather permeated with sweetness."
Lady Juliana burst into soft laughter as though this were a wonderful compliment, and she danced about the room, twirling her rose-colored skirts, her braids swinging.
Beauty watched her in wonder, her vision still unclear from her crying, and the woman seemed, like the Queen, immense and powerful. She turned her smiling face on Beauty like a light. And the glare of the torches flashed in the deep red brooch she wore at her throat, and in the jewels sewn skillfully into her heavy girdle. Her pink satin slippers had silver heels and she danced up to Beauty and kissed the top of her head lovingly.
"But you look so forlorn and that is not good. Now kneel up, fold your arms behind your back to show your exquisite breasts, that's it, and arch your back more becomingly. Her hair, Felix, brush it."
And as the Page hastened to obey, gently untangling Beauty's long locks down her back, Beauty saw the Lady Juliana take from a chest nearby a long oval paddle.
It was very like the paddle used on the Bridle Path, but nothing as big or as heavy. In fact it was so flexible that Lady Juliana, setting down her basket of flowers, could make it vibrate when she pressed the tip of it with her thumb. It was white, smooth, and limber.
It will sting, Beauty realized, but it will not truly hurt as badly as the Queen's hand, and it will hurt nothing as badly as that weapon on the Bridle Path, yet she realized her buttocks were so thoroughly welted that each light blow would enkindle a certain amount of pain in her.
Lady Juliana, laughing, whispering with the Queen in her girlish manner, turned back as Felix finished. Beauty knelt waiting.
"And so our gracious Sovereign spanked you over her lap, did she? And you have had the Bridle Path, and you have learned something of grooming. And then there has been your Lord and Master's temper and demands, and now and then a little routine smacking from your groom or Lord Gregory."
"My groom has never smacked me," Beauty thought crossly, but she merely replied, "Yes, my Lady ..." as was expected.
"But now you shall learn some actual discipline, for in the little game I devise your will to please is direly tested. And don't think you shan't profit from it. Now ..." She lifted a handful of roses from the basket. "I shall scatter these about the room, and do you know what you shall do, my precious girl, you shall run very fast to gather each one in your teeth and place in the lap of your Sovereign. And when she has quite finished with you, you shall go to fetch another, and another, and another. And you shall do that as fast as you can, and do you know why, because you are commanded to do so, and you shall be much punished if you do not rush to obey as we command you."
She raised her eyebrows, smiling at Beauty.
"Yes, my Lady," Beauty replied, unable to think, though the thought of having to hurry to obey struck a strange new note of apprehension in her. Gracelessness. She dreaded it. On the Bridle Path there had been much gracelessness when she was running so fast and out of breath.... O, but she must not think of anything save what she had to do now.
"And on your hands and knees of course, my girl, and be very very quick!"
Lady Juliana at once scattered the little pink rosebuds with their waxed stems everywhere.
Beauty bent forward and was grasping the nearest in her teeth when she realized that Lady Juliana was right behind her. The handle of the oval paddle was so long, Lady Juliana did not even bend over as she spanked Beauty and with a start, Beauty dropped the flower.
"Pick it up at once!" Lady Juliana cried out, and Beauty's lips scraped the carpet before she had it.
The paddle came down with a fearful zinging sound, smacking her sore welts as she rushed on her hands and knees to the Queen, and Lady Juliana managed some seven or eight good blows before Beauty had dropped the flower in the Queen's lap obediently.
"Now turn around at once," commanded the Lady, "and off you go." But she was already spanking Beauty fiercely as Beauty ran in search of another flower. As soon as she had it in her lips, she ran to the Queen, but the blows followed her. And Beauty wanted to cry out for patience as she went for another.
She gathered a fourth, a fifth, a sixth, depositing each in the Queen's lap, yet there was no escape from the paddle, from its persistence, nor Lady Juliana's voice urging her on crossly.
"Quickly, my girl, quickly, get it in your lips and back again." It seemed her flaring pink skirt was everywhere in Beauty's eyes, and Beauty was surrounded by the flashing of her little silver-heeled slippers. Beauty's knees burned from the rough wool of the carpet, yet she went breathlessly on in her search seeing the tiny pink roses everywhere.
And yet no matter how she gasped for breath, no matter how moist were her face and her limbs, she could not blot from her mind the thought of what she was doing. She could see her own buttocks splotched with white welts, her reddened thighs, and her breasts dangling between her arms as she rushed across the floor like a pitiable animal. There was no mercy for her, and the worst of it was that she could not please Lady Juliana, Lady Juliana goaded her, even kicked her now with the toe of her slipper. Beauty's cries were wordless pleas, but Lady Juliana's tone was angry, unsatisfied.
It was dreadful to be struck in anger.
"Hurry! Do you hear me!" Lady Juliana sounded almost contemptuous, spanking Beauty all the harder, and making little tisks now of impatience. Beauty's nipples scraped the carpet as she bent to obey, and with a shock she felt the toe of Lady Juliana's slipper in her pubis. She gave a startled cry and rushed back to the Queen with the rose as all about her it seemed was the muted laughter of the Pages and the Queen's higher laughter. But Lady Juliana had found the tender spot again, forcing that long pointed satin slipper right into Beauty's vagina.
Suddenly as Beauty turned and saw yet more roses scattered before her, her sobs went into muffled shrieks and she turned to Lady Juliana even as the paddle spanked her thighs and her calves, and kissed and kissed those pink satin slippers.
"What?" Lady Juliana said with genuine outrage. "You dare beg me for mercy before the Queen? Wretched, wretched girl!" She smacked Beauty's buttocks, but she had Beauty by the hair with her left hand and pulled her up, snapping her head back so that Beauty's knees went wide apart to keep her balance.
Beauty's open-mouthed sobs were choked and uneven. And she saw the paddle being passed to one of the Pages who offered the Lady a heavy broad leather belt immediately.
The belt struck Beauty's buttocks with a resounding wallop. Again it struck her. "Take another rose, another, two, three, four in your mouth at once and give them to your Queen immediately!"
Beauty ran to obey, and it seemed for a moment all perception left her. She was frantic to obey, to outdistance Lady Juliana's anger. It was hotter, more frenzied than the Bridle Path at its worst, and as she turned to gather more of the little roses, she felt the Queen catch her face in both hands and hold her still so that Lady Juliana could beat her.
It did not matter. She could not please. She deserved to be beaten. She quivered with every blow of the strap, yet, drenched with tears, she even lifted her buttocks to receive the punishment.
But the Queen was not satisfied still, and she turned Beauty around, her hand on Beauty's hair to pull her head back, as Lady Juliana now smacked Beauty's breasts and her belly and made the wide leather strap lick at her pubis.
The Queen held Beauty's hair fast.
"Open your legs!" Lady Juliana commanded.
"Oooooh ..." Beauty sobbed aloud, but she obeyed, and desperately she thrust her hips forward to receive the angry punishment. She must please Lady Juliana, she must show her that she had tried. Her sobs came hoarse and brokenhearted.
And the strap smacked her pubic lips again and again, and she did not know what was worse, the little shock of pain, or the violation of it.
Her head was pulled so far back she was now resting it in the Queen's lap, and she felt her own sobs rising up out of her chest and out of her lips almost languidly.
"I am defenseless, I am nothing," she found herself thinking as she had thought it on the Bridle Path in the midst of the worst exhaustion. The belt licked her breast. It was no more than she could bear, and it did not even occur to her to lift her arms, though her pubis was flooded with warm pain. Her sobs had a delicious release for her.
She felt herself growing limp, yielding. She felt the Queen's hand caressing her chin, and then she realized Lady Juliana had dropped down before her in a flurry of pink silk and was kissing her throat and her shoulders.
"There, there," said the Queen, "my brave little slave ..."
"There, there, my girl, my virtuous, lovely girl," said Lady Juliana at once as if given permission. The blows had stopped. Beauty's cries filled the room. "And you were good, very good, you tried very hard, and you struggled so hard to be graceful."
The Queen moved Beauty forward into Lady Juliana's arms, and Lady Juliana rose to her feet pulling Beauty up in her embrace, her hands pressed into Beauty's enflamed buttocks.
Lady Juliana's arms were soft and her lips were tickling Beauty, stroking her, and Beauty felt her breasts against Lady Juliana's plump breasts, and then Beauty seemed to lose all awareness of her own weight, her sense of balance.
She was drifting in Lady Juliana's arms, feeling the delicious cloth of the Lady's gown, and her rounded limbs beneath it.
"O, sweet little Beauty, my Beauty, you are so good, so very good," the Lady whispered to her. And her lips pried open Beauty's lips, and her tongue touched the inside of Beauty's mouth as her fingers pressed harder into Beauty's buttocks. Beauty's wet sex was pressed against Lady Juliana's gown, and then she felt the hard mound of Lady Juliana's sex. "Blessed Beauty, O, you do love me, don't you, I love you dearly."
Beauty could not stop herself from throwing her arms about Lady Juliana's neck. She felt the prickling of those blond braids, but Lady Juliana's skin was plump and soft, and her lips strong and silken.
They sucked at Beauty's mouth, plump lips, while Lady Juliana's teeth bit here and there as if tasting Beauty.
And then Beauty looked into Lady Juliana's eyes, so large and innocent and full of tender concern. Beauty moaned and laid her cheek against Lady Juliana's cheek.
"That is enough," said the Queen coldly.
Slowly, slowly, Beauty felt herself being released. She was being forced down, and languidly she let herself droop, until she sat on her heels on the floor, her legs parted slightly, her sex nothing but craving and pain to her.
She bowed her head. She feared above all that she would lose control of this mounting pleasure. She would blush, she would pant, she would writhe with it, unable to disguise it from those before her. So she parted her legs, feeling her pubis open and close like a hungry little mouth desperate for satisfaction.
Yet she did not care. She had known there would be no release for her.
It was enough to feel the rough wool of the carpet against her itching, stinging buttocks, and all life seemed but gradations of pain and pleasure. Her breasts seemed to be tipped with weights, and she let her head fall to the side, and a great ripple of relaxation ran through her. What more could they do to her with their games, it did not matter. "Do it," she thought, and her eyes melted into tears, the torchlight a glare before her.
She looked up.
Lady Juliana and the Queen stood side by side, the Queen's arm about Lady Juliana's shoulder. And they were both looking down at Beauty as Lady Juliana un-braided her hair and the little rosebuds fell free at her feet unheeded.
The moment seemed to go on forever.
Beauty rose to her knees again. She moved silently forward. She bent down with great delicacy and picked up one of the tiny rosebuds in her teeth, and she lifted her head in offering.
She felt the rose taken from her. And then the gentle cool kisses of both women.
"Very good, my darling," said the Queen with the first true affection.
Beauty pressed her lips to their slippers.
She heard through her drowsiness the Queen's command that she be taken by the Pages and chained to the wall of the dressing room nearby until morning.
"Spread her, and spread her wide," said the Queen.
And Beauty knew with a sweet despair that her craving would not for a long time leave her.
**WITH PRINCE ALEXI**
**T** HE QUEEN slept surely. Maybe Lady Juliana slept in her arms. The whole castle slept, and beyond it the villages and the towns, the peasants in their cottages and hovels.
And through the high, narrow window of the dressing room, the sky gave a moon-white light on the wall where Beauty was shackled, her ankles far apart, her wrists spread equally wide apart above her. She lay her head to the side, gazing at the long row of magnificent gowns, the mantles on their hooks, the circlets of gold and embroidery, the beautiful ornamental chains, and heaps upon heaps of lovely slippers.
And here she was among these things as if she were but an adornment, a possession, kept with other valuable possessions.
She sighed, and she deliberately rubbed her rear against the stone wall, wanting somehow to punish it more so that after a few seconds she could feel the relief when she stopped doing this.
Her sex would not stop its throbbing. It was sticky with its own wetness. Poor Princess Lizetta in the Hall of Punishments, did she suffer worse than this? At least she was not alone in the darkness, and suddenly even those who must pass her, taunting her, teasing her, stroking her swelling sex, seemed to Beauty a desirable company. She strained and twisted her hips. It was no comfort to her, and she did not understand why she felt this craving when only a little while ago her pain had been so great she had kissed Lady Juliana's slippers. She flushed to think of Lady Juliana's angry words, those reproving spanks that somehow hurt her worse than the others.
And how the Pages must have laughed when a dozen Princesses had probably played the little gathering game with the roses and done it better.
But why, why had Beauty at the very end picked up that last rosebud, and why had she felt her breasts swollen with warmth when Lady Juliana took it from her lips? It had seemed in that moment that Beauty's nipples were cruel little caps that prevented pleasure from breaking loose in her. Strange thought. They seemed too tight for her then, her nipples, and her sex gaped and hungered and the moisture trickled down the inside of her thighs, and when she thought of Prince Alexi's smile, and Lady Juliana's brown eyes, and her Prince's beautiful face, and even the Queen, yes, even the Queen's red lips, she felt herself burning in agony.
Prince Alexi's sex was thick and dark, like all of him, and his nipples a dark, dark rose color.
She tossed her head, rolled it against the wall. But why had she picked up the rose, offered it to pretty Lady Juliana?
She stared forward in the darkness, and hearing a creaking sound very near to her, she thought she was imagining it.
But in the darkness of the near wall, a seam of light appeared and widened. The door had been opened, and Prince Alexi slipped into the dressing room. Unbound, free, he was standing before her, and very gently, he pushed the door closed behind him.
Beauty held her breath.
He did not move, as if he must accustom himself to the darkness, and then immediately he came forward and released Beauty's wrists and ankles.
She stood trembling. And then her arms were about him. He held her against his chest, his stiff organ prodding her thighs, and she felt the silken skin of his face, and then his mouth opened over hers, hard, yes, savoring her.
"Beauty," he gave a deep sigh and she knew he was smiling. Her hand went up to feel his eyelashes. In the light of the moon she saw the planes of his face, his white teeth. She touched him all over hungrily, desperately. And then she descended upon him with loud kisses.
"Wait, wait, my lovely one, I am as anxious as you are," he whispered. But she couldn't keep her hands off his shoulders, his neck, his satin flesh.
"Come with me," he said and though it seemed an effort to disengage himself, he opened another door and took her down a long, low-roofed passage.
The moon entered windows that were no more than narrow slices out of the wall, and then he paused before one of many heavy doors, and she found herself descending a twisting stairway.
Beauty grew afraid.
"But where are we going? We'll be caught, and what will happen to us?" she whispered.
But he had opened a door and led her into a little chamber.
One little square of window gave them light, and Beauty saw a heavy straw bed covered with a white blanket. A servant's robe hung upon a hook, but all was neglected as if the room had long ago been forgotten.
Alexi bolted the door. No one could possibly open it.
"I thought you meant to escape," Beauty sighed with relief. "But will they find us here?"
Alexi was looking at her, the moon full on his face and his eyes that were filled with that strange serenity.
"The Queen sleeps every night of her life until the break of day. Felix has been dismissed. If I'm at the foot of her bed at dawn, we won't be discovered. But there is always the chance, and then we shall be punished."
"O, I don't care, I don't care." Beauty said frantically.
"Neither do I," he started to say, but his mouth was buried in Beauty's neck as Beauty flung her arms about him.
At once they were on the straw bed, against the soft blanket. Beauty's buttocks felt the prickles of the straw, but they meant nothing to her so much as Alexi's wet, hard kisses. She pressed her breasts to his chest, she wrapped her legs about his hips and strained against him.
All the long night's teasing and tormenting of her was maddening her. And then he drove into her that thick sex she had desired from the first instant she had seen it. His thrusts were brutal, strong, as if he too were overcome with denied passion. Her aching sex was filled, her tight nipples throbbing, and she snapped her hips, lifting him as she had lifted the Prince, feeling him fill her, pinion her.
At last she rose up crying out in her relief, and she felt him come with a last driving motion. Hot fluids filled her, and she lay back gasping.
She lay against his chest. He cradled her, rocked her, never stopped kissing her.
And when she sucked his nipples, bit at them playfully with her teeth, he was hard again and pushing against her.
He rose to his knees and lifted her down on his organ. She whispered her assent and then he moved her back and forth, jabbing her, working her. She had her head thrown back, her teeth clenched. "Alexi, my Prince!" she cried. And again her wet sex, stretched wide over him, throbbed in a frenzied rhythm until she was all but screaming with release as again he filled her.
It was not until after a third time that they lay still.
Yet she bit at his nipples, her hands feeling his scrotum, his penis. He rested on his elbow and smiled down at her, and let her do as she wished, even when her fingers probed his anus. She had never felt a man in this manner before. She sat up, and made him roll on his face, and then she examined all of him.
And then, overcome with shyness, she lay beside him again, nestled into his arms and buried her head in his warm, sweet smelling hair, and welcomed his gentle, deep, affectionate kisses. His lips played with hers. He whispered her name in her ear, and laying his hand between her legs sealed her tight with his palm as he clung to her.
"We must not fall asleep," he said. "I fear that for you the punishment might be too terrible."
"And not for you?" she asked.
He appeared to reflect, and then he smiled. "Probably not," he answered. "But you are a fledgling."
"And do I do so badly?" she asked.
"You are incomparable in all things," he said. "Don't let your cruel masters and mistresses deceive you. They are in love with you."
"Ah, but how should we be punished?" she asked. "Would it be the village?" She dropped her voice as she said it.
"And who has told you about the village?" he asked, a little surprised. "It could be the village ..." he was thinking ..."but no favorite of the Queen or the Crown Prince has ever been cast out into the village. But we won't be caught, and if we are I shall say I gagged you, forced you. You should suffer at most a few days in the Hall of Punishments, and what happens to me does not matter. And you must swear to me you will let me take all the blame, or I shall gag you, and carry you back and chain you up immediately."
Beauty bowed her head.
"I brought you here. I shall be punished if we're caught. That must be a rule between us. No arguments from you."
"Yes, my Prince," she whispered.
"No, don't say this to me," he pleaded. "I had not meant to command you. I'm Alexi to you, and nothing more than that, and I am sorry if I was harsh, only I cannot lead you into terrible punishment. Do as I ask because ... because ...
"Because I adore you, Alexi," she said.
"Ah, Beauty, you are my love, my love," he answered. He kissed her again. "Now you must tell me, what are your thoughts, why do you suffer so?"
"Why do I suffer? But don't you see it with your own eyes? Did I ever forget for one moment that you were watching me tonight? You see what was done to me, what is done to you, what is ..."
"Of course I watched you and was glad of the pleasure of it," he said. "Did you _not_ enjoy seeing me paddled by the Crown Prince and did you _not_ enjoy seeing me punished in the Great Hall when you were first brought in? What would you do if I told you I spilt the wine that first day so that you would notice me?"
She was stunned.
"I ask you why you suffer. I don't mean what you suffer from the paddle, or the relentless games of our Lords and Ladies. I mean what do you suffer in your heart? Why are you in such conflict? What prevents you from yielding?"
"Have you yielded?" she demanded, slightly angry.
"Of course," he said easily. "I adore the Queen and I adore pleasing her. I adore all those who torment me, because I must. It is profoundly simple."
"And you feel no pain, no humiliation?"
"I feel much pain and much humiliation. And that will never stop. If it did, even for a little while, our endlessly clever masters and mistresses would think of some new way to make us feel it. Do you think I was not humiliated in the Great Hall to be upended by Felix and spanked before an entire Court, and so casually, and for so little? I am a powerful Prince, my father is a powerful King. I never forget it. And surely it was painful to be so roughly treated by the Crown Prince for your benefit. And he thought it would make you love me less!"
"He was wrong, so wrong!" Beauty said, but she sat up and put her hands to the sides of her head in consternation. She loved them both, that was the misery of it, the Crown Prince whom even now she could envision with his lean white face and those immaculate hands and those dark eyes so full of turbulence and dissatisfaction. It had been an agony to her that he had not taken her to his bed after the Bridle Path.
"I want to help you because I love you," Alexi said. "I want to guide you. You are in rebellion."
"Yes, but not always," she admitted in a vague whisper, looking off, as if she were suddenly ashamed to admit it. "I have ... so many feelings."
"Tell me," he said with authority.
"Well, tonight ... the rose, the last little pink bud ... why did I pick it up in my teeth and offer it to Lady Juliana? Why? She had been so cruel to me."
"You wanted to please her. She is your mistress. You are a slave. The highest thing that you can do is please, so you sought to do it, and not only in response to her paddling and her commands, but in that moment _of your own will."_
"Ah, yes," said Beauty, that was it. "And ... on the Bridle Path, how can I confess it, I felt some release in myself as if I were no longer locked in struggle, I was just a slave, a poor, desperate slave who must strive, strive _purely."_
"You are eloquent," he said with feeling. "You know much already."
"But I don't want to feel this. I want to rebel in my heart, I want to steel myself against them. They torment me endlessly. My Prince, were he the only one..."
"But even if he were, he would find new ways to torment you, and he is not the only one. But tell me why you don't wish to give in to them."
"Well, surely you know. Didn't you rebel? Don't you? Why, Leon said of you there is a core in you which no one touches."
"Nonsense. I merely know and accept everything. There is no resistance."
"But how can it be?"
"Beauty, you must learn it. You must accept and yield, and then you shall see everything is simple."
"I would not be here with you if I yielded because the Prince ..."
"Yes, you could be here with me. I adore my Queen and I am here with you. I love you both. I yield to that entirely as well as everything else and even the knowledge I may be punished. And when I am punished, I shall dread it, and suffer it and understand it and accept it. Beauty, when you accept you will flower in the pain, you will flower in your suffering."
"There was a girl ahead of me in line last night who ran the Bridle Path just before me. She was resigned, wasn't she?" Beauty asked.
"No, forget about her, she is nothing, that is Princess Claire and she is silly and playful and always was and feels nothing. She has no depth, no great mystery. But you have these and you will always suffer more than she does."
"But does everyone sooner or later acquire this ability to accept?"
"No, some never do, but it is very hard to tell who has attained it. I can tell, but our masters are not always so wise, I can assure you. For example, Felix told me that yesterday you saw Princess Lizetta strung up in the Hall of Punishments. Do you think she is resigned?"
"Certainly not!"
"Ah, but she is, and she is a great and valuable slave Princess. But Princess Lizetta adores being bound up, being unable to move, and when she is greatly bored, she endures the displeasure of her betters, the better to amuse them by letting them punish her."
"Ah, no, you can't be serious."
"Yes, I can. That is her way. All slaves have their way. And you must find yours. It will never be easy for you. You will suffer much before you know it, but don't you see that on the Bridle Path and tonight when you gave the rose to Lady Juliana you felt the beginnings of it. Princess Lizetta is a struggler. You shall be a yielder, much as I am. That shall be your way, exquisite and personal devotion. Great calm, great serenity. In time perhaps you will see other slaves who are exemplary in this. Prince Tristan, for example, the slave of Lord Stefan, is incomparable. His Lord is in love with him as the Prince is with you, which makes it both difficult and simple."
Beauty gave a deep sigh. She was flooded suddenly with the sensation of kneeling before Lady Juliana and offering her the rose. She felt herself running on the Bridle Path, and the breeze touching her, and her body burning all over with her striving.
"I don't know, I feel ashamed when I give in, I feel as if I have truly lost myself."
"Yes, that is it. But listen. We have the night here together. I want to tell you the story of how I came here and how I attained the path I speak of. When I am finished, if you still feel rebellious, I ask you to think on it. I shall go on loving you, no matter, and go on striving for moments to see you in secret. But if you listen to me, you shall see that you can conquer everything about you.
"Don't try to understand all that I say at once. Merely listen and see if the story in the end does not soothe you. Remember, you cannot possibly escape this place. No matter what you do the Court will find ways to wring amusement from you. Even a wild, teeth-gnashing slave can be bound and used in an abundance of different ways to amuse everyone. So accept this limit; and then try to understand your own limits and how you must broaden them."
"O, if I know you love me I can accept, I can accept anything."
"I do love you. But the Prince loves you, too. And even so, you must seek your path of acceptance."
He embraced her, then gently forcing his tongue between her lips, kissed her violently.
He suckled her breasts until they were almost sore, as she arched her back moaning again, her passion rising. He lifted her under him and once again he drove his organ into her, turning her gently so they lay on their sides facing one another.
"They shan't rouse me tomorrow for anything and for that alone I'll be punished." He smiled. "But I do not care. It's worth it, to have you, to hold you, and to be with you."
"But I can't bear to think of you punished."
"Be comforted that I shall deserve it, and the Queen must be satisfied, and I belong to her, just as you belong to her and you belong to the Prince, and should he catch you he would have every right to punish me further."
"But how can I belong to him and to you?"
"As easily as you might belong as well to the Queen and Lady Juliana. Did you not give Lady Juliana the rose? I wager that before the month is out, you will be mad to please Lady Juliana. You will dread her displeasure; you will hunger for her paddling just as you fear it."
Beauty turned her face away and buried it in the straw because it was already true. Tonight she had been glad to see Lady Juliana. And this was the way she felt about her Prince.
"Now, listen to my story and you will understand more. It is not a neat explanation. But you will see something of a mystery unfolded."
**PRINCE ALEXI TELLS OF HIS CAPTURE AND ENSLAVEMENT**
**W** HEN IT came time to send Tributes to the Queen," Prince Alexi said, "I was not at all resigned to be chosen. There were other Princes who were brought forward to go with me, and we were told that our service with the Queen would last no more than five years at most, and that we would return greatly enhanced in wisdom, patience, self-control, and all virtues. Of course I had known others who had served, and though they are all forbidden to speak of what happens, I knew it was an ordeal and I cherished my freedom. So when my father told me I must go, I ran away from the castle and went roaming through the villages.
"I don't know how my father received this news. It was a party of the Queen's soldiers who raided the village where I was and carried me off with a number of the common boys and girls for other forms of service. These were given to minor Lords and Ladies to serve in their own manor houses. Princes and Princesses such as we serve only at the Court, as I'm sure you realize.
"It was a brilliantly sunny day. I was walking alone in a field south of the village writing poetry in my mind when I saw the Queen's soldiers. I had my broadsword, of course, but I was at once surrounded by some six horsemen. As soon as I realized they meant to take me as a slave I knew they belonged to the Queen. They threw a net over me and quickly disarmed me. I was stripped on the spot, and thrown over the Captain's saddle.
"That alone was enough to infuriate me and make me fight for my freedom. You can imagine it, my ankles tied with coarse rope, my naked buttocks in the air, my head dangling. The Captain laid his hands on me often enough when he was idle. He pinched and prodded as suited him, and seemed to enjoy his advantages."
Beauty winced at all this. She could well picture it.
"It was a long journey to the Queen's Kingdom. I was handled roughly like so much baggage, bound at night to a pole outside the Captain's tent and though no one was allowed to violate me, I was tormented by the soldiers. They would take reeds and sticks and prod my organs, touch my face, my arms and legs, whatever they could. My hands were tied over my head; I stood all that while, sleeping on my feet. The nights were warm enough but it was quite miserable.
"However, all of this had a wisdom to it. I was promised to the Queen herself, by virtue of her treaty with my father. And of course I was eager to be rid of these coarse soldiers. Each day's ride was the same, over the Captain's saddle. He often whipped me with his leather gloves playfully. He let the villagers come near the road when we passed. He taunted me, and tousled my hair, and called me pet names. But he could not really use me."
"Were you thinking of escape?" Beauty asked.
"Always," said the Prince. "But I was in the midst of the soldiers at all times, and completely naked. Even had I managed to reach a villager's cottage or a serfs hut, I would have been overpowered and returned for the ransom money. More humiliation and more degradation. I rode, bound hand and foot and ignominiously thrown over the horse, in a state of fury.
"But finally we reached the castle. I was scrubbed, then oiled and brought before her Highness. She was coldly beautiful. This made its impression upon me at once. I had never seen such pretty eyes, yet such cold eyes. And when I refused to be silent or to obey, she laughed. She ordered me gagged with a leather bit. I'm sure you've seen it. Well, mine was bound in place so I couldn't remove it. And then she had me shackled in leather so that I could not rise from my hands and knees. I could move as told, but not rise, the leather collar around my neck securely linked by leather chains to the leather cuffs on my wrists, and those to the cuffs on my legs above the knees. My ankles were linked so they couldn't be spread very wide apart. It was all quite clever.
"And then the Queen took her long lead—as she calls it—to drive me. It was a rod with a leather-encased phallus on the end of it. I shall never forget the first moment I felt it drive into my anus. She thrust it forward, and in spite of myself I moved ahead of her like an obedient pet as she commanded me. And when I lay down and refused to obey, she only laughed at this, and commenced her work with the paddle.
"Well, I was fiercely rebellious. The more she paddled me, the more I growled and refused to obey. So she had me hung upside down and paddled on and off for hours. You can well imagine the misery of it. But understand, other slaves were looking at me in utter confusion. Being stripped, being cuffed, being ordered about with the paddle was quite enough to make them obey, coupled as it was with the knowledge that they could not escape and they must serve for several years, and they were helpless.
"Yet nothing worked its magic with me. When I was taken down I was sore from the paddle on my buttocks and my legs, but I did not care. And all attempts to rouse my organ had failed. I was too stubborn.
"Lord Gregory lectured me at length. The paddle was far easier to bear with an erect organ, he told me; with passion coursing through my veins, I should see the rhyme and reason of pleasing my mistress. I wouldn't listen.
"The Queen still found me amusing. She told me I was more beautiful than any other slave sent to her. She had me bound to the wall in her chambers night and day so that she might watch me. But more truly, it was so that I might watch her and desire her.
"Well, at first, I did not look at her. But by and by I grew to studying her. I learned every detail of her, her cruel eyes, and her heavy black hair, her white breasts and her long legs, and the way that she lay abed or walked about, or ate her food so daintily. Of course, she had me paddled regularly. And a strange thing commenced to happen. The paddlings were the only things that broke the boredom of this time, aside from watching her. So that watching her and being paddled became of interest to me."
"O, she is devilish!" Beauty gasped. She could understand all this perfectly.
"Of course she is, and infinitely sure of her own beauty.
"Well, all this while, she went about the business of the Court, coming and going. I was often alone with nothing to do but struggle, and curse behind the gag. Then she would return, a vision of soft tresses and red lips. My heart started to pound when she was undressed. I loved the moment when her mantle was released from its folds and I saw her hair. Then, when she was naked and stepping into her bath, I was beside myself.
"All this was secret. I did my best to display nothing of it. I quieted my passion. But I am a man, so in a matter of days my passion commenced to build, to show itself. The Queen laughed at this. She tormented me. Then she would tell me how much less I would suffer if I were over her lap obediently accepting her paddle. This is the Queen's favorite sport, the simple spanking over the lap, as you learned painfully enough tonight. She loves the intimacy of it. All her slaves are her children."
Beauty puzzled over this, but she didn't want to interrupt Alexi, who went on.
"As I told you, she would have me paddled. And always in a most uncomfortable and cold manner. She would send for Felix, whom I then despised ..."
"You don't now?" Beauty asked. But then with a flush she remembered the scene she had witnessed on the stairway, Felix suckling the Prince so tenderly.
"I don't despise him now at all," Prince Alexi answered. "He is, of all the Pages, one of the more _interesting._ One comes to treasure that here. But in those days, I despised him as much as I did the Queen.
"She would give the order for me to be spanked. He would have me removed from the shackles that held me to the wall, as I kicked and struggled frantically. Then I'd be thrown over his knee, my legs kicked wide apart, and the spanking would go on until the Queen was tired of it. It hurt very much as I'm sure you know, and it only further humiliated me. But as I became more and more desperately bored in my hours of solitude, I commenced to look upon it as an interlude. I began to think about the pain, the various stages of it. There were the first few cracks of the paddle, not so painful at all. Then, as they came on harder and harder, the aching, the stinging, I found myself wriggling and trying to escape the blows, though I'd sworn I wouldn't. I'd remind myself to be still only to slip into writhing again, which amused the Queen immensely. When I was very sore, I felt very tired, tired of the struggle, and the Queen knew then I was most vulnerable. She would touch me. Her hands felt very delicious on my welts though I hated her. Then she'd stroke my organ, telling me in my ear what ecstasies I might enjoy in serving her. I would receive her full attention, she said, and be bathed and babied by the grooms, instead of roughly scrubbed and hung on the wall. I would weep sometimes because I couldn't stop myself. The Pages would laugh. The Queen thought it all quite laughable, too. Then I would be returned to the wall to be broken down by more interminable boredom.
"Now all this time, I never saw the other slaves punished by the Queen. She would carry out her pleasures and games in her many parlors. Sometimes I would hear cries and blows through the doors, but seldom.
"But, as I began to exhibit an erect and craving organ in spite of myself, and began to actually look forward to the terrible spankings ... in spite of myself ... these two interludes not being connected as yet in my mind ... she brought in a slave now and then for her amusement.
"I can't tell you the rages of jealousy I felt the first time I had to witness a slave punished. This was a young Prince Gerald, whom she adored in those days. He was sixteen, and had the roundest, smallest buttocks. They were irresistible to the Pages, and the grooms, as yours are ..."
Beauty blushed at this.
"Don't count yourself unlucky. Listen to what I say about the boredom," Alexi said, and he kissed her tenderly.
"As I was saying, this slave was brought in and the Queen stroked and teased him shamelessly. She placed him over her lap and proceeded to deliver a naked-handed spanking as she did to you, and I could see his erect penis, and how he tried to keep it away from her leg for fear he would spill his passion and displease her. He was utterly compliant and devoted to her. He had no dignity in his surrender at all, but scampered to obey her every command, his beautiful little face always flushed, his skin pink and white and full of blushes where he'd been punished. I couldn't take my eyes off him. I thought I can never be made to do these things. Never—I should die first. Yet I watched him, and I watched her punish him and prod him and kiss him.
"And when he had pleased her well, how she rewarded him! She had brought in six Princes and Princesses from whom he must choose with whom he would couple. Of course his choices were to please her. He chose the Princes always.
"And as she presided over him with the paddle, he would mount one of these who knelt for it obediently enough and, receiving the Queen's blows, he would achieve ecstasy. It was a tantalizing spectacle. His own plump little buttocks being soundly spanked, the red-faced submissive slave on his knees to receive Prince Gerald, and the boy's erect cock going in and out of the undefended anus. Sometimes the Queen spanked the little victim first, gave him a merry chase about the room, a chance to escape his fate if he could fetch a pair of slippers for her in his teeth before she could achieve ten good cracks of the paddle. The victim would scurry to obey. But seldom was he able to find the slippers and bring them to the proper place before the Queen had soundly paddled him. So he had to bend over for Prince Gerald, who was too well endowed for sixteen surely.
"Of course I told myself all this was disgusting and beneath me. I should never play such games." He laughed softly, and squeezed Beauty against his chest with his arm, kissing her forehead. "I've played them enough since," he said.
"But now and then, too, Prince Gerald did choose a Princess. This angered the Queen, though only slightly. She had the little girl victim perform some hopeless tasks in the hope of escape, the same game with the slippers, or the getting of a hand mirror or the like, all the while driving her mercilessly with the paddle. Then she would be thrown down on her back and taken by the lusty little Prince for the Queen's amusement. Or she might be doubled and hung as in the Hall of Punishments.
Beauty winced at this. To be taken in such a position hadn't occurred to her. But a Princess would surely be ripe and open for it.
"As you can imagine," Alexi went on, "these spectacles became a torture. In my hours alone, I longed for them. As I watched, I could feel the blows on my buttocks as if I too were being spanked, and I felt my penis stir against my will at the sight of the little girls being chased, or even Prince Gerald being stroked and sometimes suckled by a Page for the Queen's amusement.
"I should add that Prince Gerald found this very hard. He was an anxious Prince, ever striving to please the Queen, and punishing himself in his own mind, dreadfully, for failure. He never seemed to realize that many of the tasks and games were deliberately made too difficult for him. The Queen would have him brush her hair with the brush fixed in his teeth. This is most difficult. And he would be weeping when he could not brush her hair in long enough strokes, nor thoroughly enough. Of course she was annoyed. She'd throw him over her lap, and with a leather-handled brush flail at him. He wept, full of shame and misery, and feared her worst wrath: that he be given over to others for pleasure and chastisement."
"Does she ever give you to others, Alexi?" Beauty asked.
"When she's displeased with me, she gives me to others," he continued. "But I have surrendered and accepted this. It saddens me but I accept it. I am never in the frenzy in which Prince Gerald always found himself. He would beseech the Queen with silent kisses all over her slippers. It was never any use. The more he pleaded, the more she punished him."
"What became of him?"
"The time came for him to be sent back to his Kingdom. That time comes for all slaves. It will come for you, too, though when, no one can say, on account of the Prince's passion for you, and that he awakened you and claimed you. Your Kingdom was a legend here," said Prince Alexi.
"But Prince Gerald went home richly rewarded and I think most relieved to be let go. He was of course beautifully dressed before he left, and received by the Court, and then we were assembled to see him ride out. It's the custom. I think it was as humiliating for him as anything else. It was as if he remembered his nakedness and his subjugation. But other slaves suffer just as much when they are released for many reasons. Who knows, however. Maybe Prince Gerald's endless worries saved him from something worse. It's impossible to tell. Princess Lizetta is saved by her rebellion. It was _interesting_ to Prince Gerald surely ..."
Prince Alexi paused to kiss Beauty again and soothe her. "Don't try to understand all I say just now. That is, don't try to find immediate meaning in it," he said. "Merely listen and learn and perhaps what I tell you shall save you some mistakes, give you different paths for the mind later. Ah, you are so tender to me, my secret flower."
He would have embraced her again, perhaps become carried away by his passion again, but she stopped him with a touch of her fingers to his lips.
"But tell me, when you were shackled to the wall, what did you think of ... when you were alone, did you daydream, and what did you dream?"
"What a strange question," he said.
Beauty seemed very serious. "Did you think of your former life, and wish you were free for this or that pleasure?"
"Not really," he said slowly. "I thought rather of what would happen to me next, I suppose. I don't know. Why do you ask this?"
Beauty didn't reply, but she had dreamed three times since she had come and each time her old life to her had seemed grim and fraught with tiny worries. She remembered hours with her embroidery, and the endless bowing at Court to the Princes who kissed her hand. She remembered sitting quite still for hours at interminable banquets where others talked and drank, and she had felt only boredom.
"Please continue, Alexi," she said gently. "But to whom does the Queen give you when she's displeased?"
"Ah, that is a question with several answers," he said. "But let me proceed. You can well imagine what my existence was, the hours of boredom and solitude broken only by these three diversions: the Queen herself, Prince Gerald's punishment, or the fierce paddling from Felix. Well, soon, in spite of myself and my rage, I commenced to show my excitement whenever the Queen came into the chamber. She ridiculed me for it, but she marked it. And now and then, I could not conceal it when I saw Prince Gerald so boldly erect and taking his pleasure of one of the other slaves, or even taking the paddle. The Queen observed all this, and each time she saw that my organ was stiff and beyond my will, she would have Felix at once deliver a hard spanking to me. I struggled, I tried to curse her, and at first these spankings quelled my passion, but very soon they did not quell it. And the Queen added to my misery with her own hands, slapping my penis, stroking it, and then slapping it again at the very moment that Felix was punishing me. I twisted, struggled. It was no use. Very soon, I so craved the Queen's hands that I was moaning aloud and in one of these great tormented states, I did all that I could by gesture and manner to show that I would obey her.
"Of course I had no intention of doing so. I did so only long enough to be rewarded. And I wonder if you can imagine how difficult this was for me. I was put free on my hands and knees, and told to kiss her feet. It was as if I had only just been stripped naked. Never had I obeyed any command; nor been made to obey while free of shackles. And yet so tortured was I for relief, my sex so swollen with desire, that I forced myself to kneel at her feet and kiss her slippers. I shall never forget the magic of her hands when she touched me. I could feel the shock of passion through me, and as soon as she stroked and toyed with my sex, my passion was at once released, which greatly angered her.
"'You have no control,' she said crossly to me, 'and for this you will be punished. But you have tried to submit and that is something.' But at that moment, I rose up and tried to run from her. I'd never had any intention of submitting to anything.
"Of course the Pages apprehended me at once. You must never think yourself safe from them. You may be in a vast, dimly lit chamber alone with a Lord. You may think yourself quite free when he falls asleep with his wine cup. But should you try to rise and escape, at once the Pages appear to subdue you. Only now that I am the Queen's trusted valet am I allowed to sleep alone in her chamber. The Pages dare not enter the darkened room where the Queen sleeps. So they have no way of knowing that I am here with you. But this is rare, most rare. And even now we might be discovered ..."
"But what happened to you," Beauty pressed. "They apprehended you," she said fearfully.
"The Queen gave little consideration to how I should be punished. She sent for Lord Gregory and told him I was most incorrigible. That in spite of my fine hands and skin, and royal birth, I should be taken at once to the kitchen, there to serve for as long as she should decree ... and indeed, she hoped she would remember I was there and send for me.
"I was carried down to the kitchen, protesting as usual. Mind you, I had little idea what was to happen to me. But very soon I saw that I was in a dark and dirty place, full of the grease and soot of the cooking where the pots were always boiling and dozens of menials were at work at the chopping of vegetables and the cleaning or plucking of the fowl, and all the other tasks that go to produce the banquets served here.
"No sooner was I brought in than they rejoiced to have a little amusement. I was surrounded by the crudest beings I had ever seen. 'But what is this to me,' I thought. 'I obey no one.'
"But in moments, I realized these creatures were no more interested in my compliance than they were interested in the compliance of the fowl they slaughtered, or the carrots they scrubbed, or the potatoes they threw in the pot. I was a plaything to them and seldom did they even address me as though I had ears to hear or sense to comprehend what they said about me.
"I was at once collared in leather, this collar linked to the cuffs on my wrists, and my wrists to my knees so that I could not rise from a hands-and-knees position. A bit with a bridle was placed in my mouth, and bound so securely to my head that I might be pulled forward by leather straps with little ability to resist, my limbs reluctantly allowing me to follow.
"I refused to stir. I was dragged about on the dirty kitchen floor while they howled with laughter. They had their paddles out, and were soon punishing me mercilessly. Nothing was spared, of course, but my buttocks in particular delighted them. And the more I bucked or struggled, the more they found it hilarious. I was no more than a dog to them. And that was precisely how they treated me. But this was only the beginning. I was soon unshackled enough to be thrown over a great barrel. And there I was raped by one and all of the men, the women looking on with laughter. I was sore from this, and so dizzy from the motion of the barrel that I was sick, but this again they thought most amusing.
"But when they were done with me, and had to return to their work, they shackled me above the open hogshead that received the garbage. My feet were deposited firmly in the waste of cabbage leaves and carrot tops, onion peels, and chicken feathers that made up the refuse of the day's work and, as they added to it, it rose around me. The stench was terrible and when I writhed and struggled, again they laughed, and thought of other ways to torment me."
"O, this is too dreadful," Beauty gasped. Each person who had handled her or punished her had in some way admired her. And when she thought of her beautiful Alexi treated this way, she felt weak with fear.
"Of course I did not know this was to be my regular station. I was taken out hours later when after the evening meal was served they again chose to rape me. Only this time I was thrown down and spread out on a large wooden table. And for their pleasure they paddled me again this time with coarse wooden paddles, saying the leather paddles they had used earlier were now too good for me. They held my legs wide apart, they lamented they could not torture my private parts without risking punishment. But by this they did not mean my penis which they punished a great deal with slaps, and rough handling.
"I was frantic by this time. I cannot explain it. There were so many of them, they were so crude, and my movements or sounds were nothing to them. The Queen had noticed my smallest change of expression. She had scoffed at my growls and struggles, but she had savored all of it. These crude cooks and kitchen boys rubbed my hair, lifted my face, slapped my buttocks and spanked me as if I had no sense whatsoever.
"They would speak of me, 'What plump buttocks,' and 'Look at those strong legs,' and that sort of thing as if I were a mere animal. They pinched me, poking me, jabbing me as they pleased, and then they set to raping me. They greased me well with their cruel hands as they had done before, and when they had finished, they flushed me out with some crude piping attached to a wineskin filled with water. I cannot tell you the mortification of this, to be washed inside and out by them. The Queen had at least allotted to me privacy in these matters, as the needs of our bowels and bladders do not interest her. But to be emptied by this violent stream of cold water and in front of these piggish men made me weak and spiritless.
"I was limp when they hung me back in the refuse. And in the morning my arms ached, and I was sickened by the stench that rose around me. Roughly they pulled me out and shackled me on my knees again and threw down for me some food on a plate. It had been a day since I had eaten; yet I did not wish to eat for their amusement, as they would not allow me the use of my hands. It was nothing to them. I refused the meals until the third day when I could endure it no longer and I lapped up the gruel they gave me like a hungry puppy. They never took the slightest notice. When I finished my meals it was back to the heap of refuse until they had some time to make sport of me.
"In the meantime I hung there. And when they passed, they would perhaps give me some strong slap, twist the nipples of my chest, spread my legs wider with one of their paddles.
"It was an agony beyond anything I had known in the Queen's chamber. And soon, in the evening, the stable boys received word that they might come and use me as they wished. So I had them to satisfy as well.
"They were better dressed, but they smelled of the horses. They came in and took me out of the hogshead, and one of them thrust the long rounded leather handle of his whip into my anus. Lifting me up by this, he forced me into the stable. I was then laid over a barrel again and raped by all of them.
"It seemed unendurable, and yet I endured it. And as in the Queen's chamber, I could feast my eyes on my tormentors all day long though in between their wants they took little notice of me.
"One evening however, when all of them had had much to drink and had been congratulated for a very good meal upstairs, they turned for more imaginative play with me. I was terrified. I had no thoughts of dignity anymore and began to groan behind my gag as soon as they approached me. I squirmed and twisted to resist their hands.
"The games they chose were as degrading as they were disgusting. They spoke of decorating me, of improving my appearance, that I was altogether too clean and too fine an animal for my lodgings. And, spreadeagling me in the kitchen, they soon cut loose their fury on me with a dozen concoctions they made from the honey, the eggs, the various syrups and mixtures at their disposal. I was soon covered with these egregious liquids. They painted my buttocks, and laughed as I struggled. They painted my penis and balls. They decorated my face with it, and stuck back my hair with it. And when they had finished, they took the feathers from the fowl and pasted these to my body.
"I was terror-stricken, not of any real pain, but merely of their vulgarity and their meanness. I could not bear the humiliation of such disfigurement.
"Finally, one of the Pages came in, to see what was the noise, and he took pity on me. He had them release me and told them to wash me. Of course they scrubbed me roughly, and they took to paddling me again. It was then that I knew I was losing my senses. I was down on my hands and knees, though I was not shackled, and running desperately to hide from their paddles. I struggled to get under the kitchen tables, and everywhere I sought a moment's rest, they sought me out, moving the tables and chairs if need be to get at my buttocks with their paddles. Of course if I tried to rise, they pushed me down. I was desperate.
I found myself scurrying to the Page and kissing his feet just as I had seen Prince Gerald do with the Queen.
"But if he told the Queen, it was of no use to me. The next day I was shackled as before, and awaiting the boredom and restlessness of the same mistresses and masters. Sometimes passing me, they stuffed into my anus some bit of food rather than throw it away, carrots, other roots, whatever they thought liken to a penis. I was raped over and over by these things, and had to expel them with great effort. They would not have spared my mouth, I suppose, had they not been commanded to leave me gagged as all such slaves are gagged.
"And whenever I caught a glimpse of a Page I found myself pleading with him by all my gestures and manner of groaning.
"I had no real thoughts during this time. Perhaps I had begun to think of myself as the half human thing that they thought I was. I don't know. To them I was a disobedient Prince sent to them because I deserved it. Any abuse was their duty. If the flies were bad, they would paint my penis and balls with honey to attract them and think that very clever.
"Much as I feared the leather whip handles of the stable boys forced up my anus, I came to look forward to being taken to the cleaner, cooler places in the stable. Those boys at least thought it quite marvelous that they had a real Prince to torment. They rode me quite long and hard, but it was better than the kitchen.
"I don't know how long it went on. Every time they unshackled me I was terrified. They soon took to throwing about the refuse on the floor and making me gather it up as they chased me with their paddles. I had lost all sense of the wisdom of merely keeping still, and flustered and in panic I ran this way and that to finish the task as they spanked me. Prince Gerald had never been so frantic.
"Of course I thought of him as I found myself doing this. And I thought bitterly, 'He is amusing the Queen in her chambers, and I am here in this filthy place.'
"Why, to me the stable boys were royalty. And one of them in particular had become quite fascinated with me. He was big, very strong. He could mount me on his whip handle so that my feet barely touched the ground, and force me along, my back arched, my hands bound, almost carrying me. He delighted in doing this, and one day he took me off to a part of the garden alone. I tried once to struggle against him, and he simply flipped me over his knee effortlessly. He forced me down in the grass, and told me that with my teeth I was to pick the little white flowers there for him or he would take me back to the kitchen. I can't tell you how willingly I obeyed him. He kept his whip handle in me and forced me this way and that with it. And then he commenced to torment my penis. Yet even as he slapped it and abused it, he would stroke it. To my horror, I felt it swell. I wanted to stay with him forever. I thought, 'What can I do to please him?' And I was humbled by this, in despair, for I knew that this was just what the Queen had wanted in punishing me. I was convinced even in my madness that if she knew how much I suffered, she would release me. But my mind was empty of thought. I knew only that I wanted to please my stable boy lest he return me to the kitchen.
"I fetched the little flowers in my teeth and brought them back to him. He told me then I was too bad a Prince to be treated so gently by everyone, and that he knew how to punish me. He ordered me to mount a nearby table. This was a round wooden table, weathered but often draped and used when any of the Court want to take their repast in the garden.
"I obeyed at once, but I was not to kneel there, I was to squat with my legs wide apart and my hands behind my neck and to keep my eyes down. This was unbelievably degrading to me and yet all I could think of was to please him. Of course he spanked me in this position. He had a leather paddle, heavy but thin with a powerful wallop. And he commenced swatting my buttocks with it. And yet I remained there, unshackled but obedient, my legs aching as I squatted, my penis all the time swollen as he tormented me.
"It was the best thing that could have happened. Because Lord Gregory witnessed it. However, I didn't know this at the time, I knew only that others were passing near, and when I heard their voices and knew them to be Lords and Ladies, I experienced unbelievable consternation. They would see me being humiliated by this stable boy, me the proud Prince who had rebelled against the Queen. And yet all I could do was to weep, and suffer, and feel the paddle swatting me.
"I did not even think of the Queen learning of this. I was too devoid of hope. I thought only of the moment. Now, this, Beauty, is one aspect of yielding and acceptance, surely. I thought only of the stable boy and pleasing him and escaping, for a little while longer, at this terrible price, the kitchen. In other words, I thought of doing precisely what was expected of me.
"Now, my stable boy grew tired of it. He ordered me back down to the grass on my hands and knees and took me deeper into the woods. I was completely unbound, yet I was under his will utterly. Now he found a tree and told me to stand up and grasp the limb over my head. I hung by the limb, my feet off the ground, as he raped me. He thrust in deep and hard and repeatedly. I thought it would never end, and my poor penis was hard as the tree itself with suffering.
"And when he was finished, the most extraordinary thing happened. I found myself kneeling at his feet, kissing his feet and more than that, I was twisting my hips, and thrusting and doing all in my power to beg him to relieve the passion between my legs, to allow me some release, for I had known absolutely none in the kitchen.
"He laughed at all this. He pulled me up, impaled me easily on his whip handle and drove me back towards the kitchen. I was weeping as uncontrollably as ever in my life.
"The vast room was almost empty. All were out tending the vegetable gardens, or in the anterooms above as the meals were being served, and only a young serving girl remained, who climbed to her feet at once when she saw us. In a moment, the stable boy was whispering to her, and as she nodded her head, and wiped her hands on her apron, he ordered me up onto one of the square tables. I was to squat again with my hands behind my head. I obeyed without even thinking. More paddling, I thought, and for the benefit of this little girl with her wan face and brown braids. She meantime drew near and looked at me with what seemed wonder. Then for her the stable boy began to torment me. He had taken a soft little broom which was used to sweep out the inside of the oven. And with this he began to brush and stroke my penis. The more he stroked the more miserable I became, but each time it was almost too much for me he would withdraw the broom a quarter of an inch from my penis so that I struggled after it. It was more than I could bear, yet he would not allow me to move my feet, and paddled me immediately if I disobeyed him. I soon saw his game. I must thrust my hips forward as much as I could to keep my hungry penis in contact with the soft stroking bristles of the broom, and so I did, crying all the while as the girl gazed on with obvious delight. Finally, she begged to be allowed to touch me. I was so grateful for this I could not stop my sobs. The stable boy then put his broom under my chin and lifted my face. He said he wished to see me satisfy the young maid's curiosity. She had never really seen a young man spend his passion. And as he held me and scrutinized me and looked at my tear-stained face, she stroked my penis, and without pride or dignity, I felt my passion erupt into her hand, my face flushing with heat and blood as a shudder went through my loins relieving all of the days of frustration.
"I was weak when it was finished. I had no pride, nor thoughts of past and future. I made no resistance when I was manacled. I wished only that the stable boy would come again soon, and I was drowsy and afraid when all the cooks and kitchen boys returned and commenced their inevitable idle sport with me.
"The next few days were filled with the same terrible kitchen torments. I was paddled, chased, ridiculed and otherwise treated contemptuously. But I dreamed of the stable boy. Surely he would return. I don't think I even thought of her Highness. I felt only despair when I envisioned her.
"Finally one afternoon, the stable boy came in and he was finely dressed in rose-colored velvet trimmed in gold. I was aghast. He ordered me washed and scrubbed. I was too excited to fear the rough hands of the kitchen boys, though they were merciless as ever.
"My penis was rigid already at the mere sight of my stable boy Lord, but he told me quickly that it must be brought to perfect attention, and that I was to keep it thus, or be severely punished.
"I nodded all too vigorously. Then he took the gag bit out of my mouth and replaced it with an ornate one.
"How can I describe what I felt then? I didn't dare dream of the Queen. I was so bereft that any respite was wondrous to me. He led me now into the castle, and I who had rebelled against everyone was quickly scampering after him down the stone corridors past the slippers and boots of Lords and Ladies who all turned to take some notice of me and give some compliment. The stable boy was very proud.
"And then we entered a great high-ceilinged parlor. It seemed never in my life had I seen cream-colored velvet and gilt, and statues against the walls, nor the bouquets of fresh flowers everywhere. I felt myself born again with no thought of my own nakedness or subservience.
"And there sat the Queen in a high-backed chair, resplendent in purple velvet, her ermine cape over her shoulders. I scurried forward boldly, ready to offend by obsequiousness, and showered her hem and her shoes with kisses.
"At once she stroked my hair, and lifted my head. 'Have you suffered enough for your stubbornness?' she asked, and as she did not take her hands away I kissed them, kissed her soft palms and her warm fingers. The sound of her laugh was beautiful to me. I glimpsed the mounds of her white breasts, and the tight girdle about her waist. I kissed her hands until she stopped me and held my face and opened my mouth with her fingers and felt of my lips and teeth and then removed the gag, saying that I must not speak. At once I nodded.
"This will be a day of tests for you, my willful young Prince," she said. And then she put me in a paroxysm of exquisite pleasure by touching my penis. She felt of its hardness. I tried to keep my hips from moving forward towards her.
"She approved. And then she ordered my punishments. She had heard of my chastisement in the garden, she said, and would my young groom, the stable boy, please punish me so for her amusement.
"I was on the round marble table in front of her at once, squatting obediently. I remember the doors were open. I saw the distant figures of Lords and Ladies moving past. I knew there were other Ladies in this very room. I could see the soft colors of their dresses and even the shimmer of their hair. But I had no thought but to please the Queen and only hoped that I might manage to remain in this difficult squatting position for her as long as she wished, no matter how cruel the paddle. The first blows felt warm and good to me. I felt my buttocks flinch and tighten and it seemed I had never experienced such full swelling pleasure, unsatisfied as it was, in my penis.
"Of course I was soon groaning from the blows, and with my efforts to conceal the sound, the Queen kissed my face and told me that though my lips must remain sealed, I should let her know how I suffered for her. I understood her at once. My buttocks were now smarting and throbbing with pain. I arched my back, my knees opening all the more, my legs stiff and aching from the strain of the squatting, and I moaned without reservation, my moans growing louder with each crack of the paddle. Understand, nothing restrained me. I was unshackled and ungagged.
"All rebelliousness was gone from me. When next the Queen ordered me paddled about the room, I was only too willing. She threw down a handful of small gold balls the size of large purple grapes, and she bid me bring each one to her, just as you were commanded to fetch the roses. The stable boy, my groom as she called him, was to achieve no more than five cracks of the paddle before I had placed one in her hand, or she should be very displeased with me. These gold balls had rolled far and wide, and you cannot imagine how I scurried to gather them. I ran from the paddle as though it were burning me. Of course I was tender and sore by this time, and broken out in plenty of hard welts, but it was to please her that I hurried.
"I brought the first one with only three blows. I was very proud. But as I put it in her hand I saw she had put on a black leather glove, the fingers of which were traced with small emeralds. She bid me turn around and part my legs and show her my anus. I obeyed at once, and immediately felt the shock of those leather sheathed fingers opening my anus.
"As I told you, I had been raped and washed out repeatedly by my crude captors in the kitchen. Yet this was a new exposure to me, to be opened thus by her, and so simply and thoughtlessly, without the violence of rape. It made me feel softened with love and weak and totally her possession. At once I realized she was forcing the gold ball which I had retrieved into my anus. And now she instructed me that I was to hold it inside me, unless I wanted her fierce displeasure.
"I had now to fetch another. The paddle came at me quickly. I hurried, brought back another gold ball, was made to turn around, and it was forced into me.
"The game went on for a long time. My buttocks were ever the more sore. It felt quite enormous to me. I'm sure you understand the feeling. I felt swollen and huge, and very naked, and each welt was stinging under the paddle, and I was growing out of breath and desperate lest I fail, as I had to scurry ever farther away from her to retrieve the gold balls. But the new sensation was this filling of me, the stuffing of my anus, which I had now to hold tightly closed not to expel the gold balls against my will. I soon felt that my anus was widened and open, and at the same time stuffed mercilessly.
"The game grew more and more frantic. I soon glimpsed others watching from the doors. I had often to hurry past the hem of another Lady in waiting.
"I worked harder and harder, was stuffed ever the more firmly by those strong leather-sheathed fingers. And though the tears were pouring down my face, and I was breathing rapidly and hoarsely, I managed to complete the game with no more than four cracks of the paddle at any round of it.
"The Queen embraced me. She kissed me on the mouth and told me I was her loyal slave and her favorite. There was a ripple of approval throughout the Court, and she let me lie against her breasts for an instant as she held me.
"Of course I was suffering. I was struggling to hold in the gold balls, and also to not let my penis rub against her gown and disgrace me.
"She now sent for a small golden chamber pot. I knew then what was expected of me. And I know I must have blushed furiously. I had to squat over it and expel the toys I had gathered, and of course I did so.
"The day was an endless round of tasks after that.
"I shan't try to tell you all of them, save I had the Queen's absolute attention and absorption, and I intended with all my heart to keep it. I still did not know for certain that I might not be sent back to the kitchen. At any moment, I might be sent back to the kitchen.
"I remember many things. We were in the garden for a long time, the Queen walking among her roses as she much enjoys, and driving me along with that rod with the leather phallus at the end of it. It seemed sometimes she almost lifted my buttocks on the rod. My knees badly needed the relief of the soft grass after the floors of the castle. And I was so sore and tender by this time that the slightest stroke to my buttocks brought pain. But she only walked me about. And then she came to a little summer house of lattices and vines, and drove me ahead of her onto the flagstones there.
"She ordered me to rise, and a Page appeared, I don't remember if it was Felix, and he manacled my hands together over my head so that my toes barely touched the ground. The Queen seated herself right in front of me. She laid the phallus rod aside and lifted yet another rod that had been chained to her girdle. It was merely a long thin strip of wood covered with leather.
"'Now you must talk to me,' she said. 'You must address me as your Highness, and you must answer my questions very respectfully.' I felt an almost uncontrollable excitement at this. I would be allowed to speak to her. Of course I never had. In my rebelliousness I had always been gagged, and I did not know how I would feel when allowed words. I was her puppy dog, her mute slave, and now I must speak to her. She toyed with my penis; she lifted my balls on her thin leather stick and pushed them to and fro. She gave my thighs a playful smack.
"'Did you enjoy serving the crude Lords and Ladies of the kitchen?' she asked me playfully, 'or would you rather serve your Queen?'
"'I want to serve only you, your Highness, or as you wish.' I answered quickly. My own voice sounded strange to me. It was mine, but I had not heard it in so long, and when I voiced this subservience it was as if I had only just discovered it. Or rather I discovered it anew, and it produced an extraordinary outpouring of emotion in me. I wept, and hoped that it did not displease her.
"She rose then and stood very close to me. She touched my eyes and my lips. 'All this belongs to me,' she said, 'and this,' and she touched the nipples of my chest which the kitchen boys had never spared, and she touched my belly and my navel. 'And this,' she said, 'this too, belongs to me,' and she held my penis in her hand, her long nails scratching at the tip of it gently. It gave off a little fluid then and she withdrew her hand and held my scrotum in her hands and claimed that as well. 'Spread your legs,' she said and turned me on the chain that held me. 'And this is mine,' she said, touching my anus.
"I heard myself answer her, 'Yes, your Highness.' She then told me she had punishments worse, for me than the kitchen should I ever try to escape her again, or rebel or in any way displease her. But for the time being, she would be more than pleased with me, she was certain, and she would work me hard as was her pleasure. She said I had great strength for her sport which Prince Gerald had not, and she would test that strength to the limit.
"Each morning, she would spank me on the Bridle Path. At noon I would accompany her on her walks in the garden. In the late afternoons, I would play games of fetch for her. In the evening I should be spanked for her amusement while she took her supper. There were many positions for me to assume. She liked to see me opened wide in the squat, but there were even better attitudes in which she chose to study me. She squeezed my buttocks then and said these above all belonged to her, as it was her delight to punish these more than anything else. But to complete the day's regimen in the future, I should undress her for bed, and sleep in her chamber.
"To all this, I said 'Yes, your Highness.' I would have done anything to retain her favor. Now she said my buttocks would be subjected to the greatest tests to ascertain their limits.
"She had me unshackled, and drove me herself on the phallus rod through the garden and into the castle. We went into her chambers.
"I knew now she meant to place me over her lap and to spank me as intimately as she had done with Prince Gerald. I was in a welter of anticipation. I didn't know how I should keep my penis from discharging its craving. But she had thought of this. She inspected me and said that the cup needed draining just now so that it might fill again. It was not that I was to be rewarded. Yet she sent for a magnificent little Princess. The girl at once took my organ in her mouth, and as soon as she began to suck it, my passion exploded in her. The Queen observed all this; she stroked my face and examined my eyes, and my lips, and then bid the Princess to awaken me again quickly.
"This was its own form of torture. But I was soon enough as unsatisfied as before, and ready for the Queen to begin her test of endurance. I was placed over her lap, just as I had suspected I might be.
"'You've been soundly spanked by Squire Felix,' she said. 'You've been well spanked by the stable boys and the cooks in the kitchen. Do you think a woman can spank as hard as a man?' I was weeping already. I cannot anatomize the emotion I felt. Perhaps you felt it when you were over her lap earlier tonight. Or when the Prince had laid you down in the same position. It is not worse than being slung over a Page's knee, or tied with your hands over your head, or even pressed flat to a bed or table. I cannot explain it. But one feels so much more helpless across the lap of the master or mistress."
Beauty nodded. It was very true. And she had felt it when laid over the Queen's lap. All her composure had left her.
"In this position alone all obedience and subjugation can be taught, I think," said Prince Alexi. "Well, so it was with me. I lay over her lap, my head dangling, my legs spread out behind me. She wanted them slightly apart, and of course I was to arch my back and keep my hands clasped in the small of my back just as you were instructed to do. I was to see that my penis did not touch the cloth of her gown, though with all my strength I wanted it to. And then she commenced her spanking. She showed me each paddle and told me its faults and virtues. Here was one that was light; it would sting; and it was fast. A heavier one, just as thin, caused more pain and must be used carefully.
"She commenced to spank quite forcefully. And as with you, she would massage my buttocks, and pinch them when she chose. But she was steady in her work. She spanked hard and long until I was soon in terrible pain, and feeling as helpless as ever I had felt in my life.
"It seemed I felt the shock of each blow disseminated through my limbs. My buttocks of course absorbed it first. They became the center of myself in their soreness and tenderness. But the pain passed through them and into me, and all I could do was quiver beneath each blow, shudder with each sound spank, and moan ever more louder but never with the slightest hint of asking for mercy.
"The Queen was quite delighted with this show of suffering. As I told you before, she had encouraged it. She often lifted my face and wiped away my tears and rewarded me with kisses. Sometimes she would have me kneel on the floor upright. She would inspect my penis and ask if it were not hers. I would say 'Yes, your Highness, all of me is yours. I am your obedient slave.' She praised this answer, and said I must not hesitate in giving her long, devoted answers.
"But she was quite determined. She picked up the paddle again quickly enough, pressed me down again on her lap, and commenced the loud and forceful spanking. I was soon moaning loudly behind my clenched teeth. I had no pride, none of that dignity you still display unless it was completely without my knowledge. Finally she said that my buttocks were now a perfect color.
"She hated having to punish me further she so admired the color she had achieved, but she must know my limits.
"'Are you sorry you were such a disobedient little Prince?' she asked me. 'Very sorry, your Highness,' I answered through my tears. But she continued with her spanking. I could not prevent tightening my buttocks and moving as if somehow it would lessen the pain, and I could hear her laughter as though this quite delighted her.
"I was sobbing as frantically as any young Princess when she at last finished, and forced me back on my knees, ordering me to come about and kneel facing her.
"She wiped my face, blotted my eyes, and gave me a generous kiss with a great deal of sweet flattery. I would be her valet, she said, the master of her wardrobe. I alone would dress her, and brush her hair, and otherwise attend her. I would have much to learn, but she would instruct me herself. I should be very pure.
"Of course, that night I thought I had endured the worst, the abuse of common soldiers on my way to the castle, the frightful abuse in the kitchen. I had been thoroughly humbled by a coarse stable boy, and was now her abject pleasure slave with a soul that belonged to her with all the parts of my body totally. But I was very foolish in this. Much worse was to come."
Prince Alexi paused and looked down at Beauty who lay with her head against his chest.
She struggled to conceal her feelings. She did not truly know what she felt except that the story had aroused her. She could envision each humiliation Alexi described, and though her fear was aroused, so was her passion.
"It has been much easier for me," she said meekly, but this was not what she wanted to say.
"I'm not sure that is true," said Alexi. "You see, after the rough treatment of the kitchen when I became something less than an animal to my captors, I was liberated at once into being the Queen's obedient slave. You have had no such immediate liberation," he said.
"And this is what is meant by yielding," she murmured. "And I must come to it through a different path."
"Unless ... unless you do something to be vilely punished," said Alexi, "but that might take too much courage. And it might be unnecessary, for your dignity is being stripped away from you just a little already."
"I had no dignity tonight," Beauty protested.
"O yes, you did, you had a great deal of it," Alexi smiled. "But to continue, at this time, I had yielded only to my stable boy Lord and to the Queen. And once I was in her hands, I forgot my stable boy Lord completely. I was the Queen's property. I thought of my limbs, my buttocks, my penis, as hers. But I to truly yield, I had to experience much greater exposure and discipline ..."
**PRINCE ALEXI'S EDUCATION CONTINUES**
**I** WON'T tell you the details of my training with the Queen, how I learned to be her valet, my struggles with her annoyance. All this you'll learn in your training with the Prince for in his love for you he intends to make you his maidservant clearly. But these things are nothing when one is devoted to the master or mistress.
"I had to learn serenity in facing humiliations that brought others into play, and this was not easy.
"My first days with the Queen were mostly training in her bed chamber. I found myself rushing as diligently as Prince Gerald had to obey her slightest whim, and, proving very clumsy with her clothes, was often severely punished.
"But the Queen did not want me merely for these servile tasks which other slaves had been trained to perform to perfection. She wanted to study me, to break me down and make of me a toy for her complete amusement."
"A toy," Beauty whispered. She had felt like a toy in the Queen's hands exactly.
"And it amused her greatly in the first weeks to see me serve other Princes and Princesses for her pleasure. The first I had to serve was Prince Gerald. He was now nearing the end of his service, but he did not know that, and he was in a paroxysm of jealousy at my reformation. The Queen, however, had splendid ideas for rewarding him and soothing him, and at the same time developing me according to her wishes.
"Daily he was brought to her chamber and bound with his hands over his head against the wall so that he might watch me struggling with my tasks, and this was a torment to him until he realized that one of my tasks would be to give him pleasure.
"I was being driven to distraction then by the Queen's paddle, the flat of her hand, and the struggle to learn grace and accomplishment. All day I fetched, laced shoes, bound girdles, brushed hair, polished jewels, and performed any other menial task the Queen wished, my buttocks forever sore, my thighs and calves marked from the paddle, my face stained with tears as any other slave in the castle.
"And when the Queen could see that Prince Gerald's jealousy had hardened his penis to extremity, when he was all but ready to discharge his passion without the aid of any stimulant, then she set me to bathing him and satisfying him.
"I can't tell you how degrading this was to me. His body was nothing but my enemy. And yet I was to fetch a bowl of warm water, a sea sponge, and with my teeth only to hold it, bathe his genitals.
"He was positioned on a low table for this, kneeling obediently as I washed his buttocks, dipped the sponge again, bathed his scrotum and finally his penis. But the Queen wanted more than this. I must now use my tongue to cleanse him. I was horrified, and shedding tears like any Princess. But she was adamant. With my tongue I licked his penis, the balls, and then delved into the crack of his buttocks, even entering into his anus, which had a sour, almost salty taste to it.
"All the while he showed his obvious pleasure and longing.
"His buttocks were sore, of course. And it gave me great satisfaction that the Queen seldom spanked him anymore herself, but rather had it done by his groom before he was brought into her presence. So he didn't suffer for her; rather he suffered in the Slaves' Hall, ignored by those around him. Yet it was mortifying to me that my tongue stroking his welts and red marks gave him pleasure.
"Finally, the Queen ordered him to kneel up, his hands behind his back, and told me I should now fully reward him. I knew what this meant, yet I pretended I did not. She told me to take his penis in my mouth and drain it.
"I can't explain how I felt then. I felt I could not do it. And yet within seconds I had obeyed, so afraid of displeasing her as I was, and his thick penis was pushing against the back of my throat, my lips and jaws aching as I tried to suck it properly. The Queen gave me instructions, to make my strokes long, to use my tongue, and to go faster and faster. She spanked me unmercifully as I obeyed, her smacking blows in perfect rhythm with my sucking. At last his seed filled my mouth. I was commanded to swallow it.
"But the Queen was not at all pleased with my reticence. She said that I must show no disinclination towards anything."
Beauty nodded, remembering the Prince's words to her in the Inn, that even the lowly must be served for his pleasure.
"So she sent for all those Princes who had been tortured over a day's time in the Hall of Punishments, and led me to a large adjoining parlor.
"When six young men were brought in on their knees, I begged her to be merciful the only way I could, with my moans and kisses. I can't tell you how their presence affected me. I'd been mistreated by the peasants in the kitchen; I'd humbly, greedily, obeyed a rude stable boy. But these seemed both lower and higher than those others. They were Princes of my same rank in the world, haughty, proud in their own lands, and also they were abject slaves as lowly now as I was.
"I couldn't understand my own misery. I realized then that there would be endless variations in humiliation. It was not a hierarchy of punishments I faced; it was rather endless _changes._
"But I was too frightened of failing the Queen to think very much. Again I lost sight of the past and the future.
"As I knelt at her feet weeping silently, she ordered all of these Princes who were sore and aching and starved from the torture of the Hall of Punishments to take paddles from the chest she kept for the purpose.
"They formed a line of six to my right, each on his knees, his penis hardened as much by the sight of my suffering as by any pleasure that soon awaited them.
"I was told to kneel up with my hands behind my back. As I ran this gauntlet I would not even be allowed the easier and more concealing position on all fours. Rather I must struggle straight-backed, my knees apart, my own organ revealed, my progress slow as I sought to escape their paddles. They could see my face as well. I felt more exposed than I had when tethered in the kitchen.
"The Queen's game was simple. I would be made to run the gauntlet and the Prince whose paddle pleased her the most, that is, the one whose paddle struck me the hardest and fiercest, would then be rewarded before I again commenced the gauntlet and so on.
"I was urged by her to great speed; if I faltered, if my punishers accomplished too many blows, I should be delivered over to them for an hour of rough sport out of her sight, I was promised. This terrified me. She would not even be there. It would not be for her pleasure.
"I commenced at once. All their blows felt the same to me, loud, violent, and their laughter filled my ears as I struggled awkwardly in a position they had long ago learned to accomplish easily.
"Rest came only with each little session of satisfying the Prince who had marked me the most severely. I must return to him where he knelt. The others were free to watch and watch they did, and then given permission to offer their instructions.
"I had a half dozen masters eager to teach me contemptuously how to satisfy the one whom they supported with their arms as he shut his eyes and enjoyed the warm anxious sucking I afforded.
"Of course all of them prolonged it as best they could for fullest satisfaction, and the Queen who sat nearby, her elbow on the arm of her chair watched all this approvingly.
"Strange changes occurred in me as I performed my duties. There would be the frenzy of struggling past their paddles. My smarting buttocks, my sore knees, and above all the shame that they could so easily see my face, and my genitals.
"But as I took to sucking I found myself lost in contemplating the organ in my mouth, its size, its shape, its taste even and the sour salty taste of the fluids emptied into me. It was the rhythm of the sucking as much as anything else. The voices around me were a chorus that became noise at some point, and an odd feeling of being weak and abject came over me. It was very similar to the moments I'd experienced with my tall stable boy Lord when we had been alone in the garden, and he had made me squat on the table. I felt my excitement even on the surface of my skin then, and so it was now, sucking these various organs and being filled with their seed. I can't explain it. It became pleasurable. It became pleasurable because it was repeated and I was helpless. And it was repeated always as a respite from the paddle, the frenzy of the paddle. My buttocks would throb, but they felt warm; they were tingling, and I was tasting this delicious cock that was pumping its force into me.
"I found I _liked_ having so many eyes watch me. But I did not admit this to myself all at once. I felt not _liking_ so much as this weakness again, this limpness of the spirit. I was _lost_ in my suffering, my struggles, my anxiousness to please.
"Well, so it would be with each new task that lay before me. I would at first resist with terror; I would cling to the Queen with my heart; then at some point in the midst of unspeakable humiliation, I would be released into some state of calm in which my punishment became sweet to me.
"I saw myself as _one_ of these Princes, one of these slaves. When they instructed me in sucking the penis better I listened to them. When they paddled me I received the blow, bent my body in response to it.
"Perhaps it's impossible to explain. I was moving towards yielding.
"When finally the six Princes were sent away, all of them properly rewarded, the Queen took me into her arms and rewarded me with her kisses. As I lay on the pallet by her bed, I felt the most delicious exhaustion. It seemed even the air stirring around me gave me pleasure. I felt it against my skin, as if my nakedness were being stroked. And I fell asleep content that I had served her properly.
"But my next great test of strength came one afternoon when she, very cross with me for my ineptitude at brushing her hair, sent me to be the plaything of the Princesses.
"I could scarcely believe my ears. She herself would not even deign to witness it. She sent for Lord Gregory, and told him I was to be taken to the Special Punishment Hall and there given to the assembled Princesses. For one hour they could do with me what they liked. And then I was to be bound in the garden and have my thighs whipped with a leather thong, and left there until morning.
"This was my first great separation from the Queen. And I could not imagine myself, naked, helpless, and fit only for punishment, given over to the Princesses. I had dropped the Queen's hairbrush twice. I had spilled some wine earlier. All of this had seemed beyond my control and my finest efforts.
"When Lord Gregory gave me several hard spanks I was full of shame and fear. And as we neared the Special Punishments Hall, I felt I could not move by my own power.
"He had placed a leather collar around my neck. He pulled me along, spanking me only lightly as he said the Princesses must have the full enjoyment of me.
"Before we entered the room, he put a sign about my neck by means of a small ribbon. He showed it to me first, and I shuddered to see it announced me as clumsy, willful and bad, and in need of correction.
"He then exchanged my leather collar for another which had numerous small metal rings attached to it, each ring just large enough for a finger to be hooked in it. That way the Princesses could pull me this way or that, he said, and woe to me if I showed the slightest resistance.
"Cuffs with the same rings were put on my ankles and my wrists. I felt myself scarcely able to move as I was pulled towards the door.
"I did not know how to assess my emotions. As the door opened, I saw them all, some ten Princesses, a naked harem lounging about under the watchful eye of a groom, all girls being rewarded for good behavior by this hour of leisure. I learned later that if anyone is to be severely punished he or she is given to them, but on that day they had not expected anyone.
"They shrieked with delight, clapping their hands and immediately conferring with one another. All around me I saw their long hair, red, golden, raven hair in deep waves and thick curls, their naked breasts and bellies, and those hands pointing to me, and shielding their own shy and shameful whispers.
"They clustered about me. I crouched trying to conceal myself. But Lord Gregory lifted my head by means of the collar. I felt their hands all over me, feeling my skin, slapping my cock, and touching my balls as they squealed and laughed. Never had some of them seen a man so closely, save for their Lords who had complete power over them.
"I was trembling violently. I had not given way to tears and was most afraid that I would turn and try to escape, and receive only some worse punishment. I tried desperately to remain coldly indifferent. But their round naked breasts were maddening me. I could feel their thighs brush me, even their moist pubic hair as they crowded about examining me.
"And I was their complete slave, whom they scorned and admired. When I felt their fingers touching my balls, weighing them, stroking my penis, I was maddened.
"It was infinitely worse than my time with the Princes, because I could already hear their voices turning to mock contempt with the wish to discipline me, to return me to the Queen as obedient as they were. 'O, so you're a bad little Prince, are you?' said one of them in my ear, she a lovely raven-haired one with pierced ears adorned with gold. Her hair was tickling my neck, and when her fingers twisted the nipples of my chest, I felt myself losing my control.
"I feared I would break and try to flee. Lord Gregory meantime withdrew to the corner of the room. The grooms could aid them as they wished, he said, and he urged them to do their work well for the Queen's sake. This brought loud cries of delight. I was immediately slapped by several hard little hands. My buttocks were pulled open. I felt tiny fingers pushing into them.
"I squirmed, twisted, trying to hold still, trying not to look at them.
"And when I was pulled up and my hands were tied over my head from a ceiling chain, I was immensely relieved that I could no longer try to escape if I weakened.
"The grooms gave them what paddles they wanted. A number of them chose long leather straps which they tried first across their hands. In the Special Punishment Hall they had no need to stay on their knees and could stand around me as they liked. At once the round handle of a paddle was thrust in my anus. My legs were dragged wide apart. I was shuddering, and when the paddle handle proceeded to rape me with back and forth thrusts as roughly as any cock I'd ever received, I knew my face was red, and that my tears were threatening. Now and then in the midst of all this, I'd feel cool little lips pressed to my ear, my face pinched, my chin stroked, and then they would assault my nipples again.
"'Beautiful little tits,' said one of the girls as she did this. She had flaxen hair, as straight as yours. 'When I've finished my work, they'll feel like breasts,' she said, and proceeded to stretch them, and stroke them.
"All the while, to my shame, my cock was hard as if it knew its mistresses even when I refused to acknowledge them. This girl with the flaxen hair laid her thighs against mine, growing fiercer as she pulled on my nipples. I felt her wet sex against me. 'You think you are too good to suffer at our hands, Prince Alexi?' she crooned. I wouldn't answer her.
"Then the paddle handle in my anus thrust harder and more roughly. My hips were pushed forward as cruelly as they had ever been by my stable boy Lord, and I was almost lifted off the floor by the thrusts. 'You think you are too good for us to punish?' she asked again. The other girls laughed and watched her as she commenced to slap my cock hard from right to left. I winced, I could not quite control myself. I wished for all the world that I were gagged, but I wasn't. She ran her fingers over my lips and my teeth to remind me of this, and commanded me to answer her respectfully.
"And when I didn't she took her paddle now, and withdrawing the instrument of rape, she proceeded, as she kept her face next to mine, her eyelashes tickling the side of my face, to spank me soundly. Of course I was already sore as we all are, always, and her blows were very hard, and they were without rhythm. She caught me off guard and when I winced and groaned, all the girls laughed appreciatively.
"My cock was slapped by others. My nipples twisted by them, but she had clearly shown her supremacy. 'You will beg me for mercy, Prince Alexi,' she said. 'I am not the Queen, you may beg me, for all the good it will do you.' They thought all that was amusing too, and she continued to spank me harder and harder. I prayed she would break the skin before my will broke, but she was too clever for that. She spread the blows. She had them lower the chain slightly, so she could spread my legs even wider.
"She held my cock now in her left hand, tightly, roughly, running her open palm over the tip to torment me, and then tightening her grip again as she spanked furiously.
"When she slapped my nipples and my cock, and lifted my balls in her hands, I felt the tears flowing, and overcome with shame I groaned unable to conceal it. It was an astonishing moment of pain and pleasure. My buttocks were raw.
"But she had only just begun. She ordered the other Princesses to lift my legs in front of me. I felt terror to feel myself hanging from the chain above me. They did not bind my ankles to my arms; they merely held them up, in place, as she brought her blows up from under, as hard as before, and then covering my balls with her left hand, paddled me from the front as hard as she could as I struggled and moaned now uncontrollably.
"Meantime the other girls were feasting their eyes on me, touching me still, and enjoying my misery immensely. They even kissed the backs of my legs, my calves, my shoulders.
"But the blows came harder and faster. She had me set down again, spread wide again, and went to work in earnest. I think she meant to break the skin if she could, but I was now broken down and wept uncontrollably.
"This was what she wanted, and as I gave in, she applauded it. 'Very good, Prince Alexi, very good, let all of that spiteful pride go, very good, you know very well you deserve it. That's better, that is exactly what I want to see,' she said almost affectionately, 'delicious tears,' as she touched them with her fingers, her paddle never stopping.
"Then she had my hands released. I was forced down on all fours. And she drove me about the room telling me that I must move in a circle. Of course she drove me faster and faster. I didn't even realize now that I was no longer restrained. That is, I did not even realize I might have broken and run. I had been defeated. And finally, it was as always when the punishment works, I could think of nothing but escaping each blow of her paddle. And how could I do that? Merely twist, squirm, try to avoid it. She was meantime very worked up with her commands, driving me faster. I rushed past the naked feet of the other Princesses. I saw them step aside for me.
"And now she told me that crawling was indeed too good for me, that I must place my arms on the ground, and my chin, and must inch along in that manner, my buttocks high in the air where she might paddle them. 'Arch your back,' she said, 'down, I want your chest pressed to the ground,' and as skillfully as any Page or mistress, she forced me along as the others praised her and marveled at her skill and stamina. I had never been in such a position. It was so ignominious I didn't want to picture it, my knees still scraping along, my back painfully arched, my buttocks thrust as high as before. And she commanding me always to move with greater speed as my buttocks grew ever more raw. They were throbbing as the blood throbbed in my ears. And my tears were now blinding me.
"And it was then that that moment came I spoke of earlier. I belonged to this girl with the flaxen hair, this impudent, clever Princess who herself was punished as shamefully as I was day in and day out but for the moment could do as she wished with me. I struggled along, glimpsing Lord Gregory's boots, the boots of the grooms, hearing the girls' laughter. I reminded myself that I must please the Queen, I must please Lord Gregory, and finally I must please my cruel flaxen-haired mistress.
"She paused for breath. She exchanged her paddle for a leather strap and proceeded to lash me.
"At first it felt weaker than the paddle, and I felt a merciful relief. But she immediately learned to swing it with such strength it walloped the welts on my buttocks. Now she let me stop so she might feel these welts. She pinched them, and in the silence I could hear my own low crying.
"'I think he is ready, Lord Gregory,' said the Princess, and Lord Gregory said softly, he thought that I was. I thought it meant I would be returned to the Queen.
"This was very foolish of me.
"It was only that I would now be lashed swiftly into the Hall of Punishments. Of course there were a handful of Princesses chained from the ceiling, their legs tied up in front of them. Now she brought me up to the first of these.
"She told me to rise and to spread my legs very wide as I stood before her. I saw the captive Princess's pained face, her blushing cheeks, and then her naked and moist sex peering shyly from its wreath of golden pubic hair, much ready for pleasure or more pain, after days of teasing. But it hung low, at the level of my chest, I suppose, and that was just as my tormentor liked it.
"For she ordered me to bend over towards it, and to thrust my hips out behind me. 'Give me your buttocks,' she said. She stood behind me. The other girls pulled my legs wider than I myself could spread them. Again I was told to arch my back and to put my arms around the bound and doubled Princess slave hanging in front of me.
"'Now you will pleasure her with your tongue,' said my captor, 'and see you do it well as she has suffered long, and for not even half your clumsiness.'
"I looked at the bound Princess. She was mortified, though desperately hungry for pleasure. And I pressed my face into her sweet, hungry little sex, rather eager to pleasure her. But as my tongue delved into her swollen cleft, as I licked her little clitoris, and her swollen lips, I was walloped by the belt continuously. My flaxen-haired one chose one welt after another for her work, and I was in great pain as the bound Princess finally shuddered with pleasure in spite of herself.
"Of course, there were others who had been punished enough and now must be rewarded. I did my work as best I could, finding a refuge in it.
"And then with a panic I saw there were no more to be rewarded. I was again at the hands of my captor with nothing so sweet as a bound Princess in my arms.
"And again, my chest and chin pressed to the floor, I struggled along on my knees under the wallops of her strap back to the Special Punishment Hall.
"Now all the Princesses begged Lord Gregory to make me pleasure them, but Lord Gregory silenced them at once. They had their Lords and Ladies to serve, and he would not hear one word from them more unless they wanted to be hung from the ceiling of the other Hall as they deserved.
"I was now taken away, and out into the garden. As the Queen ordered, I was taken to a large tree, and my hands were tied up high so that my feet barely touched the grass beneath them. It was growing dusk and I was left there.
"It had been excruciating, but I had obeyed, I had not run away, and there had come that moment. I was tormented now only by usual needs, my aching cock which might not be rewarded for another day or more by the Queen due to her anger.
"But the garden was quiet, full of twilight sounds. The sky was purple and the trees were thickening with shadows. In a little while, they grew skeletal, the sky was white with evening, and then darkness fell around me.
"I had resigned myself to sleep in this manner. I was too far from the trunk of the tree to rub my miserable cock against it, or I would have done this, tormented as I was, for whatever pleasure the friction could give me.
"And from habit more than training, its hardness would not die away. I remained stiff and tense as though expecting something.
"Then Lord Gregory appeared. He materialized out of the dark in his grey velvet, with the gold on the edge of his cloak gleaming. I saw the shimmer of his boots, and the dull sheen of the leather strap he carried. More punishment, I thought wearily, but I must obey. I am a slave Prince and there is nothing to be done for it. Pray I have the grace to bear it in silence and without struggling.
"But he drew close to me and commenced to talk to me. He told me I had comported myself very well and asked me if I knew the name of the Princess who had tormented me. I said 'No, my Lord,' respectfully taking some relief that I had pleased him. He is very hard to please. Harder than the Queen.
"He then told me her name was Princess Lynette, and she was new and had made a great impression on everyone. She was the personal slave of the Grand Duke Andre. 'What is this to me,' I thought, 'I serve the Queen.' But he asked me pleasantly enough if I had found her pretty. I winced. How could I help it? I could remember her breasts well enough when she pressed them to me while her paddle made me smart and groan. I could remember her dark blue eyes for the one or two instants when I had not been too ashamed to look at them. 'I don't know, my Lord. I would think she would not be here,' I said, 'were she not pretty.'
"For that impertinence, he gave me at least five rapid cracks with his belt. I was sore enough to be immediately in tears. He has often said that if he had his way, he would keep all slaves that sore always. Then their buttocks would be so tender that all he would have to do was stroke them with a feather. But as I stood there, my arms stretched painfully above me, my body pushed off balance by his blows, I was aware that he was particularly angered and fascinated by me. Why else would he come here to torment me? He had a castle of slaves to torment. It gave me some strange satisfaction.
"I was conscious of my body, its obvious muscularity, what to some eyes was surely its beauty.... Well, he came around and he said to me that Princess Lynette was unsurpassed in many respects and that her attributes were fired with an unusual spirit.
"I _feigned_ boredom. I _was to hang_ in this position all night. He was a gnat, I thought. But then he told me that he had been to the Queen and told her how well Princess Lynette had punished me, that Princess Lynette showed a flair for command and shrank from nothing. I began to grow afraid. Then he assured me the Queen had been glad to hear it.
"'And so was her master, the Grand Duke Andre,' he added, 'and both were curious and somewhat regretful that such a display had not been witnessed by them, being wasted only on other slaves.' I waited. 'So a little amusement has been arranged,' he continued when I said nothing. 'You shall perform a little circus for her Highness. Surely you have seen the trainers of animals in circuses, who with deft strokes of their whips place their trained cats on stools, and force them through hoops, and other tricks for the amusements of the audience.'
"I felt desperate but I did not answer. 'Well, on the morrow, when your handsome buttocks have healed somewhat, such a little spectacle shall be arranged with the Princess Lynette and her strap to drive you through the performance.'
"I knew my face was scarlet with rage and indignation, or worse, it showed my frantic despair, but it was too dark for him to see it. I could see only the gleam of his eyes, and how I knew that he smiled I wasn't certain. 'And you shall perform your little tricks quickly and well,' he went on, 'for the Queen is eager to see you hop upon this stool and that, crouch on all fours, and then jump through the hoops that are just now being prepared for you. Because you are a two-legged pet with hands as well as feet, you can as well swing from a little trapeze that is being prepared for you, with Princess Lynette's paddle ever to spur you on, and entertain all of us as you show your agility.'
"It seemed unthinkable to me, performing this. It was not service after all, not the dressing and adorning of my Queen, not the fetching for her to show I accepted her power and worshiped her. Not suffering for her, receiving her blows. But rather a series of willfully executed ignominious positions. I couldn't endure the thought of it. But worst of all, I couldn't imagine myself managing to do it. I should be dreadfully humiliated when I failed in will, and was then dragged off again to the kitchen surely.
"I was beside myself with rage and fear, and this menace, this brutal Lord Gregory whom I hated so was smiling at me. He took hold of my cock and pulled me forward. Of course he had it at the root, not near the tip where it might have given me some pleasure. And as he tugged my hips so that I lost my footing, he said, 'This will be a grand spectacle. The Queen, the Grand Duke and others shall witness it. And Princess Lynette shall be very eager to impress the Court. See to it that she does not _outshine_ you.' "
Beauty shook her head then and kissed Prince Alexi. She now saw what he meant when he said he had only just begun to yield.
"But Alexi," she said gently, almost as if she could save him from his fate, as if it hadn't already happened long ago, "When you were brought by the stable boy to the Queen's presence, when she made you fetch the golden balls for her in the parlor, was this not something of the same thing?" She stopped. "O, how shall I ever do these things!"
"But you can do them, all of them, that's the point of my story," he said. "Each new thing seems terrible because it is new, because it is a variation. But at the heart it is all the same ultimately. The paddle, the strap, the exposure, the bending of the will. Only they infinitely vary it.
"But you do well to mention this first session with the Queen. It was similar. But remember I was raw and shaken from the kitchen, and thoughtless. I had regained my strength since then, and my strength had to be broken down again. Now perhaps had the little circus been constructed when I was fresh from the kitchen I would have taken to it eagerly then too. But I think not. It encompassed much greater exposure, much greater stamina, much greater surrender of self into positions and attitudes that appeared grotesque and inhuman.
"No wonder they need no real cruelty, no fire, no whips, to teach their lessons or amuse themselves," he sighed.
"But what happened? Did it come about?"
"Yes, of course, though Lord Gregory had no need of telling me beforehand except to rob me of sleep. I spent a painful restless night. I awoke many times thinking others were near, the stable boys, or the kitchen servants, that they had found me helpless and alone and meant to torment me. But no one approached me.
"During the night I heard whispers of conversation as Lords and Ladies walked under the stars. Now and then I even heard a slave driven past, crying fitfully under the inevitable smack of the leather. A torch would flicker under the trees, nothing more.
"When the morning came, I was bathed, and rubbed with oil, and all this time, my penis was not touched, save when it flagged. Then it was cleverly awakened.
"At twilight, the Slaves' Hall was full of talk of the circus. I was told by my groom, Leon, that the circle for performance had been prepared in a spacious hall near the Queen's apartments. There would be four rows of Lords and Ladies surrounding it, and they would bring their slaves, too, to see the amusement. The slaves were in a state of dread, lest they be made to perform. He said nothing more than that, but I knew what he was thinking. It was a grueling test of self-control. He combed my hair, rubbed much oil into my buttocks and thighs, even oiled my pubic hair a little and brushed it so it would be glossy.
"I was quiet. I was thinking.
"And when I was finally brought into the room, into the shadows near the wall from which I could see the illuminated circle, I understood what I had to do. There were stools of various heights and various circumferences. There were trapezes hung so and great hoops mounted perpendicular to the floor. Candles burned everywhere on high stands among the chairs of the Lords and Ladies who were already assembled.
"And the Queen, my cruel Queen sat in state, with the Grand Duke Andre beside her.
"Princess Lynette stood in the middle of the circle. So she would be allowed to stand, I mused, and I should be driven in on hands and knees. Well, I must make up my mind.
"And as I knelt there waiting I decided that resistance was impossible. Were I to try to hide my tears, were I to grow tense, my humiliation would only be more dreadful.
"I must make up my mind to do what I had to do. Princess Lynette looked exquisite. Her flaxen hair hung free down her back where it had been trimmed only enough to expose all of her buttocks. She had no more than a pink blush there from the paddle, and a blush on her thighs and calves too, which far from disfiguring her, appeared to shape her and improve her. It was infuriating. About her neck she wore a collar of gilded and worked leather that was a mere adornment. She wore boots as well, heavily gilded with high heels.
"And I of course was utterly naked. I did not even have a collar which meant I must control myself at her commands, I could not even be dragged this way and that.
"So I could see exactly what I must succeed in doing. She would put on a great show of inventiveness. She would be ready to vent her anger on me in commands of 'Hurry,' and 'Quickly' and scold and condemn for the slightest disobedience. She would therefore win the applause of the audience. And the more I struggled, the more she would shine, just as Lord Gregory had indicated.
"The only way I could triumph was through perfect obedience. I must execute all her commands to perfection. And I must not struggle either externally or inwardly. I must weep if I must weep, but I must do all she commanded, even if to think of it sent my heart to thudding in my wrists and temples.
"Finally everyone was ready. A handful of exquisite little Princesses had served the wine, swinging their pretty little hips and showing me some delightful sights as they bent over to fill the cups. And they too were to see me punished.
"All the Court, for the first time, was to see it.
"Then with a clap of her hands, the Queen ordered that her pet, Prince Alexi be brought in and that Princess Lynette 'tame' me and 'train' me before their very eyes.
"Lord Gregory gave me the usual quick smacks with the paddle.
"At once I was in the circle of light, my eyes hurt by it for the moment, and then I saw my trainer's high-heeled boots coming nearer. In a moment of impetuousness, I rushed to her and kissed both her shoes at once. The Court gave a loud murmur of approval.
"I continued to shower her with kisses, and I thought, 'My evil Lynette, my strong, cruel Lynette, you are my Queen now.' It was as if my passion were a fluid that coursed through all my limbs, not only my swollen cock. I arched my back and spread my legs ever so slightly without even being told to do so.
"At once the spanks commenced. But clever little demon that she was, she said, 'Prince Alexi, you will show your Queen that you are a very quick-witted pet, and you shall answer all my commands with your compliance. And you shall answer all my questions, too, with perfect courtesy.'
"So I would have to speak. I felt the blood rush to my face. But she gave no time for my terror, and I said with a quick nod of the head, 'Yes, my Princess,' to a murmur of the audience's approval.
"She was strong as I have told you. She could spank much harder than the Queen, and as hard as ever the kitchen boys or the stable boys had spanked me. I knew she meant to leave me sore if nothing else, because immediately she gave me several loud cracks, and she had that knack which some of our punishers have of lifting the buttocks with the paddle as she spanked them.
"'To that stool, there,' she commanded at once, 'at a squat with your knees wide apart and your hands behind your neck, now!' And she drove me at once to obey as I hopped up on the stool and with a great but quick effort managed to secure my balance. It was that same miserable squatting position in which my stable boy Lord had punished me. And all the Court could now see my genitals displayed if they hadn't seen them before.
"'Turn around slowly,' she commanded, in order to show me to all eyes, 'so the Lords and Ladies can see the little pet that performs for them tonight!' and again she gave me numerous exquisite cracks of the paddle. There was a sprinkling of applause from the little crowd, and the sounds of wine being poured, and no sooner had I executed a complete turn, the slapping of her paddle ringing in my ears, than she ordered me to make a quick turn around the little stage on all fours with my chin and chest on the ground as I had done for her earlier.
"It was here I had to remind myself of my intentions. I rushed fast to obey, arching my back, my knees apart, yet moving swiftly as the heels of her boots clicked beside me and my buttocks were writhing under her blows. I did not try to hold their muscles still, but let them tense, let my hips even rise and fall as they were inclined to do, shrinking from the blows, yet receiving them. And as I moved along the white marble floor, the room a blur of faces over me, I felt this is my natural state, this is what I am, there is nothing before me or after me. I could hear the responses of the Court; they laughed at this miserable position, and there was a growing excitement to their talk. The little performance had them much engaged, jaded as they were. I was being admired for my abandon. I groaned with each crack of the paddle without even thinking to stop it. I let the groans come freely, and arched my back to even greater degree.
"And when the task was complete and I was again driven into the center of the circle, I heard applause around me.
"My cruel trainer didn't pause. She commanded me at once to hop upon another stool, and from that stool to one that was even higher. I squatted on each in turn, and when her spanks caught me my hips moved forward with them without restraint and my moans, my natural moans, were surprisingly loud to me.
"'Yes, my Princess,' I said after each command, and my voice sounded tremulous to me, though deep, and full of suffering. 'Yes, my Princess,' I said again as she ordered me finally to stand before her, legs wide apart and slowly squat until I had achieved the height of which she approved. Then I must jump through the first hoop, hands behind my neck and somehow manage to squat again for her. 'Yes, my Princess,' I said and obeyed at once, and then through another hoop and another with the same compliance. I was agile and without the slightest shame, though my penis and balls moved ungracefully with my exertions.
"Her blows grew harder, less regular. My moans were very loud and sudden and provoked much laughter.
"And when she commanded me now to jump up and grab the bar of the trapeze in both hands, I felt the tears come purely from my stress and exhaustion. I hung from the trapeze as she paddled me, driving me back and forth, and then commanded me to twist and catch the chains above with my feet.
"This was quite impossible and as I struggled to obey, the hall echoed with laughter. Felix stepped forward and at once lifted my ankles until I was swinging as she had wished and I had to bear her spanks in this position.
"And as soon as she tired of this, I was ordered to drop to the ground, at which point she came forward with a long thin leather strap, and buckling the end of it around my penis, she now pulled me, on my knees towards her. I had never been so led or pulled before, by the very root of my cock, and my tears flowed copiously. My whole body was hot and trembling, and my hips were being tugged ahead of me so that no thought of grace could possibly exist even had I the presence of mind for it. She pulled me to the Queen's feet, and then turning, pulled me along, running on her clicking heels so that I struggled and groaned and cried behind my closed lips to keep up with her.
"I was wretched. The circle seemed endless. The strap around my penis constricted it, and my buttocks were so painfully tender now that they ached even when she was not striking them.
"But we'd soon completed the circle. I knew she had exhausted her inventiveness. She had relied upon my disobedience and reluctance, and encountering none, her show lacked any real feature save my complete obedience.
"But she had now a subtle test for me for which I was unprepared.
"She ordered me to stand up, spread my legs and then place my hands flat on the floor before her. I did so, my buttocks facing the Queen and the Grand Duke, a position which again, even in the midst of this, reminded me of my nakedness.
"She put aside her paddle, and picked her favorite toy then, the leather strap, and gave my legs a heavy strapping on both thighs and calves, letting the leather curl about me, and ordered me to move forward a few inches so that I might place my chin upon a high stool there. My hands must go behind my back, my back must be arched. I did as I was told and stood, spread-legged, bent at the waist, my face tipped up for all to see my miserable expression.
"As you can imagine my buttocks hung free in the air, and she commenced to shower them with compliments. 'Very pretty hips, Prince Alexi, very pretty buttocks, tough and round and muscular, and very pretty indeed when you squirm to escape my strap and my paddle.' She illustrated all this as I did with her strap, and I was crying softly now between my moans.
"It was then she gave a command which surprised me. 'But the Court wants to see you display your buttocks. They want to see you move them,' she said. 'Not merely to escape the punishment you so richly deserve and richly need, but to see some real show of humility.' I didn't know what she meant. She spanked me hard as if I meant to be stubborn, while I answered through my tears, 'Yes, my Princess.' 'But you do not obey!' she cried out. She had commenced what she really wanted, and as soon as she said this, I began to sob in spite of myself. What could I say to her? 'I want to see your buttocks move, Prince,' she said. 'I want to see them dance, while your feet remain still.' I heard the Queen laugh. And suddenly overcome with shame and fear, I knew the seemingly small thing she wanted of me was too much for me. I moved my hips, I moved them from side to side as she spanked me and my chest shook with another sob that I could scarce keep quiet.
"'No, Prince, nothing so simple as that, a real dance for the Court,' she said, 'your reddened and punished buttocks must do something besides sleep under my blows!' and she placed her hands on my hips then and slowly moved them not only from side to side, but down and around and up, so that I had to bend my knees. She rotated them. It seems a small thing as I say it. But to me it was unspeakably shameful, to have to swing my hips and rotate them, to put all my strength and spirit into this seemingly vulgar display of my buttocks. And yet she meant for me to do this, she had commanded it, and I could do nothing but obey, and my tears flowed and my sobs caught in my throat, as I rotated my buttocks as she commanded. 'Bend your knees deeper, I want to see a dance,' she said with a loud wallop of her strap. 'Bend your knees and move those hips more to the side, more to the left!' her voice rose angrily. 'You resist me, Prince Alexi, you don't amuse!' she said, and rained her smacking wallops on me as I strove to obey. 'Move!' she cried out. She was triumphing. All my composure was truly lost. She knew it.
"'So you dare reserve yourself in the presence of the Queen and her Court,' she scolded me, and then with both her hands she pulled my hips this way and that, making a greater rotation. I could endure it no longer. There was but one way to best her and that was to twist in this shameful position more wildly even than she guided me. And shaking with choked sobs, I obeyed her. There was immediate applause as I did this dance, my buttocks twisted from side to side and up and down, my knees bending deeply, my back arched, my chin resting painfully on the stool so all could see the tears coursing down my face, and my obvious destruction of spirit.
"'Yes, Princess,' I struggled to articulate in my supplicating voice, and I obeyed with all my strength putting on such a good performance that the applause continued.
"'That's good, Prince Alexi, very good,' she said. 'Spread your legs wider apart, wider and move your hips even more!' I obeyed at once. I was now snapping my hips, and I was overcome with the greatest shame I had known since I had been captured and brought to the castle. Not even the first stripping by the soldiers in the field, not even being thrown over the Captain's saddle, nor the raping in the kitchen compared to the degradation I knew now, because I performed all this gracelessly and obsequiously.
"Finally, she was finished with my little display. The Lords and the Ladies were talking among themselves, commenting, talking of all manner of things as they always do at such things, but the murmur was full of a certain restlessness, which meant their passions were aroused, and I did not have to look up to see they were all looking at the central circle no matter how they might have feigned boredom. Princess Lynette now ordered me to turn slowly, keeping my chin in the center of the stool, but moving my legs in a circle, all the while swinging my buttocks, so that all the Court might see this display of obedience equally.
"My own sobs were deafening to me. I struggled to obey without losing my balance. If I flagged in the slightest with the broad rotation of my buttocks, the Princess had an opportunity again to upbraid me.
"Finally, she raised her voice and announced to the Court that we had here an obedient Prince capable of even more imaginative amusements in the future. The Queen clapped her hands. The assembly could now rise and disperse, but they did so very slowly, and Princess Lynette, to continue the performance for the very last onlookers, quickly ordered me to grasp the trapeze over my head, and as she spanked me relentlessly, I was ordered to lift my chin and to march in place on my toes for her.
"Pain shot through my calves and thighs, but the worst as always was the burning and swelling of my buttocks. Yet I marched with my chin up as the hall emptied. The Queen had gone first. Finally all the Lords and Ladies were gone.
"Princess Lynette gave over her paddle and her strap to Lord Gregory.
"I stood holding the trapeze, my chest heaving, my limbs tingling. I had the pleasure of seeing Princess Lynette stripped of her boots and her collar by a Page who threw her over his shoulder, and then she was carried out, but I couldn't see her face, and did not know what she was feeling. Her buttocks were up in the air over the Page's shoulder; her pubic lips were long and thin, and her pubic hair reddish.
"I was alone, damp all over with sweat, and exhausted. Lord Gregory was standing there. And he came and lifted my chin and said, 'You are unconquerable, aren't you?' I was astonished. 'Miserable, proud, rebellious, Prince Alexi!' he said furiously. I tried to show my consternation. 'Tell me how I've displeased?' I begged, having heard Prince Gerald say that enough in the Queen's chamber.
"'You know you take pleasure in all of it. There is nothing that is too ungraceful for you, too undignified, too difficult. You play with all of us!' he said. Again, I was astonished.
"'Well, you will measure my cock for me now,' he said, and ordered the last Page to leave us. I still held the trapeze as commanded. The room was dark save for the luminous night sky through the windows. I heard him opening his clothes, I felt the nudge of his penis. And then he thrust it into my buttocks.
"'Damnable little Prince,' he said, as he drove into me.
"When it was finished, Felix slung me over his shoulder as unceremoniously as the other Page had carried Princess Lynette. My cock swelled against him, but I tried to control it.
"When he set me down in the Queen's chamber, she sat at her dresser filing her nails. 'I've missed you,' she said. I hurried to her on my hands and knees and kissed her slippers. She took a white silk handkerchief and wiped my face.
"'You please me very well,' she said. I was puzzled. What did Lord Gregory see in me that she did not see?
"But I was too relieved to ponder this. Had she greeted me with anger, ordered more punishments and amusements, I would have wept with despair. As it was, she was all beauty and softness. She ordered me to undress her and to turn down her bed. I obeyed as well as I could. But she refused the silk dressing gown.
"And for the first time, she stood before me naked.
"I had not been told I could look up. I was crouched at her feet. Then she said that I might look. As you can imagine, she was unspeakably lovely. She has a firm body, powerful somewhat, with shoulders just a little too strong for a woman, and long legs, but her breasts were magnificent, and her sex was a gleaming nest of black hair. I found myself breathless.
"'My Queen,' I whispered, and after I kissed her feet, I kissed her ankles. She did not protest. I kissed her knees. She did not protest. I kissed her thighs, and then impulsively I buried my face in that nest of perfumed hair, finding it hot, so hot, and she lifted me up until I was standing. She lifted my arms and I embraced her, and felt for the first time, her full womanly form, and also that no matter how strong and powerful she appeared, she was small next to me, and yielding. I moved to kiss her breasts, and she bid me silently do it, and I suckled them until she was sighing. They tasted so sweet, and they were so soft, yet plump at the same time and resistant to my respectful fingers.
"She sank down on the bed, and I on my knees buried my face between her legs again. But she said she wanted my cock now and that I must not 'come' until she allowed it.
"I moaned to show how difficult this would be out of love for her. But she lay back on her pillows, opening her legs, and I saw for the first time the pink lips there.
"She pulled me down. I could not quite believe it when I felt the sheath of her hot vagina. It had been so long since I had felt such satisfaction with a woman. Not since I had been taken prisoner by her soldiers had I felt it. I struggled not to consummate my passion at once, and when she commenced to move her hips I thought surely I would lose the struggle. She was so wet and hot and tight and my penis ached from punishment. All my body ached and the aching was delicious to me. Her hands caressed my buttocks. She pinched the welts. She spread my buttocks apart, and as this hot sheathing tightened on my penis, as the roughness of her pubic hair stroked me and tantalized me, she put her fingers into my anus.
"'My Prince, my Prince, you pass all tests for me,' she whispered. Her movements grew swifter, wilder. I saw her face and breasts suffused with scarlet. 'Now.' She commanded, and I pumped my passion into her.
"I rocked with the pumping of it, my hips snapping as wildly as they had in the little circus performance. And when I was emptied and quiet, I lay covering her face and her breasts with languid and sleepy kisses.
"She sat up in bed, and ran her hands all over me. She told me I was her loveliest possession. 'But there are many cruelties in store for you,' she said. I felt myself grow hard again. She said I should be subjected to a daily discipline far worse than any she had before invented.
"'I love you, my Queen,' I whispered. And had no thought other than serving her. Yet of course I was afraid. Though I felt powerful in all I had endured and accomplished.
"'Tomorrow,' she said, 'I go to review my armies. I must ride before them in an open coach, as much so they can see their Queen as I can see them, and after that I must proceed through the villages nearest the castle.
"'All the Court rides with me according to rank. And all the slaves, naked, and collared in leather, march on foot with us. You shall march at the side of my carriage for all eyes to see. I shall have the finest collar for you, and your anus shall be opened with a leather phallus. You shall wear a bit in your mouth and I shall hold the bridle. You will hold your head high before soldiers, officers, the common people. And for the pleasure of the people, I shall have you displayed in the villages in the main square long enough for all to admire before we continue the procession.'
"'Yes, my Queen,' I answered silently. I knew it would be a terrible ordeal, and yet I was thinking of it with curiosity, and wondering when and how my feeling of helplessness and yielding would visit me. Would it come before the villagers, or the soldiers, or when I trotted along with my head held high, my anus tortured by this phallus. Each detail she had described excited me.
"I slept deeply and well. When Leon awakened me, he groomed me as carefully as he had for the little circus.
"There was a huge commotion outside the castle. It was the first time I had seen the front gates of the courtyard, the drawbridge and moat and all the soldiers assembled. The Queen's open coach stood in the courtyard, and she was already seated surrounded by her footmen and her Pages who rode on the sides, and her coachmen with their fine caps, their plumes and their gleaming spurs. A great mounted force of soldiers was ready.
"Before being led out, I was fitted with the bit by Leon, who gave my hair a last thorough combing. He wedged the leather bit well back into my mouth, wiped my lips and then told me the hardest thing would be to keep my chin raised. I must never let it drop to a normal position. The bridle, which the Queen would hold idly in her lap could of course keep my head raised, but I must never lower my head. She would feel it if I did and be in a fury.
"Then he showed me the leather phallus. It had no straps, no belts attached to it. It was as big as a man's erect cock, and I was afraid. How should I ever keep it in? From it hung a horse's tail of thin black leather thongs for a mere decoration. He told me to spread my legs. He forced it up into my anus and told me I must keep it in place, as the Queen would suffer me to be covered with nothing. The thin leather thongs hung down and stroked my thighs. They would swing like a horse's tail when I trotted along but they were short, they would conceal nothing.
"Then he oiled my pubic hair again, my cock and my balls. He rubbed some oil into my belly. I had my hands clasped behind my back and he gave me a small leather-covered bone to hold with them saying it would make it easier to keep them clasped. But my tasks were these: to keep my chin raised, to keep the phallus in place, and to keep my own penis hard and presentable to the Queen.
"Then I was led out by the little bridle into the courtyard. The bright noonday sun flashed on the spears of the Knights and the soldiers. The horses' hooves made a loud clatter on the stones.
"The Queen who was in fast conversation with the Grand Duke at her side scarcely noticed me. She threw me one quick smile. The bridle was given her. It went up over the door of the coach and kept my head quite turned up.
"'Keep your eyes down at all times, respectfully,' Leon said.
"And soon the carriage moved out of the gates and over the drawbridge.
"Well, you can imagine what that day was like. You were brought here naked through the villages of your own Kingdom. You know what it is like to be gazed at by all, soldiers, Knights, commoners.
"It was small consolation to me that other naked slaves would follow. I was alone by the Queen's coach, and I thought only of pleasing her, and of appearing as she wanted me to appear to others. I held my head up, I contracted my buttocks to hold in the painful phallus. And soon, as we passed before hundreds upon hundreds of soldiers, I thought again, 'I am her servant, her slave, and this is my life. I have no other.'
"Perhaps the most excruciating part of the day for me was the villages. You have been through the villages. I had not. The only common people I had seen were in the kitchen.
"But this day of military parade was also the opening of the fairs in the villages. The Queen visited each of several, and after that the fair would open.
"There was a platform in the center of the square of each, and when the Queen went inside of the house of the Lord of the village to drink a cup of wine with him, I was left on display as she had told me I would be.
"But I was not to stand gracefully as I might have hoped. And the villagers knew this, though I didn't. When we reached the first village, the Queen went away, and as soon as my feet hit the platform, a great roar went up from the crowd who knew they were to see something amusing.
"I had my head down, glad of the opportunity to move the rigid muscles of my throat and shoulders. And I was quite astonished when Felix removed the phallus from my anus. Of course the crowd cheered at this. I was then made to kneel up, hands behind my neck on a turntable.
"Felix operated it with his foot. And telling me to spread my legs wide, he turned the turntable. I was perhaps more afraid in these first few moments than ever before, but never once did the fear of rising and trying to escape come to me. I was virtually helpless. Naked, a slave of the Queen, I was in the midst of hundreds of common people who would have overpowered me at once, and cheerfully for all the sport it would have given them. It was then that I realized escape was quite impossible. Any naked Prince or Princess fleeing the castle would have been apprehended by these villagers. They would have given no shelter.
"Now Felix commanded me to show to the crowd all my private parts that were in the service of the Queen, and that I was her slave, and her animal. I did not understand these words, which were spoken ceremoniously. So he told me politely enough that I must part the cheeks of my buttocks as I bent over and display for them my open anus. Of course this was a symbolic gesture. It meant I was ever to be violated. And nothing more than that which could be violated.
"But my face aflame, my hands trembling, I obeyed. A great cheer went up from the crowd. Tears slipped down my face. With a long cane, Felix lifted my balls for them to see, and pushed my penis this way and that to display its defenselessness, and all the while I had to hold my buttocks apart and display my anus. Whenever I relaxed my hands, he commanded me sharply to pull the flesh wider apart and threatened me with chastisement. 'That will infuriate her Highness,' he said, 'and amuse the crowd immensely.' Then to a loud approving cry, the phallus was shoved securely back into my anus. I was made to press my lips to the wood of the turntable. And I was led back to my position beside the Queen's coach, Felix pulling my bridle over his shoulder as I trotted with my head lifted behind him.
"By the last village I was no more used to it than at the first. But by this time Felix had assured the Queen that I displayed all conceivable humility. My beauty was unrivaled by that of any past Prince. Half the village youth of both sexes was in love with me. The Queen kissed my eyelids when I received those compliments.
"There was a grand banquet that night at the castle. You've seen such a banquet as there was one held for you at your presentation. I had not seen it before. And I had my first experience of serving wine for the Queen and for the others to whom she sent me ceremoniously as a gift now and then. When my eyes caught those of Princess Lynette I smiled at her without thinking about it.
"I felt I could do anything I was commanded to do. I had no fear of anything. And so I can say by then I had yielded. But the truest indication of my yielding was that both Leon and Lord Gregory—when they had the chance—told me I was obdurate and rebellious. They said I mocked everything. I said this was not true when I had the opportunity to answer, but I seldom had such an opportunity.
"Many other things have happened to me since then, but the lessons learned in those early months were most important.
"Princess Lynette is still here, of course. You'll come to know who she is in time, and though I can bear anything from my Queen, from Lord Gregory, and from Leon, I still find it difficult to bear Princess Lynette. But I should stake my life on it that no one knows this.
"Now, it's almost morning. I must return you to the dressing room, and also bathe you, so no one knows we've been together. But I've told you my story so that you can understand what it means to yield, and that each of us must find his or her own path to acceptance.
"There is more, however, to my story which will only reveal itself to you in time. But let it stand by you in simple ways now. If you must bear a punishment which seems too great for you, think to yourself, 'Ah, but Alexi bore this so therefore it can be borne.' "
Beauty did not mean to silence him but she could not restrain her embraces. She was as hungry for him now as she had been earlier, but it was too late.
And as he led her back to the dressing room, she wondered whether or not he could guess the true effect of his words upon her. Could he know that he had enflamed and fascinated her, and amplified for her an understanding of resignation and yielding which she had already felt?
As he bathed her, wiping away all evidence of his love, she remained still, caught in her thoughts.
What had she felt earlier this night when the Queen had said that she wanted to send her home on account of the Crown Prince's excessive devotion? Had she wanted to leave?
A horrid thought obsessed her. She saw herself asleep in that dusty chamber that had been her prison for a hundred years, she heard whispers all about her. The old witch with the spindle that had pricked Beauty's finger was laughing through her toothless gums; and lifting her hand to Beauty's breasts, she exuded some lewd sensuality.
Beauty shuddered. She winced and struggled as Alexi tightened the shackles.
"Don't be afraid. We've had the night together undiscovered," he assured her.
She stared at him as if she did not know him because she was not afraid of anyone in the castle, not of him, the Prince, the Queen. It was her mind that frightened her.
The sky was paling. Alexi embraced her. She was now bound to the wall, her long hair pressed between her back and the stones behind her. And she could not get out of that dusty chamber in her homeland, and it seemed to her she was traveling up through layers and layers of sleep, and this dressing room about her in this cruel country had lost its substantiality.
A Prince had come into her sleeping chamber. A Prince had lowered his lips to her. But it was only Alexi kissing her, wasn't it? Alexi kissing her here?
But when she opened her eyes on that ancient bed and looked at the one who now broke her spell, she saw some bland and innocent countenance! It was not her Crown Prince. It was not Alexi. It was some pristine soul liken to her own who now stood back from her in astonishment. Brave he was, yes, brave, and without complexity!
She cried out. "No!"
But Alexi's hand was over her mouth. "Beauty, what is it?"
"Don't kiss me!" she whispered.
But when she saw the pain in his face, she opened her mouth and felt his lips sealed over it. His tongue filled her. She pressed her hips against him.
"Ah, it is you, only you ..." she whispered.
"And what did you think it was? Were you dreaming?"
"It seemed for a moment all this was a dream," she confessed. But the stone was too real, his touch too real.
"And why should it be a dream? Is it such a nightmare?"
She shook her head. "You love it, all of it, you love it," she whispered in his ear. She saw his eyes linger languidly on her and then drift away. "And it seemed a dream because all the past, the real past, has lost its luster!"
But what was she saying? That in these few days she had not once longed for her homeland, she had not once longed for what her youth had been and the sleep of a hundred years had given her no wisdom?
"I love it. I loathe it," Alexi said. "I am humiliated by it, and recreated by it. And yielding means to feel all those things at once and yet to be of one mind and one spirit."
"Yes," she sighed, as though she had falsely accused him. "Wicked pain, wicked pleasure."
And he gave her his smile of approbation. "We'll be together soon again ..."
"Yes ..."
"... be sure of it. And until then, my darling, my love, belong to everyone."
**THE VILLAGE**
**T** HE NEXT few days passed as quickly for Beauty as those before them. No one discovered that she and Alexi had been together.
The following night the Prince told her she had gained his mother's approval. She would now be trained by him as his little maid, to sweep his quarters, to keep his wine cup always filled, and to perform all those duties that Alexi performed for her Highness.
And from then on Beauty would sleep in the Prince's quarters.
She found herself envied by everyone, and it was the Prince and the Prince alone who prescribed her daily punishments.
Each morning she was given to Lady Juliana for the Bridle Path. Then Beauty would serve the wine at the noon meal and woe to her if she spilt a drop of it.
Then she would sleep in the afternoons so she might be fresh to attend the Prince in the evenings. And next Festival Night she would be entered in a race of Bridle Path slaves which he expected her to win after her daily training.
All this Beauty heard out with flushes and tears, again and again stooping to kiss the Prince's boots as he gave his orders. He seemed still troubled in his love, and while the castle slept, he frequently awakened her with rough embraces. She could scarcely think of Alexi at these times, the Prince so frightened her and scrutinized her.
And when each day dawned she was brought out in her leather horseshoe boots for Lady Juliana.
Beauty was frightened but she was ready. Lady Juliana was a vision of loveliness in her crimson riding dress, and Beauty ran fast on the soft gravel path, the sun often causing her to squint as it flashed in the overhanging trees, and she was weeping when it was finished.
Then she and Lady Juliana would be alone together in the garden. Lady Juliana carried a leather strap, but seldom did she use it, and the garden was soothing to Beauty. They would sit down on the grass, Lady Juliana's skirts a wreath of embroidered silk about her, and quite suddenly Lady Juliana might give Beauty a deep kiss that startled Beauty and weakened her. Lady Juliana stroked Beauty all over. She lavished her with kisses and compliments, and when she did beat her with the leather strap, Beauty cried softly with deep moaning breaths and a languid sense of abandon.
Very soon she was gathering little flowers in her teeth for Lady Juliana, or with great grace kissing the hem of her skirts, or even her white hands, all of these gestures delighting her mistress.
"Ah, am I becoming what Alexi wanted me to become," Beauty thought. But most of the time she did not think at all.
At meals she took great care to serve the wine gracefully.
Yet there came that moment when she spilt the wine, and must take her punishment dangling from the Page's strong grip, scampering afterwards to the Prince's boots to beg silently for forgiveness. The Prince was furious with her, and when he ordered her spanked again, she was scalded with humiliation.
That night, he whipped her mercilessly with his belt before taking her. He told her he loathed the slightest imperfection in her. And she was chained to the wall to spend the night in weeping and misery.
She dreaded new and frightening punishments. Lady Juliana hinted that Beauty was but a virgin in some respects, that she was being tried very slowly.
And Beauty feared Lord Gregory too, who was always watching her.
One morning when she stumbled on the Bridle Path, Lady Juliana threatened her with the Hall of Punishments.
Beauty fell to her hands and knees at once, kissing Lady Juliana's slippers. And though Lady Juliana relented at once with a smile and a toss of her pretty braids, Lord Gregory, nearby, showed his disapproval.
Beauty's heart was a throbbing pain in her chest as she was led away for grooming. If only she could see Alexi, she mused, and yet he had lost some of his charm for her, and why, she was not certain. Even as she lay on her bed that afternoon, she thought of the Prince, and Lady Juliana. "My Lords and Masters," she whispered to herself, and wondered why Leon had given her nothing to make her sleep when she was not tired at all and tortured by the little throb of passion between her legs as always.
But she had been resting only an hour when Lady Juliana came for her.
"I don't much approve of it myself," Lady Juliana said, as she forced Beauty out into the garden, "but his Highness must let you see those poor slaves being packed off to the village."
Again, the village. Beauty tried to conceal her curiosity. Lady Juliana thrashed her idly with the leather belt, light but stinging blows, as they moved down the path together.
Finally they reached an enclosed garden full of low limbed flowering trees, and on a stone bench Beauty saw the Prince and a handsome young Lord at his side who was talking to the Prince earnestly.
"That is Lord Stefan," Lady Juliana confided in a hushed voice, "and you must show him the utmost respect. He is the Prince's favorite cousin. Besides, he is quite miserable today. It is his precious and disobedient Prince Tristan who is the cause of it."
"Ah, and if I could only see Prince Tristan," Beauty thought. She had not forgotten Alexi's mention of him, an incomparable slave who knew the meaning of yielding. So he had caused trouble, had he? She could not help but observe that Lord Stefan was very handsome. Golden-haired and gray-eyed, his youthful face was heavy with brooding and unhappiness.
His eyes rested on Beauty only for a second as she drew near, and though he seemed to acknowledge her charms, he lapsed again into listening to the Prince, who lectured him sternly.
"You bear him too much love, it's the same with me and this Princess you see before you. You must curb your love as I must curb mine. Believe me, I understand even as I condemn you."
"O, but the village," murmured the young Lord.
"He must go and he will be the better for it!"
"O, heartless Prince," whispered Lady Juliana. She urged Beauty forward to kiss Lord Stefan's boots as she took her side beside both of them. "Poor Prince Tristan will be in the village the whole summer."
The Prince lifted Beauty's chin and bent to take a kiss from her lips, which filled Beauty with a softening torment. But she was too curious about all that was being said and dared not make the slightest movement to attract him.
"I must ask you ..." Lord Stefan began. "Would you send Princess Beauty to the village if you felt she deserved it?"
"Of course I would," said the Prince. But he did not sound convincing. "I would do it in an instant."
"O, but you couldn't!" Lady Juliana protested.
"She doesn't deserve it, so it does not matter," the Prince insisted. "But we are talking about Prince Tristan, and Prince Tristan, for all the abuse and punishments he has endured, remains a mystery to everyone. He needs the rigors of the village just as Prince Alexi once needed the kitchen to teach him humility."
Lord Stefan was deeply troubled, and the words rigor and humility seemed to pierce him. He rose and begged the Prince to come with him and make a better judgment of it.
"They go tomorrow. The weather is already very warm and the villagers are already preparing for the auction. I've sent him to the prisoners' yard to wait there."
"Come, Beauty," the Prince said, rising. "It will be good for you to see this and come to understand it."
Beauty was intrigued and followed eagerly. But the Prince's coldness and sternness made her uneasy. She tried to remain close to Lady Juliana as they proceeded along a pathway, out of the gardens, past the kitchen and stables to a plain dirty yard in which she saw a great cart, without its horse, standing on four wheels against the walls that surrounded the castle.
There were common soldiers here, menials. She felt her nakedness as she was made to follow the brightly dressed trio. Her welts and cuts smarted anew and fearfully she looked up to see a small pen, fenced in crude sticks, in which a gaggle of naked Princes and Princesses stood with their hands bound to the backs of their necks, milling as if it were less exhausting to walk than to stand by the hour.
A common soldier dealt a blow now through the fence with a heavy leather belt that sent a squealing Princess towards the middle of the group for cover. And, catching other naked buttocks, he walloped them as well, producing a groan from a young Prince who turned to him resentfully.
It infuriated Beauty to see this lowly soldier abusing such lovely white legs and bottoms. Yet she could not keep her eyes off the slaves who backed away from the fence only to be tormented from the other side by another idle, devilish boy who struck much harder and with much more deliberation.
But now the soldiers saw the Prince and bowed at once, showing him the strictest attention.
And it seemed at that same moment the slaves saw the little group approaching. Moans and whimpers commenced from those who struggled in spite of their gags to make their plight known, and their muffled cries became a lamentation.
They seemed as beautiful as any slaves Beauty had seen, and as they writhed now, some of them dropping on their knees before the Prince, she saw here and there a lovely peach-colored sex beneath curls of pubic hair, or breasts quivering with crying. The Princes were many of them painfully erect as if they could not control it. And one of them had pressed his lips to the rough ground as the Prince and Lord Stefan, and Lady Juliana with Beauty at her side drew up to the little fence to inspect them.
The Prince's eyes were angry and cold, but Lord Stefan appeared shaken. And Beauty perceived that his gaze was fixed on one very dignified Prince who neither wailed or bowed, nor in any way begged for mercy. He was as fair, as was the young lord, his eyes very blue, and though the mean little gag distorted his mouth, his face was otherwise serene as ever she had seen Prince Alexi's. He looked down humbly enough, and Beauty tried to conceal her fascination with his exquisitely sculpted limbs and his swelling organ. He seemed in great distress, however, behind his indifferent expression.
Lord Stefan suddenly turned his back as if he could not quite contain himself.
"Don't be so sentimental. He deserves his time in the village," the Prince said coldly. And with an imperious gesture he ordered the other wailing Princes and Princesses to be silent.
The guards watched all with folded arms, smiling at the spectacle, and Beauty dared not look at them for fear their eyes would meet hers, giving further humiliation.
But the Prince ordered her to come forward and to kneel up and listen to his instruction.
"Beauty, look on these unfortunates," the Prince said with obvious disapproval. They are going to the Queen's Village, which is the largest and most prosperous in the country. It houses the families of all those who serve here; the craftsmen there make our linen, our simple furniture, supply us with wine, food, milk, and butter. There is the dairy there and the fowl are raised on the little farms, and there are all those who make up a town in any location."
Beauty stared at the captive Princes and Princesses, who though they could no longer beg with groans and cries, still bowed before the Prince who seemed indifferent to them.
"It is perhaps the loveliest village of the realm," the Prince went on, "with a stern Lord Mayor and many Inns and taverns that are the favorites of the soldiers. But it is allowed one special privilege that no other village enjoys, and that is to purchase at auction for the warm months those Princes and Princesses in need of dire punishment. Anyone in the village may purchase a slave if he or she has the gold for it."
It seemed at this some of the captives could not prevent themselves from imploring the Prince, and with a snap of his fingers he ordered the guards to go to work with their belts and long paddles, causing an immediate uproar. The miserable, desperate slaves huddled together, turning their vulnerable breasts and organs towards their tormentors, as if at all costs they must protect their sore backsides.
But the tall, yellow-haired Prince Tristan made no move to protect himself, merely allowing himself to be jostled by the others. His eyes had never left his Lord, but now slowly they turned and fixed upon Beauty.
Beauty's heart contracted. She felt a slight dizziness. She stared straight into those unreadable blue eyes while at the same time she thought, "Ah, this is the village."
"It is wretched service," Lady Juliana went on, obviously imploring the Prince. "The auction itself takes place as soon as the slaves arrive and you can well suppose that even the beggars and common louts about town are there to witness it. Why, the whole village declares a holiday. And each poor slave is carried off by his or her master not only to degradation and punishment, but miserable labor. Mind you, the crude practical people of the village do not keep even the loveliest Prince or Princess for mere pleasure."
Beauty was remembering Alexi's description of his exposure in the villages, the high wooden platform in the marketplace, the crude crowd, and their celebration of his humiliation. She felt her sex secretly ache with desire, and yet she was horrified.
"Ah, but for all its roughness and cruelty," said the Prince, now glancing at the inconsolable Lord Stefan who stood still with his back to the unfortunates, "it is a sublime punishment. Few slaves can learn from a year in the castle what they learn from the warm months in the village. And of course, they cannot be really hurt, any more than slaves here. The same strict rules apply: no cutting, no burning, no real wounding. And each week, they are herded to a slaves' hall for bathing and oiling. But when they return to the castle they are more than sweet or meek; they have been reborn with incomparable strength and beauty."
"Yes, as Prince Alexi was reborn," Beauty thought, her heart pounding. She wondered if anyone could see her perplexity and excitement. She saw the distant Prince Tristan among the others, his blue eyes calmly fixed to the back of his master, Lord Stefan.
Her mind was filled with lurid visions. And what was it Alexi had said, that such a punishment had been merciful and that if she found it too difficult to learn slowly, she might make herself ripe for some heavier punishment?
Lady Juliana was shaking her head and making little tisks. "But it is only Spring now," she said. "Why, the poor darlings will be there forever. Ah, the heat, the flies, and the labor. You cannot imagine how they are used, and the soldiers crowding the taverns and the Inns, at last able to buy for a few coins a lovely Prince or Princess that they should never possess in a lifetime."
"You make too much of it," the Prince insisted.
"But would you send your own slave!" Lord Stefan appealed to him again. "I don't want him to go!" he murmured, "and yet I condemned him and before the Queen."
"Then you have no choice, and yes, I would send my own slave, though no slave of the Queen or the Crown Prince has ever been so punished." The Prince turned his back to the slaves almost contemptuously. But Beauty continued to look, as the beautiful Prince Tristan commenced to push his way forward.
He reached the fence and though a haughty guard who was having much sport with the group flailed at him with the leather belt, he did not move nor show the slightest discomfort.
"Ah, he is appealing to you," Lady Juliana sighed, and at once Lord Stefan turned and the two young men faced each other.
Beauty watched as if in a trance as Lord Tristan knelt now slowly and gracefully and kissed the ground before his master.
"It's too late," said the Prince, "and this little sign of affection and humility counts for nothing."
Prince Tristan rose and stood with his eyes down in perfect patience. And Lord Stefan rushed forward and reaching over the fence embraced him immediately. He crushed Prince Tristan to his chest and kissed him all over his face and his hair. The captive Prince, his hands bound to the back of his neck, quietly returned the kisses.
The Prince was in a rage. Lady Juliana was laughing. The Prince pulled Lord Stefan away and said they must leave these miserable slaves now. Tomorrow they would be in the village.
Beauty lay on her bed afterwards unable to think of anything but the little group in the prison yard. Yet she saw too the narrow crooked streets of the villages she'd passed on her journey. She remembered the Inns with their painted signs over the gates, the half-timbered houses shadowing her path, and those tiny, diamond-pained windows.
She would never forget the men and women in coarse breeches and white aprons, with sleeves rolled to the elbows. How they had gaped at her, enjoyed her helplessness.
She could not sleep. And she was filled with a strange new terror.
It was dark when the Prince at last sent for her, and as soon as she reached the door of his private dining room, she saw that he was with Lord Stefan.
It seemed in that moment her fate was decided. She smiled as she thought of all his boasting to Lord Stefan, and she wanted to enter quickly now, but Lord Gregory held her back at the threshold.
Beauty let her eyes mist over. She did not see the Prince in his velvet tunic emblazoned with the coat of arms. Rather she saw those village cobblestone streets, the wives with their wicker brooms, the common lads in the tavern.
But Lord Gregory was speaking to her. "Don't you think I see the change in you!" he hissed low in her ear, so that it seemed part of her imagination.
Her eyebrows knit in a frown of annoyance and then she dropped her eyes.
"You're infected with the same poison as Prince Alexi. I see it working on you every day. You will soon make a mockery of everything."
Her pulse quickened. Lord Stefan, at the supper table, looked so forlorn. And the Prince was as proud as ever.
"What you need is a severe lesson ..." Lord Gregory continued in his acid whisper.
"My Lord, you can't mean the village!" Beauty shuddered.
"No, I don't mean the village!" He was obviously shocked. "And don't be flippant and bold with me. You know what I mean. The Hall of Punishments."
"Ah, your domain, where you are Prince," Beauty whispered. But he did not hear her.
And the Prince, with an air of indifference, had snapped his fingers for her to enter.
She approached on her hands and knees. But she had come only a few paces into the room when she stopped.
"Go on!" Lord Gregory hissed at her angrily; the Prince had not yet noticed.
But when he turned and looked at her crossly, still she did not move, her head bowed, her eyes fixed on him. And when she saw the anger and outrage in his face, she turned suddenly and ran on her hands and knees past Lord Gregory and into the passage.
"Stop her, stop her!" the Prince cried out before he could prevent himself. And when Beauty saw Lord Gregory's boots beside her, she rose to her full height and ran faster. He caught her by the hair and she screamed as she felt herself pulled back and thrown over his shoulder.
She beat on his back with her fists, kicking, as he held her knees tight, and she wept hysterically.
She could hear the Prince's angry voice, but she could not make out the words, and when let down again at his feet, she ran again so that two Pages came pounding after her.
She struggled as she was gagged and bound, and she did not know where she was being taken. It was dark and they were descending stairs, and she knew an appalling moment of regret and panic.
They would hang her in the Hall of Punishments and if she could not endure even that, how would she endure the village?
But a strange calm came over her even before her captors had reached the Slaves' Hall, and when she was thrust in a dark cell to lie on the cold stone floor with her bonds cutting into her flesh, she knew a quiet exhilaration.
Yet she continued to weep, her sex pulsing it seemed with her sobs and there was only silence around her.
It was almost morning when she was roused. Lord Gregory snapped his fingers as the Pages undid her fetters and lifted her to her feet on legs that were weak and unsteady. She felt the wallop of Lord Gregory's belt.
"Spoilt, disgraceful Princess!" he hissed between his teeth, but she was drowsy, softened with desire and dreaming of the village. She gave a little cry as she felt his angry blows, but she realized with wonder that the Pages were gagging her again and binding her hands to the back of her neck roughly. She was going to the village!
"O Beauty, Beauty," came Lady Juliana's voice crying beside her. "Why did you become afraid, why did you try to run, you had been so good and strong, my darling."
"Spoilt, arrogant one," Lord Gregory cursed her again as she was driven towards the open doorway. She could see the morning sky over the treetops. "You did it deliberately!" Lord Gregory whispered in her ear as he whipped her onto the garden path. "Well, you shall rue the day, and how bitterly you will weep and there will be no one there to hear you."
Beauty struggled to keep from smiling. But could they have seen a smile behind the cruel leather bit in her teeth? It did not matter. She was running fast, with her knees lifted, around the side of the castle as Lord Gregory pointed the way, his blows quick and smarting, and Lady Juliana wept as she ran along, too. "O, Beauty, I can't bear it."
The stars were not yet faded away, yet the air was already warm and caressing. They crossed the empty prison yard, entering the courtyard between the great doors, and the lowered drawbridge of the castle.
And there stood the huge cart of slaves, already tethered to the heavy white mares who would pull it down to the village.
For one moment Beauty knew terror. But a delicious abandon took hold of her.
The slaves wailed as they huddled together behind the low railing, and the driver had already taken his place while the cart was surrounded by mounted soldiers.
"One more," Lord Gregory called to the Captain of the Guard, and Beauty heard the cries of the slaves grow louder.
She was lifted by heavy hands, her legs dangling in the air.
"All right, little Princess," the Captain laughed as he set her down in the cart, and Beauty felt its rough wood beneath her feet as she struggled to keep her balance. For one instant, she glanced back and saw the tear-stained face of Lady Juliana. "Why, she is actually suffering," Beauty thought in amazement.
And high above she suddenly saw the Prince and Lord Stefan in the only torchlit window of the dark castle. It seemed the Prince saw her look up; and the slaves about her, seeing the window as well, set up a chorus of vain pleading. The Prince turned away miserably just as Lord Stefan had turned his back on the captives earlier.
Beauty felt the cart move. The great wheels creaked and the horses' hooves rang on the cobblestones. All about her the frantic slaves tumbled against one another. She looked before her and almost at once saw the calm blue eyes of Prince Tristan.
He struggled towards her as she moved towards him, though around them the slaves flinched and squirmed to avoid the spirited thrashing from the guards who rode along beside them. Beauty felt the deep cut of a strap on her calf, but Prince Tristan was now pressed against her.
Her breasts were sealed to his warm chest and her cheek rested against his shoulder. His thick rigid organ passed between her wet thighs and stroked her sex roughly. Struggling not to fall, she mounted the organ and felt it slip inside her. She thought of the village, the auction soon to begin, all the terrors that awaited her. And when she thought of her dear defeated Prince and her poor, grieving Lady Juliana she was again smiling.
But Prince Tristan filled her mind as he struggled, it seemed, with his whole body to pierce her and enfold her.
Even among the cries of the others, she heard his whisper behind his gag: "Beauty, are you frightened?"
"No!" she shook her head. She pressed her tortured mouth to his, and as he lifted her with his thrusts, she felt his heart pounding against her.
• • •
[For a complete list of this author's books click here or visit
www.penguin.com/ricechecklist](http://www.penguin.com/ricechecklist?CMP=CKL-RICE)
|
{
"redpajama_set_name": "RedPajamaBook"
}
| 3,411
|
{"url":"https:\/\/www.zbmath.org\/serials\/?q=se%3A269","text":"# zbMATH \u2014 the first resource for mathematics\n\n## Beitr\u00e4ge zur Algebra und Geometrie\n\n Short Title: Beitr. Algebra Geom. Parallel Title: Contributions to Algebra and Geometry Publisher: Springer, Berlin\/Heidelberg ISSN: 0138-4821; 2191-0383\/e Online: http:\/\/link.springer.com\/journal\/volumesAndIssues\/13366http:\/\/www.heldermann.de\/BAG\/bagcover.htmhttp:\/\/www.emis.de\/journals\/BAG\/http:\/\/www.digizeitschriften.de\/main\/dms\/toc\/?PPN=PPN301416052 Comments: Indexed cover-to-cover\n Documents Indexed: 1,714 Publications (since 1971) References Indexed: 532 Publications with 8,482 References.\nall top 5\n\n#### Latest Issues\n\n 62, No. 3 (2021) 62, No. 2 (2021) 62, No. 1 (2021) 61, No. 4 (2020) 61, No. 3 (2020) 61, No. 2 (2020) 61, No. 1 (2020) 60, No. 4 (2019) 60, No. 3 (2019) 60, No. 2 (2019) 60, No. 1 (2019) 59, No. 4 (2018) 59, No. 3 (2018) 59, No. 2 (2018) 59, No. 1 (2018) 58, No. 4 (2017) 58, No. 3 (2017) 58, No. 2 (2017) 58, No. 1 (2017) 57, No. 4 (2016) 57, No. 3 (2016) 57, No. 2 (2016) 57, No. 1 (2016) 56, No. 2 (2015) 56, No. 1 (2015) 55, No. 2 (2014) 55, No. 1 (2014) 54, No. 2 (2013) 54, No. 1 (2013) 53, No. 2 (2012) 53, No. 1 (2012) 52, No. 2 (2011) 52, No. 1 (2011) 51, No. 2 (2010) 51, No. 1 (2010) 50, No. 2 (2009) 50, No. 1 (2009) 49, No. 2 (2008) 49, No. 1 (2008) 48, No. 2 (2007) 48, No. 1 (2007) 47, No. 2 (2006) 47, No. 1 (2006) 46, No. 2 (2005) 46, No. 1 (2005) 45, No. 2 (2004) 45, No. 1 (2004) 44, No. 2 (2003) 44, No. 1 (2003) 43, No. 2 (2002) 43, No. 1 (2002) 42, No. 2 (2001) 42, No. 1 (2001) 41, No. 2 (2000) 41, No. 1 (2000) 40, No. 2 (1999) 40, No. 1 (1999) 39, No. 2 (1998) 39, No. 1 (1998) 38, No. 2 (1997) 38, No. 1 (1997) 37, No. 2 (1996) 37, No. 1 (1996) 36, No. 2 (1995) 36, No. 1 (1995) 35, No. 2 (1994) 35, No. 1 (1994) 34, No. 2 (1993) 34, No. 1 (1993) 33 (1992) 32 (1991) 31 (1991) 30 (1990) 29 (1989) 28 (1989) 27 (1988) 26 (1987) 25 (1987) 24 (1987) 23 (1986) 22 (1986) 21 (1986) 20 (1985) 19 (1985) 18 (1984) 17 (1984) 16 (1983) 15 (1983) 14 (1983) 13 (1982) 12 (1982) 11 (1981) 10 (1980) 9 (1980) 8 (1979) 7 (1978) 6 (1977) 5 (1976) 4 (1975) 3 (1974)\nall top 5\n\n#### Authors\n\n 23 Martini, Horst 19 Bezdek, K\u00e1roly 19 Wei\u00dfbach, Bernulf 18 Drechsler, Konrad 17 Sterz, Ulrich 14 Geise, Gerhard 14 Stammler, Ludwig 13 Ali, Majid M. 12 Ballico, Edoardo 12 Pambuccian, Victor V. 11 Lassak, Marek 11 Schneider, Rolf G. 10 Bohne, Erhart 10 Gu\u00e9d\u00e9non, Thomas 9 Hajja, Mowaffaq 9 Richter, Gerd 9 Stern, Manfred 8 Kr\u00f6tenheerdt, Otto 8 Naoum, Adil G. 8 Richter, Christian 8 Schiemann, G\u00fcnther 8 Soltan, Valeriu 8 Szirmai, Jen\u00f3 8 Vogel, Wolfgang 7 Bezdek, Andr\u00e1s 7 Breen, Marilyn 7 D\u2019Azevedo Breda, Ana M. R. 7 Dress, Andreas W. M. 7 Gr\u00e4be, Hans-Gert 7 Hertel, Eike 7 Juhnke, Friedrich 7 Quaisser, Erhard 7 St\u00fcckrad, J\u00fcrgen 7 Widiger, Alfred 6 B\u00f6hm, Johannes 6 Breda d\u2019Azevedo, Antonio J. 6 Chen, Bang-Yen 6 Craveiro de Carvalho, Francisco Jos\u00e9 6 de Lima, Henrique F. 6 Hecker, Hans-Dietrich 6 Kurusa, \u00c1rp\u00e1d 6 Pfister, Gerhard 6 Renschuch, Bodo 6 Schr\u00f6cker, Hans-Peter 6 Szab\u00f3, S\u00e1ndor 5 Alexandrov, Victor A. 5 Ashraf, Mohammad 5 Bannuscher, Wolfgang 5 B\u00f6rner, Walter 5 Brass, Peter 5 Connelly, Robert 5 Havlicek, Hans 5 Horv\u00e1th, \u00c1kos G. 5 Leichtweiss, Kurt 5 Liu, Huili 5 Moln\u00e1r, Emil 5 Oukhtite, Lahcen 5 Pasini, Antonio 5 Patil, Dilip P. 5 Pottmann, Helmut 5 S\u00e1nchez, Cristi\u00e1n U. 5 Schmitz, Michael 5 Smith, David J. 5 Steinke, G\u00fcnter F. 5 Struve, Rolf 5 Thron, Reinhard 5 V\u00e4is\u00e4l\u00e4, Jussi 5 Van Maldeghem, Hendrik J. 5 Wegert, Elias 5 Wills, J\u00f6rg M. 4 Averkov, Gennadiy 4 Bisztriczky, Tibor 4 Boltyanskij, Vladimir Grigor\u2019evich 4 B\u00f6r\u00f6czky, Karoly jun. 4 Brehm, Ulrich 4 Br\u00fcckmann, Peter 4 Cahen, Benjamin 4 Carter, Sheila 4 Datta, Basudeb 4 Denecke, Klaus Dieter 4 Dhara, Basudeb 4 Fitzner, Heinz-J\u00f6rg 4 Fodor, Ferenc 4 Garc\u00eda, Alicia N. 4 Hertrich-Jeromin, Udo J. 4 Jones, Gareth A. 4 Jo\u00f3s, Antal 4 Keller, Ott-Heinrich 4 Kleinert, Werner 4 Kreuzer, Martin 4 Krieg, David 4 Kripfganz, Anita 4 Kuku, Aderemi O. 4 Leemans, Dimitri 4 Linhart, Johann 4 Maity, Dipendu 4 McMullen, Peter 4 M\u00f6ller, Reinhard 4 Mozgawa, Witold 4 Nishimura, Hirokazu ...and 1,496 more Authors\nall top 5\n\n#### Fields\n\n 468 Convex and discrete geometry\u00a0(52-XX) 351 Geometry\u00a0(51-XX) 260 Differential geometry\u00a0(53-XX) 228 Algebraic geometry\u00a0(14-XX) 212 Group theory and generalizations\u00a0(20-XX) 190 Commutative algebra\u00a0(13-XX) 165 Associative rings and algebras\u00a0(16-XX) 108 Combinatorics\u00a0(05-XX) 66 Number theory\u00a0(11-XX) 59 Manifolds and cell complexes\u00a0(57-XX) 58 Order, lattices, ordered algebraic structures\u00a0(06-XX) 49 Linear and multilinear algebra; matrix theory\u00a0(15-XX) 46 Nonassociative rings and algebras\u00a0(17-XX) 41 General algebraic systems\u00a0(08-XX) 37 Global analysis, analysis on manifolds\u00a0(58-XX) 30 Functional analysis\u00a0(46-XX) 27 Category theory; homological algebra\u00a0(18-XX) 25 Field theory and polynomials\u00a0(12-XX) 25 Computer science\u00a0(68-XX) 24 Mathematical logic and foundations\u00a0(03-XX) 23 General topology\u00a0(54-XX) 21 Topological groups, Lie groups\u00a0(22-XX) 19 Several complex variables and analytic spaces\u00a0(32-XX) 16 Probability theory and stochastic processes\u00a0(60-XX) 15 Operations research, mathematical programming\u00a0(90-XX) 13 Algebraic topology\u00a0(55-XX) 11 Functions of a complex variable\u00a0(30-XX) 11 Dynamical systems and ergodic theory\u00a0(37-XX) 9 Numerical analysis\u00a0(65-XX) 9 Information and communication theory, circuits\u00a0(94-XX) 8 Calculus of variations and optimal control; optimization\u00a0(49-XX) 7 $$K$$-theory\u00a0(19-XX) 7 Real functions\u00a0(26-XX) 7 Measure and integration\u00a0(28-XX) 6 Optics, electromagnetic theory\u00a0(78-XX) 6 Quantum theory\u00a0(81-XX) 5 History and biography\u00a0(01-XX) 4 General and overarching topics; collections\u00a0(00-XX) 4 Partial differential equations\u00a0(35-XX) 4 Mechanics of particles and systems\u00a0(70-XX) 3 Special functions\u00a0(33-XX) 3 Difference and functional equations\u00a0(39-XX) 3 Operator theory\u00a0(47-XX) 3 Statistics\u00a0(62-XX) 2 Ordinary differential equations\u00a0(34-XX) 2 Approximations and expansions\u00a0(41-XX) 2 Abstract harmonic analysis\u00a0(43-XX) 2 Statistical mechanics, structure of matter\u00a0(82-XX) 2 Relativity and gravitational theory\u00a0(83-XX) 2 Systems theory; control\u00a0(93-XX) 2 Mathematics education\u00a0(97-XX) 1 Mechanics of deformable solids\u00a0(74-XX)\n\n#### Citations contained in zbMATH Open\n\n886 Publications have been cited 2,938 times in 2,496 Documents Cited by Year\nThe projective interpretation of the eight 3-dimensional homogeneous geometries.\u00a0Zbl\u00a00889.51021\nMoln\u00e1r, Emil\n1997\nThe Bellows conjecture.\u00a0Zbl\u00a00939.52009\nConnelly, R.; Sabitov, I.; Walz, A.\n1997\nFour dimensional symplectic Lie algebras.\u00a0Zbl\u00a01155.53042\nOvando, Gabriela\n2006\nA broken circuit ring.\u00a0Zbl\u00a01095.13024\nProudfoot, Nicholas; Speyer, David\n2006\nThe densest geodesic ball packing by a type of Nil lattices.\u00a0Zbl\u00a01167.52016\nSzirmai, Jen\u0151\n2007\nMax-min representation of piecewise linear functions.\u00a0Zbl\u00a00996.26007\nOvchinnikov, Sergei\n2002\nHochschild cohomology rings of algebras $$k[X]\/(f)$$.\u00a0Zbl\u00a00961.13006\nHolm, Thorsten\n2000\nGr\u00f6bner bases of lattices, corner polyhedra, and integer programming.\u00a0Zbl\u00a00863.90115\nSturmfels, Bernd; Weismantel, Robert; Ziegler, G\u00fcnter M.\n1995\nFlag-transitive extensions of dual affine spaces.\u00a0Zbl\u00a00957.51004\nHuybrechts, C\u00e9cile; Pasini, Antonio\n1999\nOn conformally flat hypersurfaces and Guichard\u2019s nets.\u00a0Zbl\u00a00820.53003\nHertrich-Jeromin, Udo\n1994\nMultigraded regularity: syzygies and fat points.\u00a0Zbl\u00a01095.13012\n2006\nAsymptotics of cross sections for convex bodies.\u00a0Zbl\u00a00983.52004\nBrehm, Ulrich; Voigt, J\u00fcrgen\n2000\nA new algebraic criterion for shellability.\u00a0Zbl\u00a00780.52012\nDress, Andreas\n1993\nThe optimal ball and horoball packings to the Coxeter honeycombs in the hyperbolic $$d$$-space.\u00a0Zbl\u00a01126.52020\nSzirmai, Jen\u0151\n2007\nMultisymmetric functions.\u00a0Zbl\u00a00953.05077\nDalbec, John\n1999\nHyperplane sections of convex bodies in isotropic position.\u00a0Zbl\u00a00966.52004\n1999\nA candidate for the densest packing with equal balls in Thurston geometries.\u00a0Zbl\u00a01301.52035\nSzirmai, Jen\u0151\n2014\n$$1\\tfrac 12$$-generation of finite simple groups.\u00a0Zbl\u00a00924.20027\nStein, Alexander\n1998\nCurves in the lightlike cone.\u00a0Zbl\u00a01063.53001\nLiu, Huili\n2004\n$$\\mathbb F_1$$-schemes and toric varieties.\u00a0Zbl\u00a01152.14001\nDeitmar, Anton\n2008\nSpecial curves and ruled surfaces.\u00a0Zbl\u00a01035.53024\nIzumiya, Shyuichi; Takeuchi, Nobuko\n2003\nAdjacency preserving mappings of rectangular matrices.\u00a0Zbl\u00a01084.15007\nHuang, Wenling; Wan, Zhexian\n2004\nThe densest translation ball packing by fundamental lattices in Sol space.\u00a0Zbl\u00a01262.51006\nSzirmai, Jen\u0151\n2010\nA note on Rees algebras and the MFMC property.\u00a0Zbl\u00a01114.13007\nGitler, Isidoro; Valencia, Carlos E.; Villarreal, Rafael H.\n2007\nExamples of K\u00e4hler-Einstein toric Fano manifolds associated to non-symmetric reflexive polytopes.\u00a0Zbl\u00a01225.14042\nNill, Benjamin; Paffenholz, Andreas\n2011\nAn approach to Hopf algebras via Frobenius coordinates.\u00a0Zbl\u00a01001.16024\n2001\nA measure of asymmetry for domains of constant width.\u00a0Zbl\u00a00996.52001\nGroemer, H.; Wallen, L. J.\n2001\nGeodesic ball packings in $$S^{2} \\times R$$ space for generalized Coxeter space groups.\u00a0Zbl\u00a01235.52029\nSzirmai, Jen\u0151\n2011\nMinimal translation surfaces in hyperbolic space.\u00a0Zbl\u00a01220.53013\nL\u00f3pez, Rafael\n2011\nAsymptotic mean values of Gaussian polytopes.\u00a0Zbl\u00a01082.52003\nHug, Daniel; Munsonius, G\u00f6tz Olaf; Reitzner, Matthias\n2004\nSpaces of stresses, projections and parallel drawings for spherical polyhedra.\u00a0Zbl\u00a00819.52018\nCrapo, Henry; Whiteley, Walter\n1994\nSpectrum of the quantized Weyl algebra. (Spectre de l\u2019alg\u00e8bre de Weyl quantique.)\u00a0Zbl\u00a00876.17012\nRigal, Laurent\n1996\nA reverse isoperimetric inequality for convex plane curves.\u00a0Zbl\u00a01121.52023\nPan, Shengliang; Zhang, Hong\n2007\nIteration of involutes of constant width curves in the Minkowski plane.\u00a0Zbl\u00a01305.52006\nCraizer, Marcos\n2014\nFinite $$p$$-groups of class 3 have noninner automorphisms of order $$p$$.\u00a0Zbl\u00a01269.20021\nAbdollahi, Alireza; Ghoraishi, Mohsen; Wilkens, Bettina\n2013\nThe classification of generalized quadrangles with two translation points.\u00a0Zbl\u00a01018.51005\nThas, Koen\n2002\nOn three-dimensional space groups.\u00a0Zbl\u00a00991.20036\nConway, John H.; Delgado Friedrichs, Olaf; Huson, Daniel H.; Thurston, William P.\n2001\nCoincidences of simplex centers and related facial structures.\u00a0Zbl\u00a01093.51014\nEdmonds, Allan L.; Hajja, Mowaffaq; Martini, Horst\n2005\nCombinatorial $$3$$-manifolds with $$10$$ vertices.\u00a0Zbl\u00a01151.57025\nLutz, Frank H.\n2008\nIdeal structure of Hurwitz series rings.\u00a0Zbl\u00a01114.13016\nBenhissi, Ali\n2007\nOn the symmetric difference metric for convex bodies.\u00a0Zbl\u00a00951.52004\nGroemer, H.\n2000\nThe finite Moufang hexagons coordinatized.\u00a0Zbl\u00a00794.51004\nDe Smet, Veerle; Van Maldeghem, Hendrik\n1993\nQuasi-hereditary algebras which are twisted double incidence algebras of posets.\u00a0Zbl\u00a00818.06002\nDeng, Bangming; Xi, Changchang\n1995\nGaussian marginals of convex bodies with symmetries.\u00a0Zbl\u00a01162.60006\nMeckes, Mark W.\n2009\nOptimal packings of up to five equal circles on a square flat torus.\u00a0Zbl\u00a01231.52013\nDickinson, William; Guillot, Daniel; Keaton, Anna; Xhumari, Sandi\n2011\nThe multiple conical surfaces.\u00a0Zbl\u00a00987.14005\nSchicho, Josef\n2001\nThe trace decomposition problem.\u00a0Zbl\u00a00839.15024\nKrupka, Demeter\n1995\nThe Riemann surface of a uniform dessin.\u00a0Zbl\u00a01064.14030\nSingerman, David; Syddall, Robert I.\n2003\nOn equilateral simplices in normed spaces.\u00a0Zbl\u00a00961.51016\nBra\u00df, Peter\n1999\nConnectivity and planarity of Cayley graphs.\u00a0Zbl\u00a00917.05036\nDroms, Carl; Servatius, Brigitte; Servatius, Herman\n1998\nA quantitative Steinitz\u2019 theorem.\u00a0Zbl\u00a00806.52009\nOnn, Shmuel; Sturmfels, Bernd\n1994\nAn example of a flexible polyhedron with nonconstant volume in the spherical space.\u00a0Zbl\u00a00881.52007\nAlexandrov, Victor\n1997\nIndecomposable racks of order $$p^2$$.\u00a0Zbl\u00a01076.57011\nGra\u00f1a, Mat\u00edas\n2004\nAssociated prime ideals of skew polynomial rings.\u00a0Zbl\u00a01167.16020\nBhat, V. K.\n2008\nA duality for conformally flat hypersurfaces.\u00a0Zbl\u00a01332.53078\nHertrich-Jeromin, Udo; Suyama, Y.; Umehara, M.; Yamada, Kotaro\n2015\nBi-null Cartan curves in semi-Euclidean spaces of index 2.\u00a0Zbl\u00a01257.53031\nSakaki, Makoto\n2012\nTropical cycles and Chow polytopes.\u00a0Zbl\u00a01282.14106\nFink, Alex\n2013\nDistance-preserving maps in generalized polygons. II: Maps on points and\/or lines.\u00a0Zbl\u00a01023.51004\nGovaert, Eline; Van Maldeghem Hendrik\n2002\nA Gel\u2019fand model for a Weyl group of type $$B_n$$.\u00a0Zbl\u00a01063.20008\nAraujo, J. O.\n2003\nClassification of dodecahedral space forms.\u00a0Zbl\u00a00926.52021\nProk, Istv\u00e1n\n1998\nDihedral f-tilings of the sphere by spherical triangles and equiangular well-centered quadrangles.\u00a0Zbl\u00a01084.52011\nBreda, Ana M.d\u2019Azevedo; Santos, Altino F.\n2004\nIllumination and visibility problems in terms of closure operators.\u00a0Zbl\u00a01074.52001\nMartini, Horst; Wenzel, Walter\n2004\nThe strong independence theorem for automorphism groups and congruence lattices of finite lattices.\u00a0Zbl\u00a00817.06005\nGr\u00e4tzer, G.; Schmidt, E. T.\n1995\nInequalities for the lattice width of lattice-free convex sets in the plane.\u00a0Zbl\u00a01236.52011\n2012\n$$K$$-contact metrics as Ricci solitons.\u00a0Zbl\u00a01241.53038\nGhosh, Amalendu; Sharma, Ramesh\n2012\nNew extensions of Napoleon\u2019s theorem to higher dimensions.\u00a0Zbl\u00a01142.51015\nHajja, Mowaffaq; Martini, Horst; Spirova, Margarita\n2008\nBlock-diagonalized rigidity matrices of symmetric frameworks and applications.\u00a0Zbl\u00a01342.52023\nSchulze, Bernd\n2010\nSpaces of sums of powers and real rank boundaries.\u00a0Zbl\u00a01403.14090\nMicha\u0142ek, Mateusz; Moon, Hyunsuk\n2018\nOn unions and intersections of nested families of cones.\u00a0Zbl\u00a01358.52008\nLawrence, Jim; Soltan, Valeriu\n2016\nFormulas and ultraproducts in categories.\u00a0Zbl\u00a00531.03042\nAndr\u00e9ka, Hajnalka; N\u00e9meti, Istv\u00e1n\n1979\nDie dichteste Packung von 36 Kreisen in einem Quadrat. (On the denest packing of 36 circles in a square).\u00a0Zbl\u00a00647.52002\nKirchner, Kerstin; Wengerodt, Gerhard\n1987\nBemerkungen zur Theorie der formal $$p$$-adischen K\u00f6rper.\u00a0Zbl\u00a00245.12101\nRoquette, Peter\n1971\nOn converses of Napoleon\u2019s theorem and a modified shape function.\u00a0Zbl\u00a01116.51013\nHajja, Mowaffaq; Martini, Horst; Spirova, Margarita\n2006\nOn finitely generated multiplication ideals in commutative rings.\u00a0Zbl\u00a00589.13011\nNaoum, Adil G.; Balboul, Majid M.\n1985\nA proof of Lov\u00e1sz\u2019s theorem on maximal lattice-free sets.\u00a0Zbl\u00a01262.90107\n2013\nThe largest intersection lattice of a discriminantal arrangement.\u00a0Zbl\u00a00965.52008\n1999\nA criterion for the ideal of a projectively embedded toric surface to be generated by quadrics.\u00a0Zbl\u00a00781.14025\nKoelman, Robert Jan\n1993\nMultivariate Descartes\u2019 rule.\u00a0Zbl\u00a00870.14038\nItenberg, Ilia; Roy, Marie-Fran\u00e7oise\n1996\nA new intrinsic curvature invariant for centroaffine hypersurfaces.\u00a0Zbl\u00a00884.53013\nScharlach, Christine; Simon, Udo; Verstraelen, Leopold; Vrancken, Luc\n1997\nA formula for angles between subspaces of inner product spaces.\u00a0Zbl\u00a01087.15001\nGunawan, Hendra; Neswan, Oki; Setya-Budhi, Wono\n2005\nGeneralized maximum principles and the unicity of complete spacelike hypersurfaces immersed in a Lorentzian product space.\u00a0Zbl\u00a01306.53056\nde Lima, Henrique F.; Lima, Eraldo A. jun.\n2014\nOn offsets and curvatures for discrete and semidiscrete surfaces.\u00a0Zbl\u00a01293.53005\nKarpenkov, Oleg; Wallner, Johannes\n2014\nCriteria for two-dimensional circle planes.\u00a0Zbl\u00a00821.51013\nPolster, Burkhard; Steinke, G\u00fcnter F.\n1994\nCan you recognize the shape of a figure from its shadows?\u00a0Zbl\u00a00828.52001\nKincses, J.; Kurusa, \u00c1.\n1995\nOn the structure of convex sets with applications to the moduli of spherical minimal immersions.\u00a0Zbl\u00a01167.52008\nToth, Gabor\n2008\nGeneralized derivations acting as a homomorphism or anti-homomorphism in semiprime rings.\u00a0Zbl\u00a01242.16039\nDhara, Basudeb\n2012\nNeighborliness of marginal polytopes.\u00a0Zbl\u00a01193.60012\nKahle, Thomas\n2010\nCharacterizing certain staircase convex sets in $$\\mathbb R^d$$.\u00a0Zbl\u00a01204.52008\nBreen, Marilyn\n2010\nPolyhedral Voronoi cells.\u00a0Zbl\u00a01342.52021\nVoigt, Ina; Weis, Stephan\n2010\nBlocks for symmetric groups and their covering groups and quadratic forms.\u00a0Zbl\u00a00894.20014\nErdmann, Karin; Michler, Gerhard O.\n1996\nOn the hyperbolic Pascal pyramid.\u00a0Zbl\u00a01354.52024\nN\u00e9meth, L\u00e1szl\u00f3\n2016\nCluster algebras of type $$D_4$$, tropical planes, and the positive tropical Grassmannian.\u00a0Zbl\u00a01401.13063\nBrodsky, Sarah B.; Ceballos, Cesar; Labb\u00e9, Jean-Philippe\n2017\nDie dichteste Packung von 14 Kreisen in einem Quadrat. (On the densest packing of fourteen circles in a square).\u00a0Zbl\u00a00647.52001\nWengerodt, Gerhard\n1987\nBiharmonic anti-invariant submanifolds in Sasakian space forms.\u00a0Zbl\u00a01123.53028\nArslan, Kadri; Ezentas, Ridvan; Murathan, Cengizhan; Sasahara, Toru\n2007\nOn minimum size blocking sets of external lines to a quadric in $$\\text{PG}(3, q)$$.\u00a0Zbl\u00a01121.51006\nBiondi, P.; Lo Re, P. M. L.; Storme, L.\n2007\nOrthogonality as single primitive notion for metric planes.\u00a0Zbl\u00a01161.51004\nPambuccian, Victor\n2007\nControllability of systems of a nilpotent Lie group.\u00a0Zbl\u00a00579.22010\nHilgert, Joachim; Hofmann, Karl Heinrich; Lawson, Jimmie D.\n1985\nOn angular measures in Minkowski planes.\u00a0Zbl\u00a01246.46011\nFankh\u00e4nel, Andreas\n2011\nAlgebraic, tropical, and fuzzy geometry.\u00a0Zbl\u00a01246.14079\nDress, Andreas; Wenzel, Walter\n2011\nRegular polyhedra of index two. I.\u00a0Zbl\u00a01223.51030\nCutler, Anthony M.; Schulte, Egon\n2011\nAlgebraic aspects of hypergeometric differential equations.\u00a0Zbl\u00a007341118\nReichelt, Thomas; Schulze, Mathias; Sevenheck, Christian; Walther, Uli\n2021\nLagrangian submanifolds in complex space forms satisfying equality in the optimal inequality involving $$\\delta (2, \\ldots, 2)$$.\u00a0Zbl\u00a007341122\nChen, Bang-Yen; Vrancken, Luc; Wang, Xianfeng\n2021\nNote on integer-valued polynomials on a residually cofinite subset.\u00a0Zbl\u00a007375625\nTamoussit, Ali\n2021\nGroups with the weak minimal condition on non-normal non-abelian subgroups.\u00a0Zbl\u00a01442.20022\nDe Mari, Fausto\n2020\n$$\\psi$$-second submodules of a module.\u00a0Zbl\u00a01461.13007\n2020\nUniserial Artinian centrally essential rings.\u00a0Zbl\u00a007164018\nMarkov, V. T.; Tuganbaev, A. A.\n2020\nThe isotopy problem for the phase tropical line.\u00a0Zbl\u00a01441.14206\nCaputo, Raffaele\n2020\nCoordinate-wise powers of algebraic varieties.\u00a0Zbl\u00a01447.51027\nDey, Papri; G\u00f6rlach, Paul; Kaihnsa, Nidhi\n2020\nCyclic order: a geometric analysis.\u00a0Zbl\u00a01448.51009\nStruve, Rolf\n2020\nRealization of graded matrix algebras as Leavitt path algebras.\u00a0Zbl\u00a007261511\nVa\u0161, Lia\n2020\n2-AB rings and 2-absorbing rings.\u00a0Zbl\u00a01434.13007\nSana, Hizem; Sihem, Smach\n2020\nAlgebraic conditions for the positivity of sectional curvature.\u00a0Zbl\u00a01440.53014\nFodor, Dan Gregorian\n2020\nDiameter, width and thickness of spherical reduced convex bodies with an application to Wulff shapes.\u00a0Zbl\u00a01446.52007\nLassak, Marek\n2020\nSemistable subcategories for tiling algebras.\u00a0Zbl\u00a007164020\nGarcia, Monica; Garver, Alexander\n2020\nCorrection to: \u201cSemi-equivelar and vertex-transitive maps on the torus\u201d.\u00a0Zbl\u00a01447.52023\nDatta, Basudeb; Maity, Dipendu\n2020\nRings with polynomial identity and centrally essential rings.\u00a0Zbl\u00a01451.16019\nMarkov, V. T.; Tuganbaev, A. A.\n2019\nSkew Poincar\u00e9-Birkhoff-Witt extensions over weak zip rings.\u00a0Zbl\u00a01451.16021\nReyes, Armando; Su\u00e1rez, H\u00e9ctor\n2019\nOn Betti numbers of edge ideals of crown graphs.\u00a0Zbl\u00a01408.13038\n2019\nLippmann\u2019s axiom and Lebesgue\u2019s axiom are equivalent to the Lotschnittaxiom.\u00a0Zbl\u00a01429.51004\nPambuccian, Victor; Schacht, Celia\n2019\nClassifying character degree graphs with six vertices.\u00a0Zbl\u00a007089226\nBissler, Mark W.; Laubacher, Jacob; Lewis, Mark L.\n2019\nParallelisms of $$\\mathrm{PG}(3,\\mathbb{R})$$ admitting a 3-dimensional group.\u00a0Zbl\u00a01417.51013\nL\u00f6wen, Rainer\n2019\nPolygons with prescribed edge slopes: configuration space and extremal points of perimeter.\u00a0Zbl\u00a01409.58005\nGordon, Joseph; Panina, Gaiane; Teplitskaya, Yana\n2019\nGeneralization of a counterexample to Derksen\u2019s theorem in characteristic two.\u00a0Zbl\u00a01441.14201\nHakuta, Keisuke\n2019\nA sharp dual $$L_p$$ John ellipsoid problem for $$p\\le -n-1$$.\u00a0Zbl\u00a01431.52008\nLv, Songjun\n2019\nConstruction of cubic curves with a node.\u00a0Zbl\u00a01429.51010\nRomakina, Lyudmila N.\n2019\nA homological dimension related to AB rings.\u00a0Zbl\u00a01412.13018\nAraya, Tokuji\n2019\nTorsion functors, small or large.\u00a0Zbl\u00a01427.13011\nRohrer, Fred\n2019\nA note on flat nonholonomic Riemannian structures on three-dimensional Lie groups.\u00a0Zbl\u00a007089219\nBarrett, Dennis I.; Remsing, Claudiu C.\n2019\nFormulas for calculating the dimensions of the sums and the intersections of a family of linear subspaces with applications.\u00a0Zbl\u00a01425.15001\nTian, Yongge\n2019\nSemistar-super potent domains.\u00a0Zbl\u00a01423.13110\nWafa, Gmiza; Sana, Hizem\n2019\nThe Brauer group of Azumaya-Poisson $$S$$-algebras.\u00a0Zbl\u00a01429.16013\nGu\u00e9d\u00e9non, T.\n2019\nRibaucour coordinates.\u00a0Zbl\u00a01433.53010\nBurstall, F.; Hertrich-Jeromin, U.; Miro, M. Lara\n2019\nDirect images of vector bundles and connections.\u00a0Zbl\u00a01420.14036\nBiswas, Indranil; Gangopadhyay, Chandranandan; Wong, Michael Lennox\n2019\nSpaces of sums of powers and real rank boundaries.\u00a0Zbl\u00a01403.14090\nMicha\u0142ek, Mateusz; Moon, Hyunsuk\n2018\nOn the ACCP in skew Poincar\u00e9-Birkhoff-Witt extensions.\u00a0Zbl\u00a01433.16030\nReyes, Armando; Su\u00e1rez, Y\u00e9sica\n2018\nInduced structures on Golden Riemannian manifolds.\u00a0Zbl\u00a01404.53040\nGherici, Beldjilali\n2018\nRings in which every 2-absorbing ideal is prime.\u00a0Zbl\u00a01415.13005\nBennis, Driss; Fahid, Brahim\n2018\nChannel surfaces in Lie sphere geometry.\u00a0Zbl\u00a01408.53014\nPember, Mason; Szewieczek, Gudrun\n2018\nOn base radical and semisimple operators for a class of finite algebras.\u00a0Zbl\u00a01428.16019\nThornton, L. K.\n2018\nAuslander modules.\u00a0Zbl\u00a01402.13012\nNasehpour, Peyman\n2018\nAnnihilators of local cohomology modules and simplicity of rings of differential operators.\u00a0Zbl\u00a01461.13018\nBoix, Alberto F.; Eghbali, Majid\n2018\nSimple transitive 2-representations of some 2-categories of projective functors.\u00a0Zbl\u00a01403.18005\nZimmermann, Jakob\n2018\nGenus-minimal crystallizations of PL 4-manifolds.\u00a0Zbl\u00a01396.57036\nBasak, Biplab\n2018\nA formula for the $$g$$-angle between two subspaces of a normed space.\u00a0Zbl\u00a01392.46017\nNur, M.; Gunawan, H.; Neswan, O.\n2018\nHomogeneous quasi-translations in dimension 5.\u00a0Zbl\u00a01403.14094\nde Bondt, Michiel\n2018\nIrreducibility of the Cayley-Menger determinant and of a class of related polynomials.\u00a0Zbl\u00a01400.12002\nHajja, Mowaffaq; Hayajneh, Mostafa; Nguyen, Bach; Shaqaqha, Shadi\n2018\nDeformation and singularities of maximal surfaces with planar curvature lines.\u00a0Zbl\u00a01396.53081\nCho, Joseph; Ogata, Yuta\n2018\nThe exponential pencil of conics.\u00a0Zbl\u00a01398.51001\nHalbeisen, Lorenz; Hungerb\u00fchler, Norbert\n2018\nCorrection to: \u201cAnnihilators of local cohomology modules and simplicity of rings of differential operators\u201d.\u00a0Zbl\u00a01441.13043\nBoix, Alberto F.; Eghbali, Majid\n2018\nSome alternating and symmetric groups and related graphs.\u00a0Zbl\u00a01387.20007\nAsboei, Alireza Khalili; Amiri, Seyed Sadegh Salehi\n2018\nA module theoretic characterization of the prime radical of near-rings.\u00a0Zbl\u00a01441.16051\nRavi, Srinivasa Rao; Koduru, Naga Koteswara Rao; Korrapati, Siva Prasad\n2018\nGeneralized skew derivations with centralizer conditions on multilinear polynomials.\u00a0Zbl\u00a01441.16021\nEro\u011flu, M\u00fcnevver P\u0131nar\n2018\nA theory of duality in Euclidean geometry.\u00a0Zbl\u00a01393.51011\nStruve, Rolf\n2018\nGeneral position of points on a rational ruled surface.\u00a0Zbl\u00a01395.14035\nAlzati, Alberto; Tortora, Alfonso\n2018\nSome results on almost Ricci solitons and geodesic vector fields.\u00a0Zbl\u00a01393.53039\nSharma, Ramesh\n2018\nGalois lines for space elliptic curve with $$j=12^3$$.\u00a0Zbl\u00a01396.14027\nKanazawa, Mitsunori; Yoshihara, Hisao\n2018\nComplex conference matrices and equi-isoclinic planes in Euclidean spaces.\u00a0Zbl\u00a01395.15030\nBlokhuis, Aart; Brehm, Ulrich; Et-Taoui, Boumediene\n2018\nAnnihilators and centralizers of generalized skew derivations on multilinear polynomials.\u00a0Zbl\u00a01441.16044\nYarbil, Nihan Baydar; De Filippis, Vincenzo; Scudo, Giovanni\n2018\nCluster algebras of type $$D_4$$, tropical planes, and the positive tropical Grassmannian.\u00a0Zbl\u00a01401.13063\nBrodsky, Sarah B.; Ceballos, Cesar; Labb\u00e9, Jean-Philippe\n2017\nMultiplicative (generalized)-derivation in semiprime rings.\u00a0Zbl\u00a01405.16051\nTiwari, S. K.; Sharma, R. K.; Dhara, B.\n2017\nOn the relation between hyperrings and fuzzy rings.\u00a0Zbl\u00a01388.14018\nGiansiracusa, Jeffrey; Jun, Jaiung; Lorscheid, Oliver\n2017\nThe middle hedgehog of a planar convex body.\u00a0Zbl\u00a01380.52003\nSchneider, Rolf\n2017\nMaximal covers of chains of prime ideals.\u00a0Zbl\u00a01387.16015\nSarussi, Shai\n2017\nOn the algebraicity of generalized power series.\u00a0Zbl\u00a01435.13026\nKedlaya, Kiran S.\n2017\nPre-kites: simplices having a regular facet.\u00a0Zbl\u00a01384.52009\nHajja, Mowaffaq; Hammoudeh, Ismail; Hayajneh, Mostafa\n2017\nAnother equivalent of the Lotschnittaxiom.\u00a0Zbl\u00a01362.51004\nPambuccian, Victor\n2017\nChiral 4-polytopes in ordinary space.\u00a0Zbl\u00a01385.52010\nPellicer, Daniel\n2017\nSihem, Smach; Hizem, Sana\n2017\nFaithful tropicalization of Mumford curves of genus two.\u00a0Zbl\u00a01401.14243\nWagner, Till\n2017\nPolygons and iteratively regularizing affine transformations.\u00a0Zbl\u00a01365.51012\nRoeschel, Otto\n2017\nCharacterizations of zero-dimensional complete intersections.\u00a0Zbl\u00a01387.14126\nKreuzer, Martin; Long, Le Ngoc\n2017\nOn the construction of Lie-algebras of type $$E_6(K)$$ for fields $$K$$ of characteristic two.\u00a0Zbl\u00a01378.17023\nAl-dhufeeri, Shuaa; Ata, Mashhour Bani\n2017\nFormally dual subsets of cyclic groups of prime power order.\u00a0Zbl\u00a006774995\nSch\u00fcler, Robert\n2017\nIncompressibility of domain-filling circle packings.\u00a0Zbl\u00a01378.52023\nKrieg, David; Wegert, Elias\n2017\nSemi-equivelar and vertex-transitive maps on the torus.\u00a0Zbl\u00a01375.52018\nDatta, Basudeb; Maity, Dipendu\n2017\nGeneralized derivations and commutativity of prime Banach algebras.\u00a0Zbl\u00a01391.16050\nSharma, R. K.; Prajapati, B.\n2017\nClosed chains of conics carrying Poncelet triangles.\u00a0Zbl\u00a01380.51003\nHalbeisen, Lorenz; Hungerb\u00fchler, Norbert\n2017\nOn the Lie nilpotency indices of modular group algebras.\u00a0Zbl\u00a01387.16020\nSiwach, Reetu; Sharma, R. K.; Sahai, Meena\n2017\nMultisemigroups with multiplicities and complete ordered semi-rings.\u00a0Zbl\u00a01376.18002\nForsberg, Love\n2017\nTopological properties of semigroup primes of a commutative ring.\u00a0Zbl\u00a01393.13004\nFinocchiaro, Carmelo A.; Fontana, Marco; Spirito, Dario\n2017\nObservations on circumcenters in normed planes.\u00a0Zbl\u00a01387.52005\nV\u00e4is\u00e4l\u00e4, Jussi\n2017\nOn units in loop algebra $$F[M(\\mathrm{Dih}(C_p^2),2)]$$.\u00a0Zbl\u00a01404.20053\nSidana, Swati\n2017\nOn Kodaira type vanishing for Calabi-Yau threefolds in positive characteristic.\u00a0Zbl\u00a01391.14032\nTakayama, Yukihide\n2017\nOn certain finite dimensional algebras over finite fields.\u00a0Zbl\u00a01418.17009\nBani-Ata, Mashhour; Al-Rashed, Mariam\n2017\nAsymmetry of Reuleaux polygons.\u00a0Zbl\u00a01378.52006\nJin, HaiLin\n2017\nOn the generalization of Cayley graphs of commutative rings.\u00a0Zbl\u00a01369.05100\nAfkhami, Mojgan; Hamidizadeh, K.; Khashyarmanesh, K.\n2017\nComputational aspects of Burnside rings. I: The ring structure.\u00a0Zbl\u00a01390.19001\nKreuzer, Martin; Patil, Dilip P.\n2017\nInvariable generation of iterated wreath products of cyclic groups.\u00a0Zbl\u00a01395.20017\nLucchini, Andrea\n2017\nOn unions and intersections of nested families of cones.\u00a0Zbl\u00a01358.52008\nLawrence, Jim; Soltan, Valeriu\n2016\nOn the hyperbolic Pascal pyramid.\u00a0Zbl\u00a01354.52024\nN\u00e9meth, L\u00e1szl\u00f3\n2016\nIdentities related to generalized derivation on ideal in prime rings.\u00a0Zbl\u00a01356.16043\nTiwari, Shailesh Kumar; Sharma, Rajendra K.; Dhara, Basudeb\n2016\nHamiltonian Floer homology for compact convex symplectic manifolds.\u00a0Zbl\u00a01339.53083\nLanzat, Sergei\n2016\nOn weakly semiprime ideals of commutative rings.\u00a0Zbl\u00a01349.13005\n2016\nOn a conjecture referring to orthic quadrilaterals.\u00a0Zbl\u00a01344.51006\nPierro, Francesco; Vincenzi, Giovanni\n2016\nInvolutes of fronts in the Euclidean plane.\u00a0Zbl\u00a01352.58007\nFukunaga, Tomonori; Takahashi, Masatomo\n2016\nOn $$n$$-absorbing submodules of modules over commutative rings.\u00a0Zbl\u00a01362.13002\nKant Dubey, Manish; Aggarwal, Pakhi\n2016\nThe Steiner-Lehmus theorem and \u201ctriangles with congruent medians are isosceles\u201d hold in weak geometries.\u00a0Zbl\u00a01367.51008\nPambuccian, Victor; Struve, Horst; Struve, Rolf\n2016\nRigidity of complete spacelike hypersurfaces with constant weighted mean curvature.\u00a0Zbl\u00a01356.53064\nde Lima, Henrique F.; Oliveira, Arlandson M. S.; Santos, M\u00e1rcio S.\n2016\nSmooth quotients of bi-elliptic surfaces.\u00a0Zbl\u00a01375.14147\nYoshihara, Hisao\n2016\nStrongly filial rings.\u00a0Zbl\u00a01349.16001\nMendes, D. I. C.\n2016\n...and 786 more Documents\nall top 5\n\n#### Cited by 2,874 Authors\n\n 41 Martini, Horst 32 Szirmai, Jen\u00f3 17 Liu, Huili 14 Steinke, G\u00fcnter F. 14 Taniguchi, Hiroaki 13 Hajja, Mowaffaq 12 Ballico, Edoardo 12 Havlicek, Hans 12 Naoum, Adil G. 12 Pambuccian, Victor V. 12 Thas, Koen 12 Tuganbaev, Askar Akanovich 12 Van Maldeghem, Hendrik J. 10 Balestro, Vitor 10 Chen, Bang-Yen 10 Jin, Hailin 10 Santos, Francisco 10 Soltan, Valeriu 10 Struve, Rolf 9 Averkov, Gennadiy 9 Breen, Marilyn 9 Cahen, Benjamin 9 Goberna, Miguel Angel 9 Merino, Bernardo Gonz\u00e1lez 9 Moln\u00e1r, Emil 9 Sturmfels, Bernd 8 Bhat, Vijay Kumar 8 Calvaruso, Giovanni 8 Herzog, J\u00fcrgen 8 Mahdou, Najib 8 Richter, Christian 8 Sakaki, Makoto 8 Schulze, Bernd 8 Seeger, Alberto 8 Spirova, Margarita Georgieva 8 Stroppel, Markus Johannes 8 Struve, Horst 7 Bezdek, K\u00e1roly 7 Blunck, Andrea 7 Bobkov, Sergey Germanovich 7 Breda d\u2019Azevedo, Antonio J. 7 De Bruyn, Bart 7 Gaifullin, Alexander A. 7 Ghosh, Amalendu 7 Hayajneh, Mostafa 7 Huang, Liping 7 \u0130larslan, Kazim 7 Jo\u00f3s, Antal 7 Jun, Jaiung 7 Kreuzer, Martin 7 Lorscheid, Oliver 7 Mozgawa, Witold 7 Nedela, Roman 7 Ng\u00f4 Vi\u00eat Trung 7 Folgado Santos, Altino 7 Swanepoel, Konrad J. 7 Th\u00e4le, Christoph 7 U\u00e7um, Ali 7 Villarreal, Rafael Heraclio 7 Vogel, Wolfgang 7 Wang, Changping 7 Whiteley, Walter John 6 Alexandrov, Victor A. 6 Connelly, Robert 6 Ghoraishi, Seyed Mohsen 6 Guo, Qi 6 Hertrich-Jeromin, Udo J. 6 Horv\u00e1th, \u00c1kos G. 6 Jung, Seoung Dal 6 Kabluchko, Zakhar A. 6 Kurusa, \u00c1rp\u00e1d 6 Lassak, Marek 6 Mazorchuk, Volodymyr 6 Nara, Chie 6 Pankov, Mark 6 Reyes, Armando 6 Sahoo, Binod Kumar 6 Torki, Mounir 6 Toth, Gabor 6 Van Tuyl, Adam 5 Alonso-Guti\u00e9rrez, David 5 Ca\u00f1ete, Antonio 5 de Lima, Henrique F. 5 Demaine, Erik D. 5 Guardo, Elena 5 Huynh, Dinh Van 5 Itoh, Jin-ichi 5 Izumiya, Shyuichi 5 Jones, Gareth A. 5 Kadison, Lars 5 Klartag, Bo\u2019az 5 Knarr, Norbert 5 L\u00e1ngi, Zsolt 5 Leng, Gangsong 5 Lima, Eraldo Almeida jun. 5 Maity, Dipendu 5 Markov, Viktor T. 5 Micale, Vincenzo 5 Murathan, Cengizhan 5 Nasz\u00f3di, M\u00e1rton ...and 2,774 more Authors\nall top 5\n\n#### Cited in 422 Journals\n\n 131 Beitr\u00e4ge zur Algebra und Geometrie 101 Journal of Algebra 86 Journal of Geometry 67 Discrete & Computational Geometry 64 Journal of Pure and Applied Algebra 54 Communications in Algebra 53 Results in Mathematics 48 Discrete Mathematics 47 European Journal of Combinatorics 43 Linear Algebra and its Applications 38 Advances in Mathematics 36 Journal of Mathematical Sciences (New York) 35 Transactions of the American Mathematical Society 33 Journal of Algebra and its Applications 32 Geometriae Dedicata 30 Differential Geometry and its Applications 29 Aequationes Mathematicae 28 Archiv der Mathematik 27 Periodica Mathematica Hungarica 26 Proceedings of the American Mathematical Society 25 Journal of Mathematical Analysis and Applications 25 Journal of Geometry and Physics 25 Journal of Symbolic Computation 25 Journal of Algebraic Combinatorics 24 Journal of Combinatorial Theory. Series A 24 Mediterranean Journal of Mathematics 23 Acta Mathematica Hungarica 22 Israel Journal of Mathematics 22 Monatshefte f\u00fcr Mathematik 21 Manuscripta Mathematica 18 Mathematische Zeitschrift 17 Rocky Mountain Journal of Mathematics 17 Advances in Applied Mathematics 16 Finite Fields and their Applications 15 Algebra Universalis 15 Czechoslovak Mathematical Journal 15 Rendiconti del Circolo Matem\u00e0tico di Palermo. Serie II 15 Semigroup Forum 14 Abhandlungen aus dem Mathematischen Seminar der Universit\u00e4t Hamburg 13 Mathematische Annalen 13 Acta Mathematica Sinica. English Series 13 Advances in Geometry 12 Computational Geometry 11 Mathematische Nachrichten 11 Annals of Global Analysis and Geometry 10 Bulletin of the Australian Mathematical Society 10 Ricerche di Matematica 10 International Journal of Geometric Methods in Modern Physics 9 Discrete Applied Mathematics 9 Annali di Matematica Pura ed Applicata. Serie Quarta 9 Siberian Mathematical Journal 9 Expositiones Mathematicae 9 Advances in Applied Clifford Algebras 9 Journal of High Energy Physics 9 Asian-European Journal of Mathematics 9 Journal of Commutative Algebra 8 Annales de l\u2019Institut Fourier 8 Inventiones Mathematicae 8 Journal of Soviet Mathematics 8 Theoretical Computer Science 8 Computer Aided Geometric Design 8 The Journal of Geometric Analysis 8 Designs, Codes and Cryptography 8 Indagationes Mathematicae. New Series 8 Algebras and Representation Theory 8 Lobachevskii Journal of Mathematics 7 Duke Mathematical Journal 7 International Journal of Mathematics and Mathematical Sciences 7 Tohoku Mathematical Journal. Second Series 7 International Journal of Mathematics 7 Mathematical Programming. Series A. Series B 7 Turkish Journal of Mathematics 7 Annals of Combinatorics 7 Palestine Journal of Mathematics 6 Acta Mathematica Vietnamica 6 Applied Mathematics and Computation 6 Journal of Functional Analysis 6 Journal of Optimization Theory and Applications 6 Memoirs of the American Mathematical Society 6 Rendiconti del Seminario Matematico della Universit\u00e0 di Padova 6 Bulletin of the Korean Mathematical Society 6 Science in China. Series A 6 Proceedings of the Indian Academy of Sciences. Mathematical Sciences 6 Bulletin of the Brazilian Mathematical Society. New Series 6 International Electronic Journal of Geometry 6 Ars Mathematica Contemporanea 6 Science China. Mathematics 6 SIAM Journal on Applied Algebra and Geometry 5 Acta Mathematica Academiae Scientiarum Hungaricae 5 American Mathematical Monthly 5 Journal of Mathematical Physics 5 Ukrainian Mathematical Journal 5 Journal of Combinatorial Theory. Series B 5 Journal of the Korean Mathematical Society 5 Kybernetika 5 Mathematica Slovaca 5 Nonlinear Analysis. Theory, Methods & Applications. Series A: Theory and Methods 5 Osaka Journal of Mathematics 5 Topology and its Applications 5 Elemente der Mathematik ...and 322 more Journals\nall top 5\n\n#### Cited in 61 Fields\n\n 607 Convex and discrete geometry\u00a0(52-XX) 466 Differential geometry\u00a0(53-XX) 362 Geometry\u00a0(51-XX) 330 Commutative algebra\u00a0(13-XX) 321 Algebraic geometry\u00a0(14-XX) 315 Combinatorics\u00a0(05-XX) 280 Group theory and generalizations\u00a0(20-XX) 255 Associative rings and algebras\u00a0(16-XX) 98 Linear and multilinear algebra; matrix theory\u00a0(15-XX) 89 Computer science\u00a0(68-XX) 86 Nonassociative rings and algebras\u00a0(17-XX) 85 Manifolds and cell complexes\u00a0(57-XX) 84 Number theory\u00a0(11-XX) 69 Functional analysis\u00a0(46-XX) 65 Operations research, mathematical programming\u00a0(90-XX) 54 Probability theory and stochastic processes\u00a0(60-XX) 53 Category theory; homological algebra\u00a0(18-XX) 53 Global analysis, analysis on manifolds\u00a0(58-XX) 46 Order, lattices, ordered algebraic structures\u00a0(06-XX) 39 Mathematical logic and foundations\u00a0(03-XX) 38 Several complex variables and analytic spaces\u00a0(32-XX) 37 General algebraic systems\u00a0(08-XX) 37 Field theory and polynomials\u00a0(12-XX) 37 Topological groups, Lie groups\u00a0(22-XX) 35 Numerical analysis\u00a0(65-XX) 32 Quantum theory\u00a0(81-XX) 30 Operator theory\u00a0(47-XX) 29 Statistics\u00a0(62-XX) 27 Dynamical systems and ergodic theory\u00a0(37-XX) 26 Functions of a complex variable\u00a0(30-XX) 24 General topology\u00a0(54-XX) 24 Algebraic topology\u00a0(55-XX) 21 Information and communication theory, circuits\u00a0(94-XX) 20 Calculus of variations and optimal control; optimization\u00a0(49-XX) 18 Measure and integration\u00a0(28-XX) 18 Mechanics of particles and systems\u00a0(70-XX) 16 Partial differential equations\u00a0(35-XX) 14 Relativity and gravitational theory\u00a0(83-XX) 13 Real functions\u00a0(26-XX) 12 $$K$$-theory\u00a0(19-XX) 11 History and biography\u00a0(01-XX) 10 Special functions\u00a0(33-XX) 9 Difference and functional equations\u00a0(39-XX) 9 Approximations and expansions\u00a0(41-XX) 8 Statistical mechanics, structure of matter\u00a0(82-XX) 8 Game theory, economics, finance, and other social and behavioral sciences\u00a0(91-XX) 8 Systems theory; control\u00a0(93-XX) 7 Biology and other natural sciences\u00a0(92-XX) 6 Ordinary differential equations\u00a0(34-XX) 6 Harmonic analysis on Euclidean spaces\u00a0(42-XX) 5 General and overarching topics; collections\u00a0(00-XX) 5 Mechanics of deformable solids\u00a0(74-XX) 4 Potential theory\u00a0(31-XX) 4 Fluid mechanics\u00a0(76-XX) 3 Integral transforms, operational calculus\u00a0(44-XX) 2 Optics, electromagnetic theory\u00a0(78-XX) 2 Classical thermodynamics, heat transfer\u00a0(80-XX) 1 Sequences, series, summability\u00a0(40-XX) 1 Abstract harmonic analysis\u00a0(43-XX) 1 Integral equations\u00a0(45-XX) 1 Mathematics education\u00a0(97-XX)","date":"2021-09-21 11:18:17","metadata":"{\"extraction_info\": {\"found_math\": true, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 0, \"math_alttext\": 0, \"mathml\": 0, \"mathjax_tag\": 0, \"mathjax_inline_tex\": 0, \"mathjax_display_tex\": 1, \"mathjax_asciimath\": 0, \"img_math\": 0, \"codecogs_latex\": 0, \"wp_latex\": 0, \"mimetex.cgi\": 0, \"\/images\/math\/codecogs\": 0, \"mathtex.cgi\": 0, \"katex\": 0, \"math-container\": 0, \"wp-katex-eq\": 0, \"align\": 0, \"equation\": 0, \"x-ck12\": 0, \"texerror\": 0, \"math_score\": 0.4975345730781555, \"perplexity\": 6920.890596655081}, \"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-2021-39\/segments\/1631780057202.68\/warc\/CC-MAIN-20210921101319-20210921131319-00381.warc.gz\"}"}
| null | null |
\section{Introduction}
Most real-world networks have power-law degrees, so that the proportion of nodes having $k$ neighbors scales as $k^{-\tau}$ with exponent $\tau$ between 2 and 3 \cite{albert1999,faloutsos1999,jeong2000,vazquez2002}. Power-law degrees imply various intriguing scale-free network properties, such as ultra-small distances~\cite{hofstad2007, newman2001} and the absence of percolation thresholds when $\tau<3$~\cite{janson2009b, pastor2001}. Empirical evidence has been matched by random graph null models that are able to explain mathematically why and how these properties arise. This paper deals with another fundamental property observed in many scale-free networks related to three-point correlations that suppress the creation of triangles and signal the presence of hierarchy. We quantify this property in terms of the {\it clustering spectrum}, the function $k\mapsto \bar c(k)$ with $\bar c(k)$ the probability that two neighbors of a degree-$k$ node are neighbors themselves.
In {\it uncorrelated} networks the clustering spectrum $\bar c (k)$ remains constant and independent of $k$. However, the majority of real-world networks have spectra that decay in $k$, as first observed in technological networks including the Internet~\cite{pastor2001b,ravasz2003}. Figure~\ref{fig:chas} shows the same phenomenon for a social network: YouTube users as vertices, and edges indicating friendships between them~\cite{snap}.
\begin{figure}[h]
\centering
\includegraphics[width=0.8\linewidth]{youtubech.pdf}
\caption{$\bar c(k)$ for the YouTube social network}
\label{fig:chas}
\end{figure}
Close inspection suggests the following properties, not only in Fig.~\ref{fig:chas}, but also in the nine further networks in Fig.~\ref{fig:ch}. The right end of the spectrum appears to be of the power-law form $k^{-\alpha}$; approximate values of $\alpha$ give rise to the dashed lines; (ii) The power law is only approximate and kicks in for rather large values of $k$. In fact, the slope of $\bar c (k)$ decreases with $k$; (iii) There exists a transition point: the minimal degree as of which the slope starts to decline faster and settles on its limiting (large $k$) value.
\begin{figure*}
\subfloat[]{%
\centering
\includegraphics[width=0.33\linewidth]{hudongch.pdf}
\label{fig:chhudong}
}%
\subfloat[]{%
\centering
\includegraphics[width=0.33\linewidth]{baiduch.pdf}
\label{fig:chbaidu}
}
\subfloat[]{
\centering
\includegraphics[width=0.33\linewidth]{wordnetch.pdf}
\label{fig:chwordnet}
}
\subfloat[]{
\centering
\includegraphics[width=0.33\linewidth]{trecch.pdf}
\label{fig:chtrec}
}
\subfloat[]{
\centering
\includegraphics[width=0.33\linewidth]{webgooglech.pdf}
\label{fig:chgoogle}
}
\subfloat[]{
\centering
\includegraphics[width=0.33\linewidth]{asskitch.pdf}
\label{fig:chwiki}
}
\subfloat[]{%
\centering
\includegraphics[width=0.33\linewidth]{catsterch.pdf}
\label{fig:chcatster}
}%
\subfloat[]{
\centering
\includegraphics[width=0.33\linewidth]{gowallach.pdf}
\label{fig:chgowalla}
}
\subfloat[]{
\centering
\includegraphics[width=0.33\linewidth]{wikitalkch.pdf}
\label{fig:chyou}
}
\caption{$\bar c(k)$ for several information (red), technological (green) and social (blue) real-world networks. (a) Hudong encyclopedia~\cite{niu2011}, (b) Baidu encyclopedia~\cite{niu2011}, (c) WordNet~\cite{miller1998}, (d) TREC-WT10g web graph~\cite{bailey2003}, (e) Google web graph~\cite{snap}, (f) Internet on the Autonomous Systems level~\cite{snap}, (g) Catster/Dogster social networks~\cite{konect}, (h) Gowalla social network~\cite{snap}, (i) Wikipedia communication network~\cite{snap}. The different shadings indicate the theoretical boundaries of the regimes as in Fig.~\ref{fig:curve}, with $N$ and $\tau$ as in Table~\ref{tab:data}.
}
\label{fig:ch}
\end{figure*}
For scale-free networks a decaying $\bar c(k)$ is taken as an indicator for the presence of modularity and hierarchy~\cite{ravasz2003}, architectures that can be viewed as collections of subgraphs with dense connections within themselves and sparser ones between them. The existence of clusters of dense interaction signals hierarchical or nearly decomposable structures. When the function $\bar c(k)$ falls off with $k$, low-degree vertices have relatively high clustering coefficients, hence creating small modules that are connected through triangles. In contrast, high-degree vertices have very low clustering coefficients, and therefore act as bridges between the different local modules. This also explains why $\bar c(k)$ is not just a local property, and when viewed as a function of $k$, measures crucial mesoscopic network properties such as modularity, clusters and communities. The behavior of $\bar c(k)$ also turns out to be a good predictor for the macroscopic behavior of the network. Randomizing real-world networks while preserving the shape of the $\bar c(k)$ curve produces networks with very similar component sizes as well as similar hierarchical structures as the original network~\cite{colomer2013}. Furthermore, the shape of $\bar c(k)$ strongly influences the behavior of networks under percolation~\cite{serrano2006}. This places the $\bar c(k)$-curve among the most relevant indicators for structural correlations in network infrastructures.
In this paper, we obtain a precise characterization of clustering in the hidden variable model, a tractable random graph null model. We start from an explicit form of the $\bar c(k)$ curve for the hidden variable model~\cite{boguna2003,serrano2007,dorogovtsev2004}.
We obtain a detailed description of the $\bar c(k)$-curve in the large-network limit that provides rigorous underpinning of the empirical observations (i)-(iii). We find that the decay rate in the hidden variable model is significantly different from the exponent $\bar c(k)\sim k^{-1}$ that has been found in a hierarchical graph model~\cite{ravasz2003} as well as in the preferential attachment model~\cite{krot2015} and a preferential attachment model with enhanced clustering~\cite{szabo2003}. Furthermore, we show that before the power-law decay of $\bar c(k)$ kicks in, $\bar c(k)$ first has a constant regime for small $k$, and a logarithmic decay phase. This characterizes the entire clustering spectrum of the hidden variable model.
This paper is structured as follows. Section \ref{sec:hidden} introduces the random graph model and its local clustering coefficient. Section \ref{sec:three} presents the main results for the clustering spectrum. Section \ref{sec:deep} explains the shape of the clustering spectrum in terms of an energy minimization argument, and Section \ref{sec:deep2} quantifies how fast the limiting clustering spectrum arises as function of the network size. We conclude with a discussion in Section \ref{sec:disc} and present all mathematical derivations of the main results in the appendix.
\section {Hidden variables}\label{sec:hidden}
As null model we employ the hidden variable model~ \cite{boguna2003,park2004,bollobas2007,britton2006,norros2006}.
Given $N$ nodes, hidden variable models are defined as follows. Associate to each node a hidden variable $h$ drawn from a given probability distribution function
\begin{equation}\label{eq:rhoh}
\rho(h)=Ch^{-\tau}
\end{equation}
for some constant $C$. Next join each pair of vertices independently according to a given probability $p(h,h')$ with $h$ and $h'$ the hidden variables associated to the two nodes.
Many networks can be embedded in this hidden-variable framework, but particular attention goes to the case in which the hidden variables have themselves the structure of the degrees of a real-world network. In that case the hidden-variable model puts soft constraints on the degrees, which is typically easier to analyze than hard constraints as in the configuration model~\cite{clauset2009,newman2003book,vazquez2002,dhara2016}.
Chung and Lu \cite{chung2002} introduced the hidden variable model in the form
\begin{equation}\label{c1}
p(h,h')\sim \frac{h h'}{N \mean{h}},
\end{equation}
so that the expected degree of a node equals its hidden variable.
We now discuss the structural and natural cutoff, because both will play a crucial role in the description of the clustering spectrum. The structural cutoff is defined as the largest possible upper bound on the degrees required to guarantee single edges, while the natural cutoff characterizes the maximal degree in a sample of $N$ vertices. For scale-free networks with exponent $\tau\in(2,3]$ the structural cutoff scales as $\sqrt{N}$ while the natural cutoff scales as $N^{1/(\tau-1)}$, which gives rise to structural negative correlations and possibly other finite-size effects. If one wants to avoid such effects, then the maximal value of the product $h h'$ should never exceed $N \mean{h}$, which can be guaranteed by the assumption that the hidden degree $h$ is smaller than the structural cutoff $h_s=\sqrt{N\mean{h}}$.
While this restricts $p(h,h')$ in \eqref{c1} within the interval $[0,1]$, banning degrees larger than the structural cutoff strongly violates the reality of scale-free networks, where degrees all the way up to the
natural cutoff $(N\mean{h})^{1/(\tau-1)}$ need to be considered. We therefore work with (although many asymptotically equivalent choices are possible; see \cite{hofstad2017b} and Appendix \ref{sec:chcomp})
\begin{equation}\label{c11}
p(h,h')= \min\Big(1,\frac{h h'}{N \mean{h}}\Big),
\end{equation}
putting no further restrictions on the range of the hidden variables (and hence degrees).
In this paper, we shall work with $c(h)$, the local clustering coefficient of a randomly chosen vertex with hidden variable $h$. However, when studying local clustering in real-world data sets, we can only observe $\bar c(k)$, the local clustering coefficient of a vertex of degree $k$. In Appendix \ref{sec:hk} we show that the approximation $\bar{c}(h)\approx c(h)$ is highly accurate.
We start from the explicit expression for $c(h)$ \cite{boguna2003}, which measures the probability that two randomly chosen edges from $h$ are neighbors, i.e.,
\begin{equation}\label{int1}
c(h)=\int_{h'}\int_{h''} p(h'|h)p(h',h'')p(h''|h){\rm d} h''{\rm d} h',
\end{equation}
with $p(h'|h)$ the conditional probability that a randomly chosen edge from an $h$-vertex is connected to an $h'$-vertex and $p(h,h')$ as in \eqref{c11}. The goal is now to characterize the $c(h)$-curve (and hence the $\bar c(k)$-curve).
\section{Universal clustering spectrum}\label{sec:three}
The asymptotic evaluation of the double integral \eqref{int1} in the large-$N$ regime reveals three different ranges, defined in terms of the scaling relation between the hidden variable $h$ and the network size $N$. The three ranges together span the entire clustering spectrum as shown in Fig.~\ref{fig:curve}. The detailed calculations are deferred to Appendix \ref{sec:chcomp}.
\begin{figure}[tb]
\centering\includegraphics[width=0.4\textwidth]{curve.pdf}
\caption{Clustering spectrum $h\mapsto c(h)$ with three different ranges for $h$: the flat range, logarithmic decay, and the power-law decay.}
\label{fig:curve}
\end{figure}
The first range pertains to the smallest-degree nodes, i.e., vertices with a hidden variable that does not exceed $N^{\beta(\tau)}$ with $\beta(\tau)= \frac{\tau-2}{\tau-1}$. In this case we show that
\begin{equation}\label{eq:r1}
c(h)\propto N^{2-\tau}\ln N, \quad h\leq N^{\beta(\tau)}.
\end{equation}
In particular, here the local clustering does not depend on the degree and in fact corresponds with the large-$N$ behavior of the global clustering coefficient \cite{hofstad2017b,colomer2012}. Note that the interval $[0,\beta(\tau)]$ diminishes when $\tau$ is close to 2, a possible explanation for why the flat range associated with Range I is hard to recognize in some of the real-world data sets.
Range II considers nodes with hidden variables (degrees) above the threshold $N^{\beta(\tau)}$, but below the structural cutoff $\sqrt{N}$. These nodes start experiencing structural correlations, and close inspection of the integral \eqref{int1} yields
\begin{equation}\label{eq:r2}
c(h)\propto N^{2-\tau}\Big(1+\ln \Big(\frac{\sqrt{N}}{h}\Big)\Big), \quad N^{\beta(\tau)}\leq h \leq \sqrt{N}.
\end{equation}
This range shows relatively slow, logarithmic decay in the clustering spectrum, and is clearly visible in the ten data sets.
Range III considers hidden variables above the structural cutoff, when the restrictive effect of degree-degree correlations becomes more evident. In this range we find that
\begin{equation}\label{eq:r3}
c(h)\propto \frac{1}{N}\Big(\frac{h}{N}\Big)^{-2(3-\tau)}, \quad h\geq \sqrt{N}
\end{equation}
hence power-law decay with a power-law exponent $\alpha=2(3-\tau)$.
Such power-law decay has been observed in many real-world networks
\cite{vazquez2002,ravasz2003,serrano2006b,catanzaro2004, leskovec2008,krioukov2012}, where most networks were found to have the power-law exponent close to one. The asymptotic relation \eqref{eq:r3} shows that the exponent $\alpha$ decreases with $\tau$ and takes values in the entire range $(0,2)$. Table~\ref{tab:data} contains estimated values of $\alpha$ for the ten data sets.
\section{Energy minimization}\label{sec:deep}
\begin{figure}[tb]
\centering
\subfloat[]{
\centering
\includegraphics[width=0.2\textwidth]{triangsmall.pdf}
\label{fig:contsmall}
}
\hspace{0.2cm}
\subfloat[]{
\centering
\includegraphics[width=0.2\textwidth]{trianglarge.pdf}
\label{fig:contlarge}
}
\caption{Orders of magnitude of the major contributions in the different $h$-ranges. The highlighted edges are present with asymptotically positive probability. (a) $h<\sqrt{N}$ (b) $h>\sqrt{N}$.}
\label{fig:major}
\end{figure}
We now explain why the clustering spectrum splits into three ranges, using an argument that minimizes the energy needed to create triangles among nodes with specific hidden variables.
In all three ranges for $h$, there is one type of `most likely' triangle, as shown in Fig.~\ref{fig:major}. This means that most triangles containing a vertex $v$ with hidden variable $h$ are triangles with two other vertices $v'$ and $v''$ with hidden variables $h'$ and $h''$ of specific sizes, depending on $h$. The probability that a triangle is present between $v$, $v'$ and $v''$ can be written as
\begin{equation}\label{eq:probtr}
\min\left(1,\frac{hh'}{N\mean{h}}\right)\min\left(1,\frac{hh''}{N \mean{h} }\right)\min\left(1,\frac{h'h''}{N\mean{h} }\right).
\end{equation}
While the probability that such a triangle exists among the three nodes thus increases with $h'$ and $h''$, the number of such nodes decreases with
$h'$ and $h''$ because vertices with higher $h$-values are rarer. Therefore, the maximum contribution to $c(h)$ results from a trade-off between large enough $h',h''$ for likeliness of occurrence of the triangle, and $h',h''$ small enough to have enough copies. Thus, having $h'> N\mean{h}/h$ is not optimal, since then the probability that an edge exists between $v$ and $v'$ no longer increases with $h'$. This results in the bound
\begin{equation}\label{eq:b1}
h',h''\leq \frac{N\mean{h} }{h}.
\end{equation}
Similarly, $h'h''> N\mean{h}$ is also suboptimal, since then further increasing $h'$ and $h''$ does not increase the probability of an edge between $v'$ and $v''$. This gives as a second bound
\begin{equation}\label{eq:b2}
h'h''\leq N\mean{h}.
\end{equation}
In Ranges I and II, $h<\sqrt{N\mean{h}}$, so that $N\mean{h}/h>\sqrt{N\mean{h}}$. In this situation we reach bound~\eqref{eq:b2} before we reach bound~\eqref{eq:b1}. Therefore, the maximum contribution to $c(h)$ comes from $h'h''\approx N$, where also $h',h''<N\mean{h}/h$ because of the bound~\eqref{eq:b1}. Here the probability that the edge between $v'$ and $v''$ exists is large, while the other two edges have a small probability to be present, as shown in Fig.~\ref{fig:contsmall}. Note that for $h$ in Range I, the bound \eqref{eq:b1} is superfluous, since in this regime $N\mean{h}/h>h_c$, while the network does not contain vertices with hidden variables larger than $h_c$. This bound indicates the minimal values of $h'$ such that an $h$-vertex is guaranteed to be connected to an $h'$-vertex. Thus, vertices in Range I are not even guaranteed to have connections to the highest degree vertices, hence they are not affected by the single-edge constraints. Therefore the value of $c(h)$ in Range I is independent of $h$.
In Range III, $h>\sqrt{N\mean{h}}$, so that $N\mean{h}/h<\sqrt{N\mean{h}}$. Therefore, we reach bound~\eqref{eq:b1} before we reach bound~\eqref{eq:b2}. Thus, we maximize the contribution to the number of triangles by choosing $h',h''\approx N\mean{h}/h$. Then the probability that the edge from $v$ to $v'$ and from $v$ to $v''$ is present is large, while the probability that the edge between $v'$ and $v''$ exists is small, as illustrated in Fig.~\ref{fig:contlarge}.
\section{Convergence rate}\label{sec:deep2}
We next ask how large networks should be, or become, before they reveal the features of the universal clustering spectrum.
In other words, while the results in this paper are shown for the large-$N$ limit, for what finite $N$-values can we expect to see the different ranges and clustering decay?
To bring networks of different sizes $N$ on a comparable footing, we consider
\begin{equation}\label{eq:sigmam}
\sigma_N(t)=\frac{\ln\left(c(h)/c(h_c)\right)}{\ln(N\mean{h})}, \quad h=(N\mean{h})^t,
\end{equation}
for $0\leq t \leq \tfrac{1}{\tau-1}$. The slope of $\sigma_N(t)$ can be interpreted as a measure of the decay of $c(h)$ at $h=(N\mean{h})^t$, and all curves share the same right end of the spectrum; see Appendix~\ref{sec:hcder} for more details.
Figure~\ref{fig:chfinite} shows this rescaled clustering spectrum for synthetic networks generated with the hidden variable model with $\tau=2.25$. Already $10^4$ vertices reveal the essential features of the spectrum: the decay and the three ranges. Increasing the network size further to $10^5$ and $10^6$ nodes shows that the spectrum settles on the limiting curve. Here we note that the real-world networks reported in Figs.~\ref{fig:chas} and~\ref{fig:ch} are also of order $10^5$-$10^6$ nodes, see Table~\ref{tab:data}.
\begin{figure}[ht]
\centering\includegraphics[width=0.4\textwidth]{clustN.pdf}
\caption{$\sigma_N(t)$ for $N=10^4,10^6$ and $10^8$ together with the limiting function, using $\tau=2.25$, for which $\tfrac{1}{\tau-1}=0.8$. }
\label{fig:chfinite}
\end{figure}
\begin{table}[htbp]
\centering
\begin{ruledtabular}
\begin{tabular}{lrrrr}
& $N$ & $\tau$ & g.o.f. &$\alpha$ \\
\textbf{Hudong } & 1.984.484 & 2,30 & 0.00 &0,85 \\
\textbf{Baidu} & 2.141.300 & 2,29 & 0.00 & 0,80 \\
\textbf{Wordnet} & 146.005 & 2,47 & 0.00 &1,01 \\
\textbf{Google web} & 875.713 & 2,73 & 0.00 & 1,03 \\
\textbf{AS-Skitter} & 1.696.415 & 2,35 & 0.06 & 1,12 \\
\textbf{TREC-WT10g} & 1.601.787 & 2,23 & 0.00 & 0,99 \\
\textbf{Wiki-talk} & 2.394.385 & 2,46 & 0.00 &1,54 \\
\textbf{Catster/Dogster} & 623.766 & 2,13 & 0.00 & 1,20 \\
\textbf{Gowalla} & 196.591 & 2,65 & 0.80 &1,24 \\
\textbf{Youtube} & 1.134.890 & 2,22& 0.00 &1,05
\end{tabular}
\end{ruledtabular}%
\caption{Data sets. $N$ denotes the number of vertices, $\tau$ the exponent of the tail of the degree distribution estimated by the method proposed in~\cite{clauset2009} together with the goodness of fit criterion proposed in~\cite{clauset2009} (when the goodness of fit is at least 0.10, a power-law tail cannot be rejected), and $\alpha$ denotes the exponent of $c(k)$.}
\label{tab:data}%
\end{table}%
Figure~\ref{fig:chfinite} also brings to bear a potential pitfall when the goal is to obtain statistically accurate estimates for the slope of $c(h)$. Observe the extremely slow convergence to the limiting curve for $N=\infty$; a well documented property of certain clustering measures~\cite{boguna2009,colomer2012,janssen2015,hofstad2017b}. In Appendix~\ref{sec:hcder}
we again use the integral expression \eqref{int1} to characterize the limiting curve for $N=\infty$ and
the rate of convergence as function of $N$, and indeed extreme $N$-values are required for statistically reliable slope estimates for e.g.~$t$-values of $\tfrac12$ and $\tfrac{1}{\tau-1}$; this is also apparent from visual inspection of Fig.~\ref{fig:chfinite}.
Therefore, the estimates in Table~\ref{tab:data} only serve as indicative values of $\alpha$. Finally, observe that Range II disappears in the limiting curve, due to the rescaling in \eqref{eq:sigmam}, but again only for extreme $N$-values. Because this paper is about structure rather than statistical estimation, the slow convergence in fact provides additional support for the persistence of Range II in Figs.~\ref{fig:chas} and~\ref{fig:ch}.
Table~\ref{tab:data} also shows that the relation $\alpha=-2(3-\tau)$ is inaccurate for the real-world data sets, in turn affecting the theoretical boundaries of the three regimes indicated in Fig.~\ref{fig:ch}. One explanation for this inaccuracy is that the real-world networks might not follow pure power-law distributions, as measured by the goodness of fit criterion in Table~\ref{tab:data}, and visualized in Appendix~\ref{sec:degree}. Furthermore, real-world networks are usually highly clustered and contain community structures, whereas the hidden variable model is locally tree-like. These modular structure may explain, for example, why the power-law decay of the hidden variable model is less pronounced in the three social networks of Fig.~\ref{fig:ch}. It is remarkable that despite these differences between hidden variable models and real-world networks, the global shape of the $c(k)$ curve of the hidden variable model is still visible in these heavy-tailed real-world networks.
\section{Discussion}\label{sec:disc}
The hidden variable model gives rise to {\it single-edge} networks in which pairs of vertices can only be connected once. Hierarchical modularity and the decaying clustering spectrum have been contributed to this restriction that no two vertices have more than one edge connecting them~\cite{pastor2001b, maslov2004,park2003,newman2002assortative,newman2003}. The physical intuition is that the single-edge constraint leads to far fewer connections between high-degree vertices than anticipated based on randomly assigned edges.
We have indeed confirmed this intuition, not only through analytically revealing the universal clustering curve, but also by providing an alternative derivation of the three ranges based on energy minimization and structural correlations.
We now show that the clustering spectrum revealed using the hidden variable model, also appears for a second widely studied null model. This second model cannot be the Configuration Model (CM), which preserves the degree distribution by making connections between vertices in the most random way possible~\cite{bollobas1980, newman2001}. Indeed, because of the random edge assignment, the CM has no degree correlations, leading in the case of scale-free networks with diverging second moment to uncorrelated networks with non-negligible fractions of self-loops (a vertex joined to itself) and multiple connections (two vertices connected by more than one edge). This picture changes dramatically when self-loops and multiple edges are avoided,
a restriction mostly felt by the high-degree nodes, who can no longer establish multiple edges among each other.
We therefore consider the Erased Configuration Model (ECM) that takes a sample from the CM and then erases all the self-loops and multiple edges. While this removes some of the edges in the graph, thus violating the hard constraint, only a small proportion of the edges is removed, so that the degree of vertex $j$ in ECM is still close to $D_j$~\cite[Chapter 7]{hofstad2009}. In the ECM, the probability that a vertex with degree $D_i$ is connected to a vertex with degree $D_j$ can be approximated by $1-{\rm e}^{-D_iD_j/\mean{D}N}$~\cite[Eq.(4.9)]{hofstad2005}. Therefore, we expect the ECM and the hidden variable model to have similar properties (see e.g.~\cite{hofstad2017b}) when we choose
\begin{equation}\label{eq:conecm}
p(h,h')= 1-{\rm e}^{-h h'/N\mean{h}}\approx\frac{h h'}{N\mean{h}}.
\end{equation}
Figure~\ref{fig:ECMhidden} illustrates how both null models generate highly similar spectra, which provides additional support for the claim that the clustering spectrum is a universal property of simple scale-free networks. The ECM is more difficult to deal with compared to hidden variable models, since edges in ECM are not independent. In particular, we expect that these dependencies vanish for the $k\mapsto\bar{c}(k)$ curve. Establishing the universality of the $k\mapsto \bar c(k)$ curve for other random graph null models such as ECM, networks with an underlying geometric space~\cite{serrano2008} or hierarchical configuration models~\cite{stegehuis2015} is a major research direction.
\begin{figure}[t]
\centering
\includegraphics[width=0.4\textwidth]{ECMhidden.pdf}
\caption{$\bar c(k)$ for a hidden variable model with connection probabilities~\eqref{eq:conecm} (solid line) and an erased configuration model (dashed line). The presented values of $\bar c(k)$ are averages over $10^4$ realizations of networks of size $N=10^5$. }
\label{fig:ECMhidden}
\end{figure}
The ECM and the hidden variable model are both null models with soft constraints on the degrees. Putting hard constraints on the degrees with the CM, has the nice property that simple graphs generated using this null model are uniform samples of all simple graphs with the same degree sequence. Dealing with such uniform samples is notoriously hard when the second moment of the degrees is diverging, for example since the CM will yield many edges between high-degree vertices. This makes sampling uniform graphs difficult~\cite{milo2003, viger2005,delgenio2010}. Thus, the joint requirement of hard degree and single-edge constraints, as in the CM, presents formidable technical challenges. Whether our results for the $k\mapsto\bar{c}(k)$ curve for soft-constraint models also carry over to these uniform simple graphs is a challenging open problem.
In this paper we have investigated the presence of triangles in the hidden variable model.
We have shown that by first conditioning on the node degree, there arises a unique
`most likely' triangle with two other vertices of specific degrees. We have not only explained this insight heuristically, but it is also reflected in the elaborate analysis of the double integral for $c(h)$ in Appendix \ref{sec:chcomp}. As such, we have introduced an intuitive and tractable mathematical method for asymptotic triangle counting. It is likely that the method carries over to counting other motifs, such as squares, or complete graphs of larger sizes. For any given motif, and first conditioning on the node degree, we again expect to find specific configuration that are most likely. Further mathematical challenges need to be overcome, though, because we expect that the `most likely' configurations critically depend on the precise motif topologies and the associated energy minimization problems.
\acknowledgements
This work is supported by NWO TOP grant 613.001.451 and by the NWO Gravitation Networks grant 024.002.003.
The work of RvdH is further supported by the NWO VICI grant 639.033.806. The work of JvL is further supported by an NWO TOP-GO grant and by an ERC Starting Grant.
|
{
"redpajama_set_name": "RedPajamaArXiv"
}
| 5,502
|
{"url":"https:\/\/www.acmicpc.net\/problem\/16034","text":"\uc2dc\uac04 \uc81c\ud55c\uba54\ubaa8\ub9ac \uc81c\ud55c\uc81c\ucd9c\uc815\ub2f5\ub9de\ud78c \uc0ac\ub78c\uc815\ub2f5 \ube44\uc728\n2 \ucd08 512 MB60373662.069%\n\n## \ubb38\uc81c\n\nMr.\u00a0Port plans to start a new business renting one or more floors of the new skyscraper with one giga floors, MinatoHarukas. He wants to rent as many vertically adjacent floors as possible, because he wants to show advertisement on as many vertically adjacent windows as possible. The rent for one floor is proportional to the floor number, that is, the rent per month for the n-th floor is n times that of the first floor. Here, the ground floor is called the first floor in the American style, and basement floors are out of consideration for the renting. In order to help Mr.\u00a0Port, you should write a program that computes the vertically adjacent floors satisfying his requirement and whose total rental cost per month is exactly equal to his budget.\n\nFor example, when his budget is 15 units, with one unit being the rent of the first floor, there are four possible rent plans, 1+2+3+4+5, 4+5+6, 7+8, and 15. For all of them, the sums are equal to 15. Of course in this example the rent of maximal number of the floors is that of 1+2+3+4+5, that is, the rent from the first floor to the fifth floor.\n\n## \uc785\ub825\n\nThe input consists of multiple datasets, each in the following format.\n\nb\n\nA dataset consists of one line, the budget of Mr.\u00a0Port b as multiples of the rent of the first floor. b\u00a0 is a positive integer satisfying 1 < b < 109.\n\nThe end of the input is indicated by a line containing a zero. The number of datasets does not exceed 1000.\n\n## \ucd9c\ub825\n\nFor each dataset, output a single line containing two positive integers representing the plan with the maximal number of vertically adjacent floors with its rent price exactly equal to the budget of Mr.\u00a0Port. The first should be the lowest floor number and the second should be the number of floors.\n\n## \uc608\uc81c \uc785\ub825 1\n\n15\n16\n2\n3\n9699690\n223092870\n847288609\n900660121\n987698769\n999999999\n0\n\n\n## \uc608\uc81c \ucd9c\ub825 1\n\n1 5\n16 1\n2 1\n1 2\n16 4389\n129 20995\n4112949 206\n15006 30011\n46887 17718\n163837 5994","date":"2022-08-16 15:25:33","metadata":"{\"extraction_info\": {\"found_math\": true, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 0, \"math_alttext\": 0, \"mathml\": 0, \"mathjax_tag\": 0, \"mathjax_inline_tex\": 0, \"mathjax_display_tex\": 0, \"mathjax_asciimath\": 1, \"img_math\": 0, \"codecogs_latex\": 0, \"wp_latex\": 0, \"mimetex.cgi\": 0, \"\/images\/math\/codecogs\": 0, \"mathtex.cgi\": 0, \"katex\": 0, \"math-container\": 0, \"wp-katex-eq\": 0, \"align\": 0, \"equation\": 0, \"x-ck12\": 0, \"texerror\": 0, \"math_score\": 0.27703019976615906, \"perplexity\": 1081.0799172200689}, \"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\/1659882572408.31\/warc\/CC-MAIN-20220816151008-20220816181008-00013.warc.gz\"}"}
| null | null |
Q: Having problems linking table at a specific webpage in Excel I am trying to pull live data from http://www.wdol.gov/sca.aspx into an Excel spreadsheet. Specifically the SCA Wage Determined Rates. I want to be able to pull in the table that is generated from the State/County into my workbook; however, I am having trouble even linking the website.
I seem to be able to link everything on the website into my workbook except for the table that the website generates.
Any help would be greatly appreciated
|
{
"redpajama_set_name": "RedPajamaStackExchange"
}
| 3,392
|
Two parents and a child were found dead in a New Jersey apartment building on Christmas in what officials say appears to be a murder-suicide.
Bergen County Prosecutor John Molinelli said Michael Stasko, 53, fatally shot his wife Melissa Stasko, 49, and their 8-year-old daughter Mellie Stasko before turning the gun on himself.
The shootings occurred in the couple's bedroom at a luxury high rise apartment in Edgewater, New Jersey, where presents were still sitting under a Christmas tree, Molinelli told a news conference Friday night.
"We don't know why he shot his wife and daughter," Molinelli said. "It does not appear that they tried to defend themselves. They were both lying in the bed when it happened and they might have both been sleeping."
Police received a call just after 7 p.m. ET on Christmas from a Missouri man concerned that his sister had not arrived for the holiday. When police checked on the woman, they found the three bodies, which had been lying dead in the same bed for more than a day.
Autopsies will be performed Friday night into Saturday.
|
{
"redpajama_set_name": "RedPajamaC4"
}
| 2,388
|
Q: не работает cкрипт авторизации php Не работает скрипт. Данные из базы выводит, но дальше-стоп. Не сравнивает данные.
если распечатать, то получаю-
81dc9bdb52d04dc20036dbd8313ed055 - klimff@mail.ru - klimff@mail.ruпароль введён неверно .Что не так в ск
<?php
$db = @mysqli_connect('localhost','root', '', 'car.local') or die('ошибка соединения');
mysqli_set_charset($db, "utf8") or die('Не установлена кодировка');
echo '<pre>';
var_dump($_POST);
echo '</pre>';
if (!empty($_POST["button_auth"])) {
$email = htmlspecialchars($_POST['email']);
$password = htmlspecialchars($_POST['password']);
}
//$query = mysqli_query($db, "SELECT * FROM users WHERE email='$email'");
$query = $db->query("SELECT * FROM users WHERE email='$email'");
$myrow = mysqli_fetch_array($query);
$email = $myrow['email'];
$password = $myrow['password'];
//echo $myrow['password'], ' - ', $password;
//echo $myrow['email'], ' - ', $email;
//print_r($myrow);
if($email == $myrow['email']){
if($password == $myrow['password']){
echo "Привет!";
}
else{
echo"пароль введён неверно";
}
}
?>
<h2>Заполните форму</h2>
<form name="auth" action="" method="post">
<table>
<tr>
<td>E-mail:</td>
<td>
<input type="text" name="email">
</td>
</tr>
<tr>
<td>Пароль</td>
<td>
<input type="Password" name="passwod">
</td>
</tr>
<tr>
<td colspan="4">
<input type="submit" name="button_auth" value="Войти">
</td>
</tr>
</table>
</form>
рипте?
A: Ачипатку устраните, да? И все заработает...
<input type="Password" name="passwod">
но
$password = htmlspecialchars($_POST['password']);
Нескладушки, однако.
P.S. Если что не так работает, как ожидается, - первым делом надо глянуть входящие данные. Для этого прекрасно подходит функция var_dump(). Вставили бы Вы в начало кода для отладки
echo '<pre>';
var_dump($_POST);
echo '</pre>';
и смогли бы самостоятельно увидеть несовпадение имени поля input и индекса $_POST.
Update 1
По обновленному коду - удалите строки
$email = $myrow['email'];
$password = $myrow['password'];
$email и $password - это данные, введенные пользователем. Зачем же их затирать данными из базы, а потом сравнивать с самими собой? Когда делаете правку кода, даже если по ответу на свой вопрос здесь, всегда переспрашивайте себя: зачем? что это изменит в работе скрипта? Не стоит бездумно копипастить из ответа в свой код, так можно такого наворотить, что 5 экспертов (это я не про себя, если что!) не разрулят.
Update 2
Во вставленном в вопрос коде Вы снова ачипатались в имени поля <input> для ввода пароля. Ниже приведен код, который должен работать. Я изменил форматирование кода, выстроил правильную "лесенку" отступов, и код стало читать намного проще. В нем есть еще как минимум один "подводный камень" - запрос к базе и вывод сообщения об успешном или неуспешном вводе пароля выполняется не только когда нажата кнопка "войти", а в любом случае при выполнении скрипта. Но это Вы поправите сами, не буду лишать Вас маленьких радостей.
<?php
echo '<pre>$_POST = ';
var_dump($_POST);
$db = @mysqli_connect('localhost','root', '', 'car.local') or die('ошибка соединения');
mysqli_set_charset($db, "utf8") or die('Не установлена кодировка');
if (!empty($_POST["button_auth"])) {
$email = htmlspecialchars($_POST['email']);
$password = htmlspecialchars($_POST['password']);
}
$query = $db->query("SELECT * FROM users WHERE email='$email'");
$myrow = mysqli_fetch_array($query);
echo "\nEmail из базы ", $myrow['email'], ', Email из запроса ', $email;
echo "\nПароль из базы ", $myrow['password'], ', пароль из запроса ', $password;
if ($email === $myrow['email']) {
if ($password === $myrow['password']) {
echo "\nПривет!";
} else {
echo "\nпароль введён неверно";
}
}
echo '</pre>';
?>
<h2>Заполните форму</h2>
<form name="auth" action="" method="post">
<table>
<tr>
<td>E-mail:</td>
<td>
<input type="text" name="email">
</td>
</tr>
<tr>
<td>Пароль</td>
<td>
<input type="Password" name="password">
</td>
</tr>
<tr>
<td colspan="4">
<input type="submit" name="button_auth" value="Войти">
</td>
</tr>
</table>
</form>
A: Возможно , у тебя произошел конфликт с переменными , посмотри внимательней как ты написал условия
if($email == $myrow['email']){
if($password == $myrow['password']){
echo "Привет!";
}
else{
echo"пароль введён неверно";
}
}
то есть $email == $myrow['email'] , то есть в $email у тебя хранится email и в этой же переменной хранится пост запрос с формы!
И вообще для чего тебе 1-я проверка??? Там можно и без этого ты уже запрос сделал!
$email = htmlspecialchars($_POST['email']);
$password = htmlspecialchars($_POST['password']);
$result_email = $myrow['email'];
$result_password = $myrow['password'];
if($result_password == $password){
echo "Привет!";
}
else{
echo"пароль введён неверно";
}
Вроде бы так , попробуй!:)
A: Не было выполнено присвоение данных вытащенных из БД переменным $email и $password.
$email = $myrow['email'];
$password = $myrow['password'];
|
{
"redpajama_set_name": "RedPajamaStackExchange"
}
| 3,299
|
{"url":"http:\/\/math.stackexchange.com\/questions\/269327\/four-color-theorem","text":"# Four Color Theorem\n\nMy wife is making a quilt. She has a whole bunch of colors and is making a very simple pattern. I enjoy watching the whole process because it's very mathematical, but it has made me question the four color theorem because I've come up with a counterexample that indicates that I either:\n\n1. Do not understand the problem.\n2. Am missing a solution for this example.\n\nThe \"counterexample\" is this:\n\n++++\n+wx+\n+yz+\n++++\n\n\nUnless w andz or x and y are allowed to be the same color you need 5 colors to make this work.\n\n-\nThat is correct: it does not include single-point adjacency. Otherwise the four-corners states would provide a familiar core for a potential counterexample. \u2013\u00a0 Brian M. Scott Jan 2 '13 at 19:08\nRight, that's what made me curious... If you did decide that single-point adjacency (correct term?) was adjacent how many colors would you need then? \u2013\u00a0 Christopher Pfohl Jan 2 '13 at 19:10\nIf single-point adjacency were admitted for consideration, then no number of colors would suffice, and the problem wouldn't be very interesting. \u2013\u00a0 Austin Mohr Jan 2 '13 at 19:11\nIf you allow single-point adjacencies then you can create maps that require arbitrarily many colors by cutting up a circle with all the pieces meeting at the center (like a pie). \u2013\u00a0 Jonathan Christensen Jan 2 '13 at 19:12\nAlso not considered are multiple-component regions, eg Michigan. \u2013\u00a0 alancalvitti Jan 2 '13 at 19:12","date":"2014-10-22 12:51:06","metadata":"{\"extraction_info\": {\"found_math\": true, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 0, \"math_alttext\": 0, \"mathml\": 0, \"mathjax_tag\": 0, \"mathjax_inline_tex\": 0, \"mathjax_display_tex\": 0, \"mathjax_asciimath\": 1, \"img_math\": 0, \"codecogs_latex\": 0, \"wp_latex\": 0, \"mimetex.cgi\": 0, \"\/images\/math\/codecogs\": 0, \"mathtex.cgi\": 0, \"katex\": 0, \"math-container\": 0, \"wp-katex-eq\": 0, \"align\": 0, \"equation\": 0, \"x-ck12\": 0, \"texerror\": 0, \"math_score\": 0.5087053179740906, \"perplexity\": 1754.0733765428256}, \"config\": {\"markdown_headings\": true, \"markdown_code\": true, \"boilerplate_config\": {\"ratio_threshold\": 0.18, \"absolute_threshold\": 10, \"end_threshold\": 15, \"enable\": true}, \"remove_buttons\": true, \"remove_image_figures\": true, \"remove_link_clusters\": true, \"table_config\": {\"min_rows\": 2, \"min_cols\": 3, \"format\": \"plain\"}, \"remove_chinese\": true, \"remove_edit_buttons\": true, \"extract_latex\": true}, \"warc_path\": \"s3:\/\/commoncrawl\/crawl-data\/CC-MAIN-2014-42\/segments\/1413507446943.4\/warc\/CC-MAIN-20141017005726-00346-ip-10-16-133-185.ec2.internal.warc.gz\"}"}
| null | null |
Nucor to construct a new $1.3 billion steel mill in Meade County
Saipriya Iyer March 29, 2019 Construction & Building Materials
Trump called Bevin to congratulate him and Ferriola for setting up the new steel mill and creating jobs for the U.S.
The Charlotte-based steel producer Nucor Corp. has announced that it will invest about $1.35 billion in the new sheet mill in Meade county along the Ohio River. Reportedly, this project will create more than 400 well-paying full-time jobs.
The facility, spread across 1.5 million-square-foot, will be built on 900 acres of land along the Ohio River in the Buttermilk Falls Industrial Park. The construction is anticipated to begin by end of the year with facility opening by 2022.
As reported by Kentucky Today, the jobs created by this project will pay an average wage of $72,000 annually. The project will also create around 2,000 construction jobs additionally with 400 permanent full-time jobs over 15 years. The Kentucky Economic Development Finance Authority has offered about $40 million in state incentives to Nucor, $30 million through the Kentucky Business Investment program and another $10 million through the Kentucky Enterprise Initiative Act.
Gov. Matt Bevin reportedly said that Nucor has been a proven, longtime corporate citizen in Kentucky and an important partner in world-class primary metals industry. The project will transform the economy of the region and help with high-quality jobs to the people in Kentucky and the generations to come, he further added.
Nucor's Ferriola and Gov. Matt Bevin reportedly praised President Donald Trump's decision to impose steel tariffs on China which helped U.S. to boost its manufacturing. At the ceremony, Trump called Bevin to congratulate him and Ferriola for setting the new mill and creating jobs for the U.S.
For the record, Nucor is headquartered in Charlotte, North Carolina. The company already has a sheet steel mill in Gallatin County. Last year, the company committed to invest $650 million in the plant to increase production capacity by 3 million tons annually.
Diatreme Resources unearths silica deposit in North Queensland
Saipriya Iyer August 13, 2018
Nico Steel invests ¥3.83M to boost production capacity at Suzhou plant
Saipriya Iyer February 1, 2019
Coca-Cola opens biggest manufacturing plant in Australia for AUD 165M
Sunil Hebbalkar October 4, 2018
Leading PCB manufacturer Orchem to pour $50M to establish China HQ
|
{
"redpajama_set_name": "RedPajamaCommonCrawl"
}
| 131
|
Home News Reports Ram Mandir case: Muslim parties apologise to Supreme Court for questioning ASI report
LawNews Reports
Ram Mandir case: Muslim parties apologise to Supreme Court for questioning ASI report
The court had said that the ASI report has become part of court records, and it can't be questioned now.
Supreme Court of India
A day after Muslim parties questioned the authorship of 2003 report of the Archaeological Survey of India (ASI) report that had confirmed the presence of a large structure at the Ram Janambhoomi site, it took a U-turn on Thursday to apologise to the Supreme Court for wasting its time in the Ayodhya land dispute case.
According to the reports, senior advocate Rajeev Dhavan, who is representing the Muslim parties apologised to the court after another advocate representing Muslim parties – Meenakshi Arora had questioned the ASI report. "Yesterday's argument was futile. Authorship can't be questioned. Authenticity not in doubt," said Dhawan.
Senior advocate Rajeev Dhavan told to the five-judge constitution bench headed by Chief Justice Ranjan Gogoi that they do not wish to question the authorship of the summary of the ASI report.
"It is not expected that every page is to be signed. The authorship of the report and the summary need not be questioned. If we had wasted my lords time, then we apologise for that. There is no point going into that. The report in question has an author and we are not questioning the authorship," Dhavan, representing the Muslim parties, said.
On Wednesday, senior advocate Meenakshi Arora, also representing the Muslim parties, questioned the ASI report saying every chapter is attributed to an author but the summary has not been attributed to anyone. Arora had also raised her doubts on the authorship of the Archeological Survey of India's (ASI) report, which had excavated the disputed site on Allahabad High Court's orders. She had questioned the credibility of archaeology as a scientific discipline.
"Archaeology is at best a social science. It is not a natural science like physics or chemistry since its findings cannot be verified and may differ from the opinion of one person to another," Arora submitted before the Constitution Bench led by Chief Justice of India Ranjan Gogoi.
However, an unimpressed Supreme Court while responding to Arora's argument, had said the mosque side should have questioned this during the trial, and not now during the appeal. The court had said that the report has become part of court records, and it can't be questioned now.
In 2003, the Archaeological Survey of India (ASI) had excavated the Babri Mosque site at the direction of the Lucknow Bench of the Allahabad High Court. The archaeologists had reported that evidence of a large structure pre-existed the Babri Masjid.
Further, the Supreme Court on Thursday asked parties in the Ramjanmabhoomi-Babri Masjid case to finish their arguments by October 18. The Supreme Court also warned that the parties will not be given any extra day. The apex court continued its hearing on the 32nd day.
"If arguments can't get over by October 18, there won't be an extra day," said Chief Justice of India (CJI) Ranjan Gogoi, who is heading a five-judge bench. The bench also comprises Justices S A Bobde, D Y Chandrachud, Ashok Bhushan, and S Abdul Nazeer.
Chief Justice Gogoi also pointed out that the bench will have just have four weeks to write and deliver the judgment once arguments conclude on October 18. "It will be miraculous if the top court can write judgment in the remaining time given the amount of materials parties have given," he said.
rajiv dhawan
Ram mandir-Babri masjid dispute
Shri Ram
|
{
"redpajama_set_name": "RedPajamaCommonCrawl"
}
| 5,378
|
\section{Introduction}
Let $\mathbb{B}_ n$ be the open unit ball in $\mathbb{C}^n$ and $dv$ the usual Lebesgue volume measure on $\mathbb{B}_ n$, normalized so that the volume of $\mathbb{B}_ n$
is one. Given a parameter $\alpha>-1$ we write
$$dv_{\alpha}(z)=c_{\alpha}\,(1-|z|^2)^{\alpha}dv(z),$$
where $c_{\alpha}$ is a positive constant such that $v_{\alpha}(\mathbb{B}_ n)=1$.
Denote by $H({{\mathbb B}_n})$ the space of holomorphic functions on ${{\mathbb B}_n}$.
For $0<p<\infty$ the weighted Bergman space $A^p_{\alpha}:=A^p_{\alpha}(\mathbb{B}_ n)$
consists of functions $f\in H({{\mathbb B}_n})$ that are
in the Lebesgue space $L^p_{\alpha}:=L^p(\mathbb{B}_ n,dv_{\alpha})$. The corresponding norm is given by
$$\|f\|_{p,\alpha}=\left(\int_{\mathbb{B}_ n}|f(z)|^p\,dv_{\alpha}(z)\right)^{1/p}.$$
When $p=2$, the space $A^2_{\alpha}$ is a reproducing kernel Hilbert space: for each $z\in \mathbb{B}_ n$ there is a function $K^{\alpha}_ z\in
A^2_{\alpha}$ such that $f(z)=\langle f,K^{\alpha}_ z \rangle _{\alpha}$ whenever $f\in A^2_{\alpha}$. Here
$$\langle f,g \rangle_{\alpha}=\int_{\mathbb{B}_ n} f \bar{g}\,dv_{\alpha}$$
is the natural inner product in $L^2_{\alpha}$. $K^{\alpha}_ z$ is called the reproducing kernel of the Bergman space $A^2_{\alpha}$.
It is explicitly given by the formula
$$K^{\alpha}_ z(w)=\frac{1}{(1-\langle w,z\rangle )^{n+1+\alpha}},\quad z,w\in \mathbb{B}_ n.$$
We also let $k^{\alpha}_ z$ denote the normalized reproducing kernel at $z$. Thus
$$k^\alpha_z(w)=K^\alpha_z(w)/\sqrt{K^\alpha_z(z)}=\frac{(1-|z|^2)^{(n+1+\alpha)/2}}{(1-\langle w,z\rangle)^{n+1+\alpha}}.$$
The orthogonal projection $P_{\alpha}:L^2_{\alpha}\rightarrow A^2_{\alpha}$ is an integral operator given by
$$P_{\alpha} f(z)=\int_{\mathbb{B}_ n} \frac{ f(w)\,dv_{\alpha}(w)}{(1-\langle z,w\rangle )^{n+1+\alpha}},\qquad f\in L^2(\mathbb{B}_ n,dv_{\alpha}).$$
The (big) Hankel operator $H_ f^{\beta}$ with symbol $f$ is defined by
$$H^{\beta}_f g=(I-P_{\beta})(fg).$$
We are interested in the mapping properties of $H^\beta_f$ between different Lebesgue spaces.
Hankel operators are closely related to Toeplitz operators and have been extensively studied by many authors in recent decades.
For analytic $f$, Axler \cite{Ax} first characterized the boundedness and compactness of $H_{\bar f}$ on the unweighted Bergman space
of the unit disk. Later on, Axler's result was generalized in \cite{AFP, AJFP} to weighted Bergman spaces of the unit ball in $\mathbb{C}^n$.
For general symbol functions, Zhu \cite{Zhu-VMO} first established the connection between size estimates of Hankel operators and
the mean oscillation of the symbols in the Bergman metric. This idea was further investigated in a series of papers \cite{BCZ0}, \cite{BCZ},
and \cite{BBCZ} in the context of bounded symmetric domains, and in \cite{Li1, Li2} in the context of strongly pseudo convex domains.
The main purpose of this paper is to characterize real-valued functions $f\in L^q_{\beta}$ such that $H^\beta_f$ is bounded or compact
from $A^p_{\alpha}$ to $L^q_{\beta}$ with $1<p\le q<\infty$. This is equivalent to characterizing complex-valued functions $f\in L^q_{\beta}$
such that both $H^\beta_f$ and $H^\beta_{\bar f}$ are bounded or compact between the above spaces.
As a consequence, we will characterize holomorphic symbols $f\in A^1_{\beta}$ such that $H^\beta_{\bar f}$ is bounded or compact
from $A^p_{\alpha}$ to $L^q_{\beta}$ with $1<p\le q<\infty$. Our characterizations are based on a family of weighted BMO spaces in
the Bergman metric.
Most previous results of this type are for bounded and compact Hankel operators from $A^p_{\alpha}$ to $L^p_{\alpha}$.
When $f$ is holomorphic, Janson \cite{J} and Wallst\'{e}n \cite{Wall} characterized bounded and compact Hankel operators
between weighted Bergman spaces (in the Hilbert space case) with different weights on the unit disk and the unit ball, respectively.
Our results generalize theirs and solve two cases left open by them.
In the following, the notation $A\lesssim B$ means
that there is a positive constant $C$ such that $A\leq CB$,
and the notation $A\asymp B$ means that both $A\lesssim B$ and $B\lesssim A$ hold.
\section{Preliminaries and auxiliary results}
In this section we collect some preliminary results that are needed for the proof of the main theorems. We begin with notation for
the rest of the paper. For any two points $z=(z_1,\dots,z_n)$ and $w=(w_1,\dots,w_n)$ in $\mathbb{C}^n$, we write
$$\langle z,w\rangle=z_1\bar w_1+\cdots+z_n\bar w_n,$$
and
$$|z|=\sqrt{\langle z,z\rangle}=\sqrt{|z_1|^2+\cdots+|z_n|^2}.$$
For any $a\in \mathbb{B}_ n$ with $a\neq 0$ we denote by $\varphi_a(z)$ the M\"obius transformation on $\mathbb{B}_ n$
that interchanges the points $0$ and $a$. It is known that
$$\varphi_a(z)=\frac{a-P_a(z)-s_aQ_a(z)}{1-\langle z,a\rangle},\qquad z\in\mathbb{B}_ n,$$
where $s_a =1-|a|^2$ , $P_a$ is the orthogonal projection from ${\mathbb C}^n$ onto the one dimensional subspace $[a]$ generated by $a$,
and $Q_a$ is the orthogonal projection from ${\mathbb C}^n$ onto the orthogonal complement of $[a]$. When $a=0$, $\varphi_a(z)=-z$.
It is known that $\varphi_a$ satisfies the following properties:
\begin{equation}\label{eq-pa}
\varphi_a\circ\varphi_a(z)=z,\qquad 1-|\varphi_a(z)|^2=\frac{(1-|a|^2)(1-|z|^2)}{|1-\langle z,a\rangle|^2}.
\end{equation}
For $z,w\in\mathbb{B}_ n$, the distance between $z$ and $w$ induced by the Bergman metric is given by
$$\beta(z,w)=\frac12 \,\log\frac{1+|\varphi_z(w)|}{1-|\varphi_z(w)|}.$$
For $z\in\mathbb{B}_ n$ and $r>0$, the Bergman metric ball at $z$ is given by
$$D(z,r)=\big \{w\in\mathbb{B}_ n:\,\beta(z,w)<r \big \}.$$
We refer to \cite{ZhuBn} for more information about automorphisms and the Bergman metric on $\mathbb{B}_ n$.
A sequence $\{a_k\}$ of points in $\mathbb{B}_ n$ is called a \emph{separated sequence} (in the Bergman metric) if there exists a positive
constant $\delta>0$ such that $\beta(a_i,a_j)>\delta$ for any $i\neq j$. The following result is Theorem 2.23 in \cite{ZhuBn}.
\begin{lemma}\label{covering}
There exists a positive integer $N$ such that for any $0<r<1$ we can find a sequence $\{a_k\}$ in $\mathbb{B}_ n$ with the following properties:
\begin{itemize}
\item[(i)] $\mathbb{B}_ n=\cup_{k}D(a_k,r)$.
\item[(ii)] The sets $D(a_k,r/4)$ are mutually disjoint.
\item[(iii)] Each point $z\in\mathbb{B}_ n$ belongs to at most $N$ of the sets $D(a_k,4r)$.
\end{itemize}
\end{lemma}
Any sequence $\{a_k\}$ satisfying the conditions of the above lemma will be called an $r$-\emph{lattice}
in the Bergman metric. Obviously any $r$-lattice is separated. The following integral estimate is well known and can
be found in \cite[Theorem 1.12]{ZhuBn} for example.
\begin{lemma}\label{Ict}
Let $t>-1$ and $s>0$. There is a positive constant $C$ such that
$$\int_{\mathbb{B}_ n} \frac{(1-|w|^2)^t\,dv(w)}{|1-\langle z,w\rangle |^{n+1+t+s}}\le C\,(1-|z|^2)^{-s}$$
for all $z\in \mathbb{B}_ n$.
\end{lemma}
We also need a well-known variant of the previous lemma.
\begin{lemma}\label{l2}
Let $\{z_ k\}$ be a separated sequence in $\mathbb{B}_ n$ and let $n<t<s$. Then
$$\sum_{k=1}^{\infty}\frac{(1-|z_ k|^2)^t}{|1-\langle z,z_ k \rangle |^s}\le C\,(1-|z|^2)^{t-s},\qquad z\in \mathbb{B}_ n.$$
\end{lemma}
Lemma~\ref{l2} above can be deduced from Lemma \ref{Ict} after noticing that, if a sequence $\{z_ k\}$ is separated, then there is a
constant $r>0$ such that the Bergman metric balls $D(z_ k,r)$ are pairwise disjoint. The following result is from \cite{Zhao-Schur}.
\begin{lemma}\label{Z-T}
Given real numbers $b$ and $c$, consider the integral operator on $\mathbb{B}_ n$ defined by
$$S_{b,c} f(z)=\int_{\mathbb{B}_ n} \frac {f(w)(1-|w|^2)^b\,dv(w)}{|1-\langle z,w\rangle |^c}.$$
Let $1< p\le q<\infty$, $\alpha>-1$, $\beta>-1$, and
$$\lambda=\frac{n+1+\beta}{q}-\frac{n+1+\alpha}{p}.$$
Then the operator $S_{b,c}$ is bounded from $L^p_{\alpha}$ to $L^q_{\beta}$ if and only if
$$\alpha+1<p(b+1),\qquad c\le n+1+b+\lambda.$$
\end{lemma}
We show that, under the same conditions, with an extra (unbounded) factor $\beta(z,w)$ in the integrand, the modified operator
is still bounded from $L^p_{\alpha}$ to $L^q_{\beta}$. Thus we consider the operator
$$T_{b,c}f(z)=\int_{{{\mathbb B}_n}}\frac{f(w)\,\beta(z,w)}{|1-\langle z,w\rangle|^c}(1-|w|^2)^b\,dv(w).$$
\begin{proposition}\label{proj-ge1-beta}
Let $b$ and $c$ be real numbers. Let $1< p\le q<\infty$, $\alpha>-1$, $\beta>-1$, and
$$\lambda=\frac{n+1+\beta}{q}-\frac{n+1+\alpha}{p}.$$
If $\alpha+1<p(b+1)$ and $c\le n+1+b+\lambda$, then $T_{b,c}$ is bounded from $L^p_{\alpha}$ to $L^q_{\beta}$.
\end{proposition}
\begin{proof}
Pick $\varepsilon>0$ so that $\alpha+1<p(b+1-\varepsilon)$ and $\beta-q\varepsilon>-1$. Since $\beta(z,w)$ grows logarithmically, we have
$$\beta(z,w)=\beta(0,\varphi_ z(w))\le C (1-|\varphi_ z(w)|^2)^{-\varepsilon}.$$
It follows from \eqref{eq-pa} that
$$|T_{b,c} f(z)| \le C (1-|z|^2)^{-\varepsilon}\int_{\mathbb{B}_ n} \frac{(1-|w|^2)^{b-\varepsilon}}{|1-\langle z,w\rangle|^{c-2\varepsilon}}\,|f(w)|\,dv(w).$$
Thus $T_{b,c}$ is bounded from $L^p_{\alpha}$ to $L^q_{\beta}$ if the operator $S_{b-\varepsilon,c-2\varepsilon}$ is bounded
from $L^p_{\alpha}$ to $L^q_{\beta-\varepsilon q}$. The desired result then follows from the previous lemma.
\end{proof}
In a similar manner, the following version of Lemma \ref{Ict} can be obtained. The proof is left to the interested reader.
\begin{lemma}\label{Ict-beta}
Let $t>-1$, $s>0$, and $d>0$. There is a positive constant $C$ such that
$$\int_{\mathbb{B}_ n} \frac{(1-|w|^2)^t\,\beta(z,w)^d\,dv(w)}{|1-\langle z,w\rangle |^{n+1+t+s}}\le C\,(1-|z|^2)^{-s}$$
for all $z\in \mathbb{B}_ n$.
\end{lemma}
The rest of this section is devoted to the proof of Theorem \ref{5}, which can be interpreted as some sort of
tangential maximum principle. We begin with the following elementary fact.
\begin{lemma}
Suppose $F$ and $G$ are holomorphic functions on ${\mathbb B}_2$. If
\begin{equation}
F(z_1,z_2)=G(z_1,z_2),\qquad z_k=(u/\sqrt2)e^{i\theta_k},
\label{eq1}
\end{equation}
where $\theta_k$ are arbitrary real numbers and $u$ is arbitrary from the unit disk ${\mathbb D}$, then $F=G$ on ${\mathbb B}_2$.
\label{4}
\end{lemma}
\begin{proof}
Suppose
$$F(z_1,z_2)=\sum_{k,l=0}^\infty a_{kl}z_1^kz_2^l,
\qquad G(z_1,z_2)=\sum_{k,l=0}^\infty b_{kl}z_1^kz_2^l.$$
Then we have
$$\sum_{k,l=0}^\infty a_{kl}\left(\frac u{\sqrt2}\right)^{k+l}e^{ik\theta_1}e^{il\theta_2}
=\sum_{k,l=0}^\infty b_{kl}\left(\frac u{\sqrt2}\right)^{k+l}e^{ik\theta_1}e^{il\theta_2}$$
for all $u\in{\mathbb D}$ and all real $\theta_1$ and $\theta_2$. By the uniqueness of Fourier coefficients
on the torus, we must have $a_{kl}=b_{kl}$ for all $k$ and $l$. This shows that $F=G$ on ${\mathbb B}_2$.
\end{proof}
For $n>1$, $f\in H({{\mathbb B}_n})$, and $z\in{{\mathbb B}_n}-\{0\}$ we will write
$$|\nabla_tf(z)|=\sup\left\{\left|\frac{\partial f}{\partial u}(z)\right|:\|u\|=1,u\in[z]^\perp\right\}$$
and call it the complex tangential gradient of $f$ at $z$.
\begin{theorem}\label{5}
Let $n>1$ and $f\in H({{\mathbb B}_n})$. If $|\nabla_tf(z)|\to0$ as $|z|\to1^-$, then $f$ is constant.
\end{theorem}
\begin{proof}
We will first prove the case $n=2$. In this case, the condition
\begin{equation}
|\nabla_tf(z)|\to0,\qquad |z|\to1^-,
\label{eq2}
\end{equation}
is equivalent to
$$\lim_{|z|\to1^-}\left(\overline z_2\frac{\partial f}{\partial z_1}(z)-
\overline z_1\frac{\partial f}{\partial z_2}(z)\right)=0.$$
Let
$$z_1=\frac{u}{\sqrt2}e^{i\theta_1},\qquad z_2=\frac u{\sqrt2}e^{i\theta_2},$$
where $u\in{\mathbb D}$ and $\theta_k$ are arbitrary real numbers. Since
$$|z_1|^2+|z_2|^2=|u|^2,$$
we see that $|z|\to1^-$ if and only if $|u|\to1^-$. Thus condition (\ref{eq2}) implies that
$$\lim_{|u|\to1^-}\left[e^{-i\theta_2}\frac{\partial f}{\partial z_1}\left(\frac u{\sqrt2}
e^{i\theta_1},\frac u{\sqrt2}e^{i\theta_2}\right)-e^{-i\theta_1}\frac{\partial f}{\partial z_2}
\left(\frac u{\sqrt2}e^{i\theta_1},\frac u{\sqrt2}e^{i\theta_2}\right)\right]=0.$$
For any fixed $\theta_k$ the expression inside the brackets above is an analytic function $F(u)$ on
the unit disk. By the classical maximum principle for analytic functions on the unit disk, we conclude that the
condition in (\ref{eq2}) implies that $F(u)$ is identically zero on ${\mathbb D}$. Therefore, the condition in (\ref{eq2}) implies that
$$e^{i\theta_1}\frac{\partial f}{\partial z_1}\left(\frac u{\sqrt2}e^{i\theta_1},
\frac u{\sqrt2}e^{i\theta_2}\right)=e^{i\theta_2}\frac{\partial f}{\partial z_2}
\left(\frac u{\sqrt2}e^{i\theta_1},\frac u{\sqrt2}e^{i\theta_2}\right)$$
for all $u\in{\mathbb D}$. Multiply the above equation by $u/\sqrt2$, we conclude that
\begin{equation}
z_1\frac{\partial f}{\partial z_1}(z_1,z_2)=z_2\frac{\partial f}{\partial z_2}(z_1,z_2)
\label{eq3}
\end{equation}
whenever
$$z_k=\frac u{\sqrt2}e^{i\theta_k},\qquad k=1,2.$$
By Lemma \ref{4}, we see that the condition in (\ref{eq2}) implies
that the identity in (\ref{eq3}) must hold for all $z=(z_1,z_2)$ in the unit ball.
Write
$$f(z)=\sum_{i,j=0}^\infty a_{ij}z_1^iz_2^j$$
and assume that the identity in (\ref{eq3}) holds for all $z=(z_1,z_2)\in{\mathbb B}_2$. Then
$$\sum_{i,j=0}^\infty ia_{ij}z_1^iz_2^j=\sum_{i,j=0}^\infty ja_{ij}z_1^iz_2^j.$$
This gives $ia_{ij}=ja_{ij}$ for all $i$ and $j$, which implies that $a_{ij}=0$ whenever $i\not=j$. Writing $a_j=a_{jj}$, we obtain
$$f(z)=\sum_{j=0}^\infty a_j(z_1z_2)^j.$$
In this case, we have
$$\overline z_2\frac{\partial f}{\partial z_1}(z)-\overline z_1
\frac{\partial f}{\partial z_2}(z)=(|z_2|^2-|z_1|^2)\sum_{j=1}^\infty ja_j(z_1z_2)^{j-1}.$$
Consider the case in which
$$z_1=u\sin\theta,\qquad z_2=u\cos\theta,$$
where $u\in{\mathbb D}$ is arbitrary and $0<\theta<\pi/4$ is fixed. Then
$$\overline z_2\frac{\partial f}{\partial z_1}(z)-\overline z_1\frac{\partial f}{\partial z_2}(z)
=|u|^2\cos(2\theta)\sum_{j=1}^\infty ja_j(u^2\sin\theta\cos\theta)^{j-1}.$$
Let $|u|\to1^-$ and apply the classical maximum principle on the unit disk, we must have $a_j=0$ for all $j\ge1$,
namely, $f$ is constant. This completes the proof of the theorem in the case $n=2$.
Next let us assume that $n\ge3$ and $|\nabla_tf(z)|\to0$ as $|z|\to1^-$ for some $f\in H({{\mathbb B}_n})$. We want to show that $f$ is
constant. For any $z=(z_1,z_2,z_3,\cdots,z_n)\in{{\mathbb B}_n}$, the vector $(z_2,-z_1,0,\cdots,0)$ is perpendicular to $z$. Therefore,
$$\lim_{|z|\to1^-}\left(\overline z_2\frac{\partial f}{\partial z_1}(z)-\overline z_1
\frac{\partial f}{\partial z_2}(z)\right)=0.$$
We proceed to show that this implies that $f$ is independent of the first two variables.
Fix $z=(z_1, z_2, z_3,\cdots,z_n)\in{{\mathbb B}_n}$ (we specifically mention that it is OK if some of the $z_k$ are $0$) and write
$$1-r^2=|z_3|^2+\cdots+|z_n|^2,$$
where $0<r\le1$. Consider the function
$$g(w_1,w_2)=f(rw_1,rw_2,z_3,\cdots,z_n),$$
where $w=(w_1,w_2)\in{\mathbb B}_2$. It is clear that $|w|\to1^-$ if and only if
$$|rw_1|^2+|rw_2|^2+|z_3|^2+\cdots+|z_n|^2\to1^-.$$
So we also have
$$\lim_{|w|\to1^-}\left[\overline w_2\frac{\partial g}{\partial w_1}(w)-\overline w_1
\frac{\partial g}{\partial w_2}(w)\right]=0.$$
By the $n=2$ case that we have already proved, $g$ must be constant. If we choose
$w\in{\mathbb B}_2$ such that $rw_k=z_k$ for $k=1,2$. Then
$$f(z_1,z_2,z_3,\cdots,z_n)=g(w_1,w_2)=g(0,0)=f(0,0,z_3,\cdots,z_n).$$
Repeat the argument for the first and $k$th variable, where $k\ge3$, and let $k$ run from
$3$ to $n$. The result is
$$f(z_1,z_2,z_3,\cdots,z_n)=f(0,0,0,\cdots,0).$$
Since $z=(z_1,\cdots,z_n)\in{{\mathbb B}_n}$ is arbitrary, we have shown that $f$ is constant.
\end{proof}
For $f\in H({{\mathbb B}_n})$ we write
$$\nabla f(z)=\left(\frac{\partial f}{\partial z_1}(z),\cdots,\frac{\partial f}{\partial z_n}(z)\right),\qquad z\in \mathbb{B}_ n$$
and call $|\nabla f(z)|$ the complex gradient of $f$ at $z$. As a consequence of Theorem \ref{5}, we obtain the following maximum
principle in terms of the invariant gradient $\widetilde{\nabla} f(z)=\nabla (f\circ \varphi_ z)(0)$.
\begin{coro}\label{C5}
Let $n>1$ and $f\in H({{\mathbb B}_n})$. If $(1-|z|^2)^{-1/2}|\widetilde\nabla f(z)|\to0$ as $|z|\to1^-$, then $f$ is constant.
\end{coro}
\begin{proof}
This follows from \cite[Theorem 7.22]{ZhuBn} and Theorem \ref{5}.
\end{proof}
\section{A family of weighted BMO spaces}
Let $\gamma \in \mathbb{R}$. For any positive radius $r$ and every exponent $p$ with $1\le p<\infty$, the space $BMO^p_{r,\gamma}$
consists of those functions $f\in L^p_{loc}(\mathbb{B}_ n)$ (the space of locally $L^p$ integrable functions on $\mathbb{B}_ n$) such that
$$\|f\|_{BMO^p_{r,\gamma}}=\sup \big \{ (1-|z|^2)^{\gamma} MO_{p,r}(f)(z):z\in \mathbb{B}_ n \big \}<\infty,$$
where
$$MO_{p,r}(f)(z)= \left [\frac{1}{v_{\sigma}(D(z,r))}\int_{D(z,r)} |f(w)-\widehat{f_ r}(z)|^p\,dv_{\sigma}(w) \right ]^{1/p}$$
is the $p$-mean oscillation of $f$ at $z$ in the Bergman metric. Here
$$\widehat{f_ r}(z)=\frac{1}{v_{\sigma}(D(z,r))} \int_{D(z,r)} f(w)\,dv_{\sigma}(w)$$
is the averaging function of $f$ and $dv_\sigma(z)=(1-|z|^2)^\sigma\,dv(z)$. At first glance, the function $MO_{p,r}(f)$ seems to depend
on the real parameter $\sigma$, but the weight factor $(1-|z|^2)^{\sigma}$ in $dv_{\sigma}$ is essentially canceled out by the extra factor
$(1-|z|^2)^{\sigma}$ in $v_{\sigma}(D(z,r))\asymp (1-|z|^2)^{n+1+\sigma}$. As a consequence, the space $BMO^p_{r,\gamma}$ is actually
independent of the weight parameter $\sigma$. In particular, this independence on $\sigma$ is a consequence of the following lemma.
\begin{lemma}\label{Lem-1}
Let $1\le p<\infty$, $f\in L^p_{loc}(\mathbb{B}_ n)$, and $r>0$. Then $f\in BMO^p_{r,\gamma}$ if and only if there exists some constant $C>0$ such
that for any $z\in \mathbb{B}_ n$, there is a constant $\lambda_ z$ satisfying
\begin{equation}\label{Eq-L1}
\frac{(1-|z|^2)^{\gamma p}}{v_{\sigma}(D(z,r))}\int_{D(z,r)} |f(w)-\lambda_ z|^p\,dv_{\sigma}(w) \le C.
\end{equation}
\end{lemma}
\begin{proof}
If $f\in BMO^p_{r,\gamma}$, then \eqref{Eq-L1} holds with $C=\|f\|_{BMO^p_{r,\gamma}}$ and $\lambda_ z=\widehat{f_ r}(z)$. Conversely,
if \eqref{Eq-L1} is satisfied, then by the triangle inequality for the $L^p$-norm, $MO_{p,r}(f)(z)$ is less than or equal to
$$\left[\frac{1}{v_{\sigma}(D(z,r))}\int_{D(z,r)}|f(w)-\lambda_ z|^p\,dv_{\sigma}(w)\right ]^{\frac1p}
+|\widehat{f_ r}(z)-\lambda_ z|.$$
By H\"{o}lder's inequality,
\begin{eqnarray*}
|\widehat{f_ r}(z)-\lambda_ z|&=&\left | \frac{1}{v_{\sigma}(D(z,r))}\int_{D(z,r)}\!\! (f(w)-\lambda_ z)\,dv_{\sigma}(w)\right |\\
&\le& \left[\frac{1}{v_{\sigma}(D(z,r))}\int_{D(z,r)}\!\! |f(w)-\lambda_ z|^p\,dv_{\sigma}(w)\right ]^{\frac1p}.
\end{eqnarray*}
It follows that
\[ (1-|z|^2) ^{\gamma} MO_{p,r}(f)(z) \le 2 C^{1/p}\]
for all $z\in \mathbb{B}_ n$, so that $f\in BMO^p_{r,\gamma}$.
\end{proof}
For a continuous function $f$ on $\mathbb{B}_ n$ let
$$\omega_ r(f)(z)=\sup\big \{ |f(z)-f(w)|:w\in D(z,r)\big \}.$$
The function $\omega_ r(f)(z)$ is called the oscillation of $f$ at the point $z$ in the Bergman metric. For any $r>0$ and $\gamma \in
\mathbb{R}$, let $BO_{r,\gamma}$ denote the space of continuous functions $f$ on $\mathbb{B}_ n$ such that
$$\|f\|_{BO_{r,\gamma}}=\sup_ {z\in\mathbb{B}_ n} (1-|z|^2)^{\gamma} \omega_ r(f)(z)<\infty.$$
\begin{lemma}\label{L-CBO}
Let $r>0$, $\gamma \in \mathbb{R}$, and $f$ be a continuous function on $\mathbb{B}_ n$. If $\gamma\ge0$, then
$f\in BO_{r,\gamma}$ if and only if there is a constant $C>0$ such that
$$|f(z)-f(w)|\le C \frac{\beta(z,w)+1}{\min (1-|z|,1-|w|)^{\gamma}}$$
for all $z$ and $w$ in ${{\mathbb B}_n}$. If $\gamma<0$, then $f\in BO_{r,\gamma}$ if and only if there is a constant $C>0$ such that
$$|f(z)-f(w)|\le C \frac{\beta(z,w)+1}{\big [\max (1-|z|,1-|w|)\big ]^{-\gamma}}\,|1-\langle z,w \rangle |^{-2\gamma}$$
for all $z$ and $w$ in $\mathbb{B}_ n$.
\end{lemma}
\begin{proof}
Assume that $f\in BO_{r,\gamma}$. If $\beta(z,w)\le r$, the result is clear, because then
$$|1-\langle z,w \rangle | \asymp 1-|z| \asymp 1-|w|.$$
Fix any $z,w\in \mathbb{B}_ n$ with $\beta(z,w)>r$. Let $\lambda(t)$, $0\le t\le 1$, be the geodesic in the Bergman metric from $z$ to $w$.
Let $N=[\beta(z,w)/r]+1$ and $t_ i=i/N$, $0\le i\le N$, where $[x]$ denotes the largest integer less than or equal to $x$.
Set $z_i=\lambda (t_ i)$, $0\le i\le N$. Then
$$\beta(z_ i,z_{i+1})=\frac{\beta(z,w)}{N}\le r.$$
Therefore,
\begin{eqnarray*}
|f(z)-f(w)|&\le& \sum_{i=1}^{N} |f(z_{i-1})-f(z_ i)| \le \sum_{i=1}^{N} \omega_ r(f)(z_ i)\\
& \le &\|f\|_{BO_{r,\gamma}} \sum_{i=1}^{N} (1-|z_ i|)^{-\gamma} .
\end{eqnarray*}
If $\gamma \ge 0$, it follows from the obvious inequality
$$(1-|z_ i|)\ge \min(1-|z|,1-|w|)$$
that
\begin{eqnarray*}
|f(z)-f(w)| &\le& \|f\|_{BO_{r,\gamma}} \frac{N} {\min (1-|z|,1-|w|)^{\gamma}}\\
&\le& \|f\|_{BO_{r,\gamma}}\,\frac{\beta(z,w)/r+1}{\min (1-|z|,1-|w|)^{\gamma}}\\
&\le& \max(1,1/r)\, \|f\|_{BO_{r,\gamma}}\, \frac{\beta(z,w)+1}{\min (1-|z|,1-|w|)^{\gamma}}.
\end{eqnarray*}
If $\gamma<0$, the result is proved in the same way once the inequality
\begin{equation}\label{eq-geo}
1-|z_ i| \le \frac{2\,|1-\langle z,w \rangle |^2}{\max(1-|z|^2,1-|w|^2)}
\end{equation}
is established. To prove this, simply note that the M\"{o}bius transformation $\varphi_ z$ sends the geodesic joining $z$ and $w$ to
the geodesic joining $0$ and $\varphi_ z(w)$. This gives
$$1-|\varphi_ z(w)|^2 \le 1-|\varphi_ z(z_ i)|^2.$$
Developing this inequality using \eqref{eq-pa}, we get
$$\frac{1-|w|^2}{|1-\langle z,w \rangle |^2} \le \frac{1-|z_ i|^2}{|1-\langle z,z_ i \rangle |^2}\le \frac{2}{|1-\langle z,z_ i \rangle |},$$
which gives
$$|1-\langle z,z_ i \rangle | \le \frac{2\, |1-\langle z,w \rangle |^2}{1-|w|^2}.$$
Interchanging the roles of $z$ and $w$, we get \eqref{eq-geo}.
The converse implication is obvious.
\end{proof}
A consequence of the above lemma is that the space $BO_{r,\gamma}$ is independent of the choice of $r$. So we will simply
write $BO_{\gamma}=BO_{1,\gamma}$ and
$$\|f\|_{BO_{\gamma}}=\|f\|_{BO_{1,\gamma}}=\sup_ {z\in\mathbb{B}_ n} (1-|z|^2)^{\gamma} \omega_ 1(f)(z).$$
Let $0<p<\infty$, $\gamma \in \mathbb{R}$, and $r>0$. We say that $f\in BA^p_{r,\gamma}$ if $f\in L^p_{loc}(\mathbb{B}_ n)$ and
$$\|f\|_{BA^p_{r,\gamma}}=\sup_{z\in \mathbb{B}_ n} (1-|z|^2)^{\gamma} \left [ \widehat{|f|^p_ r}(z)\right ]^{1/p}<\infty.$$
We proceed to show that the space $BA^p_{r,\gamma}$ is also independent of $r$.
For $\sigma>-1$ and $c>0$ the generalized Berezin transform $B_{c,\sigma}(\varphi)$ of a function $\varphi\in L^1(\mathbb{B}_ n,dv_{\sigma})$
is defined as
$$B_{c,\sigma} (\varphi) (z)= (1-|z|^2)^{c}\int_{\mathbb{B}_ n} \frac{\varphi (w)}{|1-\langle w,z\rangle|^{n+1+c+\sigma}} \,dv_ {\sigma}(w).$$
In the case when $c=n+1+\sigma$, this coincides with the ordinary Berezin transform
$B_{\sigma} \varphi (z)= \langle \varphi k^{\sigma}_ z,k^{\sigma}_ z\rangle_{\sigma}$.
\begin{lemma}\label{CMf}
Let $0<p\le q<\infty$, $\alpha,\beta>-1$, and $f\in L^q_{loc}(\mathbb{B}_ n)$. Set
$$\gamma=(n+1+\beta)/q-(n+1+\alpha)/p,\qquad d\mu_{f,\beta}=|f|^q dv_{\beta}.$$
The following conditions are equivalent:
\begin{itemize}
\item[(i)] The embedding $i:A^p_{\alpha}\rightarrow L^q(\mathbb{B}_ n,d\mu_{f,\beta})$ is bounded.
\item [(ii)] $f\in BA^q_{r,\gamma}$ for some (or all) $r>0$.
\item[(iii)] $(1-|z|^2)^{\gamma q} B_{c,\sigma}(|f|^q)\in L^{\infty}(\mathbb{B}_ n)$ for all $\sigma>-1+\gamma q$ and all $c>\max(0,-\gamma q)$.
\end{itemize}
\end{lemma}
\begin{proof}
By \cite[Theorem 50]{ZZ}, condition (i) is equivalent to
$$\mu_ {f,\beta} (D(z,r))\le C (1-|z|^2)^{(n+1+\alpha)q/p}$$
for some (or all) $r>0$. Since
$$\widehat{|f|^q_ r}(z)\asymp\frac{\mu_ {f,\beta} (D(z,r))}{(1-|z|^2)^{n+1+\beta}},$$
it follows that (i) and (ii) are equivalent.
Since $|1-\langle z,w \rangle |\asymp (1-|z|^2)$ for $w\in D(z,r)$, we have
\begin{eqnarray*}
B_{c,\sigma}(|f|^q)(z)&=&(1-|z|^2)^{c}\int_{\mathbb{B}_ n}\frac{|f(w)|^q\,dv_{\sigma}(w)}{|1-\langle z,w \rangle |^{n+1+c+\sigma}}\\
&\ge& (1-|z|^2)^{c}\int_{D(z,r)}\frac{|f(w)|^q\,dv_{\sigma}(w)}{|1-\langle z,w \rangle |^{n+1+c+\sigma}}\\
&\asymp& \widehat{|f|^q_ r}(z).
\end{eqnarray*}
This proves that (iii) implies (ii).
To finish the proof, let $\{a_j\}$ be an $r$-lattice. Then
\begin{eqnarray*}
B_{c,\sigma}(|f|^q)(z)&=&(1-|z|^2)^{c}\int_{\mathbb{B}_ n}\frac{|f(w)|^q\,dv_{\sigma}(w)}{|1-\langle z,w \rangle |^{n+1+c+\sigma}}\\
& \le& (1-|z|^2)^{c}\sum_{j=1}^\infty\int_{D(a_ j,r)}\frac{|f(w)|^q\,dv_{\sigma}(w)}{|1-\langle z,w \rangle |^{n+1+c+\sigma}}.
\end{eqnarray*}
By the estimate in (2.20) on page 63 of \cite{ZhuBn}, we have
$$|1-\langle z,w \rangle |\asymp |1-\langle z,a_ j \rangle |,\qquad w\in D(a_ j,r).$$
Thus
$$B_{c,\sigma}(|f|^q)(z) \lesssim (1-|z|^2)^{c}\sum_{j=1}^\infty\frac{(1-|a_ j|^2)^{\sigma-\beta}}{|1-\langle z,a_ j
\rangle |^{n+1+c+\sigma}}\,\,\mu_ {f,\beta} (D(a_ j,r)).$$
So condition (ii) implies that
$$B_{c,\sigma}(|f|^q)(z) \lesssim (1-|z|^2)^{c}\sum_{j=1}^\infty\frac{(1-|a_ j|^2)^{n+1+\sigma-\gamma q}}{|1-\langle z,a_ j
\rangle |^{n+1+c+\sigma}}.$$
Since $\sigma>-1+\gamma q$, or $n+1+\sigma-\gamma q>n$, an application of Lemma \ref{l2} shows that
condition (ii) implies (iii).
\end{proof}
As a consequence of the previous result, we see that the space $BA^p_{r,\gamma}$ is independent of the choice of $r$.
Thus we will simply call it $BA^p_{\gamma}$.
\begin{lemma}\label{CBO}
Let $f\in L^1_{loc}(\mathbb{B}_ n)$ and $r>0$. Then $\widehat{f}_ r$ is continuous.
\end{lemma}
\begin{proof}
The proof is elementary and we omit the details here.
\end{proof}
\begin{theorem}\label{BMO-S}
Suppose $r>0$, $\gamma \in \mathbb{R}$, $1\le p<\infty$, and $f\in L^p_{loc}(\mathbb{B}_ n)$. The following conditions are equivalent:
\begin{itemize}
\item[(a)] $f\in BMO^p_{r,\gamma}$.
\item[(b)] $f=f_ 1+f_ 2$ with $f_ 1\in BO_{\gamma}$ and $f_ 2 \in BA^p_{\gamma}$.
\item[(c)] For some (or all) $\sigma>\max(-1,-1+\gamma p)$ and for each $c>\max(0,-2\gamma p)$ we have
$$\sup_{z\in \mathbb{B}_ n} \int_{\mathbb{B}_ n} |f(w)-\widehat{f_ r}(z)|^p \frac{(1-|z|^2)^{c+\gamma p}}{|1-\langle z,w \rangle |^{n+1+c+\sigma}}
\,dv_{\sigma}(w)<\infty.$$
\item[(d)] For some (or all) $\sigma>\max(-1,-1+\gamma p)$ and for each $c>\max(0,-2\gamma p)$ there is a function
$\lambda_ z$ such that
$$\sup_{z\in \mathbb{B}_ n} \int_{\mathbb{B}_ n} |f(w)-\lambda_ z|^p \frac{(1-|z|^2)^{c+\gamma p}}{|1-\langle z,w \rangle |^{n+1+c+\sigma}}
\,dv_{\sigma}(w)<\infty.$$
\end{itemize}
\end{theorem}
\begin{proof}
That (c) implies (d) is obvious, and the implication (d) $\Rightarrow$ (a) is a consequence of Lemma \ref{Lem-1} and the inequality
\begin{eqnarray*}
&&\frac{(1-|z|^2)^{\gamma p}}{v_{\sigma}(D(z,r))}\int_{D(z,r)} \!\! |f(w)-\lambda_ z|^p\,dv_{\sigma}(w)\\
&\lesssim&\int_{\mathbb{B}_ n} \! |f(w)-\lambda_ z|^p\frac{(1-|z|^2)^{c+\gamma p}}{|1-\langle z,w \rangle |^{n+1+c+\sigma}} dv_{\sigma}(w),
\end{eqnarray*}
which follows from the well-known facts that
$$|1-\langle z,w\rangle |\asymp (1-|z|^2),\qquad v_{\sigma}(D(z,r))\asymp (1-|z|^2)^{n+1+\sigma},$$
for all $z\in\mathbb{B}_ n$ and $w\in D(z,r)$.
The proof of (a) $\Rightarrow$ (b) can be done as in \cite[Theorem 5]{Zhu-Pac}. Indeed, since $r$ is arbitrary, it suffices to show that
$$BMO^p_{2r,\gamma}\subset BO_{\gamma}+ BA^p_{\gamma}.$$
Given $f\in BMO^p_{2r,\gamma}$ and points $z,w\in \mathbb{B}_ n$ with $\beta(z,w)\le r$, we have
\begin{eqnarray*}
|\widehat{f_ r}(z)-\widehat{f_ r}(w)| &\le& |\widehat{f_ r}(z)-\widehat{f}_ {2r}(z)|+|\widehat{f}_{ 2r}(z)-\widehat{f_ r}(w)|\\
& \le& \frac{1}{v_{\sigma}(D(z,r))}\int_{D(z,r)} \!\!|f(u)-\widehat{f}_ {2r}(z)|\,dv_{\sigma}(u)\\
&&\quad +\frac{1}{v_{\sigma}(D(w,r))}\int_{D(w,r)}\!\! |f(u)-\widehat{f}_ {2r}(z)|\,dv_{\sigma}(u).
\end{eqnarray*}
Since $v_{\sigma}(D(w,r))\asymp v_{\sigma}(D(z,r))$ for $w\in D(z,r)$, and since $D(z,r)$ and $D(w,r)$ are both contained in
$D(z,2r)$, it follows from H\"{o}lder's inequality that the two integral summands above are both bounded by constant times
$(1-|z|^2)^{-\gamma}\|f\|_{BMO^p_{2r,\gamma}}$. This together with Lemma \ref{CBO} proves that $\widehat{f_ r}$ belongs to
$BO_ {r,\gamma}=BO_{\gamma}$. On the other hand, we can prove that the function $g=f-\widehat{f_ r}$ is in $BA^p_{\gamma}$
whenever $f\in BMO^p_{2r,\gamma}$. In fact, it is rather easy to see that $f\in BMO^p_{2r,\gamma}$ implies that
$f\in BMO^p_{r,\gamma}$. By the triangle inequality for $L^p$ integrals,
$$\big[\widehat{|g|^p}_ r (z)\big ]^{1/p}\le\left[\frac{1}{v_{\sigma}(D(z,r))}\int_{D(z,r)} \!\!|f(u)-\widehat{f}_ {r}(z)|^p
\,dv_{\sigma}(u)\right]^{\frac1p}+ \omega_ r(\widehat{f_ r})(z).$$
Since $ \widehat{f_ r} \in BO_{r,\gamma}$ and $f\in BMO^p_{r,\gamma}$, we deduce that $g$ belongs to $BA^p_{\gamma}$.
To show that (b) implies (c), first observe that it follows from Lemma \ref{Ict} that the integral appearing in part (c) is dominated by
$$(1-|z|^2)^{\gamma p} \Big ( B_{c,\sigma}(|f|^p)(z)+ |\widehat{f}_ r(z)|^p\Big ),$$
and by H\"{o}lder's inequality, we have $|\widehat{f}_ r(z)|^p\le \widehat{|f|^p_ r} (z)$. Thus Lemma~\ref{CMf} shows that
$f\in BA^p_\gamma$ implies condition (c). On the other hand, if $f\in BO_{\gamma}$, we write
$$f(w)-\widehat{f_ r}(z) =\frac{1}{v_{\sigma}(D(z,r))} \int_{D(z,r)}\big (f(w)- f(\zeta)\big )\,dv_{\sigma}(\zeta)$$
and use Lemma \ref{L-CBO} and the triangle inequality to obtain
$$|f(w)-\widehat{f_ r}(z)|\le C \|f\|_{BO_{\gamma}} \,\frac{\beta(z,w)+1}{\min (1-|z|,1-|w|)^{\gamma}}, \qquad \gamma\ge 0,$$
and
$$|f(w)-\widehat{f_ r}(z)|\le C \|f\|_{BO_{\gamma}} \,\frac{(\beta(z,w)+1)\,|1-\langle z,w\rangle |^{-2\gamma}}{(1-|z|)^{-\gamma}},
\qquad \gamma < 0.$$
In both cases, the integral estimates in Lemmas \ref{Ict} and \ref{Ict-beta} show that (c) holds if $f\in BO_{\gamma}$. This
shows that condition (b) implies (c) and completes the proof of the theorem.
\end{proof}
One of the consequences of the above result is that the space $BMO^p_{r,\gamma}$ is also independent of $r$. So from now on
it will simply be denoted by $BMO^p_{\gamma}$.
Next we are going to identify the space of all holomorphic functions in $BMO^p_{\gamma}$ with certain Bloch-type spaces.
Recall that for $\alpha\ge0$, the Bloch type space $\mathcal{B}^{\alpha}=\mathcal{B}^{\alpha}(\mathbb{B}_ n)$ consists of holomorphic functions
$f$ in $\mathbb{B}_ n$ for which
$$\|f\|_{\mathcal{B}^{\alpha}}=|f(0)|+\sup_{z\in \mathbb{B}_ n} (1-|z|^2)^{\alpha} |\nabla f(z)|<\infty.$$
Note that the complex gradient $\nabla f(z)$ can be replaced by the radial derivative $Rf(z)$. We will simply obtain equivalent norms.
When $\alpha>1/2$, a description can also be obtained using the invariant gradient $\widetilde{\nabla} f(z)=
\nabla (f\circ \varphi_ z)(0)$. That is, for $\alpha>1/2$, a holomorphic function $f$ belongs to $\mathcal{B}^{\alpha}$ if and only if
$$|f(0)|+\sup_{z\in \mathbb{B}_ n} (1-|z|^2)^{\alpha-1}|\widetilde{\nabla} f(z)|<\infty,$$
and this quantity defines an equivalent norm in $\mathcal{B}^{\alpha}$. Note that when $n=1$, this is true for all $\alpha\ge0$,
because in this case we actually have $\widetilde{\nabla} f(z)=(1-|z|^2) f'(z)$.
When $0<\alpha<1$, the Bloch type space $\mathcal{B}^{\alpha}$ coincides (with equivalent norms) with the holomorphic Lipschitz
space $\Lambda_{1-\alpha}=\Lambda_{1-\alpha}(\mathbb{B}_ n)$ consisting of all holomorphic functions $f$ in $\mathbb{B}_ n$ such that
$$\|f\|_{\Lambda_{1-\alpha}}=|f(0)|+\sup \left \{ \frac{|f(z)-f(w)|}{|z-w|^{1-\alpha}}:\,z,w\in \mathbb{B}_ n,\,z\neq w\right \}<\infty .$$
Note that when $\alpha=0$, the space ${\mathcal B}^\alpha$ consists of holomorphic functions $f$ with bounded partial derivatives.
Equivalently, ${\mathcal B}^0$ consists of all holomorphic functions $f$ such that
$$\sup\left\{\frac{|f(z)-f(w)|}{|z-w|}:z\not=w\right\}<\infty.$$
This space is not what is usually called the Lipschitz space $\Lambda_1$.
We refer to \cite[Chapter 7]{ZhuBn} for all these properties of Bloch and Lipschitz type spaces.
\begin{proposition}\label{Ba}
Let $\gamma \in \mathbb{R}$, $1\le p<\infty$, and $f\in H(\mathbb{B}_ n)$. Then
$f\in BMO^p_{\gamma}$ if and only if
$$\|f\|_{\gamma,*}=\sup_{z\in \mathbb{B}_ n} (1-|z|^2)^{\gamma} |\widetilde{\nabla }f(z)|<\infty.$$
\end{proposition}
\begin{proof}
We will show that condition (d) of Theorem \ref{BMO-S} is satisfied with $\lambda_ z=f(z)$ if $\|f\|_{\gamma,*}<\infty$.
To this end, let $\sigma>\max(-1,-1+p\gamma)$ and $c>\max(0,-\gamma p)$. By \cite[Lemma 7]{PZ-1} and Lemma \ref{Ict}, we have
\begin{eqnarray*}
&&\int_{\mathbb{B}_ n} |f(w)-f(z)|^p \frac{(1-|z|^2)^{c+\gamma p}}{|1-\langle z,w \rangle |^{n+1+c+\sigma}} dv_{\sigma}(w)\\
&\le& C \int_{\mathbb{B}_ n} |\widetilde{\nabla} f(w)|^p \frac{(1-|z|^2)^{c+\gamma p}}{|1-\langle z,w \rangle |^{n+1+c+\sigma}} dv_{\sigma}(w)\\
&\le& C \|f\|^p_{\gamma,*} \,(1-|z|^2)^{c+\gamma p} \int_{\mathbb{B}_ n} \frac{dv_{\sigma-\gamma p}(w)}{|1-\langle z,w \rangle |^{n+1+c+\sigma}}\\
&\le& C.
\end{eqnarray*}
Thus $\|f\|_{\gamma,*}<\infty$ implies that $f\in BMO^p_\gamma$.
To prove the other implication, we use the inequality
$$|\widetilde{\nabla}f(z)|^p \lesssim \frac{1}{v_{\sigma}(D(z,r))} \int_{D(z,r)} |f(w)-f(z)|^p\,dv_{\sigma}(w),$$
which appears on page 182 of \cite{ZhuBn}. By the triangle inequality for $L^p$ spaces,
$$|\widetilde\nabla f(z)|\le MO_{p,r}(f)(z)+|f(z)-\widehat f_r(z)|.$$
Applying Lemma 2.24 of \cite{ZhuBn} to the function $g(w)=f(w)-\widehat f_r(z)$ and the point $z$, we find a constant $C$ such that
$$|f(z)-\widehat f_r(z)|\le C\,MO_{p,r}(f)(z)$$
for all $z\in\mathbb{B}_ n$. Thus $f\in BMO^p_\gamma$ implies $\|f\|_{\gamma,*}<\infty$.
\end{proof}
\begin{coro}
Let $\gamma \in \mathbb{R}$ and $1\le p<\infty$. Then
\begin{enumerate}
\item[(a)] If $n=1$, we have
$$H(\mathbb{B}_ n)\cap BMO^p_{\gamma}=\begin{cases}
\mathcal{B}^{1+\gamma}, & \gamma \ge -1,\cr {\mathbb C}, & \gamma<-1.\end{cases}$$
\item[(b)] If $n>1$, we have
$$H(\mathbb{B}_ n)\cap BMO^p_{\gamma}=\begin{cases}
\mathcal{B}^{1+\gamma}, &\gamma > -1/2,\cr {\mathbb C}, & \gamma<-1/2.\end{cases}$$
\item[(c)] If $n>1$ and $\gamma=-\frac12$, the space $H(\mathbb{B}_ n)\cap BMO^p_{\gamma}$ consists of those holomorphic
functions $f$ on $\mathbb{B}_ n$ with
$$\sup_{z\in \mathbb{B}_ n} (1-|z|^2)^{-1/2} |\widetilde{\nabla }f(z)|<\infty.$$
\end{enumerate}
\end{coro}
\begin{proof}
Parts (a) and (b) are consequence of Proposition \ref{Ba}, the remarks preceding it, and Corollary \ref{C5}. Part (c) is just a
restatement of Proposition \ref{Ba}.
\end{proof}
According to \cite[Theorem 7.2]{ZhuBn}, the space $BMO^p_{-\frac12}$ contains non-constant functions. In fact, any function
in $\mathcal{B}^{\alpha}$ with $0<\alpha<1/2$ is in $BMO^p_{-\frac12}$. However, this space differs from $\mathcal{B}^{1/2}$.
See Corollary \ref{C5}.
In the next theorem we record some characterizations obtained for Bloch type spaces, which are of some independent interest.
\begin{theorem}\label{bmo}
Suppose $r>0$, $1\le p<\infty$, $\sigma>-1$, and $f\in H(\mathbb{B}_ n)$. Let $\gamma\ge -1$ if $n=1$; and $\gamma>-1/2$ if $n>1$.
Then the following statements are equivalent.
\begin{itemize}
\item[(i)] $f\in \mathcal{B}^{1+\gamma}$.
\item[(ii)] There is a constant $C>0$ such that
$$\frac{(1-|z|^2)^{\gamma p}}{v_{\sigma}(D(z,r))}\int_{D(z,r)}|f(w)-f(z)|^p\,dv_{\sigma}(w)\le C$$
for all $z\in\mathbb{B}_ n$.
\item[(iii)] There is a constant $C>0$ such that
$$\frac{(1-|z|^2)^{\gamma p}}{v_{\sigma}(D(z,r))}\int_{D(z,r)}|f(w)-\widehat{f_{r}}(z)|^p\,dv_{\sigma}(w)\le C$$
for all $z\in\mathbb{B}_ n$.
\item[(iv)] For each $z\in\mathbb{B}_ n$ there exists a complex number $\lambda_ z$ such that
$$\sup_{z\in\mathbb{B}_ n}\frac{(1-|z|^2)^{\gamma p}}{v_{\sigma}(D(z,r))}\int_{D(z,r)}|f(w)-\lambda_ z|^p\,dv_{\sigma}(w)<\infty.$$
\item[(v)] For some (or all) $\eta >\max(-1,-1+\gamma p)$ and for each $c>\max(0,-2\gamma p)$ we have
$$\sup_{z\in\mathbb{B}_ n}\int_{\mathbb{B}_ n}|f(w)-f(z)|^p\,\frac{(1-|z|^2)^{c+\gamma p}}{|1-\langle z,w\rangle|^{n+1+c+\eta}}\,dv_{\eta}(w)<\infty.$$
\end{itemize}
\end{theorem}
\begin{proof}
In the proof of Proposition \ref{Ba}, we proved the implications (i)$\Rightarrow$(v)$\Rightarrow$(ii)$\Rightarrow$(i). The other
equivalences are obtained from Theorem \ref{BMO-S} and Lemma \ref{Lem-1}.
\end{proof}
When $\gamma=0$, the equivalences of (i)-(iv) in Theorem \ref{bmo} is just \cite[Theorem 5.22]{ZhuBn}, and the equivalence
with (v) appears in \cite{LW}.
\section{Bounded Hankel operators}
The main result of this section is the following result, which characterizes bounded Hankel operators induced by real-valued
symbols between weighted Bergman spaces.
\begin{theorem}\label{mt1}
Let $1<p\le q<\infty$, $\alpha>-1$, $\beta>-1$, $f\in L^q_{\beta}$, and
$$\gamma=\frac{n+1+\beta}{q}-\frac{n+1+\alpha}{p}.$$
Then $H_{f}^{\beta},H_{\bar{f}}^{\beta}:A^p_{\alpha}\rightarrow L^q_{\beta}$ are both bounded if and only if $f\in BMO^q_{\gamma}$.
\end{theorem}
We are going to prove Theorem \ref{mt1} with a series of lemmas and propositions. For $t\ge 0$ let
$$K_ z^{\beta,t}(w)=\frac{1}{(1-\langle w,z\rangle)^{n+1+\beta+t}}.$$
Also, for $f\in L^q_{\beta}$ and $z\in \mathbb{B}_ n$, we consider the function $MO_{\beta,q,t} f$ defined by
$$MO_{\beta,q,t} f (z)=\big \|fh^t_z- \overline{g_ z(z)} h^t _z \big \|_{q,\beta},$$
where, for $z\in \mathbb{B}_ n$, the function $g_ z$ (that depends on $f$ and $t$) is given by
$$g_ z(w)=\frac{P_{\beta}(\bar{f} h^t_ z)(w)}{h^t_ z(w)},\qquad w\in \mathbb{B}_ n,$$
with the function $h^t_ z$ defined by
$$h^t_z(w)=h_ z^{\alpha,\beta,t}(w)=\frac{K_z^{\beta,t}(w)}{\|K_z^{\beta,t}\|_{p,\alpha}},\qquad w\in \mathbb{B}_ n.$$
Clearly, $\|h^t_z\|_{p,\alpha}=1$. When $t=0$, it is easily seen that $\overline{g_ z(z)}=B_{\beta} f(z)$ is the Berezin transform of $f$ at
the point $z$. Also, since $h^t_ z(w)$ never vanishes on $\mathbb{B}_ n$, the function $g_ z$ is holomorphic on $\mathbb{B}_ n$.
\begin{lemma}\label{R-MO}
Let $1<p\le q<\infty$ and $\alpha,\beta>-1$. Let $\gamma$ be as in Theorem \ref{mt1} and $t\ge 0$ such that
$$n+1+\beta+t>(n+1+\alpha)/p.$$
If $MO_{\beta,q,t} f\in L^{\infty}(\mathbb{B}_ n)$, then $f\in BMO^q_{\gamma}$.
\end{lemma}
\begin{proof}
Since $n+1+\beta+t>(n+1+\alpha)/p$, Lemma \ref{Ict} gives us the estimate
$$\|K_z^{\beta,t}\|_{p,\alpha}\asymp(1-|z|^2)^{(n+1+\alpha)/p-(n+1+\beta+t)}.$$
It follows that $\big [MO_{\beta,q,t} f (z)\big ]^q$ is comparable to
$$(1-|z|^2)^{\gamma q+(n+1+\beta)(q-1)+tq}\int_{\mathbb{B}_ n} \frac{|f(w)-\overline{g_ z(z)}\,|^q}{|1-\langle z,w \rangle|^{(n+1+\beta+t)q}}
\,dv_{\beta}(w),$$
which dominates
$$\frac{(1-|z|^2)^{\gamma q}}{|D(z,r)|}\int_{D(z,r)}|f(w)-\overline{g_z(z)}|^q\,dv(w).$$
This shows that, if $MO_{\beta,q,t} f\in L^{\infty}(\mathbb{B}_ n)$, the condition in Lemma \ref{Lem-1} is satisfied with
$\lambda_ z=\overline{g_ z(z)}$, so $f\in BMO^q_{\gamma}$.
\end{proof}
The following result gives the necessity in Theorem \ref{mt1}. It generalizes, in several directions, Proposition 8.19 in \cite{Zhu}, where the
method of proof is based on Hilbert space techniques. A different method was used in \cite{Zhu-Pac} to deal with the case $\alpha=\beta$
and $p=q$. Our method here is more flexible and allows us to obtain the result in much more generality.
\begin{proposition}\label{necessity}
Let $1<p,q<\infty$, $\alpha,\beta>-1$, and $t\ge 0$. Then for any $f\in L^q_{\beta}$ we have
$$
MO_{\beta,q,t}f(z)
\lesssim \big \|H^{\beta}_f h^t_z\big \|_{q,\beta}+ \big \|H^{\beta}_{\bar f} h^t_z\big \|_{q,\beta}.
$$
\end{proposition}
\begin{proof}
By the triangle inequality and the definition of Hankel operators, we have
\begin{eqnarray*}
MO_{\beta,q,t}f(z)&=&\big \|fh^t_ z-\overline{g_ z(z)}\, h^t_ z \big \|_{q,\beta}\\
&\le& \big \|fh^t_ z -P_{\beta}(f h^t_ z)\big \|_{q,\beta}+\big \|P_{\beta}(f h^t_ z)-\overline{g_ z(z)}\, h^t_ z\big \|_{q,\beta}\\
&=&\big \|H^{\beta}_ f h^t_ z\big \|_{q,\beta}+\big \|P_{\beta}(f h^t_ z)-\overline{g_ z(z)}\, h^t_ z\big \|_{q,\beta}.
\end{eqnarray*}
For any $g\in A^1_{\beta}(\mathbb{B}_ n)$ it is easy to check that
\begin{equation}\label{Eq-H1}
\overline{g(z)} h^t_ z =P_{\beta+t}(\bar{g} h^t_ z).
\end{equation}
This together with the boundedness of $P_{\beta+t}$ on $L^q_{\beta}$ yields
\begin{eqnarray*}
\big\|P_{\beta}(fh^t_ z)-\overline{g_ z(z)}\,h^t_ z\big\|_{q,\beta}&=&\big\|P_{\beta}(fh^t_ z)-P_{\beta+t}(\overline{g_ z} \,h^t_ z)\big\|_{q,\beta}\\
&=&\big \|P_{\beta+t} \big (P_{\beta}(f h^t_ z)-\overline{g_ z} \,h^t_ z)\big )\big \|_{q,\beta}\\
&\le& \big \|P_{\beta+t}\big \|_{L^q_{\beta}} \cdot \big \|P_{\beta}(f h^t_ z)-\overline{g_ z} \,h^t_ z\big \|_{q,\beta}.
\end{eqnarray*}
Finally,
\begin{eqnarray*}
\big \|P_{\beta}(f h^t_ z)-\overline{g_ z} \,h^t_ z\big \|_{q,\beta}& \le &\big \|f h^t_ z -P_{\beta}(f h^t_ z)\big \|_{q,\beta}+
\big \|fh^t_ z -\overline{g_ z} \,h^t_ z\big \|_{q,\beta}\\
&=&\big \|H^{\beta}_ f h^t_ z\big \|_{q,\beta}+\big \|\bar{f}\,h^t_ z -g_ z \,h^t_ z\big \|_{q,\beta}\\
&=&\big \|H^{\beta}_ f h^t_ z\big \|_{q,\beta}+\big \|\bar{f}\,h^t_ z -P_{\beta}(\bar{f}\,h^t_ z)\big \|_{q,\beta}\\
&=&\big \|H^{\beta}_ f h^t_ z\big \|_{q,\beta}+\big \|H^{\beta}_{\bar{ f}} h^t_ z\big \|_{q,\beta}.
\end{eqnarray*}
This proves the result with constant $C=\big (1+\big \|P_{\beta+t}\big \|_{L^q_{\beta}}\big )$.
\end{proof}
Note that the proposition above does not require $p\le q$.
The next two propositions, which require the condition $p\le q$, will
establish the sufficiency of Theorem \ref{mt1}.
\begin{proposition}\label{L-BA}
Let $1<p\le q <\infty$, $\alpha>-1$, $\beta>-1$, and
$$\gamma=(n+1+\beta)/q-(n+1+\alpha)/p.$$
If $f\in BA^q_{\gamma}$, then $H_ f^{\beta}:A^p_{\alpha}\rightarrow L^q_{\beta}$ is bounded.
\end{proposition}
\begin{proof}
Since $q>1$, the Bergman projection $P_{\beta}$ is bounded on $L^q_{\beta}$. Thus
$$
\big \|H^{\beta}_ f g \big \|_{q,\beta} \le \|fg\|_{q,\beta} +\|P_{\beta}(fg)\| _{q,\beta}
\lesssim \|fg\|_{q,\beta}=\|g\|_{L^q(d\mu_{f,\beta})}.
$$
The result then follows from Lemma \ref{CMf}.
\end{proof}
We note that the proof of the previous proposition also works for $1=p<q<\infty$. In order to show that $H^{\beta}_ f$ is bounded if
$f\in BO_{\gamma}$ with $\gamma<0$, we need the following result.
\begin{lemma}\label{LHb1}
Let $s\ge \beta>-1$, $1<q<\infty$, $f\in L^q_{\beta}$, and $g\in H^{\infty}$. Then
$$\big \|H_ f^{\beta} g \big \|_{q,\beta} \le C \,\big \|H_ f^{s} g \big \|_{q,\beta}.$$
\end{lemma}
\begin{proof}
Since $g\in H^{\infty}$, we have $gf\in L^q_{\beta}$. Also,
\begin{eqnarray*}
\big\| H_ f^{\beta} g \big \|_{q,\beta}&=& \big \|(I-P_{\beta})(gf) \big \|_{q,\beta}\\
&\le& \big \|(I-P_{s})(gf) \big \|_{q,\beta}+\big \|(P_ s-P_{\beta})(gf) \big \|_{q,\beta}\\
& =&\big \|H_ f^{s} g \big \|_{q,\beta}+\big \|(P_ {\beta}-P_{s})(gf) \big \|_{q,\beta}.
\end{eqnarray*}
Since $P_ s$ is bounded on $L^q_{\beta}$, the reproducing formula yields $P_{\beta} P_ s (gf)=P_ s(gf)$. Thus
$$(P_ {\beta}-P_{s})(gf)=(P_ {\beta}-P_{\beta} P_{s})(gf)=P_{\beta}(I-P_ s)(gf)=P_{\beta} (H_ f^s g).$$
This gives
$$\big \|(P_ {\beta}-P_{s})(gf) \big \|_{q,\beta} \le \big \|P_{\beta} \big \| \cdot \big \|H_ f^{s} g \big \|_{q,\beta},$$
so we obtain the desired inequality with $C=1+ \|P_{\beta} \|$.
\end{proof}
\begin{proposition}\label{L-BO}
Let $1<p\le q <\infty$, $\alpha>-1$, $\beta>-1$, and
$$\gamma=(n+1+\beta)/q-(n+1+\alpha)/p.$$
If $f\in BO_{\gamma}$, then $H_ f^{\beta}:A^p_{\alpha}\rightarrow L^q_{\beta}$ is bounded.
\end{proposition}
\begin{proof}
We first consider the case $\gamma \ge 0$. For $g\in H^{\infty}$, which is dense in $A^p_{\alpha}$, we have
\begin{eqnarray*}
\|H_ f ^{\beta} g\|^q_{q,\beta}& =&\int_{\mathbb{B}_ n} |H_ f^{\beta} g(z)|^q\, dv_{\beta}(z)\\
&=&\int_{\mathbb{B}_ n} \left |\int_{\mathbb{B}_ n}\frac{(f(z)-f(w))\,g(w)}{(1-\langle z,w \rangle )^{n+1+\beta}} \,dv_{\beta}(w)\right |^q\, dv_{\beta}(z)\\
&\le& \int_{\mathbb{B}_ n} \left (\int_{\mathbb{B}_ n}\frac{|f(z)-f(w)|\,|g(w)|}{|1-\langle z,w \rangle |^{n+1+\beta}} \,dv_{\beta}(w)\right )^q\, dv_{\beta}(z).
\end{eqnarray*}
By Lemma \ref{L-CBO},
$$\|H_ f ^{\beta} g\|^q_{q,\beta}\lesssim \int_{\mathbb{B}_ n} \left (\int_{\mathbb{B}_ n}\frac{(1+\beta(z,w))\,|g(w)| \,dv_{\beta}(w)}{|1-\langle z,w
\rangle |^{n+1+\beta}\min (1-|z|,1-|w|)^{\gamma}} \right )^q\, dv_{\beta}(z).$$
Write
$$I_ 1(g)=\int_{\mathbb{B}_ n} \left (\int_{\mathbb{B}_ n}\frac{|g(w)|\,dv_{\beta}(w)}{|1-\langle z,w \rangle |^{n+1+\beta}\min (1-|z|,1-|w|)^{\gamma}}\right )^q
\, dv_{\beta}(z),$$
and split the inner integral in two parts, $I_{1,1}(g)$ over $|w|\le |z|$ and $I_{1,2}(g)$ over $|w|>|z|$. Since
$$\min (1-|z|,1-|w|)^{\gamma}= (1-|z|)^{\gamma}$$
for $|w|\le |z|$, we have
\begin{eqnarray*}
I_ {1,1}(g)&\lesssim&\int_{\mathbb{B}_ n} \left (\int_{\mathbb{B}_ n}\frac{|g(w)|\,dv_{\beta}(w)}{|1-\langle z,w \rangle |^{n+1+\beta}}\right )^q
\, dv_{\beta-q\gamma}(z)\\
&=&\int_{\mathbb{B}_ n} \big |S_{b,c}(|g|)(z)\big |^q \, dv_{\beta-q\gamma}(z),
\end{eqnarray*}
where $S_{b,c}$ is the integral operator appearing in Theorem \ref{Z-T} with $b=\beta$ and $c=n+1+\beta$. Notice that
$\beta-q\gamma>-1$ is equivalent to
$$n\left(\frac1q-\frac1p\right)<\frac{1+\alpha}{p},$$
which is automatically satisfied since $p\le q$. Applying Theorem \ref{Z-T}, we obtain $I_{1,1}(g) \lesssim \|g\|_{p,\alpha}^q$,
provided $1+\alpha<p(1+b)$ and $c\le n+1+b+\lambda$. Since $b=\beta$ and
$$\lambda=\frac{n+1+(\beta-q\gamma)}{q}-\frac{n+1+\alpha}{p}=0,$$
the condition $c\le n+1+b+\lambda$ is satisfied with equality. It remains to check that the condition
\begin{equation}\label{Cab}
1+\alpha<p(1+b)=p(1+\beta)
\end{equation}
is satisfied. Since $\gamma \ge 0$ and $q\ge p$, we have
$$0\le \gamma =(n+1+\beta)/q-(n+1+\alpha)/p \le (\beta-\alpha)/p.$$
This gives $\alpha \le \beta$, so\eqref{Cab} holds since $p> 1$.
Similarly, we have
$$I_ {1,2}(g)\lesssim \int_{\mathbb{B}_ n} \big |S_{b,c}(|g|)(z)\big |^q \, dv_{\beta}(z)$$
with $b=\beta-\gamma$, $c=n+1+\beta$, and $\lambda=\gamma$. We want to apply Theorem \ref{Z-T} to estimate $I_{1,2}(g)$.
In this case, the condition $c\le n+1+b+\lambda$ in Theorem~\ref{Z-T} holds with equality.
The other condition in Theorem \ref{Z-T} is $\alpha+1<p(1+\beta-\gamma)$, which is equivalent to
$$\frac{p(n+1+\beta)}{q}<p\beta+p+n.$$
If $q'$ is the conjugate exponent of $q$, the above condition is equivalent to
$$n\left(\frac1q-\frac1p\right)<\frac{1+\beta}{q'},$$
which is automatically satisfied since $p\le q$. Hence, by Theorem \ref{Z-T}, we have $I_{1,2}(g)\lesssim \|g\|^q_{p,\alpha}$. This
together with the previous estimate yields $I_ 1(g) \lesssim \|g\|_{p,\alpha}^q$. The remaining estimate
$I_ 2(g) \lesssim \|g\|_{p,\alpha}^q$ with
$$I_ 2(g):=\int_{\mathbb{B}_ n} \left (\int_{\mathbb{B}_ n}\frac{|g(w)|\,\beta(z,w)\,dv_{\beta}(w)}{|1-\langle z,w \rangle |^{n+1+\beta}
\min (1-|z|,1-|w|)^{\gamma}}\right )^q\, dv_{\beta}(z),$$
can be proved in a similar manner, using Proposition \ref{proj-ge1-beta} instead of Theorem \ref{Z-T}. The proof of the case
$\gamma\ge 0$ is now complete.
If $\gamma<0$ and $g\in H^{\infty}$, we use Lemma \ref{LHb1}, with $s\ge \beta$ big enough so that $p(s+\gamma+1)>\alpha+1$,
to obtain
$$\big \|H_ f ^{\beta} g \big \|^q_{q,\beta}\lesssim \big \|H_ f ^{s} g\big \|^q_{q,\beta}\le \int_{\mathbb{B}_ n} \left[\int_{\mathbb{B}_ n}\frac{|f(z)-f(w)|\,|g(w)|}
{|1-\langle z,w \rangle |^{n+1+s}} \,dv_{s}(w)\right]^q\, dv_{\beta}(z).$$
By Lemma \ref{L-CBO},
$$|f(z)-f(w)|\le C \frac{\beta(z,w)+1}{ (1-|w|)^{-\gamma}}\,|1-\langle z,w \rangle |^{-2\gamma}.$$
Therefore,
$$\big \|H_ f ^{\beta} g \big \|^q_{q,\beta}\lesssim \int_{\mathbb{B}_ n} \left (\int_{\mathbb{B}_ n}\frac{(\beta(z,w)+1)\,|g(w)|}{|1-\langle z,w
\rangle |^{n+1+s+2\gamma}} \,dv_{s+\gamma}(w)\right )^q\, dv_{\beta}(z),$$
and the boundedness of $H_f^\beta:A^p_\alpha\to L^q_\beta$ follows from Theorem \ref{Z-T} and
Proposition \ref{proj-ge1-beta} again.
\end{proof}
The proof of Theorem \ref{mt1} is now complete: the necessity of the condition $f\in BMO^q_{\gamma}$ follows from Lemma \ref{R-MO}
and Proposition \ref{necessity}. Since $f\in BMO^q_{\gamma}$ if and only if $\overline{f}\in BMO^q_{\gamma}$, the sufficiency is a
consequence of Theorem \ref{BMO-S}, Proposition \ref{L-BA}, and Proposition \ref{L-BO}.
As an immediate consequence of Theorem \ref{mt1} and Proposition \ref{Ba}, we obtain the following result that characterizes the
boundedness of Hankel operators with conjugate holomorphic symbols.
\begin{coro}
Let $f\in A^1_{\beta}$, $1<p\le q<\infty$, $\alpha>-1$, $\beta>-1$, and
$$\gamma=\frac{n+1+\beta}{q}-\frac{n+1+\alpha}{p}.$$
\begin{enumerate}
\item For $n=1$ we have
\begin{enumerate}
\item[(a)] If $\gamma \ge -1$, then $H_{\bar{f}}:A^p_{\alpha}\rightarrow L^q_{\beta}$ is bounded if and only if $f\in \mathcal{B}^{1+\gamma}$.
\item[(b)] If $\gamma < -1$, then $H_{\bar{f}}:A^p_{\alpha}\rightarrow L^q_{\beta}$ is bounded if and only if $f$ is constant.
\end{enumerate}
\item For $n>1$ we have
\begin{enumerate}
\item[(a)] If $\gamma > -1/2$, then $H_{\bar{f}}:A^p_{\alpha}\rightarrow L^q_{\beta}$ is bounded if and only if $f\in \mathcal{B}^{1+\gamma}$.
\item[(b)] If $\gamma < -1/2$, then $H_{\bar{f}}:A^p_{\alpha}\rightarrow L^q_{\beta}$ is bounded if and only if $f$ is constant.
\item[(c)] If $\gamma = -1/2$, then $H_{\bar{f}}:A^p_{\alpha}\rightarrow L^q_{\beta}$ is bounded if and only if
$$\sup_{z\in \mathbb{B}_ n} (1-|z|^2)^{-1/2} |\widetilde{\nabla }f(z)|<\infty.$$
\end{enumerate}
\end{enumerate}
\end{coro}
\begin{proof}
Since $f\in A^1_{\beta}$, the Hankel operator $H_{\bar{f}}$ is densely defined. If $H_{\bar{f}}:A^p_{\alpha}\rightarrow L^q_{\beta}$ is bounded,
by testing the boundedness on the function 1 we see that $f\in A^q_{\beta}$. Since $\mathcal{B}^{1+\gamma}\subset A^q_{\beta}$, the
result follows from Theorem \ref{mt1} and Proposition \ref{Ba}.
\end{proof}
In the case $q=p=2$, this recovers the results of Janson and Wallst\'{e}n \cite{J,Wall}, where the case $\gamma=-1$ for $n=1$ and the
case $\gamma=-1/2$ for $n>1$ were left open. Thus we have resolved these open cases.
\section{Weighted VMO spaces}
Let $\gamma\in\mathbb{R}$. For any positive radius $r$ and every exponent $p$ with $1\le p<\infty$, the space $VMO^p_{r,\gamma}$
consists of those functions $f$ in $BMO^p_{r,\gamma}$ such that
$$\lim_{|z|\to1}(1-|z|^2)^{\gamma} MO_{p,r}(f)(z)=0.$$
Again, the space $VMO^p_{r,\gamma}$ is actually independent of the weight parameter $\sigma$. Similarly as before, for $r>0$, we
define $VO_{r,\gamma}$ as the space of functions $f$ in $BO_{r,\gamma}$ satisfying
$$\lim_{|z|\to 1} (1-|z|^2)^{\gamma}\omega_r(f)(z)=0,$$
and $VA^p_{r,\gamma}$ as the space of functions $f$ in $BA^p_{r,\gamma}$ satisfying
$$\lim_{|z|\to 1} (1-|z|^2)^{\gamma}\left[\widehat{|f|^p_ r}(z)\right]^{1/p}=0.$$
The following result shows that $VA^p_{r,\gamma}$ does not depend on $r$.
\begin{lemma}\label{VA}
Let $0<p\le q<\infty$, $\alpha>-1$, $\beta>-1$, $f\in L^q_{loc}(\mathbb{B}_ n)$, and
$$\gamma=(n+1+\beta)/q-(n+1+\alpha)/p,\qquad d\mu_{f,\beta}=|f|^q dv_{\beta}.$$
The following are equivalent:
\begin{itemize}
\item[(i)] If $\{f_k\}$ is a bounded sequence in $A^p_{\alpha}$ and $f_k\to0$ uniformly on every compact subset of $\mathbb{B}_ n$, then
$$\lim_{k\to\infty}\int_{\mathbb{B}_ n}|f_k(z)|^q\,d\mu_{f,\beta}(z)=0.$$
\item [(ii)] $f\in VA^q_{r,\gamma}$ for some (or all) $r>0$.
\item[(iii)] The condition
$$\lim_{|z|\to1}(1-|z|^2)^{\gamma q} B_{c,\sigma}(|f|^q)(z)=0$$
holds for all $\sigma>\max(-1,-1+\gamma q)$ and all $c>\max(0,-\gamma q)$.
\end{itemize}
\end{lemma}
\begin{proof}
By the corresponding little-oh result of Theorem 50 in \cite{ZZ}, we know that (i) is equivalent to
$$\lim_{|z|\to1}\frac{\mu_{f,\beta}(D(z,r))}{(1-|z|^2)^{(n+1+\alpha)q/p}}=0$$
for some (or all) $r>0$.
The equivalence of (i) and (ii) is a consequence of this result and the fact that
$$\widehat{|f|^q_ r}(z)\asymp \frac{\mu_ {f,\beta} (D(z,r))}{(1-|z|^2)^{n+1+\beta}}.$$
That (iii) implies (ii) follows from the fact that
$$\widehat{|f|^q_ r}(z)\lesssim B_{c,\sigma}(|f|^q)(z),$$
which has been shown in the proof of Lemma~\ref{CMf}.
It remains to prove that (ii) implies (iii). Let $f\in VA^q_{r,\gamma}$. By definition, we have
$$(1-|z|^2)^{\gamma q} B_{c,\sigma}(\widehat{|f|^q_ r})(z)=(1-|z|^2)^{c+\gamma q}
\int_{\mathbb{B}_ n}\frac{\widehat{|f|^q_ r}(w)\,dv_{\sigma}(w)}{|1-\langle z,w \rangle |^{n+1+c+\sigma}}.$$
For $0<s<1$ let
$$I_1(s)=(1-|z|^2)^{c+\gamma q}\int_{|w|\le s}\frac{\widehat{|f|^q_ r}(w)\,dv_{\sigma}(w)}{|1-\langle z,w \rangle |^{n+1+c+\sigma}},$$
and
$$I_2(s)=(1-|z|^2)^{c+\gamma q}\int_{s<|w|<1}\frac{\widehat{|f|^q_ r}(w)\,dv_{\sigma}(w)}{|1-\langle z,w \rangle |^{n+1+c+\sigma}}.$$
Let $\varepsilon$ be an arbitrary positive number. By (ii), and then by Lemma~\ref{Ict}, there exists an $s>0$ such that
$$I_2(s)\lesssim \,\varepsilon \, (1-|z|^2)^{c+\gamma q}\int_{s<|w|<1}\frac{(1-|w|^2)^{\sigma-\gamma q}
\,dv(w)}{|1-\langle z,w \rangle |^{n+1+c+\sigma}}\lesssim \, \varepsilon.$$
Since $f\in VA^q_{r,\gamma}\subset BA^q_{r,\gamma}$, we know that
$$\widehat{|f|^q_ r}(w)\lesssim (1-|w|^2)^{-q\gamma}.$$
Since $|1-\langle z,w \rangle| \gtrsim (1-|w|^2)$, we obtain
\begin{eqnarray*}
I_1(s)&\le& (1-|z|^2)^{c+\gamma q}\int_{|w|\le s}\frac{dv_\sigma(w)}{(1-|w|^2)^{n+1+c+\sigma+\gamma q}}\\
&\lesssim& \frac{(1-|z|^2)^{c+\gamma q}}{(1-s^2)^{n+1+c+\gamma q}}.
\end{eqnarray*}
Hence, we can find a $\delta\in(0,1)$ such that $I_1(s)\lesssim \,\varepsilon$ whenever $1-\delta<|z|<1$.
Combining the above two inequalities for $I_1(s)$ and $I_2(s)$ we deduce for $1-\delta<|z|<1$ that
$$(1-|z|^2)^{\gamma q} B_{c,\sigma}(\widehat{|f|^q_ r})(z)\lesssim \, \varepsilon.$$
Therefore,
$$\lim_{|z|\to1}(1-|z|^2)^{\gamma q} B_{c,\sigma}(\widehat{|f|^q_ r})(z)=0.$$
Let $d\mu_{f,\sigma}=|f|^q\,dv_{\sigma}$. Since
\begin{equation}\label{ave}
\widehat{|f|^q_r}(z)\asymp\widehat{\mu_{f,\sigma}}(z):=\frac{\mu_{f,\sigma}(D(z,r))}{(1-|z|^2)^{n+1+\sigma}},
\end{equation}
the above equation is equivalent to
$$\lim_{|z|\to1}(1-|z|^2)^{\gamma q} B_{c,\sigma}(\widehat{\mu_{f,\sigma}})(z)=0.$$
By \cite[Lemma 52]{ZZ}, we have
$$B_{c,\sigma}(\mu_{f,\sigma})(z)\lesssim B_{c,\sigma}(\widehat{\mu_{f,\sigma}})(z),$$
where
$$B_{c,\sigma}(\mu_{f,\sigma})(z)=(1-|z|^2)^{c}\int_{\mathbb{B}_ n}\frac{d\mu_{f,\sigma}(w)}{|1-\langle z,w \rangle |^{n+1+c+\sigma}}.$$
Thus we obtain
$$\lim_{|z|\to1}(1-|z|^2)^{\gamma q} B_{c,\sigma}(\mu_{f,\sigma})(z)=0,$$
which is the same as
$$\lim_{|z|\to1}(1-|z|^2)^{\gamma q} B_{c,\sigma}(|f|^q)(z)=0.$$
This proves (ii) implies (iii) and completes the proof of the lemma.
\end{proof}
The next result shows that $VO_{r,\gamma}$ does not depend on $r$.
\begin{lemma}\label{VO}
Let $\gamma \in \mathbb{R}$ and $r_1, r_2>0$. If $f\in VO_{r_1,\gamma}$, then $f\in VO_{r_2,\gamma}$.
\end{lemma}
\begin{proof}
If $r_1>r_2$, the result is obvious. So we assume that $r_1<r_2$ and fix $z\in\mathbb{B}_ n$. It follows from the continuity of $f$ on $\mathbb{B}_ n$ that
$$\omega_{r_2}(f)(z)=\sup\{|f(z)-f(\zeta)|,\, \zeta\in\overline{D(z,r_2)}\},$$
and we can find $w\in\overline{D(z,r_2)}$ such that
$$|f(z)-f(w)|=\omega_{r_2}(f)(z).$$
Let $\lambda=\lambda(t)$, $0\le t\le 1$, be the geodesic in the Bergman metric from $z$ to $w$.
Then $\lambda$ lies entirely in $\overline{D(z,r_2)}$. As in the proof of Lemma~\ref{L-CBO},
we let $N=[r_2/r_1]+2$ and $t_i=i/N$, $0\le i\le N$, where $[x]$ denotes the largest integer less than or equal to $x$.
Set $z_i=\lambda(t_i)$, $0\le i\le N$. Since $N\ge r_2/r_1+1>r_2/r_1$, we have
$$\beta(z_{i-1},z_{i})=\frac{\beta(z,w)}{N}\le \frac{r_2}{N}<r_1.$$
Because $z_i$ is in the closure of $D(z,r_2)$, there exists a constant $K>0$, independent of $i$, such that
$$\frac1{K}(1-|z|^2)^{\gamma}\le (1-|z_i|^2)^{\gamma}\le K(1-|z|^2)^{\gamma}.$$
Since $f\in VO_{r_1,\gamma}$, we know that
$$\lim_{|z_i|\to1}(1-|z_i|^2)^{\gamma}\omega_{r_1}(f)(z_i)=0.$$
But $|z_i|\to1$ as $|z|\to1$. So for any $\varepsilon>0$ there exists $\delta>0$ such that
$$(1-|z_i|^2)^{\gamma}\omega_{r_1}(f)(z_i)<\frac{\varepsilon}{NK}$$
whenever $1-|z|<\delta$. Thus
\begin{eqnarray*}
|f(z)-f(w)|&\le&\sum_{i=1}^{N}|f(z_{i-1})-f(z_{i})|\le\sum_{i=1}^{N}\omega_{r_1}(f)(z_i)\\
&\le& \frac{\varepsilon}{K(1-|z_i|^2)^{\gamma}}\le\frac{\varepsilon}{K}\cdot\frac{K}{(1-|z|^2)^{\gamma}}\\
&=&\frac{\varepsilon}{(1-|z|^2)^{\gamma}}.
\end{eqnarray*}
Therefore,
$$\omega_{r_2}(f)(z)=|f(z)-f(w)|\le\frac{\varepsilon}{(1-|z|^2)^{\gamma}}$$
for $1-\delta<|z|<1$, which shows that
$$\lim_{|z|\to1}(1-|z|^2)^{\gamma}\omega_{r_2}(f)(z)=0,$$
or $f\in VO_{r_2,\gamma}$.
\end{proof}
Because of Lemmas~\ref{VA} and \ref{VO}, we can denote $VA^p_{r,\gamma}$ and $VO_{r,\gamma}$ by $VA^p_{\gamma}$
and $VO_{\gamma}$, respectively. Just as in the big-oh case, we have the following result for $VMO^p_{r,\gamma}$.
\begin{theorem}\label{VMO-S}
Suppose $r>0$, $\gamma \in \mathbb{R}$, $1\le p<\infty$, and $f\in BMO^p_{\gamma}$.
The following conditions are equivalent:
\begin{itemize}
\item[$(a)$] $f\in VMO^p_{r,\gamma}$.
\item[$(b)$] $f=f_ 1+f_ 2$ with $f_1\in VO_{\gamma}$ and $f_ 2 \in VA^p_{\gamma}$.
\item[$(c)$] For some (or all) $\sigma >\max(-1,-1+\gamma p)$ and for each $c\ge\max(n+1+\sigma, n+1+\sigma-2\gamma)$,
we have
$$\lim_{|z|\to1} \int_{\mathbb{B}_ n} |f(w)-\widehat{f_ r}(z)|^p\frac{(1-|z|^2)^{c+\gamma p}}{|1-\langle z,w \rangle |^{n+1+c+\sigma}}
\,dv_{\sigma}(w)=0.$$
\item[$(d)$] For some ( or all) $\sigma >\max(-1,-1+\gamma p)$ and for each $c\ge\max(n+1+\sigma, n+1+\sigma-2\gamma)$,
there is a function $\lambda_ z$ such that
$$\lim_{|z|\to1} \int_{\mathbb{B}_ n} |f(w)-\lambda_ z|^p\frac{(1-|z|^2)^{c+\gamma p}}{|1-\langle z,w \rangle |^{n+1+c+\sigma}} dv_{\sigma}(w)=0.$$
\item[$(e)$] For some (or all) $\sigma >-1$ there is a function $\lambda_ z$ such that
$$\lim_{|z|\to1}\frac{(1-|z|^2)^{\gamma p}}{v_{\sigma}(D(z,r))}\int_{D(z,r)} |f(w)-\lambda_ z|^p\,dv_{\sigma}(w)=0.$$
\end{itemize}
\end{theorem}
\begin{proof}
That (c) implies (d) is obvious. That (d) implies (e) follows from the simple inequality
\begin{eqnarray*}
&&\frac{(1-|z|^2)^{\gamma p}}{v_{\sigma}(D(z,r))}\int_{D(z,r)} \!\! |f(w)-\lambda_ z|^p\,dv_{\sigma}(w)\\
&\lesssim&\int_{\mathbb{B}_ n} \! |f(w)-\lambda_ z|^p\frac{(1-|z|^2)^{c+\gamma p}}{|1-\langle z,w \rangle |^{n+1+c+\sigma}} dv_{\sigma}(w).
\end{eqnarray*}
An easy modification of the proof of Lemma~\ref{Lem-1} shows that (e) implies (a). That (a) implies (b) follows easily from the
proof of (a) implying (b) in Theorem~\ref{BMO-S}.
Thus we only need to prove that (b) implies (c). Suppose that (b) holds. As in the proof of Theorem \ref{BMO-S}, from
Lemma \ref{VA} it is not difficult to see that (c) is satisfied for $f\in VA^p_{\gamma}$. Now, for $f\in VO_{\gamma}$, it is obvious
that $f\in BO_{\gamma}$. Set
$$I(z)=\int_{\mathbb{B}_ n} |f(w)-\widehat{f_ r}(z)|^p\frac{(1-|z|^2)^{c+\gamma p}}{|1-\langle z,w \rangle |^{n+1+c+\sigma}} dv_{\sigma}(w).$$
Making the change of variables $w=\varphi_z(\zeta)$, we obtain
\begin{equation}\label{EqIZ0}
I(z)=\int_{\mathbb{B}_ n}|f\circ\varphi_z(\zeta)-\widehat{f_ r}(z)|^p\frac{(1-|z|^2)^{\gamma p}}{|1-\langle z,\zeta \rangle |^{n+1+\sigma-c}}
\,dv_{\sigma}(\zeta).
\end{equation}
In the case $\gamma\ge 0$, it follows from the proof of Theorem~\ref{BMO-S} and the invariance of the Bergman metric that
\begin{eqnarray*}
|f\circ\varphi_z(\zeta)-\widehat{f_ r}(z)|&\lesssim& \|f\|_{BO_{\gamma}} \,\frac{\beta(\varphi_z(\zeta), z)+1}{\min (1-|z|,
1-|\varphi_z(\zeta)|)^{\gamma}}\\
&\lesssim& \|f\|_{BO_{\gamma}} \,\frac{\beta(\zeta,0)+1}{\min (1-|z|,1-|\varphi_z(\zeta)|)^{\gamma}}.
\end{eqnarray*}
Let
$$
E=\{\zeta\in\mathbb{B}_ n:\, |\varphi_z(\zeta)|\le|z|\}.
$$
For $\zeta\in E$ we have $1-|\varphi_z(\zeta)|^2\ge 1-|z|^2$ and
\begin{equation}\label{I1}
|f\circ\varphi_z(\zeta)-\widehat{f_ r}(z)|^p(1-|z|^2)^{\gamma p}\lesssim (\beta(\zeta,0)+1)^p.
\end{equation}
For $\zeta\in\mathbb{B}_ n\setminus E$ we have $1-|\varphi_z(\zeta)|^2\le 1-|z|^2$ and
\begin{eqnarray*
|f\circ\varphi_z(\zeta)-\widehat{f_ r}(z)|^p(1-|z|^2)^{\gamma p}
&\lesssim& \frac{(\beta(\zeta,0)+1)^p(1-|z|^2)^{\gamma p}}{(1-|\varphi_z(\zeta)|^2)^{\gamma p}}\\
&\lesssim& (\beta(\zeta,0)+1)^p(1-|\zeta|^2)^{-\gamma p}
\end{eqnarray*}
Since $\sigma>-1+\gamma\, p\ge -1$, we have
$$\int_{\mathbb{B}_ n}(\beta(\zeta,0)+1)^p\,dv_{\sigma}(\zeta)<\infty,$$
and
$$\int_{\mathbb{B}_ n}(\beta(\zeta,0)+1)^p(1-|\zeta|^2)^{-\gamma p}\,dv_{\sigma}(\zeta)<\infty.$$
Let
$$H(\zeta)=\begin{cases}(\beta(\zeta,0)+1)^p,& \zeta\in E\\
(\beta(\zeta,0)+1)^p(1-|\zeta|^2)^{-\gamma p}),& \zeta\in \mathbb{B}_ n\setminus E
\end{cases}$$
The above argument shows that $H(\zeta)$ is in $L^1(\mathbb{B}_ n,dv_{\sigma})$ and, since $c\ge n+1+\sigma$, we have
\begin{equation}\label{EqIZ1}
|f\circ\varphi_z(\zeta)-\widehat{f_ r}(z)|^p(1-|z|^2)^{\gamma p}\,|1-\langle z,\zeta\rangle |^{c-(n+1+\sigma)}\lesssim
H(\zeta)
\end{equation}
for $\gamma\ge0$ and all $z\in \mathbb{B}_ n$.
If $\gamma <0$, it follows from the proof of Theorem~\ref{BMO-S} again that
\begin{eqnarray*}
|f\circ\varphi_z(\zeta)-\widehat{f_ r}(z)|&\lesssim& \|f\|_{BO_{\gamma}} \,\frac{\big (\beta(\varphi_z(\zeta), z)+1\big )}
{ (1-|z|^2)^{-\gamma}}\big | 1-\langle z,\varphi_ z(\zeta)\rangle |^{-2\gamma}\\
&=& \|f\|_{BO_{\gamma}} \,\frac{(\beta(\zeta,0)+1)(1-|\zeta|^2)^{\gamma}}{(1-|\varphi_ z(\zeta)|^2)^{\gamma}}\\
&=&\|f\|_{BO_{\gamma}} \,\frac{(\beta(\zeta,0)+1)\,|1-\langle z,\zeta\rangle|^{2\gamma}}{(1-|z|^2)^{\gamma}}.
\end{eqnarray*}
Since $c\ge n+1+\sigma-2\gamma p$, we also have
\begin{equation}\label{EqIZ2}
|f\circ\varphi_z(\zeta)-\widehat{f_ r}(z)|^p(1-|z|^2)^{\gamma p}\,|1-\langle z,\zeta\rangle |^{c-(n+1+\sigma)}\lesssim G(\zeta)
\end{equation}
for all $z\in \mathbb{B}_ n$, where $G(\zeta)=(\beta(\zeta,0)+1)^p$ is in $L^1(\mathbb{B}_ n,dv_{\alpha})$.
Fix any $\zeta\in \mathbb{B}_ n$ and let $t=\beta(\zeta,0)$. Since $\beta(\varphi_z(\zeta),z)=\beta(\zeta,0)=t$ and $f\in VO_{\gamma}$,
we get
\begin{equation}\label{L0-Eq1}
\lim_{|z|\to1}|f\circ\varphi_z(\zeta)-f(z)|^p(1-|z|^2)^{\gamma p}\le \lim_{|z|\to1}(1-|z|^2)^{\gamma p}\omega_t(f)(z)^p=0.
\end{equation}
On the other hand, we have
\begin{eqnarray*}
|f\circ\varphi_z(\zeta)&-&\widehat{f_ r}(z)|^p\lesssim |f\circ\varphi_z(\zeta)-f(z)|^p+|f(z)-\widehat{f_ r}(z)|^p \\
&\lesssim& |f\circ\varphi_z(\zeta)-f(z)|^p+\\
&&+\ \frac1{v_{\sigma}(D(z,r))^p}\int_{D(z,r)}|f(z)-f(t)|^p\,dv_{\sigma}(t)\\
&\lesssim& |f\circ\varphi_z(\zeta)-f(z)|^p+\omega_r(f)(z)^p.
\end{eqnarray*}
Therefore,
\begin{eqnarray*}
&&|f\circ\varphi_z(\zeta)-\widehat{f_ r}(z)|^p(1-|z|^2)^{\gamma p}\\
&\lesssim& |f\circ\varphi_z(\zeta)-f(z)|^p(1-|z|^2)^{\gamma p}+\omega_r(f)(z)^p(1-|z|^2)^{\gamma p},
\end{eqnarray*}
which tends to zero as $|z|\to 1$, because $f\in VO_{\gamma}$ and \eqref{L0-Eq1}. Since $c\ge n+1+\sigma$, we also have
$$\lim_{|z|\rightarrow 1^{-}}|f\circ\varphi_z(\zeta)-\widehat{f_ r}(z)|^p(1-|z|^2)^{\gamma p}\,|1-\langle z,
\zeta\rangle |^{c-(n+1+\sigma)}=0.$$
Thus in all cases, due to \eqref{EqIZ1} and \eqref{EqIZ2}, we can apply Lebesgue's Dominated Convergence Theorem (bearing
in mind the expression for $I(z)$ given in \eqref{EqIZ0}) to obtain $I(z)\to0$ as $|z|\to1^-$, which is (c). This completes the
proof of the theorem.
\end{proof}
Since condition (b) in the theorem above is independent of $r$, we see that the space $VMO^p_{\gamma,r}$ is actually
independent of $r$. Thus we will simply use the notation $VMO^p_\gamma$.
Notice that in (c) and (d) of Theorem~\ref{VMO-S} we require a somehow stronger condition $c\ge n+1+\sigma$ than $c>0$ in
Theorem~\ref{BMO-S}. It would be nice to know whether it is possible to replace condition $c\ge n+1+\sigma$ by $c>0$ with
$c>-\gamma p$ in (c) and (d) here. Anyway, for our main purpose here (to characterize compactness of Hankel operators)
condition $c\ge n+1+\sigma$ is enough.
For $\alpha>0$ let $\mathcal{B}^{\alpha}_ 0=\mathcal{B}^{\alpha}_ 0(\mathbb{B}_ n)$ denote the closure of the set of polynomials in
$\mathcal{B}^{\alpha}$. The space $\mathcal{B}^{\alpha}_ 0$ consists exactly of those holomorphic functions $f$ such that
$$\lim_{|z|\to1^-}(1-|z|^2)^{\alpha} \,|\nabla f(z)|=0.$$
As before, the complex gradient can be replaced by the radial derivative $Rf$. Furthermore, for $\alpha>1/2$, a function $f$ is in
$\mathcal{B}^{\alpha}_ 0$ if and only if the function $(1-|z|^2)^{\alpha-1} \,|\widetilde{\nabla} f(z)|$ belongs to ${\mathbb C}_ 0(\mathbb{B}_ n)$.
Again, we refer to \cite[Chapter 7]{ZhuBn} for all these facts. With minor modifications in the proof of Proposition \ref{Ba} together
with Corollary \ref{C5} we obtain the following result.
\begin{proposition}\label{Ba-0}
Let $\gamma \in \mathbb{R}$ and $1\le p<\infty$. Then $H(\mathbb{B}_ n)\cap VMO^p_{\gamma}=\mathcal{B}^{1+\gamma}_{0}$
for $n=1$ and $\gamma >-1$, or for $n>1$ and $\gamma>-1/2$. In all other cases, the space $H(\mathbb{B}_ n)\cap VMO^p_\gamma$
consists of only constants.
\end{proposition}
\begin{proof}
The details are left to the interested reader.
\end{proof}
\section{Compact Hankel operators}
In this section we prove the following characterization of compact Hankel operators between weighted Bergman spaces.
\begin{theorem}\label{compact}
Let $1<p\le q<\infty$, $\alpha,\beta>-1$, $f\in L^q_{\beta}$, and
$$\gamma=\frac{n+1+\beta}{q}-\frac{n+1+\alpha}{p}.$$
Then both $H^{\beta}_f$, $H^{\beta}_{\overline{f}}:\,A^p_{\alpha}\rightarrow L^q_{\beta}$ are compact if and only if
$f\in VMO^q_{\gamma}$.
\end{theorem}
Again, we are going to prove Theorem~\ref{compact} with several lemmas. We begin with the necessity.
\begin{lemma}\label{comp-n}
Let $p,q,\alpha,\beta$ and $\gamma$ be as in Theorem~\ref{compact}. If both $H^{\beta}_f$, $H^{\beta}_{\overline{f}}:\,A^p_{\alpha}
\rightarrow L^q_{\beta}$ are compact, then $f\in VMO^q_{\gamma}$.
\end{lemma}
\begin{proof}
Fix a nonnegative $t$ such that $n+1+\beta+t>(n+1+\alpha)/p$. It is easy to see that $h_ z^t$ converges to zero uniformly on compact subsets of
$\mathbb{B}_ n$ as $|z|\rightarrow 1^{-}$. Since each $h_z^t$ is a unit vector in $A^p_\alpha$, we conclude that $h_z^t\to0$ weakly in
$A^p_\alpha$ as $|z|\to1^-$. It follows from the compactness of $H^\beta_f$ that
$$\lim_{|z|\rightarrow 1^{-}}\|H_ f^{\beta}h^t_ z\|_{q,\beta}=0.$$
The same is true if $f$ is replaced by $\overline{f}$. By Proposition~\ref{necessity} we have
$$\lim_{|z|\rightarrow 1^{-}} MO_{q,\beta,t}(f)(z)=0.$$
In other words, we have
$$\lim_{|z|\rightarrow 1^{-}} (1-|z|^2)^{c+\gamma q}\int_{\mathbb{B}_ n} \big|f(w)-\overline{g_ z(z)}\,\big |^q \,\frac{dv_{\beta}(w)}
{|1-\langle z,w \rangle |^{n+1+c+\beta}}=0,$$
where $c=(q-1)(n+1+\beta)+tq$. This implies condition (e) in Theorem 5.3 with $\lambda_ z=\overline{g_ z(z)}$,
so $f\in VMO^q_{\gamma}$.
\end{proof}
The sufficiency will follow from the next two results.
\begin{lemma}\label{CVA}
Let $p,q,\alpha,\beta$ and $\gamma$ be as in Theorem~\ref{compact}.
If $f\in VA^q_{\gamma}$, then $H^{\beta}_f:\,A^p_{\alpha}\rightarrow L^q_{\beta}$ is compact.
\end{lemma}
\begin{proof}
Let $\{g_n\}$ be a bounded sequence in $A^p_{\alpha}$ converging to zero uniformly on compact subsets of $\mathbb{B}_ n$. We must
prove that $\|H^{\beta}_f g_n\|_{q,\beta} \rightarrow 0$. Following the proof of Proposition~\ref{L-BA}, we know that
$$\big \|H^{\beta}_f g_n\big \|_{q,\beta} \lesssim \|g_n\|_{L^q(d\mu_{f,\beta})}$$
with $d\mu_{f,\beta}=|f|^q dv_{\beta}$. The desired result then follows from Lemma \ref{VA}.
\end{proof}
\begin{lemma}\label{CVO}
Let $p,q,\alpha,\beta$ and $\gamma$ be as in Theorem~\ref{compact}. If $f\in VO_{\gamma}$,
then $H^{\beta}_f:\,A^p_{\alpha}\rightarrow L^q_{\beta}$ is compact.
\end{lemma}
\begin{proof}
Let $\{g_n\}$ be a bounded sequence in $A^p_{\alpha}$ converging to zero uniformly on compact subsets of $\mathbb{B}_ n$.
By Lemma \ref{LHb1} and the density of $H^{\infty}$ in $A^p_{\alpha}$, for any $\delta\ge\beta$
we have
$$\big \|H^{\beta}_fg \big \|_{q,\beta}\le C \,\big \|H^{\delta}_fg \big \|_{q,\beta},\qquad g\in A^p_{\alpha}.$$
We will be done if we can prove that
$$\lim_{n\to\infty} \big \|H^{\delta}_f g_n \big \|_{q,\beta}=0$$
for some $\delta\ge \beta$.
Since $\beta>-1$, we can find some $\eta>0$ satisfying $\beta-\eta \max(q,q')>-1$. We then choose some $\delta\ge \beta$ large
enough so that $c=\eta q +\delta-\beta$ satisfies
$$c\ge n+1+\sigma+\max(0,-2\gamma q),$$
with $\sigma =\beta-\eta q$. In fact, this is the same as
$$\delta \ge n+1+2\beta+\max(0,-2\gamma q) -2\eta q.$$
So the choice $\delta=n+1+2\beta+\max(0,-2\gamma q)$ works. Let $\varepsilon>0$ be arbitrary. Since $VO_{\gamma}
\subset VMO^q_{\gamma}$, by part (c) of Theorem~\ref{VMO-S} and with the above $c$ and $\sigma$,
we may choose $t_ 1$ sufficiently close to 1 so that
\begin{equation}\label{eps}
(1-|w|^2)^{(\eta q+\delta-\beta)+\gamma q}\int_{\mathbb{B}_ n} \frac{|f(z)-\widehat{f}_ r(w)|^q}
{|1-\langle z,w \rangle |^{n+1+\delta}} \,dv_{\beta-\eta q}(z)<\varepsilon
\end{equation}
for all $t_1<|w|<1$.
Fix $r>0$. By the definition of $ VO_{\gamma}$, there exists $t_ 2$, $0<t_ 2<1$, such that
$$\omega_{r}(f)(w)<\varepsilon (1-|w|^2)^{-\gamma},\qquad |w|>t_ 2.$$
We have
$$\|H^{\delta}_f g_n\|^q_{q,\beta}\le \int_{\mathbb{B}_ n} \left (\int_{\mathbb{B}_ n}\frac{|f(z)-f(w)|\,|g_ n(w)|}
{|1-\langle z,w \rangle |^{n+1+\delta}}\,dv_{\delta}(w)\right )^q\, dv_{\beta}(z).$$
Let $t=\max(t_ 1,t_ 2)$ and split the inner integral above in two parts: one for $|w|\le t$ and the other for $|w|>t$.
The integral on $|w|\le t$ can be made as small as we want because of the uniform convergence to zero on compact subsets
of $g_ n$. For the other, we will use our assumption $f\in VO_{\gamma}$.
Since
$$|f(z)-f(w)|\le |f(z)-\widehat{f}_r(w)|+|f(w)-\widehat{f}_r(w)|,$$
we get two integrals. The first one involves the function
$$I_1(z)=\int_{|w|>t} \frac{|f(w)-\widehat{f}_r(w)|\,|g_n(w)|}{|1-\langle z,w \rangle |^{n+1+\delta}} \,dv_{\delta}(w).$$
Since
$$f(w)-\widehat{f}_r(w)=\frac{1}{v_{\delta}(D(w,r))}\int_{D(w,r)}\big (f(w)-f(\zeta)\big )\,dv_{\delta}(\zeta),$$
we obtain for $|w|>t$ that
$$|f(w)-\widehat{f}_r(w)|\le \omega_{r}(f)(w)< \varepsilon \,(1-|w|^2)^{-\gamma}.$$
Therefore,
\begin{eqnarray*}
I_ 1&:=&\int_{\mathbb{B}_ n} I_ 1(z)^q \,dv_{\beta}(z)\lesssim \varepsilon^q\, \int_{\mathbb{B}_ n} \left(\int_{\mathbb{B}_ n}
\frac{|g_n(w)| \,dv_{\delta-\gamma}(w)}{|1-\langle z,w \rangle |^{n+1+\delta}} \right )^q \,dv_{\beta}(z)\\
&=& \varepsilon^q \int_{\mathbb{B}_ n} S_{b,d} (|g_ n|)(z)^q \,dv_{\beta}(z),
\end{eqnarray*}
with $d=n+1+\delta$ and $b=\delta-\gamma$. Now we want to apply Theorem~\ref{Z-T} to show that
$S_{b,d}:L^p_{\alpha}\rightarrow L^q_{\beta}$ is bounded. In the notation of Theorem~\ref{Z-T} we have $\lambda=\gamma$
and
$$n+1+b+\lambda=n+1+\delta=d.$$
It remains to check the condition
$$\alpha+1<p(b+1)=p(1+\delta-\gamma),$$
which is easily seen to be equivalent to
\begin{equation}\label{delta}
n\left(\frac1q-\frac1p\right)<(1+\delta)-\frac{1+\beta}{q},
\end{equation}
By the proof of Proposition~\ref{L-BO}, we have
$$n\left(\frac1q-\frac1p\right)<\frac{1+\beta}{q'}=(1+\beta)-\frac{1+\beta}{q},$$
where $q'$ is the conjugate exponent of $q$. Since $\beta\le \delta$, we see that (\ref{delta}) is indeed true.
Therefore, by Theorem~\ref{Z-T}, we have
$$I_ 1 \lesssim \varepsilon ^q\,\|g_ n\|^q_{p,\alpha} \le C \,\varepsilon^q.$$
It remains to deal with
$$I_2:=\int_{\mathbb{B}_ n} I_ 2(z)^q \,dv_{\beta}(z),$$
where
$$I_ 2(z)=\int_{|w|>t} \frac{|f(z)-\widehat{f}_ r(w)|\,\,|g_ n(w)|}{|1-\langle z,w \rangle |^{n+1+\delta}} \,dv_{\delta}(w).$$
By H\"{o}lder's inequality and Lemma~\ref{Ict},
\begin{eqnarray*}
I_ 2(z)^q&\lesssim& \left[\int_{|w|>t} \frac{|f(z)-\widehat{f}_ r(w)|^q\,\,|g_ n(w)|^q}
{|1-\langle z,w \rangle |^{n+1+\delta}} \,dv_{\delta+\eta q}(w)\right]\,\cdot\\
&&\ \cdot\,\left[\int_{\mathbb{B}_ n} \frac{ dv_{\delta-\eta q'}(w)}{|1-\langle z,w \rangle |^{n+1+\delta}} \right ]^{q/q'}\\
& \lesssim& (1-|z|^2)^{-\eta q} \left (\int_{|w|>t}
\frac{|f(z)-\widehat{f}_ r(w)|^q\,\,|g_ n(w)|^q}{|1-\langle z,w \rangle |^{n+1+\delta}} \,dv_{\delta+\eta q}(w)\right).
\end{eqnarray*}
Thus $I_ 2$ is dominated by
$$\int_{|w|>t} |g_ n(w)|^q \left[(1-|w|^2)^{\eta q}\int_{\mathbb{B}_ n} \frac{|f(z)-\widehat{f}_ r(w)|^q}
{|1-\langle z,w \rangle |^{n+1+\delta}} \,dv_{\beta-\eta q}(z) \right]\,dv_{\delta}(w).$$
By \eqref{eps}, we get
$$I_2\lesssim \varepsilon \int_{|w|>t} |g_n(w)|^q\, dv_{\beta-\gamma q}(w)\lesssim \varepsilon \,\|g_n\|^q_{A^p_{\alpha}}.$$
For the last inequality we used the fact that $A^p_{\alpha}\subseteq A^{q}_{\beta-\gamma q}$, which can be obtained from
Theorem 69 in \cite{ZZ}. Putting everything together we conclude that $\|H^{\delta}_f g_n \big \|_{q,\beta}\rightarrow 0$ as
$n\to\infty$. This finishes the proof.
\end{proof}
To summarize, the necessity of Theorem \ref{compact} is proved by Lemma~\ref{comp-n}. Since $f\in VMO^q_{\gamma}$ if
and only if $\overline{f}\in VMO^q_{\gamma}$, the sufficiency is a consequence of Theorem \ref{VMO-S}, Lemma \ref{CVA},
and Lemma \ref{CVO}.
As a direct consequence of Theorem \ref{compact} and Proposition \ref{Ba-0}, we obtain the following characterization of compactness
of Hankel operators with conjugate holomorphic symbols.
\begin{coro}
Let $f\in A^1_{\beta}$, $1<p\le q<\infty$, $\alpha,\beta>-1$, and
$$\gamma=\frac{n+1+\beta}{q}-\frac{n+1+\alpha}{p}.$$
If $n=1$ and $\gamma>-1$, or if $n>1$ and $\gamma>-1/2$, then $H^\beta_{\bar{f}}:A^p_{\alpha}\rightarrow L^q_{\beta}$
is compact if and only if $f\in \mathcal{B}^{1+\gamma}_{0}$. In all other cases, $H^\beta_{\bar f}:A^p_\alpha\to L^q_\beta$
is compact if and only if $f$ is constant.
\end{coro}
|
{
"redpajama_set_name": "RedPajamaArXiv"
}
| 8,365
|
{"url":"https:\/\/www.biglist.com\/lists\/lists.mulberrytech.com\/dssslist\/archives\/199901\/msg00045.html","text":"Meaning of the language characterisc with a #f value ? (pb with jadetex)\n\n Subject: Meaning of the language characterisc with a #f value ? (pb with jadetex) From: kubek@xxxxxxxxxxxx (kubek) Date: Wed, 13 Jan 1999 18:00:22 +0100\nHi,\nI'm giving a try to the jadetex backend in order to get pdf\noutput, and get a problem with the language characteristic.\n\nI have some part of my documents that I don't want associated\nto any language, mainly in order to avoid spelling error notification\nwhen using the rtf backend. e.g.\n(element email\n(make paragraph\nlanguage: #f))\n\nHowever, the tex backend emits some strange character when such a\nfalse characteristic is encountered.\n\n{\\def\\Language%\n{}}\n\nI wonder if this problem comes from a jadetex backend failure\nor if I misunderstood the meaning of the language characteristic (i.e.\nno language is associated to the flow object when its value is\nfalse). But surely the both are true ....\n\nJ.-M. Kubek\nComputing Center\nINSA Toulouse\n\nDSSSList info and archive: http:\/\/www.mulberrytech.com\/dsssl\/dssslist","date":"2022-05-20 13:08:49","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\": 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\": 0.6443763375282288, \"perplexity\": 8820.393815572808}, \"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-2022-21\/segments\/1652662532032.9\/warc\/CC-MAIN-20220520124557-20220520154557-00091.warc.gz\"}"}
| null | null |
{"url":"http:\/\/wikinotes.ca\/MATH_318\/summary\/fall-2011\/htsefp-prenex-normal-form","text":"# HTSEFP: Prenex normal form\n\n## 1Equivalent formulas in Prenex normal form\u00b6\n\nGiven a formula not in Prenex normal form, find an equivalent formula that is and justify your answer.\n\n### 1.1General solution\u00b6\n\nUse the rules mentioned in class on Wednesday, November 23.\n\n#### 1.1.1Tips\u00b6\n\n\u2022 The existential quantifier distributes over $\\lor$.\n\u2022 Make sure you don't have a clash of variables. To confirm, count the free variables, then count the quantifiers. The original formula and the formula in prenex form should have the same number of each, except if you use the above fact about the existential quantifier.\n\u2022 Use the distributive laws to move negations inward.\n\u2022 When \"swallowing up\" the clause on the other side of the $\\land$ or $\\lor$, make sure that the correct quantifier is on the outside.\n\n### 1.2Examples\u00b6\n\n\u2022 Assignment 9, question 7\n\u2022 Fall 2009 final exam, question 8\n\u2022 Fall 2010 final exam, question 8","date":"2018-08-19 17:38:29","metadata":"{\"extraction_info\": {\"found_math\": true, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 0, \"math_alttext\": 0, \"mathml\": 0, \"mathjax_tag\": 0, \"mathjax_inline_tex\": 1, \"mathjax_display_tex\": 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\": 0.8049402832984924, \"perplexity\": 1081.4879595198988}, \"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-2018-34\/segments\/1534221215261.83\/warc\/CC-MAIN-20180819165038-20180819185038-00367.warc.gz\"}"}
| null | null |
{"url":"https:\/\/git.eclipse.org\/c\/jubula\/org.eclipse.jubula.core.git\/tree\/org.eclipse.jubula.documentation\/manual\/en\/tex\/Tasks\/Preferences\/gdprefs.tex?h=multipleTCB&id=ff620e1b6e72272464068773f31251578d2a7bcc","text":"1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 % $Id: gdprefs.tex 11355 2010-06-16 14:48:31Z alexandra$ %ID:prefPageBasicContextId % Local Variables: % ispell-check-comments: nil % Local IspellDict: american % End: % -------------------------------------------------------- % User documentation % copyright by BREDEX GmbH 2004 % -------------------------------------------------------- \\index{Preferences!Test} \\index{Test!Preferences} \\label{gdprefs} \\begin{figure}[h] \\begin{center} \\includegraphics[width=0.60\\textwidth]{Tasks\/Preferences\/PS\/gdprefs} \\caption{\\app{} Preference Dialog} \\label{gdprefs} \\end{center} \\end{figure} In the preferences dialog, select \\bxname{Test} from the tree on the left hand side. From this page, you can configure your preferences for: \\begin{description} \\item [Auto-scrolling and -expanding:]{When the checkbox is marked, the views and browsers in \\app{} automatically scroll in the direction you move the mouse when you are dragging and dropping. Trees will also be automatically expanded when you hover over them while dragging items.} \\item [Minimizing the \\ite{}:]{When the checkbox is marked, \\app{} is automatically minimized when test execution begins. This is useful if you are letting tests run on the same machine you are specifying on.} \\item [\\gdaut{} confirmation dialog:]{When the checkbox is marked, a dialog appears to check if you are sure when you click the \\bxcaption{stop \\gdaut{}} button.} \\item [Original \\gdcase{} name:]{When the checkbox is activated, you can see the original name of a referenced \\gdcase{} in brackets behind a new name you enter for the \\gdcase{}. This can help you to see and search for \\gdcases{} you have reused. } \\item[\\gdstep{} information:]{When this checkbox is activated, you see the details about the \\gdstep{} (the component name and type, and the action) in square brackets behind the \\gdstep{} name. If you do not want to see these details, you can deactivate this checkbox.} \\item[Show referenced children:]{When this option is not active, you can only see referenced parent \\gdcases{} in the \\gdtestcasebrowser{}. The referenced \\gdcases{} contained in these \\gdcases{} are not displayed. This action can be useful if you want to improve the speed of working with the \\ite{}.} \\item [Switching to the \\execpersp{}:]{When the test begins, \\app{} can automatically change to the \\execpersp{}. You can choose to always be asked, to always change, or to never change. } \\item [Data files location:]{You can specify a location where external data files (e.g. Excel files) are held, or use the workspace directory as the base location. } \\bxtipp{The advantage of using the workspace as a location for your data files is that you can view these in the navigator view directly in the \\ite{}. Windows users can even open Excel files from the \\ite{} using the In-Place editor.} \\end{description}","date":"2022-06-26 17:24:18","metadata":"{\"extraction_info\": {\"found_math\": true, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 0, \"math_alttext\": 0, \"mathml\": 0, \"mathjax_tag\": 0, \"mathjax_inline_tex\": 1, \"mathjax_display_tex\": 0, \"mathjax_asciimath\": 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\": 0.8665573596954346, \"perplexity\": 2415.1550958883136}, \"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\/1656103271763.15\/warc\/CC-MAIN-20220626161834-20220626191834-00508.warc.gz\"}"}
| null | null |
Q: Isso é uma property, atributo de classe ou o quê? Tenho isso em meu código:
public ICommand ChangePositionCommand => new Command((obj) =>
{
_positionIndex++;
if (_positionIndex >= Positions.Count)
_positionIndex = 0;
Position = Positions[_positionIndex];
RaisePropertyChanged(nameof(Position));
});
posso afirmar que isso é um atributo de classe, uma property ou o que?
A: A resposta original está abaixo e está errada porque a forma nova de escrever deu margem para entender errado (ninguém mais percebeu - curioso como coisas certas recebem negativos e as erradas não). A certa é:
Isso é uma propriedade. É bem óbvio, não tem parênteses, e todo método tem seu identificador seguido de parênteses. Como isto poderia ser um método?
Essa é uma propriedade que está sendo inicializada com um valor, no caso uma lambda e que você só pode pegar este valor já que não foi criado nem um getter nem um setter. Neste caso o compilador cria o getter para acessar este valor, caso contrário seria melhor eliminar tudo já que seria inacessível.
Veja como realmente é assim no SharpLab.
Não deixa de se transformar em um método get, mas no seu código é uma propriedade.
Eu ainda acho que poderia ser feito de forma mais simplificada, mas deve ser exigência de algum framework.
Isso é um método. O uso da notação => neste caso não é adequada já que existe mais de uma linha efetivamente, embora seja um único statement.
Tenho dúvidas se o código assim é necessário ou deveria ser desta forma, mas não é possível afirmar sem contexto.
Mais legível:
public ICommand ChangePositionCommand {
return new Command((obj) => {
_positionIndex++;
if (_positionIndex >= Positions.Count)
_positionIndex = 0;
Position = Positions[_positionIndex];
RaisePropertyChanged(nameof(Position));
});
}
Coloquei no GitHub para referência futura.
Atributo garanto que não é.
A: Isso é um método GET introduzido nas versões mais recente do C#:
public int X { get {return A + B;}}
Equivale a isso:
public int X => A + B;
|
{
"redpajama_set_name": "RedPajamaStackExchange"
}
| 4,871
|
/**
* @author Valentin Al. Sitnick
* @version $Revision: 1.1 $
*
*/
/* *********************************************************************** */
#include "events.h"
#include "utils.h"
#include "fake.h"
static bool test = false;
static bool util = false;
static bool flag = false;
const char test_case_name[] = "SetFieldAccessWatch0101";
/* *********************************************************************** */
JNIEXPORT jint JNICALL Agent_OnLoad(prms_AGENT_ONLOAD)
{
Callbacks CB;
check_AGENT_ONLOAD;
jvmtiEvent events[] = { JVMTI_EVENT_EXCEPTION, JVMTI_EVENT_VM_DEATH, JVMTI_EVENT_FIELD_ACCESS };
cb_exc;
cb_acc;
cb_death;
return func_for_Agent_OnLoad(vm, options, reserved, &CB, events,
sizeof(events)/4, test_case_name, DEBUG_OUT);
}
/* *********************************************************************** */
void JNICALL callbackException(prms_EXCPT)
{
check_EXCPT;
if (flag) return;
jvmtiError result;
jclass myclass = NULL;
jfieldID myfield = NULL;
/*
* Function separate all other exceptions in all other method
*/
if (!check_phase_and_method_debug(jvmti_env, method, SPP_LIVE_ONLY,
"special_method", DEBUG_OUT)) return;
flag = true;
util = true;
if (!is_needed_field_found(jvmti_env, "SetFieldAccessWatch0101.java", "first_field", &myclass, &myfield, DEBUG_OUT))
return;
result = jvmti_env->SetFieldAccessWatch(myclass, myfield);
fprintf(stderr, "\tnative: SetFieldAccessWatch result = %d (must be zero) \n", result);
fprintf(stderr, "\tnative: class is %p \n", myclass);
fprintf(stderr, "\tnative: field is %p \n", myfield);
fflush(stderr);
}
void JNICALL callbackFieldAccess(prms_FLD_ACCESS)
{
check_FLD_ACCESS;
test = true;
fprintf(stderr, "\tnative: FieldAccess event was received\n");
fprintf(stderr, "\tnative: JVMTI Env is %p\n", jvmti_env);
fprintf(stderr, "\tnative: JNI Env is %p\n", jni_env);
fprintf(stderr, "\tnative: current thread %p\n", thread);
fprintf(stderr, "\tnative: current method %p\n", method);
fprintf(stderr, "\tnative: current location %lld\n", location);
fprintf(stderr, "\tnative: current class %p \n", field_klass);
fprintf(stderr, "\tnative: current object %p\n", object);
fprintf(stderr, "\tnative: current field %p\n", field);
}
void JNICALL callbackVMDeath(prms_VMDEATH)
{
check_VMDEATH;
func_for_callback_VMDeath(jni_env, jvmti_env, test_case_name, test, util);
}
/***************************************************************************/
|
{
"redpajama_set_name": "RedPajamaGithub"
}
| 2,344
|
{"url":"https:\/\/pretextbook.org\/doc\/guide\/html\/localization-contributions.html","text":"We rely heavily on those fluent in other languages to provide translations of terms like Chapter and Example. See Section\u00a04.36 for more about this feature. This chapter is meant to help you get started if you would like to contribute to PreTeXt development in this way.\n\u2022 We know it is hard to translate subtleties of one language into another. Do the best you can! Something is better than nothing. And somebody may come along later with an improvement. See the en-US file for explanations, this is the only documentation about each term that gets translated.\n\u2022 Do nt copy the documentation from the en-US file into your file. Add comments if you feel your work needs some explanations. Do try to keep the order and organization the same.\n\u2022 Explore the files in xsl\/localizations, including the README file, which should provide you enough to get started.","date":"2023-03-30 08:53:15","metadata":"{\"extraction_info\": {\"found_math\": true, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 0, \"math_alttext\": 0, \"mathml\": 0, \"mathjax_tag\": 0, \"mathjax_inline_tex\": 0, \"mathjax_display_tex\": 0, \"mathjax_asciimath\": 1, \"img_math\": 0, \"codecogs_latex\": 0, \"wp_latex\": 0, \"mimetex.cgi\": 0, \"\/images\/math\/codecogs\": 0, \"mathtex.cgi\": 0, \"katex\": 0, \"math-container\": 0, \"wp-katex-eq\": 0, \"align\": 0, \"equation\": 0, \"x-ck12\": 0, \"texerror\": 0, \"math_score\": 0.24825087189674377, \"perplexity\": 865.8128392901272}, \"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\/1679296949107.48\/warc\/CC-MAIN-20230330070451-20230330100451-00480.warc.gz\"}"}
| null | null |
Kilis es una ciudad y distrito situada al sureste de Turquía, capital de la provincia de Kilis. Tiene una población de 93266 habitantes.
Geografía
La frontera con Siria se encuentra a 5 km al sur y la ciudad de Gaziantep está a 60 km al norte. Hasta 1996, Kilis fue un distrito de la provincia de Gaziantep hasta que Tansu Çiller la convirtió en provincia tras las elecciones de 1995.
Kilis en la actualidad
Kilis es una pequeña ciudad con ambiente rural. El tráfico consiste sobre todo en gente joven en moto. La población de la ciudad era de 20.000 personas en 1927, 45.000 en 1970, 60.000 en 1980, 85.000 en 1990, 77.706 en 2007 y 93.266 en 2015.
Al encontrarse próxima a la frontera, Kilis ha sido conocida durante mucho tiempo por el contrabando y el tráfico de drogas. Aunque actualmente se ha reducido, aún se puede comprar tabaco, bebidas alcohólicas y electrodomésticos en efectivo a bajo precio.
La gastronomía local es una mezcla de platos turcos y árabes; es conocido el kebab local, los panes, el baklava y las verduras rellenas.
Lugares de interés
Los principales lugares de la ciudad incluyen diferentes mezquitas del periodo otomano y casas de piedra con patios y elaborados muebles de madera.
Referencias
Enlaces externos
Información sobre Kilis
Capitales de provincia de Turquía
Localidades de la provincia de Kilis
|
{
"redpajama_set_name": "RedPajamaWikipedia"
}
| 8,682
|
{"url":"http:\/\/quant.stackexchange.com\/questions?page=32&sort=newest","text":"# All Questions\n\n39 views\n\n### Simulated bond index returns based on term structure changes\n\nThe J.P. Morgan European Monetary Union Index (EMU) 5-7Y (which is rebalanced monthly) currently (1-Jan-16) has the following characteristics: ...\n92 views\n\n### Forecasting conditional variance using fGARCH\n\nI am forecasting the conditional standard deviation using ARMA(1,0)-GJRGARCH(1,1) in R using the fGarch package. Here is a sample code: ...\n32 views\n\n### Portfolio Analytic Metrics for Portfolios with Serially Correlated Returns\n\nI just read Andrew Lo's paper from 2002 \"The Statistics of Sharpe Ratios\" and am wondering if anyone knows of any other papers\/docs\/resources that explore the impact of serially correlated returns on ...\n248 views\n\nI saw this question as an interview, and to be honest, I have no idea what it's even asking for: Write a function (in R or Python) that finds the stock drawdown which will trigger a rebalance, ...\n268 views\n\n### What are the canonical global-macro investing books?\n\nWhat is a good reading list for Global Macro investing? What does Bridgewater, Bervan Howard, Soros, AQR etc. use to teach their staff about macroeconomic investing? Let us assume a top-class ...\n86 views\n\n### Portfolio of sum of two Bachelier processes\n\nSuppose you construct a portfolio of two stocks, whose values $A$ and $B$ are modelled as a Bachelier process: $$dA = \\sigma_A dW_A(t) \\text{ and } dB = \\sigma_B d W_B(t).$$ Each of the stock prices ...\n48 views\n\n41 views\n\n### SAS code for Brownian Motion\n\nI want to simulate call options using monte carlo algorithm. I am a noob SAS user but i know that i need to: -simulate random stock prices with no dividend in respect to different parameters(...\n58 views\n\n### Sharpe ratio with leveraged ETFs\n\nThere has been a discussion about how leverage affects Sharpe Ratios, but not in the context of leveraged ETFs (such as 2x or 3x). I'm just wondering how leveraged ETFs, if at all, change the ...\n560 views\n\n### What time series database can be used with Python and Pandas?\n\nI'm looking for a time series database that can be easily used with Python and Pandas objects such as DataFrame, Panel... But these objects will always contains time series. Ideally I'm looking for ...\n61 views\n\n104 views\n\n### How to estimate today's closing price?\n\nI'm working on interday trading algorithm and I have a basic question: How can I estimate today's closing price? I need it to predict tomorrow closing price. Should I use the price few moments before ...\n35 views\n\n### Inflow outflow of money into a stock\n\nhttp:\/\/www.barrons.com\/mdc\/public\/page\/2_3022-mfsctrscan-moneyflow.html#weekly Interestingly, for the 1-month DJIA total index, the balance is very close to one. Is there a mathematical explanation ...\n145 views\n\n### Notional Value in Equity Options\n\nI have calculated the NPV of an Equity option and need to account the notional for it and have issues understanding the NPV <-> notional relation. Example: Strike price 100 Spot rate: 107.41 NPV ...\n38 views\n\n### Do I calculate weights of assets correctly?\n\nI solved attached question but I am not sure whether I did part a and c correctly. Is there a way to calculate weights of A and B by just knowing their standard deviation and correlation's value?\n41 views","date":"2016-07-26 00:43:30","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\": 0.7961195707321167, \"perplexity\": 3303.5947093979207}, \"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-2016-30\/segments\/1469257824499.16\/warc\/CC-MAIN-20160723071024-00246-ip-10-185-27-174.ec2.internal.warc.gz\"}"}
| null | null |
Q: Add custom tag to span in Jaeger when using in Istio service mesh I want to add a custom tag in Jaeger span, I tried to install Istio on my Kubernetes cluster with this command:
istioctl install --set profile=demo --set meshConfig.accessLogFile=/dev/stdout --set meshConfig.outboundTrafficPolicy.mode=REGISTRY_ONLY -f .\tracing.yaml -y
and this is tracing.yaml:
apiVersion: install.istio.io/v1alpha1
kind: IstioOperator
spec:
meshConfig:
enableTracing: true
defaultConfig:
tracing:
sampling: 100.0
custom_tags:
my_tag_header:
header:
name: test
defaultValue: test
but there is no test tag in spans:
What should I do?
A: When using Istio operator you have to initialize it first with
istioctl operator init
By default Istio operator installs and watches only istio-operator namespace. If you want to specify another namespaces for it to watch (for example default), add --watchedNamespaces argument
istioctl operator init --watchedNamespaces=default
With the operator installed, you can now create a mesh by deploying an IstioOperator resource. To install the Istio demo configuration profile using the operator, run the following command:
kubectl apply -f - <<EOF
apiVersion: install.istio.io/v1alpha1
kind: IstioOperator
metadata:
namespace: istio-system
name: example-istiocontrolplane
spec:
profile: demo
EOF
Now, with the controller running, you can change the Istio configuration by editing or replacing the IstioOperator resource. The controller will detect the change and respond by updating the Istio installation correspondingly.
|
{
"redpajama_set_name": "RedPajamaStackExchange"
}
| 3,288
|
Mom update: Doing well. Sat up in bed yesterday, and is to start solids today. :fingers crossed: for coming home soon.
School: going well. Himself is back in the G&T classes. Herself is in Pre-Algebra, and she HATES it! "It's too Hard!" :bangs head: It's only the 2nd day of school, how can it be too hard????
Critters: Doing well. Chickens like their coop....at least 1 of the Wyandottes is laying now. (Cute li'l brown eggs....:grin:) Horses - all contained, at least for the moment.
|
{
"redpajama_set_name": "RedPajamaC4"
}
| 3,574
|
Сейчас идет снег в Méribel ?Thank you!
This page shows the current snow webcam for the ski resort of Méribel. Some cams are not operational during the summer, however if you find one that is not working, or know of a better Méribel webcam, please let us know. Each day, we archive the webcam image at around 2pm (i.e. in good daylight) and present a rolling archive of images for the past four weeks. This provides skiers with a simple way to view when and how dumps of snow fell.
|
{
"redpajama_set_name": "RedPajamaC4"
}
| 5,215
|
{"url":"https:\/\/archive.lib.msu.edu\/crcmath\/math\/math\/p\/p097.htm","text":"## Parametric Latitude\n\nAn Auxiliary Latitude also called the Reduced Latitude and denoted or . It gives the Latitude on a Sphere of Radius for which the parallel has the same radius as the parallel of geodetic latitude and the Ellipsoid through a given point. It is given by\n\nIn series form,\n\nwhere\n\nAdams, O.\u00a0S. Latitude Developments Connected with Geodesy and Cartography with Tables, Including a Table for Lambert Equal-Area Meridional Projections.'' Spec. Pub. No. 67. U.\u00a0S. Coast and Geodetic Survey, 1921.","date":"2021-10-16 05:58:11","metadata":"{\"extraction_info\": {\"found_math\": true, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 0, \"math_alttext\": 0, \"mathml\": 0, \"mathjax_tag\": 0, \"mathjax_inline_tex\": 0, \"mathjax_display_tex\": 0, \"mathjax_asciimath\": 1, \"img_math\": 0, \"codecogs_latex\": 0, \"wp_latex\": 0, \"mimetex.cgi\": 0, \"\/images\/math\/codecogs\": 0, \"mathtex.cgi\": 0, \"katex\": 0, \"math-container\": 0, \"wp-katex-eq\": 0, \"align\": 0, \"equation\": 0, \"x-ck12\": 0, \"texerror\": 0, \"math_score\": 0.8955762982368469, \"perplexity\": 2990.5476897435974}, \"config\": {\"markdown_headings\": true, \"markdown_code\": true, \"boilerplate_config\": {\"ratio_threshold\": 0.18, \"absolute_threshold\": 10, \"end_threshold\": 15, \"enable\": true}, \"remove_buttons\": true, \"remove_image_figures\": true, \"remove_link_clusters\": true, \"table_config\": {\"min_rows\": 2, \"min_cols\": 3, \"format\": \"plain\"}, \"remove_chinese\": true, \"remove_edit_buttons\": true, \"extract_latex\": true}, \"warc_path\": \"s3:\/\/commoncrawl\/crawl-data\/CC-MAIN-2021-43\/segments\/1634323583423.96\/warc\/CC-MAIN-20211016043926-20211016073926-00016.warc.gz\"}"}
| null | null |
With luck, I found a little room with a shared kitchen and toilet at Ikoyi. It appeared to be quarters for staff of a government parastatal and the staff was renting out his room. I hated to share toilets but I had to endure it. The other occupants were men so I had to be extra careful. They were good men though but I didn't leave anything to chance. I locked my door and reinforced it at night. This was my first paid rental and I was paying. I felt empowered somewhat.
In two months, we had a book deal. Stephanie had changed the title of the book from 'Sarah's Escapades' to 'Five Things You Must Do Before Marriage'. I was sceptical about the title but people were curious. She also did a lot of work, tweaked the book and added a new angle. Her name came first on the title to increase market value. We also had a movie deal.
Stephanie made a lot of money from this book but so did I. It definitely made sense to leverage on her popularity as a successful Actress. The movie was also a success. I was worried that I would have to spend a lot to look great in front of the TV during the launching and premier but Stephanie helped me out there. She showed me the ropes and how to make money from my creative work.
I had to move houses but I needed to also have money to generate more. I told Stephanie that I wanted to buy a property so that I could have some sense of security as well as increase my assets however renting on the island was going to be too much for me. Stephanie let me move to her Lekki place and I was still serving her while I devoted time to my writing. I also started investing my time in learning the art of public speaking because I realized that it was a necessity in the world I delved into.
In six months, I was able to buy a property at Idado town, which was after Lekki phase 1. I still couldn't believe my luck. The owner of the house was deceased and the children were based abroad. They just wanted to sell off desperately because it was tying them down. The price was a giveaway considering the cost of estate in that area. I was able to make a down payment. Almost all my resources went towards it. Almost everything I had made from the movie deal and sales went into it. Stephanie loaned me some money to get by. I still had to cough up the rest of the money before I could get the house papers from the children who were very accommodating. The rest of the money I made, I used to buy Treasury bills. My banker was very helpful in advising.
My mother started calling me again. She would ask how I was doing and how I was getting by. I didn't give her too much information about the jobs I had done or what I was doing at the moment. I didn't want any distractions. She told me to come home. I just couldn't believe it. I didn't want to though. I was already on a path that I really wanted to be.
I took the time to visit home one day. It was a strange experience. My family treated me as if we had no squabbles. I soon realized that they had all missed me and had no clue where I was. Fadeke did not give them any information or indication that she housed me. My Dad spoke to me as he normally did but I could still see the sadness in his eyes. I wondered if I would ever make him happy again.
Stephanie asked me to update my story as I moved on because she had a sixth sense that it would be the best ever and she didn't want me to miss any details.
"Sade, there is no romance in this your story Again. We have to stir things up." Stephanie said.
I had no inkling that she was up to something. By the end of the day, I saw Charles after a long time.
We hugged and talked about everything.
"So, have you heard from Tito since that day?" Charles asked.
"Not at all. That day was the last time I saw him." I looked past him, sadness filled my heart for a moment.
I smiled. I wasn't sure that I want to hear about Tito.
"Anyway, it was good to see you again." Charles said. I saw him often after that.
One day, I came home to meet Charles and Tito in the living room.
"Tito. Charles." I whispered. I was shocked. Tito stood up and he appeared slightly uncomfortable.
"Charley, wehdone sah." I performed a fake salute.
Charles excused himself. Tito and I stared at each other. He smiled and shook his head.
I smiled. My emotions were in disarray. I tried to recall the content of all the emotional Intelligence books I had read and tried to apply it but I was just shocked.
"Well, since you are here now. Let me just say what has been on my mind for a while now." Tito said to me moving close to where I sat.
"I'm sorry. I can't do this. I can't have this conversation with you." I tried to get up but he held me down.
I tried to respond but I was too overcome with anger, sadness, regret and many other emotions I couldn't even understand. Finally I spoke to him.
"I was told they threw you out of the house. I was worried but I couldn't call you. I didn't feel I had the right to you after that incident. At first, I was angry but when I sat myself down, I saw how I contributed to destroying that day." Tito looked apologetic.
"… for those who trust in the lord." Tito finished my statement and we both smiled.
I found out that he was a youth pastor in his church. It was hilarious thinking of the old Tito as a pastor. Pastor Tito Masha was not ringing a bell at all.
"I'm a changed man but you can laugh since you know my history." he said.
"So, are you married now?" I asked him.
"Oh no. Not married. It's not in my plans for now. Or let me just say I have not found a reason for it in a while." he responded looking at my fingers.
Months later, we were still in touch but I didn't tell my family this.
My family and I had a different kind of relationship than we had previously. My brother was relieved that I was allowed back into the family fold, and I was able to also help him handle some little financial responsibilities that he allowed. My sisters knew that I was making some progress in life because the movie and book was very successful. As a result, they were a bit cool towards me. Sope mentioned that they were probably being that way because of my new fame and they did not want it to appear that things were getting back to normal because of my success. I was not even popular; a couple of television and social media interviews as well as some print media interviews do not really make me a celebrity even if I get fame for a few months.
My mother was kind to me; always asking if I had enough food stuff but we still had not really got back to the place we were but we spoke all the time and she hugged closely every time I was leaving. My Dad was not speaking to me still. In fact, he made himself scarce any time I informed them that I would be visiting.
Tito and I became friends again. It was so easy. The edge of secrecy around him was gone. He was as open as a book. I started to see him like someone I had never met. It was interesting to be around him again and I began to heal gradually. All the hurt I felt disappeared before long and I was longing for him to ask for my hand again but he didn't. I got frustrated. One day, I challenged him.
Tito laughed. "Kai, this woman, you will not kill me. Yes, my initial mission was for restitution sake but there was a reason I fell in love with you before. I never stopped really. In fact, I feel more in love with you than I've ever been. It's as if I loved a princess but now I'm looking at a queen that I love and respect and would do anything for." he said with so much emotion.
I was blown away by his frankness.
"Hmm, so how's it going to be?" I asked impatiently.
Tito went down on one knee to my astonishment.
"Marry me?" He brought out a box with a gold ring.
"Sharpaly sharpaly! So you've had all this planned before." I asked in amazement.
I laughed and gave him my hand. "Most certainly. I have no choice but to marry you." Tito laughed. It was just a miracle that everything was falling into place.
Stephanie was ecstatic. She wanted me to finish writing the script for my story. She felt that this end was enough. I was happy to write it and she polished the manuscript and added her own angles to it till the story was just perfect.
I told my mother that Tito and I were getting married. She was speechless.
I dispelled her concerns and told her to inform my Dad. My father sent for me immediately.
"See, let me tell you. I've been so depressed. Why? Because I brought all this on your head. It's my reward for how I've treated your mother. I'm sure it's not news that I cheated on her severally during our marriage. In fact, I bullied her into thinking that it was normal. You know our age difference, I was able to ride her. Now it has come to bite me. This is God punishing me." he paused for a minute.
Tito and I went to his parent's house. That was another kettle of fish. His mother was very hostile and his Father practically asked me to leave. This was after Tito had met them personally to declare his intentions. We had to sit and think of how we were going to handle this new situation. Tito suggested that we should just go the the court and get married but I reminded him about the importance of family. If we were going to start over and be happy, we needed to make things right with our families. He agreed.
Stay tuned for the Concluding Part of Changing lanes!
|
{
"redpajama_set_name": "RedPajamaC4"
}
| 5,157
|
Dibamus tiomanensis – gatunek jaszczurki z rodziny Dibamidae, opisany w 2004 roku przez Diaza i współpracowników. Nazwa gatunkowa pochodzi od nazwy wyspy Tioman, na której został odkryty. Jest endemitem, występuje tylko na Tioman i sąsiedniej wyspie Tulai (Malezja). Jej łuski mają brązowy kolor, pysk ma nieco jaśniejszy od reszty ciała. Długość ciała nieznacznie przekracza 10 centymetrów.
D. tiomanensis różni się od pozostałych przedstawicieli rodzaju Dibamus okrągłymi łuskami, które u dorosłych osobników są delikatnie karbowane z tyłu. Młode osobniki wyróżniają się płaskimi, okrągłymi łuskami grzbietowymi z kremowymi obwódkami. Liczba kręgów przedkrzyżowych wynosi 124, a ogonowych 23.
Przypisy
Bibliografia
Dibamidae
Łuskonośne Azji
Gatunki i podgatunki zwierząt nazwane w 2004 roku
|
{
"redpajama_set_name": "RedPajamaWikipedia"
}
| 9,097
|
Family and Kids Activities
Towns and Places
Beaches and Lakes
Historical Sites and Buildings
Romance and Nightlife
Shopping and Style
Midwest Destinations
Northeast Destinations
Southeast Destinations
Southwest Destinations
West Destinations
By Map
USA Travel Guide
10 Most Beautiful Small Towns in Illinois
Illinois is one of the beautiful states in the midwest region of the United States. It is surrounded by five states and borders the Great Lakes, which makes it the perfect travel destination for thousands of people. As beautiful as the large city of Chicago is the smaller towns that share the best of what rural Illinois has to offer. Here are the 10 most beautiful and charming small towns to visit across the state.
Best Small Towns in Illinois
Table of Contents [Show]
Mount Carroll
Elsah
Affiliate links may be used in this post. I may receive a small commission at no extra cost to you if you use my affiliate link.
1: Galena
Flickr/SD Dirk
Rate attraction:
Situated in the very top Northwestern corner of the state is the town of Galena. The town is named after the mineral that the Native Americans mined in the area hundreds of years ago. Inside the Galena Historic District, there are more than eight hundred properties that are all included in the National Register of Historic Places.
Guests can tour the homes of former president Ulysses S. Grant and congressman Elihu B. Washburne, which lies within the Galena historic district. Visitors can also learn more about the social history of the town when they tour the Old Market House State Historic Site. The Old Market House was built in 1845 and is now used as a museum.
Accomodation: Where to stay in Galena
Address: Galena, Illinois
Website: www.visitgalena.org
2: Mount Carroll
Wikimedia/PeteBobb
Mount Carroll is one of the coldest places in all of Illinois due to its elevation and location. Despite the freezing temperatures, people flock to this town for the architecture, art, and culture. Inside the local post office, there is a canvas mural, Rural Scene Wakarusa Valley, that was painted with oils by Irene Bianucci in 1941. The local downtown area looks as magnificent as it did decades ago, and the bricks that line the streets give it a real old-time country feel.
Every year, locals perform a play at the Timber Lake Playhouse. The original building was completed in 1961 but was destroyed by a fire on July 22, 1974. The current building was erected later that year and is surrounded by fourteen more buildings that are used to keep the plays successful.
Address: Mount Carroll, Illinois
Website: www.mtcarrollil.org
3: Elmhurst
Flickr/Teemu008
It would not be too surprising if everyone in Elmhurst has a sweet tooth, since Sunshine Biscuits and Famous Amos cookies are distributed from this small town. Keebler cookies used to claim this town as their home as well until they were bought out by Kellogg. Elmhurst focuses on community, which is evident with their parades and other events.
Their Memorial Parade has been a tradition since 1918, and their St. Patrick's Day Parade is third in popularity just behind Chicago's and the South Side's. The limestone quarry that is located just West of downtown is quite the sight, especially since it is one-hundred-fifty feet deep.
Accomodation: Where to stay in Elmhurst
Address: Elmhurst, Illinois
Website: www.elmhurst.org
4: Fulton
Flickr/Adam Moss
Robert Fulton invented the steamboat and people decided that they needed to name this town after this ingenious man. The town is filled with people of Dutch descent, and they love to show how much they love their heritage. To express their Dutch culture, even more, the town ordered a Dutch windmill that native millwrights in the Netherlands constructed and shipped over to Fulton.
The windmill, called De Immigrant, was completed on November 19, 1999, and began grinding wheat, buckwheat, cornmeal, and rye in 2001. One of the best times to be in the town of Fulton is the first weekend of May when the Dutch Days Festival is held. The weekend is filled with arts and crafts, music, food, dancing, windmill tours, Dutch cultural displays, and a parade.
Accomodation: Where to stay in Fulton
Address: Fulton, Illinois
Website: www.cityoffulton.us
5: Nauvoo
Flickr/Ken Lund
Back in 1844, the town of Nauvoo had a population of 12,000 people, which was approximately the same amount of people that were in the city of Chicago. The population declined shortly after that, and now it is a small town that is rich in history. A few of the best historic sites are the Joseph Smith Homestead, the Nauvoo House, the Red Brick Store, and the Smith Family Cemetery.
Accomodation: Where to stay in Nauvoo
Address: Nauvoo, Illinois
6: Greenville
Flickr/Jeff Sharp
There are quite a few museums in the town of Greenville which shows that they are proud of the history of their little town. Visitors can walk through the Richard Brock Museum, the American Farm Heritage Museum, the Armed Forces Museum, and the Demoulin Museum.
The Greenville Public Library was built in 1905 with the assistance of Andrew Carnegie's money, and it is now listed on the National Register of Historic Places. Any history buff will be thrilled with the idea of being able to read every issue of the Greenville Advocate, from 1858 to today, since the library has them all.
Accomodation: Where to stay in Greenville
Address: Greenville, Illinois
7: Sycamore
Sycamore could potentially be known as the pumpkin town since they have hosted the annual Sycamore Pumpkin Festival since 1962. The festival begins on a Wednesday and concludes on Sunday and is filled with arts and crafts, carnivals, pumpkin displays, food, fireworks, and a parade. Walking tours of the Sycamore Historic District are offered every summer, and people can learn about many of the buildings from a local historian who guides the tours.
Accomodation: Where to stay in Sycamore
Address: Sycamore, Illinois
Website: www.cityofsycamore.com
8: Woodstock
Not too many towns or cities have town squares anymore, but the one in Woodstock has both the Woodstock Opera House and the Old McHenry County Courthouse nearby. Once people arrive in this town, they feel like it looks or seems familiar to them, and this is because they probably remember it from some scenes in the movies Groundhog Day and Planes, Trains, and Automobiles. After the popularity of the movie Groundhog Day, the town decided that they needed to celebrate this unique holiday every year, and they now have their own honorary groundhog, Woodstock Willie.
Accomodation: Where to stay in Woodstock
Address: Woodstock, Illinois
Website: www.woodstockil.gov
9: Elsah
Flickr/welcometoalville
With a population of just about 600 people, the entire town of Elsah is listed on the National Register of Historic Places. Four of the original taverns in the town can be found on LaSalle Street, along with all but one building that was constructed before 1861. While history is prominent in this town, many people find themselves visiting here when they are trying to spot a bald eagle or riding the Sam Vadalabene Bike Trail.
Accomodation: Where to stay in Elsah
Address: Elsah, Illinois
10: Lebanon
Wikimedia/Robert Lawton
The oldest college in Illinois, McKendree University, is in the cute little town of Lebanon. Students, locals, or visitors will want to see the Emerald Mound and Village Site. This archaeological site has five mounds, and the village remains. The Mermaid House Hotel was built in 1830, and visitors are now able to tour the building and see what Charles Dickens was talking about when he mentioned the hotel in his American Notes book after his visit there in 1842.
Address: Lebanon, Illinois
15 Most Unusual Things To Do in Chicago
10 Best Family-Friendly Hotels in Chicago
10 Best Day Trips from Chicago
Top 10 Weekend Getaways in Illinois
10 Best Things To Do in Illinois
Top 10 Tourist Attractions in Springfield, Illinois
Top 10 Tourist Attractions in Chicago, Illinois
10 Best Hot Springs in the US
Top 10 Things To Do in the USA
Top 25 Most Beautiful Cities in the USA
Top 25 Tourist Attractions in Sacramento, California
Top 10 Tourist Attractions in Santa Fe, New Mexico
Top 10 Tourist Attractions in Baltimore, Maryland
10 Best Things To Do In Florida
50 States Attractions Map
Attractions By Regions
As an Amazon Associate, Attractions of America earns from qualifying purchases.
Copyright © 2012-2022 AttractionsofAmerica.Com. All Rights Reserved.
|
{
"redpajama_set_name": "RedPajamaCommonCrawl"
}
| 4,264
|
\section{Introduction}
\label{sec: intro}
We propose a model for time-series data that is characterized by two consecutive regimes, which correspond to a highly nonstationary and nonlinear trend period and a stable, equilibrium period.
Often, researchers are interested in estimating the features of each regime, as well as the timing of the transition or change point between the two.
We are motivated by the problem of detecting contamination of mammalian cell cultures by mycoplasma using electric cell-substrate impedance sensing (ECIS) data.
Contamination of mammalian cell cultures is pervasive, costly, and can be challenging to detect \citep{gustavsson2019}.
Specifically, contamination by mycoplasma is especially prevalent, occurring in up to 20\% of cell cultures, while also expensive and time consuming to detect.
As a result, there is a pressing need for the development of additional methods for detecting contamination by mycoplasma.
ECIS is a relatively new non-invasive method used to study cell attachment, growth, morphology, function and motility \citep{ecis2019}.
ECIS measurements have been used in numerous cell biology studies, from cancer biology and cytotoxicity \citep{opp2009,hong2011}.
Because ECIS measurements have been used to differentiate between cancerous and noncancerous cells and to classify cell lines \citep{lovelady2007,gelsinger2017}, it is hypothesized that they may also be used to identify cell cultures contaminated by mycoplasma.
ECIS measurements are obtained by growing cells in a well on top of small gold-film electrodes, between which alternating current is applied and electrical impedance is measured.
As cells grow, they cover the electrode and resistance, a component of impedence, increases.
Eventually, the cells fill the well and growth ceases. In some cases, cell death occurs due to overcrowding, causing a small drop in resistance measurements after the peak.
After this point, an equilibrium period begins. Resistance fluctuations during equilibrium are caused by cell micromotion.
The equilibrium period is sometimes called confluence in the ECIS literature, and it continues until the cells exhaust their resources and begin to die.
The first row of Figure \ref{fig: norinf} shows a subset of resistance measurements for two different cell types, Madin-Darby Canine Kidney (MDCK) cells and epithelial cells of African green monkey kidney origin (BSC-1 cells). All show a nonlinear trend period followed by an equilibrium period, with a more visually obvious change point present in the MDCK cells. A color version of the figures can be found in the electronic version of the article.
\begin{figure}
\centering
\includegraphics[width=0.7\linewidth]{./g2cd_fig/exploratory_plots}
\caption{The first row shows examples of resistance measurements at 4000 hertz for MDCK and BSC cell line samples cultivated in gel. For each cell line, one example of resistance measurements displaying shorter memory and one example of resistance measurements displaying longer memory are selected. A gray vertical line at 50 hours is provided to indicate a conservative estimate of the onset of the equilibrium regime. The second row shows the corresponding sample autocorrelations for resistance measurements after 50 hours. Approximate 95\% intervals for sample autocorrelations in the absence of dependence across time during the equilibrium regime are given in gray.}
\label{fig: norinf}
\end{figure}
Equilibrium measurements are especially informative. They are believed to be less sensitive to initial conditions than features of the trend period, and can be characterized parsimoniously by a very simple three-parameter long memory time series model \citep{lovelady2007, tarantola2010}.
Equilibrium measurements display long-range dependence; the correlations between successive measurements decay very slowly over time.
The second row of Figure \ref{fig: norinf} shows sample autocorrelation functions for the subset of resistance measurements shown in the first row resistance measurements after 50 hours, at which point equilibrium has been achieved for all four selected time series. The selected time series were chosen to show how the rate of different resistance time series may display shorter or longer memory, i.e. weaker or stronger long-range dependence.
Long-range dependence has also been observed in wind speed and inflation data \citep{haslett1989, jurgen2004}, and can modeled as a Gaussian fractionally integrated (FI) or long-memory process.
The FI process has three parameters, an overall mean $\mu$, variance $\sigma^2$, and a scalar long-memory (fractional differencing) parameter $d$ that governs how quickly auto-correlations decay.
Ideally, if these parameters could be estimated well, they could be used to quantify features of the equilibrium regime in the context of ECIS data.
Unfortunately, the long-memory parameter $d$ is notoriously difficult to estimate in finite samples.
Furthermore, the change point from trend to confluence phase, which determines the amount of data available to estimate $d$, is typically not precisely known in practice.
Standard practice is use a fixed time point, e.g., 20 hours, as a conservative estimate of the start of the confluence regime \citep{tarantola2010}.
This under-utilizes the data, potentially resulting in poorer estimates of the parameters of interest. Furthermore, such a conservative estimate could incorrectly characterize the preceding trend phase.
This suggests the need for a change point detection method which can identify when the trend phase gives way to confluence, specifically an unsupervised method that can detect the transition from a nonstationary model to a FI model.
To our knowledge, existing methods for change-point detection are not appropriate.
Some existing methods assume a short-memory autoregressive moving average (ARMA), long-memory FI models, or other restrictive parametric models both before and after the change point
\citep{chen1993,gilles2008,killick12}.
Others assume that measurements between change points are independent or identically distributed, or assume that change points strictly correspond to level shifts or isolated outliers \citep{matteson2014,zhang2019}.
Alternative methods in the biomedical fields tend to be too domain-specific to apply to the the problem of detecting the change point between the trend and confluence phase in ECIS data \citep{olshen04, nika2014}.
In this paper, we develop a novel method for estimating a change point between a highly nonstationary and nonlinear trend period and a stable, equilibrium period that is characterized by an FI process. We simultaneously obtain estimates of the nonlinear trend function and the FI parameters.
We apply this method to the detection of contamination by M. hominis, a species of mycoplasma, in MDCK cells and BSC-1 cells using ECIS measurements.
The available data consists of four experiments per cell type.
Each experiment corresponds to ECIS measurements on cells on a single tray of 96 wells obtained over the course of at least 72 hours. Of the 96 wells, 16 are left empty, 32 contain uncontaminated cells and 48 contain cells contaminated by mycoplasma.
In order to mimic lab-to-lab variability in cell culture preparation, wells were prepared using either of two different types of media. Half were prepared using bovine serum albumin (BSA), and half were prepared using gel.
Within an experiment, wells containing the same media and cells with the same contamination status can be thought of as replicates.
In Section~\ref{sec:methodology}, we propose a model which we call Trend-to-Confluence Detector (T2CD) for data which display highly nonstationary and nonlinear trend period followed by a stable, equilibrium period with long-range dependence.
In Section~\ref{sec:estimation}, we discuss estimation of the parameters of the model introduced in Section~\ref{sec:methodology}. We consider both an exact estimation procedure which we call T2CD-step, as well as an generalized estimation procedure which has greater computational scalability for longer time series which we call T2CD-sigmoid.
We demonstrate the performance of T2CD-step and T2CD-sigmoid in simulations in Section~\ref{sec:simulations}. We apply T2CD-step and T2CD-sigmoid to the ECIS data shown in Figure \ref{fig: norinf} and use the estimated change points and FI parameters to better classify cells by contamination status in Section~\ref{sec:ecis}.
\section{Trend-to-Confluence Detector (T2CD) Model}
\label{sec:methodology}
\subsection{Overview}
\label{subsec:overview}
Let $y_1, y_2, \dots, y_T \in \mathbb{R}$ be a sequence of time-ordered observations at $t = 1, 2, \dots, T$, respectively. We assume that the measurements $y_t$ belong to two successive regimes, a trend regime and an equilibrium or confluent regime.
Let $\tau$ denote the change point time index. We assume that
\begin{align}
y_t &= f\paren{t; \mathbf{\beta}} + \eta_t & \quad \text{for $t < \tau$}\\
y_t &= g\paren{y_1, \dots, y_{t-1}; \mu, d, \tau} + \epsilon_t
& \quad \text{for $t \geq \tau$}
\end{align}
where $\eta_t \sim \N{0}{\text{exp}\{h\paren{t; \mathbf{\theta}}\}}$ and $\epsilon_t \sim \N{0}{\nu^2}$, $\tau_a \leq \tau \leq \tau_b$, and $\tau_a$ and $\tau_b$ are the prespecified minimum and maximum values of the change point chosen according to a priori knowledge of the change point location. In the absence of a priori information, $\tau_a = 0$ and $\tau_b = T$. The noise terms $\eta_t$ and $\epsilon_t$, which encompass measurement errors and random fluctuations due to continuous cell growth, motility, and death, are assumed to be independent within and across the two regimes.
During the first regime $( t < \tau )$, the measurement at time $t$ will be centered around a trend curve $f\paren{t; \mathbf{\beta}}$ which is a function of time $t$ and fixed but unknown parameters $\mathbf{\beta}$. The noise terms $\{ \eta_t \}$
are possibly heteroscedastic with variance $\text{exp}\{h\paren{t; \mathbf{\theta}\}}$, to reflect different degrees of uncertainty in the measurements when the cell culture undergoes different rates of growth and death.
During the second (equilibrium) regime $( t \ge \tau )$, the measurement at time $t$ will be centered about a function $g\paren{y_{1}, \dots, y_{t-1}; \mu, d, \tau}$ of previous measurements $y_{1}, \dots, y_{t-1}$ and fixed but unknown parameters $\mu$, and $d$. The noise terms $\epsilon_t \sim \N{0}{\nu^2}$ are homoscedastic with fixed but unknown variance $\nu^2$, since the cell culture is in equilibrium and not undergoing drastic changes. We describe our modeling choices of the two regimes in the following sections.
\subsection{Trend}
\label{subsec:trend}
Resistance measurements in the first regime, or the trend phase, are characterized by a trend of initial steep increase sometimes followed by a slight drop after the peak, as well as heteroscedasticity with higher variance at the stage of rapid cell growth.
As mentioned above, the trend curve is denoted as $f\paren{t;\beta}$.
Depending on the trend, any appropriate parametric, semi-parametric or nonparametric model can be used to fit the first regime. The exact formulation of the trend curve can depend on the application domain and the choice of the user.
For the ECIS application that we focus on in this paper, we assume a smooth trend curve. This is in line with visual inspection of real ECIS data in Figure \ref{fig: norinf}, and that cell growth, motility, death and other functions are continuous processes.
We utilize penalized splines \citep{ruppert2003} for their flexibility to capture the ECIS trend phase, since it is highly nonstationary. We similarly use penalized splines in modeling the noise variance.
We denote the matrix of B-spline basis functions $B_{i,D}(u)$ of degree $D_f$ evaluated on time indices for the trend as $\mathbf{X} = \left(\mathbf{x}_1', \mathbf{x}_2', \dots, \mathbf{x}_{T}' \right) \in \mathbbm{R}^{T \times (Q_f+D_f+1)}$, where $Q_f$ is the number of distinct interior knots.
Similarly, we denote $\mathbf{V} = \left(\mathbf{v}_1', \mathbf{v}_2', \dots, \mathbf{v}_{T}' \right)$ as the $T\times \paren{Q_h + D_h + 1}$ matrix of B-spline basis functions for the log variances of the noise terms.
The model for the first regime takes the form
\begin{align*}
y_t = \mathbf{x}_{t}'\mathbf{\beta} + \eta_t,
\end{align*}
where $\eta_t \sim N(0, \text{exp}\{ \mathbf{v}_{t}'\mathbf{\theta}\})$.
Let the fitted spline for the trend be $s(t) = \mathbf{x}_t' \beta$.
We impose the smoothness penalty $\lambda_f \int \hat{s}''(u)^2 du = \lambda_f \boldsymbol \beta'\boldsymbol M_f \boldsymbol \beta$ on the spline estimate to prevent overfitting,
where $\lambda_f > 0$ is a scalar that determines the smoothness of the fitted spline and $\boldsymbol M_f$ is a matrix with elements that are fixed given the matrix of B-spline basis functions $\mathbf{X}$.
An equivalent smoothness penalty $\lambda_h \boldsymbol \theta'\boldsymbol M_h \boldsymbol \theta$ is imposed on the fit for the log variances of the noise terms, where $\lambda_h$ is another smoothness parameter and $\boldsymbol M_h$ is a matrix with elements that are fixed given the matrix of B-spline basis functions $\mathbf{V}$.
\subsection{Equilibrium}
Starting at time index $\tau$, measurements are centered about a function $g\paren{y_{1}, \dots, y_{t-1}; \mu, d, \tau}$ of previous measurements $y_{1}, \dots, y_{t-1}$ and fixed but unknown parameters $\mu$, and $d$ that corresponds to the conditional mean function of a fractionally integrated (FI) process:
\begin{align}\label{eq: fi}
g\paren{y_1, \dots, y_{t-1}; \mu, d, \tau} = \mu - \sum_{i=1}^{t-1} {d \choose i} (-1)^{i} \paren{y_{t-i} - \mu}\mathbbm{1}_{t-i \geq \tau},
\end{align}
This captures long-range dependence of the measurements in confluence.
The parameter $d$ plays the role of the long memory parameter in a FI model \citep{sowell1992}. The FI model assumes that observed values of a time series $y_t$ satisfy $(1 - B)^d y_t = \epsilon_t$, where $B$ is the differencing operator $B^k y_t = y_{t-k}$ and $\epsilon_t \sim \N 0 \nu^2$.
Values of $d > 0$ correspond to processes that are said to have long memory, with
larger values of $d$ indicating more slowly decaying autocorrelations over time.
Specifically, the autocorrelation function $Corr(y_t, y_{t-k})$ exhibits hyperbolic decay: as $k \rightarrow \infty$, $Cor(y_t, y_{t-k}) \rightarrow (\Gamma(1 - d)/\Gamma(d))k^{2d-1}$ \citep{baillie1996}.
When $d < 1$, the FI process is mean reverting, and when $d < 0.5$ the FI process is stationary.
\subsection{An Extension to Multivariate Data}
\label{sec: multivariate}
To accommodate settings where $p$ related time series may be observed contemporaneously, we provide an extension to multivariate time series data $\Y \in \mathbb{R}^{T\times p}$.
We assume that all $p$ time series share a common long-memory parameter $d$, but have their own change point $\tau_j$, trend parameters $\mathbf{\beta}_j$ and $\mathbf{\theta}_j$, and equilibrium mean and variance $\mu_j$ and $\nu^2_j$.
Specifically, we assume
\begin{align}
y_{t,j} &= f\paren{t; \mathbf{\beta}_j} + \eta_{t,j} & \quad \text{for $t < \tau_j$}\\
y_{t,j} &= g\paren{y_{1,j}, \dots, y_{t-1,j}; \mu_j, d, \tau_j} + \epsilon_{t,j} & \quad \text{for $t \geq \tau_j$}
\end{align}
where $\eta_{t,j} \sim \N{0}{\text{exp}\{h\paren{t; \mathbf{\theta}_j}\}}$ and $\epsilon_{j} \sim \N{0}{\nu^2_j}$.
This is motivated by the ECIS measurements described in Section~\ref{sec: intro}, where the $p$ related time series correspond to wells containing cells of the same type, contamination status, and media in the same experiment which may have varying initial conditions but common equilibrium behavior. We account for varying initial conditions, such as the number of cells deposited, by allowing each well to have its own varying change point $\tau_j$, trend parameters $\mathbf{\beta}_j$ and $\mathbf{\theta}_j$, and equilibrium mean and variance $\mu_j$ and $\nu^2_j$. A shared long-memory parameter $d$ reflects the cells' common equilibrium or confluence behavior.
\section{Estimation}
\label{sec:estimation}
\subsection{Exact Estimation for Univariate Data: T2CD-step}
First, we introduce a strategy for estimating the T2CD parameters that we call T2CD-step, because it performs a complete search over the change point location space $[\tau_a, \tau_b]$.
We find the change point $\tau_a \leq \hat{\tau}\leq \tau_b$ which maximizes the penalized log-likelihood:
\begin{align}
\label{eqn: loglik}
&
-\sum_{t = 1}^{\tau - 1} \left(\frac{\boldsymbol v_t'\boldsymbol \theta}{2} +\frac{\left(y_t - \boldsymbol x_t' \boldsymbol \beta\right)^2}{2\text{exp}\left\{\boldsymbol v_t'\boldsymbol \theta\right\}}\right) - \frac{1}{2} \lambda_f \boldsymbol \beta'\boldsymbol M_f \boldsymbol \beta - \frac{1}{2} \lambda_h \boldsymbol \theta'\boldsymbol M_h \boldsymbol \theta + \\ \nonumber
&\hspace{2cm} -\frac{1}{2\nu^2}\sum_{t = \tau}^{T} \left(y_t - g\left(y_{1},\dots, y_{t-1}; \mu, d, \tau\right) \right)^2 - \left(\frac{T - \tau + 1}{2}\right)\text{log}\left(\nu^2\right) + constant.
\end{align}
Given a candidate change location, the penalized log-likelihood can be decomposed into one component that involves the values of the time series during the trend regime $y_1,\dots, y_{\tau-1}$ and the parameters of the trend regime, $\boldsymbol \beta$ and $\boldsymbol \theta$, and the smoothness parameters, $\lambda_f$ and $\lambda_h$ and another component that involves the values of the time series during the equilibrium period $y_{\tau},\dots, y_T$ and the parameters of the equilibrium period, $\mu$, $d$, and $\nu^2$.
It follows that the parameters of the trend and equilibrium regime can be estimated simultaneously from the trend and equilibrium data, respectively.
We estimate the parameters of the trend regime using an iterative Feasible Generalized Least Squares (FGLS) procedure \citep{kuan2004} to estimate the spline coefficients $\mathbf{\beta}$ and $\mathbf{\theta}$, with $\lambda_f$ and $\lambda_h$ chosen according to leave-one-out cross validation as implemented in \texttt{smooth.spline} in \texttt{R} \citep{R} which selects the smoothness penalties by golden-section search. A more detailed explanation of the FGLS procedure is provided in Web Appendix A.
We estimate the parameters of the equilibrium regime by computing two estimates of the long memory parameter $d$, one by maximizing \eqref{eqn: loglik} over the range $d\in(-0.5, 0.5)$ and another by maximizing \eqref{eqn: loglik} over the range $d \in(0.5, 1.5)$. The conditional mean function $g\left(y_{1},\dots, y_{t-1}; \mu, d, \tau\right)$ takes a first difference when $d\in(0.5, 1.5)$. As a result, discontinuities can occur at $d = 0.5$. For this reason, we choose the estimate of the the long-memory parameter parameter $d$ that is further from the boundary of $0.5$.
\subsection{Generalized Estimation for Univariate Data: T2CD-sigmoid}
In practice, maximizing the penalized log-likelihood \eqref{eqn: loglik} can be prohibitively computationally demanding and time consuming if there are many candidate change points, as is the case when the observed time series is long. Accordingly, we introduce a generalization to the estimation procedure that we call T2CD-sigmoid.
Let $w\paren{t;\boldsymbol \alpha}$ denote a transition function that takes on values in the interval $\left[0, 1\right]$, then we can define the mean function in the second regime $g\paren{y_1,\dots, y_{t-1}; \mu, d, \tau}$ as defined in \eqref{eq: fi} as a special case of
\begin{align} \label{eq: figen}
\mu - \sum_{i=1}^{t-1} {d \choose i} (-1)^{i} \paren{y_{t-i} - \mu}w\paren{t-i; \boldsymbol \alpha},
\end{align}
where $w\paren{t;\boldsymbol \alpha}$ has a single parameter $\alpha$ that corresponds to the change point $\tau$ and
$w\paren{t; \alpha}=\mathbbm{1}_{t-i \geq \alpha}$ that takes the form of a step function.
This suggests that an alternative approach would be to replace the discrete step transition function $\mathbbm{1}_{t-i \geq \tau}$
with a continuous sigmoid transition function $w\paren{t; \boldsymbol \alpha} = \paren{1 + \text{exp}\{-\alpha_0 - \alpha_1 t\}}^{-1}$,
which is parameterized by a pair of real-valued parameters $\boldsymbol \alpha = \set{\alpha_0, \alpha_1}$.
We denote the corresponding second regime mean function as
\begin{align*}
\tilde{g}\paren{y_1, \dots, y_{t-1}; \mu, d, \alpha_0, \alpha_1} = \mu -
\sum_{i=1}^{t-1} {d \choose i} (-1)^{i} \paren{y_{t-i} - \mu}\paren{1 + \text{exp}\{-\alpha_0 - \alpha_1 (t-i)\}}^{-1}.
\end{align*}
The parameters $\alpha_0$ and $\alpha_1$ determine the timing of the transition from trend to equilibrium phase,
which corresponds to the inflection point of the transition function $\paren{1 + \text{exp}\{-\alpha_0 - \alpha_1 t\}}^{-1}$.
The change point as estimated as when the transition function is at $0.5$, that is, $\hat{\tau} = -\frac{\hat{\alpha}_0}{\hat{\alpha}_1}$.
The timing of the transition can be constrained to the interval $\left[\tau_a, \tau_b\right]$ by adding a penalty $C(w(\tau_b;\alpha_0, \alpha_1) - w(\tau_a;\alpha_0, \alpha_1))$ with fixed penalty parameter $C > 0$ to the objective function.
Using a smooth transition function can offer computational speed-ups because the log-likelihood can be differentiated with respect to the parameters that determine the timing of the transition, $\alpha_0$ and $\alpha_1$, and accordingly does not require an exhaustive search over all candidate change points.
The penalized log-likelihood used for T2CD-sigmoid is
\begin{align}
\label{eqn: loglikapp}
& -\sum_{t = 1}^{T}
\paren{1-w(t;\alpha_0, \alpha_1)} \left(\frac{\boldsymbol v_t'\boldsymbol \theta}{2} +\frac{\left(y_t - \boldsymbol x_t' \boldsymbol \beta\right)^2}{2\text{exp}\left\{\boldsymbol v_t'\boldsymbol \theta\right\}}\right)
- \frac{1}{2} \lambda_f \boldsymbol \beta'\boldsymbol M_f \boldsymbol \beta - \frac{1}{2} \lambda_h \boldsymbol \theta'\boldsymbol M_h \boldsymbol \theta + \\ \nonumber
&\hspace{2cm} -\frac{1}{2\nu^2}\sum_{t = 1}^{T} w(t;\alpha_0, \alpha_1) \left(y_t - \tilde{g}\left(y_{1},\dots, y_{t-1}; \mu, d,\alpha_0, \alpha_1\right) \right)^2 + \nonumber \\
&\hspace{2cm}
- \frac{1}{2} \sum_{t = 1}^{T}
w(t;\alpha_0, \alpha_1) \text{log}\left(\nu^2\right) + C\paren{w(\tau_b;\alpha_0, \alpha_1) - w(\tau_a;\alpha_0, \alpha_1)} + constant\nonumber
\end{align}
where $C > 0$ is a constant that can be set to ensure that the inflection point of the smooth transition function occurs between $\tau_a$ and $\tau_b$. The intuition for weighing the log-likelihood can also be found in tempered likelihoods for Bayesian inference where a model likelihood is down-weighted if model misspecification is suspected \citep{Thomas2019DiagnosingMM}.
The penalized log-likelihood used by T2CD-sigmoid cannot be decomposed into two components, one of which involves the values of the time series during the trend period and corresponding parameters and another component that involves the values of the time series during the equilibrium period and the corresponding parameters.
Fortunately, B-spline bases are flexible enough to fit local trends. Accordingly, the first step of T2CD-sigmoid is to estimate the trend regime parameters from the entire time series by maximizing the penalized log likelihood
\begin{align}
\label{eqn: trendonly}
& -\sum_{t = 1}^{T} \left(\frac{\boldsymbol v_t'\boldsymbol \theta}{2} +\frac{\left(y_t - \boldsymbol x_t' \boldsymbol \beta\right)^2}{2\text{exp}\left\{\boldsymbol v_t'\boldsymbol \theta\right\}}\right) - \frac{1}{2} \lambda_f \boldsymbol \beta'\boldsymbol M_f \boldsymbol \beta - \frac{1}{2} \lambda_h \boldsymbol \theta'\boldsymbol M_h \boldsymbol \theta.
\end{align}
Again, we use an iterative Feasible Generalized Least Squares (FGLS) procedure \citep{kuan2004} to estimate the spline coefficients $\mathbf{\beta}$ and $\mathbf{\theta}$, with $\lambda_f$ and $\lambda_h$ chosen according to leave-one-out cross validation as implemented in \texttt{smooth.spline} in \texttt{R} \citep{R}.
Via simulations provided in Web Appendix A, we show that estimates of the trend regime parameters obtained from this procedure are comparable to estimates of the trend regime parameters obtained by estimating the trend regime parameters from the true trend regime data alone.
Having obtained estimates of $\mathbf{\beta}$ and $\mathbf{\theta}$, we can set $C$ to be on the order of the log-likelihood component in Equation \eqref{eqn: loglikapp} at the estimated values of $\mathbf{\beta}$ and $\mathbf{\theta}$ in order to place approximately equal weight on model fitting and change point regularization.
While alternative procedures such as using cross-validation to choose $C$ can be used, we find that this simpler strategy performs well empirically by encouraging the inflection point of the transition function to occur in the interval $[\tau_a, \tau_b]$.
Having now also fixed $C$, we can maximize
\eqref{eqn: loglikapp} with respect to $\mathbf{\alpha}$, $d$, $\mu$, and $\nu^2$. As in T2CD-step, we maximize \eqref{eqn: loglikapp} twice, once for $-0.5 \leq d \leq 0.5$, and a second time for $0.5 \leq d \leq 1.5$, and choose the maximizing set of values of $\mathbf{\alpha}$, $d$, $\mu$, and $\nu^2$ that includes an estimate of $d$ that is further from $0.5$.
\subsection{Exact and Generalized Estimation for Multivariate Data}
\label{sec: multivariate_est}
When there are $p$ replicates of the sequences, the penalized log-likelihood function is a sum of the penalized log-likelihoods of the individual sequences. The only constraint is that the long-memory parameter $d$ is shared across dimensions as described in Section \ref{sec: multivariate}.
Recall that the change locations are allowed to differ across replicates, the number of possible combinations for change locations is $m^p$, where $m$ is the number of time indices in $[\tau_a, \tau_b]$. An exhaustive search for the best combination is often computationally prohibitive.
For this reason, we use the following two-step procedure. First, we run either T2CD-step or T2CD-sigmoid on each univariate sequence to obtain estimates of $\mathbf{\beta}_j$, $\mathbf{\theta}_j$, $d_j$, $\mu_j$, $\nu^2_j$, and $\tau_j$ for T2CD-step or $\mathbf{\alpha}_j$ for T2CD-sigmoid. Fixing the estimates of $\mathbf{\beta}_j$, $\mathbf{\theta}_j$, and $\tau_j$ for T2CD-step or $\mathbf{\alpha}_j$ for T2CD-sigmoid, we then optimize over $d$, $\mu_j$, $\nu^2_j$, initializing $d$ at the mean univariate estimate across all of the time series $p^{-1}\sum_{j = 1}^p \hat{d}_j$ and $\mu_j$ and $\nu^2_j$ at the univariate estimates.
\section{Simulation Study}
\label{sec:simulations}
We evaluate the performance of T2CD-step and T2CD-sigmoid for estimating $\tau$ and $d$ under several different scenarios, using both univariate and multivariate time series data. We set up the simulations to be similar to the ECIS data described in Section \ref{sec: intro}.
First, we consider one simple scenario and compare estimates of the change point $\tau$ obtained by T2CD-step and T2CD-sigmoid, in order to examine how generalizing the discrete transition using a smooth transition function affects change point estimation.
We then consider a broader set of scenarios and compare T2CD-step and T2CD-sigmoid not only to each other but also to several alternative methods.
We simulate univariate time series for comparing T2CD-step and T2CD-sigmoid as follows. Given a fixed change point $\tau$, we simulate trend curves $\boldsymbol f = (f_1, \dots, f_\tau)$ from a mean zero Gaussian process with squared exponential kernel $Cov[f_t, f_s] = 10 \text{exp}\paren{-0.5(s-t)^2}$. We simulate trend regime measurements $y_t = f_t + \eta_t$, where $\eta_t$ are mean zero heteroscedastic measurement errors with standard deviation $\sigma_t = \frac{2-0.1}{\max \set{f_s}_{s=1}^\tau - \min \set{f_s}_{s = 1}^\tau}\left[ f_t - \min \set{f_s}_{s=1}^\tau \right] + 0.1$.
We simulate equilibrium measurements $y_{\tau + 1}, \dots, y_T$ according to a mean-zero FI model with noise variance $\nu = 0.5$ and long memory parameter $d$: $(1 - B)^d y_t = \epsilon_t$, where $\epsilon_t \sim \text{N}(0, 0.25)$.
For comparison with the observed ECIS data, we simulate univariate time series of length $T = 400$, which we can think of as 70 hours of data.
For each combination of true change points $\tau$ set to values in the interval $[85,258]$ chosen to correspond to change points at $\{15, 20, \dots, 45\}$ hours and long memory parameter $d \in \left\{-0.25, -0.05, \dots, 1.45\right\}$, we simulate 100 univariate time series.
When applying T2CD-step and T2CD-sigmoid to each simulated univariate time series, we set the candidate range of $\tau$ to $[\tau_a = 10, \tau_b = 50]$, use spline basis of degree 3 with knots at every integer value of $t$ when fitting $\beta$, and knots at every integer multiple of 5 when fitting $\theta$.
For T2CD-sigmoid, we fix $C=1000$ throughout.
We check the choice of these hyperparameters in Figure \ref{fig: mdck} through residual analysis. Extensive studies on hyperparameter tuning is beyond the scope of this work.
\begin{figure}
\begin{subfigure}[b]{\linewidth}
\includegraphics[width=\linewidth]{./g2cd_fig/univgp_stationary_tau}
\caption{Estimates of $\tau$ when the second regime is stationary at $d<0.5$.}
\end{subfigure}
\begin{subfigure}[b]{\linewidth}
\includegraphics[width=\linewidth]{./g2cd_fig/univgp_nonstationary_tau}
\caption{Estimates of $\tau$ when the second regime is nonstationary at $d>0.5$. T2CD-step tends to overestimate $\tau$ because nonstationarity can be mistaken for the first regime. The overestimation issue is less severe for T2CD-sigmoid because the smooth transition function accommodates uncertainty about the change point.}
\end{subfigure}
\caption{T2CD estimates of $\tau$ for simulation setup where the first regime is generated via Gaussian process with squared exponential kernel and the second regime generated via FI($d$).}
\label{fig: g2cd_tau}
\end{figure}
The performance of estimates of $\tau$ are shown in Figure \ref{fig: g2cd_tau}. Estimated change points for T2CD-sigmoid are set to the time index when the smooth transition function is equal to 0.5. Both T2CD-step and T2CD-sigmoid estimate the change point $\tau$ well when $d$ is much smaller than $0.5$.
We hypothesize that when $d$ is close to or larger than $0.5$, the change point is more difficult to recover because the long-range autocorrelations between equilibrium measurements can yield smoothly varying time trends during the equilibrium period, which can be mistaken for a continuation of the trend period.
Surprisingly, when $d$ is closer to or greater than $0.5$ T2CD-step tends to overestimate $\tau$ while T2CD-sigmoid continues to estimate $\tau$ well on average.
\begin{figure}
\begin{subfigure}[b]{\linewidth}
\includegraphics[width=0.49\linewidth]{./g2cd_fig/fitwt_smalld}
\includegraphics[width=0.49\linewidth]{./g2cd_fig/wt_smalld}
\caption{Estimates by T2CD-sigmoid for simulation with $d=-0.45$.}
\end{subfigure}
\begin{subfigure}[b]{\linewidth}
\includegraphics[width=0.49\linewidth]{./g2cd_fig/fitwt_larged}
\includegraphics[width=0.49\linewidth]{./g2cd_fig/wt_larged}
\caption{Estimates by T2CD-sigmoid for simulation with $d=1.35$. }
\end{subfigure}
\caption{T2CD-sigmoid estimates for simulation setup where the first regime is generated via Gaussian process with squared exponential kernel and the second regime generated via FI($d$). The blue and green overlaid lines are the fit by T2CD-sigmoid for the trend and confluence phase, respectively. The vertical red dashed line marks the time index when the regime transition function is estimated to cross 0.5. The estimated transition is less abrupt for large $d$.}
\label{fig: g2cdfast_wt}
\end{figure}
In order to understand why T2CD-sigmoid provides better change point estimates than T2CD-step when $d$ is close to or greater than $0.5$, we zoom in on a pair of estimated smooth transition functions from simulations with $d = -0.45$ and $d = 1.35$ in Figure \ref{fig: g2cdfast_wt}.
We observe that the estimated transition function is much steeper and more similar to the discrete transition function assumed by T2CD-step when $d = -0.45$.
By allowing a smooth transition function, T2CD-sigmoid can accommodate greater uncertainty about the change point when $d$ is close to or greater than $0.5$.
Having shown that using T2CD-sigmoid and generalizing the discrete transition function assumed in T2CD-step with a smooth transition function can actually result in improved estimation of the true change point, we examine the relative performance of T2CD-step and T2CD-sigmoid with respect to estimating the long memory parameter $d$ in Figure~\ref{fig: univgp_eaft}.
For context, we also consider estimation of $d$ using a procedure that fixes the change point $\tau =50$ (FixedTau) and a procedure that fixes the change point $\tau$ at its true value (TrueTau).
FixedTau sets the bar for estimating $d$ with conservative data usage, whereas TrueTau gives the best $d$ estimate that can be attained if the true change point were known.
\begin{figure}
\centering
\includegraphics{./g2cd_fig/simresults_diff_gp_eaft}
\caption{Performance of estimates for change location $\tau$ and long-memory parameter $d$ obtained using T2CD-step, T2CD-sigmoid, FixedTau, and TrueTau across 100 simulated series per each combination of $\tau$ and $d$ for each simulation configuration. The first regime is generated via Gaussian process with squared exponential kernel; the second regime generated via FI($d$).}
\label{fig: univgp_eaft}
\end{figure}
We see that estimation of the change point $\tau$ and estimation of the long memory parameter $d$ are closely related. When the estimated change point occurs too early, we tend to overestimate the long-memory parameter. When the estimated change point occurs too late, we tend to underestimate the long-memory parameter $d$.
This pattern is most apparent when T2CD-step is used.
Both T2CD-step and T2CD-sigmoid provide better estimates of $d$ than FixedTau as long as the true change point occurs before 40 hours. We also observe that T2CD-step provides only slightly poorer estimation of $d$ than TrueTau. We further investigate the relative performance of T2CD-step and T2CD-sigmoid in Figure~\ref{fig: univgp_ea}.
\begin{figure}
\centering
\includegraphics{./g2cd_fig/simresults_percent_gp_exact_vs_approx}
\caption{Relative performance of estimates for change location $\tau$ and long-memory parameter $d$ obtained using T2CD-step and T2CD-sigmoid across 100 simulated series per each combination of $\tau$ and $d$ for each simulation configuration. The first regime generated via Gaussian process with squared exponential kernel; the second regime generated via FI($d$).}
\label{fig: univgp_ea}
\end{figure}
We see that T2CD-step and T2CD-sigmoid provide comparably accurate estimates of the differencing parameter $d$ when the change point occurs early. When the equilibrium process is non-stationary with true long memory parameter $d > 0.5$, the improved estimation of the change point T2CD-sigmoid relative to T2CD-exact also results in improved estimation of the long-memory parameter $d$.
Now we compare the performance of T2CD-step and T2CD-sigmoid for estimating the change point $\tau$ and long memory parameter $d$to the performance of two procedures that use the popular E-Divisive algorithm introduced in \cite{matteson2014} to estimate the change point (ECP and ECP.diff).
The popular E-Divisive algorithm is a nonparametric procedure which uses the energy statistics as a distance metric for binary segmentation \citep{matteson2014}.
E-Divisive can be used to find multiple change points. In our comparison, we use E-Divisive to find a maximum of $3$ change points and use the most significant change point within the candidate range $[\tau_a, \tau_b]$.
We consider two different procedures based on E-Divisive: ECP applies the E-Divisive algorithm to the observed time series $\boldsymbol y$, whereas ECP.diff applies the E-Divisive algorithm to the first difference of the observed time series data.
Once an estimated change point $\tau$ has been obtained, both ECP and ECP.diff procedures estimate the parameters of the FI model for the equlibrium period using maximum likelihood.
The relative performance of T2CD-step and T2CD-sigmoid compared to ECP and ECP.diff is shown in Figure~\ref{fig: univgp_ecp}.
\begin{figure}
\centering
\includegraphics{./g2cd_fig/simresults_percent_gp_vs_ecp.eps}
\caption{Relative performance of estimates for change location $\tau$ and long-memory parameter $d$ obtained using T2CD-step, T2CD-sigmoid, ECP, and ECP.diff across 100 simulated series per each combination of $\tau$ and $d$ for each simulation configuration. The first regime generated via Gaussian process with squared exponential kernel; the second regime generated via FI($d$).}
\label{fig: univgp_ecp}
\end{figure}
When compared to alternative methods ECP and ECP.diff, both T2CD-step and T2CD-sigmoid estimate the change point $\tau$ better for all true change points when the equilibrium process is stationary with $d < 0.5$, and for late true change points $\tau > 35$ when the equilibrium process is non-stationary with $d \geq 0.5$.
Careful examination of the change point estimates indicates that ECP and ECP.diff tend to underestimate the change point, which is likely due to the fact that both assume that observations between change points are independently and identically distributed.
This does not hold for data that we simulated, nor do we expect it to hold for the ECIS data described in Section~\ref{sec: intro}.
Relative performance of the long memory parameter $d$ mirrors the relative performance of the change point $\tau$.
T2CD-step and T2CD-sigmoid tend to perform comparably, with slightly better estimates of the long memory parameter $d$ from T2CD-step when the equilibrium process is stationary with $d < 0.5$ and slightly better estimates of the long memory parameter $d$ from T2CD-sigmoid when the equilibrium process is non-stationary with $d \geq 0.5$.
ECP and ECP.diff produce much poorer estimates of the long memory parameter $d$ than both T2CD-step and T2CD-sigmoid for all true change point and long memory parameter values, which is unsurprising given we observed poorer estimates of the change point $\tau$ from ECP and ECP.diff.
However, the performance advantages of T2CD-step and T2CD-sigmoid do come at a computational price. For the first univariate experiment where the trend regime is generated via Gaussian processes, on average on a 2.7 GHz CPU, ECP and ECP.diff both take 1.20 seconds, T2CD-step takes 196 seconds and T2CD-sigmoid takes 19.2 seconds.
While both of the T2CD methods are slower than the alternatives, T2CD-sigmoid is roughly 10 times faster than T2CD-step on average.
This makes T2CD-sigmoid a competitive option in providing balance between the quality of estimation and computational speed.
Next, we consider multivariate time series data made up of $p$ individual time series with unique change points $\tau_1, \dots, \tau_p$ and common long memory parameter $d$.
We simulate $100$ multivariate time series of length $T = 400$ with $p = 3$ for each value of the long memory parameter $d \in \left\{-0.25, -0.05, \dots, 1.45\right\}$. For each value of $d$, a single simulated multivariate time series is comprised of three individual time series with different change points $\tau_1 = 15$, $\tau_2 = 25$, and $\tau_3 = 45$. As in the univariate simulations, trend curves $\boldsymbol f_j = (f_{j1}, \dots, f_{j\tau_j})$ are simulated from a mean zero Gaussian process with squared exponential kernel $Cov[f_t, f_s] = 10 \text{exp}\paren{-0.5(s-t)^2}$.
We simulate trend regime measurements $y_{jt} = f_{jt} + \eta_{jt}$, where $\eta_{jt}$ are mean zero heteroscedastic measurement errors with standard deviation $\sigma_{jt} = \frac{2-0.1}{\max \set{f_{js}}_{s=1}^{\tau_j} - \min \set{f_{js}}_{s = 1}^{\tau_j}}\left[ f_{jt} - \min \set{f_{js}}_{s=1}^{\tau_j} \right] + 0.1$.
We simulate equilibrium measurements $y_{\tau_j + 1}, \dots, y_T$ according to a mean-zero FI model with noise variance $\nu = 0.5$ and long memory parameter $d$: $(1 - B)^d y_{jt} = \epsilon_{jt}$, where $\epsilon_{jt} \sim \text{N}(0, 0.25)$.
Again, we set the candidate range of $\tau$ to $[\tau_a = 10, \tau_b = 50]$, use spline basis of degree 3 with knots at every integer value of $t$ when fitting $\beta$, and knots at every integer multiple of 5 when fitting $\theta$.
For T2CD-sigmoid, we fix $C=1000$ throughout.
Estimates of the change point $\tau$ and long-memory parameter $d$ are summarized in Figure \ref{fig: mv_eaft}.
\begin{figure}
\centering
\includegraphics{./g2cd_fig/simresults_diff_gp_mv_eaft}
\caption{Performance of estimates for change location $\tau$ and long-memory parameter $d$ obtained using T2CD-step, T2CD-sigmoid, FixedTau, and TrueTau across 100 simulated series per each combination of $\tau$ and $d$. Multivariate simulations with $p = 3$, with change points at 15, 25, and 45. For each series, the first regime is generated via Gaussian process with squared exponential kernel, and the second regime is generated via FI($d$).}
\label{fig: mv_eaft}
\end{figure}
The multivariate results shown in Figure~\ref{fig: mv_eaft} mirror the univariate results shown in Figure~\ref{fig: univgp_eaft}.
Both T2CD-step and T2CD-sigmoid tend to overestimate earlier changepoints and underestimate the latest changepoint.
Also, both T2CD-step or T2CD-sigmoid slightly overestimation the differencing parameter $d$.
The performance of both T2CD-step and T2CD-sigmoid is on par with the conservative and oracle methods FixedTau and TrueTau. T2CD-step and T2CD-sigmoid provide better estimates of the long-memory parameter $d$ than FixedTau as long as the true long-memory parameter is not close to $d = 0.5$, and only slightly worse estimates of the long-memory parameter $d$ than TrueTau.
\begin{figure}
\centering
\includegraphics{./g2cd_fig/simresults_percent_gp_mv_exact_vs_approx}
\caption{Relative performance of estimates for change location $\tau$ and long-memory parameter $d$ obtained using T2CD-step and T2CD-sigmoid across 100 simulated series per each combination of $\tau$ and $d$. Multivariate simulations with $p = 3$, with change points at 15, 25, and 45. For each series, the first regime is generated via Gaussian process with squared exponential kernel, and the second regime is generated via FI($d$).}
\label{fig: mv_ea}
\end{figure}
Figure~\ref{fig: mv_ea} zooms in on the relative performance of T2CD-step and T2CD-sigmoid.
T2CD-sigmoid tends to provide better estimation of the change points $\tau_1$, $\tau_2$, and $\tau_3$. Better estimation of the differencing parameter $d$ is provided by T2CD-step when the equilibrium process is more stationary.
\begin{figure}
\centering
\includegraphics{./g2cd_fig/simresults_percent_gp_mv_vs_ecp}
\caption{Relative performance of estimates for change location $\tau$ and long-memory parameter $d$ obtained using T2CD-step, T2CD-sigmoid, ECP, and ECP.diff across 100 simulated series per each combination of $\tau$ and $d$. Multivariate simulations with $p = 3$, with change points at 15, 25, and 45. For each series, the first regime is generated via Gaussian process with squared exponential kernel, and the second regime is generated via FI($d$).}
\label{fig: mv_vs_ecp}
\end{figure}
Figure~\ref{fig: mv_vs_ecp} examines the relative performance of T2CD-step and T2CD-sigmoid compared to ECP and ECP.diff. T2CD-step and T2CD-sigmoid provide better estimates of the change points $\tau_1$, $\tau_2$, and $\tau_3$ compared to ECP and ECP.diff, as long as the equilibrium process is stationary or the change point occurs late.
Regarding estimation of the long-memory parameter $d$, we observe consistently better performance of T2CD-step and T2CD-sigmoid estimates relative to ECP and ECP.diff estimates.
\section{Application to ECIS Data}
\label{sec:ecis}
Now we apply the T2CD-step and T2CD-sigmoid to the MDCK and BSC cell data described in Section~\ref{sec: intro}.
ECIS resistance measurements were obtained at several frequencies, however we focus on resistance measured at the frequency of 500 hertz. We also exclude wells that are mechanically disrupted to create a ``wound-healing'' assay and a single well containing MDCK cells that displayed evidence of instrument failure.
In order to assess whether or not cell culture preparation affects our ability to identify cells contaminated with mycoplasma,
we analyze data from BSA and gel wells separately.
For model fitting, we use spline basis of degree 3 with knots at every integer value of $t$ when fitting $\beta$, and knots at every integer multiple of 5 when fitting $\theta$.
As in Section~\ref{sec:simulations}, we set $C=1000$ when implementing T2CD-sigmoid.
Based on visual inspection of the MDCK and BSC data, we set the candidate range of $\tau$ is set to $[\tau_a = 10, \tau_b = 50]$ for MDCK cells and $[\tau_a = 5, \tau_b = 45]$ for BSC cells.
To check the choice of hyperparameters, we plot in Figure \ref{fig: mdck} the time series for a MDCK sample, as well as standardized residuals from model fitting with T2CD-step. The residuals from the first regime are scaled by $\sigma_t$ estimated. The plots show that our choice of model parameters give reasonable fits to both regimes of the data.
\begin{figure}
\begin{subfigure}[b]{\linewidth}
\centering
\includegraphics[width=0.45\linewidth]{./g2cd_fig/mdck_inf_res}
\caption{Time series of resistance measurements recorded at 500 hertz. The blue and green overlaid lines are the fit by T2CD-step for the trend and confluence phase, respectively.}
\end{subfigure}
\begin{subfigure}[b]{\linewidth}
\includegraphics[width=0.32\linewidth]{./g2cd_fig/mdck_inf_residual1_sq}
\includegraphics[width=0.32\linewidth]{./g2cd_fig/mdck_inf_qqplot1_sq}
\includegraphics[width=0.32\linewidth]{./g2cd_fig/mdck_inf_acf1_sq}
\caption{Standardized residuals for the trend phase.}
\end{subfigure}
\begin{subfigure}[b]{\linewidth}
\includegraphics[width=0.32\linewidth]{./g2cd_fig/mdck_inf_residual2_sq}
\includegraphics[width=0.32\linewidth]{./g2cd_fig/mdck_inf_qqplot2_sq}
\includegraphics[width=0.32\linewidth]{./g2cd_fig/mdck_inf_acf2_sq}
\caption{Standardized residuals for the confluence phase.}
\end{subfigure}
\caption{MDCK cell; infected and cultivated in gel.}
\label{fig: mdck}
\end{figure}
\subsection{MDCK cell line}
From Figure \ref{fig: norinf}, we see that the resistance measurements for MDCK cells tend to peak before slightly decreasing and stablizing. The start of confluence or equilibrium is hypothesized to be at or slightly after the peak, and as a result is visually distinct.
Figure \ref{fig: mdck_exact} plots estimates of the change points $\tau$ and long memory parameters $d$ estimated by applying T2CD-step to each well as a univariate time series, and by applying T2CD-step to all replicate wells within the same experiment as a multivariate time series.
The estimated change points are scattered within the candidate range of $[10, 50]$, signifying varied initial conditions even in the same batch.
We observe clear evidence of long-range dependence at confluence, with all estimates of the long memory parameter above $0.5$.
Experiments 1, 3, and 4 suggest that MDCK cells that are contaminated by mycoplasma tend to show longer memory than MDCK cells that are uncontaminated.
Experiment 2 suggests the opposite, but this may be a consequence of batch effects.
Web Appendix C contains more a detailed review of estimates of the change point $\tau$ and the long memory parameter $d$ estimated by T2CD-step and T2CD-sigmoid across experiments, serum types and infection status.
\begin{figure}
\centering
\begin{subfigure}[b]{\linewidth}
\includegraphics{./g2cd_fig/mdck_exact_optimizeall_jan11_v2}
\caption{MDCK}
\label{fig: mdck_exact}
\end{subfigure}
\begin{subfigure}[b]{\linewidth}
\includegraphics{./g2cd_fig/bsc_exact_optimizeall_jan11_v2}
\caption{BSC}
\label{fig: bsc_exact}
\end{subfigure}
\caption{T2CD-step estimates for $\tau$ and $d$. Points are estimates from the univariate version of the method, and horizontal lines mark estimates from the multivariate version.}
\end{figure}
\subsection{BSC Cells}
\label{sec: bsc}
From Figure \ref{fig: norinf}, we see that the resistance measurements for the BSC cell line tend to increase sharply before plateauing. As compared to the MDCK cell line, the end of the BSC trend phase is less visually distinct. This makes change point detection and subsequent estimation of the long-memory parameter more difficult.
Figure \ref{fig: bsc_exact} plots estimates of the change points $\tau$ and long memory parameters $d$ estimated by applying T2CD-step to each well as a univariate time series, and by applying T2CD-step to all replicate wells within the same experiment as a multivariate time series.
We observe evidence of long memory regardless of contamination status, with most univariate and multivariate estimates of the long memory parameter $d$ exceeding one.
We do not observe distinct separation between the contaminated and uncontaminated cells.
However, we do observe some evidence that contaminated BSC cells tend to have shorter memory, corresponding to lower estimates of $d$, than uncontaminated cells in Experiments 1 and 2.
See Web Appendix C for a more detailed review of estimates of the change point $\tau$ and the long memory parameter $d$ estimated by T2CD-step and T2CD-sigmoid across experiments, serum types and infection status.
\subsection{Mycoplasma Contamination Classification}
To demonstrate the quality and utility of our change point $\tau$ and long memory parameter $d$ estimates, we incorporate the estimates as features in a downstream task of classifying cells by their mycoplasma contamination status.
We build on the linear discriminant analysis (LDA) and quadratic discriminant analysis (QDA) classifiers built to to classify cell lines using ECIS measurements in \cite{gelsinger2017}.
Let $c$ indicate the possible classes of observations, which in this application corresponds to whether or not a well contains cells contaminated by mycoplasma. $\boldsymbol z$ a vector of features and $\bar{\boldsymbol z}_c$ be the average feature vector across all observations in class $c$, LDA and QDA class discriminant scores can both be written as special cases of
\begin{align}
\delta_c\paren{\boldsymbol z} &= \paren{\boldsymbol z-\widebar{\boldsymbol z}_c}^T \widehat{\Sigma}_c^{-1}(\rho)\paren{\boldsymbol z-\widebar{\boldsymbol z}_c} + \log |\widehat{\Sigma}_c(\rho)|\\
\widehat{\Sigma}_c(\rho) &= (1-\rho)\widehat{\Sigma}_c + \rho\widehat{\Sigma} \nonumber
\end{align}
LDA is obtained by setting $\rho = 1$ and QDA is obtained by setting $\rho = 0$.
For each cell line we train four LDA and QDA classifiers, training each classifier on data from three experiments and computing classification accuracy on data from the remaining experiment.
The average classification accuracy across all four classifiers is provided in Table \ref{tab: classify}, along with the correspond standard deviations.
We compare classifiers trained using the original features described in \cite{gelsinger2017} to classifiers trained using the best feature from among the original features described in \cite{gelsinger2017} as well as estimates of the change point $\tau$ and long memory parameter $d$, obtained by applying either T2CD-step or T2CD-sigmoid to data from each well as a univariate time series.
A more detailed description of how we constructed the original features described in \cite{gelsinger2017} for our ECIS measurements is given in the Web Appendix D.
\begin{table}[]
\centering
\begin{tabular}{llllll}
\hline
\multirow{2}{*}{Cell line} & \multirow{2}{*}{Features} & \multicolumn{2}{l}{LDA} & \multicolumn{2}{l}{QDA} \\ \cline{3-6}
& & Mean & SD & Mean & SD \\ \hline
\multirow{3}{*}{MDCK} & Original & 0.743 & 0.272 & 0.580 & 0.237 \\ \cline{2-6}
& T2CD-step & 0.880 & 0.091 & 0.862 & 0.136 \\ \cline{2-6}
& T2CD-sigmoid & 0.962 & 0.033 & 0.975 & 0.020 \\ \hline
\multirow{3}{*}{BSC} & Original & 0.563 & 0.060 & 0.588 & 0.072 \\ \cline{2-6}
& T2CD-step & 0.650 & 0.098 & 0.630 & 0.113 \\ \cline{2-6}
& T2CD-sigmoid & 0.675 & 0.108 & 0.644 & 0.085 \\ \hline
\end{tabular}
\caption{Classification accuracy for infection status. Average is taken by taking each of the 4 experiments as the test set, and the other 3 as training set. Parameters $\tau$ and $d$ estimated by T2CD increased classification accuracy for both MDCK and BSC cell line.}
\label{tab: classify}
\end{table}
From Table \ref{tab: classify}, it is evident that the $\tau$ and $d$ estimates from T2CD-step and T2CD-sigmoid are useful features that increase classification accuracy for both cell lines.
For MDCK cells, LDA using the original features has a mean classification accuracy of 0.743. Including T2CD-step or T2CD-sigmoid features improved the mean classification accuracy by 18.4\% and 29.4\%, respectively. For BSC cells, QDA using the original features has a mean classification accuracy of 0.588. Including T2CD-step and T2CD-sigmoid features improved the mean classification accuracy by 7.1\% and 9.5\%, respectively.
The smaller improvements in classification accuracy for BSC cells are likely a consequence of less visually obvious change points in the ECIS measurements for BSC cells, as noted in Section \ref{sec: bsc}.
\section{Conclusion}
\label{sec: conclusion}
In this paper, we propose a model called T2CD for estimating a change point between a smooth, nonlinear trend period and a long-memory equilibrium period and for quantifying features of the trend and equilibrium periods. We provide exact and generalized estimation strategies, T2CD-step and T2CD-sigmoid. Via simulations, we show that T2CD-step outperforms a two step comparison method based on the popular E-Divisive algorithm for change point detection when the equilibrium period can be characterized by a long memory time series model. Compared to E-Divisive, T2CD-step tends to produce better estimates of the change points and long memory parameters. We also show that T2CD-sigmoid offers computational efficiency gains over T2CD-step with minimal reductions and even occasional improvements in performance.
Practical usage on the MDCK and BSC cell lines shows that T2CD recovers meaningful estimates of change points and long-memory parameters during confluence confluence phase. Importantly, using T2CD reduces the amount of human supervision needed to manually identify change points, ensures that the change points are identified using the same logic, and makes full use of the available data. Furthermore, we show that estimates of the change points and long memory parameters improve classification performance downstream.
\bibliographystyle{apalike}
|
{
"redpajama_set_name": "RedPajamaArXiv"
}
| 1,616
|
#define DT_DRV_COMPAT renesas_smartbond_uart
#include <errno.h>
#include <zephyr/drivers/uart.h>
#include <zephyr/drivers/pinctrl.h>
#include <zephyr/kernel.h>
#include <zephyr/spinlock.h>
#include <zephyr/sys/byteorder.h>
#include <DA1469xAB.h>
#include <zephyr/irq.h>
#define IIR_NO_INTR 1
#define IIR_THR_EMPTY 2
#define IIR_RX_DATA 4
#define IIR_LINE_STATUS 5
#define IIR_BUSY 7
#define IIR_TIMEOUT 12
#define STOP_BITS_1 0
#define STOP_BITS_2 1
#define DATA_BITS_5 0
#define DATA_BITS_6 1
#define DATA_BITS_7 2
#define DATA_BITS_8 3
#define RX_FIFO_TRIG_1_CHAR 0
#define RX_FIFO_TRIG_1_4_FULL 1
#define RX_FIFO_TRIG_1_2_FULL 2
#define RX_FIFO_TRIG_MINUS_2_CHARS 3
#define TX_FIFO_TRIG_EMPTY 0
#define TX_FIFO_TRIG_2_CHARS 1
#define TX_FIFO_TRIG_1_4_FULL 2
#define TX_FIFO_TRIG_1_2_FULL 3
#define BAUDRATE_CFG_DLH(cfg) (((cfg) >> 16) & 0xff)
#define BAUDRATE_CFG_DLL(cfg) (((cfg) >> 8) & 0xff)
#define BAUDRATE_CFG_DLF(cfg) ((cfg) & 0xff)
struct uart_smartbond_baudrate_cfg {
uint32_t baudrate;
/* DLH=cfg[23:16] DLL=cfg[15:8] DLF=cfg[7:0] */
uint32_t cfg;
};
static const struct uart_smartbond_baudrate_cfg uart_smartbond_baudrate_table[] = {
{ 2000000, 0x00000100 },
{ 1000000, 0x00000200 },
{ 921600, 0x00000203 },
{ 500000, 0x00000400 },
{ 230400, 0x0000080b },
{ 115200, 0x00001106 },
{ 57600, 0x0000220c },
{ 38400, 0x00003401 },
{ 28800, 0x00004507 },
{ 19200, 0x00006803 },
{ 14400, 0x00008a0e },
{ 9600, 0x0000d005 },
{ 4800, 0x0001a00b },
};
struct uart_smartbond_cfg {
UART2_Type *regs;
int periph_clock_config;
const struct pinctrl_dev_config *pcfg;
bool hw_flow_control_supported;
#ifdef CONFIG_UART_INTERRUPT_DRIVEN
void (*irq_config_func)(const struct device *dev);
#endif
};
struct uart_smartbond_data {
struct uart_config current_config;
struct k_spinlock lock;
#ifdef CONFIG_UART_INTERRUPT_DRIVEN
uart_irq_callback_user_data_t callback;
void *cb_data;
uint32_t flags;
uint8_t rx_enabled;
uint8_t tx_enabled;
#endif
};
static int uart_smartbond_poll_in(const struct device *dev, unsigned char *p_char)
{
const struct uart_smartbond_cfg *config = dev->config;
struct uart_smartbond_data *data = dev->data;
k_spinlock_key_t key = k_spin_lock(&data->lock);
if ((config->regs->UART2_USR_REG & UART2_UART2_USR_REG_UART_RFNE_Msk) == 0) {
k_spin_unlock(&data->lock, key);
return -1;
}
*p_char = config->regs->UART2_RBR_THR_DLL_REG;
k_spin_unlock(&data->lock, key);
return 0;
}
static void uart_smartbond_poll_out(const struct device *dev, unsigned char out_char)
{
const struct uart_smartbond_cfg *config = dev->config;
struct uart_smartbond_data *data = dev->data;
k_spinlock_key_t key = k_spin_lock(&data->lock);
while (!(config->regs->UART2_USR_REG & UART2_UART2_USR_REG_UART_TFNF_Msk)) {
/* Wait until FIFO has free space */
}
config->regs->UART2_RBR_THR_DLL_REG = out_char;
k_spin_unlock(&data->lock, key);
}
static int uart_smartbond_configure(const struct device *dev,
const struct uart_config *cfg)
{
const struct uart_smartbond_cfg *config = dev->config;
struct uart_smartbond_data *data = dev->data;
uint32_t baudrate_cfg = 0;
k_spinlock_key_t key;
uint32_t reg_val;
int err;
int i;
if ((cfg->parity != UART_CFG_PARITY_NONE && cfg->parity != UART_CFG_PARITY_ODD &&
cfg->parity != UART_CFG_PARITY_EVEN) ||
(cfg->stop_bits != UART_CFG_STOP_BITS_1 && cfg->stop_bits != UART_CFG_STOP_BITS_2) ||
(cfg->data_bits != UART_CFG_DATA_BITS_5 && cfg->data_bits != UART_CFG_DATA_BITS_6 &&
cfg->data_bits != UART_CFG_DATA_BITS_7 && cfg->data_bits != UART_CFG_DATA_BITS_8) ||
(cfg->flow_ctrl != UART_CFG_FLOW_CTRL_NONE &&
cfg->flow_ctrl != UART_CFG_FLOW_CTRL_RTS_CTS)) {
return -ENOTSUP;
}
/* Flow control is not supported on UART */
if (cfg->flow_ctrl == UART_CFG_FLOW_CTRL_RTS_CTS &&
!config->hw_flow_control_supported) {
return -ENOTSUP;
}
/* Lookup configuration for baudrate */
for (i = 0; i < ARRAY_SIZE(uart_smartbond_baudrate_table); i++) {
if (uart_smartbond_baudrate_table[i].baudrate == cfg->baudrate) {
baudrate_cfg = uart_smartbond_baudrate_table[i].cfg;
break;
}
}
if (baudrate_cfg == 0) {
return -ENOTSUP;
}
key = k_spin_lock(&data->lock);
CRG_COM->SET_CLK_COM_REG = config->periph_clock_config;
config->regs->UART2_SRR_REG = UART2_UART2_SRR_REG_UART_UR_Msk |
UART2_UART2_SRR_REG_UART_RFR_Msk |
UART2_UART2_SRR_REG_UART_XFR_Msk;
config->regs->UART2_LCR_REG |= UART2_UART2_LCR_REG_UART_DLAB_Msk;
config->regs->UART2_IER_DLH_REG = BAUDRATE_CFG_DLH(baudrate_cfg);
config->regs->UART2_RBR_THR_DLL_REG = BAUDRATE_CFG_DLL(baudrate_cfg);
config->regs->UART2_DLF_REG = BAUDRATE_CFG_DLF(baudrate_cfg);
config->regs->UART2_LCR_REG &= ~UART2_UART2_LCR_REG_UART_DLAB_Msk;
/* Configure frame */
reg_val = 0;
switch (cfg->parity) {
case UART_CFG_PARITY_NONE:
break;
case UART_CFG_PARITY_EVEN:
reg_val |= UART2_UART2_LCR_REG_UART_EPS_Msk;
/* no break */
case UART_CFG_PARITY_ODD:
reg_val |= UART2_UART2_LCR_REG_UART_PEN_Msk;
break;
}
if (cfg->stop_bits == UART_CFG_STOP_BITS_2) {
reg_val |= STOP_BITS_2 << UART2_UART2_LCR_REG_UART_STOP_Pos;
}
switch (cfg->data_bits) {
case UART_CFG_DATA_BITS_6:
reg_val |= DATA_BITS_6 << UART2_UART2_LCR_REG_UART_DLS_Pos;
break;
case UART_CFG_DATA_BITS_7:
reg_val |= DATA_BITS_7 << UART2_UART2_LCR_REG_UART_DLS_Pos;
break;
case UART_CFG_DATA_BITS_8:
reg_val |= DATA_BITS_8 << UART2_UART2_LCR_REG_UART_DLS_Pos;
break;
}
config->regs->UART2_LCR_REG = reg_val;
/* Enable hardware FIFO */
config->regs->UART2_SFE_REG = UART2_UART2_SFE_REG_UART_SHADOW_FIFO_ENABLE_Msk;
config->regs->UART2_SRT_REG = RX_FIFO_TRIG_1_CHAR;
config->regs->UART2_STET_REG = TX_FIFO_TRIG_1_2_FULL;
data->current_config = *cfg;
k_spin_unlock(&data->lock, key);
err = pinctrl_apply_state(config->pcfg, PINCTRL_STATE_DEFAULT);
if (err < 0) {
return err;
}
return 0;
}
#ifdef CONFIG_UART_USE_RUNTIME_CONFIGURE
static int uart_smartbond_config_get(const struct device *dev,
struct uart_config *cfg)
{
struct uart_smartbond_data *data = dev->data;
*cfg = data->current_config;
return 0;
}
#endif /* CONFIG_UART_USE_RUNTIME_CONFIGURE */
static int uart_smartbond_init(const struct device *dev)
{
struct uart_smartbond_data *data = dev->data;
return uart_smartbond_configure(dev, &data->current_config);
}
#ifdef CONFIG_UART_INTERRUPT_DRIVEN
static inline void irq_tx_enable(const struct device *dev)
{
const struct uart_smartbond_cfg *config = dev->config;
config->regs->UART2_IER_DLH_REG |= UART2_UART2_IER_DLH_REG_PTIME_DLH7_Msk |
UART2_UART2_IER_DLH_REG_ETBEI_DLH1_Msk;
}
static inline void irq_tx_disable(const struct device *dev)
{
const struct uart_smartbond_cfg *config = dev->config;
config->regs->UART2_IER_DLH_REG &= ~(UART2_UART2_IER_DLH_REG_PTIME_DLH7_Msk |
UART2_UART2_IER_DLH_REG_ETBEI_DLH1_Msk);
}
static inline void irq_rx_enable(const struct device *dev)
{
const struct uart_smartbond_cfg *config = dev->config;
config->regs->UART2_IER_DLH_REG |= UART2_UART2_IER_DLH_REG_ERBFI_DLH0_Msk;
}
static inline void irq_rx_disable(const struct device *dev)
{
const struct uart_smartbond_cfg *config = dev->config;
config->regs->UART2_IER_DLH_REG &= ~UART2_UART2_IER_DLH_REG_ERBFI_DLH0_Msk;
}
static int uart_smartbond_fifo_fill(const struct device *dev,
const uint8_t *tx_data,
int len)
{
const struct uart_smartbond_cfg *config = dev->config;
struct uart_smartbond_data *data = dev->data;
int num_tx = 0;
k_spinlock_key_t key = k_spin_lock(&data->lock);
while ((len - num_tx > 0) &&
(config->regs->UART2_USR_REG & UART2_UART2_USR_REG_UART_TFNF_Msk)) {
config->regs->UART2_RBR_THR_DLL_REG = tx_data[num_tx++];
}
if (data->tx_enabled) {
irq_tx_enable(dev);
}
k_spin_unlock(&data->lock, key);
return num_tx;
}
static int uart_smartbond_fifo_read(const struct device *dev, uint8_t *rx_data,
const int size)
{
const struct uart_smartbond_cfg *config = dev->config;
struct uart_smartbond_data *data = dev->data;
int num_rx = 0;
k_spinlock_key_t key = k_spin_lock(&data->lock);
while ((size - num_rx > 0) &&
(config->regs->UART2_USR_REG & UART2_UART2_USR_REG_UART_RFNE_Msk)) {
rx_data[num_rx++] = config->regs->UART2_RBR_THR_DLL_REG;
}
if (data->rx_enabled) {
irq_rx_enable(dev);
}
k_spin_unlock(&data->lock, key);
return num_rx;
}
static void uart_smartbond_irq_tx_enable(const struct device *dev)
{
struct uart_smartbond_data *data = dev->data;
k_spinlock_key_t key = k_spin_lock(&data->lock);
data->tx_enabled = 1;
irq_tx_enable(dev);
k_spin_unlock(&data->lock, key);
}
static void uart_smartbond_irq_tx_disable(const struct device *dev)
{
struct uart_smartbond_data *data = dev->data;
k_spinlock_key_t key = k_spin_lock(&data->lock);
irq_tx_disable(dev);
data->tx_enabled = 0;
k_spin_unlock(&data->lock, key);
}
static int uart_smartbond_irq_tx_ready(const struct device *dev)
{
const struct uart_smartbond_cfg *config = dev->config;
return (config->regs->UART2_USR_REG & UART2_UART2_USR_REG_UART_TFNF_Msk) != 0;
}
static void uart_smartbond_irq_rx_enable(const struct device *dev)
{
struct uart_smartbond_data *data = dev->data;
k_spinlock_key_t key = k_spin_lock(&data->lock);
data->rx_enabled = 1;
irq_rx_enable(dev);
k_spin_unlock(&data->lock, key);
}
static void uart_smartbond_irq_rx_disable(const struct device *dev)
{
struct uart_smartbond_data *data = dev->data;
k_spinlock_key_t key = k_spin_lock(&data->lock);
irq_rx_disable(dev);
data->rx_enabled = 0;
k_spin_unlock(&data->lock, key);
}
static int uart_smartbond_irq_tx_complete(const struct device *dev)
{
const struct uart_smartbond_cfg *config = dev->config;
return (config->regs->UART2_USR_REG & UART2_UART2_USR_REG_UART_TFE_Msk) != 0;
}
static int uart_smartbond_irq_rx_ready(const struct device *dev)
{
const struct uart_smartbond_cfg *config = dev->config;
return (config->regs->UART2_USR_REG & UART2_UART2_USR_REG_UART_RFNE_Msk) != 0;
}
static void uart_smartbond_irq_err_enable(const struct device *dev)
{
k_panic();
}
static void uart_smartbond_irq_err_disable(const struct device *dev)
{
k_panic();
}
static int uart_smartbond_irq_is_pending(const struct device *dev)
{
k_panic();
return 0;
}
static int uart_smartbond_irq_update(const struct device *dev)
{
const struct uart_smartbond_cfg *config = dev->config;
bool no_intr = false;
while (!no_intr) {
switch (config->regs->UART2_IIR_FCR_REG & 0x0F) {
case IIR_NO_INTR:
no_intr = true;
break;
case IIR_THR_EMPTY:
irq_tx_disable(dev);
break;
case IIR_RX_DATA:
irq_rx_disable(dev);
break;
case IIR_LINE_STATUS:
case IIR_TIMEOUT:
/* ignore */
break;
case IIR_BUSY:
/* busy detect */
/* fall-through */
default:
k_panic();
break;
}
}
return 1;
}
static void uart_smartbond_irq_callback_set(const struct device *dev,
uart_irq_callback_user_data_t cb,
void *cb_data)
{
struct uart_smartbond_data *data = dev->data;
data->callback = cb;
data->cb_data = cb_data;
}
static void uart_smartbond_isr(const struct device *dev)
{
struct uart_smartbond_data *data = dev->data;
if (data->callback) {
data->callback(dev, data->cb_data);
}
}
#endif /* CONFIG_UART_INTERRUPT_DRIVEN */
static const struct uart_driver_api uart_smartbond_driver_api = {
.poll_in = uart_smartbond_poll_in,
.poll_out = uart_smartbond_poll_out,
#ifdef CONFIG_UART_USE_RUNTIME_CONFIGURE
.configure = uart_smartbond_configure,
.config_get = uart_smartbond_config_get,
#endif
#ifdef CONFIG_UART_INTERRUPT_DRIVEN
.fifo_fill = uart_smartbond_fifo_fill,
.fifo_read = uart_smartbond_fifo_read,
.irq_tx_enable = uart_smartbond_irq_tx_enable,
.irq_tx_disable = uart_smartbond_irq_tx_disable,
.irq_tx_ready = uart_smartbond_irq_tx_ready,
.irq_rx_enable = uart_smartbond_irq_rx_enable,
.irq_rx_disable = uart_smartbond_irq_rx_disable,
.irq_tx_complete = uart_smartbond_irq_tx_complete,
.irq_rx_ready = uart_smartbond_irq_rx_ready,
.irq_err_enable = uart_smartbond_irq_err_enable,
.irq_err_disable = uart_smartbond_irq_err_disable,
.irq_is_pending = uart_smartbond_irq_is_pending,
.irq_update = uart_smartbond_irq_update,
.irq_callback_set = uart_smartbond_irq_callback_set,
#endif /* CONFIG_UART_INTERRUPT_DRIVEN */
};
#ifdef CONFIG_UART_INTERRUPT_DRIVEN
#define UART_SMARTBOND_CONFIGURE(id) \
do { \
IRQ_CONNECT(DT_INST_IRQN(id), \
DT_INST_IRQ(id, priority), \
uart_smartbond_isr, \
DEVICE_DT_INST_GET(id), 0); \
\
irq_enable(DT_INST_IRQN(id)); \
} while (0)
#else
#define UART_SMARTBOND_CONFIGURE(id)
#endif
#define UART_SMARTBOND_DEVICE(id) \
PINCTRL_DT_INST_DEFINE(id); \
static const struct uart_smartbond_cfg uart_smartbond_##id##_cfg = { \
.regs = (UART2_Type *)DT_INST_REG_ADDR(id), \
.periph_clock_config = DT_INST_PROP(id, periph_clock_config), \
.pcfg = PINCTRL_DT_INST_DEV_CONFIG_GET(id), \
.hw_flow_control_supported = DT_INST_PROP(id, hw_flow_control_supported), \
}; \
static struct uart_smartbond_data uart_smartbond_##id##_data = { \
.current_config = { \
.baudrate = DT_INST_PROP(id, current_speed), \
.parity = UART_CFG_PARITY_NONE, \
.stop_bits = UART_CFG_STOP_BITS_1, \
.data_bits = UART_CFG_DATA_BITS_8, \
.flow_ctrl = UART_CFG_FLOW_CTRL_NONE, \
}, \
}; \
static int uart_smartbond_##id##_init(const struct device *dev) \
{ \
UART_SMARTBOND_CONFIGURE(id); \
return uart_smartbond_init(dev); \
} \
DEVICE_DT_INST_DEFINE(id, \
uart_smartbond_##id##_init, \
NULL, \
&uart_smartbond_##id##_data, \
&uart_smartbond_##id##_cfg, \
PRE_KERNEL_1, CONFIG_SERIAL_INIT_PRIORITY, \
&uart_smartbond_driver_api); \
DT_INST_FOREACH_STATUS_OKAY(UART_SMARTBOND_DEVICE)
|
{
"redpajama_set_name": "RedPajamaGithub"
}
| 2,020
|
\section{Introduction}
The last two decades have seen the birth and growth of the space borne
photometry due to missions like MOST \citep{Walker2003}, CoRoT
\citep{Baglin2006a}, Kepler \citep{Borucki2010} and TESS
\citep{Ricker2014TESS}. These instruments provided for the first time
long, continuous high-quality photometric data, enabling the detection
of thousands of transiting planets, but also opened a new window on
the dynamic and evolution of stars. For example, one of the most
remarkable achievements of the Kepler space instrument is the
discovery that older low-mass stars rotate too fast compared to
theoretical expectations \citep{Saders2016}. This could only be
established by the combined analysis of the stellar photometric
variability due to spots and to the stellar pulsations \citep[see
e.g.][]{Kjeldsen1995,JCD1996Science,2010ApJ...713L.169C}. In
general, the study of the rotation-age relation
\citep{Skumanich1972ApJ,Kawaler1988ApJ,MacGregor1991ApJ} is an
important tool to evaluate the age of stars. The rotation also plays
an important role on the solar and stellar dynamo
\citep{Ossendrijver2003,Varela2016}, itself believed to be important
for sustaining a latitudinal differential rotation.
Observationally, the stellar rotation period can be estimated from a
few independent methods. First, one can combine the equatorial
rotational velocity from Doppler broadening and the stellar
radius. The spectroscopically derived rotation period, however,
depends on the assumed model for the turbulence, and also requires the
values of the stellar radius and inclination that are not
well-determined in general\citep{Kamiaka2018}. Second, the
asteroseismic analysis of the stellar pulsation can estimate the
rotation period and the stellar inclination simultaneously. The
asteroseismology, however, also required various model
assumptions in the analysis, and is applicable only to a relatively
small fraction of stars that exhibit measurable oscillations
\citep[e.g.,][]{Appourchaux2008,Huber2013b,Benomar2014b,Lund2017,
Kamiaka2018,Kamiaka2019}. Finally, the photometric variation of
lightcurves is by far the most widely used method to estimate the
stellar rotation period, and has been intensively applied for the
Kepler data \citep[e.g.,][]{McQuillan2014,Mazeh2015b,Angus2018}.
The photometric variation is induced by star-spots corotating with the
star. If those spots do not dynamically evolve on the stellar
surface, it is relatively easy to estimate the stellar rotation
period. In reality, however, the spots have individual lifetimes and
even move on the stellar surface, and stars are not necessarily rigid
rotators \citep[e.g.,][]
{Donati1997,Barnes2005,Donati2010,Roettenbacher2013,Walkowicz2013,Brun2017,
Benomar2018,Basri2020}. The formation and dissipation of spots on
the differentially rotating stars, therefore, complicate the
interpretation of the photometrically estimated rotation period
$P_{\rm photo}$. Furthermore, we cannot exclude a possibility that
that starspots and stellar pulsations may have similar time scales
in some stellar types, even if not so likely.
Properties of spots have been extensively studied in the past
literature for the Sun
\citep[e.g.,][]{Maunder1904,Zharkov2005,Mandal2021}, and also for
other stars \citep{Morris2020}. \citet{Roettenbacher2013}, for
instance, achieved a wonderful lightcurve inversion to predict the
starspot evolution on Kepler target KIC 5110407. Nevertheless, it is
not easy to accurately predict the nature of spots in general. On the
other hand, the photometric rotation periods combined with the
spectroscopic Doppler broadening have been extensively used to infer
the inclination angle of stars hosting planets
\citep{Sanchis-Ojeda2011a,Sanchis-Ojeda2011b,Hirano2012,
Louden2021,Albrecht2021}, which have profound implications for the
spin-orbit architecture of exoplanetary
systems\citep{Queloz2000,Ohta2005,SS2021}. Therefore, it is still
useful to have parameterized templates for the photometric variation
of stellar lightcurves due to non-evolving starspots. This is the
purpose of this paper. We present an analytic model of the photometric
lightcurves induced by starspots on a differentially rotating stellar
surface {\it assuming that they do not evolve during the finite
observing duration}. We compute mock lightcurves based on the
multi-spot model, and address how to interpret the measured
distribution of peaks in the Lomb-Scargle periodogram in terms of the
stellar differential rotation law.
The rest of the paper is organized as follows. We derive the
photometric variation pattern due to a single infinitesimal spot in
section 2. The resulting lightcurve modulation including the limb
darkening effect is expressed in the Fourier series expansion. In
section 3, we apply the analytic model for multispots on a
differentially rotating star. Then we generate simulated lightcurves
adopting the statistical distribution model of the Sun spots,
perform the Lomb-Scargle analysis, and examine the information content
of the resulting power spectra. Final section is devoted to summary
and conclusion of the paper. The Fourier expansion coefficients in
our analytic model are given in Appendix.
\section{Photometric variation due to a single starspot}
As illustrated in Figure \ref{fig:obs-spot}, we consider a spherical
star with radius $R_{\star}$, and parameterize a position vector on
the stellar surface in terms of its latitude $\ell$ and
longitude $\varphi$:
\begin{eqnarray}
\label{eq:rspot}
\bm{r}_{\star} =
\left( \begin{array}{c} x_{\star} \\ y_{\star} \\ z_{\star}
\end{array} \right)
= R_{\star}
\left( \begin{array}{c}
\cos \ell \cos\varphi \\
\cos \ell \sin\varphi\\
\sin \ell \end{array} \right) ,
\end{eqnarray}
where the $z$-axis is chosen to be the direction of the stellar
rotation. In what follows, we assume that the surface angular
velocity $\omega(\ell)$ at $\ell$ is given
by the following parameterized model for the
latitudinal differential rotation:
\begin{equation}
\label{eq:diff-law}
\omega (\ell) = \omega_0 (1 - \alpha_2 \sin^2\ell -
\alpha_4 \sin^4\ell).
\end{equation}
For the Sun, $\omega_{0\odot}\approx 2.972\times
10^{-6}\mathrm{~rad~s^{-1}}$, $\alpha_{2\odot} \approx 0.163$, and
$\alpha_{4\odot} \approx 0.121$ \citep{1990ApJ...351..309S}.
Thus, the angular velocity at $\ell=30^\circ$ is about 5 percent
smaller than its equatorial value.
Without loss of generality, we consider a distant observer located at
$\varphi_{\rm o}=0$ and $\ell_{\rm o}$. Thus the unit vector toward the observer is
\begin{equation}
\label{eq:eo}
{\bm e}_{\rm o}=(\cos\ell_{\rm o}, 0, \sin\ell_{\rm o}).
\end{equation}
According to equation
(\ref{eq:diff-law}), the longitude of the starspot located at the
latitude $\ell_{\rm s}$ at epoch $t$ becomes
\begin{equation}
\label{eq:varphis-t}
\varphi_{\rm s}(t) = \varphi_{\rm s0} +\omega_{\rm s} t
\end{equation}
due to the stellar surface rotation, where $\varphi_{\rm s0}$
is the longitude at which the spot is located on the stellar surface
initially ($t=0$), and $\omega_{\rm s} \equiv \omega(\ell_{\rm s})$ is
the angular velocity of the spot at $\ell_{\rm s}$ defined as equation
(\ref{eq:diff-law}).
\begin{figure}
\begin{center}
\includegraphics[width=10cm, bb = 0 0 723 451]{obs-spot.pdf}
\end{center}
\caption{Schematic illustration of the observer and a starspot. The
stellar rotation axis is chosen to be the $Z$-axis. The
location of the spot is specified by the latitude $\ell_{\rm s}$ and the
azimuth angle $\varphi_{\rm s}$, and the direction of the observer's
line-of-sight is defined by the latitude $\ell_{\rm o}$ on the $X-Z$
plane ($\varphi=0$).}
\label{fig:obs-spot}
\end{figure}
\subsection{A single infinitesimal starspot without limb darkening
\label{subsec:singlespot-wo-ld}}
A normalized lightcurve of the stellar surface is
\begin{equation}
\label{eq:lightcurve}
L(t) = \frac{\displaystyle
\int \cos\ell d\ell d\varphi ~ A(\ell, \varphi)
K(\ell, \varphi; \ell_{\rm o}, \varphi_{\rm o})}
{\displaystyle \int \cos\ell d\ell d\varphi
~ K(\ell, \varphi; \ell_{\rm o}, \varphi_{\rm o})}.
\end{equation}
where the integration is over the stellar surface, $A$ indicate the
surface intensity distribution, and $K$ is the weighting kernel of the
surface visible for the observer
\citep[e.g.,][]{Fujii2010,Fujii2011,Farr2018,Haggard2018,Nakagawa2020}.
In the case of a single infinitesimal starspot at $(\ell_{\rm s},
\varphi_{\rm s})$ on a homogeneous sphere, we set
\begin{equation}
\label{eq:intensity-spot}
A(\ell, \varphi) = 1 - \frac{b_{\rm spot}}{\cos\ell}
\delta(\ell-\ell_{\rm s}, \varphi-\varphi_{\rm s}).
\end{equation}
The dimensionless parameter $b_{\rm spot}$ represents the amplitude of the
photometric modulation \citep{Dorren1987,Haggard2018}.
Sunspots consist of the central darker part ({\it umbra}), and the
surrounding lighter part ({\it penumbra}). In addition, there is a
type of brighter spots ({\it faculae}). Note that it is not
necessary to specify separately the temperature and area of spots,
since the amplitude of the photometric variations of stars depends
on their flux (i.e., $b_{\rm spot}$) alone. As described in \S
\ref{subsec:mock-LS}, we consider the spot distribution directly
derived from the observed properties using the photometric variation
data of the Sun.
Our analytic formulation, however, is general and applicable to
various types of spots including umbra, penumbra and faculae
($b_{\rm spot}<0$), if we employ their distribution function properly.
Without loss of generality, we can define the initial phase of the
single starspot to be $\varphi_{\rm s}(t=0)=\varphi_{\rm s0}=0$. For an
isotropically emitting stellar surface, the weighting kernel $K$ is
equivalent to the visibility computed from the direction cosine $\mu
\equiv {\bm e}_{\rm o}\cdot{\bm e}_{\star}$ between the stellar
surface $\bm{r}_\star=R_{\star} \bm{e}_\star$ and the observer ${\bm e}_{\rm
o}$. The spot is visible (invisible) to the observer if $\mu>0$
($\mu<0$). Thus the weighting kernel is simply computed from
equations (\ref{eq:rspot}) and (\ref{eq:eo}) as
\begin{eqnarray}
K(\ell, \varphi; \ell_{\rm o}, \varphi_{\rm o})
&=& \max (\mu, 0 ) \cr
&=& \max ( \cos\ell_{\rm o} \cos\ell \cos\varphi+\sin\ell_{\rm o}\sin\ell, 0 )\cr
\label{eq:kernel}
&=& \cos\ell_{\rm o} \cos\ell
\max(\cos\varphi + \tan\ell_{\rm o} \tan\ell, 0).
\end{eqnarray}
Substituting equations (\ref{eq:intensity-spot}) and
(\ref{eq:kernel}) into equation (\ref{eq:lightcurve}), one obtains a
normalized lightcurve modulation due to a single starspot on an
otherwise homogeneous spherical surface:
\begin{equation}
\label{eq:Lt-spot}
L_{\rm s}(t) \equiv L(t)-1 = - \frac{b_{\rm spot} \cos\ell_{\rm o} \cos\ell_{\rm s}}{\pi}
\max(\tan\ell_{\rm o} \tan\ell_{\rm s} + \cos\varphi_{\rm s}(t), 0).
\end{equation}
For $\ell_{\rm o}=\ell_{\rm s}=\varphi_{\rm s}(t)=0$, equation (\ref{eq:Lt-spot}) reduces
to $L_{\rm s}(t) = - b_{\rm spot}/\pi$, and the denominator $\pi$ indeed
corresponds to the visible projected area of the stellar surface $\pi
R_{\star}^2$. Thus, we note that $b_{\rm spot}$ represents the {\it effective area}
of the spot $A_{\rm spot}$ in units of $R_{\star}^2$, instead of $\pi R_{\star}^2$. Since
$b_{\rm spot}$ in equation (\ref{eq:intensity-spot}) is defined with respect to
the flux, $A_{\rm spot}$ is equivalent to the geometric area of the spot $S$
only when it is completely black. In general, $A_{\rm spot}$ should be
interpreted to represent an flux-weighted area of the spot.
In what follows, we adopt a parameterized model of the
distribution of $A_{\rm spot}$ that is directly estimated from the observed
photometric variations of the Sun (see \S
\ref{subsec:mock-LS}). Then we will compute the dimensionless parameter
$b_{\rm spot} \equiv A_{\rm spot}/R_{\star}^2$. If the black-body approximation for the
stellar surface and the spot is valid, the effective and geometric
areas of the spot are related as $A_{\rm spot} \approx (1-T_{\rm
spot}^4/T_\star^4)S$ with $T_\star$ and $T_{\rm spot}$ being the
temperatures of the star and the spot.
Equation (\ref{eq:Lt-spot}) indicates that the starspot is visible at
$t$ if
\begin{eqnarray}
\label{eq:visible-criterion}
\cos\varphi_{\rm s}(t) + \tan\ell_{\rm o} \tan\ell_{\rm s} > 0 .
\end{eqnarray}
For convenience, let us introduce a parameter
\begin{eqnarray}
\label{eq:def-gamma}
\Gamma \equiv \tan\ell_{\rm o} \tan\ell_{\rm s}.
\end{eqnarray}
A starspot with $\Gamma>1$ is always visible
to the observer, and equation (\ref{eq:Lt-spot}) reduces to
\begin{equation}
\label{eq:Lt-spot-visible}
L_{\rm s}(t) = - \frac{b_{\rm spot} \cos\ell_{\rm o} \cos\ell_{\rm s}}{\pi}
\left[\cos\varphi_{\rm s}(t)+\Gamma\right].
\end{equation}
If $\Gamma < -1$, on the other hand, the starspot is totally invisible
and $L_{\rm s}(t)=0$.
A starspot with $|\Gamma| \leq 1$ becomes visible periodically as
the stellar surface rotation. In this case, equation
(\ref{eq:Lt-spot}) is expanded analytically in the Fourier series.
The result is
\begin{eqnarray}
\label{eq:Lt-spot-Fourier}
L_{\rm s}(t) = - \frac{b_{\rm spot} \cos\ell_{\rm o} \cos\ell_{\rm s}}{\pi^2}
&&\Big[
\left(\sin\theta_{\rm c}-\theta_{\rm c}\cos\theta_{\rm c}\right)
+\left(\theta_{\rm c} -\sin\theta_{\rm c}\cos\theta_{\rm c}\right)\cos\omega_{\rm s} t\cr
&& + \sum_{n=2}^\infty
\left(\frac{\sin (n-1)\theta_{\rm c}}{n(n-1)}
-\frac{\sin (n+1)\theta_{\rm c}}{n(n+1)}\right)\cos n\omega_{\rm s} t
\Big],
\end{eqnarray}
where the parameter $\theta_{\rm c}$ is defined through $\Gamma =
\tan\ell_{\rm o} \tan\ell_{\rm s} \equiv -\cos\theta_{\rm c} (0<\theta_{\rm c}<\pi)$; see
Appendix \ref{sec:FT-ld} for the derivation of equation
(\ref{eq:Lt-spot-Fourier}).
The visibility of a single starspot is determined by the
parameter $\Gamma$ or equivalently $\theta_{\rm c}$. We plot the contours
of $\Gamma$ and $\theta_{\rm c}$ on $\ell_{\rm o}$ -- $\ell_{\rm s}$ plane in the left
and right panels of Figure \ref{fig:gamma-thetac}, respectively.
For a roughly edge-on view observer ($|\ell_{\rm o}| \ll 1$), spots located
near the equatorial plane ($|\ell_{\rm s}| \ll 1$) correspond to $|\Gamma|
\approx |\ell_{\rm o}\ell_{\rm s}| \ll 1$, and $\theta_{\rm c} \approx \pi/2+\ell_{\rm o}\ell_{\rm s}$.
\begin{figure}
\begin{center}
\includegraphics[width=8cm, bb = 0 0 576 576]{gamma-lols.pdf}
\includegraphics[width=8cm, bb = 0 0 576 576]{thetac-lols.pdf}
\end{center}
\caption{Contours of $\Gamma (\equiv \tan\ell_{\rm o} \tan\ell_{\rm s})$ and
$\theta_{\rm c} (\equiv - \cos^{-1} \Gamma$ for $|\Gamma|<1)$
on $\ell_{\rm o}$ -- $\ell_{\rm s}$ plane. Red and blue solid lines in both
panels indicate contours for $0 \leq \Gamma \leq 1$ ($90^\circ \leq
\theta_{\rm c} \leq 180^\circ$) and $-1 \leq\Gamma <0$ ($0^\circ \leq
\theta_{\rm c} < 90^\circ$), for which the corresponding starspot
becomes visible periodically to the observer as the star
rotates. The orange and black lines correspond to those spots
that are always visible and invisible to the observer,
respectively.}
\label{fig:gamma-thetac}
\end{figure}
\subsection{A single infinitesimal starspot with limb darkening}
The stellar limb darkening produces an additional modulation to the
photometric variation due to the starspot. Adopting the quadratic
limb darkening law, the normalized stellar surface intensity at
$\bm{r}_\star$ is characterized by the two limb darkening parameters
$u_1$ and $u_2$ as
\begin{eqnarray}
I(\mu)=1-u_1(1-\mu)-u_2(1-\mu)^2 = (1-u_1-u_2) +(u_1+2u_2)\mu - u_2\mu^2,
\end{eqnarray}
where $\mu= {\bm e}_{\rm o}\cdot{\bm e}_{\star}$ is the direction
cosine that we defined before. We adopt the values of $u_1$ and
$u_2$ from the Sun \citep{Cox2000book}: $u_1 = 0.47$ and $u_2 =
0.23$ at $550$ nm (they become $0.42$ and $0.23$, respectively, at
$600$ nm).
Including the limb darkening effect, equation
(\ref{eq:lightcurve}) is generalized to be
\begin{equation}
\label{eq:lightcurve-ld}
L(t) = \frac{\displaystyle
\int \cos\ell d\ell d\varphi ~ I(\mu) A(\ell, \varphi)
K(\ell, \varphi; \ell_{\rm o}, \varphi_{\rm o})}
{\displaystyle \int \cos\ell d\ell d\varphi
~ I(\mu) K(\ell, \varphi; \ell_{\rm o}, \varphi_{\rm o})}.
\end{equation}
Since the denominator of equation (\ref{eq:lightcurve-ld}) is
\begin{eqnarray}
&&\int \cos\ell d\ell d\varphi
~ I(\mu) K(\ell, \varphi; \ell_{\rm o}, \varphi_{\rm o})
= \int_0^{2\pi}d\phi\int_0^1 \mu d\mu \left[1 - u_1(1-\mu)
- u_2 (1-\mu)^2\right]\cr
=&& 2\pi\left(\int_0^1\mu d\mu - u_1 \int_0^1\mu(1-\mu) d\mu
- u_2 \int_0^1\mu(1-\mu)^2 d\mu\right)\cr
=&& \pi\left(1 - \frac{u_1}{3} - \frac{u_2}{6}\right),
\end{eqnarray}
equation (\ref{eq:Lt-spot}) is now written as
\begin{equation}
\label{eq:Lt-spot-ld}
L_{\rm s}(t) = - \frac{b_{\rm spot}}{\pi}
\left(1 - \frac{u_1}{3} - \frac{u_2}{6}\right)^{-1}
\max(\mu_{\rm s}, 0) I(\mu_{\rm s}).
\end{equation}
where $\mu_{\rm s}= \cos\ell_{\rm o} \cos\ell_{\rm s}(\cos\omega_{\rm s} t + \Gamma)$.
Similarly to the previous subsection, equation (\ref{eq:Lt-spot-ld})
for $|\Gamma|\le 1$ is expanded analytically in the Fourier series.
The derivation is explicitly given in Appendix \ref{sec:FT-ld},
and the normalized lightcurve modulation including the limb
darkening effect is summarized in the following expression:
\begin{eqnarray}
\label{eq:FTLt-spot-ld}
L_{\rm s}(t) = - \frac{b_{\rm spot}}{\pi}
\left(1 - \frac{u_1}{3} - \frac{u_2}{6}\right)^{-1}
\times \left\{
\frac{A_0}{2} + \sum_{n=1}^{\infty} A_n \cos n \omega_{\rm s} t \right\},
\end{eqnarray}
where
\begin{equation}
\label{eq:A-Gamma<1}
A_n \equiv (1 - u_1 - u_2) (\cos \ell_{\rm o} \cos \ell_{\rm s}) a_n + (u_1 + 2 u_2)
(\cos \ell_{\rm o} \cos \ell_{\rm s})^2 b_n - u_2 (\cos \ell_{\rm o} \cos \ell_{\rm s})^3 c_n,
\end{equation}
and the coefficients $a_n$, $b_n$, and $c_n$ are explicitly given in
Appendix \ref{sec:FT-ld}.
For spots with $\Gamma>1$, $\mu_{s}$ is always positive, and the
corresponding lightcurve is written in the same form as equation
(\ref{eq:Lt-spot-ld}) by replacing $A_n$ by $\tilde{A}_n$, which are
given in Appendix \ref{sec:FT-ld-2}.
Figures \ref{fig:ello0}, \ref{fig:ello30}, and \ref{fig:ello60} show
the trajectories of a single spot on a rotation stellar surface and
the corresponding normalized lightcurves $L_s(t)/b_{\rm spot}$ against
$t/P_{\rm spin}(\ell_{\rm s})$ for an observer located at $\ell_{\rm o}=0^\circ$,
$30^\circ$, and $60^\circ$, respectively. Black, red, blue and orange
curves indicate the results for the spot at the latitude of
$\ell_{\rm s}=0^\circ$, $15^\circ$, $45^\circ$, and $75^\circ$. Solid and
dashed lines in the right panels indicate the lightcurves with and
without limb-darkening (LD).
The right panel of Figure \ref{fig:ello0} shows that the modulation
amplitude $|L_s(t)/b_{\rm spot}|$ without the limb darkening effect becomes
$1/\pi$ for $\ell_{\rm o}=\ell_{\rm s}=0^\circ$ (black-dashed curve) at
$\varphi_{\rm s}(t)=0$. Limb darkening decreases the effective visible
area of the entire surface by a factor of $(1 - u_1/3 -
u_2/6)^{-1}$, while that of the starspot by a factor of $I(\mu_{\rm
s})$. Depending on the location of the spot, $\ell_{\rm o}$,
$\ell_{\rm s}$, and $\varphi_{\rm s}(t)$, the resulting $|L_s(t)/b_{\rm spot}|$ with limb
darkening becomes either smaller or larger than that without limb
darkening; see Figures \ref{fig:ello0}, \ref{fig:ello30} and
\ref{fig:ello60}.
\begin{figure}
\begin{center}
\includegraphics[width=6cm, bb = 0 0 432 432]{00.pdf}
\includegraphics[width=9cm, bb = 0 0 670 531]{ello0.pdf}
\end{center}
\caption{Photometric modulation due to a single spot at $\ell_{\rm s}$
viewed from the observer at $\ell_{\rm o}=0^\circ$. {\it left:}
trajectories of four spots at $\ell_{\rm s}=0^\circ$ (black), $15^\circ$
(red), $45^\circ$ (blue), and $75^\circ$ (orange) on the stellar
surface. {\it right:} modulation curves $L_s/b_{\rm spot}$ over the one
rotation period of each spot $P_{\rm spin}(\ell_{\rm s})$ in the left
panel. Solid and dashed lines correspond to those with and
without limb darkening (LD) for the differential rotation
parameters of $\alpha_2=\alpha_{2\odot}$ and
$\alpha_4=\alpha_{4\odot}$; see equation (\ref{eq:diff-law}).}
\label{fig:ello0}
\end{figure}
\begin{figure}
\begin{center}
\includegraphics[width=6cm, bb = 0 0 432 432]{30.pdf}
\includegraphics[width=9cm, bb = 0 0 670 531]{ello30.pdf}
\end{center}
\caption{Same as Figure \ref{fig:ello0} but viewed from the observer
at $\ell_{\rm o}=30^\circ$.}
\label{fig:ello30}
\end{figure}
\begin{figure}
\begin{center}
\includegraphics[width=6cm, bb = 0 0 432 432]{60.pdf}
\includegraphics[width=9cm, bb = 0 0 670 531]{ello60.pdf}
\end{center}
\caption{Same as Figure \ref{fig:ello0} but viewed from the observer
at $\ell_{\rm o}=60^\circ$.}
\label{fig:ello60}
\end{figure}
Figure \ref{fig:a234} plots the ratios of Fourier coefficients of the
single spot modulation, $A_n/A_1$ ($n=2, 3, 4$). If limb darkening is
neglected, they reduce to $a_n/a_1$ that are a function of $\Gamma
\equiv \tan\ell_{\rm o}\tan\ell_{\rm s}$ (or $\theta_{\rm c}$) alone, which are
plotted in dotted lines. When the limb darkening effect is taken
into account, $A_n/A_1$ depends on both $\ell_{\rm o}$ and $\ell_{\rm s}$. As
Figure \ref{fig:a234} implies, however, difference among the three
curves for $\ell_{\rm o}=10^\circ$, $30^\circ$, and $60^\circ$ is
small. Thus, $A_n/A_1$ is still largely determined by the value of
$\Gamma$ (or $\theta_{\rm c}$) even with limb darkening.
This result suggests that $A_n/A_1$ may be used to
examine if the periodic signals detected from the observed
photometric lightcurve are due to starspots, instead of other
sources. It may be even possible to put a constraint on $\Gamma$
from $A_n/A_1$ in principle. Since $\ell_{\rm o}$ is equivalent to the
stellar inclination for the observer that can be independently
measured from either spectroscopy or asteroseismology
\citep[e.g.,][]{Kamiaka2018,Kamiaka2019,SS2021}, the constraint on
$\Gamma$ is translated to that on the spot latitude $\ell_{\rm s}$. In
reality, it is feasible to derive a robust constraint on $\Gamma$
only for a single spot case. The statistical distribution of
$A_n/A_1$ for multi-spots is more useful to constrain the
differential rotation as discussed below.
\begin{figure}
\begin{center}
\includegraphics[width=8cm, bb = 0 0 576 576]{a234-thetac.pdf}
\includegraphics[width=8cm, bb = 0 0 576 576]{a234-gamma.pdf}
\end{center}
\caption{Ratios of Fourier coefficients of the photometric
modulation due to a single spot, $A_n/A_1$, plotted against
$\theta_{\rm c}$ ({\it left}) and $\Gamma$ ({\it right}). Red, blue,
and orange lines indicate $A_2/A_1$, $A_3/A_1$, and $A_4/A_1$,
respectively. We assume the limb darkening parameters of
$\alpha_2=\alpha_{2\odot}$ and $\alpha_4=\alpha_{4\odot}$, and
plot those ratios in solid ($\ell_{\rm o}=10^\circ$), dashed
($\ell_{\rm o}=30^\circ$), and dot-dashed ($\ell_{\rm o}=60^\circ$) lines. For
reference, the results without limb darkening (w/o LD) are
plotted in dotted lines.}
\label{fig:a234}
\end{figure}
\section{Multiple starspots: model predictions and mock data analysis}
\subsection{Superposition of multiple starspots}
If more than one starspots are involved, we have to take into account
their relative phases, namely $\varphi_{\rm s0}$ in equation
(\ref{eq:varphis-t}), as well. In that case, equation
(\ref{eq:FTLt-spot-ld}) can be generalized to
\begin{eqnarray}
\label{eq:FTLt-spot-ld-2}
&& L_{\rm s}(t) = - \frac{b_{\rm spot}}{\pi} \left(1 - \frac{u_1}{3} -
\frac{u_2}{6}\right)^{-1} \Big[ \frac{A_0}{2} +
\sum_{n=1}^{\infty} A_n \cos n (\omega_{\rm s} t+\varphi_{\rm s0}) \Big] \cr
&& \quad= - \frac{b_{\rm spot}}{\pi} \left(1 - \frac{u_1}{3} -
\frac{u_2}{6}\right)^{-1} \Big[ \frac{A_0}{2} +
\sum_{n=1}^{\infty} A_n \left(\cos n\varphi_{\rm s0} \cos n \omega_{\rm s} t
-\sin n\varphi_{\rm s0} \sin n \omega_{\rm s} t
\right) \Big] .
\end{eqnarray}
Thus, the lightcurve due to multispots becomes the superposition of
the following form:
\begin{eqnarray}
\label{eq:FTLt-multispot-ld}
L_{\rm s}(t) && = - \frac{1}{\pi} \left(1 - \frac{u_1}{3}
-\frac{u_2}{6}\right)^{-1}\cr
&& \times \Big\{\sum_{i=1}^{N_s} b_{\rm spot,i}
\Big[ \frac{A_{0,i}}{2} +
\sum_{n=1}^{\infty} A_{n,i} \left(\cos n\varphi_{\rm s0,i}
\cos n \omega_{{\rm s},i} t
-\sin n\varphi_{\rm s0,i} \sin n \omega_{{\rm s},i} t
\right) \Big] \cr
&& ~ + \sum_{i=N_s+1}^{N_s+\tilde{N_s}} b_{\rm spot,i}
\Big[ \frac{\tilde{A}_{0,i}}{2} +
\sum_{n=1}^{3} \tilde{A}_{n,i} \left(\cos n\varphi_{\rm s0,i}
\cos n \omega_{{\rm s},i} t
-\sin n\varphi_{\rm s0,i} \sin n \omega_{{\rm s},i} t
\right) \Big] \Big\} .
\end{eqnarray}
In the above equation, $N_s$ and $\tilde{N}_s$ denote the number of
spots with $|\Gamma|<1$ and $\Gamma>1$, respectively, $b_{\rm spot,i}$ is the
amplitude of the photometric variation,
$\varphi_{\rm s0,i}$ is the initial phase, $\omega_{{\rm s},i}=\omega(\ell_{{\rm
s},i})$ is the angular frequency, and $A_{n,i}$ and
$\tilde{A}_{n,i}$ are the Fourier components, of the $i$-th starspot.
\subsection{Mock lightcurves and the Lomb-Scargle power spectra
\label{subsec:mock-LS}}
In order to examine to what extent one can extract the characteristic
signature of starspots from photometric stellar lightcurves, we create
mock lightcurves in the time domain, and compute the Lomb-Scargle
power spectra. Our fiducial set of parameters is listed in Table
\ref{tab:parameter}.
The key parameter characterizing the spot in our model
is $b_{\rm spot}$. As described in subsection \ref{subsec:singlespot-wo-ld},
$b_{\rm spot}$ is defined as $A_{\rm spot}/R_{\star}^2$ in our model.
\citet{2015ApJ...800...48M} found that the flux-weighted effective
area, $A_{\rm spot}$, for the Solar spot empirically obeys the
Weibull distribution:
\begin{equation}
\label{eq:weibull}
f(A_{\rm spot}; k, \lambda) dA_{\rm spot} = k \left( \frac{A_{\rm spot}}{\lambda} \right)^{k-1}
e^{-(A_{\rm spot}/\lambda)^k} \frac{dA_{\rm spot}}{\lambda}
\end{equation}
from the observed photometric variation over years. The
Weibull distribution is written in terms of $A_{\rm spot}/\lambda$, the
amplitude of the resulting spot modulation is simply scaled to the
adopted value of $\lambda$.
The expectation value of $A_{\rm spot}$ from equation (\ref{eq:weibull}) is
\begin{equation}
\label{eq:mean-weibull}
\langle A_{\rm spot} \rangle = \int_{A_{\rm th}}^\infty A_{\rm spot} f(A_{\rm spot}; k, \lambda) dA_{\rm spot}
= \lambda \Gamma(1+\frac{1}{k}),
\end{equation}
where $\Gamma(x)$ denotes the Gamma function and
$\Gamma(1+\frac{1}{k}) \approx 1.75$ for the solar value of $k=0.54$.
Also the corresponding cumulative number distribution of $A_{\rm spot}$
exceeding the threshold value $A_{\rm th}$ is
\begin{equation}
\label{eq:cumulative-weibull}
P(A_{\rm spot}>A_{\rm th}) = \int_{A_{\rm th}}^\infty f(A_{\rm spot}; k, \lambda) dA_{\rm spot}
=e^{-(A_{\rm th}/\lambda)^k}.
\end{equation}
For instance, the top 10 percentile of spots have
$A_{\rm spot}> 1.47\lambda$.
The best-fit values of the two parameters, $k$ and $\lambda$, vary for
different definitions of spots and different datasets
\citep{2015ApJ...800...48M}. For definiteness, we adopt ``Sunspot
Umbral Area'' from the Helio-seismic and Magnetic Imager on the Solar
Dynamics Observatory (see their Table 1 ), and adopt $k=0.54$ and
$\lambda = 2.88 ~{\rm\mu Hem}=2.88\times10^{-6} (2\pi R_{\star}^2)$. It is
likely that different stars may have different values of $k$ and
$\lambda$. Since our model is fully analytical, however, it is
readily applicable for other choices. Thus, we fix their values below,
and generate mock data for multi-spots.
Equation (\ref{eq:mean-weibull}) suggests that a characteristic
amplitude of the dimensionless parameter $b_{\rm spot}$ of our spots is
\begin{equation}
\label{eq:bspot-lambda}
\langleb_{\rm spot}\rangle \equiv \frac{\langleA_{\rm spot}\rangle}{R_{\star}^2}
\approx \Gamma(1+1/k) \frac{\lambda}{R_{\star}^2} =
2\pi \left(\frac{\lambda}{1 ~{\rm\mu Hem}}\right)\Gamma(1+1/k)~
{\rm ppm},
\end{equation}
where the factor of $2\pi$ comes from the fact that $\lambda$ is
given relative to the area of hemisphere (Hem), $2\pi R_{\star}^2$. Thus
we can safely neglect the finite size effect of an individual spot,
which is consistent with the assumptions of our analytic model, for
our adopted value of $\lambda =2.88~{\rm\mu Hem}$).
In order to understand the meaning of equation (\ref{eq:bspot-lambda}),
let us define the effective radius of the spot $r_{\rm spot}$ through
\begin{equation}
\label{eq:def-rspot}
A_{\rm spot} = \pi r_{\rm spot}^2.
\end{equation}
Substituting equation (\ref{eq:bspot-lambda}) into equation
(\ref{eq:def-rspot}), one obtains
\begin{equation}
\label{eq:rspot-Rs}
\frac{r_{\rm spot}}{R_{\star}} = \sqrt{\frac{b_{\rm spot}}{\pi}}
\approx 1.8\times 10^{-3}\sqrt{\frac{b_{\rm spot}}{10~{\rm ppm}}}
\approx 0.1^\circ \sqrt{\frac{b_{\rm spot}}{10~{\rm ppm}}},
\end{equation}
or equivalently
\begin{equation}
\label{eq:rspot-size}
r_{\rm spot} \approx 0.19R_\oplus \sqrt{\frac{b_{\rm spot}}{10~{\rm ppm}}}
\left(\frac{R_{\star}}{R_\odot}\right) .
\end{equation}
Equations (\ref{eq:rspot-Rs}) and (\ref{eq:rspot-size}) correspond to
the angular and real size corresponding to $r_{\rm spot}$ in terms of $b_{\rm spot}$.
We generate $N_{\rm tot}$ spots with $b_{\rm spot}$ following the Weibull
distribution, equation (\ref{eq:weibull}). We adopted $N_{\rm
tot}=30$ for definiteness so as to roughly reproduce the Solar
spots. The corresponding fraction of spots over the entire stellar
surface may be computed from equation (\ref{eq:mean-weibull}):
\begin{equation}
\label{eq:area-fraction}
F(k,\lambda) = N_{\rm tot} \frac{\langle A_{\rm spot} \rangle}{4\pi R_{\star}^2}
= 7.6\times10^{-5} \left(\frac{N_{\rm tot}}{30}\right)
\left(\frac{\lambda}{2.88~\mu{\rm Hem}}\right)
\left(\frac{\Gamma(1+1/k)}{1.75}\right).
\end{equation}
The value of $N_{\rm tot}$ is sensitive to the threshold value $A_{\rm
th}$ in identifying a single spot even for the Sun, and moreover is
not clear for other stars. Our analytic formulation can be applied to
a different choice of $N_{\rm tot}$ in a straightforward manner.
The latitudes of spots $\ell_{\rm s}$ are drawn from the isotropic
distribution function ($\propto |\sin\ell_{\rm s}|$) but over the restricted
range of $-\ell_{\rm s,max}<\ell_{\rm s}<\ell_{\rm s,max}$. We choose
$\ell_{\rm s,max}=30^\circ$ as our fiducial value, but consider
$75^\circ$ as well to examine its impact. The initial phases are
selected randomly for $0<\varphi_{\rm s}<2\pi$.
For a given value of the observer's latitude $\ell_{\rm o}$, we classify
each spot according to $|\Gamma|<1$ and $\Gamma>1$, and compute the
number of such spots $N_{\rm s}$ and $\tilde{N}_{\rm s} (=N_{\rm
tot}-N_{\rm s})$, respectively. Then the lightcurve modulation
due to those spots is computed from equation
(\ref{eq:FTLt-multispot-ld}).
We generate the mock lightcurves with cadence $T_{\rm samp}$ over the
duration of $T_{\rm obs}$. We set the fiducial values as $T_{\rm
samp}=30$ mins and $T_{\rm obs}=90$ days, following the long cadence
observation for one single quarter of the Kepler dataset.
Finally, we add the Gaussian noise to the lightcurves:
\begin{equation}
f(x)= \frac{1}{\sqrt{2\pi\sigma_{\rm n}^2}}
\exp\left(-\frac{x^2}{2\sigma_{\rm n}^2}\right).
\end{equation}
In what follows, we consider two cases, $\sigma_{\rm n}=0$ (noiseless)
and $\sigma_{\rm n}= 35 {\rm ppm}$ as a typical value for
the Kepler data \citep[c.f.,][]{Walkowicz2013,Basri2020}, for
simplicity. Equation (\ref{eq:weibull}) implies that the flux
modulation induced by a single spot is typically much smaller than
the noise:
\begin{equation}
\label{eq:typical-amplitude}
\frac{\langleb_{\rm spot}\rangle}{\pi} = 10
\left(\frac{1.75 \lambda}{5.04 ~{\rm\mu Hem}}\right) {\rm ppm}.
\end{equation}
Thus, in the case of $\sigma_{\rm n}=35$ ppm, the clear periodic
signal is visible only for a relatively big spot ($A_{\rm spot}>5\lambda$,
roughly corresponds to the top 10 percentile) or a clustered group of
nearby spots.
\begin{table}
\tbl{Fiducial parameters for mock lightcurves}{%
\begin{tabular}{llll}
symbol & range & note \\
\hline
$b_{\rm spot}$ & $k=0.54$, $\lambda=2.88\mu$Hem
& the Weibull distribution \\
$N_{\rm tot}$ & 30& total number of generated starspots\\
$\varphi_s$ & $[0, 2\pi]$ & uniform \\
$\ell_{\rm s}$ & $|\ell_{\rm s}|<\ell_{\rm s,max}=30^\circ$
& $P(\ell_{\rm s}) \propto |\sin\ell_{\rm s}|$ \\
$\ell_{\rm o}$ & $0^\circ$
& stellar inclination relative to the observer's line-of-sight \\
$\alpha_2$ & $\alpha_{2\odot}=0.163$ & differential rotation coefficient \\
$\alpha_4$ & $\alpha_{4\odot}=0.121$ & differential rotation coefficient \\
$u_1$ & 0.47 & linear limb-darkening parameter \\
$u_2$ & 0.23 & quadratic limb-darkening parameter \\
$2\pi/\omega_0$ & 10 days & equatorial rotation period \\
$T_{\rm samp}$ & 30 mins & cadence of the observation \\
$T_{\rm obs}$ & 90 days & duration of the observation \\
\end{tabular}}\label{tab:parameter}
\begin{tabnote}
\end{tabnote}
\end{table}
Figure \ref{fig:mock-lo0-Ntot30} shows the mock data for two different
sets of realizations of starspots ($N_{\rm tot}=30$ $\ell_{\rm
s,max}=30^\circ$) for an observer at $\ell_{\rm o}=0^\circ$. The
fractional area covered by spots varies from 0.3 to 1.3 times the
expectation value of equation (\ref{eq:area-fraction}).
We search for periodic signals of an angular
frequency $\omega$:
\begin{eqnarray}
\label{eq:LS-nterms}
S(t) = S_0 +
\sum_{n=1}^{nterms} (S_{2n-1} \sin n \omega t + S_{2n} \cos n \omega t)
\end{eqnarray}
embedded in the mock lightcurves ({\it center panels}) using the
Lomb-Scargle (LS) method \citep{Lomb1976,Scargle1982}. The
conventional LS adopts $nterms=1$, and we compute the normalized
power spectrum:
\begin{eqnarray}
\label{eq:LSpower}
P(\omega) = 1- \frac{\chi^2(\omega)}{\chi_{\rm ref}^2},
\end{eqnarray}
where $\chi^2(\omega)$ is the residuals of the fit with $\chi_{\rm
ref}^2$ being the reference value for a constant model.
Left panels of Figure \ref{fig:mock-lo0-Ntot30} plot the spot
distribution at $t=0$ for two different realizations. The area of
each circle is plotted in proportion to $b_{\rm spot}$, and approximately
represents the true ratio of the spot area and the entire stellar
surface (but neglecting the distortion due to the projection onto
the plane). Since those spots span a range of latitudes, the
resulting lightcurves (center panels) are not exactly periodic in the
time domain due to the latitudinal surface differential rotation.
Right panels of Figure \ref{fig:mock-lo0-Ntot30} plot the
corresponding LS power. The highest peaks around 10 days are located
over a range of rotation periods spanning $P_{\rm
spin}(\ell_{\rm s}=0^\circ)$ and $P_{\rm spin}(\ell_{\rm s}=\ell_{\rm s,max})$ due
to the differential rotation. The secondary peaks around 5 days are
the second harmonics. The ratios of those amplitudes carry important
information on $\ell_{\rm s}$ and $\ell_{\rm o}$, and will be discussed later
(subsection \ref{subsec:harmonics}) using the LS analysis with
$nterms=2$.
Our mock data completely neglect the dynamics of spots (their creation
and dissipation, and motion on the stellar surface) over the duration
of the observation $T_{\rm obs}=90$days, that corresponds to the
duration of a single quarter of the Kepler dataset. In order to
empirically evaluate the effects of the spot dynamic, we create 10
totally independent realizations drawn from the statistically same
spot distribution, and compute each LS power and the average over the
10 realizations. The latter may be interpreted as the average LS power
of the entire Kepler observing period, that is made of up to 10
quarters.
The results are shown in Figure \ref{fig:LSpower-lo0-Ntot30}.
Possible signatures of differential rotation may be found in the
variance among the LS power spectra for different quarters. The width
of a peak with a detected period is determined by the entire duration
of the observation $T_{\rm obs}$, instead of the cadence $T_{\rm
samp}$ in the present examples. For instance, one can resolve
the periods for different spots only if they are static over $T_{\rm
obs}=900$ days, but cannot for $T_{\rm obs}=90$ days. While the
non-evolving spots over $T_{\rm obs}=900$ days may not be so
realistic in general, a small fraction of stars may have such
spots. Therefore our study suggests that it is worthwhile to attempt
searching for such signatures in the Kepler archive data.
\begin{figure}
\begin{center}
\includegraphics[width=16cm, bb = 0 0 829 550]{mock-lo0-Ntot30.pdf}
\end{center}
\caption{Lightcurve modulations due to starspots ($N_{\rm tot}=30$,
$\ell_{\rm s,max}=30^\circ$) for an observer at $\ell_{\rm o}=0^\circ$.
Left, center, and right panels show the spot distribution on the
stellar surface, normalized lightcurve, and the corresponding
Lomb-Scargle power spectrum. Results for $\sigma_n=0$ and $35$ppm
are plotted in blue and red, respectively. Upper and lower panels
are different realizations of the statistically same model. The
vertical dotted lines indicate the range of the differentially
rotation periods; $P_{\rm spin}(\ell_{\rm s}=0^\circ)/2$, $P_{\rm
spin}(\ell_{\rm s}=\ell_{\rm s,max})/2$, $P_{\rm spin}(\ell_{\rm s}=0^\circ)$,
and $P_{\rm spin}(\ell_{\rm s}=\ell_{\rm s,max})$.}
\label{fig:mock-lo0-Ntot30}
\end{figure}
\begin{figure}
\begin{center}
\includegraphics[width=12cm, bb = 0 0 825 834]{LSpower-lo0-Ntot30.pdf}
\end{center}
\caption{LS power spectra for different realizations corresponding to
Figure \ref{fig:mock-lo0-Ntot30}. Upper and lower panels are for
$\sigma_n=0$ and $35$ppm, and left and right panels are without
and with differential rotation. Thin curves (10 in total) indicate
the results for different realizations of the spots for $T_{\rm
samp}=30$ mins and $T_{\rm obs}=90$ days, with thick red lines
being their average. Thick blue lines show the LS power for one
realization but observed for $T_{\rm obs}=900$days. }
\label{fig:LSpower-lo0-Ntot30}
\end{figure}
Figures \ref{fig:mock-lo45-Ntot30} and \ref{fig:LSpower-lo45-Ntot30}
show the same plots as Figures \ref{fig:mock-lo0-Ntot30} and
\ref{fig:LSpower-lo0-Ntot30}, but for a wider distribution of spots
($\ell_{\rm s,max}=75^\circ$) observed from an observer located far
outside the stellar equatorial plane ($\ell_{\rm o}=45^\circ$). As expected,
the effect of differential rotation is more visible than that for
$\ell_{\rm s,max}=30^\circ$ and $\ell_{\rm o}=0^\circ$.
The visible periodicity of the lightcurve modulation in the center
panels of Figures \ref{fig:mock-lo0-Ntot30} and
\ref{fig:mock-lo45-Ntot30} seems to be generated by a relatively
small number of large spots. To clarify this point, we repeated the
analysis by dividing the 30 spots in the two realizations of Figures
\ref{fig:mock-lo0-Ntot30} and \ref{fig:mock-lo45-Ntot30} separately
into two groups; the top 10 spots and the remaining 20 spots.
The resulting plots are shown in Figures \ref{fig:mock-lo0-10-20}
and \ref{fig:mock-lo45-10-20}. While those small spots still show
periodic signals in the {\it noiseless} lightcurve, they are
substantially buried in the case of our adopted noise of
$\sigma_n=35$ppm. In other words, the peaks in the LS power spectra
are dominated by a small fraction of spots, and should represent
mostly their properties (size, latitude, and rotation velocity), as
long as the Weibull distribution is a good approximation for the spot
distribution for stars other than the Sun. The above result also
implies that our basic conclusion is not so sensitive to the choice
of $N_{\rm tot}$; see Figure \ref{fig:mock-lo0-Ntot10}.
\begin{figure}
\begin{center}
\includegraphics[width=16cm, bb = 0 0 829 549]{mock-lo45-Ntot30.pdf}
\end{center}
\caption{Same as Figure \ref{fig:mock-lo0-Ntot30}, but for
$\ell_{\rm s,max}=75^\circ$ and $\ell_{\rm o}=45^\circ$. }
\label{fig:mock-lo45-Ntot30}
\end{figure}
\begin{figure}
\begin{center}
\includegraphics[width=10cm, bb = 0 0 825 834]{LSpower-lo45-Ntot30.pdf}
\end{center}
\caption{LS power spectra for different realizations corresponding to
Figure \ref{fig:mock-lo45-Ntot30}.}
\label{fig:LSpower-lo45-Ntot30}
\end{figure}
\begin{figure}
\begin{center}
\includegraphics[width=16cm, bb = 0 0 829 549]{mock-lo0-10.pdf}
\includegraphics[width=16cm, bb = 0 0 829 549]{mock-lo0-20.pdf}
\end{center}
\caption{Same as Figure \ref{fig:mock-lo0-Ntot30},
but computed for the top 10 spots ({\it upper two panels})
and the remaining 20 spots ({\it lower two panels}).}
\label{fig:mock-lo0-10-20}
\end{figure}
\begin{figure}
\begin{center}
\includegraphics[width=16cm, bb = 0 0 829 549]{mock-lo45-10.pdf}
\includegraphics[width=16cm, bb = 0 0 829 549]{mock-lo45-20.pdf}
\end{center}
\caption{Same as Figure \ref{fig:mock-lo45-Ntot30},
but computed for the top 10 spots ({\it upper two panels})
and the remaining 20 spots ({\it lower two panels}).}
\label{fig:mock-lo45-10-20}
\end{figure}
\begin{figure}
\begin{center}
\includegraphics[width=16cm, bb = 0 0 829 549]{mock-lo0-Ntot10.pdf}
\end{center}
\caption{Same as Figure \ref{fig:mock-lo0-Ntot30}, but for
$N_{\rm tot}=10$. }
\label{fig:mock-lo0-Ntot10}
\end{figure}
\subsection{Extracting the spot signature from the amplitude
ratios of harmonics
\label{subsec:harmonics}}
We have shown that a single spot leaves a distinctive modulation
pattern in the Fourier coefficients of the harmonics (Figure
\ref{fig:a234}). In principle, the signature is important to
distinguish between the true and false rotation periods from
photometry. Nevertheless, it may be weakened for more realistic cases
of multispots, in particular under the presence of the stellar
differential rotation. We consider this question in detail using mock
data analysis.
Consider a single spot case. Figure \ref{fig:harmonic-ratio-single}
compares the theoretical model predictions $(A_2/A_1)^2$ (solid lines)
for the stellar rotation period against the measurement from the mock
data for a single spot located at a given $\ell_{\rm s}$ viewed from a
line-of-sight direction of $\ell_{\rm o}$. We choose the value of $b_{\rm spot}
\approx 85$ ppm so that it corresponds to the top 10 percent of the
whole spot distribution, {\it i.e.,} $A_{\rm spot}=4.7\lambda$ from equations
(\ref{eq:bspot-lambda}) and (\ref{eq:cumulative-weibull}).
Incidentally, the theoretical curve is invariant with respect to the
transformation of $(\ell_{\rm o}, \ell_{\rm s}) \leftrightarrow (\ell_{\rm s}, \ell_{\rm o})$ as
equation (\ref{eq:Lt-spot-visible}) indicates.
We adopt two different estimators. One is based on the standard
Fourier power spectrum, and plots the corresponding amplitude ratio
$P_2/P_1$ (left panel). The other is based on the LS analysis. In
this case, we first identify the best-fit angular frequency
$\omega_{\rm fit}$ from the LS power spectra using $nterms=1$ in
equation (\ref{eq:LS-nterms}). Then we fit the data to equation
(\ref{eq:LS-nterms}) with $nterms=2$ by setting $\omega=\omega_{\rm
fit}$, and obtain the Fourier coefficients $S_1$, $S_2$, $S_3$, and
$S_4$ simultaneously. The symbols in the right panel plot the ratio
$(S_3^2+S_4^2)/(S_1^2+S_2^2)$.
In the noiseless case, the Fourier power spectrum recovers the
theoretical predictions very well, but the LS analysis seems to
slightly but systematically underestimate the theoretical values.
We do not understand why, but the fit to equation
(\ref{eq:LS-nterms}) with $nterms=2$ might be too restrictive and
thus very sensitive to the best-fit value of $\omega_{\rm fit}$
estimated from that with $nterms=1$.
In any case, the ratios estimated for data with $\sigma_{\rm n}=35$
ppm are not so accurate especially when the latitude of the spot is
significantly different from the observer's line-of-sight (with
different signs of $\ell_{\rm s}$ and $\ell_{\rm o}$, for instance). Therefore,
Figure \ref{fig:harmonic-ratio-single} implies that it is possible to
constrain $\ell_{\rm s}$ and $\ell_{\rm o}$ from the harmonic amplitude ratio for a
single spot at least for $\sigma_{\rm n}=35$ ppm. For multi-spots
cases, however, we find that the amplitude ratio varies significantly
due to the differential rotation. Thus this methodology seems to be
useful to constrain the spot parameter only when the photometric
signal is dominated by a single prominent region.
\begin{figure}
\begin{center}
\includegraphics[width=7cm, bb = 0 0 536 536]{harmonic-ratio-FFT.pdf}
\includegraphics[width=7cm, bb = 0 0 536 536]{harmonic-ratio-LS.pdf}
\end{center}
\caption{Ratios of the second to fundamental Fourier coefficients for
a single spot. Theoretical predictions $(A_2/A_1)^2$ for
$\ell_{\rm s}=-45^\circ$ (green yellow), $-30^\circ$ (purple), $0^\circ$
(green), $30^\circ$ (gray), and $45^\circ$ (orange) are plotted
against $\ell_{\rm o}$ in solid ($\ell_{\rm s} \ell_{\rm o}\ge 0$) and dashed ($\ell_{\rm s}
\ell_{\rm o}<0$) lines. Open circles and crosses indicate the ratios
estimated from the mock lightcurves for $b_{\rm spot}\approx 85$ ppm with
$\sigma=0$, and $35$ ppm, respectively. Left and right panels plot
the result based on the Fourier power spectrum and the LS analysis
with $nterms=2$.}
\label{fig:harmonic-ratio-single}
\end{figure}
\subsection{Photometric rotation period for differentially rotating stars
\label{subsec:Prot}}
Time-dependent distribution of multi-spots over a stellar
surface leads to complex photometric modulation signals. Combined
with the effect of latitudinal differential rotation, the peak of
the rotation period would vary at different observing epochs. In
turn, the variation of the rotation period among different quarters
may constrain the degree of the differential rotation.
In order to examine to what extent such signatures are indeed
detectable from the Kepler data, we perform the LS analysis for
seven different sets of mock data and plot distribution of the peak
rotation period $P_{\rm rot, LS}$ in Figure
\ref{fig:Prot-distribution}. Basically we adopt the fiducial values
for parameters in Table \ref{tab:parameter}; the equatorial rotation
period of 10 days, $N_{\rm tot}=30$ spots following the Weibull
distribution with $k=0.54$ and $\lambda=2.88 \mu$Hem, the cadence of
$T_{\rm samp}=30$ mins over an observing period of $T_{\rm obs}=90$
days corresponding to one quarter of the Kepler long-cadence data.
The latitude of the observer's line-of-sight $\ell_{\rm o}$ and the range of the
spot latitude $\ell_{\rm s,max}$ are indicated in each panel.
The left panel of Figure \ref{fig:Prot-distribution} shows the
histograms of the identified rotation period $P_{\rm rot, LS}$,
while the right panel plots histograms of the corresponding
harmonic amplitude ratio $(A_2/A_1)^2$. Each histogram for seven
models is computed from 300 realizations. The first three panels
(a), (b) and (c) assume the differential rotation law and spot
pattern similar to the Sun. They use the same 300 realizations of
the spot pattern over $ -30^\circ \leq \ell_{\rm s} \leq 30^\circ$, but
viewed from $\ell_{\rm o}=0^\circ$, $45^\circ$, and $75^\circ$,
respectively. Similarly, panels (d) and (e) share the same set of
300 realizations with $-75^\circ \leq \ell_{\rm s} \leq 75^\circ$ but
viewed from $\ell_{\rm o}=0^\circ$ and $45^\circ$, respectively.
According to equation (\ref{eq:diff-law}), the rotation period of
the surface is longer than its equatorial value (10 days), and the
width of the distribution reflects the observed range of the spot
latitudes $\ell_{\rm s}$ and the values of $\alpha_2$ and $\alpha_4$.
Difference among panels (a), (b), and (c) is simply due to the fact
that the observer at higher $\ell_{\rm o}$ preferentially sees the spots
located at higher $\ell_{\rm s}$ as clearly illustrated in Figures
\ref{fig:ello0}, \ref{fig:ello30}, and \ref{fig:ello60}. Since the
rotation periods estimated by observers at high $\ell_{\rm o}$ should be
dominated by a small number of big spots around $\ell_{\rm s} > 0^\circ$,
their distribution is shifted towards the larger $P_{\rm rot, LS}$
due to the differential rotation, and the corresponding amplitude
ratio becomes smaller as qualitatively expected from Figure
\ref{fig:harmonic-ratio-single}. A fraction of spot patterns may
exhibit an approximate symmetry between $\varphi_{\rm s}$ and
$\varphi_{\rm s}+\pi$ by chance, which would be interpreted as $P_{\rm rot,
LS}=5$ days. Such symmetric patterns are more likely to be visible
from the edge-on view ($\ell_{\rm o}=0^\circ$), which explains the fraction
of the second peak around $P_{\rm rot, LS}=5$ days in panels (a),
(b) and (c).
The next two panels (d) and (e) consider the case for the broader
spot distribution over $-75^\circ \leq \ell_{\rm s} \leq 75^\circ$. Because
of the presence of a few spots located at higher latitudes, the
differential rotation becomes more important, and the distribution
of $P_{\rm rot, LS}$ becomes even broader towards its larger value.
The last two panels are shown just for comparison purpose; panel (f)
is for the stronger differential rotation case
($\alpha_2=3\alpha_{2\odot}$ and $\alpha_4=3\alpha_{4\odot}$), and
panel (g) is for rigid rotation. Given the same spot distribution
pattern, comparison among panels (a), (f) and (g) indicates how the
differential rotation law affects the distribution of the rotation
period at different quarters of the Kepler data, for instance. This
is expected to be directly applicable to put statistical constraints
on the degree of latitudinal differential rotation of a population of
stars, or to estimate the parameter $\alpha_2$ (and even $\alpha_4$)
for stars exhibiting clear photometric lightcurve modulations.
While the harmonic ratios shown in the right panels reflect the
statistical distribution of the spot latitudes to some extent, they
are sensitive to the spot area distribution and do not seem to provide
quantitatively useful information. Nevertheless, the histograms are
qualitatively consistent with the expected range of the ratios
plotted as the vertical dotted lines.
\begin{figure}
\begin{center}
\includegraphics[width=15cm, bb = 0 0 741 744]{Prot-distribution.pdf}
\end{center}
\caption{Distribution of the photometric rotation period and the
Fourier coefficient ratio estimated from 300 realizations of
$N_{\rm tot}=30$ spots with $\sigma_{\rm n}=35$ppm. We adopt the
fiducial parameter set of the equatorial rotation period $10$ days,
$T_{\rm samp}=30$ mins, $T_{\rm obs}=90$ days. We compute the
histograms by varying the observer's line-of-sight, and the range
of the spot latitudes $-\ell_{\rm s,max}<\ell_{\rm s}<\ell_{\rm s,max}$,
and the differential rotation parameters, which are indicated in
each panel. Vertical dotted lines indicate the range of the
differentially rotation periods as in Figure
\ref{fig:mock-lo0-Ntot30} (left panels), and the predicted ratio
for $\ell_{\rm s,max}$ (right panels). Panels (a), (b) and (c)
use the same set of 300 realizations but viewed from the
different observer's latitudes.}
\label{fig:Prot-distribution}
\end{figure}
\section{Summary and conclusion}
We have presented an analytic model of the lightcurve variation due to
starspots on a differentially rotating surface. If the dynamics of the
spots over the timescale of the observing period is neglected,
the Fourier coefficients of the harmonics of the rotation period are
written primarily in terms of the latitude of spots and the observer's
line-of-sight direction angle.
In order to understand the
resulting lightcurve variations, we generate various realizations of
starspots according to the analytic model, and
compute the Lomb-Scargle power spectra for the mock datasets.
Even though our analytical model neglects the evolution of spots on
the stellar surface (dynamical motion, creation and annihilation), its
prediction provides a useful framework to interpret the photometric
variation of stars, in particular from the existing Kepler data and
the future space-born mission. The conclusion and implications of the
paper are summarized below.
1) If a photometric lightcurve of a star exhibits a clear single
peak in the LS periodogram, the star may be well approximated as a
rigid rotator, and the peak should correspond to the rotation
period.
2) For those stars that have multiple peaks in the LS periodogram,
the distribution of the peaks estimated in different quarters may be
used to put constraints on parameters characterizing the
differential rotation law.
3) In principle, the ratio of harmonics for the rotation period may
constrain the spot latitude $\ell_{\rm s}$ and stellar inclination $\ell_{\rm o}$
given a limb darkening law. The constraint, however, is sensitive to
the spot distribution, and seems to be useful only for a single spot
dominated case. Nevertheless, joint analysis with independent
constraints on $\ell_{\rm o}$ from spectroscopic and/or asteroseismic
measurements may improve the constraint. We have not explored this
possibility in the present paper, but it is worthwhile to pursue in
future.
4) The analytical model presented in the paper is based on the
distribution of the {\it effective} area of spots $b_{\rm spot}$ alone, and
does not require the information of the geometric area and
temperature simultaneously. Thus it is applicable not only for
spots on main-sequence stars, but for other inhomogeneities on
rotating systems. For instance, the recent discovery of
the fastest-period white dwarf \citep{Kilic2021} indicates that the
interpretation of the photometric modulation of white dwarfs is
crucial in extracting their rotation period. Since it is likely to
originate from the small hot spot around the polar region, the
determination of $\ell_{\rm s}$ and $\ell_{\rm o}$ with respect to our
line-of-sight may be more promising for white dwarfs than for stars
with many different spots, as long as the modulation signal-to-noise
ratio is sufficiently high.
The above findings may have numerous useful applications even in the
existing Kepler data that cover a wide variety of stars with
different properties of spots on their surface. We are currently
working on the joint analysis of photometric and asteroseismic
measurements of Kepler stars selected by \citet{Kamiaka2018}, and
plan to present the results elsewhere in due course (Y.Lu et al. in
preparation).
\bigskip
\section*{Acknowledgements}
We thank an anonymous referee for various constructive comments on
the manuscript. Simulations and analyses in this paper made use of
a community-developed core Python package for Astronomy, {\tt
Astropy}. This work is supported by Grants-in Aid for Scientific
Research by the Japan Society for Promotion of Science (JSPS)
No.18H012 and No.19H01947, and from JSPS Core-to-core Program
``International Network of Planetary Sciences''.
\bibliographystyle{apj}
|
{
"redpajama_set_name": "RedPajamaArXiv"
}
| 3,118
|
class CreateAddresses < ActiveRecord::Migration
def self.up
create_table :addresses do |t|
t.references :contact
t.string :tag
t.string :street
t.timestamps
end
end
def self.down
drop_table :addresses
end
end
|
{
"redpajama_set_name": "RedPajamaGithub"
}
| 6,434
|
NGC 6408 (другие обозначения — UGC 10930, MCG 3-45-7, ZWG 112.17, NPM1G +18.0520, IRAS17366+1854, PGC 60637) — спиральная галактика с перемычкой (SBa) в созвездии Геркулес.
Этот объект входит в число перечисленных в оригинальной редакции «Нового общего каталога».
Примечания
6408
Геркулес (созвездие)
|
{
"redpajama_set_name": "RedPajamaWikipedia"
}
| 532
|
Q: Comparing two strings in Fortran What is the correct way to compare two strings say abc and bcd depending on the alphabetic order? Is there a built in command to do so? Or would > or .lt. do the work without any problems?
A: The intrinsic relational operators .lt. and < (along with the "equal" and "greater than" friends) indeed may be used to compare character variables.
We see the definition (Fortran 2018, 10.1.5.5.1):
the character operand x1 is considered to be less than x2 if the character value of x1 at this position precedes the value of x2 in the collating sequence
where the comparison is done with the first character part in the corresponding strings which differ.
The collating sequence tells you whether, for example, 'a' precedes 'b'. So, if 'abc' is compared with 'bcd' then the comparison is between 'a' and 'b'.
If the two strings to be compared are of different lengths, then the comparison is performed as though the shorter string is padded with blanks (spaces) on the right to make it the same length of the longer. This means that when comparing 'ab' and 'abc' we look at 'ab ' and 'abc': 'ab'<'abc' if and only if ' '<'c'.
|
{
"redpajama_set_name": "RedPajamaStackExchange"
}
| 9,866
|
Q: How can I share user profiles between different subsites? I have a home site based on Sharepoint Foundation 2010 with FBA and many different subsites.
I have defined all users profiles in the home site, but now I have the problem that I need to add again users for each subsites.
In addition doing this cause that user profiles information are not inherited from the home site and I need to insert again all personal data.
Is it possible share user profiles between different subsites?
A: By subsites do you mean you have
*
*1 site collection with many subsites
*or 1 site collection at the root and many site collections underneath that
The first one shares profile information already, the second one doesn't (in foundation).
The profile sharing feature is pretty much what you are paying for when you go with the standard/enterprise sharepoint editions.
You could either buy a third party component to do it for you:
http://store.bamboosolutions.com/sharepoint-user-profile-sync.aspx
Or you could build a SPJobDefinition that keeps all the information in the "User Information List" synchronized between your site collections.
|
{
"redpajama_set_name": "RedPajamaStackExchange"
}
| 7,974
|
{"url":"https:\/\/www.varsitytutors.com\/hotmath\/hotmath_help\/topics\/properties-of-square-roots.html","text":"# Properties of Square Roots\n\n## PRODUCT PROPERTY OF SQUARE ROOTS\n\nFor all positive real numbers $a$ and $b$ ,\n\n$\\sqrt{a}\\cdot \\sqrt{b}=\\sqrt{a\\cdot b}$\n\nThat is, the square root of the product is the same as the product of the square roots.\n\n## QUOTIENT PROPERTY OF SQUARE ROOTS\n\nFor all positive real numbers $a$ and $b$ , $b\\ne 0$ :\n\n$\\frac{\\sqrt{a}}{\\sqrt{b}}=\\sqrt{\\frac{a}{b}}$\n\nThe square root of the quotient is the same as the quotient of the square roots.","date":"2019-01-22 04:07:28","metadata":"{\"extraction_info\": {\"found_math\": true, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 0, \"math_alttext\": 0, \"mathml\": 7, \"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\": 0.9132863283157349, \"perplexity\": 206.30330175317064}, \"config\": {\"markdown_headings\": true, \"markdown_code\": true, \"boilerplate_config\": {\"ratio_threshold\": 0.18, \"absolute_threshold\": 10, \"end_threshold\": 15, \"enable\": false}, \"remove_buttons\": true, \"remove_image_figures\": true, \"remove_link_clusters\": true, \"table_config\": {\"min_rows\": 2, \"min_cols\": 3, \"format\": \"plain\"}, \"remove_chinese\": true, \"remove_edit_buttons\": true, \"extract_latex\": true}, \"warc_path\": \"s3:\/\/commoncrawl\/crawl-data\/CC-MAIN-2019-04\/segments\/1547583826240.93\/warc\/CC-MAIN-20190122034213-20190122060213-00471.warc.gz\"}"}
| null | null |
Q: Vue Js fetch data onclick and append inside a modal I have two components, one is parent component and another one is child component which is a modal popup.
Parent component
<parent-component>
<!-- this is a modal popup -->
<child-component v-bind:message="childMessage"></child-component>
<a href="#" @click="openModal(5)">Open model</a>
</parent-component>
<script>
export default {
data: function() {
return {
childMessage:{}
}
},
methods:{
openModal: function(id){
axios.get('api/message/'+id)
.then(response => {
this.childMessage = response.data;
})
.catch(error => {
console.log(error);
})
this.showModal = true
}
}
}
</script>
Child component
<!-- this is a popup modal -->
<child-component>
<h1>{{ message.title }}</h1>
</child-component>
<script>
export default {
props:{
message: {},
}
</script>
In parent component, I trigger the modal and request ajax at the same time.
And I can pass the ajax data to child component correctly.
But if I open the console, there is an error
Cannot read property 'title' of undefined
(Although I can see the data is working fine and it's already in html page)
It seems the appending data {{ childMessage.title }} run first (before the ajax request).
1 - How can I append the data correctly, probably after the ajax request.
2 - Do I need to check the condition for undefined value?
A: In this case, you have to check the condition for undefined value because child component is being rendered before the message API call ends. You can do it in this way,
<child-component>
<h1 v-if = "message">{{ message.title }}</h1>
</child-component>
A: I don't see where you use showModal but I suppose it's a sort of switch to display or not the child-component. If it's the case the error can come from the fact that you set showModal to true just after the call to the API. But this call is asynchronous you should probably move this.showModal = true in the success callback under this.childMessage = response.data;.
If you do that the message prop will be initialize at the moment the child component is rendered.
Also pay attention to your prop type, as @ittus mention message seems to be a String according to the default value in the child-component but you use it like an object in the template.
|
{
"redpajama_set_name": "RedPajamaStackExchange"
}
| 2,020
|
/*
* Licensed under the Apache License, Version 2.0 (the "License");
* you may not use this file except in compliance with the License.
* You may obtain a copy of the License at
*
* http://www.apache.org/licenses/LICENSE-2.0
*
* Unless required by applicable law or agreed to in writing, software
* distributed under the License is distributed on an "AS IS" BASIS,
* WITHOUT WARRANTIES OR CONDITIONS OF ANY KIND, either express or implied.
* See the License for the specific language governing permissions and
* limitations under the License.
*/
package com.facebook.presto.server;
import com.facebook.drift.codec.ThriftCodec;
import com.facebook.drift.codec.metadata.ThriftType;
import com.facebook.drift.protocol.TProtocolReader;
import com.facebook.drift.protocol.TProtocolWriter;
import com.facebook.presto.spi.function.SqlFunctionId;
// TODO: convert SqlFunctionId to be a native Thrift struct
public class SqlFunctionIdCodec
implements ThriftCodec<SqlFunctionId>
{
public ThriftType getType()
{
return new ThriftType(ThriftType.STRING, SqlFunctionId.class);
}
@Override
public SqlFunctionId read(TProtocolReader protocol)
throws Exception
{
return SqlFunctionId.parseSqlFunctionId(protocol.readString());
}
@Override
public void write(SqlFunctionId value, TProtocolWriter protocol)
throws Exception
{
protocol.writeString(value.toJsonString());
}
}
|
{
"redpajama_set_name": "RedPajamaGithub"
}
| 9,528
|
\section{Introduction} \label{sec:intro}
Type II supernovae (SNe) originate from massive stars with $ M_{ZAMS} > 8 \mbox{M$_{\odot}$} $ \citep{2013RvMP...85..245B} which have retained substantial hydrogen in the envelope at the time of explosion. They belong to a subclass of core-collapse SNe (CCSNe), which collapse under their own gravity at the end of the nuclear burning phase, having insufficient thermal energy to withstand the collapse.
The most common subtype among hydrogen rich supernovae is type IIP. At the time of shock breakout almost the entire mass of hydrogen is ionized. Type IIP SNe have an extended hydrogen envelope, which recombines slowly over a prolonged duration sustaining the plateau phase. During this phase the SN light curve shows almost constant brightness lasting for 80-100 days. At the end of plateau phase the SN experiences a sudden drop in luminosity, settling onto the slow declining radioactive tail, also known as nebular phase, which is mainly powered by gamma rays released from the decay of \mbox{$^{56}$Co}\ to \mbox{$^{56}$Fe}, which in turn depends upon the amount of \mbox{$^{56}$Ni}\ synthesized at the time of explosion.
{The plateau slope of SN type II light curve primarily depends on the amount of hydrogen present in the ejecta. If hydrogen content is high, as in type IIP, the initial energy deposited from shock and decay of freshly produced \mbox{$^{56}$Ni}\, shall be released slowly over a longer period of time. On the other hand if hydrogen content is relatively low, the light curve will decline fast but with higher peak luminosity. Thus if hydrogen content is low enough, one would expect a linear decline in the light curve classifying it as type IIL.
By the historical classification, type IIL \citep{1979A&A....72..287B} shows linear decline in light curve over 100 days until it reaches the radioactive tail phase. \cite{2012ApJ...756L..30A} claimed to find type IIP and IIL as to distinct group of events which may further indicate their distinct class of progenitors. However, recent studies by \cite{2014ApJ...786...67A} and \cite{2015ApJ...799..208S} on large sample of type II SNe do not favor any such bi-modality in the diversity, rather they found continuum in light curve slopes as well as in other physical parameters. The continuous distribution of plateau slopes in type II events is rather governed by variable amount of hydrogen mass left in the envelope at the time of explosion. Based on a sample of 11 type IIL events, \cite{2014MNRAS.445..554F} proposed that any event having decline of 0.5mag in V band light curve in first 50 days can be classified as type IIL. In light of these recent developments a large number of type IIP SNe classified earlier may now fall under IIL class. Thus many of the past studies collectively on samples of type IIP SNe, which we shall be referring in this work may include both IIP as well as IIL.}
Extensive studies have been done to relate observable parameters and progenitor properties of IIP SNe \citep[e.g.,][]{1985SvAL...11..145L,2003ApJ...582..905H}. Stellar evolutionary models suggest that these SNe may originate from stars with zero-age-main-sequence mass of 9-25\mbox{M$_{\odot}$}\ \citep[e.g.,][]{2003ApJ...591..288H}. However, progenitors directly recovered for a number of nearby IIP SNe, using the pre-SN \textit{HST} archival images, are found to lie within $ 8-17 \mbox{M$_{\odot}$}$ RSG stars \citep{2009ARA&A..47...63S}. Recent X-ray study also infers an upper mass limit of $ <19\mbox{M$_{\odot}$} $ for type IIP progenitors \citep{2014MNRAS.440.1917D}, which is in close agreement to that obtained from direct detection of progenitors.
The geometry of the explosion and presence of pre-existent circumstellar medium (CSM), often associated with progenitor mass loss during late stellar evolutionary phase, can significantly alter the observables even though originating from similar progenitors. There are number of recent studies of II SNe, like 2007od \citep{2011MNRAS.417..261I}, 2009bw \citep{2012MNRAS.422.1122I} and 2013by \citep{2015arXiv150106491V} which show signature of such CSM interactions during various phases of evolution.
SN 2013ej\ is one of the youngest detected type II SN which was discovered soon after its explosion. The earliest detection was reported on July 24.125 UTC, 2013 by C. Feliciano in \textit{Bright Supernovae\footnote{http://www.rochesterastronomy.org/supernova.html}} and subsequent independent detection on July 24.83 UTC by \cite{2013ATel.5466....1L} at \textit{V}-band magnitude of $ \sim $14.0. The last non-detection was reported on July 23.54 UTC, 2013 by All Sky Automated Survey
for Supernovae \citep{2013ATel.5237....1S} at a \textit{V}-band detection limit of $ > 16.7 $ mag. Therefore, we adopt an explosion epoch (0d) of July 23.8 UTC (JD $ =2456497.3\pm0.3 $), which is chosen in between the last non-detection and first detection of SN 2013ej. This being one of the nearest and brightest events, it provides us with an excellent opportunity to study the origin and evolution of type II SN. Some of the basic properties of SN 2013ej\ and its host galaxy are listed in Table~\ref{tab:host}.
\cite{2014MNRAS.438L.101V} presented early observations of SN 2013ej\ and using temperature evolution for the first week, they estimated a progenitor radius of 400-600 \mbox{R$_{\odot}$}. \cite{2014MNRAS.439L..56F} used high resolution archival images from \textit{HST} to examine the location of SN 2013ej\ and identified the progenitor candidate to be a supergiant of mass $ 8-15.5\mbox{M$_{\odot}$} $. \cite{2013ATel.5275....1L} reported unusually high polarization using spectropolarimetric observation for the week old SN, as implying substantial asymmetry in the scattering atmosphere of ejecta. {X-ray emission has also been detected by \textit{Swift} XRT \citep{2013ATel.5243....1M}, which may indicate SN 2013ej\ has experienced CSM interaction.}
In this work we present photometric and spectroscopic observation of SN 2013ej, and carry out qualitative as well as quantitative analysis of the various observables through modelling and comparison with other archetypal SNe. The paper is organized as follows. In section \ref{sec:obs} we describe photometric and spectroscopic observations and data reduction. The estimation of line of sight extinction is discussed in section \ref{sec:ext}. In section \ref{sec:lc} we analyze the light curves, compare absolute magnitude light curves and color curves. We also derive bolometric luminosities and estimate nickel mass from the tail luminosity. Optical spectra are analyzed in section \ref{sec:sp}, where we model and discuss evolution of various spectral features and compare velocity profile with other type II SNe. In section \ref{modelling}, we model the bolometric light curve of SN 2013ej\ and estimate progenitor and explosion parameters. Finally in section \ref{sec:sum}, we summarize the results of this work.
\input{./host.tex}
\section{Observation and data reduction} \label{sec:obs}
\subsection{Photometry} \label{sec:obs.phot}
Broadband photometric observations in \textit{UBVRI} filters have been carried out from 2.0m IIA Himalayan Chandra Telescope (HCT) telescope at Hanle and ARIES 1.0m Sampurananand (ST) and 1.3m Devasthal Fast Optical (DFOT) telescopes at Nainital. Additionally SN 2013ej\ has been also observed with \textit{Swift} Ultraviolet/optical (UVOT) telescope in all six bands.
Photometric data reductions follows the same procedure as described in \cite{2013MNRAS.433.1871B}. Images are cleaned and processed using standard procedures of IRAF software. DAOPHOT routines have been used to perform PSF photometry and extracting differential light-curves.
To standardize the SN field, three Landolt standard fields (PG~0231, PG~2231 and SA~92) were observed on October 27, 2013 with 1.0-m ST under good photometric night and seeing (typical FWHM $ \sim$2\arcsec.1 in \textit{V} band) condition. For atmospheric extinction measurement, PG~2231 and PG~0231 were observed at different air masses. The SN field has been also observed in between standard observations. The standardization coefficients derived are represented in the following transformation equations,
\begin{figure}
\centering
\includegraphics[width=8.4cm]{./id.eps}
\caption{SN 2013ej\ in NGC 0628. The $BR$-band composite image taken from 104-cm Sampurnanad telescope, covering an area of
about 13\arcmin$\times$13\arcmin\, is shown. Eight local field standards and SN are marked in the image.}
\label{fig:snid}
\end{figure}
\input{./photstar.tex}
\input{./photsn.tex}
\begin{eqnarray*}
u &=& U + (7.800\pm0.005) - (0.067\pm0.009) \cdot (U-B)\\
b &=& B + (5.269\pm0.007) - (0.060\pm0.009) \cdot (B-V)\\
v &=& V + (4.677\pm0.004) - (0.056\pm0.005) \cdot (B-V)\\
r &=& R + (4.405\pm0.005) - (0.038\pm0.010) \cdot (V-R)\\
i &=& I + (4.821\pm0.006) - (0.048\pm0.006) \cdot (V-I)
\end{eqnarray*}
\noindent
where $u$, $b$, $v$, $r$ and $i$ are instrumental magnitudes corrected for
time, aperture and airmass; $U$, $B$, $V$, $R$ and $I$ are standard magnitude.
The standard-deviation of the difference between the calibrated and the standard magnitudes of the observed Landolt stars are
found to be $\sim$ 0.03 mag in $U$, $\sim$ 0.02 mag in $BR$ and $\sim$ 0.01 mag in $VI$. The transformation coefficients
were then used to generate eight local standard stars in the field of SN 2013ej, which are verified to be non-variable and have brightness similar to SN. These stars are identified in Fig.\ref{fig:snid} and the calibrated \textit{UBVRI} magnitudes are listed in Table~\ref{tab:photstar}. These selected eight local standards were further used to standardize the instrumental light curve of the SN. One of these stars (star B) is common to that used in the study by \cite{2014JAVSO.tmp..275R}, and its \textit{BVRI} magnitudes are found to lie within 0.03 mag of our calibrated magnitudes. {Our calibrated magnitudes for SN 2013ej\ are also found to be consistent within errors to that presented in earlier studies of the event \citep{2014MNRAS.438L.101V,2014JAVSO.tmp..275R}.}
The standard photometric magnitudes of SN 2013ej\ are listed in Table~\ref{tab:photsn}.
This supernova was also observed with the Ultra-Violet/Optical Telescope \citep[UVOT;][]{2005SSRv..120...95R} in six bands (viz. uvw2, uvm2, uvw1, uvu, uvb, uvv) on the Swift spacecraft \citep{2004ApJ...611.1005G}. The UV photometry was obtained from the Swift Optical/Ultraviolet Supernova Archive\footnote{http://swift.gsfc.nasa.gov/docs/swift/sne/swift\_sn.html} (SOUSA; \citealp{2014Ap&SS.354...89B}). The reduction is based on that of \citet{2009AJ....137.4517B}, including subtraction of the host galaxy count rates and uses the revised UV zeropoints and time-dependent sensitivity from \citet{2011AIPC.1358..373B}. The UVOT photometry is listed in Table.~\ref{tab:photsn}. The first month of UVOT photometry was previously presented by \cite{2014MNRAS.438L.101V}.
\subsection{Spectroscopy} \label{sec:obs.spec}
Spectroscopic observations have been carried out at 10 phases during 12 to 125d. Out of these, nine epochs of low resolution spectra are obtained from Himalaya Faint Object Spectrograph and Camera (HFOSC) mounted on 2.0m HCT.
Spectroscopy on the HCT/HFOSC was done using a slit width of 1.92 arcsec, and grisms with resolution $ \lambda/\Delta\lambda = 1330$ for Gr7 and $2190$ for Gr8, and bandwidth coverage of $0.38 - 0.64$ $\mu m$ and $0.58 - 0.84$ $\mu m$ respectively.
One high resolution spectrum is obtained from the ARC Echelle Spectrograph (ARCES) mounted on 3.5m ARC telescope located at Apache Point Observatory (APO).
ARCES is a high resolution cross-dispersion echelle spectrograph, the spectrum is recorded in 107 echelle orders covering a wavelength range of $\lambda\sim$ 0.32-1.00\mbox{$\mu{\rm m}$}, at resolution of $R\sim31500$ \citep{2003SPIE.4841.1145W}. Summary of spectroscopic observations is given in Table.~\ref{tab:speclog}.
\input{./speclog.tex}
Spectroscopic data reduction was done under the \texttt{IRAF}\ environment. Standard reduction procedures are followed for bias subtraction and flat fielding. Cosmic ray rejections are done using a Laplacian kernel detection algorithm for spectra, L.A.Cosmic \citep{2001PASP..113.1420V}. One dimensional low resolution spectra were extracted using the \textsc{apall} task. Wavelength calibration was done using the \textsc{identify} task applied on FeNe and FeAr (for HCT) arc spectra taken during observation.
Wavelength calibration was crosschecked against the [\ion{O}{I}] $ \lambda5577 $ sky line in the sky spectrum, and it was found to lie within 0.3 to 4.5 \AA\ of the actual value.
Spectra were flux calibrated using \textsc{standard, sensfunc} and \textsc{calibrate} tasks in \texttt{IRAF}. For flux calibration, spectrophotometric standards were used which were observed on the same nights as the SN spectra were recorded. All spectra were tied to absolute flux scale using the observed flux from \textit{UBVRI} photometry of SN. To perform the tying, individual spectrum is multiplied by a wavelength dependent polynomial, which is convolved with \textit{UBVRI} filters and then the polynomial is tuned to match the convolved flux with observations. The one dimensional calibrated spectra were corrected for heliocentric velocity of host galaxy (658 \mbox{$\rm{\,km\,s^{-1}}$}; Table~\ref{tab:host}) using \textsc{dopcor} task.
\section{Distance and Extinction} \label{sec:ext}
We adopt a distance of $ 9.57\pm0.70$ Mpc which is a mean value of four different distance estimation techniques used for NGC 0628, viz., 9.91 Mpc applying Standard Candle Method (\textsc{scm}) to SN 2003gd by \cite{2010ApJ...715..833O}; 10.19 Mpc using the Tully-Fisher method (\texttt{HyperLeda}\footnote{http://leda.univ-lyon1.fr/}); 9.59 Mpc using brightest supergiant distance estimate by \cite{2005MNRAS.359..906H}; and planetary nebula luminosity function distance 8.59 Mpc \citep{2008ApJ...683..630H}. Although for each of these methods number of distance estimates exists in literature, we tried to select only most recent estimates. \cite{2014JAVSO.tmp..275R} estimated a distance of $ 9.1\pm0.4 $ Mpc by applying Expanding Photosphere Method (\textsc{EPM}) to SN 2013ej, which we find consistent to that we adopted.
One of the most reliable and well accepted method for SNe line-of-sight reddening estimation is using the \Nai~D absorption feature. The equivalent width (EW) of \Nai~D doublet (\mbox{$\lambda\lambda$}~5890, 5896) is found to be correlated with the reddening, estimated from the tail color curves of type Ia SNe \citep{1990A&A...237...79B,2003fthp.conf..200T}. However, \cite{2011MNRAS.415L..81P} suggested that although \Nai~D EW is weakly correlated with \mbox{$E(B-V)$}, the EWs estimated from low resolution spectra is a bad estimator of \mbox{$E(B-V)$}. \cite{2012MNRAS.426.1465P} used a larger sample of data and presented a more precise and rather different functional form of the correlation than that was derived earlier. Our high resolution echelle spectra at 79.5d provided an excellent opportunity to investigate the line-of-sight extinction.
\begin{figure}
\centering
\includegraphics[width=\linewidth]{./NaID_echelle.eps}
\caption{Echelle spectra at 79.5d showing the \Nai~D doublet for Milky-way while no impression for NGC 0628\ is detected.}
\label{fig:naid.ech}
\end{figure}
The resolved \Nai~D doublet for Milky-way is clearly visible in the high-resolution spectra (recorded on 79.5d) as shown in Fig.\ref{fig:naid.ech}. Whereas no impression of \Nai~D for NGC 0628\ is detected at the expected redshifted position relative to Milky-way. This indicates that the reddening due to host is negligible, only Galactic reddening will contribute to the total line of sight extinction. A similar conclusion has also been inferred by \cite{2014MNRAS.438L.101V} from their high resolution spectra obtained at 31d. Thus, we adopt a total $ \mbox{$E(B-V)$}=0.060\pm0.001 $ mag, which is entirely due to Galactic reddening \citep{2011ApJ...737..103S} and assuming total-to-selective extinction at V band as $ R_V=3.1 $, it translates into $ A_V=0.185\pm0.004 $ mag.
\section{Light curve} \label{sec:lc}
\subsection{Light curve evolution and comparison} \label{sec:lc.app}
The optical light curves of SN 2013ej\ in \textit{UBVRI} and six UVOT bands are shown in Fig.~\ref{fig:lc.app}. \textit{UBVRI} photometric observations were done at 38 phases during 12 to 209d (from plateau to nebular phase). The duration of plateau phase is sparsely covered, while denser follow-up initiated after 68d. The plateau phase lasted until $ \sim85 $d with an average decline rate of 6.60, 3.57, 1.74, 1.07 and 0.74 mag (100 d)$ ^{-1} $ in \textit{UBVRI} bands respectively. Since 95d, the light curve declines very fast until 115d, after which it settles to a relatively slow declining nebular phase. During this phase the decline rates for \textit{UBVRI} bands are 0.98, 1.22, 1.53, 1.42 and 1.55 mag (100 d)$ ^{-1} $ respectively.
\begin{figure}
\centering
\includegraphics[width=\linewidth]{./lc_all_color.eps}
\caption{The photometric light curves in Johnson-Cousins \textit{UBVRI} and \textit{Swift}~UVOT bands. The light curves are vertically shifted for clarity. The line joining the data points of light curves is for visualization purpose only.}
\label{fig:lc.app}
\end{figure}
SN 2013ej\ has been also observed by \textit{Swift}~UVOT at 35 phases during 7 to 139d. The UVOT \textit{UV} band light curves declines steeply during the first 30d at a rate of 0.182, 0.213, 0.262 mag d$ ^{-1} $ in \textit{uvw1, uvw2} and \textit{uvm2} bands respectively, thereafter settling into a slow declining phase until it reaches the end of plateau.
SN 2013ej\ experience a steeper plateau decline than that observed for SN 1999em \citep{2002AAS...201.2303L}, SN 1999gi \citep{2002AJ....124.2490L}, SN 2012aw \citep{2013MNRAS.433.1871B} and SN 2013ab \citep{2015arXiv150400838B}. For example, SN 2012aw plateau declines at a rate of 5.60, 1.74, 0.55 mag (100 d)$ ^{-1} $ in $ UBV $-bands, similarly for SN 2013ab decline rates in \textit{UBVRI} are 7.60, 2.72, 0.92, 0.59 and 0.30 mag (100 d)$ ^{-1} $ and 0.169, 0.236, 0.257 mag d$ ^{-1} $ in UVOT \textit{uvw1, uvw2} and \textit{uvm2} bands (during first 30d).
{The absolute \textit{V}-band ($ M_V $) light curve of SN 2013ej\ is plotted in Fig.~\ref{fig:lc.abs} and is compared with other well studied type II SNe (after correcting for extinction and distance).
In Table~\ref{tab:slopendrop} we list the plateau slope of all compared type II events. The comparison shows that the decline rate of SN 2013ej\ during this phase is highest (1.74 mag (100 d)$ ^{-1} $) among most other SNe, except three type IIL SNe 1980K, 2000dc and 2013by, where SN 1980 is among the very first observed prototypical type IIL event. The early plateau ($ <40 $d) light curve of SN 2013ej\ is identical to SN 2009bw. However, unlike most other IIP SNe, e.g. 2009bw and 2013ab, which becomes flatter during late plateau, SN 2013ej\ continues to decline almost at a steady rate until the end of plateau ($ \sim $ 85d). The mid-plateau $ M_V=-14.7 $ mag for SN 2013ej, which places it in the class of normal luminous type II events.
SN 2013ej\ is comparable with fast declining and short plateau SNe in the sample of \cite{2014ApJ...786...67A}.
Following the plateau phase, $ V $-band light drops very fast to reach slow declining nebular phase (1.53 mag (100 d)$ ^{-1} $), which is powered by the radioactive decay of $ ^{56} $Co to $ ^{56} $Fe. The fall of $ M_V $ during the plateau nebular transition is $ \sim 2.4$ mag, which is on the higher side of the compared events. The closest comparison is SNe 2009bw and 2012A which exhibits a drop of $ \sim $2.4 mag and $ \sim $2.5 mag respectively.} This also indicates low amount of \mbox{$^{56}$Ni}\ mass synthesized during the explosion which we shall further discuss in the next section.
\begin{figure*}
\centering
\includegraphics[width=16cm]{./Mv_comparison.eps}
\caption{
The M$_{V}$ light curve of SN 2013ej\ is compared with other type II SNe.
The exponential decline of the tail light curve following the radioactive decay law for \mbox{$^{56}$Co}$ \rightarrow $\mbox{$^{56}$Fe}\ is shown with a dashed line. On the bottom left side, pair of dotted lines in each gray and green colors represent the slope range for type IIP and IIL SNe templates as given by \citet{2014MNRAS.445..554F}. The adopted explosion time in JD-2400000,
distance in Mpc, \mbox{$E(B-V)$}\ in mag and
the reference for observed V-band magnitude, respectively, are :
SN 1980K -- 44540.5, 5.5, 0.30; \citet{1982A&A...116...35B}, NED database;
SN 1987A -- 46849.8, 0.05, 0.16; \citet{1990AJ.....99.1146H};
SN 1999em -- 51475.6, 11.7, 0.10; \citet{2002PASP..114...35L,2003MNRAS.338..939E};
SN 1999gi -- 51522.3, 13.0, 0.21; \citet{2002AJ....124.2490L};
SN 2000dc -- 51762.4, 49.0, 0.07; \citet{2014MNRAS.445..554F}, NED database;
SN 2003hn -- 52866.5, 17.0, 0.19; \citet{2009AJ....137...34K,2014ApJ...786...67A};
SN 2004et -- 53270.5, 5.4, 0.41; \citet{2006MNRAS.372.1315S};
SN 2005cs -- 53549.0, 7.8, 0.11; \citet{2009MNRAS.394.2266P};
SN 2009N -- 54848.1, 21.6, 0.13; \citet{2014MNRAS.438..368T};
SN 2009bw -- 54916.5, 20.2, 0.31; \citet{2012MNRAS.422.1122I};
SN 2012A -- 55933.5, 9.8, 0.04; \citet{2013MNRAS.434.1636T};
SN 2012aw -- 56002.6, 9.9, 0.07; \citet{2013MNRAS.433.1871B};
SN 2013ab -- 56340.0, 24.0, 0.04; \citet{2015arXiv150400838B};
SN 2013by -- 56404.0, 14.8, 0.19; \citet{2015arXiv150106491V}.}
\label{fig:lc.abs}
\end{figure*}
\input{./slopendrop.tex}
\textit{Swift}~UVOT absolute magnitude light curves of SN 2013ej\ are shown in Fig.~\ref{fig:uv.abs} and compared with other well observed type II SNe. The sample is selected in such a way that SNe have at least a month of observations. Most SNe are not followed for more than a month by \textit{Swift}, mainly because of the large distances or high extinction values. However, both these factors work in favor of SN 2013ej\, making it possible to have about four months of observations. Moreover, the location of the SN being in the outskirt of a spiral arm of NGC 0628, the background flux contamination is also negligible. The comparison shows that the SN 2013ej\ UV light curves are identical to SN 2012aw. SN 2013ej\ also shows a similar UV plateau trend as observed in SN 2012aw \citep{2013ApJ...764L..13B}, which is although expected but rarely detected for IIP/L SNe.
\begin{figure}
\centering
\includegraphics[width=\linewidth]{./UVOT_comp.eps}
\caption{Comparison of the \textit{Swift}~UVOT UV absolute light curves of SN 2013ej, with other well observed II SNe from UVOT. For the compared SNe, references for UVOT data, extinction and distance are: SN 2005cs -- \citet{2009AJ....137.4517B,2009MNRAS.394.2266P}, SN 2006at -- \citet{2009AJ....137.4517B}; Distance 65 Mpc; $ \mbox{$E(B-V)$}=0.031 $ mag \citep[only Galactic reddening][]{2011ApJ...737..103S}, SN 2006bp -- \citet{2008ApJ...675..644D}, SN 2012aw -- \citet{2013ApJ...764L..13B,2013MNRAS.433.1871B}, SN 2013ab -- \citet{2015arXiv150400838B}, SN 2013by -- \citet{2014Ap&SS.354...89B,2015arXiv150106491V}. Some late data points for SN 2013ab with large errors has been omitted from the plot.}
\label{fig:uv.abs}
\end{figure}
Broadband color provides important information to study the temporal evolution of SN envelope. In Fig.~\ref{fig:cc.abs}, we plot the intrinsic colors \textit{U-B, B-V, V-R} and \textit{V-I} for SN 2013ej\ and compare its evolution with type II-pec SN 1987A, and type IIP SNe 1999em, 2004et, 2012aw and 2013ab. All the colors show generic signature of fast cooling ejecta until the end of plateau ($ \sim110 $d). With the start of the nebular phase it continues to cool at a much slower rate in \textit{V-I} and \textit{V-R} colors, whereas \textit{U-V} and \textit{B-V} shows a bluer trend. This is because, as the SN enters the nebular phase, the ejecta become depleted of free electrons, thereby making the envelope optically thin, and so unable to thermalize the photons from radioactive decay of $ ^{56} $Co to $ ^{56} $Fe.
\begin{figure}
\centering
\includegraphics[width=8.75cm]{./color_comp.eps}
\caption{The intrinsic colors evolution of SN 2013ej\ is compared with other well-studied
type IIP SNe 1987A, 1999em, 2004et, 2012aw and 2013ab. The reference for the data is same as in Fig.~\ref{fig:lc.abs}.}
\label{fig:cc.abs}
\end{figure}
\subsection{Bolometric light-curve} \label{sec:lc.bol}
We compute the pseudo-bolometric luminosities following the method described in \cite{2013MNRAS.433.1871B}; which include SED integration over the semi-deconvolved photometric fluxes after correcting for extinction and distance. Supernova bolometric luminosities during early phases ($ \le30 $d) are dominated by ultraviolet fluxes, while after mid-plateau ($ \sim50 $d) UV contribution becomes insignificant as compared to optical counterpart \citep[e.g., as seen in SNe 2012aw, 2013ab;][]{2013MNRAS.433.1871B,2015arXiv150400838B}. Similarly, during late phases $ >100 $d NIR becomes dominant over optical fluxes. However, during most of the light curve evolution, optical fluxes still provide significant contribution. We compute pseudo-bolometric luminosities in the wavelength range of \textit{U} to \textit{I} band (3335-8750\AA). We also computed UV-optical pseudo-bolometric light curve with wavelength starting from \textit{uvw2} band (wavelength range of 1606-8750\AA). The UV contribution enhances the luminosity significantly during early phases, whereas it is almost negligible after mid-plateau.
In Fig.~\ref{fig:lc.bol}, we plot pseudo bolometric light curve for SN 2013ej\ and compare it with other SNe light curves computed using the same technique. We also include UV-optical bolometric light curve for SNe 2012aw and 2013ab along with SN 2013ej for comparison. Although the UV-optical light curve is initially brighter than the optical light curve, they completely coincide by the end of plateau phase (85d). It is evident from the comparison that SN 2013ej\ experienced a steep decline during the plateau phase, but with a much shorter duration. This is consistent with the anti-correlation observed between plateau slope and duration for type II SNe \citep{1993A&A...273..106B,2014ApJ...786...67A}.
The UV-optical bolometric light decreases by 0.83 dex during plateau phase (from 12 to 85d), followed by an even faster drop by 0.76 dex in a short duration of 21 days (from 90 to 111d). Thereafter, the SN settles in a slow declining nebular phase. The tail luminosities are significantly lower than other normal luminosity IIP events, e.g., SN 2013ej\ luminosities are lower by $ \sim0.5 $ dex (at 200d) than that of type II SNe 1987A, 1999em, 2004et and 2012aw, but higher than subluminous events like SN 2005cs. Another noticeable dissimilarity of the tail light curve is its high decline rate. SN 2013ej\ tail luminosity declines at a rate of 0.55 dex 100 d$ ^{-1} $, which is much higher than that expected from radioactive decay of \mbox{$^{56}$Co}\ to \mbox{$^{56}$Fe}. This is possibly because of inefficient gamma-ray trapping in the ejecta, and thus incomplete thermalization of the photons. We shall further explore this in \S\ref{modelling} in context of modeling the light curve.
\begin{figure}
\includegraphics[width=8.75cm]{./bol_lc_multi.eps}
\caption{The \textit{UBVRI} bolometric light-curve of {SN 2013ej} is compared with other well studied supernovae. Light
curves with added UVOT UV contributions are also shown for SNe 2013ej, 2013ab and 2012aw (labeled as UVO).
The adopted values of distances, reddening and explosion time are same as in Fig.~\ref{fig:lc.abs}. The exponential decline of the tail light curve following the radioactive decay law is shown with a dashed line.}
\label{fig:lc.bol}
\end{figure}
\subsection {Mass of nickel} \label{sec:lc.nick}
During the explosive nucleosynthesis of silicon and oxygen, at the time of shock-breakout in CCSNe, radioactive \mbox{$^{56}$Ni}\ is produced. The nebular-phase light-curve is mainly powered by the radioactive decay of \mbox{$^{56}$Ni}\ to \mbox{$^{56}$Co}\ and \mbox{$^{56}$Co}\ to \mbox{$^{56}$Fe}\ with half-life times
of 6.1d and 77.1d respectively
emitting $\gamma$-rays and
positrons. Thus the tail luminosity will be proportional to the amount of radioactive \mbox{$^{56}$Ni}\ synthesized at the time of explosion.
We determine the mass of \mbox{$^{56}$Ni}\ using following two methods.
For SN 1987A, one of the most well studied and well observed event, the mass of \mbox{$^{56}$Ni}\ produced in the explosion has been estimated quite accurately, to be $ 0.075\pm0.005 $ \mbox{M$_{\odot}$}\ \citep{1996snih.book.....A}.
By comparing the tail luminosities of SN 2013ej\ and SN 1987A at similar phases, it is possible to estimate the \mbox{$^{56}$Ni}\ mass for SN 2013ej.
In principle true bolometric luminosities (including UV, optical and IR) are to be used for this purpose, which are available for SN 1987A, whereas for SN 2013ej\ we have only UV and optical observations. Thus, in order to have uniformity in comparison, we used only the \textit{UBVRI} bolometric luminosities for both SNe and computed using the same method and wavelength range.
We estimate the tail \textit{UBVRI} luminosity at 175d, by making a linear fit over 155 to 195d, to be $ 2.90\pm0.43\times 10^{40} $ erg s$^{-1}$. Likewise, SN 1987A luminosity is estimated to be $ 9.60\pm0.06\times 10^{40} $ erg s$^{-1}$ at similar phase. Thus, the ratio of SN 2013ej\ to SN 1987A luminosity is $0.302\pm0.044$, which corresponds to a \mbox{$^{56}$Ni}\ mass of $ 0.023\pm0.003 \mbox{M$_{\odot}$}$ for SN 2013ej.
Assuming the $\gamma$-photons emitted from radioactive decay of \mbox{$^{56}$Co}\ thermalize the ejecta, \mbox{$^{56}$Ni}\ mass can be independently estimated from the tail luminosity as described by \cite{2003ApJ...582..905H}.
\begin{eqnarray*}
M_{\rm Ni} = 7.866\times10^{-44} \times L_{t} \exp\left[ \frac{(t_{t}-t_{0})/(1+z)-6.1}{111.26}\right]\mbox{M$_{\odot}$}
\end{eqnarray*}
where $t_{0}$ is the explosion time, 6.1d is the half-life time of \mbox{$^{56}$Ni}\ and 111.26d is the e-folding time of the \mbox{$^{56}$Co}\ decay. We compute tail luminosity $L_{t}$ at 6 epochs within 153 to 185d from the $ V $ band data corrected for distance, extinction and bolometric correction factor of $0.26 \pm 0.06$ mag during nebular phase \citep{2003ApJ...582..905H}. The weighted mean value of $L_{\rm t}$ is found to be $5.45\pm0.35\times10^{40}\,$\mbox{$\rm{\,erg\,s^{-1}}$} corresponding to mean phase of 170d. This tail luminosity corresponds to a value of $M_{\rm Ni} =0.019\pm0.002$\mbox{M$_{\odot}$}.
We take the weighted mean of the estimated values from above two methods, and adopt a \mbox{$^{56}$Ni}\ mass of $ 0.020\pm0.002 \mbox{M$_{\odot}$}$ for SN 2013ej.
{\cite{2003ApJ...582..905H} found a strong correlation between the \mbox{$^{56}$Ni}\ mass and the mid plateau (at 50d) $ V $ band absolute magnitude for type II SNe and this correlation was further confirmed by \cite{2014MNRAS.439.2873S} specifically for low luminous events. Fig.~\ref{fig:nicomp} shows the correlation of mid plateau M$ _V $ versus \mbox{$^{56}$Ni}\ mass for 34 events, including SN 2013ej. The SN lies within the scatter relation, but towards the lower mass range of \mbox{$^{56}$Ni}\ than where most of the events cluster around (top right).}
\begin{figure}
\centering
\includegraphics[width=0.9\linewidth]{./Ni_comp.eps}
\caption{The plot of absolute $ V $ band magnitude at 50 day versus \mbox{$^{56}$Ni}\ mass for 34 type II SNe. Data taken from \citet{2003ApJ...582..905H} and \citet{2014MNRAS.439.2873S}. The position of SN 2013ej\ on the correlation is shown with filled red circle.}
\label{fig:nicomp}
\end{figure}
\section{Optical spectra} \label{sec:sp}
\subsection{Key spectral features} \label{sec:sp.key}
The spectroscopic evolution of SN 2013ej\ is presented in Fig.~\ref{fig:sp.all}. Preliminary identifications of spectral features has been done as per previously studied type IIP SNe \citep[e.g.,][]{2002PASP..114...35L,2013MNRAS.433.1871B}. The spectrum at 12d shows broad \ha, \hb\ and \Hei\ features on top of a hot blue continuum. The 35d spectrum shows a relatively flat continuum with well developed features of \ha, \hb, \Feii\ along with blends of other heavier species \Tiii\ and \Baii. \Hei\ line is no longer detectable, instead \Nai~D features start to appear at similar location. The spectra from 35 to 80d represent the cooler photospheric phase, where the photosphere starts to penetrate deeper layers rich in heavier elements like \Feii\ and \Scii. During these phases we see the emergence and development of various other heavy atomic lines and their blends like \Tiii, \Baii, \Nai~D and \Caii.
Fig.~\ref{fig:sp.lit} shows the comparison of three plateau phase spectra, viz. 12, 35 and 68d with other well studied type IIP SNe at similar epochs. The comparison shows the spectra of SN 2013ej\ is broadly identical to others in terms of observable line features and their evolution.
A notable feature during early spectrum (12d) is the dip on the bluer wing of \ha\ profiles near 6170 \AA\, which can be attributed as the \Siii\ feature.
{\cite{2013ATel.5275....1L} also identified this feature at $ \sim 9$d spectra of SN 2013ej\, however, due to unlikeliness of such a strong \Siii\ feature at such early epochs, a possiblity of non-standard red supergiant envelope or CSM interaction was suggested.}
However, such dips are detectable in 35 and 42d spectra, which we identify as \Siii\ feature in \textsc{synow}\ modeling.
\begin{figure*}
\centering
\includegraphics[width=14cm]{./spec_all_color.eps}
\caption{The redshift corrected spectra of SN 2013ej\ are plotted for 10 phases during 12d to 125d. The prominent P-Cygni
profiles of hydrogen (\ha, \hb, \hg) and helium (\Hei\ \ld5876) are marked. The telluric absorption features of O$ _2 $
are marked with $ \oplus $, symbol. {Portion of spectra in extreme blue or red ends have low SNR. Individual spectra with with overall low SNR has been binned for better visualization.}}
\label{fig:sp.all}
\end{figure*}
\begin{figure}
\centering
\includegraphics[width=\linewidth]{./spec_compare.eps}
\caption{Comparison of early (12d) and plateau (35d, 68d) phase spectra of {SN 2013ej} with
other well-studied type IIP SNe 1999em \citep{2002PASP..114...35L}, 1999gi \citep{2002AJ....124.2490L},
2004et \citep{2006MNRAS.372.1315S,2010MNRAS.404..981M}, 2012aw \citep{2013MNRAS.433.1871B} and 2013ab \citep{2015arXiv150400838B}. All
comparison spectra are corrected for extinction and redshift (adopted values are same as in Fig.~\ref{fig:lc.abs}).}
\label{fig:sp.lit}
\end{figure}
The spectra at 96 and 97d represents the plateau-nebular transition phase. Thereafter, spectra at 109 and 125d represents the nebular phase, where the ejecta has become optically thin. {These spectra shows the emergence of some emission features from forbidden lines of \Oia\ \mbox{$\lambda\lambda$}~6300, 6364 and \Caiia\ \mbox{$\lambda\lambda$}~7291, 7324\AA, as well as previously evolved permitted lines
of \Hi, and the \Nai\ \ld5893 doublet (see Fig.~\ref{fig:sp.neb}).}
\begin{figure}
\centering
\includegraphics[width=8.5cm]{./spec_nebular.eps}
\caption{The nebular phase spectrum of SN 2013ej\ at 125d. Prominent emission and absorption features are marked and labeled.}
\label{fig:sp.neb}
\end{figure}
\cite{2014ApJ...786L..15G} found correlations between \ha\ absorption to emission strengths and light curve parameters, i.e. plateau slope and duration of optically thick phase. Following their selection criteria for choosing phase of SN spectra, i.e. ten days after start of recombination, we selected 42d spectrum as the closet available phase to the criteria. The \ha\ absorption to emission ratio of equivalent widths for SN 2013ej\ is found to be $ 0.23\pm0.02 $, the optically thick phase is $ \sim85 $d and \textit{B}-band late plateau (40 to 85d) slope is $ \sim0.27 $ mag (100 d)$ ^{-1} $. The correlation for optically thick phase duration is found to follow that presented by \cite{2014ApJ...786L..15G}. For the plateau slope, the correlation also hold true, but here SN 2013ej\ lies in the border line position of the scattered relation.
However, it may be noted that \ha\ profiles are possibly contaminated by high velocity features as we describe in next sections, which may result in deviation from correlation.
\subsection{\textsc{SYNOW} modelling of spectra} \label{sec:synow}
SN 2013ej\ spectra has been modeled with \textsc{synow}\footnote{https://c3.lbl.gov/es/\#id22} \citep{1997ApJ...481L..89F,1999MNRAS.304...67F,2002ApJ...566.1005B} for line identification and its velocity estimation. \textsc{synow}\ is a highly parametrized spectrum synthesis code which employs the Sobolev approximation to simplify radiation transfer equations assuming a spherically symmetric supernova expanding homologously. The strength of the \textsc{synow}\ code is its capability to reproduce P-Cygni profiles simultaneously in synthetic spectra for a given set of atomic species and ionization states.
The applicability of \textsc{synow}\ is well tested in various core-collapse SNe studies \citep[e.g.][]{2012MNRAS.422.1178I,2013MNRAS.433.1871B,2013ApJ...767...71M,2014ApJ...782...98B, 2014MNRAS.438..368T,2014ApJ...781...69M} for velocity estimation and analysis of spectral lines.
To model the spectra we tried various optical depth profiles (viz. gaussian, exponential and power law) with no significant difference among them, however we find exponential profile ($\tau\propto exp[-v/v_e]$) marginally better suited to match the absorption trough of observed spectra, where $v_{e}$ the e-folding velocity, is a fitted parameter. While modeling spectra, \Hi\ lines are always dealt as detached scenario. This implies the velocity of hydrogen layer is significantly higher and is thus detached from photospheric layer, close to which most heavier atomic lines form, as assumed in \textsc{synow}\ code. As a consequence to this, the \ha\ lines in synthetic spectrum, which are highly detached, has flat topped emissions with blue shifted absorption counter parts.
SN 2013ej\ spectra are dereddened and approximate blackbody temperature is supplied in the model to match the spectral continuum. For early spectrum (12d), local thermodynamic equilibrium (LTE) assumption holds good and thus \textsc{synow}\ could fit the continuum well, whereas at later epochs it fails to fit properly. The set of atomic species incorporated to generate the synthetic model spectrum are \Hi, \Hei; \Feii; \Tiii; \Scii; \Caii; \Baii; \Nai\ and \Siii. The photospheric velocity $ v_{\rm ph} $ is optimized to simultaneously fit the \Feii\ (\mbox{$\lambda\lambda$}~4924, 5018, 5169) P-Cygni profiles and \Hi\ lines are treated as detached. The optical depths and optical depth profile parameters, e-folding velocity are varied for individual species to fit respective line profiles. In Fig.~\ref{fig:sp.synph} we show the model fit of 71d spectrum. Most of the observable spectral features are reproduced well and are identified in the figure.
\begin{figure*}
\centering
\includegraphics[width=0.8\linewidth]{./plot_single.eps}
\caption{\textsc{synow}\ modelling of SN 2013ej\ spectrum at 71d. Model spectrum is shown with thick solid line (blue),
while the observed one is shown with thin solid line (red). Observed fluxes are corrected for extinction.}
\label{fig:sp.synph}
\end{figure*}
Similarly all spectra during 12 to 97d are modeled with \textsc{synow}. The model fits for \Feii\ (\mbox{$\lambda\lambda$}~4924, 5018, 5169), \hb\ and \ha\ spectral sections are shown in Fig.~\ref{fig:sp.synall}. The atomic species which are important to model these features are \Hi, \Feii, \Baii, \Tiii, \Scii\ and \Nai. In addition to these \Siii\ is also used to model the dips in the blue wing of \ha\ P-Cygni during 12 to 42d. While modeling the \ha\ and \hb\ profiles, \textsc{synow}\ was unable to properly fit the broad and extended P-Cygni absorption troughs with single regular component. In order to fit these extended troughs, we invoke high-velocity (HV) component of \Hi. Although no separate dip is seen, possibly due to low spectral resolution and overlapping of broad P-Cygni profiles, the HV component can well reproduce the observed features in synthetic model spectrum. The implication and interpretation of these HV components are further discussed in \S\ref{sec:sp.vel}. The \textsc{synow}-derived velocities for \Feii, \ha, \hb\ lines and corresponding HV components are listed in Table~\ref{tab:synow}. The nebular spectra during 109 to 125d have not been modeled primarily due to limitations of the LTE assumption of \textsc{synow}, and also because nebular phase spectra are dominated by emission lines rather than P-Cygni profiles.
\input{./synow.tex}
\begin{figure}
\centering
\includegraphics[height=7.75cm]{./synow_iron_combi.eps}
\includegraphics[height=7.75cm]{./synow_ha_combi.eps}
\caption{\textsc{synow}\ modelling of SN 2013ej\ spectra at 8 phases during 12d to 97d for \hb, \Feii\ multiplet (left) and \ha\ (right) profiles. Model spectra are shown with thick solid line (blue), while the observed ones are shown with thin solid line (red).
In the model, \Hi\ lines are treated as detached to fit the absorption troughs.
Along with \Feii\ and \Hi; other ions (\Scii, \Baii, \ion{Si}{ii} and \Nai,
\Tiii) are also incorporated in model to fit some weaker features, specially at later phases. In addition to this, high-velocity \Hi\ lines are also incorporated (42d onwards) to fit the extended \ha\ and \hb\ absorption troughs.
The 97d spectrum do not have \hb\ and \Feii\ wavelength region, hence it is not shown here.
}
\label{fig:sp.synall}
\end{figure}
\subsection{Evolution of spectral lines} \label{sec:sp.line}
Investigation of the spectral evolution sheds light on various important aspects of the SN, like interaction of ejecta
with the circumstellar material, geometrical distribution of expanding shell of ejecta and formation of dust during late time. SN spectra are dominated by P-Cygni profiles which are direct indicators of expansion velocities and they evolve with the velocity of photosphere. As ejecta expands and opacity decreases allowing photons to escape from deeper layers rich in heavier elements, we are able to see emergence and growth of various spectral lines.
To illustrate the evolution of \ha\ line,
in Fig.~\ref{fig:sp.line} partial region of spectra is plotted in velocity domain corresponding to rest wavelengths of \ha. At 12d broad P-Cygni profile (FWHM $ \sim9500 $ \mbox{$\rm{\,km\,s^{-1}}$}) is visible which becomes narrower with time as the expansion slows down. The blue-shifted absorption troughs are direct estimator of expansion velocity of the associated line forming layer. The emission peaks are found to be blue-shifted (by $ \sim3200 $ \mbox{$\rm{\,km\,s^{-1}}$} at 12d), which progressively decreases with decrease in expansion velocity and almost settling to zero velocity when the SN starts to enter nebular phase (97d). Such blue-shifted emission peaks, especially during early phases are generic features observable in SN spectra, e.g., SNe 1987A \citep{1987A&A...182L..29H}, 1998A \citep{2005MNRAS.360..950P}, 1999em \citep{2003MNRAS.338..939E}, 2004et \citep{2006MNRAS.372.1315S}, 2012aw \citep{2013MNRAS.433.1871B}, 2013ab \citep{2015arXiv150400838B}. These features are tied with the density structure of the ejecta, which in turn controls the amount of occultation of the receding part of ejecta, resulting in biasing of the emission peak \citep{2014MNRAS.441..671A}, which are not limited to \ha\ but applicable to all spectral lines. However, such a blue-shift is clearly detected for \ha\, whereas for most other lines, emission profiles are weak and peaks are contaminated by adjacent P-Cygni profiles. Detailed SN spectral synthesis code like \textsc{cmfgen}\ \citep{2005ASPC..332..415D} is capable of reproducing such blue-shifted emission peaks.
\begin{figure}
\centering
\includegraphics[width=0.7\linewidth]{./spec_all_line_6563.eps}
\caption{Evolution of \ha\ line profile at 10 phases during 12d to 125d.
A zero-velocity line is plotted with a dashed line corresponding to the rest
wavelength of \ha\ \ld6563.}
\label{fig:sp.line}
\end{figure}
As inferred from Fig.~\ref{fig:sp.lit}, the spectral evolution of SN 2013ej\ is almost identical to other typical IIP SNe. However, the comparison of 35 and 68d spectra indicates \Feii\ lines are somewhat under developed as compared to other SNe at similar phase. As seen in the 68d comparison, the \Feii\ (\mbox{$\lambda\lambda$}~4924, 5018, 5169) absorption dips are significantly weaker in comparison to that seen in other SNe.
Another prominent and unusual feature is seen in nebular spectra at 109d and 125d, on top of \ha\ emission, and the same is marked as feature A in Fig.~\ref{fig:sp.line}. This unusual dip is resulting into an apparent blue-shift of the emission peak, which is in fact larger than that seen in the last plateau spectra at 97d. Such evolution is unexpected and against the general trend of emission peak evolution in SNe. The low resolution of these spectra prohibits us from investigating this feature in detail.
{This feature can be split into two emission components, one redshifted at 1200 \mbox{$\rm{\,km\,s^{-1}}$}\ and another blueshifted by 1300 \mbox{$\rm{\,km\,s^{-1}}$}\ (see \S\ref{app:nebular_ha} for further explanation) with respect to \ha\ rest position. Such an asymmetric or double peaked \ha\ nebular emission has been observed in a number of SNe, e.g. SN 1999em \citep{2002PASP..114...35L} and SN 2004dj \citep{2005AstL...31..792C}. \cite{2002PASP..114...35L} identified such a dip or notch in \ha\ emission profile only during nebular phase of SN 1999em, which they suggested as possible ejecta-CSM interaction or asymmetry in line emitting region. In SN 2004dj, the asymmetry in nebular \ha\ spectra identified by \cite{2005AstL...31..792C} has been explained by bipolar distribution of \mbox{$^{56}$Ni}\ with a spherical hydrogen envelope \citep{2006AstL...32..739C}.}
\subsection{Ejecta velocity} \label{sec:sp.vel}
Progenitor stars prior to explosion develop stratified layers of different elements, which are generally arranged in an elemental sequence, hydrogen being abundant in the outermost shell, whereas heavier metals like iron predominate at deeper layers. However at the time of shock breakout significant mixing of layers may occur. Spectral lines originating from different layers of the ejecta attains different characteristic velocities. Thus study of velocity evolution provides important clues to the explosion geometry and the characteristics of various layers. Evolution of photospheric layer is of special interest as it is directly connected to the kinematics and other related properties. Photosphere represents the layer of SN atmosphere where optical depth attains a value of $\sim~^2/_3 $ \citep{2005A&A...437..667D}. Due to complex mixing of layers and continuous recession of the recombination front, no single spectral line can represent the true photospheric layer. During the plateau phase, \Feii\ or \Scii\ lines are the best estimator of photospheric velocity ($v_{\rm ph}$). In early phases when \Feii\ lines are not strongly detectable, the best proxy for $v_{\rm ph}$ is \Hei, or \hb\ \citep{2012MNRAS.419.2783T} in even earlier phases.
Line velocities can either be estimated by directly locating the P-Cygni absorption troughs, as done using \textsc{splot} task of \texttt{IRAF}, or by by modeling the line profiles with velocity as one of the input, as we do in \textsc{synow}. In Fig.~\ref{fig:sp.velall}, we plot the line velocities of \ha, \hb, \Feii\ (\mbox{$\lambda\lambda$}~4924, 5018, 5169) and \Scii\ (\mbox{$\lambda\lambda$}~4670, 6247), using the absorption minima method. It is evident that \Feii\ and \Scii\ line velocities are very close to each other and they are formed at deeper layers, whereas \ha\ and \hb\ line velocities are consistently higher at all phases as they form at larger radii. The \textsc{synow}\ estimated photospheric velocities are also plotted for comparison, which is very close to the \Feii\ and \Scii\ velocities estimated from absorption minima method. Here the \textsc{synow}-derived photospheric velocities are estimated by modelling \Hei\ line for 12d spectrum and \Feii\ lines for rest of the spectra. Velocities for various lines estimated using \textsc{synow}\ are tabulated in Table~\ref{tab:synow}.
\begin{figure}
\includegraphics[width=8.5cm]{./vel_profile.eps}
\caption{Velocity evolution of \ha, \hb, \Hei, \Scii\ and \Feii\ lines. The velocities are estimated using
blueshift of the absorption minima. The expansion velocity of photosphere ($v_{\rm phm}$) estimated from
\textsc{synow}\ modeling of \Hei\ line at 12d and \Feii\ lines at later phases
(see Table~\ref{tab:synow}) are also overplotted for comparison.}
\label{fig:sp.velall}
\end{figure}
Fig.~\ref{fig:sp.velph} shows the comparison of photospheric velocity of SN 2013ej\ with other well-studied type II
SNe 1987A, 1999em, 1999gi, 2004et, 2005cs, 2012aw and 2013ab.
For the purpose of comparison the absorption trough velocities have been used, taking the mean of \Feii\ line triplet, or \Hei\ lines at early phases where \Feii\ lines are not detectable. The velocity profile of SN 2013ej\
is very similar to other normal IIP SNe 1999em, 1999gi, 2004et, 2012aw and 2013ab, on the other hand velocities of SN 2005cs and 1987A are significantly lower. The velocity profile of SN 2013ej\ is almost identical with SNe 2004et, 2012aw and 2013ab, whereas it is consistently higher than SNe 1999gi and 1999em by $ \sim800-900 $\mbox{$\rm{\,km\,s^{-1}}$}.
For comparison of \Hi\ (\ha\ and \hb) velocities, we have chosen all those events which are at least photometrically and spectroscopically similar to SN 2013ej. Comparison reveals that, H velocities during later phases (60-100 d) are consistently higher than all comparable events. SNe 2012aw and 2013ab, have photospheric velocities identical to SN 2013ej, but their H velocities are significantly lower by large values, e.g., for SN 2013ej\ the \ha\ velocity at 80d is higher by 1500 \mbox{$\rm{\,km\,s^{-1}}$}\ and \hb\ is higher by 2400 \mbox{$\rm{\,km\,s^{-1}}$}. Likewise, H velocities for SNe 1999em and 1999gi are even lower at similar phases. Although SN 2004et \Hi\ velocities are somewhat on higher end, they are still significantly less than those of SN 2013ej. It is also to be noted that, at 12d SN 2013ej\ \Hi\ velocities are consistent and similar to those of other normal SNe, but as it evolves these velocities decline relatively slowly, ultimately turning out into a higher velocity profile after $ \sim40 $d.
\begin{figure}
\centering
\includegraphics[width=0.7\linewidth]{./HVLV_comp.eps}
\caption{For 68d spectrum, the \hb\ profile is fitted using \textsc{synow}\ with various velocity components. (a) The fit only with a single high velocity component to match the blue wing of the absorption dip, (b) with a single low velocity component to match the red wing, (c) with single velocity to only fit the trough, (d) with two velocity components to fit entire absorption profile.}
\label{fig:synow.hvlv}
\end{figure}
\subsection{High velocity components of \Hi\ and CSM interaction}
\label{sec:hvcsm}
As discussed in \S\ref{sec:synow}, the broad and extended \ha\ or \hb\ absorption profiles are not properly reproduced using single \Hi\ velocity component in \textsc{synow}, and those profiles can only be fitted by incorporating a high-velocity (HV) components along with the regular one.
{Fig.~\ref{fig:synow.hvlv} shows the comparison of \textsc{synow}\ fits for 68d \hb\ profile with various single velocity as well as for combined two velocity components. A single velocity component at 5600 \mbox{$\rm{\,km\,s^{-1}}$}\ can match the blue wing well and partially the trough, whereas, it does not match the red side at all. Similarly, with a single velocity component at 4000 \mbox{$\rm{\,km\,s^{-1}}$}\ can partially match the red slope of the trough, but does not include the trough as well as the extended blue wing. By only matching the trough position, the model fits for a single velocity of 5300 \mbox{$\rm{\,km\,s^{-1}}$}, which does not fit either of the blue or red wing. Even-though the `detachment' of \Hi\ from photosphere in \textsc{synow}\ model makes the fit of red wing worse by steepening it further, but it is still conclusive that none of these single velocity component can properly reproduce the absorption profile. It is only by including two velocity components together in the model could reproduce the entire \hb\ profile. Such a scenario start to appear from 42d spectrum which only becomes stronger as the line evolves until 97d. The \ha\ troughs are also reproduced in a similar fashion.} {However, it may be noted, that such an extended \Hi\ feature may also be explained as a possible outcome of a different (complex and extended) density profile which \textsc{synow}\ can not reproduce.}
The comparison of \ha\ and \hb\ velocities with other normal IIP SNe (see Fig.~\ref{fig:sp.velph}), estimated by directly locating the P-Cygni absorption troughs, shows that SN 2013ej\ velocities are significantly higher and declines relatively slowly (especially during later phases; 60-100d) as compared to those seen in typical IIP SNe, e.g., 1999em, 1999gi, 2012aw or 2013ab. On the other hand the photospheric velocity comparison with other IIP SNe does not show any such anomaly. This, we suggest as the effect of blending with \Hi\ HV components in \ha\ and \hb, which we could separate out while modeling these broad features with \textsc{synow}\ having two velocity components. The regular \ha\ and \hb\ velocities estimated from \textsc{synow}\ declines at a normal rate consistent to that seen in other SNe (see Fig.\ref{fig:sp.velph}), whereas the HV components remains at higher velocities by $ 1000 - 2000 $ \mbox{$\rm{\,km\,s^{-1}}$}, declining at relatively slower rate. It is also interesting to note that the velocity difference between the regular and HV component for \ha\ and \hb\ is similar at same epochs. \cite{2007ApJ...662.1136C} identified similar HV absorption features associated close to \ha\ and \hb\ troughs in SNe 1999em and 2004dj, which remained constant with time. Presence of such HV features has also been detected in SN 2009bw \citep{2012MNRAS.422.1122I} and SN 2012aw \citep{2013MNRAS.433.1871B} which is suggestive of interaction of SN ejecta with pre-existent CSM. Similar to SN 2013ej, HV signatures has been detected all throughout the plateau phase evolution of SN 2009bw, while in SN 2012aw such features were only detected at late plateau phase (55 to 104d). Although, we found HV components in SN 2013ej\ by modeling the extended P-Cygni troughs, we are unable to visually detect such two individual velocity components, this is possibly because of our signal-to-noise-ratio limited spectra and weaker strength of HV components.
\cite{2007ApJ...662.1136C} argued that SN ejecta can interact with the cooler dense shell of CMS material, which might have originated from the pre-supernova mass loss in the form of stellar winds. Their analysis showed that such interaction can led to the detection of HV absorption features on bluer wings of Balmer lines due to enhanced excitation of the outer layers of unshocked ejecta. We, therefore suggest weak or moderate ejecta-CSM interaction in SN 2013ej.
{X-ray emission from SN 2013ej\ has also been reported by \cite{2013ATel.5243....1M}, which they measured a 0.3-10 keV count-rate of 2.7$ \pm $0.5 cps, translating into a flux of $ \sim1.1\times10^{-13} $ erg~s$ ^{-1} $cm$ ^{-2} $ (assuming simple power-law spectral model with photon index Gamma $ =2 $). Such X-ray emission may also indicate ejecta-CSM interaction suffered by SN 2013ej.}
\begin{figure}
\centering
\includegraphics[width=8.5cm]{./vel_comparison.eps}
\caption{The photospheric velocity (top) evolution ($v_{\rm ph}$) of {SN 2013ej} is compared with other well-studied type II SNe.
The $v_{\rm ph}$ plotted here are the absorption trough velocities (average of \Feii\ lines at late
phases and \Hei\ at early phases). Similar comparison of P-Cygni absorption velocities, but for \ha\ and \hb\ are shown in middle and bottom panels respectively. The regular velocity component for \ha\ and \hb\ estimated from \textsc{synow}\ (without HV components; see Table.~\ref{tab:synow}) are also plotted for comparison.}
\label{fig:sp.velph}
\end{figure}
\section{Status of SN 2013ej\ in type II diversity}
\subsection{Factors favoring SN 2013ej\ as type IIL}
{Having characterized the event both photometrically and spectroscopically, we may now revisit the aspects which favor SN 2013ej\ as type IIL event.
The SN was originally classified as type IIP \citep{2013ATel.5228....1V} based on spectroscopic similarity to SN 1999gi. Due to same underlying physical mechanisms which govern both type IIP and IIL SNe, early spectra may not clearly distinguish these sub classes of SN type II. The distinguishing factor among IIP and IIL is nominal and mainly depend upon light curve characteristics.
SN 2013ej\ shows a decline of 1.74 \mbox{mag (100 d)$ ^{-1} $}\ (see Table~\ref{tab:slopendrop}) or $ \sim0.87 $ mag in 50 days, which definitely falls in the criteria of type IIL SNe as proposed by \cite{2014MNRAS.445..554F}. In Fig.~\ref{fig:lc.abs}, the spread of template light curves for type IIP and IIL \citep{2014MNRAS.445..554F} is shown along with M$ _V $ light curves of SNe sample. It is evident that under this scheme of classification, SN 2013ej\ is not a type IIP, rather it is marginally within the range of type IIL template light curves. This is also justified from the point of basic idea behind these classifications, that type IIP must show a `plateau' of almost constant brightness for some time ($ \sim90$d), which is not the case with SN 2013ej.} {Due to the very fact that SN type II light curves and physical properties exhibit a continuum distribution rather than a bi-modality \citep{2014ApJ...786...67A}, SN 2013ej\ shows intermediate characteristic in the SN type II diversity.}
{One distinguishing spectroscopic property \cite{2014MNRAS.445..554F} found for type IIL SNe is the overall higher photospheric (\Feii~\ld5196) velocity and flatter \Hi\ (\hb\ and \ha) velocity profiles as compared to type IIP counterpart. Although \Feii\ velocities are on the higher end as compared to typical IIP SNe velocities, we do not find it as a remarkable deviation to distinguish SN 2013ej\ from IIP sample. However, we do see a anomaly in \ha, \hb\ absorption minima velocity profiles, as they start off with velocities consistent with those of type IIP but declines relatively slowly (see \S\ref{sec:sp.vel} for more description of this feature) ultimately surpassing faster declining IIP velocity profiles after 50 days. This characteristic feature of \Hi\ velocities for SN 2013ej\ is typical for most IIL SNe as found by \cite{2014MNRAS.445..554F}.}
\subsection{CSM interaction and type IIL}
{\cite{2014MNRAS.445..554F} proposed a possible explanation for the flatter velocity profiles in IIL SNe, which is due the lack of hydrogen in deeper and slow expanding layers of ejecta, resulting into higher \Hi\ absorption velocities arising mostly from outer layer. However, for SN 2013ej\ we suggest the flattening of \ha\ and \hb\ velocity profiles are due to the contamination of HV component of \Hi\ (see \S\ref{sec:hvcsm}). Indication of CSM interaction in SN 2013ej\ may also be inferred from X-ray detection by \cite{2013ATel.5243....1M}.
\cite{2015arXiv150106491V} found SN 2013by, a type IIL SN, to be moderately interacting with CSM. This led them to ask the prevalence of CSM interaction among IIL SNe in general. Type IIL SNe originate from progenitors similar to IIPs, but have lost a significant fraction of hydrogen before explosion during pre SN evolution.
Hence it may not be usual to detect HV \Hi\ signatures in \ha, \hb\ absorption profiles as a consequence of ejecta-CSM interaction. A moderate or weak interaction may produce a HV component blending with \ha, \hb\ profiles, which may result into shift in absorption minima, rather than a prominent secondary HV dip.
Such a scenario may perfectly explain the relatively higher and flatter \Hi\ velocity profiles of most type IIL SNe as compared to IIP counterparts, found by \cite{2014MNRAS.445..554F} based on direct velocity estimates of absorption minima.}
{Another example of CSM interaction in type IIL is SN 2008fq, which does show strong interaction signature like a type IIn \citep{2013A&A...555A..10T}, but also shows a steep decline like IIL during first 60 days \citep{2014MNRAS.445..554F}. {Supernova PTF11iqb \citep{2015MNRAS.449.1876S} is also a type IIn SN, having prominent CSM interaction signatures, but with IIL like steeper light curve. Initial spectra of this SN showed IIn characteristics, however late plateau spectra revealed features similar to type IIL. PTF11iqb originated from a progenitor identical to type IIP/L, instead of a LBV as expected for a typical IIn.} However, because of rare detection of type IIL events and its fast decline in magnitudes we do not have sufficient information to investigate CSM interaction in all such objects. Thus, the question still remains open if all or most IIL SNe interact with CSM and whether the flatter \Hi\ absorption minima velocity profiles is a consequence of interaction.}
\section{Light curve modelling}\label{modelling}
To determine the explosion parameters of SN 2013ej, the observed light curve is modeled following the semi-analytical approach originally developed by \cite{1980ApJ...237..541A} and further refined in \cite{1989ApJ...340..396A}.
More appropriate and accurate approach would have been detailed hydrodynamical modeling \citep[e.g.][]{1977ApJS...33..515F,2007A&A...461..233U,2011ApJ...729...61B,2011ApJ...741...41P} to determine explosion properties, however application of simple semi-analytical models \citep{1980ApJ...237..541A,1982ApJ...253..785A,1989ApJ...340..396A,1993ApJ...414..712P,2003MNRAS.338..711Z,2012ApJ...746..121C} can be useful to get preliminary yet reliable estimates of the parameters without running resource intensive and time consuming hydrodynamical codes. \cite{2014A&A...571A..77N} also followed the original semi-analytical formulation presented by \cite{1989ApJ...340..396A} and modeled a few well studied II SNe. The results are compared with hydrodynamical models from the literature and are found to be in good agreement.
The model light-curve is computed by solving the energy balance of the spherically symmetric supernova envelope, which is assumed to be in homologous expansion having spatially uniform density profile.
The temperature evolution is given as \citep{1980ApJ...237..541A},
\[ T(x,t)^4=T_0^4\psi(x)\phi(t)\left(\frac{R_0}{R(t)}\right)^4 ,\]
where $ x $ is defined as dimensionless co-moving radius relative to the mass of the envelope and, $ \psi(x) $ is the radial component of temperature profile which falls off with radius as $ sin(\pi x)/\pi x $.
Here we incorporate the effect of recombination, as shock heated and ionized envelope expands and cools down to recombine at temperature $ T_{rec} $. We define $ x_i $ as the co-moving radius of the recombination front and the opacity ($ \kappa $) changes very sharply at this layer such that $ \kappa \approx 0 $ for the ejecta above $ x_i $. Following the treatment of \cite{1989ApJ...340..396A} the
temporal component of temperature, $ \phi(t) $ can be expressed as \citep{2014A&A...571A..77N},
\[ \frac{d\phi(t)}{dz}= \frac{R(t)}{R_0 x_i^3}\left[p_1\zeta(t)-p_2\phi(t)x_i-2 x_i^2 \phi(t) \frac{R_0}{R(t)}\frac{dx_i}{dz}\right] ,\]
here $ \zeta(t) $ is the total radioactive energy input from decay chain of unit mass of \mbox{$^{56}$Ni}, which is normalized to the energy production rate of \mbox{$^{56}$Ni}. The rest of the parameters in the equation have usual meaning and can be found in aforementioned papers. From this ordinary differential equation we can find out the solution of $ \phi(t) $ using Runge-Kutta method. The treatment adopted to determine $ x_i $ is somewhat similar to \cite{2014A&A...571A..77N}, where we numerically determine the radius $ x_i $ (to an accuracy of $ 10^{-12} $) for which the temperature of the layer reaches $ T_{rec} $. Once we find out the solution of $ \phi(t) $ and $ x_i $, the total bolometric luminosity is calculated as the sum of radioactive heating and rate of energy released due to recombination,
\[ L(t)=x_i\frac{\phi(t)E_{th}(0)}{\tau_d}\left(1-e^{-A_g/t^2}\right)+4\pi r_i^2 Q\rho(x_i,t)R(t)\frac{dx_i}{dt} , \]
here, $ d(x_i)/dt $ is the inward velocity of co-moving recombination front and
the term $ [1-exp(-A_g/t^2)] $, takes into account of gamma-ray leakage from the ejecta.
The factor $ A_g $ is the effectiveness of gamma ray trapping {\citep[see e.g., ][]{1997ApJ...491..375C,2012ApJ...746..121C}}, where large $ A_g $ means full trapping of gamma rays, this factor is particularly important to model the SN 2013ej\ tail light curve. In this relation we also modified the second term to correctly account for the amount of envelope mass being recombined.
\begin{figure}
\centering
\hspace{-0.5cm}
\includegraphics[width=1.05\linewidth]{./model.eps}
\caption{Model fit (solid line) on the observed bolometric light curve (open circles) of SN 2013ej. The green solid line follows only the radioactive decay law, where the recombination front has completely disappeared.}
\label{fig:model}
\end{figure}
To model SN light curves it is essential to obtain the true bolometric luminosity from observations. Since our data is limited only to optical and UV bands, we adopt the prescription for color dependent bolometric corrections by \cite{2009ApJ...701..200B} to obtain bolometric light curve for SN 2013ej. Figure~\ref{fig:model} shows the model fit with the observed bolometric light curve of the SN. We estimate an ejecta mass of 12 \mbox{M$_{\odot}$}, progenitor radius of 450 \mbox{R$_{\odot}$}\ and explosion energy (kinetic + thermal) of 2.3 foe ($ 10^{51} $ erg). The uncertainty in mass and radius is about 25\%. We find that the plateau duration is strongly correlated with explosion energies (especially kinetic), and also with $ \kappa $ and $ T_{rec} $. Thus depending upon these parameters our model is consistent with a wide range of explosion energies, with 2.3 foe towards the lower end and energies up to 4.5 foe at higher end. Assuming the mass of the compact remnant to be 1.5-2.0 \mbox{M$_{\odot}$}, the total progenitor mass adds up to be 14\mbox{M$_{\odot}$}.
The mass of radioactive \mbox{$^{56}$Ni}\ estimated from the model is 0.018\mbox{M$_{\odot}$}, which primarily governs the tail light curve of the SN. As discussed in \S\ref{sec:lc.bol}, the slope of the tail light curve observed for SN 2013ej\ is significantly higher than other typical IIP SNe and also to that expected from radioactive decay of \mbox{$^{56}$Co}\ to \mbox{$^{56}$Fe}.
The light curve powered by full gamma-ray trapping from radioactive decay chain of $ \mbox{$^{56}$Ni} \rightarrow \mbox{$^{56}$Co} \rightarrow \mbox{$^{56}$Fe} $ results in a slower decline and does not explain the steeper tail observed in SN 2013ej.
In the model we decreased the gamma-ray trapping effectiveness parameter $ A_g $ to $ 3\times10^4 $ day$ ^2 $, which matches the steeper radioactive tail. The gamma-ray optical depth can be related to this parameter as $ \tau_g\sim A_g/t^2 $. This implies that the gamma-ray leakage in SN 2013ej\ is significantly higher than other typical type IIP SNe.
\cite{2014MNRAS.438L.101V} using early temperatures ($ <5 $ days) of SN 2013ej\ provided a preliminary estimate of the progenitor radius as $ 400-600 $ \mbox{R$_{\odot}$}, which is in good agreement with our result. Our progenitor mass estimate is also consistent with that reported by \cite{2014MNRAS.439L..56F} from direct observational identification of the progenitor using \textit{HST} archival images, which is $ 8 - 15.5 $ \mbox{M$_{\odot}$}.
\section {Summary} \label{sec:sum}
We present photometric and spectroscopic observations of SN 2013ej. Despite low cadence optical photometric follow up during photospheric phase, we are able to cover most of the important phases and features of light curve.
Our high resolution spectrum at 80d shows the presence of \Nai~D (\mbox{$\lambda\lambda$}~5890, 5896) doublet for Milky Way, while no impression for host galaxy NGC 0628. This indicates that SN 2013ej\ suffers minimal or no reddening due to its host galaxy.
The optical light curves are similar to type IIL SNe, with a relatively short plateau duration of 85d and steeper decline rates of 6.60, 3.57, 1.74, 1.07 and 0.74 mag 100 day$ ^{-1} $ in \textit{UBVRI} bands respectively. The comparison of absolute \textit{V} band light curves shows that SN 2013ej\ suffers the higher decline rate than all type IIP SNe, but similar to type IIL SNe 1980k, 2000dc and 2013by.
The drop in luminosity during the plateau-nebular transition is also higher than most type II SNe in our sample, which is 2.4 mag in \textit{V} band.
The UVOT UV optical light curves shows steep decline during first 30 days at a rate of 0.182, 0.213, 0.262 mag d$ ^{-1} $ in \textit{uvw1, uvw2} and \textit{uvm2} bands respectively. The absolute UV light curves are identical to SN 2012aw and also shows a similar UV-plateau trend as observed in SN 2012aw.
Owing to the large drop in luminosity during plateau-nebular transition, the light curve settles to a significantly low luminous tail phase as compared to other normal IIP SNe. The mass of radioactive \mbox{$^{56}$Ni}\ estimated from the tail bolometric luminosity is $ 0.020\pm0.002 $ \mbox{M$_{\odot}$}, which is in between normal IIP SNe (e.g., 1999em, 2004et, 2012aw) and subluminous events, like SN 2005cs.
The spectroscopic features and their evolution is similar to normal type II events. Detailed \textsc{synow}\ modelling has been performed to identify spectral features and to estimate velocities for \ha, \hb, \Feii\ (\mbox{$\lambda\lambda$}~4924, 5018, 5169) and \Scii\ (\mbox{$\lambda\lambda$}~4670, 6247) lines. The photospheric velocity profile of SN 2013ej, which is represented by \Feii\ lines and \Hei\ line at 12d, is almost identical to SNe 2004et, 2012aw and 2013ab.
The \ha, \hb\ velocities estimated by directly locating the absorption troughs are significantly higher and slow declining as compared to other normal IIP events. However, such \Hi\ velocity profiles are typical for type IIL SNe.
{The P-Cygni absorption troughs of \ha\ and \hb\ are found to be broad and extended which a single \Hi\ component in \textsc{synow}\ model could not fit properly. However, these extended features are fitted well with \textsc{synow}\ by incorporating a high velocity \Hi\ component.
These HV components can be traced throughout the photospheric phase which may indicate possible ejecta-CSM interaction.
Our inference is also supported by the detection of X-ray emission from the SN 2013ej\ \citep{2013ATel.5243....1M} indicating possible CSM interaction, and the unusually high polarization reported by \cite{2013ATel.5275....1L} may also further indicate asymmetry in environment or ejecta of the SN.
Such CSM interaction and their signature in \ha, \hb\ profiles has also been reported for SNe 2009bw \citep{2012MNRAS.422.1122I} and 2012aw \citep{2013MNRAS.433.1871B}.}
Nebular phase spectra during 109 to 125d phases are dominated by characteristic emission lines, however the \ha\ line shows an unusual notch, which may be explained by superposition of HV emission on regular \ha\ profile. Although, the origin of the feature is not fully explained, it may indicate bipolar distribution of \mbox{$^{56}$Ni}\ in the core.
We modeled the bolometric light curve of SN 2013ej\ and estimated a progenitor mass of $ \sim14 $\mbox{M$_{\odot}$}, radius of $ \sim450 $\mbox{R$_{\odot}$}\ and explosion energy of $ \sim2.3$ foe. These progenitor property estimates are consistent to those given by \cite{2014MNRAS.439L..56F} and \cite{2014MNRAS.438L.101V} for mass and radius respectively. The tail bolometric light curve of SN 2013ej, is found to be significantly steeper than that expected from decay chain of radioactive \mbox{$^{56}$Ni}. Thus, in the model we decreased the effectiveness of gamma ray trapping, which could explain the steeper slope of tail light curve.
\acknowledgments
We are thankful to the observing staffs and technical assistants of ARIES 1.0-m and 1.3-m telescopes and we also express our thanks to 2-m HCT telescope staffs for their kind cooperation in observation of SN 2013ej. We also express our thanks to Mr. Shashank Shekhar for his sincere efforts and co-operation during observations at ARIES 1.3m telescope. Authors gratefully acknowledge the services of the
NASA ADS and NED databases which are used to access data and references in this paper. SOUSA is supported by NASA's Astrophysics Data Analysis Program through grant NNX13AF35G. VVDs work on Type IIP SNe is supported by the NASA through Chandra Award Number GO2-13092B issued by the Chandra X-ray Observatory Center, which is operated by the Smithsonian Astrophysical Observatory for and on behalf of the NASA under contract NAS8-03060.
We also thank the anonymous referee for detailed and insightful comments which helped in significant improvement of the manuscript.
|
{
"redpajama_set_name": "RedPajamaArXiv"
}
| 5,215
|
namespace LiveSplit.AutoSplitting.Editors
{
partial class AutoSplitEditor
{
/// <summary>
/// Required designer variable.
/// </summary>
private System.ComponentModel.IContainer components = null;
/// <summary>
/// Clean up any resources being used.
/// </summary>
/// <param name="disposing">true if managed resources should be disposed; otherwise, false.</param>
protected override void Dispose(bool disposing)
{
if (disposing && (components != null))
{
components.Dispose();
}
base.Dispose(disposing);
}
#region Windows Form Designer generated code
/// <summary>
/// Required method for Designer support - do not modify
/// the contents of this method with the code editor.
/// </summary>
private void InitializeComponent()
{
this.tlpMain = new System.Windows.Forms.TableLayoutPanel();
this.label1 = new System.Windows.Forms.Label();
this.txtName = new System.Windows.Forms.TextBox();
this.label2 = new System.Windows.Forms.Label();
this.cbEvent = new System.Windows.Forms.ComboBox();
this.flpConfirmButtons = new System.Windows.Forms.FlowLayoutPanel();
this.btnOK = new System.Windows.Forms.Button();
this.btnCancel = new System.Windows.Forms.Button();
this.gbVariables = new System.Windows.Forms.GroupBox();
this.tlpConditions = new System.Windows.Forms.TableLayoutPanel();
this.tlpListBtn = new System.Windows.Forms.TableLayoutPanel();
this.btnAdd = new System.Windows.Forms.Button();
this.btnRemove = new System.Windows.Forms.Button();
this.lstVariables = new System.Windows.Forms.ListBox();
this.tlpMain.SuspendLayout();
this.flpConfirmButtons.SuspendLayout();
this.gbVariables.SuspendLayout();
this.tlpConditions.SuspendLayout();
this.tlpListBtn.SuspendLayout();
this.SuspendLayout();
//
// tlpMain
//
this.tlpMain.ColumnCount = 2;
this.tlpMain.ColumnStyles.Add(new System.Windows.Forms.ColumnStyle());
this.tlpMain.ColumnStyles.Add(new System.Windows.Forms.ColumnStyle(System.Windows.Forms.SizeType.Percent, 100F));
this.tlpMain.Controls.Add(this.label1, 0, 0);
this.tlpMain.Controls.Add(this.txtName, 1, 0);
this.tlpMain.Controls.Add(this.label2, 0, 1);
this.tlpMain.Controls.Add(this.cbEvent, 1, 1);
this.tlpMain.Controls.Add(this.flpConfirmButtons, 0, 3);
this.tlpMain.Controls.Add(this.gbVariables, 0, 2);
this.tlpMain.Dock = System.Windows.Forms.DockStyle.Fill;
this.tlpMain.Location = new System.Drawing.Point(0, 0);
this.tlpMain.Name = "tlpMain";
this.tlpMain.Padding = new System.Windows.Forms.Padding(3, 0, 3, 0);
this.tlpMain.RowCount = 4;
this.tlpMain.RowStyles.Add(new System.Windows.Forms.RowStyle());
this.tlpMain.RowStyles.Add(new System.Windows.Forms.RowStyle());
this.tlpMain.RowStyles.Add(new System.Windows.Forms.RowStyle(System.Windows.Forms.SizeType.Percent, 100F));
this.tlpMain.RowStyles.Add(new System.Windows.Forms.RowStyle());
this.tlpMain.Size = new System.Drawing.Size(334, 206);
this.tlpMain.TabIndex = 0;
//
// label1
//
this.label1.Anchor = ((System.Windows.Forms.AnchorStyles)((System.Windows.Forms.AnchorStyles.Left | System.Windows.Forms.AnchorStyles.Right)));
this.label1.AutoSize = true;
this.label1.Location = new System.Drawing.Point(6, 6);
this.label1.Name = "label1";
this.label1.Size = new System.Drawing.Size(38, 13);
this.label1.TabIndex = 0;
this.label1.Text = "Name:";
//
// txtName
//
this.txtName.Anchor = ((System.Windows.Forms.AnchorStyles)((System.Windows.Forms.AnchorStyles.Left | System.Windows.Forms.AnchorStyles.Right)));
this.txtName.Location = new System.Drawing.Point(50, 3);
this.txtName.Name = "txtName";
this.txtName.Size = new System.Drawing.Size(278, 20);
this.txtName.TabIndex = 0;
//
// label2
//
this.label2.Anchor = ((System.Windows.Forms.AnchorStyles)((System.Windows.Forms.AnchorStyles.Left | System.Windows.Forms.AnchorStyles.Right)));
this.label2.AutoSize = true;
this.label2.Location = new System.Drawing.Point(6, 33);
this.label2.Name = "label2";
this.label2.Size = new System.Drawing.Size(38, 13);
this.label2.TabIndex = 2;
this.label2.Text = "Event:";
this.label2.Visible = false;
//
// cbEvent
//
this.cbEvent.Anchor = ((System.Windows.Forms.AnchorStyles)((System.Windows.Forms.AnchorStyles.Left | System.Windows.Forms.AnchorStyles.Right)));
this.cbEvent.DropDownStyle = System.Windows.Forms.ComboBoxStyle.DropDownList;
this.cbEvent.Enabled = false;
this.cbEvent.FormattingEnabled = true;
this.cbEvent.Location = new System.Drawing.Point(50, 29);
this.cbEvent.Name = "cbEvent";
this.cbEvent.Size = new System.Drawing.Size(278, 21);
this.cbEvent.TabIndex = 1;
this.cbEvent.TabStop = false;
this.cbEvent.Visible = false;
//
// flpConfirmButtons
//
this.flpConfirmButtons.Anchor = System.Windows.Forms.AnchorStyles.None;
this.flpConfirmButtons.AutoSize = true;
this.flpConfirmButtons.AutoSizeMode = System.Windows.Forms.AutoSizeMode.GrowAndShrink;
this.tlpMain.SetColumnSpan(this.flpConfirmButtons, 2);
this.flpConfirmButtons.Controls.Add(this.btnOK);
this.flpConfirmButtons.Controls.Add(this.btnCancel);
this.flpConfirmButtons.Location = new System.Drawing.Point(86, 174);
this.flpConfirmButtons.Name = "flpConfirmButtons";
this.flpConfirmButtons.Size = new System.Drawing.Size(162, 29);
this.flpConfirmButtons.TabIndex = 0;
//
// btnOK
//
this.btnOK.Location = new System.Drawing.Point(3, 3);
this.btnOK.Name = "btnOK";
this.btnOK.Size = new System.Drawing.Size(75, 23);
this.btnOK.TabIndex = 0;
this.btnOK.Text = "OK";
this.btnOK.UseVisualStyleBackColor = true;
this.btnOK.Click += new System.EventHandler(this.btnOK_Click);
//
// btnCancel
//
this.btnCancel.DialogResult = System.Windows.Forms.DialogResult.Cancel;
this.btnCancel.Location = new System.Drawing.Point(84, 3);
this.btnCancel.Name = "btnCancel";
this.btnCancel.Size = new System.Drawing.Size(75, 23);
this.btnCancel.TabIndex = 5;
this.btnCancel.Text = "Cancel";
this.btnCancel.UseVisualStyleBackColor = true;
//
// gbVariables
//
this.tlpMain.SetColumnSpan(this.gbVariables, 2);
this.gbVariables.Controls.Add(this.tlpConditions);
this.gbVariables.Dock = System.Windows.Forms.DockStyle.Fill;
this.gbVariables.Location = new System.Drawing.Point(6, 56);
this.gbVariables.Name = "gbVariables";
this.gbVariables.Size = new System.Drawing.Size(322, 112);
this.gbVariables.TabIndex = 2;
this.gbVariables.TabStop = false;
this.gbVariables.Text = "Conditions";
//
// tlpConditions
//
this.tlpConditions.ColumnCount = 2;
this.tlpConditions.ColumnStyles.Add(new System.Windows.Forms.ColumnStyle(System.Windows.Forms.SizeType.Percent, 100F));
this.tlpConditions.ColumnStyles.Add(new System.Windows.Forms.ColumnStyle());
this.tlpConditions.Controls.Add(this.tlpListBtn, 1, 0);
this.tlpConditions.Controls.Add(this.lstVariables, 0, 0);
this.tlpConditions.Dock = System.Windows.Forms.DockStyle.Fill;
this.tlpConditions.Location = new System.Drawing.Point(3, 16);
this.tlpConditions.Name = "tlpConditions";
this.tlpConditions.RowCount = 1;
this.tlpConditions.RowStyles.Add(new System.Windows.Forms.RowStyle(System.Windows.Forms.SizeType.Percent, 100F));
this.tlpConditions.Size = new System.Drawing.Size(316, 93);
this.tlpConditions.TabIndex = 0;
//
// tlpListBtn
//
this.tlpListBtn.AutoScroll = true;
this.tlpListBtn.AutoSize = true;
this.tlpListBtn.AutoSizeMode = System.Windows.Forms.AutoSizeMode.GrowAndShrink;
this.tlpListBtn.ColumnCount = 1;
this.tlpListBtn.ColumnStyles.Add(new System.Windows.Forms.ColumnStyle());
this.tlpListBtn.Controls.Add(this.btnAdd, 0, 0);
this.tlpListBtn.Controls.Add(this.btnRemove, 0, 1);
this.tlpListBtn.Dock = System.Windows.Forms.DockStyle.Top;
this.tlpListBtn.Location = new System.Drawing.Point(282, 0);
this.tlpListBtn.Margin = new System.Windows.Forms.Padding(0);
this.tlpListBtn.Name = "tlpListBtn";
this.tlpListBtn.RowCount = 2;
this.tlpListBtn.RowStyles.Add(new System.Windows.Forms.RowStyle());
this.tlpListBtn.RowStyles.Add(new System.Windows.Forms.RowStyle());
this.tlpListBtn.Size = new System.Drawing.Size(34, 62);
this.tlpListBtn.TabIndex = 1;
//
// btnAdd
//
this.btnAdd.Anchor = System.Windows.Forms.AnchorStyles.None;
this.btnAdd.BackgroundImageLayout = System.Windows.Forms.ImageLayout.Zoom;
this.btnAdd.FlatAppearance.BorderSize = 0;
this.btnAdd.FlatAppearance.MouseDownBackColor = System.Drawing.Color.Transparent;
this.btnAdd.FlatAppearance.MouseOverBackColor = System.Drawing.Color.Transparent;
this.btnAdd.FlatStyle = System.Windows.Forms.FlatStyle.Flat;
this.btnAdd.Location = new System.Drawing.Point(3, 0);
this.btnAdd.Margin = new System.Windows.Forms.Padding(3, 0, 3, 3);
this.btnAdd.Name = "btnAdd";
this.btnAdd.Size = new System.Drawing.Size(28, 28);
this.btnAdd.TabIndex = 0;
this.btnAdd.Text = "+";
this.btnAdd.UseVisualStyleBackColor = true;
this.btnAdd.Click += new System.EventHandler(this.btnAdd_Click);
//
// btnRemove
//
this.btnRemove.Anchor = ((System.Windows.Forms.AnchorStyles)((((System.Windows.Forms.AnchorStyles.Top | System.Windows.Forms.AnchorStyles.Bottom)
| System.Windows.Forms.AnchorStyles.Left)
| System.Windows.Forms.AnchorStyles.Right)));
this.btnRemove.BackgroundImageLayout = System.Windows.Forms.ImageLayout.Zoom;
this.btnRemove.FlatAppearance.BorderSize = 0;
this.btnRemove.FlatAppearance.MouseDownBackColor = System.Drawing.Color.Transparent;
this.btnRemove.FlatAppearance.MouseOverBackColor = System.Drawing.Color.Transparent;
this.btnRemove.FlatStyle = System.Windows.Forms.FlatStyle.Flat;
this.btnRemove.Location = new System.Drawing.Point(3, 31);
this.btnRemove.Margin = new System.Windows.Forms.Padding(3, 0, 3, 3);
this.btnRemove.Name = "btnRemove";
this.btnRemove.Size = new System.Drawing.Size(28, 28);
this.btnRemove.TabIndex = 1;
this.btnRemove.Text = "-";
this.btnRemove.UseVisualStyleBackColor = true;
this.btnRemove.Click += new System.EventHandler(this.btnRemove_Click);
//
// lstVariables
//
this.lstVariables.Dock = System.Windows.Forms.DockStyle.Fill;
this.lstVariables.FormattingEnabled = true;
this.lstVariables.IntegralHeight = false;
this.lstVariables.Location = new System.Drawing.Point(3, 3);
this.lstVariables.Margin = new System.Windows.Forms.Padding(3, 3, 0, 3);
this.lstVariables.Name = "lstVariables";
this.lstVariables.Size = new System.Drawing.Size(279, 87);
this.lstVariables.TabIndex = 0;
this.lstVariables.DoubleClick += new System.EventHandler(this.lstVariables_DoubleClick);
//
// AutoSplitEditor
//
this.AutoScaleDimensions = new System.Drawing.SizeF(6F, 13F);
this.AutoScaleMode = System.Windows.Forms.AutoScaleMode.Font;
this.AutoSize = true;
this.AutoSizeMode = System.Windows.Forms.AutoSizeMode.GrowAndShrink;
this.CancelButton = this.btnCancel;
this.ClientSize = new System.Drawing.Size(334, 206);
this.ControlBox = false;
this.Controls.Add(this.tlpMain);
this.MinimumSize = new System.Drawing.Size(350, 222);
this.Name = "AutoSplitEditor";
this.ShowIcon = false;
this.SizeGripStyle = System.Windows.Forms.SizeGripStyle.Hide;
this.StartPosition = System.Windows.Forms.FormStartPosition.CenterParent;
this.Text = "AutoSplit Editor";
this.tlpMain.ResumeLayout(false);
this.tlpMain.PerformLayout();
this.flpConfirmButtons.ResumeLayout(false);
this.gbVariables.ResumeLayout(false);
this.tlpConditions.ResumeLayout(false);
this.tlpConditions.PerformLayout();
this.tlpListBtn.ResumeLayout(false);
this.ResumeLayout(false);
}
#endregion
private System.Windows.Forms.TableLayoutPanel tlpMain;
private System.Windows.Forms.Label label1;
private System.Windows.Forms.TextBox txtName;
private System.Windows.Forms.Label label2;
private System.Windows.Forms.ComboBox cbEvent;
private System.Windows.Forms.FlowLayoutPanel flpConfirmButtons;
private System.Windows.Forms.Button btnOK;
private System.Windows.Forms.Button btnCancel;
private System.Windows.Forms.GroupBox gbVariables;
private System.Windows.Forms.TableLayoutPanel tlpConditions;
private System.Windows.Forms.TableLayoutPanel tlpListBtn;
private System.Windows.Forms.Button btnAdd;
private System.Windows.Forms.Button btnRemove;
private System.Windows.Forms.ListBox lstVariables;
}
}
|
{
"redpajama_set_name": "RedPajamaGithub"
}
| 5,487
|
{"url":"https:\/\/papers.nips.cc\/paper\/2021\/hash\/76f1cfd7754a6e4fc3281bcccb3d0902-Abstract.html","text":"#### Authors\n\nMingguo He, Zhewei Wei, zengfeng Huang, Hongteng Xu\n\n#### Abstract\n\nMany representative graph neural networks, $e.g.$, GPR-GNN and ChebNet, approximate graph convolutions with graph spectral filters. However, existing work either applies predefined filter weights or learns them without necessary constraints, which may lead to oversimplified or ill-posed filters. To overcome these issues, we propose $\\textit{BernNet}$, a novel graph neural network with theoretical support that provides a simple but effective scheme for designing and learning arbitrary graph spectral filters. In particular, for any filter over the normalized Laplacian spectrum of a graph, our BernNet estimates it by an order-$K$ Bernstein polynomial approximation and designs its spectral property by setting the coefficients of the Bernstein basis. Moreover, we can learn the coefficients (and the corresponding filter weights) based on observed graphs and their associated signals and thus achieve the BernNet specialized for the data. Our experiments demonstrate that BernNet can learn arbitrary spectral filters, including complicated band-rejection and comb filters, and it achieves superior performance in real-world graph modeling tasks. Code is available at https:\/\/github.com\/ivam-he\/BernNet.","date":"2022-05-28 14:25:18","metadata":"{\"extraction_info\": {\"found_math\": true, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 0, \"math_alttext\": 0, \"mathml\": 0, \"mathjax_tag\": 0, \"mathjax_inline_tex\": 1, \"mathjax_display_tex\": 0, \"mathjax_asciimath\": 0, \"img_math\": 0, \"codecogs_latex\": 0, \"wp_latex\": 0, \"mimetex.cgi\": 0, \"\/images\/math\/codecogs\": 0, \"mathtex.cgi\": 0, \"katex\": 0, \"math-container\": 0, \"wp-katex-eq\": 0, \"align\": 0, \"equation\": 0, \"x-ck12\": 0, \"texerror\": 0, \"math_score\": 0.6742882132530212, \"perplexity\": 2129.3814073408093}, \"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\/1652663016853.88\/warc\/CC-MAIN-20220528123744-20220528153744-00640.warc.gz\"}"}
| null | null |
\section{Introduction}
\label{sec1}
The SARS-CoV-2 virus was first registered in humans in Wuhan, China by December 2019. Since then, it has reached almost all countries around the globe with pandemic proportions and devastating effects. This contagion seems unprecedented and can already be rules as the definite health crisis of the $21^{\text{rst}}$ century. The SARS-CoV-2 virus causes a severe acute respiratory syndrome, which can become potentially fatal. This virus spreads rapidly and efficiently: by mid-June the virus had already infected over $6$ million people. As of mid-July, over $13$ million COVID-19 cases have been confirmed.
To tackle and extenuate the effects caused by this virus, global scientific efforts have been called out, being urgently necessary \cite{bedford19}. Unfortunately, vaccines take quite some time to be procedure and are, \textit{a priori}, previewed to be ready only by mid-$2021$. To retain the diffuse of this disease, most countries have chosen to adopt social distancing measures (in different levels and with diverse strategies) since March, $2020$ \cite{Adam2020}. This has been explicitly pointed out as the most pertinent control option for the COVID-19 outbreak, including the cases of countries with large social inequalities, as Brazil \cite{baumgartner2020social}. It should be mentioned that the concept underneath social distancing is to impede the saturation of health systems due to large amounts of active COVID-19 infections, which would require treatment at the same time. In this way, when social distancing policies are enacted, the demands for treatment become diluted over time, and the health systems do not have to deal with hospital bed shortages associated with a large peak of active infections.
Brazil has shown itself as quite a particular case regarding COVID-19 \cite{werneck2020covid}: the country is very large, with $26$ federated states, and each federated state has had autonomy to choose its own health policies, which has lead to different levels of social distancing measures for each state. Furthermore, there has been no coordinated nation-wide public health policy to address the viral spread by the federal government, which is very reluctant to do so, claiming that the negative economic effects are too steep and that social distancing is an erroneous choice \cite{THELANCET20201461,zacchi:hal-02881690}. The country currently ranks as second with respect to numbers of cases and deaths. The expectations disclosed on the recent literature suggest catastrophic scenarios for the next few months \cite{rocha2020expected, morato2020optimal}, which might pursue until mid-$2021$.
Regarding this discussion, we consider the COVID-19 contagion data from Brazil in this paper. In order to better illustrate the Brazilian scenario with respect to other countries, Figure \ref{DoubleFig} depicts the evolution of the SARS-CoV-2 contagion curves in different countries of the world, considering the cumulative cases confirmed according to the pandemic period of each country (\ref{Cases_Country}). We note that, as of July $7$th, $2020$, Brazil counts over $1.5$ million confirmed cases of the SARS-CoV-2 virus. The right-side of this Figure (\ref{Cases_map}) maps the concentration level of COVID-19 cases, per country. Complementary, Figure \ref{googlecovidbr} shows the ``hot-spots" of COVID-19 cases in Brazil. The state of S\~ao Paulo concentrates the most number of infections.
\begin{figure}[htb]
\centering
\begin{subfigure}{.5\textwidth}
\centering
\includegraphics[width=1\linewidth]{./Cases_World.pdf}
\caption{The COVID-19 contagion curve worldwide (in days, since the $100$th confirmed case).}
\label{Cases_Country}
\end{subfigure}%
\begin{subfigure}{.5\textwidth}
\centering
\includegraphics[width=1\linewidth]{./COVID_map.pdf}
\caption{World map COVID-19 case concentration level.}
\label{Cases_map}
\end{subfigure}
\caption{COVID-19 in the world. Data published by the European Center for Disease Prevention and Control (ECDC), \cite{owidcoronavirus}.}
\label{DoubleFig}
\end{figure}
\begin{figure}[htb]
\centering
\includegraphics[width=0.5\linewidth]{./googlecovidbr.png}
\caption{Brazilian map w.r.t. to ``hot-spots" of COVID-19 cases, from Google, dating $30/06/2020$.}
\label{googlecovidbr}
\end{figure}
Brazil is currently facing many issues due to the SARS-CoV-2 contagion, despite having a strong universal health system. The current situation is near-collapsing, since the majority of Intense Care Unit (ICU) hospital beds are occupied with COVID-19 patients, all around the country. In addition, the virus is progressing to farthest western cities of the country, away from urban areas, where medical care is somehow less present. It can also be noted that SARS-CoV-2 is currently posing great threat to indigenous communities, such as the Yanomami and Ye'kwana ethnicities\footnote{The Brazilian
Socioenvironmental Institute (ISA, \textit{Instituto Socioambiental}, see \url{https://www.socioambiental.org/en}) has released a technical note \cite{ferrante2020protect} which warns for the contagion of COVID-19 of up to $40 \, \%$ of Yanomami Indigenous Lands, amid the states of Amazonas and Roraima and a long the border between Brazil and Venezuela, due to the presence of approximately $20000$ illegal mining prospectors. Datasets regarding the COVID-19 spread amid indigenous communities are available in \url{https://covid19.socioambiental.org}.}. Overall, the current situation in the country is very critical.
The first official death due to the SARS-CoV-2 virus in Brazil was registered in March, $17$th $2020$. Anyhow, through inferential statistics, \citet{delatorre2020tracking} acknowledge the fact that community transmission has been ongoing in the state of S\~ao Paulo since the beginning of February (over one month before the first official reports). This points to empirical evidence that the true amount of infected individuals, and possibly registered deaths, are actually very under-reported. Moreover, due to the absence of mass testing, the country is only accounting for COVID-19 patients with moderate to severe symptoms. People with mild or no symptoms are not being accounted for, following guidelines from the federal Ministry of Health. Additionally, we know that the SARS-CoV-2 virus is, for a large number of individuals, an
asymptomatic disease (yet transmissible). For these reasons, the scientific community has been warning for a possibly huge margins of underestimated cases in
Brazil \cite{silva2020bayesian, rocha2020expected, delatorre2020tracking, paixo2020estimation,bastos2020covid19}.
Regarding this issue, it becomes fundamental to perform coordinated social distancing interventions, as soon as possible. Furthermore, these public health interventions should be put in practice at the correct time and for the correct duration. Well-designed social distance guidelines should help mitigating the contagion and thus avoiding the saturation of the heath systems, while, at the same time, trying to balance social and economic side-effects by releasing/relaxing the quarantine measures as soon as possible. Modeling and identifying the COVID-19 epidemiological process, and designing an optimal controller that addresses this issue are the main focuses of this paper.
Regarding model frameworks for this contagion, recent literature has demonstrated how the SARS-CoV-2 viral contagion dynamics can be very appropriately described by Susceptible-Infected-Recovered-Deceased (SIRD) models \cite{peng2020epidemic,kucharski2020early}. These SIRD models comprise coupled nonlinear differential equations, as originally presented for population dynamics in work of \citet{Kermack1927}. Therefore, the Nonlinear Model Predictive Control (NMPC) \cite{camacho2013model} framework shows itself as a rather convenient approach to guide model-based public health policies, given the fact that it can adequately consider the nonlinear dynamic of the virus spread (through these SIRD equations) together with the effect of lockdown/quarantine measures, introduced as constraints of the optimization problem. Recent literature has indeed shown how SIRD-based NMPCs can be applied to plan COVID-19 social distancing measures in three novel works:
\begin{itemize}
\item \citet{alleman2020covid} consider the application of a NMPC algorithm regarding the data from Belgium. The control input is the actual isolation parameter that is plugged into the SIRD model;
\item \citet{morato2020optimal} consider an optimal On-Off MPC design, which is formulated as mixed-integer problem with dwell-time constraints. Furthermore, the response of the population to social isolation rules with an additional dynamic variable, which is then incorporated to the SIRD dynamics;
\item \citet{kohler2020robust} applied the technique for the German scenario. The approach considers that the control input is a variable that directly affects the infection and transmission rates (parameters of the SIRD model).
\end{itemize}
Building upon this previous results, herein we propose an NMPC algorithm formulated on the basis of a SIRD-kind model, which is obtained by the means of an identification algorithm. Following the lines of \citet{morato2020optimal}, we assumed that the control action is a social isolation guideline, which is enacted and passed on to the population. Then, the population responds in some time to these measures, which can be mathematically described as a dynamic social isolation factor. Furthermore, in accordance with the discussion presented by \citet{kohler2020robust}, we also introduce an additional dynamic nonlinear model, which gives the input/output (I/O) relationship between the COVID-19 contagion infection and transmission rates according to the stage of the pandemic, as suggested by \citet{bastos2020covid19}. Complementary, we use auto-regressive equations for the epidemiological parameters of the disease (transmission factor, infection factor, lethality rate) directly related to social distancing as control input. These time-varying regressions are used to provide better long-term forecasts than the regular SIRD equations. Regarding the NMPC framework, the novelty in this paper resides in the formulation of the control input as a finitely parametrized variable, which makes the NMPC implementation persuadable through search mechanisms, which run much faster than the Nonlinear Programming methods from the prior.
Motivated by the previous discussion, the problem of how optimal predictive control can be used to formulate adequate social distancing policies, regarding Brazil, is investigated in this work. The main contribution of our approach comprises the following ingredients:
\begin{itemize}
\item Firstly, an adapted SIRD model is proposed, which incorporates delayed auto-regressive dynamics for the transmission, infection and lethality rates of the SARS-CoV-2 virus, which vary according to the stage of the pandemic. The adapted model also incorporates the dynamic response of the population to a given social distancing guideline, which is the NMPC control input.
\item In order to identify the SIRD model parameter auto-regressive dynamics (for the epidemiological parameters), we use a three-layered identification procedure, which concatenates analytical expansions and Least-Squares optimization procedures. The identified models are validated with regard to the available set of data from Brazil, which includes the social distancing factor observed in the country.
\item The main innovation of the paper is the proposed Parametrized Nonlinear Predictive Control algorithm, which is based on the identified adapted SIRD model with auto-regressive epidemiological parameters. The control input (social distancing guideline) is finitely parametrized over the NMPC prediction horizon, at each sampling instant. Then, an explicit nonlinear programming solver is simulated for all possible input sequences along time. The NMPC solution, then, is simply found through a search mechanism regarding these simulated sequences, which is rather numerically cost-efficient.
\item Finally, we present results considering the application of this NMPC algorithm to the COVID-19 scenario in Brazil. These results are a twofold: a) those that regard the application of the optimal strategy to control the pandemic \textit{since its beginning}, comparing the simulation results to those seen in practice; and b) applying the control method from $30$th of July onward, aiming to mitigate and revert the current health crisis catastrophe.
\end{itemize}
This paper is structured as follows. Section \ref{sec2} presents the new SIRD model with auto-regressive epidemiological parameters, the respective identification procedure and validation results. Section \ref{sec3} discusses the proposed NMPC strategy. Section \ref{sec4} depicts the obtained control results, regarding the COVID-19 contagion mitigation for Brazil. General conclusions are drawn in Section \ref{sec5}.
\section{Model, Identification and Validation}
\label{sec2}
In this Section, we describe the used modeling framework for the COVID-19 contagion dynamics in Brazil. We build upon the SIRD models for the SARS-CoV-2 virus from works of \citet{peng2020epidemic, kucharski2020early, ndairou2020mathematical}. The model adaptations are done such that the epidemiological parameters are taken as time-varying piece-wise constant functions of the stage of the pandemic. This adaptation is in accordance with recent immunology
results \cite{sun2020understanding, dowd2020demographic, he2020temporal}, which discuss the transmission and reproduction rate of this virus. It is also taken into account two different sampling periods, to include the issue of the incubation period of the virus in human, reportedly, in
average, as $5.1$ days \cite{lauer2020incubation}.
\subsection{SIRD Epidemiological Model}
The SIRD model describes the spread of a given disease with respect to a population split into four non-intersecting classes, which stand for:
\begin{itemize}
\item The total amount of susceptible individuals, that are prone to contract the disease at a given (discrete) sample of time $k$, denoted through the dynamic variable $S(k)$;
\item The individuals that are currently infected with the disease (active infections at a given sample of time $k$), denoted through the dynamic variable $I(k)$;
\item The total amount of recovered individuals, that have already recovered from the disease, from an initial instant $k_0$ until the current sample $k$, denoted through the dynamic variable $R(k)$;
\item Finally, the total amount of deceased individuals due to a fatal SARS-CoV-2 infection, from an initial instant $k_0$ until the current sample $k$, is denoted $D(k)$.
\end{itemize}
Due to the evolution of the spread of the disease, the size of each of these classes change over time. Therefore, the total population size \(N\) is the sum of the first three classes as follows:
\begin{eqnarray}
N(k)&=&S(k)+I(k)+R(k)\label{eq:Nconstant} \quad \text{.}
\end{eqnarray}
Since the Brazilian government discloses \textbf{daily samples} of total infections and accumulated deaths, we consider that these discrete-time dynamics samples $k$, given each $T_1 \, = \, 1$ day. Furthermore, to account for the average incubation period of the disease, we consider that the epidemiological parameters vary weekly (each $T_2 \, = \, 7$ days), kept as zero-order-held/piece-wise-constant samples. We will further assess this matter in the sequel.
In the SIRD model, there are three major epidemiological parameters, which express the specific dynamics of the SARS-CoV-2 virus in the population set. The dynamics of these parameters are given in the sparser discrete-time \textbf{weekly samples}, since they change according to the incubation of the virus:
\begin{itemize}
\item The transmission rate parameter \(\beta\) stands for the average number of contacts that are sufficient for transmission of the virus from one individual. According to the detailed classes of individuals, then, it follows that \(T_1\beta(k) I(k)/N(k)\)
determines the number of contacts that are sufficient for transmission from infected individuals to one susceptible
individual; and \((T_1\beta (k) I(k)/N(k))S(k)\) determines the number of new cases (per day) due to the amount of \(S(k)\) susceptible individuals (these are the ones ``available for infection'').
\item The infectiousness Poisson parameter \(\gamma\) denotes the inverse of the period of time a given infected individual is indeed infectious. Consequently, $\gamma$ affects the rate of recovery (or death) of an infected person. This parameter directly quantifies the amount of individuals that ``leaves'' the infected class, in a given sample.
\item The mortality rate parameter \(\rho\) stands for the
observed mortality rate of the COVID-19 contagion. We model the amount of deceased individuals due to the SARS-CoV-2 infection following the lines of \citep{keeling2011}: the new number of deaths, at each day, can be accounted for through the following expressions: $\frac{T_1\rho(k)}{1-\rho(k)} \gamma(k) I(k)$
\end{itemize}
Considering these explanations, the ``{\bf SIRD}''
(Susceptible-Infected-Recovered-Dead) model is expressed through the
following nonlinear discrete-time difference equations:
\begin{equation}
\left\{\begin{array}{rcl}
S(k+1) &=& S(k) - T_1\left(1-\psi(k)\right)\frac{\beta (k) I(k) S(k)}{N(k)} \\[3mm]
I(k+1) &=& I(k) + T_1\left(1-\psi(k)\right)\frac{\beta (k) I(k)
S(k)}{N(k)} - T_1\gamma (k)\frac{I(k)}{1-\rho(k)}
\\[3mm]
I_S (k+1) &=& p_{\text{sym}}I(k+1) \\[3mm]
I_{c} (k+1) &=& \left(I(k+1) + R(k+1) + D(k+1) \right) \\[3mm]
R(k+1) &=& R(k) + T_1\gamma (k)I(k) \\[3mm]
D(k+1) &=& D(k) + T_1\frac{\rho(k)}{1-\rho(k)}\gamma (k) I_S (k)
\\[3mm]\end{array}\right.\;\;\;\textrm{\bf [SIRD]} \quad \text{,}
\label{eqSIRDmodel}
\end{equation}
where $I_S$ denotes the portion of the infected individuals which in fact display symptoms. This ``symptomatic" class has been accounted for in previous papers \cite{bastos2020modeling,morato2020optimal}. We note that only these symptomatic individuals will require possible hospitalization and may die. The remainder $(1-p_{\text{sym}})I$ are asymptomatic or lightly-symptomatic, which do transmit the virus but do not die or require hospitalization. The class of recovered individuals considers immunized people, encompassing those that displayed symptoms and those that did not. The symptomatic ration parameter $p_{\text{sym}}$ is constant and borrowed from previous papers.
The cumulative number of cases $I_c$ denotes the total number of people that have been infected by the SARS-CoV-2 virus until a given day. In fact, this cumulative variable is equivalent to those that are currently infected $I$, summed with those that have recovered $R$ and those that have died $D$.
We must also stress that, in the SIRD model equations given above, parameter $\psi$ represents a transmission rate mitigation factor. This factor accounts for the observed social isolation factor with the population set $N$. It follows that $\psi \, = \, 0$ denotes the situation where the whole population set has sustained social interactions. As discussed in previous papers \citep{keeling2011,bastos2020covid19}, there exists some "natural" $\psi \, = \, \underline{\psi}$ factor, which stands for a normal/nominal conditions, still with ``no control'' of the viral spread (with no social isolation guidelines, this kind of situation has been observed in Brazil in the first weeks of the contagion, end of February, beginning of March). In contrast, $\psi \, = \, 1$ represents a complete lockdown scenario, where there are no more social interactions (this conditions has not been seen and is, in practice, unattainable). In practice, there is some maximal social isolation factor $\psi \, = \, \overline{\psi}$ that can be put in practice. In this paper, we consider the values for social isolation in Brazil from the work of \citet{bastos2020covid19}, which discloses weekly estimates for $\psi$.
Another essential information in epidemiology theory is the basic reproduction number, usually denoted by $R_0$. This factor is able to measure the average potential transmissibility of the disease. In practice, this value is fixed/constant and inherent to a given disease. Through the sequel, a time-varying version of this basic reproduction number is considered, denoted $R_t$, which represents how many COVID-19 cases could be expected to be generated due to a single primary case, in a population for which all individuals are susceptible. From a control viewpoint, $R_t$ represents the epidemic spread velocity: if $R_t> 1$, the infection is spreading and the number of infected people increases along time (this typically happens at the beginning of the epidemic), otherwise, if $R_t<1$, it means that more individuals ``leave'' from the infected class, either recovering or dying, and, thereby, the epidemic ceases. The reproduction number $R_0$ is affected by different factors, including immunology of the virus, biological characteristics, but also governments policies to control the number of susceptible population. Since the epidemic parameters change along the time, we process by considering the effective reproduction number $R_0$ also as a dynamic variable.
The underlying assumption used to calculate $R_t$, is that, at the beginning of the pandemic, $S\approx N$. Considering parameters $\beta$, $\gamma$, $\rho$ and $\psi$ from the SIRD model Eq. \eqref{eqSIRDmodel}, $R_t$ can be roughly given by:
\begin{eqnarray}
\label{R0eq} R_t(k) \approx \frac{(1-\psi (k))\beta (k)(1-\rho (k))}{\gamma
(k)} \quad \text{.}
\end{eqnarray}
\begin{remark}
Regarding the SIRD model given in Eq. \eqref{eqSIRDmodel}, it
follows that $N(0) = N_0$ is the initial population
size (prior to the viral infection). Furthermore, we stress that, in SIRD-kind models, $I(k)$
represents the \textbf{active} infections at a given
moment, while $D(k)$ represents the \textbf{total} amount of deaths until this
given moment; for this reason, it follows that $D(k+1) - D(k)$ is
proportionally dependent to $I(k)$.
\end{remark}
\begin{remark}
We note that we are not able to use more ``complex'' descriptions
of the COVID-19 contagion in Brazil, such
as the ``SIDARTHE'' model used by \citet{kohler2020robust} (that
splits the infections into (symptomatic, asymptomatic) detected,
undetected, recovered, threatened and extinct) because we have
insufficient amount of data. The Ministry of Health only disclosed
the total amount of infections ($I(k) + D(k) + R(k)$) and the total
amount of deaths ($D(k)$) at each day. Due to the
absence of mass (sampled) testing, there is no data regarding
detected asymptomatic individuals, for instance, as it is available
in Germany (where \citep{kohler2020robust} originate). To choose a
more complex model may only decrease the truthfulness/validity of
the identification results, since the parameters can only represent
a singular combination that matches the identification datasets and
cannot be used for forecasting/prediction purposes.
\end{remark}
To illustrate the dynamics of a contagion like the pandemic outbreak
of COVID-19, in Figure \ref{forecasts} we present long-term forecast using a regular SIRD model
with constant epidemiological parameters, following the methodology
presented by \citet{bastos2020covid19}. These predictions were
computed on $11/06/2020$ and are here shown to demonstrate the
qualitative evolution of the $I$ and $D$ curves. The active
infections curve $I$ shows a increase-peak-decrease characteristic,
while the total number of deaths $D$ shows an asymptotic behavior to some steady-state condition. We note that, as of $30/06$, the number of infections had already significantly increased, which means that the most recent forecasts preview an even worse catastrophe.
\begin{figure}[htb]
\centering
\includegraphics[width=\linewidth]{./forecasts.pdf}
\caption{Long-term SIRD Forecasts for Brazil (from $11/06/2020$). Consolidated datasets ($I(t)$ and $D(t)$): a)
No-Control Situation (\textcolor{red}{$\mathbf{-}$}), b)
Hard social distancing ($\psi \, = \, 0.6$) from $11/06/2020$ onwards
(\textcolor{blue}{$\mathbf{-}$}), c) $\psi = 0.6$ Hard social distancing applied from March
(\textcolor{green}{$\mathbf{-}$}) and d) Real
data ($\times$). Shades represent total
variation over a $95\%$ confidence interval, solid lines represent mean values.}
\label{forecasts}
\end{figure}
\subsection{Model Extensions}
\label{modelextensionsec}
In this paper, the SIRD dynamics from Eq. \eqref{eqSIRDmodel} are adapted. These adaptations are included for three main reasons:
\begin{enumerate}
\item In accordance with the discussions presented by \citet{morato2020optimal},
we understand that it is imperative to embed to the SIRD model how
the population responds to public health policies, that is, to
incorporate the dynamics and effects of social distancing. When a
government enacts some social distancing policy, the population
takes some time to adapt to it and to, in fact, exhibit the
expected social distancing factor $\psi$. This is, to take $\psi$ as a time-varying dynamic map of the enacted social distancing guidelines (which will, later on, be defined by an optimal controller).
\item Many results seen in the literature \cite{bastos2020modeling,
ImperialCollegeNew, ndairou2020mathematical} consider constant
parameter values for $\gamma$, $\beta$ and $\rho$, understanding
that these factor are inherent to the disease. Anyhow, recent
results \cite{bastos2020covid19, kucharski2020early} indicate that these parameters are indeed time-varying, achieve steady-state values
at ending stages of the pandemic. It is reasonable that when more
active infections occur, the virus tends to spread more
efficiently, for instance. Moreover, the mortality rate should stabilize after
the infection curve has decreased. To incorporate this issue in the
model, auto-regressive moving-average time-varying dynamics for the viral
parameters are proposed, which stabilize according to stage of the pandemic. Furthermore, by doing so, the
inherent incubation characteristic of the virus is also taken into
account, since the viral transmission, lethality and infection
rates should vary with dynamics coherent with the average
incubation period.
\item Recent papers provide forecasts
\cite{bastos2020modeling, ImperialCollegeNew} for the evolution of
the virus assuming that the epidemiological parameters remain
constant along the prediction horizon. This is not reasonable, due
to the fact that such parameters are inherently time-varying, given
according to the exhibited pandemic level. Thus, if parameters are
held constant for forecasts, these forecasts are only qualitative
and allow short-term conclusions. Since dynamic models
for the epidemiological parameters are proposed, make more
coherent long-term forecasts can be derived. Of course, it must be stressed that these
forecasts will still be qualitative, since one cannot account for
perfect accurateness regarding the number of infection and deaths
due to the absence of mass testing in Brazil. Furthermore, the effect of future unpredicted phenomena cannot be accounted for, such as the
early development of an effective vaccine, which would certainly
make the infections drop largely.
\end{enumerate}
These model adaptations are discussed individually in the sequel.
\subsubsection{Social Distancing Guidelines / The Control Input}
In order to design and synthesize effective control strategies for
social distancing (public) policies, to be oriented to the population
by local governments, the social distancing factor $\psi$ is further
exploited.
In what concerns the available data from Brazil (that is used for identification of the model parameters), the social distancing factor $\psi$ is a known variable, given in weekly piece-wise constant samples. Later on, regarding the control procedure (Section \ref{sec4}), this isolation factor will
vary according to the enacted social distancing policy $u$, as defined by
a nonlinear optimal predictive control algorithm.
The differential equation that models the response of the susceptible population to quarantine guidelines is taken as suggests \cite{morato2020optimal}, this is:
\begin{equation}
\left\{ \begin{array}{rcl}
\psi (k+1) & = & \psi (k) +
T_2\varrho_{\psi}\left(K_\psi(k)u(k)
- \psi(k)\right) \end{array}\right. \quad \text{,}
\label{TheControlPsi}
\end{equation}
where $u(k)$ is the actual control input, the guideline that defines the social isolation factor goal, as regulated by the government (this signal will be later on determined by the proposed optimal controller), $\varrho_{\psi} \, \, = \,\, 0.4317 \,\, \rm{day^{-1}}$ is a settling-time parameter, which is related to the average time the population takes to respond to the enacted social isolation measures, and
$K_\psi$ is a time-varying stating gain relationship between $\psi$ and $u$.
As recommend \citet{morato2020optimal}, we assume that when more infected cases have been reported, and when the hospital bed occupation surpasses $70 \%$, the population will be more prone to follow the social distancing guidelines, with larger values for the gain relationship $K_\psi$:
\begin{eqnarray}
\label{TheKPsiEq}
K_\psi(k) &=& \max\left\{1 \,\text{,} \,\frac{p_{\text{sym}}I(k)}{0.7n_{ICU}} \right\} \quad \text{,}
\end{eqnarray}
being $n_{ICU}$ the total number of ICU beds in the country. We recall that $p_{\text{sym}}$ is a parameter which gives the amount of infected individuals that in fact display symptoms and may need to be hospitalized ($I_s \, = \, p_{\text{sym}}I$).
\begin{remark}
We note that Brazil has, as of February, $45,848$ ICU beds. This number is estimated to increase in up to $80 \, \rm{\%}$ with field hospitals that were built specifically for the COVID-19 contagion (i.e. $n_{ICU} \, = \, 82,526$). The percentage of symptomatic individuals is taken as $16 \, \rm{\%}$, according to the suggestions of \citet{bastos2020modeling}. This value is coherent with the available information regarding this virus, concerning multiple countries \cite{lima2020information}. The percentage of symptomatic considered groups the infections with severe/acute symptoms, which will, indeed, most possibly require treatment.
\end{remark}
\begin{remark}
In practice, Eq. \eqref{TheControlPsi} is bounded to the minimal and maximal values for the social distancing factor $\underline{\psi}$ and $\overline{\psi}$, respectively. These values are the same are that used as saturation constraints on $u$, as discussed in the sequel.
\end{remark}
We proceed by considering that $u$ is a finitely parametrized control input. This is: the enacted social distancing guidelines can only be given within a set of pre-defined values. This approach is coherent with possible ways to enforce and put in practice social distancing measures. Pursuing this matter, the actual control signal, that is to be defined by the proposed optimal automatic controller,
must abide to a piece-wise constant characteristic with a possible increase/decrease of $5\%$ of isolation per week. This percentage is the average increase of social isolation, in the beginning of the pandemic in Brazil, when social distancing measures were strengthened significantly over a small time period. The values for social isolation $\psi(k)$ in Brazil have been estimated by \citet{InLoco} and are presented in Figure \ref{psi_br}.
\begin{figure}[htb]
\centering
\includegraphics[width=\linewidth]{./Psi.pdf}
\caption{Social distancing factor in Brazil from February $26$th, $2020$ to June, $30$th $2020$ \cite{InLoco}.}
\label{psi_br}
\end{figure}
Moreover, $u$ is considered to be defined within the admissibility set $[\underline{\psi} \, , \, \overline{\psi}]$. As gives Figure \ref{psi_br}, which shows the observed social isolation factor $\psi$ in Brazil, the ``natural" isolation factor is of $\underline{\psi} \, = \, 0.3$ (which stands for a situation of no-isolation guidelines). Furthermore, $\overline{\psi}$ is the maximal attainable
isolation factor for the country; the maximal observed value in Brazil is of roughly $53 \, \%$ (see Figure \ref{psi_br}). Anyhow, coordinated ``lockdown" measures were not forcefully enacted. For this matter, we consider $\overline{\psi} \,=\,
0.7$, which would represent that the population can be restricted to, at most, a $70\%$ reduction on the level of social interactions. As reported by \citet{bastos2020covid19}, it seems unreasonable to consider larger values for social isolation in the country, due to multiple reasons (hunger, social inequalities, laboring
necessities, lack of financial aid from the government for people to
stay home, \textit{etc.}).
\begin{remark}
It must noted that the SIRD model considers some ``natural" isolation factor (which is the lower bound on $u$). As seen in Figure \ref{psi_br}, the social isolation factor of $30\,\rm{\%}$ is the observed lower bound for $\psi$ seen in Brazil.
\end{remark}
Mathematically, these constraints are expressed through the following Equation set:
\begin{eqnarray}
\label{UConstraints} \left\{\begin{array}{rcl}
u(k) &=& u(k-1) + \delta u(k) \\
\delta u (k) &=& \{-0.05 \text{ or } 0 \text{ or } 0.05\}\\
u(k) &\in& \left[\begin{array}{ccccccc}0.30 & 0.35& 0.40& \dots &
0.60
&0.65&
0.70 \end{array}\right] \end{array}\right. \quad \text{.}
\end{eqnarray}
This constraints given in Eq. \eqref{UConstraints} are, in fact, very
interesting from an implementation viewpoint, as the government could
convert the finitely parametrized values for $u$ into actual
practicable measures, as illustrates Table
\ref{TheTableUParametre}. We remark that this Table is only
illustrative. Epidemiologists and viral specialists should be the ones
to formally discuss the actual implemented measures that ensure the
social distancing factor guideline given by $u$.
\begin{table}[htbp]
\caption{\label{TheTableUParametre} Illustrative Example of
Finitely Parametrized Social Distancing Measures.}
\centering
\begin{tabularx}{\textwidth}{|c | X | c |}
\hline \hline
Control Signal / Social Distancing Guideline ($u$) & Implemented
measures &
Infection
Risk \\\hline
$u \, = \, 0.3$ & No public health emergency. All economy
sectors can return to their normal activities. & Controlled Contagion \\ \hline
$\vdots$ & $\vdots$ & \\\hline
$u \, = \, 0.35$ & Low restriction levels. Use of masks to go
outside. Public transport functioning. Limited opening of
shops and small public spaces. & Low Risk \\ \hline
$\vdots$ & $\vdots$ & \\\hline
$u \, = \, 0.4$ & Moderate restrictions. Use of masks to go
outside. Closed public spaces. Restricted openings only. & Moderate Risk \\ \hline
$\vdots$ & $\vdots$ & \\\hline
$u \, = \, 0.45$ & Very restrictive policies. Reduced public
transport. Urge for people to stay home at all times. Very
restrictive openings only. & High Risk \\ \hline
$\vdots$ & $\vdots$ & \\\hline
$u \, = \, 0.7$ & Severe restrictive policies. No public
transport. Urge for people to stay home at all times. Only
basic services may open, with reduced capacities. & Very High Risk \\ \hline
\end{tabularx}
\end{table}
\subsubsection{Dynamic Epidemiological Parameter Models}
The second main modification of the original SIRD model is to consider
dynamic models for the epidemiological parameters of the SARS-CoV-2
virus, $\gamma$, $\beta$ and $\rho$. These models are taken as auto-regressive moving average functions, which converge as the pandemic progresses. Therefore, the following dynamics are considered:
\begin{eqnarray}
\label{TheDynamicsEpModel}
\left\{\begin{array}{rcl}
\beta (k) &=& f_\beta \left(\beta (k-1)\, \dots, \beta (k-n_\beta)\right) \\
\gamma (k) &=& f_\gamma \left(\gamma (k-1)\, \dots, \gamma (k-n_\gamma)\right) \\
\rho (k) &=& f_\rho \left(\rho (k-1)\, \dots, \rho (k-n_\rho)\right) \\ \end{array}\right. \quad \text{.}
\end{eqnarray}
The models given in Eq. \eqref{TheDynamicsEpModel} are possibly delayed and auto-regressive. Anyhow, despite the
parameters $\beta$, $\gamma$ and $\rho$ being time-varying, the
model functions $f_\beta$, $f_\gamma$ and $f_\rho$ are constant. The order and number of regressions are found through an optimization procedure, which is further detailed in Section \ref{Identifsec}.
\subsubsection{The Complete COVID-19 Model}
The complete model used in this work to describe the COVID-19
contagion outbreak in Brazil is illustrated through the
block-diagram of Figure \ref{modeltotal}. We note that the
``COVID-19 Epidemiological Parameter Models'', represented through
Eq. \eqref{TheDynamicsEpModel}, enable the complete model to offer
long-term predictions with more accurateness, as discussed in the
beginning of this Section. We denote as the ``SIRD$+$ARIMA" model, henceforth, the cascade of the population response from Eq. \eqref{TheControlPsi} to the SIRD model, with auto-regressive time-varying epidemiological parameters as give Eq. \eqref{TheDynamicsEpModel}.
\begin{figure}[htb]
\centering
\includegraphics[width=\linewidth]{./modeltotal2.png}
\caption{The ``SIRD$+$ARIMA" model for COVID-19 in Brazil.}
\label{modeltotal}
\end{figure}
\subsection{Identification Procedure}
\label{Identifsec}
Recent numerical algorithms have been applied to estimate the model parameters of the COVID-19 pandemic \cite{bastos2020covid19,SARKAR,SUN2020,KENNEDY20,Caccavo2020,Oliveira2020,Villaverde2020estimating}. The majority of these papers indicates that SIRD models provide the best model-data fits.
We note, anyhow, that the SIRD model offers three degrees-of-freedom at each instant $k$ (i.e. $\beta(k)$, $\gamma (k)$ and $\rho(k)$), as gives Equation \ref{eqSIRDmodel}. This means that different instantaneous combinations of these parameters can yield the same numerical values for $S(k)$, $I(k)$, $R(k)$ and $D(k)$. Therefore, although mathematical and graphical criteria have been used to validate these dynamic models when compared to real data, the estimated values for these parameters should be coherent with biological characteristics of this viral pandemic. An indicative of badly adjusted SIRD model parameters is the effective reproduction number of the disease $R_0$, which should naturally decrease to a threshold smaller than one as the pandemic ceases.
Thus, in order to identify a SIRD model that is indeed able to describe the pandemic contagion accurately (which is especially important for forecast and control purposes), we proposed a two step identification method to determine the time-varying dynamics for $\beta(k)$, $\gamma(k)$ and $\rho(k)$, with respect to the dynamic parameter models in Eq. \eqref{TheDynamicsEpModel}.
Regarding datasets, we must comment that the Brazilian Ministry of Health discloses, daily, the values for the cumulative number of confirmed cases $I_{c}(k)$ and the cumulative number of deaths $D(k)$. Furthermore, we recall that, through the sequel, the values of the observed average social isolation in the country, $\psi(k)$, are assumed to be known (as given in Figure \ref{psi_br}). The symptomatic percentage is assumed as constant along the prediction horizons.
Then, we proceed by depicting the used identification procedure, which is performed in three consecutive steps/layers, as follows:
\begin{itemize}
\item The first step resides in analytically solving the SIRD regressions from Equation \ref{eqSIRDmodel} for a fixed interval of points (days).
\item Then, the derived values from the analytical solutions are passed as initial conditions to an optimization layer, which solves a constrained Ordinary Least Square minimization problem, aiming to adjust the parameter values to pre-defined (biologically coherent) sets, so that the identified model yields small error with adequate parameter choices. The output of this second step stands for the time-series vectors regarding the SIRD parameters.
\item Finally, these time-series are used to fit auto-regressive models, via Least Squares, as gives Eq. \eqref{TheDynamicsEpModel}.
\end{itemize}
By following this three-layered procedure, the estimated parameter equations (Eq. \eqref{TheDynamicsEpModel}) are found in accordance with biological conditions, which are described as the pre-defined optimization sets. We note that we embed to the optimization layer sets that are also in accordance with previous results for SIRD model estimations for Brazil \cite{bastos2020modeling,morato2020optimal}.
\begin{figure}[htb]
\centering
\includegraphics[width=1\linewidth]{./Iden_flux_isa.png}
\caption{Parameter Identification Procedure.}
\label{ident_algorithm}
\end{figure}
The complete identification procedure used to estimate the SIRD model dynamics follows the lines illustrated in Figure \ref{ident_algorithm}.
\subsection{First Layer: Analytical Solutions}
We note that, in order to simplifying the formulation of the optimization algorithm, the difference Equations for $I(k+1)$ and $D(k+1)$ are modified in order to decouple parameters related to the number of fatal and related to the number of recovered individuals, with regard to $I(k)$. This is:
\begin{eqnarray}
I(k+1) &=& I(k) + T_1\left(1-\psi(k)\right)\frac{\beta (k) I(k)S(k)}{N(k)} - T_1\gamma (k)I(k) - T_1\alpha (k)I(k) \\[3mm]
D(k+1) &=& D(k) + T_1\alpha(k) I (k)
\end{eqnarray}
\noindent where $\alpha(k) = \rho(k) \frac{\gamma(k)}{(1-\rho(k))}$. Therefore, instead of identifying $\beta$, $\gamma$ and $\rho$, we pursue the identification of $\beta$, $\gamma$ and $\alpha$, and, then, $\rho$ is computed as follows:
\begin{eqnarray}
\label{rhoalfarelationship}
\rho (k) &=& \frac{\frac{\alpha(k)}{\gamma (k)}}{1+\frac{\alpha(k)}{\gamma (k)}}\quad \text{.}
\end{eqnarray}
The identification procedure starts by collecting the available datasets (regarding the cumulative number of COVID-19 cases $I_c$ and deaths $D$) inside a fixed interval $k \, \in \,[k_{i}, k_{f}]$. Then, the first layer computes ``exact", analytical values for the epidemiological parameters, denoted $\tilde{\beta}$, $\tilde{\gamma}$ and $\tilde{\alpha}$ according to the following discrete analytical expansions:
\begin{eqnarray}
\tilde{\gamma} &=& \frac{R(k_f)-R(k_i)}{\sum_{i=k_i}^{i=k_f}I(i)} \, \text{,}\\[3mm]
\tilde{\alpha} &=& \frac{D(k_f)-D(k_i)}{\sum_{i=k_i}^{i=k_f}I(i)} \, \text{,}\\[3mm]
\tilde{\beta}&=&\frac{1}{(1-\psi(k_f))} \frac{I(k_{f})-I(k_{i})+(\tilde{\alpha}+\tilde{\gamma})\sum_{i=k_{i}}^{i=k_{f}} I(i)}{\sum_{i=k_{i}}^{i=k_{f}}S(i)I(i)/N(i)} \, \text{.}
\end{eqnarray}
\subsection{Second Layer: Ordinary Least Squares Optimization}
It is assumed that the used datasets might be corrupted by series of issues, such as cases that are not reported on given day $k$ and simply accounted for on the following days, or sub-reported cases, as discussed by \citet{bastos2020covid19}. Therefore, we adjust these parameters through the optimization layer, in order to improve the reliability of the identification.
Once again, the available data from the same interval discrete $[k_{i}, k_{f}]$ is used. The optimization procedure minimizes a constrained Ordinary Least Squares problem, whose solution comprises the following vectors: $\boldsymbol{S}$, $\boldsymbol{I}$, $\boldsymbol{R}$ and $\boldsymbol{D}$. These vectors comprise the model-based outputs within the considered discrete interval. The decision variables for the optimization procedure are the degrees-of-freedom of the SIRD model, in this layer denoted $\hat{\beta}$, $\hat{\gamma}$, $\hat{\alpha}$. The Ordinary Least Square criterion ensures that the optimization minimizes the error between the real data and the estimated values from the SIRD model, by choosing coherent values for the decision variables.
The quadratic model-data error variables terms used in the optimization layer are the following:
\begin{eqnarray} \label{eq:err}
Er_{I}(i) &=& (I(i) - \hat{I}(i,\hat{\beta} (1-\psi),\hat{\gamma},\hat{\alpha}))^2\text{,} \\ \label{eq:err2}
Er_{R}(i) &=& (R(i) - \hat{R}(i,\hat{\beta}(1-\psi),\hat{\gamma},\hat{\alpha}))^2 \text{,}\\ \label{eq:err3}
Er_{D}(i) &=& (D(i) - \hat{D}(i,\hat{\beta}(1-\psi),\hat{\gamma},\hat{\alpha}))^2 \text{,}
\end{eqnarray}
\noindent for which the variables $\hat{I}, \hat{R}, \hat{D}$ are estimated according to the SIRD model equations. The complete optimization problem is formulated as follows:
\begin{eqnarray}
\nonumber
&&\underset{\beta,\gamma,\alpha}{\min} J = \sum_{i=k_i}^{i=k_f} \left(w_1Er_I(i) + w_2 Er_R(i) + w_3 Er_D(i)\right) \text{,} \\ \label{eq:opt}
\text{s.t.:} && \underline{\delta} \tilde{\beta}(1-\psi(i)) \leq \beta (1-\psi(i)) \leq \overline{\delta} \tilde{\beta}(1-\psi(i)) \text{,}\\ \nonumber
&& \underline{\delta}\tilde{\gamma} \leq \gamma \leq \overline{\delta}\tilde{\gamma} \text{,}\\ \nonumber
&& \underline{\delta}\tilde{\alpha} \leq \alpha \leq \overline{\delta} \tilde{\alpha}\text{.} \\ \nonumber
&& \begin{bmatrix}
\beta_0\\
\gamma_0\\
\alpha_0 \end{bmatrix} = \begin{bmatrix}
\tilde{\beta} \\
\tilde{\gamma} \\
\tilde{\alpha}
\end{bmatrix} \text{.}
\end{eqnarray}
\noindent wherein $\underline{\delta}$ and $\overline{\delta}$ are uncertainty interval margins used to define the lower and upper bound of each decision variable on the optimization problem, $\beta_0$, $\gamma_0$ and $\alpha_0$ are the initial conditions of the optimization problem and $w_1$, $w_2$ and $w_3$ are taken as positive weighting values (tuning parameters), used to normalize the magnitude order of the total cost and adjust the model fit with respect to variables $Er_I$, $Er_R$ and $Er_D$.
With respect to the discussion regarding the average incubation period of the SARS-CoV-2 virus, the optimization procedure compute piece-wise constant values for $\beta(k)$, $\gamma(k)$ and $\alpha(k)$ each $T_2 \, = \, 7$ days. This is also done to improve the model-data fitting results, as suggest \citet{morato2020optimal}.
Considering the application of this second layer to a complete data-set, with new values for the epidemiological parameter each $T_2$ days, the output of this layer stands for the epidemiological time-series vectors denoted $\beta_{opt}$, $\gamma_{opt}$ and $\alpha_{opt}$. These time-series are, then, used to fit the auto-regressive models in Eq. \eqref{TheDynamicsEpModel}, in the third layer. We note that these SIRD parameter dynamics are the ones that can be can be used for forecasting and control purposes (and also to calculate the effective reproduction number $R_t$).
\subsection{Third Layer: ARIMA fits}
The third layer of the identification procedure resides in fitting an auto-regressive model to the time-series derived from the optimization $\beta_{opt}$, $\gamma_{opt}$ and $\rho_{opt}$ (note that $\rho_{opt}$ is given as a function of $\beta_{opt}$, $\gamma_{opt}$ and $\alpha_{opt}$, as gave Eq. \eqref{rhoalfarelationship}). In this paper, the auto-regressive functions in Eq. \eqref{TheDynamicsEpModel} are of Auto-Regressive Integrated Moving Average (ARIMA) kind. The ARIMA approach is widely used for prediction of epidemic diseases, as shown by \citet{Kirbas2020}. It follows that, from a time-series viewpoint, the ARIMA model can express the evolving of a given variable (in this case, the epidemiological parameters) based on prior values. Such models, then, are coherent with the prequel discussion regarding the convergence of these parameters to steady-state conditions (Sec. \ref{modelextensionsec}).
For simplicity, instead of presenting the ARIMA fits for the three epidemiological time-series ($\beta_{opt}$, $\gamma_{opt}$ and $\rho_{opt}$), we proceed by focusing on the SARS-CoV-2 transmission factor $\beta$. We note that equivalent steps are pursued for the other parameters. The main purpose of this third layer is to model the trends of the SIRD epidemiological parameters (as provided by the two previous layers) and use these trends, in the fashion of Eq. \eqref{TheDynamicsEpModel}, to improve the forecasting of the SIRD$+$ARIMA model, making it more coherent and consistent for feedback control strategies.
It is worth mentioning that this layer is an innovative and important advantage of the SIRD model identification proposed in this work. As depicted by \citet{Villaverde2020estimating}, using a SIRD model with time-varying epidemiological parameters allows one to provide forecasts that differ at the initial and long-term moments of the COVID-19 contagion.
\subsubsection{ARIMA Fit for the Viral Transmission Rate $\beta$}
As exploited in Section \ref{sec2}, the transmission rate parameter $\beta$ gives an important measure to analyze the pandemic panorama. It has been shown that this parameter varies according to health measures applied to the prone population \cite{Villaverde2020estimating,Oliveira2020,Caccavo2020}. The ARIMA expression is given as follows:
\begin{eqnarray}\label{ARIMA_model}
\left(a_{\beta_0} + a_{\beta_1}\beta(k-1)+a_{\beta_2}\beta(k-2) + \dots + a_{\beta_{n_\beta}}\beta(k-n_\beta + \epsilon(k)\right)\boldsymbol{\delta \beta}_{k} &=& \\ \nonumber \left(b_{\beta_0}\epsilon(k) + b_{\beta_1}\epsilon(k-1)+b_{\beta_2}\epsilon(k-2) + \dots + b_{\beta_{n_\beta}}\epsilon(k-n_\beta)\right) && \quad \text{,}
\end{eqnarray}
\noindent where $\boldsymbol{\delta \beta}_{k} \,=\ [\delta \beta(k) \; \delta \beta (k-1) \; \delta \beta (k-2) \; \dots \; \delta \beta (k-n_\beta)]^T$ is the vector of each incremental difference $\delta \beta (k) \, = \,\beta(k) - \beta(k-1)$. Notice that, regarding the total regression order of this model, in accordance with Eq. \eqref{TheDynamicsEpModel}.
Then, the ARIMA fit is performed minimizing an Extended Least-Squares (ELS) Procedure, used to find depolarized estimations for the ARIMA parameter values ($a_{\beta_0}$, $\dots$, $b_{\beta_{n_\beta}}$). This procedure is based on the following steps:
\begin{enumerate}[label =(\roman*)]
\item Concatenate the ARIMA the regression term from Eq. \eqref{ARIMA_model} as $\beta(k) = \omega^T(k-1)\Theta + \epsilon(k)$, where $\omega(k-1)$ concatenates the input values from the previous layers (in this case, the time-series $\beta_{opt}$) and $\Theta$ concatenates the ARIMA coefficients;
\item Determine a constrained Least-Squares estimation $\hat{\Theta}$, w.r.t. Eq. \eqref{ARIMA_model};
\item Compute the Least-Squares residue $\epsilon = \beta_{opt} - \omega\hat{\Theta}$ and
use it as the initial guess for $\epsilon$ for the next iteration;
\item Iterate until convergence or some ($2$-norm wise) small residue is achieved.
\end{enumerate}
\subsection{Model Validation Results}
With respect to the detailed identification procedure, the datasets provided by Brazilian Ministry of Health are used for validation purposes, considering the interval from the first confirmed COVID-19 case in the country, dating February $26$th, until the data from June $30$th, $2020$. The Ministry of Health disclosed daily values for the cumulative number of cases and deaths. The data is sufficient to obtain the behaviors for $S$, $I$, $R$, and $D$. Note that the total population size of Brazil is used as the initial condition $N_0 \, \approx \, 210$ million.
Regarding the second layer, the optimization (normalization) weights are presented in Table \ref{tabelOpti}. Moreover, the uncertainties bounds taken as $\underline{\delta}=0.95$ and $\overline{\delta}=1.05$.
\begin{table}[htbp]
\caption{Optimization Weights.}
\centering
\begin{tabular}{c | c c c }
Parameter & $w_1$ & $w_2$ & $w_3$ \\ \hline
Value & $1$ & $10$ & $2$ \\
\end{tabular}
\label{tabelOpti}
\end{table}
We proceed the model validation in a twofold: a) first we consider the first $100$ data points to tune the SIRD$+$ARIMA model and demonstrate its validity, globally well representing the following points; and b) we tune the SIRD$+$ARIMA model by identifying its parameters for the whole dataset; this is the model used for control. We note that the values used for the social isolation factor $\psi$ are those as exhibited in Figure \ref{psi_br}.
\subsection{a) Validation}
Regarding the first validation part, Figure \ref{parameters} shows the estimated time-series values from the optimization output. Recall that these parameters are piece-wise constant for periods of $T_2 \, = \, 7$ days. As discussed in previous literature, these parameters tend to follow stationary trends as the pandemic progresses.
\begin{figure}[htb]
\centering
\includegraphics[width=\linewidth]{./parameters.pdf}
\caption{$\beta$, $\gamma$, $\alpha$ and $\rho$ parameters identified by the two layer approach - from February $26$th, $2020$ to June, $04$th $2020$.}
\label{parameters}
\end{figure}
In order to illustrate the ARIMA fits, the identified auto-regressive model for SARS-CoV-2 transmission parameter $\beta$ is provided in Figure \ref{fit_beta}. The ELS procedure yields a regression $f_\beta(\cdot)$ with $n_\beta \, = \, 21$ daily steps (i.e. $3$ weeks). Figure \ref{fit_beta} demonstrates how the ARIMA is indeed able to describe the time-series $\beta_{opt}$, globally catching the time-varying behavior with $99.04$ \% accuracy.
\begin{figure}[htb]
\centering
\includegraphics[width=\linewidth]{./FIT_beta2.pdf}
\caption{Comparison between ARIMA model and previous identified $\beta$. FIT - 99.04\%}
\label{fit_beta}
\end{figure}
The first $100$ data points comprise the period from February $26$th to June, $04$th $2020$. Using the resulting SIRD$+$ARIMA model identified in this period, we proceed by extending the model forward, making forecasts from June, $05$th until June $30$th, $2020$, in order to demonstrate the validity of the identification. The forecasts are made using Eqs. \eqref{eqSIRDmodel} wherein parameters $\gamma$, $\beta$ and $\rho$ are given by the ARIMA regressions model in Eq. \eqref{TheDynamicsEpModel}. Figures \ref{I_SIRD}, \ref{R_SIRD} and \ref{D_SIRD} show the model forecasts compared with real data for active infections, recovered individuals and deaths, respectively, considering a interval of confidence of $95$\%. Clearly, the global behavior of the COVID-19 contagion is well described by the proposed models. To further illustrate this fact, Figure \ref{error} depicts the model-data error terms (from Eqs. \eqref{eq:err}-\eqref{eq:err3}), given in percentage. The coefficient of determination for these identification results are all above $0.99$.
\begin{figure}[htb]
\centering
\includegraphics[width=\linewidth]{./I_SIRD.pdf}
\caption{Validation of the SIRD$+$ARIMA model using estimated parameters with official data - Active Infected Curve.}
\label{I_SIRD}
\end{figure}
\begin{figure}[htb]
\centering
\includegraphics[width=\linewidth]{./R_SIRD.pdf}
\caption{Validation of the SIRD$+$ARIMA model using estimated parameters with official data - Recovered Curve.}
\label{R_SIRD}
\end{figure}
\begin{figure}[htb]
\centering
\includegraphics[width=\linewidth]{./D_SIRD.pdf}
\caption{Validation of the SIRD$+$ARIMA model using estimated parameters with official data - Fatal Cases Curve.}
\label{D_SIRD}
\end{figure}
\begin{figure}[htb]
\centering
\includegraphics[width=\linewidth]{./Error.pdf}
\caption{Error between the simulated model and official data (in percentage).}
\label{error}
\end{figure}
\subsection{b) Data-driven SIRD$+$ARIMA Model}
We note that the data-driven model, that is used for the NMPC strategy, is the based on the whole set of data. The simulation of this model is given in Figure \ref{Model_SIRD}.
\begin{figure}[htb]
\centering
\includegraphics[width=\linewidth]{./Model_SIRD2.pdf}
\caption{SIRD model curves used for NMPC framework. Right axis for Susceptible individuals (S). Left axis for Active Infections (I), Recovered (R) and Fatal (D) Cases.}
\label{Model_SIRD}
\end{figure}
Based on the prior developments, given the high coefficient of determination and observing Figures \ref{fit_beta}-\ref{error} (in which a $95$\% confidence interval is considered), it can be noticed that the identified model SIRD$+$ARIMA describes very well the observed COVID-19 pandemic behavior in Brazil and, in addition, it can forecast the SIRD curve with relatively small error. It is worth mentioning that this model identification approach needs to be updated regularly. From Figure \ref{error}, one notes that $20$-days-ahead forecasts can be derived with model-data errors below $10$\%. This error margins may increase when considering larger forecast horizons. Nevertheless, regarding the control purpose of this work and taking into account prediction horizons of roughly one month, the proposed SIRD$+$ARIMA is indeed able to be highly representative regarding the COVID-19 spread. For perfectly coherent forecasts, it is recommendable for the identification to be re-performed each week, and the ARIMA fits updated. This recommendation has also been provided by \citet{morato2020optimal}.
We also remark that the ARIMA fits for the epidemiological parameters are given in weekly samples i.e. $k_{\text{week}}$, while the SIRD variables are given for each day i.e. $k_{\text{day}}$. Regarding this matter, we can represent the weekly-sampled variables, from the viewpoint of the daily-sampled variables, as:
\begin{eqnarray} \nonumber
\beta\left(k_{\text{day}}\right) &=& \beta\left(k_{\text{week}}T_2\right) \, \, \, \forall \, k_{\text{day}} \, \in \, \left[k_{\text{week}}T_2 \, , \, (k_{\text{week}}+1)T_2\right) \quad \text{,}\\ \ \nonumber
\gamma\left(k_{\text{day}}\right) &=& \gamma\left(k_{\text{week}}T_2\right) \, \, \, \forall \, k_{\text{day}} \, \in \, \left[k_{\text{week}}T_2 \, , \, (k_{\text{week}}+1)T_2\right) \quad \text{,} \\
\rho\left(k_{\text{day}}\right) &=& \rho\left(k_{\text{week}}T_2\right) \, \, \, \forall \, k_{\text{day}} \, \in \, \left[k_{\text{week}}T_2 \, , \, (k_{\text{week}}+1)T_2\right) \quad \text{,} \\ \nonumber
\psi\left(k_{\text{day}}\right) &=& \psi\left(k_{\text{week}}T_2\right) \, \, \, \forall \, k_{\text{day}} \, \in \, \left[k_{\text{week}}T_2 \, , \, (k_{\text{week}}+1)T_2\right) \quad \text{,} \\ \nonumber
u\left(k_{\text{day}}\right) &=& u\left(k_{\text{week}}T_2\right) \, \, \, \forall \, k_{\text{day}} \, \in \, \left[k_{\text{week}}T_2 \, , \, (k_{\text{week}}+1)T_2\right) \quad \text{,}
\end{eqnarray}
where $T_2 \, = \, 7$ days.
Finally, considering this validated SIRD$+$ARIMA model, the basic reproduction number $R_t$ of the SARS-CoV-2 virus in Brazil can be inferred through Eq. \eqref{R0eq}. In Figure \ref{r0}, the evolution of the viral reproduction number $R_t$ for each week of the pandemic in show. As it can be seen, this reproduction number presents stronger variations at the beginning stage of, but then tends to converge to a steadier values, which corroborates with the model extensions considering the epidemiological parameters. Moreover, notice that the reproduction number for since June $4$th, $2020$ is somewhat steady near $1.603$, which indicates that the COVID-19 pandemic is still spreading in the country.
\begin{figure}[htb]
\centering
\includegraphics[width=\linewidth]{./Rt.pdf}
\caption{Basic reproduction number $R_t$ calculated by the multi-layered identification framework.}
\label{r0}
\end{figure}
\FloatBarrier
\section{The NMPC strategy}
\label{sec3}
In this Section, we detail the proposed NMPC strategy used to determine public health guidelines (regarding social isolation policies) to mitigate the spread of the COVID-19 contagion in Brazil. This NMPC algorithm is conceived under the feedback structure illustrated in Figure \ref{modeltotal}.
We note that the generated control action $u(k)$ represents the input to the population's response to social distancing measures, as gives the differential Eq. \eqref{TheControlPsi}. For this reason, it follows that the proposed NMPC algorithm operates with a weekly sampling period ($T_2$). Implementing a new social distancing guideline every week is coherent with previous discussions in the literature \cite{morato2020optimal}. It does not seem reasonable to change the distancing measures every few days.
\subsection{Possible Control Sequences}
Before presenting the actual implementation of NMPC tool, we note that its generated control sequence must be given in accordance with the constraints expressed in Eq. \eqref{UConstraints}.
Considering that the NMPC has a horizon of $N_p$ steps (given in weekly samples) from the viewpoint of each week $k$, the generated control signal is\footnote{Notation $\chi(k+j|k)$ denotes the predicted values for variable $\chi(k+j)$, computed at the discrete instant $k$.}:
\begin{eqnarray}
\label{BigUSequence}
U_{k} &=& \left[\begin{array}{cccc} u(k|k) & u(k+1|k) & \cdots & u(k+N_p-1|k)\end{array}\right] \quad \text{.}
\end{eqnarray}
Since the variations from each quarantine guideline $u (k-1)$ to the following $u(k)$, denoted $\delta u(k)$ are equal to $\pm \, 0.05$ or $0$, it follows that all possible control sequences can be described as, from the viewpoint of sample $k$:
\begin{eqnarray}
\label{BigUSequence2}
U_{k} &=& \left[\begin{array}{ccc} \left(u(k-1) + \delta u(k)\right) & \cdots & \left(u(k-1) \overbrace{+ \delta u(k) \dots + \delta u(k+N_p-1)}^{N_p\text{ times}}\right) \end{array}\right] \, \text{.}
\end{eqnarray}
From Eq. \eqref{BigUSequence2}, we can conclude that, in the considered settling, there are $3^{N_p}$ possible control sequences.
\subsection{Control Objectives}
The main purpose of social isolation is to distribute the demands for hospital bed over time, so that all infected individuals can be treated. It seems reasonable to act though social isolation to minimize the number of active infections ($I$), while ensuring that this class of individuals remains smaller than the total number of available ICU beds $n_{ICU}$.
Moreover, it seems reasonable to ensure that social isolation measures are enacted for as little time as possible, to mitigate the inherent economic backlashes.
This trade-off objective (mitigating $I$ against relaxing social isolation) is expressed mathematically as:
\begin{eqnarray}
\label{TheCO}
\left|\left|I(k)\right|\right|_{\frac{Q}{n_I^2}} + \left|\left|u(k)\right|\right|_{(1-Q)} \,\,=\,\, I(k)^T\frac{Q}{n_I^2}I(k) + u(k)^T(1-Q)u(k) \quad \text{,}
\end{eqnarray}
where $n_I$ is a nominal limit for $I$ (i.e. the initial population size $N_0$) included for a magnitude normalization of the trade-off objective. Note that $u$ is given within $[0.3 \, \, 0.7]$, so there is no need for normalization.
\subsection{Process Constraints}
The proposed NMPC algorithm must act to address the Control Objective given in Eq. \eqref{TheCO} while ensuring some inherent process constraints. These are:
\begin{enumerate}
\item Ensure that the control signal has the piece-wise constant characteristic, as gives Eq. \eqref{UConstraints}; and
\item Ensuring that the number of infected people with active acute symptoms does not surpass the total number of available ICU hospital beds in Brazil, this is:
\begin{eqnarray}
\label{constInICU}
p_{\text{sym}}I(k) &\leq& n_{ICU} \text{.}
\end{eqnarray}
\end{enumerate}
\subsection{The Complete NMPC Optimization}
Bearing in mind the previous discussions, the NMPC procedure is formalized under the following optimization problem, which is set to mitigate the SARS-CoV-2 viral contagion spread with respect to the trade-off control objective of Eq. \eqref{TheCO} and the constraints from Eqs. \eqref{UConstraints}-\eqref{constInICU}, for a control horizon of $N_p$ (weekly) steps, from the viewpoint of each sampling instant $k$:
\begin{eqnarray}
\label{TheCompleteNMPC}
\underset{U_{k}}{\min} \, J(\cdot) &=& \underset{U_{k}}{\min} \sum_{i=1}^{N_p} \left( \left(I(k+i)^T\frac{Q}{n_I^2}I_j(k+i)\right) + \left(u(k+i-1)^T(1-Q)u(k+i-1)\right)\right) \text{,} \\ \nonumber
\text{s.t.:} && \text{SIRD$+$ARIMA Model} \,\,\, \forall \, i \in \, \mathbb{N}_{[1\, , \, N_p]} \,\text{,} \\ \nonumber
&& \underline{\psi} \leq u(k+i-1) \leq \overline{\psi} \,\text{,}\\ \nonumber
&& u(k+i-1) \, = \, u(k+i-2) + \delta u(k+i-1) \, \text{,} \\ \nonumber
&& \delta u(k+i-1) \, = \, -0.05\text{ or }0\text{ or }0.05 \,\text{,}\\ \nonumber
&& p_{\text{sym}}I(k+i) \leq n_{ICU} \,\text{.}
\end{eqnarray}
\subsection{Finitely Parametrized NMPC Algorithm}
The finitely parametrized NMPC methodology has been elaborated by \citet{alamir2006stabilization}. This paradigm has recently been extended to multiple applications \cite{rathai2018parameterized,rathai2019gpu}, with successive real-time results.
Given that this control framework offers the tool to formulate (finitely parametrized) social distancing guidelines for the COVID-19 spread in Brazil, with regard to the discussions in the prior, we proceed by detailing how it is implemented at each sampling instant $k$, ensuring that the Nonlinear Optimization Problem in Eq. \eqref{TheCompleteNMPC} is solved.
Basically, the parametrized NMPC algorithm is implemented by simulating the SIRD$+$ARIMA model with an explicit nonlinear solver, testing it according to all possible control sequences (as gives Eq. \eqref{BigUSequence2}). Then, the predicted variables are used to evaluate the cost function $J(\cdot)$. The control sequences that imply in the violation of constraints are neglected. Then, the resulting control sequence is the one that yields the minimal $J(\cdot)$, while abiding to the constraints. Finally, the first control signal is applied and the horizon slides forward. This paradigm is explained in the Algorithm below. Figure \ref{ThepNMPCAlgoCovid} illustrates the flow of the implementation of the proposed Social Distancing control methodology for the COVID-19 viral spread in Brazil.
\begin{algorithm}{\textbf{Finitely Parametrized NMPC for Social Distancing Guidelines}} \label{ThepNMPCAlgoCovidalgo}\\ \noindent\makebox[\linewidth]{\rule{\linewidth}{0.4pt}}
\textbf{Initialize}: $N(0)$, $S(0)$, $I(0)$, $R(0)$ and $D(0)$. \\
\textbf{Require}: $Q$, $n_I$, $n_{ICU}$ \\
\textbf{Loop every $T_1$ days}:
\begin{itemize}
\item Step (\textit{i}): "Measure" the available contagion data ($N(k)$, $S(k)$, $I(k)$, $R(k)$ and $D(k)$);
\item Step (\textit{ii}): \textbf{Loop every $T_2$ days}:
\begin{itemize}
\item Step (a): \textbf{For each sequence $j \, \in \, 3^{N_p}$}:
\begin{itemize}
\item Step (1): Build the control sequence vector according to Eq. \eqref{BigUSequence2};
\item Step (2): Explicitly simulate the future sequence of the SIRD variables;
\item Step (3): Evaluate if constraints are respected. If they are not, end, else, compute the cost function $J(\cdot)$ value.
\end{itemize}
\item \textbf{end}
\item Step (b): Choose the control sequence $U_{k}$ that corresponds to the smallest $J(\cdot)$.
\item Step (c): Increment $k$, i.e. $k \leftarrow k+1$.
\end{itemize}
\item \textbf{end}
\item Step (\textit{iii}): Apply the local control policy $u(k)$ as gives Eq. \eqref{UConstraints};
\item Step (\textit{iv}): Simulate the SIRD$+$ARIMA model, as give Eqs. \eqref{eqSIRDmodel}-\eqref{TheControlPsi}.
\item Step (\textit{v}): Increment $k$, i.e. $k \leftarrow k+1$.
\end{itemize}
\textbf{end}
\end{algorithm}
\begin{figure}[htb]
\centering
\includegraphics[width=\linewidth]{./ThepNMPCAlgoCovid.png}
\caption{Algorithm Implementation Flowchart.}
\label{ThepNMPCAlgoCovid}
\end{figure}
\FloatBarrier
\section{Simulation Results}
\label{sec4}
Bearing in mind the previous discussions, the proposed SIRD$+$ARIMA model and the finitely parametrized NMPC toolkit, this Section is devoted in presenting the simulation results regarding the COVID-19 contagion spread in Brazil. The following results were obtained using Matlab. The average computation time needed to solve the solution of the proposed NMPC procedure is of $2$ ms\footnote{The implementation was performed on a i5 $2.4$ GHz Macintosh computer.}.
\subsection{Simulation Scenarios}
We proceed by depicting two simulation scenarios, which account for the following situations:
\begin{enumerate}[label=(\alph*)]
\item What would be the case if the NMPC generated social distancing policy was enacted back in the beginning of the COVID-19 pandemic in Brazil.
\item How plausible is it to apply the NMPC technique from now on ($30/06/2020$) and reduce and mitigate the effected of the spread.
\end{enumerate}
In order to provide these scenarios, we consider an ``open-loop" comparison condition, which represents the simulation of the SIRD$+$ARIMA model with the social distancing factor in fact observed in Brazil (see Figure \ref{psi_br}). The results are based on $126$ days of data. From the $127^{th}$ sample on, the open-loop simulation takes $\psi = 0.3$ (no isolation).
The proposed NMPC is designed with a cost function $J(\cdot)$ with tuning weight $Q \, = \, 0.7$ (which means reducing the number of infections is prioritized over relaxing social distancing). Furthermore, the NMPC optimization is set with a prediction horizon of $N_p \, = \, 4$ weeks ($28$ days), which means that the controller makes its decision according to model-based forecasts of the SIRD$+$ARIMA model for roughly one month ahead of each sample $k$. For simplicity $n_I$ is taken as $\frac{n_{ICU}}{p_{\text{sym}}}$, since the main goal of the MPC is to ensure $p_{\text{sym}}I \, \leq \, n_{ICU}$.
\subsection{Scenario (a): NMPC since the beginning}
We recall that the COVID-19 pandemic in Brazil formally ``started" $26/02/2020$, when the first case was officially registered.
This first simulation scenario considers the application of the NMPC control strategy to guide social distancing from $30$ days after the first case ($27/03/2020$). We choose this date because, in Brazil, the majority of states started social distancing measures (in different levels) around the period of late-March/mid-April. We do not deem it reasonable to consider the application of the NMPC strategy from ``day $1$", since this was not seen anywhere in Brazil.
Considering this control paradigm, Figure \ref{WhatIfActiveU} depicts the predicted evolution of severe/acute symptomatic COVID-19 infections over time. These active symptomatic infections stand for those that may need ICU hospitalization. The results shows that the NMPC is able to thorough attenuate the peak of infections, ensuring that it always stays below the ICU threshold. This is a quite significant results, which shows that the proposed feedback framework is able to offer an enhanced paradigm, with time-varying social distancing measures, such that the COVID-19 spread curve is indeed flattened, never posing serious catastrophic difficulties to Brazilian hospitals (and health system overall). The result also indicated that the social distancing should vary with relaxing and strengthening periods over the years of $2020$, $2021$ and $2022$. We note, once again, that health professionals should better qualify the relationship between the social distancing factor and actual economic/social restrictions, as illustrated in Table \ref{TheTableUParametre}. The results also indicate that if no coordinated action is deployed by the Brazilian federal government, the number of active symptomatic infections at a given day could be up to $540000$ individuals, with this peak forecast to October $16$th, $2020$. If the NMPC strategy was indeed applied, two peaks would have been seen, with $80\, \rm{\%}$ of total ICU capacity, previewed for September $25$th, $2020$ and March, $31$rst, $2021$.
Regarding this scenario, Figure \ref{WhatIfAllIAllD} shows the evolution of the total amount of infections and cumulative number of deaths. This Figure also places the real data points against the simulated model. The possibilities to come are catastrophic, as also previewed by \citet{morato2020optimal}. We note, once again, that the SIRD$+$ARIMA models offers qualitative results, the magnitude of $1.5$ million deaths seems quite alarming, but the model is identified considering a large margin for sub-reports (in number of deaths and confirmed cases). Not all deaths due to COVID-19 are currently being accounted for in Brazil, as discuss \citet{THELANCET20201461,zacchi:hal-02881690}.
\begin{figure}[htb]
\centering
\includegraphics[width=\linewidth]{./WhatIfActiveU.pdf}
\caption{Scenario (a): Symptomatic Active Infections $p_{\text{sym}}I$ and Control Input $u$ (Social Distancing, $\psi$).}
\label{WhatIfActiveU}
\end{figure}
\begin{figure}[htb]
\centering
\includegraphics[width=\linewidth]{./WhatIfAllIAllD3.pdf}
\caption{Scenario (a): Total Active Infections $I$ and Total Deaths $D$.}
\label{WhatIfAllIAllD}
\end{figure}
\FloatBarrier
\subsection{Scenario (b): NMPC from the now on}
The second simulation scenario that we consider is to control the situation from the current moment ($30/06/2020$) to avoid a total collapse of the Brazilian health system. We note that as of this date, the country counts over $1.4$ million confirmed COVID-19 cases and more than $59500$ deaths due to the SARS-CoV-2 virus.
Figure \ref{FromNowActiveInfControl} shows the main results regarding this second scenario, considering the active symptomatic infections, that may require ICU hospitalization. Even though a partial catastrophe is already under course in Brazil, if social distancing are guided through the proposed NMPC, a total collapse of the public health system can still be avoided. Figure \ref{FromNowIcumvsR} shows the evolution for recovered individuals and cumulative number of cases $I_c$, Figure \ref{FromNowActiveI} shows the evolution of all active infections (symptomatic and asymptomatic), while Figure \ref{FromNowDeaths} shows the mounting number of deaths. The proposed NMPC, if rapidly put in practice, could still be able to slow the viral spread, saving $25 \, \rm{\%}$ of lives w.r.t. the open-loop condition. The peak of infections, if such technique is applied, has its forecast previewed to September $2$nd, 2020, being anticipated in $17$ days.
\begin{figure}[htb]
\centering
\includegraphics[width=\linewidth]{./FromNowActiveInfControl.pdf}
\caption{Scenario (b): Symptomatic Active Infections $p_{\text{sym}}I$ and Control Input $u$ (Social Distancing, $\psi$).}
\label{FromNowActiveInfControl}
\end{figure}
\begin{figure}[htb]
\centering
\includegraphics[width=\linewidth]{./FromNowIcumvsR.pdf}
\caption{Scenario (b): Cumulative Cases $I_c$ and Recovered Individuals $R$.}
\label{FromNowIcumvsR}
\end{figure}
\begin{figure}[htb]
\centering
\includegraphics[width=\linewidth]{./FromNowActiveI.pdf}
\caption{Scenario (b): Total Active Cases $I$.}
\label{FromNowActiveI}
\end{figure}
\begin{figure}[htb]
\centering
\includegraphics[width=\linewidth]{./FromNowDeaths3.pdf}
\caption{Scenario (b): Total Deaths $D$.}
\label{FromNowDeaths}
\end{figure}
\FloatBarrier
\section{Conclusions}
\label{sec5}
In this paper, an optimal control procedure is proposed for ruling social isolation guidelines in Brazil, in order to mitigate the spread of the SARS-CoV-2 virus. To do so, a new model is proposed, based on extensions of the SIRD equations. The proposed model embeds weekly auto-regressive dynamics for the epidemiological parameters and also takes a dynamic social distancing factor (as seen in previous papers \citep{morato2020optimal,bastos2020covid19}). The social distancing factor measures and expresses the population's response to quarantine measures, guided by the Nonlinear Model Predictive Control procedure. The NMPC strategy is designed within a finitely parametrized input paradigm, which enables its fast realization.
In this work, some key insights were given regarding the future panoramas for the COVID-19 pandemic in Brazil. Below, some of the main findings of this paper are summarized:
\begin{itemize}
\item The presented results corroborate the hypothesis formulated in many of the previous papers regarding the COVID-19 pandemic in Brazil \cite{hellewell2020feasibility, THELANCET20201461, morato2020optimal}: herd immunity is not a plausible option for the country\footnote{Previous paper have also elaborated on the fact that vertical isolation is also not an option for the time being, since Brazil does not have the means to pull of efficient public policies to separate the population at risk from those with reduced risk, due to multiple social-economical issues of the country \cite{silva2020bayesian,rocha2020expected,rodriguez2020covid}.}; if no coordinates social distancing action is enforced, the ICU threshold will be largely surpassed, which can lead to elevated fatality.
\item The prediction of evolution of the viral spread is relatively accurate with the proposed adapted SIRD model for up to $20$ days. Larger prediction horizons can be considered, but daily model-updates are recommended.
\item The simulation forecasts derived with the NMPC strategy and with an open-loop condition (no social distancing) indicate that social distancing measures should still be maintained for a long time. The strength of these measures will be diluted as time progresses. The forecasts indicate an infection peak of over $600000$ symptomatic individuals to late September, $2020$, in the current setting. If model-based control is enacted, the peak could be anticipated and the level of infections could be contained below the ICU hospital bed threshold. The NMPC could save over $400000$ lives if enacted from now (July, $2020$).
\item The results also indicate that if such coordinated control strategy was applied since the first month of COVID-19 infections in Brazil, a more relaxed social distancing paradigm would be possible as of late $2020$. Since this has not been pursued, the social distancing measures may go up until late $2021$ if no vaccine is made available.
\end{itemize}
These results presented in this paper are qualitative. Brazil has not been testing enough its population (neither via mass testing or sampled testing), which means that the data regarding the number of infections is very inconsistent. As \citet{bastos2020covid19} thoroughly details, the uncertainty margin associated to the available data (in terms of case sub-reporting) is very significant. Anyhow, the results presented herein can help guiding long-term regulatory decision
policies in Brazil regarding COVID-19. One must note that social distancing measures, in different levels, will be recurrent and ongoing for a long time. Due to this fact, compensatory social aid policies should also be developed in order to reduce the effects of a possibly long-lasting economic turn-down. Recent papers have discussed this matter \cite{ahamed2020role, zacchi:hal-02881690}.
\section*{Acknowledgments}
The Authors acknowledge the financial support of National Council for Scientific and Technological Development (CNPq, Brazil) under grants $304032/2019-0$ and $201143/2019-4$ (PhD Program Abroad). The Authors also thank Saulo B. Bastos and Daniel O. Cajueiro for previous collaborations and discussions.
\subsection*{Notes}
The authors report no financial disclosure nor any potential conflict of interests.
\bibliographystyle{model5-names}
|
{
"redpajama_set_name": "RedPajamaArXiv"
}
| 510
|
Ernst Decani, Szinnyeinél Decani Ernő (Beszterce, 1801. május 28. – Buda, 1860. augusztus 20.) orvos.
Élete
A gymnasiumot szülővárosában járta s a bécsi egyetem hallgatója volt, azután 1827-ben a paduai egyetemen nyert orvosdoktori oklevelet. Hazájába visszatérve, 1829-ben Nagysinken lett városi és 1831-ben kamarai orvos az abrudbányai kincstárnál. 1842 októberében zalatnai királyi bánya-kincstári orvossá nevezték ki.
Munkái
Dissertatio inaug. medica de graviditate extrauterina, accedit Descriptio memorandae cujusdam graviditatis tubae dextrae. Petavii, 1827.
Források
Erdélyi szász orvosok
Beszterceiek
1801-ben született személyek
1860-ban elhunyt személyek
|
{
"redpajama_set_name": "RedPajamaWikipedia"
}
| 1,446
|
Europaparlamentsvalet i Danmark 1994 ägde rum torsdagen den 9 juni 1994. Knappt fyra miljoner personer var röstberättigade i valet om de sexton mandat som Danmark hade tilldelats innan valet. I valet tillämpade landet ett valsystem med partilistor och d'Hondts metod, utan någon spärr för småpartier. d'Hondts metod användes dock på valalliansbasis, det vill säga mandaten fördelades först mellan valallianserna och därefter mellan partierna inom varje allians. I valet ställde bland annat Konservative Folkeparti, Centrum-Demokraterne och Venstre respektive Junibevægelsen mod Union och Folkebevægelsen mod EU upp som valallianser.
Valets största vinnare var det nystartade euroskeptiska partiet Junibevægelsen mod Union, som erhöll över 15 procent av rösterna och två mandat. Deras framgång skedde till stor del på bekostnad av det sedan tidigare etablerade euroskeptiska partiet Folkebevægelsen mod EU, som tappade två mandat. Även Socialdemokraterne backade kraftigt och tappade ett mandat. Således kunde Venstre inta positionen som största parti. Partiet vann fyra mandat, ett mer än i valet 1989. Konservative Folkeparti hamnade oväntat på andra plats med 17,74 procent av rösterna och tre mandat. Bland de mindre partierna ökade Radikale Venstre kraftigt och kunde därmed säkra ett mandat i Europaparlamentet. Samtidigt tappade Centrum-Demokraterne nästan hela sitt stöd från valet 1989 och förlorade därmed båda sina mandat. Socialistisk Folkeparti var det enda parti som behöll ungefär samma andel röster som i det föregående valet.
Valdeltagandet uppgick till 52,92 procent, en ökning med 6,75 procentenheter jämfört med valet 1989. Det var dock fortfarande under genomsnittet för hela unionen och ett mycket lågt valdeltagande för att vara ett danskt val.
Valresultat
Se även
Europaparlamentsvalet 1994
Referenser
Noter
1994 i Danmark
1994
Danmark
|
{
"redpajama_set_name": "RedPajamaWikipedia"
}
| 7,008
|
Dance-hall Crew president Michael Mugwanya commonly known as DJ Michael's fiancee Rhona Ndagire has given birth to their first kid a beautiful baby gal. Gorgeous Rhona gave birth on Friday from Nsambya hospital.
She gave birth by caesarian. Both m,um and baby were in good shape and they have actually been discharged from hospital. DJ Michael told us ion Friday: "I'm over the moon. My official girl Rhona has given birth to a beautiful girl. This is definitely the best day of my life."
Now knowing our Ugandan Celebrity biff, we're not surprised at learning that DJ Michael and DJ Shiru are feuding. Whereas insiders have intimated to us that he biff between the two stems from a babe only identified as Brenda, who the two are said to have juggled at one time, other sources have it that they are feuding over Shiru not playing Michael's music.
We've learnt that for quite sometime, DJ Michael and Shiru have been at loggerheads, after Shiru has stubbornly refused to play DJ Michael's music, arguing that it's not worth Shiru's standards.
We've learnt that when Michael got to know of this through his Dance-hall crew members, he got so furious that when he got a chance to, he rudely told off Shiru to f**k off if he doesn't want to play his music.
The two were recently smoked out exchanging bitter words outside one f the city spots.
Dancehall Crew President DJ Michael has moved in with her preggie fiancee despite recent rumours about another gal claiming he destroyed her future. A few weeks back, former singer Brenda claimed the Kwata Kwata singer had shattered her education and dumped her.
But despite the claims, preggie Ndagire has moved in with Michael. The pair now stay together in Makindye. Meanwhile DJ Michael's new song Kili Okay is currently number one on most of the local charts.
And it has been earning him lots of dime lately because of its appeal.
Kwata Kwata singer and former Lido Beach disc spinner, DJ Michael has bought a speed boat worth $ 7,000 (around Shs 12 Million). We hear the mega speed boat was shipped into the country from Japan by James Car Company.
He tested the machine on Sunday during his Kwata Kwata album launch at Lido. His pals told us that during his recently concluded tour of Sweden, DJ Michael earned lots of dime. Even his album launch was quite a success. The singer already owns a toyota surf vehicle.
Sensitive word reaching our snoops shows that 'Muko Muko' star DJ Michael has been frequenting Difra Language services premises of late. Our snoops have exclusively learnt the Ragga ace decided to go back to school some time recently to help improve his English language skills.
Having made it big in Luganda dance-hall Ragga, we now hear DJ Michael wants to venture into English Ragga, to cater for his fans that don't understand Luganda. He's been smoked out on several times at Difra Language Services and we hear he pursues German Computer lessons and English Language. Wish him luck.
|
{
"redpajama_set_name": "RedPajamaC4"
}
| 6,485
|
// Torsion TorqueScript IDE - Copyright (C) Sickhead Games, LLC
// This file is subject to the terms and conditions defined in
// file 'LICENSE.txt', which is part of this source code package.
#ifndef _FINDINFILESDLG_H_
#define _FINDINFILESDLG_H_
#if defined(__GNUG__) && !defined(__APPLE__)
#pragma interface "FindInFilesDlg.cpp"
#endif
/*!
* Includes
*/
////@begin includes
////@end includes
/*!
* Forward declarations
*/
class FindThread;
////@begin forward declarations
////@end forward declarations
/*!
* Control identifiers
*/
////@begin control identifiers
#define SYMBOL_FINDINFILESDLG_STYLE wxCAPTION|wxSYSTEM_MENU|wxCLOSE_BOX
#define SYMBOL_FINDINFILESDLG_TITLE _("Find in Files")
#define SYMBOL_FINDINFILESDLG_IDNAME wxID_ANY
#define SYMBOL_FINDINFILESDLG_SIZE wxSize(266, 184)
#define SYMBOL_FINDINFILESDLG_POSITION wxDefaultPosition
#define ffID_BROWSE 10012
#define ffID_MATCHCASE 10013
#define ffID_WHOLEWORD 10014
////@end control identifiers
/*!
* Compatibility
*/
#ifndef wxCLOSE_BOX
#define wxCLOSE_BOX 0x1000
#endif
#ifndef wxFIXED_MINSIZE
#define wxFIXED_MINSIZE 0
#endif
/*!
* FindInFilesDlg class declaration
*/
class FindInFilesDlg: public wxDialog
{
DECLARE_DYNAMIC_CLASS( FindInFilesDlg )
DECLARE_EVENT_TABLE()
public:
/// Constructors
FindInFilesDlg();
FindInFilesDlg( wxWindow* parent, wxWindowID id = SYMBOL_FINDINFILESDLG_IDNAME, const wxString& caption = SYMBOL_FINDINFILESDLG_TITLE, const wxPoint& pos = SYMBOL_FINDINFILESDLG_POSITION, const wxSize& size = SYMBOL_FINDINFILESDLG_SIZE, long style = SYMBOL_FINDINFILESDLG_STYLE );
/// Creation
bool Create( wxWindow* parent, wxWindowID id = SYMBOL_FINDINFILESDLG_IDNAME, const wxString& caption = SYMBOL_FINDINFILESDLG_TITLE, const wxPoint& pos = SYMBOL_FINDINFILESDLG_POSITION, const wxSize& size = SYMBOL_FINDINFILESDLG_SIZE, long style = SYMBOL_FINDINFILESDLG_STYLE );
/// Creates the controls and sizers
void CreateControls();
////@begin FindInFilesDlg event handler declarations
/// wxEVT_COMMAND_BUTTON_CLICKED event handler for ffID_BROWSE
void OnBrowseClick( wxCommandEvent& event );
/// wxEVT_COMMAND_BUTTON_CLICKED event handler for wxID_OK
void OnFindClick( wxCommandEvent& event );
/// wxEVT_UPDATE_UI event handler for wxID_OK
void OnFindUpdate( wxUpdateUIEvent& event );
////@end FindInFilesDlg event handler declarations
////@begin FindInFilesDlg member function declarations
/// Retrieves bitmap resources
wxBitmap GetBitmapResource( const wxString& name );
/// Retrieves icon resources
wxIcon GetIconResource( const wxString& name );
////@end FindInFilesDlg member function declarations
void AddLookIn( const wxString& path );
void AddWhat( const wxString& what );
void AddType( const wxString& type );
wxArrayString GetPathsFromLookIn( const wxString& lookin ) const;
wxString GetWhat() const;
wxString GetLookIn() const;
wxString GetTypes() const;
FindThread* m_FindThread;
/// Should we show tooltips?
static bool ShowToolTips();
////@begin FindInFilesDlg member variables
wxComboBox* m_WhatCtrl;
wxComboBox* m_LookInCtrl;
wxComboBox* m_TypeCtrl;
wxCheckBox* m_MatchCaseCtrl;
wxCheckBox* m_WholeWordCtrl;
////@end FindInFilesDlg member variables
};
#endif
// _FINDINFILESDLG_H_
|
{
"redpajama_set_name": "RedPajamaGithub"
}
| 50
|
\section{Introduction}
During the past three years, I have been teaching the course of concepts in physics and problems solving in physics at the
African Institute for Mathematical Sciences (AIMS) Centres at Mbour/Senegal (2018) and Accra/Ghana (2019, 2020). The course is basically about Newtonian physics. Like the students, I found the basic astronomy concepts fascinating.
Defining the universe as all of space and time and their contents, including planets, stars, galaxies, and all other forms of matter and energy, some natural questions arise about the beginning, the existence of the universe, its present constituents and how will be the life in the future. The field of cosmology is the best candidate to respond to those questions as cosmology seeks to answer to these questions that are the oldest of the mankind. The study of the cosmos is as old as humanity and has always been fascinating.
Physical cosmology is the scientific study of the universe as a whole based on the laws of physics, to which Einstein's equations are very likely to give a correct description. General relativity describes the observed universe very well.
However, it is a complicated theory and often it is difficult to find solutions. The Newtonian theory, on the other hand, is in many ways far simpler and is very often able to satisfactorily model the universe.
The Newtonian cosmology began with the work of McCrea and Milne in 1934 in which simple isotropic and homogeneous universe were studied \cite{milne, crea}. The two assumptions fit well with the observable universe, namely the ex\-pan\-ding universe free of rotation and shear.
There are two approaches to Newtonian cosmology: the first one
who deals with potential function worked out by McCrea and Milne \cite{ milne, crea}, G\"odel \cite{godel}, Herkmann and Sch\"ucking \cite{herk1,herk2}, Raychaudhury \cite{ray}, Zeldovich \cite{zeldo}.
The second approach who uses gravitational force
is given by Narlikar \cite{narli} .
Summary of Newton cosmology is given by Bondi \cite{bondi}, North \cite{north}, Ellis \cite{ellis}.
The application of quantum theory to the description of the universe as a whole is called quantum cosmology. Quantum Newtonian cosmology was proposed by Freedman et al.\cite{freedman}. It is possible to construct a wave function for Newtonian cosmology in the framework of non-relativistic quantum mechanics \cite{jnorb, juan, bras2, bras3}. We are interested in quantum cosmology and a good way to start is to revisit quantum Newtonian cosmology in the sense of \cite{bras2}.
Our work is presented as follows:
In section (\ref{sec2}) we present the classical model, where we start by recalling the cosmological principle and then formulate the Lagrangian like in \cite{bras1}. In section (\ref{sec3}), we perform the affine quantization procedures and formulate the quantum Hamiltonian operator. In section (\ref{sec4}) the wave functions of the Newtonian universe and the energy levels are given. The section (\ref{sec5}) is about some concluding remarks. An appendix about Biconfluent Heun equation and its polynomial form of solution is added after the concluding remarks.
\section{The classical model}\label{sec2}
\subsection{Cosmological principle}
In modern physics, cosmology begins with the application of Einstein's theory of gravity, or general relativity, to the universe. This is a difficult task and would probably not be possible without a basic assumption about the universe called the cosmological principle that says that `` On large (but not too large) scales, the universe is homogeneous and isotropic ". The statement of isotropy means that the universe is the same in all directions (the universe looks the same whether you are looking directly outward from the North Pole or the South Pole. The Homogeneity means that the universe is the same at all points.
The statements of homogeneity and isotropy are distinct but closely related: for example a universe which is isotropic will be homogeneous while a universe that is homogeneous may not be isotropic. A universe which is only isotropic around one point is not homogeneous. Without the cosmological principle, much of our presumed understanding of the workings of the universe would be invalid.
\subsection{Lagrangian formulation}
We consider a cosmological model based on Newtonian dynamics. It uses an approach worked out by Milne and McCrea in which it has no sense to associate the gravitational phenomena to the effects of spacetime curvature \cite{milne, crea}. In this context, it was shown that the universe behavior could be understood on the basis of classical physics, which does not use the mathematical complexity in the study of the universe as in the Einsteinian cosmology.
The simplest model of the universe one can think of is that of a universe filled with dust (non-relativistic pressureless matter, $p=0$). We can think of this dust as a collection of point particles, and on the cosmological scales of approximation 10-1000Mpc. These point particles are a sufficiently good approximation for galaxies or even galaxy clusters.
Consider the evolution of a small spherical portion of the universe, where Newton's theory applies: then the behavior of this portion will reflect the evolution of the universe.
The cosmological principle states that there is no prefered place or direction in the universe on large scales, so we can pick any coordinate system with respect to which we can measure the positions and velocities of these test particles.
Let us consider a galaxy that we consider as a particle of mass $m$ for our Newtonian system located at a radius $a(t)$ from an arbitrary defined origin, where $a(t)$ is a distance we
refer to as the scale factor. This scale factor acts to simply scale up or scale down our Newtonian universe. We assume that the motion about the origin must be spherically symmetric.
The force of gravity on the particle (galaxy) at distance $a(t)$ coming from the mass of the homogeneous universe inside the sphere of radius $a(t)$ is the same as if all the mass were at the center of the sphere. There is no force arising from the region outside the sphere.
In a fixed rectangular coordinates (comoving), the kinetic energy for the motion of the particle (galaxy ) is
given by
\begin{equation}
T = \frac{1}{2}m\dot{a}^2\,.
\end{equation}
We assume that the particle (galaxy) moves in a conservative force field so the potential energy is given by
\begin{equation}
U = - \frac{GMm}{a}\,.
\end{equation}
In both Newtonian and relativistic cosmology, the universe is unstable to gravitational collapse. Both Newton and Einstein believed the universe is static. In order to obtain this, Einstein introduced a repulsive gravitational force
\begin{equation}
F_\Lambda = \frac{\Lambda}{3} m a,
\end{equation}
where the constant $\Lambda$ is called the cosmological constant. When $\Lambda > 0$, this force is pointed radially outward, repulsive relative to the point origin. If $\Lambda < 0$, the force is attractive relative to the point origin. We can also introduce in the Newtonian approach to cosmology a term containing the cosmological constant, associated with a kind of cosmological force. We assume then that there exists a global cosmological force that affects the particle (galaxy). We take this force to be the repulsive gravitational force that yields the potential
\begin{equation}
U_\Lambda = -\int_0^a F_\Lambda da' = -\frac{1}{6}\Lambda m a^2.
\end{equation}
In a fixed rectangular coordinates (comoving) the Lagrangian of the system is
\begin{equation}\label{clagrang}
L (a, \dot{a}) = \frac{1}{2}m\dot{a}^2 + \frac{GMm}{a} +
\frac{1}{6}\Lambda m a^2.
\end{equation}
The Lagrangian depends only on the scale factor and its derivative, and is independent of the time because the system is under the action of a uniform force field.
\section{Affine quantization and Hamiltonian operator}\label{sec3}
The constant quantity of the motion is the classical Hamiltonian of the system defined by
\begin{equation}\label{chamil1}
H = \frac{\partial L}{\partial \dot{a}}\dot{a} - L\,.
\end{equation}
Inserting equation (\ref{clagrang}) into the equation (\ref{chamil1}) we got the Hamiltonian at the classical level
\begin{equation}\label{chamil2}
H(a, p_a) = \frac{1}{2m}p_a^2 - \frac{G\;M\; m}{a} - \frac{1}{6}\Lambda m a^2\,,
\end{equation}
where $p_a$ is considered as the corresponding momentum
\begin{equation}\label{mom}
p_a = \frac{\partial L}{\partial \dot{a}} = m\dot{a}\,.
\end{equation}
We consider a fixed rectangular comoving coordinate in which we can define the phase space as $(a, p_a)$ with $p_a$ in (\ref{mom})
being the linear momentum.
The associate Hamiltonian in terms of the phase space coordinates is given by (\ref{chamil2}).
Since the variable $a$ is positive, we consider the positive axis. In that sense we choose to perform the affine quantization procedure due to J. R. Klauder \cite{klauder1, klauder2, klauder3} that received some recent applications \cite{fanuel,bergeron, zonetti, almeida, laure1, laure2, riccardo}. Here we choose to pay a careful attention to the quantization procedures.
The non-vanishing Poisson bracket of the canonical coordinates is given by
\begin{equation}\label{npb}
\left\lbrace a,\; p_a\right\rbrace = 1\,.
\end{equation}
Multiplying the equation (\ref{npb}) by $a$, we have
\begin{equation}
a \left\lbrace a,\;p_a\right\rbrace = a\,,
\end{equation}
and setting $d_a = a p_a $, we have
\begin{equation}
\left\lbrace a,\;d_a\right\rbrace = a,
\end{equation}
and
\begin{equation}
H (a, d_a)= \frac{1}{2}d_a\; a^{-2} d_a - \frac{G\,M\,m}{a} - \frac{\Lambda}{6}\; m\; a^2\,.
\end{equation}
The variables $a,\; d_a$ are not canonical variables but
form a Lie algebra and then worthy to be considered as a new
pair of classical variables. The canonical quantization involves
$\hat a,\; \hat p_a$ which are self adjoint operators that satisfies the commutation relations
\begin{equation}
\left[ \hat{a}, \hat{p}_a\right] = i\hbar,
\end{equation}
and from the canonical quantization, it follows that
\begin{equation}
\left[ \hat a, \hat{d}_a \right] = i\hbar \hat a,
\end{equation}
$\hat d$ is the dilation operator and the operators $\hat a$ and $\hat{d}_a$ are realized as follows
\begin{equation}
\hat a \psi (a) = a\psi(a);\quad
\hat{d}_a \psi (a) = -i\hbar\left( a\frac{d}{d a} +\frac{1}{2}\right)\psi(a)\,,
\end{equation}
where $\psi(a)$ is the wave function of the Newtonian universe. The Hamiltonian operator for a particle (galaxy) moving in the Newtonian universe is then given by
\begin{equation}
\hat{H} = -\frac{\hbar^2}{2m}\frac{d^2}{da^2} +
\frac{\hbar^2}{2m}\frac{3}{4}\frac{1}{a^2} -\frac{G\;M\;m}{a}
- \frac{\Lambda}{6}ma^2\,.
\end{equation}
\section{The Wave functions of the Newtonian universe and the energy levels}\label{sec4}
Once we have the Hamiltonian operator of the system, it is possible to find the wave functions of the Newtonian universe \cite{juan, bras2, bras3}, by solving the time-independent Schr\"odinger equation
\begin{equation}\label{tise}
\hat H \psi(a) = E \psi(a)\,,
\end{equation}
where $\psi(a)$ corresponds to the eigenvalues $E$, with
$\Psi(a,t) = \psi(a)e^{-iEt/\hbar}$ being the general wave function solution of the time-dependent Schr\"odinger equation.
The equation (\ref{tise}) is equivalent to
\begin{equation}\label{equat1}
-\frac{\hbar^2}{2m}\frac{d^2\psi(a)}{da^2} + \left(
\frac{\hbar^2}{2m}\frac{3}{4}\frac{1}{a^2} -G M m\frac{1}{a}
- \frac{\Lambda}{6}ma^2\right) \psi(a) = E\psi(a)\,.
\end{equation}
Let's solve the equation (\ref{equat1}) that is equivalent by dividing by $\frac{\hbar^2}{2m}$ to
\begin{equation}
-\frac{d^2\psi(a)}{da^2} + \frac{2m}{\hbar^2}\left(
\frac{\hbar^2}{2m}\frac{3}{4}\frac{1}{a^2} -G M m\frac{1}{a}
- \frac{\Lambda}{6}ma^2\right) \psi(a) = \frac{2m}{\hbar^2}E\psi(a)\,,
\end{equation}
labelling the parameters as follows
\begin{equation}
\alpha = \frac{3}{4};\:\: \delta = -\frac{4GMm^2}{\hbar^2};\:\:
\omega^2 = (\frac{-\Lambda}{3})\frac{m^2}{\hbar^2};\:\: k^2 = \frac{2m E}{\hbar^2}\,,
\end{equation}
we have
\begin{equation}\label{equat2}
-\frac{d^2\psi(a)}{da^2} + \left(
\frac{\alpha}{a^2} + \frac{1}{2}\frac{\delta}{a} + \omega^2 a^2 -k^2
\right) \psi(a) = 0.
\end{equation}
The sign of the cosmological constant is for the moment left arbitrary.
In search of solution of the equation (\ref{equat2}), we set the Antsatz
\begin{equation}
\psi(a) = a^{\beta + 1}e^{-\frac{\omega}{2}a^2}v(a)\,,
\end{equation}
where $\beta(\beta + 1) = \alpha$ and the function $v(a)$ is an auxiliary function satisfying the equation
\begin{equation}\label{aeq1}
v''(a)+ \left( 2(\beta +1)\frac{1}{a} - 2\omega a\right)v' (a)
+ \left( k^2 -\omega(2\beta +3) -\frac{1}{2}\frac{\delta}{a}\right) v (a) = 0\,.
\end{equation}
Performing the change of variable $x = w^{1/2}a$ in (\ref{aeq1}), we have
\begin{equation}\label{equat3}
v''(x) + \left(\frac{2(\beta +1)}{x} -2x \right)v'(x) +
\left( \frac{k^2}{\omega} -(2\beta +3) -\frac{1}{2}\frac{\delta \omega^{-1/2}}{x}\right) v(x) = 0.
\end{equation}
The equation (\ref{equat3}) is a particular case of the biconfluent Heun equation \cite{ronv} and its solution is the biconfluent Heun function
\begin{equation}
v(x) = \rm{HeunB}\left( 2\beta +1;\; 0;\; k^2\omega^{-1};\; \delta\omega^{-1/2};\;x\right)\,.
\end{equation}
Some details about biconfluent Heun equation are given in the Appendix (see after section (\ref{sec6})).
From the relation $\beta (\beta +1) = \alpha $ and $\alpha = 3/4$,
there are two possibilities $\beta = 1/2$ or $\beta = -3/4$.
For $(2\beta + 1)$ not a negative integer, the biconfluent Heun functions $v(x)$ can be written in series form as
\begin{eqnarray}
v(x) & =& \rm{HeunB}\left( 2\beta +1;\; 0;\; k^2\omega^{-1};\; \delta\omega^{-1/2};\;x\right);\\
&=& \sum_{p =0}^\infty\frac{A_p}{(1 + (2\beta +1))_p}\frac{x^p}{p!}\,,
\end{eqnarray}
where
\begin{equation}
A_{p+2} = \frac{1}{2}\delta\omega^{-1/2}A_{p+1} -
(p+1)(p +1 + (2\beta +1))(k^2\omega^{-1} - (2\beta + 1) -2- 2p)A_p\,.
\end{equation}
We consider the value $\beta = 1/2$ that gives $2\beta + 1 = 2$ and then
\begin{eqnarray}
v(x) & =& \rm{HeunB}\left( 2;\; 0;\; k^2\omega^{-1};\; \delta\omega^{-1/2};\;x\right);\\
&=& \sum_{p = 0}^\infty\frac{A_p}{(3)_p}\frac{x^p}{p!},
\end{eqnarray}
where
\begin{equation}\label{receq}
A_{p+2} = \frac{1}{2}\delta\omega^{-1/2}A_{p+1} -
(p+1)(p + 3)(k^2\omega^{-1} - 4 - 2p)A_p.
\end{equation}
From the recursion equation (\ref{receq}), the function $v(x)$ becomes a polynomial of degree n and the series terminate if and only if the two following conditions are fulfilled
\begin{enumerate}
\item $k^2\omega^{-1} - 4 = 2n $;
\item $A_{n +1}$ = 0,
\end{enumerate}
where $A_{n+1}$ is a polynomial of degree $(n +1)$ in $\tilde{\delta} = \delta \omega^{-1/2}$. There are at most $(n+1)$ suitable values of value $ \tilde\delta $ usually labelled as $\tilde{\delta}_\mu^n, \: 0\le \mu \le n$. In that sense, we have
\begin{eqnarray}
v_n(x) & =& \rm{HeunB}\left( 2;\; 0;\; 2(n+2);\; \tilde{\delta}^n_\mu;\;x\right)\,;\\
&=& P_{n,\mu}(2,0,x) = \sum_{p =0}^n\frac{A_p}{(3)_p}\frac{x^p}{p!}\,,
\end{eqnarray}
with
\begin{equation}
A_0 = 1\,;\; A_1 = \frac{1}{2}\tilde{\delta}\,;
\end{equation}
\begin{equation}
A_{p+2} = \frac{1}{2}\tilde{\delta}A_{p+1} -2
(p+1)(p + 3)(n-p)A_p\,;\quad A_{n+1} = 0.
\end{equation}
The eigenvalues are determined from the condition $k^2\omega^{-1} -4 = 2n$ as follows
\begin{equation}
E_n = (n +2)\hbar \left(\frac{-\Lambda}{3}\right)^{\frac{1}{2}}\,.
\end{equation}
The associated physically acceptable wave functions $\psi_n(a)$ are
given by
\begin{equation}\label{wavfunc}
\psi_n(a) = C_n a^{3/2}e^{-\frac{\omega}{2}a^2}\rm{HeunB}
\left( 2 ;\; 0;\;2(n + 2) ;\tilde\delta_\mu^n;\;\omega^{\frac{1}{2}}a\right)\,.
\end{equation}
The constant $C_n$ are arbitrary factors that can be determined under the normalization condition
\begin{equation}
\int_0^\infty \vert \psi_n(a)\vert ^2 da = 1\,.
\end{equation}
We have
\begin{equation}
C_n^2 \int_0^\infty a^3 e^{-\omega a^2} \sum_{p =0}^n\frac{A_p}{(3)_p}\frac{\omega^{\frac{p}{2}}a^p}{p!}\sum_{s =0}^m\frac{A_s}{(3)_s}\frac{\omega^{\frac{s}{2}}a^s}{s!}da = 1\,,
\end{equation}
\begin{equation}
C_n^2\sum_{p=0}^n\sum_{s =0}^m\frac{A_p\omega^{\frac{p}{2}}}{(3)_p p!}\frac{A_s\omega^{\frac{s}{2}}}{(3)_s s!}
\int_0^\infty a^{3+p+s}e^{-\omega a^2} da = 1\,.
\end{equation}
For $ p = s$, $\mu = \nu$ and $m=n$, we have
\begin{equation}\label{formu1}
C_n^2 \sum_{p= 0}^n\frac{A_p^2\omega^p}{[(3)_p p!]^2}\int_0^\infty a^{3+2p}e^{-\omega a^2}da = 1.
\end{equation}
Let's use now the formula
\begin{equation}
\int_0^\infty y^{\lambda -1} e^{-\eta y^u}dy = \frac{1}{u}\eta^{-\lambda/u}\Gamma(\frac{\lambda}{u})\,,
\end{equation}
so we have
\begin{equation}\label{formu2}
\int_0^\infty a^{3+2p}e^{-\omega a^2}da = \frac{1}{2}\omega^{-(2 +p)}\Gamma(2 +p)\,,
\end{equation}
and inserting (\ref{formu2}) into (\ref{formu1}), we have
\begin{equation}\label{formu3}
C_n^2 \sum_{p= 0}^n\frac{A_p^2\omega^p}{[(3)_p p!]^2}
\frac{1}{2}\omega^{-(2 +p)}\Gamma(2 +p) = 1\,,
\end{equation}
that is
\begin{equation}
C_n^2\sum_{p = 0}^n \frac{A_p^2}{ [(3)_p p!]^2 }\frac{1}{2\omega^{2}}\Gamma(2 +p) = 1\,,
\end{equation}
so
\begin{equation}\label{intconst}
C_n =\left[ \sum_{p = 0}^n \frac{A_p^2}{ [(3)_p p!]^2 }\frac{1}{2\omega^{2}}\Gamma(2 +p)\right]^{-\frac{1}{2}}\,.
\end{equation}
Inserting the constant of integration $C_n$ in equation (\ref{intconst}) into the equation (\ref{wavfunc}), we have
\begin{eqnarray}\nonumber
\psi_n(a) &=& \left[ \sum_{p = 0}^n \frac{A_p^2}{ [(3)_p p!]^2 }\frac{1}{2\omega^{2}}\Gamma(2 +p)\right]^{-\frac{1}{2}}\\
&\times & a^{\frac{3}{2}}e^{-\frac{\omega}{2}a^2}\rm{HeunB}
\left( 2 ;\; 0;\;2(n +2) ;\tilde\delta_\mu^n;\;\omega^{\frac{1}{2}}a\right)\,.
\end{eqnarray}
We consider now the case of matter dominated by a negative cosmological constant, so $\Lambda = - |\Lambda|$. The wave functions for the Newtonian universe with negative cosmological constant are given by
\begin{eqnarray}\nonumber
\psi_n(a) = \left[ \sum_{p = 0}^n \frac{A_p^2}{ [(3)_p p!]^2 }\frac{1}{2\omega^{2}}\Gamma(2 +p)\right]^{-\frac{1}{2}}\\\label{leq}
a^{\frac{3}{2}}e^{-\frac{\omega}{2}a^2}\rm{HeunB}
\left( 2 ;\; 0;\;2(n +2) ;\tilde\delta_\mu^n;\;\omega^{\frac{1}{2}}a\right)\,;
\end{eqnarray}
\begin{equation}
E_n = (n +2)\hbar \left(\frac{|\Lambda|}{3}\right)^{\frac{1}{2}},\: n= 0, 1, 2 \ldots;
\end{equation}
with the parameters given by
\begin{equation}
\omega^2 = \frac{|\Lambda|}{3}\frac{m^2}{\hbar^2};\quad
\omega = \left(\frac{|\Lambda|}{3}\right)^{\frac{1}{2}}\frac{m}{\hbar};\quad
\omega^{\frac{1}{2}} = \left( \frac{|\Lambda|}{3}\right)^{\frac{1}{4}}\left( \frac{m}{\hbar}\right)^{\frac{1}{2}}\,;
\end{equation}
\begin{equation}
\tilde{\delta}= -4GM \left( \frac{|\Lambda|}{3}\right)^{-\frac{1}{4}}\left( \frac{m}{\hbar}\right)^{\frac{3}{2}}\,.
\end{equation}
\section{Concluding remarks}\label{sec5}
We consider a small spherical portion of the universe, where Newton's theory applies and the behavior of this portion may reflect the evolution of the universe.
Since the variable $a(t)$ that is referred to as the scale factor is positive, we choose to pay attention to the affine quantization procedures. The solution of the time independent Schr\"odinger equation is of type biconfluent Heun function where the two first parameters are fixed to $2$ and $0$ respectively. Our results are quite similar to the ones in \cite{bras2} with the difference that performing affine quantization, the Hamiltonian operator gains an extra term that is proportial to $1/a^2$ and that extra term fixes the first parameter of the biconfluent Heun function to the value of $2$. In a case of matter dominated by negative cosmological constant, $\Lambda = - |\Lambda|$, the eigenvalues are positive equally spaced and non-degenerated.
The Newtonian approach is much simple from the conceptual and mathematical point of views however it met also problems and criticisms \cite{prob1, prob2, prob3, prob4, prob5}. It is interesting to work out under which conditions Newtonian cosmology applies. Recent works in Newtonian cosmology include \cite{barrow, casadio}.
\section*{Appendix: Biconfluent Heun Equation}\label{sec6}
A biconfluent Heun equation, denoted by
$\rm{BHE}(\alpha,\beta,\gamma,\delta)$ is the equation of form
\begin{equation}
xu''(x) + \left[ 1 +\alpha -\beta x -2 x^2 \right]u'(x)
+ \left\lbrace
\left( \gamma -\alpha -2 \right) x -
\frac{1}{2}\left(\delta + (\alpha +2)\beta \right)
\right\rbrace u(x) = 0
\end{equation}
in which $(\alpha, \beta,\gamma,\delta) \in \mathbb{C}^4$. It has a regular singular point at $0$ and an irregular singular point at
$\infty$. The biconfluent Heun equation is well known and studied in a mathematical point of view (\cite{ronv}, \cite{math1, math2, math3, math4} ) and has some applications in different areas of physics \cite{phys1, phys2, phys3}. Refering to [ \cite{ronv}, pp. 203 - 206], if the biconfluent Heun equation admits a polynomial solution then it is necessary that $\gamma -\alpha -2p = 2n$, where $n$ is some non-negative integer, holds. When $\alpha$ is not a negative integer, one can denote by $N(\alpha,\beta, \delta, x)$ a power series (analytic) solution that can be written as
\begin{equation}
N(\alpha,\beta,\gamma, \delta, x) = \sum_{p =0}^\infty \frac{A_p}{(1 +\alpha)_p}\frac{x^p}{p!},
\end{equation}
with $(\alpha)_p = \frac{\Gamma(\alpha + p)}{\Gamma(\alpha)},\; p\ge 0$, and satisfies the three -term recursion formula
\begin{eqnarray}\nonumber
A_{p+2} & =& \left\lbrace
(p + 1)\beta + \frac{1}{2}[\delta + \beta (1 +\alpha)]
\right\rbrace A_{p+ 1} \\ \label{eqrec}
&-& (p +1)(p+1+\alpha)(\gamma -\alpha -2-2p)A_p = 0\,,
\end{eqnarray}
where $A_0 = 1,\: A_1 = \frac{1}{2}[\delta + \beta (1 + \alpha)]$. The three recursion term in (\ref{eqrec}) terminates if and only if
$\gamma -\alpha -2 = 2n,$ and $ A_{n +1} = 0$ simultaneously where $n$ is some non negative integer. Performing induction $A_{n+1}(\delta)$ is a polynomial of $\delta$ of degree $n +1$ hence possessing at most $n+1$ roots $\delta_\mu^n, \mu = 0,1,2,\ldots$ When the series solution terminates, then we write
\begin{equation}
P_{n,\mu}(\alpha,\beta; x) = N(\alpha, \beta, \alpha + 2(n +1),\delta_\mu^n;x), \quad 0 \le \mu \le n,\: n = 0,1,2,\ldots
\end{equation}
When $\alpha +1 > 0$ and $\beta \in \mathbb{R}$, then the $(n+1) $
roots are real. When all the roots are simple, then the polynomial solutions described are precisely orthogonal polynomials (see \cite{ronv}, pargraph 3.3).
\section*{Acknowledments}
I am grateful to Sana Khadim PhD student at NUST-Islamabad/Pakistan whose interest in quantization of black holes inspired and motivated me to include quantum cosmology in my research interests.
|
{
"redpajama_set_name": "RedPajamaArXiv"
}
| 3,485
|
Q: PostgreSQL: could not connect to server: Connection refused I have a Django project, I want to switch from SQLite to PostgreSQL.
After installation, I can't run the command psql
Here is the traceback:
psql: could not connect to server: Connection refused (0x0000274D/10061)
Is the server running on host "localhost" (::1) and accepting
TCP/IP connections on port 5432?
could not connect to server: Connection refused (0x0000274D/10061)
Is the server running on host "localhost" (127.0.0.1) and accepting
TCP/IP connections on port 5432?
OS: Windows
A: TCP/IP connections are not enabled by default, so you probably have to edit a file called postgres.conf:
vi /etc/postgresql/9.4/main/postgresql.conf
For you it may reside in a different location. Look for a line saying:
#listen_addresses = '' # what IP address(es) to listen on;
Change it to this:
listen_addresses = '*' # what IP address(es) to listen on;
Right under this there's the port setting. For me it reads:
port = 5432 # (change requires restart)
Higher up in the same file there is a reference to another config file:
hba_file = '/etc/postgresql/9.4/main/pg_hba.conf' # host-based authentication file
Go ahead and edit that file. You should insert a line like this:
host all all 192.168.1.0 255.255.255.0 trust
Your IP may be different. (Once you ensure this is working, you can change "trust" to "md5" for better security.) After doing this, you need to restart the postgres server.
/usr/lib/postgresql/9.4/bin/pg_ctl restart
|
{
"redpajama_set_name": "RedPajamaStackExchange"
}
| 2,484
|
Cornelis Jan Dirk (Cees) Waal (Soest, 15 september 1943 – Leiden, 11 juli 2011) was een Nederlandse jurist en politicus voor de PvdA.
Leven en werk
Waal volgde het Stedelijk Gymnasium Leiden en studeerde vanaf 1960 Nederlands Recht aan de Universiteit Leiden. Na het voltooien van zijn studie werd hij vervolgens wetenschappelijk medewerker Romeins recht te Leiden, de stad waar hij ook politiek actief werd voor de PvdA. In 1970 kwam hij in de gemeenteraad. Eind jaren zeventig en begin jaren tachtig was hij wethouder en locoburgemeester van Leiden. In 1984 volgde zijn benoeming tot burgemeester van Deventer, een functie die hij ruim negen jaar zou vervullen, maar die hij in 1993 verwisselde voor het voorzitterschap van het college van bestuur van de Hogeschool Rotterdam. Na het neerleggen van deze functie in 1998 vervulde hij enkele waarnemend burgemeesterschappen in Castricum en Sassenheim. In 2006 werd hij benoemd tot waarnemend burgemeester van Delfzijl, als opvolger van Maritje Appel, die in conflict was gekomen met haar wethouders en was afgetreden als burgemeester. De commissaris der Koningin, Alders, belastte hem met het "weer in het gareel krijgen" van deze gemeente, "een rol die hij vaker had vervuld".
Waal vervulde diverse nevenfuncties o.a. het voorzitterschap van de raad van toezicht van de VARA. Tevens was hij raadsheer-plaatsvervanger bij het gerechtshof Amsterdam en vicevoorzitter van de commissie bezwaarschriften van het ministerie van OCW.
Hij werd op 26 september 2008 benoemd tot Officier in de Orde van Oranje-Nassau. Waal ontving de onderscheiding uit handen van commissaris van de koningin Max van den Berg tijdens zijn afscheidsreceptie als waarnemend burgemeester van Delfzijl.
Publicaties over Cees Waal
Frits van Oosten: 'De stad en de wethouder. Hoe Cees Waal de binnenstad van Leiden vernieuwde'. Leiden, 2017, .
Hans Blom: 'Cornelis Dirk Jan Waal'. In: Jaarboek van de Maatschappij der Nederlandse Letterkunde te Leiden. 2011-2012, pag. 138-148.
Burgemeester van Castricum
Burgemeester van Delfzijl
Burgemeester van Deventer
Burgemeester van Sassenheim
Gemeenteraadslid van Leiden
Nederlands rechter
PvdA-politicus (Nederland)
Wethouder van Leiden
|
{
"redpajama_set_name": "RedPajamaWikipedia"
}
| 6,713
|
\section{Introduction}
As an emerging technology, visible light communication (VLC) use the license-free light spectrum to provide high speed communication and network connectivity, which can appreciably meet the unprecedented demands of communication and connection in the era of Internet of things \cite{wu2014visible}.
Over the decades, the user mobility problem is one of the challenges that both the research community and industry have to tackle to make VLC really commercial. On the account of high directionality and attenuation of light, in mobile scenarios, the received signal amplitude is easily either too large causing ADC clipping or too small resulting in quantization noise, and thus BER deterioration. Similar to a mature RF system, the solution to this is to introduce automatic gain control (AGC), which can stabilize the input amplitude of ADC regardless of the signal variation, into the VLC system\cite{cuailean2017current}.
For AGC, the signal strength is detected and then used to control the signal gain, where the strength detection process can be accomplished either digitally, namely, ADC sampling and digital calculation, or with analog circuit. Digital AGCs are widely used in RF systems for its flexibility of adjustment and high reliability, where the ADC quantizer combined with AGC were analyzed in \cite{yeh2004agc,sun2011particle,narieda2013agc}. While in the VLC research, \cite{okada2009vehicle} first realized the importance of AGC while studying the VLC vehicular communication. In \cite{cailean2013robust}, the authors conducted preliminary experiments of AGC for VLC of different distances, where the signal gain was set by switches. Due to the ability to mitigate the amplitude attenuation, an analog AGC amplifier was used to extend the LED bandwidth in \cite{chow2015practical}. Besides, in terms of AGC circuit design towards VLC system, certain papers were found\cite{fuada2017automatic,zhang2008analog}.
Until now, effects of AGC has not been fully studied in VLC situations, both theoretically and experimentally. In this paper, starting from a simple amplifier model, we derived the SNR model for AGC. Besides, combined with VLC front-end reception model, some AGC practical design insights for VLC systems were given. Finally, experiments were conducted to evaluate the effectiveness of AGC.
\begin{figure}[!htbp]
\begin{center}
\includegraphics[height=3.1cm]{SystemBlocksV6.pdf}
\makeatletter\def\@captype{figure}\makeatother
\caption{A typical mobile VLC scenario}\label{fig:sysBlocks}
\end{center}
\end{figure}
\vspace{-0.5cm}
\section{System Model}
\subsection{Signal propagation} \label{suc:A}
For the VLC transmitting-end, signals are firstly generated by the programmable hardware, e.g., FPGA. Then proper power amplification and DC bias are given to drive the LED. In practice, due to the DAC quantization noise and the power amplifier noise, we consider that the transmitted signals are noisy, and for a certain discrete time, the transmitted optical signal of LED was denoted by random variable
\begin{equation}\label{eq:LED_output}
b=\alpha(s+n_t+v_b)
\end{equation}
where $b$ and $s$ are random variables, representing the optical power and electrical signal of transmitter; $n_t\sim N(0,\sigma^2_t)$ is the transmitter noise, assumed as zero-mean Gaussian white noise and to be independent of $s$; $v_b$ is the DC bias voltage; $\alpha$ is the electro-optical conversion coefficient.
Through channel propagation, the converted electrical current at the receiving-end is\cite{Kahn1994Wireless}
\begin{equation}\label{eq:APD_output}
x=\beta(hb+b_0)+n_i+n_d
\end{equation}
where $x$ is the electrical current of front-end; $\beta$ is the APD responsivity; $b_0$ is the optical power of ambient lights, which usually is considered as a constant; $h$ is the Lambertian channel gain, given by \cite{Kahn1994Wireless}
\begin{equation}\label{eq:h}
h=
\begin{cases}
\displaystyle\frac{(n+1)A}{2\pi d^2}cos^n\phi T_s(\psi)g(\psi)cos\psi & 0\leqslant\psi\leqslant\Psi_c \\
\displaystyle 0 & \Psi_c\leqslant\psi
\end{cases}
\end{equation}
where the Lambertian index $n$ depends on the half-power angle of transmitter $\phi_{1/2}$, given by $n=-1/log_2(cos\phi_{1/2})$; $A$ is the receiver area of photodiode; $d$ is the communication distance; $\phi$ is the emission angle; $\psi$ is the incident angle; $T_s(\psi)$ is the optical filter gain; $\Psi_c$ is the angle of half field of view (FOV); $g(\psi)=\chi^2/sin^2\Psi_c$ is the concentrator gain, where $\chi$ is the refractive index.
In (\ref{eq:APD_output}$), n_i \sim N(0,\sigma^2_i)$ is the input-independent noise, representing both the circuit noise and the ambient light induced shot noise, given by
\begin{equation}\label{eq:idpnoise}
\sigma^2_i=\sigma^2_{c}+2qMF_A\Delta f\beta b_0
\end{equation}
where $\sigma^2_{c}$ is the circuit noise; $q$ is the electron charge; $M$ is the APD multiplication factor; $F_A$ is the excess noise factor; $\Delta f$ is the system bandwidth.
$n_d \sim N(0,\sigma^2_d)$ is the input-dependent noise, representing the signal induced shot noise, whose average variance $\bar{\sigma^2_d}$ can be denoted by\cite{Ming2015Coding}
\begin{equation}\label{eq:shotNoise}
\bar{\sigma^2_d}=\mathbb{E}_s[\sigma^2_d]=
2qMF_A\Delta fh\alpha\beta(\mathbb{E}[s]+v_b)
\end{equation}
Considering DC components are blocked by capacitors, the total front-end electrical AC power is thus given by
\begin{equation}\label{eq:frontEndElecPower}
\begin{split}
&p_x=\mathbb D[x]r_l \\
&=\{\mathbb D[\beta hb]+\bar{\sigma^2_d}+2\beta h\operatorname{Cov}[b,n_d]+\sigma^2_i\}r_l
\end{split}
\end{equation}
where $p_x$ represents the converted electrical power; $r_l$ is the front-end load resistor. Despite the signal dependency between $b$ and $n_d$, the covariance term $\operatorname{Cov}[\cdot]$ is $0$. Using (\ref{eq:LED_output}), (\ref{eq:APD_output}) and (\ref{eq:shotNoise}), we have
\begin{equation}\label{eq:finalPowerEq}
p_x=p_s+p_n \\
\end{equation}
where $p_s$ represents the electrical power of signal and $p_n$ is the total electrical power of front-end noise, denoted by
\begin{gather}\label{eq:ps}
p_s=(h\alpha\beta)^2\mathbb{D}[s]r_l \\
p_n=\lambda p_s+2qMF_A\Delta f\sqrt{(p_sr_l)/\mathbb{D}[s]}(\mathbb{E}[s]+v_b)+\sigma^2_ir_l \label{eq:pn}
\end{gather}
where $\lambda$, defined as $\sigma^2_t/\mathbb{D}[s]$, is the transmitted noise signal ratio. The transmitter is noiseless once $\lambda=0$.
The front-end input SNR thus is denoted by
\begin{equation}\label{eq:SNR}
SNR_i=\frac{p_s}{p_n}
\end{equation}
\subsection{Automatic gain control}
After the front-end reception, signals are fed to AGC to prevent signal fluctuations caused by relative movements of the VLC transmitter and receiver. The essence of an AGC amplifier is a variable gain amplifier (VGA) that can adjust gains dynamically according to the input strength, signal amplitude or power, so that the output remains stable (see AGC diagram in Fig.(\ref{fig:AGC})). Here $x(t)$ and $y(t)$ are the input and output signal of continuous time, respectively. $g(v_c)$ is the VGA gain, and usually is a exponential function of control voltage $v_c$, leading a constant setting time of AGC loop that is independent of input signal $x(t)$\cite{Prez2011Automatic}. The VGA output strength is then detected by a loop detector, whose output $d(y(t))$ is further compared with a external reference voltage $v_{ref}$. Finally, through a loop filter, the comparison difference is smoothed into a DC voltage $v_c$ to control the VGA's gain, where we have \cite{Prez2011Automatic}
\begin{equation}\label{eq:loopDynamic}
v_c(t)=k_2\int_{0}^{t}(k_1v_{ref}-d(y(\tau)))\, d\tau
\end{equation}
where $k_1$ and $k_2$ are the circuit scaling factors. The strength detection can be implemented digitally using ADC or by analog detectors, such as the square-law detector\cite{whitlow2003design}.
Apparently, from (\ref{eq:loopDynamic}), as the input strength rises, the detector output becomes larger than the reference level, namely, $k_1v_{ref}-d(y(\tau))<0$. Through the time integral, then $v_c$ will decline correspondingly, resulting in a smaller amplifier gain $g(v_c)$, which will lower the output $y(\tau)$. Subsequently, the loop reaches the equilibrium-state as $k_1v_{ref}=d(y(\tau))$ and outputs a constant power that can be adjusted by $v_{ref}$. This process of loop settling is the same vice versa when the input strength declines.
Though such a AGC system is inherently non-linear, for small changes of input strength, after logarithmized formulation with proper approximation technique, namely, Taylor series expansion, we can operate and analyze the AGC system as a first order linear system in decibel (dB)\cite{khoury1998design}.
The dynamic performance of first-order linear system is described by the time constant $\tau$\cite{kuo1995automatic}. For instance, for a unit step input, it takes $t=3\tau$ for the system to rise from 0\% to 95\% of the final steady amplitude. In practical mobile VLC scenarios, the designed time constant thus should be much smaller than the power fluctuation period.
\begin{figure}[!htbp]
\vspace{-0.4cm}
\begin{center}
\includegraphics[height=4cm]{AGCBlocksV4-eps-converted-to.eps}
\makeatletter\def\@captype{figure}\makeatother
\caption{A typical feedback AGC diagram}\label{fig:AGC}
\end{center}
\end{figure}
Next, we give a signal SNR model for the settled AGC amplifier. Basically, for a fixed gain amplifier, after a power amplification of $g$, we have\cite{niknejad2007electromagnetics}
\begin{equation}\label{eq:ampModel}
p_y=gp_x+p_a
\end{equation}
where $p_a$ is the AGC noise power, which is independent of the input. Using (\ref{eq:finalPowerEq}), we have the output SNR
\begin{equation}\label{eq:SNRo}
SNR_o=\frac{gp_s}{gp_n+p_{a}}=\frac{SNR_i}{1+p_{a}/gp_n}
\end{equation}
As discussed earlier, the output of the equilibrium-state AGC is a stable power $p_e$. While in practice, on account of the limit gain range of the VGA component, once the gain that needed to amplify the input to the equilibrium level is out of range, the AGC amplifier will behave as a fixed gain amplifier with the boundary gain $g_{max}$ or $g_{min}$. Thus $g$ was denoted by a piecewise function
\begin{equation}
g=
\begin{cases}\label{eq:AGCgain}
g_{max} & p_{x}<p_{l} \\
\displaystyle\frac{p_e-p_{a}}{p_{x}} & p_{l}\leqslant p_{x}\leqslant p_{u} \\
g_{min} & p_{x}>p_{u}
\end{cases}
\end{equation}
where $g_{max}$ and $g_{min}$ are the maximum and minimum accessible gain of VGA; $p_e$ is the equilibrium output power of AGC; two equilibrium thresholds $p_{l}$ and $p_{u}$ were given by
\begin{gather}\label{eq:Pth1}
p_{l}=\frac{p_e-p_{a}}{g_{max}} \\
p_{u}=\frac{p_e-p_{a}}{g_{min}} \label{eq:Pth2}
\end{gather}
Applying (\ref{eq:AGCgain}) to (\ref{eq:SNRo}), defining the AGC index $m=p_e/{p_{a}}$, we have the output SNR of AGC
\begin{equation}\label{eq:AGCTotalSNRo}
SNR_o=
\begin{cases}
\displaystyle\frac{SNR_i}{1+p_a/g_{max}p_n} & p_x< p_l \\
\displaystyle\frac{(m-1)SNR_i}{m+SNR_i} & p_l\leqslant p_x\leqslant p_u \\
\displaystyle\frac{SNR_i}{1+p_a/g_{min}p_n} & p_x< p_u
\end{cases}
\end{equation}
\section{Analytical Results and Analysis}
\subsection{Effect of AGC index $m$}\label{subsec:1}
As shown in (\ref{eq:AGCTotalSNRo}), when AGC is in equilibrium-state ($p_l\leqslant p_x\leqslant p_u$), the output SNR is determined by both the input SNR and the AGC index $m$. In Fig.\ref{fig:AGC_ioSNR}, we first investigated the output SNR performance of equilibrium-state AGC with different $m$, where $m$ was presented in dB like SNR.
\begin{figure}[!htbp]
\begin{center}
\includegraphics[height=6cm]{AGCSNRioV4-eps-converted-to.eps}
\makeatletter\def\@captype{figure}\makeatother
\caption{AGC SNR performance with respect to different $m$}\label{fig:AGC_ioSNR}
\end{center}
\end{figure}
It should be noted in the beginning that, $SNR_o$ is monotonically increasing with respect to $m$. As illustrated in Fig.\ref{fig:AGC_ioSNR}, the SNR curve can be analyzed from three different regions (see the example of region under curve of $m$ = 10 dB). Firstly, in the low SNR region, namely, $SNR_i\ll m$, the output SNR approaches to the input SNR, and is barely improved as we increase $m$. On the contrary, in the high SNR region, namely, $SNR_i\gg m$, certain SNR floors of $m - 1$ occur.
Overall, while the AGC is in equilibrium-state, the system SNR benefits from a larger $m$, avoiding the floor effect. From definition, large $m$ means increasing $p_e$. Nevertheless, the AGC equilibrium thresholds, i.e., $[p_{l},p_{u}]$, is controlled by $p_e$ (see (\ref{eq:Pth1}) and (\ref{eq:Pth2})), simply increasing $p_e$ will result in higher thresholds, which implies a possible trade-off while designing practical systems.
\subsection{Effect of VGA gain $g_{max}$}
\begin{table}
\centering
\caption{Simulation Parameters}\label{tb:Simulation Parameters}
\begin{tabular}{|c|c|c|c|}
\hline
Transmitted signal power ($\mathbb{D}[s]$)&0.08 A$^2$ \\ \hline
LED DC bias ($v_b$)&8 V \\ \hline
LED threshold voltage&6 V \\ \hline
Transmitted noise signal ratio ($\lambda$)&-30 dB \\ \hline
LED conversion coefficient ($\alpha$)&0.125 A/W \\ \hline
Electron charge ($q$)&$1.6\times10^{-19}$ C \\ \hline
Multiplication factor ($M$)&30 \\ \hline
Excess noise factor ($F_A$)&4.77 \\ \hline
System bandwidth ($\Delta f$)&12.5 MHz \\ \hline
Independent noise density&$6.654\times10^{-15}$ mW/Hz \\ \hline
AGC noise density&$2.71\times10^{-12}$ mW/Hz \\ \hline
Load resistor ($r_l$)&$50 \Omega$ \\ \hline
VGA adjustable gain range&48 dB \\ \hline
AGC equilibrium output ($p_e$)&0 dB \\ \hline
APD responsivity ($\beta$)& 460 W/A \\ \hline
Half-power angle ($\phi_{1/2}$)& 60 degree \\ \hline
Optical filter gain ($T_s(\psi)$)& 1 \\ \hline
Half-FOV ($\Psi_c$)& 60 degree \\ \hline
Reflective index ($\chi$)& 1.5 \\ \hline
\end{tabular}
\end{table}
Next, performances of the AGC output SNR with different VGA maximum gains $g_{max}$ under certain front-end situation was given (see Fig.\ref{fig:CombinedAGC_gmax}). In this simulation, the transmitted signal is the zero-mean bipolar OOK, where $\mathbb E[s]=0$. The transmitted noise signal $\lambda$ was set to -30 dB without loss generality. APD parameters are with Hamamatsu C12702-04 and we measured the module open-circuit output in indoor with curtains open to approximate the noise power term $\sigma^2_ir_l$. Also, the AGC amplifier open-circuit output was measured as the AGC noise $p_{a}$. The VGA gain range is fixed by 48 dB, where the VGA minimum gain is then $g_{min}=g_{max}-48$ in dB. The $p_e$ was set to 0 dBm considering the actual ADC input limits (detail system parameters see Table \ref{tb:Simulation Parameters}).
By simple calculation, we have $m$ = 44.7 dB and thresholds were plotted with vertical lines. In Fig.\ref{fig:CombinedAGC_gmax}, the input SNR of AGC is the front-end SNR calculated by (\ref{eq:finalPowerEq})-(\ref{eq:SNR}). Here, the front-end SNR is upper bounded by the transmitted SNR $1/\lambda$ as we included transmitter noise.
\begin{figure}[!htbp]
\begin{center}
\includegraphics[height=6cm]{CombinedAGC_differentGmaxV4-eps-converted-to.eps}
\makeatletter\def\@captype{figure}\makeatother
\caption{AGC output SNR performance with different VGA maximum gain}\label{fig:CombinedAGC_gmax}
\end{center}
\end{figure}
When $p_{x}<p_1$, the output SNR is improved by increasing $g_{max}$. In this interval, AGC behaves as a fixed gain amplifier with gain $g_{max}$. The output SNR described in (\ref{eq:SNRo}), is mainly determined by the term $p_a/gp_n$ while given the input SNR. In fact, since the front-end input noise power $p_n$ is lower bounded by the term $\sigma_c^2r_l$, careful $g_{max}$ design
\begin{equation}\label{eq:gmaxDesign}
g_{max}\gg\frac{p_{a}}{\sigma_c^2r_l}=26.1\;dB
\end{equation}
make $p_{a}/(g_{max}\sigma_i^2r_l)\rightarrow0$ and thus $SNR_i\approx SNR_o$ (see the curve of $g_{max2} = 40$ dB in Fig.\ref{fig:CombinedAGC_gmax}). In addition, aside from the SNR loss caused by small gain, it is noteworthy that as $p_x$ decreases, quantization noise will occur at ADCs stage.
When $p_1\geq p_{x}\geq p_2$, since the input SNR is bounded by 30dB, where $SNR_i\ll m$, the low SNR region conclusion can be applied in this interval. When $p_{x}>p_1$, AGC is a fixed gain amplifier with the bounded input SNR. Same analysis can be applied to this situation using (\ref{eq:SNRo}). Besides, a amplifier with fixed gain of 10 dB was plotted as reference in Fig.\ref{fig:CombinedAGC_gmax}.
\subsection{Analysis of channel dynamic range}
Furthermore, the dynamic range of the received optical power corresponding to the AGC equilibrium range were analyzed. From Section.\ref{suc:A}, ignoring the ambient light $b_0$, the average received optical power is denoted by
\begin{equation}
\bar{o}=\mathbb{E}[hb]=\alpha hv_b
\end{equation}
In our simulation, considering the SNR of the equilibrium thresholds $p_l$ and $p_u$ are higher than 10dB (see Fig.\ref{fig:CombinedAGC_gmax}), we have $p_x\approx p_s$. Using (\ref{eq:ps}), the AGC equilibrium range can thus be noted by
\begin{equation}
DR=10\,lg\frac{p_u}{p_l}\approx 20\,lg\frac{h_u}{h_l}
\end{equation}
where $DR$ is the equilibrium range of AGC; $h_u$ and $h_l$ are the channel gain when the received electrical power are $p_u$ and $p_l$, respectively. The dynamic range for the received optical power is then calculated by
\begin{equation}
10\,lg\frac{\bar{o_u}}{\bar{o_l}}=10\,lg\frac{h_u}{h_l}=\frac{1}{2}DR=24\;dB
\end{equation}
To be more specific, using (\ref{eq:h}), (\ref{eq:ps}) and parameters given in Table.\ref{tb:Simulation Parameters}, considering the emission and incident angle are both 90 degree, the distance $d_l$ corresponding to $p_l$ is calculated as 1.57 m. And considering distance at 1 m, the deviation angle $\phi_l$ corresponding to $p_l$ is calculated as 66 degree, covering the whole FOV of 60 degree.
\section{Experimental Results}
\subsection{AGC experimental transmission characteristic}
Numerous integrated analog AGC amplifiers are commercially available, and we implemented the circuit using AD8362 and AD8331 (from Analog Devices Inc.), to evaluate the effectiveness. The gain range for our VGA is -4.5 dB to 43.5 dB, and the AGC equilibrium output power was set as 0 dBm approximately. The AGC transmission characteristics with OOK signal was given in Fig. \ref{fig:AGCtransCharac}. The OOK signal was generated by arbitrary waveform generator (AWG), and we measured the AGC output directly using oscilloscope in 50 $\Omega$ (see experiment setup in Fig.\ref{fig:ExperimentSetup}.(a)).
\begin{figure}\centering
\subfigure[System setup for AGC transmission and BER experiments]{
\begin{minipage}[b]{0.9\columnwidth}
\includegraphics[height=3cm]{ex_AGCBlocksV3-eps-converted-to.eps}
\end{minipage}}
\subfigure[System diagram of real-time VLC mobile tracking platform]{
\begin{minipage}[b]{0.9\columnwidth}
\includegraphics[height=3.8cm]{TrackingPlatformV2-eps-converted-to.eps}
\end{minipage}}
\caption{Systems diagram for experiments } \label{fig:ExperimentSetup}
\end{figure}
As depicted in Fig.\ref{fig:AGCtransCharac}, due to the gain of the extra loop amplifier (approximately 6dB), two threholds are $p_l=-49.5$ dB and $p_u=-1.5$ dB, respectively. On account of the frequency response of the chosen AGC components, the equilibrium power declines with the increasing bandwidth of input signals. While $p_{x}<p_1$, the AGC behaves as a fixed gain with maximum gain approximately 49.5 dB. On the other side, when $p_{x}>p_u$, we detected large harmonic distortions for this circuit, which makes the AGC unusable in this input region.
\begin{figure}[!htbp]
\begin{center}
\includegraphics[height=6cm]{AGCTransmissionCharacteristicV3-eps-converted-to.eps}
\makeatletter\def\@captype{figure}\makeatother
\caption{Measured transmission characteristics for AGC amplifier}\label{fig:AGCtransCharac}
\end{center}
\end{figure}
\subsection{AGC experimental BER performance}
Next, to evaluate the AGC BER performance, a 1Mbps OOK signal, whose transmitted data is pseudo-random sequence, with different level Gaussian noise was intentionally generated by AWG. Through AGC amplifier, the output then was demodulated offline. While changing the AGC equilibrium power by adjusting the $v_{ref}$, the input signal amplitude was also adjusted to adapt to the AGC input thresholds. Using the AGC noise measured in Table \ref{tb:Simulation Parameters}, we further calculated $m$. In addition, the hard-decision demodulation (HDD) without AGC was conducted as the reference (results see Fig.\ref{fig:AGCberTest}).
\begin{figure}[!htbp]
\begin{center}
\includegraphics[height=6cm]{AGCberTestV3-eps-converted-to.eps}
\makeatletter\def\@captype{figure}\makeatother
\caption{Measured BER performance for different $p_e$}\label{fig:AGCberTest}
\end{center}
\end{figure}
In Fig.\ref{fig:AGCberTest}, coincided with our AGC model analysis earlier, in low SNR region ($SNR_i\ll m$), the output BER approaches to the HDD result (see curves of NoAGC, $m$=21.8 dB,43.7 dB), and while in high SNR region ($SNR_i\gg m$), certain BER floors occur (see curves of $m$=10.6 dB,7.5 dB,4.6 dB). Note that due to the AGC inherent non-linearity (non-linear gain, detector function and etc.), the signal quality is slightly deteriorated after passing AGC. This phenomenon can be effectively improved when applied equalization techniques.
\subsection{Mobile tracking experiment with AGC}
Finally, the AGC function in mobile situation was tested in our real-time VLC mobile tracking platform\cite{liu2019simple} (system see Fig.\ref{fig:ExperimentSetup}.(b)). In this platform, the transmitting-end consists of a information LED, which used for data transmission, and a beacon LED, which used for position detection. The lights moved back and forth with different speeds on a rail. At the receiving-end, the light position was first detected by a high-speed camera using image processing technique and then sent to the motor to finish the light tracking process. Data demodulation was completed by APD.
We compared the AGC amplifier with a fixed gain amplifier with a power gain of 4.5 dB, where both the outputs of AGC and fixed gain amplifier are within the ADC input range. The static BER and output powers of two amplifiers at different rail position are given in Fig.\ref{fig:staticBenchmark} as benchmark. The transmitted signal is 25 Mbps OOK signal of 25 bits pseudo-random sequence and the communication distance is 3 m, with the AGC index $m$ = 43.7 dB.
\vspace{-0.3cm}
\begin{figure}[!htbp]
\begin{center}
\includegraphics[height=6cm]{TrackingBenchmarkPeBER-eps-converted-to.eps}
\makeatletter\def\@captype{figure}\makeatother
\caption{Static BER benchmark}\label{fig:staticBenchmark}
\end{center}
\end{figure}
\vspace{-0.3cm}
As shown in Fig.\ref{fig:staticBenchmark}, when the transmitter places from rail sides to the rail center, the AGC is in equilibrium, and both the BER and output power stay relatively stable. The slight BER and power asymmetry phenomena arise from the asymmetry placements of the experimental equipment.
Considering the values of $m$ and the AGC BER results, we know that the input SNR must be in low SNR region ($SNR_i\ll m$), but still the BER floor occurs. We may explain this as the effect of transmitted noise or LED non-linearity, since the non-linearity can be considered as one type of noise. Also, for the fixed gain amplifier, as discussed earlier, we ascribed the BER loss to the small gain. Real-time tracking experiments at moving speeds of 0.25 m/s, 0.5 m/s and 1 m/s were further performed to evaluate the AGC dynamic performance (see Fig.\ref{fig:moblieAGC}).
As demonstrated in Fig.\ref{fig:moblieAGC}, when moving speeds are low (cases of 0.25 m/s and 0.5 m/s), the BER results are close to the static benchmarks. However, as the transmitter moving speed is up to 1 m/s (tracking angular speed of 18\textdegree/s), due to the mechanical delay of tracking, the light was slightly out of the plane of the APD, causing the power loss, thus BER degradation.
Also note that, since the rise time of our AGC loop, provided by chip datasheet, is roughly 1ms (time of rise from 10\% to 90\% for a step input). And the power changing period in the 1 m/s case is 4 s. Thus the effect of loop dynamic setting is negligible. In practical vehicular scenario, one should also be considered is the power fluctuation caused by road irregularities and vehicle vibrations, whose vibration frequency are mainly low frequency component less than 20Hz\cite{kinoshita2014motion}.
\vspace{-0.3cm}
\begin{figure}[!htbp]
\begin{center}
\includegraphics[height=6cm]{AGCmobile25-eps-converted-to.eps}
\makeatletter\def\@captype{figure}\makeatother
\caption{Tracking BER performance at 25Mbps}\label{fig:moblieAGC}
\end{center}
\end{figure}
\vspace{-0.6cm}
\section{Conclusion}
In this paper, we investigate the effect of AGC amplifier, both theoretical and experimentally.
Our key contributions and findings are as following: We modeled the VLC transmitter as a noisy transmitter, which can explain the BER floor in the experiment while the received signal power varied. A theoretical SNR model for the settled AGC was derived. On the basis of that, the effect of AGC index $m$ and the VGC gain $g_{max}$ were studied:
Firstly, when the AGC is in the equilibrium-state, the output SNR is monotonically increasing with respect to $m$. In the high SNR region ($SNR_i\gg m$), a certain SNR floor occurs and in the low SNR region ($SNR_i\ll m$), the input SNR approaches the output SNR. We also pointed out a possible trade-off between the equilibrium output $p_e$ and the equilibrium thresholds $[p_l,p_u]$ while adjusting $m$.
Secondly, we found that proper VGA gain $g_{max}$ designing, namely, $g_{max}\gg p_{a}/\sigma_i^2r_l$, can avoid the undesired output SNR loss throughout the interval of $p_x<p_l$.
Mobile VLC tracking experiments with AGC was conducted, which proved the effectiveness of AGC in dynamic scenario. Practical considerations such as the time constant of AGC was also mentioned, e.g., power fluctuations caused by road irregularities and vehicles vibrations.
{\small
\bibliographystyle{ieee_fullname}
|
{
"redpajama_set_name": "RedPajamaArXiv"
}
| 4,382
|
{"url":"https:\/\/www.deepdyve.com\/lp\/springer_journal\/an-upper-bound-on-the-double-roman-domination-number-sR52rHRro7","text":"# An upper bound on the double Roman domination number\n\nAn upper bound on the double Roman domination number A double Roman dominating function (DRDF) on a graph $$G=(V,E)$$ G = ( V , E ) is a function $$f : V \\rightarrow \\{0, 1, 2, 3\\}$$ f : V \u2192 { 0 , 1 , 2 , 3 } having the property that if $$f(v) = 0$$ f ( v ) = 0 , then vertex v must have at least two neighbors assigned 2 under f or one neighbor w with $$f(w)=3$$ f ( w ) = 3 , and if $$f(v)=1$$ f ( v ) = 1 , then vertex v must have at least one neighbor w with $$f(w)\\ge 2$$ f ( w ) \u2265 2 . The weight of a DRDF f is the value $$f(V) = \\sum _{u \\in V}f(u)$$ f ( V ) = \u2211 u \u2208 V f ( u ) . The double Roman domination number $$\\gamma _{dR}(G)$$ \u03b3 dR ( G ) of a graph G is the minimum weight of a DRDF on G. Beeler et al. (Discrete Appl Math 211:23\u201329, 2016) observed that every connected graph G having minimum degree at least two satisfies the inequality $$\\gamma _{dR}(G)\\le \\frac{6n}{5}$$ \u03b3 dR ( G ) \u2264 6 n 5 and posed the question whether this bound can be improved. In this paper, we settle the question and prove that for any connected graph G of order n with minimum degree at least two, $$\\gamma _{dR}(G)\\le \\frac{8n}{7}$$ \u03b3 dR ( G ) \u2264 8 n 7 . http:\/\/www.deepdyve.com\/assets\/images\/DeepDyve-Logo-lg.png Journal of Combinatorial Optimization Springer Journals\n\n# An upper bound on the double Roman domination number\n\n, Volume 36 (1) \u2013 Apr 2, 2018\n9 pages\n\n\/lp\/springer_journal\/an-upper-bound-on-the-double-roman-domination-number-sR52rHRro7\nPublisher\nSpringer Journals\nSubject\nMathematics; Combinatorics; Convex and Discrete Geometry; Mathematical Modeling and Industrial Mathematics; Theory of Computation; Optimization; Operations Research\/Decision Theory\nISSN\n1382-6905\neISSN\n1573-2886\nD.O.I.\n10.1007\/s10878-018-0286-6\nPublisher site\nSee Article on Publisher Site\n\n### Abstract\n\nA double Roman dominating function (DRDF) on a graph $$G=(V,E)$$ G = ( V , E ) is a function $$f : V \\rightarrow \\{0, 1, 2, 3\\}$$ f : V \u2192 { 0 , 1 , 2 , 3 } having the property that if $$f(v) = 0$$ f ( v ) = 0 , then vertex v must have at least two neighbors assigned 2 under f or one neighbor w with $$f(w)=3$$ f ( w ) = 3 , and if $$f(v)=1$$ f ( v ) = 1 , then vertex v must have at least one neighbor w with $$f(w)\\ge 2$$ f ( w ) \u2265 2 . The weight of a DRDF f is the value $$f(V) = \\sum _{u \\in V}f(u)$$ f ( V ) = \u2211 u \u2208 V f ( u ) . The double Roman domination number $$\\gamma _{dR}(G)$$ \u03b3 dR ( G ) of a graph G is the minimum weight of a DRDF on G. Beeler et al. (Discrete Appl Math 211:23\u201329, 2016) observed that every connected graph G having minimum degree at least two satisfies the inequality $$\\gamma _{dR}(G)\\le \\frac{6n}{5}$$ \u03b3 dR ( G ) \u2264 6 n 5 and posed the question whether this bound can be improved. In this paper, we settle the question and prove that for any connected graph G of order n with minimum degree at least two, $$\\gamma _{dR}(G)\\le \\frac{8n}{7}$$ \u03b3 dR ( G ) \u2264 8 n 7 .\n\n### Journal\n\nJournal of Combinatorial OptimizationSpringer Journals\n\nPublished: Apr 2, 2018\n\n## You\u2019re reading a free preview. Subscribe to read the entire article.\n\n### DeepDyve is your personal research library\n\nIt\u2019s your single place to instantly\nthat matters to you.\n\nover 18 million articles from more than\n15,000 peer-reviewed journals.\n\nAll for just $49\/month ### Explore the DeepDyve Library ### Search Query the DeepDyve database, plus search all of PubMed and Google Scholar seamlessly ### Organize Save any article or search result from DeepDyve, PubMed, and Google Scholar... all in one place. ### Access Get unlimited, online access to over 18 million full-text articles from more than 15,000 scientific journals. ### Your journals are on DeepDyve Read from thousands of the leading scholarly journals from SpringerNature, Elsevier, Wiley-Blackwell, Oxford University Press and more. All the latest content is available, no embargo periods. DeepDyve ### Freelancer DeepDyve ### Pro Price FREE$49\/month\n\\$360\/year\n\nSave searches from\nPubMed\n\nCreate lists to\n\nExport lists, citations","date":"2018-10-18 22:22:51","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\": 0.730640172958374, \"perplexity\": 531.1272232366331}, \"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-2018-43\/segments\/1539583512015.74\/warc\/CC-MAIN-20181018214747-20181019000247-00292.warc.gz\"}"}
| null | null |
Q: partially downloaded images Hi I have some simple buttons that when clicked fire some js to add an image. My problem is and I'm not sure but might be to do with the internet connection is that every now and again the image will only partially download (see image). Once this has happened the partially downloaded image will be stored in the cache so you cannot click again to get the full image from the server.
All the images are good. What I need to know if possible is how to fix this. Firstly I have tried to detect when an image is only partially downloaded with http://desandro.github.io/imagesloaded/ but this does not fire an error. Once I can detect the error I need to reload the image from the server not the cache.
Sometimes in firefox the error "image corrupt or truncated" will show, all the images are good.
Any help with these two points will be great thanks.
JS: (I have tried to just change the src value but the same error happens)
$('#Canvas_' + ParentName).empty().prepend('<img '
+ 'class="' + Fabric
+ '" match="' + that.attr('match')
+ '" title="' + that.attr('title')
+ '" cost="' + that.attr('cost')
+ '" src="'+imgFolder+imgViews+imgSmall+that.parent().parent().attr('id') + '_' + that.attr('class').replace(toolRepalce, "") + '.' + that.parent().parent().attr('alt')
+ '" height="305px"'
+ ' width="645px" >');
|
{
"redpajama_set_name": "RedPajamaStackExchange"
}
| 8,712
|
Look like a starlet ready to walk the red carpet in this Tarik Ediz 92804 evening gown. This dress is off-the-shoulder with fine sparkling beadwork throughout. A U-shaped detail at back creates a showstopping look. The skirt is smooth with a flared silhouette and finishes with a full length hem. It's a decadent style that's sure to make everyone take note.
|
{
"redpajama_set_name": "RedPajamaC4"
}
| 4,279
|
A greater meaning
July 19, 2015 - By Beth Geraci
Photo: AMTOPP
In the Treasure State, one association is enriching the lives of veteran families, the hungry and the young.
In Montana, where the population hovers around 1 million, people have to drive far to make a difference. And those in the landscape industry do drive far—up to eight hours, in fact—every year to honor veterans.
In an annual outing inspired by Renewal & Remembrance, the annual landscape industry volunteer event at Arlington National Cemetery, the Association of Montana Turf, Ornamental & Pest Professionals (AMTOPP) has groomed two Montana cemeteries alternately every year for the last 10 years.
Called Summer Field Day, the event is held every August at one of two Montana veterans cemeteries: the Columbia Falls Veterans Home, Hospital & Cemetery in Columbia Falls, Mont., and the Eastern Montana State Veterans Cemetery in Miles City, Mont.
As with Renewal & Remembrance, "we prune, fertilize, mulch, kill weeds, do any type of maintenance," says Kari Brittain, the longtime, now former executive director of AMTOPP.
All materials for Summer Field Day are donated by distributors, and equipment is loaned by participating landscape firms. Together, volunteers enhance grounds, remove declining trees, prune, fertilize, groom beds, control weeds and more.
The monetary value of the cemetery projects typically ranges from $4,000 to $6,000 annually. Since Summer Field Day began, the association has donated about 600 man hours and $15,770 in materials to the cause.
Graphic: LM Staff
"It feels awesome to give to people who have sacrificed so much for us," Brittain says. "It's an honor for us to be able to do that."
Every year since its inception, Summer Field Day attendance has grown. Last year, about 20 people participated at the Columbia Falls Veterans Home, Hospital & Cemetery. What started as simply fertilizing and mowing around headstones has become " a little bit more involved," Brittain says.
In a meaningful tribute, the association now plants a tree at Summer Field Day each year, and it's planted by a tree grower who is a veteran. The association then commemorates the day with a special plaque.
In addition to the heartfelt nature of the project, says Charles Cherry, who organizes the Miles City event and owns the Miles City business Evergreen Landscaping & Liquid Lawn Fertilizer Service, "networking together as a group lets us form a tighter bond."
Cherry says his own experience with Renewal & Remembrance gave greater meaning to his involvement in Summer Field Day. At least two AMTOPP members continue to attend Renewal & Remembrance annually, usually the association's president and vice president.
Digging deep to make it happen
Mobilizing for a volunteer event in Montana is much more labor intensive than it would be in a more populous state. It takes effort. And the association, in the midst of its 25th anniversary year, has shown it has as much staying power as it does altruistic chops.
It coordinates its statewide volunteer efforts through mailed newsletters, emails and phone calls. And in a world where electronic communication is the driver, it's not so in Montana, where snail mail reigns.
AMTOPP members spruce up veterans cemeteries at its annual Summer Field Day. Photo: AMTOPP
"A lot of our members do not have email addresses, so we depend a lot on paper," Brittain says.
The association uses that paper every year to organize an annual food drive that it has strengthened in the last three years.
"We felt that we needed to do a better job in the last few years, so we really promote it, and whoever brings the most food gets a free membership for the following year," Brittain says.
In 2014, Jason Stringer, an arborist with Tree Amigos in Great Falls, Mont., offered customers a 10 percent discount if they donated food. Consequently, Stringer brought to the association's annual conference 958 pounds of food, helping the association generate 788 more pounds of food than it did during the previous year's drive.
The association's members also fund three youth scholarships: two $250 4-H Club scholarships that center on green industry projects like community gardens and tree plantings, and a $500 scholarship to a college-bound student. "The goal is to promote the green industry in the state of Montana and to encourage kids to think about improving their communities and how they want to do that," Brittain says.
Brittain says the association's members strive to make a difference in whatever they do. "A lot of us volunteer doing other things," Brittain says. "I really hadn't done a lot of volunteering until I was part of AMTOPP, but once I started volunteering, I saw I could make a difference."
NALP initiates national search for next CEO
NALP Field Trip: Frank & Marty's Excellent Adventure
NALP: Join for as little as $32 a month
Hardscape Solutions: Play on
This article is tagged with AMTOPP, Arlington National Cemetery, Association of Montana Turf Ornamental & Pest Professionals, Beth Geraci, community service, Kari Brittain, NALP, National Association of Landscape Professionals, Renewal & Remembrance, State Association and posted in 0715, Featured
About the Author: Beth Geraci
Geraci is a freelance writer based in Cleveland. She has worked as a professional journalist for more than 15 years, including six years as a writer for the Chicago Tribune. A graduate of Allegheny College and Northwestern University's Medill School of Journalism, Geraci began her career as an editor at a newswire service in Washington, D.C., where she edited and distributed press releases from the White House and congressional leaders. She went on to become the community news reporter at the Jackson Hole Guide newspaper, winning two national feature writing awards. Her other experience includes working as a book editor in Chicago and as a professor of business communications at Cleveland State University.
Subscribe to Landscape Management
If you enjoyed this article, subscribe to Landscape Management to receive more articles just like it.
|
{
"redpajama_set_name": "RedPajamaCommonCrawl"
}
| 733
|
Q: Spirit Qi sequence parsing issues I have some issues with parser writing with Spirit::Qi 2.4.
I have a series of key-value pairs to parse in following format <key name>=<value>.
Key name can be [a-zA-Z0-9] and is always followed by = sign with no white-space between key name and = sign. Key name is also always preceded by at least one space.
Value can be almost any C expression (spaces are possible as well), with the exception of the expressions containing = char and code blocks { }.
At the end of the sequence of the key value pairs there's a { sign.
I struggle a lot with writing parser for this expression. Since the key name always is preceded by at least one space and followed by = and contains no spaces I defined it as
KeyName %= [+char_("a-zA-Z0-9_") >> lit("=")] ;
Value can be almost anything, but it can not contain = nor { chars, so I defined it as:
Value %= +(char_ - char_("{=")) ;
I thought about using look-ahead's like this to catch the value:
ValueExpression
%= (
Value
>> *space
>> &(KeyName | lit("{"))
)
;
But it won't work, for some reason (seems like the ValueExpression greedily goes up to the = sign and "doesn't know" what to do from there). I have limited knowledge of LL parsers, so I'm not really sure what's cooking here. Is there any other way I could tackle this kind of sequence?
Here's example series:
EXP1=FunctionCall(A, B, C) TEST="Example String" \
AnotherArg=__FILENAME__ - 'BlahBlah' EXP2= a+ b+* {
Additional info: since this is a part of a much larger grammar I can't really solve this problem any other way than by a Spirit.Qi parser (like splitting by '=' and doing some custom parsing or something similar).
Edit:
I've created minimum working example here: http://ideone.com/kgYD8
(compiled under VS 2012 with boost 1.50, but should be fine on older setups as well).
A: I'd suggest you have a look at the article Parsing a List of Key-Value Pairs Using Spirit.Qi.
I've greatly simplified your code, while
*
*adding attribute handling
*removing phoenix semantic actions
*debugging of rules
Here it is, without further ado:
#define BOOST_SPIRIT_DEBUG
#include <boost/fusion/adapted.hpp>
#include <boost/spirit/include/qi.hpp>
#include <map>
namespace qi = boost::spirit::qi;
namespace fusion = boost::fusion;
typedef std::map<std::string, std::string> data_t;
template <typename It, typename Skipper>
struct grammar : qi::grammar<It, data_t(), Skipper>
{
grammar() : grammar::base_type(Sequence)
{
using namespace qi;
KeyName = +char_("a-zA-Z0-9_") >> '=';
Value = qi::no_skip [+(~char_("={") - KeyName)];
Sequence = +(KeyName > Value);
BOOST_SPIRIT_DEBUG_NODE(KeyName);
BOOST_SPIRIT_DEBUG_NODE(Value);
BOOST_SPIRIT_DEBUG_NODE(Sequence);
}
private:
qi::rule<It, data_t(), Skipper> Sequence;
qi::rule<It, std::string()> KeyName; // no skipper, removes need for qi::lexeme
qi::rule<It, std::string(), Skipper> Value;
};
template <typename Iterator>
data_t parse (Iterator begin, Iterator end)
{
grammar<Iterator, qi::space_type> p;
data_t data;
if (qi::phrase_parse(begin, end, p, qi::space, data)) {
std::cout << "parse ok\n";
if (begin!=end) {
std::cout << "remaining: " << std::string(begin,end) << '\n';
}
} else {
std::cout << "failed: " << std::string(begin,end) << '\n';
}
return data;
}
int main ()
{
std::string test(" ARG=Test still in first ARG ARG2=Zombie cat EXP2=FunctionCall(A, B C) {" );
auto data = parse(test.begin(), test.end());
for (auto& e : data)
std::cout << e.first << "=" << e.second << '\n';
}
Output will be:
parse ok
remaining: {
ARG=Test still in first ARG
ARG2=Zombie cat
EXP2=FunctionCall(A, B C)
If you really wanted '{' to be part of the last value, change this line:
Value = qi::no_skip [+(char_ - KeyName)];
|
{
"redpajama_set_name": "RedPajamaStackExchange"
}
| 6,436
|
Фифи́ () — один из видов улитов семейства бекасовых, ранее род относили к ржанковым.
Фифи — птица с чёрно-белым оперением, на спине есть коричневые или бурые крапления. Длина тела 15—25 см, у летящей птицы ноги заметно выдаются за край хвоста. По внешности больше всего напоминает черныша и улита-отшельника, однако не является их близким родственником. Из всех улитов ближе всего к нему стоит травник.
Гнездится фифи на земле в тундре, лесотундре, иногда в лесной зоне от горной Шотландии, Северной Европе, в Азии вплоть до Командорских и Алеутских островов. Иногда может строить гнёзда в дуплах или старых гнёздах рябинников. На зиму улетает в Африку, Индию, Юго-Восточную Азию, встречалась также в Австралии. В кладке обычно 3—4 яйца, которые высиживаются самцом и самкой попеременно 22—23 дня.
Питаются на мелководьях и влажной грязи в основном насекомыми и червями, а также мелкими моллюсками.
Галерея
Примечания
Ссылки
Позвоночные животные России: Фифи
Фифи на сайте «Птицы Алтая»
Фифи Tringa glareola (со звуковым файлом)
Фотографии Фифи
Улиты
Животные, описанные в 1758 году
Птицы Евразии
Птицы Африки
Птицы Австралии
|
{
"redpajama_set_name": "RedPajamaWikipedia"
}
| 1,783
|
Read the ingredients, they should read "sauerkraut and salt." It's better to find a type that contains no preservatives, but these may be difficult to find depending on your location.
Even though sauerkraut is very beneficial to the flora in your system, a lot of people still receive a reaction from sauerkraut, so if you should do so, discontinue use for at least two to four weeks, then if you choose to you could try it again. Remember to start with only a few bites until you know you're not receiving reactions from it.
|
{
"redpajama_set_name": "RedPajamaC4"
}
| 9,969
|
Haloperidol plus promethazine for psychosis-induced aggression
Overview of attention for article published in Cochrane database of systematic reviews, November 2016
In the top 25% of all research outputs scored by Altmetric
High Attention Score compared to outputs of the same age (84th percentile)
Above-average Attention Score compared to outputs of the same age and source (54th percentile)
13 tweeters
3 Facebook pages
1 Wikipedia page
dimensions_citation
17 Dimensions
167 Mendeley
Summary Twitter Facebook Wikipedia Dimensions citations
Cochrane database of systematic reviews, November 2016
10.1002/14651858.cd005146.pub3
Gisele Huf, Jacob Alexander, Pinky Gandhi, Michael H Allen
Health services often manage agitated or violent people, and such behaviour is particularly prevalent in emergency psychiatric services (10%). The drugs used in such situations should ensure that the person becomes calm swiftly and safely. To examine whether haloperidol plus promethazine is an effective treatment for psychosis-induced aggression. On 6 May 2015 we searched the Cochrane Schizophrenia Group's Register of Trials, which is compiled by systematic searches of major resources (including MEDLINE, EMBASE, AMED, BIOSIS, CINAHL, PsycINFO, PubMed, and registries of clinical trials) and their monthly updates, handsearches, grey literature, and conference proceedings. All randomised clinical trials with useable data focusing on haloperidol plus promethazine for psychosis-induced aggression. We independently extracted data. For binary outcomes, we calculated risk ratio (RR) and its 95% confidence interval (CI), on an intention-to-treat basis. For continuous data, we estimated the mean difference (MD) between groups and its 95% CI. We employed a fixed-effect model for analyses. We assessed risk of bias for included studies and created 'Summary of findings' tables using GRADE. We found two new randomised controlled trials (RCTs) from the 2015 update searching. The review now includes six studies, randomising 1367 participants and presenting data relevant to six comparisons.When haloperidol plus promethazine was compared with haloperidol alone for psychosis-induced aggression for the outcome not tranquil or asleep at 30 minutes, the combination treatment was clearly more effective (n=316, 1 RCT, RR 0.65, 95% CI 0.49 to 0.87, high-quality evidence). There were 10 occurrences of acute dystonia in the haloperidol alone arm and none in the combination group. The trial was stopped early as haloperidol alone was considered to be too toxic.When haloperidol plus promethazine was compared with olanzapine, high-quality data showed both approaches to be tranquillising. It was suggested that the combination of haloperidol plus promethazine was more effective, but the difference between the two approaches did not reach conventional levels of statistical significance (n=300, 1 RCT, RR 0.60, 95% CI 0.22 to 1.61, high-quality evidence). Lower-quality data suggested that the risk of unwanted excessive sedation was less with the combination approach (n=116, 2 RCTs, RR 0.67, 95% CI 0.12 to 3.84).When haloperidol plus promethazine was compared with ziprasidone all data were of lesser quality. We identified no binary data for the outcome tranquil or asleep. The average sedation score (Ramsay Sedation Scale) was lower for the combination approach but not to conventional levels of statistical significance (n=60, 1 RCT, MD -0.1, 95% CI - 0.58 to 0.38). These data were of low quality and it is unclear what they mean in clinical terms. The haloperidol plus promethazine combination appeared to cause less excessive sedation but again the difference did not reach conventional levels of statistical significance (n=111, 2 RCTs, RR 0.30, 95% CI 0.06 to 1.43).We found few data for the comparison of haloperidol plus promethazine versus haloperidol plus midazolam. Average Ramsay Sedation Scale scores suggest the combination of haloperidol plus midazolam to be the most sedating (n=60, 1 RCT, MD - 0.6, 95% CI -1.13 to -0.07, low-quality evidence). The risk of excessive sedation was considerably less with haloperidol plus promethazine (n=117, 2 RCTs, RR 0.12, 95% CI 0.03 to 0.49, low-quality evidence). Haloperidol plus promethazine seemed to decrease the risk of needing restraints by around 12 hours (n=60, 1 RCT, RR 0.24, 95% CI 0.10 to 0.55, low-quality evidence). It may be that use of midazolam with haloperidol sedates swiftly, but this effect does not last long.When haloperidol plus promethazine was compared with lorazepam, haloperidol plus promethazine seemed to more effectively cause sedation or tranquillisation by 30 minutes (n=200, 1 RCT, RR 0.26, 95% CI 0.10 to 0.68, high-quality evidence). The secondary outcome of needing restraints or seclusion by 12 hours was not clearly different between groups, with about 10% in each group needing this intrusive intervention (moderate-quality evidence). Sedation data were not reported, however, the combination group did have less 'any serious adverse event' in 24-hour follow-up, but there were not clear differences between the groups and we are unsure exactly what the adverse effect was. There were no deaths.When haloperidol plus promethazine was compared with midazolam, there was clear evidence that midazolam is more swiftly tranquillising of an aggressive situation than haloperidol plus promethazine (n=301, 1 RCT, RR 2.90, 95% CI 1.75 to 4.8, high-quality evidence). On its own, midazolam seems to be swift and effective in tranquillising people who are aggressive due to psychosis. There was no difference in risk of serious adverse event overall (n=301, 1 RCT, RR 1.01, 95% CI 0.06 to 15.95, high-quality evidence). However, 1 in 150 participants allocated haloperidol plus promethazine had a swiftly reversed seizure, and 1 in 151 given midazolam had swiftly reversed respiratory arrest. Haloperidol plus promethazine is effective and safe, and its use is based on good evidence. Benzodiazepines work, with midazolam being particularly swift, but both midazolam and lorazepam cause respiratory depression. Olanzapine intramuscular and ziprasidone intramuscular do seem to be viable options and their action is swift, but resumption of aggression with subsequent need to re-inject was more likely than with haloperidol plus promethazine. Haloperidol used on its own without something to offset its frequent and serious adverse effects does seem difficult to justify.
The data shown below were collected from the profiles of 13 tweeters who shared this research output. Click here to find out more about how the information was compiled.
United Kingdom 2 15%
Spain 2 15%
Australia 2 15%
United States 1 8%
Venezuela, Bolivarian Republic of 1 8%
Practitioners (doctors, other healthcare professionals) 2 15%
The data shown below were compiled from readership statistics for 167 Mendeley readers of this research output. Click here to see the associated Mendeley record.
Spain 1 <1%
Australia 1 <1%
Unknown 165 99%
Unspecified 30 18%
Student > Master 30 18%
Student > Ph. D. Student 23 14%
Other 21 13%
Researcher 18 11%
Unknown 1 <1%
Medicine and Dentistry 59 35%
Nursing and Health Professions 19 11%
Psychology 18 11%
Social Sciences 9 5%
This research output has an Altmetric Attention Score of 11. This is our high-level measure of the quality and quantity of online attention that it has received. This Attention Score, as well as the ranking and number of research outputs shown below, was calculated when the research output was last mentioned on 31 October 2018.
Outputs from Cochrane database of systematic reviews
of 10,454 outputs
Outputs of similar age from Cochrane database of systematic reviews
Altmetric has tracked 12,867,507 research outputs across all sources so far. Compared to these this one has done well and is in the 89th percentile: it's in the top 25% of all research outputs ever tracked by Altmetric.
So far Altmetric has tracked 10,454 research outputs from this source. They typically receive a lot more attention than average, with a mean Attention Score of 20.3. This one has gotten more attention than average, scoring higher than 61% of its peers.
Older research outputs will score higher simply because they've had more time to accumulate mentions. To account for age we can compare this Altmetric Attention Score to the 372,907 tracked outputs that were published within six weeks on either side of this one in any source. This one has done well, scoring higher than 84% of its contemporaries.
We're also able to compare this research output to 161 others from the same source and published within six weeks on either side of this one. This one has gotten more attention than average, scoring higher than 54% of its contemporaries.
|
{
"redpajama_set_name": "RedPajamaCommonCrawl"
}
| 5,362
|
Q: Login issue for this web site one of my friend has login for this forum but he was out of town for long time and did not access this forum for a long time and that is why he forgot his userid which he use to login to this web site. probably he use yahoo or gmail accound to login to this web site but now he could not remind his user id by which he login to this web site. so anyone can tell me how he can know what user id or mail id he used to login to this web site. how to request moderator of this web site to retrieve his login details.....at least his login id.
my friend profile is https://stackoverflow.com/users/750398/keith-costa looking for help. thanks
A: This is the List of Moderators at Stack Overflow. Only Moderators can see a user's email address. Try sending one of them a message with his user-id (750398), or contacting StackOverflow directly via Contact Us.
That user Keith Costa (750398) hasn't logged in since Jun 20 2012, so I would also recommend he checks all of his email accounts for mail from around that time.
Good luck!
|
{
"redpajama_set_name": "RedPajamaStackExchange"
}
| 8,640
|
Another Ticket ist das siebte Studioalbum des britischen Rockgitarristen Eric Clapton. Es erschien am 17. Februar 1981. Es war sein letztes Album für RSO Records. Am 25. Oktober 1990 erschien das Album auch auf CD.
Titelliste
"Something Special" (Clapton) – 2:36
"Black Rose" (Troy Seals, Eddie Setser) – 3:44
"Blow Wind Blow" (Muddy Waters) – 2:58
"Another Ticket" (Clapton) – 5:43
"I Can't Stand It" (Clapton) – 4:07
"Hold Me Lord" (Clapton) – 3:27
"Floating Bridge" (Sleepy John Estes) – 6:32
"Catch Me If You Can" (Gary Brooker, Clapton) – 4:24
"Rita Mae" (Clapton) – 5:03
Rezeption und Auszeichnungen
Die Musikwebsite Allmusic vergab drei von fünf möglichen Sternen für das Album. Kritiker William Ruhlmann fasste es wie folgt zusammen: "es war kein brillanter Clapton, aber ein guter". Rolling Stone zeichnete das Album mit vier von fünf Sternen aus. Am 25. Juni 1981 bezeichnete Kritiker John Piccarella Claptons Version von Floating Bridge als "den vielleicht subtilsten und raffiniertesten Blues, den Clapton jemals gespielt hat". Piccarella fand, dass Clapton erreicht hat, ein populäres Album aus sowohl authentischem als auch tragischem Blues aufzunehmen. Er "adressiere Charts und Herz mit einem Geschoss". Das Album erreichte Platz 26 der deutschen Charts und blieb dort 16 Wochen lang. Auf den Rängen sieben und 18 platzierte sich das Album in den amerikanischen und britischen Albumcharts. Die Singleauskopplung I Can't Stand It erreichte Platz 10 der Billboard Hot 100 und Platz eins der Mainstream-Rock-Chart. Die Single Another Ticket positionierte sich auf Rang 78 der US-amerikanischen Singlecharts. Die weiteren Auskopplungen Blow Wind Blow, Catch Me If You Can und Rita Mae erreichten die Plätze 24, 23 und 18 der Mainstream-Rock-Charts. Das Album wurde in den USA und Kanada mit "Gold" ausgezeichnet und verkaufte sich weltweit mehr als 550.000-mal.
Verkaufszahlen
Einzelnachweise
Album (Rock)
Album 1981
Eric-Clapton-Album
|
{
"redpajama_set_name": "RedPajamaWikipedia"
}
| 303
|
Stop by California State University San Marcos from 12pm to 2pm to win 91X prizes and swag! 333 N Twin Oaks Valley Rd, San Marcos, CA 92069. Calling all intellect-seekers. Now that you are back to school, are you looking for an affordable and convenient way to get to campus in north San Diego?
|
{
"redpajama_set_name": "RedPajamaC4"
}
| 5,284
|
# Fix-It and Forget-It
## Holiday Main Dishes and Sides
Phyllis Pellman Good
_Fix-It and Forget-It Holiday Main Dishes and Sides_ is based on
_Fix-It and Forget-It Christmas Cookbook: 600 Slow Cooker Holiday Recipes_ ,
published by Good Books, 2010.
Cover illustration and illustrations throughout the book by Cheryl Benner
Design by Cliff Snyder
Copyright © 2010 by Good Books, Intercourse, PA 17534
978-1-4532-7696-9
Limit of Liability/Disclaimer of Warranty: The publisher and author have made their best effort in preparing this book with care and accuracy, and it is presented in good faith. But they make no representations or warranties with respect to the completeness or accuracy of the contents of this book. Sales representatives may not create or extend any warranty either verbally or in sales materials about this book. The advice and strategies contained in these materials may not be suitable for your situation. Consult with a professional where appropriate. Neither the author nor the publisher shall be liable for any commercial damages or loss of profit, including but not limited to special, consequential, or incidental damages.
All rights reserved. Printed in the United States of America.
No part of this book may be reproduced in any manner, except for brief quotations in critical articles or reviews, without permission.
# **Contents**
**Welcome to Fix-It and Forget-It Holiday Main Dishes and Sides**
**Main Dishes**
* Wine Tender Roast
* Machaca Beef
* Pot-Roast Complete
* Stuffed Flank Steak
* Beef Burgundy
* Barbecued Ribs
* Autumn Harvest Pork Loin
* No-Fuss Pork and Sauerkraut
* Oxford Canal Chops Deluxe
* Raspberry-Glazed Ham
* Ham with Sweet Potatoes and Oranges
* Verenike (or Creamy Lasagna)
* Sunday Roast Chicken
* Chicken Cacciatore with Fettuccine
* Spicy Chicken Curry
* Garlic with Lime Chicken
* Pineapple-Glazed Chicken with Sweet Potatoes
* Szechwan-Style Chicken and Broccoli
* Indonesian Peanut Chicken
* Chicken Azteca
* Herb-Roasted Turkey Breast
* Turkey Breast with Orange Sauce
* Turkey Thighs, Acorn Squash, and Apples
* Barbecued Turkey Cutlets
* Turkey Fajitas
* Turkey Lasagna
* Holiday Meatloaf
* Company Seafood Pasta
* Oyster and Potato Filling Yuletide
* Arroy Con Queso
* Minestra Di Ceci
**Sides**
* Easy Flavor-Filled Green Beans
* Baked Corn
* Quick Broccoli Fix
* Orange Glazed Carrots
* Sweet-Sour Red Cabbage
* Stuffed Acorn Squash
* Stewed Tomatoes
* Stuffed Mushrooms
* Easy Olive Bake
* Caponata
* Orange Yams
* Barbecued Black Beans with Sweet Potatoes
* Potatoes Perfect
* Herbed Potatoes
* Holiday Wild Rice
* Mushroom Stuffing
* Old-Fashioned Stuffing
* Sweet Potato Stuffing
* Scalloped Pineapple
**Tips**
# **Welcome to _Fix-It and Forget-It_**
# **_Holiday Main Dishes and Sides_**
If you've invited your family or your special friends or neighbors to your home for a holiday meal, you might be caught somewhere between anticipation and dread right now.
Who doesn't like to have loved ones around the table? But which of us hasn't panicked on the way to that lovely moment?
Or maybe your energy and your cooking ideas left you before your holiday guests did. How do you prepare without running out of time—or graciousness?
Never fear. You absolutely can make holiday-worthy dishes in a slow cooker. Here are 50 easy-to-make, manageable recipes, ranging from elegant to comfortable.
Now you can join the parties!
—Phyllis Pellman Good
# **Main Dishes**
**Wine Tender Roast**
**Rose Hankins**
Stevensville, MD
_Makes 8-10 servings_
**_Prep. Time: 10 minutes_**
**_Cooking Time: 8-10 hours_**
**_Ideal slow cooker size: 4- to 5-qt._**
**2½-3-lb. chuck roast**
**1 cup thinly sliced onion**
**½ cup chopped apple, peeled,** _or_ **unpeeled**
**3 cloves garlic, chopped**
**1 cup red wine**
**salt and pepper**
1. Put roast in slow cooker. Layer onions, apples, and garlic on top of roast.
2. Carefully pour wine over roast without disturbing its toppings.
3. Sprinkle with salt and pepper.
4. Cover. Cook on Low 8-10 hours, or until meat is tender but not dry.
**Machaca Beef**
**Jeanne Allen** , Rye, CO
_Makes 10-12 servings_
**_Prep. Time: 5-7 minutes_**
**_Cooking Time: 10-12 hours_**
**_Ideal slow cooker size: 3½-qt._**
**1½-lb. beef roast**
**1 large onion, sliced**
**4-oz. can chopped green chilies, undrained**
**2 beef bouillon cubes**
**1½ tsp. dry mustard**
**½ tsp. garlic powder**
**1 tsp. seasoning salt**
**½ tsp. pepper**
**1 cup salsa**
1. Combine all ingredients except salsa in slow cooker. Add just enough water to cover meat.
2. Cover cooker and cook on Low 10-12 hours, or until beef is tender but not dry. Drain and reserve liquid.
3. Shred beef using two forks to pull it apart.
4. Combine beef, salsa, and enough of reserved liquid to have the consistency you want.
5. Use as filling for burritos, chalupas, quesadillas, or tacos.
**Note:**
_After living in New Mexico for the past 30 years, I get homesick for New Mexican cuisine now that I live in Colorado. I keep memories of New Mexico alive by cooking foods that remind me of home._
**Pot-Roast Complete**
**Naomi E. Fast** , Hesston, KS
_Makes 6-8 servings_
**_Prep. Time: 15 minutes_**
**_Cooking Time: 6½-7½ hours_**
**_Ideal slow cooker size: 4-qt._**
**3-3½-lb. arm roast, boneless**
**2 large onions, sliced**
**½ cup brown sugar**
**⅓ cup soy sauce**
**⅓ cup cider vinegar**
**2 bay leaves**
**2-3 garlic cloves, minced**
**1 tsp. grated fresh ginger,** _or_ **¼ tsp. ground ginger**
**1 cup julienned carrots, matchstick size,** _or_ **baby carrots**
**2 cups sliced button mushrooms**
**2-3 cups fresh spinach leaves,** _or_ **2 10-oz. pkgs. frozen spinach, thawed and squeezed dry**
**2 Tbsp. cornstarch**
1. Place meat in slow cooker. Top with onions.
2. Combine brown sugar, soy sauce, and vinegar in a small bowl. Spoon over beef.
3. Scatter bay leaves, garlic, and ginger over roast.
4. Cover. Cook on Low 6-7 hours.
5. Spread carrots, mushrooms, and spinach over beef.
6. Cover. Cook on High 20 minutes.
7. In small bowl, mix cornstarch with ½ cup broth from slow cooker. Return to slow cooker.
8. Cover. Cook 10 minutes more.
_Serve over **rice**._
**Note:**
_I can't count how many times I have used this recipe over the last 15-20 years as a guest meal._
**Stuffed Flank Steak**
**Renee Baum**
Chambersburg, PA
_Makes 6 servings_
**_Prep. Time: 30 minutes_**
**_Cooking Time: 4½-5½ hours_**
**_Ideal slow cooker size: 4- to 5-qt._**
**8-oz. pkg. crushed cornbread stuffing**
**1 cup chopped onion**
**1 cup chopped celery**
**¼ cup minced fresh parsley**
**2 eggs**
**1¼ cups beef broth**
**5⅓ Tbsp. (⅓ cup) butter, melted**
**½ tsp. seasoned salt,** _optional_
**½ tsp. pepper**
**1½ lbs. flank steak**
1. Combine stuffing, onion, celery, and parsley in large bowl.
2. In a small bowl, beat eggs. Stir in broth and butter. Pour over stuffing mixture. Sprinkle with seasoned salt if you wish, and pepper. Stir well.
3. Pound steak to ¼"- thickness.
4. Spread 1½ cups stuffing mixture over steak. Roll up, starting with short side. Tie with string.
5. Place steak in slow cooker.
6. Wrap remaining stuffing tightly in foil and place on top of rolled steak.
7. Cover. Cook on Low 4½-5½ hours, or until meat thermometer reaches 165 degrees.
8. Remove string before slicing.
**Note:**
_If you have a helpful butcher, ask him/her to pound the steak for you._
**Beef Burgundy**
**Joyce Kaut** , Rochester, NY
_Makes 6 servings_
**_Prep. Time: 30 minutes_**
**_Cooking Time: 3¼-4¼ hours_**
**_Ideal slow cooker size: 3- to 4-qt._**
**2 slices bacon, cut in squares**
**2 lbs. sirloin tip,** _or_ **round, steak, cubed**
**¼ cup flour**
**½ tsp. salt**
**¼ tsp. seasoning salt**
**¼ tsp. dried marjoram**
**¼ tsp. dried thyme**
**¼ tsp. pepper**
**1 garlic clove, minced**
**1 beef bouillon cube, crushed**
**1 cup burgundy wine**
**¼ lb. fresh mushrooms, sliced**
**1-1½ cups ketchup**
**2 Tbsp. cornstarch,** _optional_
**2 Tbsp. cold water,** _optional_
1. Cook bacon in skillet until crisp and browned. Remove bacon, reserving drippings.
2. Coat beef with flour and brown on all sides in bacon drippings. (Don't crowd the skillet so the beef browns rather than steams.)
3. Combine steak, bacon drippings, bacon, seasonings and herbs, garlic, bouillon, and wine in slow cooker.
4. Cover. Cook on Low 4 hours (or 3 hours, if you've browned the beef well in Step 2), or until beef is just tender.
5. Stir in mushrooms and ketchup.
6. Dissolve cornstarch in water in a small bowl. Stir into slow cooker—if sauce is not as thick as you wish.
7. Cover. Cook on High 15 minutes, until sauce thickens.
_Try serving this over **noodles**._
**Barbecued Ribs**
**Virginia Bender**
Dover, DE
_Makes 6 servings_
**_Prep. Time: 10 minutes_**
**_Cooking Time: 8-10 hours_**
**_Ideal slow cooker size: 6-qt._**
**4 lbs. pork ribs**
**½ cup brown sugar**
**12-oz. jar chili sauce**
**¼ cup balsamic vinegar**
**2 Tbsp. Worcestershire sauce**
**2 Tbsp. Dijon mustard**
**1 tsp. hot sauce**
1. Place ribs in slow cooker.
2. Combine remaining ingredients in a good-sized bowl.
3. Pour half of sauce over ribs.
4. Cover. Cook on Low 8-10 hours.
5. Serve with remaining sauce.
**Autumn Harvest Pork Loin**
**Stacy Schmucker Stoltzfus**
Enola, PA
_Makes 4-6 servings_
**_Prep. Time: 30 minutes_**
**_Cooking Time: 5-6 hours_**
**_Ideal slow cooker size: 5-qt._**
**1 cup cider,** _or_ **apple juice**
**1½-2-lb. pork loin**
**salt**
**pepper**
**2 large Granny Smith apples, peeled and sliced**
**1½ whole butternut squashes, peeled and cubed**
**½ cup brown sugar**
**¼ tsp. cinnamon**
**¼ tsp. dried thyme**
**¼ tsp. dried sage**
1. Heat cider in hot skillet. Sear pork loin on all sides in cider.
2. Sprinkle meat with salt and pepper on all sides. Place in slow cooker, along with pan juices.
3. In a good-sized bowl, combine apples and squash. Sprinkle with sugar and herbs. Stir. Spoon around pork loin in cooker.
4. Cover. Cook on Low 5-6 hours.
5. Remove pork from cooker. Let stand 10-15 minutes. Slice into ½"-thick slices.
6. Serve topped with apples and squash.
**No Fuss Pork and Sauerkraut**
**Vera M. Kuhns**
Harrisonburg, VA
_Makes 12 servings_
**_Prep. Time: 10-15 minutes_**
**_Cooking Time: 4-5 hours_**
**_Ideal slow cooker size: 5-qt._**
**3-lb. pork roast**
**3 2-lb. pkgs. sauerkraut (drain off juice from 1 pkg.)**
**2 apples, peeled and sliced**
**½ cup brown sugar**
**1 cup apple juice**
1. Place meat in large slow cooker.
2. Place sauerkraut on top of meat.
3. Add apples and brown sugar. Add juice.
4. Cover. Cook on High 4-5 hours.
_I think this is made to be eaten with **mashed potatoes**._
**Tip:**
If your slow cooker isn't large enough to hold all the ingredients, cook one package of sauerkraut and half the apples, brown sugar, and apple juice in another cooker. Mix the ingredients of both cookers together before serving.
**Variations:**
1. Replace brown sugar with ¼ cup honey.
2. Sprinkle 2 Tbsp. dried basil over contents of slow cooker at end of Step 3.
**Jean Harris Robinson**
Pemberton, NJ
**Oxford Canal Chops Deluxe**
**Willard E. Roth** , Elkhart, IN
_Makes 6 servings_
**_Prep. Time: 25 minutes_**
**_Cooking Time: 4 hours_**
**_Ideal slow cooker size: 5-qt._**
**6 6-oz. boneless pork chops**
**¼ cup flour**
**1 tsp. powdered garlic**
**1 tsp. sea salt**
**1 tsp. black pepper**
**1 tsp. dried basil and/or dried oregano**
**2 medium onions, sliced**
**2 Tbsp. oil**
**1 cup burgundy wine**
**14½-oz. can beef broth**
**1 soup can water**
**6-oz. can tomato sauce**
**8 oz. dried apricots**
**½ lb. fresh mushroom caps**
1. Shake chops in bag with flour and seasonings.
2. Glaze onions in oil in medium hot skillet. Add chops and brown on both sides.
3. Pour any remaining flour over chops in skillet.
4. In large bowl mix together wine, broth, water, and tomato sauce. Pour over meat. Bring to boil.
5. Remove chops from skillet and place in cooker.
6. Layer in apricots and mushrooms. Pour heated broth over top.
7. Cover. Cook on Low 2½ hours, and then on High 1½ hours, or until chops are just tender.
_This is a great dish to serve with the Celtic speciality Bubble and Squeak— **Irish potatoes mashed with green cabbage or brussels sprouts**._
**Note:**
_My favorite memory with this recipe was the time I prepared it in the tiny kitchen of a houseboat on the Oxford Canal and then shared it with five friends. It was a hit!_
**Raspberry-Glazed Ham**
**Gloria Frey**
Lebanon, PA
_Makes 16-20 servings_
**_Prep Time: 10-15 minutes_**
**_Cooking Time: 4 hours_**
**_Ideal slow-cooker size: 6-qt._**
**8-10-lb. boneless ham, fully cooked**
**¼ cup apple juice**
**2 Tbsp. lemon juice**
**2 tsp. cornstarch**
**⅓ cup seedless raspberry jam,** _divided_
**1 Tbsp. butter**
1. Place ham in slow cooker. Cover. Cook on Low 2 hours.
2. While ham is cooking, blend apple juice, lemon juice, and cornstarch together in saucepan.
3. Stir in about half of jam after liquid is well blended.
4. Cook and stir until hot and bubbly. Add butter. Stir in remaining jam.
5. Spoon glaze over ham after it has cooked 2 hours.
6. Cover. Cook 2 more hours on Low.
7. Slice ham and serve.
**Tip:**
If you wish, double the amount of glaze. Place half in a serving bowl. Pass alongside ham slices so individuals can add more glaze to the meat if they wish.
**Ham with Sweet Potatoes and Oranges**
**Esther Becker** , Gordonville, PA
_Makes 4 servings_
**_Prep. Time: 15 minutes_**
**_Cooking Time: 7-8 hours_**
**_Ideal slow cooker size: 3-qt._**
**2-3 sweet potatoes, peeled and sliced ¼" thick**
**1 large ham slice**
**3 seedless oranges, peeled and sliced**
**3 Tbsp. orange juice concentrate**
**3 Tbsp. honey**
**½ cup brown sugar**
**2 Tbsp. cornstarch**
1. Place sweet potato slices in slow cooker.
2. Arrange ham and orange slices on top of potatoes.
3. Combine remaining ingredients in a small bowl. Drizzle over ham and oranges.
4. Cover. Cook on Low 7-8 hours.
_Delicious served with a **fruit salad**._
**Verenike (or Creamy Lasagna)**
**Jennifer Yoder Sommers**
Harrisonburg, VA
_Makes 8-10 servings_
**_Prep. Time: 10-15 minutes_**
**_Cooking Time: 5-6 hours_**
**_Ideal slow cooker size: 5-qt._**
**24 ozs. cottage cheese**
**3 eggs**
**1 tsp. salt**
**½ tsp. pepper**
**1 cup sour cream**
**2 cups evaporated milk**
**2 cups cubed cooked ham**
**7-9 uncooked lasagna noodles**
1. Combine all ingredients except noodles in a good-sized mixing bowl.
2. Place half of creamy ham mixture in bottom of cooker.
3. Stack in uncooked noodles. Break them to fit if you need to.
4. Cover with remaining half of creamy ham sauce. Push noodles down so that they are fully submerged in the sauce.
5. Cover. Cook on Low 5-6 hours, or until noodles are tender but not mushy.
_Serve it with **green salad** , **peas** , and **zwiebach** (a favorite Mennonite food), or small **dinner rolls**._
**Note:**
_This is an easy way to make the traditional Russian Mennonite dish—verenike._
**Sunday Roast Chicken**
**Ruth A. Feister**
Narvon, PA
_Makes 4-5 servings_
**_Prep. Time: 30-35 minutes_**
**_Cooking Time: 6 hours_**
**_Ideal slow cooker size: 6- to 7-qt._**
**Seasoning Mix:**
**1 Tbsp. salt**
**2 tsp. paprika**
**1½ tsp. onion powder**
**1½ tsp. garlic powder**
**1½ tsp. dried basil**
**1 tsp. dry mustard**
**1 tsp. cumin**
**2 tsp. pepper**
**½ tsp. dried thyme**
**½ tsp. savory**
**¼ stick (2 Tbsp.) butter**
**2 cups chopped onions**
**1 cup chopped green bell pepper**
**1 roasting chicken**
**¼ cup flour**
**1-2 cups chicken stock**
1. Combine Seasoning Mix ingredients in small bowl.
2. Melt butter over high heat in skillet. When butter starts to sizzle, add chopped onions and peppers, and 3 Tbsp. seasoning mix.
3. Cook until onions are golden brown. Cool.
4. Stuff cavity of chicken with cooled vegetables.
5. Sprinkle outside of chicken with 1 Tbsp. seasoning mix. Rub in well so it sticks.
6. Place chicken in large slow cooker.
7. Cover. Cook on Low 6 hours.
8. Empty vegetable stuffing and juices into saucepan. Whisk in flour and 1 cup stock from slow cooker.
9. Cook over high heat until thickened. Add more stock if you prefer a thinner gravy.
**Note:**
_The first time I served this dish was when we had family visiting us from Mississippi. We had a wonderful time sitting around a large table, sharing many laughs and catching up on the years since our last visit._
**Chicken Cacciatore with Spaghetti**
**Phyllis Pellman Good**
Lancaster, PA
_Makes 4-5 servings_
**_Prep. Time: 15 minutes_**
**_Cooking Time: 4 hours_**
**_Ideal slow cooker size: 4- to 5-qt._**
**2 onions, sliced**
**2½-3 lbs. chicken legs and thighs, skin removed**
**2 garlic cloves, minced**
**16-oz. can stewed tomatoes**
**8-oz. can tomato sauce**
**1 tsp. salt**
**¼ tsp. pepper**
**1-2 tsp. dried oregano**
**½ tsp. dried basil**
**1 bay leaf**
**¼ cup white wine**
1. Place onions in bottom of slow cooker.
2. Lay chicken over onions.
3. Combine remaining ingredients in a good-sized bowl. Pour over chicken.
4. Cover. Cook on Low 4 hours, or until chicken is tender but not dry.
5. Remove bay leaf before serving.
_Serve over hot **buttered spaghetti** , **linguini** , or **fettucini**._
**Dad's Spicy Chicken Curry**
**Tom & Sue Ruth**
Lancaster, PA
_Makes 8 servings_
**_Prep. Time: 25 minutes_**
**_Cooking Time: 6-8 hours_**
**_Ideal slow cooker size: 6- to 7-qt._**
**4 lbs. chicken pieces, with bones in**
**water**
**2 onions, diced**
**10-oz. pkg. frozen chopped spinach, thawed and squeezed dry**
**1 cup plain yogurt**
**2-3 diced red potatoes**
**3 tsp. salt**
**1 tsp. garlic powder**
**1 tsp. ground ginger**
**1 tsp. ground cumin**
**1 tsp. ground coriander**
**1 tsp. pepper**
**1 tsp. ground cloves**
**1 tsp. ground cardamom**
**1 tsp. ground cinnamon**
**½ tsp. chili powder**
**1 tsp. red pepper flakes**
**3 tsp. turmeric**
1. Place chicken in large slow cooker. Cover with water.
2. Cover. Cook on High 2 hours, or until tender.
3. Drain chicken. Remove from slow cooker. Cool briefly and cut/shred into small pieces.
4. Return chicken to slow cooker.
5. Stir in all remaining ingredients.
6. Cover. Cook on Low 4-6 hours, or until potatoes are tender.
_Serve on **rice**. Accompany with **fresh mango slices** or **mango chutney**._
**Variation:**
Substitute 5 tsp. curry powder for the garlic powder, ginger, cumin, coriander, and pepper.
**Garlic Lime Chicken**
**Loretta Krahn**
Mountain Lake, MN
_Makes 5 servings_
**_Prep. Time: 10 minutes_**
**_Cooking Time: 4-8 hours_**
**_Ideal slow cooker size: 4-qt._**
**5 skinless chicken breast halves**
**½ cup soy sauce**
**¼-⅓ cup lime juice, according to your taste preference**
**1 Tbsp. Worcestershire sauce**
**2 garlic cloves, minced,** _or_ **1 tsp. garlic powder**
**½ tsp. dry mustard**
**½ tsp. ground pepper**
1. Place chicken in slow cooker.
2. Combine remaining ingredients in a bowl. Pour over chicken.
3. Cover. Cook on High 4-6 hours or on Low 6-8 hours, or until chicken is tender but not dry.
**Pineapple-Glazed Chicken with Sweet Potatoes**
**Katrina Eberly**
Wernersville, PA
_Makes 4 servings_
**_Prep. Time: 25-30 minutes_**
**_Cooking Time: 3½-4 hours_**
**_Ideal slow cooker size: 4- to 5-qt._**
**3 medium sweet potatoes, peeled and sliced into 1" chunks**
**⅔ cup, plus 3 Tbsp., flour,** _divided_
**1 tsp. salt**
**1 tsp. onion powder**
**1 tsp. ground nutmeg**
**1 tsp. ground cinnamon**
**1 tsp. pepper**
**4 5-oz. boneless skinless chicken breast halves**
**¼ stick (2 Tbsp.) butter**
**10¾-oz. can cream of chicken soup**
**¾ cup pineapple juice**
**2 tsp. brown sugar**
**1 tsp. grated orange peel**
1. Layer sweet potatoes into slow cooker.
2. In a large resealable plastic bag, combine ⅔ cup flour and seasonings. Add chicken, one piece at a time, and shake to coat.
3. In a large skillet over medium heat, cook chicken in butter for 3 minutes on each side, or until lightly browned. (Don't crowd the skillet or the chicken will steam rather than brown.) Arrange browned chicken over sweet potatoes.
4. Place remaining flour in a medium-sized mixing bowl. Stir in soup, pineapple juice, brown sugar, and orange peel until blended. Pour over chicken.
5. Cover. Cook on Low 3½-4 hours, or until potatoes are soft and chicken is cooked through but not dry.
_Serve over **rice** if you wish._
**Szechwan-Style Chicken and Broccoli**
**Jane Meiser** , Harrisonburg, VA
_Makes 4 servings_
**_Prep. Time: 20 minutes • Cooking Time: 1-3 hours_**
**_Ideal slow cooker size: 4-qt._**
**2 whole boneless, skinless chicken breasts**
**oil**
**½ cup picante sauce**
**2 Tbsp. soy sauce**
**½ tsp. sugar**
**½ Tbsp. quick-cooking tapioca**
**1 medium onion, chopped**
**2 garlic cloves, minced**
**½ tsp. ground ginger**
**2 cups broccoli florets**
**1 medium red bell pepper, cut into pieces**
1. Cut chicken into 1" cubes. Brown lightly in oil in skillet. Place meat in slow cooker.
2. Stir in remaining ingredients.
3. Cover. Cook on High 1-1½ hours or on Low 2-3 hours.
**Indonesian Peanut Chicken**
**Naomi Ressler**
Harrisonburg, VA
_Makes 6 servings_
**_Prep. Time: 15-20 minutes_**
**_Cooking Time: 4 hours_**
**_Ideal slow cooker size: 4-qt._**
**1½ lbs. boneless, skinless chicken breasts, cut into chunks**
**⅓ cup chopped onions**
**⅓ cup water**
**¼ cup creamy,** _or_ **chunky, peanut butter**
**3 Tbsp. chili sauce**
**¼ tsp. salt**
**¼ tsp. pepper**
**¼ tsp. cayenne pepper**
**cooked rice,** _or_ **soba noodles**
**6 Tbsp. chopped peanuts**
**6 Tbsp. chopped sweet red peppers**
1. Place chicken in slow cooker.
2. In a small bowl, combine onions, water, peanut butter, chili sauce, salt, and peppers. Pour over chicken.
3. Cover. Cook on Low 4 hours, or until chicken is tender.
4. Serve over rice or soba noodles. Sprinkle with peanuts and red peppers.
**Chicken Azteca**
**Katrine Rose**
Woodbridge, VA
_Makes 10-12 servings_
**_Prep. Time: 15-20 minutes_**
**_Cooking Time: 2½-6½ hours_**
**_Ideal slow cooker size: 6- to 7-qt._**
**2 15-oz. cans black beans, drained**
**4 cups frozen corn kernels**
**2 garlic cloves, minced**
**¾ tsp. ground cumin**
**2 cups chunky salsa,** _divided_
**10 boneless, skinless chicken breast halves**
**2 8-oz. pkgs. cream cheese, cubed**
1. Combine beans, corn, garlic, cumin, and half of salsa in slow cooker.
2. Arrange chicken breasts over top. Pour remaining salsa over top.
3. Cover. Cook on High 2-3 hours or on Low 4-6 hours, or until chicken is tender but not dry.
4. Remove chicken and cut into bite-sized pieces. Return to cooker.
5. Stir in cream cheese. Cook on High until cream cheese melts.
_Spoon chicken and sauce over cooked **rice**. Top with **shredded cheese**._
**Herb-Roasted Turkey Breast**
**Kristi See**
Weskan, KS
_Makes 6 servings_
**_Prep. Time: 15 minutes_**
**_Cooking Time: 4-6 hours_**
**_Ideal slow cooker size: 5- to 6-qt._**
**5 tsp. lemon juice**
**1 Tbsp. olive oil**
**1-2 tsp. pepper**
**1 tsp. dried rosemary, crushed**
**1 tsp. dried thyme**
**1 tsp. garlic salt**
**6-7-lb. bone-in turkey breast**
**1 medium onion, cut into wedges**
**1 celery rib, cut into 2"-thick pieces**
**½ cup white wine,** _or_ **chicken broth**
1. Spray slow cooker with nonstick cooking spray.
2. In a small bowl, combine lemon juice and olive oil. In another bowl, combine pepper, rosemary, thyme, and garlic salt.
3. With your fingers, carefully loosen skin from both sides of breast. Brush oil mixture under skin. Rub herb-seasoning mixture under and on top of skin.
4. Arrange onion and celery in slow cooker. Place turkey breast, skin-side up, on top of vegetables.
5. Pour wine around breast.
6. Cover. Cook on Low 4-6 hours, or until meat is tender but not dry.
**Note:**
_I made this turkey recipe for our first Thanksgiving after we were married. We really enjoyed the taste. Clean-up was easy, and because I prepared the turkey in the slow cooker, I wasn't short on oven space._
**Variations:**
1. Add carrot chunks to Step 4 to add more flavor to the turkey broth.
2. Reserve broth for soups, or thicken with flour-water paste and serve as gravy over sliced turkey.
3. Freeze broth in pint-sized containers for future use.
4. Debone turkey and freeze in pint-sized containers for future use. Or freeze any leftover turkey.
**Liz Ann Yoder**
Hartville, OH
See Turkey Cacciatore, page 188, for a great dish with cut-up cooked turkey!
**Turkey Breast with Orange Sauce**
**Jean Butzer**
Batavia, NY
_Makes 4-6 servings_
**_Prep. Time: 10 minutes_**
**_Cooking Time: 7-8 hours_**
**_Ideal slow cooker size: 6-qt._**
**1 large onion, chopped**
**3 garlic cloves, minced**
**1 tsp. dried rosemary**
**½ tsp. pepper**
**2-3-lb. boneless, skinless turkey breast**
**1½ cups orange juice**
1. Place onions in slow cooker.
2. Combine garlic, rosemary, and pepper in a small bowl.
3. Make gashes in turkey, about ¾ of the way through at 2" intervals. Stuff with herb mixture. Place turkey in slow cooker.
4. Pour juice over turkey.
5. Cover. Cook on Low 7-8 hours, or until turkey is no longer pink in center.
**Note:**
_This very easy, impressive-looking and -tasting recipe is perfect for company._
**Turkey Thighs, Acorn Squash, and Apples**
**Mary E. Wheatley**
Mashpee, MA
_Makes 6-8 sevings_
**_Prep. Time: 35 minutes_**
**_Cooking Time: 6-8 hours_**
**_Ideal slow cooker size: 6-qt._**
**2 lbs. acorn squash, peeled, seeded, and cut into 1"- thick rings**
**6 medium-sized Granny Smith,** _or_ **other tart, apples cored and cut into ½"-thick rings**
**4 turkey thighs, skin and excess fat removed**
**salt and pepper to taste**
**1 shallot,** _or_ **small onion, chopped**
**½ cup apple juice,** _or_ **cider**
**1 Tbsp. apple brandy**
**3 Tbsp. brown sugar**
**1 tsp. ground cinnamon**
**½ tsp. ground allspice**
1. Spray inside of slow cooker with non-stick spray. Layer in squash, followed by apple rings.
2. Place turkey thighs on top. Sprinkle with salt, pepper, and onion or shallot.
3. In a small bowl, combine apple juice, brandy, brown sugar, cinnamon, and allspice. Pour over turkey.
4. Cover. Cook on Low 6-8 hours, or just until turkey and squash are tender.
**Barbecued Turkey Cutlets**
**Maricarol Magill**
Freehold, NJ
_Makes 6-8 servings_
**_Prep. Time: 10 minutes • Cooking Time: 3 hours_**
**_Ideal slow cooker size: 4- to 5-qt._**
**6-8 (1½-2 lbs.) turkey cutlets**
**¼ cup molasses**
**¼ cup cider vinegar**
**¼ cup ketchup**
**3 Tbsp. Worcestershire sauce**
**1 tsp. garlic salt**
**3 Tbsp. chopped onion**
**2 Tbsp. brown sugar**
**¼ tsp. pepper**
1. Place turkey cutlets in slow cooker.
2. Combine remaining ingredients in bowl. Pour over turkey.
3. Cover. Cook on Low 3 hours.
_Serve over **white** or **brown rice**._
**Turkey Fajitas**
**Carol Ambrose**
McMinnville, OR
_Makes 8 servings_
**_Prep. Time: 10-15 minutes_**
**_Cooking Time: 3-4 hours_**
**_Ideal slow cooker size: 2½-qt._**
**2½ lbs. turkey tenderloins**
**1¼-oz. envelope taco seasoning mix**
**1 celery rib, chopped**
**1 onion, chopped**
**14½-oz. can mild diced tomatoes and green chilies, undrained**
**1 cup (4 oz.) shredded cheddar cheese**
**8 (7½") flour tortillas**
**Toppings:
lettuce
sour cream
sliced olives
chopped tomatoes**
1. Cut turkey into 2½"-long strips. Place in zip-top plastic bag.
2. Add taco seasoning to bag. Seal and shake to coat meat.
3. Empty seasoned turkey into slow cooker. Add celery, onion, and tomatoes. Stir together gently.
4. Cover. Cook on High 3-4 hours, or just until turkey is cooked through and tender.
5. Stir in cheese.
6. Warm tortillas according to package directions. Spoon turkey mixture evenly into center of each tortilla, and roll up.
7. Serve with Toppings.
**Tip:**
Put this recipe in your cooker. Then take your kids or grandkids out to cut down the tree or to do their holiday shopping. When you get back, your meal is ready!
**Turkey Lasagna**
**Rhoda Atzeff**
Lancaster, PA
_Makes 8-10 servings_
**_Prep. Time: 20-30 minutes_**
**_Cooking Time: 5 hours_**
**_Ideal slow cooker size: 5-qt._**
**1 lb. lean ground turkey**
**1 onion, chopped**
**⅛ tsp. garlic powder**
**2 15-oz. cans tomato sauce**
**6-oz. can tomato paste**
**½-1 tsp. salt**
**1 tsp. dried oregano,** _or_ **½ tsp. dried oregano and ½ tsp. dried basil**
**12 oz. fat-free cottage cheese**
**½ cup grated Parmesan cheese**
**12 oz. shredded non-fat mozzarella cheese**
**12 oz. lasagna noodles, uncooked,** _divided_
1. Brown ground turkey and onions in skillet. Drain off any drippings.
2. Stir garlic powder, tomato sauce, tomato paste, salt, and herbs into browned turkey in skillet.
3. In a good-sized mixing bowl, blend together cottage cheese, Parmesan cheese, and mozzarella cheese.
4. Spoon ⅓ of meat sauce into slow cooker.
5. Add ⅓ of uncooked lasagna noodles, breaking them to fit.
6. Top with ⅓ of cheese mixture. You may have to use a knife to spread it.
7. Repeat layers two more times.
8. Cover. Cook on Low 5 hours.
9. Allow to stand 10 minutes before serving.
**Note:**
_I tried this on my brothers. It is a delicious dish, but I thought their raves were maybe a bit overdone. But it was a good feeling to know it pleased them. I overheard my one brother calling another brother in Virginia and telling him about it!_
**Holiday Meatloaf**
**Wafi Brandt**
Manheim, PA
_Makes 4-6 servings_
**_Prep. Time: 25 minutes_**
**_Cooking Time: 4 hours_**
**_Ideal slow cooker size: 4-qt._**
**Meat Loaf:**
**2 eggs**
**1 envelope dry onion soup mix**
**½ cup seasoned bread crumbs**
**¼ cup chopped dried cranberries**
**1 tsp. parsley**
**1½ lbs. ground beef**
**Sauce:**
**16-oz. can whole berry cranberry sauce**
**¾ cup ketchup**
**½ cup beef broth**
**3 Tbsp. brown sugar**
**3 Tbsp. finely chopped onion**
**2 tsp. cider vinegar**
1. Mix all Meat Loaf ingredients together in a good-sized bowl. Shape into loaf and place in lightly greased slow cooker.
2. Mix Sauce ingredients together in another bowl. Pour over meat.
3. Cover. Cook on High 4 hours.
4. Allow to stand 10 minutes before lifting out of cooker and slicing.
**Tip from Tester:**
After slicing and arranging the Meat Loaf on a platter, I ladled the Sauce over the slices. There was more Sauce than fit on the platter, so I put the remainder in a dish to pass. Then individuals could add more to their slices as a condiment. We loved it!
**Company Seafood Pasta**
**Jennifer Yoder Sommers**
Harrisonburg, VA
_Makes 4-6 servings_
**_Prep. Time: 15 minutes_**
**_Cooking Time: 1-2 hours_**
**_Ideal slow cooker size: 4-qt._**
**2 cups sour cream**
**3 cups shredded Monterey Jack cheese**
**¼ stick (2 Tbsp.) butter, melted**
**½ lb. crabmeat,** _or_ **imitation flaked crabmeat**
**⅛ tsp. pepper**
**½ lb. bay scallops, lightly cooked**
**1 lb. medium shrimp, cooked and peeled**
1. Combine sour cream, cheese, and butter in slow cooker.
2. Stir in remaining ingredients.
3. Cover. Cook on Low 1-2 hours.
_Serve immediately over **linguine**. Garnish with **fresh parsley**._
**Oyster and Potato Filling Yuletide**
**Jane Geigley**
Honey Brook, PA
_Makes 3-4 servings_
**_Prep. Time: 10 minutes if mashed potatoes are prepared; 50 minutes if you need to make the mashed potatoes_**
**_Cooking Time: 1½-3 hours_**
**_Ideal slow cooker size: 2-qt._**
**1 small onion, minced**
**½ cup celery, diced**
**¼ stick (2 Tbsp.) butter, melted**
**2 cups very moist mashed potatoes**
**1 egg, well beaten**
**1 Tbsp. minced parsley**
**1-1½ tsp. salt**
**dash of black pepper**
**1 qt. stale bread, cubed**
**1 doz. oysters with liquid**
**milk, _if needed_**
1. Sauté onion and celery in butter in skillet.
2. Blend together mashed potatoes and egg in slow cooker. Pour in sautéed onion and celery. Mix together well.
3. Stir in parsley, salt, pepper, bread cubes, and oysters with their liquid. If mixture is stiff, add several Tbsp. milk.
4. Cover cooker.
5. Cook on High 1½ hours, or on Low 3 hours.
**Arroz Con Queso**
**Nadine L. Martinitz**
Salina, KS
_Makes 6-8 servings_
**_Prep. Time: 15 minutes_**
**_Cooking Time: 6-9 hours_**
**_Ideal slow cooker size: 4-qt._**
**14½-oz. can whole tomatoes, mashed**
**15-oz. can Mexican-style beans, undrained**
**1½ cups uncooked long-grain rice**
**1 cup grated Monterey Jack cheese**
**1 large onion, finely chopped**
**1 cup cottage cheese**
**4¼-oz. can chopped green chili peppers, drained**
**1 Tbsp. oil**
**3 garlic cloves, minced**
**1 tsp. salt**
**1 cup grated Monterey Jack cheese**
1. Combine all ingredients except final cup of cheese in well greased slow cooker.
2. Cover. Cook on Low 6-9 hours, or until rice is fully cooked but dish is not dry.
3. Sprinkle with remaining cheese before serving.
_We eat this with **salsa** on the side._
**Minestra Di Ceci**
**Jeanette Oberholtzer**
Manheim, PA
_Makes 4-6 servings_
**_Prep. Time: 25 minutes_**
**_Soaking Time: 8 hours, or overnight_**
**_Cooking Time: 5½-6 hours_**
**_Ideal slow cooker size: 4-qt._**
**1 lb. dry chickpeas**
**1 sprig fresh rosemary**
**10 leaves fresh sage**
**2 Tbsp. salt**
**1-2 large garlic cloves, minced**
**olive oil**
**1 cup uncooked small pasta, your choice of shape,** _or_ **uncooked penne**
1. Wash chickpeas. Place in slow cooker. Cover with water. Stir in rosemary, sage, and salt. Soak 8 hours, or overnight.
2. Drain water. Remove herbs.
3. Refill slow cooker with peas and fresh water to 1" above peas.
4. Cover. Cook on Low 5 hours.
5. Sauté garlic in olive oil in skillet until clear.
6. Purée half of peas, along with several cups of broth from cooker, in blender. Return purée to slow cooker.
7. Add garlic and oil.
8. Boil pasta in saucepan until al dente, about 5 minutes. Drain. Add to beans.
9. Cover. Cook on High 30-60 minutes, or until pasta is tender and heated through, but not mushy.
**Variation:**
Add ½ tsp. black pepper in Step 1, if you like.
# **Sides**
**Easy Flavor-Filled Green Beans**
**Paula Showalter**
Weyers Cave, VA
_Makes 10 servings_
**_Prep. Time: 5-10 minutes_**
**_Cooking Time: 2-4 hours_**
**_Ideal slow cooker size: 3- to 3½-qt._**
**2 qts.,** _or_ **4 14½-oz. cans, green beans, drained**
**⅓ cup chopped onions**
**4-oz. can mushrooms, drained**
**2 Tbsp. brown sugar**
**3 Tbsp. butter**
**pepper to taste**
1. Combine beans, onions, and mushrooms in slow cooker.
2. Sprinkle with brown sugar.
3. Dot with butter.
4. Sprinkle with pepper.
5. Cover. Cook on Low 2-4 hours, or until onions are tender.
6. Stir just before serving.
**Baked Corn**
**Velma Stauffer** , Akron, PA
_Makes 8 servings_
**_Prep. Time: 5-10 minutes_**
**_Cooking Time: 3¾ hours_**
**_Ideal slow cooker size: 2-qt._**
**1 qt. fresh,** _or_ **2 1-lb. bags frozen, corn**
**2 eggs, beaten**
**1 tsp. salt**
**1 cup milk**
**⅛ tsp. pepper**
**2 tsp. oil**
**3 Tbsp. sugar**
**3 Tbsp. flour**
1. Combine all ingredients well in greased slow cooker.
2. Cover. Cook on High 3 hours and then on Low 45 minutes.
**Tip:**
If you use home-grown sweet corn, you could reduce the amount of sugar.
**Quick Broccoli Fix**
**Willard E. Roth** , Elkhart, IN
_Makes 6 servings_
**_Prep. Time: 15 minutes_**
**_Cooking Time: 5-6 hours_**
**_Ideal slow cooker size: 3½-qt._**
**1 lb. fresh,** _or_ **frozen, broccoli, cut up**
**10¾-oz. can cream of mushroom soup**
**½ cup mayonnaise**
**½ cup plain yogurt**
**½ lb. sliced fresh mushrooms**
**1 cup shredded cheddar cheese,** _divided_
**1 cup crushed saltine crackers**
**sliced almonds,** _optional_
1. Microwave broccoli for 3 minutes. Place in greased slow cooker.
2. Combine soup, mayonnaise, yogurt, mushrooms, and ½ cup cheese in mixing bowl. Pour over broccoli. Stir together gently but well
3. Cover. Cook on Low 5-6 hours.
4. Top with remaining cheese and crackers for last half hour of cooking time.
5. If you wish, top with sliced almonds, for a special touch, before serving.
**Orange-Glazed Carrots**
**Barbara Smith**
Bedford, PA
_Makes 3-4 servings_
**_Prep. Time: 15 minutes_**
**_Cooking Time: 2½ -3½ hours_**
**_Ideal slow cooker size: 2½-qt._**
**3 cups thinly sliced carrots**
**2 cups water**
**¼ tsp. salt**
**2-3 Tbsp. butter**
**3 Tbsp. orange marmalade**
**2 Tbsp. chopped pecans,** _optional_
1. Combine carrots, water, and salt in slow cooker.
2. Cover. Cook on High 2-3 hours, or until carrots are as tender as you like them.
3. Drain. Stir in butter and marmalade.
4. Cover. Cook on High 30 minutes.
**Sweet-Sour Red Cabbage**
**Kaye Taylor**
Florissant, MO
_Makes 6-8 servings_
**_Prep. Time: 30-45 minutes_**
**_Cooking Time: 3-4 hours_**
**_Ideal slow cooker size: 3½-qt._**
**4 slices bacon, diced**
**¼ cup brown sugar**
**2 Tbsp. flour**
**1 tsp. salt**
**⅛ tsp. pepper**
**½ cup water**
**¼ cup vinegar**
**1 medium head red cabbage shredded (6-8 cups)**
**1 small onion, finely chopped**
1. Sauté bacon in skillet until crisp. Set bacon aside. Reserve 1 Tbsp. drippings.
2. Combine 1 Tbsp. bacon drippings in slow cooker with sugar, flour, salt, and pepper. Stir in water and vinegar.
3. Add cabbage and onion. Mix together well.
4. Cover. Cook on Low 3-4 hours, or until cabbage and onion are as tender as you like them.
5. Sprinkle cooked bacon on top just before serving.
**Stuffed Acorn Squash**
**Jean Butzer**
Batavia, NY
_Makes 6 servings_
**_Prep. Time: 15 minutes_**
**_Cooking Time: 2½ hours_**
**_Ideal slow cooker size: 5- to 6-qt._**
**3 small carnival,** _or_ **acorn, squash**
**5 Tbsp. instant brown rice, uncooked**
**3 Tbsp. dried cranberries**
**3 Tbsp. diced celery**
**3 Tbsp. minced onion**
**pinch of ground,** _or_ **dried, sage**
**1 Tbsp. butter,** _divided_
**3 Tbsp. orange juice**
**½ cup water**
1. Slice off points on bottoms of squash so they will stand in slow cooker. Slice off tops and discard. Scoop out seeds. Place squash side by side in slow cooker.
2. Combine rice, cranberries, celery, onion, and sage in bowl. Stuff into squash centers.
3. Dot with butter.
4. Pour 1 Tbsp. orange juice into each squash
center.
5. Pour water into bottom of slow cooker.
6. Cover. Cook on Low 2½ hours.
_Serve with cooked **turkey breast** or **pork** or **ham**._
**Tip:**
To make squash easier to slice, microwave whole squash on High 5 minutes to soften skin.
**Stewed Tomatoes**
**Michelle Showalter**
Bridgewater, VA
_Makes 10-12 servings_
**_Prep. Time: 10 minutes_**
**_Cooking Time: 3-4 hours_**
**_Ideal slow cooker size: 3-qt._**
**2 qts. canned,** _or_ **4 14½-oz. cans diced** _or_ **stewed, tomatoes**
**⅓ cup sugar**
**1½ tsp. salt**
**dash of pepper**
**3 Tbsp. butter**
**2 cups soft bread cubes**
1. Place tomatoes in slow cooker.
2. Sprinkle with sugar, salt, and pepper.
3. Lightly toast bread cubes in melted butter on baking sheet in oven, or in large skillet.
4. Spread over tomatoes.
5. Cover. Cook on High 3-4 hours.
**Variation:**
If you prefer bread that is less moist and soft, add bread cubes 15 minutes before serving and continue cooking without lid.
**Stuffed Mushrooms**
**Melanie L. Thrower**
McPherson, KS
_Makes 4-6 servings_
**_Prep. Time: 15-20 minutes_**
**_Cooking Time: 2-4 hours_**
**_Ideal slow cooker size: 5-qt._**
**8-10 large mushrooms**
**¼ tsp. minced garlic**
**1 Tbsp. oil**
**dash of salt**
**dash of pepper**
**dash of cayenne pepper**
**¼ cup grated Monterey Jack cheese**
1. Remove stems from mushrooms and dice.
2. Heat oil in skillet. Sauté diced stems with garlic until softened. Remove skillet from heat.
3. Stir in seasonings and cheese. Stuff into mushroom caps.
4. Place in slow cooker.
5. Cover. Heat on Low 2-4 hours.
**Variations:**
1. Add 1 Tbsp. minced onion to Step 2.
2. Use Monterey Jack cheese with jalapeños.
**Easy Olive Bake**
**Jean Robinson**
Cinnaminson, NJ
_Makes 8 servings_
**_Prep. Time: 15 minutes_**
**_Cooking Time: 3 hours_**
**_Ideal slow cooker size: 3½-qt._**
**1 cup uncooked long-grain rice**
**2 medium onions, chopped**
**1 stick (8 Tbsp.) butter, melted**
**2 cups, or 1 14½-oz. can, stewed tomatoes**
**2 cups water**
**1 cup black olives, quartered**
**½-¾ tsp. salt**
**½ tsp. chili powder**
**1 Tbsp. Worcestershire sauce**
**4-oz. can mushrooms with juice**
**½ cup grated cheese of your choice**
1. Place rice in slow cooker.
2. Add all remaining ingredients, except cheese. Mix well.
3. Cover. Cook on High 1 hour, and then on Low 2 hours, or until rice is tender but not mushy.
4. Stir in cheese just before serving.
_This is a good accompaniment to **baked ham**._
**Caponata**
**Katrine Rose**
Woodbridge, VA
_Makes 8-10 servings_
**_Prep. Time: 20 minutes_**
**_Cooking Time: 7-8 hours_**
**_Ideal slow cooker size: 3½- to 4-qt._**
**1 medium eggplant, peeled and cut into ½" cubes**
**14½-oz. can diced tomatoes**
**1 medium onion, chopped**
**1 red bell pepper, cut into ½" pieces**
**¾ cup salsa, your choice of heat**
**¼ cup olive oil**
**2 Tbsp. capers, drained**
**3 Tbsp. balsamic vinegar**
**3 garlic cloves, minced**
**1¼ tsp. dried oregano**
**⅓ cup chopped fresh basil, packed in measuring cup**
**toasted, sliced French bread**
1. Combine all ingredients except basil and bread in slow cooker.
2. Cover. Cook on Low 7-8 hours, or until vegetables are tender.
3. Stir in basil. Serve on toasted bread.
**Orange Yams**
**Gladys Longacre**
Susquehanna, PA
_Makes 6-8 servings_
**_Prep. Time: 15 minutes_**
**_Cooking Time: 3 hours_**
**_Ideal slow cooker size: 3½-qt._**
**40-oz. can yams,** _or_ **sweet potatoes, drained**
**2 apples, peeled if you wish and thinly sliced**
**3 Tbsp. butter, melted**
**2 tsp. orange zest**
**1 cup orange juice**
**2 Tbsp. cornstarch**
**½ cup brown sugar**
**1 tsp. salt**
**dash of ground cinnamon and/or nutmeg**
1. Place yams and apples in slow cooker.
2. Stir in butter and orange zest.
3. Combine remaining ingredients in small bowl. Pour over yams and apples.
4. Cover. Cook on High 1 hour and on Low 2 hours, or until apples are tender.
**Variation:**
Substitute 6-8 medium-sized cooked sweet potatoes, or approximately 4 cups cubed butternut squash, for yams.
**Barbecued Black Beans with Sweet Potatoes**
**Barbara Jean Fabel**
Wausau, WI
_Makes 4-6 servings_
**_Prep. Time: 10-15 minutes_**
**_Cooking Time: 2-4 hours_**
**_Ideal slow cooker size: 3-qt._**
**4 large sweet potatoes, peeled and cut into 8 chunks each**
**15-oz. can black beans, rinsed and drained**
**1 medium onion, diced**
**2 ribs celery, sliced**
**9 oz. Sweet Baby Ray's Barbecue Sauce**
1. Place sweet potatoes in slow cooker.
2. Stir in additional ingredients.
3. Cover. Cook on High 2-3 hours, or on Low 4 hours.
**Potatoes Perfect**
**Naomi Ressler**
Harrisonburg, VA
_Makes 4-6 servings_
**_Prep. Time: 15 minutes_**
**_Cooking Time: 3-10 hours_**
**_Ideal slow cooker size: 3½-qt._**
**¼ lb. bacon, diced and browned in a skillet until crisp,** _divided_
**2 medium-sized onions, thinly sliced,** _divided_
**6-8 medium-sized potatoes, thinly sliced,** _divided_
**½ lb. cheddar cheese, thinly sliced,** _divided_
**salt to taste**
**pepper to taste**
**¼-½ stick (2-4 Tbsp.) butter**
1. Layer half of bacon, onions, potatoes, and cheese in greased slow cooker. Season to taste.
2. Dot with butter. Repeat layers.
3. Cover. Cook on Low 8-10 hours or on High 3-4 hours, or until potatoes are soft.
**Herbed Potatoes**
**Jo Haberkamp**
Fairbank, IA
_Makes 6 servings_
**_Prep. Time: 10 minutes_**
**_Cooking Time: 2½-3 hours_**
**_Ideal slow cooker size: 3-qt._**
**1½ lbs. small new potatoes**
**¼ cup water**
**half a stick (4 Tbsp.) butter, melted**
**3 Tbsp. chopped fresh parsley**
**1 Tbsp. lemon juice**
**1 Tbsp. chopped fresh chives**
**1 Tbsp. dill weed**
**¼-½ tsp. salt, according to your taste preference**
**⅛-¼ tsp. pepper, according to your taste preference**
1. Wash potatoes. Peel a strip around the middle of each potato. Place prepared potatoes in slow cooker.
2. Add water.
3. Cover. Cook on High 2½-3 hours. Drain well.
4. In saucepan, heat butter, parsley, lemon juice, chives, dill, salt, and pepper.
5. Pour over potatoes.
_Serve with **ham** or any **meat dish** that does not make its own gravy._
**Holiday Wild Rice**
**Susan Kasting** , Jenks, OK
_Makes 4 servings_
**_Prep. Time: 10 minutes_**
**_Cooking Time: 2½-3 hours_**
**_Ideal slow cooker size: 4-qt._**
**1½ cups wild rice, uncooked**
**3 cups chicken stock**
**3 Tbsp. orange zest**
**2 Tbsp. orange juice**
**½ cup raisins (I like golden raisins)**
**1½ tsp. curry powder**
**1 Tbsp. butter, softened**
**½ cup fresh parsley**
**½ cup chopped pecans**
**½ cup chopped green onion**
1. Mix rice, chicken stock, orange zest, orange juice, raisins, curry powder, and butter in slow cooker.
2. Cover and cook on High 2½-3 hours, or until rice is tender and has absorbed most of the liquid, but is not dry.
3. Stir in parsley, pecans, and green onion just before serving.
**Mushroom Stuffing**
**Laverne Stoner**
Scottdale, PA
_Makes 7-8 cups stuffing_
**_Prep. Time: 15 minutes_**
**_Cooking Time: 2-3 hours_**
**_Ideal slow cooker size: 5-qt._**
**1 stick (8 Tbsp.) butter**
**1 cup finely chopped onions**
**1 cup finely chopped celery**
**8-oz. can sliced mushrooms, drained**
**¼ cup chopped parsley**
**1½-2 tsp. poultry seasoning**
**½ tsp. salt**
**⅛ tsp. pepper**
**12 cups toasted bread cubes***
**2 eggs, well beaten**
**1½ cups chicken broth**
1. Sauté onions and celery in butter in skillet until cooked. Stir in mushrooms and parsley.
2. Combine seasonings and sprinkle over bread cubes in large mixing bowl.
3. Gently add remaining ingredients. Spoon lightly into slow cooker.
4. Cover. Cook on High 1 hour, and then reduce to Low and cook 1-2 hours.
***** Lay 18-22 slices of bread on baking sheets. Toast in oven for 15 minutes at 300°.
**Tip:**
This is not as much a time-saver as it is a space-saver. If your oven is full, make your stuffing in your slow cooker.
**Variations:**
1. Add extra flavor to your stuffing by adding 1 tsp. dried sage, ¾ tsp. dried thyme, and ¼ tsp. dried marjoram to Step 2.
**Mary H. Nolt** , East Earl, PA
**Jean Turner** , Williams Lake, BC
**Mary Rogers** , Waseca, MN
**Kristi See** , Weskan, KS
2. Add 1 lb. loose sausage, browned and drained, to Step 3.
**Dede Peterson**
Rapid City, SD
**Old-Fashioned Stuffing**
**Elaine Rineer**
Lancaster, PA
**Rhonda Freed**
Croghan, NY
_Makes 6 servings_
**_Prep. Time: 20-30 minutes_**
**_Cooking Time: 5 hours_**
**_Ideal slow cooker size: 4-qt._**
**1 stick (8 Tbsp.) butter**
**1½ cups chopped celery**
**1 small onion, chopped**
**12-oz. pkg. bread cubes,** _or_ **about 15 slices stale bread**
**½-¾ tsp. salt**
**⅛ tsp. pepper**
**1-2 Tbsp. fresh parsley, chopped**
**2 eggs**
**1½ cups milk**
1. Melt butter in skillet. Sauté celery and onion in it.
2. Meanwhile, spray interior of slow cooker with non-stick cooking spray. Place bread cubes, salt, pepper, and parsley in cooker.
3. Pour in sautéed vegetables, eggs, and milk. Stir together gently until well mixed.
4. Cover. Cook on High 1 hour. Stir.
5. Cover. Cook on High 4 more hours.
**Tip:**
This is delicious and an easy way to free up oven space for the rest of your meal.
**Sweet Potato Stuffing**
**Tina Snyder**
Manheim, PA
_Makes 8 servings_
**_Prep. Time: 15 minutes_**
**_Cooking Time: 4 hours_**
**_Ideal slow cooker size: 4-qt._**
**½ cup chopped celery**
**½ cup chopped onions**
**half a stick (4 Tbsp.) butter**
**6 cups dry bread cubes**
**1 large sweet potato, cooked,* peeled, and cubed**
**½ cup chicken broth**
**¼ cup chopped pecans**
**½ tsp. poultry seasoning**
**½ tsp. rubbed sage**
**½ tsp. salt**
**¼ tsp. pepper**
1. Sauté celery and onions in skillet in butter until tender. Pour into greased slow cooker.
2. Add remaining ingredients. Toss gently.
3. Cover. Cook on Low 4 hours.
***** To prepare a cooked sweet potato, wash, dry, and prick it at about 4 different places with a sharp fork. Lay on a paper towel in the microwave. Cook on High 1½ minutes. Turn over. Cook on High another minute. Jag with fork to see if it's tender. If not, continue cooking for 30-second intervals on High, turning over each time, until tender.
Allow to cool. When you're able to handle it, gently peel and cube.
**Scalloped Pineapple**
**June Groff**
Denver, PA
_Makes 8 servings_
**_Prep. Time: 10 minutes_**
**_Cooking Time: 3 hours_**
**_Ideal slow cooker size: 2- to 3-qt._**
**1 stick (8 Tbsp.) butter, softened**
**½ cup sugar**
**4 eggs**
**20-oz. can crushed pineapple, drained**
**5 slices fresh bread, cubed**
1. Cream butter and sugar together in a medium-sized mixing bowl.
2. Add eggs. Mix well.
3. Add drained pineapple. Blend.
4. Fold bread into mixture.
5. Pour into slow cooker.
6. Cover. Cook on High 2 hours and then on Low 1 hour.
_This is a good accompaniment to **turkey** , **ham** , and **chicken** , especially at holiday-time. Small individual servings add a bright sweetness to special meals._
# **Tips**
* I like to keep a basket full of small gifts by the door to give to guests as they leave. These are simple gifts like home-made dip mixes in jars or freshly baked cookies, wrapped in colorful papers or gift bags. Sending friends home with a little something extra is a fun way to end a get-together.
**Mary Rogers**
Waseca, MN
* Bauer 24-Day Countdown to Christmas: Our family hangs a string of our own stockings in our living room each December. There are three of us, so we each choose eight socks and hang them using clothes pins. Each morning we take turns checking a sock for a surprise. We may find a small gift or something as simple as a note that says "Will cook supper for Mom tonight." We have fun thinking of unique ideas for each other, and before you know it, Christmas is here.
**Amy Bauer**
New Ulm, MN
* I have made fabric gift bags for my friends. They each know which one is theirs (the bags are all different). I get them back every year at the end of the holiday season, ready to use again the next year. It sure saves on wrapping paper.
**Sharon Easter**
Yuba City, CA
* Make sure your party invitations inform your guests about whether your party is informal or formal. It's really uncomfortable to be over-dressed at an informal party or vice versa.
**Anita Troyer**
Fairview, MI
* Picking a color theme and re-purposing everyday items makes holiday decorating easy. Fill glass flower vases with candy (red, green, white) and tie red ribbons around knobs and handles. Your kitchen will look festive without losing work space.
**Susan Tjon**
Austin, TX
* Sometime between Christmas and New Year's we have a neighborhood dinner, which we follow with singing hymns or Christmas songs. One neighbor brings her autoharp and another brings her accordion.
**Sara Kinsinger**
Stuarts Draft, VA
* I put up a small tree in my kitchen and decorate it with my mother's old cookie cutters and some of her recipe cards. I enjoy it while I'm in the kitchen during the holidays—which is a lot of time!
**Carol L. Miller**
Lockport, NY
* Our out-of-state relatives always bring gifts specifically made in their states to share with others.
**Annie C. Boshart**
Lebanon, PA
* We call the local assisted-living facility and ask for names and a list of gift ideas for 3 people who live in the facility but who don't usually get visitors at Christmas-time. Then on Christmas Day, our family delivers the gifts and visits with each person. Our kids are learning that Christmas is not all about them.
**Kendra Dreps**
Liberty, PA
# **FIX-IT AND FORGET-IT EBOOKS**
**FROM GOOD BOOKS**
**AND OPEN ROAD MEDIA**
**Available wherever ebooks are sold**
Good is our family name. We hope that word also represents the quality of the books we publish. And we also promise that "good" is the type of service you will receive from our staff.
Many of our books have been written by experts on our various staffs. Many were written by other outstanding authors. In any case, we want quality, reliability, and readability to be the hallmarks of Good Books.
**FIND OUT MORE AT**
**WWW.GOODBOOKS.COM**
Good Books is one of a select group of
publishing partners of Open Road Integrated Media, Inc.
**Open Road Integrated Media** **is a digital publisher and multimedia content company. Open Road creates connections between authors and their audiences by marketing its ebooks through a new proprietary online platform, which uses premium video content and social media.**
**Videos, Archival Documents, and New Releases**
**Sign up for the Open Road Media newsletter and get news delivered straight to your inbox.**
**Sign up now at**
**www.openroadmedia.com/newsletters**
**FIND OUT MORE AT**
**WWW.OPENROADMEDIA.COM**
**FOLLOW US:**
**@openroadmedia** **and**
**Facebook.com/OpenRoadMedia**
|
{
"redpajama_set_name": "RedPajamaBook"
}
| 9,677
|
Students with career ambitions in the logistics industry will be better placed than ever thanks to a new partnership between Cardiff University and NOVUS.
Sector-specific guest lectures, mentoring, career coaching and industrial placement opportunities are just some of the benefits in store for undergraduates specialising in Logistics and Operations Management on BSc Business Management at Cardiff Business School.
NOVUS works with universities across the UK to support the development of students and address skills shortages within the logistics industry.
The latest to join the Novus Lite scheme, Cardiff University joins Aston University, University of Derby, University of Hull and Northumbria University to put work-ready logistics and operations graduates at the forefront of their learning and teaching ambitions.
Cardiff Business School has one of the largest groupings of academic staff in logistics and operations management in the world, with experts in mathematics, economics, engineering, geography, law, management, and psychology.
Discover how their teaching aims to raise and nurture the next generation of logistics and operations management leaders on the BSc Business Management.
NOVUS is a not-for-profit organisation operating under the umbrella of The Chartered Institute of Logistics and Transport in the UK (CILT (UK)).
CILT (UK) is the membership organisation for professionals involved in the movement of goods and people and their associated supply chains.
|
{
"redpajama_set_name": "RedPajamaC4"
}
| 6,832
|
Der Delage Type S war ein Rennwagen der französischen Marke Delage.
Beschreibung
Die Homologation erfolgte am 14. Mai 1914, nachdem das Fahrzeug mit der Fahrgestellnummer 5795 und der Motornummer 101 geprüft worden war.
Delage setzte das Modell in Europa nur 1914 ein, in den Vereinigten Staaten dagegen bis 1917. Vorgänger war der Delage Type Y.
Ein Vierzylindermotor trieb die Fahrzeuge an. Er hatte 94 mm Bohrung und 160 mm Hub. Das ergab 4441 cm³ Hubraum. Der Motor war mit 26 Cheval fiscal eingestuft und leistete 115 PS. Die Vierventiltechnik des Vorgängermodells wurde beibehalten.
Als Höchstgeschwindigkeit sind 170 km/h genannt.
Das Fahrgestell hatte 1350 mm Spurweite und 2700 mm Radstand. Das Leergewicht war mit 950 kg angegeben.
Insgesamt entstanden vermutlich vier Fahrzeuge.
Renneinsätze
Quelle:
Literatur
Weblinks
Einzelnachweise
S
Phaeton
|
{
"redpajama_set_name": "RedPajamaWikipedia"
}
| 5,741
|
Q: Intro to Logic - help Hello Mathematics community,
As finals are approaching, my professor (regardless of how good is he) decided to just skim through the material that is going to be on the test. However, I wrote down two questions he constantly asked...
1) What are the advantages and disadvantages of Categorical Syllogism?
2) What are some of the advantages and disadvantages of Natural Deduction?
I have searched and read multiple papers and none of them had a clear cut on what these advantages and disadvantages are.
From what I have read thus far from my textbook and papers was the difference between Symbolic logic and Aristotelian logic?
Traditional Aristotelian logic (also called Classical Deductive Logic or Categorical Syllogisms) is an ancient method of deductive reasoning. Historians say that Aristotle, that ancient Greek philosopher guy, was the first to talk about syllogisms. In ancient Greece, men often first presented their arguments in syllogisms, and then they gave their arguments in a more rhetorical form. Men were held to a higher standard back then.
Modern Symbolic Logic is another method of deductive reasoning developed after the seventeenth century. Gottfried Leibniz and other logicians had a vision for a more simple and useful method for translating ordinary reasoning into a universal language of symbols. Among other things, this method can more quickly analyze longer arguments to see if they are valid. This method of deductive reasoning made logic even more "mathematical." It also made it uglier.
This is for two weeks from now so take your time answering and thank you so much for you help and making me try to understand the difference.
A: That's a very broad question with a wide variety of possible answers, but here is just my 2 cents:
Aristotelean Categorical Logic
Advantages:
*
*Can be analyzed using handy visual Venn diagrams
*Syllogisms not uncommon in real life, including the fallacious versions
*You can potentially memorize all the valid syllogisms ... there are not so many of them
Disadvantages:
*
*Limited applicability; hard to deal with 'or', for example
Natural Deduction
Advantages:
*
*Much more powerful and much wider applicability. Can do propositional logic, predicate logic, and other logics yet
Disadvantages:
*
*Trying to fit reasoning into the 'straightjacket' of formal logic can take away from understanding of logic at more intuitive level
|
{
"redpajama_set_name": "RedPajamaStackExchange"
}
| 7,496
|
Love Who You Love! Celebrating 12 Years of the International Day Against Homophobia!
Tomorrow people all across the world will be celebrating the International Day Against Homophobia, Transphobia and Biphobia (IDAHOT), including us here at Ben & Jerry's! We've been supporting equal rights for all for over 37 years now, and our commitment to the LGBTI community is grounded in our company's core values.
Back in 1989, we were the first major employer in Vermont to offer health insurance to domestic partners of employees, which included same sex couples, and we haven't spent one minute regretting it! This has progressed to supporting equal marriage legislation in a number of countries across the world, including most recently in Ireland where marriage equality became law by popular vote for the first time!
We believe that everyone should love who they love, no matter what their orientation or gender, which is just why we're supporting this special day.
What is the International Day Against Homophobia?
The IDAHOT started out back in 2004, aiming to draw the attention of policy makers, opinion leaders, social movements, the public and the media to the unacceptable violence and discrimination experienced by LGBTI people all over the world. Since then, it has gone on to establish itself as the single most important date for LGBTI people communities to mobilise on an international scale.
Where Did It All Start?
Coordinated by the Paris-based IDAHO Committee, IDAHOT was founded by French academic, Louis-Georges Tin. For those wondering the significance of the date, there is in fact an important piece of history behind 17th May. This date marks the day that the World Health Organisation removed homosexuality from the International Classification of Diseases in 1990… It's pretty shocking to think that just over 25 years ago homosexuality was considered an illness by such a reputable institution, and whilst great progress has been made in many countries around the world, we still face a situation where people suffer from discrimination simply for loving who they love.
Why Is It so Important?
IDAHO is celebrated in over 130 countries all over the world, however despite the day's global recognition, loving someone of the same sex is still considered illegal in almost 80 countries. In addition to this, in many countries citizens are still denied the right to live according to their preferred gender identity.
Here in the UK, there are all sorts of activities taking place up and down the country – from a special film screening in Shrewsbury, to 'one minute of noise' in commemoration of the victims of LGBTI violence in Brighton, and the Big Rainbow Picnic in Warwickshire, there's something for everyone! Check out the full list of events here.
Here at Ben & Jerry's we commend the organisers of this inspirational event, and look forward to the day that all of us, no matter who we are or who we love, can have the freedom to be who we want to be!
|
{
"redpajama_set_name": "RedPajamaCommonCrawl"
}
| 1,568
|
Why Marriages End In Divorce By Fulton Sheen
Love is primarily in the will, not in the emotions or the glands. The will is like the voice; the emotions are like the echo. The pleasure associated with love, or what is today called "sex," is the frosting on the cake; its purpose is to make us love the cake, not ignore it.
The greatest illusion of lovers is to believe that the intensity of their sexual attraction is the guarantee of the perpetuity of their love. It is because of this failure to distinguish between the glandular and spiritual–or between sex which we have in common with animals, and love which we have in common with God—that marriages are so full of deception.
What some people love is not a person, but the experience of being in love. The first is irreplaceable; the second is not. As soon as the glands cease to react with their pristine force, couples who identified emotionalism and love claim they no longer love one another. If such is the case they never loved the other person in the first place; they only loved being loved, which is the highest form of egotism. Marriage founded on sex passion alone lasts only as long as the animal passion lasts. Within two years the animal attraction for the other may die, and when it does, law comes to its rescue to justify the divorce with the meaningless words "incompatibility," or "mental torture." Animals never have recourse to law courts, because they have no will to love; but man, having reason, feels the need of justifying his irrational behavior when he does wrong.
There are two reasons for the primacy of sex over love in a decadent civilization. One is the decline of reason. As humans give up reason, they resort to their imaginations. That is why motion pictures and picture magazines enjoy such popularity. As thinking fades, unrestrained desires come to the fore. Since physical and erotic desires are among the easiest to dwell upon, because they require no effort and because they are powerfully aided by bodily passions, sex begins to be all-important. It is by no historical accident that an age of anti-intellectualism and irrationalism, such as our own, is also an age of carnal license. The second factor is egotism. As belief in a Divine Judgment, a future life, heaven and hell, a moral order, is increasingly rejected,
the ego becomes more and more firmly enthroned as the source of its morality. Each person becomes a judge in his own case. With this increase of selfishness, the demands for self-satisfaction become more and more imperious, and the interests of the community and the rights of others have less and less appeal. All sin is self-centeredness, as love is otherness and relatedness. Sin is the infidelity of man to the image of what he ought to be in his eternal vocation as an adopted son of God: the image God sees in Himself when He contemplates His Word.
There are two extremes to be avoided in discussing married love: one is the refusal to recognize sexual love, the other is the giving of primacy to sexual attraction. The first error was Victorian; the second is Freudian. To the Christian, sex is inseparable from the person, and to reduce the person to sex is as silly as to reduce personality to lungs or a thorax. Certain Victorians in their education practically denied sex as a function of personality; certain sexophiles of modern times deny personality and make a god of sex. The male animal is attracted to the female animal, but a human personality is attracted to another human personality.
The attraction of beast-to-beast is physiological; the attraction of human-to-human is physiological, psychological, and spiritual The human spirit has a thirst for the infinite which the quadruped has not. This infinite is really God. But man can pervert that thirst, which the animal cannot because it has no concept of the infinite.
Infidelity in married life is basically the substitution for an infinite of a succession of finite carnal experiences. The false infinity of succession takes the place of the Infinity of Destiny, which is God. The beast is promiscuous for an entirely different reason than man. The false pleasure given by new conquests in the realm of sex is the ersatz for the conquest of the Spirit in the Sacrament! The sense of emptiness, melancholy, and frustration is a consequence of the failure to find infinite satisfaction in what is carnal and limited. Despair is disappointed hedonism The most depressed spirits are those who seek God in a false god!
If love does not climb, it falls. If, like the flame, it does not burn upward to the sun, it burns downward to destroy. If sex does not mount to heaven, it descends into hell. There is no such thing as giving the body without giving the soul. Those who think they can be faithful in soul to one another, but unfaithful in body, forget that the two are inseparable. Sex in isolation from personality does not exist! An arm living and gesticulating apart from the living organism is an impossibility. Man has no organic functions isolated from his soul.
There is involvement of the whole personality. Nothing is more psychosomatic than the union of two in one flesh; nothing so much alters a mind, a will, for better or for worse. The separation of soul and body is death. Those who separate sex and spirit are rehearsing for death. The enjoyment of the other's personality through one's own personality, is love. The pleasure of animal function through another's animal function is sex separated from love.
Sex is one of the means God has instituted for the enrichment of personality. It is a basic principle of philosophy that there is nothing in the mind which was not previously in the senses. All our knowledge comes from the body. We have a body, St. Thomas tells us, because of the weakness of our intellect. Just as the enrichment of the mind comes from the body and its senses, so the enrichment of love comes through the body and its sex. As one can see a universe mirrored in a tear on a cheek, so in sex can be seen mirrored that wider world of love. Love in monogamous marriage includes sex; but sex, in the contemporary use of the term, does not imply either marriage or monogamy.
Every woman instinctively realizes the difference between the two, but man comes to understand it more slowly through reason and prayer. Man is driven by pleasure; woman by the meaning of pleasure. She sees pleasure more as a means to an end, namely, the prolongation of love both in herself and in her child.
But when sex is divorced from love there is a feeling that one has been stopped at the vestibule of the castle of pleasure; that the heart has been denied the city after crossing the bridge. Sadness and melancholy result from such a frustration of destiny, for it is the nature of man to be sad when he is pulled outside himself, or exteriorized without getting any nearer his goal. There is a closer correlation between mental instability and the animal view of sex than many suspect.
Happiness consists in interiority of the spirit, namely, the development of personality in relationship to a heavenly destiny. He who has no purpose in life is unhappy; he who exteriorizes his life and is dominated, or subjugated, by what is outside himself, or spends his energy on the external without understanding its mystery, is unhappy to the point of melancholy. There is the feeling of being hungry after having eaten, or of being disgusted with food, because it has nourished not the body, in the
case of an individual, or another body, in the case of marriage. In the woman, this sadness is due to the humiliation of realizing that where marriage is only sex, any other woman could fulfill her role; there is nothing personal, incommunicable, and therefore nothing dignified. Summoned by her God-implanted nature to be ushered into the mysteries of life which have their source in God, she is condemned to remain on the threshold as a tool or an instrument of pleasure alone, and not as a companion of love.
Two glasses that are empty cannot fill up one another. There must be a fountain of water outside the glasses, in order that they may have communion with one another. It takes three to make love. Every person is what he loves. Love becomes like unto that which it loves. If it loves heaven, it becomes heavenly; if it loves the carnal as a god, it becomes corruptible.
–F. Sheen
Tags: Body, Divine Judgment, God, Human, Human sexual activity, Marriage, Pleasure, Sexual attraction, Sexual intercourse, Sexuality
Categories : Marriage, Sexuality
The Pill Made Same Sex Marriage Inevitable
Opponents of legalized same-sex marriage say they're trying to protect a beleaguered institution, but they're a little late.
The walls of traditional marriage were breached 40 years ago; what we are witnessing now is the storming of the last bastion.
Marriage is primarily a social institution, not a religious one. That is, marriage is a universal phenomenon of human cultures in all times and places, regardless of the religion of the people concerned, and has taken the same basic form in all those cultures. Marriage existed long before Abraham, Jesus or any other religious figure. The institution of marriage is literally prehistoric.
The three monotheistic faiths (Judaism, Christianity and Islam) actually recognize this explicitly in their holy writings. The book of Genesis ascribes the foundation of marriage in the very acts of God himself in the creation of the world: "It is not good that man should be alone; I will make him a helper comparable to him. . . . A man shall leave his father and mother and be joined to his wife, and they shall become one flesh" (Genesis 2:18, 24).
The three great religions base their definition of marriage on these verses and others that echo them. In Christian theological terms, the definition of marriage is part of the natural law of the creation; therefore, the definition may not be changed by human will except in peril to the health of human community.
Psychobiologists argue that marriage evolved as a way of mediating the conflicting reproductive interests of men and women. It was the means by which a woman could guarantee to a specific man that the children she bore were his. In biological terms, men can sire hundreds of children in their lives, but this biological ability is limited by the fact that no one woman can keep pace.
Siring kids by multiple women is the only way men can achieve high levels of reproduction, but there is no adaptive advantage for women in bearing children by men who are simply trying to sire as many children as possible. For a mother, carrying and raising a child is a resource-intensive, years-long business. Doing it alone is a marked adaptive disadvantage for single mothers and their children.
So the economics of sex evolved into a win-win deal. Women agreed to give men exclusive sexual rights and guaranteed paternity in exchange for their sexual loyalty and enduring assistance with childbearing and -rearing. The man's promise of sexual loyalty meant that he would expend his labor and resources supporting her children, not another woman's. For the man, this arrangement lessens the number of potential children he can sire, but it ensures that her kids are his kids. Guaranteed sex with one woman also enabled him to conserve his resources and energies for other pursuits than repetitive courtship, which consumes both greatly.
Weddings ceremoniously legitimated the sexual union of a particular man and woman under the guidance of the greater community. In granting this license, society also promised structures beneficial to children arising from the marriage and ensuring their well-being.
Society's stake in marriage as an institution is nothing less than the perpetuation of the society itself, a matter of much greater than merely private concern. Yet society cannot compel men and women to bring forth their replacements. Marriage as conventionally defined is still the ordinary practice in Europe, yet the birthrate in most of Europe is now less than the replacement rate, which will have all sorts of dire consequences for its future.
Today, though, sexual intercourse is delinked from procreation. Since the invention of the Pill some 40 years ago, human beings have for the first time been able to control reproduction with a very high degree of assurance. That led to what our grandparents would have called rampant promiscuity. The causal relationships between sex, pregnancy and marriage were severed in a fundamental way. The impulse toward premarital chastity for women was always the fear of bearing a child alone. The Pill removed this fear. Along with it went the need of men to commit themselves exclusively to one woman in order to enjoy sexual relations at all. Over the past four decades, women have trained men that marriage is no longer necessary for sex. But women have also sadly discovered that they can't reliably gain men's sexual and emotional commitment to them by giving them sex before marriage.
Nationwide, the marriage rate has plunged 43% since 1960. Instead of getting married, men and women are just living together, cohabitation having increased tenfold in the same period. According to a University of Chicago study, cohabitation has become the norm. More than half the men and women who do get married have already lived together.
The widespread social acceptance of these changes is impelling the move toward homosexual marriage. Men and women living together and having sexual relations "without benefit of clergy," as the old phrasing goes, became not merely an accepted lifestyle, but the dominant lifestyle in the under-30 demographic within the past few years. Because they are able to control their reproductive abilities–that is, have sex without sex's results — the arguments against homosexual consanguinity began to wilt.
When society decided — and we have decided, this fight is over — that society would no longer decide the legitimacy of sexual relations between particular men and women, weddings became basically symbolic rather than substantive, and have come for most couples the shortcut way to make the legal compact regarding property rights, inheritance and certain other regulatory benefits. But what weddings do not do any longer is give to a man and a woman society's permission to have sex and procreate.
Sex, childbearing and marriage now have no necessary connection to one another, because the biological connection between sex and childbearing is controllable. The fundamental basis for marriage has thus been technologically obviated. Pair that development with rampant, easy divorce without social stigma, and talk in 2004 of "saving marriage" is pretty specious.
There's little there left to save. Men and women today who have successful, enduring marriages till death do them part do so in spite of society, not because of it.
If society has abandoned regulating heterosexual conduct of men and women, what right does it have to regulate homosexual conduct, including the regulation of their legal and property relationship with one another to mirror exactly that of hetero, married couples?
I believe that this state of affairs is contrary to the will of God. But traditionalists, especially Christian traditionalists (in whose ranks I include myself) need to get a clue about what has really been going on and face the fact that same-sex marriage, if it comes about, will not cause the degeneration of the institution of marriage; it is the result of it.
Rev. Sensing is pastor of the Trinity United Methodist Church in Franklin, Tenn. He writes at DonaldSensing.com. Donald Sensing.
Tags: Human, Marriage, Pennsylvania, Reproduction, Same-sex marriage, Sexual intercourse, Uncategorized, United States, University of Chicago
Why doesn't everybody believe that there is a purpose in Life?
Because some people think there is no real purpose or destiny to human life! They believe that only the things we make, like cars and watches, have design and purpose in them. We know what the purposes of these objects are because we designed them. (For instance, we know that the purpose of a car is transportation, and the purpose of a watch is to tell time.) But the things in nature, like trees and stars, were not designed by any human beings, so we do not know their purposes as we know the purposes of the things we design. So some people believe that there are no real purposes in the things in nature, but only in humanly designed artificial objects.
But one of the things in nature is human beings. They are not artificial objects! They are not artifacts like cars or watches. We did not design human nature; we only carry it on, by reproduction.
So the people who deny that human life has any real purpose argue this way:
If only artifacts have purposes, while things in nature do not; And if we are things in nature rather than artifacts; Then we have no real purpose.
So the answer to the question "What is the purpose of my existence?" is that there is no real purpose; we can imagine or make up any subjective purposes we want, but there is no objectively real purpose to human life. Life is purposeless, pointless, meaningless, in vain. "Vanity of vanities! All is vanity" (Eccl 1:2).
This is the worst philosophy in the world. For it denies us the things we need most: meaning and purpose; a reason to live, learn, grow, and endure.
Meaninglessness is unendurable. Even pain isn't as bad as meaninglessness. We can accept pains if they are meaningful: for instance, the pains of childbirth, or the pains of sacrificing for someone you love, or even the pains of martyrdom for a good cause. But we cannot accept meaninglessness. Even pleasures are not worthwhile if they are meaningless. (That's why a billionaire can choose to commit suicide.) And even pains are worthwhile if they are meaningful. (That's why a woman wants to give birth to a baby.)
The idea that objective things have no purpose is really atheism. For if God is real and if He created and designed everything, then everything has a purpose.
We can see some of the purpose of the things in nature. For instance, we can see that one of the purposes of stars is to enable us to think. For (a) if we did not breathe and bring oxygen to our brains, we could not think; and (b) if there were no green plants, we could not breathe, since their photosynthesis replaces carbon dioxide with oxygen; and (c) if there were no sun, there could be no green plants, for green plants need sunlight and heat, and (d) if there were no stars, there would be no sun, for the sun is a star. Therefore, if there were no stars, we could not think.
But many of the things in nature have designs and purposes that are not clear to us. They do not seem to be useful for us. (For instance, we wonder why God made so many mosquitoes.) So it takes a little faith, a little trust, to believe that everything has a purpose and that "all things work together for good to those who love God, who are called according to His purpose" (Rom, 8:28), though we do not see this. This is especially true of things that make us suffer. We do not always see how suffering has a good purpose.
But if the Creator is all-wise, all-good, and all-powerful, then the quotation above from Romans 8:28 is true. If He is all-good, He wants what is best. If He is all-powerful, He is able to bring about what is best, in the end. And if He is all-wise, He knows what is best.
And since we are not all-wise, we do not know what is best in the long run. That is why we have to trust Him with all those mosquitoes and even with much worse things, like cancers. He knows how to bring greater goods out of great evils. That is what He did two thousand years ago on the Cross of Calvary when He brought about the greatest good for us, the greatest gift we have ever been given–salvation from sin and the ability to enter Heaven–through the greatest evil that ever happened, the torture and murder of Jesus Christ, the only perfect man who ever lived, the man who was God Himself.
Christians believe this. Many people don't. Can Christian give them any reason to believe their religion's answer to the question "Why do I exist"?
The best reason we can give them is ourselves: our love and our joy. You can't argue with the happiness of a saint.
The greatest love, and the greatest joy, is mutual: it comes from both loving and being loved. The next-greatest joy comes from loving, even without being loved back. Even this second-best joy of loving without being loved back is greater and deeper than the third joy, the joy of being loved without loving. That is why saints are so happy: they are never in the third level of joy but always in the second or the first. (In fact, since they know God always loves them, you could say they are always in the first.) That's why the prayer attributed to Saint Francis says:
Lord, make me an instrument of Your peace. Where there is hatred, let me sow love; where there is injury, pardon; where there is doubt, faith; where there is despair, hope; where there is darkness, light and where there is sadness, joy. O divine Master, may I always seek not so much to be condemned as to console, to be understood as to understand, to be loved as to love. For it is in giving that we receive, it is in pardoning that we are pardoned, and it is in dying that we are born to eternal life. Amen.
By Peter Kreeft
Tags: Christianity, Ecclesiastes, God, Human, Jesus, Jesus Christ, Lord, Meaning of life, Uncategorized
Universal Morality? How absurd!
I know that some people say the idea of a morality known to all men is unsound, because different civilizations and different ages have had quite different moralities. But this is not true. I have never known a country, age or civilization where men were admired for running away in battle, or were a man felt proud of double-crossing all the people who had been kindest to him.
Some people may ask, "Isn't what you call the Moral Law simply our herd instinct and hasn't it been developed just like all our other instincts?" Now I do not deny that we may have a herd instinct: but that is not what I mean by the Moral Law. We all know what it feels like to be prompted by instinct—by mother love, or sexual instinct, or the instinct for food. It means that you feel a strong want or desire to act in a certain way. And, of course, we sometimes do feel just that sort of desire to help another person: and no doubt that desire is due to the herd instinct. But feeling a desire to help is quite different from feeling that you ought to help whether you want to or not. Supposing you hear a cry for help from a man in danger.
You will probably feel two desires—one a desire to give help (due to your herd instinct), the other a desire to keep out of danger (due to the instinct for self-preservation). But you will find inside you, in addition to these two impulses, a third thing which tells you that you ought to follow the impulse to help, and suppress the impulse to run away. Now this thing that judges between two instincts, that decides which should be encouraged, cannot itself be either of them. You might as well say that the sheet of music which tells you, at a given moment, to play one note on the piano and not another, is itself one of the notes on the keyboard. The Moral Law tells us the tune we have to play: our instincts are merely the keys.
Another way of seeing that the Moral Law is not simply one of our instincts is this. If two instincts are in conflict, and there is nothing in a creature's mind except those two instincts, obviously the stronger of the two must win. But at those moments when we are most conscious of the Moral Law, it usually seems to be telling us to side with the weaker of the two impulses. You probably want to be safe much more than you want to help the man who is drowning: but the Moral Law tells you to help him all the same. And surely it often tells us to try to make the right impulse stronger than it naturally is? I mean, we often feel it our duty to stimulate the herd instinct, by waking up our imaginations and arousing our pity and so on, so as to get up enough steam for doing the right thing. But clearly we are not acting from instinct when we set about making an instinct stronger than it is. The thing that says to you, "Your herd instinct is asleep. Wake it up," cannot itself be the herd instinct. The thing that tells you which note on the piano needs to be played louder cannot itself be that note.
Here is a third way of seeing it If the Moral Law was one of our instincts, we ought to be able to point to some one impulse inside us which was always what we call "good," always in agreement with the rule of right behaviour. But you cannot. There is none of our impulses which the Moral Law may not sometimes tell us to suppress, and none which it may not sometimes tell us to encourage. It is a mistake to think that some of our impulses— say mother love or patriotism—are good, and others, like sex or the fighting instinct, are bad. All we mean is that the occasions on which the fighting instinct or the sexual desire need to be restrained are rather more frequent than those for restraining mother love or patriotism. But there are situations in which it is the duty of a married man to encourage his sexual impulse and of a soldier to encourage the fighting instinct. There are also occasions on which a mother's love for her own children or a man's love for his own country have to be suppressed or they will lead to unfairness towards other people's children or countries. Strictly speaking, there are no such things as good and bad impulses. Think once again of a piano. It has not got two kinds of notes on it, the "right" notes and the "wrong" ones. Every single note is right at one time and wrong at another. The Moral Law is not any one instinct or any set of instincts: it is something which makes a kind of tune (the tune we call goodness or right conduct) by directing the instincts.
By the way, this point is of great practical consequence. The most dangerous thing you can do is to take any one impulse of your own nature and set it up as the thing you ought to follow at all costs. There is not one of them which will not make us into devils if we set it up as an absolute guide. You might think love of humanity in general was safe, but it is not. If you leave out justice you will find yourself breaking agreements and faking evidence in trials "for the sake of humanity," and become in the end a cruel and treacherous man.
Other people wrote to me saying, "Isn't what you call the Moral Law just a social convention, something that is put into us by education?" I think there is a misunderstanding here. The people who ask that question are usually taking it for granted that if we have learned a thing from parents and teachers, then that thing must be merely a human invention. But, of course, that is not so. We all learned the multiplication table at school. A child who grew up alone on a desert island would not know it. But surely it does not follow that the multiplication table is simply a human convention, something human beings have made up for themselves and might have made different if they had liked?
I fully agree that we learn the Rule of Decent Behaviour from parents and teachers, and friends and books, as we learn everything else. But some of the things we learn are mere conventions which might have been different—we learn to keep to the left of the road, but it might just as well have been the rule to keep to the right—and others of them, like mathematics, are real truths. The question is to which class the Law of Human Nature belongs.
There are two reasons for saying it belongs to the same class as mathematics. The first is, as I said in the first chapter, that though there are differences between the moral ideas of one time or country and those of another, the differences are not really very great—not nearly so great as most people imagine—and you can recognise the same law running through them all: whereas mere conventions, like the rule of the road or the kind of clothes people wear, may differ to any extent. The other reason is this.
When you think about these differences between the morality of one people and another, do you think that the morality of one people is ever better or worse than that of another? Have any of the changes been improvements? If not, then of course there could never be any moral progress. Progress means not just changing, but changing for the better. If no set of moral ideas were truer or better than any other, there would be no sense in preferring civilised morality to savage morality, or Christian morality to Nazi morality. In fact, of course, we all do believe that some moralities are better than others. We do believe that some of the people who tried to change the moral ideas of their own age were what we would call Reformers or Pioneers—people who understood morality better than their neighbours did. Very well then.
The moment you say that one set of moral ideas can be better than another, you are, in fact, measuring them both by a standard, saying that one of them conforms to that standard more nearly than the other. But the standard that measures two things is something different from either. You are, in fact, comparing them both with some Real Morality, admitting that there is such a thing as a real Right, independent of what people think, and that some people's ideas get nearer to that real Right than others. Or put it this way. If your moral ideas can be truer, and those of the Nazis less true, there must be something—some Real Morality—for them to be true about.
The reason why your idea of New York can be truer or less true than mine is that New York is a real place, existing quite apart from what either of us thinks. If when each of us said "New York" each meant merely "The town I am imagining in my own head," how could one of us have truer ideas than the other? There would be no question of truth or falsehood at all. In the same way, if the Rule of Decent Behaviour meant simply "whatever each nation happens to approve," there would be no sense in saying that any one nation had ever been more correct in its approval than any other; no sense in saying that the world could ever grow morally better or morally worse.
I conclude then, that though the differences between people's ideas of Decent Behaviour often make you suspect that there is no real natural Law of Behaviour at all, yet the things we are bound to think about these differences really prove just the opposite. But one word before I end. I have met people who exaggerate the differences, because they have not distinguished between differences of morality and differences of belief about facts. For example, one man said to me, "Three hundred years ago people in England were putting witches to death. Was that what you call the Rule of Human Nature or Right Conduct?" But surely the reason we do not execute witches is that we do not believe there are such things.
If we did—if we really thought that there were people going about who had sold themselves to the devil and received supernatural powers from him in return and were using these powers to kill their neighbours or drive them mad or bring bad weather, surely we would all agree that if anyone deserved the death penalty, then these filthy quislings did. There is no difference of moral principle here: the difference is simply about matter of fact. It may be a great advance in knowledge not to believe in witches: there is no moral advance in not executing them when you do not think they are there. You would not call a man humane for ceasing to set mousetraps if he did so because he believed there were no mice in the house.
Professor C.S Lewis
Tags: C.S Lewis, Christian ethics, Convention (norm), Herd behavior, Human, Libido, Moral absolutism, Moral Law, Morality, Uncategorized
Who invented "Fair Play"?
Every one has heard people quarreling. Sometimes it sounds funny and sometimes it sounds merely unpleasant; but however it sounds, I believe we can learn something very important from listening to the kind of things they say. They say things like this: 'How'd you like it if anyone did the same to you?'–'That's my seat, I was there first'–'Leave him alone, he isn't doing you any harm'–Why should you shove in first?'–'Give me a bit of your orange, I gave you a bit of mine'–'Come on, you promised.' People say things like that every day, educated people as well as uneducated, and children as well as grown-ups.
Now what interests me about all these remarks is that the man who makes them is not merely saying that the other man's behaviour does not happen to please him. He is appealing to some kind of standard of behaviour which he expects the other man to know about. And the other man very seldom replies: 'To hell with your standard.' Nearly always he tries to make out that what he has been doing does not really go against the standard, or that if it does there is some special excuse. He pretends there is some special reason in this particular case why the person who took the seat first should not keep it, or that things were quite different when he was given the bit of orange, or that something has turned up which lets him off keeping his promise. It looks, in fact, very much as if both parties had in mind some kind of Law or Rule of fair play or decent behaviour or morality or whatever you like to call it, about which they really agreed. And they have. If they had not, they might, of course, fight like animals, but they could not quarrel in the human sense of the word. Quarreling means trying to show that the other man is in the wrong. And there would be no sense in trying to do that unless you and he had some sort of agreement as to what Right and Wrong are; just as there would be no sense in saying that a footballer had committed a foul unless there was some agreement about the rules of football.
Now this Law or Rule about Right and Wrong used to be called the Law of Nature. Nowadays, when we talk of the 'laws of nature' we usually mean things like gravitation, or heredity, or the laws of chemistry. But when the older thinkers called the Law of Right and Wrong 'the Law of Nature,' they really meant the Law of Human Nature. The idea was that, just as all bodies are governed by the law of gravitation, and organisms by biological laws, so the creature called man also had his law–with this great difference, that a body could not choose whether it obeyed the law of gravitation or not, but a man could choose either to obey the Law of Human Nature or to disobey it.
We may put this in another way. Each man is at every moment subjected to several different sets of law but there is only one of these which he is free to disobey. As a body, he is subjected to gravitation and cannot disobey it; if you leave him unsupported in mid-air, he has no more choice about falling than a stone has. As an organism, he is subjected to various biological laws which he cannot disobey any more than an animal can. That is, he cannot disobey those laws which he shares with other things; but the law which is peculiar to his human nature, the law he does not share with animals or vegetables or inorganic things, is the one he can disobey if he chooses.
This law was called the Law of Nature because people thought that every one knew it by nature and did not need to be taught it. They did not mean, of course, that you might not find an odd individual here and there who did not know it, just as you find a few people who are colour-blind or have no ear for a tune. But taking the race as a whole, they thought that the human idea of decent behaviour was obvious to every one. And I believe they were right. If they were not, then all the things we said about the war were nonsense. What was the sense in saying the enemy were in the wrong unless Right is a real thing which the Nazis at bottom knew as well as we did and ought to have practised? If they had had no notion of what we mean by right, then, though we might still have had to fight them, we could no more have blamed them for that than for the colour of their hair.
I know that some people say the idea of a Law of Nature or decent behaviour known to all men is unsound, because different civilizations and different ages have had quite different moralities.
But this is not true. There have been differences between their moralities, but these have never amounted to anything like a total difference. If anyone will take the trouble to compare the moral teaching of, say, the ancient Egyptians, Babylonians, Hindus, Chinese, Greeks and Romans, what will really strike him will be how very like they are to each other and to our own.
I need only ask the reader to think what a totally different morality would mean. Think of a country where people were admired for running away in battle, or where a man felt proud of double-crossing all the people who had been kindest to him. You might just as well try to imagine a country where two and two made five. Men have differed as regards what people you ought to be unselfish to–whether it was only your own family, or your fellow countrymen, or every one. But they have always agreed that you ought not to put yourself first. Selfishness has never been admired. Men have differed as to whether you should have one wife or four. But they have always agreed that you must not simply have any woman you liked.
But the most remarkable thing is this. Whenever you find a man who says he does not believe in a real Right and Wrong, you will find the same man going back on this a moment later. He may break his promise to you, but if you try breaking one to him he will be complaining 'It's not fair' before you can say Jack Robinson. A nation may say treaties do not matter; but then, next minute, they spoil their case by saying that the particular treaty they want to break was an unfair one. But if treaties do not matter, and if there is no such thing as Right and Wrong–in other words, if there is no Law of Nature–what is the difference between a fair treaty and an unfair one? Have they not let the cat out of the bag and shown that, whatever they say, they really know the Law of Nature just like anyone else?
It seems, then, we are forced to believe in a real Right and Wrong. People may be sometimes mistaken about them, just as people sometimes get their sums wrong; but they are not a matter of mere taste and opinion any more than the multiplication table. Now if we are agreed about that, I go on to my next point, which is this. None of us are really keeping the Law of Nature. If there are any exceptions among you, I apologise to them. They had much better read some other book, for nothing I am going to say concerns them. And now, turning to the ordinary human beings who are left:
I hope you will not misunderstand what I am going to say. I am not preaching, and Heaven knows I do not pretend to be better than anyone else. I am only trying to call attention to a fact; the fact that this year, or this month, or, more likely, this very day, we have failed to practise ourselves the kind of behaviour we expect from other people. There may be all sorts of excuses for us. That time you were so unfair to the children was when you were very tired. That slightly shady business about the money–the one you have almost forgotten-came when you were very hard up. And what you promised to do for old So-and-so and have never done–well, you never would have promised if you had known how frightfully busy you were going to be. And as for your behaviour to your wife (or husband) or sister (or brother) if I knew how irritating they could be, I would not wonder at it–and who ens am I, anyway? I am just the same. That is to say, I do not succeed in keeping the Law of Nature very well, and the moment anyone tells me I am not keeping it, there starts up in my mind a string of excuses as long as your arm. The question at the moment is not whether they are good excuses. The point is that they are one more proof of how deeply, whether we like it or not, we believe in the Law of Nature. If we do not believe in decent behaviour, why should we be so anxious to make excuses for not having behaved decently? The truth is, we believe in decency so much–we feel the Rule or Law pressing on us so–that we cannot bear to face the fact that we are breaking it, and consequently we try to shift the responsibility. For you notice that it is only for our bad behaviour that we find all these explanations. It is only our bad temper that we put down to being tired or worried or hungry; we put our good temper down to ourselves.
These, then, are the two points I wanted to make. First, that human beings, all over the earth, have this curious idea that they ought to behave in a certain way, and cannot really get rid of it. Secondly, that they do not in fact behave in that way. They know the Law of Nature; they break it. These two facts are the foundation of all clear thinking about ourselves and the universe we live in.
Tags: Abolition of Man, Ancient Egypt, C.S Lewis, Ethics, Human, Law of Nature, Mere Christianity, Natural law, Nature, Newton's law of universal gravitation, Uncategorized
|
{
"redpajama_set_name": "RedPajamaCommonCrawl"
}
| 1,886
|
NUREYEV: LEAPS OF FAITH
Rudi made me cry.
I don't cry at or during films anymore, mostly because the varying levels of discomfort my eyes are usually in, make it hard for me to escape entirely into the visual world before me. But the life of Rudolf Khametovich Nureyev contained within a single powerful documentary, All The World His Stage, had me in tears for much of the second half.
Exile is a common enough story for many of us. Voluntary or involuntary. But in Rudi's case he was a pawn of a nation, a government, an ideology that his body and will refused to succumb to. Of course, his defection in Paris in 1961 had nasty consequences for his friends, the 'kitchen culture' crowd, who had, in secrecy, danced and recited poetry and played under the wrap of darkness. But they separated him from his mother, and more than anyone else in his family, I think that may have been the worst of it, though in the end they let him back for the days before her death. They say that although she was failing to recognise anyone else, she knew he had been to see her, but what was that one moment against all the years - the millions of yearnings, and achings for home?
And then there was Dame Margot - La Fonteyn - the substitute mother figure? No, she was more, she was everything to him; after she died, he had no one; he would call friends before dawn, and say nothing, only cry. She was on the verge of asking her husband Tito for a divorce, and then he got shot, and Fonteyn's mother said how will it look if you don't go immediately to his side? How will it look? Even the greatest prima donna ballerina obeys when her mother says those four sinister words. That was the end of Nureyev and Fonteyn, so said the documentary, although in reality they danced for years afterwards, and stayed close until her death.
The violinist Yehudi Menuhin called Nureyev a panther. Parkinson asked him to describe how things had been in Russia when Nureyev was a child. 'Bad,' replied the dancer. 'But how bad?' pressed Parky. Because people always want to know how bad, from the safety and comfort of their own lives. Richard Avedon photographed Nureyev's leaps into air as though he were challenging gravity to call him merely human, but Bob Dylan wrote 'No one is free, even the birds are chained to the sky.'
The documentary was at once the embodiment of freedom and simultaneously a chaining down, a weighing down of things that are bigger than us, wider than we can control. Politics, AIDS. Being called Russian when you are really Tatar, but no one knows of Tatars or their complicated history with Russia...
Ah... go watch the documentary if you can. Also Lady Gaga: Five Foot Two. Also M.I.A, about the Sri Lankan Tamil English singer and activist Mathangi 'Maya' Arulpragasam. Also, the manager of the Cambridge Arts Picturehouse has promised me the Nureyev poster when the film has finished running... these are moments worth living for.
Labels: dance, film review, Heroes, inspiration, love, movies
Now I want to watch ALL these films. How wonderful this post is with exciting stories to pursue. I always know I will like films and books you like, we share a worldview I believe. I am so drawn to true stories of peoples' lives and how they rise in spite of their hard journeys. Thank you for this post, my friend.
Buy zopiclone online cheap
vaiybora said...
Very great post. I simply stumbled upon your blog and wanted to say that I have really enjoyed browsing your weblog posts.
rallentanda said...
We were still giggly but precocious little school girls in the sixties. Our parents ( relieved that we were interested in high culture and not those awful Beatle boys:) allowed us to camp out over night to buy tickets to see Nureyev making his first visit down under.
My best friend and I were both mad about him. We had pictures of him inside our desks at school and all over our bedroom walls.We were lucky enough to get tickets and I will never forget that performance. Now what follows is even more amazing. The Prime Minister and his retinue were present at this performance. We just tagged on to the group ( looking as if we belonged) and merged with them back stage. In those days security was not an issue here( obviously)
We met Margot Fonteyn and Nureyev. My friend asked for his autograph...did not have anything for him to write on except for a newspaper (we could not afford programs, the tickets cost a fortune)...He did... somewhat bemused... made some sort of joke and then we left...He was gorgeous . Thank you for reminding me..
|
{
"redpajama_set_name": "RedPajamaCommonCrawl"
}
| 4,802
|
\section{Introduction}
Voronoi diagrams and their dual Delaunay triangulations are fundamental constructions with
numerous associated guarantees, and extensive application in
practice (for a thorough review consult~\cite{Aurenhammer13} and references therein).
At their heart is the use of a distance between points, which in the original
version is taken to be Euclidean.
This suggests that, by considering
distances other than Euclidean,
it may be possible to obtain variants which can be well-suited to
a wider range of applications.
Attempts in this direction have been met with some success.
Power diagrams~\cite{power} generalize Euclidean distance by associating
a {bias-term}
to each
site. The duals of these diagrams
are
guaranteed to be embedded triangulations, in any number of
dimensions.
Although this is a strict generalization of Euclidean distance, it is a somewhat
limited one. The effect of the bias term is to locally enlarge or shrink the
region associated to each site, loosely-speaking ``equally in every
direction". It allows some freedom in choosing local scale, with no
preference for specific directions.
Two related, and relatively recent generalizations of Voronoi diagrams and Delaunay triangulations have been proposed,
independently, by Labelle and Shewchuk~\cite{LS}, and Du and Wang~\cite{DW}.
Although their associated anisotropic Voronoi diagrams are, in general, no longer orphan-free
(i.e.~they may have disconnected Voronoi regions),
Labelle and Shewchuk show that a set of sites exists
with an orphan-free diagram, whose dual is embedded, in two dimensions.
They accomplish this by proposing an iterative site-insertion algorithm
that, for any given metric, constructs one such set of sites.
Note that this is a property of the output of the algorithm, and not a general condition for obtaining embedded triangulations.
The recent work of~\cite{Bregman} discusses Voronoi diagrams and their duals with respect to Bregman divergences.
They show that Bregman Voronoi diagrams of the \emph{first kind} are simply power diagrams,
whose duals are known to always be embedded~\cite{powerdiag}.
Bregman diagrams of the \emph{second kind} are power diagrams in the dual (gradient) space,
but, prior to this work, no results for them were available in the primal space.
In this paper we discuss properties of Voronoi diagrams and Delaunay triangulations for
a general class of divergences,
including Bregman, quadratic, and all distances derived from strictly convex $\mathcal{C}^1$ norms.
We show that,
given a divergence $D$ that is convex in the first argument and continuously differentiable in the second,
and under a \emph{bounded anisotropy} assumption on the divergence,
if a set of sites produces an orphan-free
Voronoi diagram with respect to $D$,
then its dual is always an embedded triangulation
(or an embedded polygonal mesh with convex faces in general),
in two dimensions (theorem~\ref{th:main}).
This effectively states that, regardless of the sites' positions, if the primal
is well-behaved, then the dual is also well-behaved.
Further, in a way that parallels the ordinary Delaunay case, the dual has no
degenerate elements (proposition~\ref{prop:ECB}), its elements (vertices, edges, faces)
are unique (Cor.~\ref{cor:VorI}), and the dual is guaranteed to cover the convex hull of the sites (theorem~\ref{th:main}).
\section{Voronoi diagrams
with respect to divergences}\label{sec:setup}
The class of divergences that we consider in this work are non-negative functions
$D:\mathbb{R}^2\times\mathbb{R}^2\rightarrow\mathbb{R}$
which are strictly convex in the first argument and continuously differentiable in the second,
and such that $\D{x}{x}=0$ for all $x\in\mathbb{R}^2$.
Following~\cite{Bregman}, we let
\begin{equation}\label{eq:defball}
B_1(p; \rho) = \{v\in\mathbb{R}^2 : \D{v}{p} \le \rho\}, \quad\quad B_2(p;\rho) = \{v\in\mathbb{R}^2 : \D{p}{v} \le \rho\}
\end{equation}
be, respectively, balls of the \emph{first} and \emph{second kind}, centered at $p\in\mathbb{R}^2$ of radius $\rho$.
Note that balls of the first kind are necessarily convex since $\D{\cdot}{p}$ is convex.
We also assume that $D$ satisfies what we term a \emph{bounded anisotropy} condition,
defined in assumption~\ref{ass:BAA} below.
Given a set $S=\{s_1,\dots,s_n\}\subset\mathbb{R}^2$ of
$n$ distinct sites on the plane, and a divergence $D:\mathbb{R}^2\times\mathbb{R}^2\rightarrow\mathbb{R}$,
the Voronoi regions
of the first and second kinds~\cite{Bregman} are:
\begin{eqnarray}
\label{eq:defvor1} {\text{Vor}^1_i} = \{p\in\mathbb{R}^2 : \D{p}{s_i} \le \D{p}{s}, \forall s\in S \}, \\
\label{eq:defvor2} {\text{Vor}^2_i} = \{p\in\mathbb{R}^2 : \D{s_i}{p} \le \D{s}{p}, \forall s\in S \},
\end{eqnarray}
respectively, and are indexed by the site its points are closest to.
Of course, the two kinds of Voronoi diagrams are different because $D$ is in general not symmetric.
In the sequel, and whenever not otherwise specified, we will assume that balls are of the \emph{first kind} (convex),
and Voronoi diagrams, and their dual Delaunay triangulations are of the \emph{second kind}.
For instance, we will use the convexity of balls (of the first kind) to prove
that every face in a Delaunay triangulation (of the second kind)
satisfies an \emph{Empty Circum-Ball} property (proposition~\ref{prop:ECB})
that parallels the empty circumcircle property of Euclidean Delaunay triangulations.
Consider the following definition of Voronoi element:
\begin{definition}\label{def:VorI}
For each subset $I\subseteq\{1,\dots,n\}$, the set $\text{Vor}_I=\cap_{i\in I}\text{Vor}^2_i \setminus \cup_{j\not\in I}\text{Vor}^2_j$ is a Voronoi element of order $|I|$.
Elements of orders $1$, $2$, and $|I|\ge 3$ are denoted regions, edges, and vertices, respectively.
\end{definition}
\begin{remark}
The set of all Voronoi elements $\text{Vor}_I$ forms a partition of the plane.
\end{remark}
\begin{table}[htbp]\caption{Notation}\label{table:notation}
\begin{center}
\begin{tabular}{r c p{11cm} }
\toprule
$\D{\cdot}{\cdot}$ && A non-negative divergence strictly convex in its first argument and continuously differentiable in the second. \\
$D_F(\cdot \parallel \cdot)$ && Bregman divergence (section~\ref{sec:DF}). \\
$D_f(\cdot \parallel \cdot)$ && Csisz\'ar divergence (section~\ref{sec:Df}).\\
$D_Q(\cdot \parallel \cdot)$ && Quadratic divergence (seciton~\ref{sec:DQ}).\\
$\gamma$ && Global lower bound on the ratio of eigenvalues
of metric $Q$ (quadratic divergence, lemma~\ref{lem:DQgamma})
or of the Hessian of $F$ (Bregman divergence, lemma~\ref{lem:DFgamma}). \\
$S = \{s_1,\dots,s_n\}$ && Set of $n$ sites. \\
$L_{ij}$ && The supporting line of sites $s_i$, $s_j$. \\
${conv}\left\{\Sites\right\}$ && Convex hull of $S$. \\
$W=\{w_i\in S : i=1,\dots,m\}$ && Subset of sites on the boundary of ${conv}\left\{\Sites\right\}$, in clock-wise order. \\
$B(\cdot\, ;\cdot)$ && Convex ball of the first kind (equation~\eqref{eq:defball}). \\
$\theta_p(v)$ && The ball (of the first kind) $B(v;\D{p}{v})$ centered at $v$ with $p$ in its boundary. \\
$\text{Vor}_i$ && Voronoi region of the second kind corresponding to site $s_i$ (equation~\eqref{eq:defvor2}). \\
$\text{Vor}_I$ && Voronoi element of order $|I|=1$ (Voronoi region), $|I|=2$ (Voronoi edge), or $|I|\ge 3$ (Voronoi vertex).\\
$G=(S,E,F)$ && The straight-edge dual triangulation with vertices at the sites. \\
$B$ && The edges in the topological boundary of $G$ (incident to one face). \\
$\mathcal{B}=(w_i,w_{i\oplus 1})_{i=1}^{|W|}$ && The edges in the boundary of ${conv}\left\{\Sites\right\}$. \\
$\pi$ && Projection from $C(\sigma)$ onto $\partial{conv}\left\{\Sites\right\}$ (section~\ref{sec:boundary}). \\
$\nu_\sigma$ && Projection function onto a circle of radius $\sigma$ (section~\ref{sec:boundary}). \\
$H^{+}_{ij}, H^{-}_{ij}$ && The half-spaces on either side of $L_{ij}$, chosen so $H^{+}_{ij}\capS=\phi$ (fig.~\ref{fig:pinu}). \\
$C(\sigma)$ && The origin-centered circle of radius $\sigma$ (with respect to the natural metric). \\
\bottomrule
\end{tabular}
\end{center}
\end{table}
The following ``bounded anisotropy" condition is assumed to hold.
It is written in its most general (but very technical) form below,
but it becomes much simpler in particular cases, as shown in Section~\ref{sec:summary}.
Typically, it can be rewritten as a simple regularity condition on a symmetric positive definite matrix,
such that its ratio of minimum to maximum eigenvalues
(a measure of anisotropy) is globally bounded away from zero.
\begin{figure}[thb]
\centering
\includegraphics[width=2.1in]{boundedanisotropy.pdf}
\caption{The \emph{bounded anisotropy asumption} ensures that balls of the first kind are globally well-behaved. }
\label{fig:gamma}
\end{figure}
\begin{assumption}[Bounded anisotropy]\label{ass:BAA}
For every two points $p,q\in\mathbb{R}^2$ with supporting line $L_{pq}$, and every point $r\not\in L_{pq}$,
there is a sufficiently large value $\mu > 0$ such that
for every point $c\in\mathbb{R}^2$ lying on the same side of $L_{pq}$ as $r$,
such that $\|c\| > \mu$, and
whose closest point $m$ in $L_{pq}$ lies in the segment $\overline{pq}$,
it is $\D{r}{c} < \D{m}{c}$.
\end{assumption}
\begin{remark}
Note that the condition $\|c\|<\mu$ depends on the (arbitrary) choice of origin.
Assumption~\ref{ass:BAA} is, however, independent of this choice.
\end{remark}
Loosely speaking, this condition ensures that balls of the first kind are
not just convex, but also ``sufficiently round".
For instance, it is satisfied by all the $L_p$ distances with $1<p<\infty$,
but not for $p=1,\infty$, since (aside from not being strictly convex) the corresponding balls have ``kinks".
\begin{assumption}[Extremal gradients]\label{ass:EGA}
For each Voronoi vertex $\text{Vor}\{i_1,\dots,i_m\}$ with $m\ge 3$,
the gradients $g_j(p)\equiv\nabla_p \D{s_{i_j}}{p}, j=1,\dots,m$, at $p\in\text{Vor}_I$ are distinct and extremal,
i.e.\ they are vertices of the convex hull: $\CH\{g_1(p),\dots,g_m(p)\}$.
\end{assumption}
\begin{remark}
In the ``typical" case that $m=3$, the above simply means that $g_1,g_2,g_3$ are not colinear.
Given two distinct gradients $g_1\ne g_2$, requiring $g_3$ not to be colinear only constraints it
to be outside a line.
If $D$ is the $L_p$ distance (or any other non-spatially-varying divergence),
the extremal gradient assumption can be shown to be always automatically satisfied at Voronoi vertices.
Finally, the extremal gradient assumption will be shown to imply that Voronoi vertices are composed of isolated points,
and therefore, when satisfied, the assumption only needs to be enforced at a discrete set of points.
\end{remark}
\subsection{Orphan-free Voronoi diagrams and dual triangulations}\label{sec:simpleplanar}
As described in the classic survey by Aurenhammer~\cite{Aurenhammer:1991},
planar Voronoi diagrams and Delaunay triangulations are duals in a graph theoretical sense.
Associated to the ordinary Voronoi diagram is a simple, planar (primal) graph with vertices at
points equidistant to three or more sites (Voronoi vertices),
and edges composed of line segments
equidistant to two sites (Voronoi edges).
Because edges are always line segments, the graph is simple (has no multi-edges or self-loops),
and this construction provides an embedding of the graph, which must therefore be planar.
For Voronoi diagrams defined by divergences, the situation is markedly different.
The incidence relations between Voronoi elements cannot be so easily established.
For instance, Voronoi edges may be disconnected and incident to any number of Voronoi vertices.
For this reason, we begin our proof by constructing an embedding of a primal graph
from the incidence relations of the Voronoi diagram (definition~\ref{def:incidence}),
in a way that generalizes ordinary Voronoi diagrams,
and show that this graph is simple and planar (section~\ref{sec:planar}).
This primal graph is then dualized into a simple, planar graph.
The dual graph is denoted the Delaunay \emph{triangulation} because, as will be shown,
it is composed of convex faces which can be triangulated without breaking any of
its important properties, such as embeddability or the \emph{empty circum-ball} property (property~\ref{cor:VorI}).
The rest of the paper makes heavy use of the following trivial lemmas, which we include here for convenience.
The first follows directly from the properties of $D$, while the second
is a direct consequence of the strict convexity of $\D{\cdot}{p}$ and the continuity of $D$
(note that $D$ is globally continuous since it is continuous in the second argument
and convex in the first, and therefore it is also continuous in the first argument~\cite{rockafellar1997convex}).
\begin{lemma}\label{lem:site}
Every site $s_i\inS$ is an interior point of its corresponding Voronoi region $\text{Vor}_i$.
\end{lemma}
\begin{lemma}\label{lem:midpoint}
Given two sites $s_i,s_j\in S$ with supporting line $L_{ij}$,
all points $p\in L_{ij}$ that are equidistant to $s_i$ and $s_j$
belong to the segment $\overline{s_i s_j}$.
Furthermore, there is always at least one such point.
\end{lemma}
\section{Summary of results}\label{sec:summary}
\begin{figure}[ht]
\begin{center}
\subfloat{\includegraphics[height=3.5cm]{c09.png}}\quad
\subfloat{\includegraphics[height=3.5cm]{c10.png}}
\caption{If all sites are colinear, the dual is always a chain connecting consecutive sites along their supporting line.
This structure is independent of the divergence, and doesn't require assumption~\ref{ass:BAA}. }
\label{fig:colinear}
\end{center}
\end{figure}
Consider first the special case that all sites in $S$ are colinear.
The structure of the Voronoi diagram and the Delaunay triangulation is very simple in this case.
If we order the sites $s_1,\dots,s_n$ sequentially along their supporting line,
lemma~\ref{lem:midpoint} shows that there must be Delaunay edges between successive sites,
while the strict convexity of the balls implies that these are the only edges
(all points in $\overline{s_i s_{i\oplus 1}}$ are strictly closer to $s_i,s_{i\oplus 1}$ than to any other site),
and that there are no Delaunay faces
(since three colinear points cannot be in the boundary of a strictly convex ball).
The following proposition does not require assumption~\ref{ass:BAA} nor~\ref{ass:EGA}.
\begin{proposition}
{For all divergences} $D$,
the Delaunay triangulation of a set of colinear sites is a chain connecting successive sites $s_i,s_{i\oplus 1}$, $i=1,\dots,n-1$
along their supporting line.
\end{proposition}
With the colinear site case covered, we assume in the remainder
that \emph{not all} sites are colinear,
and that $D$ satisfies assumptions~\ref{ass:BAA} and~\ref{ass:EGA}.
We begin, in section~\ref{sec:planar}, by constructing a primal graph from the incidence relations between Voronoi elements,
and dualize it to obtain a simple, planar graph.
\begin{theorem}\label{th:simpleplanar}
The dual of the primal Voronoi graph of an orphan-free Voronoi diagram
is a simple, connected, planar graph.
\end{theorem}
\begin{remark}
Note that the differentiability of $D$ with respect to the second argument is only used in (a small neighborhood around) Voronoi vertices
(a set of isolated points).
Everywhere else, it suffices that $D$ is continuous in its second argument.
\end{remark}
While this dual graph is an embedded planar graph with curved edges, we then show
that it is also an embedded planar graph with vertices at the sites and straight edges.
\begin{theorem}\label{th:main}
The straight-edge dual of a primal Voronoi graph
(obtained from an orphan-free Voronoi diagram of a set of sites $S$)
is embedded with vertices at the sites,
has (non-degenerate) strictly convex faces, and covers the convex hull of $S$.
\end{theorem}
As described in Section~\ref{sec:simpleplanar}, lemmas~\ref{lem:regionSC} and~\ref{lem:SCedges}
can be used in conjunction with theorem~\ref{th:main} to
conclude that orphan-freedom
is a sufficient condition
for the well-behavedeness of not just the
dual, but also of the primal Voronoi diagram.
Note that this excludes isolated Voronoi edges (those not incident to any Voronoi vertex),
which are shown to be contained in Voronoi regions,
and are considered part of their containing regions (section~\ref{sec:propedges}).
\begin{corollary}\label{cor:VorI}
All the elements of an orphan-free Voronoi diagram are connected,
with the exception of isolated Voronoi edges.
\end{corollary}
\begin{remark}
Isolated edges are connected components of a Voronoi edge
which are incident to a single Voronoi region.
Since they do not affect the construction of the primal Voronoi graph,
they can be safely discarded, as shown in section~\ref{sec:propedges}.
\end{remark}
Perhaps the most fundamental property of the diagrams that we use in the proofs is that
every dual face has an ``empty" circumscribing \emph{convex} ball.
This empty circum-ball (ECB) property
is analogous to the empty circumcircle property of ordinary Voronoi diagrams:
\begin{proposition}[Empty Circum-Ball property]\label{prop:ECB}
For every dual face with vertices $s_{i_1},\dots,s_{i_k}$ there is a convex ball
that circumscribes $s_{i_1},\dots,s_{i_k}$ and contains no site in its interior.
\end{proposition}
Indeed, since to every dual face $f$ with vertices $s_{i_1},\dots,s_{i_k}$ ($k\ge 3$) corresponds
a Voronoi element $\text{Vor}_{\{i_1,\dots,i_k\}}$,
any point $c\in\text{Vor}_{\{i_1,\dots,i_k\}}$ serves as center of an empty circumscribing ball of $f$.
To see that this ball must be ``empty", note that
no site $s'$
can be strictly inside the circumscribing ball (certainly not $s_{i_1},\dots,s_{i_k}$, since they are in the boundary),
or $c$ would be closer to $s'$ than to $s_{i_1},\dots,s_{i_k}$,
and therefore it would not be $c\in\text{Vor}_{\{i_1,\dots,i_k\}}$.
Notice that, although we consider Voronoi diagrams of the second kind,
it is the convexity of balls of the \emph{first kind} that establishes the ECB condition.
The ECB property is, in general, not satisfied by Delaunay triangulations of the first kind.
\begin{figure}[htbp]
\begin{center}
\includegraphics[width=2.5in]{outline.pdf}
\caption{
We prove that the Delaunay triangulation is embedded (theorem~\ref{th:main})
by showing that its boundary is simple and convex (corollary~\ref{cor:boundary}),
and its interior is a ``flat sheet": it has no edge fold-overs (green edge) (lemma~\ref{lem:ef}).
We use a discrete version of the Poincar\'e-Hopf index theorem (lemma~\ref{lem:ph}) to prove that
an edge fold-over would create a ``wrinkle" (circled) somewhere in the triangulation (lemma~\ref{lem:index-1}),
which in turn would force some vertex (blue) to ``invade" a face (red) (lemma~\ref{lem:non-negative}),
breaking the face's empty circum-ball (grey, dotted) condition (proposition~\ref{prop:ECB}).}
\label{fig:outline}
\end{center}
\end{figure}
After establishing that a Voronoi diagram can be associated with an embedded planar primal graph
which can be dualized into a planar dual graph (section~\ref{sec:planar}),
the rest of the paper is concerned with the proof of our main claim (theorem~\ref{th:main}),
whose structure is outlined in figure~\ref{fig:outline}.
The proof of embeddability of the straight-edge dual is divided in two parts.
In the first part (section~\ref{sec:boundary}), we use the bounded anisotropy assumption (assumption~\ref{ass:BAA})
to show that the
\empty{topological} boundary of the straight-edge dual Delaunay triangulation
(the set of edges shared by only one face) coincides with the boundary of the convex hull of the sites,
and therefore is a simple, closed polygonal chain, a fact necessary for the second part of the proof to proceed.
Section~\ref{sec:boundary} is the more technical part of the proof; at its heart it is an application of Brouwer's fixed point theorem.
In section~\ref{sec:interior}, we use the theory of discrete one-forms~\cite{1form}
to show that the Delaunay triangulation has no fold-overs (is a ``flat sheet")
and is therefore a single-cover of the convex hull of $S$.
Note that these two results, along with the ECB property, mirror similar properties of ordinary Delaunay triangulations.
The above results can be particularized to a number of existing divergences and metrics.
We briefly discuss next a few of them, as well as simple conditions for assumption~\ref{ass:BAA} to hold
for some of them (with proofs in Appendix A).
\subsection{Bregman divergences}\label{sec:DF}
Given a strictly convex, everywhere differentiable function $F:\mathbb{R}^2\rightarrow\mathbb{R}$,
the Bregman divergence
\begin{equation}\label{eq:defDF}
D_F(p \parallel q) \equiv F(p) - F(q) - \langle p-q, \nabla F(q)\rangle
\end{equation}
is the (non-negative) difference between $F(p)$ and the first-order Taylor approximation of $F(p)$ around $q$
(the first order Lagrange remainder).
Bregman divergences are widely used in statistics and
include the Kullback-Leibler divergence.
By the (strict) convexity of $F$, and the definition of $D_F$ it
it is clear that, whenever $F$ is twice continuously differentiable,
$D_F$ is (strictly) convex in the first argument and continuously differentiable in the second.
From the definition of $D_F$, it is clear that Bregman Voronoi diagrams of the first kind are composed of regions
\[ \text{Vor}^1_{F,i}
= \{ p\in\mathbb{R}^2 : \langle p, \nabla F(s) - \nabla F(s_i)\rangle \le
F(s_i)-\langle s_i, \nabla F(s_i)\rangle +
F(s) - \langle s, \nabla F(s)\rangle , \forall s\in S \}, \]
which are intersections of half-spaces of the form $\{p\in\mathbb{R}^2 : \langle p, a\rangle \le b\}$.
Furthermore,
Bregman Voronoi diagrams of the first kind are simply power diagrams~\cite{Bregman},
and thus their dual Delaunay triangulations of the first kind are always embedded~\cite{powerdiag,DMG}.
On the other hand, Bregman diagrams of the \emph{second} kind can be shown to be affine diagrams
only in the dual (gradient) space~\cite{Bregman}.
In the original space, the cells $\text{Vor}^2_{F,i}$ are not simple intersections of half-spaces and, in general, they have curved boundaries.
Prior to this work, no guarantees concerning Bregman Delaunay triangulations of the second kind were available.
\begin{lemma}[Bounded anisotropy for Bregman divergences]\label{lem:DFgamma}
If $F\in\mathcal{C}^2$ and there is $\gamma > 0$ such that the Hessian of $F$ has ratio of eigenvalues bounded by $\lambda_{\text{min}}/\lambda_{\text{max}}\ge \gamma$,
then assumption~\ref{ass:BAA} holds.
\end{lemma}
\subsection{Quadratic divergences}\label{sec:DQ}
As is well known, the approximation efficiency of a
piecewise-linear function supported on a triangulation can be greatly improved by
adapting the shape and orientation of its elements to the target function~\cite{triangle,DAzevedo,DBLP:conf/imr/CanasG06}.
An effective way to construct such anisotropic triangulations is to dualize a Voronoi diagram
derived from an anisotropic divergence~\cite{LS,DW}.
By considering a $\mathcal{C}^1$ metric
(in coordinates: a function $Q:\mathbb{R}^2\rightarrow\mathbb{R}^{2\times 2}$ that is symmetric, positive definite),
we define the quadratic divergence as:
\begin{equation}\label{eq:defDQ}
D_Q(p \parallel q) \equiv \left[ (p-q)^t Q(q) (p-q) \right]^{1/2},
\end{equation}
which is clearly strictly convex in the first argument and continuously differentiable in the second.
Voronoi diagrams and Delaunay triangulations with respect to $D_Q$, of the first and seconds kinds, have been considered in the literature.
The diagram and the dual triangulation of the \emph{first kind} were proposed by Labelle and Shewchuk~\cite{LS},
while those of the second kind were discussed by Du and Wang~\cite{DW}.
While the work of Du and Wang does not provide theoretical guarantees,
that of Labelle and Shewchuk provides an algorithm that is guaranteed to output \emph{a} set of sites for which the
Voronoi diagram of the \emph{first kind} is orphan-free, and whose corresponding Delaunay triangulation is embedded.
\begin{lemma}[Bounded anisotropy for quadratic divergences]\label{lem:DQgamma}
If there is $\gamma > 0$ such that $Q$ has ratio of eigenvalues bounded by $\lambda_{\text{min}}/\lambda_{\text{max}}\ge \gamma$,
then assumption~\ref{ass:BAA} holds.
\end{lemma}
Note that the above condition on the bounded anisotropy of $Q$ may commonly hold in practice,
for instance if the metric is sampled on a compact domain and continuously extended to the plane by reusing sampled values only.
In the case of quadratic divergences, there already exists sufficient conditions to generate orphan-free Voronoi diagrams.
In particular, it has been shown that if $\sigma$ is a bound on a certain measure of variation of $Q$,
then any (asymmetric) $\epsilon$-net with respect to $D_Q$ that satisfies $\epsilon\sigma \le 0.098$
(corresponding to a roughly $10\%$ variation of eigenvalues between Voronoi-adjacent sites)
is guaranteed to be orphan-free~\cite{avd}.
\subsection{Normed spaces}\label{sec:Lp}
Our results also cover all normed spaces with a continuously differentiable, strictly convex norm,
including the $L_p$ spaces,
but excluding the cases $p=1$ and $p=\infty$.
\begin{lemma}[Bounded anisotropy for normed spaces
\label{lem:Lpgamma}
Distances derived from strictly convex $\mathcal{C}^1$ norms satisfy assumption~\ref{ass:BAA}.
\end{lemma}
\subsection{Csisz\'ar f-divergences}\label{sec:Df}
Given a convex real function $f$ with $f(1)=0$ and two measures $\rho,\mu$ over a probability space $\Omega$,
Csisz\'ar's f-divergence~\cite{CsiszarTutorial} is
\begin{equation}\label{eq:defDf}
D_f(\rho\parallel \mu) \equiv \int_{\Omega}d\mu\, f\left(\frac{d\rho}{d\mu}\right)
\end{equation}
where $\rho$ is absolutely continuous with respect to $\mu$,
and therefore has a Radon-Nikodym derivative $d\rho/d\mu$.
If $f$ is strictly convex, then the f-divergence is strictly convex in the first argument and continuously differentiable in the second
(in this case it is also jointly convex).
For instance, the strictly convex function $f:x\mapsto \left(\sqrt x - 1\right)^2$ generates the Hellinger distance.
F-divergences are functions of measures, and thus often in practice restricted to the probability simplex.
\begin{remark}The limitation of our work to two dimensions implies that results for f-divergences
are limited to probability measures supported on just three atoms.
Their applicability is thus somewhat limited, and are only included for completeness.
\end{remark}
\section{Primal Voronoi diagram and dual Delaunay triangulation}\label{sec:planar}
In this section we use the definition of Voronoi diagram (definition~\ref{def:VorI})
to construct an embedded simple planar
graph whose incidence relations match
those of the Voronoi diagram.
We then dualize this graph to obtain an embedded simple planar
graph
with vertices at the sites and curved edges.
Section~\ref{sec:dual} will then show that the dual graph is also embedded when replacing curved edges by straight segments.
Recall that we have assumed that not all sites are colinear (the colinear case is described in section~\ref{sec:summary}).
\subsection{Assumptions}\label{sec:assumptions}
We begin by making the following two technical assumptions.
\vspace*{0.1in}\noindent{\bf Path-connectedness.}
Assume that all connected components of Voronoi elements are also path-connected.
In fact, given the assumption below, as well as assumptions~\ref{ass:BAA} and~\ref{ass:EGA},
we only need to further assume that connected components of Voronoi \emph{edges} are path-connected.
Indeed, Voronoi regions are open and Voronoi vertices will be shown to be composed of isolated points,
and therefore their connected components are automatically path-connected~\cite[p.\ 158]{munkres2000topology}.
\vspace*{0.1in}\noindent{\bf Boundaries of Voronoi regions.}
Further assume that the boundary of bounded, simply-connected Voronoi regions are simple, closed (Jordan) curves.
For unbounded regions $U$, we assume that they can be first mapped
through a continuous transformation $T:U\rightarrow U'$
onto a bounded set $U'$, for instance through an appropriate M{\"o}bius transformation.
Bounded simply-connected sets whose boundary is a Jordan curve
are those that are uniformly connected \emph{im kleinen}~\cite{Moore1918}\footnote{
A space $M$ is uniformly connected \emph{im kleinen} if for every $\varepsilon>0$ there is $\delta_\varepsilon>0$
such that for every pair of points $p,q\in M$ with $\|p-q\|_2<\delta_\varepsilon$
there is a connected subset $V\subseteq M$ with $p,q\in V$ and $V\subseteq B_2(p;\varepsilon)$.
}.
\subsection{Properties of Voronoi elements}\label{sec:properties}
Before constructing an appropriate primal graph from the connectivity relations of the Voronoi diagram,
we first establish some relevant properties of the diagram's elements.
We say that Voronoi element $\text{Vor}_I$ is incident to Voronoi element $\text{Vor}_J$
(denoted $\text{Vor}_I\rightsquigarrow\text{Vor}_J$)
if their closures overlap
and $\overline{\text{Vor}_I}\cap\overline{\text{Vor}_J}\subseteq\overline{\text{Vor}_J}$.
From this incidence relation we build a primal Voronoi graph, whose dual is the Delaunay triangulation with respect to $D$.
Since ``planar graphs, and graphs embeddable on the sphere are one and the same''~\cite[p.\ 247]{bondy2008graph},
we consider incidence relations
on the Riemann sphere (by stereographically projecting the plane onto $\mathbb{S}^2$),
where the added vertex at infinity is defined to be incident to unbounded elements on the plane.
Geometric constructions will, however, typically be carried out on the plane for convenience.
\subsubsection{Incident elements}\label{sec:incidence}
\begin{figure}[htbp]
\centering
\includegraphics[width=2.5in]{incidence.pdf}
\caption{A portion of a Voronoi diagram, with highlighted incidence relations between Voronoi elements.
The incidence relation (definition~\ref{def:incidence}) forms a directed acyclic graph. }
\label{fig:incidence}
\end{figure}
Consider the following definition of incidence between Voronoi regions (or between connected components of Voronoi regions):
\begin{definition}\label{def:incidence}
Given $I,J\subseteq S$, we say that $\text{Vor}_I$ is incident to $\text{Vor}_J$ (written $\text{Vor}_I \rightsquigarrow \text{Vor}_J$) iff
$\overline{\text{Vor}_I}\cap\overline{\text{Vor}_J}\ne\phi$
\emph{and} $I\subset J$.
\end{definition}
\begin{remark}
By orphan-freedom, and lemma~\ref{lem:connectededges}, both Voronoi regions and edges are connected
(except for isolated edges, which are defined in section~\ref{sec:propedges}).
For simplicity, in the sequel we refer to connected components of Voronoi vertices simply as ``Voronoi vertices",
except for the statement of lemma~\ref{lem:vertexincidence}, which makes this distinction explicit.
\end{remark}
Note that this definition and the one in section~\ref{sec:properties} are equivalent since,
for distinct sets $I\ne J$, and by the continuity of $D$,
$\overline{\text{Vor}_I}\cap\overline{\text{Vor}_J}\subseteq\overline{\text{Vor}_J}$ implies $I\subset J$
(and viceversa).
Given the following substitution rules:
\begin{eqnarray*}
A,B\rightsquigarrow C ~&\Rightarrow~ A\rightsquigarrow C \text{ and } B\rightsquigarrow C \\
A \rightsquigarrow B,C ~&\Rightarrow~ A\rightsquigarrow B \text{ and } A\rightsquigarrow C,
\end{eqnarray*}
the following are the incidence relations depicted in figure~\ref{fig:incidence}:
\begin{eqnarray*}
\text{Vor}_i,\text{Vor}_j &\rightsquigarrow& \text{Vor}_{ij},\text{Vor}_{ijk} \\
\text{Vor}_j,\text{Vor}_k &\rightsquigarrow& \text{Vor}_{jk},\text{Vor}_{ijk} \\
\text{Vor}_k,\text{Vor}_i &\rightsquigarrow& \text{Vor}_{ki},\text{Vor}_{ijk} \\
\text{Vor}_{ij},\text{Vor}_{jk},\text{Vor}_{ki} &\rightsquigarrow& \text{Vor}_{ijk},
\end{eqnarray*}
where we often write $\text{Vor}_{ij}$ instead of $\text{Vor}_{\{i,j\}}$ for simplicity.
\begin{property}\label{prop:boundaryincidence}
All points in the boundary of a Voronoi element $\text{Vor}_I$ belong to either $\text{Vor}_I$,
or to an element
that $\text{Vor}_I$ is incident to.
\end{property}
\begin{proof}
Let $p\in\partial\text{Vor}_I$, and $J$ be the set of sites that $p$ is equidistant to.
Since $p\in\partial\text{Vor}_I$, by the continuity of $D$, $p$ is equidistant to all sites in $I$,
and therefore $I\subseteq J$.
The property follows from the definition of incidence.
\end{proof}
\begin{property}
From the properties of strict set containment,
it follows that the incidence relation $\rightsquigarrow$ forms a directed acyclic graph
(a cycle $\text{Vor}_I\rightsquigarrow\text{Vor}_J\rightsquigarrow\text{Vor}_K\rightsquigarrow\text{Vor}_I$ would imply $I\subset I$, a contradiction).
\end{property}
From property~\ref{prop:boundaryincidence} it follows that closed Voronoi elements
are those with zero out-degree in the incidence graph (e.g.\ $\text{Vor}_{ijk}$ in figure~\ref{fig:incidence}),
and that open Voronoi elements (i.e.\ Voronoi regions) are those with zero in-degree
(e.g.\ $\text{Vor}_i,\text{Vor}_j,\text{Vor}_k$ in figure~\ref{fig:incidence}).
\subsubsection{Properties of Voronoi vertices}
The main properties at Voronoi vertices are derived from the two assumptions in section~\ref{sec:setup}.
Assumptions~\ref{ass:BAA} and~\ref{ass:EGA} are useful when deriving properties of the vertex at infinity,
and bounded vertices (all other vertices), respectively.
Given the set negated gradients $g_1,\dots,g_m$ at a bounded vertex point (eq.~\ref{eq:neggrad}),
by assumption~\ref{ass:EGA} they are distinct vertices of their convex hull.
It is then possible to define ``outward" vectors $d_1,\dots,d_m$ (eq.~\ref{eq:dkdef})
such that eq.~\ref{eq:extremal1} holds.
This is because, for each $k=1,\dots,m$,
eq.~\ref{eq:extremal1} simply requires all gradients other than $g_k$
to be below the (red dotted) line orthogonal to $d_k$ passing through $g_k$
(as shown in fig.~\ref{fig:EGA.a} for $d_1$),
which is possible because $g_1,\dots,g_m$ are the distinct vertices of $\CH\{g_1,\dots,g_m\}$.
Figure~\ref{fig:EGA.b} shows that eq.~\ref{eq:extremal2} holds for the same reason as above.
Given two gradients that are adjacent vertices of $\CH\{g_1,\dots,g_m\}$
(for instance $g_1,g_2$),
eq.~\ref{eq:extremal2} (in this case with $k=1, k\oplus 1=2$)
is possible whenever all gradients different from $g_1,g_2$
are simultaneously below two lines, both passing through $g_1$ and orthogonal to $d_1$ and $d_2$
(the gray area).
This holds because the outward directions $d_k$ can be chosen to form an obtuse angle with
both segments $g_k,g_{k\oplus 1}$ and $g_k,g_{k\ominus 1}$.
The same argument applies to eq.~\ref{eq:extremal3}.
\begin{figure}[htbp]
\centering
\subfloat[]{\label{fig:EGA.a}\includegraphics[width=2.2in]{EGA.pdf}}\quad\quad\quad
\subfloat[]{\label{fig:EGA.b}\includegraphics[width=2.4in]{EGA2.pdf}}
\caption{Diagrams used in the proof of lemma~\ref{lem:vertexincidence}.
Assumption~\ref{ass:EGA} ensures that for all $k$ there is a vector $d_k$
with $\left<d_k,s_k-s_j\right> > 0, j\ne k$ (a),
and such that all intermediate direction vectors $d$ between $d_k$ and $d_{k\oplus 1}$
satisfy $\left<d,s_k-s_j\right> >0, j\ne k,k\oplus 1$ (blue lines)
and $\left<d,s_{k\oplus 1}-s_j\right> >0, j\ne k,k\oplus 1$ (red lines) (b). }
\end{figure}
\begin{lemma}[Incidence at Voronoi vertices]\label{lem:vertexincidence}
A Voronoi vertex $\text{Vor}_I$
is a collection of discrete points,
at each of which there is an ordered set of indices $i_1,\dots,i_m$
such that $I=\{i_i,\dots,i_m\}$ and the following incidence relations hold:
\begin{eqnarray*}
\text{Vor}_{i_1},\dots,\text{Vor}_{i_m} &\rightsquigarrow& \text{Vor}_I ~~~~~~~~~~~~~ \text{(region-vertex incidence)}\\
\text{Vor}_{\{i_1,i_2\}},\text{Vor}_{\{i_2,i_3\}},\dots,\text{Vor}_{\{i_m,i_1\}} &\rightsquigarrow& \text{Vor}_I ~~~~~~~~~~~~~ \text{(edge-vertex incidence)}.
\end{eqnarray*}
Additionally, if an edge $\text{Vor}_{jk}$ is incident to a vertex $\text{Vor}_I$, then $\text{Vor}_j,\text{Vor}_k \rightsquigarrow \text{Vor}_{jk}$.\\
If $\text{Vor}_I$ is the vertex at infinity ($\text{Vor}_\infty$), then $i_1,\dots,i_m$
are the indices of the sites in the boundary of the convex hull ${conv}\left\{\Sites\right\}$,
in either clockwise or counter-clockwise order.
\end{lemma}
\begin{proof}
\noindent{\bf [Bounded vertices, $\text{Vor}_I$]}.
Let $\text{Vor}_I$ be a Voronoi vertex not at the point at infinity and
$v$ be a point in $\text{Vor}_I$.
By the extremal gradient assumption (assumption~\ref{ass:EGA}),
the negated gradients
\begin{equation}\label{eq:neggrad}
g_k\equiv -\nabla_p \D{s_{i_k}}{p}\big|_v ~,~~ i_k \in I
\end{equation}
are distinct vertices of their convex hull.
Let $i_1,\dots,i_m$ be the indices in $I$ ordered (for instance clockwise)
around $\partial \CH\{g_1,\dots,g_m\}$,
as shown in figure~\ref{fig:EGA.a}.
Since $g_k$, with $k=1,\dots,m$ are distinct vertices of their convex hull,
it is easy to show that there are direction (unit) vectors $d_k$, with $k=1,\dots,m$,
such that for all $k$ it holds:
\begin{equation} \label{eq:extremal1}
\left< g_k - g_j, d_k \right> > 0 ~,~~ j\in\{1,\dots,m\}, j \ne k.
\end{equation}
For instance
\begin{equation}\label{eq:dkdef}
d_k\equiv \frac{(g_k-g_{k\oplus 1})+(g_k-g_{k\ominus 1})}{\|g_k-g_{k\oplus 1}+g_k-g_{k\ominus 1}\|}.
\end{equation}
By the multivariate version of Taylor's theorem~\cite[p.\ 68]{konigsberger2006analysis},
for each $k,j$, and $p\in\mathbb{R}^2$, we may write:
\[ -\D{s_k}{p} + \D{s_j}{p} = \left<g_k - g_j, p-v\right> + o\left(\|p-v\|\right). \]
For each $k=1,\dots,m$, and $j=1,\dots,m$ with $j\ne k$, let $p-v = \mu d_k$, with $\mu > 0$,
and let $\alpha_{k,j} \equiv \left<g_k - g_j, d_k\right> / 2 > 0$.
It then follows that:
\[ \left[-\D{s_k}{v+\mu d_k} + \D{s_j}{v+\mu d_k}\right]/\mu = 2\alpha_{k,j} + f(\mu), \]
where $\lim_{\mu\rightarrow 0} f(\mu)=0$.
Note that, crucially, $f$ depends on $\mu$ but not on the direction $d_k$.
Since $f(\mu)\rightarrow 0$ with $\mu\rightarrow 0$,
we can pick constants $\varepsilon_{k,j}>0$ sufficiently small so that
for all $\mu < \varepsilon_{k,j}$ it holds $|f(\mu)| < \alpha_{k,j}$,
and therefore $\left[-\D{s_{i_k}}{v+\mu d_k} + \D{s_{i_j}}{v+\mu d_k}\right]/\mu > \alpha_{k,j}$.
Let $\varepsilon >0$ be the minimum of all $\varepsilon_{k,j}$, with $j,k=1,\dots,m$, and $j\ne k$.
Since $v$ is strictly closest to sites $s_{i_1},\dots,s_{i_m}$,
let $\delta$ be small enough so all points $p\in\mathbb{R}^2$ with $\|p-v\|<\delta$
are closest only to sites in $s_{i_1},\dots,s_{i_m}$ (which is possible since $D$ is continuous).
Consider the set of points in a small circle of radius $0 < \mu < \min\{\delta,\varepsilon\}$ around $v$.
From the above, we have that at the point $v + \mu d_k$, it holds:
\[ \left[-\D{s_k}{v+\mu d_k} + \D{s_j}{v+\mu d_k}\right]/\mu > \alpha_{k,j} > 0 ~,~~ j\in\{1,\dots,m\}, j \ne k, \]
from which it follows that $v+\mu d_k$ is strictly closer to $s_{i_k}$ than to any other site.
Since this is true for all $k=1,\dots,m$ and for all sufficiently small $0<\mu<\min\{\delta,\varepsilon\}$, the incidence relations
\[ \text{Vor}_{i_1},\dots,\text{Vor}_{i_m} \rightsquigarrow \text{Vor}_I \]
follow.
Because $g_1,\dots,g_m$ are vertices of $\CH\{g_1,\dots,g_m\}$,
it is clear, as shown in figure~\ref{fig:EGA.b},
that for each $k=1,\dots,m$ there are constants $\beta_{k,j},\beta_{k\oplus 1,j} > 0$, with $j\ne k$ and $j\ne k\oplus 1$,
such that, for every unit vector $d_{k,k\oplus 1}$ intermediate between $d_k$ and $d_{k\oplus 1}$,
it holds:
\begin{align}
\label{eq:extremal2} \left< g_k - g_j, d_{k,k\oplus 1} \right> &> 2\beta_{k,j} > 0 ~,~~~~~ j\ne k, j \ne k\oplus 1 \\
\label{eq:extremal3} \left< g_{k\oplus 1} - g_j, d_{k,k\oplus 1} \right> &> 2\beta_{k \oplus 1,j} > 0 ~,~~ j\ne k, j \ne k\oplus 1.
\end{align}
Let $\xi_k > 0$ be small enough such that for all $0 < \mu < \xi$,
it holds $f(\mu) < \min\{\min_j \beta_{k,j}, \min_j \beta_{k\oplus 1,j}\}$.
Let $\xi\equiv \min_k \xi_k$,
then for all $0 < \mu < \min\{\delta,\varepsilon,\xi\}$,
and every point $v+\mu d_k$ it holds:
\begin{align*}
-\D{s_k}{v+\mu d_{k,k\oplus 1}} + \D{s_j}{v+\mu d_{k,k\oplus 1}} &> \beta_{k,j} > 0 ~,~~~~~ j\ne k, j \ne k\oplus 1 \\
-\D{s_{k\oplus 1}}{v+\mu d_{k,k\oplus 1}} + \D{s_j}{v+\mu d_{k,k\oplus 1}} &> \beta_{k\oplus 1,j} > 0 ~,~~~~~ j\ne k, j \ne k\oplus 1,
\end{align*}
and therefore $v+\mu d_{k,k\oplus 1}$ is closest to either $s_k,s_{k\oplus 1}$, or to both.
For each such $\mu$, and for each $k=1,\dots,m$,
by the intermediate value theorem,
there is a direction vector $d$ between $d_k,d_{k\oplus 1}$ such that $v+\mu d$
is in $\text{Vor}_{k,k\oplus 1}$.
Note that, by the above construction, for every such sufficiently small $\mu$,
$\text{Vor}_{k,k\oplus 1}$, with $k=1,\dots,m$, are the only Voronoi edges inside the ball of radius $\mu$ around $v$.
From this it directly follows that:
\begin{enumerate}
\item since all points $v+\mu d$, with unit vector $d$ and sufficiently small $\mu$
have been shown to be in a Voronoi region or edge,
$v$ is an isolated point of $\text{Vor}_I$;
since $v$ is a generic point of $\text{Vor}_I$, it follows that $\text{Vor}_I$ is composed of isolated points;
\item it holds $\text{Vor}_{\{i_1,i_2\}},\text{Vor}_{\{i_2,i_3\}},\dots,\text{Vor}_{\{i_m,i_1\}} \rightsquigarrow \text{Vor}_I $; and
\item if a Voronoi edge $\text{Vor}_{jk}$ is incident to $\text{Vor}_I$, then $\text{Vor}_j,\text{Vor}_k\rightsquigarrow \text{Vor}_{jk}$,
since the only edges incident to $\text{Vor}_I$ are $\text{Vor}_{i_k,i_{k\oplus 1}}$, with $k=1,\dots,m$.
\end{enumerate}
\vspace*{0.1in}\noindent{\bf [Vertex at infinity, $\text{Vor}_\infty$]}.
Incidence to the vertex at infinity is dealt with in section~\ref{sec:boundary},
where lemma~\ref{lem:VW} shows that the only unbounded elements
are of the form $\text{Vor}_I$ where all $i_k\in I$ are vertices of ${conv}\left\{\Sites\right\}$,
and lemmas~\ref{boundary_easy} and~\ref{lem:hard} show that,
if $s_{i_1},\dots,s_{i_m}$ are the vertices on the boundary of ${conv}\left\{\Sites\right\}$ (whether on an edge or vertex of $\partial{conv}\left\{\Sites\right\}$),
ordered around $\partial{conv}\left\{\Sites\right\}$,
then $\text{Vor}_{i_1},\dots,\text{Vor}_{i_m}$ and $\text{Vor}_{i_1,i_2},\dots,\text{Vor}_{i_m,i_1}$ are the only unbounded elements
(and therefore incident to $\text{Vor}_\infty$).
In this sense we can say that the vertex at infinity $\text{Vor}_\infty$ is the Voronoi vertex $\text{Vor}_{i_1,\dots,i_m}$.
The proofs in section~\ref{sec:boundary} show that points $p$ in any circle of sufficiently large radius are incident
only to sites in $s_{i_1},\dots,s_{i_m}$,
that $p$ cannot be incident to more than two sites simultaneously (lemma~\ref{lem:contrad}),
and therefore $p$ cannot belong to a Voronoi vertex,
and finally that $p$ can \emph{only} be simultaneously closest to two consecutive sites of the form $s_{i_k},s_{i_{k\oplus 1}}$
(page~\pageref{text:boundary}).
Note that the relevant proofs of section~\ref{text:boundary} use the bounded anisotropy assumption (assumption~\ref{ass:BAA}),
but do not use any result from this section.
\end{proof}
From the proof of lemma~\ref{lem:vertexincidence},
it is clear that
the bounded anisotropy assumption (assumption~\ref{ass:BAA})
is constructed so that lemma~\ref{lem:vertexincidence} holds for the vertex at infinity,
while the extremal gradient assumption (assumption~\ref{ass:EGA})
is meant to ensure that lemma~\ref{lem:vertexincidence} holds for regular (bounded) vertices.
\subsubsection{Properties of Voronoi edges}\label{sec:propedges}
We begin by considering (isolated) Voronoi edges that are bounded and not incident to any Voronoi vertex.
Since, as will be shown in lemma~\ref{lem:SCedges},
Voronoi edges are simply connected,
it is easy to see that for any Voronoi edge $\text{Vor}_{ij}$
that is not incident to any bounded Voronoi vertex,
it can only be $\text{Vor}_i\rightsquigarrow\text{Vor}_{ij}$ or $\text{Vor}_j\rightsquigarrow\text{Vor}_{ij}$,
and $\text{Vor}_{ij}$ cannot be involved in any other incidence relation.
To see this, first note that an isolated component of $\text{Vor}_{ij}$ has,
by definition, zero out-degree, and therefore it is closed.
Because $\text{Vor}_{ij}$ is not incident to the vertex at infinity, it is bounded.
Since $\text{Vor}_{ij}\rightsquigarrow\text{Vor}_{kl}$ implies
that their common boundary belongs to vertex $\text{Vor}_{ijkl}$ (where it may be $k=l$),
$\text{Vor}_{ij}$ is not incident to any Voronoi edge.
$\text{Vor}_{ij}$ cannot be incident to a region $\text{Vor}_k$ with $k\notin\{i,j\}$,
or else their common boundary would belong to vertex $\text{Vor}_{ijk}$.
Finally, we show that it cannot be
both $\text{Vor}_i\rightsquigarrow\text{Vor}_{ij}$ and $\text{Vor}_j\rightsquigarrow\text{Vor}_{ij}$.
Because $\text{Vor}_{ij}$ is closed, simply connected, and bounded,
by the continuity of $D$,
we can consider a sufficiently small $\varepsilon > 0$
such that every $\varepsilon$-offset of its outer boundary cannot be
closest to any site $s_k$ with $k\notin\{i,j\}$.
If $\text{Vor}_i,\text{Vor}_j\rightsquigarrow\text{Vor}_{ij}$, then
there must be $0 < \mu < \varepsilon$
such that the $\mu$-offset $\nu_\mu$ of $\text{Vor}_{ij}$'s outer boundary
has at least one point closest to $s_i$, and one point closest to $s_j$,
and therefore, by continuity of $D$, at least one point equally close to $s_i,s_j$.
Since all points in $\nu_\mu$ are closest to $s_i,s_j$ only, then $\nu_\mu$ has
been shown to have a point in $\text{Vor}_{ij}$,
contradicting the fact that $\nu_\mu$ is a $\mu$-offset of $\text{Vor}_{ij}$'s outer boundary,
and therefore outside $\text{Vor}_{ij}$.
Let $\text{Vor}_{ij}$ be an bounded isolated Voronoi edge
such that $\text{Vor}_i\rightsquigarrow\text{Vor}_{ij}$.
Because they are not incident to any Voronoi vertex,
bounded isolated edges will not be considered part of the primal Voronoi graph.
For simplicity, we consider all points of an isolated edge $\text{Vor}_{ij}$ to be part of its containing
Voronoi region (say $\text{Vor}_i$),
and therefore to be (by definition) strictly closer to $s_i$ than to any other site.
This is not just a simplification (which does not affect the final Voronoi graph),
but will allow us to prove that Voronoi regions are simply connected.
We begin by proving the following technical lemma.
\begin{figure}[htbp]
\centering
\subfloat[]{\label{fig:RSC.a}\includegraphics[width=2.5in]{SC1.pdf}}
\quad\quad\quad
\subfloat[]{\label{fig:RSC.b}\includegraphics[width=2.5in]{SC2.pdf}}
\caption{Diagrams used in the proof of lemma~\ref{lem:RSC}. }
\label{fig:RSC}
\end{figure}
\begin{lemma}\label{lem:RSC}
Let the boundary $\partial R$ of $R\subset\mathbb{R}^2$ be a simple, closed path,
and $\text{Vor}_I$ be a Voronoi element of an orphan-free diagram.
If $\partial R\subseteq\text{Vor}_I$,
then $R\subseteq\text{Vor}_I$.
\end{lemma}
\begin{proof}
Let $I=i_1,\dots,i_m$, and $\gamma\equiv\partial R$.
We begin by showing that $R$ does not contain any site $s_i$ whenever $m>1$ or $i\ne i_1$.
Let $m=1$, and $s_i\in R$ with $i\ne i_1$, as in figure~\ref{fig:RSC.a}.
Let $r$ be the ray starting from $s_i$ in the direction of $s_i-s_{i_1}$
(note that $s_{i_1}$ may be inside or outside $R$).
Since $r$ is unbounded and $R$ is bounded, then part of $r$ is outside $R$ and,
by the Jordan curve theorem, it must intersect $\gamma$ at some point $q_{i_1}$.
Since $\gamma\subset\text{Vor}_{i_1}$,
$q_{i_1}$ is closest to $s_{i_1}$, while $s_i$ is closest to $s_i$
(since $\D{s_i}{s_i}=0$ and $\D{\cdot}{s_i}$ is non-negative and convex).
By the continuity of $D$, there is an intermediate point $q_{i,i_1}$ between $s_i$ and $q_{i_1}$
that is equidistant to $s_{i_1}$ and $s_i$, contradicting lemma~\ref{lem:midpoint}.
Let $m>1$, and let $s_i$ be any site (figure~\ref{fig:RSC.b}).
Pick $j\ne i$ among $j\in\{i_1,\dots,i_m\}$,
which is always possible because $m>1$.
The argument is identical in this case, except that,
because $\gamma\subset\text{Vor}_I$, then $q_j\in\gamma$ is closest and equidistant to $\{i_1,\dots,i_m\}$,
and therefore closer to $s_i$ than to $s_j$, and the same argument holds.
\vspace*{0.05in}\noindent{\bf [Voronoi regions]}.
We now prove that no point $p\in R$ belongs to a Voronoi region $\text{Vor}_i$ if $m>1$ or $i\ne i_1$.
Let $p\in R$ belong to $\text{Vor}_i$, with $m>1$ or $i\ne i_1$, we show that this leads to a contradiction.
We first show that $\text{Vor}_i\subset R$.
Assume otherwise. Since $\text{Vor}_i$ is open and connected (by the orphan-freedom property), it is path connected.
Let $\Gamma\subset\text{Vor}_i$ be a simple path from $p$ to a point $q\in\text{Vor}_i$ outside $R$.
By the Jordan curve theorem, $\Gamma\subset\text{Vor}_i$ intersects $\gamma\subset\text{Vor}_I$,
which leads to a contradiction whenever $m>1$ or $i\ne i_1$.
Since $\text{Vor}_i\subset R$ and,
by lemma~\ref{lem:site},
$s_i\in\text{Vor}_i$, then $s_i\in R$, contradicting the fact that $R$ does not contain any site $s_i$ if $m>1$ or $i\ne i_1$.
\vspace*{0.05in}\noindent{\bf [Voronoi vertices]}.
If $R$ contains a point $p$ that belongs to a Voronoi vertex $\text{Vor}_J$ with $J\ne I$,
then $p$ must be in the interior of $R$, since its boundary $\gamma$ is in $\text{Vor}_I$.
By lemma~\ref{lem:vertexincidence}, $p$ is incident to $\text{Vor}_{j_1},\dots,\text{Vor}_{j_k}$,
where $J=j_1,\dots,j_k$ and $k\ge 3$.
Since $p$ is in the interior of $R$, then there are points
$p_{j_i},\dots,p_{j_k}\in R$ that belong to $\text{Vor}_{j_1},\dots,\text{Vor}_{j_k}$, respectively.
If $m>1$, then this contradicts the fact that $R$ does not have any point in a Voronoi region.
If $m=1$, since $k\ge 3$, then one of $j_1,\dots,j_k$ must be different from $i_1$,
contradicting the fact that $R$ does not have any point in a Voronoi region different from $\text{Vor}_{i_1}$.
\vspace*{0.05in}\noindent{\bf [Voronoi edges]}.
Let $\text{Vor}'_{ij}$ be a connected component of a Voronoi edge, with $\{i,j\}\ne I$.
If some point $p\in\text{Vor}'_{ij}$ is in $R$,
then $\text{Vor}'_{ij}\subset R$, or else since, by the assumption in section~\ref{sec:assumptions},
$\text{Vor}'_{ij}$ is path connected, there would be a path $\Gamma\subset\text{Vor}'_{ij}$ connecting
$p$ to a point of $\text{Vor}'_{ij}$ outside $R$.
By the Jordan curve theorem $\Gamma\subset\text{Vor}'_{ij}$ would intersect $\gamma\subset\text{Vor}_I$, a contradiction.
Since we have already discarded isolated Voronoi edges that are not incident to any Voronoi vertex,
a Voronoi edge is always incident to a Voronoi vertex and,
since $\text{Vor}'_{ij}$ is in the interior of $R$, then its incident Voronoi vertex is in $R$,
a contradiction.
Finally, since we have shown that there cannot be any Voronoi vertices, edges, or regions $\text{Vor}_J$
with $J\ne I$ in $R$, then it must be $R\subset\text{Vor}_I$.
\end{proof}
\begin{figure}[htbp]
\centering
\subfloat[]{\label{fig:connectededges.a}\includegraphics[width=2.7in]{conedges.pdf}}
\quad\quad\quad
\subfloat[]{\label{fig:connectededges.b}\includegraphics[width=2.7in]{Vinf.pdf}}
\caption{Diagrams used in the proof of lemmas~\ref{lem:connectededges} (a),
and~\ref{lem:val_ge2} (b).}
\end{figure}
\begin{lemma}\label{lem:connectededges}
Voronoi edges of an orphan-free diagram are connected.
\end{lemma}
\begin{proof}
Let $\text{Vor}^1_{ij},\text{Vor}^2_{ij}$ be two disconnected pieces of a Voronoi edge $\text{Vor}_{ij}$,
as shown in figure~\ref{fig:connectededges.a}.
Since we have discarded (bounded) isolated edges, we assume that
$\text{Vor}_{ij}$ is incident to at least one vertex, and therefore
by lemma~\ref{lem:vertexincidence},
it is $\text{Vor}_i,\text{Vor}_j\rightsquigarrow\text{Vor}^1_{ij}$ and $\text{Vor}_i,\text{Vor}_j\rightsquigarrow\text{Vor}^2_{ij}$.
Since $\text{Vor}_i,\text{Vor}_j$ are incident to both $\text{Vor}^1_{ij},\text{Vor}^2_{ij}$,
the boundaries of $\text{Vor}_i$ and $\text{Vor}_{ij}^1$ overlap
(and likewise $\partial\text{Vor}_i \cap \partial\text{Vor}_{ij}^2, \partial\text{Vor}_j \cap \partial\text{Vor}_{ij}^1, \partial\text{Vor}_j \cap \partial\text{Vor}_{ij}^2 \ne \phi$).
Let $p^1_i\in\partial\text{Vor}_i\cap\partial\text{Vor}^1_{ij}$
be a point in the common boundary between $\text{Vor}_i$ and $\text{Vor}^1_{ij}$,
and $p^1_j\in\partial\text{Vor}_j\cap\partial\text{Vor}^1_{ij}$ be a point in the common boundary between $\text{Vor}_j$ and $\text{Vor}^1_{ij}$,
and define the points $p^2_i,p^2_j$ analogously.
Since $\text{Vor}^1_{ij},\text{Vor}^2_{ij}$ are disjoint,
it holds $p^1_i\ne p^2_i$ and $p^1_j\ne p^2_j$,
and therefore by lemma~\ref{lem:regionpath}
there are non-crossing simple paths $\gamma^1_i,\gamma^2_i\subset\text{Vor}_i$
from $s_i\inS$ to $p^1_i,p^2_i$, respectively,
and non-crossing simple paths $\gamma^1_j,\gamma^2_j\subset\text{Vor}_j$
from $s_j\inS$ to $p_{j,1},p_{j,2}$, respectively.
Additionally, since by the assumption in section~\ref{sec:assumptions} $\text{Vor}^1_{ij},\text{Vor}^2_{ij}$ are path connected,
there are simple paths $\gamma^1_{ij}\subset\text{Vor}^1_{ij}$ and $\gamma^2_{ij}\subset\text{Vor}^2_{ij}$
connecting $p^1_i$ to $p^1_j$, and $p^2_i$ to $p^2_j$, respectively.
Let $\gamma^1$ be the concatenation of paths $\gamma^1_i,\gamma^1_{ij},\gamma^1_j$,
and $\gamma^2$ be the concatenation of paths $\gamma^2_i,\gamma^2_{ij},\gamma^2_j$.
By construction, and since $\text{Vor}^1_{ij},\text{Vor}^2_{ij}$ are disjoint,
the simple paths $\gamma^1,\gamma^2$ only meet at their endpoints $s_i,s_j$.
Let $\gamma$ be the simple closed curve resulting from concatenating $\gamma^1,\gamma^2$.
By the Jordan curve theorem, $\gamma$ divides the plane into an interior ($Int$) and exterior regions, bounded by $\gamma$.
We first show that $Int$ does not contain any sites (other than $s_i,s_j$).
\vspace*{0.1in}\noindent{\bf [$Int$ contains no sites]}.
We first divide $Int$ in three parts, as shown in figure~\ref{fig:connectededges.a}:
\begin{enumerate}
\item the region $U$ bounded by $\overline{s_i p^1_i}$, $\gamma^1_{ij}$, and $\overline{p^1_j s_j}$,
\item the region $L$ bounded by $\overline{s_i p^2_i}$, $\gamma^2_{ij}$, and $\overline{p^2_j s_j}$,
\item and $R\equiv Int \setminus \left(U \cup L\right)$.
\end{enumerate}
We begin by observing that if $w_{ij}\in \text{Vor}_{ij}$, then the triangle $\triangle{s_i w_{ij} s_j}$
cannot contain any site (other than $s_i,s_j$)
because 1) $w_{ij}$ is closest and equidistant to $s_i,s_j$,
and 2) the ball of the first kind $\theta_{w_{ij}}(s_i)$
centered at $w_{ij}$ with $s_i,s_j$ in its boundary (see table~\ref{table:notation})
is convex and therefore contains $\triangle{s_i w_{ij} s_j}$.
Since the sides of $\triangle{s_i w_{ij} s_j}$ are line segments, and $\theta_{w_{ij}}(s_i)$ is strictly convex,
the only points of $\triangle{s_i w_{ij} s_j}$ touching the boundary of $\partial\theta_{w_{ij}}(s_i)$
are $s_i,s_j$, and therefore a site at any other point in $\triangle{s_i w_{ij} s_j}$ would be strictly closer to $w_{ij}$ than $s_i,s_j$,
a contradiction.
Since $U$ can be written as the union of triangles with vertices $s_i,w_{ij},s_j$ with $w_{ij}\in\gamma^1_{ij}\subset\text{Vor}^1_{ij}\subset\text{Vor}_{ij}$,
then $U$ does not contain any site other than $s_i,s_j$.
An analogous argument proves that $L$ does not contain any site other than $s_i,s_j$.
We split the remaining region $R$ into four parts $R_{1,i},R_{1,j},R_{2,i},R_{2,j}$.
Let $R_{1,i}$ be the part of $R$ bounded by the segment $\overline{s_i p^1_i}$ and the curve $\gamma^1_i$.
Let $R'_{1,i} \equiv \displaystyle{\cup_{r\in \gamma^1_i} \overline{s_i r}}$ be the union of segments connecting $s_i$ to points in $\gamma^1_i$.
Clearly, it is $R_{1,i}\subset R'_{1,i}$. We show that $R'_{1,i}$ cannot contain any site other than $s_i$,
and thus the same is true of $R_{1,i}$.
Let $z\in R'_{1,i}$ be a site, and let $r\in\gamma^1$ be the point such that $z\in\overline{s_i r}$.
Because $r\in\gamma^1_{i}\subset\overline{\text{Vor}_i}$, $r$ is closest and equidistant to $s_i$ (and possibly also to $s_j$),
that is: $\D{s_i}{r} \le \D{s_k}{r}$ for all $k=1,\dots,n$.
Since $z\in\overline{s_i r}$ and $z\ne s_i$, we can write $z=\lambda s_i + (1-\lambda) r$, with $0 \le \lambda < 1$,
and therefore by the strict convexity of $\D{\cdot}{r}$ it holds:
\[ \D{z}{r} = \D{\lambda s_i + (1-\lambda) r}{r} < \lambda\D{s_i}{r} + (1-\lambda)\D{r}{r} = \lambda\D{s_i}{r} < \D{s_i}{r}, \]
where the last equality follows from $\D{r}{r}=0$, and the last inequality follows from the non-negativity of $D$.
This shows that the site $z$ is \emph{strictly} closer to $r$ than $s_i$, a contradiction.
Therefore there are no sites in $R'_{1,i}$, and thus no sites in $R_{1,i}\subseteq R'_{1,i}$ either.
Applying an identical argument to $R_{1,j},R_{2,i},R_{2,j}$ shows that $R$ cannot contain any sites other than $s_i,s_j$.
\vspace*{0.05in}\noindent{\bf [Points in $Int$ can only be closest to $s_i$ and/or $s_j$]}.
We begin by showing that there is no point $p\in Int$ that is
\emph{strictly} closer to a site $s_k\notin \{s_i,s_j\}$ than to any other site ($p\in\text{Vor}_k$).
If $p\in Int$ is closest to $s_k\notin\{s_i,s_j\}$, then we first show that $\text{Vor}_k$ is wholly contained in $Int$.
Assume otherwise, and pick a point $q\in\text{Vor}_k$ outside $Int$.
Since Voronoi regions are path-connected, let $\Gamma_{pq}\subset\text{Vor}_k$ be a path connecting $p,q$.
By the Jordan curve theorem, $\Gamma_{pq}$
crosses the boundary $\gamma\subset \text{Vor}_i\cup \text{Vor}_j\cup \text{Vor}_{ij}$,
contradicting the fact that $\Gamma_{pq}\subset\text{Vor}_k$.
Since $\text{Vor}_k$ is completely inside $Int$ then, by lemma~\ref{lem:site}, it is $s_k\in\text{Vor}_k\subset Int$,
contradicting the fact the $Int$ contains no sites other than $s_i,s_j$,
and therefore $\text{Vor}_k\cap Int=\phi$ with $k\notin\{i,j\}$.
We now show that no point $p\in Int$ can be closest to $s_k\notin\{s_i,s_j\}$, even if it is also simultaneously closest to
$s_i$ and/or $s_j$.
Since $p$ is closest to $s_k$,
and the boundary of $Int$ is $\gamma\subset \text{Vor}_i\cup\text{Vor}_j\cup\text{Vor}_{ij}$, then $p$ belongs to the interior of $Int$.
By definition, $p$ belongs to a Voronoi edge or vertex. If it belongs to a Voronoi vertex and is closest to $s_k\inS$ then,
by lemma~\ref{lem:vertexincidence}, and since Voronoi vertices are composed of isolated points,
$p$ is incident to $\text{Vor}_k$, a contradiction since $\text{Vor}_k\cap Int=\phi$ whenever $k\notin\{i,j\}$.
Therefore $Int$ does not contain any Voronoi vertices.
Finally, we show that no point $p\in Int$ can be closest to a site $s_k\notin\{s_i,s_j\}$
and belong to a Voronoi edge $\text{Vor}_E$.
Since $p$ is in the interior of $Int$, the connected component $\text{Vor}'_E$ of $\text{Vor}_E$ that $p$ belongs to
must be fully contained in $Int$,
or else by the Jordan curve theorem $\text{Vor}'_E$ would be separated by the boundary $\gamma\subset \text{Vor}_i\cup\text{Vor}_j\cup\text{Vor}_{ij}$ of $Int$.
Since we have discarded connected components of Voronoi edges not incident to any Voronoi vertex,
then $\text{Vor}'_E$ is incident to some vertex $\text{Vor}_I$.
Since $\text{Vor}'_E$ is in the interior of $Int$, then $\text{Vor}_I$ must be contained in $Int$.
As we have shown above, $Int$ does not contain any Voronoi vertex, and therefore $p$ cannot be closest to $s_k\notin\{s_i,s_j\}$.
\vspace*{0.1in}\noindent{\bf [$\text{Vor}_{ij}$ is connected]}.
Finally, we show that there is a path in $\text{Vor}_{ij}$ connecting $\text{Vor}^1_{ij}$ to $\text{Vor}^2_{ij}$,
and therefore $\text{Vor}_{ij}$ is connected.
Recall that all points in $Int$ can only be closest to $s_i$ and/or $s_j$,
that $\gamma^1,\gamma^2$ are simple paths from $s_i$ to $s_j$,
and that, by construction, they do not meet except at their endpoints.
Clearly, $\gamma^1,\gamma^2$ are path homotopic~\cite[p.\ 323]{munkres2000topology},
for instance via the straight-line homotopy.
We begin by constructing a path homotopy $F$ between $\gamma^1$ and $\gamma^2$
(a continuous function $F:[0,1]\times [0,1]\rightarrow \mathbb{R}^2$
such that $F(\cdot,0)=\gamma^1(\cdot)$ and $F(\cdot,1)=\gamma^2(\cdot)$)
contained in $Int$.
Since $\gamma$ is a Jordan curve, and $Int$ is simply connected, by Carath\'eodory's theorem~\cite{conformal},
there is a homeomorphism $h$ from $\overline{Int}$ to the closed unit disk $D_2$ that maps $\gamma$ to the unit circle.
Since $\gamma^1,\gamma^2\subset\gamma$ and $D_2$ is convex, the straight-line homotopy $F'$
between $h(\gamma^1)$ and $h(\gamma^2)$ is contained in $D_2$.
We can now inversely map this homotopy through $h^{-1}$ to obtain a path homotopy $F=h^{-1}\circ F'$
between $\gamma^1$ and $\gamma^2$ which is contained in $Int$
(i.e.\ $F(\cdot,\alpha)\subset\overline{Int}$ with $0\le\alpha\le 1$).
Since every path $F(\cdot,\alpha)$ with $0\le\alpha\le 1$ starts at $s_i$ and ends at $s_j$,
and $D$ is continuous, there is $0< t_\alpha < 1$ such that $F(t_\alpha,\alpha)\in\overline{Int}$ is equidistant to $s_i,s_j$.
Since we have shown above that all points in $\overline{Int}$ are closest to $s_i$ and/or $s_j$,
then $F(t_\alpha,\alpha)\in\text{Vor}_{ij}$ for $0\le\alpha\le 1$.
By the continuity of $D$ and $F$, is it possible to choose $t_\alpha$ to be continuous with $\alpha$,
and such that the path $\Phi:[0,1]\rightarrow\mathbb{R}^2$
with $\Phi(\alpha)=F(t_\alpha,\alpha)$ is $\Phi([0,1])\subset\overline{Int}\cap\text{Vor}_{ij}$.
Since the path $\Phi$ is defined to start at $\text{Vor}^1_{ij}$ and end at $\text{Vor}^2_{ij}$, then
$\text{Vor}^1_{ij}$ and $\text{Vor}^2_{ij}$ are connected,
and therefore $\text{Vor}_{ij}$ must be connected.
\end{proof}
\begin{lemma}\label{lem:SCedges}
Voronoi edges of orphan-free diagrams are simply connected.
\end{lemma}
\begin{proof}
Recall that, by the assumption in section~\ref{sec:assumptions},
connected Voronoi edges are also path connected.
Let $\text{Vor}_{ij}$ be a Voronoi edge, and $\gamma\subset\text{Vor}_{ij}$ be a simple path not contractible to a point.
By the Jordan curve theorem, $\gamma$ divides the plane into an exterior (unbounded),
and an interior (bounded) region $R$.
By lemma~\ref{lem:RSC}, $R\subset\text{Vor}_{ij}$, and therefore $\gamma$ is contractible to a point.
\end{proof}
\subsubsection{Properties of Voronoi regions}
\begin{lemma}\label{lem:regionSC}
Voronoi regions of orphan-free diagrams are simply connected.
\end{lemma}
\begin{proof}
Let $\text{Vor}_i$ be a Vornoi region, which must be connected since the diagram is orphan-free.
Since $\text{Vor}_i$ is open, it is path connected~\cite[p.\ 158]{munkres2000topology}.
Assume that $\text{Vor}_i$ is not simply connected,
and therefore has a closed simple path $\gamma\subset\text{Vor}_i$ that is not contractible to a point.
By the Jordan curve theorem the path $\gamma$ separates the plane into an exterior and an interior region $R$.
By lemma~\ref{lem:RSC}, $R\subset\text{Vor}_{i}$, and therefore $\gamma$ is contractible to a point.
\end{proof}
\begin{lemma}\label{lem:regionpath}
For every Voronoi region $\text{Vor}_i$ of an orphan-free Voronoi diagram,
there is a collection of simple paths
connecting the site $s_i$ to each point in the boundary of $\text{Vor}_i$,
such that:
\begin{enumerate}
\item all paths are contained in $\overline{\text{Vor}_i}$,
\item paths intersect the boundary $\partial\text{Vor}_i$ only at the final endpoint, and
\item two paths meet only at the starting point $s_i$.
\end{enumerate}
\end{lemma}
\begin{proof}
By the assumption in section~\ref{sec:assumptions}, the boundary of Voronoi regions are simple closed paths.
Since a Voronoi region $\text{Vor}_i$ is also simply connected (lemma~\ref{lem:regionSC}), we may use Carath\'eodory's theorem~\cite{conformal}
to map $\overline{\text{Vor}_i}$ to the closed unit disk $D_2$ through a homeomorphism $h$
that maps the boundary $\partial\text{Vor}_i$ to the unit circle.
Since, by lemma~\ref{lem:site}, $s_i$ is an interior point of $\text{Vor}_i$,
then $s'_i\equiv h(s_i)$ is an interior point of $D_2$.
We now simply construct a set of straight paths
from $s'_i$ to each point in the unit circle.
These paths are contained in $D_2$, and meet only at the starting point.
We map them back through $h^{-1}$ to obtain the desired set of paths.
\end{proof}
\subsection{Voronoi edges are incident to two and only two Voronoi vertices}
\begin{lemma}\label{lem:val_ge2}
No Voronoi edge is incident to just one Voronoi vertex.
\end{lemma}
\begin{proof}
Let $\text{Vor}_{ij}$ be a Voronoi edge incident to just one Voronoi vertex $\text{Vor}_I$.
By lemma~\ref{lem:vertexincidence}, it is $\text{Vor}_i\rightsquigarrow\text{Vor}_{ij}$,
and therefore $\text{Vor}_{ij}$ has a common boundary with $\text{Vor}_i$.
Recall from property~\ref{prop:boundaryincidence} that the boundary $\partial\text{Vor}_i$ belongs
to Voronoi edges and vertices to which $\text{Vor}_i$ is incident.
Since, by lemma~\ref{lem:vertexincidence}, Voronoi vertices are isolated points,
and two Voronoi edges $\text{Vor}_{ij},\text{Vor}_{kl}$ can only meet at a Voronoi vertex $\text{Vor}_I$ (with $\{i,j,k,l\}\subset I$),
we can enumerate an alternating sequence of Voronoi edges and vertices
$\left[\dots,\text{Vor}_{ij},\text{Vor}_I,\text{Vor}_{kl},\text{Vor}_K,\dots\right]$ in clockwise order around $\partial\text{Vor}_i$,
in which every edge is incident to the previous and next vertices in the sequence.
Therefore, a Voronoi edge can only be incident to one Voronoi vertex if the sequence is $\left[\text{Vor}_{ij},\text{Vor}_I\right]$.
If $\text{Vor}_I$ is not the vertex at infinity, then we can show that the above is not possible
with an argument identical to the proof of lemma~\ref{lem:RSC} (figure~\ref{fig:RSC}).
Note that $\text{Vor}_{ij}\rightsquigarrow\text{Vor}_I$ implies $\{i,j\}\subset I$,
and therefore all points in $\partial\text{Vor}_i$ are equidistant to $s_i,s_j$.
Let $\gamma\equiv\partial\text{Vor}_i$, and consider the ray $r$ from $s_i$ in the direction $s_i-s_j$
which, since $r$ is unbounded and $\text{Vor}_i$ is bounded (since it is not incident to $\text{Vor}_\infty$),
it must cross $\gamma$ at some point $q$.
Since $q\in\gamma$, $q$ is equidistant to $s_i,s_j$, contradicting lemma~\ref{lem:midpoint}.
If $\text{Vor}_\infty$ is the vertex at infinity, then $\text{Vor}_{ij}$ is not incident to any Voronoi vertex, and is unbounded.
Therefore, $\text{Vor}_{ij}$ does not cross any Voronoi edge,
or else $\text{Vor}_{ij}$ would be incident to their intersection point (a Voronoi vertex).
Recall from lemma~\ref{lem:midpoint}
that $\text{Vor}_{ij}$ can never intersect the supporting line $L_{ij}$ of $s_i,s_j$ outside the segment $\overline{s_i,s_j}$.
Let $L^i_{ij}$ ($L^j_{ij}$) be the ray starting at $s_i$ ($s_j$) with direction $s_i-s_j$ ($s_j-s_i$),
as shown in figure~\ref{fig:connectededges.b}.
It can be easily shown that every point in $L^i_{ij}$ ($L^j_{ij}$) is strictly closer to $s_i$ ($s_j$) than to $s_j$ ($s_i$).
Since, regardless of the choice of origin,
\emph{every} origin-centered circle $C(\sigma)$ of sufficiently large radius $\sigma$ intersects $L_{ij}$ at
exactly one point $p_i$ in $L^i_{ij}$, and one point $p_j$ in $L^j_{ij}$,
the following holds.
Let $L_{ij}$ divide $\mathbb{R}^2$ into two half spaces $H^{+}_{ij},H^{-}_{ij}$,
and let $C^{+}(\sigma)\equiv C(\sigma)\cap H^{+}_{ij}$ and
$C^{-}(\sigma)\equiv C(\sigma)\cap H^{-}_{ij}$.
Since $p_i$ ($p_j$) is closer to $s_i$ ($s_j$) than to $s_j$ ($s_i$),
and $p_i,p_j$ are the endpoints of $C^{+}(\sigma), C^{-}(\sigma)$,
by the continuity of $D$,
there are points $p^{+}_{ij}\in C^{+}_{ij}$ and $p^{-}_{ij}\in C^{-}_{ij}$ equidistant to $s_i,s_j$.
Since $\text{Vor}_{ij}$ does not intersect any Voronoi element, then $p^{+}_{ij},p^{-}_{ij}$ are also closest to $s_i,s_j$.
Because this holds for all sufficiently large $\sigma$,
then both $\text{Vor}_{ij}\cap H^{+}_{ij}$ and $\text{Vor}_{ij}\cap H^{-}_{ij}$ are unbounded,
contradicting lemma~\ref{lem:contrad},
which states that every point $p^{-}_{ij}\in H^{-}_{ij}$ that is sufficiently far from the origin
and equidistant to $s_i,s_j$ (and therefore its closest point in $L_{ij}$ lies in $\overline{s_i,s_j}$)
is closer to a site in $S\setminus\{s_i,s_j\}$ than to $s_i,s_j$.
\end{proof}
\begin{figure}[htbp]
\centering
\subfloat[]{\label{fig:closureedge.a}\includegraphics[width=2.7in]{W.pdf}}\quad\quad
\subfloat[]{\label{fig:closureedge.b}\includegraphics[width=2.7in]{closureedge.pdf}}
\caption{From assumption~\ref{ass:EGA}, it follows that every Voronoi edge $\text{Vor}_{ij}$,
in the close vicinity of a Voronoi vertex can be written as the graph of a function $f(x)$ with an endpoint at $v$ (a).
Figure (b) shows a hypothetical Voronoi edge $\text{Vor}_{ij}$ that breaks assumption~\ref{ass:EGA},
for which lemma~\ref{lem:closureedge} does not hold. }
\label{fig:closureedge}
\end{figure}
\begin{lemma}\label{lem:closureedge}
Let $\text{Vor}_{ij}$ be a Voronoi edge.
For every $r\in\text{Vor}_{ij}$ and $q\in\overline{\text{Vor}_{ij}}$ there is a simple path $\gamma:[0,1]\rightarrow\overline{\text{Vor}_{ij}}$
such that $\gamma(0)= r$, $\gamma(1)= q$, and $\gamma((0,1))\subset\text{Vor}_{ij}$.
\end{lemma}
\begin{proof}
\noindent{{\bf [Case $q\in\text{Vor}_{ij}$]}}.
Recall that connected components Voronoi edges are assumed to be path-connected (section~\ref{sec:assumptions}).
Since Voronoi edges are connected (lemma~\ref{lem:connectededges}), they are path-connected.
Therefore, if $q\in\text{Vor}_{ij}$, there is always a path $\gamma:[0,1]\rightarrow\text{Vor}_{ij}$ connecting $r,q$. \\
\noindent{{\bf [Case $q\in\partial\text{Vor}_{ij}\setminus\text{Vor}_{ij}$]}}.
In this case, by property~\ref{prop:boundaryincidence}, $q$ must belong to a Voronoi element of higher order than $\text{Vor}_{ij}$
(a Voronoi vertex $\text{Vor}_I$), to which $\text{Vor}_{ij}$ is incident (with $\text{Vor}_{ij}\rightsquigarrow\text{Vor}_I$).
Since, by lemma~\ref{lem:vertexincidence}, Voronoi vertices are composed of isolated points,
then $q$ is a connected component of $\text{Vor}_I$ (possibly the vertex at infinity).
Consider separately whether $q$ is the vertex at infinity. \\
\noindent{{\bf [Case $q\in\partial\text{Vor}_{ij}\setminus\text{Vor}_{ij}$ and $q$ is not the vertex at infinity]}}.
Recall that the proof of lemma~\ref{lem:vertexincidence}
defines an ordering of $I=i_1,\dots,i_m$, and a set of associated direction vectors $d_1,\dots,d_m$.
Let $g_k=\nabla_p\D{s_i}{p}\big|_{p=q}$, with $k=1,\dots,m$,
and let $d_{ij}$ be the unit vector orthogonal to $g_i-g_j$
in the direction outgoing from $\CH\{-g_1,\dots,-g_m\}$
(which exists since, by assumption~\ref{ass:EGA}, it is $g_i\neq g_j$).
We assume, without loss of generality, that the coordinate representation of $d_{ij}$
is $[\left(d_{ij}\right)_x, \left(d_{ij}\right)_y] = [1,0]$.
Since $D\in\mathcal{C}^1$ and
$g_i\ne g_j$, by the implicit function theorem,
there is an open $L_2$ ball $B_2(q;\xi)$ around $q$ in which the implicit equation $ \D{s_i}{p} = \D{s_j}{p} $
can be written as $y=f(x)$, with $f'(0)=0$,
as shown in figure~\ref{fig:closureedge.a}.
Since $\text{Vor}_{ij}$ is incident to $\text{Vor}_I$ at $q$,
there is $k\in\{1,\dots,m\}$ such that $i=i_k, j=i_{k\oplus 1}$.
Choose $0 < \mu < \xi$ to be sufficiently small for the conditions of the proof of lemma~\ref{lem:vertexincidence} to apply
(in particular $\mu < \min\{\delta,\varepsilon,\xi\}$, as defined in the proof).
Let $W$ be a circular wedge contained in the $L_2$ ball $B_2(q;\mu)$,
and bounded by the rays $q+\mu d_k$ and $q+\mu d_{k\oplus 1}$ which,
aside from $q$, only contains points strictly closer to $\{s_i,s_j\}$ than to all other sites.
From the definition of $d_k,d_{k\oplus 1}$ it is clear that the segment $\overline{q, q+\mu d_{ij}}$
is contained in $W$.
Since $\mu<\xi$, and inside $W$ all points with the exception of $q$ are closest only to $s_i,s_j$,
the implicit equation $ \D{s_i}{p} = \D{s_j}{p} $ represents the set of points in $W\cap \text{Vor}_{ij}$.
Since $ \D{s_i}{p} = \D{s_j}{p} $ can be written in coordinates as $y=f(x)$ inside $W$,
it is clear that, inside $W$, $\text{Vor}_{ij}$ is a simple curve,
and that this is the only part of $\text{Vor}_{ij}$ incident to $q$.
Given $r\in\text{Vor}_{ij}$, find any point $v\in W\cap\text{Vor}_{ij}$ that is closer to $q$ than $r$.
Because $v\in\text{Vor}_{ij}$, there is a simple path $\gamma_1\subset\text{Vor}_{ij}$ connecting $r$ to $v$ and,
because $v$ is in $W$, there is also a simple path
$\gamma_2\subset\text{Vor}_{ij}$ from $v$ to $q$
(part of the curve $y=f(x)$ of figure~\ref{fig:closureedge}).
Finally, because $v$ is closer to $q$ than $r$ is,
the paths $\gamma_1$ and $\gamma_2$ do not cross, and therefore the
concatenation of $\gamma_1$ and $\gamma_2$
meets the requirements of the lemma.
\noindent{{\bf [Case $q\in\partial\text{Vor}_{ij}\setminus\text{Vor}_{ij}$ and $q$ is the vertex at infinity]}}.
Since $\text{Vor}_{ij}\rightsquigarrow\text{Vor}_{\infty}$ then, by definition, $\text{Vor}_{ij}$ is unbounded.
Let $r_0\equiv r$ and, for each $k\in\mathbb{N}$, let $r_k\in\text{Vor}_{ij}$ be at distance $\|r-r_k\|_2=k$.
One can always find such a sequence of points because $\text{Vor}_{ij}$ is unbounded and path-connected
(if there is no $r_k\in\text{Vor}_{ij}$ at distance $\|r-r_k\|_2=k$ then the circle with center at $r$ and radius $k$ would disconnect $\text{Vor}_{ij}$).
Let $\gamma'_k:[0,1]\rightarrow\text{Vor}_{ij}$ be paths connecting $r_{k-1}$ to $r_k$,
and $\gamma':\mathbb{R}^+\rightarrow\text{Vor}_{ij}$ be the concatenation of $\gamma'_1,\gamma'_2,\dots$,
where $\gamma'(k+t)\equiv\gamma'_k(t)$, with $k\in\mathbb{N}$ and $t\in[0,1]$.
Define $\gamma:[0,1]\rightarrow\text{Vor}_{ij}\cup\text{Vor}_\infty$ as $\gamma(t) \equiv \gamma'(1/(1-t))$.
Consider $\gamma$ on the Riemann sphere, transformed through a stereographic projection.
Since $\gamma'$ is continuous and $\gamma$ has an accumulation point at the point at infinity (north pole on the sphere),
it is continuous on the sphere.
If $\gamma$ is not simple, it can be appropriately cut and reparametrized until it is
(i.e.\ by tracing the path and, upon arrival to a point $c$ where the path crosses itself,
cutting out the next portion up to the highest $t$ for which $\gamma(t)=c$,
and proceeding this way to the end of the path).
\end{proof}
Note that for lemma~\ref{lem:closureedge} to hold it is crucial that edges $\text{Vor}_{ij}$ are incident to vertices $\text{Vor}_I$
as a curve arriving at $v\in\text{Vor}_I$ from a single direction,
as illustrated in figure~\ref{fig:closureedge.a}.
To see that assumption~\ref{ass:EGA} is required,
consider figure~\ref{fig:closureedge.b},
which depicts an edge $\text{Vor}_{ij}$ incident to two vertices $\text{Vor}_I,\text{Vor}_J$
which do not satisfy assumption~\ref{ass:EGA},
in which \emph{every} path connecting the two disks passes through either $\text{Vor}_I$ or $\text{Vor}_J$,
and therefore for which lemma~\ref{lem:closureedge} does not hold.
\begin{figure}[htbp]
\centering
\subfloat[]{\label{fig:tree.a}\includegraphics[width=3.0in]{tree1.pdf}}
\quad
\subfloat[]{\label{fig:tree.b}\includegraphics[width=3.0in]{tree2.pdf}}
\caption{The construction of a tree (blue) inside an edge $\text{Vor}_{ij}$ (green region), with root $r$ and leafs
at its incident Voronoi vertices $\text{Vor}_{I_1},\dots,\text{Vor}_{I_m}$.}
\label{fig:tree}
\end{figure}
\begin{lemma}\label{lem:tree}
In an orphan-free diagram,
for every Voronoi edge $\text{Vor}_{ij}$ that is incident to Voronoi vertices $\text{Vor}_{I_1},\dots,\text{Vor}_{I_m}$,
there is an embedded tree graph in $\overline{\text{Vor}_{ij}}$ whose leafs are $\text{Vor}_{I_1},\dots,\text{Vor}_{I_m}$.
\end{lemma}
\begin{proof}
Unless otherwise specified, we assume in this proof that all paths are simple, contained in $\overline{\text{Vor}_{ij}}$,
parametrized over the unit interval $[0,1]$,
and that, using lemma~\ref{lem:closureedge},
there is a path connecting any two points in $\overline{\text{Vor}_{ij}}$ that does not intersect a Voronoi vertex
(expect perhaps at the endpoints).
We use throughout the fact that Voronoi edges are path connected
(lemma~\ref{lem:SCedges} and section~\ref{sec:assumptions}).
If $m=1$, pick a point $r\in\text{Vor}_{ij}$ as root and,
using lemma~\ref{lem:regionpath},
consider a simple path $\gamma_{r,1}\subset\overline{\text{Vor}_{ij}}$ connecting $r$ to $\text{Vor}_{I_1}$,
then the tree with vertex set $V=\{r,\text{Vor}_{I_1}\}$, and edge set $E=\{\gamma_{r,1}\}$ meets the requirements of the lemma.
For each $k\ge 2$, assume that there is an embedded tree graph $T_{k-1}\subset\overline{\text{Vor}_{ij}}$
with $\text{Vor}_{I_1},\dots,\text{Vor}_{I_{k-1}}$ as leafs.
We construct a new embedded tree $T_k$ as follows (figure~\ref{fig:tree}).
Let $r\in\text{Vor}_{ij}$ be the root of $T_{k-1}$,
and let $\gamma$ be a simple path connecting $r$ to $\text{Vor}_{I_k}$ which,
making use of lemma~\ref{lem:closureedge}, is chosen such that it does not intersect
any Voronoi vertex (other than the final endpoint).
Let
\[ t_k \equiv \max \{t\in [0,1] ~:~ \gamma(t)\in T_{k-1} \}, \]
which always exists because $T_{k-1}$ is closed
and $\gamma(0)=r\in T_{k-1}$.
Let $v_k\equiv \gamma(t_k)$ be the ``last" point along $\gamma$ that belongs to $T_{k-1}$.
Because $\gamma(1)=\text{Vor}_{I_k}\notin T_{k-1}$
then it must be $t_k <1$.
Additionally, $v_k$ cannot be a Voronoi vertex,
since $\gamma$ doesn't intersect Voronoi vertices except at the final endpoint $\text{Vor}_{I_k}$.
Let $\gamma_k$ be the path $\{\gamma(t) ~:~ t\in [t_k,1]\}$,
that is, the part of $\gamma$ from $v_k$ to $\text{Vor}_{I_k}$.
We construct a new tree graph $T_k\subset\overline{\text{Vor}_{ij}}$ as follows.
Begin by setting $T_k$ equal to $T_{k-1}$.
We then insert a new vertex $\text{Vor}_{I_k}$ into $T_k$.
Next, we proceed differently depending on whether $v_k\in T_{k-1}$
is a vertex, or it belongs to an edge of $T_{k-1}$
(note that, since $v_k$ is not a Voronoi vertex, it cannot be a leaf vertex of $T_{k-1}$).
If $v_k$ is an internal vertex of $T_{k-1}$,
as in figure~\ref{fig:tree.a},
then we add a new edge $\gamma_k$ to $T_k$ connecting vertices $v_k$ and $\text{Vor}_{I_k}$.
Since, by construction, $\gamma_k$ does not cross any edge in $T_k$, the tree graph remains embedded.
If, on the other hand, $v_k$ belongs to an edge $e$ of $T_{k-1}$ connecting vertices $v_1,v_2$,
as shown in figure~\ref{fig:tree.b},
then:
\begin{enumerate}
\item we insert a new (internal) vertex $v_k$ into $T_k$;
\item we split $e$ into two edges:
$e_1$ and $e_2$, connecting $v_1,v_k$, and $v_k,v_2$, respectively;
\item we insert a new edge $\gamma_k$ connecting vertices $v_k$ and $\text{Vor}_{I_k}$.
\end{enumerate}
Note that
the edge $e$ is split into two edges that represent the same set of points,
and therefore, since $\gamma_k$ didn't cross any edges of $T_{k-1}$,
then $\gamma_k$ does not cross any edge of $T_k$.
Hence, since $T_{k-1}$ is an embedded tree graph,
the new tree $T_k$ is also embedded
and has $\text{Vor}_{I_1},\dots,\text{Vor}_{I_k}$ as leafs.
The lemma follows by induction on $m$.
\end{proof}
\begin{figure}[!h]
\centering
\includegraphics[width=3.5in]{K3_graph2.pdf}
\caption{By assuming that a Voronoi edge $\text{Vor}_{ij}$ is incident to three Voronoi vertices $\text{Vor}_{I_1},\text{Vor}_{I_2},\text{Vor}_{I_3}$,
we can construct a planar embedding of the non-planar graph $K_{3,3}$, a contradiction.
The more general figure~\ref{fig:planarity.a}
further illustrates the proof of lemma~\ref{lem:val_le2}. }
\label{fig:K33}
\end{figure}
The final lemma of this section can be used in conjunction with lemma~\ref{lem:val_ge2}
to establish that Voronoi edges are incident to exactly two Voronoi vertices.
We sketch here the argument that shows that a Voronoi edge $\text{Vor}_{ij}$ cannot
be incident to three vertices $\text{Vor}_{I_1},\text{Vor}_{I_2},\text{Vor}_{I_3}$ (figure~\ref{fig:K33}).
The general case in the proof of lemma~\ref{lem:val_le2} follows a similar argument.
We first use lemma~\ref{lem:tree} to build a tree inside $\text{Vor}_{ij}$ with leafs at $\text{Vor}_{I_1},\text{Vor}_{I_2},\text{Vor}_{I_3}$,
and show that it can be collapsed into a star-graph with a vertex $r\in\text{Vor}_{ij}$, and non-crossing edges
$(r,\text{Vor}_{I_1}), (r,\text{Vor}_{I_2}), (r,\text{Vor}_{I_3})$, as shown in the figure.
The incidence rules of lemma~\ref{lem:vertexincidence}, as well as lemma~\ref{lem:regionpath} allows us
to construct six non-crossing edges from $s_i$ and $s_j$, to $\text{Vor}_{I_1},\text{Vor}_{I_2},\text{Vor}_{I_3}$, respectively.
We have just constructed an embedding of a graph which can be easily shown to be the non-planar graph $K_{3,3}$,
thereby reaching a contradiction.
\begin{figure}[!h]
\centering
\subfloat[]{\label{fig:planarity.a}\includegraphics[width=2.8in]{graph1.pdf}}
\quad\quad
\subfloat[]{\label{fig:planarity.b}\includegraphics[width=2.8in]{graph2.pdf}}
\caption{Every Voronoi edge $\text{Vor}_{ij}$ (green region) incident to $m$ Voronoi vertices
allows the construction of an embedded planar graph $G_m$ connecting
a tree $T_m$ inside $\text{Vor}_{ij}$, to the sites $s_i,s_j$ (a).
This graph has a minor $G'_m$ obtained from $G_m$ by contracting edges of $T_m$.
$G'_m$ can be shown not to be planar for $m>2$, and therefore Voronoi edges are incident to no more than two Voronoi vertices.
}
\label{fig:planarity}
\end{figure}
\begin{lemma}\label{lem:val_le2}
Voronoi edges of an orphan-free diagram are incident to no more than two Voronoi vertices.
\end{lemma}
\begin{proof}
Let $\text{Vor}_{ij}$ be a Voronoi edge incident to Voronoi vertices $\text{Vor}_{I_1},\dots,\text{Vor}_{I_m}$.
Since $\text{Vor}_{ij} \rightsquigarrow \text{Vor}_{I_1},\dots,\text{Vor}_{I_m}$,
and Voronoi vertices are of higher order ($|I_k| > 2$) than Voronoi edges,
by the definition of incidence (definition~\ref{def:incidence}), it is $\{i,j\} \subset \text{Vor}_{I_k}$, with $k=1,\dots,m$.
We prove the lemma on the sphere $\mathbb{S}^2$, where any of the Voronoi vertices may be the vertex at infinity.
Note also that some of the sets $I_k$ with $k=1,\dots,m$ may be equal,
since Voronoi vertices have not yet been shown to be connected.
By lemma~\ref{lem:vertexincidence},
the vertices $\text{Vor}_{I_k}$ are isolated points
(possibly the point at infinity),
and $\text{Vor}_i,\text{Vor}_j\rightsquigarrow \text{Vor}_{I_1},\text{Vor}_{I_2},\dots,\text{Vor}_{I_m}$.
We begin by assuming that $m>2$, and build an embedded planar graph $G_m$ (figure~\ref{fig:planarity.a}).
We then show that $G_m$ can only be planar if $m\le 2$, reaching a contradiction.
By lemma~\ref{lem:tree}, there is an embedded tree graph $T_m\subset\overline{\text{Vor}_{ij}}$
with $\text{Vor}_{I_1},\dots,\text{Vor}_{I_m}$ as leafs.
We begin by setting $G_m$ equal to $T_m$.
We then insert the vertices $s_i$ and $s_j$ in $G_m$ (as shown in figure~\ref{fig:planarity.a}).
Since $\text{Vor}_i,\text{Vor}_j \rightsquigarrow \text{Vor}_{I_1},\text{Vor}_{I_2},\dots,\text{Vor}_{I_m}$,
by lemma~\ref{lem:regionpath},
there are non-crossing paths $\gamma_{i,k}\subset\overline{\text{Vor}_i}$, with $k=1,\dots,m$,
connecting $s_i$ to $\text{Vor}_{I_k}$
and non-crossing paths $\gamma_{j,k}\subset\overline{\text{Vor}_j}$, with $k=1,\dots,m$,
connecting $s_j$ to $\text{Vor}_{I_k}$.
We insert the above paths $\gamma_{i,k},\gamma_{j,k}$, $k=1,\dots,m$, as edges of $G_m$.
Aside from all paths $\gamma_{i,k}$ ($\gamma_{j,k}$) only crossing
at their starting point,
all paths $\gamma_{i,k}$ ($\gamma_{j,k}$) are, by lemma~\ref{lem:regionpath},
contained (except for their final endpoint)
in the interior of $\text{Vor}_i$ ($\text{Vor}_j$),
and therefore they can only cross an edge of $T_m\subset\overline{\text{Vor}_{ij}}$ at an endpoint.
$G_m$ is therefore embedded in $\mathbb{S}^2$, and so it is a planar graph.
Recall that the minors of a graph are obtained by erasing vertices, erasing edges, or contracting edges,
and that minors of planar graphs are themselves planar~\cite[p.\ 269]{bondy2008graph}.
We now construct an appropriate minor $G'_m$ of the planar graph $G_m$,
shown in figure~\ref{fig:planarity.b},
and prove that it is non-planar whenever $m > 2$, creating a contradiction.
Clearly, every tree $T_m$ satisfying the conditions of lemma~\ref{lem:tree} has a minor $T'_m$
directly connecting the root to each leaf $\text{Vor}_{I_k}$, $k=1,\dots,m$ (see figure~\ref{fig:planarity.b}),
which is obtained by successively contracting every edge of $T_m$ that connects two internal vertices.
We apply the same sequence of edge contractions to obtain $G'_m$ from $G_m$, as shown in figure~\ref{fig:planarity}.
Let $r'$ be the root of $T'_m$, and $\gamma_{r',k}$ be edges from $r'$ to $\text{Vor}_{I_k}$, with $k=1,\dots,m$.
The minor $G'_m$ has vertex set
\[ V=\{s_i,s_j,r', ~\text{Vor}_{I_1},\dots,\text{Vor}_{I_m}\},\]
and edge set
\[ E=\{ \gamma_{i,1},\dots,\gamma_{i,m}, ~ \gamma_{j,1},\dots,\gamma_{j,m}, ~ \gamma_{r',1},\dots,\gamma_{r',m}\},\]
and therefore $G'_m$ has $v=m+3$ vertices and $e=3m$ edges.
Since (as is easily verified) every cycle in $G'_m$ has length four or more, and $G'_m$ is planar,
then it holds $2e \ge 4f$, where $f$ is the number of faces.
Using Euler's identity for planar graphs, $v-e+f=2$~\cite{bondy2008graph},
and the fact that $2e \ge 4f$, $v=m+3$, and $e=3m$, it follows that $m \le 2$,
and therefore $G'_m$ is not planar whenever $m> 2$
(for instance, $G'_3$ is the utility graph $K_{3,3}$).
Since $m> 2$ leads to a contradiction, it follows that every Voronoi edge is incident to at most two Voronoi vertices.
\end{proof}
\subsection{Primal Voronoi graph and dual Delaunay triangulation}\label{primaldual}
We use the results in this section to construct a graph from the
incidence relations of an orphan-free Voronoi diagram,
and dualize it into a planar embedded graph.
Let the \emph{primal Voronoi graph} $\tilde{P}=(\tilde{P}_V, \tilde{P}_E)$
of an orphan-free Voronoi diagram be defined as follows.
The vertices $\tilde{P}_V$ are the connected components of Voronoi vertices.
Since, by lemma~\ref{lem:vertexincidence},
Voronoi vertices are composed of isolated points, then $\tilde{P}_V$ is a collection of isolated points.
By lemmas~\ref{lem:val_ge2} and~\ref{lem:val_le2},
Voronoi edges that are incident to some Voronoi vertex are incident to exactly two Voronoi vertices.
For each Voronoi edge $\text{Vor}_{ij}$ incident to some Voronoi vertex,
we include in $\tilde{P}_E$ an edge connecting the vertices in $\tilde{P}_V$
corresponding to the connected components of Voronoi vertices that $\text{Vor}_{ij}$ is incident to.
By lemma~\ref{lem:closureedge}, for each such Voronoi edge $\text{Vor}_{ij}$ there is a simple path in $\text{Vor}_{ij}$
connecting the two Voronoi vertices incident to $\text{Vor}_{ij}$, and therefore $\tilde{P}$ is an embedded planar graph.
\begin{figure}[htbp]
\centering
\subfloat[]{\label{fig:planar.a}\includegraphics[width=2.3in]{planar1.pdf}}
\quad\quad
\subfloat[]{\label{fig:planar.b}\includegraphics[width=2.3in]{planar2.pdf}}
\caption{Diagrams used in the proof of theorem~\ref{th:simpleplanar}. }
\end{figure}
\vspace*{0.1in}\noindent{\bf Theorem~\ref{th:simpleplanar}}. \emph{
Let $\tilde{G}=(\tilde{V}, \tilde{E})$ be the dual of the primal Voronoi graph corresponding to an orphan-free Voronoi diagram,
then $\tilde{G}$ is a simple, connected, planar graph.
}
\begin{proof}
The dual graph $\tilde{G}$ is constructed by dualizing $\tilde{P}$
and using the natural embedding described in~\cite[p.\ 252]{bondy2008graph},
in which dual vertices are placed inside primal faces (at the sites in this case),
and dual edges cross once their corresponding primal edges.
From this construction, $\tilde{G}$ is an embedded planar graph~\cite[p.\ 252]{bondy2008graph},
and is connected by virtue of being the dual of a planar graph~\cite[p.\ 253]{bondy2008graph}.
We show that $\tilde{G}$ is simple (edges have multiplicity one, and there are no loops: edges incident to the same vertex).
Edges of $\tilde{G}$ are one-to-one with edges of $\tilde{P}$.
In turn, edges of $\tilde{P}$ correspond to Voronoi edges, and these are,
by lemma~\ref{lem:connectededges},
connected.
Therefore the edges of $\tilde{G}$ have multiplicity one.
Since loops and cut edges (those whose removal disconnects the graph)
are duals of each other~\cite[p.\ 252]{bondy2008graph},
we now show that $\tilde{P}$ has no cut edges, and therefore $\tilde{G}$ has no loops.
By~\cite[p.\ 86]{bondy2008graph},
an edge of $\tilde{P}$ is a cut edge iff it belongs to no cycle of $\tilde{P}$.
To every edge of $\tilde{P}$ corresponds an Voronoi edge $\text{Vor}_{ij}$ that is incident to two Voronoi vertices.
By lemma~\ref{lem:vertexincidence},
$\text{Vor}_{ij}$ is incident to at least one Voronoi region $\text{Vor}_i$.
We next show that the Voronoi elements in the boundary of every Voronoi region $\text{Vor}_i$ form a cycle,
and therefore $\text{Vor}_{ij}$ belongs to a cycle, so it cannot be a cut edge.
Clearly, the boundary $\partial \text{Vor}_i$ of $\text{Vor}_i$ is composed of Voronoi edges and Voronoi vertices,
since $\text{Vor}_i\rightsquigarrow\text{Vor}_j$ is not possible because $\{i\}\not\subset\{j\}$ (see section~\ref{sec:incidence}).
Let $C=\left[\text{Vor}_{i,j_1},\text{Vor}_{i,j_1,j_2}, \text{Vor}_{i,j_2}, \dots\right]$ be the sequence of elements around the boundary of $\text{Vor}_i$,
with $\text{Vor}_{ij}\in C$.
We show that $C$ is a cycle.
\vspace*{0.05in}\noindent{\bf [$C$ has no repeated Voronoi vertices]}.
By the assumption of section~\ref{sec:assumptions},
Voronoi regions have boundaries that are simple closed curves (in $\mathbb{S}^2$).
Note that, because vertices are isolated points, there are no repeated vertices in $C$ since
the boundary of $\text{Vor}_i$ is a simple curve.
\vspace*{0.05in}\noindent{\bf [$C$ has no repeated Voronoi edges]}. \\
Let $\text{Vor}_{ij}$ appear twice in $C$ as $\left[\dots, \text{Vor}_I, \text{Vor}_{ij}, \text{Vor}_J, \dots, \text{Vor}_{ij}, \dots\right]$,
where $\text{Vor}_I,\text{Vor}_J$ are Voronoi vertices, as in figure~\ref{fig:planar.a}.
Let $p,q$ be two points in each of the two common boundaries between $\text{Vor}_i$ and $\text{Vor}_{ij}$.
By lemma~\ref{lem:regionpath}, there are simple paths $\gamma_{i,p},\gamma_{i,q}\subset\text{Vor}_i$ from $s_i$ to $p,q$, respectively,
which only meet at the initial endpoint (figure~\ref{fig:planar.a}).
Since $\text{Vor}_{ij}$ is simply connected, we can consider a simple path $\gamma_{ij}\subset\text{Vor}_{ij}$ connecting $p,q$.
Let $\gamma$ be the simple closed path obtained by concatenating $\gamma_{i,p},\gamma_{ij},\gamma_{i,q}$
which, by the Jordan curve theorem divides the plane into a bounded region $Int$, and an unbounded region.
Since it must be $\text{Vor}_I\in Int$ or $\text{Vor}_J\in Int$,
assume without loss of generality that $\text{Vor}_I\in Int$, and note that it cannot be $\text{Vor}_I=\text{Vor}_\infty\in Int$,
since $Int$ is bounded.
We show that $\text{Vor}_I\in Int$ is not possible,
and therefore that $C$
has no repeated elements.
Let $\text{Vor}_I\in Int$ and let $k\in I$ be $k\ne i,j$, which always exists because $|I|\ge 3$.
By lemma~\ref{lem:vertexincidence}, there is a point $p\in\text{Vor}_k\cap Int$.
Since $\text{Vor}_k$ is path connected, and the boundary of $Int$ is $\gamma\subset\text{Vor}_{i}\cup\text{Vor}_{ij}$, then $\text{Vor}_k\subset Int$,
and therefore $s_k\in Int$.
We show that $Int$ cannot contain any sites other than $s_i$, reaching a contradiction.
Recall that the boundary $\gamma$ of $Int$ is the concatenation of
$\gamma_{i,p}\subset\text{Vor}_i$, $\gamma_{ij}\subset\text{Vor}_{ij}$, and $\gamma_{i,q}\subset\text{Vor}_i$,
and that $s_i\in\gamma$, as in figure~\ref{fig:planar.b}.
Let $Int'$ be the union of segments from $s_i$ to every point in $\gamma$:
\[ Int'\equiv \displaystyle{\left(\cup_{r\in\gamma_{i,p}} \overline{s_i, r}\right) \cup
\left(\cup_{r\in\gamma_{ij}} \overline{s_i, r}\right) \cup
\left(\cup_{r\in\gamma_{i,q}}\overline{s_i,r}\right)}. \]
Since it is clearly $Int\subset Int'$, it suffices to show that $Int'$ does not contain any site $s_k$ different from $s_i$.
Every segment of the form $\overline{s_i,r}$
with $r\in\gamma_{i,p}\subset\text{Vor}_i$ or $r\in\gamma_{i,q}\subset\text{Vor}_i$ cannot contain a site $s_k$
or else, by the convexity of $D$, $r$ would be closer to $s_k$ than to $s_i$.
Similarly, every segment of the form $\overline{s_i,r}$ with
$r\in\gamma_{ij}$ cannot contain a site $s_k$, or else by the convexity of $D$, $r$ would be closer to $s_k$ than to $s_i,s_j$.
Since every Voronoi edge $\text{Vor}_{ij}$ is part of a cycle, it cannot be a cut edge, and therefore its dual has no loops.
\end{proof}
\section{Embeddability of the Delaunay triangulation}\label{sec:dual}
Let $\tilde{G}=(\tilde{V}, \tilde{E})$ be the dual of the primal Voronoi graph corresponding to an orphan-free Voronoi diagram,
as defined in section~\ref{sec:planar}.
By theorem~\ref{th:simpleplanar}, $\tilde{G}$ is simple and planar with vertices at the sites.
Let $G=(S,E,F)$ be the planar graph obtained by replacing curved edges by straight segments.
Recall from section~\ref{sec:planar} that,
while Voronoi regions and edges are connected, Voronoi vertices may have multiple connected components,
and therefore $G$ can have duplicate faces in $F$.
We only show after this section that faces have multiplicity one by virtue of $G$ being embedded.
\vspace*{0.08in}\noindent{\bf Faces with more than three vertices}.
Every face $f\in F$ is dual to a Voronoi element $\text{Vor}_I$ of order $|I|=k\ge 3$,
to which corresponds (proposition~\ref{prop:ECB}) a convex ball $B(c;\rho)$, with $c\in\text{Vor}_I$,
that circumscribes the sites $(s_i)_{i\in I}$ incident to $f$.
Due to the planarity of $G$, we can assume the sites $(s_i)_{i\in I}$ to be ordered around $f$.
In order to find whether a point $p\in\mathbb{R}^2$ belongs to $f$,
we simply triangulate $f$ in a fan arrangement:
$\tau_1=\{s_{i_1},s_{i_2},s_{i_k}\}; \tau_2=\{s_{i_2},s_{i_3},s_{i_k}\}; \dots$,
and consider that $p\in f$ iff it lies in any of the resulting $\tau_j$.
Note that this arrangement does not interfere with the original edges in $E$ (other than creating new ones),
all new edges are incident to two faces (they are not in the topological boundary of $G$),
and most importantly,
every $\tau_j$, with $j=1,\dots,k-2$ satisfies the empty circum-ball property with the same {witness} ball $B(c;\rho)$
as $f$.
We assume in the sequel that $G$ has been triangulated in this way.
The fact that this triangulated $G$ will be shown to be embedded will imply that every face $f$ is in fact convex.
For convenience in the remainder of this section
we name $W=\{w_i\in S : i=1,\dots,m\}$ the sites that are part of the boundary of
the convex hull ${conv}\left\{\Sites\right\}$, and order them
in clock-wise order around ${conv}\left\{\Sites\right\}$.
\subsection{Boundary}\label{sec:boundary}
In this section, we assume that the divergence $D$ satisfies the bounded anisotropy assumption~\ref{ass:BAA},
and conclude that the boundary of the dual triangulation of an orphan-free diagram is the same as the boundary of the convex hull of the sites (and in particular it is simple and closed).
The vertices in the \emph{topological boundary} of $G$ are those
whose corresponding primal regions are unbounded,
while topological boundary edges are those connecting topological boundary vertices.
For convenience, we call $B\subseteq E$ the set of topological boundary edges of $G$.
The boundary $\mathcal{B}$ of the convex hull is a simple circular chain
$\mathcal{B} = \{(w_i, w_{i\oplus 1}) : i=1,\dots,m\}$.
We prove that
it is $B=\mathcal{B}$
(loosely speaking: the topological, and geometric boundaries of $G$ are the same and coincide with the boundary of ${conv}\left\{\Sites\right\}$),
which implies that $G$ covers
the convex hull of the sites, and its topological boundary edges form
a simple, closed polygonal chain.
All the proofs of this section are in Appendix B.
\begin{lemma}[$B\subseteq\mathcal{B}$]\label{boundary_easy}
To every topological boundary edge of $G$ corresponds a segment in the boundary of ${conv}\left\{\Sites\right\}$.
\end{lemma}
We now turn to the converse claim: that to every segment
in $\mathcal{B}$ corresponds one in $B$.
Since $B$ is the set of boundary edges of $G$, whose corresponding primal edges
are unbounded, the claim is equivalent to proving that, to every segment
in $\mathcal{B}$
corresponds a boundary edge $(w_i,w_j)\in E$ of $G$ whose corresponding primal edge $\text{Vor}_{ij}$ is unbounded.
The proof proceeds as follows. First, assume without loss of generality that the origin is in the interior of ${conv}\left\{\Sites\right\}$.
Let $C(\sigma)=\{x\in\mathbb{R}^2 : \|x\|=\sigma\}$ be an origin-centered circle
of radius $\sigma$ large enough so that lemmas~\ref{lem:VW} and~\ref{lem:contrad} hold in $C(\sigma)$.
We define two functions:
\begin{eqnarray}\label{eq:pinu}
&\nu_\sigma &: \partial{conv}\left\{\Sites\right\}\rightarrow C(\sigma), ~~~~~~~~~~~~~\nu_\sigma: r\mapsto \sigma \cdot r / \|r\|, \\
&\pi &: C(\sigma)\rightarrow\partial{conv}\left\{\Sites\right\},
\end{eqnarray}
$\nu_\sigma$ simply projects points in the boundary of ${conv}\left\{\Sites\right\}$ out to their closest point in $C(\sigma)$
(using the natural metric; note that $\sigma$ can always be chosen large enough so this projection is unique).
$\pi$ is constructed as follows.
\begin{figure}[htbp]
\centering
\includegraphics[width=2.7in]{pi.pdf}
\caption{The construction of the projection function $\pi:C(\sigma)\rightarrow \partial{conv}\left\{\Sites\right\}$.
Note that in this case the region-to-site-index function is simply $j(i)=i$, but this cannot be assumed in general. }
\label{fig:pi}
\end{figure}
\refstepcounter{foo}\thefoo\label{text:boundary}
Consider the situation illustrated in figure~\ref{fig:pi}.
By lemma~\ref{lem:VW}, all points in $C(\sigma)$ are closer to $W$ than to any interior site $S\setminus W$.
We split $C(\sigma)$ into
a sequence $(C_j)$ of \emph{connected} parts closest
to the same boundary site $w_{i(j)}$
(the function $i(\cdot)$ is used to map part indices to the index of their closest site).
By the convexity of balls,
adjacent regions \emph{must} be closest to (circularly) consecutive sites in $W$
(e.g.~if regions $C_1,C_2$ had $i(1)=1$ and $i(2)=3$,
by the continuity of $D$, the point $p$ where $C_1,C_2$ meet would be closest to $w_1,w_3$;
however, since the sites $w_i$ are in cyclic order around $\partial{conv}\left\{\Sites\right\}$,
$p$ would be closer to $w_2$ than to $w_1,w_3$, a contradiction).
Pick one point $p_j$ for each region $C_j$, and let $\pi(p_j) \equiv w_{i(j)}$.
For each pair of consecutive regions $C_j, C_{j\oplus 1}$ meeting at $p_{j,j\oplus 1}$,
let $\pi(p_{j,j\oplus 1}) \equiv (w_{i(j)} + w_{i(j\oplus 1)})/2$ (the midpoint of two consecutive boundary sites).
The remaining values of $\pi$ are filled using simple linear interpolation.
By construction, the following holds:
\begin{property}\label{prop:pi}
$i\cdot$ $\pi:C(\sigma)\rightarrow\partial{conv}\left\{\Sites\right\}$ is continuous. \\
\hspace*{0.91in}$ii\cdot$ Given $p\in C(\sigma)$ and consecutive boundary sites $w_i, w_{i\oplus 1}$,
then $p\in\text{Vor}_{i, i\oplus 1}$ iff $\pi(p)=(w_i + w_{i\oplus 1})/2$.
\end{property}
By the convexity of ${conv}\left\{\Sites\right\}$, $\nu_\sigma$ is continuous in $\partial{conv}\left\{\Sites\right\}$.
Note that, because ${conv}\left\{\Sites\right\}$ is assumed to contain the origin then,
as shown in figure~\ref{fig:pinu}, $\nu_\sigma$ projects
every point $\pi(p)\in (w_i,w_j)$ lying on a segment of $\partial{conv}\left\{\Sites\right\}$,
\emph{outwards} from the convex hull (and on the \emph{empty} side of
$(w_i,w_j)$); that is, so that $\nu_\sigma(\pi(p))\in H^{+}_{ij}\cap C(\sigma)$
(i.e.~$\nu_\sigma(\pi(p))$ is in the empty half-space of $(w_i,w_j)$).
\begin{figure}[htbp]
\centering
\includegraphics[width=2.7in]{pinu.eps}
\caption{The construction for the proof of lemma~\ref{lem:hard}.}
\label{fig:pinu}
\end{figure}
The claim now reduces to showing that for each segment
$(w_i,w_j)$ of $\partial{conv}\left\{\Sites\right\}$, and for \emph{every} sufficiently large $\sigma$,
there is $p\in C(\sigma)$
with $p\in\text{Vor}_{ij}$
(i.e.~$\pi(p)=(w_i+w_j)/2$).
Since this implies that $\text{Vor}_{ij}$ is unbounded,
it means that the corresponding edge $(w_i,w_j)$ is in $B$ (the topological boundary of $G$).
The proof is by contradiction.
Lemma~\ref{lem:Sn} uses Brouwer's fixed point theorem to show that, for every segment $(w_i,w_j)$ of $\mathcal{B}$,
if there were no $p\in C(\sigma)$ closest to $w_i,w_j$, then
the function $\nu_\sigma\circ\pi:C(\sigma)\rightarrow C(\sigma)$
(in fact a slightly different but related function)
would have a point $q\in C(\sigma)$ such that
$\nu_\sigma(\pi(q))=-q$, that is, such that
$q$ is ``behind" the segment $(w_i,w_j)\in\partial{conv}\left\{\Sites\right\}$ to which it is
closest ($q\in H^{-}_{ij}$).
On the other hand, lemma~\ref{lem:contrad} shows that, for all sufficiently large circles $C$, no
point $q\in C(\sigma)$ can be closest to a segment
$(w_i,w_j)\in\partial{conv}\left\{\Sites\right\}$ it is \emph{behind} of, creating a contradiction.
The next Lemma is used to create a contradiction, and relies on assumption~\ref{ass:BAA}.
Lemma~\ref{lem:Sn} is the key lemma in this section, and is a simple application of Brouwer's fixed point theorem.
\begin{lemma}\label{lem:contrad}
There is $\rho > 0$ such that, for any segment $(w_i,w_j)\in\mathcal{B}$ with supporting line $L_{ij}$,
every $p\in H^{-}_{ij}$ with $\|p\| > \rho$ whose closest point in $L_{ij}$ belongs to $\overline{w_i w_j}$ is
closer to a site in $S\setminus\{w_i,w_j\}$ than to $L_{ij}$.
\end{lemma}
\begin{lemma}\label{lem:Sn}
Every continuous function $F:\mathbb{S}^n\rightarrow\mathbb{S}^n$ that is not onto has a fixed point.
\end{lemma}
\begin{lemma}[$B\supseteq\mathcal{B}$]\label{lem:hard}
To every segment in the boundary of
${conv}\left\{\Sites\right\}$ corresponds a boundary edge of $G$.
\end{lemma}
Finally, since we have shown that the topological boundary of the dual triangulation is the
same as the boundary of the convex hull of the sites, we can conclude that:
\begin{corollary}\label{cor:boundary}
The topological boundary of the dual of an orphan-free Voronoi diagram
is the boundary of the convex hull ${conv}\left\{\Sites\right\}$, and is therefore simple and closed.
\end{corollary}
\subsection{Interior}\label{sec:interior}
This section concludes the proof of Theorem~\ref{th:main} by showing that,
if the topological boundary of $G$ is simple and closed, then $G$ must be embedded.
The main argument in the proof
uses proposition~\ref{prop:ECB} and~\ref{cor:boundary},
as well as the theory of discrete one-forms on graphs,
to show that there are no
``edge fold-overs" in $G$
(edges whose two incident faces are on the same side of its supporting line),
and uses this to conclude that the interior of $G$ is a single ``flat sheet", and therefore it is embedded.
The following definition, from~\cite{1form}, assumes that, for each edge $(s_i,s_j)\in E$ of $G$,
we distinguish the two opposing half-edges $(s_i,s_j)$ and $(s_j,s_i)$.
\begin{definition}[Gortler et al.\ \cite{1form}]\label{def:1form}
A non-vanishing (discrete) one-form $\xi$ is an assignment of a real value
$\xi_{ij} \neq 0$ to each half edge $(s_i,s_j)$ in $G$, such that
$\xi_{ji} = -\xi_{ij}$.
\end{definition}
We can construct a non-vanishing one-form
over $G$ as follows.
Given some unit direction vector $n\in\mathbb{S}^1$
(in coordinates $n=\left[n_1,n_2\right]^t$),
we assign a real
value $z(v) = n^t v$ to each vertex $v$ in $G$, and define
$\xi_{ij} \equiv z(s_i) - z(s_j)$, which clearly satisfies
$\xi_{ji} = -\xi_{ij}$. The one-form, denoted by $\xi^n$,
is non-vanishing if, for all edges $(s_i,s_j)\in E$,
it is $\xi_{ij} = n^t (s_i - s_j) \neq 0$,
that is,
if $n$ is not orthogonal to any edge.
The set of edges has finite cardinality $|E| \le |S| (|S|-1)/2$,
so \emph{almost all} directions $n\in\mathbb{S}^1$ generate a non-vanishing one-form $\xi^n$.
Since $G=(S,E,F)$ is a planar graph with a well-defined face structure,
there is, for each face $f\in F$, a cyclically ordered set
$\partial f$ of half-edges around the face.
Likewise, for each vertex $v\in S$, the set $\delta v$ of cyclically ordered
(oriented) half-edges emanating from each vertex is well-defined.
\begin{definition}[Gortler et al.\ \cite{1form}]
Given
non-vanishing one-form $\xi$,
the index of vertex $v$ with respect to $\xi$ is \[\mathbf{ind}_{_{\xi}}(v) \equiv 1 - \mathbf{sc}_{_{\xi}}(v) / 2, \]
where $\mathbf{sc}_{_{\xi}}(v)$ is the number of sign changes of $\xi$
when visiting the half-edges of $\delta v$ in order.
The index of face $f$ is \[\mathbf{ind}_{_{\xi}}(f) \equiv 1 - \mathbf{sc}_{_{\xi}}(f)/2\] where $\mathbf{sc}_{_{\xi}}(f)$ is the number
of sign changes of $\xi$ as one visits the half-edges of $\partial f$ in order.
\end{definition}
Note that, by definition, it is always $\mathbf{ind}_{_{\xi^n}}(v) \le 1$.
A discrete analog of the Poincar\'e-Hopf index theorem relates
the two indices above:
\begin{theorem}[Gortler et al.\ \cite{1form}]\label{lem:ph}
For any non-vanishing one-form $\xi$, it is
\[ \displaystyle{\sum_{v\in S} \mathbf{ind}_{_{\xi}}(v) + \sum_{f\in F} \mathbf{ind}_{_{\xi}}(f) = 2} \]
\end{theorem}
Note that this follows from Theorem 3.5 of~\cite{1form} because the unbounded,
outside face, which is not in $G$, is assumed in this section to be closed and simple
(corollary~\ref{cor:boundary}), and therefore has null index.
Note that the machinery from~\cite{1form} to deal with degenerate cases
isn't needed here because vertices, by definition, cannot coincide ($S$ is not a multiset).
All proofs in this section, except for that of theorem~\ref{th:main}, are in Appendix C.
The one-forms $\xi^n$ constructed above satisfy the following property:
\begin{lemma}\label{lem:non-negative}
Given a non-vanishing one-form $\xi^n$, the sum of indices of interior vertices ($S\setminus W$) of $G$ is non-negative.
\end{lemma}
The next two lemmas
relate the presence of edge fold-overs and
the ECB
property (proposition~\ref{prop:ECB}) to the indices of vertices in $G$.
\begin{lemma}\label{lem:index-1}
If $G$ has an edge fold-over, then there is $n\in\mathbb{S}^1$ and non-vanishing one-form $\xi^n$ such
that $\mathbf{ind}_{_{\xi^n}}(v) < 0$ for some interior vertex $v\in S\setminus W$.
\end{lemma}
\begin{lemma}\label{lem:index1}
Given $n\in\mathbb{S}^1$ and non-vanishing one-form $\xi^n$, if $G$ has an interior vertex $v\in S\setminus W$ with index
$\mathbf{ind}_{_{\xi^n}}(v)=1$, then there is a face $f$ of
$G$ that does not satisfy the empty circum-ball
property (proposition~\ref{prop:ECB}).
\end{lemma}
The above provides the necessary tools to prove the following key lemma.
\begin{lemma}\label{lem:ef}
$G$
has no edge fold-overs.
\end{lemma}
Finally, the absence of edge fold-overs,
together with a simple and closed boundary,
is sufficient to show that $G$ is
embedded.
\begin{lemma}\label{lem:interior}
If its (topological) boundary is simple and closed,
then the straight-line dual of an orphan-free Voronoi diagram,
with vertices at the sites,
is an embedded triangulation.
\end{lemma}
\section{Proof-of-concept implementation}\label{sec:implementation}
\begin{figure}[ht]
\centering
\subfloat[]{\includegraphics[height=2.6cm]{c07.png}\label{fig:img_a}}
\subfloat[]{\includegraphics[height=2.6cm]{c08.png}\label{fig:img_b}}
\subfloat[]{\includegraphics[height=2.6cm]{c03.png}\label{fig:img_c}}
\subfloat[]{\includegraphics[height=2.6cm]{c04.png}\label{fig:img_d}}\quad
\subfloat[]{\includegraphics[height=2.6cm]{c00.png}\label{fig:img_e}}
\subfloat[]{\includegraphics[height=2.6cm]{c01.png}\label{fig:img_f}
\subfloat[]{\includegraphics[height=2.6cm]{c05.png}\label{fig:img_g}}
\subfloat[]{\includegraphics[height=2.6cm]{c06.png}\label{fig:img_h}}
\caption{
Anisotropic Voronoi diagrams, and their duals generated by our
proof-of-concept implementation.
Voronoi vertices are marked as red dots, while dual vertices (sites) and edges are drawn
in black.}
\label{fig:test}
\end{figure}
Though not aiming for an efficient implementation,
we tested a simple proof-of-concept that constructs anisotropic Voronoi diagrams
(using a quadratic divergence $D_Q$ of the type discussed in section~\ref{sec:DQ})
and their duals
(figure~\ref{fig:test}).
A closed-form metric, which
has bounded ratio of eigenvalues
(and therefore by lemma~\ref{lem:DQgamma} satisfies assumption~\ref{ass:BAA}),
is discretized on a fine regular grid, and linearly interpolated inside grid elements, resulting in a
continuous metric. The sites are generated randomly (figures~\ref{fig:img_a}
and~\ref{fig:img_b}), or using a combination of random, and equispaced
points forming an (asymmetric) $\epsilon$-net~\cite{avd} (remaining figures).
The primal diagram was obtained using front propagation from the sites
outwards, until fronts meet at Voronoi edges.
The runtime is proportional to the grid size, since every grid-vertex is visited exactly six times (equal to their valence),
and so linear in the resolution of the sampled divergence $D_Q$.
The implementation does not
guarantee the correctness of the diagram unless it \emph{is} orphan-free, and serves to verify the claims of the paper since well-behave-ness of the dual is predicated on that of the primal.
The two main claims of the paper (that orphan-freedom is sufficient to ensure well-behavedeness of
both the dual and the primal) are clearly illustrated in these examples.
In all examples,
the dual covers the convex hull of the vertices
(corollary~\ref{cor:boundary}), is a
single cover, embedded with straight edges without edge crossings
(lemma~\ref{lem:interior}),
and has no degenerate faces
(since, by proposition~\ref{prop:ECB}, the vertices of a face lie on the boundary of a strictly convex ball).
By focusing on the primal diagrams (second and fourth column), further claims in
the paper become apparent, namely that Voronoi regions
(Voronoi elements of order one according to definition~\ref{def:VorI}) are simply connected (lemma~\ref{lem:regionSC}),
and Voronoi edges (order two), and vertices (order three or higher) are connected (corollary~\ref{cor:VorI}).
\section{Conclusion and open problems}
We studied the properties of duals of orphan-free Voronoi diagrams with respect to divergences, for the
purposes of constructing triangulations on the plane.
The main result (Theorems~\ref{th:main}) is that
the dual, with straight edges and vertices at the sites, is embedded
and covers the convex hull of the sites, mirroring similar results for
ordinary Voronoi diagrams and their duals.
Additionally, the primal is composed of connected elements (corollary~\ref{cor:VorI}).
\begin{figure}[ht]
\centering
\includegraphics[width=5.6cm]{3dbreaks.pdf}
\caption{
The main proof of this paper does not work as is in higher dimensions.
This arrangement of tetrahedra is not embedded:
the red tetrahedra has been ``inverted"
(the green dotted edge is \emph{behind} the solid blue edge),
``invading" the two front tetrahedra (closest to the viewer),
as well as the two back tetrahedra (farthest from the viewer).
However, it does not violate the ECB condition (proposition~\ref{prop:ECB}).
}
\label{fig:3dbreaks}
\end{figure}
Perhaps the most important outstanding question is whether these results
extend to higher dimensions.
The proofs in Secs.~\ref{sec:boundary} and~\ref{sec:interior},
except for lemma~\ref{lem:regionSC}, can be trivially extended to n dimensions.
Section~\ref{sec:boundary} has been written only for the two-dimensional case,
but a similar construction, and the same argument would work in higher
dimensions (lemma~\ref{lem:Sn} being a hint of this).
It is the argument in section~\ref{sec:interior}, and described in figure~\ref{fig:outline}, that becomes problematic.
While the ECB property is shown to be sufficient to prevent fold-overs in the
triangulation, it is not sufficient in
higher dimensions. In particular, fixing the boundary to be simple and convex,
there are simple arrangements of tetrahedra in $\mathbb{R}^3$ that contain
face fold-overs but do not break the ECB property.
In particular, the arrangement of tetrahedra of figure~\ref{fig:3dbreaks} is not embedded:
the red tetrahedra has been ``inverted"
(indicated by the green dotted edge being \emph{behind} the solid blue edge);
its interior overlaps that of the two front tetrahedra (closest to the viewer),
as well as the two back tetrahedra (those farthest from the viewer).
However, this arrangement does not break the ECB condition (proposition~\ref{prop:ECB}, which holds in any dimension),
and therefore the same argument used in this work would not create a
contradiction in higher dimensions.
\bibliographystyle{plain}
|
{
"redpajama_set_name": "RedPajamaArXiv"
}
| 7,652
|
<?php
namespace Concerto\PanelBundle\Controller;
use Symfony\Component\HttpFoundation\Request;
use Symfony\Component\HttpFoundation\Response;
use Symfony\Bundle\FrameworkBundle\Templating\EngineInterface;
use Concerto\PanelBundle\Service\TestSessionLogService;
use Symfony\Component\Routing\Annotation\Route;
use Symfony\Component\Translation\TranslatorInterface;
use Sensio\Bundle\FrameworkExtraBundle\Configuration\Security;
/**
* @Route("/admin")
* @Security("has_role('ROLE_TEST') or has_role('ROLE_SUPER_ADMIN')")
*/
class TestSessionLogController extends ASectionController
{
const ENTITY_NAME = "TestSessionLog";
public function __construct(EngineInterface $templating, TestSessionLogService $service, TranslatorInterface $translator)
{
parent::__construct($templating, $service, $translator);
$this->entityName = self::ENTITY_NAME;
}
/**
* @Route("/TestSessionLog/collection", name="TestSessionLog_collection")
* @param string $format
* @return Response
*/
public function collectionAction($format = "json")
{
return parent::collectionAction($format);
}
/**
* @Route("/TestSessionLog/Test/{test_id}/collection", name="TestSessionLog_collection_by_test")
* @param $test_id
* @return Response
*/
public function collectionByTestAction($test_id)
{
return $this->templating->renderResponse('ConcertoPanelBundle::collection.json.twig', array(
'collection' => $this->service->getLatestByTest($test_id)
)
);
}
/**
* @Route("/TestSessionLog/{object_ids}/delete", name="TestSessionLog_delete", methods={"POST"})
* @param Request $request
* @param string $object_ids
* @return Response
*/
public function deleteAction(Request $request, $object_ids)
{
return parent::deleteAction($request, $object_ids);
}
/**
* @Route("/TestSessionLog/Test/{test_id}/clear", name="TestSessionLog_clear", methods={"POST"})
* @param $test_id
* @return Response
*/
public function clearAction($test_id)
{
$this->service->clear($test_id);
$response = new Response(json_encode(array("result" => 0)));
$response->headers->set('Content-Type', 'application/json');
return $response;
}
}
|
{
"redpajama_set_name": "RedPajamaGithub"
}
| 9,700
|
California Updates Law to Protect Dependent Adults from Caregiver Marriages: Updates to Probate Code 21380, 21382, and 21611
While several financial elder abuse cases involve family members, there are many that include caregivers. Caregivers often spend many hours alone with vulnerable dependent adults. As the law currently stands, a lifetime gift or beneficiary provision in a will or trust made by a dependent adult to a "care custodian" is presumed to be a result of fraud or undue influence. However, there are exceptions. In particular, spouses, domestic partners, and cohabitants who receive the gifts are exempt from the presumption of undue influence. This leads to the possibility that a strategic caregiver could marry the dependent adult to avoid the presumption of fraud.
The new law introduced as Assembly Bill No. 328, signed by Governor Newsom on June 26, 2019 and effective January 1, 2020, expands the presumption of fraud to include coverage for: (1) omitted spouse claims by caregivers, and (2) donative gifts to caregivers.
Under the law, a dependent adult is defined as an adult of any age who cannot provide properly for his or her basic needs, has difficulty managing his or her financial resources, or is unable to resist fraud or undue influence.
Omitted Spouse Claims by Caregivers
As the average age in marriage increases, it is not uncommon for single persons to have an estate plan and not get around to updating it when they get married or remarried. As a result, California existing law provides that if a married person passes away without providing for their spouse in their estate plan, the surviving spouse may claim a part of the estate, even if they were not named in the will or trust. This is called a "omitted spouse" claim.
Existing law provides that if the surviving spouse is not provided for in his or her will or trust after the execution of the document, absent an exception where the surviving spouse is provided for or agrees otherwise, the omitted spouse is required to receive one-half of the decedent's community property and a share of the separate property.
While this law on its face is simple enough, a caregiver could financially abuse an elderly adult by marrying him or her near the end of their lifetime and then make a claim for a portion of the decedent's estate as an omitted spouse.
To address this type of financial of abuse by caregivers, the new law prohibits a surviving spouse from receiving a share of the decedent's estate as an omitted spouse if (1) the spouse was a caregiver of the decedent who was a dependent adult and the marriage commenced while the caregiver was providing services or within 90 days after those services were last provided, and (2) the decedent dies less than 6 months after the date of marriage.
In order to overcome the new presumption under the law, a spouse can prove by clear and convincing evidence that the marriage was not a product of fraud or undue influence.
Expansion of Presumption of Fraud in Gifts to Caregivers
California law has presumed that a dependent adult who signs an instrument benefitting a caregiver does so as a result of fraud or undue influence. Under Probate Code 21380, there is a presumption of fraud or undue influence when a dependent adult makes a will, a trust, or a donative transfer or gift to the person who is the caregiver; as a result, the gift is presumptively invalid.
However, there are exemptions to this law. For example, existing law exempts spouses and cohabitants of the transferor from the presumption of fraud or undue influence. To take advantage of the exemptions to the law, some conniving caregivers have married dependent adults to take advantage of the marriage exemption and avoid the presumption of fraud.
Assembly Bill 328 expands the presumption of fraud to caregivers who marry a dependent adult while providing services to the adult, or within 90 days after those services were last provided to the dependent adult if the donative transfer occurred, or the instrument was executed less than six months after the marriage started.
The presumption of invalidity is rebuttable. If a caregiver can prove by clear and convincing evidence that the gift was not the product of fraud or undue influence, then a court will validate the gift. However, to deter caregivers from making fraudulent claims, if the caregiver is unsuccessful in attempting to rebut the presumption, the caregiver will be responsible for all costs of the proceeding, including reasonable attorney's fees.
You Created A Trust, Now What?
Planning for a Major Health Event
|
{
"redpajama_set_name": "RedPajamaCommonCrawl"
}
| 2,458
|
DYLAN, BOB
Chronicles: Volume One (Chronicles)
Simon & Schuster, September 2005. Trade Paperback. WINNER OF THE NOBEL PRIZE IN LITERATURE The celebrated first memoir from arguably the most influential singer-songwriter in the country, Bob Dylan. "I'd come from a long ways off and had started a long ways down. But now destiny was about to manifest..... More
Jonathan Cott
Doubleday, September 1985. Oversize Softcover. More
RINZLER, ALAN.
Bob Dylan: The Illustrated Record
Outlet, October 1978. Oversize Softcover. More
SANTELLI, ROBERT, DYLAN, BOB
Bob Dylan Scrapbook : 1956-1966
N/A, September 2005. Novelty. THE BOB DYLAN SCRAPBOOK is the highly collectable illustrated biography of Dylan's life during the 1950s and 60s. Created in association with Bob Dylan, the SCRAPBOOK is crammed with features including rare photographs, facsimiles of handwritten lyrics and rare memorabilia. The text includes interviews with Dylan..... More
|
{
"redpajama_set_name": "RedPajamaCommonCrawl"
}
| 4,856
|
Q: Is there any ignoring mechanism in PhpStorm similar to .gitignore Is there any ignoring mechanism in PhpStorm similar to .gitignore to exclude indexing and inception on some files/folders? So I can share that file among the team and commit its changes?
(btw, I already know the possible GUI methods to do that.)
A: *
*Mark directory as/Excluded for folders. Information about excluded stuff is stored in .idea/<project name>.iml file
*Ignore files and folders in Settings/Editor/File types - stored in Webstorm configuration directory/options/filetypes.xml
|
{
"redpajama_set_name": "RedPajamaStackExchange"
}
| 2,423
|
\section{Introduction}
The geometry of level sets has been a vital and fruitful topic of investigations involving variety of function and space settings. In particular, in the Euclidean spaces the convexity properties and curvature estimates of level sets of harmonic functions and their generalizations (such as $p$-harmonic functions and mappings and second order elliptic PDEs) have been studied by many researchers, for instance by Alessandrini~\cite{al, al2}, Caffarelli--Spruck~\cite{cs}, Chang--Ma--Yang~\cite{cmy}, Gabriel~\cite{gab}, Jost--Ma--Ou~\cite{jmo}, Kawohl~\cite{kaw}, Laurence~\cite{la}, Lewis~\cite{lew3}, Longinetti~\cite{long}, Ma--Ou--Zhang~\cite{moz}, Talenti~\cite{tal}; see also~\cite{ad}. Similar studies have been conducted for harmonic functions in space forms by Ma--Zhang~\cite{mz} and on Riemannian surfaces with constant Gaussian curvature by Wang--Wang~\cite{ww}.
The main goal of this note is to extend number of the aforementioned results to the setting of Riemannian $2$-manifolds with not necessarily constant curvature, including the surfaces of bounded integral curvature (Alexandrov surfaces). The latter one is a class of singular spaces which includes the polyhedral surfaces, the surfaces with conical singularities as well as all the topological metric surfaces with (one sided) bounds on the curvature in the sense of Alexandrov, such as $CBB(k)$ and $CAT(k)$, or $RCD(k,2)$ surfaces. This topic has been attracting an increasing interest in the recent decades, for instance due to its connections with the study of Gromov--Hausdorff limits of manifolds with bounded curvature.
Moreover, we present a number of growth estimates and maximum (minimum) principles for the geodesic curvature of level sets and the curvature of the steepest descent for harmonic functions.
Let us now present the main results and the organization of the paper. In {\bf Section 2} we consider the harmonic Dirichlet problem on an annular domain of a Riemannian surface $M^2$:
\begin{equation}\label{DP0}\tag{DP}
\begin{cases}
\Delta_{M} u=0 & \hbox{in } \Om,\\
u|_{\Gamma_1}=t_1, & u|_{\Gamma_2}=t_2,
\end{cases}
\end{equation}
assuming constant data $t_1$ and $t_2$ on the two boundary components $\Gamma_1$ and $\Gamma_2$, respectively. We show that in the smooth setting the \emph{isoperimetric inequality}
\begin{equation}\label{convex intro}
(\ln L(t))''\geq 0,\quad t_1\leq t \leq t_2,
\end{equation}
for the length of the level curves $L(t)$ characterizes the surfaces with nonpositive Gaussian curvature. Moreover, the equality in \eqref{convex intro} forces the domain to be a flat round annulus; see Theorems~\ref{thm-main1} and~\ref{thm-main2}. The fact that the key differential inequalities studied here become equalities depending on the geometry of domains, motivates the name isoperimetric inequality in the title of our work. Indeed, we follow the terminology in~\cite{al2, long, la}, where such results for~\eqref{DP} are shown in the Euclidean spaces; see also~\cite{lew3} for importance of the capacitary estimates related to the problem~\eqref{DP}. The generalization to non-flat surfaces requires obtaining integral formulas for the derivatives $L'$ and $L''$: see Lemma~\ref{lem: L''} whose proof relies on the coarea formula, the Bochner and the Kato identities. Our discussion is complemented by
refined lower estimates on $(\ln L)''$ for surfaces with bounds for the Gaussian curvature (e.g. surfaces with pinched curvatures), see Propositions 2.9 and~\ref{thm-main2-2}. We illustrate the presentation by analyzing some level curves in a hyperbolic surface; see Example 1. An alternative more analytic proof of Theorem \ref{thm-main1}, based on a conformal approach, is presented in Appendix A. We also remark here that inequalities similar to~\eqref{convex intro} appear in the context of the Hadamard's three-circles theorems (see Remark \ref{rmk:AG}) and studies of the K\"ahler manifolds, see e.g.~\cite[Theorems 4.2 and 2.2]{mac}.
Furthermore, for non-smooth \emph{surfaces with non-negative bounded integral curvature} we show that the function $t\to \ln L(t)$ is convex (Theorem~\ref{thm-reshetnyak} in {\bf Section 3}). Since $L(t)$ is no more a smooth function in this setting, such a result requires a different approach based on the measure-theoretic results due to Reshetnyak~\cite{re} and Troyanov~\cite{tr-AnnIHP}. As a byproduct of our studies we obtain that in the special case of zero curvature, Theorem~\ref{thm-reshetnyak} allows to study also annuli with non-smooth boundary components (for example von Koch snowflakes) and thus the result is new also in the Euclidean spaces; see Remark~\ref{r: Jordan bdy}. Among the surfaces with bounded integral curvature to which our result applies, let us point out in particular the important class of the \emph{surfaces with conical singularities} whose angles at the vertices are greater than $2\pi$ (i.e. of non-positive curvature); see Section 3.
The aim of {\bf Section 4} is to extend some results in~\cite{al, tal}, obtained for domains in $\R^n$, to the setting of domains in $2$-manifolds and for harmonic functions with no critical points in the underlying domain (see the discussion following Theorem~\ref{thm-pdes} for the feasibility of this assumption). Moreover, we generalize certain results proved in~\cite{mz, ww} for surfaces with constant Gaussian curvature, by allowing the curvature to vary.
Specifically, let $u$ be a harmonic function in a domain $\Om$ in a surface $M^2$ with Gaussian curvature $K=K(x)$ for $x\in M^2$, such that $u$ has no critical points in $\Om$. Then, Theorem~\ref{thm-pdes} says that the geodesic curvature $k$ of the level curves of $u$ satisfies:
\begin{align*}
&\Delta \left(\frac{k}{|\nabla u|}\right)+2K\frac{k}{|\nabla u|}=\frac{\langle \nabla K,\nabla u\rangle}{|\nabla u|^2}, \\
&-\Delta \ln |k| \geq K-\frac{1}{|k|}\langle \nabla K,\frac{\nabla u}{|\nabla u|}\rangle,\quad k\not=0.
\end{align*}
Similar differential (in)equalities hold true replacing $k$ with the curvature $h$ of the steepest descent of $u$. Namely:
\begin{align*}
& \Delta \left(\frac{h}{|\nabla u|}\right)+2K\frac{h}{|\nabla u|}=-\frac{\langle \nabla K,\star\nabla u\rangle}{|\nabla u|^2},
\\
&-\Delta \ln |h| \geq K+\frac{1}{|h|}\langle \nabla K,\frac{\star\nabla u}{|\nabla u|}\rangle,\quad h\not=0,
\end{align*}
where $\star$ stands for the Hodge star operator (so that with respect to a local orthonormal frame one has $\star \nabla u=(u_2,-u_1)$). Our Theorem~\ref{thm-pdes} generalizes Theorem 1.3 in~\cite{ww} and Theorem 3 in~\cite{tal} to the non-constant curvature setting. Furthermore, we study more types of curvatures of level sets than in~\cite{ww}, see the discussion before the statement of Theorem~\ref{thm-pdes} for a detailed presentation of the novelties obtained in Section 4. The corollaries of Theorem~\ref{thm-pdes} encompass weak and strong maximum and minimum principles for the curvatures of the level curves (Corollaries~\ref{c: k}\,--\,\ref{c:cor3.7}). The proof of Theorem~\ref{thm-pdes} is presented in Appendix B.
\vspace{0.2cm}
{\bf Acknowledgements.} Part of the work was conducted during the Simons semester in \emph{Geometry and analysis in function and mapping theory on Euclidean and metric measure spaces} at IMPAN in fall 2019 partially supported by the grant 346300 for IMPAN from the Simons Foundation and the matching 2015-2019 Polish MNiSW fund.
The second author is member of INdAM-GNAMPA. The authors would like to thank Luciano Mari and Tommaso Pacini for their comments about the manuscript and for pointing some literature.
\section{Isoperimetric inequality}
The goal of this section is to show a counterpart of Alessandrini's isoperimetric inequality result~\cite[Theorem 1.1]{al2} for harmonic functions on two-dimensional Riemannian manifolds. However, we will formulate the main problem for all dimensions $n\geq 2$, as some of our results below can be applied in the general case of smooth Riemannian manifolds.
Let $(M^n,g)$ be an $n$-dimensional Riemannian manifold with the Ricci curvature bounded from below: $\Ric\geq c$, for some fixed $c\in \R$. Moreover, let $\Om_1, \Om_2$ with $\Om_1\Subset \Om_2\subset M^n$ be two simply connected domains with boundaries denoted by $\Gamma_1$ and $\Gamma_2$, respectively. In what follows we will either assume that $\Gamma_1$ and $\Gamma_2$ are Jordan curves (mostly for $2$-dimensional results) or require
them to be $C^{1,\alpha}$ (mostly for $n>2$), see Remark~\ref{rem-c1a-reg} for a further discussion. We will specify the boundary regularity assumptions when stating the results. We define a topological annulus in $M^n$ (i.e. a ring domain, sometimes also called in the literature a $2$-connected domain when $n=2$) by
\begin{equation}\label{def-top-ring}
\Om:=\Om_2\setminus \overline{\Om_1}.
\end{equation}
Furthermore, let $t_1,t_2\in \R$ be such that $t_1<t_2$ and let us consider a continuous up to the boundary solution of the following Dirichlet problem in $\Om$ for the Laplace--Beltrami harmonic operator $\Delta_M$ on $M^n$:
\begin{equation}\label{DP}\tag{DP}
\begin{cases}
\Delta_{M} u=0 & \hbox{in } \Om,\\
u|_{\Gamma_1}=t_1, & u|_{\Gamma_2}=t_2.
\end{cases}
\end{equation}
Since now on we adopt a convention that for a fixed manifold $M$ and a fixed metric $g$ we will omit the subscript $M$ in $\Delta_M$ and write $\Delta$ for simplicity. Furthermore, when discussing $2$-dimensional manifolds we use the Gaussian curvature $K$ instead of the Ricci curvature, since $\Ric(\nabla u, \nabla u)\equiv K|\nabla u|^2$ for $n=2$.
The following auxiliary result is a counterpart of the well-known subharmonicity property for harmonic functions in the Euclidean setting.
\begin{lem}\label{lem:log}
Let $\Om$ be a domain in a $2$-dimensional Riemannian manifold and $u$ be a harmonic function in $\Om$. Then,
\begin{equation}\label{log-sub-ineq}
\Delta (\log|\nabla u|)=
\end{equation}
at points, where $|\nabla u|\neq 0$. In particular, if $K|_{\Om}\geq 0$, then $
\Delta(\log|\nabla u|)\geq 0$, while if $K|_{\Om}\leq 0$, then $
\Delta(\log|\nabla u|)\leq 0$.
\end{lem}
Before proving the lemma we recall the refined Kato (in)equality, a standard tool in geometric analysis (see for instance \cite[p. 520]{lw} or \cite[Proposition 1.3]{PRS} and references therein). In dimension $2$, the refined Kato's inequality turns out to be an equality; see \eqref{refined-Kato2}. Although this observation might be known to experts, we were not able to find a reference in the literature. Accordingly, we decided to provide a complete proof.
\begin{lem}\label{lem: kato}
Let $\Om$ be a domain in an $n$-dimensional Riemannian manifold and $u$ be harmonic in $\Om$. Then, at points where $|\nabla u|\neq 0$, it holds that
\begin{equation}\label{refined-Kato}
|\nabla^2u|^2\geq \frac{n}{n-1}\left|\nabla|\nabla u|\right|^2,\quad n\geq 2.
\end{equation}
Moreover, if $n=2$ then one has indeed
\begin{equation}\label{refined-Kato2}
|\nabla^2u|^2=2\left|\nabla|\nabla u|\right|^2.
\end{equation}
\end{lem}
\begin{proof}
We prove \eqref{refined-Kato} and \eqref{refined-Kato2} at an arbitrary point $p\in \Omega$. Let us consider a normal coordinate system $(x_1,\dots,x_n)$ defined in a neighbourhood of $p$. Since Christoffel's symbols vanish at $p$, we have that $(\nabla^2 u)_{ij}(p)= \partial_i\partial_j u$ is a symmetric matrix. Accordingly, there exists an orthonormal basis $\{e_j\}_{j=1}^n$ of $T_pM$ which diagonalizes $(\nabla^2 u)_{ij}(p)$. Note that $\{e_j\}_{j=1}^n$ can be obtained by $\{\partial_j\}_{j=1}^n$ via an orthonormal matrix $A$. In particular, composing with $A$ we get a new normal coordinate system $(y_1,\dots,y_n)$ defined in a neighbourhood of $p$ with respect to which:
\begin{itemize}
\item $(0,\dots,0)$ represents $p$,
\item $g_{ij}(p)=\delta_{ij}$ and $g^{ij}(p)=\delta^{ij}$. Hence, in particular $|\nabla u|^2(p)=\sum_{j=1}^n(\partial_ju)^2=:\sum_{j=1}^nu_j^2$.
\item $\partial_ig_{jk}(p)=0$ and $\Gamma_{jk}^i(p)=0$. Hence, in particular $(\nabla^2 u)_{ij}(p)= \partial_i\partial_j u=:u_{ji}$.
\item $(\nabla^2 u)_{ij} (p) =u_{ij}(p)= \lambda_i\delta_{ij}$ for real numbers $\lambda_1,\dots,\lambda_n$ satisfying $\sum_{j=1}^n\lambda_j =0$ (because $\Delta u=0$).
\end{itemize}
In any point of this coordinate system we have $|\nabla u|^2=\sum_{i,j=1}^nu_iu_j g^{ij}$, so that at $p$
\begin{align*}
(\nabla |\nabla u|^2)_k(p)&=\sum_{i,j=1}^n\Big(u_{ik}(p)u_{j}(p) g^{ij}(p)+u_{i}(p)u_{jk}(p) g^{ij}(p)+u_{i}(p)u_{j}(p) \partial_k g^{ij}(p)\Big)\\
&=\sum_{i,j=1}^n\Big(\lambda_k\delta_{ik}u_{j}(p) \delta^{ij}+u_{i}(p)\lambda_k\delta_{jk}(p) \delta^{ij}\Big)\\
&=2\lambda_k u_k(p),
\end{align*}
and
\begin{equation}\label{hess1}
|\nabla u|^2(p)|\nabla |\nabla u||^2(p)=\frac{1}{4}|\nabla |\nabla u|^2|^2(p)= \sum_{k=1}^n \lambda_k^2 u^2_k(p).
\end{equation}
On the other hand
\[
|\nabla^2u|^2(p)=\sum_{i,j,k,l=1}^n (\nabla^2u)_{ij}(p)(\nabla^2u)_{kl}(p) g^{ik}(p)g^{jl}(p)=\sum_{k=1}^n \lambda_k^2,
\]
so that
\begin{equation}\label{hess2}
|\nabla u|^2(p)|\nabla^2 u|^2(p) = \sum_{k=1}^n\lambda_k^2\sum_{k=1}^n u_k^2(p).
\end{equation}
Now, if $n=2$, we have $\lambda_2^2=(-\lambda_1)^2$ and so
\[
|\nabla u|^2(p)|\nabla^2 u|^2(p) = 2\lambda_1^2(u_1^2(p)+u_2^2(p))=2(\lambda_1^2 u^2_1(p)+\lambda_2^2 u^2_2(p))= 2|\nabla u|^2(p)|\nabla |\nabla u||^2(p).
\]
From this, identity~\eqref{refined-Kato2} follows immediately. For general $n>2$, since $\sum_{k=1}^n\lambda_k=0$, we get for every $j$,
\[
\lambda_j^2=\frac{n-1}{n}\lambda_j^2 + \frac{1}{n}\Big(\sum_{k\neq j}\lambda_k\Big)^2 \leq \frac{n-1}{n}\lambda_j^2 + \frac{1}{n}(n-1)\sum_{k\neq j}\lambda^2_k = \frac{n-1}{n}\sum_{k=1}^n\lambda^2_k.
\]
By combining these inequalities with \eqref{hess1} and \eqref{hess2} we obtain estimate~\eqref{refined-Kato} and conclude the proof.
\end{proof}
\begin{proof}[Proof of Lemma \ref{lem:log}]
Since all assertions of the lemma are pointwise, we may apply the Korn--Lich\-ten\-stein theorem and locally change the coordinates to the conformal (isothermal) ones. The existence of the conformal coordinates on the Riemannian surfaces can be proven, e.g. by employing the Beltrami equation. Then, locally the given metric $g$ reads $g=\lambda(z)^2 dz^2$ for a smooth $\lambda>0$ and planar Euclidean coordinates $z=(x,y)$. Recall that the harmonicity of $u$ in dimension $2$ and the fact that $\nabla u$ is non-vanishing are conformal invariants. Thus, upon denoting the Laplacian and, respectively, the length of the gradient in the Euclidean coordinates by $\Delta_0$, respectively $|\nabla \cdot|_{0}$, the following computations hold true, completing the proof of the lemma:
\begin{align*}
\Delta (\log|\nabla u|)=\lambda^{-2} \Delta_0 \log (\lambda^{-1} |\nabla u|_{0})&=-\lambda^{-2} \Delta_0 \log (\lambda)+\lambda^{-2} \Delta_0 \log (|\nabla u|_{0}) \\
&=-\lambda^{-2} \lambda^{2} (-K)+0 =K.
\end{align*}
\end{proof}
\begin{rem} In dimensions $n>2$ the above approach is not efficient. However, one may instead prove that $\Delta |\nabla u|^\alpha\geq 0$ for $\alpha\geq \frac{n-2}{n-1}$ and $n\geq 2$ when the Ricci curvature is non-negative. Indeed, a direct computation using the curvature assumption, the Bochner formula \eqref{formula-Bochner} and the refined Kato inequality gives
\begin{align*}
\Delta |\nabla u|^\alpha &= \alpha |\nabla u|^{\alpha -2}\left[(\alpha -1)|\nabla|\nabla u||^2+|\nabla u|\Delta|\nabla u|\right]\\
& =\alpha |\nabla u|^{\alpha -2}\left[(\alpha -2)|\nabla|\nabla u||^2+
|\nabla^2 u|^2 + \Ric(\nabla u, \nabla u)
\right] \\
&\geq \alpha |\nabla u|^{\alpha -2}\Ric(\nabla u, \nabla u) \geq 0.
\end{align*}
The relation $\Delta |\nabla u|^\alpha\geq \alpha \kappa |\nabla u|^\alpha$ on manifolds with Ricci curvature lower bounded by $\kappa$ was used for instance in \cite[Theorem 2.1]{lw} to obtain some structure theorems and in \cite{dindos, MeVe} to study the Hardy spaces.
\end{rem}
\begin{rem}\label{rem: hd}
Suppose that in the setting of $n$-dimensional manifolds, for $n\geq 3$, one can prove the following variant of the Kato estimate
\begin{equation}\label{oq}
|\nabla^2u|^2\leq 2\left|\nabla|\nabla u|\right|^2+k|\nabla u|^2.
\end{equation}
Then, one could assume that $\Ric\leq k$ to obtain a variant of Lemma~\ref{lem:log} with the upper bound $\Delta(\log|\nabla u|)\leq 2k$. For this reason, we wonder whether there exists some class of assumptions implying the validity of \eqref{oq}.
\end{rem}
Recall that the measure of a level set of a function $v:\Om\to\R$ for a domain $\Om\subset M^n$ is given by
\begin{equation}\label{iso-in-length}
L(t)=\int_{\{x\in \Om\,:\,v(x)=t\}}1\,{\rm d}\mathcal{H}^{n-1},
\end{equation}
where ${\rm d}\mathcal{H}^{n-1}$ stands for the {$(n-1)$-Hausdorff} measure.
In the next lemma we prove formulas allowing us to compute the first and the second derivatives of $L$ with respect to the height of the level curve of harmonic functions in a topological annulus $\Om\subset M^n$. The lemma generalizes Lemma 2.1 in~\cite{al2}.
\begin{lem}\label{lem: L''}
Suppose that $u:\Om\to\R$ is a harmonic function satisfying $|\nabla u|>0$ in a topological annulus $\Om\subset M^n$ and that $u$ attains constant boundary values, respectively, $u|_{\Gamma_1}=t_1$ and $u|_{\Gamma_2}=t_2$.
Then, the following holds for all $t_1<t<t_2$:
\begin{align}
L'(t)&=\int_{\{x\in \Om\,:\,u(x)=t\}} \div\left(\frac{\nabla u}{|\nabla u|}\right)\frac{{\rm d}\mathcal{H}^{n-1}}{|\nabla u|}
=\int_{\{x\in \Om\,:\,u(x)=t\}} \left \langle -\frac{\nabla u}{|\nabla u|}, \frac{ \nabla |\nabla u|}{|\nabla u|^2} \right \rangle {\rm d}\mathcal{H}^{n-1}. \label{L-der1}
\end{align}
Moreover,
\begin{align}\label{L'' n}
L''(t) &\leq \int_{\{x\in \Om\,:\,u(x)=t\}} \left(\frac{2n-3}{n-1}\right)\frac{\big|\nabla |\nabla u|\big|^2}{|\nabla u|^4} - \frac{\Ric(\nabla u, \nabla u)}{|\nabla u|^4} {\rm d}\mathcal{H}^{n-1}.
\end{align}
Additionally, if $\Omega$ is a domain in a $2$-dimensional manifold with Gauss curvature $K=K(x)$, inequality \eqref{L'' n} holds in the following stronger form
\begin{align}
L''(t) &= \int_{\{x\in \Om\,:\,u(x)=t\}} \frac{\big|\nabla |\nabla u|\big|^2}{|\nabla u|^4} - \frac{K}{|\nabla u|^2} {\rm d}\mathcal{H}^1. \label{L-der2}
\end{align}
\end{lem}
In the proof of the lemma we employ the following Bochner formula for harmonic functions:
\begin{equation}\label{formula-Bochner}
\Delta \frac{|\nabla u|^2}{2}=|\nabla^2u|^2+\Ric(\nabla u, \nabla u).
\end{equation}
Moreover, upon noticing that
\[
\Delta \frac{|\nabla u|^2}{2}=\div\left(\nabla \frac{|\nabla u|^2}{2}\right)=|\nabla u|\Delta|\nabla u|+\left|\nabla|\nabla u|\right|^2
\]
we obtain
\begin{equation}\label{cor-Bochner}
\frac{\Delta|\nabla u|}{|\nabla u|}=\frac{|\nabla^2u|^2-\left|\nabla|\nabla u|\right|^2
}{|\nabla u|^2}+\frac{\Ric(\nabla u, \nabla u)}{|\nabla u|^2}.
\end{equation}
\begin{proof}
Since $|\nabla u|>0$ by assumptions, then $\nu=\frac{\nabla u}{|\nabla u|}$ is a unit vector normal to the level sets of $u$. Therefore, by the definition of the function $L$ in~\eqref{iso-in-length} and the Stokes theorem, we have that
\begin{align*}
L'(t)=\lim_{\ep \to 0}\frac{1}{\ep}\left(\int_{\{u=t+\ep\}} 1- \int_{\{u=t\}} 1\right) &=\lim_{\ep \to 0}\frac{1}{\ep}\left(\int_{\{u=t+\ep\}} \left \langle \nu, \frac{\nabla u}{|\nabla u|} \right \rangle - \int_{\{u=t\}} \left \langle \nu, \frac{\nabla u}{|\nabla u|} \right \rangle \right) \\
&= \lim_{\ep \to 0}\frac{1}{\ep} \int_{\{t<u<t+\ep\}} \div\left(\frac{\nabla u}{|\nabla u|}\right).
\end{align*}
At this stage we notice that the following integrals converge in $L^1$, as $\ep\to 0$:
\begin{equation}\label{conv-level-sets}
\frac{1}{\ep}\int_{\{t<u<t+\ep\}} 1 \to \int_{\{u=t\}}\frac{1}{|\nabla u|}.
\end{equation}
Indeed, since the limit on the left hand side above equals $\frac{d}{dt}\left(\int_{\{u<t\}} 1\right)$, the claim follows from the coarea formula; see e.g. Proposition 3 and its proof in~\cite[Chapter 3.4.4]{EGbook} or Exc. III.12(c) in \cite[Ch. 3]{chavel}. Similarly, the proof of the same proposition implies that
\[
\lim_{\ep \to 0}\frac{1}{\ep} \int_{\{t<u<t+\ep\}} \div\left(\frac{\nabla u}{|\nabla u|}\right)=\int_{\{u=t\}} \frac{1}{|\nabla u|}\div\left(\frac{\nabla u}{|\nabla u|}\right).
\]
Hence, the first equation in \eqref{L-der1} is proved. In order to see the second representation formula in~\eqref{L-der1} we directly compute that
\begin{equation} \label{lem:L''-ident}
\div\left(\frac{\nabla u}{|\nabla u|}\right)\frac{1}{|\nabla u|}=\frac{1}{|\nabla u|^3}(\Delta u |\nabla u|-\langle \nabla u, \nabla|\nabla u|\rangle).
\end{equation}
From this the aforementioned formula follows immediately for harmonic function $u$.
In order to show assertions~\eqref{L'' n} and \eqref{L-der2} we begin with the following computations involving~\eqref{L-der1} and the Stokes theorem:
\begin{align*}
L''(t)&=\lim_{\ep \to 0}\frac{L'(t+\ep)- L'(t)}{\ep} \\
&=\lim_{\ep \to 0}\frac{1}{\ep}\left(\int_{\{u=t+\ep\}} \left \langle -\frac{\nabla u}{|\nabla u|}, \frac{ \nabla |\nabla u|}{|\nabla u|^2} \right \rangle- \int_{\{u=t\}} \left \langle -\frac{\nabla u}{|\nabla u|}, \frac{ \nabla |\nabla u|}{|\nabla u|^2} \right \rangle \right)\\
&= \lim_{\ep \to 0}\frac{1}{\ep} \int_{\{t<u<t+\ep\}} -\div\left(\frac{\nabla |\nabla u|}{|\nabla u|^2}\right).
\end{align*}
We continue our computations as in the proof of~\eqref{L-der1} and employ the Bochner identity~\eqref{formula-Bochner} together with~\eqref{cor-Bochner} to obtain that:
\begin{align}
& \lim_{\ep \to 0}\frac{1}{\ep} \int_{\{t<u<t+\ep\}} -\div\left(\frac{\nabla |\nabla u|}{|\nabla u|^2}\right) \nonumber \\
&= \lim_{\ep \to 0}\frac{1}{\ep} \int_{\{t<u<t+\ep\}} \left[-\frac{\Delta |\nabla u|}{|\nabla u|^2} + 2 \frac{\left|\nabla |\nabla u|\right|^2}{|\nabla u|^3}\right] \nonumber \\
&=\lim_{\ep \to 0}\frac{1}{\ep} \int_{\{t<u<t+\ep\}} \left[-\frac{|\nabla^2 u|^2-\left|\nabla |\nabla u|\right|^2}{|\nabla u|^3} - \frac{\Ric(\nabla u, \nabla u)}{|\nabla u|^3}+ 2 \frac{\left|\nabla |\nabla u|\right|^2}{|\nabla u|^3}\right] \nonumber \\
&=\lim_{\ep \to 0}\frac{1}{\ep} \int_{\{t<u<t+\ep\}} \left[-\frac{|\nabla^2 u|^2-2\left|\nabla |\nabla u|\right|^2}{|\nabla u|^3} - \frac{\Ric(\nabla u, \nabla u)}{|\nabla u|^3}+ \frac{\left|\nabla |\nabla u|\right|^2}{|\nabla u|^3}\right]. \label{L-der2-est}
\end{align}
By applying the $n$-dimensional refined Kato inequality at \eqref{L-der2-est} we get
\[
-\frac{|\nabla^2 u|^2}{|\nabla u|^3} - \frac{\Ric(\nabla u, \nabla u)}{|\nabla u|^3}+ 3 \frac{\left|\nabla |\nabla u|\right|^2}{|\nabla u|^3}\leq (2-\frac{1}{n-1}) \frac{\left|\nabla |\nabla u|\right|^2}{|\nabla u|^3} - \frac{\Ric(\nabla u, \nabla u)}{|\nabla u|^3}.
\]
Upon applying this inequality in \eqref{L-der2-est} and using the coarea formula, conclusion \eqref{L'' n} follows readily, cf. the discussion following derivation of~\eqref{conv-level-sets}.
Now assume that $n=2$, so that $\Ric(\nabla u, \nabla u)=K|\nabla u|^2$. By applying the $2$-dimensional refined Kato identity~\eqref{refined-Kato2} we find that
\[
L''(t)= \lim_{\ep \to 0}\frac{1}{\ep} \int_{\{t<u<t+\ep\}} \frac{\left|\nabla |\nabla u|\right|^2}{|\nabla u|^3}- \frac{K}{|\nabla u|}.
\]
Using again the coarea formula we arrive at the estimate~\eqref{L-der2}.
\end{proof}
We are now in a position to present one of the main results of this section and of the whole paper. Namely, in Theorems~\ref{thm-main1} and~\ref{thm-main2} we provide a characterization of annular domains with Gaussian curvature $K\leq 0$ in terms of the log-convexity of level sets for harmonic functions. The corresponding results for non-smooth surfaces with bounded integral curvature are discussed in Section 3.
The following theorem gives a counterpart of Alessandrini's result~\cite[Theorem 1.1]{al2} for harmonic functions on non-positively curved Riemannian $2$-manifolds. Recall that we do not assume that the curvature is constant, i.e. we present the argument for $K=K(x)$ for $x\in M^2$. Notice that, following \cite{al2,long}, the second part of the assertion justifies the name isoperimetric inequality.
\begin{theorem}[Isoperimetric inequality]\label{thm-main1}
Let $\Omega$ be a $C^{1,\alpha}$-topological annular domain in a $2$-di\-men\-sional Riemannian manifold $(M^2, g)$ of non-positive curvature $K|_\Om \leq 0$. Let $t_1,t_2\in \R$ be such that $t_1<t_2$ and let us consider a continuous up to the boundary harmonic solution $u$ of the Dirichlet problem~\eqref{DP} in $\Om$.
Then
\[
(\ln L(t))''\geq 0\quad\hbox{for all } t\in (t_1,t_2).
\]
Moreover, the equality in the assertion holds on $(t_1,t_2)$ if and only if $K\equiv 0$, in which case all the level curves of $u$ are concentric circles and $\Om$ is the regular (circular) annulus in the plane.
\end{theorem}
\begin{rem}\label{rem-c1a-reg}
The $C^{1,\alpha}$-regularity assumption on the boundaries of topological annuli in subject is a consequence of the interior ball condition assumed in the Hopf lemma used in the proof of Theorem~\ref{thm-main1}, cf.~\cite[3.71]{Aubin} and the discussion in~\cite[Chapter 3.2]{gt}. It is known that the $C^{1,\alpha}$-regularity characterizes domains with both interior- and exterior- ball conditions. Notice further that \cite{al2} assumes the $C^{2,\alpha}$-regularity and \cite{mz} the $C^{\infty}$-regularity of the boundary. However, in the general setting of BIC surfaces, see Theorem~\ref{thm-reshetnyak} below, we may weaken the regularity assumption and prove Theorem \ref{thm-main1} assuming that $\Gamma_i$ for $i=1,2$ are merely Jordan curves (see Remark \ref{r: Jordan bdy}). In this section we decided to assume higher regularity on boundary components of annuli for two reasons: (1) it gives a simpler proof than the one in Theorem~\ref{thm-reshetnyak} based on geometric arguments and so it may be more appealing to the reader not interested in the BIC setting; (2) the $C^{1, \alpha}$-regularity assumption would be more natural in higher dimensions, and with different operators such as the $p$-Laplacian, and thus allows to approach generalizations of Theorem~\ref{thm-main1}.
\end{rem}
Before presenting the proof of Theorem~\ref{thm-main1} we would like to comment that an important part of its proof, as in Alessandrini's result~\cite[Theorem 2.1]{al2}, is showing that the gradient of harmonic function in subject does not vanish. In the manifold setting of space forms a similar observation is proven in~\cite[Prop 3.2]{mz}, but it relies on some other auxiliary results. Moreover, the proof of~\cite[Theorem 1.1]{al2} could be obtained in our case by the analytic approach via the uniformization result for annuli (see Lemma~\ref{unif-annulus}) and harmonic functions in the plane, see Appendix A. The proof below is self-contained and refers to simple geometric arguments instead.
\begin{proof}
Let us notice that by the maximum and minimum principles for the harmonic function $u$ we have that $\max_{\overline{\Om}} u=t_2$ and
$\min_{\overline{\Om}} u=t_1$. Moreover, by the Hopf lemma one gets that $|\nabla u|\geq const>0$ on $\partial \Om$, see~\cite[3.71, pg. 96]{Aubin}.
We first show that $|\nabla u|\geq const>0$ on $\Omega$. On the contrary, let us suppose that there exists $x_0\in \Om$ such that $\nabla u(x_0)=0$. Consider the corresponding level curve $\gamma=\{x\in \Om: u(x)=u(x_0)\}$.
\emph{Claim:} There exists at least two simple closed curves $\gamma'_i\subset \gamma$, $i=1,2$.
\emph{Proof of the claim:} The discussion can be localized in a neighbourhood $U_{x_0}\subset\Omega$ of the point $x_0$. Therefore, we introduce isothermal coordinates $(x,y)$ induced by the conformal chart $\phi:U_{x_0}\to\R^2$ (cf. also the discussion in the proof of Lemma~\ref{lem:log}). Since the harmonicity of $u$ is a conformal invariant in dimension $2$, we have that $\Delta_0 u=\Delta u=0$, where $\Delta_0$ denotes the planar Laplacian in the coordinates $(x,y)$. Since $\phi(x_0)$ remains a critical point for $u\circ \phi^{-1}$, we know by the theory of planar harmonic functions that the level curve in the neighbourhood of $\phi(x_0)$ forms a finite family consisting of at least two arcs intersecting at $\phi(x_0)$.
This follows from the analyticity of harmonic functions in $\R^2$: indeed, the Taylor expansion of a harmonic function in the neighbourhood of a critical point starts with terms of order determined by number of derivatives vanishing at this point, see~\cite{hawi} and~\cite[Section 2.1]{dur}.
Thus, for points on the level curve $\gamma$ we obtain, via $\phi^{-1}$, that there are at least two curves passing through $x_0$ contained in $\gamma$. If any of those branches would intersect $\partial \Om$, then by the assumption of continuity of $u$ up to the boundary, it would hold that $u(x_0)=t_1$ (or $u(x_0)=t_2$), hence the maximum of $u$ (or, respectively, minimum of $u$) would be attained in the interior of $\Om$, forcing $u=const$ by the strong maximum (respectively, minimum) principle. This is impossible, since $t_1\not =t_2$.
Next, we rule out the possibility that the level curve $\gamma$ terminates at a point inside $\Om$. Indeed, suppose that there exists $y_0\in \gamma \cap \Om$, where $\gamma$ terminates, and consider two cases.
If $\nabla u(y_0)\not=0$, then the implicit function theorem implies that $\gamma$ can not terminate inside $\Om$, since it must be at least $C^1$ in a neighbourhood of $y_0$.
If $\nabla u(y_0)=0$, then by the discussion above, $\gamma$ would branch at $y_0$, contradicting assumption that it terminates there.
To summarize, since $\gamma$ does not intersect $\partial \Om$ and does not terminate in $\Om$, it must contain at least two simple closed curve, denoted $\gamma'_i$, $i=1,2$, obtained by gluing regular curves (contained in $\gamma$). This ends the proof of the claim.
Since none of the curves $\gamma'_1$ and $\gamma'_2$ touches the boundary of $\Omega$, a topological argument together with the maximum principle allow us to infer that at least one of them bounds a domain $\Om'\Subset \Om$. Hence, we conclude that $\nabla u\not=0$ in $\Om$.
We can thus apply Lemma~\ref{lem:log} obtaining that $\Delta (\log|\nabla u|)\leq 0$, and so the minimum principle together with the Hopf lemma implies that
$ \min_{\Om} |\nabla u|\ge \min _{\partial \Om} |\nabla u|>0$.
We are in a position to complete the proof of the first part of the theorem. By Lemma~\ref{lem: L''} and the Cauchy--Schwarz inequality we have that
\begin{align}
(L'(t))^2&\leq L(t) \int_{\{x\in \Om\,:\,u(x)=t\}} \left|\left \langle -\frac{\nabla u}{|\nabla u|}, \frac{ \nabla |\nabla u|}{|\nabla u|^2} \right \rangle\right|^2 {\rm d}\mathcal{H}^1 \nonumber \\
&\leq L(t) \int_{\{x\in \Om\,:\,u(x)=t\}} \frac{ |\nabla |\nabla u||^2}{|\nabla u|^4} {\rm d}\mathcal{H}^1 \label{step-main-thm1} \\
&\leq L(t)L''(t). \nonumber
\end{align}
In order to show the second part of the assertion, suppose that $(\ln L(t))''=0$. This is equivalent to $LL''=(L')^2$ which then by the Cauchy--Schwarz and the H\"older inequalities reads
\begin{align*}
L(t) \left(\int_{\{x\in \Om\,:\,u(x)=t\}} \frac{\big|\nabla |\nabla u|\big|^2}{|\nabla u|^4} - \frac{K}{|\nabla u|^2} {\rm d}\mathcal{H}^1\right)&=\left(\int_{\{x\in \Om\,:\,u(x)=t\}} \left \langle -\frac{\nabla u}{|\nabla u|}, \frac{ \nabla |\nabla u|}{|\nabla u|^2} \right \rangle {\rm d}\mathcal{H}^1\right)^2\\
& \leq L(t)\left(\int_{\{x\in \Om\,:\,u(x)=t\}} \frac{ |\nabla |\nabla u||^2}{|\nabla u|^4}{\rm d}\mathcal{H}^1\right)
\end{align*}
Since $K\leq 0$, this inequality may hold only when $K\equiv 0$ in which case we reduce the discussion to the planar case and so Theorem 3.1 in~\cite{long} gives the second assertion of the theorem (see also \cite[Theorem 1.1]{al2}).
\end{proof}
Next we prove that the assertion of Theorem~\ref{thm-main1} in fact characterizes open surfaces with negative curvature.
\begin{theorem}\label{thm-main2}
Let $(M^2,g)$ be an open surface. Then $K\leq 0$ in $M^2$ if and only if for every topological annulus domain $\Omega$ in $M$ with $C^{1,\alpha}$-boundary, and every real constants $t_1<t_2$ the following property is satisfied:
given the continuous up to the boundary harmonic solution $u$ of the Dirichlet problem~\eqref{DP}
it holds $(\ln L(t))''\geq 0$ for all $t\in (t_1, t_2)$.
\end{theorem}
The $C^{1,\alpha}$-regularity assumption on boundary of annuli follows from the analogous assumption in Theorem~\ref{thm-main1}, see Remark~\ref{rem-c1a-reg} following the statement of that theorem.
\begin{proof}
The sufficiency of the condition $K\leq 0$ is proven in Theorem~\ref{thm-main1}. In order to show its necessity, we will argue by contradiction and suppose that there exists a point $x_0\in M$ where $K(x_0)>0$. This will imply that there exist a harmonic function and an annular domain $\Om$, determined at the end of this proof, with $(\ln L(t))''< 0$.
By the Korn-Lichtenstein theorem, we can choose an isothermal chart $(U,\phi)$ where $U$ is an open set of $M$ containing $x_0$ and $\phi:U\to V:=\phi(U)\subset \R^2$ is a diffeomorphism such that $\phi(x_0)=0$. The metric $g$ pulled back to this coordinate system writes $(\phi^{-1})^\ast g(y)=\lambda^2(y)dy^2$ for some smooth conformal factor $\lambda:V\to (0,\infty)$. Moreover, up to a composition with a linear isometry of $\R^2$, we can suppose that $\lambda(0)=1$ and $\nabla\lambda(0)=0$. In particular,
\begin{equation*}
\lambda(y)=1+\sum_{i,j=1}^2A_{ij}y_iy_j+o(|y|^2),
\end{equation*}
where $2A$ is the Hessian matrix $\nabla^2\lambda (0)$ written in this coordinate system.
For later purposes, we compute
\[
\partial_r\lambda(y)=\nabla\lambda (y) \cdot \frac y{|y|} = \frac{2}{|y|} y^TAy + o(|y|) = \nabla^2 \lambda(0) \left(y,\frac y{|y|}\right)+ o(|y|),
\]
where $\displaystyle \lim_{y\to 0}\frac {o(|y|)}{|y|}=0$. Similarly,
\[
\partial^2_{rr}\lambda(y)=\nabla(\partial_r\lambda(y)) \cdot \frac y{|y|} = \frac{2}{|y|^2} y^TAy + o(1)= \nabla^2 \lambda(0) \left(\frac y{|y|},\frac y{|y|}\right)+ o(1).
\]
Let us consider on $V\setminus \{0\}$ the (Euclidean) harmonic function $-\ln(|y|)$, and observe that $x\mapsto -\ln(|\phi(x)|)$ is harmonic on $(U,g)$ by the conformal invariance of the harmonicity on surfaces.
Using polar coordinates in $\R^2$, we express
\[\begin{aligned}
L(t)&=\int_{\{u=t\}}\lambda\,d\mathcal H^1=\int_{\partial B_{e^{-t}}(0)} \lambda\,d\mathcal H^1=
e^{-t}\int_{0}^{2\pi}\lambda (e^{-t},\theta) \,d\theta,\\
L'(t)&=-e^{-t}\int_{0}^{2\pi}\lambda (e^{-t},\theta) \,d\theta - e^{-2t}\int_{0}^{2\pi}\partial_r\lambda (e^{-t},\theta)\, d\theta,\\
L''(t)&=e^{-t}\int_{0}^{2\pi}\lambda (e^{-t},\theta) \,d\theta +3 e^{-2t}\int_{0}^{2\pi}\partial_r\lambda (e^{-t},\theta)\, d\theta + e^{-3t}\int_{0}^{2\pi}\partial^2_{rr}\lambda (e^{-t},\theta)\, d\theta.
\end{aligned}
\]
By the asymptotic behaviour of $\lambda$, we compute
\[
\begin{aligned}
\int_{0}^{2\pi}\lambda (e^{-t},\theta) \,d\theta &= 2\pi+\frac12e^{-2t}\int_{0}^{2\pi} \nabla^2 \lambda(0)(\theta,\theta) \,d\theta + o(e^{-2t})\\
\int_{0}^{2\pi}\partial_r\lambda (e^{-t},\theta)\, d\theta &= e^{-t}\int_{0}^{2\pi} \nabla^2 \lambda(0)(\theta,\theta) \,d\theta + o(e^{-t})\\
\int_{0}^{2\pi}\partial^2_{rr}\lambda (e^{-t},\theta)\, d\theta &= \int_{0}^{2\pi} \nabla^2 \lambda(0)(\theta,\theta) \,d\theta + o(1)
\end{aligned}\]
as $t\to\infty$. In the equalities above, with an abuse of notation we write $\theta$ for a point of polar coordinates $(1,\theta)$ when $\theta$ appears as an argument of $\nabla^2\lambda$. By inserting the latter equations in the previous relations, we obtain that
\begin{equation}\label{eq-main-thm2}
(L(t))^2(\ln L(t))'' = L(t)L''(t)- (L'(t))^2 = 4\pi e^{-4t}\int_{0}^{2\pi} \nabla^2 \lambda(0)(\theta,\theta) \,d\theta + o(e^{-4t}).
\end{equation}
Now, we note the following facts:
\begin{itemize}
\item by a direct computation, $\int_{0}^{2\pi} \nabla^2 \lambda(0)(\theta,\theta) \,d\theta = \pi\Delta\lambda (0)$,
\item by the conformal change formula for the Gaussian curvature we have $\Delta(\ln\lambda)=-\lambda^2K$, so that $\Delta\lambda(0)=-K(x_0)$.
\end{itemize}
Recall that we assumed $K(x_0)>0$, and thus obtain $(\ln L(t_0))''\leq 0$ for $t_0$ large enough by~\eqref{eq-main-thm2}.
Choosing $t_1<t_0<t_2$, $\Omega:= \phi^{-1}(B_{e^{-t_1}}(0)\setminus B_{e^{-t_2}}(0))$ and $u=-\ln(|\phi(x)|)$ gives the contradiction.
\end{proof}
Theorem~\ref{thm-main2} can be further refined if the Gauss curvature satisfies $K\leq \kappa\leq 0$.
\begin{prop}\label{prop sharp K}
Let $(M^2,g)$ be an open surface. Then $K\leq \kappa \leq 0$ in $M^2$
if and only if for every topological annulus domain $\Omega$ in $M$ with $C^{1,\alpha}$-boundary, and every real constants $t_1<t_2$ the following property is satisfied:
given the continuous up to the boundary harmonic solution $u$ of the Dirichlet problem~\eqref{DP} it holds that
\begin{align}\label{est sharp K}
(\ln L(t))''&\geq
-\frac{\kappa }{L(t)}\int_{\{u=t\}}\frac{1}{|\nabla u|^2} d\mathcal H^1,\quad \hbox{for } t_1<t<t_2.
\end{align}
\end{prop}
\begin{rem}
The above proposition is sharp at least in the case of surfaces with constant curvature. In Example~\ref{ex hyperb sf} below, we construct a harmonic function $u$ on a hyperbolic surface for which the equality holds in~\eqref{est sharp K}.
\end{rem}
\begin{proof}
For the sufficiency part of the proposition, notice that the assertion follows by repeating the steps of the proof for Theorem~\ref{thm-main1} and by using equation~\eqref{L-der2} in ~\eqref{step-main-thm1}.
In order to show the necessity of the assertion let us follow the steps of the corresponding part of the proof for Theorem~\ref{thm-main2} and argue by contradiction. Then, equation~\eqref{eq-main-thm2} together with facts following it allows us to obtain that
\[
L(t)L''(t)- (L'(t))^2 = - 4\pi^2 e^{-4t}K(x_0).
\]
On the other hand $|\nabla u| (e^{-t},\theta)= \lambda (e^{-t},\theta) e^{t}$, so that
\[
\int_{u=t}\frac{1}{|\nabla u|^2}\, d\mathcal H^1 =
e^{-t} \int_{0}^{2\pi} \frac{e^{-2t}}{\lambda^2 (e^{-t},\theta)}\,d\theta=
2\pi e^{-3t} + o(e^t) .
\]
Hence,
\[
L(t)L''(t)- (L'(t))^2 \geq - \kappa L(t)\int_{u=t}\frac{1}{|\nabla u|^2}\, d\mathcal H^1
=
-\kappa 4\pi^2 e^{-4t} + o(e^{-4t}) .
\]
This gives a contradiction if $K(x_0)>\kappa $ at some point $x_0\in M^2$.
\end{proof}
It turns out that the inequality in Theorem~\ref{thm-main1} can be quantified in the setting of surfaces with pinched curvature, provided that the harmonic function defined on the annular domain is the restriction of a globally defined function which is harmonic on the whole surface.
We first prove the following proposition and then discuss an example of class of surfaces and harmonic functions satisfying the assumptions of the proposition.
\begin{prop}\label{thm-main2-2}
Let $(M^2,g)$ be a complete non-compact surface whose Gauss curvature satisfies
\begin{equation}\label{assm-thm-main2-2}
-\kappa _1\leq K \leq -\kappa_2 \leq 0,
\end{equation}
for some $\kappa _1\geq \kappa_2\ge 0$ and let $u>0$ be harmonic on $M^2$. Then for every topological annulus domain $\Omega$ in $M$ with $C^{1,\alpha}$-boundary, and such that $u$ takes constant values $0<t_1<t_2$ on the boundary components of $\Om$, it holds that
\[
(\ln L(t))''\geq \frac{\kappa_2 }{\kappa _1}\frac{1}{t^2},\quad \hbox{for } t_1<t<t_2.
\]
\end{prop}
\begin{proof} Let $u>0$ be harmonic in $M^2$ satisfying~\eqref{assm-thm-main2-2} and $\Om\Subset M^2$ be an annulus such that $u$ takes constant values, respectively $t_1$ and $t_2$, on the boundary components of $\Om$, cf.~\eqref{def-top-ring}. By Proposition \ref{prop sharp K}
\begin{align}\label{com-est}
(L'(t))^2&\leq
L(t)\left(L''(t)-\kappa_2 \int_{u=t}\frac{1}{|\nabla u|^2} d\mathcal H^1\right).
\end{align}
Then, the second claim of Thm 1.1 in~\cite{pli} for $M^2$ and $\lambda=0$ and $m=n=2$ asserts that
$|\nabla u|^2\leq \kappa _1 |u|^2$ for all points in $M^2$. Therefore,
\[
-\kappa_2\int_{u=t}\frac{1}{|\nabla u|^2} d\mathcal H^1\leq -\frac{\kappa_2 }{\kappa _1}\int_{u=t}\frac{1}{|u|^2} d\mathcal H^1=-\frac{\kappa_2 }{\kappa _1}\frac{1}{t^2}L(t).
\]
This combined with inequality~\eqref{com-est} gives the assertion.
\end{proof}
\begin{ex}\label{ex hyperb sf}
In order to construct an example of $2$-manifolds satisfying the assumptions of Proposition~\ref{thm-main2-2}, let us consider the $2$-dimensional infinite cylinder endowed with a warped Riemannian metric defined by
\[
(S, g)=(\R\times\mathbb{S}^1, \ud t^2+\left(\frac{\ln\lambda}{2\pi}\cosh t\right)^2
\ud \theta^2).
\]
By standard computations, the Gaussian curvature of $S$ equals $K=-\frac{(\cosh t)''}{\cosh t}=-1$, so that $(S,g)$ is a hyperbolic surface. Indeed, it is isometric to the quotient of the
hyperbolic plane $H^2:=(\R\times (0,\infty), \frac{dx^2+dy^2}{y})$ by the cyclic group of isometries generated by the isometry $f:H^2\to H^2$ defined as $f(x):=\lambda x$ for a given $\lambda>1$.
A positive non-constant harmonic function $u:S\to (0,\pi)$ can be explicitly defined as $u(t,\theta):=2\arctan(e^t)$. Indeed, upon computing directly the Laplace--Beltrami operator $\Delta_Su$ we find that
$$
\Delta_Su=\left(\partial^2_{tt}+\frac{(\cosh t)'}{\cosh t}\partial_t+\frac{4\pi^2}{\ln^2\lambda\cosh^2t}\partial^2_{\theta \theta}\right)u=0.
$$
Notice that for annular domains in $S$ defined as $[a,b]\times \mathbb{S}^1$ with $-\infty<a<b<\infty$, $u$ is constant at each boundary component of such an annulus.
Finally we observe that $L(s)=\ln\lambda\cosh (u^{-1}(s))=\frac{\ln\lambda}{\sin s}$ and so it holds that
$$
(\ln L(s))''=\frac{1}{\sin^2s}=\frac{1}{L(s)}\int_{\{u=s\}}\frac{1}{|\nabla u|^2} d\mathcal H^1\geq 1.
$$
Notice that this also proves the sharpness of Proposition~\ref{prop sharp K}.
More generally, the existence of a non-trivial positive harmonic function can be proved by using a theory introduced by P. Li and L.-F. Tam.
By Theorem 2.1 in~\cite{li-tam} a surface $S$ supports a non-constant global positive harmonic function provided that $S$ has at least two ends, and at least one of them is non-parabolic (cf. Definitions 0.3-0.5 in~\cite{li-tam}). In the example above, as we will now argue, $S$ has exactly two non-parabolic ends. Indeed, the set $S\setminus B^S_R((0,0))$, for large enough $R$, consists of two components. Moreover, one can estimate that ${\rm Vol}(B^S_R((0,0)))\gtrsim \ln\lambda e^R+const$ and therefore
$$
\int^{\infty} \frac{R}{{\rm Vol}(B^S_R((0,1)))}\,\ud R<\infty,
$$
which implies that both ends are non-parabolic (see ~\cite{holop} for the details).
This in particular suggests that the example above is stable under small perturbations of the metric.
\end{ex}
\section{Surfaces with bounded integral curvature}
In this section we focus our attention on surfaces with bounded integral curvature (\textsl{BIC surfaces} in short), also known in the literature as Alexandrov surfaces. This class represents a very important example of non-smooth (singular) spaces. Indeed, among BIC surfaces we find, for instance, the polyhedral surfaces (both Euclidean, hyperbolic, spherical or more generally Riemannian), the metric surfaces with curvature either lower or upper bounded in the sense of Alexandrov, and thus in particular the Gromov--Hausdorff limits of surfaces with lower bounded Gaussian curvature and $RCD(k,2)$ surfaces. Moreover, the BIC surfaces have a number of nice structure properties which permit to develop a differential calculus, including a singular (in some sense) Riemannian metric. We refer to~\cite{re, tr-AnnIHP, tr} for comprehensive introductions to the theory of Alexandrov surfaces, see also the more recent ~\cite{ab, fi, klp} and references therein. A significant class of BIC surfaces consists of surfaces with conical singularities, i.e.~surfaces which are smooth Riemannian except for a discrete set of singular points where the metric structure is at an infinitesimal level the one of a Euclidean cone; see for instance~\cite{bdm, ch, de, tr2} and their references. The results of this section apply to general Alexandrov surfaces with non-positive curvature measure (see details below), including surfaces with conical singularities whose angles at the vertices are greater than $2\pi$, and surfaces of $CAT(0)$ type.
We start with recalling the basic definition of a \emph{surface of bounded integral curvature}. Let $(S,d)$ be a compact topological surface $S$ endowed with a metric function $d:S\times S \to [0,+\infty)$ which induces the topology of $S$ and is geodesic, i.e. $(S,d)$ is a geodesic metric space.
The surface $(S,d)$ is said to have \emph{bounded integral curvature}, if there exists a sequence of smooth Riemannian metrics $(g_j)$ on $S$ such that all $\int_S |K_{g_j}|\,d\mu_j$ are uniformly bounded and $d_j\to d$ uniformly in $j$. Here $K_{g_j}$ and $\mu_j$ denote, respectively, the Gaussian curvature and the Riemannian area measure of $(S, g_j)$, and $d_j$ is the distance induced by $g_j$ on $S$.
The \textsl{a posteriori} unique weak limit of the sequence of measures $(K_{g_j}\mu_j)$, denoted by $\omega$, is called the \emph{curvature measure of $(S,d)$}.
There exist several non-trivially equivalent definitions of BIC surface which are based for instance on polyhedral approximations (instead of Riemannian ones), or on a direct definition of $\omega$ as an outer measure exploiting the Gauss-Bonnet theorem at the infinitesimal level. Here, we are in particularly interested in the following analytic characterization.
Following \cite{tr}, let $S$ be a closed surface (i.e. compact surface without boundary) and $V (S, h)$ be the space of functions $u:S\to\R$ such that $\mu=\Delta_hu$ is a measure (note that for us a measure is a signed measure, i.e. not necessarily positive). For every $v\in V (S, h)$ and every $x,y \in S$, we set
\[
d_{h,v}(x, y) = \inf\left\{ \int_0^1 e^{v(\alpha(t))}\sqrt{h(\dot\alpha(t),\dot\alpha(t))} \,dt\right\},
\]
where the infimum is taken over all the Lipschitz paths $\alpha: [0,1] \to S$ such that $\alpha(0) = x$ and $\alpha(1) = y$. It turns out that $d_{h,v}$ is indeed a distance \cite[Proposition 5.3]{tr}. In analogy with the smooth case, by using a common notation, we say that the distance $d$ on $S$ is induced by the (singular) Riemannian metric $e^{2v}h$; see also \cite{ab} for results about the regularity of this latter Riemannian metric. Since the harmonicity of a function is a conformal invariant in dimension two, the above relation permits to define harmonic functions on open subsets of $(S,d)$ as the functions which are harmonic with respect to the metric $h$. More precisely, we define the Laplace-Beltrami operator of $(S,d)$ by
\[
\Delta_S:=\Delta_{(S,d)}:=e^{-2v}\Delta_h.
\]
Recall that a surface $(S,d)$ with bounded integral curvature is said to be without cusps, if $\omega(\{x\})< 2\pi$ for all $x\in S$. In particular, both the $BIC$ surfaces with non-positive curvature measure and all the smooth surfaces are automatically without cusps. Therefore, surfaces studied in Section 2 under the assumption $K\leq 0$ are covered by our discussion here as well.
We have the following theorem due to \cite{re-Sib} (see also~\cite[Theorem 7.1]{tr} and \cite{hu}).
\begin{theorem}\label{BIC-riem}
Let $(S,d)$ be a closed surface with bounded integral curvature without cusps. Then, there exist a smooth Riemannian metric $h$ on $S$ and a function $v\in V(S,h)$ such that $d=d_{h,v}$.
\end{theorem}
Here we obtain the following generalization of Theorem~\ref{thm-main1} to the non-smooth case.
\begin{theorem}\label{thm-reshetnyak}
Let $(S,d)$ be a surface of bounded integral curvature $(S,d)$ with non-positive curvature measure $\omega$, and let $\Omega$ be a non-degenerate topological annulus domain in $(S,d)$ whose boundary components are the Jordan curves $\Gamma_i$, $i=1,2$. Let $t_1,t_2\in \R$ be such that $t_1<t_2$ and let us consider a continuous up to the boundary solution of the following Dirichlet problem in $\Om$ for the Laplace--Beltrami harmonic operator $\Delta_S$ on $S$:
\[
\begin{cases}
\Delta_{S} u=0 & \hbox{in } \Om,\\
u|_{\Gamma_1}=t_1, & u|_{\Gamma_2}=t_2.
\end{cases}
\]
Then $t\mapsto \log L(t)$ is a convex function on $[t_1,t_2]$.
\end{theorem}
\begin{rem}\label{r: Jordan bdy}
As a special case of the Theorem~\ref{thm-reshetnyak}, we get a generalization of Theorems~\ref{thm-main1} and \ref{thm-main2} to non-degenerate annular domains in smooth Riemannian manifolds whose boundary consists of Jordan curves, i.e. no further smoothness of boundary curves is assumed. According to our knowledge this result is new also in the Euclidean case.
\end{rem}
\begin{rem} It is natural to conjecture that the converse of Theorem \ref{thm-reshetnyak} is also true, i.e.~that the function $L(t)$ is convex for every solution of the Dirichlet problem on topological annulus domain only if the BIC surface has non-positive curvature. In particular, this would generalize Theorem \ref{thm-main2} to the singular setting. This appear as a quite difficult question, due to the possible presence of a purely singular positive part of the curvature measure (i.e.,~supported on a set of Hausdorff dimension strictly smaller than $2$).
\end{rem}
In the proof of the theorem we need the following uniformization lemma for a non-degenerate topological annulus on a BIC surface. Recall that, by \textsl{non-degenerate} we mean an annulus not homeomorphic to a punctured disc.
\begin{lem}\label{unif-annulus}
Let $(S,d)$ a compact surface with bounded integral curvature without cusp. Let $\Om\Subset S$ be a non-degenerate topological annulus in $(S,d)$. Then $(\Om, d_\Om)$ is isometric to the annulus $A_{1, R}:=\{1< |z|< R\}\subset \C$, for some $R>1$, endowed with a conformally flat metric. More precisely, there exists a homeomorphism $\phi:A_{1, R}\to \Omega$ and a function $\bar{v}\in V(A_{1, R}, g_0)$ such that
\[
d_\Omega(\phi(x),\phi(y))=d_{g_0,\bar{v}}(x,y),\quad \hbox{for all }x,y\in A_{1,R},
\]
where $g_0$ stands for the Euclidean metric $g_0=\ud x ^2+\ud y^2$.
\end{lem}
Here $d_\Omega$ stands for the intrinsic metric induced by $d$ on $\Omega$ and, as above, $V(A_{1, R},g_0)$ is the space of functions $u:A_{1,R}\to\R$ such that $\mu=\Delta_hu$ is a (signed) measure. Moreover, for every $v\in V (A_{1, R},g_0)$ and every $x,y \in S$, we set
\[d_{g_0,v}(x, y) = \inf\left\{ \int_0^1 e^{v(\alpha(t))}\sqrt{g_0(\dot\alpha(t),\dot\alpha(t))} \,dt\right\},
\]
where the infimum is among all the Lipschitz paths $\alpha$ in $A_{1,R}$ connecting $x$ to $y$.
\begin{proof}[Proof of Lemma~\ref{unif-annulus}]
By Theorem \ref{thm-reshetnyak}, there exist a smooth Riemannian metric $h$ on $S$ and a function $v\in V(S,h)$ such that $d=d_{h,v}$. Since $\Omega$ is non-degenerate, by approximation, we can find an open collared neighbourhood $\Omega'$ of $\Omega$ such that $\Omega\Subset \Omega'\subset S$ and $\partial \Omega'$ is smooth in $(S,h)$. Then, there exists a smooth Riemannian metric $\tilde h$ on $\mathbb S^2$ such that $(\Omega',h)$ embeds isometrically in $(\mathbb S^2,\tilde h)$, see e.g. Theorem A in \cite{pv}.
According to the uniformization theorem for surfaces, there exists a $\psi\in C^\infty(\mathbb S^2)$ such that $\tilde h= e^{2\psi}g_1$, $g_1$ being the round metric on $\mathbb S^2$ of constant Gaussian curvature $1$. Hence, up to isometries, $(\Omega,d_\Omega)$ can be identified with a non-degenerate topological annular domain $\Omega_{\mathbb S^2}$ of $\mathbb S^2$ endowed with the metric $e^{2(\psi+v)}g_1$.
Finally, let $P\in \mathbb S^2\setminus \overline{\Omega_{\mathbb S^2}}$. The stereographic projection from $P$ induces a conformal diffeomorphism of $(\Omega_{\mathbb S^2},g_1)$ onto $(\Omega_{\C},g_0)$, for some non-degenerate topological annular domain $\Omega_{\C}\Subset \C$. For later purposes, note that this conformal diffeomorphism can be defined on a slightly larger open set properly containing $\Omega_{\mathbb{S}^2}$ giving rise via the aforementioned stereographic projection to an open set $\hat\Omega_\C\Supset \Omega_\C$. In order to complete the proof, we apply a version of the Riemann mapping theorem to obtain that $\Omega_{\C}$ is conformal to $A_{1,R}\subset \C$ for some $R>1$, see Theorem 10 in \cite[Chapter 6]{ah}. All together, we have obtained the existence of an isometry $\phi:(A_{1,R},e^{2\bar{v}}g_0)\to (\Omega,d)$ for some conformally flat metric $e^{2\bar{v}}g_0$, where $\bar{v}:A_{1,R}\to \R$ is such that
$\Delta_{g_0}\bar{v}$ is a measure on $A_{1,R}$. Note that $\bar{v}$ is non-smooth in general, so that $e^{2\bar{v}}g_0$ is not Riemannian. However, it is a well-defined length metric in which the length
of any constant-speed curve $\alpha:[0.1]\to A_{1,R}$ is given by the formula $\int_0^1 e^{\bar{v}(\alpha(t))}\sqrt{g_0(\dot\alpha(t),\dot\alpha(t))}\,dt$.
\end{proof}
\begin{rem}
We encourage readers less familiar with the BIC setting to see the presentation in the analytic proof of Theorem~\ref{thm-main1} in Appendix A, especially computations for $L$.
\end{rem}
\begin{proof}[Proof of Theorem~\ref{thm-reshetnyak}]
By Lemma~\ref{unif-annulus}, we have the existence of an isometry $\phi$ from the annulus $A_{1,R}:=\{1<|z|<R\}\subset \mathbb C = \R^2$ endowed with the conformally flat metric $e^{2\bar{v}} g_0$ onto the topological annular domain $(\Omega, h)$. Here $g_0:=dx^2+dy^2$, $R>1$ and $\bar{v}:A_{1,R}\to\R$ is such that $\bar{v}\in V(A_{1,R},g_0)$, i.e. $\Delta_{g_0}\bar{v}$ is a measure. More precisely, $\bar v$ is obtained as $\bar v = \hat v\circ F$ where $F$ is a conformal univalent mapping from $A_{1,R}$ to $\Omega_\C$ and $\hat v\in V(\hat\Omega_\C,g_0)$; see the proof of Lemma~\ref{unif-annulus}.
According to \cite[Corollary 6.3]{tr}
, we know that the curvature measure $\omega$ of $(S,d)$ is pulled-back via the isometry $\phi$ precisely to the measure $(\phi)^\ast(\omega)=-e^{-2\bar v}\Delta_{g_0}\bar{v}$ on $A_{1,R}$. With an abuse of notation from now on, for the sake of simplicity, we will denote by $\omega$ the measure $(\phi)^\ast(\omega)$ defined on $A_{1,R}$.
By a similar reasoning, $(\hat\phi)^\ast(\omega)=-e^{-2\hat v}\Delta_{g_0}\hat{v}$ on $\hat \Omega_\C$, where $\hat\phi$ is the conformal map from $\hat \Omega_\C$ to $(S,d)$.
Let $u$ be the harmonic function in the statement of the theorem. Up to a vertical shift, we can assume without loss of generality that $t_2=0$. Then, up to a multiplicative constant, we can assume that $t_1 = -\ln R$. Note that both of these transformations do not affect the conclusion of the theorem. Thus, $u$ is the unique harmonic function on $A_{1,R}$ with boundary data $0$ and $-\ln R$, namely $u=-\ln |z|$.
If the curvature measure $\omega$ of $(S,d)$ is non-positive, then $\bar{v}$ verifies
\[
\Delta_0\bar{v} \geq 0,
\]
in the weak sense. More precisely, by definition of the distributional Laplacian \cite[Chapter 4.3]{ArGa}, one has
\[
\int_{A_{1,R}} \bar{v} \Delta_0\psi \,d\lambda \ge0
\]
for all $0\le \psi\in C^\infty_c(A_{1,R})$.
Furthermore, \cite[Theorem 4.3.10]{ArGa} implies that $\bar v$ is subharmonic (see also \cite[p. 99]{re}), meaning that $\bar{v}$ is upper semicontinuous in $A_{1,R}$, $\bar{v}\not\equiv-\infty$ on $S$, and that $\bar{v}$ satisfies the subharmonic mean value property for each ball compactly contained in $A_{1,R}$, cf. \cite[Definition 3.1.2]{ArGa}. Similarly, $\hat v$ is subharmonic on $\hat \Omega_\C$, and it is there upper semicontinuous.
We are going to prove that $L(t)$ is a convex function on $[t_1, 0]$.
First, we \textsl{claim} that $L(t)$ is convex on any subinterval $I_\epsilon:=(t_1+\epsilon,-\epsilon)$, for $0<\epsilon<\frac12\ln R$. Note that $L(t)$ can be expressed in the integral form as follows
\begin{equation}\label{eq:lengthBIC}
L(t) = \int_{|z|=e^{-t}}e^{\bar v}\,d\sigma,
\end{equation}
see, for instance, \cite[Example 1.5]{de} and the corresponding computations for $L(t)$ in Appendix A.
By definition of $u$ it holds that $\Omega_{\epsilon/2}:=u^{-1}(I_{\epsilon/2})\Subset A_{1,R}$.
Then, by \cite[Theorem 3.3.3]{ArGa}, there exists a decreasing sequence $(s_j)$ of subharmonic functions defined in $A_{1,R}$ and smooth in $\Omega_{\epsilon/2}$ for sufficiently large $j$, constructed by mollifications (cf. \cite[(3.3.1)]{ArGa}), such that $(s_j)$ converges pointwise to $\bar v$ on $\Omega_{\epsilon/2}$. We associate with $(s_j)$ a sequence $(g_j)$ of smooth Riemannian metrics on $\Omega_{\epsilon/2}$ conformally defined by $g_j := e^{2s_j}g_0$. By the proof of~\cite[Theorem 3.3.3]{ArGa} we have that $s_j$ are subharmonic for large enough $j$ such that $1/j<\dist(\Om_\epsilon, \partial A_{1,R})\leq e^{-t_1}(e^{\epsilon/2}-1)$. Let us denote the smallest of such $j$ by $j_0$ and note that $j_0$ depends only on $\epsilon$. Therefore, for $j\geq j_0$ the smooth metrics $g_j$ have nonpositive Gaussian curvatures $K_{g_j}$:
\[
K_{g_j}=(-\Delta_{0}s_j)\, e^{-2s_j}\leq 0.
\]
Since the harmonicity is a conformal invariant in dimension $2$, we get that the harmonic function $u$ given by assumptions of Theorem~\ref{thm-reshetnyak} is harmonic in $\Om_\epsilon$ also with respect to metric $g_j$ for every $j\geq j_0$. Moreover, the annular domain $\Omega_\epsilon$ has smooth boundary in $(\Omega_{\epsilon/2},g_j)$. Applying Theorem~\ref{thm-main1}, we get that $t\mapsto L_j(t)$ is a convex function on $I_\epsilon$. Here,
\[
L_j(t) = \int_{|z|=e^{-t}}e^{s_j}\,d\sigma
\]
is the length of the level set $\{u=t\}$ with respect to the metric $g_j$. Observe that the sequence of functions $(e^{s_j})$ is monotone non-increasing, locally uniformly bounded above by $e^{s_{j_0}}$ and trivially bounded below by $0$, and converges pointwise to $e^{\bar v}$.
Hence, by the monotone convergence theorem we get that $L_j(t)\to L(t)$ pointwise on $I_\epsilon$.
To conclude the proof of the claim, we observe that the pointwise limit of a sequence of convex functions is convex as well.
Since $\epsilon>0$ is arbitrary we straightforwardly deduce that $L(t)$ is convex on open interval $(t_1,0)$.
It remains to prove the convexity of $L(t)$ at the boundary points $t_1$ and $0$, hence on the whole interval $[t_1,0]$. To this end, we note that $\bar v$ (so far defined on $A_{1,R}$) is indeed well-defined and upper semicontinuous on the closure $\overline{A_{1,R}}$. Indeed, $F$ can be continuously extended to a map $\bar F:\overline{A_{1,R}}\to\overline{\Omega_\C}$ by a version of the Osgood--Carath\'eodory Theorem, see \cite[Section 8]{OsTa} or \cite[Theorem 2.8]{GaSh}. Since, $\hat v$ is upper semicontinuous on $\hat\Omega_\C\supset \overline{\Omega_\C}$, we may define $\bar v := \hat v\circ \bar F$ on $\overline{A_{1,R}}$. The convexity at the boundary points is thus a simple consequence. Namely, for $t<0$ we have
\begin{align*}
L(0)&=\int_{|z|=1} e^{\bar v(z)}\,d\sigma
\\&= \int_{|z|=1} e^{\bar v(z)}\,d\sigma -\int_{|z|=e^{-t}} e^{\bar v(z/|z|)}\,d\sigma + \int_{|z|=e^{-t}} e^{\bar v(z/|z|)}\,d\sigma - \int_{|z|=e^{-t}} e^{\bar v(z)}\,d\sigma +\int_{|z|=e^{-t}} e^{\bar v(z)}\,d\sigma \\
& = \int_{|z|=1} e^{\bar v(z)}\,d\sigma -\int_{|z|=1} e^{-t}e^{\bar v(z)}\,d\sigma + \int_{|z|=e^{-t}} \left[e^{\bar v(z/|z|)}- e^{\bar v(z)}\right]\,d\sigma + L(t).
\end{align*}
Taking the $\limsup$ as $t\to 0^-$ gives $L(0)\ge \limsup_{t\to 0^-} L(t)$. Thus $L(t)$ is convex on $(t_1,0]$. A similar argument permits to deal with the other boundary point $t_1$.
\end{proof}
\begin{rem}\label{rmk:AG}
Notice that, since the integrand in \eqref{eq:lengthBIC} is the exponential of a subharmonic function, one can deduce the log-convexity of $L$ in $I_\epsilon$ from \cite[Theorem 3.5.7(ii)]{ArGa}. However, for the readers convenience and in order to present a self-contained argument, we decided to provide the complete discussion in the proof of Theorem \ref{thm-reshetnyak}.
\end{rem}
\section{Curvature of level sets and PDEs}
The purpose of this section is to study some curvature and length estimates for the level curves of harmonic functions on Riemannian $2$-manifolds. Similar studies in the setting of planar harmonic functions and more general quasilinear equations have been conducted by e.g. Alessandrini~\cite{al} and Talenti~\cite{tal}. In the setting of manifolds, let us mention results by Ma--Qu--Zhang~\cite{moz}, Ma--Zhang~\cite{mz} and Wang--Wang~\cite{ww}. The key results in those papers are obtained assuming that the Gaussian curvature of the surface is constant, or on higher-dimensional space-forms . One of the main novelties of our work is to allow the curvature to vary. Also, we study more types of curvatures of level sets than in the aforementioned papers; see the discussion before the statement of the main result of this section, Theorem~\ref{thm-pdes}. In its corollaries we investigate the weak and strong maximum and minimum principles for curvatures of level curves (Corollaries~\ref{c: k}\,--\,\ref{c:cor3.7}) and further estimates for the length of level curves and its derivative (Corollary~\ref{c:cor3.8-2ndversion}).
We begin the presentation of the results with a technical lemma, which provides the key identities for computations in the proof of Theorem~\ref{thm-pdes}. The second assertion of the following lemma generalizes~\cite[Lemma 2.3]{ww} to the setting of non-constant sectional curvature (see identities (2.3) and (2.10) in~\cite[Lemma 2.1]{cm} for the proof of the lemma).
\begin{lem}\label{lem-rr}
Let $u:M^n\to \R$ be a smooth function on a Riemannian manifold $M^n$ (with possibly non-constant sectional curvature). Then
\begin{align}
u_{ijk}&=u_{ikj}+u_l R_{lijk} \label{lem-rr1}\\
\sum_k u_{ijkk}&=\sum_ k u_{kkij}-\sum_{l,m} 2u_{lm}R_{iljm}\nonumber\\
&+\sum_l\left[\Ric_{il}u_{lj}+\Ric_{jl}u_{li}+u_l(\nabla_i\Ric)_{lj}+u_l(\nabla_j\Ric)_{li}-u_l(\nabla_l\Ric)_{ji}\right],
\label{lem-rr2}
\end{align}
where $R_{ijkl}$ and $\Ric_{ij}$ stand, respectively, for the coefficients of the curvature tensor and of the Ricci tensor, while a function with subscript indices, e.g. in the expression $u_{ij}$, denote the (higher order) covariant derivative of the function with respect to a local orthonormal frame.
\end{lem}
Next, we generalize the main result of Wang--Wang~\cite[Theorem 1.3]{ww} and also Theorem 3 in Talenti~\cite{tal} in several different directions:
\begin{itemize}
\item[(1)] we study Riemannian manifolds with non-constant Gauss curvature $K=K(x)$ for $x\in M^2$, whereas in~\cite{ww} it is assumed that $K\equiv const$.
\item[(2)] While~\cite{ww} investigates only the case of \emph{the curvature of curves of the steepest descent} $h$ we study both $h$ and \emph{the curvature of the level curves} $k$:
\begin{equation}\label{eq-k-h}
h:=\div\left(\frac{\star \nabla u}{|du|}\right),\qquad k:=-\div\left(\frac{\nabla u}{|\nabla u|}\right).
\end{equation}
Here $\star$ denotes the Hodge star operator acting on tangent vectors. If we fix a local orthonormal frame $\{e_1,e_2\}$ and write $u_i=e_i(u)$, $i=1,2$, then $h$ can be written in the more friendly equivalent form
\[
h:=-\div\left(\frac{(u_2,-u_1)}{|\nabla u|}\right).
\]
\item[(3)] Equations~\eqref{pde1} and~\eqref{pde1-star} generalize the harmonicity result obtained for the planar case and $K=0$, see in~\cite[Theorem 3(i)]{tal}, while inequalities~\eqref{pde2} and \eqref{pde2-star} generalize the subharmonicity property for $h$ and $k$ in~\cite[Theorem 3(ii)]{tal}.
\item[(4)] The maximum and minimum principles for $\frac{k}{|\nabla u|}$ and $\frac{h}{|\nabla u|}$ presented in Corollaries~\ref{c: k} and~\ref{cor-up-lo-bound_2ndversion} below, generalize Corollary 1.1 in~\cite{ww} to the setting of surfaces with non-constant Gauss curvature.
\end{itemize}
\begin{theorem}\label{thm-pdes}
Let $\Om\subset M^2$ be a domain in the Riemannian manifold with Gaussian curvature $K$ (not necessarily constant). Let further $u:\Om\to \R$ be a harmonic function with no critical points in $\Om$. Then the function $\phi=\frac{k}{|\nabla u|}$ satisfies the following differential equation:
\begin{equation}\label{pde1}
\Delta \phi+2K\phi=\frac{\langle \nabla K,\nabla u\rangle}{|\nabla u|^2}.
\end{equation}
A similar equation holds true for $\frac{h}{|\nabla u|}$:
\begin{equation}\label{pde1-star}
\Delta \left(\frac{h}{|\nabla u|}\right)+2K\frac{h}{|\nabla u|}=-\frac{\langle \nabla K,\star\nabla u\rangle}{|\nabla u|^2}.
\end{equation}
Moreover, it holds that
\begin{equation}\label{pde2}
-\Delta \ln |k| \geq K-\frac{1}{|k|}\langle \nabla K,\frac{\nabla u}{|\nabla u|}\rangle,\quad k\not=0.
\end{equation}
A similar inequality holds true for $|h|$ at points where $h\not=0$:
\begin{equation}\label{pde2-star}
-\Delta \ln |h| \geq K+\frac{1}{|h|}\langle \nabla K,\frac{\star \nabla u}{|\nabla u|}\rangle,\quad h\not=0.
\end{equation}
\end{theorem}
The proof of the above theorem is largely computational and, therefore, we present it in Appendix B. Here instead we focus on discussing some consequences and applications of Theorem~\ref{thm-pdes}.
First, let us comment on conditions implying that a harmonic function has no critical points in the domain of its definition.
As observed in the previous section, a harmonic solution $u$ to the Dirichlet problem with constant boundary data on an annular domain satisfies $|\nabla u|>0$. Moreover, let $u$ solve the harmonic Dirichlet problem in a domain $\Om\subset M^2$ (not necessarily an annulus) and $u\in C^1(\overline{\Om})$. According to the discussion following Theorem 3.2 in~\cite{mag} in the planar Euclidean case (zero curvature), one can estimate the number of critical points of $u$ inside $\Om$ in terms of information given by prescribed tangential, normal, co-normal, partial, or radial derivatives of $u$. This is due to the flexibility of assumptions in \cite[Theorem 3.2]{mag}; we also refer to Theorems 2.1 and 2.2 in~\cite{almag} for further details. In particular, the appropriate Neumann condition may result in no critical points of $u$. For instance, consider the planar unit disc, a vector field $l=z/|z|$ and any boundary data for a harmonic Dirichlet problem such that $u_l$ has two zeros on the unit circle. Then, according to ~\cite[Theorem 3.2]{mag} (and with its notation), it holds that $M^+=M^-=1$ and by the argument principle for $l$ we have $D=1$. Thus, $u$ has no critical points inside the disc.
As commented in \cite[Section 3.2]{mag}, corresponding results can be obtained for harmonic functions defined on domains in Riemannian $2$-manifolds $M^2$.
We proceed now to present some of the consequences of Theorem~\ref{thm-pdes}.
\begin{cor}\label{c: k}
Let $u$ be harmonic in a domain $\Om\subset M^2$ and $\Om'\Subset \Om$ be any subdomain such that $\overline{\Om'}$ does not contain critical points of $u$. If $K\geq 0$ and $\langle \nabla K,\nabla u\rangle\leq 0$ in $\Om'$, then $|k|$ attains its minimum on the boundary of $\Om'$ (provided that $k\neq 0$ in $\Omega$).
Similarly, if $K\geq 0$ and $\langle \nabla K,\star \nabla u\rangle\geq 0$ in $\Om'$, then $|h|$ attains its minimum on the boundary of $\Om'$ (provided that $h\neq 0$ in $\Omega$).
\end{cor}
\begin{proof}
By assumptions of the corollary, we may infer from~\eqref{pde2} that
\[
0\le K\leq -\Delta \ln |k| + \frac{\langle \nabla K,\nabla u\rangle}{|\nabla u|\,|k|}\leq -\Delta \ln |k|.
\]
Then, the minimum principle for the inequality $\Delta \ln |k| \leq 0$ implies that $|k|$ attains its minimum on the boundary of $\Om'$. The assertion involving $h$ is proved similarly.
\end{proof}
The minimum and the maximum principles for equation~\eqref{pde1} imply the following observation. Note that by assumptions of the corollary below, $|\nabla u|$ is bounded from below and above in $\overline{\Om'}$ for $\Om'\Subset \Om$, since $\Om$ is a domain and thus bounded by the definition.
\begin{cor}\label{cor-up-lo-bound_2ndversion}
Let $u$ be harmonic in a domain $\Om\Subset M^2$ such that $\overline{\Om}$ does not contain critical points of $u$.
\begin{enumerate}
\item Let $K\le 0$ and $\langle\nabla K, \nabla u\rangle\!\ge\! 0$. If the maximum of $|\nabla u|^{-1}k$ is non-negative, then it is attained on $\partial \Omega$.
\item Let $K\le 0$ and $\langle\nabla K, \nabla u\rangle \le 0$. If the minimum of $|\nabla u|^{-1}k$ is non-positive, then it is attained on $\partial \Omega$.
\item Let $K\ge 0$ and $\langle\nabla K, \nabla u\rangle \ge 0$. If the maximum of $|\nabla u|^{-1}k$ is non-positive, then it is attained on $\partial \Omega$.
\item Let $K\ge 0$ and $\langle\nabla K, \nabla u\rangle\!\le 0$. If the minimum of $|\nabla u|^{-1}k$ is non-negative, then it is attained on $\partial \Omega$.
\end{enumerate}
The same conclusions hold for $|\nabla u|^{-1}h$ up to replacing $\langle\nabla K, \nabla u\rangle$ with $\langle\nabla K, -\star \nabla u\rangle $ in the assumptions.
\end{cor}
The above corollary together with Corollary~\ref{c: k} generalize~\cite[Corollary 1.1]{ww}, where the case of constant $K$ and only the curvatures $h$ and $\frac{h}{|\nabla u|}$ are considered.
\begin{ex}
The class of surfaces and harmonic functions satisfying assumptions of Corollary~\ref{cor-up-lo-bound_2ndversion}~(1) contains, for instance, all conformally Euclidean annular domains, with the conformal factor satisfying $\Delta\varphi\geq 0 (\Leftrightarrow K\leq 0)$ and $\partial_r (e^{2\varphi}\Delta\varphi)\leq 0$. The latter condition is equivalent to $\langle \nabla u, \nabla K \rangle\geq 0$, as upon a conformal deformation and for a constant boundary data, it holds that $\nabla u$ is parallel to $\partial_r$ resulting in $\partial_r K=\partial_r (-e^{2\varphi}\Delta\varphi)$. Similar classes of examples can be constructed also for the assumptions of Corollary~\ref{cor-up-lo-bound_2ndversion}~(2), (3) and (4).
\end{ex
Next, we obtain a minimum principle for the functions $k$ and $h$.
\begin{cor}\label{c:cor3.7}
Let $u$ be harmonic in a domain $\Om\Subset M^2$ such that $\overline{\Om}$ does not contain critical points of $u$. If nonconstant $k$ attains a minimum at a point $y$ in the interior of $\Omega$, then at $y$ one has necessarily $k\le |\nabla K|/K$.
Similarly, if nonconstant $h$ attains a minimum at a point $y$ in the interior of $\Omega$, then at $y$ one has necessarily $h\le |\nabla K|/K$.
\end{cor}
Let us note that this corollary generalizes the second assertion in~\cite[Corollary 1.1]{ww} to the setting of non-constant Gaussian curvature and curvature $k$. In particular, our observation gives the minimum principle for $k$ for harmonic functions on sphere $\mathbb{S}^2$ not covered by~\cite{ww}.
\begin{proof}
The Schwartz inequality implies that~\eqref{pde2} reads ($k>0$ by our assumptions):
\[
-\Delta \ln k +\frac{|\nabla K|}{k}\geq K.
\]
We proceed by contradiction and suppose that $k>|\nabla K|/K$ in a neighbourhood of the minimum $y$. Set $v:=\frac{1}{k}>0$. Then $v \in C^{\infty}$ and upon substituting $v$ in the differential inequality above, one gets
\begin{equation}
0\leq \Delta v - \frac{|\nabla v|^2}{v}+|\nabla K|v^2 -Kv
\leq \Delta v \end{equation}
Thus by the Hopf strong maximum, see~\cite[Theorem 3.5]{gt}, we conclude that $k$ is constant, contradicting assumptions of the corollary. Hence, we get the assertion of the corollary for $k$. The reasoning for $h$ follows the same lines.
\end{proof}
The maximum principle allows us to infer also the following upper bound for the growth of the function $L$.
\begin{cor}\label{c:cor3.8-2ndversion}
Let $u$ be a harmonic solution to the Dirichlet problem~\eqref{DP} in a topological $C^{1,\alpha}$-annulus domain $\Om\subset M^2$ with constant boundary data $t_1$ and $t_2$, respectively.
If $K\leq 0$ and $\langle\nabla K,\nabla u\rangle\leq 0$ in $\Omega$,
then for every level curve $\{u=t\}$ contained it the interior of $\Om$ it holds
\[
\left(\ln L(t) \right)' \leq \max\{-\inf_{\partial \Om} (|\nabla u|^{-1}k);0\}.
\]
Similarly, if $K\geq 0$, $\langle\nabla K,\nabla u\rangle\leq 0$ and $k\ge 0$ in $\Omega$,
then for every level curve $\{u=t\}$ contained it the interior of $\Om$ it holds
\[
\left(\ln L(t) \right)' \leq -\inf_{\partial \Om'} (|\nabla u|^{-1}k).
\]
\end{cor}
\begin{proof}
Suppose first that the minimum of $|\nabla u|^{-1}k$ is non-positive. Then, by assertion (2) of Corollary~\eqref{cor-up-lo-bound_2ndversion} we have that $\inf_{\Om} |\nabla u|^{-1}k=\inf_{\partial \Om} |\nabla u|^{-1}k\leq 0$. Hence, by formula~\eqref{L-der1} for $L'$, we have
\[
L'(t)=\int_{\{x\in \Om\,:\,u(x)=t\}} \div\left(\frac{\nabla u}{|\nabla u|}\right)\frac{{\rm d}\mathcal{H}^1}{|\nabla u|}=
\int_{\{x\in \Om\,:\,u(x)=t\}} \left(-\frac{k}{|\nabla u|}\right) {\rm d}\mathcal{H}^1\leq -\inf_{\partial \Om} (|\nabla u|^{-1}k) L(t).
\]
If the minimum of $|\nabla u|^{-1}k$ is positive, then $\sup(-|\nabla u|^{-1}k)<0$, and so in the above estimate we get that $L'(t)<0$ for all $t_1<t<t_2$.
In order to prove the second assertion we apply part (4) of Corollary~\eqref{cor-up-lo-bound_2ndversion} and the similar reasoning as above.
\end{proof}
We finish this section by applying to our setting a maximum principle due to Mugnai-Pucci~\cite{mupu}. With respect to our previous results the conclusion is somehow weaker. However, here no sign assumptions on $K$ or $\nabla K$ are required.
\begin{cor}\label{cor-up-lo-bound2}
Let $u$ be harmonic in a domain $\Om\Subset M^2$ such that $\overline{\Om}$ does not contain critical points of $u$.
If $\|K\|_{L^{\infty}(\Om)}, \|\nabla K\|_{L^{\infty}(\Om)}<\infty$ and $|\nabla u|$ is bounded from below in $\Om$, then the following weak maximum principle holds for $|\nabla u|^{-1} k$:
\begin{equation}
\sup_{\Om} (|\nabla u|^{-1}k) \leq \left(1+c_1(|\Om|)\|K\|_{L^{\infty}(\Om)}\right) \sup_{\partial \Om} (|\nabla u|^{-1}k)+c_1(|\Om|)c_2, \label{weak-max-k2}
\end{equation}
where $c_2=\|(|\nabla u|^{-1} k)^{+}\|_{L^2(\Om)}+\||\nabla K|\|_{L^{\infty}(\Om)}\|\frac{1}{|\nabla u|}\|_{L^{\infty}(\Om)}+\|K\|_{L^{\infty}(\Om)}^{\frac12}$.
Moreover, the corresponding weak maximum principles hold also for $|\nabla u|^{-1} h$ with estimate~\eqref{weak-max-k2} modified accordingly, i.e.
\[
\inf_{\Om} (|\nabla u|^{-1}k) \geq \left(1+c_1(|\Om|)\|K\|_{L^{\infty}(\Om)}\right) \inf_{\partial \Om} (|\nabla u|^{-1}k)-c_1(|\Om|)c_2.
\]
Similarly, the analogous weak minimum principles corresponding to estimate~\eqref{weak-max-k2} hold for $|\nabla u|^{-1} k$ and $|\nabla u|^{-1} h$ as well.
\end{cor}
In the planar setting, estimate \eqref{weak-max-k2} for $|\nabla u|^{-1}k$, as well as the corresponding one for $|\nabla u|^{-1}h$, hold in a stronger version without the constants due to harmonicity of these expressions, cf.~\cite[Theorem 3(i)]{tal} .
\begin{proof}
Let us discuss the case of $\phi:=|\nabla u|^{-1} k$, keeping in mind that the discussion for $\phi:=|\nabla u|^{-1} h$ is analogous.
By \cite[Theorem 3.1]{mupu}, and following the notation in~\cite{mupu}, we have that ${\bf A}(x, \phi,\nabla \phi)=A(\nabla \phi):=\nabla \phi$ and by smoothness of $\phi$ it holds that $A(\nabla \phi)\in L_{loc}^p(\Om, T\Om)$ for any $p>1$, hence in particular $\phi$ is $2$-regular. Moreover,
\[
\langle A(\zeta), \zeta \rangle = |\zeta|^2 \quad \hbox{ and }\quad
B(x,z,\zeta):=2K(x)z+\frac{|\nabla K(x)|}{|\nabla u(x)|}\leq b_2|z|+b,
\]
with $b_2, b$ defined accordingly in terms of the $L^{\infty}$-norms of $K, \nabla K$ and $|\nabla u|^{-1}$. Then, the assertion of \cite[Theorem 3.1]{mupu} says that
\begin{align*}
\sup_{\Om} |\nabla u|^{-1}k &\leq \left(1+C(|\Om|)\|K\|_{L^{\infty}(\Om)}\right) \sup_{\partial \Om} |\nabla u|^{-1}k \\
&\,\,+C(|\Om|)\left(\|(|\nabla u|^{-1} k)^{+}\|_{L^2(\Om)}+\||\nabla K|\|_{L^{\infty}(\Om)}\|\frac{1}{|\nabla u|}\|_{L^{\infty}(\Om)}+\|K\|_{L^{\infty}(\Om)}^{\frac12}\right).
\end{align*}
For the weak minimum principle for $\phi=|\nabla u|^{-1}k$ observe that $-\phi$ satisfies equation~\eqref{pde1} with the negative right-hand side. We apply the maximum principle to $-\phi$ obtaining \eqref{weak-max-k2} with constant $-c_1(|\Om|) c_2$.
\end{proof}
\section*{Appendix A: an analytic proof of Theorem \ref{thm-main1}}
By a version of the uniformization theorem (see Lemma~\ref{unif-annulus}), we have that the annular domain $\Omega$ is isometric to the annulus $\{1< |z|< R\}\subset \mathbb C = \R^2$ endowed with the conformally flat metric $g=e^{2\varphi(z)}dzd\bar z = e^{2\varphi(x,y)}(dx^2 + dy^2)$. Here $R>1$ and $\phi$ is some smooth real function defined on $\{1\leq |z|\leq R\}$.
By the conformal invariance of harmonicity in dimension $2$, in this isometrically equivalent representation $u$ is a solution to
\[
\begin{cases}
\Delta_0 u=0,\\
u|_{\{|z|=1\}}=t_1,\\
u|_{\{|z|=R\}}=t_2.
\end{cases}
\]
Up to a horizontal translation we can assume that $t_1=0$ and up to a multiplicative constant we can assume that $t_2 = -\ln R$, cf. the corresponding discussion in the proof of Theorem~\ref{thm-main2}). Also, note that both these transformations do not affect the conclusion of the theorem. Accordingly, we can suppose without the loss of generality that $u(z)=-\ln |z|$.
Hence, we may compute $L(t)$ and its derivatives in terms of the conformal factor $\vp$ only. Namely, let $d\si$ denote the $1$-dimensional Hausdorff measure and $d\lambda$ the Lebesgue measure in $\R^2$. Moreover, set $r(z)=|z|$ and $\partial_rf=\langle \nabla_0f, \nabla_0r \rangle$, where $\nabla_0$ and $\langle\cdot,\cdot\rangle$ are the Euclidean gradient and the Euclidean inner product, respectively. Then
\[
L(t)= \int_{\{|z|=e^{-t}\}}e^{\vp} d\si = e^{-t}\int_{\{|z|=1\}}e^{\varphi(e^{-t}\theta)}\, d\si(\theta).
\]
As observed in Remark \ref{rmk:AG}, at this point Theorem \ref{thm-main1} could be deduced from an abstract result in subharmonic functions theory, \cite[Theorem 3.5.7 ii]{ArGa}. However, since $\varphi$ is smooth, we present here a simple self-contained proof.
First, we compute
\[\begin{aligned}
L'(t)&=-e^{-t}\int_{\{|z|=1\}}e^{\varphi(e^{-t}\theta)}\, d\si(\theta)-e^{-2t}\int_{\{|z|=1\}}e^{\varphi(e^{-t}\theta)}\partial_r\vp(e^{-t}\theta)\, d\si(\theta)\\
&=-\int_{\{|z|=e^{-t}\}}e^{\varphi(\theta)}\, d\si(\theta)-\int_{\{|z|=e^{-t}\}}re^{\varphi(\theta)}\partial_r\vp(\theta)\, d\si(\theta)\\
&= \int_{\{|z|=e^{-t}\}} r^2e^{2\vp(\theta)}\Big\langle\nabla_0r, \nabla_0\left(\frac{e^{-\vp(\theta)}}{r}\right)\Big \rangle\,d\si(\theta).
\end{aligned}
\]
Moreover,
\[
\begin{aligned}
L''(t)&=\lim_{\epsilon\to 0}\frac{1}{\epsilon}\left[\int_{\{|z|=e^{-t-\epsilon}\}} r^2e^{2\vp}\left \langle \nabla_0r, \nabla_0\left(r^{-1} e^{-\vp}\right)\right \rangle\,d\si - \int_{\{|z|=e^{-t}\}} r^2e^{2\vp}\left \langle\nabla_0r, \nabla_0\left(r^{-1} e^{-\vp}\right)\right \rangle\,d\si\right]\\
&=\lim_{\epsilon\to 0}\frac{1}{\epsilon}\int_{\{e^{-t-\epsilon}\leq|z|\leq e^{-t}\}}-\div_0\left(r^2e^{2\vp}\nabla_0\left(r^{-1} e^{-\vp}\right)\right)\,d\lambda\\
&=\lim_{\epsilon\to 0}\frac{1}{\epsilon}\int_{\{e^{-t-\epsilon}\leq|z|\leq e^{-t}\}} e^{\vp}\left(\frac{1}{r} + 2\partial_r\vp+r|\nabla_0\vp|^2 + r \Delta_0\vp\right)\,d\lambda\\
&=\lim_{\epsilon\to 0}\frac{1}{\epsilon}\int_{\{e^{-t-\epsilon}\leq|z|\leq e^{-t}\}} e^{3\vp}\left( r^3 \left|\nabla_0\left(r^{-1} e^{-\vp}\right)\right|^2-rK\right)\,d\lambda\\
&=\int_{\{|z|=e^{-t}\}}e^{3\vp}\left( r^4 \left|\nabla_0\left(r^{-1} e^{-\vp}\right)\right|^2-r^2K\right)\,d\sigma
\end{aligned}
\]
with $\div_0$ standing for the Euclidean divergence.
Combining the formulas above and using the Cauchy-Schwartz inequality we obtain
\[
\begin{aligned}
(L'(t))^2&\leq \left(\int_{\{|z|=e^{-t}\}} e^{\vp}\,d\si\right)\left( \int_{\{|z|=e^{-t}\}} r^4e^{3\vp}\left| \nabla_0\left(r^{-1} e^{-\vp}\right)\right|^2\,d\si\right)\\
&= L(t)\left(L''(t)+ \int_{\{|z|=e^{-t}\}}e^{3\vp}r^2K\,d\sigma \right),
\end{aligned}
\]
which concludes the proof, since $K\leq 0$.
\section*{Appendix B: Proof of Theorem~\ref{thm-pdes}}
For the proof of equation~\eqref{pde1} we closely follow the discussion in~\cite{ww}. Therefore, since the original proof is based on direct computations for $h$ and $K\equiv const$, we will focus on emphasizing differences and novelties in our computations for $k$ (and $h$) and non-constant $K$. Moreover, our approach is slightly more direct and the proof of inequality~\eqref{pde2} is shorter than the corresponding one in the proof of~\cite[Theorem 1.3]{ww}.
Since our aim is to show that equation~\eqref{pde1} holds pointwise, we may localize the discussion and introduce an orthonormal frame in a neighbourhood $U$ of $x\in \Om$. Covariant derivatives are thus computed with respect to this frame. Without the loss of generality we can suppose that
\begin{equation}\label{norm-coord1}
|\nabla u|(x)=u_2(x)>0,\quad u_1(x)=0.
\end{equation}
Moreover, $\det g(x)=1$ and the matrix of metric $g$ at $x$ is the identity.
The harmonicity of $u$ implies that $u_{22}=-u_{11}$ in $U$. Then, by~\eqref{eq-k-h} one computes that
\[
k=-\frac{u_{11}(u_2^2-u_{1}^2)-2u_1u_2u_{12}}{|\nabla u|^3}:=\frac{f}{|\nabla u|^3}.
\]
In the chosen coordinate system, the following identities hold at $x$:
\begin{equation}\label{norm-coord2}
k=-\frac{u_{11}}{u_2},\quad f=-u_2^2u_{11},\quad |\nabla u|_{i}=u_{2i},\quad \left(\frac{1}{|\nabla u|}\right)_{i}=-\frac{u_{2i}}{u_2^2},\quad |\nabla u|_{ii}=\frac{u_{1i}^2}{u_2}+u_{2ii}\quad\hbox{for }i=1,2.
\end{equation}
Let $\phi:=|\nabla u|^{-4}f$. Then
\[
\Delta \phi=f\Delta(|\nabla u|^{-4})+2\langle \nabla(|\nabla u|^{-4}), \nabla f\rangle +|\nabla u|^{-4}\Delta f:=A+B+C.
\]
Denote the covariant derivatives of $\phi$ with respect to the chosen orthonormal frame as follows:
\[
\phi_{i}:=(|\nabla u|^{-4})_{i}f+|\nabla u|^{-4}f_{i},\quad \hbox{ for }i=1,2.
\]
Upon direct computations of $\phi_1$ and $\phi_2$ we get the following formulas at $x$
\begin{equation}\label{u-id2}
u_2u_{111}=-u_2^{3}\phi_1+4u_{11}u_{12},\quad u_2u_{112}=-u_2^{3}\phi_2+2u_{11}u_{22}+2u_{12}^2.
\end{equation}
Notice that by~\eqref{lem-rr1}, harmonicity of $u$ and by symmetries of the Riemann tensor we have the following identities:
\begin{equation}
u_{111}=-u_{221},\quad u_{121}=u_{112}+u_2R_{2121}=-u_{222}+u_2R_{2121},\quad u_2u_{121}+u_2u_{222}=u_2^2R_{2121}=u_2^2K. \label{u-id}
\end{equation}
Then, by using computations for $\beta=-2$ in \cite[(3.11)]{ww} we obtain that
\begin{align*}
A&=-u_{11}u_2^2 \Delta (|\nabla u|^{-4})=-16u_2^{-4}(u_{12}^2u_{11}+u_{11}^3)+4u_2^{-4}(u_2u_{121}+u_2u_{222})u_{11}\\
&=-16u_2^{-4}(u_{12}^2u_{11}+u_{11}^3)+4u_2^{-2}u_{11}K.
\end{align*}
Similarly to \cite[(3.14)]{ww}, we obtain that
\begin{equation*}
B=2\big(f_1 (|\nabla u|^{-4})_1+f_2 (|\nabla u|^{-4})_2\big)=-8u_2^{-1}(u_{21}\phi_1-u_{11}\phi_2)+32u_2^{-4}(u_{11}u_{12}^2+u_{11}^3).
\end{equation*}
Finally, we compute
\begin{align*}
-\Delta f&=-(f_{11}+f_{22})\\
&=u_{1111}u_2^2+2u_2u_{12}u_{111}+u_{2211}u_1^2+2u_1u_{11}u_{221}-2u_{11}^3-4u_1u_{11}u_{111}-2u_{11}u_{12}^2 -6u_1u_{12}u_{121}\\
&-2u_{2}u_{11}u_{121}-2u_1u_2u_{1211}+u_{1122}u_2^2-6u_2u_{11}u_{112}+u_{2222}u_1^2+2u_{11}^3+2u_1u_{12}u_{222}+2u_{12}^2u_{11}\\
&+2u_1u_{11}u_{122}-6u_{2}u_{12}u_{122}-2u_1u_2u_{1222},
\end{align*}
which evaluated at $x$ results in the following formula ($u_1(x)=0$):
\begin{equation}\label{lapl-f}
\Delta f(x)=-u_2^2(u_{1111}+u_{1122})-2u_2u_{12}u_{111}+2u_2u_{11}u_{121}+6u_2u_{11}u_{112}+6u_2u_{12}u_{122}.
\end{equation}
By applying~\eqref{lem-rr2} we have:
\begin{equation}\label{u4-id}
u_{1111}=-u_{2211}=-u_{1122}+4u_{11}R_{2121}-(\nabla_2\Ric)_{22}u_2.
\end{equation}
Thus, by appealing to~\eqref{lem-rr1}, \eqref{u-id}, \eqref{u-id2} and simplifying the arising expression, we get
\begin{align}
\Delta f(x)=&-u_2^2(4u_{11}K-(\nabla_2\Ric)_{22}u_2)-2u_2u_{12}u_{111}+2u_2u_{11}(u_{112}+u_2K) \nonumber \\
&+6u_2u_{11}u_{112}+6u_2u_{12}(-u_{111}) \nonumber \\
=&-u_2^2(2u_{11}K-(\nabla_2\Ric)_{22}u_2)-8u_2u_{12}u_{111}+8u_2u_{11}u_{112} \label{lapl-f-aux}\\
=&-u_2^2(2u_{11}K-(\nabla_2\Ric)_{22}u_2)+8u_2^{3}(u_{12}\phi_1-u_{11}\phi_2) -16u_{11}u_{12}^2-16u_{11}^3. \nonumber
\end{align}
By adding up $A,B$ and $C=u_2^{-4}\Delta f$ we arrive at assertion~\eqref{pde1}
\begin{equation}\label{phi-k-eq}
\Delta \phi=2u_2^{-2}u_{11}K+u_2^{-1}(\nabla_2\Ric)_{22}=-2\phi K+\frac{(\nabla_2\Ric)_{22}}{|\nabla u|}.
\end{equation}
In order to show equation~\eqref{pde1-star} we use the same orthonormal frame as in the reasoning for $k$. Then
\begin{equation}\label{h-eq}
h:=-\div\left(\frac{(u_2,-u_1)}{|\nabla u|}\right)=-\frac{u_{12}(u_1^2-u_2^2)-u_1u_2(u_{11}-u_{22})}{|\nabla u|^3},\quad h(x)=\frac{u_{12}}{u_2}.
\end{equation}
The proof of~\eqref{pde1-star} follows from direct computations similar to the above reasoning for $|\nabla u|^{-1}k$ and, therefore, we will omit it. The counterpart of~\eqref{phi-k-eq} for $\phi=|\nabla u|^{-1}h$ reads
\begin{align*}
\Delta \phi&=u_2^{-1}(4u_{12}K+u_2(\nabla_1\Ric)_{22})+u_2^{-3}(-6u_{21}u_2K)=4\frac{u_{12}}{u_2^2}K+\frac{(\nabla_1\Ric)_{22}}{u_2}-6\frac{u_{21}}{u_2^2}\\
&=-2\phi K-\frac{\langle \nabla K, (u_2,-u_1)\rangle}{|\nabla u|^2}\quad\hbox{at }x.
\end{align*}
Let us now turn to the proof of inequality~\eqref{pde2}.
We employ the Bochner formula~\eqref{formula-Bochner}, the Kato equality~\eqref{refined-Kato2} and argue as follows
\begin{align*}
-\Delta\ln k &= -\Delta\ln\phi -\frac 12 \Delta \ln |\nabla u|^2 \\
&= -\frac{\Delta\phi}{\phi} +\frac{|\nabla\phi|^2}{\phi^2}-\frac 12 \frac{\Delta|\nabla u|^2}{|\nabla u|^2} + \frac 12 \frac{|\nabla|\nabla u|^2|^2}{|\nabla u|^4}\\
&=2K - \frac 1\phi \frac{\langle\nabla K,\nabla u\rangle}{|\nabla u|^2}+\frac{|\nabla\phi|^2}{\phi^2} - \frac{|\nabla^2u|^2}{|\nabla u|^2} - K\frac{|\nabla u|^2}{|\nabla u|^2}+ 2 \frac{|\nabla|\nabla u||^2}{|\nabla u|^2}\qquad (\hbox{by }\eqref{pde1} \hbox{ and }~\eqref{formula-Bochner})\\
&= K - \frac{\langle\nabla K,\nabla u\rangle}{k|\nabla u|}+\frac{|\nabla\phi|^2}{\phi^2}\qquad (\hbox{by }\eqref{refined-Kato2}).
\end{align*}
This shows estimate~\eqref{pde2} for curvature $k$. Similar reasoning gives the inequality for the curvature of the steepest descent $h$, upon setting $\phi:=\frac{h}{|\nabla u|}$ for $h$ as in~\eqref{h-eq} and repeating the above computations with~\eqref{pde1-star} applied instead of~\eqref{pde1}.
|
{
"redpajama_set_name": "RedPajamaArXiv"
}
| 518
|
Lutró (grec Λουτρό, AFI [lu'tro]) és un llogaret del municipi de Sfakià, a la costa sud de la prefectura de Chanià, a l'oest l'illa de Creta. Lutró és en una badia encarada al sud-est i protegida del vent de Mestral (el Meltemi) a l'oest del nucli principal del municipi que és Khora Sfakion (Chora Sfakion), i a l'est del poblet d'Hàgia Rumeli (conegut per ser al final de la famosa Gorja de Samarià).
Els esfaquians tenen fama de feréstecs habitants de les muntanyes. Han participat en cada una de les revoltes contra els successius invasors de Creta: francs, venecians, turcs, alemanys...
Història
Lutró és un lloc xic però ha tingut una història moguda.
És al costat de Phoinix o Phoinikous, l'antic port d'Anopolis, un lloc important a les èpoques hel·lenística i romana.
Va ser una base dels pirates sarraïns per rapinyar tot el sud de l'illa.
Els venecians hi van fer un petit fortí a la punta del cap que domina la badia, les seves ruïnes encara hi són actualment.
Els turcs hi van fer una altra fortalesa al damunt del turó que separa els nuclis actuals de Lutró i Fínix.
Com arribar-hi
L'única manera d'arribar a Lutró és en vaixell (des de Khora Sfakion) o a peu per caminets de cabres. No hi arriba cap carretera, no hi ha carrers, només dues fileres de cases baixes encarades a la badia, sense trànsit, això és part del seu encant únic.
El ferry surt de Khora Sfakion, passa per davant d'una costa rocallosa de cales i penya-segats de fins a 700 m d'alt aixecant-se sobre un mar blau, i arriba a la tranquil·la badia de Lutró, on espera una filera de casetes blanques a tocar d'una estreta platgeta de grava, una joieta amagada entre les muntanyes esfaquianes.
Les coordenades nàutiques aproximades són 35° 11′ 53″ N i 24° 4′ 48″ E.
Unes poques dotzenes de residents (la majoria estacionals) viuen principalment del turisme.
Hi ha uns quants apartaments i petits hotels, i un grapat de petits restaurants i bars.
Els subministraments venen en ferry des de Sfakià. Els residents solen utilitzar el seu propi vaixell per a anar i venir.
Lutró com a destinació turística
Lutró és una destinació turística només per als que busquen tranquil·litat i silenci: no hi ha cap dels entreteniments habituals del turisme de massa, cap discoteca, el més entretingut que es pot fer és:
Seure en una terrassa davant del mar i prendre's un Cafè frappé sense presses, contemplar l'horitzó, el mar o els vòltors planejant per sobre d'Anòpolis. Esperar l'arribada del pròxim ferry carregat de xirucaires que han baixat la gorja de Samarià per veure quina mena de fauna en baixa. Ja cap al tard, contemplar com baixen les cabres de la muntanya per rebre un sac de garrofes al moll.
Fer caminades pels réssecs voltants tot buscant ombres, com a cas d'escarràs extrem es pot pujar caminant el camí de mules fins al poblet d'Anòpolis (lloc de dalt), 700 m més amunt del nivell del mar. Un altre possible itinerari amb vistes impressionants és explorar la Gorja d'Aradena i el seu pont famós entre els aficionats al salt de pont.
Llogar una canoa i fer una escapada a alguna de les platges veïnes de sorra fina, com Màrmara (platja de marbre) al final de la gorja d'Aradena, o a la platja de Glyka Nera (aigua dolça) freqüentada per naturistes.
Enllaços externs
Welcome to Loutro (en anglès)
Prefectura de Khanià
|
{
"redpajama_set_name": "RedPajamaWikipedia"
}
| 8,448
|
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.