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{"url":"https:\/\/tex.stackexchange.com\/questions\/257467\/latex-external-document-referencing-from-main-tex-file","text":"# LaTeX external document referencing from Main tex file\n\nI have a file structure like this\n\n\u2022 Main.tex\n\u2022 chapters (folder)\n\u2022 Chapter_1 (folder)\n\u2022 chap1.tex\n\u2022 Chapter_2 (folder)\n\u2022 chap2.tex\n\nTo refence diagrams and equations from chap1.tex in chap2.tex, I have used\n\n \\usepackage{xr}\n\\externaldocument[I-]{..\/Chapter_1\/chap1}\n\n\nas mentioned in this_question and it works when I compile chap2.tex separately.\n\nHowever, in Main.tex, I have added all the chapters using\n\n \\cfchapter{Chapter 1}{chapters\/Chapter_1}{chap1.tex}\n\\cfchapter{Chapter 2}{chapters\/Chapter_2}{chap2.tex}\n\n\nNow when I compile main.tex, the cross chapter references are not happening as I am getting Latex errors that externaldocument command needs to be declared in the preamble. Adding the externaldocument commands (with corrected paths) in Main.tex does not work. Can someone provide me with a fix ? Thanks\n\n\u2022 What is wrong with plain \\include{...} or \\input{...}? Jan 24, 2016 at 22:37\n\nI just found out the answer. I need to remove the reference specifier within chap2.tex and using plain reference name. Example: instead of specifying\n\n \\ref{I-figure1}\n\n\nI need to just use\n\n \\ref{figure1}\n\n\nwithout the need for \\externaldocument and when I compile Main.tex, the correct reference is shown! But I think I need to make sure that label names are unique to prevent clashes.\n\n\u2022 I havent used those packages, I just use \\input{chapters\/chapter_1\/chap1.tex} in main.tex. What this does (and I think your commands work the same way) is to read that section as if it was written in the main.tex document. And referencing other files or things is done as if you are in the working directory of the main file. Hope that helps! Jul 28, 2015 at 13:42\n\u2022 You might prefer \\ref{fig:descriptive_name} if you aren't doing that already en.wikibooks.org\/wiki\/LaTeX\/Labels_and_Cross-referencing Jul 28, 2015 at 13:45\n\u2022 Thanks guys but I found the solution. As I have mentioned, the \\externaldocument command is only required to link between unrelated tex files. Since I use a main.tex file to add all the chapters (which are individual tex files) there is no need to use \\externaldocument` command .. Jul 29, 2015 at 16:11","date":"2022-08-18 17:00:39","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\": 0.9781102538108826, \"perplexity\": 3107.987194430846}, \"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\/1659882573242.55\/warc\/CC-MAIN-20220818154820-20220818184820-00583.warc.gz\"}"} | null | null |
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"redpajama_set_name": "RedPajamaCommonCrawl"
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\section{Introduction and general result}\label{s:intro}
nontrivial
Let $Y:=(Y_t)_{t\ge0}$ be a nontrivial strictly $\beta$-stable process on ${\mathbb R}^d$ with $0<\beta \le 2$, in the sense of \cite[Definition 13.1]{Sa13}.
Assume that none of the one dimensional projections of $Y$ is a subordinator, and $Y$ has no drift when $\beta=1$ (namely, $\tau=0$ in \cite[(14.16)]{Sa13}). Then $Y$ satisfies the following Chung-type liminf LIL: There exists a constants $C \in (0,\infty)$ such that
\begin{align}
\label{e:ieq0}
\liminf_{t \to 0 \;\; (\text{resp. } t\to \infty) } \frac{\sup_{0<s\le t} |Y_s|}{(t/\log|\log t|)^{1/\beta}}
=C \;\; \mbox{ a.s.}
\end{align}
See, e.g. \cite[Chapters 47, 48]{Sa13}.
The liminf LIL \eqref{e:ieq0} was established for random walks on $\Z$ by Chung \cite{Ch48} under the assumption that their i.i.d. increments have a finite third moment and expectation zero.
The liminf LIL in \cite{Ch48}
was improved to a finite second moment assumption by Jain and Pruitt \cite{JP75}.
For some related results, we refer to \cite{Wi74, EM94, Ke97}.
Chung also showed the large time result of \eqref{e:ieq0} for a Brownian motion in ${\mathbb R}$. The liminf LIL has been extended to non-Cauchy $\beta$-stable processes on ${\mathbb R}^d$ with $\beta<d$ by Taylor \cite{T}, increasing random walks and subordinators by Fristedt and Pruitt \cite{FP71}, and symmetric L\'evy processes in ${\mathbb R}$ by Dupuis \cite{Du74}. Then Wee \cite{Wee88} succeeded in obtain liminf LILs for numerous non-symmetric
L\'evy processes in ${\mathbb R}$. See also \cite{ADS13, BM} and the references therein. Recently, Knopova and Schilling \cite{KS} extended liminf LIL at zero to non-symmetric L\'evy-type processes in ${\mathbb R}$. Also, very recently, the second named author, jointly with Kumagai and Wang \cite{KKW17} extended liminf LIL for symmetric mixed stable-like Feller processes on metric measure spaces.
The purpose of this paper is to understand asymptotic behaviors of a given Markov process by establishing liminf law of iterated logarithms for both near zero and near infinity under some minimal assumptions. In particular, we introduce new but general version of it. See Theorem \ref{inf} below.
Our assumptions are weak enough so that our results cover a lot of Markov processes including jump processes with diffusion part, jump processes with small jumps of slowly varying intensity,
some non-symmetric processes, processes with singular jumping kernels and random conductance models with long range jumps. See the examples in Sections \ref{s:Feller}-\ref{s:hunt}, and the references therein. In particular, the class of Markov processes considered in this paper extends the results of \cite{KKW17}.
Moreover, metric measure spaces in this paper can be random, disconnected and highly space-inhomogeneous (see Definition \ref{d:VD}).
\medskip
Throughout this section, Section \ref{s:proof} and Appendix \ref{s:A},
we assume that $(M,d,\mu)$ is a locally compact separable metric measure space where $\mu$ is a positive Radon measure on $M$ with full support.
We add a cemetery point $\partial$ to $M$ and denote $M_\partial=M \cup \{\partial\}$. We consider a Borel standard Markov process $X = (\Omega, {\cal F}_t, X_t, \theta_t, t \ge 0; {\mathbb P}^x, x \in M_\partial)$ on $M_\partial$ with the lifetime $\zeta:=\inf\{t>0:X_t = \partial\}$. Here $(\theta_t)_{t\ge0}$ is the shift operator with respect to $X$ which is defined as $\theta_t X_s = X_{s+t}$ for all $t,s \ge 0$.
Since $X$ is a Borel standard process, $X$ has a L\'evy system in the sense of \cite[Theorem 1.1]{BJ73}. In this paper, we always assume that $X$ admits a L\'evy system of the form $(J(x,\cdot),ds)$ so that
for any $z \in M$, $t>0$ and nonnegative Borel function $F$ on $M \times M_\partial$ vanishing on the diagonal,
\begin{equation*}
\E^z \bigg[ \,\sum_{s \le t} F(X_{s-}, X_s)\, \bigg] = \E^z \left[ \int_0^t \int_{M_\partial} F(X_s,y)J(X_s,dy)ds \right].
\end{equation*}
The measure $J(x,dy)$ on $M_\partial$ is called the \textit{L\'evy measure} of the process $X$. Here we emphasize that the killing term $J(x, \partial)$ is included in the L\'evy measure.
For $x \in M$ and $r\in(0,\infty]$, set $B(x,r):= \{ y \in M : d(x,y) <r \}$ and $V(x,r):=\mu(B(x,r))$ with a convention $B(x,\infty)=M$. For a subset $U \subset M$, we denote $\updelta_U(x)$ for the distance between $x$ and $M \setminus U$, namely,
\begin{equation*}
\updelta_U(x)=\inf \{ d(x,y) : y \in M \setminus U\}, \quad x \in M.
\end{equation*}
We fix a base point $o \in M$ and define
\begin{equation*}
{\mathsf{d}} (x) = d(x,o) + 1, \quad x \in M.
\end{equation*}
Since ${\mathsf{d}} (x) \ge 1$, the map ${\upsilon} \to {\mathsf{d}}(x)^{\upsilon}$ is nondecreasing on $(0,\infty)$.
For a Borel set $D\subset M$, we denote
$$\tau_D:= \inf \{ t>0 : X_t \in M_\partial \setminus D \}$$ for the first exit time of $X$ from $D$.
We are now ready to introduce our assumptions. Our assumptions are given in terms of mean exit times, tails of L\'evy measures and survival probabilities on balls. Our assumptions are weaker than those in \cite{CKL}, see Lemmas \ref{l:NDL} and \ref{l:hkendl} in Appendix \ref{s:A}.
Here are our assumptions for liminf LIL at zero. Let $U\subset M$ be an open subset of $M$.
\bigskip
\setlength{\leftskip}{5mm}
\textit{There exist constants $R_0>0$, $C_0\in(0,1)$, $C_1>1$, $C_i>0$, $2\le i\le 7$ such that for every $x \in U$ and $0<r<R_0 \wedge (C_0\updelta_U(x))$,}
\begin{equation}\label{A1}\tag{A1}
C_1^{-1} \E^y[\tau_{B(y,r)}]\le \E^x[\tau_{B(x,r)}] \le C_1 \E^y[\tau_{B(y,r)}]\quad \text{for all} \; y \in B(x,r),
\end{equation}\\[-6mm]
\begin{equation}\label{A2}\tag{A2}
\lim_{r \to 0} \E^x[\tau_{B(x,r)}] = 0, \quad \E^x[\tau_{B(x,r)}] \le C_2 \E^x[\tau_{B(x,r/2)}],
\end{equation}\\[-6mm]
\begin{equation}\label{A3}\tag{A3}
J(x,M_\partial \setminus B(x,r)) \le \frac{C_3}{ \E^x[\tau_{B(x,r)}]},
\end{equation}\\[-5mm]
\begin{equation}\label{A4}\tag{A4}
C_4e^{-C_5 n} \le {\mathbb P}^x\big(\tau_{B(x,r)} \ge n \E^x[\tau_{B(x,r)}]\big) \le C_6e^{-C_7 n} \quad \text{for all} \; n \ge 1.
\end{equation}
\bigskip
\setlength{\leftskip}{0mm}
Next, we give assumptions for liminf LIL at infinity.
\bigskip
\setlength{\leftskip}{5mm}
\textit{There exist constants $R_\infty\ge 1$, ${\upsilon}\in(0,1)$, $\ell,C_1>1$, $C_i>0$, $2\le i\le 7$ such that for every $x \in M$ and $r>R_\infty {\mathsf{d}}(x)^{\upsilon}$,}
\begin{equation}\label{B1}\tag{B1}
C_1^{-1}\E^o[\tau_{B(o,r)}] \le \E^x[\tau_{B(x,r)}] \le C_1 \E^o[\tau_{B(o,r)}],
\end{equation}\\[-6mm]
\begin{equation}\label{B2}\tag{B2}
2 \E^o[\tau_{B(o,s/\ell)}] \le \E^o[\tau_{B(o,s)}] \le C_2 \E^o[\tau_{B(o,s/2)}] \quad \text{for all} \; s >R_\infty,
\end{equation}\\[-6mm]
\begin{equation}\label{B3}\tag{B3}
J(x,M_\partial \setminus B(x,r)) \le \frac{C_3}{\E^x[\tau_{B(x,r)}]},
\end{equation}\\[-5mm]
\begin{equation}\label{B4}\tag{B4}
C_4e^{-C_5 n} \le {\mathbb P}^x\big(\tau_{B(x,r)} \ge n\E^x[\tau_{B(x,r)}]\big) \le C_6e^{-C_7 n} \quad \text{for all} \; n \ge 1.
\end{equation}
\medskip
We recall the following figure from \cite[Figure 1]{CKL} which shows ranges of $r$ in our conditions.
\begin{figure}[h!]\label{f:1}
\includegraphics[width=0.36\columnwidth]{int.png} \hspace{20mm} \includegraphics[width=0.32
\columnwidth]{ext.png} \vspace{-2mm}
\caption{Range of $r$ in local conditions}
\end{figure}
\setlength{\leftskip}{0mm}
\begin{remark}\label{r:basic}
{\rm (i) Note that, like \cite{CKL}, we impose conditions at infinity \eqref{B1}-\eqref{B4} only for $r>R_\infty {\mathsf{d}}(x)^{\upsilon}$. By considering such weak assumptions at infinity, our LILs cover some random conductance models. See Section \ref{s:RCM} below.
\noindent (ii) The assumptions \eqref{A3} and \eqref{B3} are quite mild and natural.
Let $x \in U$.
Suppose \eqref{A2} holds for $U$ and for $0<r<R_0 \wedge (C_0\updelta_U(x))$
$$J(y,M_\partial \setminus B(x,r)) \ge c J(x,M_\partial \setminus B(x,r)) \quad \text{ for all } y \in B(x, r/2).$$
Then, by the L\'evy system, we have that for $0<r<R_0 \wedge (C_0\updelta_U(x))$
\begin{align*}
1 &\ge {\mathbb P}^x(X_{\tau_{B(x,r/2)}} \in M_\partial \setminus B(x,r) ) = \E^x \left[\int_0^{\tau_{B(x,r/2)}} \int_{M_\partial \setminus B(x,r)} J(X_s,dy) ds \right] \\ &\ge
c \,\E^x[\tau_{B(x,r/2)}] J(x,M_\partial \setminus B(x,r)) \ge c \, C_2^{-1} \E^x[\tau_{B(x,r)}] J(x,M_\partial \setminus B(x,r)).
\end{align*}
\noindent (iii) The lower inequality in \eqref{B2} implies that $\lim_{r \to \infty} \E^o[\tau_{B(o,r)}] = \infty$. Hence, under \eqref{B1} and \eqref{B2}, we have $\lim_{r \to \infty} \E^x[\tau_{B(x,r)}] = \infty$ for all $x \in M$.
\noindent (iv) We need the lower inequality in \eqref{B2} because the space $M$ may be highly inhomogeneous. This inequality implies that the map $r \mapsto \E^o[\tau_{B(x,r)}]$ grows at least polynomially for large values of $r$. Hence under this condition, with a high probability, $X_t$ is not located in regions extremely far away from the base point $o$ in view of the time $t$ where we do not impose any conditions.
}
\end{remark}
From now on, {\it whenever condition \eqref{A1}-\eqref{A3} are assumed with $R_0>0$ for an open set $U$, we let $\phi(x,r)$ be
any increasing function
defined on $U \times (0,R_0)$ which satisfies the following condition: There is a constant $C \ge 1$ such that for all $x \in U$ and $0<r<R_0 \wedge (C_0\updelta_U(x))$,
\begin{equation}\label{e:phi}
C^{-1}\E^x[\tau_{B(x,r)}] \le \phi(x,r) \le C\E^x[\tau_{B(x,r)}].
\end{equation} }
Also, {\it whenever condition \eqref{B1}-\eqref{B3} are assumed with $R_\infty \ge 1$ and ${\upsilon} \in (0,1)$, we let $\phi(r)$ be any function on $M \times (0,\infty)$ satisfying \eqref{e:phi} for all $x \in M$ and $r>R_\infty {\mathsf{d}}(x)^{\upsilon}$, with $\phi(r)$ instead of $\phi(x,r)$. }
Then by condition \eqref{A2}, we see that there exist constants $\beta_2>0$ and $C_U\ge 1$ such that
\begin{equation}\label{e:scaling-zero}
\frac{\phi(x,r)}{\phi(x,s)} \le C_U \bigg(\frac{r}{s}\bigg)^{\beta_2} \quad \text{for all} \; x \in U, \; 0<s\le r<R_0 \wedge (C_0\updelta_U(x)),
\end{equation}
and by \eqref{B2}, there exist constants $\beta_1,\beta_2>0$ and $C_U\ge 1 \ge C_L >0$ such that
\begin{equation}\label{e:scaling-infty}
C_L \bigg(\frac{r}{s}\bigg)^{\beta_1} \le \frac{\phi(r)}{\phi(s)} \le C_U \bigg(\frac{r}{s}\bigg)^{\beta_2} \quad \text{for all} \; r \ge s> R_\infty {\mathsf{d}}(x)^{\upsilon}.
\end{equation}
Now, we give our results in full generality. Our first result is the liminf law of LIL at zero. Note that we don't put any extra assumptions on our metric measure space such as volume doubling property.
\begin{thm}\label{inf}
Suppose that \eqref{A1}-\eqref{A4} hold for an open subset $U \subset M$. Then, there are constants $a_2 \ge a_1>0$ such that for all $x \in U$, there exists a constant $a_x \in [a_1,a_2]$ satisfying
\begin{equation}\label{1}
\liminf_{t\to0} \frac{\phi\big(x,\sup_{0<s\le t} d(x, X_s)\big)}{t/\log|\log t|}=a_x,\qquad
{\mathbb P}^x\mbox{-a.s.}
\end{equation}
\end{thm}
Note that in Theorem \ref{inf}, $r \mapsto \phi(x,r)$ for $x \in U$ can be slowly varying at zero so that we can cover jump processes whose jumping measures have logarithmic tails.
When $\phi(x,r)$ is a mixed polynomial type near zero (i.e., both \eqref{e:scaling-zero} and \eqref{e:scaling-zero-lower} hold), we recover the classical form of the liminf LIL at zero.
We denote $\phi^{-1}(x,t):=\sup\{r>0:\phi(x,r)\le t\}$ for the right continuous inverse of $\phi(x,\cdot)$.
\begin{cor}\label{c:inf}
Suppose that \eqref{A1}-\eqref{A4} hold for an open subset $U \subset M$ and there exist constants $\beta_1,C_L>0$ such that
\begin{equation}\label{e:scaling-zero-lower}
\frac{\phi(x,r)}{\phi(x,s)} \ge C_L \bigg(\frac{r}{s}\bigg)^{\beta_1} \quad \text{for all} \; x \in U, \; 0<s\le r<R_0 \wedge (C_0\updelta_U(x)).
\end{equation}
Then, there are constants $\widetilde a_2 \ge \widetilde a_1>0$ such that for all $x \in U$, there exists a constant $\widetilde a_x \in [\widetilde a_1,\widetilde a_2]$ satisfying
\begin{equation}\label{c1}
\liminf_{t\to0} \frac{\sup_{0<s\le t} d(x, X_s)}{\phi^{-1}(x,t/\log|\log t|)}=\widetilde a_x,\qquad
{\mathbb P}^x\mbox{-a.s.}
\end{equation}
\end{cor}
\bigskip
Our second result is the liminf law of LIL at infinity.
\begin{thm}\label{t:infinf}
Suppose that \eqref{B1}-\eqref{B4} hold. Then, there are constants $b_2 \ge b_1>0$ such that
\begin{equation}\label{20}
\liminf_{t\to \infty} \frac{\sup_{0<s\le t} d(x, X_s)}{\phi^{-1}(t/\log\log t)} \in [b_1, b_2],\qquad
{\mathbb P}^y\mbox{-a.s.} \;\; \forall x,y \in M.
\end{equation}
\end{thm}
The $\liminf$ in \eqref{20} may not be deterministic. In \cite{CKL}, we have obtained a zero-one law
for shift-invariant events
under volume of doubling assumptions
and near diagonal lower estimates on heat kernel (see Proposition \ref{p:law01} below).
Using the same zero-one law, in this paper we also establish the deterministic limit in liminf law.
We recall following versions of volume doubling property from authors' previous paper \cite{CKL}.
\begin{defn}\label{d:VD}
{\rm (i) For an open set $U \subset M$ and $R_0' \in (0,\infty]$, we say that \textit{the interior volume doubling and reverse doubling property} ${\mathrm {VRD}}_{R_0'}(U)$ \ holds if there exist constants $C_V \in (0,1)$, $d_2 \ge d_1>0$ and $C_\mu \ge c_\mu>0$ such that for all $x \in U$ and $0<s\le r<R_0' \wedge (C_V \updelta_U(x))$,
\begin{equation}\label{e:VRD}
c_\mu \left( \frac{r}{s} \right)^{d_1} \le \frac{V(x,r)}{V(x,s)} \le C_\mu \left( \frac{r}{s} \right)^{d_2}.
\end{equation}
\noindent (ii) For $R_\infty'\ge1$ and ${\upsilon} \in (0,1)$, we say that \textit{a weak volume doubling and reverse doubling property at infinity} ${\mathrm {VRD}}^{R_\infty'}(\up)$ \ holds if there exist constants $d_2 \ge d_1>0$ and $C_\mu \ge c_\mu>0$ such that \eqref{e:VRD} holds for all $x \in M$ and $r \ge s > R_\infty' {\mathsf{d}}(x)^{\upsilon}$.
} \end{defn}
For an open set $D\subset M$, let $X^D$ be the part process of $X$ defined as $X_t^D := X_t \, {\bf 1}_{\{\tau_D > t\}} + \partial \, {\bf 1}_{\{\tau_D \le t\}}$. Then $X_t$ is a Borel standard process on $D$. See, e.g. \cite[Section 3.3]{CF}.
Let $(P^D_t)_{t \ge 0}$ be the semigroup associated with $X^D$,
namely, $P^D_t f(x) := \E^x [f(X^D_t)]$. We call a Borel measurable function $p^D: (0,\infty) \times D \times D \to [0,\infty]$ the \textit{heat kernel} (transition density) of $(P^D_t)_{t \ge 0}$ (or $X^D$) if the followings hold:
\medskip
\setlength{\leftskip}{4mm}
\noindent (i) $P^D_t f(x) = \int_{D} p^D(t,x,y)f(y) \mu(dy)$ for all $t>0$, $x \in D$ and $f \in L^\infty(D;\mu)$.
\smallskip
\noindent (ii) $p^D(t+s,x,y) = \int_{D} p^D(t,x,z) p^D(s,z,y) \mu(dz)$ for all $t,s>0$ and $x,y \in D$.
\medskip
\setlength{\leftskip}{0mm}
\noindent
We simply write $P_t$ for $P_t^M$ and $p(t,x,y)$ for $p^M(t,x,y)$.
We now consider the following near diagonal lower estimates on heat kernels:
\begin{align*}
&\textit{There exist constants $R_\infty' \ge 1$, ${\upsilon},\eta \in (0,1)$, $c_*>0$ such that for all $x \in M$ and}\nonumber\\
&\textit{$r>R_\infty' {\mathsf{d}}(x)^{\upsilon}$, the heat kernel $p^{B(x,r)}(t,x,y)$ of $X^{B(x,r)}$ exists and}
\end{align*}
\begin{equation}\label{B4+}
p^{B(x,r)}( \phi(\eta r),y,z) \ge \frac{c_*}{V(x, r)} \quad \text{for all} \;\, y,z \in B(x, \eta^2r).\tag{B4+}
\end{equation}
\medskip
Under \eqref{B1} and ${\mathrm {VRD}}^{R_\infty'}(\up)$, the above condition \eqref{B4+} is stronger than \eqref{B4}. See Proposition \ref{r:E} below.
\begin{remark} \label{r:NDL} {\rm
The standard version of near diagonal lower estimates on heat kernels (without the restriction $r>R_\infty {\mathsf{d}}(x)^{\upsilon}$) has been studied a lot.
In particular, in \cite{CKW16b} and \cite{CKW19}, it is shown that, for a large class of symmetric Hunt process, the standard version of near diagonal lower estimates on heat kernels can be obtained under \eqref{B2} and a H\"older-type regularity of corresponding harmonic functions
See. \cite[Proposition 4.9]{CKW16b} and its proof.}
\end{remark}
\begin{cor}\label{c:infinf}
Suppose that ${\mathrm {VRD}}^{R_\infty'}(\up)$ \ holds. If \eqref{B1}, \eqref{B2}, \eqref{B3} and \eqref{B4+} hold, then there exists a constant $b_\infty \in (0, \infty)$ such that
\begin{equation}\label{2}
\liminf_{t\to \infty} \frac{\sup_{0<s\le t} d(x, X_s)}{\phi^{-1}(t/\log\log t)} = b_\infty,\qquad
{\mathbb P}^y\mbox{-a.s.} \;\; \forall x,y \in M.
\end{equation}
\end{cor}
\begin{remark}\label{r:a}
{\rm (i)
Once we prove that the liminf LILs \eqref{1} and \eqref{c1} hold true with $\phi(x,r)=\E^x[\tau_{B(x,r)}]$, by the Blumenthal's zero-one law, they hold true with general $\phi$ satisfying \eqref{e:phi} after redefining constants $a_1,a_2, \widetilde a_1, \widetilde a_2$ by $C^{-1}a_1,Ca_2, C^{-1}\widetilde a_1, C\,\widetilde a_2$, respectively, with the constant $C \ge 1$ in \eqref{e:phi}. Similarly, thanks to the zero-one law given in Proposition \ref{p:law01}, it suffices to prove Theorem \ref{t:infinf} and Corollary \ref{c:infinf} with a particular function $\phi(r):=\E^o[\tau_{B(o,r)}]$.
\noindent (ii) Using the zero-one law in Proposition \ref{p:law01} again, we see that the liminf LIL \eqref{2} remains true even if the function $\phi$, which comes from condition \eqref{B4+}, is replaced by any function $\widetilde \phi$ comparable to $\phi$.}
\end{remark}
Let us also consider the following
counterpart of \eqref{B4+}:
\begin{align*}
&\textit{For a given open set $U \subset M$, there exist constants $R_0'>0$, $C_0',\eta \in (0,1)$ and $c_*>0$ such that}\nonumber\\
&\textit{for all $x \in U$ and $0<r<R_0' \wedge (C_0' \updelta_U(x))$, the heat kernel $p^{B(x,r)}(t,x,y)$ of $X^{B(x,r)}$ exists and}
\end{align*}
\begin{equation}\label{A4+}
p^{B(x,r)}(\phi(x,\eta r),y,z) \ge \frac{c_*}{V(x, r)} \quad \text{for all} \;\, y,z \in B(x, \eta^2r).\tag{A4+}
\end{equation}
\begin{prop}\label{r:E}
(i) Suppose that ${\mathrm {VRD}}_{R_0'}(U)$ \ holds. If \eqref{A2} and \eqref{A4+} hold, then \eqref{A4} holds with some $R_0>0$ and $C_0\in (0,1)$.
\noindent (ii) Suppose that ${\mathrm {VRD}}^{R_\infty'}(\up)$ \ holds. If \eqref{B2} and \eqref{B4+} hold, then \eqref{B4} holds with some $R_\infty\ge1$.
\end{prop}
\noindent{\bf Proof. } (i) By following the proof of \cite[Proposition 4.3(i)]{CKL} and using $\phi(x,r)$ instead of $\phi(r)$ therein, we can deduce that there exist constants $R_1,c_2,c_3>0$, $c_1>1$ and $C_0 \in (0,1)$ such that for all $x \in U$, $0<c_1r<R_1 \wedge (C_0 \updelta_U(x))$ and $n \ge 1$,
\begin{equation*}
e^{-c_2n} \le {\mathbb P}^x\big(\tau_{B(x,r)} \ge n\E^x[\tau_{B(x,c_1r)}](x,c_1r)\big) \le e^{-c_3 n}.
\end{equation*}
By taking $R_1$ small enough if needed, we may assume that \eqref{A2} holds with $R_0=R_1$. Using \eqref{e:scaling-zero}, it follows that for all $x \in U$, $0<r<R_1 \wedge (C_0 \updelta_U(x))$ and $n \ge 1$,
\begin{align*}
e^{-c_2n} &\le {\mathbb P}^x\big(\tau_{B(x,r)} \ge n\phi(x,c_1r)\big) \le {\mathbb P}^x\big(\tau_{B(x,r)} \ge n\phi(x,r)\big)\\
& \le {\mathbb P}^x\big(\tau_{B(x,r)} \ge c_1^{-\beta_2}C_U^{-1}n\phi(x,c_1r)\big) \le e^{-c_3( c_1^{-\beta_2}C_U^{-1}n -1 )}.
\end{align*}
(ii) Analogously, we can deduce the result by following the proof of \cite[Proposition 4.3(ii)]{CKL}. {\hfill $\Box$ \bigskip}
The rest of the paper is organized as follows. In Sections \ref{s:Feller}--\ref{s:hunt}, we
show that the conditions
\eqref{A1}-\eqref{A4}, \eqref{B1}-\eqref{B4} and \eqref{B4+}
can be checked for important classes of Markov jump processes, which may be non-symmetric and space-inhomogeneous. Thus we can
apply our main theorems to get explicit liminf LILs for them.
Precisely, in Section \ref{s:Feller}, we consider general (non-symmetric) Feller processes on ${\mathbb R}^d$ whose domain of the generator contains $C_c^\infty({\mathbb R}^d)$
and introduce some local assumptions (see \eqref{O1}-\eqref{O4} below).
Under these assumptions, we establish liminf LIL at zero for Feller processes on ${\mathbb R}^d$.
Then, combining results in this paper and \cite{CKL}, we present concrete examples of non-symmetric Feller processes and Feller processes with singular L\'evy measures
for which both liminf LILs and limsup LILs hold.
In the remainder of Section \ref{s:Feller}, we give another assumption (see {\bf (S)} below), which can be checked directly from the symbols of Feller processes. As a consequence, we show that liminf LILs at zero holds for Feller processes with symbols of varying order.
Section \ref{s:RCM} revisits \cite[Section 3]{CKL} and discuss liminf LILs for the random conductance model with long range jumps studied in \cite{CKW18, CKW20-1}.
Our conditions at infinity is motivated by the random conductance model therein.
In Section \ref{s:hunt}, we deal with subordinate processes and symmetric Hunt processes whose tail of the L\'evy measure decays in (mixed) polynomial order. We assume that there is a Hunt process $Z$ enjoying sub-Gaussian heat kernel estimates. Then we show that general liminf LIL holds true for every subordinate process of $Z$ if the corresponding subordinator is not a compound Poisson process. In particular, we get liminf LILs for jump processes with low intensity of small jumps such as geometric stable processes. Using local stability theorems obtained in \cite{CKL}, we also get liminf LIL for symmetric Hunt processes associated with a regular Dirichlet form.
Section \ref{s:proof} is devoted to the proofs of our main theorems. We follow the well-known arguments in
\cite[Chapter 3]{Du74}, \cite[Theorem 2]{KS} and \cite[Theorem 3.7]{KKW17}.
But non-trivial modifications are required since we allow our processes and state spaces to be highly space-inhomogeneous.
The paper ends with Appendix \ref{s:A} which contains some comparisons between conditions of the current paper and \cite{CKL} and a simple lemma about lower heat kernel estimates for Dirichlet heat kernel.
\medskip
{\bf Notations}: Values of capital letters with subscripts $C_i$, $i=0,1,2,...$ are fixed throughout the paper
both at zero and at infinity.
Lower case letters with subscripts $a_i$, $c_i$, $i=0,1,2,...$ denote positive real constants and are fixed in each statement and proof, and the labeling of these constants starts anew in each proof.
We use the symbol ``$:=$'' to denote a definition,
which is read as ``is defined to be.''
Recall that $a\wedge b:=\min\{a,b\}$ and $a\vee b:=\max\{a,b\}$. We denote by $\overline{A}$ the closure of $A$. We extend a function $f$ defined on $M$ to $M_\partial$ by setting $f(\partial)=0$. The notation $f(x) \asymp g(x)$ means that there exist constants $c_2 \ge c_1>0$ such that $c_1g(x)\leq f (x)\leq c_2 g(x)$ for specified range of $x$.
For $D \subset M$, denote by $C_c(D)$ the space of all continuous functions with compact support in $D$.
\section{LIL for Feller processes on ${\mathbb R}^d$}\label{s:Feller}
Throughout this section, we assume that $X$ is a Feller process on ${\mathbb R}^d$ with the generator $({\cal L}, {\cal D}({\cal L}))$ such that $C_c^\infty({\mathbb R}^d) \subset {\cal D}({\cal L})$. It is well known that the generator ${\cal L}$ restricted to $C_c^\infty({\mathbb R}^d)$ is a \textit{pseudo-differential operator}, which has the following representation (see \cite{Co65}):
\begin{equation*}
{\cal L} u(x)= - (2\pi)^{-d} \int_{{\mathbb R}^d} e^{i {\langle} x, \xi {\rangle}}q(x, \xi) \int_{{\mathbb R}^d} e^{- i {\langle} y, \xi{\rangle}}u(y)dyd \xi, \quad u \in C_c^\infty({\mathbb R}^d),
\end{equation*}
where the function $q:{\mathbb R}^d \times {\mathbb R}^d \to \mathbb{C}$, which is called the {\it symbol} of $X$ (or ${\cal L}$), enjoys the following {L\'evy-Khinchine formula}
\begin{equation*}
q(x, \xi) = {\mathbf c}(x) - i {\langle} {\mathbf b}(x), \xi {\rangle} + {\langle} \xi, {\mathbf a}(x)\xi {\rangle} + \int_{{\mathbb R}^d \setminus \{0\}} (1- e^{i{\langle} z, \xi {\rangle}} + i {\langle} z, \xi {\rangle} {\bf 1}_{\{| z| \le 1\}})\nu(x, dz).
\end{equation*}
Here $({\mathbf c}(x), {\mathbf b}(x), {\mathbf a}(x), \nu(x, dz))_{x \in {\mathbb R}^d}$ is a family of the L\'evy characteristics, that is, ${\mathbf c}:{\mathbb R}^d \to [0,\infty)$ and ${\mathbf b}:{\mathbb R}^d \to {\mathbb R}^d$ are measurable functions, ${\mathbf a}:{\mathbb R}^d \to {\mathbb R}^{d \times d}$ is a nonnegative definite matrix-valued function, and $\nu(x,dz)$ is a nonnegative, $\sigma$-finite kernel on ${\mathbb R}^d \times \sB({\mathbb R}^d \setminus \{0\})$ such that $\int_{{\mathbb R}^d \setminus \{0\}}(1 \land |z|^2) \nu(x, dz)<\infty$ for every $x \in {\mathbb R}^d$.
{\it Throughout this section, we always assume that ${\mathbf c}(x)$ is identically zero} so that $X$ has no killing inside.
\medskip
Define for $x \in {\mathbb R}^d$ and $r>0$,
\begin{equation}\label{e:def-Phi}
\Phi(x,r) = \bigg(\sup_{|\xi| \le 1/r} \text{Re} \, q(x,\xi) \bigg)^{-1}.
\end{equation}
For example, if $q(x,\xi)=|\xi|^{\alpha(x)}(\log(1+|\xi|))^{\gamma(x)}$, then $\Phi(x,r)=r^{\alpha(x)} (\log ( 1+ 1/r))^{-\gamma(x)}$.
\medskip
Here are our assumptions on the Feller process $X$. Let $U \subset {\mathbb R}^d$ be an open subset.
\medskip
\setlength{\leftskip}{5mm}
\textit{There exist constants $R_0, \varepsilon_0>0$, $C_i\in(0,1)$, $8 \le i \le 11$ such that for every $x \in U$ and $\xi \in {\mathbb R}^d$ with $1/|\xi|<R_0 \wedge (C_8 \updelta_U(x))$}, the following hold
\begin{equation}\label{O1}\tag{O1}
\lim_{r \to 0} \Phi(x,r)=0 \quad \text{(or, equivalently, either ${\mathbf a}(x) \neq 0$ or $\nu(x, {\mathbb R}^d)=\infty$)},
\end{equation}\\[-5mm]
\begin{equation}\label{O2}\tag{O2}
\sup_{|\xi'|\le |\xi|}\text{Re} \, q(x, \xi') \ge C_9 | \, \text{Im} \, q(x, \xi) |,
\end{equation}\\[-5mm]
\begin{equation}\label{O3}\tag{O3}
\inf_{| x-y | \le 1/|\xi|} \textrm{Re}\,q(y,\xi) \ge C_{10} \sup_{| x-y|\le 1/|\xi|} \textrm{Re}\,q(y,\xi),
\end{equation}\\[-5mm]
\begin{equation}\label{O4}\tag{O4}
{\mathbb P}^x\big(2{\langle} X_t- x, z {\rangle} \le -| X_t -x | \big) \ge C_{11} \quad \text{for all} \;\;|z|=1 \; \text{and} \; 0<t<\varepsilon_0\Phi\big(x, R_0 \wedge (C_8 \updelta_U(x))\big).
\end{equation}
\setlength{\leftskip}{0mm}
\medskip
Under \eqref{O1}, the probability that the process $X$ starting from $x\in U$ stays at $x$ for a positive time is zero.
\eqref{O2} is not only a local formulation of \textit{the sector condition} but also a weaker version of it since we take the supremum in the left-hand side. \eqref{O3} and \eqref{O4} give a weak spatial homogeneity of the process in $U$. Similar conditions are appeared in \cite{KS} (see (A2)-(A3) therein) where small time Chung-type LILs for one-dimensional L\'evy-type processes were studied.
\begin{remark}\label{r:O4}
{\rm When $d=1$, \eqref{O4} is equivalent to ${\mathbb P}^x(X_t-x \ge 0) \in [C_{11}, 1-C_{11}]$ for all $x \in U$ and $t \le \Phi\big(x, R_0 \wedge (C_8 \updelta_U(x))\big)$. Thus, if $d=1$ and $X$ is symmetric, then \eqref{O4} holds with $C_{11}=1/2$.
}
\end{remark}
Following \cite{Pr81}, we set for $x \in {\mathbb R}^d$ and $r>0$,
\begin{equation}\label{e:def-h}
h(x,r)= \frac{1}{r^2} \lVert {\mathbf a}(x) \rVert + \int_{{\mathbb R}^d} \left(\frac{ |z |^2}{r^2} \land 1 \right) \nu(x, dz),
\end{equation}
where $\lVert {\mathbf a}(x)\rVert:=\sup_{\xi\in {\mathbb R}^d, \, |\xi|\le 1} {\langle} \xi, {\mathbf a}(x)\xi{\rangle}$. The function $h$ frequently appears in maximal inequalities for L\'evy-type processes. See, e.g. \cite{Du74, KS}.
The following result is well-known. We give a full proof for the reader's convenience.
\begin{lem}\label{l:Phi-1}
There exists a constant $C_{12}>1$ which depends only on the dimension $d$ such that
\begin{equation*}
\sup_{| \xi| \le 1/r} \text{\rm Re} \, q(x, \xi ) \le 2h(x,r) \le C_{12} \sup_{1/(2r) \le | \xi| \le 1/r} \text{\rm Re} \, q(x, \xi ) \quad \text{for all} \;\; x \in {\mathbb R}^d, \; r>0.
\end{equation*}
In particular, it holds that
\begin{equation}\label{e:Phi=h}
\frac{1}{2h(x,r)} \le \Phi(x,r) \le \frac{C_{12}}{2h(x,r)} \quad \text{for all} \;\; x \in {\mathbb R}^d, \; r>0.
\end{equation}
\end{lem}
\noindent{\bf Proof. } Using the inequality $1-\cos y \le y^2 \wedge 2$ for $y \in {\mathbb R}$, we get that for all $x \in {\mathbb R}^d$ and $r>0$,
\begin{equation*}
\sup_{|\xi| \le 1/r}\text{Re}\,q(x, \xi) \le \frac{1}{r^2} \lVert {\mathbf a}(x) \rVert + \sup_{|\xi| \le 1/r} \int_{{\mathbb R}^d } (1- \cos {\langle} z, \xi {\rangle}) \nu(x, dz) \le 2 h(x,r).
\end{equation*}
Next, it is clear that $\sup_{1/(2r)\le | \xi| \le 1/r} \text{Re} \, q(x, \xi ) \ge r^{-2} \lVert {\mathbf a}(x) \rVert$. Moreover, using Tonelli's theorem and the inequality $1-\cos y \ge y^2/4$ for $|y| \le 1$, we get that for all $x \in {\mathbb R}^d$ and $r>0$,
\begin{align*}
&\sup_{1/(2r)\le | \xi| \le 1/r} \text{Re} \, q(x, \xi ) \ge c_1 \int_{1/2 \le |\rho|\le 1} \text{Re} \, q(x, \rho/r) d \rho \ge c_1 \int_{{\mathbb R}^d} \int_{1/2 \le |\rho|\le 1} (1- \cos {\langle} z, \rho/r {\rangle}) d \rho \, \nu(x,dz)\nonumber\\
& \ge c_1 \left( \int_{|z| \le r} \frac{|z|^2}{r^2} \, \int_{1/2 \le |\rho| \le 1} \frac{{\langle} z/|z|, \rho {\rangle}^2}{4} d\rho \, \nu(x, dz) +\int_{|z|>r } \int_{ 1/2 \le |\rho | \le 1} (1- \cos {\langle} z/r, \rho {\rangle} ) d \rho \, \nu(x,dz) \right)\\
&\ge c_1 \int_{{\mathbb R}^d} \left(\frac{ |z |^2}{r^2} \land 1 \right) \nu(x, dz) \inf_{y \in {\mathbb R}^d, \, |y|=1} \left( \int_{ 1/2 \le |\rho| \le 1} \frac{{\langle} y, \rho {\rangle}^2}{4} d\rho \wedge \int_{1/2 \le |\rho| \le 1} (1- \cos {\langle} y, \rho {\rangle}) d\rho \right).
\end{align*}
Let $\mathbf e_1:=(1,0,...,0)\in {\mathbb R}^d$. By the symmetry, we see that for all $y \in {\mathbb R}^d$ with $|y|=1$,
\begin{align*}
\int_{ 1/2 \le |\rho| \le 1}{\langle} y, \rho {\rangle}^2 d\rho\ge \int_{ 1/2 \le |\rho| \le 1, \, {\langle} \mathbf e_1, \rho {\rangle} \ge |\rho|/2} {\langle} \mathbf e_1, \rho {\rangle}^2 d\rho \ge \frac{1}{16}\int_{ 1/2 \le |\rho| \le 1, \, {\langle} \mathbf e_1, \rho {\rangle} \ge |\rho|/2} d\rho=c_2
\end{align*}
and
\begin{align*}
&\int_{1/2 \le |\rho| \le 1} (1- \cos {\langle} y, \rho {\rangle}) d\rho = \inf_{a>1} \int_{1/2 \le |\rho| \le 1} (1- \cos a{\langle} \mathbf e_1, \rho {\rangle}) d\rho \\
&\ge 2\inf_{a>1} \int_{0}^{1/2} \int_{\widetilde \rho \in {\mathbb R}^{d-1}, |\widetilde \rho| \le 1/2} (1- \cos a\rho_1 ) d\widetilde\rho d\rho_1 \ge c_3\inf_{a>1} \int_{0}^{1/2} (1- \cos a\rho_1 ) d\rho_1.
\end{align*}
Since $\lim_{a \to \infty} \int_{0}^{1/2} (1- \cos a\rho_1 ) d\rho_1= 1/2$, it holds that $\inf_{a>1}\int_{0}^{1/2} (1- \cos a\rho_1 ) d\rho_1>0$. Therefore, using the inequality $a \vee b \ge (a+b)/2$ for $a,b \in {\mathbb R}$, we get that $\sup_{1/(2r)\le | \xi| \le 1/r} \text{Re} \, q(x, \xi ) \ge c_4 h(x,r)$ and finish the proof. {\hfill $\Box$ \bigskip}
Recall that $\Phi$ is defined in \eqref{e:def-Phi}.
It is clear that $s^2h(x,s) \le r^2h(x,r)$ for all $x \in{\mathbb R}^d$ and $0<s\le r$. Thus, by \eqref{e:Phi=h}, there exists a constant $C_U'>0$ depends only on $d$ such that
\begin{equation}\label{e:Phi-upper}
\frac{\Phi(x,r)}{\Phi(x,s)} \le C_U' \bigg(\frac{r}{s}\bigg)^{2} \quad \text{for all} \; x \in {\mathbb R}^d, \; 0<s\le r.
\end{equation}
\begin{lem}\label{l:Phi-2}
Suppose that \eqref{O3} holds for an open subset $U \subset {\mathbb R}^d$. Then there exists a constant $C_{13} \in (0,1)$ such that
$$
\inf_{|x-y|\le 2r} \Phi(y,r) \ge C_{13} \sup_{|x-y| \le 2r} \Phi(y,r) \quad \text{for all} \;\; x \in U, \; 0<4r<R_0 \wedge (C_8 \updelta_U(x)).
$$
\end{lem}
\noindent{\bf Proof. } By \eqref{e:Phi-upper}, Lemma \ref{l:Phi-1} and \eqref{O3}, we get that for all $x \in U$ and $0<4r<R_0 \wedge (C_8 \updelta_U(x))$,
\begin{align*}
&\inf_{|x-y|\le 2r} \Phi(y,r) \ge c_1 \inf_{|x-y| \le 2r} \bigg( \sup_{|\xi|\le 1/(2r)} \text{Re}\, q(y,\xi) \bigg)^{-1} \ge c_2 \inf_{|x-y| \le 2r} \bigg( \sup_{1/(4r) \le |\xi|\le 1/(2r)} \text{Re}\, q(y,\xi) \bigg)^{-1}\\
&= c_2 \inf_{1/(4r) \le |\xi|\le 1/(2r)}\, \inf_{|x-y| \le 2r} \text{Re}\, q(y,\xi)^{-1} \ge c_2 C_{10} \inf_{1/(4r) \le |\xi|\le 1/(2r)}\, \sup_{|x-y| \le 2r} \text{Re}\, q(y,\xi)^{-1}\\
&\ge c_2 C_{10} \sup_{|x-y| \le 2r} \, \inf_{1/(4r)\le |\xi|\le 1/(2r)} \text{Re}\, q(y,\xi)^{-1} \ge c_2 C_{10} \sup_{|x-y| \le 2r} \Phi(x,r).
\end{align*}
{\hfill $\Box$ \bigskip}
As an application of our Theorem \ref{inf}, we obtain the following LIL for Feller processes. See \cite[Theorem 2]{KS} for a 1-dimensional result under similar assumptions.
\begin{thm}\label{t:Feller1}
Let $X$ be a Feller process on ${\mathbb R}^d$ with symbol $q$. Suppose that \eqref{O1}-\eqref{O4} hold for an open subset $U \subset {\mathbb R}^d$. Then, there are constants $a_2 \ge a_1>0$ such that for all $x \in U$, there exists a constant $a_x \in [a_1,a_2]$ satisfying
\begin{equation}\label{eq:Feller1}
\liminf_{t\to0} \frac{\Phi (x, \,\sup_{0<s\le t} | X_s - x | )}{t/\log|\log t|}=a_x,\qquad
{\mathbb P}^x\mbox{-a.s.}
\end{equation}
Moreover, if there exist constants $\beta_1',C_L'>0$ such that
\begin{equation}\label{e:Phi-lower}
\frac{\Phi(x,r)}{\Phi(x,s)} \ge C_L' \bigg(\frac{r}{s}\bigg)^{\beta_1'} \quad \text{for all} \; x \in U, \; 0<s\le r<R_0 \wedge (C_8\updelta_U(x)),
\end{equation}
then there are constants $\widetilde a_2 \ge \widetilde a_1>0$ such that for all $x \in U$, there exists a constant $\widetilde a_x \in [\widetilde a_1,\widetilde a_2]$ satisfying
\begin{equation}\label{eq:Feller2}
\liminf_{t\to0} \frac{\sup_{0<s\le t} d(x, X_s)}{\Phi^{-1}(x,t/\log|\log t|)}=\widetilde a_x,\qquad
{\mathbb P}^x\mbox{-a.s.}
\end{equation}
\end{thm}
\begin{remark}\label{r:a2}
{\rm Let $\widetilde \Phi$ be any function on $U \times (0,1)$ such that $\widetilde \Phi(x,r) \asymp \Phi(x,r)$ for $x \in U$ and $r \in (0,1)$. Thanks to the Blumenthal's zero-one law, the liminf LILs \eqref{eq:Feller1} and \eqref{eq:Feller2} hold true with $\widetilde \Phi$ instead of $\Phi$. Cf. Remark \ref{r:a}. }
\end{remark}
To prove Theorem \ref{t:Feller1}, we need the following two lemmas.
The first one is a consequence of \cite[Section 5]{BSW}. Since we only put assumptions on the symbol $q$ locally, we carefully check ranges of variables in the proof of the following lemma.
\begin{lem}\label{l:Feller0}
Suppose that \eqref{O1}, \eqref{O2} and \eqref{O3} hold for an open subset $U \subset {\mathbb R}^d$. Then there exist constants $C_{14}, C_{15}>0$ and $C_{16}>1$ such that for all $x \in U$, $0< \frac{8C_{12}}{C_9} r < R_0 \wedge (C_8 \updelta_U(x))$, $w \in B(x,r)$ and $t>0$,
\begin{equation}\label{e:Pruitt1}
{\mathbb P}^w(\tau_{B(w,r)} \le t ) \le \frac{C_{14}t}{\Phi(x,r)},
\end{equation}
\begin{equation}\label{e:Pruitt2}
{\mathbb P}^w(\tau_{B(w,r)} \ge t ) \le \exp \bigg( - \frac{C_{15}t}{\Phi(x,r)} +1\bigg),
\end{equation}
and
\begin{equation}\label{e:Pruitt3}
C_{16}^{-1} \Phi(x,r) \le \E^w[\tau_{B(w,r)}] \le C_{16} \Phi(x,r),
\end{equation}
where $C_9,C_{10}$ and $C_{12}$ are constants in \eqref{O2}, \eqref{O3} and Lemma \ref{l:Phi-1} respectively.
\end{lem}
\noindent{\bf Proof. } Fix $x\in U$. Let $r_0:=R_0 \wedge (C_8 \updelta_U(x))$ and $k:=2C_{12}/C_9>2$. Note that for all $0<4kr<r_0$ and $w \in B(x,2r)$, by the triangle inequality,
\begin{equation*}
C_8\updelta_U(w) \ge C_8(\updelta_U(x) - r) >r_0 - 2r >3kr.
\end{equation*}
Hence, by Lemma \ref{l:Phi-1} and \eqref{O2}, it holds that for all $0<4kr<r_0$ and $w \in B(x,r)$,
\begin{equation}\label{e:estimate-range}
\sup_{1/(4kr)\le |\xi| \le 1/(2kr)} \sup_{|y-w| \le r} \frac{\text{Re} \, q(y,\xi)}{|\xi||\text{Im}\,q(y,\xi)|} \ge \frac{2kr}{C_{12}} \frac{ \sup_{|\xi| \le 1/(kr)} \text{Re} \, q(w,\xi)}{ \sup_{|\xi| \le 1/(kr)} |\text{Im}\,q(w,\xi)|} \ge \frac{2C_9kr}{C_{12}} = 4r.
\end{equation}
Using \eqref{O3}, Lemma \ref{l:Phi-1} and the monotone property of $\Phi$, we get that for all $0<4kr < r_0$ and $w \in B(x,r)$,
\begin{align}\label{e:infsup_q}
&\sup_{1/(4kr)\le |\xi| \le 1/(2kr)} \, \inf_{|y-w| \le 3r} \text{Re} \, q(y, \xi) \nonumber\\
&\ge \sup_{1/(4kr)\le |\xi| \le 1/(2kr)} \, \inf_{|y-x| < 4r} \text{Re} \, q(y, \xi) \ge C_{10} \sup_{1/(4kr)\le |\xi| \le 1/(2kr)} \text{Re} \, q(x, \xi) \ge \frac{C_{10}}{C_{12}} \Phi(x,r).
\end{align}
On the other hand, by Lemma \ref{l:Phi-1} and \eqref{O3}, we also get that for all $0<4r < r_0$ and $w \in B(x,r)$,
\begin{equation}
\sup_{|y-w|\le r} \, \sup_{|\xi| \le 1/r} \text{Re}\,q(y, \xi) \le C_{12} \sup_{1/(2r) \le |\xi| \le 1/r} \, \sup_{|y-w|\le r} \text{Re}\,q(y, \xi)\le \frac{C_{12}}{C_{10}} \Phi(x,r)^{-1}.
\end{equation}
Moreover, we get from Lemma \ref{l:Phi-2} and \eqref{O2} that for all $0<4kr < r_0$ and $w \in B(x,r)$,
\begin{align*}
\sup_{|y-w| \le r} \,\sup_{4/r_0<|\xi|\le 1/r} |\text{Im}\, q(y,\xi)| \le \frac{1}{C_9} \sup_{|y-x|<2r} \,\sup_{|\xi|\le 1/r} |\text{Re}\, q(y,\xi)| \le \frac{C_{13}}{C_9} \Phi(x,r)^{-1}.
\end{align*}
Using the triangle inequality several times, \eqref{O2} in the second inequality, the inequality $|a-\sin a|\le |a|^2$ for $a \in {\mathbb R}$ in the third, Lemma \ref{l:Phi-1} in the fourth, and the monotone property of $\Phi$ and Lemma \ref{l:Phi-2} in the last, we get that for all $y,\xi\in {\mathbb R}^d$ such that $|y-x| < 2r$ and $|\xi| \le 4/r_0$,
\begin{align*}
|\,\text{Im} \, q(y,\xi)| & \le \frac{r_0|\xi|}{4} \big|\,\text{Im} \, q(y,\frac{4}{r_0|\xi|}\xi)\big| + \bigg| \int_{{\mathbb R}^d } \bigg( \frac{r_0|\xi|}{4} \sin {\langle} z, \frac{4}{r_0|\xi|}\xi{\rangle}- \sin {\langle} z, \xi {\rangle}\bigg)\nu(y, dz) \bigg| \\
& \le \frac{r_0|\xi|}{4C_9} \Phi(y,4/r_0)^{-1}+ \frac{r_0|\xi|}{4} \int_{|z| < r_0/4 } \bigg| {\langle} z, \frac{4}{r_0|\xi|}\xi {\rangle} - \sin {\langle} z, \frac{4}{r_0|\xi|}\xi{\rangle} \bigg| \,\nu(y, dz) \\
& \quad + \int_{|z| < r_0/4} \big|{\langle} z, \xi {\rangle}- \sin {\langle} z, \xi {\rangle} \big|\nu(y, dz) + \bigg( \frac{r_0|\xi|}{4} + 1 \bigg) \int_{|z| \ge r_0/4} \nu(y, dz) \\
& \le \frac{1}{C_9} \Phi(y,4/r_0)^{-1}+ \bigg(\frac{16}{r_0^2} + \frac{16}{r_0^2}\bigg) \int_{|z| < r_0/4 } |z|^2\nu(y, dz) + 2 \nu\big(y,{\mathbb R}^d \setminus B(0, r_0/4) \big) \\
& \le c_1 \Phi(y,4/r_0)^{-1} \le c_2 \Phi(x,r)^{-1}.
\end{align*}
Therefore, we deduce that for all $0<4kr<r_0$ and $w \in B(x,r)$,
\begin{align}\label{e:supsup_q}
\sup_{|y-w|\le r} \sup_{|\xi| \le 1/r} |q(y, \xi)| &\le \sup_{|y-w|\le r} \Big( \sup_{|\xi| \le 1/r} \text{Re}\,q(y, \xi)+ \sup_{4/r_0<|\xi| \le 1/r} |\text{Im}\,q(y, \xi)| + \sup_{|\xi| \le 4/r_0} |\text{Im}\,q(y, \xi)| \Big)\nonumber\\
&\le c_3\Phi(x,r)^{-1}.
\end{align}
Finally, by \eqref{e:estimate-range}, \eqref{e:infsup_q} and \eqref{e:supsup_q}, we obtain the results from \cite[Theorem 5.1, Corollary 5.3 and Theorem 5.9]{BSW}. {\hfill $\Box$ \bigskip}
\begin{lem}\label{l:Feller1}
Suppose that \eqref{O1}-\eqref{O4} hold for an open subset $U \subset {\mathbb R}^d$. Then \eqref{A4} holds for $U$.
\end{lem}
\noindent{\bf Proof. } By following the proof of \cite[Proposition 5.2]{GRT19}, one can deduce \eqref{A4} from \eqref{e:Pruitt1}, \eqref{e:Pruitt2} and \eqref{O4}. Below, we give the proof of this conclusion in details for the reader's convenience.
Choose any $x \in U$ and set $r_0:= R_0 \wedge (C_8 \delta_U(x))$. For all $w \in B(x, r_0/8)$, since $\updelta_U(w) \ge \updelta_U(x)-r_0/8>\updelta_U(x)/5$, we get from Lemma \ref{l:Phi-2} and \eqref{e:Phi-upper} that
\begin{equation}\label{e:Feller1-1}
\Phi\big(w, R_0 \wedge (C_8\updelta_U(w))\big) \ge \Phi(w, r_0/5 ) \ge C_{13} \Phi(x, r_0/5 ) \ge c_1 \Phi(x,r_0),
\end{equation}
where the constant $c_1>0$ is independent of $x$.
Let $0<\frac{8C_{12}}{C_9}r <r_0$.
By \eqref{e:Pruitt1} and \eqref{e:Pruitt3}, there is a constant $\varepsilon_1 \in (0,c_1\varepsilon_0/C_{16})$ independent of $x$ and $r$ such that for all $w \in B(x,r/2)$,
\begin{equation}\label{e:def-unittime}
{\mathbb P}^w(\tau_{B(w,r/2)} \le \varepsilon_1 \E^x[\tau_{B(x,r)}] ) \le \frac{C_{11}}{2},
\end{equation}
where $\varepsilon_0$, $C_{11}$ and $C_{16}$ are the constants in \eqref{O4} and \eqref{e:Pruitt3}. We set $t_0:= \varepsilon_1 \E^x[\tau_{B(x,r)}]$ and for $k \ge 0$,
$$
S_k:= \left\{ \sup_{kt_0 \le u \le (k+1)t_0}|X_u-X_{kt_0}|<\frac{r}{2}, \, 2{\langle} X_{(k+1)t_0}-X_{kt_0}, X_{kt_0} {\rangle} \le -|X_{(k+1)t_0}-X_{kt_0}| |X_{kt_0}| \right\}.
$$
Note that, for any $y,z \in {\mathbb R}^d$, if $|y|, |z| < r/2$ and $2{\langle} y,z {\rangle} \le -|y||z|$, then $|y+z| < r/2$. Thus, by the Markov property, we see that for all $n \ge 1$,
\begin{align}\label{e:SP_low}
{\mathbb P}^{x}(\tau_{B(x,r)} \ge n t_0) &\ge {\mathbb P}^{x}\Big( \cap_{k=0}^{n-1} S_k\Big) = {\mathbb P}^{x}\E\Big[ \cap_{k=0}^{n-1} S_k \, \big| \, {\mathcal F}_{(n-1)t_0} \Big] \nonumber\\
& \ge \inf_{w \in B(x,r/2)} {\mathbb P}^w(S_0) \cdot {\mathbb P}^{x}\Big( \cap_{k=0}^{n-2} S_k\Big) \ge ... \ge \Big( \inf_{w \in B(x,r/2)} {\mathbb P}^w(S_0) \Big)^n. \qquad
\end{align}
For all $w \in B(x,r/2)$, since $\varepsilon_0 \Phi\big(w, R_0 \wedge (C_8\updelta_U(w))\big) > c_1 \varepsilon_0 C_{16}^{-1}\E^x[\tau_{B(x,r)}]>t_0$ by \eqref{e:Feller1-1} and \eqref{e:Pruitt3}, using
\eqref{O4} and \eqref{e:def-unittime}, we get that
\begin{align*}
{\mathbb P}^w(S_0) &\ge 1 -{\mathbb P}^w(2{\langle} X_{t_0}-w, w {\rangle} > -|X_{t_0}-w| |w|)-{\mathbb P}^w(\tau_{B(w, r/2)} \le t_0) \ge \frac{C_{11}}{2}.
\end{align*}
It follows that for all $n \ge 1$,
$$
{\mathbb P}^x(\tau_{B(x,r)} \ge n \E^x[\tau_{B(x,r)}]) = {\mathbb P}^x(\tau_{B(x,r)} \ge n\varepsilon_1^{-1} t_0) \ge \bigg( \frac{C_{11}}{2} \bigg)^{-\varepsilon_1^{-1}n +1}.
$$
On the other hand, by \eqref{e:Pruitt2} and \eqref{e:Pruitt3}, we get that for all $n \ge 1$,
$$
{\mathbb P}^x(\tau_{B(x,r)} \ge n \E^x[\tau_{B(x,r)}]) \le e^{-\frac{C_{15}}{C_{16}} n+1 }.
$$
The proof is complete. {\hfill $\Box$ \bigskip}
\noindent \textbf{Proof of Theorem \ref{t:Feller1}.} With a redefined $R_0$, in view of \eqref{e:Pruitt3}, we get \eqref{A1} from Lemma \ref{l:Phi-2}, \eqref{A2} from \eqref{O1} and \eqref{e:Phi-upper}, \eqref{A3} from Lemma \ref{l:Phi-1}, and \eqref{A4} from Lemma \ref{l:Feller1}. Then by the proof of Theorem \ref{inf}, one can see that \eqref{e:inf3} below holds for all $x \in U$ with $\Phi(x, \cdot)$ instead of $\phi(x, \cdot)$. Hence, by the Blumenthal's zero-one law, we deduce \eqref{eq:Feller1}. Moreover, if \eqref{e:Phi-lower} also holds true, then we arrive at \eqref{eq:Feller2} by a similar argument to that in the proof of Corollary \ref{c:inf}. We omit details here. {\hfill $\Box$ \bigskip}
We now give concrete examples of Feller processes which satisfy both liminf LIL at zero and limsup LIL at zero with help from the paper \cite{CKL}. In the following two examples, we use conditions Tail, E and NDL introduced in \cite{CKL}. See Definitions \ref{d:0}--\ref{d:inf} in Appendix for their definitions.
\begin{example}\label{E:non}{\bf (Non-symmetric Feller processes)}
{\rm
Let $\nu$ be a nonincreasing nonnegative function on $(0,\infty)$ satisfying $\int_0^\infty (r^{d-1} \land r^{d+1}) \nu(r)dr < \infty$.
Define for $r>0$,
$$\mathcal{G}(r)= 1\bigg/\int_{|z| \ge r} \nu(|z|) dz \quad \text{and} \quad \mathcal{H}(r)=1\bigg/ \int_{{\mathbb R}^d } \left( \frac{|z|^2}{r^2} \land 1 \right) \nu(|z|) dz.$$
Then $\mathcal H \le \mathcal G$, $\mathcal H$ is increasing and $\sH(r)/\sH(s) \le (r/s)^2$ for all $0<s\le r$. We assume that there exist constants $0<\beta_1\le\beta_2 \le 2$ and $c_1,c_2>0$ such that
\begin{equation}\label{e:sH-scaling}
c_1\bigg(\frac{r}{s}\bigg)^{\beta_1}\le \frac{\sH(r)}{\sH(s)} \le c_2\bigg(\frac{r}{s}\bigg)^{\beta_2} \quad \text{for all} \;\, 0<s \le r \le 1.
\end{equation}
Let $J(z)$ be a nonnegative function on ${\mathbb R}^d$ comparable to $\nu(|z|)$ and $\kappa(x,z)$ be a Borel function on ${\mathbb R}^d \times {\mathbb R}^d$ such that for some constants $a_1,a_2,a_3>0$ and $\beta \in (0,1)$,
\begin{equation}\label{e:non-kappa}
a_1 \le \kappa(x,z) \le a_2 \quad \text{and} \quad |\kappa(x,z) - \kappa(y,z)| \le a_3 |x-y|^\beta \quad \text{for all} \;\, x,y,z \in {\mathbb R}^d.
\end{equation}
In this example, we always suppose that one of the following assumptions holds true:
\smallskip
(P1) \eqref{e:sH-scaling} holds with $\beta_1>1$,
(P2) \eqref{e:sH-scaling} holds with $\beta_2<1$,
(P3) $J(z)=J(-z)$ and $\kappa(x,z)=\kappa(x,-z)$ for all $x,z \in {\mathbb R}^d$.
\medskip
\noindent In each case when (P1), (P2) and (P3) holds, respectively, we consider an operator
\begin{align*}
&{\cal L}^\kappa f(x) :=
\begin{cases}
\int_{{\mathbb R}^d} \big( f(x+z) - f(x) - {\bf 1}_{|z| \le 1} \langle z, \nabla f(x) \rangle \big) \kappa(x,z) J(|z|) dz, & \text{when (P1) holds}; \\
\int_{{\mathbb R}^d} \big( f(x+z) - f(x) \big) \kappa(x,z) J(|z|)dz, & \text{when (P2) holds}; \\
\frac{1}{2} \int_{{\mathbb R}^d} \big( f(x+z) + f(x-z) - 2f(x) \big) \kappa(x,z)J(|z|)dz & \text{when (P3) holds}.
\end{cases}
\end{align*}
According to \cite[Theorem 1.3 and Remark 1.5]{GS19}, if \eqref{e:non-kappa} and one among (P1)-(P3) hold, then
there exists a Feller process $X$ on ${\mathbb R}^d$ whose infinitesimal generator is an extension of $({\cal L}^\kappa, C_c^2({\mathbb R}^d))$. Indeed, the process $X$ is the unique solution to the martingale problem for $({\cal L}^\kappa, C_c^\infty({\mathbb R}^d))$.
By Lemma \ref{l:Phi-1} and \eqref{e:non-kappa}, since $J(z)$ is comparable to $\nu(|z|)$, the symbol $q$ of $X$ satisfies that
\begin{equation}\label{e:sH-Phi}
\sup_{| \xi| \le 1/r} \text{\rm Re} \, q(x, \xi ) \asymp 1/\sH(r) \quad \text{for} \;\, x \in {\mathbb R}^d, \; r>0
\end{equation}
In the followings, we check that $X$ satisfies conditions \eqref{O1}-\eqref{O4} for $U={\mathbb R}^d$.
First, we note that, by \eqref{e:sH-scaling} and \eqref{e:non-kappa}, $\int_{{\mathbb R}^d} \kappa(x,z) J(z)dz \ge c\int_{{\mathbb R}^d}\nu(|z|)dz= c\lim_{r \to 0} 1/\sH(r) =\infty$ for all $x \in {\mathbb R}^d$. Hence, \eqref{O1} holds for $U={\mathbb R}^d$.
When (P3) holds, the symbol $q(x,\xi)$ is a real number for all $x, \xi$ so that \eqref{O2} for $U={\mathbb R}^d$ immediately follows.
We now check \eqref{O2} for the cases (P2) and (P3) separately.
Suppose (P1) holds. Using the triangle inequality, Taylor expansion for the sine function and \eqref{e:sH-scaling}, since $\kappa$ is bounded above, $J(z)$ is comparable to $\nu(|z|)$. Since $\nu(|z|)dz$ is a L\'evy measure on ${\mathbb R}^d$ and $\beta_1>1$, we get that for all $x\in {\mathbb R}^d$ and $\xi \in {\mathbb R}^d$ with $|\xi|\ge 1$,
\begin{align*}
|\text{Im}\, q(x,\xi) | &= \bigg| \int_{{\mathbb R}^d \setminus\{0\}} \big( \langle z,\xi \rangle {\bf 1}_{\{|z| \le 1\}}- \sin \langle z, \xi \rangle \big) \kappa(x,z)J(z)dz \bigg| \\
&\le \bigg| \int_{|z| \le 1} \big( \langle z,\xi \rangle - \sin \langle z, \xi \rangle \big) \kappa(x,z)J(z)dz \bigg| + \bigg| \int_{|z| > 1} \sin \langle z, \xi \rangle \kappa(x,z)J(z)dz \bigg|\\&\le c_2 \int_{|z| \le 1} \big( (|z||\xi|)^3 \wedge (|z||\xi|) \big) \nu(|z|)dz + c_2 \int_{|z| > 1} \nu(|z|)dz \\
&\le c_2 \int_{|z| \le 1/|\xi|} (|z||\xi|)^2 \nu(|z|)dz + c_2 |\xi| \int_{1/|\xi|<|z| \le 1} |z|^2 \nu(|z|)dz +c_3 \\
&\le \frac{c_4}{\sH(1/|\xi|)} + \frac{c_4|\xi|+ c_3\sH(1)}{\sH(1)} \le \frac{c_4}{\sH(1/|\xi|)} + \frac{(c_4+ c_3\sH(1)) |\xi|}{c_1 |\xi|^{\beta_1}\sH(1/|\xi|)} \le \frac{c_5}{\sH(1/|\xi|)}.
\end{align*}
Suppose (P2) holds. Using \eqref{e:sH-scaling}, since $\kappa$ is bounded above, $J(z)$ is comparable to $\nu(|z|)$. Using this fact and $\beta_2<1$, we see that for all $x \in {\mathbb R}^d$ and $\xi \in {\mathbb R}^d$ with $|\xi|\ge 1$,
\begin{align*}
|\text{Im}\, q(x,\xi) | &=\bigg| \int_{{\mathbb R}^d \setminus \{0\} } \sin \langle z, \xi \rangle \kappa(x,z)J(z)dz\bigg|\le c_6|\xi| \int_{|z|<1/|\xi|} |z| \nu(|z|)dz + c_6\int_{|z| \ge 1/|\xi|} \nu(|z|)dz\\
&\le c_6 \sum_{n \ge 1} \int_{2^{-n}/|\xi|\le |z|<2^{-n+1}/|\xi|} 2^{-n+1} \nu(|z|)dz + \frac{c_6}{\sH(1/|\xi|)}\\
&\le c_6\sum_{n \ge 1} \frac{ 2^{-n+1}}{\sH(2^{-n}/|\xi|)} + \frac{c_6}{\sH(1/|\xi|)} \le \frac{c_2c_6}{\sH(1/|\xi|)} \sum_{n \ge 1} 2^{-n+1+n\beta_2} + \frac{c_6}{\sH(1/|\xi|)}=\frac{c_7}{\sH(1/|\xi|)}.
\end{align*}
Therefore, by \eqref{e:sH-Phi}, we deduce that \eqref{O2} always holds true for $U={\mathbb R}^d$.
\eqref{O3}
immediately
follows from the fact that $\kappa(x,z)$ is bounded above and below by positive constants.
For \eqref{O4}, we see from \cite[(84)]{GS19} that the heat kernel $p(t,x,y)$ of $X$ satisfies that for all $t \le 1$ and $x,y \in {\mathbb R}^d$ with $|x-y| \le \sH^{-1}(t)$,
$$ p(t,x,y) \ge c_8 \sH^{-1}(t)^{-d}. $$
It follows that for all $t \le 1$, $x \in {\mathbb R}^d$ and $z \in {\mathbb R}^d$ with $|z|=1$,
\begin{align*}
&{\mathbb P}^x\big(2{\langle} X_t- x, z {\rangle} \le -| X_t -x | \big) \ge \int_{B(x,\sH^{-1}(t)) \cap \{ y : 2{\langle} y- x, z {\rangle} \le -| y -x | \}} p(t,x,y) dy \nonumber\\
& \ge c_8 \sH^{-1}(t)^{-d} \int_{B(x,\sH^{-1}(t)) \cap \{ y : 2{\langle} y- x, z {\rangle} \le -| y -x | \}} dy = c_8 \int_{B(0,1) \cap \{ y : 2{\langle} y, z {\rangle} \le -| y| \}} dy =c_9.
\end{align*}
Therefore, \eqref{O4} holds true for $U={\mathbb R}^d$.
Now, using \eqref{e:sH-scaling} and \eqref{e:sH-Phi}, we conclude from Theorem \ref{t:Feller1} and Remark \ref{r:a2} that the liminf LIL at zero \eqref{eq:Feller2} holds true with $U={\mathbb R}^d$ and $\Phi^{-1}(x, t/\log|\log t|)$ replaced by $\sH^{-1}(t/\log|\log t|)$.
\medskip
To obtain a limsup LIL at zero for the Feller process $X$, we also assume that ${\mathrm{U}}_{r_1}(\mathcal G, \gamma, c)$
(in Definition \ref{d:ws})
holds for some $\gamma>0$ and $r_1, c\in (0,1)$. Here, we emphasize that $\gamma$ may not be smaller than $2$. By \eqref{e:non-kappa},
$\mathrm{Tail}_\infty(\mathcal G,{\mathbb R}^d)$ and $\mathrm{Tail}_\infty(\sH,{\mathbb R}^d, \le)$
(in Definition \ref{d:inf}(i))
hold. Moreover, by \cite[Theorem 1.2(4) and Lemma 4.11]{GS19} and our Lemma \ref{l:hkendl},
$\mathrm{NDL}_{R_0'}(\sH,{\mathbb R}^d)$ (in Definition \ref{d:inf}(iii)) holds for some $R_0'>0$.
Therefore, using \eqref{e:sH-scaling}, we deduce that for all $x \in {\mathbb R}^d$, the limsup LIL at zero given in \cite[Theorem 1.11(i-ii)]{CKL} holds true for $X$ with functions $\phi=\sH$ and $\psi=\mathcal G$.
{\hfill $\Box$ \bigskip}
}
\end{example}
In the following example, we directly check that conditions \eqref{A1}-\eqref{A4}, \eqref{B1}-\eqref{B3} and \eqref{B4+} hold.
\begin{example}\label{E:KKK19}{\bf (Singular L\'evy measure)}
{\rm
Let $\alpha \in (0,2)$, $d \ge 2$ and ${\mathbb R}_i := \{(x_1,\dots,x_d) \in {\mathbb R}^d: x_j = 0 \mbox{ if } j \neq i \}$ for $1\le i \le d$. Denote by $e^i$, $1\le i \le d$ the standard unit vectors in ${\mathbb R}^d$. Define a kernel $J(x,y)$ on ${\mathbb R}^d \times {\mathbb R}^d$ by
\begin{equation}\label{e:axis}
J(x,y) = \begin{cases}
b(x,y)|x-y|^{-1-\alpha}, \;\; &\mbox{if} \;\; y-x \in \cup_{i=1}^d {\mathbb R}_i \setminus \{0\},\\
0, \;\; &\mbox{otherwise},
\end{cases}
\end{equation}
where $b(x,y)$ is a symmetric function on ${\mathbb R}^d \times {\mathbb R}^d$ that is bounded between two positive constants. Using this kernel, define a symmetric form $({\mathcal E}, {\mathcal F})$ on $L^2({\mathbb R}^d;dx)$ as
\begin{equation*}
\begin{split}
{\mathcal E}(u,v) &= \int_{{\mathbb R}^d} \Big( \sum_{i=1}^d \int_{{\mathbb R}} (u(x+e^i \tau) - u(x) )(v(x+e^i \tau) - v(x)) J(x,x+e^i \tau)d\tau \Big)dx,\\
{\mathcal F} &= \{ u \in L^2({\mathbb R}^d; dx) \, | \, {\mathcal E}(u,u) < \infty \}.
\end{split}
\end{equation*}
According to \cite[Theorem 3.9 and Corollary 4.15]{Xu13}, the above form $({\mathcal E}, {\mathcal F})$ is a regular Dirichlet form and the associated Hunt process $X$ is a strong Feller process in ${\mathbb R}^d$. Below, we get liminf and limsup LILs for $X$, both at zero and at infinity.
From the definition \eqref{e:axis}, we see that $\mathrm{Tail}_\infty(r^\alpha, {\mathbb R}^d)$ holds true. Indeed, for all $x \in {\mathbb R}^d$ and $r>0$, $\int_{B(x,r)^c} J(x,dy) = \sum_{i=1}^d \int_{|\tau| \ge r} J(x,x+\tau e^i) d\tau \asymp d\int_{|\tau| \ge r} \tau^{-1-\alpha}d\tau =c r^{-\alpha}$.
By \cite[Proposition 4.4 and the proof of Theorem 4.6]{Xu13}, there exist $c_1,c_2>0$ such that for all $x \in {\mathbb R}^d$, $r,t>0$ and $n \in {\mathbb N}$,
\begin{equation}\label{e:KKK191}
{\mathbb P}^x(\tau_{B(x,r)} <t) \le c_1tr^{-\alpha} \quad \text{and} \quad \; {\mathbb P}^x(\tau_{B(x,r)} \ge c_2nr^\alpha) \le 2^{-n}.
\end{equation}
By \cite[Proposition 4.18]{Xu13}, there exist $c_3,c_4>0$ such that
\begin{equation}\label{e:LHK-singular}
p(t,x,y) \ge c_3 t^{-d/\alpha} \quad \text{for all} \;\, t>0, \; x,y \in {\mathbb R}^d \text{ with } |x-y| \le c_4t^{1/\alpha}.
\end{equation}
It follows that for all $x \in {\mathbb R}^d \setminus \{0\}$ and $t>0$,
\begin{equation}\label{e:KKK192}
{\mathbb P}^x\big(2{\langle} X_t-x, x {\rangle} \le - |X_t-x||x| \big) \ge c_3t^{-d/\alpha} \int_{2{\langle} y-x, x {\rangle} \le - |y-x||x|, \, |y-x| \le c_4t^{1/\alpha}} dy \ge c_3 c_5(d),
\end{equation}
for a constant $c_5(d)>0$ which only depends on the dimension $d$.
Now, by using the first inequality in \eqref{e:KKK191} and \eqref{e:KKK192}, one can repeat the proof of Lemma \ref{l:Feller1} and deduce that for all $x \in {\mathbb R}^d$, $r>0$ and $n \in {\mathbb N}$,
\begin{equation}\label{e:KKK191-2}
{\mathbb P}^x(\tau_{B(x,r)} \ge c_6nr^\alpha) \ge c_7e^{-c_8n}
\end{equation}
with some $c_6,c_7,c_8>0$. From the latter inequality in \eqref{e:KKK191} and \eqref{e:KKK191-2}, we get that $\E^x[\tau_{B(x,r)}] \asymp r^\alpha$ for $x \in {\mathbb R}^d$ and $r>0$, and conditions \eqref{A4} with $U={\mathbb R}^d$ and \eqref{B4} hold true. Consequently, all conditions \eqref{A1}-\eqref{A3} (with $U={\mathbb R}^d$) and \eqref{B1}-\eqref{B3} are satisfied since we already checked that $\mathrm{Tail}_\infty(r^\alpha, {\mathbb R}^d)$ holds true. Moreover, using H\"older continuity of the heat kernel given in \cite[Corollary 4.19]{Xu13} and \eqref{e:LHK-singular}, one can repeat the proof of \cite[Proposition 4.15]{CKL} and deduce that the zero-one law for shift-invariant events stated in Proposition \ref{p:law01} holds true.
Eventually, from Corollaries \ref{c:inf} and \ref{c:infinf}, and Remark \ref{r:a},
we conclude that for all $x,y \in {\mathbb R}^d$, both liminf LILs \eqref{c1} and \eqref{2} hold with $\phi(x,r)=\phi(r) = r^\alpha$. Also, we conclude from \cite[Theorems 1.11-1.12]{CKL} that the limsup LILs \cite[(1.12) and (1.15)]{CKL} hold with $\phi(r)=r^\alpha$.
{\hfill $\Box$ \bigskip}
}
\end{example}
Using { \textit{the local symmetrization} } which has been introduced in \cite{SW13}, we can obtain a sufficient condition for \eqref{O4} in terms of the symbol $q(\cdot,\xi)$. We introduce the following condition:
\medskip
\setlength{\leftskip}{3mm}
\noindent {\bf (S)} $C_c^\infty({\mathbb R}^d)$ is an operator core for $({\cal L}, {\cal D}({\cal L}))$, i.e. $\overline{{\cal L}|_{C_c^\infty({\mathbb R}^d)}}={\cal L}$, and there exist constants $R_0, A_0\in (0,1)$, $K_0 \ge 1$ and $c_L,c_U>0$ such that the following conditions hold for every $x \in U$:
\smallskip
\begin{enumerate}[(i)]
\setlength{\leftskip}{2mm}
\item There exists an increasing function $g(x,\cdot)$ and constants $0<\alpha(x)\le \beta(x)$ such that
\begin{equation}\label{e:alphabeta}
\frac{1}{\alpha(x)}-\frac{1}{\beta(x)} < \frac{1}{d^2+d},
\end{equation}
\begin{equation}\label{e:g-scaling}
c_L \bigg(\frac{r}{s} \bigg)^{\alpha(x)} \le \frac{g(x,r)}{g(x,s)} \le c_U \bigg(\frac{r}{s} \bigg)^{\beta(x)} \quad \text{for all} \;\; r \ge s> 1/(R_0 \wedge (A_0 \updelta_U(x))).
\end{equation}
and
\begin{equation}\label{e:g-comp}
\hspace{15mm} K_0^{-1}g(x,|\xi|) \le \text{Re}\, q(x, \xi) \le K_0 g(x,|\xi|)\; \text{ for all } \xi \in {\mathbb R}^d, \; |\xi|>1/(R_0 \wedge (A_0 \updelta_U(x))).
\end{equation}
\item For every $0<r<R_0 \wedge (A_0 \updelta_U(x))$, there exists a Feller process $Y=Y^{x,r}$ with symbol $q_Y(\cdot,\xi)$ such that
\begin{align}
&(a) \; q(y, \xi)=2 \text{Re}\,q_Y(y,\xi/2) \; \text{ for all } y \in \overline{B(x,r)} \text{ and } \xi\in {\mathbb R}^d,\label{a:as2}\\[5pt]
&(b) \; K_0^{-1} \inf_{|z-x| \le r} \textrm{Re}\,q_Y(z,\xi) \le \textrm{Re}\,q_Y(y,\xi) \le K_0 \sup_{|z-x|\le r} \textrm{Re }q_Y(z,\xi) \\
&\quad\;\; \text{for all } y \in {\mathbb R}^d \setminus \overline{B(x,r)} \text{ and } \xi\in {\mathbb R}^d, \; |\xi|> 1/(R_0 \wedge (A_0 \updelta_U(x))). \label{a:as3}
\end{align}
\end{enumerate}
\setlength{\leftskip}{0mm}
The condition {\bf (S)} looks complicated but it is quite straightforward to check when some concrete form of $q(x,\xi)$ is given. See Examples \ref{E:Feller} and \ref{E:singular} below.
\begin{remark}\label{r:well-posed}
{\rm The assumption that $C_c^\infty({\mathbb R}^d)$ is an operator core for $({\cal L}, {\cal D}({\cal L}))$ is equivalent to the well-posedness of the martingale problem for $(-q(\cdot, D), C_c^\infty({\mathbb R}^d))$. See \cite[Proposition 4.6]{SW13}.
}
\end{remark}
\begin{lem}\label{l:(S)}
Suppose that \eqref{e:g-comp} holds. Then there exists a constant $K_1>1$ such that
\begin{equation*}
\frac{K_1^{-1}}{\Phi(x,r)} \le g(x,1/r) \le \frac{K_1}{\Phi(x,r)}\; \text{ for all } \; x\in U \; \text{and } \; 0<2r<R_0 \wedge (A_0 \updelta_U(x)).
\end{equation*}
\end{lem}
\noindent{\bf Proof. } Using \eqref{e:g-comp}, Lemma \ref{l:Phi-1} and the monotonicity of $g$, we get that for all $x \in U$ and $0<2r<R_0 \wedge (A_0 \updelta_U(x))$,
\begin{equation*}
\frac{1}{\Phi(x,r)} \ge \sup_{|\xi|=1/r}\text{Re} \, q(x, \xi) \ge K_0^{-1} g(x,1/r)
\end{equation*}
and
\begin{equation*}
\frac{1}{\Phi(x,r)} \le C_{12} \sup_{1/(2r) \le |\xi|\le 1/r}\text{Re} \, q(x, \xi) \le C_{12}K_0 g(x,1/r).
\end{equation*}{\hfill $\Box$ \bigskip}
\begin{prop}\label{p:FellerO4}
Suppose that \eqref{O3} and {\bf (S)} hold. Then \eqref{O1}, \eqref{O2} and \eqref{O4} hold.
\end{prop}
\noindent{\bf Proof. } The first inequality in \eqref{e:g-scaling} implies that $\lim_{r \to \infty} g(x,r)=\infty$ for all $x \in U$. Hence, we get \eqref{O1} from Lemma \ref{l:(S)}. \eqref{O2} is obvious because $q(x,\xi)$ is real for all $x \in U$ and $\xi \in {\mathbb R}^d$ by \eqref{a:as2}. Thus it remains to prove \eqref{O4}.
Fix $x_0 \in U$ and set $r_0:=8^{-1} (R_0 \wedge (A_0 \updelta_U(x_0))$.
Let $\eta_0 \in (0,1)$ be a constant which will be chosen later. Pick any $0<t_1< \eta_0\Phi(x_0, r_0)$ and then define $r_1=\Phi^{-1}(x_0,\eta_0^{-1}t_1) \in (0, r_0)$. Let $Y=Y^{x_0,2r_1}$ be a Feller process on ${\mathbb R}^d$ satisfying \eqref{a:as2} and \eqref{a:as3} with $x=x_0$ and $r=2r_1$. Denote by $Y'$ an independent copy of $Y$ and set $ Y^S_t:=\frac{1}{2} (Y_t+2Y'_0-Y'_t)$. Then according to \cite[Lemma 2.8]{SW13}, $Y^S$ is a Feller process with symbol $2\text{Re}\,q_Y(\cdot, \xi/2)$ and its characteristic function $\lambda_t(y,\xi):=\E^{y}[e^{i{\langle} Y^S_t-y, \xi{\rangle}}]$
is nonnegative for every $t \ge 0$ and $y,\xi \in {\mathbb R}^d$.
Since the martingale problem for $(-q(\cdot, D), C_c^\infty({\mathbb R}^d))$ is well-posed (Remark \ref{r:well-posed}), by \cite[Theorem 5.1]{Hoh}, the stopped martingale problem for $(-q(\cdot, D), C_c^\infty({\mathbb R}^d))$ and $B(x_0, 2r_1)$ is also well-posed. Therefore, by constructing $X$ and $Y^S$ in the same probability space, we may assume that $X_s$ and $Y^S_s$ have the same distribution for $0\le s<\tau_{B(x_0,r_1)}$ under ${\mathbb P}^{x_0}$.
Then using \eqref{e:Pruitt1}, we get that for all $z \in {\mathbb R}^d$ with $|z|=1$,
\begin{align}\label{e:O4-main}
{\mathbb P}^{x_0}\big(2{\langle} X_{t_1}- x_0, z {\rangle} \le -| X_{t_1} -x_0 | \big) &\ge {\mathbb P}^{x_0}\big(2{\langle} Y^S_{t_1}- x_0, z {\rangle} \le -| Y^S_{t_1} -x_0 |, \; t_1<\tau_{B(x_0,r_1)} \big) \nonumber\\
& \ge {\mathbb P}^{x_0}\big(2{\langle} Y^S_{t_1}- x_0, z {\rangle} \le -| Y^S_{t_1} -x_0 | \big) - {\mathbb P}^{x_0} \big( \tau_{B(x_0, r_1)} \le t_1\big)\nonumber\\
& \ge {\mathbb P}^{x_0}\big(2{\langle} Y^S_{t_1}- x_0, z {\rangle} \le -| Y^S_{t_1} -x_0 | \big) - C_{14} \eta_0.
\end{align}
For simplicity, we denote $\alpha$ for $\alpha(x_0)$ and $\beta$ for $\beta(x_0)$. Using \eqref{a:as3}, \eqref{e:g-comp}, \eqref{a:as2}, \eqref{e:g-scaling}, and Lemmas \ref{l:(S)} and \ref{l:Phi-2}, we get that for all $u<2r_1$,
\begin{align}\label{e:H-W-condition}
&\inf_{z \in {\mathbb R}^d} \inf_{|\xi|=1/u} \text{Re} \,q_Y(z, \xi) \ge \frac{1}{K_0} \inf_{z \in B(x_0, 2r_1)} \inf_{|\xi|=1/u} \text{Re} \,q_Y(z, \xi) \ge\frac{1}{2K_0^2} \inf_{z \in B(x_0, 2r_1)} g(z, 2/u) \nonumber \\
&\ge \frac{c_L}{2K_0^2} \left( \frac{2r_1}{u}\right)^\alpha\inf_{z \in B(x_0, 2r_1)} g(z, 1/r_1) \ge \frac{c_L}{2K_0^2K_1} \left( \frac{2r_1}{u}\right)^\alpha\inf_{z \in B(x_0, 2r_1)}\frac{1}{\Phi(z, r_1)}\nonumber\\
& \ge \frac{c_LC_{13}}{2K_0^2K_1} \left( \frac{2r_1}{u}\right)^\alpha \frac{1}{\Phi(x_0, r_1)} = \frac{c_LC_{13}}{2K_0^2K_1} \left( \frac{2r_1}{u}\right)^\alpha \frac{\eta_0}{t_1}.
\end{align}
In particular, we have
\begin{equation*}
\lim_{|\xi| \to \infty} \frac{\inf_{z \in{\mathbb R}^d} \text{Re} \,q_Y(z, \xi) }{\log(1+|\xi|)} \ge c_1 \lim_{u \to 0}\frac{ u^{-\alpha} }{\log(1+1/u)} =\infty.
\end{equation*}
Thus, by \cite[Theorem 1.2]{SW13} and the Fourier inversion theorem, $Y^S$ has a transition density function $p_S(t,x,y)$ which is given by
\begin{align*}
p_S(t,x,y) = (2\pi)^{-d} \int_{{\mathbb R}^d} e^{-i{\langle} \xi, y-x{\rangle}} \lambda_t(x,\xi) d \xi, \quad t>0,\; x,y \in {\mathbb R}^d.
\end{align*}
By \cite[Theorem 2.7]{SW13} and \eqref{e:H-W-condition}, we see that for all $y \in {\mathbb R}^d$,
\begin{align*}
& |p_S(t_1,x_0,x_0)-p_S(t_1,x_0,x_0+y)|\\
&\le (2\pi)^{-d} \int_{{\mathbb R}^d} |1-e^{-i{\langle} \xi, y{\rangle}} | \lambda_{t_1}(x_0,\xi) d \xi \le (2\pi)^{-d} |y| \int_{{\mathbb R}^d} |\xi| \lambda_{t_1}(x_0,\xi) d \xi\\
& \le (2\pi)^{-d} |y| \int_{{\mathbb R}^d} |\xi| \exp \bigg(-\frac{t_1}{8} \inf_{z \in {\mathbb R}^d} \text{Re} \, q_Y(z,\xi)\bigg) d \xi\\
&\le (2\pi)^{-d} |y| \bigg( \int_{|\xi| \le \eta_0^{-1/\alpha}r_1^{-1}} |\xi| d \xi+ \int_{|\xi| >\eta_0^{-1/\alpha}r_1^{-1}} |\xi| \exp \bigg(-\frac{t_1}{8} \inf_{z \in {\mathbb R}^d} \text{Re} \, q_Y(z,\xi)\bigg) d \xi\bigg) \\
& \le c_2 |y| \bigg(\eta_0^{-(d+1)/\alpha}r_1^{-(d+1)}+ \int_{\eta_0^{-1/\alpha}r_1^{-1}}^\infty s^d \exp \big( - c_3 \eta_0 r_1^\alpha s^\alpha \big) ds\bigg).
\end{align*}
Using the inequality $e^{-s} \le r^r s^{-r}$ for all $s,r>0$, we obtain
\begin{align*}
\int_{\eta_0^{-1/\alpha}r_1^{-1}}^\infty s^d \exp \big( - c_3 \eta_0 r_1^\alpha s^\alpha \big) ds \le c_4\eta_0^{-(d+2)/\alpha}r_1^{-(d+2)}\int_{\eta_0^{-1/\alpha}r_1^{-1}}^\infty s^{-2} ds=c_5 \eta_0^{-(d+1)/\alpha}r_1^{-(d+1)}.
\end{align*}
Therefore, we deduce that
\begin{equation}\label{e:pY-gradient}
|p_S(t_1,x_0,x_0)-p_S(t_1,x_0,x_0+y)| \le c_6 |y| \eta_0^{-(d+1)/\alpha}r_1^{-(d+1)} \quad \text{for all} \;\; y \in {\mathbb R}^d.
\end{equation}
On the other hand, similar to \eqref{e:H-W-condition}, using \eqref{a:as3}, \eqref{e:g-comp}, \eqref{a:as2}, \eqref{e:g-scaling} and Lemmas \ref{l:(S)} and \ref{l:Phi-2}, we get that for all $u<2r_1$,
\begin{align*}
\sup_{z \in {\mathbb R}^d} \sup_{|\xi|=1/u} 2 \text{Re} \, q_Y(z, \xi/2)& \le K_0^2 \sup_{z \in B(x_0,2r_1)} g(z, 1/u) \le c_U K_0^2 \bigg( \frac{r_1}{u}\bigg)^\beta \sup_{z \in B(x_0,2r_1)} g(z, 1/r_1) \\
& \le c_U K_0^2 K_1 \bigg( \frac{r_1}{u}\bigg)^\beta \sup_{z \in B(x_0,2r_1)} \frac{1}{\Phi(z, r_1)} \\
&\le \frac{c_U K_0^2 K_1}{C_{13}} \bigg( \frac{r_1}{u}\bigg)^\beta \frac{1}{\Phi(x_0, r_1)} = \frac{c_U K_0^2 K_1}{C_{13}} \bigg( \frac{r_1}{u}\bigg)^\beta \frac{\eta_0}{t_1}.
\end{align*}
Put $c_7:= c_U K_0^2 K_1/C_{13}>1$. By taking $\eta_0$ small enough, we may assume $4c_7\eta_0<1$. Then by the second display in \cite[p.3265]{SW13}, it holds that for all $2^{-1}r_1^{-1}<|\xi|<(4c_7\eta_0)^{-1/\beta}r_1^{-1}$,
\begin{equation*}
\text{Re} \, \lambda_{t_1}(x_0, \xi) \ge 1- 2t_1 \sup_{z \in {\mathbb R}^d} 2\text{Re}\, q_Y(z, \xi/2) \ge 1-2c_7\eta_0r_1^\beta |\xi|^{\beta} \ge 2^{-1}.
\end{equation*}
Since $\lambda_{t_1}(x_0,\xi)\ge 0$ for every $\xi\in {\mathbb R}^d$, it follows that
\begin{align*}
p_S(t_1, x_0,x_0) \ge (2\pi)^{-d} \int_{2^{-1}r_1^{-1}<|\xi|<(4c_7\eta_0)^{-1/\beta}r_1^{-1}} 2^{-1} d\xi \ge c_8 \eta_0^{-d/\beta}r_1^{-d}.
\end{align*}
Combining with \eqref{e:pY-gradient}, we obtain that for all $y \in {\mathbb R}^d$ with $ |y| \le 2^{-1}c_6^{-1}c_8 \eta_0^{ - d/\beta + (d+1)/\alpha} r_1$,
\begin{equation*}
p_S(t_1,x_0, x_0+y) \ge c_8 \eta_0^{-d/\beta} r_1^{-d} -2^{-1}c_8 \eta_0^{-d/\beta} r_1^{-d} =2^{-1}c_8 \eta_0^{-d/\beta} r_1^{-d} .
\end{equation*}
Therefore, it holds that for every $z \in {\mathbb R}^d$ with $|z|=1$,
\begin{align*}
&{\mathbb P}^{x_0}\big(2{\langle} Y^S_{t_1}- x_0, z {\rangle} \le -| Y^S_{t_1} -x_0 | \big) \ge \int_{ 2 {\langle} y, z{\rangle} \le -|y|, \, |y| \le 2^{-1}c_6^{-1}c_8 \eta_0^{ - d/\beta + (d+1)/\alpha} r_1} p_S(t_1, x_0, x_0+y) dy\\
& \ge 2^{-1}c_8 \eta_0^{-d/\beta} r_1^{-d} \int_{ 2 {\langle} y, z{\rangle} \le -|y|, \, |y| \le 2^{-1}c_6^{-1}c_8 \eta_0^{ - d/\beta + (d+1)/\alpha} r_1} dy \\
&=2^{-1}c_8 \eta_0^{-d/\beta} r_1^{-d} \big(2^{-1}c_6^{-1}c_8 \eta_0^{ - d/\beta + (d+1)/\alpha} r_1 \big)^d \int_{ 2 {\langle} y, z{\rangle} \le -|y|, \, |y| \le 1} dy= c_9 \eta_0^{(d^2+d)(1/\alpha - 1/\beta)}
\end{align*}
and hence by \eqref{e:O4-main},
$$
{\mathbb P}^{x_0}\big(2{\langle} X_{t_1}- x_0, z {\rangle} \le -| X_{t_1} -x_0 | \big) \ge \eta_0 \big( c_9\eta_0^{-1+(d^2+d)(1/\alpha - 1/\beta)} - C_{14} \big).
$$
Note that the above constants $c_9$ and $C_{14}$ are independent of $x_0$ and $t_1$. In view of \eqref{e:alphabeta}, we conclude \eqref{O4} by taking $\eta_0$ sufficiently small. {\hfill $\Box$ \bigskip}
Below, we give two concrete examples. In the following examples, we assume that $U \subset {\mathbb R}^d$, $d \ge 1$ is an open set and $C_c^\infty({\mathbb R}^d)$ is an operator core for the generator of the Feller process $X$.
\begin{example}\label{E:Feller}
{\rm {\bf (Symbols of varying order)} Suppose that there are H\"older continuous functions $\alpha:U \to (0,2)$ and $\gamma:U \to (-1,1)$ such that $\inf_{x \in U} \alpha(x)>0$, $ \alpha(x)/2 + \gamma(x) \in [0,1]$ for all $x \in U$, and that
$$
q(x, \xi)=|\xi|^{\alpha(x)}(\log (1+|\xi|))^{\gamma(x)} \quad \text{for all} \;\; x \in U, \; \xi \in {\mathbb R}^d.
$$
By H\"older continuities of $\alpha(x)$ and $\gamma(x)$, there exist constants $c_1>0$ and $\theta \in (0,1]$ such that $|\alpha(x)-\alpha(y)|+|\gamma(x)-\gamma(y)|\le c_1|x-y|^\theta$ for all $x,y \in U$. Since $\lim_{r \to 0} r^{c_1 r^\theta}=\lim_{r \to 0} (\log(1+1/r))^{-c_1 r^\theta}=1$, we see that for all $x \in U$ and $\xi\in {\mathbb R}^d$ with $r:=1/|\xi|<1 \wedge \updelta_U(x)$,
\begin{align*}
\inf_{|x-y| \le r} \text{Re} \, q(y,\xi) &\ge r^{-\alpha(x)}(\log (1+1/r))^{\gamma(x)} \inf_{|x-y| \le r} r^{\alpha(x)-\alpha(y)} (\log(1+1/r))^{\gamma(y)-\gamma(x)} \\
&\ge r^{-\alpha(x)}(\log (1+1/r))^{\gamma(x)} r^{ c_1 r^\theta} (\log(1+1/r))^{-c_1 r^\theta} \\[6pt]
&\ge c_2r^{-\alpha(x)}(\log (1+1/r))^{\gamma(x)}
\end{align*}
and
\begin{align*}
\sup_{|x-y| \le r} \text{Re} \, q(y,\xi) &\le r^{-\alpha(x)}(\log (1+1/r))^{\gamma(x)} r^{ -c_1 r^\theta} (\log(1+1/r))^{c_1 r^\theta} \le c_3r^{-\alpha(x)}(\log (1+1/r))^{\gamma(x)}.
\end{align*}
Hence, \eqref{O3} holds.
Now, we check that {\bf (S)} is fulfilled.
Define $g(x,r)=r^{\alpha(x)} (\log (1+r))^{\gamma(x)}$ for $x \in U$, $r>0$. Since for any $\varepsilon>0$, there is a constant $c_4=c_4(\varepsilon)>0$ such that
\begin{equation*} \frac{\log(1+r)}{\log(1+s)} \le c_4 \bigg(\frac{r}{s} \bigg)^{\varepsilon} \quad \text{for all} \;\; r \ge s \ge 1,
\end{equation*}
one can see that $g(x,\cdot)$ satisfies {\bf (S)}(i).
Next, fix any $x_0 \in U$ and $0<2r<1 \wedge \updelta_U(x_0)$. Let $\widetilde \alpha:{\mathbb R}^d \to (0,2]$ and $\widetilde \gamma:{\mathbb R}^d \to (-1,1)$ be H\"older continuous functions such that (i) for every $x \in \overline{B(x_0,r)}$, $\widetilde\alpha(x)=\alpha(x)$ and $\widetilde \gamma(x)=\gamma(x)$ and (ii) for every $x \in {\mathbb R}^d \setminus \overline{B(x_0,r)}$, $\widetilde \alpha(x)/2+\widetilde \gamma(x) \in [0,1]$ and
\begin{equation*}
\frac{1}{2}\inf_{|y-x_0| \le r} u^{\alpha(y)}(\log (1+u))^{\gamma(y)} \le u^{\alpha(x)}(\log (1+u))^{\gamma(x)}\le 2\sup_{|y-x_0| \le r} u^{\alpha(y)}(\log (1+u))^{\gamma(y)} \;\; \text{for all} \; u>16.
\end{equation*}
According to \cite[Theorem 3.3 and Extension 3.13]{Ku}, there exists a Feller process $Y$ on ${\mathbb R}^d$ having the symbol $q_Y(x,\xi)=2^{\widetilde \alpha(x)-1} |\xi|^{\widetilde \alpha(x)} (\log (1+2|\xi|))^{\widetilde \gamma(x)}$. Hence {\bf (S)}(ii) holds.
Note that $\Phi(x,r)=r^{\alpha(x)} (\log ( 1+1/r))^{-\gamma(x)}$ for $x \in U$ and $r>0$ in this case. Hence,
\begin{equation}\label{e:Phi-inverse}
\lim_{t \to 0} \frac{\Phi^{-1}(x, t/\log|\log t|)}{t^{1/\alpha(x)} |\log t|^{\gamma(x)/\alpha(x)} (\log|\log t|)^{-1/\alpha(x)}}=\alpha(x)^{\gamma(x)} \quad \text{for all} \;\; x \in U.
\end{equation}
Finally, since $\inf_{y \in U} \alpha(y) \wedge 2^{-1} \le \alpha(x)^{\gamma(x)} \le 2$ for all $x \in U$, using
Proposition \ref{p:FellerO4}, Theorem \ref{t:Feller1} and \eqref{e:Phi-inverse}, we conclude that there are constants
$a_2 \ge a_1>0$ such that for all $x \in U$, there exists a constant $a_x \in [a_{1}, a_{2}]$ such that
\begin{equation}\label{E:Feller-result1}
\liminf_{t\to0} \frac{ \sup_{0<s\le t} | X_s - x | }{t^{1/\alpha(x)} |\log t|^{\gamma(x)/\alpha(x)} (\log|\log t|)^{-1/\alpha(x)}}=a_x,~\quad
{\mathbb P}^x\mbox{-a.s.}
\end{equation} {\hfill $\Box$ \bigskip}
}
\end{example}
Let $\mathbb S^{d-1}:=\{y \in {\mathbb R}^d: |y|=1\}$ and $\mathbf e_i=\mathbf e_i(d)$, $1\le i\le d$ denote the standard basis of ${\mathbb R}^d$.
\begin{example}\label{E:singular}
{\rm {\bf (Cylindrical stable-like processes)} Suppose that $d \ge 2$ and there exists a H\"older continuous function $\alpha:U \to (0,2)$ with $\inf_{x \in U} \alpha(x)>0$ such that
\begin{equation*}
q(x, \xi)=\sum_{i=1}^d |\xi_i|^{\alpha(x)} \;\; \text{for all $x \in U$ and $\xi=(\xi_1,...,\xi_d)\in {\mathbb R}^d$}.
\end{equation*}
Note that for every $x \in U$, the L\'evy measure $\nu(x,dz)$ is a stable kernel of the form
\begin{equation}
\nu(x,A) = \frac{\alpha(x) 2^{\alpha(x)-1} \Gamma((1+\alpha(x))/2)}{\pi^{1/2}\Gamma(1-\alpha(x)/2)} \int_0^\infty \int_{\mathbb S^{d-1}} {\bf 1}_A(r \theta) r^{-1-\alpha(x)} \sum_{i=1}^d \delta_{\{\mathbf e_i\}}(\theta) dr,
\end{equation}
where $\Gamma(z):=\int_0^\infty u^{z-1}e^{-u}du$ is the gamma function and $\delta_{\{\mathbf e_i\}}$ is a Dirac measure on $\{\mathbf e_i\}$. Since $|\xi|^{\alpha(x)} \le q(x,\xi) \le d|\xi|^{\alpha(x)}$ for all $x \in U$ and $\xi\in {\mathbb R}^d$, using the H\"older continuity of $\alpha$, one can see that \eqref{O3} holds as in Example \ref{E:Feller}. Clearly, {\bf (S)}(i) holds with $g(x,r)=r^{\alpha(x)}$. Choose any $x_0 \in U$ and $0<2r<1 \wedge \updelta_U(x_0)$, and let $\widetilde \alpha:{\mathbb R}^d \to (0,2)$ be a H\"older continuous function such that for every $x \in \overline{B(x_0,r)}$, $\widetilde\alpha(x)=\alpha(x)$ and for every $x \in {\mathbb R}^d \setminus \overline{B(x_0,r)}$,
\begin{equation*}
\frac{1}{2}\inf_{|y-x_0| \le r} u^{\alpha(y)} \le u^{\alpha(x)} \le 2\sup_{|y-x_0| \le r} u^{\alpha(y)} \;\; \text{for all} \; u>16.
\end{equation*}
According to \cite[Theorem 3.1]{KKS}, since the measure $\sum_{i=1}^d \delta_{\{\mathbf e_i\}}$ on $\mathbb S^{d-1}$ is nondegenerate in the sense of \cite[{\bf (M1)}]{KKS}, there exists a Feller process $Y$ on ${\mathbb R}^d$ having the symbol $q_Y(x,\xi)=2^{\widetilde \alpha(x)-1} \sum_{i=1}^d |\xi_i|^{\widetilde \alpha(x)}$. Thus, {\bf (S)}(ii) is satisfied.
In the end, using
Proposition \ref{p:FellerO4} and Theorem \ref{t:Feller1} again,
we get a similar equation to \eqref{e:Phi-inverse} and we can deduce that for all $x \in U$, the LIL \eqref{E:Feller-result1} holds with $\gamma=0$. {\hfill $\Box$ \bigskip}
}
\end{example}
\section{Liminf LILs at infinity for random conductance model with long range jumps}\label{s:RCM}
In \cite[Section 3]{CKL}, we have obtained limsup LILs at infinity for random conductance model with long range jumps using results in \cite{CKW18, CKW20-1}. In this section, we give liminf LILs at infinity for such models. We repeat the setting of the random conductance models in \cite[Section 3]{CKL} here
for the readers' convenience.
Let $G= (\mathbb{L}, E_\mathbb{L})$ be a locally finite connected infinite undirected graph, where $\mathbb{L}$ is the set of vertices, and $E_\mathbb{L}$ the set of edges. For $x,y \in \mathbb{L}$, we denote $d(x,y)$ for the graph distance, namely, the length of the shortest path joining $x$ and $y$. Let $\mu_c$ be the counting measure on $\mathbb{L}$. We assume that for some constant $d>0$,
\begin{equation}\label{(3.1)}
\mu_c(B(x,r)) \asymp r^d \quad \mbox{for} \,\, x \in \mathbb{L}, \, r>10
\end{equation}
A \textit{random conductance} $\boldsymbol{\eta}=(\eta_{xy} : x,y \in \mathbb{L})$ on $\mathbb{L}$ is a family of nonnegative random variables defined on some probability space $(\Omega,{\bf F}, {\bf P})$ such that $\eta_{xx} = 0$ and $\eta_{xy} = \eta_{yx}$ for all $x,y \in \mathbb{L}$. We set $\nu_x:=\sum_{y \in \mathbb{L}} \eta_{xy}$ for $x \in \mathbb{L}$ and denote $\mathbf E$ for the expectation with respect to $\mathbf{P}$.
For each $\omega \in \Omega$, the \textit{variable speed random walk} (VSRW) $X^{\omega} = (X^\omega_t, t \ge 0; {\mathbb P}^x_\omega, x \in \mathbb{L})$ (associated with $\boldsymbol{\eta}$) is defined by the symmetric Markov process on $\mathbb{L}$ with $L^2(\mathbb{L}, \mu_c)$-generator
$$ {\cal L}_V^\omega f(x) = \sum_{y \in \mathbb{L}} \eta_{xy}(\omega)(f(y) - f(x)), \quad x \in \mathbb{L}, $$
and the \textit{constant speed random walk} (CSRW) $Y^\omega = (Y^\omega_t, t \ge 0; {\mathbb P}^x_\omega, x \in \mathbb{L})$ (associated with $\boldsymbol{\eta}$) is the symmetric Markov process on $\mathbb{L}$ with $L^2(\mathbb{L}, \nu)$-generator
$$ {\cal L}_C^\omega f(x) = \nu_x^{-1}(\omega) \sum_{y \in \mathbb{L}} \eta_{xy}(\omega)(f(y)-f(x)), \quad x \in \mathbb{L}. $$
Let $\alpha \in (0,2)$ and $\boldsymbol{\eta}$ be a random conductance on $\mathbb{L}$. With the constant $d>0$ in \eqref{(3.1)}, we write $w_{xy}:=\eta_{xy} |x-y|^{d+\alpha}$ for $x,y \in \mathbb{L}$ so that
\begin{equation*}
\eta_{xx}=w_{xx} = 0 \quad \mbox{and} \quad \eta_{xy}(\omega) = \frac{w_{xy}(\omega)}{|x-y|^{d+\alpha}}, \quad x \neq y, \quad x,y \in \mathbb{L},
\end{equation*}
Suppose that $d > 4-2\alpha$,
\begin{equation*}
\sup_{x,y \in \mathbb{L}, x\neq y} {\bf P}(w_{xy} = 0)=\sup_{x,y \in \mathbb{L}, x\neq y} {\bf P}(\eta_{xy} = 0)< 1/2 \quad \mbox{and} \quad \sup_{x,y \in \mathbb{L}, x\neq y} \big( {\bf E}[w_{xy}^p] + {\bf E}[w_{xy}^{-q} {\bf 1}_{\{ w_{xy} > 0 \}}] \big) < \infty
\end{equation*}
with some constants
\begin{equation*}
p> \frac{d+2}{d} \lor \frac{d+1}{4-2\alpha} \quad \mbox{and} \quad q > \frac{d+2}{d}.
\end{equation*}
When we consider the CSRW $Y^\omega$, we also assume that there exist constants $m_2 \ge m_1 >0$ such that for ${\bf P}$-a.s. $\omega$,
\begin{equation*}
\eta_{xy}(\omega) > 0 \;\; \mbox{for all} \; x,y \in \mathbb{L}, \, x \neq y \quad \mbox{and} \quad m_1 \le \sum_{y \in \mathbb{L}, y \neq x} \eta_{xy}(\omega)\le m_2 \;\; \mbox{for all} \; x \in \mathbb{L}.
\end{equation*}
According to the proof of \cite[Theorem 3.1]{CKL}, for ${\bf P}$-a.s. $\omega$, there are constants ${\upsilon} \in (0,1)$ independent of $\omega$ and $r_1(\omega), r_2(\omega) \ge 1$ such that conditions ${\rm Tail}^{r_1(\omega)}(r^\alpha,{\upsilon})$ and ${\rm NDL}^{r_2(\omega)}(r^\alpha,{\upsilon})$ (in Definitions \ref{d:inf})
hold for both $X^\omega$ and $Y^\omega$. Then, since ${\rm VRD}^{10}({\upsilon})$ holds by \eqref{(3.1)}, using Lemma \ref{l:NDL}(ii), we conclude from Corollary \ref{c:infinf} that
there exist constants $0<a_1 \le a_2 <\infty$ such that for $\mathbf P$-a.s. $\omega$, there exist $a_3(\omega),a_4(\omega) \in [a_1,a_2]$ so that for all $x,y \in {\mathbb L}$,
\begin{align*}
\liminf_{t \to \infty} \frac{ \sup_{0<s \le t} d(x,X^\omega_s)}{t^{1/\alpha} (\log\log t)^{-1/\alpha}} = a_3(\omega), \quad \liminf_{t \to \infty} \frac{ \sup_{0<s \le t} d(x,Y^\omega_s)}{t^{1/\alpha} (\log\log t)^{-1/\alpha}} = a_4(\omega), \quad\;\, {\mathbb P}^y_\omega\text{-a.s.}
\end{align*}
Moreover, when $\alpha \in (0,1)$, the above LILs still hold true for $d > 2-2\alpha$, if $p>\max\{ (d+2)/d, (d+1)/(2-2\alpha) \}$ and $q>(d+2)/d$, by the proof for the latter statement of \cite[Theorem 3.1]{CKL}.
\medskip
\section{Liminf LILs for subordinate processes and symmetric Hunt processes}\label{s:hunt}
In this section, we give liminf LILs for subordinate processes and symmetric Hunt processes. See \cite[Section 2]{CKL} for detailed descriptions and limsup LILs for such processes.
\medskip
Recall that $(M,d,\mu)$ is a locally compact separable metric space with a positive Radon measure $\mu$ on $M$ with full support. Let $\bar R := \sup_{y,z \in M} d(y,z)$ and $F$ be an increasing and continuous function on $(0,\infty)$ such that for some constants $\gamma_2 \ge \gamma_1 > 1$ and $c_L,c_U>0$,
\begin{equation}\label{e:scaling-F} c_L \bigg( \frac{R}{r} \bigg)^{\gamma_1} \le \frac{F(R)}{F(r)} \le c_U \bigg( \frac{R}{r} \bigg)^{\gamma_2} \quad \mbox{for all} \,\, 0<r \le R < \bar R.
\end{equation}
We assume that ${\rm VRD}_{\bar R}(M)$ and a chain condition ${\rm Ch}_{\bar R}(M)$ (see \cite[Definition 1.2]{CKL}) hold. We also assume that there exists a conservative Hunt process $Z = (Z_t, t \ge 0; {\mathbb P}^x, x \in M)$ on $M$ whose heat kernel $q(t,x,y)$ (with respect to $\mu$) exists and satisfies the following estimates: There are constants $R_1 \le \bar R$ and $c_1,c_2,c_3>0$ such that for all $t \in (0,F(R_1))$ and $x,y \in M$,
\begin{equation}\label{e:heatkernel-F}
\frac{c_1}{V(x,F^{-1}(t))} {\bf 1}_{\{F(d(x,y)) \le t\}} \le q(t,x,y) \le \frac{c_2}{V(x,F^{-1}(t))} \exp \big( - c_3 F_1(d(x,y),t) \big),
\end{equation}
where the function $F_1$ is defined by $\displaystyle F_1(r,t) := \sup_{s >0} \Big( \frac{r}{s} - \frac{t}{F(s)} \Big)$.
\smallskip
Let $S = (S_t)_{t \ge 0}$ be a subordinator independent of $Z$. We denote by $\phi_1$ the Laplace exponent of $S$. Then it is well known that there exist a constant $b \ge 0$ and a Borel measure $\nu$ on $(0,\infty)$ satisfying $\int_0^\infty (1 \wedge u) \nu(du)<\infty$ such that
$$ \phi_1(\lambda):=-\log \E[e^{-\lambda S_1}] = b \lambda + \int_{(0,\infty)}(1- e^{-\lambda u})\nu(du), \quad\;\, \lambda>0. $$
We assume either $b \neq 0$ or $\nu((0,\infty)) = \infty$.
Let $X=(X_t)_{t\ge 0}$ be the subordinate process defined by $X_t := Z_{S_t}$. Define
$$ \Phi(r)= \frac{1}{\phi_1(1/F(r))} \quad \mbox{ and } \quad \Pi(r)= \frac{2e}{\nu((F(r),\infty))} \quad\;\; \mbox{for} \;\, r>0.$$
Then $\Phi$ and $\Pi$ are nondecreasing and $ \Phi(r) \le \Pi(r)$ for all $r>0$. Moreover, since we have assumed that either $b\neq 0$ or $\nu((0,\infty))=\infty$, by \eqref{e:scaling-F}, we have that $\lim_{r \to 0} \Phi(r)=0$ and that
\begin{equation}\label{e:scaling-Phi}
\frac{\Phi(r)}{\Phi(s)} \le c_U \bigg( \frac{r}{s}\bigg)^{\gamma_2} \quad \mbox{for all} \,\, 0<r \le R < \bar R.
\end{equation}
See \cite[Subsection 2.1]{CKL}.
By \eqref{e:scaling-Phi} and \cite[Lemmas A.2 and A.3(i)]{CKL}, using Lemma \ref{l:NDL}(i), we see that conditions \eqref{A1}-\eqref{A4} hold for $U = M$ and that the function $\phi(x,r):=\Phi(r)$ satisfies \eqref{e:phi} for $0<r\le 1$. Therefore, we get from Theorem \ref{inf} and Corollary \ref{c:inf} that
\begin{thm}
There are constants $a_2 \ge a_1>0$ such that for all $x \in M$, there exists a constant $a_x \in [a_1,a_2]$ satisfying
\begin{equation}\label{e:Hunt-zero-pre}
\liminf_{t\to0} \frac{\Phi\big(\sup_{0<s\le t} d(x, X_s)\big)}{t/\log|\log t|}=a_x,\qquad
{\mathbb P}^x\mbox{-a.s.}
\end{equation}
Moreover, if $\phi_1$ satisfies lower scaling property ${\mathrm{L}}^1(\phi_1, \beta_1, c_1)$ (see Definition \ref{d:ws} in Appendix) for some $\beta_1,c_1>0$, then there are constants $\widetilde a_2 \ge \widetilde a_1>0$ such that for all $x \in M$, there exists a constant $\widetilde a_x \in [\widetilde a_1,\widetilde a_2]$ satisfying
\begin{equation}\label{e:Hunt-zero}
\liminf_{t\to0} \frac{\sup_{0<s\le t} d(x, X_s)}{\Phi^{-1}(t/\log|\log t|)}=\widetilde a_x,\qquad
{\mathbb P}^x\mbox{-a.s.}
\end{equation}
\end{thm}
\medskip
Here, we point out that our liminf LIL \eqref{e:Hunt-zero-pre} covers the cases when $\phi_1$ is slowly varying at infinity. Therefore, general liminf LIL \eqref{e:Hunt-zero-pre} can be applicable to some jump processes with low intensity of small jumps such as geometric $2\alpha$-stable processes on ${\mathbb R}^d$ ($0<\alpha\le 1$), namely, a L\'evy process on ${\mathbb R}^d$ with the characteristic exponent $\log(1+|\xi|^{2\alpha})$.
\medskip
To get liminf LILs at infinity, we also assume that constants $\bar R = R_1=\infty$ in \eqref{e:scaling-F} and \eqref{e:heatkernel-F}, and ${\mathrm{L}}_1(\phi_1, \beta_1, c_1)$ (see Definition \ref{d:ws} in Appendix \ref{s:A}) hold for some $\beta_1, c_1>0$.
Then by \cite[Lemma A.4(ii)]{CKL} and Lemma \ref{l:NDL}(ii), the function $\phi(x,r):=\Phi(r)$ satisfies \eqref{e:phi} for $r\ge 1$ and \eqref{B4+} holds true. Since conditions \eqref{B1}-\eqref{B3} holds by \cite[Lemmas A.2 and A.3(i)]{CKL}, we deduce from Corollary \ref{c:infinf} that
\begin{thm}
Suppose that \eqref{e:scaling-F} and \eqref{e:heatkernel-F} hold true with $\bar R = R_1=\infty$, and $\phi_1$ satisfies the lower scaling property ${\mathrm{L}}_1(\phi_1, \beta_1, c_1)$ for some $\beta_1, c_1>0$, i.e.,
$$ \frac{\phi_1(r)}{\phi_1(s)} \geq c_1 \Big(\frac{r}{s}\Big)^{\beta_1} \quad \text{for all} \quad s\leq r< 1.$$
Then, there exists a constant $b_\infty$ such that for all $x,y \in M$,
\begin{equation}\label{e:Hunt-infty}
\liminf_{t\to\infty} \frac{\sup_{0<s\le t} d(x, X_s)}{\Phi^{-1}(t/\log\log t)}=b_\infty,\qquad
{\mathbb P}^y\mbox{-a.s.}
\end{equation}
\end{thm}
\bigskip
Similar results hold for symmetric Hunt processes considered in \cite[Subsection 2.2]{CKL}. Precisely, let $X$ be a Hunt process on $M$ associated with a regular Dirichlet form $(\sE^X, {\cal F}^X)$ of the form \cite[(2.21)]{CKL} satisfying \cite[Assumption L]{CKL}. With the function $\Phi_1$ defined in \cite[(2.22)]{CKL} and open subset $\sU$ of $M$ in \cite[Assumption L]{CKL}, using \cite[(B.7), Propositions B.1 and B.13 and Lemma B.8]{CKL} and our Lemma \ref{l:NDL}, we can deduce that the function $\phi(x,r):=\Phi_1(r)$ satisfies \eqref{e:phi} for $0<r\le 1$, and conditions \eqref{A1}-\eqref{A3} and \eqref{A4+} hold for $U=\sU$. Moreover, when constants $\bar R = R_1=\infty$ in \eqref{e:scaling-F} and \eqref{e:heatkernel-F}, the function $\phi(x,r):=\Phi_1(r)$ satisfies \eqref{e:phi} for $r\ge 1$, and conditions \eqref{B1}-\eqref{B3} and \eqref{B4+} hold. Therefore, we conclude from Corollaries \ref{c:inf} and \ref{c:infinf} that the liminf LIL at zero \eqref{e:Hunt-zero} holds for $x\in \sU$ with the function $\Phi_1$ instead of $\Phi$, and if we also assume $\bar R = R_1=\infty$ in \eqref{e:scaling-F} and \eqref{e:heatkernel-F}, then the liminf LIL at infinity \eqref{e:Hunt-infty} holds with the function $\Phi_1$ instead of $\Phi$.
\section{Proof of Main theorems}\label{s:proof}
Recall that we always assume that $\phi(x,r)$ (and $\phi(r)$) satisfies \eqref{e:phi}.
Proposition \ref{p:EP}(ii) below follows from \cite[Proposition 4.9(ii) and Corollary 4.10]{CKL}.
Moreover, when $r\mapsto \phi(x,r)$ is comparable with a strictly increasing continuous function on $(0,\infty)$ independent of $x \in U$, the inequality \eqref{e:EP+} of Proposition \ref{p:EP}(i) is obtained in
\cite[Proposition 4.9(i)]{CKL}
with $\theta=1$. But since we allow $\phi(x,r)$ to depend on the space variable $x$ here, we need some significant modifications in the proof for the next proposition.
\begin{prop}\label{p:EP}
\noindent (i) Suppose that \eqref{A1}, \eqref{A2} and \eqref{A3} hold. Then there exist constants $\theta \in (0,1]$ and $c>0$ such that for all $x \in U$, $0<r<3^{-1}( R_0 \wedge (C_0\updelta_U(x)))$ and $t>0$,
\begin{equation}\label{e:EP+}
{\mathbb P}^x(\tau_{B(x,r)} \le t) \le c \left(\frac{t}{\phi(x,r)}\right)^{\theta}.
\end{equation}
\noindent (ii) Suppose that \eqref{B1}, \eqref{B2} and \eqref{B3} hold. Let ${\upsilon}_1 \in ({\upsilon}, 1)$. Then there exist constants $c>0$ and $R_1 \ge R_\infty$ such that \eqref{e:EP+} holds with $\theta=1$ and $\phi(r)$ instead of $\phi(x,r)$ for all $x \in M$, $r>R_1 {\mathsf{d}}(x)^{{\upsilon}_1}$ and $t \ge \phi(2r^{{\upsilon}/{\upsilon}_1})$. Moreover, $X$ is conservative, that is, ${\mathbb P}^x(\zeta=\infty)=1$ for all $x \in M$.
\end{prop}
Before giving the proof of Proposition \ref{p:EP}, we present some lemmas which will be used in the proof of Proposition \ref{p:EP}(i).
For $\rho>0$, let $X^{(\rho)}$ be a Borel standard Markov process on $M$
obtained from $X$ by suppressing all jumps with jump size bigger than $\rho$
so that the L\'evy measure $J^{(\rho)}(x,dy)$ of $X^{(\rho)}$ is $J^{(\rho)}(x,A)= J(x,A \cap B(x,\rho))$ for every measurable set $A \subset M$. Then the original process $X$ can be constructed from $X^{(\rho)}$ by the Meyer's construction. See \cite{Me75} and \cite[Section 3]{BGK09} for details.
Denote $\tau^{(\rho)}_D:=\inf\{t>0:X_t^{(\rho)} \in M_\partial \setminus D\}$ for the first exit time of $X^{(\rho)}$ from $D$.
We first generalize \cite[Lemma 4.7(i)]{CKL}.
\begin{lem}\label{l:3.8}
Suppose that \eqref{A1}, \eqref{A2} and \eqref{A3} hold. Then, there exist constants $\delta \in (0,1)$ and $K_1>0$ such that for all $x \in U$ and $0<\rho<3^{-1}\big( R_0 \wedge (C_0 \updelta_U(x))\big)$,
\begin{equation*}
\E^x \bigg[ \exp \Big( - \frac{K_1 }{\E^x[\tau_{B(x,r)}]} \tau^{(\rho)}_{B(x,\rho)} \Big)\bigg] \le 1-\delta.
\end{equation*}
\end{lem}
\noindent{\bf Proof. } Let $x \in U$ and $0<\rho<3^{-1}\big( R_0 \wedge (C_0 \updelta_U(x))\big)$, and denote $\psi(x,r)= \E^x[\tau_{B(x,r)}]$. We follow the proof of \cite[Lemma 4.7(i)]{CKL}. By \eqref{A1} and \eqref{A2}, we have
$$
\sup_{z\in B(x,\rho)} \E^z\tau_{ B(x,\rho)} \le \sup_{z\in B(x,\rho)} \E^z\tau_{B(z,2\rho)} \le c_1\sup_{z\in B(x,\rho)} \E^z\tau_{B(z,\rho)} \le c_2
\psi(x,\rho).
$$
Thus, by the same argument as that of \cite[(4.17)]{CKL}, there exist constants $c_3,c_4>0$ such that
\begin{equation}\label{e:l.3.8-1}
{\mathbb P}^x(\tau_{ B(x,\rho)}>t) \ge c_3-\frac{c_4t}{
\psi(x,\rho)}, \quad t>0.
\end{equation}
Moreover, by following the proof for \cite[(4.18)]{CKL}, using \eqref{A1}, one can deduce that
\begin{equation}\label{e:l.3.8-2}
\Big| {\mathbb P}^x(\tau_{ B(x,\rho)}>t) - {\mathbb P}^x(\tau_{ B(x,\rho)}^{(\rho)}>t) \Big| \le \frac{c_5t}{
\psi(x,\rho)}, \quad t>0.
\end{equation}
Let $\delta=c_3/3$, $K_1=(c_4+c_5)\delta^{-1}\log(\delta^{-1})$ and $t_\rho=\delta
\psi(x,\rho)/(c_4+c_5)$. Then by \eqref{e:l.3.8-1} and \eqref{e:l.3.8-2},
\begin{align*}
{\mathbb P}^x(\tau^{(\rho)}_{ B(x,\rho)}\le t_\rho)&=1 - {\mathbb P}^x( \tau_{ B(x,\rho)}> t_\rho) + {\mathbb P}^x( \tau_{ B(x,\rho)} > t_\rho)- {\mathbb P}^x( \tau_{ B(x,\rho)}^{(\rho)} > t_\rho) \\
& \le 1-3\delta + \frac{(c_4+c_5)t_\rho}{
\psi(x,\rho)} = 1-2\delta.
\end{align*}
Hence, by the choice of $K_1$, we get that
\begin{align*}
\E^x \bigg[ \exp \Big( - \frac{K_1 }{
\psi(x,\rho)} \tau^{(\rho)}_{B(x,\rho)} \Big)\bigg] &\le \E^x \bigg[ \exp \Big( - \frac{K_1 }{
\psi(x,\rho)} \tau^{(\rho)}_{B(x,\rho)} \Big) : \tau_{ B(x,\rho)}^{(\rho)} \le t_\rho \bigg] + \exp \Big( - \frac{K_1t_\rho }{
\psi(x,\rho)} \Big) \\
&\le {\mathbb P}^x( \tau_{ B(x,\rho)}^{(\rho)} \le t_\rho) + \exp \Big( - \frac{\delta K_1}{c_4+c_5} \Big)\le 1 - 2\delta + \delta = 1-\delta.
\end{align*}
{\hfill $\Box$ \bigskip}
Unlike \cite[Lemma 4.8(i)]{CKL}, we only get some polynomial bounds in the next lemma. But it is enough to prove Proposition \ref{p:EP} below.
\begin{lem}\label{l:3.9}
Suppose that \eqref{A1}, \eqref{A2} and \eqref{A3} hold. Then, there exist constants $a_1,\theta_1>0$ such that for all $x \in U$ and $0<\rho \le r < 3^{-1}\big( R_0 \wedge (C_0 \updelta_U(x))\big)$,
\begin{equation}\label{e:l.3.9}
\E^x \bigg[ \exp\Big(-\frac{C_1K_1}{\E^x[\tau_{B(x,\rho)}]}\tau_{B(x,r)}^{(\rho)} \Big)\bigg] \le a_1 \bigg( \frac{\rho}{r}\bigg)^{\theta_1},
\end{equation}
where $C_1>0$ and $K_1>0$ are constants in \eqref{A1} and Lemma \ref{l:3.8} respectively.
\end{lem}
\noindent{\bf Proof. }
By taking $a_1$ larger than $6^{\theta_1}$ in \eqref{e:l.3.9}, we may assume that $6\rho \le r$ without loss of generality.
Fix $x \in U$ and $0<6 \rho \le r < 3^{-1}\big( R_0 \wedge (C_0 \updelta_U(x))\big)$, and let $
\psi(x,s)=\E^x[\tau_{B(x,s)}]$ for $s>0$.
Let $\lambda=C_1K_1/
\psi(x,\rho)$, $\tau_0=\tau^{(\rho)}_{B(x,r)}$, $u(z)=\E^z[e^{-\lambda \tau_0}]$ and
\begin{equation*}
B_k=B(x, (2^k-1)\rho) \quad \text{and} \quad b_k= \sup_{y \in B_k} u(y), \quad k \ge 1.
\end{equation*}
Fix any $\delta' \in (0,\delta)$ where $\delta \in(0,1)$ is the constant in Lemma \ref{l:3.8}. For each $k \ge 1$, let $z_k \in B_k$ be a point such that $u(z_k) \ge (1-\delta') b_k$ and $\tau_k:=\tau^{(\rho)}_{B(z_k,(2^k-1)\rho)}$. Since jump sizes of $X^{(\rho)}$ are at most $\rho$, it holds that either $X^{(\rho)}_{\tau_k} \in B(z_k, 2^k\rho) \subset B_{k+1}$ or $X_{\tau_k}^{(\rho)}=\partial$. Therefore by the strong Markov property, we have that for all $k \ge 1$,
\begin{align}\label{e:l.3.9-1}
(1-\delta')b_k &\le u(z_k) = \E^{z_k}[e^{-\lambda \tau_0} \, ;\, \tau_0<\zeta]= \E^{z_k} [ e^{-\lambda \tau_k} e^{-\lambda(\tau_0-\tau_k)} \, ; \, \tau_k \le \tau_0<\zeta] \nonumber\\
&= \E^{z_k} [ e^{-\lambda \tau_k}\, \E^{X^{(\rho)}_{\tau_k}} [e^{-\lambda \tau_0}] \, ; \, \tau_k \le \tau_0<\zeta] \le b_{k+1} \E^{z_k}[e^{-\lambda \tau_k}].
\end{align}
Let $n_0 \in {\mathbb N}$ be such that $(2^{n_0}-1)\rho \le r/3 < (2^{n_0+1}-1)\rho$. Using the monotonicity of $
s \mapsto \psi(x,s)$, \eqref{A1} and Lemma \ref{l:3.8}, since $z_k \in B_k$, we get that for $k \le n_0$,
\begin{align*}
\E^{z_k}[e^{-\lambda \tau_k}]& \le \E^{z_k} \bigg[ \exp \Big( - \frac{C_1K_1 }{
\psi(x, (2^k-1)\rho)} \tau^{(\rho)}_{B(z_k,(2^k-1)\rho)} \Big)\bigg] \\
&\le \E^{z_k} \bigg[ \exp \Big( - \frac{K_1}{
\psi(z_k, (2^k-1)\rho)} \tau^{(\rho)}_{B(z_k,(2^k-1)\rho)} \Big)\bigg] \le 1-\delta.
\end{align*}
Combining with \eqref{e:l.3.9-1}, we conclude that
\begin{align*}
u(x) &\le b_1 \le \frac{1-\delta}{1-\delta'} b_2 \le... \le \left(\frac{1-\delta}{1-\delta'} \right)^{n_0} b_{n_0+1} \le \left(\frac{1-\delta}{1-\delta'} \right)^{n_0} \le \frac{1-\delta'}{1-\delta} \left(\frac{3\rho}{r}\right)^{\log\frac{1-\delta'}{1-\delta}/\log 2}.
\end{align*}
{\hfill $\Box$ \bigskip}
\noindent \textbf{Proof of Proposition \ref{p:EP}.}
(i) It suffices to prove for the case when $\phi(x,r)= \E^x[\tau_{B(x,r)}]$ in view of \eqref{e:phi}. We follow the proof of \cite[Proposition 4.9(i)]{CKL}, but with nontrivial modifications.
Choose any $x \in U$, $0< r < 3^{-1}\big( R_0 \wedge (C_0 \updelta_U(x))\big)$ and $t>0$. Let $\beta_2$ and $C_U$ be the constants from \eqref{e:scaling-zero}. If $t\ge C_U^{-1} 4^{-2\beta_2}\phi(x,r)$, then by taking $c$ larger than $C_U4^{2\beta_2}$, \eqref{e:EP+} holds true. Thus, we assume that $t<C_U^{-1} 4^{-2\beta_2}\phi(x,r)$.
Set $\rho:=r (C_Ut/\phi(x,r))^{1/(2\beta_2)}$. Then $\rho \in [\phi^{-1}(x,t), r/4)$. Indeed,
since we have assumed $t<C_U^{-1} 4^{-2\beta_2}\phi(x,r)$, by \eqref{e:scaling-zero}, it holds that
\begin{equation*}
r/4 > \rho \ge r \left(\phi^{-1}(x,t)/r\right)^{1/2} = r^{1/2} \phi^{-1}(x,t)^{1/2} \ge \phi^{-1}(x,t).
\end{equation*}
Using \eqref{e:scaling-zero} and \eqref{A1}, we see that for every $z \in B(x,2r)$,
\begin{equation}\label{e:z-rho}
\frac{1}{\phi(z,\rho)} \le \frac{C_U2^{\beta_2}}{\phi(z,2r)}\left( \frac{r}{\rho}\right)^{\beta_2} \le \frac{C_1C_U2^{\beta_2}}{\phi(x,2r)}\left( \frac{r}{\rho}\right)^{\beta_2}\le \frac{C_1C_U2^{\beta_2}}{\phi(x,r)}\left( \frac{r}{\rho}\right)^{\beta_2}.
\end{equation}
Define $J_1(x,dy)=J(x,dy) {\bf 1}_{\{\rho \le d(x,y) <r/4\}}$ and $J_2(x,dy)=J(x,dy) {\bf 1}_{\{d(x,y) \ge r/4\}}$. Then we get from
\eqref{A3} and \eqref{e:z-rho} that
\begin{equation}\label{e:p.3.5-1}
\sup_{z \in B(x,r)} J_1(z, M_\partial) \le \sup_{z \in B(x,r)} \frac{C_3}{\phi(z,\rho)} \le \frac{c_1}{\phi(x,r)} \left( \frac{r}{\rho}\right)^{\beta_2}.
\end{equation}
We also get from \eqref{A3}, \eqref{e:scaling-zero} and \eqref{A1} that
\begin{equation}\label{e:p.3.5-2}
\sup_{z \in B(x,r)} J_2(z, M_\partial) \le \sup_{z \in B(x,r)} \frac{C_3}{\phi(z,r/4)} \le \sup_{z \in B(x,r)} \frac{C_3C_U4^{\beta_2}}{\phi(z,r)} \le \frac{c_2}{\phi(x,r)}.
\end{equation}
As in \cite{CKL}, we let $Y^1:=X^{(\rho)}$, $Y^2$ be a Markov process obtained from $Y^1$ by attaching jumps coming from $J^1(x,dy)$, and $Y^3$ be a Markov process obtained from $Y^2$ by attaching jumps coming from $J^2(x,dy)$. For $n \ge 1$, denote by $T^1_n$ and $T^2_n$ the time at which $n$-th extra jump attached to $Y^1$ and $Y^2$, respectively. Let $\widetilde \tau_{B(x,r)}:= \inf\{u>0:Y^3_u \in M_\partial \setminus B(x,r)\}$. By the Meyer's construction, the law of $(Y^3_s: s<\tau_B)$ is the same as that of $(X_s: s<\tau_B)$. Therefore, it holds that
\begin{align}\label{e:4.30}
&{\mathbb P}^x(\tau_{B(x,r)} \le t) = {\mathbb P}^x(\widetilde \tau_{B(x,r)} \le t)\nonumber\\
&= {\mathbb P}^x(T^1_{2} \le \widetilde \tau_{B(x,r)} \le t, \,\widetilde \tau_{B(x,r)} < T^2_{1} ) + {\mathbb P}^x(T^2_{1} \le \widetilde \tau_{B(x,r)} \le t) + {\mathbb P}^x(\widetilde \tau_{B(x,r)} \le t, \, \widetilde \tau_{B(x,r)} <T^1_{2} \wedge T^2_{1} )\nonumber\\
&=:I_1+I_2+I_3.
\end{align}
Let $Z_1,Z_2$ and $Z_3$ be i.i.d. exponential random variables with rate parameter $1$. From the Meyer's construction, using \eqref{e:p.3.5-1} and \eqref{e:p.3.5-2}, respectively, we get that
$$
I_1 \le {\mathbb P}\big( \frac{c_1t}{\phi(x,r)} \left( \frac{r}{\rho} \right)^{\beta_2} \ge Z_1 + Z_2 \big)\le \frac{c_1^2\,t^2}{\phi(x,r)^2} \left( \frac{r}{\rho}\right)^{2\beta_2}
$$
and
$$
I_2 \le {\mathbb P}\big( \frac{c_2t}{\phi(x,r)} \ge Z_3\big) \le \frac{c_2t}{\phi(x,r)}.
$$
On the event $\{\widetilde\tau_{B(x,r)} \le t, \, \widetilde\tau_{B(x,r)} < T^1_{2} \wedge T^2_{1} \}$, using the triangle inequality, we see that
\begin{align}\label{e:I3-jump}
r \le d(x, Y^3_{\widetilde \tau_{B(x,r)}}) &\le d(x, Y^3_{ \widetilde \tau_{B(x,r)}\land T^1_{1}-}) + d( Y^3_{ \widetilde \tau_{B(x,r)} \land T^1_{1}-}, Y^3_{ \widetilde \tau_{B(x,r)} \land T^1_{1}}) + d( Y^3_{ \widetilde \tau_{B(x,r)}\land T^1_{1}}, Y^3_{ \widetilde \tau_{B(x,r)}})\nonumber\\
&\le d(x, Y^3_{ \widetilde \tau_{B(x,r)}\land T^1_{1}-}) + d( Y^3_{ \widetilde \tau_{B(x,r)}\land T^1_{1}}, Y^3_{ \widetilde \tau_{B(x,r)}}) + \frac{r}{4}.
\end{align}
In the last inequality above, we used the fact that the jump size of $Y^3$ at time $T^1_{1}$ is at most $r/4$. Denote by $\theta^{Y^3}$ the shift operator with respect to $Y^3$. In view of the Meyer's construction, using the strong Markov property, we obtain from \eqref{e:I3-jump} that
\begin{align*}
I_3 &\le {\mathbb P}^x\Big(d(x, Y^3_{ \widetilde \tau_{B(x,r)}\land T^1_{1}-})>r/3, \, \widetilde \tau_{B(x,r)} \le t, \, \widetilde \tau_{B(x,r)} <T^1_{2} \wedge T^2_{1} \Big)\nonumber\\
&\quad + {\mathbb P}^x\Big(d( Y^3_{ \widetilde \tau_{B(x,r)}\land T^1_{1}}, Y^3_{ \widetilde \tau_{B(x,r)}}) >r/3, \, \widetilde \tau_{B(x,r)} \le t, \, \widetilde \tau_{B(x,r)} <T^1_{2} \wedge T^2_{1} \Big) \\
&\le {\mathbb P}^x\Big( \tau^{(\rho)}_{B(x, r/3)} \le \widetilde\tau_{B(x,r)} \le t, \; \widetilde\tau_{B(x,r)} < T^1_{2} \land T^2_{1} \Big) \nonumber\\
&\quad + {\mathbb P}^x\Big(\tau^{(\rho)}_{B(Y^3_{\widetilde \tau_{B(x,r)} \land T^1_1}, r/3)} \circ \theta^{Y^3}_{\widetilde \tau_{B(x,r)}\land T^1_1} \le \widetilde\tau_{B(x,r)} \le t, \; \widetilde\tau_{B(x,r)} < T^1_{2} \land T^2_{1} \Big) \nonumber\\
& \le 2 \sup_{z \in B(x,5r/4)} {\mathbb P}^z \big(\tau^{(\rho)}_{B(z, r/3)} \le t\big).
\end{align*}
In the second inequality above, we used the fact that $Y^3_{\widetilde \tau_{B(x,r)}\land T^1_1} \in B(x, r + r/4)$. Therefore, we obtain from Markov inequality, \eqref{e:z-rho} and Lemma \ref{l:3.9} that
\begin{align*}
I_3& \le 2 \sup_{z \in B(x, 5r/4)} \E^z \bigg[ \exp\left(\frac{C_1K_1t}{\phi(z,\rho)} -\frac{C_1K_1}{\phi(z,\rho)}\tau_{B(z,r/3)}^{(\rho)} \right)\bigg]\\
& \le 2\exp \bigg(\frac{c_3t}{\phi(x,r)}\left(\frac{r}{\rho}\right)^{\beta_2} \bigg) \sup_{z \in B(x, 5r/4)} \E^z \bigg[ \exp\left( -\frac{C_1K_1}{\phi(z,\rho)}\tau_{B(z,r/3)}^{(\rho)} \right)\bigg]\\
&\le c_4 \bigg( \frac{\rho}{r}\bigg)^{\theta_1}\exp \bigg(\frac{c_3t}{\phi(x,r)}\left(\frac{r}{\rho}\right)^{\beta_2} \bigg),
\end{align*}
where $\theta_1,C_1, K_1>0$ are the constants in \eqref{e:l.3.9}.
Finally, since $t<C_U^{-1}\phi(x,r)$, we deduce from the definition of $\rho$ and \eqref{e:4.30} that
\begin{align*}
{\mathbb P}^x(\tau_{B(x,r)} \le t) &\le \frac{c_1^2\,t^2}{\phi(x,r)^2} \frac{\phi(x,r)}{C_Ut} + \frac{c_2t}{\phi(x,r)} + c_4 \bigg( \frac{C_Ut}{\phi(x,r)}\bigg)^{\theta_1/(2\beta_2)}\exp \bigg(\frac{c_3t}{\phi(x,r)}\left(\frac{\phi(x,r)}{C_Ut}\right)^{1/2} \bigg) \\
& \le \frac{(c_1^2C_U^{-1}+c_2)t}{\phi(x,r)} + e^{c_3} c_4 \bigg( \frac{C_Ut}{\phi(x,r)}\bigg)^{\theta_1/(2\beta_2)} \le c_5 \bigg( \frac{t}{\phi(x,r)}\bigg)^{((2\beta_2) \wedge \theta_1)/(2\beta_2)} .
\end{align*}
(ii) The result follows from
\cite[Proposition 4.9(ii) and Corollary 4.10]{CKL}.{\hfill $\Box$ \bigskip}
An event $G$ is called \textit{shift-invariant} if $G$ is a tail event (i.e. $\cap_{t>0}^\infty \sigma(X_s:s>t)$-measurable), and ${\mathbb P}^y(G)= {\mathbb P}^y(G \circ \theta_t)$ for all $y \in M$ and $t>0$.
The following zero-one law for shift-invariant events is established in \cite[Proposition 4.15]{CKL}.
\begin{prop}\label{p:law01}
Suppose that ${\mathrm {VRD}}^{R_\infty'}(\up)$ \ holds. If \eqref{B1}, \eqref{B2}, \eqref{B3} and \eqref{B4+} hold, then for every shift-invariant $G$, it holds either ${\mathbb P}^z(G)=0$ for all $z \in M$ or else ${\mathbb P}^z(G) = 1$ for all $z \in M$.
\end{prop}
Now, we are ready to prove our main results in Section \ref{s:intro}.
\bigskip
\noindent \textbf{Proof of Theorem \ref{inf}.}
In view of Remark \ref{r:a}, it suffices to prove for the case when $\phi(x,r)=\E^x[\tau_{B(x,r)}]$. We claim that there exist constants $q_2 \ge q_1>0$ such that for all $x \in U$,
\begin{equation}\label{e:inf0}
\limsup_{r\to0} \frac{\tau_{B(x,r)}}{\phi(x,r) \log|\log \phi(x,r)|} \in [q_1, q_2],\qquad {\mathbb P}^x\mbox{-a.s.}
\end{equation}
We follow the main idea of the proof in \cite[Theorem 3.7]{KKW17} and will prove upper and lower bound of the limsup behavior in \eqref{e:inf0} separately.
Pick $x \in U$. Let $C_7>0$ be the constant in \eqref{A4}. We set
\begin{equation*}
l_n:=\phi^{-1}(x,e^{-n}) \;\; \text{ and } \;\; A_n:=\Big\{ \sup_{l_{n+1}\le r\le l_n} \frac{\tau_{B(x,r)}}{\phi(x,r) \log|\log \phi(x,r)|}\ge \frac{2e }{C_7} \Big\}, \quad n \ge 3.
\end{equation*}
Since $\lim_{r \to 0} \phi(x,r)=0$ by \eqref{A2}, we have $\lim_{n \to \infty}l_n=0$. Then using \eqref{A4}, we get that for all $n$ large enough,
\begin{align*}
{\mathbb P}^x(A_n) &\le {\mathbb P}^x \bigg( \tau_{B(x,l_n)}\ge \frac{2e}{C_7} \phi(x,l_{n+1}) \log|\log \phi(x,l_{n+1})| \bigg) \le {\mathbb P}^x \bigg( \frac{\tau_{B(x,l_n)}}{\phi(x,l_n)}\ge \frac{2\log n}{C_7} \bigg) \le C_6 e^{C_7} n^{-2}.
\end{align*}
Thus, $\sum_{n=3}^\infty {\mathbb P}^x(A_n)<\infty$. Then by the Borel-Cantelli lemma, the upper bound in \eqref{e:inf0} holds with $q_2=2e/C_7$.
Now, we prove the lower bound in \eqref{e:inf0}. Let $C_1,C_5$ be the constants in \eqref{A1} and \eqref{A4}, and set
\begin{equation*}
r_n:=\phi^{-1}(x,e^{-n^2}) \;\; \text{ and } \;\; u_n:=\frac{\phi(x,r_{n}) \log|\log \phi(x,r_n)|}{8C_1C_5}, \quad n \ge 3.
\end{equation*}
We also define for $n \ge 3$,
\begin{align*}
E_n&:=\big\{ \sup_{0<s\le u_{n+1}}d(x,X_s) \ge r_n\big \}, \qquad
F_n:=\big\{ \sup_{u_{n+1} < s\le u_{n}}d(X_{u_{n+1}}, X_s) \ge r_n\big\},\\
G_n&:= \big\{ \sup_{0 < s\le u_{n} } d(x,X_s) \ge 2r_n \big\}, \qquad H_n:= \cap_{k=n}^{2n} G_k = \Big\{ \sup_{n \le k \le 2n}\frac{\tau_{B(x, 2r_k)}}{u_{k}} \le 1\Big\}.
\end{align*}
Note that $G_n \subset E_n \cup F_n$ for all $n \ge 3$ by the triangle inequality. Thus, we have
\begin{equation}\label{e:decomA}
H_n \subset \cap_{k=n}^{2n} \big( E_k \cup (F_k \setminus E_{k}) \big) \subset \big(\cup_{k=n}^{2n} E_k \big) \cup \big( \cap_{k=n}^{2n} (F_k \setminus E_k) \big).
\end{equation}
By Proposition \ref{p:EP}(i), we have that for all large enough $k$,
\begin{align}\label{e:inf1}
{\mathbb P}^x (E_k) &= {\mathbb P}^x( \tau_{B(x, r_k)} \le u_{k+1}) \le c_1 \bigg( \frac{u_{k+1}}{\phi(x,r_k)} \bigg)^\theta = c_1 \bigg( \frac{e^{-(k+1)^2} \log (k+1)}{4C_1C_5 e^{-k^2}} \bigg)^\theta = c_2 e^{-2\theta k} (\log (k+1))^\theta.
\end{align}
Next, by \eqref{A1} and \eqref{A4}, we have that for all $k$ large enough and $z \in B(x, r_k)$,
\begin{equation}\label{e:F-1}
{\mathbb P}^z (\tau_{B(z,r_k)} \ge u_{k} ) \ge {\mathbb P}^z\bigg( \tau_{B(z,r_k)}\ge \frac{ \phi(z,r_{k}) \log k }{4C_5}\bigg) \ge
C_4e^{-C_5} k^{-1/4} \ge 1-\exp \big( -C_4e^{-C_5}k^{-1/4}\big)
\end{equation}
and hence
\begin{equation}\label{e:F-2}
{\mathbb P}^z\big(\sup_{0<s\le u_{k}-u_{k+1} } d(z,X_s) \ge r_k
\big) \le 1 - {\mathbb P}^z (\tau_{B(z,r_k)} \ge u_{k} ) \le \exp \big( -C_4e^{-C_5}k^{-1/4}\big).
\end{equation}
Thus, using the Markov property, we get that for all $n$ large enough,
\begin{align}\label{e:F-3}
&{\mathbb P}^x \big(\cap_{k=n}^{2n} (F_k \setminus E_k)\big)
\le
\E^x {\mathbb P}^x\big(\cap_{k=n}^{2n} F_k, \, X_{u_{j+1}} \in B(x,r_j), \,\, n \le j \le 2n \, | \, {\mathcal F}_{u_{n+1}}\big) \nonumber\\[6pt]
& \le {\mathbb P}^x \big(\cap_{k=n+1}^{2n} F_k, \, X_{u_{j+1}} \in B(x,r_j), \,\, n+1 \le j \le 2n \big) \sup_{z \in B(x, r_n)} {\mathbb P}^z\big(\sup_{0<s\le u_n - u_{n+1} } d(z,X_s) \ge r_n \big) \nonumber\\
& \le \exp \big( -C_4e^{-C_5}n^{-1/4}\big)\,\E^x {\mathbb P}^x \big(\cap_{k=n+1}^{2n} F_k, \, X_{u_{j+1}} \in B(x,r_j), \,\, n+1 \le j \le 2n \, | \, {\mathcal F}_{u_{n+2}}\big) \nonumber\\
& \le...\le \prod_{k=n}^{2n} \exp \big( -C_4e^{-C_5}k^{-1/4}\big) \le \prod_{k=n}^{2n} \big( -C_4e^{-C_5}(2n)^{-1/4}\big) \le \exp(\, - c_3 n^{3/4}).
\end{align}
Therefore, by combining the above with \eqref{e:decomA} and \eqref{e:inf1}, we get that for all $n$ large enough,
\begin{align*}
{\mathbb P}^x (H_n) &\le \sum_{k=n}^{2n}{\mathbb P}^x(E_k)+ {\mathbb P}^x \left(\cap_{k=n}^{2n} (F_k \setminus E_k)\right) \le c_2e^{-2\theta n} (n+1)(\log (2n+1))^\theta + \exp(\, - c_3 n^{3/4}),
\end{align*}
which yields $\sum_{n=3}^\infty {\mathbb P}^x(H_n)<\infty$. By the Borel-Cantelli lemma, it follows that
\begin{equation}\label{e:liminflower}
{\mathbb P}^x \big(\limsup_{k \to \infty} \frac{\tau_{B(x,2r_k)}}{u_k}\ge 1\big)=1.
\end{equation}
Since $\lim_{k \to \infty} r_k=0$ and
$$
u_k \ge \frac{\phi(x, 2r_k) \log |\log \phi(x, 2r_k)|}{2^{3+\beta_2}C_1C_5C_U}
$$
for all $k$ large enough by \eqref{e:scaling-zero}, we conclude from \eqref{e:liminflower} that the lower bound in \eqref{e:inf0} holds.
\smallskip
Now, we claim that for all $x \in U$, it holds that
\begin{equation}\label{e:inf3}
\liminf_{t \to 0} \frac{\phi\big(x,\sup_{0<s \le t}d(x,X_s)\big)}{t/\log |\log t|} \in [e^{-1}q_2^{-1}, q_1^{-1}],\qquad
{\mathbb P}^x\mbox{-a.s.}
\end{equation}
Note that once we prove \eqref{e:inf3}, the proof is finished thanks to the Blumenthal's zero-one law. Also, since $q_1$ and $q_2$ in \eqref{e:inf3} can be chosen by $C$ and the constants $q_1$ and $q_2$ with respect to $\E^x[\tau_{B(x,r)}]$, Remark \ref{r:a} is also verified.
Here, we show \eqref{e:inf3}. Recall that $l_n:=\phi^{-1}(x,e^{-n})$. Set $t_n:=\phi(x, l_n) \log|\log \phi(x, l_n)|=e^{-n}\log n$. Choose any $\delta>0$.
By \eqref{e:inf0}, for ${\mathbb P}^x$-a.s. $\omega$, there exists $N =N(\omega)$ such that $
\tau_{B(x,l_n)} \le (q_2+\delta) t_n$ for all $n \ge N$. Thus, by \eqref{e:scaling-zero}, it holds that for ${\mathbb P}^x$-a.s. $\omega$,
\begin{align*}
\liminf_{t \to 0} \frac{\phi\big(x,\sup_{0<s \le t}d(x,X_s)\big)}{ t/\log |\log t|} &\ge \liminf_{n \to \infty} \inf_{ t \in [(q_2+\delta)t_n, (q_2+\delta)t_{n-1}]} \frac{\phi\big(x,\sup_{0<s \le t}d(x,X_s)\big)}{ t/\log |\log t|}\\
&\ge \liminf_{n \to \infty}\frac{\phi\big(x,\sup_{0<s \le (q_2+\delta)t_n}d(x,X_s)\big)}{ (q_2+\delta)t_{n-1}/ \log |\log (q_2+\delta)t_{n-1}|}\\
& \ge \liminf_{n \to \infty} \frac{\phi(x, l_n)}{ (q_2+\delta)e^{-(n-1)} \log(n-1) / \log n } = \frac{1}{e(q_2+\delta)}.
\end{align*}
On the other hand, we also get from \eqref{e:inf0} that for ${\mathbb P}^x$-a.s. $\omega$, there exists a decreasing sequence $(\widetilde r_n)_{n \ge 1} = (\widetilde r_n(\omega))_{n \ge 1}$ converging to zero such that
\begin{equation*}
\tau_{B(x,\widetilde r_n)}(\omega) \ge (q_1-\delta) \phi(x, \widetilde r_n) \log|\log \phi(x, \widetilde r_n)|=:\widetilde t_{\delta, n} \quad \text{for all} \;\; n \ge 1.
\end{equation*}
It follows that ${\mathbb P}^x$-a.s.,
\begin{align*}
\liminf_{t \to 0} \frac{\phi\big(x,\sup_{0<s \le t}d(x,X_s)\big)}{ t/\log |\log t|} &\le \liminf_{n \to \infty} \frac{\phi\big(x,\sup_{0<s \le \widetilde t_{\delta, n}}d(x,X_s)\big)}{\widetilde t_{\delta, n}/\log |\log \widetilde t_{\delta, n}|} \\
& \le \liminf_{n \to \infty} \frac{\phi(x, \widetilde r_n)}{\widetilde t_{\delta, n}/\log |\log \widetilde t_{\delta, n}|}\\
&= \liminf_{n \to \infty} \frac{\phi(x, \widetilde r_n)}{ (q_1-\delta) \phi(x, \widetilde r_n)} \frac{\log |\log \widetilde t_{\delta,n}| }{\log |\log \phi(x, \widetilde r_n)|}=\frac{1}{q_1-\delta}.
\end{align*}
Since $\delta$ can be arbitrarily small, we obtain \eqref{e:inf3}. The proof is complete. {\hfill $\Box$ \bigskip}
\noindent \textbf{Proof of Corollary \ref{c:inf}.} Using \eqref{e:scaling-zero} and \eqref{e:scaling-zero-lower}, we can see from \eqref{e:inf3} that there exist constants $c_2 \ge c_1>0$ such that for all $x \in U$, $\liminf_{t \to 0} \sup_{0<s \le t}d(x,X_s)/\phi^{-1}(x,t/\log |\log t|) \in [c_1, c_2]$, ${\mathbb P}^x$-a.s.
Then using the Blumenthal's zero-one law again, we obtain the result. {\hfill $\Box$ \bigskip}
\noindent \textbf{Proof of Theorem \ref{t:infinf}.} By \eqref{e:phi}, it suffices to prove the theorem with $\phi(r):=\E^o[\tau_{B(o,r)}]$. We follow the proof of Theorem \ref{inf} with some modifications. To obtain the desired result, by repeating the arguments for obtaining \eqref{e:inf3} and using \eqref{e:scaling-infty}, it is enough to show that there exist constants $q_4 \ge q_3>0$ such that for all $x,y \in M$,
\begin{equation}\label{e:inf0'}
\limsup_{r\to \infty} \frac{\tau_{B(x,r)}}{\phi(r) \log\log \phi(r)} \in [q_3, q_4], \qquad
{\mathbb P}^y\mbox{-a.s.}
\end{equation}
By \eqref{e:scaling-infty} and the monotone property of $\phi(r)$, we have that, for all $x,y \in M$, since $d(x,y)<\infty$,
\begin{equation*}
\limsup_{r\to \infty} \frac{\tau_{B(x,r)}}{\phi(r) \log\log \phi(r)} \le \limsup_{r\to \infty} \frac{\tau_{B(y,r+d(x,y))}}{\phi(r) \log\log \phi(r)} \le 2^{\beta_2} C_U\limsup_{r\to \infty} \frac{\tau_{B(y, 2r)}}{\phi(2r) \log\log \phi(2r)}
\end{equation*}
and
\begin{equation*}
\limsup_{r\to \infty} \frac{\tau_{B(x,r)}}{\phi(r) \log\log \phi(r)} \ge \limsup_{r\to \infty} \frac{\tau_{B(y,r-d(x,y))}}{\phi(r) \log\log \phi(r)} \ge 2^{-\beta_2} C_U^{-1}\limsup_{r\to \infty} \frac{\tau_{B(y, r/2)}}{\phi(r/2) \log\log \phi(r/2)}.
\end{equation*}
Thus, to get \eqref{e:inf0'}, it is enough to prove that for all $y \in M$,
\begin{equation}\label{e:inf0''}
\limsup_{r\to \infty} \frac{\tau_{B(y,r)}}{\phi(r) \log\log \phi(r)} \in [2^{\beta_2}C_Uq_3, \, 2^{-\beta_2}C_U^{-1}q_4],\qquad
{\mathbb P}^y\mbox{-a.s.}
\end{equation}
Let $y \in M$. With the constant $C_7$ in \eqref{B4}, we define
\begin{equation*}
\widetilde l_n=\phi^{-1}(e^{n}) \;\; \text{ and } \;\; \widetilde A_n=\Big\{ \sup_{\widetilde l_{n}\le r\le \widetilde l_{n+1}} \frac{\tau_{B(y,r)}}{\phi(r) \log\log \phi(r)}\ge \frac{2e }{C_7} \Big\}, \quad n \ge 3.
\end{equation*}
Note that $\lim_{n \to \infty}\widetilde l_n=\infty$ by \eqref{B2} (see Remark \ref{r:basic}(iii)). Hence, $\widetilde l_n>R_\infty {\mathsf{d}}(y)^{\upsilon}$ for all $n$ large enough. Then by \eqref{B4}, we get that for all $n$ large enough,
\begin{align*}
{\mathbb P}^y(\widetilde A_n) &\le {\mathbb P}^y \bigg( \tau_{B(y,\widetilde l_{n+1})}\ge \frac{2e}{C_7} \phi(\widetilde l_{n}) \log\log \phi(\widetilde l_{n}) \bigg) ={\mathbb P}^y \bigg( \frac{\tau_{B(y,\widetilde l_{n+1})}}{\phi(\widetilde l_{n+1})}\ge \frac{2\log n}{C_7} \bigg) \le C_6 e^{C_7} n^{-2}.
\end{align*}
Using the Borel-Cantelli lemma, we deduce that the upper bound in \eqref{e:inf0''} holds true.
To prove the lower bound, we set
$$
m_n:=\phi^{-1}(e^{n^2}) \;\; \text{ and } \;\; s_n:= \frac{\phi(m_n) \log \log \phi(m_n)}{8C_1C_5}, \quad n \ge 3,
$$
where $C_1,C_5$ are the constants in \eqref{B1} and \eqref{B4}. We also let
\begin{align*}
\widetilde E_n&: =\big\{\sup_{0 < s\le s_{n-1} } d(y,X_s) \ge m_n\big\}, \qquad \widetilde F_n:=\big\{\sup_{s_{n-1} < s\le s_{n}} d(X_{s_{n-1}}, X_s) \ge m_n \big\},\\
\widetilde G_n&:= \big\{ \sup_{0 < s \le s_{n} } d(y,X_s) \ge 2 m_n \big\}, \qquad \widetilde H_n:= \cap_{k=n}^{2n} \widetilde G_k = \Big\{ \sup_{n \le k \le 2n}\frac{\tau_{B(y, 2m_k)}}{s_{k}} \le 1\Big\}.
\end{align*}
Then for all $n$, $\widetilde G_n \subset \widetilde E_n \cup \widetilde F_n$ by the triangle inequality so that $\widetilde H_n \subset (\cup_{k=n}^{2n} \widetilde E_k) \cup (\cap_{k=n}^{2n} (\widetilde F_k \setminus \widetilde E_k))$.
First, using Proposition \ref{p:EP}(ii) (with ${\upsilon}_1 = \sqrt {\upsilon}<1$) and \eqref{e:scaling-infty} twice, we get that for all $n$ large enough,
\begin{align}\label{e:wtF}
&{\mathbb P}^y (\widetilde E_n) \le {\mathbb P}^x\big( \tau_{B(x, m_n)} \le s_{n-1} + \phi(2 m_n^{\sqrt{\upsilon}})\big)\le c_1\frac{s_{n-1} + \phi(2 m_n^{\sqrt{\upsilon}})}{\phi(m_n)}\nonumber\\
& \le c_2 e^{-2n} \log n + c_2 m_n^{-(1-\sqrt{\upsilon})\beta_1}\le c_2 e^{-2n} \log n + c_3 R_\infty \bigg(\frac{e^{n^2}}{\phi(2R_\infty)} \bigg)^{-(1-\sqrt {\upsilon}) \beta_1/\beta_2} \le c_4 e^{-n}.
\end{align}
Next, we note that since ${\upsilon}<1$ and $\lim_{n \to \infty}m_n=\infty$, for all $n$ large enough and $z \in B(y, m_n)$,
$$
R_\infty {\mathsf{d}}(z)^{\upsilon} \le R_\infty {\mathsf{d}}(y)^{\upsilon} + R_\infty d(y,z)^{\upsilon} < m_n/2 + R_\infty m_n^{{\upsilon}} <m_n.
$$
Hence, by following the calculations \eqref{e:F-1}, \eqref{e:F-2} and \eqref{e:F-3}, using \eqref{B1}, \eqref{B4} and the Markov property, we get that for all $n$ large enough,
\begin{align*}
&{\mathbb P}^y \big(\cap_{k=n}^{2n} (\widetilde F_k \setminus \widetilde E_{k})\big) \le {\mathbb P}^x \big(\cap_{k=n}^{2n} \widetilde F_k, \, X_{s_{j-1}} \in B(y, m_{j}), n \le j \le 2n \big) \nonumber\\[6pt]
& \le {\mathbb P}^y \big(\cap_{k=n}^{2n-1} \widetilde F_k, \, X_{s_{j-1}} \in B(y, m_{j}), n \le j \le 2n-1 \big) \sup_{z \in B(y, m_{2n})}{\mathbb P}^z\big( \sup_{0<s\le s_{2n} - s_{2n-1} } d(z,X_s) \ge m_{2n} \big) \nonumber\\
& \le...\le \prod_{k=n}^{2n} \exp(-C_4 e^{-C_5}k^{-1/4}) \le \exp(-c_5n^{3/4}).
\end{align*}
By combining the above with \eqref{e:wtF}, we get
$$\sum_{n=1}^\infty {\mathbb P}^y(\widetilde H_n) \le \sum_{n=1}^\infty (\sum_{k=n}^{2n} {\mathbb P}^y(\widetilde E_k) +{\mathbb P}^y (\cap_{k=n}^{2n} (\widetilde F_k \setminus \widetilde E_k)))<\infty.$$
Hence ${\mathbb P}^y(\limsup \widetilde H_n)=0$ by the Borel-Cantelli lemma. Since $\lim_{k \to \infty}m_k=\infty$ and
$$s_k \ge \frac{\phi(2m_k) \log \log \phi(2m_k)}{2^{4 + \beta_2}C_1C_5C_U}$$
for all $k$ large enough by \eqref{e:scaling-infty}, we get the lower bound in \eqref{e:inf0''}. The proof is complete. {\hfill $\Box$ \bigskip}
\noindent \textbf{Proof of Corollary \ref{c:infinf}.}
By Proposition \ref{r:E}(ii) and Theorem \ref{t:infinf}, the liminf law \eqref{20} holds under the current setting. Thus, by Proposition \ref{p:law01}, it suffices to show that for every $x \in M$ and $\lambda>0$,
$$E=E(x, \lambda) :=\Big\{\liminf_{t \to \infty} \frac{\phi\big(\sup_{0 < s \le t} d(x,X_s)\big)}{t/\log \log t} \ge \lambda \Big\}$$ is a shift-invariant event.
Let $\lambda, u>0$ and $x,y \in M$. Observe that by the Markov property,
$$
E \circ \theta_u=\Big\{\liminf_{t \to \infty} \frac{\phi\big(\sup_{0 < s \le t} d(x,X_{s+u})\big)}{t/\log \log t} \ge \lambda \Big\}.$$
Since $X$ is conservative by Proposition \ref{p:EP}(ii), for all $t>0$, it holds that $\sup_{0<s\le t}d(x,X_s) < \infty$, ${\mathbb P}^y$-a.s. Hence, since $\phi$ is positive, we see that for all $t>0$,
\begin{equation*}
\phi\big(\sup_{0 < s \le t+u} d(x,X_{s}) \big) = \phi\big(\sup_{s \in (u, t+u] \, \cup (0, u]} d(x,X_{s}) \big) \le \phi\big(\sup_{0 < s \le t} d(x,X_{s+u})\big) + \phi\big(\sup_{0<s \le u}d(x, X_s) \big).
\end{equation*}
Therefore, we get that for ${\mathbb P}^y$-a.s. $\omega \in E$,
\begin{align*}
&\liminf_{t \to \infty} \frac{\phi\big(\sup_{0 < s \le t} d(x,X_{s+u})\big)}{t/\log \log t} \\
&\ge \liminf_{t \to \infty} \frac{\phi\big(\sup_{0 < s \le t+u} d(x,X_{s}) \big)}{(t+u)/\log \log (t+u)}\frac{ (t+u)/\log \log (t+u)}{t/\log \log t} - \limsup_{t \to \infty} \frac{\phi\big(\sup_{0<s \le u}d(x, X_s)\big)}{t/\log \log t}\ge \lambda.
\end{align*}
On the other hand, for every $\omega \in E \circ \theta_u$, we see that
\begin{align*}
&\liminf_{t \to \infty} \frac{\phi\big(\sup_{0 < s \le t} d(x,X_{s})\big)}{t/\log \log t} \\
&= \liminf_{t \to \infty} \frac{\phi\big(\sup_{0 < s \le t+u} d(x,X_{s})\big)}{(t+u)/\log \log (t+u)}\ge \liminf_{t \to \infty} \frac{\phi\big(\sup_{0 < s \le t} d(x,X_{s+u}) \big)}{t/ \log \log t}\frac{t/\log \log t} { (t+u)/\log \log (t+u)}\ge \lambda.
\end{align*}
Hence, ${\mathbb P}^y(E_u) \le {\mathbb P}^y(E)$. Since $E$ is clearly a tail event, this completes the proof.
{\hfill $\Box$ \bigskip}
\section{Appendix}\label{s:A}
In this section, we
follow the setting in Section \ref{s:intro} and
compare the conditions in this paper with those in \cite{CKL}. We recall the conditions ${\rm Tail}$ and ${\rm NDL}$, and upper and lower scaling properties for nonnegative functions which were presented in \cite[Definitions 1.5, 1.6 and 1.9]{CKL}. We will give a sufficient condition for NDL too.
\smallskip
Throughout the appendix, we let $\varphi: (0,\infty) \to (0,\infty)$ be an increasing and continuous function such that $\displaystyle \lim_{r \to 0} \varphi(r) = 0$ and $\displaystyle \lim_{r \to \infty} \varphi(r) = \infty$.
\begin{definition}\label{d:0}
{\rm Let $R_0 \in (0,\infty]$ be a constant and $U \subset M$ be an open set.
\smallskip
\noindent (i) We say that ${\mathrm {Tail}}_{R_0}(\varphi, U)$ holds if there exist constants $C_0\in (0,1)$, $c_J>1$ such that for all $x \in U$ and $0<r<R_0 \wedge (C_0\updelta_U(x))$,
\begin{equation}\label{e:Tail_0}
\frac{c_J^{-1}}{\varphi(r)} \le J(x,M_\partial \setminus B(x,r)) \le \frac{c_J}{\varphi(r)}.
\end{equation}
We say that ${\mathrm {Tail}}_{R_0}(\varphi, U, \le)$ (resp. ${\mathrm {Tail}}_{R_0}(\varphi, U, \ge)$) holds (with $C_0$) if the upper bound (resp. lower bound) in \eqref{e:Tail_0} holds for all $x \in U$ and $0<r<R_0 \wedge (C_0\updelta_U(x))$.
\smallskip
\noindent (ii) We say that ${\mathrm {E}}_{R_0}(\varphi, U)$ holds if there exist constants $C_0 \in (0,1)$, $C_1>0$ and $c_E > 1$ such that for all $x \in U$ and $0<r<R_0 \land (C_0\updelta_U(x))$, \begin{equation}\label{e:Eo} c_E^{-1}\varphi(C_1 r) \le \E^x[\tau_{B(x,r)}] \le c_E \varphi(C_1 r). \end{equation}
\noindent (iii) We say that ${\mathrm {NDL}}_{R_0}(\varphi, U)$ holds if there exist constants $C_2, \eta \in (0,1)$ and $c_{l}>0$ such that for all $x \in U$ and $0<r<R_0 \land (C_2\updelta_U(x))$, the heat kernel $p^{B(x,r)}(t,y,z)$ of $X^{B(x,r)}$ exists and
\begin{equation}\label{e:NDL_inf}
p^{B(x,r)}(\varphi(\eta r),y,z) \ge \frac{c_{l}}{V(x, r)}, \quad \quad y,z \in B(x, \eta^2r).
\end{equation}
}
\end{definition}
\begin{definition}\label{d:inf}
{\rm Let $R_\infty \ge 1$ and ${\upsilon} \in(0,1)$ be constants.
\smallskip
\noindent (i) We say that $\mathrm{Tail}^{R_\infty}( \varphi,\up)$ \ holds if there exists a constant $c_J>1$ such that \eqref{e:Tail_0} holds for all $x \in M$ and $r >R_\infty {\mathsf{d}}(x)^{\upsilon}$. We say that $\mathrm{Tail}^{R_\infty}(\varphi,\up,\le)$ \ (resp. $\mathrm{Tail}^{R_\infty}(\varphi,\up,\ge)$) holds if the upper bound (resp. lower bound) in \eqref{e:Tail_0} holds for all $x \in M$ and $r>R_\infty {\mathsf{d}}(x)^{\upsilon}$.
\smallskip
\noindent (ii) We say that $\mathrm{E}^{R_\infty}(\varphi,\up)$ \ holds if there exist constants ${\upsilon} \in (0,1)$, $C_1 >0$ and $c_E > 1$ such that \eqref{e:Eo} holds for all $x \in M$ and $r>R_\infty {\mathsf{d}}(x)^{\upsilon}$.
\smallskip
\noindent (iii) We say that ${\mathrm{NDL}}^{R_\infty}(\varphi,\up)$ \ holds if there exist
constants $\eta \in (0,1)$ and $c_l>0$ such that for all $x \in M$ and $r >R_\infty {\mathsf{d}}(x)^{\upsilon} $, the heat kernel $p^{B(x,r)}(t,y,z)$ of $X^{B(x,r)}$ exists and satisfies \eqref{e:NDL_inf}.}
\end{definition}
\begin{defn}\label{d:ws}
{\rm For $g:(0,\infty) \to (0,\infty)$ and constants $a \in (0, \infty]$, $\beta_1, \beta_2>0$, $ c_1,c_2>0$, we say that ${\mathrm{L}}_a(g,\beta_1, c_1)$ (resp. ${\mathrm{L}}^a(g,\beta_1, c_1)$) holds if
$$ \frac{g(r)}{g(s)} \geq c_1 \Big(\frac{r}{s}\Big)^{\beta_1} \quad \text{for all} \quad s\leq r< a\;\;(\text{resp.}\,\;a < s\leq r).$$
and we say that ${\mathrm{U}}_a(g,\beta_2, c_2)$ (resp. ${\mathrm{U}}^a(g,\beta_2, c_2)$) holds if
$$ \frac{g(r)}{g(s)} \leq c_2 \Big(\frac{r}{s}\Big)^{\beta_2} \quad \text{for all} \quad s\leq r< a\;\;(\text{resp.}\;a < s\leq r).$$
We say that ${\mathrm{L}}(g,\beta_1,c_1)$ holds if ${\mathrm{L}}_\infty(g,\beta_1,c_1)$ holds, and that ${\mathrm{U}}(g,\beta_2,c_2)$ holds if ${\mathrm{U}}_\infty(g,\beta_2,c_2)$ holds.
}\end{defn}
We now show that the assumptions in this papers are weaker than those in \cite{CKL}.
\begin{lem}\label{l:NDL}
(i) Suppose that ${\rm VRD}_{R_0}(U)$, ${\rm Tail}_{R_0}(\varphi,U,\le)$, ${\rm U}_{R_0}(\varphi,\beta_2,C_U)$ and ${\rm NDL}_{R_0}(\varphi,U)$ hold. Then the function $\phi(x,r):=\varphi(r)$ satisfies \eqref{e:phi} for all $x \in U$ and $0<r<r_0 \wedge (C_0'\updelta_U(x))$ with some constants $r_0>0$ and $C_0'\in(0,1)$, and conditions \eqref{A1}, \eqref{A2}, \eqref{A3} and \eqref{A4+} hold for $U$.
\noindent (ii) Suppose that ${\rm VRD}^{R_\infty}({\upsilon})$, ${\rm Tail}^{R_\infty}(\varphi,{\upsilon}, \le)$, ${\rm U}^{R_\infty}(\varphi,\beta_2,C_U)$, ${\rm L}^{R_\infty}(\varphi,\beta_1,C_L)$ and ${\rm NDL}^{R_\infty}(\varphi,{\upsilon})$ hold. Then the function $\phi(x,r):=\varphi(r)$ satisfies \eqref{e:phi} for all $x \in M$ and $r>r_1 {\mathsf{d}}(x)^{\upsilon}$ with some constant $r_1\ge1$, and conditions \eqref{B1}, \eqref{B2}, \eqref{B3} and \eqref{B4+} hold.
\end{lem}
\noindent{\bf Proof. } (i) Under the setting, by \cite[Proposition 4.3(i)]{CKL} and ${\rm U}_{R_0}(\varphi,\beta_2,C_U)$, there exist constants $r_0\in(0,R_0)$, $C_0'\in(0,1)$ and $c_1>1$ such that $\E^x[\tau_{B(x,r)}] \asymp \varphi(r)$ for $x \in U$ and $0<r < r_0 \land C_0' \updelta_U(x)$. Hence, using ${\rm U}_{R_0}(\varphi,\beta_2,C_U)$ and the fact that $\lim_{r\to 0}\varphi(r) =0$, we see that \eqref{A1}-\eqref{A3} hold for $U$. Now \eqref{A4+} immediately follows from ${\rm NDL}_{R_0}(\varphi,U)$.
\noindent (ii) Similarly, using \cite[Proposition 4.3(ii)]{CKL}, one can deduce the desired results. {\hfill $\Box$ \bigskip}
Recall the notion of the heat kernel from Section \ref{s:intro}. In the next lemma, we let
$X$ be a strong Markov process on $M$ having the heat kernel $p(t,x,y):=p^M(t,x,y)$ such that $p(t,x,y)<\infty$ unless $x = y$. Then by the strong Markov property of $X$, one can see that for any open set $D \subset M$, the heat kernel $p^D(t,x,y)$ of $X^D$
exists and can be written as
\begin{equation}\label{e:dhk}
p^D(t,x,y) = p(t,x,y) - \E^x \Big[ \E^{X_{\tau_D}} \big[ p(t-\tau_D,X_{\tau_D},y) ; \tau_D < t \big] \Big].
\end{equation}
Using \eqref{e:dhk}, the proof of the next lemma is a simple modification of that of \cite[Proposition 2.3]{CKK09} and \cite[Proposition 2.5]{CKSV}. We give a full proof for the reader's convenience.
\begin{lem}\label{l:hkendl}
Let $U \subset M$ be an open subset. Suppose that there exist constants $R_0 \in (0,\infty]$, $C,C' \ge 1$ such that ${\mathrm {VRD}}_{R_0}(U)$ holds, and for all $t \in (0,\varphi(R_0/2))$,
\begin{equation}\label{e:uhk}
p(t,x,y) \le \frac{Ct}{V(y,d(x,y)) \varphi(d(x,y))} \quad \mbox{ for all} \;\; x \in M, \, y \in U \, \mbox{ with } \, d(x,y) > C'\varphi^{-1}(t)
\end{equation}
and
\begin{equation}\label{e:lhk}
p(t,x,y) \ge \frac{C^{-1}}{V(x,\varphi^{-1}(t))} \quad \mbox{ for all} \;\; x,y \in U \, \mbox{ with } \, d(x,y) < C'^{-1}\varphi^{-1}( t).
\end{equation}
Then ${\mathrm {NDL}}_{R_0}(\varphi, U)$ holds true.
\end{lem}
\noindent{\bf Proof. } Set $\eta:=(2C')^{-1}( 2^{d_2+1} C^2 C_\mu /c_\mu)^{-1/d_1} \in (0,1/2)$ where $d_1, d_2, c_\mu, C_\mu$ are the constants from \eqref{e:VRD}. Choose any $x \in U$, $0<r<R_0 \land (C_V \updelta_U(x))$ and $y,z \in B(x,\eta^2r)$.
We observe that $B(x, \eta^2r) \subset B(x, \updelta_U(x)) \subset U$ and $d(y,z) \le 2\eta^2 r <C'^{-1} \eta r$. Thus, by \eqref{e:lhk} and ${\mathrm {VRD}}_{R_0}(U)$, since $\eta<1/2$, it holds that
\begin{equation}\label{e:hkendl2}
p(\varphi(\eta r),y,z) \ge \frac{C^{-1}}{V(y,\eta r)} \ge \frac{C^{-1}}{V(x,\eta r + d(x,y))} \ge \frac{C^{-1}}{V(x,2\eta r )} \ge \frac{C^{-1} c_\mu (2\eta)^{-d_1}}{V(x,r)} \ge \frac{2^{d_2+1}C C_\mu }{V(x,r)}.
\end{equation}
On the other hand, for every $w \in M \setminus B(x,r)$, we see that $d(w,z) \ge d(w,x) - d(x,z) \ge 3r/4 >C' \eta r$.
Therefore, for every $0<s\le \varphi(\eta r)$ and $w \in M \setminus B(x,r)$, since $\varphi$ is increasing and $\eta<1/2$, we get from \eqref{e:uhk} and $\mathrm{VRD}_{R_0}(U)$ that
\begin{align}\label{e:hkendl1}
p(s, w,z) &\le \frac{C \varphi(\eta r)}{V(z,d(w,z))\varphi(d(w,z))} \le \frac{C \varphi(\eta r)}{V(z,3r/4)\varphi(3r/4)} \le \frac{C}{V(z,3r/4)} \nonumber\\[2pt]
&\le \frac{C}{V(x,3r/4-d(x,z))} \le \frac{C}{V(x,r/2)} \le \frac{2^{d_2}CC_\mu }{V(x,r)}.
\end{align}
Therefore, since $X_{\tau_{B(x,r)}} \in M_\partial \setminus B(x,r)$, using the formula \eqref{e:dhk}, we conclude from \eqref{e:hkendl2} and \eqref{e:hkendl1} that $p^{B(x,r)}(\varphi(\eta r),y,z) \ge 2^{d_2}CC_\mu/V(x,r)$. The proof is complete. {\hfill $\Box$ \bigskip}
\small
| {
"redpajama_set_name": "RedPajamaArXiv"
} | 3 |
Q: SQL Where between word value ranges (eg., "low" to "high") I have a field in my database that has 5 possible values: fair, good, very good, ideal, siganture ideal
I have a coldfusion form that has 2 drop-downs each with all the values. What I am looking to do is be able to have the user select a range. For example dropdown1 = Fair dropdown2 = Very Good. So this would somehow generate the SQL WHERE statement:
grade IN ('fair', 'good', 'very good')
Can you think of a smart way to program this given that the values have to be this way. I think maybe if I put them in an array and then looped through it or something. I'm a little stumped on this any help would be appreciated.
A: As others mentioned, redesigning is ultimately the better course of action, both in terms of efficiency and data integrity. However, if you absolutely cannot change the structure, a possible workaround is to create a lookup table of the allowable grade descriptions, along with a numeric rating value for each one:
GradeID | GradeText | Rating
1 | Fair | 0
2 | Good | 1
3 | Very Good | 2
4 | Ideal | 3
5 | Signature Ideal | 4
Then populate your select list from a query on the lookup table. Be sure to ORDER BY Rating ASC and use the rating number as the list value. Then on your action page, use the selected values to filter by range. (Obviously validate the selected range is valid as well)
SELECT t.ColumnName1, t.ColumnName2
FROM SomeTable t INNER JOIN YourLookupTable lt ON lt.Grade = t.GradeText
WHERE lt.Rating BETWEEN <cfqueryparam value="#form.dropdown1#" cfsqltype="cf_sql_integer">
AND <cfqueryparam value="#form.dropdown2#" cfsqltype="cf_sql_integer">
Again, I would recommend restructuring instead. However, the above should work if that is really not an option.
| {
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Statistics Canada is committed to serving its clients in a prompt, reliable and courteous manner and in the official language of their choice. To this end, the agency has developed standards of service which its employees observe in serving its clients. To obtain a copy of these service standards, please contact us.
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COVER STORY Free trade agreements leave us even more dependent on China
EDITORIAL Why Russia re-elected Vladimir Putin
CANBERRA OBSERVED Empty seat last vestige of minor parties' party
NATIONAL AFFAIRS Liberals take power but plan for none for SA
INTERNATIONAL AFFAIRS Sexual exploitation at Oxfam symptom of culture of death
RELIGIOUS FREEDOM General protection gives a false sense of security
PHILOSOPHY AND CULTURE On celestial politics
GENDER POLITICS Trans ideology awash with big money from big biomed and big pharma
REGIONAL AFFAIRS Taiwan stands up to Beijing's bullyboy tactics
CINEMA Outstanding film follows St Paul to his death in Rome
HUMOUR An Appetite for Diamonds: Porphyry Volpone investigates
MUSIC Power playing: Technique v musicality
CINEMA Peter Rabbit: More Bugs than Beatrix, but lots of fun
BOOK REVIEW We're doomed; but we're not alone
BOOK REVIEW Subcontinent set for Asian century
NATIONAL AFFAIRS The deeper causes of Australia's social malaise
GENDER POLITICS Queensland proposes transgender birth certificates
We're doomed; but we're not alone
News Weekly, April 7, 2018
CREDIT CODE RED: How Financial Deregulation and World Instability Are Exposing Australia to Economic Catastrophe
by Peter Brain and Ian Manning
Scribe, Brunswick
Reviewed by David James
The warning that Peter Brain and Ian Manning sound in Credit Code Red: How Financial Deregulation and World Instability Are Exposing Australia to Economic Catastrophe, certainly makes sense. The indebtedness that they document does sound a warning that Australia is becoming stretched beyond its means.
Trouble is, nothing much makes sense in the global financial system these days. Yes, Australia is becoming more exposed because of rising debt. But so is the rest of the developed world. Global debt is estimated to be about $US220 trillion, which equates with over 300 per cent of world gross domestic product (GDP). It is unsustainable, which is why interest rates have fallen to negligible levels, effectively kicking the can down the street.
Japan's debt-to-GDP ratio is a staggering 250 per cent, although almost all of it is owed internally. America's debt is officially $US18 trillion, about equivalent to its annual GDP.
In Australia, government debt is comparatively low, at just over 40 per cent of GDP, about half the level for most European countries. But Australian household debt, according to the International Monetary Fund (IMF), is about 100 per cent of GDP, compared with an average of 63 per cent in other developed economies. The nation's financial system, unfortunately, has become one giant property gamble.
That is bad enough, but there is worse news: a giant casino called derivatives is sitting on top of the financial world. Derivatives are based on "leverage", which is not exactly debt but is similar. The idea is that to make a bet on one of these financial instruments, the trader only has to come up with a small percentage of the amount, and the rest is notionally agreed. If the bet succeeds, the trader makes a fortune; if it fails, there is a huge loss. Notionally.
Derivatives now are valued at over $US700 trillion, according to the Bank for International Settlements: a staggering amount. Although that is sort of notional, it is also sort of debt.
Brain and Manning capture well enough the Australian leg of this insanity. They pinpoint the fundamental scam that is behind it, financial deregulation, which they rightly identify as "the extension from free trade in markets to free trade in money". As they observe, deregulation was supposed to increase the efficiency of the allocation of funds but instead it has tempted banks and other financial institutions into "the reckless misallocation of funds".
They also identify one of the great flaws behind Australia's supposedly strong economic growth. Finance appears in GDP statistics (GDP is just a record of transactions), but it does not produce anything. As Brain and Manning show, the finance sector is now "9-10 per cent of national income, more or less double the proportion it attracted during the post-war era of economic growth". That increased "growth" is not higher production, it is income transfer to a privileged, often parasitical, sector.
Their attack on the neo-liberal ideology – that "all government is bad, therefore give everything over to the private" – is entirely correct. As Robert Reich explains in his book, Saving Capitalism, you cannot deregulate markets because "the rules are the economy", they are not separate from it. This is especially so in finance, which consists entirely of rules. It is why the phrase "financial deregulation" is an oxymoron and why it is nonsense to see government as being necessarily in opposition to private industry.
The authors' heavy focus on Australia leads to a limited perspective. For one thing, their claim that Australia's indebtedness will soon mean that the country will find it hard to borrow overseas is highly questionable. The Australian dollar is the fifth most traded currency in the world, far in advance of what it should be for the size of the economy (partly because it is considered a proxy for investment in China). That does not suggest that there will be a shortage of demand for Australian financial assets.
The authors also, bizarrely, describe overseas equity investment as borrowing (p23). True, this silliness is embedded in the national accounts, which is one reason why the current account deficit/surplus is such a flawed figure. But equity is not borrowing. When you borrow, the risk is on the receiver of the capital. When you make an equity investment, the risk is on the provider of the capital.
The authors' claim that we are in the credit "red zone" is no doubt correct. But so is the rest of the world. We are not going to be very different when the music stops in this game of musical chairs.
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"redpajama_set_name": "RedPajamaCommonCrawl"
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\section{Introduction and summary}
One of the central results of Einstein gravity states that the entropy of black holes is universally given by a quarter of the area of the event horizon. When quantum effects are taken into consideration, corrections to the entropy area law arise. Of particular interest is a special type of logarithmic correction which can be computed in the low-energy effective theory (the infrared) and should be reproduced by any candidate ultraviolet complete description of the gravitational theory. Such an infrared window into the black hole microstates was emphasized by Ashoke Sen and collaborators in the context of asymptotically flat black holes and matched with the microscopic results when available \cite{Banerjee:2010qc,Banerjee:2011jp, Sen:2012cj, Sen:2012dw, Sen:2012kpz}.
In this paper we explain how to compute such logarithmic corrections to the entropy of asymptotically AdS$_4$ black holes. Although our methods apply broadly, our main focus here is in the simplest theory that admits supersymmetric black hole solutions: minimal $\mathcal{N}=2$ gauged supergravity.
There are a number of recent results that precipitate the timing of our investigations. In the context of the AdS/CFT correspondence, a microscopic foundation for the entropy of AdS$_4$ black holes has now been provided. The first explicit computation was performed for static magnetically charged black holes by Benini, Hristov and Zaffaroni in \cite{Benini:2015eyy} (see \cite{Zaffaroni:2019dhb} for a review). More recently, a microscopic description for the entropy of rotating, electrically charged AdS$_4$ black holes has been provided via the superconformal index \cite{Choi:2019zpz} and by a related localization computation \cite{Nian:2019pxj}. Similar results were obtained in \cite{Bobev:2019zmz,Benini:2019dyp}. These developments built upon the original counting for rotating, electrically charged, asymptotically AdS$_5$ black holes via the superconformal index of ${\cal N}=4$ maximally supersymmetric Yang-Mills \cite{Cabo-Bizet:2018ehj,Choi:2018hmj,Benini:2018ywd} which has also been extended to black holes in AdS$_6$ \cite{Choi:2019miv} and AdS$_7$ \cite{Kantor:2019lfo,Nahmgoong:2019hko}. These impressive results set the stage for precision entropy counting and we hope that this manuscript will serve as a handbook for computations on the gravitational side.
We use the heat kernel method to compute the logarithmic corrections to the entropy of AdS$_4$ black holes. This technique has been applied to obtain the logarithmic corrections for various flat space black holes and can be applied to extremal as well as non-extremal black holes. We are particularly interested in the result for BPS black holes since microscopic countings are available only in the BPS case. In flat space, the BPS logarithmic corrections are \emph{topological}, {\it i.e.}, they are controlled by the topological Euler density. As a result, the coefficient of the logarithm is \textit{universal} ~-~a pure number that does not depend on the black hole parameters. This universality property was explored for flat space black holes in \cite{Charles:2015eha,Charles:2017dbr,Castro:2018hsc,Larsen:2018cts}. In this paper, we show that AdS$_4$ black holes have a richer set of logarithmic corrections, which can be non-topological in the BPS case.
\subsection{Summary of results}
In this manuscript, we take a practical, bottom-up approach to the question of logarithmic contributions in four dimensions. Our main result is the computation of logarithmic correction in $\mathcal{N}=2$ minimal gauged supergravity. We also present results for minimally coupled fields as well as for the Einstein-Maxwell theory with a negative cosmological constant.
The black hole we are interested is the AdS-Kerr-Newman geometry \cite{carter1968, Plebanski:1976gy, Caldarelli:1999xj}. In the extremal case, we will also consider the near horizon geometry which includes a warped circle fibration over AdS$_2$. Our results also give the logarithmic correction to the free energy of thermal AdS$_4$. They can also be applied to the hyperbolic black hole \cite{Casini:2011kv} from which we obtain the logarithmic corrections to the corresponding entanglement entropy.
The microcanonical entropy of the black hole is given by
\be
S = {A\/4 G} + C \log \lambda + \dots~, \qquad (\lambda\to+\infty)
\ee
where $A$ is the area of the horizon and the subleading logarithmic term is the explicit quantum correction we seek. We are interested in the coefficient of the $\log \lambda$ in the ``isometric'' scaling regime where all length scales (in Planck units) are multiplied by $\lambda$ and we take $\lambda\to+\infty$.
The logarithmic correction receives two types of contributions
\be
C=C_\mathrm{local}+C_\mathrm{global}~.
\ee
The global contribution $C_\mathrm{global}$ is an integer that captures the contribution from the zero modes and from the change of ensemble from canonical to microcanonical. The more interesting local contribution, $C_\mathrm{local}$, receives contributions from the non-zero modes and can be computed using the heat kernel expansion. It is given by an integral over the Euclidean spacetime
\begin{align}
C_{\text{local}} \equiv \int d^d x \sqrt{ g} \, a_4(x)~,
\end{align}
where the so-called fourth Seeley-DeWitt coefficient is a sum of four-derivative terms
\be\label{a4general}
a_4(x) = - a_\mathrm{E} E_4 + c\, W^2 + b_1 R^2 + b_2 R F_{\mu\nu} F^{\mu\nu},
\ee
evaluated on the background. The backgrounds we consider are solutions of Einstein-Maxwell theory with a negative cosmological constant. Using the equations of motion, a general four-derivative expression such as $a_4(x)$ can always be decomposed in the above basis. The expression of Euler, $E_4$, and the Weyl tensor squared, $ W^2$, are given in \eqref{E4 and W2 definition}. The heat kernel expansion provides a way to compute these coefficients from any two-derivative action using the formula \eqref{intro:a4}. The results are summarized for the theories studied in this paper in Table \ref{tab:results}.
\begin{table}
\centering\arraycolsep=4pt\def1.4{1.4}
\begin{tabular}{|c|c|c|c|c|}
\hline
Multiplet & $a_\mathrm{E}$ & $c$ & $b_1$ & $b_2$ \cr
\hline
\hline
{\rm Free scalar} & ${1\/360}$ & ${1\/120}$ & ${1\/288}(\Delta(\Delta-3)-2)^2$ & $0$ \cr
\hline
{\rm Free fermion} & $-{11\/360}$ & $ {1\/20}$ & ${1\/72} \left( \Delta-\tfrac32\right)^2\left(\left( \Delta-\tfrac32\right)^2-2 \right)$ & $0$ \cr
\hline
{\rm Free vector} & ${31\/180} $ & $ {1\/10}$ & $0$ & $0$ \cr
\hline
{\rm Free gravitino} & $- {229\/720} $ & $ - {77\/120} $ & $- {1\/9} $ & $0$ \cr
\hline
{\rm Einstein-Maxwell} & ${53\/45}$ & $ {137\/60} $ & $-{13\/36} $ & $0$ \cr
\hline
{\rm $\mathcal{N}=2$ gravitini} & $-{589\/360}$ & $ -{137\/60}$ & $0$ & $ {13\/18}$ \cr
\hline
{\rm $\mathcal{N}=2$ gravity multiplet} & $-{11\/24} $ & $0$ & $-{13\/36} $ & $ {13\/18} $ \cr
\hline
\end{tabular}
\caption{Results for the Seeley-DeWitt coefficient $a_4$ responsible for the logarithmic corrections. The results for $a_0$ and $a_2$ are given in Table \ref{tab:a0a2} in Appendix \ref{app:renorm}. }\label{tab:results}
\end{table}
\newpage
Our final result for the Seeley-DeWitt coefficient of minimal $\mathcal{N}=2$ gauged supergravity takes the form
\be
\label{eqn: HKC}
(4\pi)^2a_4(x) = {11\/24} E_4 -{13\/36} R^2 + {13\/18} R F_{\mu\nu} F^{\mu\nu}~.
\ee
Evaluating this expression on the BPS Kerr-Newman black hole, we obtain
\be\label{ClocalSummary}
C_\mathrm{local} = {11\/6} -{26\/3}{a(\ell^2-4\ell-a^2)\/(\ell-a)(a^2+ 6 a\ell + \ell^2)},
\ee
where $a = J/M$ is the rotation parameter and $\ell$ is the AdS$_4$ radius.
The integer corrections, $C_\mathrm{global}$, are summarized in Table \ref{tab:global}.
We observe that the logarithmic correction for a BPS black hole in gauged supergravity has a richer structure than in flat space: {\it the logarithmic correction is non-topological, i.e., its coefficient is not a pure number but depends on black hole parameters.}
Our result is, to our knowledge, the first computation of the Seeley-DeWitt coefficient $a_4(x)$ in gauged supergravity. We find that the non-topological contribution comes from the additional four-derivative terms $R^2$ and $R F_{\mu\nu}F^{\mu\nu}$. In the flat space limit, these terms both vanish and the logarithmic correction becomes topological and gives $C_\mathrm{local} = {11\/6}$. This was shown in \cite{Charles:2015eha,Charles:2017dbr} and is a non-trivial consequence of supersymmetry.
We suspect that the non-topological piece can be interpreted as a contribution from the AdS boundary. It is possible to interpret the logarithmic correction as the Atiyah-Singer index of an appropriate supercharge \cite{Hristov:2021zai}. We surmise that the non-topological term should correspond to the $\eta$-invariant which is a correction due to the presence of a boundary.
We note that according to microscopic computations \cite{Liu:2017vll,Benini:2019dyp,Gang:2019uay,PandoZayas:2020iqr,Liu:2018bac}, we expect the full logarithmic entropy correction to be topological. Such expectation has been confirmed in various 11d supergravity computations \cite{Liu:2017vbl,Benini:2019dyp,Gang:2019uay,PandoZayas:2020iqr}. There is, however, no contradiction because the 4d minimal gauged supergravity is by itself not the low-energy effective theory of a UV complete theory as matter multiplets, arising from Kaluza-Klein reduction, need to be included. Nonetheless, our result shows that supersymmetry is not enough to guarantee a topological logarithmic correction. This observation suggests that the topological nature of the logarithmic correction could be used to indicate which low-energy theories admit a UV completion\footnote{The possibility of using the topological nature of logarithmic correction for such questions was emphasized to us by Alejandra Castro and was discussed in \cite{AlejandraKITP}.}.
The rest of the paper is organized as follows. We review the heat kernel formalism for the computation of logarithmic corrections to black hole entropy in section \ref{Sec:LogReview} with special emphasis on the scaling limit that we should use in AdS. After presenting the black hole backgrounds and their near horizon geometries in section \ref{Sec:Backgrounds}, we present the main results for the logarithmic corrections in section \ref{Sec:EMLambda} and \ref{Sec:N2Sugra}. We discuss the Einstein-Maxwell theory with a negative cosmological constant in section \ref{Sec:EMLambda}. In section \ref{Sec:N2Sugra} we discuss the corrections for minimal ${\cal N}=2$ gauged supergravity. We conclude with a discussion in section \ref{Sec:Discussion}. The topic is intrinsically quite technical and we relegate a number of partial results to a series of appendices to aid the reader interested in technical details.
\section{Logarithmic corrections in AdS$_4$}\label{Sec:LogReview}
In this section, we review logarithmic corrections to black hole entropy and the heat kernel method for their computation \cite{Sen:2012kpz, Sen:2012cj, Sen:2012dw}. This method has been chiefly applied to asymptotically flat black holes. We also explain how to apply it to asymptotically AdS black holes.
\subsection{Euclidean quantum gravity}
We consider theories of Einstein gravity in $D$ dimensions coupled to matter fields. We restrict to theories with a scaling property so that purely bosonic terms have two derivatives, terms with two fermions have one derivative and terms with four fermions have no derivative. This covers a wide range of theories, such as Einstein gravity with minimally coupled scalars, fermions and gauge fields, but also a variety of supergravity theories at a generic point in the moduli space. We also allow for the presence of a cosmological constant.
We now consider a black hole solution in this theory. To define the quantum entropy of the black hole, we use the fact that this black hole appears as a saddle-point of the Euclidean path integral
\be
Z(\beta, \mu_\alpha) = \int D\Psi \,e^{-S_E(\Psi)}~,
\ee
where $\mathcal{S}_E $ is the Euclidean action and the integration is done while fixing the temperature $\beta$, thermodynamically conjugate to the mass, $M$, and appropriate chemical potentials $\mu^\alpha$ associated to the $\mathrm{U}(1)$ charges $q_\alpha$.
Upon studying the black hole solutions in this paper, we probe the Euclidean spacetimes via a continuation to imaginary time and analytically continue the action. For the case of the Kerr solutions, these quasi-Euclidean metrics are complex and do indeed give appropriate thermodynamics, see for example \cite{Gibbons:1976ue}. Our computations focus on the small fluctuations around the complex saddle points and we do not expect the subtleties of analytic continuation to affect these quantum corrections. Therefore, for the sake of this paper, we consider these quasi-Euclidean solutions (which we call Euclidean) as is, and leave the subtleties of spacetimes with complex metrics for future study. We simply comment that complex solutions in Euclidean gravity is an evolving subject and we refer the reader to a few examples in the literature \cite{Louko:1995jw, Sorkin:2009ka} as well as more recent discussions on this matter \cite{Kontsevich:2021dmb, Witten:2021nzp}.
The black hole entropy is given by the Legendre transform
\be
S = \log Z+ \beta M + \sum_\alpha \mu^\alpha q_\alpha~.
\ee
At leading order, the classical approximation $\log Z = -S_E^\mathrm{classical}$ is the Euclidean on-shell action. It is a classic result of Gibbons and Hawking \cite{Gibbons:1976ue} that the transform leads to the Bekenstein-Hawking entropy formula
\be
S = {\mathrm{Area}(q_\alpha)\/4 G} + \dots~.
\ee
At one-loop order around the saddle-point, we obtain
\be
Z(\beta, \mu_\alpha) \sim {1\/\sqrt{\det \mathcal{Q}}} e^{-S_E^\mathrm{classical}},
\ee
where $\mathcal{Q} = {\delta^2 S_E \/ \delta\Psi^2}$ is the quadratic operator for the fluctuating fields on the background. This expression is divergent and needs to be regulated. The one-loop correction to the black hole entropy is
\be
\delta S = -\frac{1}{2} \log \det \mathcal{Q}.
\ee
\subsection{Scaling regime}\label{sec:scaling}
The result for the logarithmic correction is highly sensitive to the precise scaling regime we consider. To isolate the logarithmic correction, we consider a reference configuration with fixed length scales $\ell_i^{(0)}$. In the example of AdS-Schwarzschild, these length scales can be taken to be the AdS$_4$ radius $\ell$ and the horizon size $r_+$. We then consider a rescaled configuration where all length scales are multiplied by the \emph{same} factor $\lambda\gg1$: $\ell_i = \lambda \ell_i^{(0)}$. We are then interested in the coefficient of $\log\lambda$ in the one-loop correction to the entropy of the rescaled configuration.
This scaling regime is ``isometric'' because it only magnifies the geometry without deforming it. As a result, the eigenvalues of $\mathcal{Q}$ are given by
\be \label{logs:eigenchange}
\kappa_n = \lambda^2\kappa_n^{(0)},
\ee
where $\kappa_n^{(0)}$ are the eigenvalues of the reference configuration. As explained in the next section, this relation is important to ensure that the logarithmic correction depends only on the small $s$ expansion of the heat kernel.
For more general scaling regimes, there will not be any simple relation between the eigenvalues of the scaled versus reference configuration, because the geometry gets deformed. In this case, the logarithmic correction cannot be computed by the heat kernel expansion and would require knowledge of the heat kernel at general values of $s$. For a background with $k$ independent length scales $\ell_1,\dots,\ell_k$ (in Planck units), the general logarithmic correction would take the form
\be
S = {A\/4G}+ \sum_{i=1}^k C_i\log\ell_i+\dots~,
\ee
with an independent coefficient $C_i$ for each independent length scale $\ell_i$. In these terms, the heat kernel expansion can only give us the sum
\be
C=\sum_{i=1}^k C_i~,
\ee
without being able to access the individual $C_i$. Indeed, $C$ is the coefficient of $\log \lambda$ if we write $\ell_i = \lambda\ell_i^{(0)}$ with $\ell_i^{(0)}$ fixed.
Let us now contrast this regime with the flat space regime of \cite{Sen:2012kpz,Sen:2012cj,Sen:2012dw}. In flat space, we do not rescale the mass $m$ of massive fields. As a consequence, it can be shown that massive fields do not contribute to the logarithmic correction of flat space black holes. In AdS, the prescription is to fix the conformal dimension, or equivalently the combination $m \ell$, so we get non-trivial logarithmic corrections for massive fields as a function of the conformal dimension. Clearly we can see that in the flat space limit $\ell\to+\infty$, only fields with $m=0$ can contribute and in that limit, we actually reproduce the scaling regime of \cite{Sen:2012kpz,Sen:2012cj,Sen:2012dw}. We indeed see that we reproduce known results for flat space black holes by taking the flat space limit of our results.
It can also be shown that higher loops do not contribute to the logarithmic correction as they are suppressed by positive powers of $\lambda$ \cite{Sen:2012dw}. Summarizing, the logarithmic correction to the entropy arises only at one-loop from the two-derivative Lagrangian and can be unambiguously computed in the low-energy effective theory.
For extremal black holes, we need to be more careful. In particular, the thermal circle is infinite which naively makes the Euclidean on-shell action divergent. To obtain a well-defined $\beta\to+\infty$ limit, we remove a divergence that can be viewed as an infinite shift in the ground state energy. This can be made precise using the quantum entropy function \cite{Sen:2008vm} in which the quantum entropy is defined using the AdS$_2$/CFT$_1$ correspondence in the near horizon geometry. This procedure was used, for example, in \cite{Banerjee:2010qc, Banerjee:2011jp,Sen:2012cj}.
\subsection{Heat kernel expansion}\label{sec:HKexpansion}
We will now describe the main technical tool which makes possible the exact computation of the logarithmic correction for a variety of black holes: the heat kernel expansion \cite{Vassilevich:2003xt,Fursaev:2011zz,Percacci:2017fkn}.
The one-loop correction to the partition function decomposes as a contribution $Z_{\text{nz}}$ from the non-zero modes and a contribution $ Z_{\text{zm}}$ from the zero modes of the corresponding kinetic operators, so that we have
\be
Z_\text{1-loop}(\beta,\mu_\alpha)= Z_\text{nz}Z_{\text{zm}}\, e^{-\mathcal{S}_E^\mathrm{class.}}~.
\ee
The one-loop corrected Bekenstein-Hawking entropy, defined in the microcanonical ensemble, takes the form
\be
\label{eqn: BH entropy}
S = {A\/4 G} + \left( C_{\text{local}} +C_{\text{global}}\right) \log \lambda + \dots~.
\ee
Here $C_\mathrm{local}$ is the local contribution computed using the heat kernel. The global term $C_\mathrm{global}$ is an integer correction due to the zero modes and the change of ensemble from canonical to microcanonical. We now explain how to compute the local contribution. It originates from the non-zero modes
\be
\log Z_\mathrm{nz} = -{1\/2}{\sum_{n}}'\log \kappa_n~,
\ee
where $\kappa_n$ are the eigenvalues of the quadratic operator $\mathcal{Q}$ and the primed sum runs only over the non-zero eigenvalues $\kappa_n\neq 0 $. This can be computed by introducing the heat kernel
\be
K(x,s) = \sum_n e^{-\kappa_n s} f_n^\ell(x)f_n^{\ell'}(x) G_{\ell\l'}~,
\ee
where $\{f_n^\ell\}$ are the ortho-normalized eigenfunctions of $\mathcal{Q}$ with eigenvalues $\{\kappa_n\}$ and $G_{\ell\l'}$ is the metric on field space. In particular, we have
\be
\int_\mathcal{M} d^D x\sqrt{g} \,K(x,s) = \sum_n e^{-s \kappa_n}={\sum_n}' e^{-s \kappa_n} + N_\mathrm{zm}~,
\ee
where $N_\mathrm{zm}$ is the number of zero modes. We will make use of the relation
\be
\log \kappa-\log \kappa^{(0)} = -\lim_{\epsilon\to 0 }\int_{\epsilon}^{\infty} {d s\/s} \left( e^{-s \kappa }-e^{-s \kappa^{(0)} }\right) ~.
\ee
In our scaling regime, the eigenvalues are rescaled according to \eqref{logs:eigenchange}. This allows us to show that we have
\be\label{Logs:finalintZnz}
\log Z_{\text{nz}}- \log Z_{\text{nz}}^{(0)} = \frac{1}{2} \int_{\epsilon}^{\epsilon \lambda} \frac{ds}{s} \left(\int_{\mathcal{M}} d^D x \sqrt{ g}\: K(x,s) - N_{\text{zm}} \right)~.
\ee
The above expression makes it clear that only the range of very small $s$ contributes due to a cancellation between $Z_\mathrm{nz}$ and $ Z_\mathrm{nz}^{(0)}$. We can then use the heat kernel expansion which states the existence of a small $s$ expansion of the form
\be
K(x,s) = \sum_{n\geq 0} s^{n-D/2} a_{2n}(x)
\ee
where $D$ is the dimension of spacetime. The coefficients $a_{2n}(x)$ are known as Seeley-DeWitt coefficients. For smooth manifolds, $a_{2n}(x)$ is a sum of $2n$-derivative terms constructed from the fields appearing in the action \cite{Vassilevich:2003xt}.
\medskip
We are mainly interested in $D=4$ for which we have
\be
K(x,s) = s^{-2}a_0(x)+ s^{-1}a_2(x)+ s^0 a_4(x) + \mathcal{O}(s)~.
\ee
We want to compute the $\log \lambda$ contribution in $\log Z_\mathrm{nz}$. The integral \eqref{Logs:finalintZnz} makes it clear that this comes from the $a_4$ coefficient and we have
\be
\log Z_{\text{nz}} =C_\mathrm{local} \log \lambda + \dots~,
\ee
where we have defined
\begin{align}\label{Clocala4}
C_{\text{local}} \equiv \int d^4 x \sqrt{ g} \, a_4(x)~.
\end{align}
We refer to this as the local contribution as it is given by an integral over spacetime. In general spacetime dimension $D$, $a_4(x)$ should be replaced by $a_D(x)$ in the above formula. Note that this vanishes when $D$ is odd so there is no local contribution in odd-dimensional spacetimes.
The power of the heat kernel expansion lies in the fact that there is a general expression for $a_4(x)$ summarized in \cite{Vassilevich:2003xt}. This allows to compute $C_\mathrm{local}$ without computing the eigenvalues of $\mathcal{Q}$.
The other Seeley-DeWitt coefficients $a_0(x)$ and $a_2(x)$ capture one-loop corrections to the cosmological constant, Newton's constant and the other couplings in the Lagrangians. This is discussed, for completeness, in Appendix \ref{app:renorm}.
\paragraph{Bosonic fluctuations.} We write the operator of quadratic fluctuations for bosons as
\be\label{Lap}
\mathcal{Q}^n_m = (\Box) I^n_m + 2 (\omega^\mu D_\mu)^n_m + P^n_m~,
\ee
where the Latin indices $m,n$ refer to the different fields and $D_\mu$ is the spacetime covariant derivative. We define $\mathcal{D}_\mu = D_\mu + \omega_\mu$ to complete the square so that
\be
\label{eqn: Lap}
\mathcal{Q}^n_m = (\mathcal{D}^\mu \mathcal{D}_\mu)I^n_m+ E^n_m, \qquad
\ee
with
\begin{align}
\label{eqn: a4E}
E \equiv P - \omega^\mu\omega_\mu - (D^\mu\omega_\mu)~.
\end{align}
The Seeley-DeWitt coefficient $a_4(x)$ is then given explicitly by the formula
\begin{eqnarray}\label{intro:a4}
(4\pi)^2 a_4(x) \= \mathrm{Tr}\left[{1\over2} E^2 + {1\over6} R E + {1\over 12} \Omega_{\mu\nu}\Omega^{\mu\nu} \right. \-
&& \hspace{2.5cm}\left.+ {1\over360} I(5 R^2 + 2R_{{\mu\nu}{\rho\sigma}}R^{{\mu\nu}{\rho\sigma}} - 2R_{\mu\nu} R^{\mu\nu})\right]~,
\end{eqnarray}
where $\Omega_{\mu\nu} = [D_\mu+\omega_\mu,D_\nu+\omega_\nu]$ is the curvature associated to the connection $\mathcal{D}_\mu$.
\paragraph{Fermionic fluctuations.} For fermionic fields, the quadratic Lagrangian takes the form $\mathcal{L}= \bar{\psi}\mathcal{D} \psi$ where $\mathcal{D} = \slashed{D} + L$ is a Dirac-type operator and $\psi$ denotes all the fermions of the theory. The prescription is then to use the fact that
\be
\log \mathrm{det}\,\mathcal{D} = {1\/2}\log \mathrm{det}\,\mathcal{D}^\dagger\mathcal{D},
\ee
so that we can apply the heat kernel method to $\mathcal{Q} = \mathcal{D}^\dagger\mathcal{D}$. We have, more explicitly,
\be\label{omegaPforfermions}
\omega^\mu=\frac{1}{2}\big(\gamma^\mu L-L^\dagger \gamma^\mu\big),\qquad P=\mathcal{R}+\big(\slashed{D}L\big)-L^\dagger L,
\ee
where $\mathcal{R} =-{1\/4}R$ for spin ${1\/2}$ and $\mathcal{R} = -{1\/4}g_{\mu\nu} + {1\/2}\gamma^{\rho\sigma} R_{{\mu\nu} {\rho\sigma}}$ for spin ${3\/2}$.
\subsection{Global contribution}
The global contribution consists of an integer correction which is the sum of two contributions
\be \label{Cglobal}
C_\mathrm{global}=C_\mathrm{ens}+C_\mathrm{zm}.
\ee
The first term corresponds to the correction due to changing from the grand canonical to the microcanonical ensemble \cite{Sen:2012dw}.
The zero modes are associated to asymptotic symmetries: gauge transformations with parameters that do not vanish at infinity and are thus, not normalizable. In the path integral, we can treat them by making a change of variable to the parameters of the asymptotic symmetry group. For a field $\Psi$, the Jacobian of this change of variable introduces a factor
\be
\lambda^{\beta_\Psi}~,
\ee
which contributes a logarithmic correction $\beta_\Psi \log L$ to the entropy. As a result, the total contribution from the zero modes is
\be
C_{\text{zm}} = \sum_{\Psi} (\beta_\Psi -1 ) n_\Psi^0,
\ee
where we are summing over all fields $\Psi$ (including ghosts) and we denote by $n_\Psi^0$ the number of zero modes for $\Psi$. There is a $-1$ because we include here the $-N_{\text{zm}}$ which was in the non-zero mode contribution \eqref{Logs:finalintZnz} (and not included in $C_\mathrm{local}$). The value of $\beta_\Psi$ can be computed by normalizing correctly the path integral measure. We refer to \cite{Sen:2012cj} for a more detailed discussion. As an illustration, we report below the values of $\beta_\Psi$ for the gauge field, the Rarita-Schwinger field and the graviton in $D$ spacetime dimensions
\be\label{beta}
\beta_A = {D\/2}-1 ,\qquad\beta_{\psi} = D-1,\qquad \beta_{g} = {D\/2}~.
\ee
\section{Black hole backgrounds}\label{Sec:Backgrounds}
In this section, we present the background geometries for which we compute the logarithmic corrections. They are solutions of Einstein-Maxwell theory with a negative cosmological constant.
We give the integrated four-derivative terms as a precursor to the computations of the logarithmic corrections and describe the extremal limit and the near horizon geometry. At this level, we already observe that the local contribution $C_\mathrm{local}$ for extremal black holes is the same in the full geometry and in the near horizon geometry so that the only difference is due to the zero mode contribution.
In the following subsections, we review the metrics of AdS-Schwarzschild, thermal AdS$_4$ and the Reissner-Nordstr\"om AdS$_4$ black hole as simple examples before we consider the general Kerr-Newman AdS$_4$ black hole solution with particular emphasis on its BPS limit. We compute the curvature invariants in both the full solution and the near horizon before giving the general result for the logarithmic corrections to the entropy. The results are written in terms of the theory-dependent coefficients $a_\mathrm{E},c,b_1$ and $b_2$. The computation of these coefficients for the theories of interest will be the subject of subsequent sections.
\subsection{General structure}
The local contribution to the logarithmic correction is given by the Seeley-DeWitt coefficient $a_4(x)$ using \eqref{Clocala4}. For solutions of Einstein-Maxwell-AdS theory, a general four-derivative term can be decomposed as
\be\label{a4general}
(4\pi)^2 a_4(x) = - a_{\text{E}} \, E_4 + c\, W^2 + b_1 R^2 + b_2 R F_{\mu\nu} F^{\mu\nu} ~,
\ee
after using the equations of motion for the background fields.
Here we write the curvature invariants in terms of the Euler density and the Weyl tensor squared given explicitly as
\begin{eqnarray} \label{E4 and W2 definition}
E_4 & \equiv & R_{{\mu\nu}{\rho\sigma}}R^{{\mu\nu}{\rho\sigma}}-4 R_{\mu\nu} R^{\mu\nu} +R^2~, \\
W^2 & \equiv & R_{{\mu\nu}{\rho\sigma}}R^{{\mu\nu}{\rho\sigma}} - 2R_{\mu\nu} R^{\mu\nu} + {1\/3}R^2 ~.
\end{eqnarray}
Note that the equations of motion implies that $R=4\Lambda = -12/\ell^2$. The difference with the previous flat space computations lies in the last two terms in \eqref{a4general}, which vanish if $R=0$. These terms are responsible for making the logarithmic correction non-topological.
To regularize the integral over spacetime, we use the same prescription as in holographic renormalization, which gives an unambiguous finite answer. A consistency check on this procedure is that for the Euler term, the regularized integral gives
\be\label{chiGBC}
\chi = {1\/32\pi^2} \int d^4 x\sqrt{g} \,E_4, \qquad \text{(regularized)}
\ee
where $\chi$ is the Euler characteristic of spacetime. This is possible because our regularization procedure produces the same boundary as the one appearing in the Gauss-Bonnet-Chern theorem, as we explain in Appendix \ref{app:Euler}. Thus, we see that the logarithmic correction is topological if and only if $a_4(x)$ contains only the Euler term, that is, $c=b_1=b_2=0$.
\subsection{AdS-Schwarzschild black hole}
\label{subsec: AdSSS}
The Euclidean AdS-Schwarzschild black hole is described by the line element
\begin{equation}
ds^2=f(r) dt^2+\frac{dr^2}{f(r)}+r^2 d\Omega^2,\qquad f(r)=1+\frac{r^2}{\ell^2}-\frac{2 m}{r}~,
\end{equation}
where $m$ is the mass of the black hole and $\ell$ is the radius of AdS$_4$. Here-forth, Euclidean time is identified with a period proportional to the inverse Hawking temperature $\beta$,
\be
\label{eqn: AdSSST}
t \sim t+\beta ,\qquad \beta=\frac{4\pi r_+}{1+\frac{3r_+^2}{\ell^2}}~,
\ee
where $r_+$ is the position of the horizon given by the largest real root of $f(r_+)=0$. The curvature invariants in \eqref{a4general} for this solution are
\begin{equation}
\label{eqn: SS curvature invariants}
E_{4} =\frac{24}{\ell^4}+\frac{48m^2}{r^6}~,\qquad
W^2 =\frac{48m^2}{r^6}~ ,\qquad
R^2=\frac{144}{\ell^4}~,\qquad R F_{\mu\nu}F^{{\mu\nu}}= 0~.
\end{equation}
The integrated curvature invariant are divergent due to the infinite volume. To regularize these divergences, we utilize the same prescription as holographic renormalization \cite{Skenderis:2002wp,Natsuume:2014sfa}. Such choice of renormalization is natural given that the logarithmic contributions are corrections to the on-shell action and it allows us to obtain finite and unambiguous results in all cases. A more systematic understanding of this prescription would require a quantum version of holographic renormalization.
The prescription is to impose a cutoff at large $r=r_c$. At the boundary, we add a counter term written in terms of intrinsic data
\be
a_4^\mathrm{CT} = \int_{\partial M} d^3 y \sqrt{h}\, (c_1+ c_2 \mathcal{R})~,
\ee
where $\mathcal{R}$ is the Ricci curvature of the boundary $\partial M$. The coefficients $c_1,c_2$ are determined by the requirement that $a_4+ a_4^\mathrm{CT}$ remains finite as we take $r_c\to+\infty$. The regularized integrated invariants take the form
\begin{align}\notag
{1\/(4\pi)^2 }\int d^4 x\sqrt{g}\,E_4 & = 4~, & {1\/(4\pi)^2 }\int d^4 x\sqrt{g} & \,W^2 = {4(\ell^2 + r_+^2)^2 \/\ell^2(\ell^2+3 r_+^2)}~,\\
{1\/(4\pi)^2 }\int d^4 x\sqrt{g}\,R^2 & = {24 r_+^2 (\ell^2 - r_+^2) \/\ell^2(\ell^2+3 r_+^2)}~, & {1\/(4\pi)^2 }\int d^4 x\sqrt{g} & \,R F_{\mu\nu}F^{{\mu\nu}} =0~.
\end{align}
As expected from the Gauss-Bonnet-Chern theorem, the Euler characteristic is
\be
\chi = {1\/32\pi^2}\int d^4x \sqrt{g}\,E_4 = 2.
\ee
In fact, we verify that with the holographic renormalization procedure, the integral of the Euler density is always the Euler characteristic of the spacetime, for all the backgrounds considered in this paper. This suggests that the holographic counterterm reproduces exactly the boundary term comparable to that of the Gauss-Bonnet-Chern theorem. This is evidence that our renormalization procedure is correct and we refer to Appendix \ref{app:Euler} for details.
The final result for $C_{\text{local}}$ for AdS-Schwarzschild takes the form
\be
C_\mathrm{local} = {4\/\ell^2(\ell^2+3r_+^2)} \left( (c-a_\text{E})\ell^4 + (2c-3a_\text{E}+6b_1) \ell^2 r_+^2 + (c-6b_1)r_+^4\right)~.
\ee
\subsubsection{Thermal AdS$_4$} \label{subsubsec: thermalads}
We are mainly interested in logarithmic corrections to black hole entropy. However, the dominant saddle-point in the canonical ensemble is not always a black hole in AdS. For temperatures below the Hawking-Page transition \cite{Hawking:1982dh}, it is a thermal AdS. Our computation gives the logarithmic corrections to the free energy of AdS$_4$. The metric of the AdS spacetime with only radiation, {\it thermal AdS}, is given by
\begin{equation}
\label{eqn: thermalAdS}
ds^2=\left(1+\frac{r^2}{\ell^2}\right)dt^2+\frac{dr^2}{\left(1+\frac{r^2}{\ell^2}\right)}+r^2 d\Omega^2~.
\end{equation}
The curvature invariants for the thermal AdS background read
\be
E_4 =\frac{24}{\ell^4},\qquad W^2 = 0,\qquad R^2=\frac{144}{\ell^4}~,\qquad F_{\mu\nu}F^{{\mu\nu}} = 0~.
\ee
Using the same regularization procedure as above, the integrated invariants all vanish
\begin{align}\notag
{1\/(4\pi)^2 }\int d^4 x\sqrt{g}\,E_4 & = 0~, & {1\/(4\pi)^2 }\int d^4 x\sqrt{g} & \,W^2 = 0~,\\
{1\/(4\pi)^2 }\int d^4 x\sqrt{g}\,R^2 & = 0~, & {1\/(4\pi)^2 }\int d^4 x\sqrt{g} & \,R F_{\mu\nu}F^{{\mu\nu}} =0~.
\end{align}
This shows that on thermal AdS$_4$, we have $a_4(x) = 0$ so that the local contribution vanishes
\be
C_\mathrm{local} = 0~,
\ee
and the logarithmic correction comes only from the zero mode contribution. Thus we may use $\text{C}_\mathrm{local}$ as an order parameter indicating the Hawking page transition. In the case of Einstein-Maxwell theory, we must include a fixed gauge potential $\Phi$ as thermal AdS \cite{Caldarelli:1999xj,Chamblin:1999hg,Chamblin:1999tk}. Since it is a pure gauge, it does not affect the logarithmic term of the entropy.
\subsection{Reissner-Nordstr\"om black hole}\label{sec: RNBH}
We now turn to the AdS-Reissner-Nordstr\"om black hole and its extremal limit. It is important to note that this black hole is not a BPS solution of minimal gauged supergravity. A non-zero rotation is necessary to solve the BPS equations as we discuss in the next section.
\subsubsection{Non-extremal black hole}
\label{subsubsec: RNAdSFull}
The Euclidean Reissner-Nordstr\"om black hole in AdS is described by
\be
\label{eqn: RNAdS}
d s^{2}= f(r) d t^{2}+ {dr^2\/f(r)} +r^{2} d\Omega^2~,\qquad
A = \frac{i q_{e}}{r}dt - q_m \r{cos} \, \theta \, d\phi ~.
\ee
with
\be
f(r) = 1+\frac{r^2}{\ell^2}-{2m\/r}+{q^2_e+q_m^2\/r^2} ~,
\ee
where $m$, $q_e$ and $q_m$ characterize the mass, the electric charge and the magnetic charge of the black hole, respectively. The horizon $r_+$ is the largest root of $f(r)=0$ and the Hawking temperature is
\begin{equation}
T_H =\beta^{-1}= {f'(r_+)\/4\pi} = {1\/4\pi r_+}\left( 1+\frac{3 r_+^2}{\ell^2}-\frac{(q_e^2+q_m^2)}{r_+^2}\right)~.
\end{equation}
The curvature invariants are computed to be
\begin{align}\notag
R^2 &= \frac{144}{\ell^4}~, &
E_4 &=\frac{24}{\ell^4}+\frac{8 \left(6 m^2 r^2-12 m (q_e^2+q_m^2) r+5 (q_e^2+q_m^2)^2\right)}{r^8}~, \\
F_{\mu\nu} F^{\mu\nu} &=-\frac{2(q_e^2-q_m^2)}{r^4}~, &
W^2 &=\frac{48 \left(m r-(q_e^2+q_m^2)\right)^2}{r^8}~.
\end{align}
The integrated invariants can be computed using the same renormalization procedure as described above for the AdS-Schwarzschild case. The results are
\begin{eqnarray}
\label{eqn: RN curvature invariants}
{1\/(4\pi)^2} \int d^4 x \sqrt{g}\, E_4 \= 4 ~,\\
{1\/(4\pi)^2} \int d^4 x \sqrt{g}\,W^2 \= {2\/5}\left(2 - {14 r_+^2\/\ell^2} + {16\pi r_+\/\beta} + {(\ell^4+ \ell^2 r_+^2 + 4 r_+^4)\/ \pi \ell^4 r_+}\beta \right)~,\\\label{RNR2}
{1\/(4\pi)^2} \int d^4 x\sqrt{g} \, R^2 \= {12 r_+(r_+^2+\ell^2)\/\pi\ell^4}\beta - {24 r_+^2\/\ell^2}~, \\
{1\/(4\pi)^2} \int d^4 x \sqrt{g} \,R F_{\mu\nu} F^{\mu\nu} \= {6 (3 r_+^4 + \ell^2 r_+^2-2 \ell^2 q_m^2) \/\pi \ell^4 r_+} \beta- {24 r_+^2\/\ell^2}~.
\end{eqnarray}
The final result for the Reissner-Nordstr\"om black hole takes the following form
\begin{eqnarray}
C_\mathrm{local} \= {2\/5}\left( 2( c-5a_\text{E}) - {2 r_+^2\/\ell^2}(7c + 30(b_1+b_2)) + {16\pi r_+\/\beta}c \right.\-
&& \hspace{0.4cm} \left.+ {\beta\/\pi\ell^4 r_+} \Big(c \ell^4 + \big(c+30 b_1+15b_2\big) \ell^2 r_+^2 + \big(4c + 30 b_1 + 45b_2\big) r_+^4 - 30 b_2 \ell^2 q_m^2\Big) \right)~.
\end{eqnarray}
The appearance of $q_m$ indicates that if the final result has a non-vanishing $b_2$, the logarithmic correction does not preserves the electromagnetic duality. As we will see in section \ref{Sec:N2Sugra}, if we consider $\mathcal{N}=2$ supergravity, we do have a non-trivial $b_2$.
\subsubsection{Extremal limit}
\label{subsubsec: ExtLimit}
The result for the extremal black hole is obtained by taking the $T\to 0$ or $\beta\to+\infty$ limit. This limit is naively divergent and we will describe how to implement it in this context. The prescription is as follows. First, the outer horizon is a function of $\beta$, and must be substituted as an explicit expression in terms of $\beta$. We then take the $\beta \to \infty$ limit while keeping the charges fixed and subsequently impose the extremal values of the charges. The low-temperature expansion yields
\be
r_+ = r_0 + {2\pi \ell_2^2 \/\beta} + O(\beta^{-2}),
\ee
where $r_0$ is the position of the extremal horizon and $\ell_2$ is the AdS$_2$ radius and can be expressed as
\be\label{r0l2}
r_0^2 = {1\/6}\ell(\sqrt{\ell^2+12 q^2} - \ell)~,\qquad \ell_2^2 = {r_0^2\/1+ {6 r_0^2\/\ell^2}} = {\ell^2\/6}\left( 1 -{\ell\/\sqrt{\ell^2+12 q^2}}\right)~.
\ee
In the $\beta\to+\infty$, we generally have
\be
\int d^4 x\sqrt{g}\,a_4(x) = C_1 \beta + C_0 + O(\beta^{-1})~.
\ee
The first term, linear in $\beta$, is divergent. As this expression is a correction to the effective action, we can interpret this term as a shift of the ground state energy due to one-loop fluctuations. As a result, we ignore this term and define the limit $\beta\to+\infty$ to be the constant term $C_0$. The resulting four-derivative terms are
\begin{eqnarray}
\label{RNfourLimit}
\lim_{\beta\to+\infty} {1\/(4\pi)^2} \int d^4 x \sqrt{g}\, E_4 \= 4 ~,\\
\lim_{\beta\to+\infty} {1\/(4\pi)^2} \int d^4 x \sqrt{g}\,W^2 \= - {2 ( r_0^2-\ell_2^2 )^2 \/ 3r_0^2 \ell_2^2 }~,\\\label{RNR2}
\lim_{\beta\to+\infty} {1\/(4\pi)^2} \int d^4 x\sqrt{g} \, R^2 \=-{2 ( r_0^2-\ell_2^2 )^2 \/ r_0^2 \ell_2^2}~, \\
\lim_{\beta\to+\infty} {1\/(4\pi)^2} \int d^4 x \sqrt{g} \,R F_{\mu\nu} F^{\mu\nu} \= -{ (r_0^2-\ell_2^2)(r_0^4 + r_0^2 \ell_2^2 - 4q_m^2 \ell_2^2) \/r_0^4\ell_2^2}~.
\end{eqnarray}
This leads to the final result
\be\label{ClocalRNext}
\lim_{\beta\to+\infty} C_\mathrm{local} = -4\, a_\text{E} -{r_0^2-\ell_2^2 \/r_0^2\ell_2^2}\left( \left( {2\/3} c + 2 b_1\right) \Big(r_0^2-\ell_2^2\Big) + b_2 \Big(r_0^2+\ell_2^2 - {4 \ell_2^2 q_m^2\/ r_0^2}\Big) \right)~.
\ee
Note that in the flat space limit, we have $r_0=\ell_2$ and the logarithmic correction is manifestly topological, but such cancellation does not occur for AdS black holes.
\subsubsection{Near horizon geometry}
\label{subsubsec: RNAdSNH}
As we would like to investigate where the quantum degrees of freedom live for asymptotically AdS spacetimes, we compare the basis of curvature invariants of the full solution to that of the near horizon geometry. Let us first consider the extremal black hole. The near horizon geometry can be obtained using the change of coordinates
\begin{align}
r \to r_0+ \epsilon\, \tilde{r}, \qquad t \to \ell_2^2\,{\tilde{t}\/\epsilon}
\end{align}
and taking the limit $\epsilon\to0$. The result is the AdS$_2\times S^2$ geometry
\be\label{eqn: RNAdSNH}
ds^2 =\ell_2^2 \left(\tilde{r}^2d\tilde{t}^2 + {d\tilde{r}^2\/\tilde{r}^2} \right) + r_0^2\, d\Omega_2^2 ,\qquad
A = -{i \ell_2^2 q_e \/r_0^2} \tilde{r} d\tilde{t} + q_m\, \mathrm{cos}\,\theta \, d\phi~,
\ee
where $\ell_2$ and $r_0$ are defined in \eqref{r0l2}. For the gauge field, a pure gauge term needs to be added to obtain a smooth $\epsilon\to0$ limit. We can express everything in terms of the two scales $\ell_2$ and $r_0$. The AdS$_4$ radius and the extremal charges are given by
\be
{6\/\ell^2} = {1\/\ell_2^2} -{1\/r_0^2}~,\qquad q_e^2+q_m^2 = {r_0^2(r_0^2+\ell_2^2)\/2\ell_2^2}~.
\ee
In particular we see that we must have $r_0 > \ell_2$. Note that in flat space we obtain $r_0=\ell_2$.
The infinite volume of AdS$_2$ is regularized by removing the divergence through a redefinition of the ground state energy in the dual CFT$_1$ \cite{Sen:2008yk,Sen:2008vm}. This leads to a regularized volume of unit AdS$_2$ which is $-2\pi$. The integrated invariants can then be computed and we find
\be
C_\mathrm{local} = -4 a_\text{E} -{r_0^2-\ell_2^2 \/r_0^2\ell_2^2}\left( \left( {2\/3} c + 2 b_1\right) \Big(r_0^2-\ell_2^2\Big) + b_2 \left(r_0^2+\ell_2^2- {4 \ell_2^2 q_m^2\/ r_0^2}\right) \right)~.
\ee
This expression matches the result \eqref{ClocalRNext} obtained by taking the $\beta\to+\infty$ limit of the non-extremal $C_{\text{local}}$. Hence, the computation of $C_\mathrm{local}$ for an extremal black hole can be done either in the full geometry or in the near horizon region. The difference in logarithmic correction between the full geometry and the near horizon geometry come exclusively from zero modes.
\subsection{Kerr-Newman black hole}\label{sec:KN}
We now turn to the AdS-Kerr-Newman black hole \cite{Plebanski:1976gy,Caldarelli:1999xj}. This solution is particularly interesting because it has a regular BPS limit unlike the Reissner-Nordstr\"om black hole \cite{Romans:1991nq,Kostelecky:1995ei,Hristov:2010ri}.
\subsubsection{Non-extremal black hole}
As given in \cite{Caldarelli:1999xj}, the line element takes the form ,
\begin{align}
d s^{2}=-\frac{\Delta_{r}}{\rho^2}\left(d t-\frac{a~\r{sin} ^{2}\theta}{\Xi} d \phi\right)^{2}+\frac{\rho^2 d r^{2}}{\Delta_{r}}+\frac{\rho^2 d \theta^{2}}{\Delta_{\theta}}+\frac{\Delta_{\theta} \,\r{sin} ^{2} \theta}{\rho^2}\left(a \,d t-\frac{r^{2}+a^{2}}{\Xi} d \phi\right)^{2},
\end{align}
where we have defined
\be
\Delta_{r} = \left(r^{2}+a^{2}\right)\left(1+{r^2\/\ell^2}\right)-2 m r+q_{e}^{2}+q_{m}^2, \qquad \Delta_{\theta} = 1-{a^2\/\ell^2}\,\r{cos} ^{2} \theta~,
\ee
with $ \rho^2 = r^{2}+a^{2} \r{cos} ^{2} \theta$ and $\Xi= 1-{a^2\/\ell^2}$. The gauge field is given by
\begin{align}
A =-\frac{q_e r}{\rho^{2}}\left(d t-\frac{a\, \r{sin} ^{2} \theta}{\Xi} d \phi\right)-\frac{q_m\,\r{cos}\,\theta}{\rho^2}\left(adt-\frac{r^2+a^2}{\Xi}d\phi\right)~,
\end{align}
The parameters satisfy $a^2 <\ell^2$ and we take $a\geq 0$ without loss of generality.\footnote{The general result is obtained by replacing $a\rightarrow|a|$ everywhere.} The physical mass $M$, angular momentum $J$, electric charge $Q_e$ and magnetic charge $Q_m$ are given by
\begin{align}
M = \frac{m}{\Xi^2}, \qquad J = \frac{a m}{\Xi^2}, \qquad Q_e = \frac{q_e}{\Xi},\qquad Q_m=\frac{q_m}{\Xi},
\end{align}
and the inverse temperature is
\begin{align}
\beta=\frac{4 \pi\left(r_{+}^{2}+a^{2}\right)}{r_{+}\left(1+\frac{a^{2}}{\ell^{2}}+3 \frac{r_{+}^{2}}{\ell^{2}}-\frac{a^{2}+q^{2}_{e}+q^2_{m}}{r_{+}^{2}}\right)}.
\end{align}
For the non-extremal black hole, the general form is
\be
C_\mathrm{local} = -4 \,a_\mathrm{E}+ (6 A_1+c W_1) \beta + (24 A_2+cW_2) + {c\, W_3\/\beta}~,
\ee
where the logarithmic corrections depends on five independent parameters $\{r_+,\beta,\ell,a,q_m\}$. The Euler term simply gives a pure number in agreement with the formula
\be
\chi = {1\/32\pi^2} \int d^4 x\sqrt{g} E_4 = 2~.
\ee
The expressions $A_i$ and $W_i$ are independent of $\beta$ and take the form
\begin{align}
\begin{split}
A_1 &={ (2b_1+b_2)(a^2+\ell^2)r_+^3 + (2 b_1+3b_2) r_+^5 + ( (2b_1-b_2)a^2 - 2 b_2 q_m^2) r_+ \ell^2 \/ \pi\ell^2(\ell^2-a^2) (a^2+r_+^2)}~,
\\
A_2 &=-{b_1 a^2 + (b_1+b_2)r_+^2 \/\ell^2-a^2} ~,
\end{split}
\end{align}
and we have isolated the contribution $W_i$ from the Weyl squared term, explicitly given as
\begin{eqnarray}\notag
W_1\= {1\/16\pi a^5 r_+^4\ell^2 (\ell^2-a^2)(a^2+r_+^2)}\left[ 3 a r_+(a^8(\ell^2-r_+^2)^2 + r_+^8 (\ell^2+3 r_+^2)^2) \right. \-
&& - 4 a^3 r_+^3 (r_+^4(\ell^4 - 9r_+^4) + a^4 (\ell^4 + 12 \ell^2 r_+^2 + 3 r_+^4) +2 a^5 r_+^5(\ell^4 - 14\ell^2 r_+^2 + 5 r_+^4) \-
&& \left. -3 (a^2+ r_+^2) (a^2(\ell^2-r_+^2) - r_+^2(\ell^2+ 3 r_+^2) )^2 (r_+^4-a^4) \,\mathrm{arctan}(a/r_+) \right]~,\\
W_2 \= {a^2 + r_+^2\/ 2 a^5 r_+^3 (\ell^2-a^2)}\left[ 4 a^3\ell^2 r_+^3 + 3 a r_+ (a^4(\ell^2-r_+^2) - r_+^4(\ell^2+3 r_+^2)) \right. \-
&&\hspace{2.6cm} \left. -3(a^2(\ell^2-r_+^2)-r_+^2(\ell^2+ 3 r_+^2)) (r_+^4-a^4) \,\mathrm{arctan}(a/r_+) \right]~,\-
W_3 \= {\pi \ell^2 (a^2 + r_+^2)\/ a^5 r_+^2 (\ell^2-a^2)}\left[ a r_+ (3 a^4 + 2 a^2 r_+^2 +3 r_+^4 )-3 (r_+^2+a^2) (r_+^4-a^4) \,\mathrm{arctan}(a/r_+)\right]~.
\end{eqnarray}
We have checked that we reproduce the Reissner-Nordstr\"om results of section \ref{sec: RNBH} in the limit $a=0$.
\subsubsection{Extremal limit}
As was done in the Reissner-Nordstr\"om black hole, the extremal limit can be found by taking the limit $T \to 0$ or $\beta\to+\infty$ while keeping the charges fixed. To do this appropriately, we use that for small temperatures
\be
r_+ = r_0 + {2\pi \ell_2^2 \/\beta} + O(\beta^{-2})~,
\ee
and we take the $\beta\to+\infty$ limit while keeping $r_0,\ell,a ,q_m$ fixed. The procedure yields a finite piece in $\beta$ as well as a piece linear in $\beta$, which can be removed by a renormalization of the ground state energy. The final result can be written in terms of the four independent parameters $\{r_0,\ell,a,q_m\}$. It takes the form
\begin{eqnarray} \notag
C_\mathrm{local} \= -4 a_\mathrm{E} + {1\/2 a r_0^5 (\ell^2-a^2) (a^2+r_0^2)(a^2+\ell^2+6 r_0^2)} \Big[ - 3 a^7 r_0 (16 b_1 r_0^4 + c(\ell^2- r_0^2)^2). \-
&& \hspace{1.3cm}+ a^5 r_0^3(c\ell^4 + 2( 11c -12 b_2) \ell^2 r_0^2 - 3(13 c-8b_2+80 b_1) r_0^4) \-
&& \hspace{1.3cm}+a^3 r_0^5 (15 c\ell^4 + 2 (25 c + 24 b_2) \ell^2 r_0^2 -(49 c +336 b_1-48b_2) r_0^4- 48 b_2 \ell^2 q_m^2) \-
&& \hspace{1.3cm}+ 3 a r_0^7 (c\ell^4 + 2(3c-4 b_2) \ell^2 r_0^2 -(7c+48b_1 + 24b_2) r_0^4 + 16 b_2 \ell^2 q_m^2) \\\label{ClocalExtKN}
&& \hspace{1.3cm} - 3 c(a^2 + r_0^2) (a^2 (r_0^2-\ell^2)+ r_0^2(\ell^2+3 r_0^2))^2 \,\mathrm{arctan}(a/r_0) \Big] ~.
\end{eqnarray}
We can also compare with the computation performed in the near horizon geometry obtained via
\be\label{toNHGKN}
r \rightarrow r_0 + \epsilon\, \tilde{r},\qquad t \rightarrow \ell_2^2 \,{\tilde{t}\/\epsilon} ,\qquad \phi\rightarrow \phi - { i a \ell_2^2(\ell^2-a^2)\/\ell^2(a^2+r_0^2)}\, {t\/\epsilon}~,
\ee
while taking $\epsilon\to0$. This leads to
\begin{eqnarray}
d\tilde{s}^2 \= {\ell_2^2 (r_0^2 + a^2\,\mathrm{cos}^2\theta)\/ a^2 + r_0^2} \left(\tilde{r}^2 d\tilde{t}^2 + {d\tilde{r}^2\/\tilde{r}^2} \right) + {\ell^2 (r_0^2 + a^2\,\mathrm{cos}^2\theta)\/\ell^2 - a^2\,\mathrm{cos}^2\theta} d\theta^2 \-
&& + {\ell^2 (a^2+r_0^2)^2 (\ell^2 -a^2 \,\mathrm{cos}^2\theta)\,\mathrm{sin}^2\theta \/(\ell^2-a^2)^2 (r_0^2 + a^2\,\mathrm{cos}^2\theta)} \left(d\phi -{2\ell_2^2 a r_0(\ell^2-a^2)\/ \ell^2 (a^2+r_0^2)^2} \,i r dt \right)^2~,
\end{eqnarray}
where the AdS$_2$ radius is
\be\label{l2KN}
\ell_2 = \ell \sqrt{a^2+r_0^2\/a^2+\ell^2 + 6 r_0^2}~.
\ee
The near horizon geometry is a warped version of AdS$_2$ with a circle fiber, similar to the near horizon of extreme Kerr (NHEK), which we recover in the appropriate limit. The near horizon gauge field takes the form
\begin{eqnarray}
\widetilde{A} \= {1\/r_0^2 + a^2\,\mathrm{cos}^2\theta} \left[ -{i\ell_2^2\/a^2 +r_0^2} (q_e (r_0^2 -a^2\,\mathrm{cos}^2\theta) + 2 q_m \,a r_0\,\mathrm{cos}\,\theta) \tilde{r} d\tilde{t} \right.\-
&& \hspace{5cm}\left. + {\ell^2\/\ell^2-a^2}( q_e\, a r_0 \,\mathrm{sin}^2\theta + q_m (a^2+r_0^2)\,\mathrm{cos}\,\theta)d\tilde\phi \right]~.
\end{eqnarray}
We can perform more general near horizon limits by taking at the same time a near-extremal limit. Instead of setting $q_e = q_e^\ast$, we can consider a deformation $q_e = q_e^\ast + \delta q_e \epsilon^2$ parametrized by the same $\epsilon$ as in \eqref{toNHGKN}. Moreover, keeping subleading corrections in $\beta^{-1}$ would yield corrections to the entropy in the near-extremal regime. The non-zero energy associated to this large diffeomorphism can be understood in terms of the Schwarzian action of Jackiw-Teitelboim gravity \cite{Maldacena:2016upp}.
We are now in a position to compute the logarithmic corrections in the near horizon geometry and we find that the result is equal to \eqref{ClocalExtKN} obtained by taking the extremal limit appropriately, {\it i.e.}, fixing the charges while taking $\beta \to +\infty$. Thus, the local contribution is the same in the near horizon region and the full geometry.
\subsubsection{BPS limit}
\label{subsubsec: AdSKNBPS}
The BPS limit can be obtained by imposing the additional conditions to the extremal black hole
\be
r_0 = \sqrt{a \ell},\qquad q_m=0~.
\ee
The resulting black hole preserves half of the supersymmetries \cite{Kostelecky:1995ei}. Its charges are given by
\be
M =\frac{\sqrt{a\ell }}{\left(1-\frac{a}{\ell}\right)^{2}}, \qquad Q_e=\frac{\sqrt{a\ell }}{1-\frac{a}{\ell}}, \qquad Q_{m}=0, \qquad J=\frac{a\sqrt{a\ell}}{\left(1-\frac{a}{\ell}\right)^{2}}
\ee
and it satisfies a BPS bound:
\begin{align}
M=Q_e+\frac{J}{\ell}~.
\end{align}
The BPS result can be written in terms of the two independent parameters $\ell$ and $a$
\begin{eqnarray}\notag
C_\mathrm{local} \= - 4 a_\mathrm{E} + {3\ell_2^2\/2 a\ell^2(\ell^2-a^2)} \left[ (9c-8b_2) a \ell^2 - (9c +48 b_1- 8b_2)a^2 \ell - (c+16 b_1)a^3\vphantom{c (a+\ell)^4\/\sqrt{al}} \right.\\\label{ClocalBPS}
&&\hspace{6cm}\left.+ c\ell^3 - {c (a+\ell)^4\/\sqrt{al}} \,\mathrm{arctan}(\sqrt{a/\ell})\right],
\end{eqnarray}
where the AdS$_2$ radius given in \eqref{l2KN} is
\be
\ell_2= \ell\sqrt{a(a+\ell)\/a^2+ 6 a\ell+\ell^2} \qquad \text{(BPS case)}~.
\ee
It is clear from this formula that there is no non-rotating BPS solution as the limit $a\to 0$ is singular.
\subsection{AdS-Rindler geometry}\label{sec:hypBH}
Our computation of the logarithmic correction can also be applied to the so-called hyperbolic black hole of \cite{Casini:2011kv}, \emph{i.e.}, the AdS$_4$-Rindler geometry. The entropy of this black hole is the entanglement entropy
\be
S_\mathrm{EE} = -\mathrm{Tr}\,\rho_A \log\rho_A,
\ee
associated to a ball-shaped boundary subregion $A$. Here $\rho_A$ is the reduce density matrix defined by tracing over the complement $\bar{A}$
\be
\rho_A = \mathrm{Tr}_{\bar{A}} |0\rangle\langle 0|,
\ee
where $|0\rangle$ is the global vacuum. Here the only length scale is the AdS$_4$ radius $\ell$ so we are considering the regime of large $\ell$ and computing
\be
S_\mathrm{EE} = {\mathrm{Area}\/4 G}+ (C_\mathrm{local}+C_\mathrm{zm}) \log \ell+ \dots .
\ee
The geometry of the hyperbolic black hole is given by
\be
ds^2 = \left( {\rho^2\/\ell^2} - 1\right)dt^2 + {d\rho^2\/{\rho^2\/\ell^2}-1 }+\rho^2 ds^2_{H_2}~,\qquad ds^2_{H_2}=du^2 + \mathrm{sinh}^2 u\, d\phi^2,
\ee
where $\rho \geq \ell$, $u\geq 0$ and $t\sim t+\beta$. The inverse temperature is given by
\be
\beta=2\pi \ell~.
\ee
We regularize the integral over spacetime using holographic renormalization. In this case, there is also a divergence coming from the volume of $H_2$ and we take a regulator such that $\mathrm{vol}(H_2) = -2\pi$. The integrated four-derivative invariants are given by
\begin{align}\notag
{1\/(4\pi)^2 }\int d^4 x\sqrt{g}\,E_4 & = 2~, & {1\/(4\pi)^2 }\int d^4 x\sqrt{g}\,W^2 & = 0~,\\
{1\/(4\pi)^2 }\int d^4 x\sqrt{g}\,R^2 & = 12~, & {1\/(4\pi)^2 }\int d^4 x\sqrt{g}\,R F_{\mu\nu} F^{\mu\nu} & =0~.
\end{align}
This implies that we have
\be
C_\mathrm{local} = -2 a_\text{E}+ 12 b_1~.
\ee
Note that the Gauss-Bonnet-Chern theorem gives
\be
\chi = {1\/32\pi^2 }\int d^4 x\sqrt{g}\,E_4 = 1~,
\ee
as expected since $M$ is topologically $D_2\times H_2$ where $D_2$ is a disk and we have $\chi(M) = \chi(D_2)\chi(H_2) = 1$ since $\chi(D_2)=\chi(H_2)=1$. This is a non-trivial consistency check for our regularization procedure.
\subsection{Global contribution}
We now compute the global contribution \eqref{Cglobal} which comes from the zero modes and the change of ensemble. The results are summarized in Table \ref{tab:global}. In the full geometry, the contribution from the bosonic zero modes in the full asymptotically AdS$_4$ geometry vanishes \cite{Sen:2012dw}. Indeed, the fact that AdS$_4$ admits a 2-form zero mode follows from the general result of Camporesi and Higuchi who established that AdS$_{2M}$ admits a M-form zero mode \cite{CAMPORESI199457}. This 2-form zero mode is central in generating the logarithmic correction in asymptotically AdS$_4$ backgrounds embedded in eleven-dimensional supergravity \cite{Bhattacharyya:2012ye,Liu:2017vbl}. However, in the four-dimensional theories we consider in this manuscript, there is no contribution from such a 2-form zero mode.
Hence we have
\be
C_\mathrm{zm}=0\qquad \text{(full geometry)}~.
\ee
In the near horizon geometry, additional zero modes come from the AdS$_2$ factor. The metric contributes $-3$ zero modes. In the near horizon geometry of BPS black holes, we also have $8$ fermionic zero modes. The zero mode contribution for extremal black holes in the near horizon geometry is then given by
\be
C_\mathrm{zm}= - 3 + 8\, \delta_\mathrm{BPS}\qquad \text{(near horizon geometry)},
\ee
where $\delta_\mathrm{BPS}=1$ in the BPS case and $0$ otherwise. It is interesting to observe that this contribution can be interpreted in the context of nearly AdS$_2$ holography \cite{Maldacena:2016upp}. The asymptotic symmetry group of AdS$_2$ is $\mathrm{Diff}(S^1)/\mathrm{SL}(2,\mathbb{R})$. Upon a choice of configuration, the number of broken symmetries is $n_0 = +\infty - 3$, the infinite piece being absorbed in a renormalization of the energy. So the $-3$ zero modes come from the unbroken $\mathrm{SL}(2,\mathbb{R})$ symmetry of AdS$_2$. A similar argument for BPS black holes explains the $8$ fermionic zero modes as arising from the eight fermionic generators of the $\mathrm{PSU}(1,1|2)$ near horizon symmetry. These patterns of symmetry breaking can be studied using Jackiw-Teitelboim gravity \cite{Almheiri:2014cka,Maldacena:2016upp, Nayak:2018qej,Heydeman:2020hhw,Iliesiu:2020qvm,Castro:2019crn,Castro:2021csm,Castro:2021wzn}.
We also include in $C_\mathrm{global}$ the correction that comes from the change of ensemble from canonical to microcanonical. Following \cite{Sen:2012dw}, the change of ensemble gives a contribution
\be
C_\mathrm{ens}= -K ~,
\ee
where $K$ is the number of rotational symmetries of the black hole.
\begin{table}
\centering\arraycolsep=4pt\def1.4{1.2}
\begin{tabular}{|l|c|c||c|}
\hline
Background spacetime & $C_\mathrm{zm}$ & $C_\mathrm{ens}$ & $C_\mathrm{global}$ \\\hline
Schwarzschild & 0 & $-3$ & $-3$\\
Reissner-Nordstr\"om & 0 & $-3$ & $-3$\\
Kerr & 0 & $-1$ & $-1$\\
Kerr-Newman & $0$ & $-1$ & $-1$\\
BPS Kerr-Newman & 0& $-1$& $-1$\\
Reissner-Nordstr\"om near horizon & $-3$ & $-3$ & $-6$\\
Kerr-Newman near horizon & $-3$ & $-1$ & $-4$ \\
BPS Kerr-Newman near horizon & $5$ & $-1$ & $4$ \\\hline
Thermal AdS$_4$ & $0$ & $-3$ & $-3$ \\
AdS$_4$-Rindler & $0$ & $-3$ & $-3$ \\
\hline
\end{tabular}\caption{Global contribution to the logarithmic correction.}\label{tab:global}
\end{table}
\section{Minimally coupled matter}
To obtain the logarithmic corrections, we need to compute the coefficients $a_\mathrm{E},c,b_1,b_2$ that appear in the general expression \eqref{a4general}. Our ultimate aim is to evaluate logarithmic corrections in theories that can arise as consistent low-energy truncations from string and M-theory. However, in the next sections, we compute these logarithmic corrections in Einstein-Maxwell theory with a negative cosmological constant and in minimal $\mathcal{N}=2$ gauged supergravity. As a warm-up, we also present the logarithmic corrections to AdS black holes due to minimally coupled fields, as was done for flat space black holes in \cite{Sen:2012dw}.
\subsection{Minimal theories}
In this subsection, we compute $C_\mathrm{local}$ for minimal scalars, fermions, vectors and gravitini.
\paragraph{Free scalar.}
We consider a scalar field of mass $m$ described by the action
\be
S = -{1\/2}\int d^4 x\sqrt{g}
\left((\partial\phi)^2+m^2\phi^2 \right).
\ee
The result for a scalar field is obtained by setting
\be
P=E=m^2, \qquad\Omega=0~,
\ee
in equation \eqref{intro:a4}. As explained in section \ref{sec:scaling}, we consider a regime where every length scales with a factor $\lambda$. So here $m$ scales as $\lambda^{-1}$ and what is fixed is the conformal dimension
\be
\Delta = {1\/2}\left(3+\sqrt{9+4m^2\ell^2}\right)~.
\ee
This is to be contrasted with flat space where massive fields do not contribute to the logarithmic correction as explained in \cite{Sen:2012kpz,Sen:2012cj,Sen:2012dw}.
The heat kernel takes the form
\be
(4\pi)^2 a_4(x) = -{1\/360}E_4 +{1\/120}W^2+{1\/288}(\Delta(\Delta-3)-2)^2 R^2~.
\ee
The explicit result for $C_\mathrm{local}$ can be obtained using \eqref{Clocala4} and \eqref{a4general}. We report the result for the extremal black hole
\be
C_\mathrm{local} = -{1\/90} - { r_0^2\/20 \ell^4} (24 + 5(\Delta+1)\Delta(\Delta-3)(\Delta-4))~.
\ee
\paragraph{Free fermion.}
We consider a free Dirac fermion with Euclidean action
\be
S =\int d^{d+1}x \sqrt{g}\,\bar\psi\left( \gamma^\mu\nabla_\mu -m\right) \psi
\ee
This is dual to an operator with scaling dimension \cite{Henningson:1998cd}
\be
\Delta = {3\/2} + m\ell.
\ee
The result is
\be
(4\pi)^2a_4(x) = {11\/360} E_4 + {1\/20} W^2 +{1\/72} \left( \Delta-\tfrac32\right)^2\left(\left( \Delta-\tfrac32\right)^2-2 \right)R^2.
\ee
\paragraph{Free vector.}
We now consider a free Maxwell field $a^\mu$ with the Lagrangian
\be
\mathcal{L} = -{1\/4}f_{\mu\nu} f^{\mu\nu},
\ee
where $f_{\mu\nu} = \nabla_\mu a_\nu - \nabla_\nu a_\mu$. We add the gauge fixing term
\be
\mathcal{L}_\mathrm{g.f.} = {1\/2} (\nabla_\mu a^\mu)^2 ,
\ee
so that the total Lagrangian becomes
\be
\mathcal{L} + \mathcal{L}_\mathrm{g.f.} = a^\mu \Box a_\mu - a^\mu R_{\mu\nu} a^\nu~.
\ee
The gauge-fixing induces two massless scalar fields with fermionic statistics. We obtain the result
\be
(4\pi)^2 a_4(x) = -{31\/180} E_4 + {1\/10}W^2~.
\ee
\paragraph{Free Rarita-Schwinger field.}
We consider here a Majorana spin-${3\/2}$ field described by the Lagrangian
\be
\mathcal{L}_{3/2} = -\bar\psi_\mu \gamma^{\mu\rho\nu} \nabla_\rho \psi_\nu.
\ee
We use the gauge-fixing condition $\gamma^\mu\psi_\mu=0$. This is implemented with the gauge-fixing term
\be
\mathcal{L}_\text{g.f} = -{1\/2}(\bar\psi_\mu \gamma^\mu) \gamma^\rho \nabla_\rho (\gamma^\nu \psi_\nu)~,
\ee
so that the total Lagrangian is
\be
\mathcal{L}_{3/2} + \mathcal{L}_\text{g.f}= \bar{\chi}_\mu \gamma^\nu D_\nu \chi^\mu,
\ee
after using the field redefinition $\psi_\mu = \chi_\mu - {1\/2} \gamma_\mu \gamma^\nu\chi_\nu$. The gauge-fixing leads to three Majorana ghosts which are free massless fermions. We refer the reader to section \ref{sec:FPfermions} for details on the gauge-fixing procedure. Hence, we find
\be
(4\pi)^2 a_4(x) = {229\/720} E_4- {77\/120}W^2 - {1\/9}R^2.
\ee
\subsection{Logarithmic corrections}
The results for minimally coupled scalars are summarized in Table \ref{tab:results}.
\subsubsection{Massless fields}
For massless fields, we can present the result as
\begin{eqnarray}
a_\text{E} \= {1\/720}( 2 n_S + 22 n_F+124 n_V -229 n_\psi)~,\\
c\= {1\/120}( n_S+6n_F + 12 n_V- 77 n_\psi)~,\\
b_1 \= {1\/72}(n_S - 8 n_\psi)~.
\end{eqnarray}
where $n_S,n_F,n_V,n_\psi$ the number of scalars, spin-${1\/2}$ Majorana fermion, vector and gravitini. The result for AdS-Schwarzschild takes the form
\begin{eqnarray}
C_\mathrm{local}\= {1\/180\ell^2(\ell^2 + 3r_+^2)} \left[ \ell^4 (4 n_S + 14 n_F-52 n_V -233 n_\psi)\right. \-
&& \left.+ 3 r_+^2\ell^2 (22 n_S+2 n_F- 76 n_V-239 n_\psi) + 18 r_+^4 (-3 n_S +2 n_F+4n_V +n_\psi) \right].
\end{eqnarray}
It is easily seen that in the flat space limit, we have
\be
\lim_{\ell\to+\infty}C_\mathrm{local} = {1\/180} (4 n_S + 14 n_F-52 n_V -233 n_\psi).
\ee
which reproduces the results of \cite{Sen:2012dw}.
\subsubsection{Corrections to entanglement entropy}
Our result can also be applied to compute logarithmic correction to entanglement entropy. We consider a ball-shaped region $A$ in the boundary. The entanglement entropy of $A$ is given by the area of the hyperbolic black hole discussed in section \ref{sec:hypBH}. The logarithmic corrections to entanglement entropy are given by
\be
S_\mathrm{EE} = {\mathrm{Area}\/4G} + C\, \log \beta+\dots .
\ee
The contribution of a minimal scalar field of conformal dimension $\Delta$ gives
\be\label{CEEscalar}
C = {29\/180} + {1\/24}(\Delta+1)\Delta(\Delta-3)(\Delta-4)~.
\ee
We have here $C=C_\mathrm{local}$ since there is no zero mode for the scalar field. Quantum corrections to entanglement entropy can also be interpreted in terms of bulk entanglement entropy \cite{Faulkner:2013ana}. It would be interesting to see if we can understand \eqref{CEEscalar} as the logarithmic piece of the bulk entanglement entropy of a scalar field in the Rindler wedge.
\section{Einstein-Maxwell-AdS theory}
\label{Sec:EMLambda}
We now consider Einstein-Maxwell theory with a negative cosmological constant. This is the minimal theory that contains the AdS-Kerr-Newman black hole and is the bosonic part of minimal $\mathcal{N}=2$ gauged supergravity studied in section \ref{Sec:N2Sugra}. The action is given by
\begin{align}
\label{EM action}
S=\int d^{4} x \sqrt{ g}\left(R-2\Lambda-F_{\mu \nu} F^{\mu \nu}\right),
\end{align}
where $F_{\mu\nu}=\partial_\mu A_\nu-\partial_\nu A_\mu$ is the field strength with $A_\mu$ the gauge potential. Note that we find it convenient to use a convention $4\pi G=1$.
The computation is easily performed using the algorithm described in Appendix \ref{app:algo}. We have also performed an independent computation by hand, as detailed in Appendix \ref{app:bosons}.
\subsection{Bosonic fluctuations}
\label{subsec: bosonquadexpansion}
We consider variations of the metric and gauge field
\be
\delta g_{\mu\nu} =\sqrt{2} h_{\mu\nu}~,\qquad \delta A_\mu = \frac{1}{2} a_\mu~,
\ee
where $h_{\mu\nu}$ and $a_\alpha$ are the graviton and graviphoton, respectively. We impose a particular gauge to the theory by adding a suitable gauge-fixing Lagrangian
\begin{align}
\label{eqn: EMgf}
S= -\int d^{4} x \sqrt{\operatorname{det} g}\left\{\left(D^{\mu} h_{\mu \rho}-\frac{1}{2} D_{\rho} h\right)\left(D^{\nu} h_{\nu}^{\rho}-\frac{1}{2} D^{\rho} h\right)+\frac{1}{2}\left(D^{\mu} a_{\mu}\right)\left(D^{\nu} a_{\nu}\right)\right\},
\end{align}
and the corresponding ghost action to the action \eqref{EM action}. We then expand the action up to quadratic order. The linear order variation yields the equation of motion for the background fields
\begin{align}
\label{eqn: gravityEoM}
R_{\mu\nu}-\frac{1}{2}g_{\mu\nu}R+g_{\mu\nu}\Lambda=&2F_{\mu\rho}F_\nu^{~\rho}-\frac{1}{2}g_{\mu\nu}F_{\alpha\beta}F^{\alpha\beta}~,\\
\label{eqn: gaugeEoM}
D^\mu F_{\mu\nu}=&0~.
\end{align}
Note that the equations of motion implies that $R=4\Lambda = -12/\ell^2$. It is also worth mentioning the Bianchi identity for the gravitational field and gauge field
\begin{align}
\label{eqn: gravitybianchi}
D_{[\mu}F_{\nu\rho]}=&0~,\\
\label{eqn: gaugebianchi}
R_{\mu[\nu\rho\sigma]}=&0~,
\end{align}
as they serve as handy tools for simplifying our calculations. The quadratic action can be put in the canonical form \eqref{eqn: Lap}. The details can be found in appendix \ref{appendix:fluctuations} where we present the explicit form of the quadratic fluctuations. This allows us to extract the matrices $I$, $E$ and $\Omega$:
\begin{align}
\label{eqn: a4BI}
\phi_{m}I^{mn}\phi_{n}=& h_{\mu \nu}\left(\frac{1}{2}g^{\mu\alpha}g^{\nu\beta}+\frac{1}{2}g^{\mu\beta}g^{\nu\alpha}-\frac{1}{2}g^{\mu\nu}g^{\alpha\beta}\right)h_{\alpha\beta}+a_\alpha g^{\alpha\beta}a_\beta,\\
\begin{split}
\label{eqn: a4BE}
\phi_{m} E^{m n} \phi_{n}=& h_{\mu \nu}\left(R^{\mu \alpha \nu \beta}+R^{\mu \beta \nu \alpha}-g^{\mu \nu} R^{\alpha \beta}-g^{\alpha \beta} R^{\mu \nu} + \Lambda g^{\mu\nu}g^{\alpha\beta} \right) h_{\alpha \beta} \\
&+a_{\alpha}\left(\frac{3}{2} g^{\alpha \beta} F_{\mu \nu} F^{\mu \nu} - \Lambda g^{\alpha\beta}\right) a_{\beta}+\frac{\sqrt{2}}{2} h_{\mu \nu}\left(D^{\mu} F^{\alpha \nu}+D^{\nu} F^{\alpha \mu}\right) a_{\alpha} \\
&+\frac{\sqrt{2}}{2} a_{\alpha}\left(D^{\mu} F^{\alpha \nu}+D^{\nu} F^{\alpha \mu}\right) h_{\mu \nu}~,
\end{split}\\
\begin{split}
\label{eqn: a4BOmega}
\phi_{m}\left(\Omega^{\rho \sigma}\right)^{m n} \phi_{n}=& h_{\mu \nu}\left\{\frac{1}{2}\left(g^{\nu \beta} R^{\mu \alpha \rho \sigma}+g^{\nu \alpha} R^{\mu \beta \rho \sigma}+g^{\mu \beta} R^{\nu \alpha \rho \sigma}+g^{\mu \alpha} R^{\nu \beta \rho \sigma}\right)\right.\\
&\left.+\left[\omega^{\rho}, \omega^{\sigma}\right]^{\mu \nu \alpha \beta}\right\} h_{\alpha \beta}+a_{\alpha}\left\{R^{\alpha \beta \rho \sigma}+\left[\omega^{\rho}, \omega^{\sigma}\right]^{\alpha \beta}\right\} a_{\beta} \\
&+h_{\mu \nu} \left(D^{[\rho} \omega^{\sigma]}\right)^{\mu \nu \alpha} a_{\alpha}+a_{\alpha}\left( D^{[\rho}\omega^{\sigma]}\right)^{\alpha \mu \nu} h_{\mu \nu}~,
\end{split}
\end{align}
where $\omega^\rho$ is the spin-connection given by
\begin{equation}
\begin{split}
\phi_{m}\left(\omega^{\rho}\right)^{m n} \phi_{n}=&\frac{\sqrt{2}}{2} h_{\mu \nu}\Big(g^{\alpha\mu}F^{\rho\nu}+g^{\alpha\nu}F^{\rho\mu}-g^{\mu\rho}F^{\alpha\nu}-g^{\nu\rho}F^{\alpha\mu}-g^{\mu\nu}F^{\rho\alpha} \Big)a_{\alpha}\\
&-\frac{\sqrt{2}}{2} a_{\alpha}\Big(g^{\alpha\mu}F^{\rho\nu}+g^{\alpha\nu}F^{\rho\mu}-g^{\mu\rho}F^{\alpha\nu}-g^{\nu\rho}F^{\alpha\mu}-g^{\mu\nu}F^{\rho\alpha} \Big) h_{\mu \nu}~.
\end{split}
\end{equation}
We then find the trace of \eqref{eqn: a4BI}-\eqref{eqn: a4BOmega}. The computation is tedious, but it may also be illuminating for some readers. We present the intermediate steps in appendix \ref{appendix: bosontrace}. The final contribution to the heat kernel coefficient is
\begin{align}
\label{Eq:a4EM}
(4\pi)^2 a_{4}^{\text{EM}}(x) &= -{277\/180}E_4 + {38\/15}W^2 + {7\/18}R^2~.
\end{align}
The reader familiar with the literature might notice that we have not treated the {\it trace mode} in the graviton. Traditionally, as in the literature \cite{Charles:2015eha, Christensen:1979iy, Gibbons:1976ue}, one decomposes the fields appearing in the Lagrangian into the irreducible fields $\phi(A,B)$ which transform according to the irreducible $(A,B)$ representation of SO(4). For example, in \cite{Christensen:1979iy}, the authors considered the decomposition of fluctuation of geometry $h_{\mu\nu}$ into a $(1,1)$ symmetric traceless tensor, a scalar characterizing the trace part transforming in $(0,0)$ and the corresponding vector ghost field in $(\frac{1}{2},\frac{1}{2})$. Here, we choose the operator $I^{mn}$ as the effective metric, which is equivalent to making this decomposition.
\subsubsection{Ghost contribution}
\label{subsec:bosonghost}
The addition of the gauge-fixing Lagrangian \eqref{eqn: EMgf} gives an action for the ghosts
\begin{align}
\label{eqn: EMgh}
\mathcal{S}_{\mathrm{ghost}, b}=\frac{1}{2} \int d^{4} x \sqrt{g}\Big\{2 b_{\mu}\big(g^{\mu \nu} \Box+R^{\mu \nu}\big) c_{\nu}+2 b \, \Box \, c-4 b F^{\rho \nu} D_{\rho} c_{\nu}\Big\},
\end{align}
where $b_{\mu}$ and $c_{\mu}$ are vector fields and $b$ and $c$ are scalar fields. From these expression, we can extract the matrices $E$ and $\Omega$ as
\begin{align}
\begin{aligned}
\phi_{n} E_{m}^{n} \phi^{m}=& b_{\mu}\left(R_{~\nu}^{\mu}\right) b^{\nu}+c_{\mu}\left(R_{~\nu}^{\mu}\right) c^{\nu} ,
\\
\phi_{n}\left(\Omega_{\alpha \beta}\right)_{m}^{n} \phi^{m}=& b_{\mu}\left(R_{~\nu \alpha \beta}^{\mu}\right) b^{\nu}+c_{\mu}\left(R_{~\nu \alpha \beta}^{\mu}\right) c^{\nu}-\frac{1}{2}\left(b_{\mu}-i c_{\mu}\right)\left(D^{\mu} F_{\alpha \beta}\right)(b+i c)
\\
&+\frac{1}{2}(b+i c)\left(D_{\nu} F_{\alpha \beta}\right)\left(b^{\nu}-i c^{\nu}\right),
\end{aligned}
\label{eqn: bosonghost}
\end{align}
The result for the Seeley-DeWitt coefficient is
\be
a_4^\text{ghosts}(x) = {13\/36} E_4 - {1\/4} W^2-{3\/4}R^2~,
\ee
where we have already included here the minus sign due to the opposite statistics.
\subsection{Logarithmic correction}
Adding the above results, the heat kernel for Einstein-Maxwell theory takes the form,
\be\label{a4B}
(4\pi)^2 a_4^{\text{B}}(x) = -{53\/45}E_4 + {137\/60} W^2 - {13\/36} R^2 ~.
\ee
We can read off the coefficients from \eqref{a4general} to be
\be
a_\text{E} = {53\/45},\qquad c = {137\/60},\qquad b_1 = -{13\/36}~,\qquad b_2= 0.
\ee
We note that in the flat limit $\ell\to+\infty$, the coefficients $a$ and $c$ match the known flat space computations in \cite{Bhattacharyya:2012wz, Charles:2015eha, Karan:2019gyn} while the coefficients $b_1$ and $b_2$ are unique to AdS. We can also note that the result does not explicitly depend on $F^{\mu\nu}$ as $b_2=0$. This implies that the final result is invariant under electric-magnetic duality. This property has also been observed in the asymptotically flat case in \cite{Bhattacharyya:2012wz,Karan:2019gyn}. Another sanity check is to consider the truncation of the terms involving $F_{\mu\nu}$ in the fluctuations. Then \eqref{a4B} reduces to the neutral limit which was first obtained in \cite{Christensen:1979iy}; we show this in detail in Appendix \ref{sec:neutrallimit}.
We can evaluate this result for the BPS Kerr-Newman solution described in section \ref{subsubsec: AdSKNBPS}. The result is
\begin{eqnarray}
C_\mathrm{local} \= -{212\/45}+ {\ell_2^2\/120 a\ell^2(\ell^2-a^2)} \left(629 a^3 - 579 a^2\ell + 3699 a\ell^2 \vphantom{{(a+\ell)^4\/\sqrt{a\ell}}}\right. \-
&&\hspace{6cm} \left. + 411 \ell^3 - 411 {(a+\ell)^4\/\sqrt{a\ell}}\,\mathrm{arctan}(\sqrt{a/\ell}) \right).
\end{eqnarray}
\section{Minimal ${\cal N}=2$ gauged supergravity}
\label{Sec:N2Sugra}
We now consider the simplest supersymmetric theory with a consistent truncation to Einstein-Maxwell theory with a negative cosmological constant. This is minimal ${\cal N}=2$ gauged supergravity \cite{Freedman:1976aw,Fradkin:1976xz,Romans:1991nq,Caldarelli:2003pb}. In this section, we compute the logarithmic corrections in this theory. We find that in contrast to flat space, the logarithmic correction for BPS black holes is not topological. The results of this section were obtained using a Mathematica algorithm described in Appendix \ref{app:algo} which we have made publicly available \cite{github}.
Ultimately, we would like to compute the logarithmic correction for AdS black holes where a microscopic counting is available. Although the techniques of this paper are applicable in those cases, the computations are more involved due to additional matter multiplets.
\subsection{Fermionic fluctuations}
The bosonic Lagrangian of minimal $\mathcal{N}=2$ supergravity is the same as \eqref{EM action}. Hence, the result of the previous section can be applied and gives \eqref{a4B}. In this section, we will compute the contribution from the fermions. In the conventions of \cite{Caldarelli:2003pb}, the fermionic Lagrangian takes the form
\begin{eqnarray}
\mathcal{L}_f \= {1\/2} \bar{\psi}_\mu\gamma^{\mu\nu\rho} D_\nu\psi_\rho + {i\/4} F^{\mu\nu} \bar{\psi}_\rho \gamma_{\mu} \gamma^{\rho\sigma} \gamma_{\nu} \psi_\sigma -{1\/2\ell} \bar{\psi}_\mu \gamma^{\mu\nu} \psi_\nu~,
\end{eqnarray}
where the gravitino $\psi_\mu$ is a Dirac spin-${3\/2}$ field with charge one, in units of the AdS$_4$ length, under the $\mathrm{U}(1)$ gauge symmetry. The action of the covariant derivative is
\be
D_\mu \psi_\nu = \nabla_\mu \psi_\nu - {i\/\ell} A_\mu \psi_\nu~.
\ee
We now put the fermionic Lagrangian in a form suitable for the heat kernel computation. Firstly, we fix the gauge by adding the following gauge-fixing Lagrangian
\be
\label{eqn: gravitinigf}
\mathcal{L}_\mathrm{g.f.}=- {1\/4} (\bar{\psi}_{\mu} \gamma^\mu) \gamma^\nu D_\nu (\gamma^\rho \psi_\rho)~.
\ee
This choice is convenient because after we perform the field redefinition
\be
\psi_\mu = \sqrt{2}\left( \chi_\mu - {1\/2}\gamma_\mu \gamma^\nu \chi_\nu\right)~,
\ee
we obtain a simpler kinetic term. The resulting Lagrangian takes the form
\begin{eqnarray}
\mathcal{L}_f \= g^{{\mu\nu}}\bar{\chi}_\mu\gamma^{\rho} D_\rho\chi_\nu + {i\/2} F^{\mu\nu} \bar{\chi}_\rho \gamma_{\mu} \gamma^{\rho\sigma} \gamma_{\nu} \chi_\sigma -{1\/\ell} \bar{\chi}_\mu \gamma^{\mu\nu} \chi_\nu~.
\end{eqnarray}
More details on this computation are given in Appendix \ref{app:fermions}. We then write the Dirac spinor as
\be
\chi^\mu = \chi^\mu_1+i \chi^\mu_2~,
\ee
where $\chi_1$ and $\chi_2$ are Majorana spin-${3\/2}$ spinors \footnote{For definiteness, we can use here the ``really real'' representation of the Clifford algebra in which the Majorana condition is just the reality condition \cite{Freedman:2012zz}.}. We use the label $A=1,2$ for the two Majorana spinors. The covariant derivative acting on $\chi_A^\mu$ takes the form
\be
D_\mu \chi^\nu_A = \left(\delta_{AB} \nabla^\mu +{1\/\ell}\,\varepsilon_{AB} A_\mu\right) \chi^\nu_B~,
\ee
where $\varepsilon_{AB}$ is the antisymmetric symbol with $\varepsilon_{12}=1$. This is necessary if we want to preserve the reality condition. It is useful to use the Majorana flip identities \eqref{MajoranaFlip} reviewed in Appendix \ref{app:fermions}. The computation detailed there leads to the Lagrangian in terms of Majorana spinors
\begin{eqnarray}\label{LagMajoranas}
\mathcal{L}_f \= \delta_{AB} g_{{\mu\nu}}\bar{\chi}^\mu_A\gamma^{\rho} D_\rho\chi^\nu_B -{1\/2}\varepsilon_{AB} F^{\mu\nu} \bar{\chi}^\rho_A \gamma^{\mu} \gamma_{\rho\sigma} \gamma^{\nu} \chi^\sigma_B -{1\/\ell}\delta_{AB} \bar{\chi}^\mu_A \gamma_{\mu\nu} \chi^\nu_B~.
\end{eqnarray}
Finally, we reinterpret this Lagrangian as being a Euclidean Lagrangian in which $\chi_\mu^A$ are Euclidean spinors satisfying $\bar\chi_\mu^A= (\chi_\mu^A)^\dagger$. This Lagrangian can then be used in the algorithm to obtain the result for the heat kernel. Note that we can question the validity of the Wick rotation here because Majorana spinors actually do not exist in four Euclidean dimensions. This can be addressed by using symplectic Majorana spinors. We find, however, that this procedure actually gives the same result as the naive Wick rotation.
\subsubsection{Symplectic Majoranas}
The Lagrangian \eqref{LagMajoranas} is written in terms of Majorana spinors in $(1,3)$ signature. We would like to Wick rotate this Lagrangian to $(0,4)$ signature. As mentioned above, this appears problematic because Majorana spinors do not exist in $(0,4)$ signature. A better approach is to use symplectic Majorana spinors which exist in both $(1,3)$ and $(0,4)$ signature \cite{deWit:2017cle}.
It is shown in \cite{Cortes:2003zd} that one can map Majorana spinors $\chi_A^\mu$ to symplectic Majorana spinors $\lambda_A^\mu$ using
\begin{eqnarray}
\chi_1^\mu \= {1\/2}(\lambda_1^\mu +\gamma^5 \lambda_2^\mu)~,\-
\chi_2^\mu \= {i\/2}(\lambda_1^\mu -\gamma^5 \lambda_2^\mu)~.
\end{eqnarray}
This allows us to write the Lagrangian in terms of $\lambda_A^\mu$. We find that the two flavors actually decouple as
\be
\mathcal{L}_f = \mathcal{L}_1+\mathcal{L}_2,
\ee
with
\begin{eqnarray}
\mathcal{L}_1 \= g_{\mu\nu} \bar{\la}_1^\mu\gamma^\rho(\nabla_\rho + i\ell^{-1} A_\rho)\lambda_1^\nu - {i\/2} F^{\mu\nu} \bar{\la}_1^\rho \gamma_{\mu} \gamma_{\rho\sigma} \gamma_{\nu} \lambda_1^\sigma -{1\/\ell} \bar{\la}_1^\mu\gamma_{\mu\nu} \lambda_1^\nu~,\\
\mathcal{L}_2 \= g_{\mu\nu} \bar{\la}_2^\mu\gamma^\rho(\nabla_\rho - i\ell^{-1} A_\rho)\lambda_2^\nu - {i\/2} F^{\mu\nu} \bar{\la}_2^\rho \gamma_{\mu} \gamma_{\rho\sigma} \gamma_{\nu} \lambda_2^\sigma +{1\/\ell} \bar{\la}_2^\mu\gamma_{\mu\nu} \lambda_2^\nu~.
\end{eqnarray}
The Wick rotation is done by reinterpreting $\lambda_A^\mu$ as symplectic Majorana spinors in $(0,4)$ signature with $\bar{\la}_A^\mu = (\lambda_A^\mu)^\dagger$. This can then be used in the algorithm, described in Appendix \eqref{app:algo}, to compute the heat kernel\footnote{The contribution to the heat kernel of $\mathcal{L}_1$ and $\mathcal{L}_2$ are equal because the two Lagrangian differs by $\ell\rightarrow-\ell$ and the four-derivative terms are invariant under that change.}. We obtain the gravitini contribution
\be
(4\pi)^2 a_4^\text{F,gravitini}(x) = {139\/90} E_4-{32\/15}W^2 -{2\/9}R^2 + {8\/9} RF_{\mu\nu} F^{\mu\nu}~.
\ee
Note that the result is ultimately the same as what we obtain naively by using directly the Majorana Lagrangian \eqref{LagMajoranas} in the algorithm.
\subsubsection{Ghost contribution}
The gauge-fixing of the gravitini leads to three pairs of ghosts. The gauge condition $\gamma^\mu\psi_\mu^A=0$ leads to a $bc$ ghost Lagrangian given as
\be
\mathcal{L}_{bc}= \delta_{AB}\bar{b}_A \gamma_\mu\delta_c\psi^\mu_B,
\ee
where $\delta_c\psi^\mu$ is the supersymmetry transformation with parameter $c$. This gives
\be
\mathcal{L}_{bc} =\delta_{AB} \bar{b}_A\left(\gamma^\mu D_\mu + {2\/\ell}\right) c_B~.
\ee
We can get a diagonal kinetic term by a suitable redefinition. This leads to two pairs of ghosts which are charged spin-${1\/2}$ fermions with mass $m={2\/\ell}$. In addition, implementing the gauge-fixing term in the path integral leads to an additional pair of massless charged ghosts, giving us a ghost for ghost phenomena \cite{Nielsen:1978mp,Siegel:1980jj}. Details are given in Appendix \ref{app:fermions}.
The total heat kernel of the fermionic ghosts is
\be
(4\pi)^2a_4^\mathrm{F,ghosts}(x) = {11\/120}E_4 -{3\/20}W^2+{2\/9}R^2 - {1\/6} R F_{\mu\nu} F^{\mu\nu} ~,
\ee
where we have already included the minus sign due to the opposite statistics of ghosts.
\subsection{Logarithmic correction}
The total fermionic contribution is
\be\label{ResGravitini}
(4\pi)^2a_4^\mathrm{F}(x) = {589\/360}E_4 -{137\/60}W^2 + {13\/18} R F_{\mu\nu} F^{\mu\nu}~.
\ee
Adding this to \eqref{a4B} from the bosonic computation, we obtain for minimal $\mathcal{N}=2$ supergravity
\be
(4\pi)^2a_4(x) = {11\/24} E_4 -{13\/36} R^2 + {13\/18} R F_{\mu\nu} F^{\mu\nu}~.
\ee
We find that the full result is not only given by the Euler term as other four-derivative invariants are present. This indicates that the logarithmic correction is non-universal. We note that the $ W^2$ term, which would give another non-universal contribution, does cancel between bosons and fermions. This is expected from the flat space result \cite{Charles:2015eha}, which we recover in the flat limit.
\subsubsection{Evaluation}
We can evaluate the heat kernel coefficient on the backgrounds summarized in section \ref{Sec:Backgrounds}. For the non-extremal Kerr-Newman black hole, we get
\be\label{ClocalKNres}
C_\mathrm{local} = {11\/6} + {26 \left[r_+\beta (r_+^4-\ell^2(a^2+q_m^2) ) - \pi\ell^2(r_+^4-a^4) \right]\/3\pi\ell^2 (\ell^2-a^2)(a^2+r_+^2)}~.
\ee
We note that the fermionic contribution breaks electromagnetic duality as it generates a non-zero $b_2$. This is reflected by the dependence in the magnetic charge $q_m$ in the above expression.
The result for extremal Kerr-Newman takes the form
\be\label{evalClocalextKN}
C_\mathrm{local} = {11\/6} + {26 \ell_2^2 \left[ (a^2(a+\ell)+r_0^2 (3a-\ell))(a^2(a-\ell)+r_0^2 (3a+\ell))+ 2 \ell^2 q_m^2(r_0^2-a^2) \right]\/3\ell^2 (\ell^2-a^2)(a^2+r_0^2)^2},
\ee
where here the AdS$_2$ radius is $\ell_2 = \ell\sqrt{a^2+r_0^2\/a^2+\ell^2+6r_0^2}$. As explained in section \ref{sec:KN}, this is obtained by either taking the extremal limit of \eqref{ClocalKNres} or by doing the computation in the near horizon geometry.
We are particularly interested in evaluating the logarithmic corrections on BPS black holes. Rotation is necessary to have a regular BPS background in minimal gauged supergravity as the extremal AdS-Reissner-Nordstr\"om is singular in the BPS limit \cite{Caldarelli:1999xj, Caldarelli:1998hg, Kostelecky:1995ei}. We obtain the BPS result by imposing the BPS constraints $r_0=\sqrt{a\ell}$ and $q_m=0$ on \eqref{evalClocalextKN}. The contribution is
\be\label{ClocalBPSres}
C_\mathrm{local} = {11\/6} -{26\/3}{a(\ell^2-4\ell-a^2)\/(\ell-a)(a^2+ 6 a\ell + \ell^2)},
\ee
where the first term comes from the topological Euler term and the second term comes $R^2$ and $RF^2$ and constitute the non-topological piece. We discuss the significance of this non-topological term in the next subsection.
\subsection{Implications}
We shall now comment on the non-topological nature of the logarithmic correction. For the flat space Kerr-Newman black hole, the heat kernel $a_4(x)$ is the sum of only two terms: the Euler term and the Weyl squared term. Although $W^2=0$ for extremal non-rotating black holes in flat space, it is non-zero for extremal rotating black holes. It was shown in \cite{Charles:2015eha,Larsen:2018cts} that supersymmetry ensures that the coefficient $c$ multiplying $W^2$ actually cancels. This shows that supersymmetry makes the logarithmic correction topological in ungauged supergravity.
In AdS$_4$, the $ W^2$ term never vanishes in the near horizon geometry (even without rotation) and there are two additional terms. We also obtain that supersymmetry ensures $c=0$ due to cancellations between bosons and fermions. This could be expected from the flat space results of \cite{Charles:2015eha} which we reproduce in the flat limit $\ell\to+\infty$. Hence, even with a negative cosmological constant, supersymmetry makes the logarithmic correction less complicated as it removes the non-topological term $W^2$. This term is a complicated function of the black hole parameters. For the BPS Kerr-Newman AdS black hole, it takes the form
\be\label{Weyl2expr}
{1\/(4\pi)^2}\int d^2 x\sqrt{g}\,W^2 ={3\ell_2^2\/2 a\ell^2(\ell^2-a^2)} \left( (\ell-a)(\ell^2+10 a \ell + a^2) - {(a+\ell)^4 \/\sqrt{a\ell}} \mathrm{arctan}(\sqrt{a/\ell}) \right) ~.
\ee
The two other four-derivative, $R^2$ and $RF^2$, terms do not cancel so supersymmetry does not imply that the logarithmic corrections are topological. However, we see that the BPS result \eqref{ClocalBPSres} is still a simpler function of $a$ and $\ell$ as it has $c=0$. It is a rational function rather than a transcendental one.
It is natural to expect topological logarithmic corrections in the UV given the known examples of microscopic counting of black hole entropy \cite{Mandal:2010cj,Sen:2014aja,Belin:2016knb,Liu:2017vll,Benini:2019dyp,Gang:2019uay,PandoZayas:2020iqr,Liu:2018bac}. This is also automatic if the 4d theory comes from an odd-dimensional theory by Kaluza-Klein reduction because $C_\mathrm{local}=0$ in odd dimensions. Hence, the logarithmic correction can be a useful probe of whether a low-energy effective theory can have a UV completion. The idea is that from a bottom-up perspective, we should prefer low-energy theories which have topological logarithmic correction. This is only possible if $c=b_1=b_2=0$ which is a rather strong constraint on the low-energy Lagrangian, analogous to anomaly cancellation.
\section{Discussion}\label{Sec:Discussion}
In this paper we have computed the logarithmic corrections to the entropy of black holes in minimal gauged supergravity using the four dimensional heat kernel expansion. The inclusion of a negative cosmological constant leads to new features compared to the case of asymptotically flat black holes. In the especially interesting case of BPS black holes, the logarithmic corrections present a richer structure and can be non-topological.
The original explicit logarithmic corrections performed for asymptotically AdS$_4\times S^7$ black holes based on Sen's entropy function formalism, using the near horizon geometry, did not agree with the field theory computations \cite{Liu:2017vll,Jeon:2017aif}. It was only in \cite{Liu:2017vbl} that agreement was found by considering the full geometry. The results of this manuscript clarify that the difference between the two approaches comes from the contributions of the zero modes which are indeed different in the two geometries. Namely, we have shown that for extremal black holes the {\it local} contribution to the logarithmic correction, $C_\mathrm{local}$, is the same when computed either from the full AdS$_4$ asymptotic region or for the near horizon geometry. This result elucidates the question of where the degrees of freedom responsible for the quantum entropy live.
For the BPS Kerr-Newman black hole in minimally gauged supergravity, we have found that the logarithmic correction, given in \eqref{ClocalBPSres}, is non-topological. To obtain this result, we have used holographic renormalization to regularize the divergent volume integrals. This appears to be the right prescription as, for example, it gives the correct counterterm to obtain the Euler characteristic when integrating the Euler density, see Appendix \ref{app:Euler} for more details.
The non-topological nature of the logarithmic corrections suggest that they might contain more information than the flat space counterpart, providing a wider ``infrared window into the microstates''. Moreover, this non-topological nature is interesting because for all the available examples of microscopic counting and for BPS black holes in flat space \cite{Charles:2015eha}, the logarithmic correction is always topological, {\it i.e.}, the coefficient of the logarithm is a pure number. In minimal gauged supergravity, we find that it is instead a rather non-trivial function of the black hole parameters.
It is illuminating to compare this result to recent investigations using supergravity localization \cite{Hristov:2018lod,Hristov:2019xku,Hristov:2021zai}. In \cite{Hristov:2021zai}, the general structure of the logarithmic correction of 4d $\mathcal{N}=2$ gauged supergravity on BPS backgrounds was studied using index theory. It was shown that the universal piece coming from the Euler term arises from the application of the Atiyah-Singer theorem to an appropriate supercharge. We surmise that the non-universal piece that we obtained should be interpreted as the contribution from the $\eta$ invariant, not considered in \cite{Hristov:2021zai}, which is a non-topological correction due to the presence of a boundary \cite{atiyah_patodi_singer_1975}. Supergravity localization has the potential of ultimately providing the full quantum entropy of the black holes and it would be fruitful to test it against one-loop supergravity results such as ours.
Our work also clarifies the role of supersymmetry. One could think that supersymmetry guarantees that the logarithmic corrections are topological. This is suggested by the index theory interpretation \cite{Hristov:2021zai} and by results in flat spacetime \cite{Charles:2015eha,Charles:2017dbr}. In this paper, we have seen that supersymmetry is not enough to make the other terms cancel which shows the logarithmic corrections can be non-topological for BPS black holes. Nevertheless, supersymmetry does play a role in making $C_\mathrm{local}$ simpler as it cancels the coefficient, $c$, of the Weyl squared term \eqref{Weyl2expr}. This simplifies the logarithmic correction for the BPS black hole as its dependence on $a$ and $\ell$ becomes rational instead of transcendental.
We might hope to use the topological nature of logarithmic corrections as a criterion for a low-energy theory to admit a UV completion. In the available examples of microscopic countings, the logarithmic correction is indeed topological \cite{Mandal:2010cj,Sen:2014aja,Belin:2016knb,Liu:2017vll,Benini:2019dyp,Gang:2019uay,PandoZayas:2020iqr,Liu:2018bac}. Such a criteria would greatly constrain effective supergravity theories as it gives rather stringent conditions similar to anomaly cancellation. Note that in odd dimensions, the logarithmic correction is automatically topological because the local contribution is trivially zero. So if the four-dimensional theory comes from the dimensional reduction of an odd-dimensional theory, such as 11d supergravity, the logarithmic correction computed in 4d has to be topological. For ten dimensional theories, there is a local contribution in 10d and, as a result, the topological criterion should be much more constraining.
We have obtained the logarithmic correction for the simplest gauged supergravity in four dimensions. Our goal is to grow this direction towards more interesting theories and to relate our results to other approaches such as the computations performed in eleven-dimensional supergravity \cite{Liu:2017vbl,Gang:2019uay,PandoZayas:2020iqr,Benini:2019dyp}. It should be possible to perform the same computation in the gauged $\mathrm{U}(1)^4$ supergravity which comes from eleven dimensional supergravity on AdS$_4\times S^7$. Similarly, the logarithmic correction to the entropy of black holes in AdS$_4\times SE_7$ has been computed both in field theory and supergravity for a large class of Sasaki-Einstein seven-dimensional manifolds \cite{PandoZayas:2020iqr}. In both these cases, the topological nature follows from the fact that the parent theory is odd-dimensional. It would be interesting to see explicitly how this is realized from a four-dimensional perspective.
More challenging would be the cases where the AdS$_4$ black holes are embedded in ten-dimensional theories such as massive IIA supergravity. A matching of the Bekenstein-Hawking entropy at leading order was presented in \cite{Azzurli:2017kxo,Hosseini:2017fjo,Benini:2017oxt}. The available sub-leading, microscopic analysis confirms the topological nature of the logarithmic term \cite{Liu:2018bac}. However, the supergravity computations need to be in agreement with the nontrivial nature of $C_{\rm local}$. We hope to address some of these issues in the future.
\section*{Acknowledgements}
We would like to thank Rodrigo de Le\'on Ard\'on and Alberto Faraggi for discussions. We are particularly grateful to Kiril Hristov and Valentin Reys for engaging and clarifying correspondence and to Alejandra Castro and Ashoke Sen for insightful comments on the draft.
This work was supported in part by the U.S. Department of Energy under grant DE-SC0007859. M.D. was supported by the NSF Graduate Research Fellowship Program under NSF Grant Number: DGE 1256260.
VG acknowledges the postdoctoral program at ICTS for funding support through the Department of Atomic Energy, Government of India, under project no. RTI4001.
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{"url":"https:\/\/codereview.stackexchange.com\/questions\/14723\/validation-default-value","text":"# Validation default value\n\nI want to check if all items of a list meet a set of criteria:\n\nbool AreValid(list<string> vals)\n{\nbool allValid=true;\n\nforeach(string val in vals)\n{\nif(!condition1)\n...\nallValid=false;\nelse if(!condition2)\n...\nallValid=false;\n}\n\nreturn allValid;\n\n}\n\n\nThe reason I can't use LINQ All() is that for each failed condition I'm doing some job.\n\nis it safe to set allValid as true by default?\n\n\u2022 Why don't you just return false if any of the conditions fail? \u2013\u00a0Jeff Vanzella Aug 16 '12 at 1:40\n\nYes, it is fine - I mean it definitelly works and there is nothing unsafe in assigning default value of True to allValid.\n\nMy concern is the design. The name of your method AreValid suggest that it is just checking if objects meet specific criteria. You've mentioned that for all those objects that do not meet the criteria there is something being done on them. That doesn't seem to be the very greatest approach. I would suggest you consider separating checking if objects meets condition and changing state of the objects. It will make your code simpler and easier to read and understand to others. Is changing object state really what you expect from the method named AreValid? It would be quite a surprise for me if I found similar method in the code base.\n\nThe other thing is that you could use LiNQ and check for all the objects that do not meet the criteria and then perform some actions on those objects.\n\nIt's safe. A few things to consider:\n\n1. When the list is empty you might want to return false.\n\n2. Consider using an abstract data type (instead of string) which stores your data (the string) and move the validation logic to there.\n\n3. As Jeff Vanzella already suggested, you can return false immediately when you now that a value is not valid. It's faster and easier to read:\n\nbool AreValid(list<string> vals)\n{\nforeach(string val in vals)\n{\nif(!condition1)\n...\nreturn false;\nif(!condition2)\n...\nreturn false;\n}\n\nreturn true;\n\n}\n\n\nNote that you don't need the else keyword anymore.","date":"2020-03-29 10:00:22","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\": 0.2521524131298065, \"perplexity\": 1184.7096328126997}, \"config\": {\"markdown_headings\": true, \"markdown_code\": true, \"boilerplate_config\": {\"ratio_threshold\": 0.18, \"absolute_threshold\": 10, \"end_threshold\": 15, \"enable\": true}, \"remove_buttons\": true, \"remove_image_figures\": true, \"remove_link_clusters\": true, \"table_config\": {\"min_rows\": 2, \"min_cols\": 3, \"format\": \"plain\"}, \"remove_chinese\": true, \"remove_edit_buttons\": true, \"extract_latex\": true}, \"warc_path\": \"s3:\/\/commoncrawl\/crawl-data\/CC-MAIN-2020-16\/segments\/1585370494064.21\/warc\/CC-MAIN-20200329074745-20200329104745-00551.warc.gz\"}"} | null | null |
\section{Introduction}
In addition to producing kilohertz gravitational waves (GW) detectable
by ground-based interferometers \citep{Abadie_2010_LIGO}, compact
object (CO) mergers involving a neutron star (NS) are likely to emit a
variety of electromagnetic (EM) signals. Immediately post-merger, the
accretion of disrupted NS material onto the central black hole (BH) or
hypermassive NS may drive a short gamma ray burst
\citep{Paczynski_1986_GRB,Eichler_1989_sGRBs,Narayan_1992_sGRBs}. Mergers
may also produce optical/infrared transients
\citep{Li_Paz_1998,Metzger_2010,Roberts_2011,Barnes_2013} powered by
the radioactive decay of heavy elements synthesized via rapid neutron
capture~\citep[the \emph{r}-process;][]{arnould.goriely.takahashi:2007}. The \emph{r}-process~
is expected to operate in material ejected from the system
dynamically~\citep{Lattimer_1974,Lattimer_1976,Freiburghaus_1999,Korobkin_NSM_rp,Rosswog_1999,Goriely_2011},
or unbound from a remnant accretion
disk~\citep{Fernandez_Metz_2013_diskOutlfows,Perego_2014_vWinds,Just_2015_torusNucleo}.
On much longer timescales, the interaction of the ejecta with the
interstellar medium will generate a radio
signal~\citep{Nakar_Piran_2011_radio}.
Observing an EM counterpart will enhance the science returns of a GW
detection \citep{Metzger_Berger_2012} by identifying the host galaxy
and the position of the merger within the host
\citep{Nissanke_2013,Kasliwal_Niss_2014,Holz_Hughes_2005_GWEM,
Dalal_etal_2006}, constraining the neutron star equation of state
\citep{Bauswein_2013,Hotokezaka_2013,Bauswein_2015}, and confirming
low signal-to-noise GW events \citep{Kochanek_1993,
Harry_Fairhurst_followup}. Among possible counterparts, the radioactive
transients---known as ``kilonovae''---are especially
promising. Kilonova emission is roughly isotropic
\citep{Roberts_2011,Bauswein_2013}, and peaks on timescales of
days--weeks post-merger
\citep{Barnes_2013,Tanaka_Hotok_rpOps,Grossman_2014_kNe}, making it
ideal for EM follow-up of a GW trigger. Because kilonovae derive
their energy from radioactive decay, they probe nucleosynthesis in the
merger in a way other counterparts cannot, and may therefore constrain
the astrophysical origin of \emph{r}-process\ element production.
Accurate models of kilonova photometry are crucial for dual detection
efforts. Unfortunately, the exotic composition of the heavy element
ejecta, and uncertainties in \emph{r}-process\ nucleosynthesis and decay, pose
challenges to radiation transport simulations required to build these
models. Recent work \citep{Kasen_2013_AS} clarified the opacity of
\emph{r}-process\ material, reducing a major uncertainty in kilonova radiation
transport simulations, but other key inputs remain relatively
unconstrained.
Any rigorous
kilonova model must address the following aspects of radioactivity: \emph{i})
the total amount of radioactive energy released; \emph{ii}) the
decay channels that dominate the energy production during different
phases of kilonova evolution; and \emph{iii}) the efficiency with which
suprathermal radioactive decay products--- \ensuremath{\beta}-particles,
\ensuremath{\alpha}-particles, \ensuremath{\gamma}-rays, and fission fragments---transfer their energy
to the thermal background. Once thermalized, the energy is
re-radiated as thermal emission, powering the kilonova light
curve.
Although thermalization determines the kilonova's luminosity,
and is thus essential for light curve modeling, no detailed
calculation of thermalization efficiencies has been attempted.
\citet{Metzger_2010} presented analytic estimates of thermalization,
but focused on timescales shorter than those now believed to
characterize kilonova light curves. \citet{Hotokezaka_2015_heating}
studied \ensuremath{\gamma}-ray deposition in kilonovae, but did not investigate the
thermalization of charged particles, which carry a large fraction
of the radioactive energy.
Modeling the thermalization of \emph{r}-process\ decay energy in the kilonova ejecta is
challenging. Thermalization rates are sensitive to the ejecta's mass,
velocity, and composition, as well as its magnetic field structure,
which has not been definitively determined by magnetohydrodynamic
simulations. The broad range of elements synthesized by the \emph{r}-process, and
the often unknown properties of the heaviest of those elements,
complicates the situation, as does the complexity of the net emission
spectrum, which is a sum over several decay chains, each evolving on
its own timescale.
This paper addresses the issues outlined above, with special emphasis
on the key physical processes influencing the
thermalization of \emph{r}-process\ decay products in kilonovae. In \S \ref{sec:ejMod}, we describe our ejecta model and
its uncertainties. Section \ref{sec:thermRates} defines energy loss
rates for \ensuremath{\beta}-particles, $\alpha$-particles, $\gamma$-rays, and
fission fragments, and explores their sensitivity to ejecta
parameters. Analytic estimates and analytic expressions for
thermalization efficiencies are developed in \S
\ref{sec:analytics}. In \S \ref{sec:results}, we present detailed numerical calculations of
time-dependent thermalization efficiencies $f(t)$ for individual
species and
for the system as a whole, and discuss the sensitivity of
$f(t)$ to properties of the ejecta. Finally, \S \ref{sec:LCs}
evaluates the effect of thermalization on kilonova light curves, and
uses improved light curve models to estimate the mass ejected by the claimed
kilonova associated with GRB 130603B.
\section{Properties of the kilonova ejecta}
\label{sec:ejMod}
\subsection{Ejecta model}
Predictions of kilonova outflows vary, due to natural diversity in the
merging systems (e.g. different mass ratios, BHNS v. NS$^2$) and
uncertainties in the NS EOS. Recent hydrodynamic simulations
\citep{Bauswein_2013,Hotokezaka_2013, Kyutoku_2015_massEj,
Sekiguchi_2105_massEj} suggest that mergers dynamically eject
between $\sim 10^{-4}$ and a few $\times 10^{-2}$ \ensuremath{M_{\odot}}\ of material, with bulk velocities of a few tenths the speed of
light. Additional material ($\sim 10^{-3}-10^{-2} \ensuremath{M_{\odot}}$) can exit the
system at slightly lower velocities ($0.05c - 0.1c$) as a wind from an accretion torus
\citep{Fernandez_Metz_2013_diskOutlfows,Perego_2014_vWinds,
Fernandez_etal_2015_BHspin,Just_2015_torusNucleo}.
We adopt as our fiducial model a system with \ensuremath{M_{\rm ej}} = $5 \times 10^{-3}$
\ensuremath{M_{\odot}}\ and \ensuremath{v_{\rm ej}} = $0.2c$, where \ensuremath{v_{\rm ej}}\ is defined in terms of the
explosion kinetic energy, $E_{\rm k} = \ensuremath{M_{\rm ej}}\ensuremath{v_{\rm ej}}^2/2$. Since denser
ejecta configurations thermalize more efficiently than diffuse
systems, we vary these parameters over the ranges
$\ensuremath{M_{\rm ej}}/\ensuremath{M_{\odot}} \in [10^{-3}, 5\times 10^{-2}]$ and
$\ensuremath{v_{\rm ej}}/c \in [0.1, 0.3]$.
We assume the ejecta is spherical and expanding homologously, and that
the density profile follows a broken power-law, declining with
velocity coordinate $v = r/t$ as $v^{-\delta}$ in the inner regions of
the ejecta, and $v^{-n}, \: n > \delta$, in the outer regions. We set
$\delta=1$ and $n=10$. \citet{Barnes_2013} provides a complete
mathematical description of the density profile.
\subsection{Magnetic fields}
\label{subsec:Bej}
Kilonova ejecta contain a residual magnetic field, either inherited
directly from the parent neutron star(s), or seeded by amplified
fields produced by turbulence during the merger or in the resultant accretion disk
\citep{Kiuchi_2014_BAmp,Kiuchi_2015_BAmp}.
Though weakened by expansion, the fields remain strong enough to
influence charged particle motion. In a sufficiently strong field,
charged particles have Larmor radii smaller than the coherence length
of the magnetic field, and their motion is confined to ``flux tubes''
that trace the field lines.
If magnetic flux is frozen into the homologously-expanding ejecta, the field strength is related to the ejecta radius $R_{\rm ej} = \ensuremath{v_{\rm ej}} t$ by
\begin{align}
B(t) \approx \frac{B_{0} R_{0}^2}{R_{\rm ej}^2}
\approx 3.7 \times 10^{-6} B_{12} R_6^2 v_2^{-2} t_{\rm d}^{-2} \mbox{ G},
\end{align}
where $v_2 = v/0.2c$, $t_{\rm d}$ is the elapsed time in days, and
$B_0$ and $R_0$ are the magnetic field and radius at the time of mass
ejection. The quantities $B_{12} = B_0/10^{12}~$G and
$R_6 = R_0/10^6~$cm have been scaled to typical values;
$R_0 \approx 10^6 - 10^7$ cm is characteristic of the size of NSs or
the post-merger accretion disk, and $B_0$ may range from
$10^9 - 10^{15}$~G, depending on the initial NS fields and the
efficiency of magnetic field amplification.
A relativistic particle of mass $m$, charge $q$, kinetic energy $E$,
and velocity $v$ in a magnetic field $B$ has a maximum Larmor radius
(when $\mathbf{v} \perp \mathbf{B}$) of
\begin{equation*}
r_{\rm L,max} = \frac{(E + mc^2) v}{qBc},
\end{equation*}
Assuming typical emission energies of $E_{\ensuremath{\beta},0} = 0.5$ MeV for
\ensuremath{\beta}-particles, $E_{\ensuremath{\alpha},0} = 10$~MeV for \ensuremath{\alpha}-particles, and
$E_{\rm ff,0} = 150$ MeV for fission fragments, and assuming fission
fragments are singly ionized and have masses of $\sim 130~m_{\rm u}$, with $m_{\rm u}$ the
atomic mass unit, the Larmor radii are
\begin{align}
\frac{r_{\rm L,max}(t)}{R_{\rm ej}(t)} =
\begin{cases}
1.5 \times 10^{-6}~ v_2 t_{\rm d} B_{12}^{-1}R_6^{-2} &\mbox{ \ensuremath{\beta}-particles} \\
2.4 \times 10^{-4}~ v_2 t_{\rm d} B_{12}^{-1}R_6^{-2} &\mbox{ \ensuremath{\alpha}-particles} \\
1.0 \times 10^{-2}~ v_2 t_{\rm d} B_{12}^{-1}R_6^{-2} &\mbox{ fiss. fragments}.
\end{cases}
\end{align}
We will adopt the flux tube approximation for all particles. This is
clearly appropriate for \ensuremath{\alpha}- and \ensuremath{\beta}-particles, which have
$r_{\rm L}/R_{\rm ej} \ll 1$ for the duration of the kilonova. Fission
fragments may, at later times, have $r_{\rm L}$ large enough that they
jump from field line to field line. We discuss this possibility in
\ref{subsec:fragTrans}, but do not employ models of fission fragment transport beyond the flux tube approximation in this work.
The magnetic field structure determines charged particle
trajectories and so affects thermalization. Radial fields can escort
fast charged particles straight out of the ejecta, reducing
thermalization. In contrast, toroidal or tangled fields trap charged
particles, and so enhance thermalization. We consider three types of
configurations here: radial ($\mathbf{B} \propto \hat{\mathbf{r}}$),
which may be produced by the outward motion of the ejecta ``combing''
out the field lines; toroidal
($\mathbf{B} \propto \hat{\boldsymbol{\phi}}$), perhaps created by the
spiral motion of neutron stars shedding mass through tidal stripping;
and random, which may be generated by turbulent motions in the
material during mass ejection. To model the latter case, we assume
field lines re-orient on a length scale $\lambda R_{\rm ej}$, where
the dimensionless parameter $\lambda < 1$.
\subsection{Composition}
\label{subsec:comp}
The ejecta composition impacts thermalization in two ways. First, it
determines the partition of radioactive energy among different decay
channels, and the energy spectra of the decay products. Second,
it sets the properties of the background material (e.g., isotope mass,
atomic number, and ionization energy) which influence the energy loss
rates of the decay products.
To determine the ejecta composition, we calculate \emph{r}-process\ nucleosynthesis
on a set of smoothed-particle hydrodynamics (SPH) trajectories
extracted from a relativistic simulation of an equal-mass (1.35
\ensuremath{M_{\odot}}-1.35 \ensuremath{M_{\odot}}) NS$^2$ merger \citep{Goriely_2011}. At the start of
the nucleosynthesis calculation, all trajectories had temperatures of
6 GK, densities set by their hydrodynamical histories, compositions
determined by nuclear statistical equilibrium, and initial electron
fractions $Y_{\rm e,0}$ between $1.5 \times 10^{-2}$ and
$5.5 \times 10^{-2}$. These quantities were then evolved according to
the reaction network, which tracks charged particle reactions, neutron
capture, photo-dissociation, \ensuremath{\beta}- and \ensuremath{\alpha}-decay, and fission, for more
than 7300 nuclei. \citep[For a detailed description of the network, see][]{MendozaTemis_etal_rProcess}.
The hydrodynamical model predicts two classes of trajectories that
produce two distinct compositions: ``slow'' trajectories, where all
free neutrons are depleted by neutron-capture, and ``fast''
trajectories, where early rapid expansion precludes the capture of all
neutrons by seed nuclei~(see also: \citealt{Just_2015_torusNucleo,
Goriely_2014_rP_fast,
Metzger_2015_nPrecurs,MendozaTemis_etal_rProcess}). The slow
trajectories comprise $\sim 90$\% of the ejecta, and robustly produce
\emph{r}-process\ elements up to the third peak. The fast ejecta \textit{r}-pattern
is different from the slow component due to the longer neutron-capture
timescale.
In constructing our ejecta model, we assume that at times relevant for thermalization, material from slow trajectories will be located in the ejecta's interior regions (henceforth ``inner ejecta''), while material from the fast trajectories occupies the outer regions (``outer ejecta'').
We sum the trajectories in each
class to construct mass-integrated inner and outer compositions. We
also select a representative case from the set of inner trajectories,
which typifies conditions in the merger ejecta. We use this
representative trajectory to study the details of the radioactivity.
The top panel of Figure \ref{fig:comp} shows the inner ejecta
composition at $t=1$ day, based on neutron-capture and
photodissociation rates computed using the statistical model for
four different nuclear mass models: the Finite Range Droplet Model
(FRDM; \citealt{Moller_1995_FRDM}), the Hartree-Fock-Bogoliubov model
HFB21 \citep{Goriely_HFB21}, the Weizs{\"a}cker-Skyrme model (WS3;
\citealt{Liu_etal_2011_WS3}), and the Duflo-Zucker model with 31
parameters (DZ31; \citealt{Duflo_Zuker_1995_DZ31}). We find that
although the abundance pattern is similar for different mass models,
particularly around $A \sim 130$ due to fission cycling, the position
of the peak at $A\sim 195$ and the abundances for $A \gtrsim 195$ depend on the mass
model. The differences in the translead abundances impact the late time kilonova light curves,
as will be discussed in \S \ref{subsec:lateLC}.
The abundances are also influenced by the electron fraction
$Y_{\rm e,0}$ at the onset of the \emph{r}-process; neutron scarcity (high
$Y_{\rm e,0}$) suppresses the assembly of the heaviest \emph{r}-process\ elements.
Our SPH trajectories are all initially very neutron rich; however,
weak interactions in the aftermath of a merger could raise
$Y_{\rm e, 0}$
substantially~\citep{Wanajo.Sekiguchi.ea:2014,Sekiguchi.Kiuchi.ea:2015,Goriely.Bauswein.ea:2015}.
To explore this effect, we
artificially increased the initial $Y_{\rm e,0}$ of our
representative trajectory from its value of $0.04$, and reran the nuclear reaction network. As
expected, higher initial electron fractions produce fewer heavy
elements (bottom panel of Figure \ref{fig:comp}) and for
$Y_{\rm e,0} \gtrsim 0.2$ the \emph{r}-process\ fails to reach the third peak,
instead producing material with $A \sim 70 - 110$.
\begin{figure}
\includegraphics[width=3.5 in]{f1.pdf}
\caption{ Abundance yields from our nuclear network calculations at $t
= 1$ day. \textit{Top panel:} Mass-integrated abundances from the
``inner'' ejecta ($\sim 90$\% of the ejected mass) for four nuclear
mass models. The \emph{r}-process\ proceeds past the third peak, and strong
fission cycling reduces differences among nuclear mass
models. \textit{Bottom panel:} An illustration of major factors
affecting the final abundances. The red curve shows mass-integrated
abundances for the ``outer'' ejecta, using the FRDM mass
model. Rapid expansion hinders free neutron capture, decreasing
heavy element production, and creating a substantial amount of H as
uncaptured neutrons decay to protons. The cyan curve shows the
abundance yield of a representative inner ejecta trajectory, whose
initial electron fraction has been artificially increased to $Y_{\rm e,0} =
0.25$, leading to limited production of nuclei with $A > 130$.}
\label{fig:comp}
\end{figure}
The ejecta composition evolves with time as neutron-rich isotopes
gradually decay to stability. However, on timescales relevant for
kilonova light curves ($t \sim 0.1 - 10$ days), this evolution is
fairly slow, and driven primarily by \ensuremath{\alpha}- and \ensuremath{\beta}-decays, which do not
dramatically change the abundance-averaged properties of the composition. For the
purpose of calculating energy loss rates, we therefore assume that the
abundance-averaged properties are constant in time,
but vary in space, with the inner 90\% (outer 10\%) of the mass
described by the average abundance pattern of the inner (outer)
trajectories, calculated at $t=1$ day using the FRDM mass model. The
outer ejecta differs from the inner ejecta primarily in its high
abundance of hydrogen, produced by the decay of remnant free neutrons to protons.
\subsection{Radioactivity}
\label{subsec:Ej_Radio}
The energy generation rate from r-process decay has been shown to
approximately follow
$\dot{\epsilon} = \epsilon_0 t_{\rm d}^{-\alpha}$, with
$\epsilon_0 \approx 10^{11} \text{ ergs s}^{-1} \text{g}^{-1}$
and $\alpha = 1.1 - 1.4$
\citep{Metzger_2010,Roberts_2011,Goriely_2011,Korobkin_NSM_rp}.
However, the fraction of the energy supplied by each decay channel,
and the emission spectra for each decay product, are less clear.
Though \emph{r}-process\ radioactivity is most commonly associated with
\ensuremath{\beta}-decay, any translead nuclei synthesized will decay by \ensuremath{\alpha}-emission, and heavier ($A \gtrsim 250$) nuclei may undergo fission,
trends which have implications for thermalization. Radioactive
emission in kilonovae will at all times be dominated by isotopes
with half-lives $\tau_{1/2}$ of order $t_{\rm exp}$, the time since explosion. For any
particular decay channel, $\tau_{1/2}$ is strongly correlated with
the energy $Q$ emitted when a nucleus decays. However, this is not
true across decay channels; for a given $\tau_{1/2}$, \ensuremath{\beta}-decay has
lower $Q$ than \ensuremath{\alpha}-decay, which has lower $Q$ than fission. As a
result, \ensuremath{\alpha}-decay can generate a substantial fraction of the \emph{r}-process\
radioactive energy, even though \emph{r}-process\ yields are dominated by
nuclei that \ensuremath{\beta}-decay. Fission could, in principle, also be an
important source of energy, but we find almost all fissioning nuclei
have $\tau_{1/2}$ less than a day, suggesting that fission supplies
a negligible amount of energy after very early times. Since energy
from \ensuremath{\alpha}-decay thermalizes with a different efficiency than
\ensuremath{\beta}-decay energy, thermalization depends on the relative importance of these decay channels, and thus on the yields of
translead nuclei.
The top panel of Figure \ref{fig:en_gen} shows the fraction of radioactive
energy produced by \ensuremath{\alpha}-decay, \ensuremath{\beta}-decay, and fission, for the
representative trajectory introduced in \ref{subsec:comp}, calculated
for four nuclear mass models. Beta-decay is the primary source of
energy for all mass models out to late times. Fission, including
\ensuremath{\beta}-delayed, neutron-induced, and spontaneous fission, contributes
$\sim 10$\% of the energy at times $\lesssim 1$ day, and \ensuremath{\alpha}-decay
becomes significant within a few hours. The fractions for the
different nuclear mass models generally agree with each other, but the
estimates of energy generated by \ensuremath{\alpha}-decay differ by a factor of
almost ten, with DZ31 predicting the largest
contribution from \ensuremath{\alpha}-decay and FRDM predicting the least. Since
\ensuremath{\alpha}-decays release more energy, per decay, than \ensuremath{\beta}-decay, the
enhanced role of \ensuremath{\alpha}-decay predicted by the DZ31 model also results
in an increase in the total energy generated by the decay of \emph{r}-process\ isotopes. The
increase is modest early on (a factor of $\lesssim 1.2$ for
$t \leq 1$ day) but becomes more important at late times (a factor
of $\gtrsim 2$ by $t =$ 1 month.)
The bottom panel of of Figure~\ref{fig:en_gen} explores the effect of
electron fraction on the decay channels. Fission and \ensuremath{\alpha}-decay are
significant sources of energy for $t\lesssim 1$~day and
$t\gtrsim 1$~day, respectively, for ejecta with
$Y_{\rm e,0} \lesssim 0.2$, but become negligible at higher
$Y_{\rm e,0}$ because the reduced number of free neutrons chokes the
production of the heaviest nuclei.
\begin{figure}
\includegraphics[width=3.5 in]{f2.pdf}
\caption{ \textit{Top panel:} The fraction of the total radioactive energy
produced by \ensuremath{\beta}-decays, \ensuremath{\alpha}-decays, and fission in our
representative trajectory, for four nuclear mass
models. While \ensuremath{\beta}-decay dominates, fission (\ensuremath{\alpha}-decay) can be
important at early (late) times. Agreement between the four mass
models studied is within an order of magnitude.
\textit{Bottom panel:} Energy released in \ensuremath{\alpha}-decays and fission, for
the FRDM mass model, for a range of $Y_{\rm e,0}$. Lower electron
fractions favor the assembly of the heavy elements that later decay
by fission and \ensuremath{\alpha}-emission. As $Y_{\rm e,0}$ increases, these
processes become less important, and are negligible for $Y_{\rm e,0} >
0.2$.}
\label{fig:en_gen}
\end{figure}
\subsection{Emission Spectra of Decay Products}
\label{subsec:spec}
Modeling the energy spectra of \emph{r}-process~radioactive decay products is
complicated by the large number of decay chains and the uncertain
nuclear data. However, we can construct approximate spectra by
considering emission from a range of contributing decays. We calculate
emission spectra using the time-dependent composition of our
representative inner SPH trajectory. The decay energies for \ensuremath{\alpha}- and
\ensuremath{\beta}-decay were determined from experimental mass excesses (AME 2012;
\citealt{AME2012_a, AME2012_b}) when available, and theoretical (FRDM)
mass excesses otherwise. Decay data, including \ensuremath{\beta}\ endpoint energies,
\ensuremath{\gamma}-spectra, and half-lives for \ensuremath{\beta}- and \ensuremath{\alpha}-decay, were retrieved from
the Nuclear Science References database \citep{NucData_References},
accessed via the website of the International Atomic Energy Agency.
\subsubsection{Beta decay}
Energy from \ensuremath{\beta}-decay takes the form of energetic \ensuremath{\beta}-particles,
\ensuremath{\gamma}-rays, and neutrinos. (Beta-delayed fission is treated as
part of fission, and we neglect \ensuremath{\beta}-delayed neutron- and
\ensuremath{\alpha}-emission, as they are expected to be negligible for nuclei with
lifetimes longer than a day.) Following \ensuremath{\beta}-emission, nuclear
de-excitation can also emit low-energy atomic electrons, delayed
neutrons, and $\sim$ keV X-rays, but we found that these secondary
processes were negligible.
We constructed \ensuremath{\beta}- and \ensuremath{\gamma}-spectra using selected isotopes
that dominated the \ensuremath{\beta}-decay energy production. The energy generation
rate of an isotope $i$ was estimated as
$\dot{\epsilon}_{\rm \ensuremath{\beta},i} = Y_{\rm i} Q_{\rm \beta,i}/\tau_{\rm 1/2,i}$,
where $Y_{\rm i}$ is the number abundance of the isotope,
$Q_{\rm \beta,i}$ the decay energy, and $\tau_{\rm 1/2,i}$ the half
life. We used experimental values for $Q_{\rm \beta}$ and
$\tau_{\rm 1/2}$ when available, and theoretical values otherwise.
We excluded isotopes lacking decay data and those with heating rates
less than 1\% of the maximum single-isotope heating rate. The excluded
\ensuremath{\beta}-decays account for only 5-7\% of the total \ensuremath{\beta}-decay energy at all
times. The \ensuremath{\gamma}-ray intensities were taken directly from nuclear
measurements, while $\beta$-spectra were constructed from endpoint
energies and intensities assuming all decays had an allowed spectral
shape and using the simplified Fermi formula fit proposed by
\citet{Schenter_Vogel_1983_betaSpec}.
We find that roughly $20\%$ of the \ensuremath{\beta}-decay energy emerges as
\ensuremath{\beta}-particles, $45\%$ as \ensuremath{\gamma}-rays and 35\% as neutrinos. The energy
lost to neutrinos, which escape the ejecta without depositing any
energy, sets an upper limit of $\sim 65 \%$ on the \ensuremath{\beta}-decay
thermalization efficiency. The top two panels of
Figure~\ref{fig:netSpec} show the \ensuremath{\beta}- and \ensuremath{\gamma}- spectra for the composition of our neutron-rich representative SPH trajectory for $t = 1$ -- $30$ days. The
\ensuremath{\gamma}-ray spectra peak at several hundred keV and the \ensuremath{\beta}-spectra at
around 0.5 MeV.
\begin{figure}
\includegraphics[width = 3.5 in]{f3.pdf}
\caption{The emission spectra for \ensuremath{\beta}-particles (top panel), \ensuremath{\gamma}-rays (middle panel), and \ensuremath{\alpha}-particles (bottom panel) as a function of time.}
\label{fig:netSpec}
\end{figure}
The \ensuremath{\beta}-spectrum was found to be consistent across mass models, which is not surprising since \ensuremath{\beta}-decay energy is very sensitive to half-life, and \ensuremath{\beta}-emission at all times is dominated by nuclei with half-lives of order the time since explosion. However, we did find the spectrum depends mildly on electron fraction, with higher $Y_{\rm e,0}$ slightly enhancing the spectrum's high-energy tail.
This is due to differences in how \ensuremath{\beta}-decay energy is
divided among \ensuremath{\beta}-particles, \ensuremath{\gamma}-rays, and neutrinos. Compositions evolved
from higher initial $Y_{\rm e}$ impart a greater \emph{fraction} of the
total \ensuremath{\beta}-decay energy $Q_{\ensuremath{\beta}}$ to \ensuremath{\beta}-particles, at the expense of
\ensuremath{\gamma}-rays (see the lower three panels of Figure \ref{fig:YeDep}). As
shown in Figure~\ref{fig:comp}, higher electron fractions yield
compositions with lower $A$. The \ensuremath{\beta}-decays for these lighter
nuclei tend to be dominated by one or a few transitions to low-lying
nuclear energy states; the energy carried away by the \ensuremath{\beta}-particle
and the neutrino is close to $Q_{\ensuremath{\beta}}$, and the energy released in
\ensuremath{\gamma}-rays during nuclear de-excitation is reduced. In contrast, for
more massive nuclei, the excitation energy of the daughter nucleus
after emission of the \ensuremath{\beta}-particle and neutrino is more likely to be
a significant fraction of $Q_{\ensuremath{\beta}}$, and a greater portion of the
energy takes the form of \ensuremath{\gamma}-rays. Therefore, despite having similar
$Q_\ensuremath{\beta}$, nuclei synthesized in high-$Y_{\rm e,0}$ conditions
generate more energetic \ensuremath{\beta}-particles. We found these effects to be
independent of mass model.
\begin{figure}
\includegraphics[width = 3.5 in]{f4.pdf}
\caption{The effect of electron fraction $Y_{\rm e,0}$ on \ensuremath{\beta}-decay
for the FRDM nuclear mass model. (Other nuclear mass models studied
showed similar trends). \textit{Top panel:} The \ensuremath{\beta}-spectrum for a
composition with $Y_{\rm e,0}= 0.25$. The spectrum is shifted to
higher energies, relative to the low-$Y_{\rm e,0}$ case
(Figure~\ref{fig:netSpec}). \textit{Bottom panels:} The fraction of
$Q_{\beta}$, $\chi$, imparted to \ensuremath{\beta}-particles, \ensuremath{\gamma}-rays, and neutrinos, for
different values of $Y_{\rm e,0}$. As $Y_{\rm e,0}$ increases, a
greater fraction of $Q_{\ensuremath{\beta}}$ goes to \ensuremath{\beta}'s and neutrinos, while
$\chi_{\ensuremath{\gamma}}$ shrinks. This effect is particularly pronounced at late
times. }
\label{fig:YeDep}
\end{figure}
\subsubsection{Alpha decay}
While the majority of species produced by the \emph{r}-process\ stabilize through
\ensuremath{\beta}-decay, some heavier isotopes ($A \gtrsim 200$) undergo
\ensuremath{\alpha}-decay. Unlike \ensuremath{\beta}-particles, \ensuremath{\alpha}-particles are ejected from nuclei
at discrete energies that fall within the fairly narrow range
$E_{\ensuremath{\alpha}} \sim 5 - 9$ MeV. Due to the fact that alpha decay is a
tunneling process, \ensuremath{\alpha}-particles carry all of the decay energy in the
majority of decays, and the incidence of \ensuremath{\gamma}-emission is vanishingly
low.
We determined the most important sources of \ensuremath{\alpha}-emission using the
procedure detailed above for $\beta$-decays. The $\alpha$-spectrum as
a function of time is given in the bottom panel of Figure
\ref{fig:netSpec}. The energy is fairly evenly distributed in the
range 5 MeV $< E_{\alpha} < 9$ MeV.
\subsubsection{Fission}
Spontaneous, neutron-induced, and \ensuremath{\beta}-delayed fission of heavy nuclei
($A \gtrsim 250$) contribute a few percent of the total \emph{r}-process\ radioactive decay energy at
times $\lesssim 1$ day. The mass distribution and energy spectra of
the fission fragments depend sensitively on the nuclear physics
models, and a thorough exploration of these parameters is beyond the
scope of this work. We can, however, estimate the final kinetic energy
of fission as equal to the repulsive Coulomb energy between the
daughter nuclei immediately after fission occurs:
\begin{equation}
E_{\rm K,tot} = E_{\rm Coul} = \frac{Z_1 Z_2 e^2}{r_0\left( A_1^{1/3} + A_2^{1/3}\right)},
\end{equation}
where $e$ is the elementary charge, $(A_1, Z_1)$ and $(A_2, Z_2)$ are the masses and atomic numbers of the daughter nuclei, and the nuclear radius is given by $r_0 A^{1/3}$. For deformed post-scission nuclei, $r_0 \simeq 1.8$ fm.
Fission favors the production of nuclei at or near the doubly-magic nucleus $(A, Z) = (132, 50)$.
Assuming a typical parent isotope has mass and atomic numbers $A_p = 250$ and $Z_p = 100$, and that ($A_1, Z_1) = (132, 50)$, the fission daughters will have kinetic energies of order 100 MeV. We assume the fission fragment spectrum is flat, and ranges from 100--150 MeV. Given the limited role of fission at times later than 1 day, a more detailed treatment is unnecessary.
\section{Thermalization Physics}
\label{sec:thermRates}
In this section, we discuss the processes by which energetic decay products thermalize in the kilonova ejecta, and present energy loss rates for \ensuremath{\beta}-particles, \ensuremath{\alpha}-particles, and fission fragments.
\subsection{Gamma-rays}
Gamma-rays lose energy through photoionization and Compton
scattering.
We calculated
the Compton opacity from the Klein-Nishina formula and the
photoionization opacity using the Photon Cross Section Database
(\textit{XCOM}; \citealt{NIST_XCOM}) published by the National
Institute of Standards and Technology (NIST).
The total \ensuremath{\gamma}-ray opacity for our fiducial composition at $t=1$ day is shown in Figure~\ref{fig:gammaOp}.
The high-$Z$ elements produced in the \emph{r}-process\ have higher ionization thresholds ($\sim 100$~keV) than do the metals in typical astrophysical mixtures, so the photoionization cross-section in kilonovae dominates out to $\sim 1$~MeV, above which
Compton scattering takes over.
The opacity varies little between the inner and outer ejecta, and changes over time are minor, so we assume the \ensuremath{\gamma}-ray opacity to be constant.
Both photoionization and Compton scattering events produce a non-thermal electron, which loses energy by the physical processes described in the next section.
\begin{figure}
\includegraphics[width=3.5 in]{f5.pdf}
\caption{The $\gamma$-ray opacity, $\kappa$, in the inner (solid lines) and outer (dotted lines) ejecta. Photoionization opacity is plotted in green, Compton opacity in blue, and total opacity in black. Differences between the inner and outer ejecta compositions have a negligible impact on $\kappa$. The gray bar indicates the energies at which most \ensuremath{\gamma}-rays are emitted.}
\label{fig:gammaOp}
\end{figure}
\subsection{Beta particles}
Suprathermal \ensuremath{\beta}-particles lose energy primarily through Coulomb interactions with free thermal electrons (plasma losses), and by exciting or ionizing bound atomic electrons. Bremsstrahlung (free-free) emission is important for very high-energy \ensuremath{\beta}-particles. While earlier studies of thermalization assumed plasma interactions dominated the energy loss, we find \ensuremath{\beta}-particles lose most of their energy to ionization and excitation.
In the limit that the \ensuremath{\beta}-particle energy far exceeds that of thermal electrons, the plasma energy loss per unit time is \citep{Huba_NRL}
\begin{align}
\dot{E}_\ensuremath{\beta}^{\rm pl} &= 7.7 \times 10^{-15} E_\ensuremath{\beta}^{-1/2} \\
&\times
\left( \frac{n_{\rm e}}{1 \text{ cm}^{-3}} \right) \lambda_{\rm ee} \left(1.0 - \frac{3.9}{7.7}\frac{T}{E_\ensuremath{\beta}} \right)
\mbox{ MeV s}^{-1}
\end{align}
where $E_\ensuremath{\beta}$ is the \ensuremath{\beta}-particle's kinetic energy in MeV, $T$ is the ejecta temperature in MeV, $\lambda_{\rm ee} \sim 10$ is the Coulomb logarithm for electron-electron scattering, and $n_{\rm e}$ is the free electron number density.
Radioactive \ensuremath{\beta}-particles have $E_\ensuremath{\beta} \sim 1$ MeV whereas $k_{\rm B} T \sim 1$ eV in kilonova ejecta, so the assumption that $E_{\ensuremath{\beta}} \gg k_{\rm B} T$ holds.
We determine $n_{\rm e}$ assuming that all elements heavier than hydrogen are singly ionized, as expected for kilonova ejecta near peak brightness
\citep{Kasen_2013_AS}. The outer ejecta contains a substantial quantity of hydrogen, which we assume to be neutral given the low temperatures ($T \lesssim 5000$~K) expected in the ejecta periphery.
We calculate energy losses due to ionization and excitation of atomic electrons using the well-established formula (\citealt{Heitler_1954_qntmRad,Berger_Seltzer_1964,Gould_1975,Blumenthal_Gould_1970}; see also \citet{Chan_Lingen_1993} and \citealt{Milne_1999_p+})
\begin{gather}
\begin{align}
\dot{E}_\ensuremath{\beta}^{\rm IE}
&=\frac{2 \pi r_{\rm e}^2 m_{\rm e} c^3 n_{\rm e,b}}{v_\ensuremath{\beta}/c} \\
&\times \left\{ 2\ln\left(\frac{\displaystyle E_\ensuremath{\beta}}{\displaystyle \bar{I}}\right) + \ln\left(1 + \frac{\tau}{2}\right) + \left(1 - \frac{v_\beta^2}{c^2}\right)g(\tau) \right\}, \nonumber
\end{align}
\\ g(\tau) = 1 + \frac{\tau^2}{8} - (2 \tau+1)\ln 2,
\end{gather}
where
$r_{\rm e}$ is the classical electron radius,
$m_{\rm e}$ is the electron mass,
$n_{\rm e,b}$ is the number density of bound electrons,
$v_{\beta}$ is the \ensuremath{\beta}-particle's speed,
and $\tau = E_\ensuremath{\beta}/m_{\rm e} c^2$. The quantity $\bar{I}$ is an average ionization and excitation potential which can be approximated for an element of atomic number $Z$ as
\citep{Segre_NucPart}
\begin{equation}
\bar{I} = 9.1Z\left(1 + \frac{1.9}{Z^{2/3}} \right) \text{ eV}.
\end{equation}
Following \citet{Chan_Lingen_1993}, we use averaged quantities for $n_{\rm e,b}$ and $\bar{I}$,
\begin{gather}
\langle n_{\rm e,b} \rangle = \frac{\rho}{m_{\rm u}} \left\langle \frac{Z}{A} \right\rangle,\\
\left\langle \ln \frac{\bar{I}}{\text{eV}}\right\rangle = \left\langle\frac{Z}{A}\right\rangle^{-1}\sum\limits_{\rm j} \left( \frac{A}{Z}\right)_{\rm j} X_{\rm j} \ln \left(\frac{\bar{I}_{\rm j}}{\text{eV}}\right), \\
\text{where}~~ \left\langle \frac{Z}{A}\right\rangle = \sum\limits_{\rm j}\left(\frac{Z}{A}\right)_{\rm j}X_{\rm j},
\end{gather}
where $m_{\rm u}$ is the nuclear mass unit,
$X_{\rm j}$ is the mass fraction of
element $j$, and the sum runs over all species in the composition. For
the inner (outer) ejecta, we find
$\langle \ln \bar{I}/\text{eV} \rangle = 6.4$ ($4.9$), and
$\langle Z/A \rangle = 0.4$ ($0.55$).
Plasma and ionization/excitation losses are the cumulative results of many distant interactions that individually transfer very little energy. The thermal and bound electrons energized by \ensuremath{\beta}-particles through these channels have very low kinetic energies, and thermalize rapidly.
Instead of tracking secondary electrons explicitly, we assume their kinetic energy is transferred directly to the thermal pool.
Bremsstrahlung (free-free) and synchrotron emission are other possible
means of \ensuremath{\beta}-particle energy loss. The rate of cooling due to
synchrotron emission in a magnetic field $B$ is
\begin{equation}
\dot{E}_{\ensuremath{\beta}}^{\rm syn} = \frac{4}{9} r_{\rm e}^2 c\gamma^2 \left(\frac{v_{\beta}}{c}\right)^2 B^2,
\end{equation}
where $\gamma$ is the \ensuremath{\beta}-particle's Lorentz factor.
Neglecting logarithmic terms (which only increase $\dot{E}_{\ensuremath{\beta}}^{\rm
IE}$), and assuming $\langle Z/A\rangle$ = 0.4 and
$\gamma^{2}(v_{\ensuremath{\beta}}/c)^{3} \approx 10$, we estimate the ratio of
synchrotron to ionization/excitation losses as
\begin{equation}
\frac{\dot{E}_{\ensuremath{\beta}}^{\rm syn}}{\dot{E}_{\ensuremath{\beta}}^{\rm IE}} \sim 1.6 \times 10^{-15} \left( \frac{B_{\rm d}}{3.7 \times 10^{-6} \text{ G}}\right)^2 M_5^{-1} v_2^{-1} t_{\rm d}^{-1},
\end{equation}
where $M_5 = M_{ej}/(5 \times 10^{-3} \ensuremath{M_{\odot}})$
and $B_{\rm d}$ is the magnetic field at 1 day.
This is much less than unity for all parameters of interest, so we neglect synchrotron losses.
In contrast, Bremsstrahlung contributes, albeit modestly, to \ensuremath{\beta}\ energy loss for $E_\ensuremath{\beta}\gtrsim 1$ MeV. From \citet{Seltzer_Berger_Brem_1986},
\begin{equation}
\dot{E}_{\beta}^{\rm Brem} = n_{\rm i} v_{\ensuremath{\beta}} (E_\beta + m_{\rm e} c^2) Z^2 r_{\rm 0}^2 \alpha \phi_{\rm rad}
\end{equation}
where $n_i$ is the number density of the scattering species, $\alpha$ is the fine-structure constant, and $\phi_{\rm rad}$ are energy-dependent empirical fitting constants, also from \citet{Seltzer_Berger_Brem_1986}. We model Bremsstrahlung losses in the inner ejecta using characteristic values $Z = 60$ and $A = 144$,
similar to the average values of the inner composition.
For the outer ejecta, we use a two component composition, with $(Z,A)=(1,1)$ accounting for the high amount of hydrogen, and $(Z,A) = (55,133)$ representing elements with $Z > 1$.
Bremsstrahlung may produce high-energy photons that do not thermalize promptly in the ejecta, an effect of order $\lesssim 10\%$ at typical \ensuremath{\beta}-particle energies. Our treatment of Bremsstrahlung is discussed in more detail in \S \ref{subsec:partTrans}.
We plot the total energy loss rate in the inner and outer ejecta, normalized to mass density, in the top panel of Figure~\ref{fig:dEdt_all}. While the lower degree of ionization in the outer ejecta makes plasma and Bremsstrahlung losses less efficient, this is more than compensated for by enhanced ionization and excitation losses due to the greater number of bound electrons per nucleon, and to the lower average ionization potential. Overall, thermalization rates in the outer ejecta are higher by a factor of a few.
\begin{figure}
\includegraphics[width=3.25 in]{f6.pdf}
\caption{\textit{All panels}: Energy loss rates in the inner (outer) ejecta are plotted in solid (dashed) lines. Gray bars indicate typical particle energies at emission. \textit{Top panel:} The total energy loss rate for fast \ensuremath{\beta}-particles, normalized to the mass density $\rho$. Thermalization rates are higher in the outer ejecta by a factor of a few. \textit{Middle panel:} The energy loss rates for \ensuremath{\alpha}-particles in simplified \emph{r}-process\ mixtures standing in for the full inner and outer ejecta compositions (see Figure \ref{fig:rpSimpComp}), normalized to density. Alpha particle thermalization is a few to $\sim 10 \times$ more efficient in the outer ejecta. \textit{Bottom panel:} The energy loss rate for fission fragments, normalized to density and assuming most atoms in the ejecta are singly ionized. Thermalization is more efficient in the outer ejecta.}
\label{fig:dEdt_all}
\end{figure}
\subsection{Alpha-particles}
Suprathermal \ensuremath{\alpha}-particles thermalize by interacting with free and bound
electrons. Long-range interactions with ions and short-range
interactions with atomic nuclei do not significantly contribute to
\ensuremath{\alpha}-particle energy loss.
Alpha particles scattering off of free, thermal electrons lose energy at a rate given by
\citet{Huba_NRL} for fast ions in a plasma,
\begin{equation}
\begin{split}
\dot{E}_{\rm i}^{\rm pl} &= 1.7 \times 10^{-13} E_{\rm i}^{-1/2}\mu_{\rm i}^{1/2} Z_{\rm i}^2 \left( \frac{n_{\rm e}}{1 \text{ cm}^{-3}}\right) \lambda_{\rm ie} \\
&\times \left(2 - \frac{1.1\times 10^{-3}}{\mu_{\rm i}} - \frac{T}{E_{\rm i}} \right) \text{ MeV s}^{-1},
\end{split}
\label{eq:plasma_ie}
\end{equation}
where $E_{\rm i}$ is the ion's kinetic energy in MeV, $\mu_{\rm i}$ is the ion mass in $m_{\rm u}$, $Z_{\rm i}$ is the charge in units of the elementary charge, and $\lambda_{\rm ie} \sim 5 - 10$ is the Coulomb logarithm for ion-electron scattering. For \ensuremath{\alpha}-particles, $Z_{\rm i} = 2$ and $\mu_{\rm i} = 4$.
The rates of \ensuremath{\alpha}-particle energy loss due to interactions with bound electrons
are taken from NIST's \textit{ASTAR} database \citep{NIST_*Star}.
Lacking \ensuremath{\alpha}-particle stopping data for all elements in our \emph{r}-process\ mixture, we map the full inner and outer compositions onto a reduced set of elements for which \ensuremath{\alpha}-stopping powers are available (see Figure~\ref{fig:rpSimpComp}). The middle panel of Figure~\ref{fig:dEdt_all} shows the total \ensuremath{\alpha}-particle energy loss rates. Plasma losses dominate for $E_{\ensuremath{\alpha}} \lesssim 1$ MeV, while interactions with bound electrons are important at higher energies. The thermalization rate in the outer ejecta is greater than in the inner ejecta by up to an order of magnitude.
\begin{figure}
\includegraphics[width=3.5 in]{f7.pdf}
\caption{The simplified composition (circles) used to calculate the electronic stopping for \ensuremath{\alpha}-particles and fission fragments. The full compositions, shown in solid lines, were mapped onto a composition of elements for which \ensuremath{\alpha}- and proton stopping data were available through NISTS's \textit{ASTAR} database.}
\label{fig:rpSimpComp}
\end{figure}
\subsection{Fission fragments}
Interactions with free and bound electrons, and with atomic nuclei all contribute to fission fragment thermalization.
The energy loss to thermal free electrons is described by Eq. \ref{eq:plasma_ie}, where $Z_{\rm i }$ depends on the ionization state of the fission fragment and the length scale of the collision. For impact parameters greater than the size of the interacting particles, the relevant charge is the total charge carried by the fragment, $Z_{\rm ff,ion} = Z_{\rm nuc} - N_{\rm e^-,b}$, where $Z_{\rm nuc}$
is the fragment's atomic number and $N_{\rm e^-,b}$ is the number of bound electrons. At lower impact parameters, more of the nuclear charge is felt.
We calculate $Z_{\rm ff,ion}$ as a function of fission fragment energy $E_{\rm ff}$ using the formula of
\citet{Schiwietz_Grande_2001_chargeState} for ion charge state in a gaseous medium.
Since fragments with $Z_{\rm ff,ion} \gtrsim 7$ deflect thermal electrons at impact parameters greater than the fission fragment radius, we set $Z_{\rm i} \rightarrow \max\{Z_{\rm ff,ion}(E_{\rm ff}), 7\}$.
Fission fragments can scatter off thermal \emph{ions} at much lower impact parameters, in which case the full nuclear charge is felt. The energy loss from these interactions is given by the nuclear stopping formula of \citet{Ziegler_1980}.
To model the stopping of heavy particles by bound electrons, we adopt the technique of \citet{Ziegler_1980}, in which the stopping power of a heavy particle in any material is proportional to the stopping power of a proton in the same material, with the constant of proportionality given by $Z_{\rm ff,ion}^2$. We calculate the stopping power for the same simplified composition used to model \ensuremath{\alpha}-particle energy loss, using proton stopping powers extracted from NIST's \textit{PSTAR} database \citep{NIST_*Star}.
The total energy loss rate for fission fragments is presented in the bottom panel of Figure \ref{fig:dEdt_all}. Interactions with bound electrons dominate the rate at high energies, while losses to free electrons become important at energies less than $\sim 10$ MeV. Thermalization rates in the outer composition are a factor of a few higher than in the inner composition.
\section{Analytic Results}
\label{sec:analytics}
Before moving to detailed numerical calculations of kilonova thermalization, we consider simple analytic estimates of the relevant timescales and time evolution. This work extends the analytic treatments proposed by
\citet{Metzger_2010,Hotokezaka_2015_heating}.
Unless stated otherwise, our estimates describe thermalization in the ``inner'' composition, which typically makes up $\sim 90$\% or more of the ejected mass.
\subsection{Analytic estimates of thermalization timescales}
\label{subsec:anEst_Sum}
The net thermalization of the energy from the radioactive decay of \emph{r}-process\ material depends on the relative importance of each decay channel and on how efficiently the decay products thermalize in the ejecta.
Energy loss rates depend on the density of the medium, so thermalization is also a function of \ensuremath{M_{\rm ej}}\ and \ensuremath{v_{\rm ej}}. If we approximate the ejecta as a uniform density sphere of mass \ensuremath{M_{\rm ej}}\ and kinetic energy $E_{\rm k} = \ensuremath{M_{\rm ej}}\ensuremath{v_{\rm ej}}^2/2$, the density is
\begin{equation}
\rho(t) \approx 7.9 \times 10^{-15} M_{5} \; v_{2}^{-3} \; t_{\rm d}^{-3} \text{ g cm}^{-3},
\end{equation}
where again, $M_5 = \ensuremath{M_{\rm ej}}/5.0 \times 10^{-3} M_{\odot}$ and $v_2 = \ensuremath{v_{\rm ej}}/0.2c$.
Thermalization becomes inefficient at a time, $t_{\rm ineff}$, when
the timescale for a particle to thermalize becomes similar to the
ejecta expansion timescale, $t_{\rm exp}$. The inefficiency time can
be compared to the peak of the kilonova light curve,
\begin{equation}
t_{\rm peak} \sim \left(C \frac{M_{\rm ej} \kappa}{v_{\rm ej} c}\right)^{1/2} \simeq 4.3 \; M_{5}^{1/2} \; v_2^{-1/2} \: \text{days},
\end{equation}
where $\kappa$ is the opacity for optical/infrared light (we take $\kappa = 10 \text{ cm$^2$ g}^{-1}$, appropriate for an \emph{r}-process\ medium) and $C = 0.32$ is a scaling factor we estimate from kilonova radiation transport simulations (e.g. Barnes \& Kasen 2013). If $t_{\rm ineff} < t_{\rm peak}$, thermalization will impact the kilonova light curve significantly.
\vspace{\baselineskip}
\noindent $\boldsymbol{\gamma}$\textbf{-rays:} Gamma rays stop thermalizing efficiently when they can escape the ejecta without undergoing any scatters or absorptions. This occurs when the optical depth $\tau \approx \rho \kappa_\ensuremath{\gamma} R_{\rm ej}$ falls below unity.
For \ensuremath{\gamma}-rays with energies $E_{\ensuremath{\gamma}} \gtrsim 1$ MeV, the relevant opacity is the Compton opacity, $\kappa_{\rm C} \approx 5 \times 10^{-2}$ cm$^2$ g$^{-1}$ while the photoionization opacity, $\kappa_{\rm PI} \gtrsim 1$ cm$^2$ g$^{-1}$, dominates for lower-energy photons. The ejecta becomes transparent ($\tau < 1$) to \ensuremath{\gamma}-rays at a time
\begin{align}
t_{\rm ineff} \approx& \begin{cases}
0.5 \: M_5^{1/2} \: v_2^{-1} \mbox{ days} &\mbox{for } E_{\ensuremath{\gamma}} \gtrsim 1 \mbox{ MeV} \\
2.3 \: M_5^{1/2} \: v_2^{-1} \mbox{ days} &\mbox{for } E_{\ensuremath{\gamma}} \lesssim 1 \mbox{ MeV}
\end{cases}
\end{align}
In both cases, inefficiency sets in before the kilonova light curve peaks,
\begin{subnumcases}{\frac{t_{\rm ineff}}{t_{\rm peak}} \simeq }
0.12 \: v_2^{-1/2} \hfill &\mbox{ } $E_{\ensuremath{\gamma}} \gtrsim 1 \mbox{ MeV}$ \label{eq:titp_g1} \\
0.5 \: v_2^{-1/2} \hfill &\mbox{ } $E_{\ensuremath{\gamma}} \lesssim 1 \mbox{ MeV.}$ \label{eq:titp_g2}
\end{subnumcases}
\vspace{\baselineskip}
\noindent $\boldsymbol{\beta}$ \textbf{-particles:} The energy loss rate for \ensuremath{\beta}-particles, modulo mass density, has a fairly constant value $\dot{E}_{\ensuremath{\beta}} \simeq 4 \times 10^{10} \rho \text{ MeV s}^{-1} $ over a broad range of energies (see Fig. \ref{fig:dEdt_all}). The thermalization time for \ensuremath{\beta}-particles is
\begin{eqnarray}
t_{\rm th} &\approx & \frac{E_{\ensuremath{\beta},0}}{\dot{E}_{\ensuremath{\beta},0}} = \frac{E_{\ensuremath{\beta},0}}{ 4 \times 10^{10} \; \rho \text{ MeV s}^{-1}} \nonumber \\
&=& 0.02 \left(\frac{E_{\ensuremath{\beta},0}}{0.5 \text{ MeV}}\right) \; M_5^{-1} \; v_2^{3} \; t_{\rm d}^3 \text{ days},
\end{eqnarray}
where $E_{\ensuremath{\beta},0}$ is the initial \ensuremath{\beta}-particle energy.
Beta particles trapped in the ejecta fail to efficiently thermalize
when $t_{\rm th} \gtrsim t_{\rm exp}$,
which occurs at
\begin{equation}
t_{\rm ineff} \approx 7.4 \: \left(\frac{E_{\ensuremath{\beta},0}}{0.5 \text{ MeV}}\right)^{-1/2} \; M_5^{1/2} \; v_2^{-3/2} \text{ days}.
\end{equation}
For a typical initial energy, $t_{\rm ineff}$ is comparable to the rise time of the light curve,
\begin{align}
\frac{t_{\rm ineff}}{t_{\rm peak}} \approx 1.7 \left(\frac{E_{\ensuremath{\beta},0}}{0.5 \text{ MeV}}\right)^{-1/2} \; v_2^{-1}. \label{eq:titp_beta}
\end{align}
If the magnetic field is radial or only slightly tangled, \ensuremath{\beta}-particles can escape the ejecta before they thermalize, and escape will significantly reduce the thermalization efficiency. The escape time is
\begin{equation}
t_{\rm esc} \simeq \frac{R_{\rm ej}(t)}{\lambda v_{\ensuremath{\beta},\parallel}},
\end{equation}
where $\lambda R_{ej}$ is the coherence length of the magnetic field, $v_{\ensuremath{\beta},\parallel}$ is the component of the \ensuremath{\beta}-particle velocity parallel to the field lines, and we have modeled the \ensuremath{\beta}'s motion in a random field as a random walk of step size $\lambda R_{\rm ej}$.
For a \ensuremath{\beta}-particle with $E_{\rm \ensuremath{\beta}, 0} = 0.5$ MeV and pitch angle 1 ($v_{\ensuremath{\beta},\parallel} = v_{\beta}$), $t_{\rm esc}$ is less than $t_{\rm th}$ when
\begin{equation}
t \gtrsim \frac{3.5 \; M_5^{1/2} \; v_2^{-1}}{\lambda^{1/2}}~~{\rm days}.
\end{equation}
For radial fields ($\lambda = 1$), this is less than $t_{\rm peak}$, so escape is important for \ensuremath{\beta}-particle thermalization.
In contrast, for disordered fields there is a degree of randomness above which \ensuremath{\beta}-particle escape cannot significantly impact the light curve. This limit is defined by the condition $t_{\rm th}(t_{\rm peak}) < t_{\rm esc}(t_{\rm peak})$. Again considering a 0.5
MeV \ensuremath{\beta}-particle, we find
\begin{equation}
t_{\rm th}({t_{\rm peak}}) < t_{\rm esc}(t_{\rm peak}) \rightarrow \lambda \lesssim 0.8 v_2^{-1}. \\
\end{equation}
Thus, high-energy \ensuremath{\beta}-particles are effectively trapped by even a slightly tangled magnetic field.
\vspace{\baselineskip}
\noindent $\boldsymbol{\alpha}$\textbf{-particles and fission fragments:}
Fission fragments and \ensuremath{\alpha}-particles are emitted with greater energies than \ensuremath{\beta}-particles ($E_{\ensuremath{\alpha},0} \simeq 6$ MeV; $E_{\rm ff,0} \simeq 100$ MeV), but have higher energy loss rates ($\dot{E}_{\ensuremath{\alpha}}(E_{\ensuremath{\alpha},0}) \sim 5 \times 10^{11}\rho$ MeV s$^{-1}$; $\dot{E}_{\rm ff}(E_{\rm ff,0}) \sim 5 \times 10^{13} \rho$ MeV s$^{-1}$.) The efficiency of \ensuremath{\alpha}-particle thermalization is similar to that of \ensuremath{\beta}\ particles, while fission fragments thermalize efficiently out to very late times:
\begin{numcases}{\frac{t_{\rm ineff}}{t_{\rm peak}} \simeq }
1.8 \: \left(\frac{E_{\ensuremath{\alpha},0}}{6 \text{ MeV}}\right)^{-1/2} v_2^{-1} \hfill &\mbox{ \ensuremath{\alpha}-particles} \nonumber \\
\label{eq:titp_a} \\
3.9 \: \left(\frac{E_{\rm ff,0}}{125 \text{ MeV}}\right)^{-1/2} v_2^{-1} \hfill &\mbox{ fiss. fragments.} \nonumber \\
\label{eq:titp_ff}
\end{numcases}
Unlike \ensuremath{\beta}-partices, both \ensuremath{\alpha}'s and fission fragments have velocities much lower than $v_{\rm ej}$, and so in general cannot escape the ejecta. However, because
these particles are
propagating through a steep velocity gradient,
their speed relative to
the background gas continually decreases. This reduces the kinetic energy
of the particles as measured in the co-moving frame. Because the particles
have a spiraling motion about magnetic field lines, their motion is
never completely frozen out in the fluid frame. Still, these ``frame-to-frame''
effects can reduce thermalization by $\lesssim 15\%$.
\subsection{Summary of thermalization timescales}
While low-energy \ensuremath{\beta}-particles, \ensuremath{\alpha}-particles, and especially fission fragments typically thermalize efficiently at $t = t_{\rm peak}$, the thermalization at peak of high-energy \ensuremath{\beta}-particles and \ensuremath{\gamma}-rays is not robust.
Figure~\ref{fig:anEstSum} plots the ratio of thermalization time to light curve peak for all particles as a function of initial energy for a range of $v_{\rm ej}$.
For \ensuremath{\alpha}- and \ensuremath{\beta}-particles, we calculated $t_{\rm ineff}/t_{\rm peak}$ from Eq.s~\ref{eq:titp_a} and \ref{eq:titp_beta}. The \ensuremath{\gamma}-ray curve was calculated from Eq.~\ref{eq:titp_g1} for $E_{\ensuremath{\gamma}} \leq 200$ keV, \ref{eq:titp_g2} for $E_{\ensuremath{\gamma}} \geq 1$ MeV, and a simple linear interpolation for intermediate $E_{\ensuremath{\gamma}}$. For fission fragments, we modified Eq.~\ref{eq:titp_ff} slightly to account for the positive slope of $\dot{E}_{\rm ff}$ in the range $E_{\rm ff} = 100 - 150$ MeV. This renders $\dot{E}_{\rm ff}$ approximately constant, so the fission fragment curve is essentially flat.
\begin{figure}
\includegraphics[width = 3.5 in]{f8.pdf}
\caption{The ratio $t_{\rm ineff}/t_{\rm peak}$ for all particles, for \ensuremath{v_{\rm ej}}\ in the range $0.1c - 0.3c$. Fission fragments, and to a lesser extent \ensuremath{\alpha}-particles and low-energy \ensuremath{\beta}-particles, thermalize efficiently out to late times. Higher energy \ensuremath{\beta}'s and \ensuremath{\gamma}-rays are expected to become inefficient on kilonova timescales. The width of the curves is due to the range of $v_{\rm ej}$ considered, since $t_{\rm ineff}/t_{\rm peak}$ varies inversely with \ensuremath{v_{\rm ej}}. Curves for the fiducial velocity \ensuremath{v_{\rm ej}} = $0.2c$ are over-plotted in dotted black lines.}
\label{fig:anEstSum}
\end{figure}
\subsection{Analytic thermalization model}
\label{subsec:anMod}
We develop an analytic expression for time-dependent thermalization
efficiencies of massive particles under the following
assumptions: first, that the radioactive energy generation rate evolves as
$t^{-\ensuremath{\alpha}}$ with $\ensuremath{\alpha} = 1.0$ (close to the expected values
$\ensuremath{\alpha} = 1.1-1.4$); second, that the density in the
ejecta is spatially uniform; third, that energy loss rates are
independent of particle energy, and depend only on $\rho$; and fourth,
that all particles of a given type are emitted at a single energy
$E_0$. Despite these simplifications, we find our model agrees fairly
well with the detailed numerical calculations to be presented in
\S\ref{sec:results}.
The thermalization efficiency is defined as the ratio of energy
emitted by radioactive processes to energy absorbed by the ejecta at
any time $t$,
\begin{align}
f(t) = \frac{\dot{E}_{\rm th}(t)}{\dot{E}_{\rm rad}(t)}.
\end{align}
We approximate the radioactive energy generation rate by
$\dot{E}_{\rm rad} = \dot{\epsilon}_0(t_0/t)$ with $\dot{\epsilon_0} =
10^{11} M_{\rm ej}$ ergs s$^{-1}$ and $t_0 = 1$ day. Assuming charged
particle thermalization depends only on mass density (which declines
like $t^{-3}$ in a homologous flow) the energy loss is
\begin{equation}
\dot{E}_{\rm part}(t) = \psi \rho_0 \left(\frac{t}{t_0}\right)^{-3},
\end{equation}
where $\rho_0$ is the density at $t_0$, and $\psi$ is a scaling factor
such that $\psi\rho_0 = \dot{E}_{\rm part}(t_0)$, which will be unique
to each particle type. The rate at which energy is thermalized,
$\dot{E}_{\rm th}(t)$, is given by the number of live particles $N$
multiplied by the rate at which they lose energy,
\begin{align}
\dot{E}_{\rm th}(t) = N(t) \times \psi \rho_0 \left(\frac{t}{t_0}\right)^{-3}.
\end{align}
At any time $t$, the oldest live particle originates from an earlier time $t_{\rm i}$, defined by
\begin{align}
E_{\rm part}(t) &= E_0 - \int\limits_{t_{\rm i}}^t \psi \rho_0 \left(\frac{t'}{t_0}\right)^{-3} \dd{t'} = 0, \\
\intertext{ which is satisfied by }
t_{\rm i} &= \left(\frac{\psi \rho_0 t_0^3 t^2}{2 E_0 t^2 + \psi \rho_0 t_0^3}\right)^{1/2}.
\end{align}
The number of live particles at time $t$ is then
\begin{equation}
N(t) = \frac{\dot{\epsilon}_0 t_0}{2 E_0} \ln\left[1 + 2\left(\frac{t}{t_{\rm ineff}}\right)^2\right]
\end{equation}
where $t_{\rm ineff}$ is the inefficiency timescale defined in the previous section.
It is now straightforward to calculate the ratio $f_{\rm p}$ of thermalized to emitted energy for a massive particle of type $p$,
\begin{equation}
f_{\rm p}(t) = \frac{\dot{E}_{\rm th}}{\dot{E}_{\rm rad}} = \frac{\ln\left[1 + 2\left(\frac{t}{t_{\rm ineff,p}}\right)^2\right]}{2\left(\frac{t}{t_{\rm ineff,p}}\right)^2}.
\label{eq:ftAn}
\end{equation}
Eq.~\ref{eq:ftAn} can be used to estimate the thermalization efficiencies of massive particles, where the relevant timescales $t_{\rm ineff,p}$ are given by Eq.s~\ref{eq:titp_beta} (\ensuremath{\beta}-particles), \ref{eq:titp_a} (\ensuremath{\alpha}-particles), and \ref{eq:titp_ff} (fission fragments).
For \ensuremath{\gamma}-rays, the thermalization efficiency is approximately equal to the interaction probability: $f_{\ensuremath{\gamma}}(t) \approx 1 - e^{-\tau}.$
We can estimate the optical depth $\tau \approx \rho \kappa_{\ensuremath{\gamma}} R_{\rm ej}$ using $\bar{\kappa}_\ensuremath{\gamma}$, the \ensuremath{\gamma}-ray opacity averaged over the emission spectrum. Optical depth is related to $t_{\rm ineff,\ensuremath{\gamma}}$ by
\begin{gather}
\left(\frac{t_{\rm ineff,\ensuremath{\gamma}}}{t_0}\right)^{2} = \rho_0 \bar{\kappa}_\ensuremath{\gamma} R_0 = \tau_0 \nonumber \\
\rightarrow \tau(t) = \tau_0\left(\frac{t}{t_0}\right)^{-2} = \left( \frac{t_{\rm ineff,\ensuremath{\gamma}}}{t}\right)^2 \nonumber,
\intertext{so}
f_\ensuremath{\gamma}(t) = 1 - \exp\left[-\left(\frac{t}{t_{\rm ineff,\ensuremath{\gamma}}}\right)^{-2}\right] \label{eq:ftg_an}
\end{gather}
Figure \ref{fig:anSimComp} shows our analytic thermalization functions for $\ensuremath{M_{\rm ej}} = 5 \times 10^{-3} M_{\odot}$, and $\ensuremath{v_{\rm ej}} = 0.2c$, using the expressions for $t_{\rm ineff}$ derived in \S \ref{sec:thermRates}. For massive particles, we used $E_{\ensuremath{\beta},0} = 0.5$ MeV, $E_{\alpha,0} = 6$ MeV, and $E_{\rm ff,0} = 125$ MeV. For \ensuremath{\gamma}-rays, we take $\bar{\kappa} = 0.1$ cm$^2$ g$^{-1}$, which gives $t_{\rm ineff,\ensuremath{\gamma}} \approx 1.4$ days.
As we will see in \S \ref{sec:results}, the approximate analytic expressions Eq.s \ref{eq:ftAn} and \ref{eq:ftg_an} agree fairly well with our numerical results.
\begin{figure}
\includegraphics[width=3.5 in]{f9.pdf}
\caption{Analytic thermalization efficiencies, calculated with Eq.s \ref{eq:ftAn} and \ref{eq:ftg_an}. We use $t_0 = 1$ day, and $\rho_0 = 7.9 \times 10^{-15}$ cm$^{-3}$, corresponding to a uniform density ejecta with the same mass and energy as our fiducial model. For \ensuremath{\alpha}'s, \ensuremath{\beta}'s, and fission fragments we take $E_0 = 6, 1,$ and $125$ MeV, respectively.}
\label{fig:anSimComp}
\end{figure}
\section{Numerical Results}
\label{sec:results}
In this section, we present numerical calculations of thermalization efficiencies
as determined by modeling the 3-dimensional
transport of \ensuremath{\gamma}-rays, fission fragments, and \ensuremath{\alpha}- and \ensuremath{\beta}-particles in
a magnetized expanding medium.
Our calculations used the time-evolving emission spectra introduced in \S \ref{subsec:spec}, accounted for the time-dependent partition of radioactive energy among different decay products, and incorporated the detailed, energy-dependent energy loss rates derived in \S \ref{sec:thermRates}. The flux tube approximation was used to model charged particle transport, allowing us to explore the sensitivity of our results to the architecture of the ejecta's magnetic field.
Additional details of our transport method
are given in the Appendix.
\subsection{Thermalization efficiencies}
Figure \ref{fig:ft_PartFid} presents the numerically calculated thermalization efficiency, $f(t)$, of all particles for the fiducial ejecta model (\ensuremath{M_{\rm ej}} = $5\times 10^{-3} \ensuremath{M_{\odot}}$ and \ensuremath{v_{\rm ej}} = $0.2c$.) Fission fragments thermalize most efficiently, having $f(t) \gtrsim 0.5$ out to $t \sim 15$ days. Alpha- and \ensuremath{\beta}-particle thermalization is slightly lower, reaching $f(t)= 0.5$ around a week post-merger, while $f(t)$ for \ensuremath{\gamma}-rays is much lower, falling below 0.5 by $t\sim 1$ day.
For massive particles, we show $f(t)$ for radial (dotted lines), toroidal (solid lines), and lightly tangled ($\lambda = 0.25$; dashed lines) magnetic field geometries.
The magnetic field configuration
affects thermalization in three ways:
\begin{enumerate}
\item \textbf{Diffusion:} Radial or lightly tangled fields allow particles to diffuse outward into regions of lower density, and lead to lower $f(t)$.
\item \textbf{Escape:} Radial fields that allow charged particles to escape before they have completely thermalized will lower $f(t)$. This is most important for \ensuremath{\beta}-particles, which move faster than the ejecta.
\item \textbf{Frame-to-frame effects:} Particles in a homologous flow
lose energy, as measured in the co-moving frame (cmf), as they move through the ejecta. These frame-to-frame losses reduce the amount
of kinetic energy a particle has to thermalize, and therefore reduce
$f(t)$. Radial fields and lightly tangled fields, which allow particles to move fairly freely through the ejecta, facilitate frame-to-frame effects.
These losses are most important for \ensuremath{\alpha}-particles and fission
fragments, which have velocities of order \ensuremath{v_{\rm ej}}, and thus have
substantially different cmf energies in different regions of the
ejecta.
\end{enumerate}
In light of the above, it is not surprising that toroidal fields maximize $f(t)$; toroidal fields hold particles at one position in velocity space, preventing diffusion, escape, and frame-to-frame losses. Radial fields, in contrast, enhance all
three of these effects and hence minimize $f(t)$.
Thermalization in random fields
falls between these two extremes. This behavior holds for all ejecta models studied.
\begin{figure}
\includegraphics[width=3.5 in]{f10.pdf}
\caption{Thermalization efficiencies $f(t)$ for all particles in an ejecta with $M_{\rm ej} = 5\times 10^{-3} \ensuremath{M_{\odot}}$, and $v_{\rm ej} = 0.2c$ (our fiducial model). Fission fragments thermalize most efficiently, followed by \ensuremath{\alpha}-particles, \ensuremath{\beta}-particles, and \ensuremath{\gamma}-rays. For charged particles we plot $f(t)$ for radial (dotted lines), toroidal (solid lines), and moderately tangled ($\lambda = 0.25$; dashed lines) magnetic fields. Toroidal fields thermalize most efficiently, followed by random, then radial fields.}
\label{fig:ft_PartFid}
\end{figure}
While the trends shown in Figure~\ref{fig:ft_PartFid}---i.e., that
$f_{\rm ff}(t) > f_{\ensuremath{\alpha}}(t) \approx f_\ensuremath{\beta}(t) > f_\ensuremath{\gamma}(t)$---are
consistent across ejecta models, the values of $f(t)$ can
vary significantly with \ensuremath{M_{\rm ej}}\ and \ensuremath{v_{\rm ej}}. Figure \ref{fig:t50}
illustrates the variance and clarifies the dependence of $f(t)$ on the
ejecta parameters. For each point (\ensuremath{M_{\rm ej}}, \ensuremath{v_{\rm ej}}) in parameter space, and
for each particle type, we plot $t_{50}$---the time at which $f(t)$
drops to 50\%. Cases in which $f(t = 30 \: {\rm days}) > 50\%$ are
omitted from Figure \ref{fig:t50}.) To show how sensitive
thermalization is to magnetic fields, we include results for radial
(top panel) and toroidal (bottom panel) field geometries.
The thermalization of all particles increases with \ensuremath{M_{\rm ej}}\ and decreases
with \ensuremath{v_{\rm ej}}. The changes in efficiency are especially dramatic for
massive particles. For the heaviest ejecta mass considered
($\ensuremath{M_{\rm ej}} = 5 \times 10^{-2} \ensuremath{M_{\odot}}$), massive particles thermalize
efficiently out to late times regardless of $\ensuremath{v_{\rm ej}}$. The thermalization
of \ensuremath{\gamma}-ray energy is low for all models tested.
Though the results shown are for the two-component composition described above, we find that the higher energy loss rates in the outer ejecta have only a small impact on thermalization. For radial fields, where the effect is most pronounced, the two-zone model results in an increase in total thermalization of $< 5$\% relative to a one-zone model that assumes the entire ejecta has the ``inner'' ejecta composition. The $f(t)$ we calculate should be fairly insensitive to the exact division between inner and outer ejecta.
\begin{figure}
\includegraphics[width = 3.5 in]{f11.pdf}
\caption{ The time at which $f(t)$ drops below 50\% ($t_{50}$) for all
particles, for all \ensuremath{M_{\rm ej}}\ and \ensuremath{v_{\rm ej}}\ considered. Results for a radial
(toroidal) magnetic field are shown in the top (bottom)
panel. Thermalization increases with mass and decreases with
velocity. Fission fragments thermalize most efficiently, followed by
\ensuremath{\alpha}-particles and \ensuremath{\beta}-particles, and finally \ensuremath{\gamma}-rays. Toroidal
fields result in more robust thermalization of all massive
particles.}
\label{fig:t50}
\end{figure}
\subsubsection{Effect of aspherical ejecta}
The ejecta from a CO merger is likely to be aspherical, particularly in the case of NSBH merger, where most of the ejected mass is confined to the equatorial plane \citep[e.g.,][]{Hotokezaka_2013_massEj}. To estimate the effect on thermalization, we compare a spherical model to an oblate one, where both models have \ensuremath{M_{\rm ej}} = $5\times 10^{-3}\ensuremath{M_{\odot}}$ and \ensuremath{v_{\rm ej}} = $0.2c$, radial magnetic fields, and a broken power law density profile with $(\delta, n) = (1,10)$. For the oblate geometry, the density is a function of $\tilde{v}$, where
\begin{equation*}
\tilde{v} = v\sqrt{a^{-2/3}\sin^2\theta + a^{4/3}\cos^2\theta}
\end{equation*}
is chosen so that isodensity contours are oblate spheroids of aspect ratio $a$.
Figure \ref{fig:sphVOb} compares the $f(t)$ for the oblate and spherical cases, and shows
that massive particle thermalization increases with increasing asphericity. For an aspect ratio $a=4$, the $f(t)$ for \ensuremath{\alpha}'s, \ensuremath{\beta}'s, and fission fragments increase by a factor of $\sim 1.5$ relative to spherical ejecta. Gamma-ray thermalization is higher for the oblate geometry, but only slightly.
The higher $f(t)$ are due to the higher density of the oblate ejecta, which more than compensates for the increased ease of escape in directions perpendicular to the equatorial plane.
Figure~\ref{fig:sphVOb} shows $f(t)$ only for radial magnetic fields, but we found similar increases for random and toroidal fields.
\begin{figure}
\includegraphics[width = 3.5 in]{f12.pdf}
\caption{Thermalization efficiencies for oblate ejecta with aspect ratio $a=4$, compared to the standard spherical geometry, for the fiducial mass and velocity and radial magnetic fields. Thermalization increases with increasing asymmetry. We found similar increases for random and toroidal fields.}
\label{fig:sphVOb}
\end{figure}
\subsection{Total heating efficiency}
To study the net heating efficiency, we convolve $f(t)$
for each decay product with the fraction that particle contributes to the total energy generation.
The bottom panel of Figure~\ref{fig:ftTot_FRDM} shows how
the \emph{r}-process\ decay
energy is divided among different particles, while the top panel shows the energy \emph{thermalized} by each particle
type, as a fraction of the total energy emitted across all decay channels. The $f(t)$ represented in Figure \ref{fig:ftTot_FRDM} are for a fiducial ejecta model with moderately tangled ($\lambda = 0.25$) magnetic fields. The total thermalization efficiency, which is simply a sum over particle types, is plotted in black. While \ensuremath{\gamma}'s, \ensuremath{\alpha}'s, \ensuremath{\beta}'s, and fission fragments all have $f(t) \approx 1$ at very early times, the initial \emph{total} thermalization efficiency is less than one because a significant fraction of the \ensuremath{\beta}-decay energy is lost to neutrinos.
The net thermalization efficiency, in this model, drops below 0.5 by $t=1$ day, and below 0.1 by $t \sim 10$ days.
While \ensuremath{\beta}-particles and \ensuremath{\gamma}-rays dominate the energy production at all times, \ensuremath{\gamma}-rays thermalize inefficiently, and supply very little heating after $t \sim 1$ day. While \ensuremath{\alpha}-decay produces less than $\sim 10$\% percent of the total energy, the $\alpha$-particles thermalize fairly efficiently, and so
contribute a significant fraction of the total thermalized energy.
\begin{figure}
\includegraphics[width = 3.5 in]{f13.pdf}
\caption{\textit{Bottom panel:} The fractional energy generation associated with
each type of particle, from \emph{r}-process\ simulations using the FRDM mass
model. The division of \ensuremath{\beta}-decay energy among \ensuremath{\beta}-particles,
\ensuremath{\gamma}-rays, and neutrinos was calculated for our representative SPH
trajectory with $Y_{\rm e, 0} = 0.04$. \textit{Top panel:} The fractions from the bottom panel, convolved with $f(t)$ for
each particle, for the fiducial model with random magnetic
fields. The total thermalization efficiency, $f_{\rm tot}$, plotted
as a dashed black line, is the sum of the particle-specific
curves. Beta- and \ensuremath{\alpha}-particles supply most of the thermalized
energy.}
\label{fig:ftTot_FRDM}
\end{figure}
The total heating efficiency has the expected dependence on the ejecta
parameters: greater masses and lower velocities lead to higher \ensuremath{f_{\rm tot}(t)},
as shown in Figure \ref{fig:ftTot_MV}. Thermalization for the low-mass and high-velocity models falls below 0.5 within a few days, and
below 0.2 by $5-7$ days. The high-mass and low-velocity models
thermalize much more efficiently, sustaining $f_{\rm tot}(t) > 0.5$
out to $t \lesssim 1$ week, and not falling below
$f_{\rm tot}(t) = 0.2$ until $t \sim 15 - 20$ days. There is also
variation within each model (up to a factor of $\sim 2$) due to
uncertainties in the magnetic field.
\begin{figure}
\includegraphics[width = 3.5 in]{f14.pdf}
\caption{Total thermalization efficiencies for different ejecta models
(\ensuremath{M_{\rm ej}}, \ensuremath{v_{\rm ej}}) using FRDM energy generation rates. The fiducial model
is plotted in black. Other curves differ from the fiducial model in
\ensuremath{M_{\rm ej}}\ or \ensuremath{v_{\rm ej}}\ only. The width of the curves reflects the variation in
$f(t)$ for different magnetic field configurations; the curves are
bounded on top by \ensuremath{f_{\rm tot}(t)}\ for a toroidal field and on bottom by
$\ensuremath{f_{\rm tot}(t)}$ for a radial field configuration. }
\label{fig:ftTot_MV}
\end{figure}
\subsubsection{Dependence on nuclear physics}
\label{subsec:nucPhys}
The radioactive energy generation---and therefore the thermalization---depends
on \emph{r}-process\ yields, which in turn are sensitive to variations in nuclear physics models and astrophysical conditions. To explore
this effect, we consider \emph{r}-process\ yields computed for different mass
models, and for different initial $Y_{\rm e}$ of the ejected matter.
The yields differ primarily in the amount of translead nuclei synthesized relative to lighter \emph{r}-process\ elements.
\cite{MendozaTemis_etal_rProcess}
have shown that the production of translead nuclei is sensitive
to nuclear physics inputs, in particular to neutron
separation energies near $N=130$.
As discussed in \S\ref{subsec:comp}, the production of translead nuclei also depends on initial electron fraction, decreasing as $Y_{\rm e,0}$ increases.
\emph{R}-process yields could impact thermalization in two ways. First, different yields have different abundance-averaged compositional properties, and could give rise to different thermalization rates.
Second, because nuclei heavier than lead decay mainly by fission and \ensuremath{\alpha}-emission, while lighter nuclei undergo \ensuremath{\beta}-decay, the amount of translead material will alter the relative importance of \ensuremath{\alpha}- and \ensuremath{\beta}-decay. Since all \ensuremath{\alpha}-decay energy is transferred to energetic \ensuremath{\alpha}-particles, which thermalize efficiently, while $\gtrsim 70$\% of \ensuremath{\beta}-decay energy goes to \ensuremath{\gamma}-rays and neutrinos, which do not, enhanced \ensuremath{\alpha}-decay may increase thermalization. Based on these arguments,
we expect that differences in the amounts of translead nuclei
will result in different $f_{\rm tot}(t)$, and therefore, differences in predicted kilonova light curves.
To explore the strength of these effects, we compare the thermalization efficiency for three different compositions:
the reference \emph{r}-process\ yields (based on the FRDM mass model); yields for the DZ31 mass model, which predicts increased production of
translead nuclei (see Figure~\ref{fig:comp}); and yields from a calculation using the FRDM model with $Y_{e,0} = 0.25$.
We found that the DZ31 model predicts a composition whose abundance-averaged properties and emission spectra are very similar to those predicted by the FRDM model. We therefore expect that the different yields found for the DZ31 model will not significantly change $f(t)$ for individual particles. In contrast, the high-$Y_{\rm e,0}$ composition has average compositional properties and emission spectra that depart from the reference case (FRDM, $Y_{\rm e,0} = 0.04$), so we calculate for this composition $f(t)$ of all individual decay products for our fiducial ejecta
($\ensuremath{M_{\rm ej}} = 5 \times 10^{-3} \ensuremath{M_{\odot}}$, $\ensuremath{v_{\rm ej}} = 0.2c$).
The thermalization
timescales, plotted in Figure \ref{fig:t50} as
open triangles, are similar to those for the standard
low-$Y_{\rm e,0}$ composition.
For both the DZ31 and high-$Y_{\rm e,0}$ cases then, impacts on \ensuremath{f_{\rm tot}(t)}\ result from differences in the relative importance of each heating channel, not differences in how efficiently individual decay products
thermalize.
Figure \ref{fig:ft_nucDep} compares \ensuremath{f_{\rm tot}(t)}\ for the three cases studied.
In the top panel, we show $f_{\rm tot}(t)$ and the
contributions from each decay product, determined using energy generation rates from
the DZ31 nuclear mass model abundances, for which \ensuremath{\alpha}-decay dominates
the energy production at late times. The middle panel shows an
analogous calculation for the FRDM model with $Y_{\rm e,0} = 0.25$,
which has negligible late-time \ensuremath{\alpha}-decay.
In the bottom panel, we compare $f_{\rm tot}(t)$ for these models with
the fiducial FRDM model. The greater role
of \ensuremath{\alpha}-decay in the DZ31 model increases \ensuremath{f_{\rm tot}(t)}\ by a factor of
$\gtrsim 1.5$, mainly due to the fact that less energy is lost in
neutrinos and \ensuremath{\gamma}-rays, which thermalize very inefficiently. In the fiducial composition \ensuremath{\alpha}-decay and fission produce only a
small fraction of the energy, so the effect
of increasing $Y_{\rm e,0}$ is modest. A stronger effect might be seen for DZ31, which produces more translead nuclei when $Y_{\rm e,0}$ is low, and is therefore more likely to experience dramatic decreases in translead production when the initial electron fraction rises.
\begin{figure}
\includegraphics[width = 3.5 in]{f15.pdf}
\caption{The effect of nuclear physics inputs on total thermalization
efficiency. \textit{Panels 1 and 2:} $f(t)$ (fiducial \ensuremath{M_{\rm ej}}, \ensuremath{v_{\rm ej}};
random fields) convolved with fractional energy generation rates for the DZ31
nuclear mass model (top panel) and a high-$Y_{\rm e,0}$ FRDM
trajectory (middle panel). Solid lines show the fraction of emitted
energy thermalized by each particle as a function of time, and
$f_{\rm tot}(t)$ is plotted in black dashed lines. \textit{Panel 3:}
The range of $f_{\rm tot}(t)$ expected for each of the cases shown
in Panels 1 and 2. We plot $f_{\rm tot}(t)$ for the low-$Y_{\rm e,0}$ FRDM composition in black for comparison. The widths of the curves are due to
the range of possible magnetic field configurations.}
\label{fig:ft_nucDep}
\end{figure}
\begin{table}
\centering
\caption{Analytic fit parameters for $f_{\rm tot}(t)$}
\label{tab:coeffs}
\begin{tabular}{cc|ccc}
\multicolumn{2}{c|}{Model} & \multicolumn{3}{c}{Coefficients} \\
$M/M_{\odot}$ & $v_{\rm ej}/c$ & $a$ & $b$ & $d$ \\
\hline
\hline
$1 \times 10^{-3}$ & 0.1 & 2.01 & 0.28 & 1.12\\
$1 \times 10^{-3}$ & 0.2 & 4.52 & 0.62 & 1.39\\
$1 \times 10^{-3}$ & 0.3 & 8.16 & 1.19 & 1.52\\
$5 \times 10^{-3}$ & 0.1 & 0.81 & 0.19 & 0.86\\
$5 \times 10^{-3}$ & 0.2 & 1.90 & 0.28 & 1.21\\
$5 \times 10^{-3}$ & 0.3 & 3.20 & 0.45 & 1.39\\
$1 \times 10^{-2}$ & 0.1 & 0.56 & 0.17 & 0.74\\
$1 \times 10^{-2}$ & 0.2 & 1.31 & 0.21 & 1.13\\
$1 \times 10^{-2}$ & 0.3 & 2.19 & 0.31 & 1.32\\
$5 \times 10^{-2}$ & 0.1 & 0.27 & 0.10 & 0.60\\
$5 \times 10^{-2}$ & 0.2 & 0.55 & 0.13 & 0.90\\
$5 \times 10^{-2}$ & 0.3 & 0.95 & 0.15 & 1.13\\
\end{tabular}
\end{table}
\section{Effect on Kilonova Light curves}
\label{sec:LCs}
To determine the effect of thermalization on kilonova
observables, we incorporated our results for \ensuremath{f_{\rm tot}(t)}\ into the
time-dependent Monte Carlo radiation transport code \texttt{Sedona}
\citep{Kasen_MC}, and carried out light curve calculations. The calculations here resemble those of \citet{Barnes_2013}, but include thermalization effects.
\subsection{Analytic fit to thermalization efficiency}
For easy inclusion of thermalization effects in light
curve simulations, we propose a simple analytic formula for $f_{\rm
tot}(t)$ which provides a good fit to our detailed
numerical calculations,
\begin{equation}
f_{\rm tot}(t) = 0.36\left[ \exp\left( -at \right) +\\
\frac{\ln \left( 1 + 2bt^d \right) }{2b t^{d}} \right],
\label{eq:fTot}
\end{equation}
where $a$, $b$, and $d$ are fitting constants. The parameterized form
of Eq. \ref{eq:fTot} is motivated by our approximate analytic
solutions for $f(t)$ (Eq.s \ref{eq:ftAn} and \ref{eq:ftg_an}), with
slight modifications to improve the quality of the fit and account for
energy lost to neutrinos. Table~\ref{tab:coeffs} gives the best-fit
parameters for all the ejecta models considered. These fits assume the
FRDM nuclear mass model and random magnetic fields.
We found that, for the FRDM mass model, compositions from high-$Y_{\rm e,0}$ ejecta have thermalization profiles similar to compositions from initially neutron-rich ejecta.
This suggests that our thermalization models may be appropriate for material ejected dynamically and from disk winds, regardless of the initial electron fraction.
However, we note that the insensitivity of \ensuremath{f_{\rm tot}(t)}\ to $Y_{\rm e,0}$ may not be as robust for other nuclear mass models.
The effect of $Y_{\rm e,0}$ may be particularly pronounced for the DZ31 model, which produces large amounts of translead nuclei, and therefore predicts significant \ensuremath{\alpha}-decay. Changes in $Y_{\rm e,0}$ could inhibit the production of these nuclei, decrease the role of \ensuremath{\alpha}-decay, and thus alter thermalization efficiency.
The effect on \ensuremath{f_{\rm tot}(t)}\ would be much stronger than for the FRDM model, which does not produce many translead nuclei even for favorable $Y_{\rm e,0}.$
\subsection{Bolometric light curves}
The net thermalization efficiency, $f_{\rm tot}(t)$, has a significant
impact on kilonova luminosity. Figure \ref{fig:LCeffs} compares
bolometric light curves calculated using our derived $f_{\rm tot}(t)$
to those assuming 100\% thermalization. We also show results for a
treatment which propagates \ensuremath{\gamma}-rays, but assumes charged particle
energy thermalizes instantly. This was the method used to estimate
$f_{\rm tot}(t)$ in earlier \texttt{Sedona} kilonova simulations,
including \citet{Barnes_2013}.\footnote{\texttt{Sedona}'s original
treatment of thermalization assumed that \ensuremath{\beta}-decay generated 90\% of
the \emph{r}-process\ decay energy, with fission accounting for the other 10\%. Of the
\ensuremath{\beta}-decay energy, 25\% was taken to be lost to neutrinos, and the
remaining 75\% was split evenly between \ensuremath{\beta}-particles and
\ensuremath{\gamma}-rays. The energy from \ensuremath{\beta}-particles and fission fragments was
thermalized promptly, while the energy from \ensuremath{\gamma}-rays was converted
into 1 MeV photons, which were propagated through the ejecta in a
Monte Carlo transport scheme.} (A similar simplification was invoked in the discussion of net heating by \citet{Hotokezaka_2015_heating}.) For all radiation transport
simulations, we have used the simplified composition and the boosted,
synthetic \emph{r}-process\ opacities of \citet{Kasen_2013_AS}. We consider here
only models with low $Y_{\rm e,0}$, which robustly produce \emph{r}-process\
elements including Lanthanides and Actinides, making our choice of
opacity appropriate. Models with higher initial electron fractions may
fail to produce these heavy elements. The opacities for such models
would be much lower, and the associated light curves would be shorter,
brighter, and bluer \citep[e.g.][]{Metzger_2010, Barnes_2013, Kasen_Fernandez_Metzger_dwkilonova}.
Figure~\ref{fig:LCeffs} shows that our more accurate treatment of
thermalization impacts predicted photometry for all ejecta models
considered. Relative to earlier calculations with less sophisticated
thermalization schemes, we find kilonova light curves peak
slightly earlier, have lower luminosities at peak, and have much
dimmer late-time luminosities.
The effects of thermalization are most pronounced for less massive and
higher velocity ejecta models, which are dimmer and fade more quickly
than their slower, more massive counterparts.
Nuclear physics plays a role by determining the yields of translead
nuclei, and therefore the amount of energy produced by
\ensuremath{\alpha}-decay. Models for which \ensuremath{\alpha}-decay contributes prominently to the
energy generation yield somewhat higher $f_{\rm tot}(t)$. The middle
panel of Figure \ref{fig:LCeffs}, which shows light curves
corresponding to both FRDM and DZ31 (high \ensuremath{\alpha}-heating) mass models,
illustrates this point. The increased luminosity found
for the DZ31 composition is a result both of higher \ensuremath{f_{\rm tot}(t)}\ due to
more \ensuremath{\alpha}-decay, and to the greater absolute amount of energy produced by
\emph{r}-process\ decay for the DZ31 model relative to the FRDM model (see \S
\ref{subsec:Ej_Radio}.)
\begin{figure}
\includegraphics[width=3.5 in]{f16.pdf}
\caption{Synthetic bolometric light curves for our fiducial ejecta
model, calculated with \texttt{Sedona} for three different
treatments of thermalization: full thermalization (blue curve);
\texttt{Sedona}'s original thermalization scheme, which deposits
charged particle energy but explicitly tracks the deposition of
\ensuremath{\gamma}-ray energy (lime curve); and the time-dependent $f_{\rm tot}(t)$
from our numerical simulations (red curve). Accounting for
time-dependent thermalization efficiencies has a significant impact
on kilonova luminosity, particularly for models with lower masses
and higher luminosities. For our fiducial model, the predicted
luminosity is lower by a factor of $\lesssim 2$ at peak, and by 10
days is lower by an factor of 5.}
\label{fig:LCeffs}
\end{figure}
\subsection{Implications for the kilonova accompanying GRB 130603B}
An excess near infrared (NIR) flux discovered in the afterglow of the
short gamma ray burst GRB 130603B has been widely interpreted as a
kilonova \citep{Tanvir_2013_kNa,
Berger_2013_kNa}. \citet{Tanvir_2013_kNa} determined that the source
of the flux had an absolute AB magnitude in the \emph{J}-band of
-15.35 at $t \sim 7$ days.
Having incorporated $f_{\rm tot}(t)$ into kilonova light curve models,
we can more confidently constrain the mass ejected in the kilonova
associated with GRB 130603B.
In Figure \ref{fig:ir_bbs}, we compare the detected flux to
\emph{J}-band light curves for various ejecta models, and find the
observed flux is consistent with
$5 \times 10^{-2} M_{\odot} \lesssim M_{\rm ej} \lesssim 10^{-1}
\ensuremath{M_{\odot}}$.
This mass is higher than what is typically predicted for the dynamical
ejecta from a binary neutron star merger, suggesting that if the
kilonova interpretation is correct, the progenitor of GRB~130603B was
perhaps a neutron star-black hole merger, or that the mass ejected was
significantly enhanced by post-merger disk winds.
\begin{figure}
\includegraphics[width=3.5 in]{f17.pdf}
\caption{Absolute (AB) \emph{J}-band light curves for several ejecta models. The excess IR flux (gold star) suggests an ejected mass between $5 \times 10^{-2}$ and $10^{-1} \ensuremath{M_{\odot}}$. }
\label{fig:ir_bbs}
\end{figure}
Our mass estimate here is an improvement over earlier work which neglected detailed thermalization, and gives substantially different results. For example, \citet{Piran_2014_130603BkNa} suggested $\ensuremath{M_{\rm ej}} \sim 0.02 \ensuremath{M_{\odot}}$, less than half our new value.
However,
we have not accounted for viewing angle effects.
If the ejected material is mainly confined to the equatorial
plane, the emission will be brighter when the system is viewed face-on \citep{Roberts_2011}, which
would reduce the inferred mass somewhat. If the ejecta is oblate, thermalization will also be more efficient, which could have a small impact on mass estimates. Radiation transport simulations in three dimensions with time-dependent thermalization models will further constrain $M_{\rm ej}$.
\subsection{Late-time light curve}
\label{subsec:lateLC}
Late time kilonova light curves may probe the history of \emph{r}-process\
nucleosynthesis in CO mergers. At $\sim 2$ days after merger, fission
ceases to be important, and \ensuremath{\alpha}- and \ensuremath{\beta}-decay dominate the kilonova's
energy supply. Energy from \ensuremath{\alpha}-decay is transferred entirely to fast
\ensuremath{\alpha}-particles, which thermalize fairly efficiently out to late
times. Beta particles thermalize with similar efficiency, but carry
only a fraction ($\sim 25\%$) of the total \ensuremath{\beta}-decay energy, with the
rest lost to neutrinos and \ensuremath{\gamma}-rays. A kilonova's late-time luminosity
will therefore depend on the relative importance of \ensuremath{\alpha}- versus
\ensuremath{\beta}-decay. Because only nuclei with $200 \lesssim A \lesssim 250$
undergo \ensuremath{\alpha}-decay, the late time kilonova luminosity may diagnose the
presence of heavy elements in the ejecta, and therefore constrain the
neutron-rich conditions required for heavy element formation.
We gauge the relative strength of late-time kilonova light curves for
different $Y_{\rm e,0}$ by estimating the percent of energy from the decay of \emph{r}-process\ elements
emitted as fission fragments, \ensuremath{\alpha}-, and \ensuremath{\beta}-particles, time-averaged over $t=10-100$
days. (Note that while all energy from \ensuremath{\alpha}-decay emerges as \ensuremath{\alpha}-particles, \ensuremath{\beta}-\emph{particles} receive only 25-30\% of the energy from \ensuremath{\beta}-\emph{decay}.) The results for our representative SPH trajectory, for a range
of $Y_{e,0}$ and two nuclear mass models, are shown in Figure
\ref{fig:ltLCedot}. The curves suggest that systems with
$Y_{\rm e,0} \lesssim 0.17$ have more robust late-time heating, and are likely to exhibit late time light curves that are more luminous by a factor of up to $\sim 2$.
If fission is more significant at late times than our calculation predicts (e.g. \citet{Hotokezaka_2015_heating} find that fission supplies $\gtrsim 10\%$ of the total energy out to late times) the dependence of the late-time light curve on $Y_{\rm e,0}$ could be much stronger. Fission fragments thermalize extremely efficiently well past maximum light. Since very neutron-rich conditions are needed to build up the heavy nuclei ($A \gtrsim 250$) that undergo fission, in a strongly fissioning ejecta, late-time luminosity could depend sensitively on $Y_{\rm e,0}$.
A key observable for probing the nuclear physics of
CO mergers may therefore be the ratio of the kilonova luminosity measured at peak brightness to that observed
on the light curve tail. The former is powered by $\beta$-decay, while the latter is driven by $\alpha$-decay and fission, which thermalize more efficiently. This ratio could therefore constrain the composition of the ejecta, and so the conditions of nucleosynthesis. More detailed studies are needed to clarify this relationship.
\begin{figure}
\includegraphics[width=3.5 in]{f18.pdf}
\caption{The time-averaged fraction of the \emph{r}-process\ decay energy emitted as \ensuremath{\beta}- and \ensuremath{\alpha}-particles and fission, as a function of initial $Y_{\rm e}$. The data was taken from \emph{r}-process\ calculations on our representative SPH trajectory, with $Y_{\rm e,0}$ artificially altered. The lower the initial electron fraction is, the more energy from \ensuremath{\alpha}-decays is available to drive a late time light curve.
}
\label{fig:ltLCedot}
\end{figure}
\section{Conclusion}
We have shown that the radioactive energy from the decay of \emph{r}-process\ material does not completely thermalize in the ejecta from CO mergers. For the first time, we have explicitly simulated the thermalization of all suprathermal \emph{r}-process\ decay products in the heavy-element-rich kilonova ejecta. From these simulations, we derived time-dependent expressions for the net kilonova thermalization efficiency, $f_{\rm tot}(t)$, for models spanning a range of expected ejecta masses and velocities. For most parameters studied, \ensuremath{f_{\rm tot}(t)}\ drops below 0.5 within five days after the merger. At 15 days after merger, \ensuremath{f_{\rm tot}(t)}\ may be as low as $0.01-0.1$. Thermalization therefore has a significant impact on the peak luminosity of kilonovae and the late time light curve decline rate.
We have also explored the dependence of $f_{\rm tot}(t)$ on electron fraction and nuclear mass model, and outlined how variations in these parameters may systematically
affect thermalization. In general, systems that favor the production of translead nuclei have higher thermalization efficiencies at late times. This is because a greater fraction of radioactive energy is emitted through $\alpha$-decay and fission channels, which thermalize more efficiently than energy from $\beta$-decays, since less energy is lost to \ensuremath{\gamma}-rays and neutrinos.
We have presented updated radiation transport simulations that incorporate our new calculations of time-dependent thermalization, and find that thermalization has a significant effect on the predicted photometry of kilonovae. Compared to older models that neglected detailed thermalization, our new light curves peak earlier and at lower luminosities, and are much dimmer (by a factor of $\sim 10$) at late times ($\gtrsim 15$~days after peak). Our new models of kilonova with Lanthanide-rich \emph{r}-process\ ejecta keep their characteristic red color, and are much more luminous in infrared (\emph{I}-, \emph{J}-, and \emph{K}-) bands than in the optical.
Our results have consequences for detecting kilonovae, whether
blindly, or as counterparts to gravitational wave events. While our
new models retain the red color believed to be a defining kilonova
signature, the rapid decline of \ensuremath{f_{\rm tot}(t)}\ poses a challenge to detection,
and underscores the necessity of timely follow-up of GW
triggers. Since \ensuremath{f_{\rm tot}(t)}\ is lower, and declines more quickly, in less
massive systems, this is especially important for kilonovae generated
by merging neutron stars, which are expected to be less massive than
BHNS kilonovae by a factor of $\sim 10$.
The recent detection of the gravitational wave event GW150914
\citep{LIGO_150914} spurred a slew of EM follow-up activities
\citep{EM_followup_150914,DeCAM_Followup_GW150914}. While the GW
signal turned out to be the result of a binary black hole merger,
which is not expected to have an EM counterpart, the follow-up
campaign offers a sense of the prospects of detecting a kilonova
counterpart to future GW events. From a kilonova standpoint, some of
the most promising observational efforts were carried out by the Dark
Energy Camera (DECam), which had limiting (AB) magnitudes $i < 22.5$
and $z < 21.5$; the VLT Survey Telescope (VLS), which reached
$r < 22.4$, and VISTA, with $J < 20.7$.
Had the GW trigger been due to a typical NSBH merger located 100 Mpc
distant, the associated kilonova ($\ensuremath{M_{\rm ej}} = 5 \times 10^{-2} \ensuremath{M_{\odot}}$;
$\ensuremath{v_{\rm ej}} = 0.2 c$) could have been observed by DECam in \emph{i} and
\emph{z} for $t \lesssim 7$ days; by VST in \emph{r} for
$t \lesssim 5$ days; and by VISTA in \emph{J} for $t \lesssim 8$
days. The situation is less promising for a NS$^2$ merger ejecting
less mass ($\ensuremath{M_{\rm ej}} = 5 \times 10^{-3} \ensuremath{M_{\odot}}$; $\ensuremath{v_{\rm ej}} = 0.2 c$),
which will be intrinsically dimmer and suffer from less efficient
thermalization. To be visible to DECam in \emph{i} (\emph{z}) at peak,
such a system would need to be closer than $\sim 63$ ($\sim 57$) Mpc,
while VISTA (VST) could only detect it at distances less than
$\sim 52$ ($\sim 10$) Mpc.
This analysis highlights the importance of seeking optical
counterparts at early times, before they fade below detection
thresholds. It also suggests that observing strategies should focus on
depth, rather than area, to improve the chances of detecting signals
that are likely to be faint. Lastly, these findings emphasize the
criticality of developing facilities with greater IR
sensitivity. \emph{Euclid} \citep{Amendola_2012_Euclid} and
\emph{WFIRST} \citep{Green_2012_WFIRST}, each which will have an
\emph{H}-band depth of $\sim 25$, could detect a typical NS$^2$
kilonova, located at 100 Mpc, out to $t \sim 15$ days.
Our calculation of time-dependent thermalization efficiencies for
kilonovae constrains a key uncertainty in models of \emph{r}-process\
transients. Additional work can further improve these models. We have
focused on thermalization in the dynamical ejecta from compact object
mergers, but the ideas developed here could---and should---be applied to
study disk wind outflows, which may produce ejecta poor in
Lanthanides and Actinides and potentially contribute a
blue/optical component to kilonova light curves. Calculating
heating efficiencies for the multiple components believed to make up a
kilonova, and incorporating realistic models of \ensuremath{f_{\rm tot}(t)}\ in
three-dimensional radiation transport simulations of multi-component
light curves would yield the best predictions to date of kilonovae's
EM signatures.
\acknowledgments
This work is supported in part by a Department of Energy Office of Nuclear
Physics Early Career Award, and by the Director, Office of Energy
Research, Office of High Energy and Nuclear Physics, Divisions of
Nuclear Physics, of the U.S. Department of Energy under Contract No.
DE-AC02-05CH11231, and from NSF grant AST-1206097
Support for M-RW and GM-P is provided in part by the Helmholtz Association through the
Nuclear Astrophysics Virtual Institute (VH-VI-417) and the
BMBF-Verbundforschungsprojekt number 05P15RDFN1.
| {
"redpajama_set_name": "RedPajamaArXiv"
} | 39 |
\section*{Introduction}
Active particles are intrinsically out of equilibrium and exhibit
peculiar dynamical behavior ~\cite{Romanczuk:2012,Vicsek:2012, Marchetti:2013,
Elgeti:2015, Bechinger:2016} on the single as well as on the collective level.
These active agents are ubiquitous in nature and include bacteria
~\cite{Berg:1972,Berg:1990,Lauga:2006,Copeland:2009},
algae~\cite{Merchant:2007}, unicellular
protozoa~\cite{Machemer:1972,Blake:1974,Roberts:2010}
or spermatozoa~\cite{Woolley:2003,Riedel:2005}, that move due to a single or an
array of flagella pushed by molecular motors. Only recently, artificial active
particles have been synthesized and are self-propelled by either biomimetic
motors ~\cite{Dreyfus:2005,Kudrolli:2010}, or due to the response of their patterned surface
to chemical or temperature gradients, thereby converting chemical energy into
directed motion
~\cite{Howse:2007,Jiang:2010,Zheng:2013,tenHagen:2014,Lee:2014}. Furthermore,
they also move in crowded media and their effective swimming speed is
strongly determined by the viscoelasticity and geometrical constraints of the
surroundings~\cite{Martinez:2014,Brown:2016}.
To capture analytically the intricacies of the propulsion mechanisms, simple
models for single swimmers have been conceived on different levels of
coarse-graining. Microscopic theories for squirmers
\cite{Lighthill:1952,Blake:1971}, linked-bead swimmers
\cite{Golestanian:2004,Felderhof:2015, Smith:2015}, self-thermophoresis
\cite{Jiang:2010}, and, self-diffusiophoresis~\cite{Wuerger:2015} of Janus
particles have been elaborated and include the full hydrodynamic flow. On a
larger scale, effective models for individual self-propelled particles ignoring
hydrodynamics and the origin of the swimming motion are used to describe the
stochastic motion and the dynamic behavior. There, the dynamics is modeled in
terms of non-equilibrium Langevin equations~\cite{Romanczuk:2012,tenHagen:2014,
Sevilla:2014, vanTeffelen:2008} such that the noise strength is an effective
parameter unrelated to the temperature of the environment, in striking contrast
to the fluctuation-dissipation theorem for equilibrium dynamics. In
particular, these equations of motion serve as a suitable starting point for
simulations~\cite{Volpe:2014}.
The complexity of these transport properties has often been quantified
experimentally and in simulations in terms of low-order moments of the
displacements ~\cite{Howse:2007,Zheng:2013,Brown:2016} and compared to
theoretical models. For example, generically the mean-square displacement
exhibits a regime resembling ballistic motion which directly reflects the
persistent swimming. Only at longer times the motion becomes randomized and
the mean-square displacement increases as anticipated from conventional
diffusion. Higher moments can be derived \cite{Zheng:2013} in principle from
the stochastic equtions of motion, yet the calculations become more and more
cumbersome with increasing order. However, these low-order moments provide
only restricted information on the statistical properties of the random
displacements as a function of time, in particular, they are to a large extend
insensitive to the shape of the probability distribution.
More general spatio-temporal information is encoded in the intermediate
scattering function $F(k,t)$, which resolves the motion of the particle at
lag-time $t$ on a length scale $2\pi/k$, and is directly measurable in
scattering experiments~\cite{Berne:1976} such as dynamic light scattering. The
same quantity can be obtained by advanced image analysis within the recently
developed differential dynamic microscopy (DDM)~\cite{Martinez:2012,Poon:2016},
which provides direct access to the relevant length scales of active particles.
Of course, single-particle tracking also collects the full statistical
information and the intermediate scattering function can be obtained from this information,
yet often the temporal resolution is not high enough to monitor the dynamics on
small length scales. Last, the intermediate scattering function can also be
viewed as the characteristic function~\cite{Gardiner:2009} of the random displacements, which is
equivalent to the full probability distribution. In particular, the moments of
the displacements are encoded as derivatives with respect to the wavenumber.
Theoretical approaches to the intermediate scattering function for active
particles are rare \cite{Sevilla:2015} and no exact solutions appear to be
available.
\section*{Dynamics of an Active Brownian Particle \label{sec:langevin}}
\subsection*{Model}
We assume the active Brownian particle to move at constant velocity $v$ along its
instantaneous orientation $\vec{u}(t)$ subject to random fluctuations
determined by the rotational diffusion coefficient $D_\text{rot}$. This
diffusion process can geometrically be regarded as the diffusion of the
orientation $\vec{u}(t)$ on the unit sphere, as Fig.~\ref{fig:rod}.
In addition, the motion of the anisotropic active particle is
characterized by axisymmetric translational diffusion measured in terms of
the short time diffusion coefficients parallel ($D_\parallel$) and
perpendicular ($D_\perp$) to the anisotropic particle, Fig.~\ref{fig:rod}.
\begin{figure}[h]
\centering
\includegraphics[width=0.5\linewidth, keepaspectratio]{Fig-1-Franosch}
\caption{Model set up. Left: Anisotropic particle with orientation
$\vec{u}(t)$ and translational $D_\parallel,D_\perp$ and rotational
$D_\text{rot}$ diffusion coefficients. Right: Diffusion of the orientation
$\vec{u}(t)$ on the unit sphere. \label{fig:rod}}
\end{figure}
\noindent
Hence, for a three dimensional swimmer the dynamics are described by the
Langevin equations in It$\bar{\text{o}}$ form for the position $\vec{r}(t)$ and
the orientation $\vec{u}(t)$
\begin{align}
\diff \vec{u}(t) &= -2D_\text{rot}\vec{u}(t) \diff t-\sqrt{2D_\text{rot}}\vec{u}(t)\times \diff\boldsymbol{\xi}(t), \label{eq:u}\\
\diff \vec{r}(t) &= v\vec{u}(t) \diff t + \left[\sqrt{2D_\parallel}\vec{u}(t)\vec{u}(t)^{\text{T}}+\sqrt{2D_\perp}\left(\mathbb{I}-\vec{u}(t)\vec{u}(t)^\text{T}\right)\right]\diff\boldsymbol{\zeta}(t).\label{eq:r}
\end{align}
Here the diffusion coefficients $D_\parallel$ and $D_\perp$ for the motion
along and perpendicular to the axis of the swimmer encode the
translational-rotational coupling. The random fluctuations are modeled in
terms of independent white noise processes, $\boldsymbol \xi(t)$ and
$\boldsymbol \zeta(t)$ with zero mean and covariance $\langle
\xi_i(t)\xi_j(t')\rangle =
\langle\zeta_i(t)\zeta_j(t')\rangle=\delta_{ij}\delta(t-t')$ for $i,j=1,2,3$.
The drift term in Eq.~(\ref{eq:u}) ensures that the normalization
condition remains fulfilled, $\diff[\vec{u}(t)^2]/\diff t =0$. Let us
emphasize that if the Stratonovich interpretation is used, the drift term
in the equation for the orientation needs to be dropped.
The model contains two dimensionless parameters, first the translational
anisotropy $\Delta D = D_\parallel-D_\perp$ relative to the mean diffusion
coefficient $\bar{D}=(D_\parallel+2D_\perp)/3$. For passive rod-like particles
in the limit of very large aspect ratio
hydrodynamic suggests $D_\parallel = 2D_\perp$~\cite{Doi:1986}, such that $\Delta
D/\bar{D}=3/4$. Here we consider $D_\parallel$ and $D_\perp$ as effective
parameters quantifying the noise only, and the anisotropy can take arbitrary
values in $-3/2 \leq \Delta D/\bar{D} \leq 3$.
Next, the problem displays a characterstic length,
$a=\sqrt{3\bar{D}/D_\text{rot}}/2$, which corresponds to the geometric radius of a spherical
particle in the case of equilibrium diffusion coefficients $D_\text{rot}=k_\text{B}T/8\pi\eta a^3$
and $\bar{D}=k_\text{B}T/6\pi\eta a$. Then the second dimensionless
parameter is the P{\'e}clet number $\text{Pe}=va/\bar{D}$
measuring the relative importance of the active motion with respect to
diffusion.
\subsection*{Analytic solution}
From the stochastic differential equations one derives the Fokker-Planck
equation~\cite{Birkhaeuser:2009,Gardiner:2009} for the time
evolution of the probability density
$\mathbb{P}(\vec{r},\vec{u},t|\vec{r}_0,\vec{u}_0,t_0)$ to find the swimmer at
position $\vec{r}$, with orientation $\vec{u}$ at time $t$ given that it has
been at some position $\vec{r}_0$ with initial orientation $\vec{u}_0$ at an
earlier time $t_0$. Since the stochastic process is translational invariant in
time and space, only displacements $\Delta \vec{r}=\vec{r}-\vec{r}_0$ and
lag times $t$ (with $t_0=0$) have to be considered,
$\mathbb{P}\equiv\mathbb{P}(\Delta\vec{r},\vec{u},t|\vec{u}_0)$. Then the
Fokker-Planck equation assumes the form
\begin{align}
\partial_t \mathbb{P} &= -v\vec{u}\cdot\partial_\vec{r}\mathbb{P}+D_\text{rot} \Delta_{\vec{u}}\mathbb{P}+\partial_\vec{r}\cdot(\vec{D}\cdot\partial_\vec{r}\mathbb{P}),\label{eq:FP}
\end{align}
subject to the initial condition $\mathbb{P}(\Delta
\vec{r},\vec{u},t=0|\vec{u}_0)=\delta(\Delta\vec{r})\delta^{(2)}(\vec{u},\vec{u}_0)$,
where the delta function on the surface of the sphere $\delta^{(2)}(\cdot,\cdot)$
enforces both orientations to coincide.
Here, $\partial_\vec{r}$ denotes the spatial gradient, $\Delta_\vec{u}$ the
angular part of the Laplacian, reflecting the orientational diffusion, and
$\vec{D}=D_\parallel\vec{u}\vec{u}^\text{T}+D_\perp(\mathbb{I}-\vec{u}\vec{u}^\text{T})$.
The first term on the right describes the active motion, in addition to the
standard Smoluchowski-Perrin equation~\cite{Doi:1986} for the diffusion of an
anisotropic particle. The Fokker-Planck equation for $\mathbb{P}$ simplifies
upon a spatial Fourier transform
\begin{align}
\widetilde{\mathbb{P}}(\vec{k},\vec{u},t|\vec{u}_0)& = \int\!\diff^3 r \exp(-\imath\vec{k}\cdot\vec{r})\mathbb{P}(\vec{r},\vec{u}, t|\vec{u}_0),\label{eq:Fourier}
\end{align}
which solves the equation of motion
\begin{align}
\partial_t \widetilde{\mathbb{P}} &= D_\text{rot}\Delta_\vec{u}\widetilde{\mathbb{P}} -\imath v \vec{u}\cdot\vec{k}\widetilde{\mathbb{P}} -[D_\perp \vec{k}^2+\Delta D(\vec{u}\cdot\vec{k})^2]\widetilde{\mathbb{P}}.\label{eq:PDEchar}
\end{align}
The quantity of interest in scattering experiments~\cite{Berne:1976} is the
intermediate scattering function (ISF)
\begin{align}
F(\vec{k},t) &= \langle \exp[-\imath \vec{k}\cdot\Delta\vec{r}(t)]\rangle,\label{eq:ISF}
\end{align}
which is obtained by marginalizing over all final orientations $\vec{u}$ and
averaging over all initial orientations $\vec{u}_0$,
\begin{align}
F(\vec{k},t)&= \int\!\diff^2 u\!\int\!\frac{\diff^2u_0}{4\pi} \ \widetilde{\mathbb{P}}(\vec{k},\vec{u},t|\vec{u}_0).
\end{align}
The ISF can also be interpreted as the characteristic
function~\cite{Gardiner:2009} of the random displacement variable $\Delta
\vec{r}(t)$. In particular, the moments are obtained by taking derivatives
with respect to the wave vector $\vec{k}$. Since after averaging the motion is
isotropic, the ISF $ F(k,t) \equiv F(\vec{k},t)$ depends only on the magnitude of the wave vector
$k=|\vec{k}|$. Averaging over the directions of $\vec{k}$ yields the equivalent representation
\begin{align}
F(k,t) &=\left\langle \frac{\sin(k|\Delta\vec{r}(t)|)}{k|\Delta\vec{r}(t)|}\right\rangle \label{eq:sinc}
\end{align}
and the expansion of the ISF for small wavenumbers
\begin{align}
F(k,t) &= 1- \frac{k^2}{3!}\langle|\Delta\vec{r}(t)|^2\rangle+\frac{k^4}{5!}\langle|\Delta\vec{r}(t)|^4\rangle+\mathcal{O}(k^6)\label{eq:expansionSinc}
\end{align}
allows one to recover the mean-square displacement
$\langle|\Delta\vec{r}(t)|^2\rangle$ and the mean-quartic displacement $\langle
|\Delta\vec{r}(t)|^4\rangle$ by comparing the corresponding terms in the small wavenumber
expansion. More generally, even moments can be obtained numerically by taking derivatives
of the ISF with respect to the squared wavenumber,
\begin{align}
\langle |\Delta \vec{r}(t)|^{2n}\rangle &= (-1)^n \frac{(2n+1)!}{n!} \left. \frac{\partial^n}{\partial (k^2)^n} F(k,t)\right|_{k^2=0}.
\end{align}
The equation of motion Eq.~(\ref{eq:PDEchar}) is reminiscent
of a Schr\"odinger equation on the unit sphere and can be solved
by separation of variables. We parametrize the orientation
$\vec{u}=(\sin\vartheta \cos\varphi,\sin\vartheta\sin\varphi,\cos\vartheta)^T$ in terms
of its polar angles, and similarly for $\vec{u}_0$.
Then the solution is a superposition
of appropriate eigenfunctions
\begin{align
\widetilde{\mathbb{P}}(\vec{k},\vec{u},t|\vec{u}_0)=\frac{1}{2\pi}e^{-D_\perp k^2 t}\sum_{\ell=0}^\infty\sum_{m=-\infty}^\infty\! e^{\imath m (\varphi-\varphi_0)}\text{Ps}_\ell^m(c,R,\eta)\text{Ps}_\ell^m(c,R,\eta_0)e^{-A^m_{\ell} D_\text{rot}t}.\label{eq:expansionP}
\end{align}
Here we abbreviated $\eta = \cos\vartheta$, $\eta_0 = \cos\vartheta_0$, and $\text{Ps}_\ell^m(c,R,\eta)$ are
the generalized spheroidal wave functions of order $m$ and degree $\ell$ \cite{Yan:2009,NIST:online, NIST:print}.
They solve the corresponding eigenvalue problem
\begin{align}
\left[\frac{\diff}{\diff\eta}\left((1-\eta^2)\frac{\diff}{\diff \eta}\right)+R\eta-c^2\eta^2-\frac{m^2}{1-\eta^2}+A^m_{\ell}\right]\text{Ps}_\ell^m(c,R,\eta)&= 0,\label{eq:Pslm}
\end{align}
with eigenvalue $A^m_{\ell}=A^m_{\ell}(R,c)$ and we identify the dimensionless parameters $R = - \imath kv/D_\text{rot}$
and $c^2= \Delta D k^2/D_\text{rot}$. Hence,
at fixed wavenumber $k$, $R$ parametrizes the importance of active motion with
respect to orientational diffusion whereas $c$ measures the coupling of the
translational and orientational diffusion. In particular the ratio
$|R/c|=\text{Pe}\sqrt{4\Delta D/3\bar{D}}$ is wavenumber-independent.
Integrating Eq.~(\ref{eq:expansionP}) over the polar angles, only $\text{Ps}_\ell^0$ contributes and we obtain
\begin{align}
F(k,t) &=\frac{1}{2}e^{-D_\perp k^2 t}\sum_{\ell=0}^\infty e^{-D_\text{rot}A^0_{\ell}t}\Bigl[\int_{-1}^1 \diff\eta \text{Ps}_\ell^0(c,R,\eta) \Bigr]^2,\label{eq:solISF}
\end{align}
The explicit expression Eq.~(\ref{eq:solISF}) for the intermediate scattering
function $F(k,t)$ in terms of the generalized spheroidal wave functions is one
of the principal results of this work.
\subsection*{Exact low moments}
The low-order moments can be obtained upon expanding the ISF for small wave
numbers (Eq.~(\ref{eq:solISF})) such that the moments can be identified with Eq.~(\ref{eq:expansionSinc}).
Here we illustrate the derivation only for the mean-square displacement.
For $R=0$ and $c^2=0$ the spheroidal wave functions reduce to the Legendre
polynomials, $\text{Ps}_\ell^0(0,0,\eta) =
\text{P}_\ell(\eta)\sqrt{(2\ell+1)/2}$ with eigenvalues
$A^0_{\ell}(0,0)=\ell(\ell+1)$. For small dimensionless parameters $R$, $c$ the
Legendre polynomials are deformed analytically, to order $\mathcal{O}(k^2)$, as
required for the mean-square displacement Eq.~(\ref{eq:expansionSinc}),
the $\text{Ps}_\ell^0$ acquire contributions $\text{P}_\ell$,
$\text{P}_{\ell\pm1}$, and, $\text{P}_{\ell\pm2}$, concomitantly the eigenvalues $A^0_{\ell}$ shift.
The explicit expressions are lengthy and deferred to the methods section.
The integral in Eq.~(\ref{eq:solISF}) can then be performed using the orthogonality
of the Legendre polynomials and one concludes that only terms $\ell \leq 2$ need
to be taken into account to order $\mathcal{O}(k^2)$. Yet, inspection of Eq.~(\ref{eq:Psexpansion})
of the methods section shows that integration of $\text{Ps}_2^0(R,c,\eta)$ yields terms of
order $\mathcal{O}(R^2)$ and $\mathcal{O}(c^2)$ and after squaring in Eq.~(\ref{eq:solISF})
of only order $\mathcal{O}(k^4)$.
Hence, the contributing eigenfunctions for the mean-square displacement evaluate to
\begin{align}
\frac{1}{\sqrt{2}}\int_{-1}^1\diff\eta\text{Ps}_\ell^0(R,c,\eta) &=
\begin{cases}
1-R^2/24 +\mathcal{O}(\cdot) & \text{ } \ell=0, \\
-R/2\sqrt{3}+\mathcal{O}(\cdot) & \text{ } \ell=1.\\
\end{cases}
\end{align}
and the corresponding eigenvalues read
\begin{align}
A^0_{\ell}(R,c) &= \begin{cases}
c^2/3 - R^2/6 +\mathcal{O}(\cdot), & \text{ } \ell=0,\\
2+3c^2/5 +R^2/10+\mathcal{O}(\cdot), & \text{ } \ell=1.\\
\end{cases}
\end{align}
Collecting results for the ISF $F(k,t)$ to order $\mathcal{O}(k^2)$ and comparing with Eq.~(\ref{eq:expansionSinc}),
yields for the mean-square displacement
\begin{align}
\langle|\Delta\vec{r}(t)|^2\rangle &= \frac{v^2}{2D_\text{rot}^2}(e^{-2D_\text{rot}t}+2D_\text{rot}t-1) +6\bar{D}t.\label{eq:msd}
\end{align}
This expression generalizes the earlier result for the case of an isotropic
active agent~\cite{Sevilla:2014,Sevilla:2015} and anisotropic passive
particle~\cite{Doi:1986,Han:bla:2006}. It also recovers the mean-square displacement of a
freely rotating ellipsoidal particle~\cite{tenHagen:2011} obtained directly
from the Langevin equations. Alternatively $\langle |\Delta\vec{r}(t)|^2\rangle$
can be calculated by time-dependent perturbation theory from Eq.~(\ref{eq:PDEchar}) up to second
order.
The first contribution to the mean-square displacement in Eq.~(\ref{eq:msd}) reflects the
active motion, which displays directed motion $v^2t^2$ for times $t \lesssim
\tau_\text{rot}:=D_\text{rot}^{-1}$ where the particle does not
change its direction significantely. During this time the particle
covers a typical distance $L=v/D_\text{rot}$, which we refer to as the
persistence length. In contrast at times $t \gtrsim \tau_\text{rot}$ the
active contribution increases linearly $v^2 t/6D_\text{rot}$ where the
orientational degree of freedom is relaxed. The second contribution is merely
the isotropically averaged translational motion. Interestingly at the level of
the mean-square displacement there is no coupling between the translational
diffusion and the active motion induced by the orientational diffusion.
\begin{figure}[htp]
\centering
\includegraphics[width = \linewidth, keepaspectratio]{Fig-2-Franosch}
\caption{Exact low-order moments of a single
self-propelled particle subject to translational Brownian motion with hydrodynamic anisotropy $\Delta
D/\bar{D} = 3/4$. (a) Mean-square displacement $\langle |\Delta\vec{r}(t)|^2\rangle/L^2$ in
units of the persistence length $L=v/D_\text{rot}$, and, (b) non-Gaussian parameter $\alpha_2(t)$ for
different P{\'e}clet numbers, $\text{Pe} =va/\bar{D}$.
Simulation and theory results are shown using symbols and lines,
respectively.
\label{fig:moments}}
\end{figure}\noindent
From the mean-square displacement we identify three temporal windows,
Fig.~\ref{fig:moments}~(a). For short times $t \lesssim \tau_\text{diff}:=\bar{D}/v^2$
it increases linearly by the translational diffusion only, while at longer
times the persistent swimming motion dominates. At even longer times $t
\gtrsim \tau_\text{rot}$ the mean-square displacement increases again linearly
with an effective diffusion coefficient $D_\text{eff} =
\bar{D}+v^2/6D_\text{rot}$, equivalently the enhancement is
$D_\text{eff}/\bar{D}=1+2\text{Pe}^2/9$.
The crossover from persistent motion to
effective diffusion occurs at length scale
$L^2[1+\mathcal{O}(\text{Pe}^{-2})]$. The window of persistent motion is set
by the ratio of the two crossover times
$\tau_\text{rot}/\tau_\text{diff}=4\text{Pe}^2/3$ and opens upon increasing the
P{\'e}clet number.
Extending the expansion of the intermediate scattering function up to fourth
order in the wavenumber $k$ is tedious and the result is lengthy,
\begin{align}
\langle|\Delta\vec{r}(t)|^4\rangle &= \Bigl[\bigl\{8 D_\text{rot}^2 [405 D_\text{rot}^2 \bar{D}^2t^2+2 \Delta D^2(6 D_\text{rot} t-1)]
+4D_\text{rot}v^2[135D_\text{rot}\bar{D}t(2D_\text{rot}t-1)+\Delta D (60D_\text{rot}t-52)]\notag\\
& \ \ \ \ \ +v^4[107+6D_\text{rot}t(15D_\text{rot}t-26)]\bigr\}+18e^{-2D_\text{rot}t}v^2\bigl( 30D_\text{rot}^2\bar{D}t+4\Delta D D_\text{rot}(3+2D_\text{rot}t)-3v^2(2+D_\text{rot}t)\bigr)\notag\\
& \ \ \ \ \ +e^{-6D_\text{rot}t}(v^2-4\Delta D D_\text{rot})^2\Bigr]/54D_\text{rot}^4.
\end{align}
In contrast to the mean-square displacement, the mean-quartic displacement
depends explicitly on the translational anisotropy $\Delta D$ such that the
rotational-translational coupling becomes important. We shall see below that
depending on $\Delta D$ the dynamics becomes qualitatively different.
Rather then the mean-quartic displacement, we focus on the non-Gaussian
parameter~\cite{Hofling:2013}
\begin{align} \alpha_2(t) &=
\frac{3\langle|\Delta\vec{r}(t)|^4\rangle}{5\langle|\Delta\vec{r}(t)|^2\rangle^2}-1,
\end{align}
which is a sensitive indicator on how far the process deviates from
diffusion, see Fig. \ref{fig:moments}~(b).
For long times $t\gtrsim \tau_\text{rot}$
the non-Gaussian parameter approaches zero $\mathcal{O}(t^{-1})$ for all P{\'e}clet numbers as anticipated
by the central limit theorem. Interestingly, for the limiting case of a self-propelled
particle without any translational diffusion, $\text{Pe} = \infty$, one infers
$\alpha_2(t\to 0)= -2/5$, which reflects the persistent swimming motion at
short-times. In contrast, for non-vanishing translational diffusion,
$\text{Pe}<\infty$,
the non-Gaussian parameter approaches a constant
$\alpha_2(t\lesssim\tau_\text{diff})= 4\Delta D^2/45\bar{D}^2$ for
short-times, as anticipated
for anisotropic translational diffusion.
In particular, for $D_\parallel=2D_\perp$ it assumes the value $\alpha_2(t\lesssim\tau_\text{diff})=1/20$,
whereas it vanishes for isotropic diffusion.
For large P{\'e}clet number
there is an extended intermediate temporal regime,
where the non-Gaussian parameter is close to the one for infinite P{\'e}clet number,
thereby, a prominent minimum emerges. Here the negative non-Gaussian
parameter can be traced back to the directed swimming motion, which
dominates the translational diffusion of the active agent at these intermediate times.
Thus, for decreasing $\tau_\text{diff}$ the intermediate negative
plateau of directed swimming motion in the non-Gaussian parameter is observed
for longer times, see Fig.~\ref{fig:moments}~(b).
For the parameters shown in Fig. \ref{fig:moments}~(b) an additional
maximum occurs at shorter times. One can work out analytically from the initial
slope of $\alpha_2(t)$ that this
happens only for positive anisotropies $\Delta D > 0$ and P{\'e}clet numbers
$\text{Pe}> \sqrt{3 \Delta D/2D_\perp}$. Conversely, we conclude that a maximum in the
non-Gaussian parameter is a genuine fingerprint of active motion.
\subsection*{Intermediate scattering function}
We have evaluated numerically the series for the intermediate scattering
function in Eq.~(\ref{eq:solISF}) for arbitrary times and wavenumbers and compare the results
to stochastic simulations, see Fig.~\ref{fig:corr}. The natural scale for the wavenumbers
$k$ is set by the persistence length $L$, and our data cover the small length
scales resolving the persistent swimming motion as well as large length scales
where the particle undergoes a random walk. Indeed for small wavenumbers the
ISF are well approximated by an effective diffusion, $\exp(-D_\text{eff}k^2t)$
with the effective diffusion coefficient obtained from the long-time behavior of the
mean-square displacement. Increasing the wavenumber the qualitative behavior
depends on the P{\'e}clet number.
\begin{figure*}[htp]
\centering
\includegraphics[width=\linewidth, keepaspectratio]{Fig-3-Franosch}\\
\caption{Intermediate scattering function $F(k,t)$ of an active
Brownian particle subject to translational diffusion (here $\Delta D /\bar{D}=3/4$)
for the full range of wavenumbers $k$ measured in terms of the persistence length $L=v/D_\text{rot}$.
The dashed line represents relaxing exponentials $\exp(-D_\text{eff}k^2t)$ and $\exp(-\bar{D}k^2t)$ for small
and large wavenumbers, respectively. The dashed-dotted line indicates the sinc function $\sin(kvt)/kvt$.
\label{fig:corr}}
\end{figure*}
For small P{\'e}clet number (see Fig.~\ref{fig:corr}~(a)) the ISF decreases monotonically for all wave
numbers, in particular, the large wavenumbers approach again an exponential
$\exp(-\bar{D}k^2t)$ characterized by the mean translational diffusion
coefficient $\bar{D}$. This behavior is consistent with the linear increase of
the mean-square displacement, Fig.~\ref{fig:moments}~(a), for small P{\'e}clet numbers. For
intermediate wavenumbers (Fig.~\ref{fig:corr}~(b)-(c)) the shape of the ISF is no longer a pure exponential
since the rotational-translational coupling becomes relevant at time scales
$t\lesssim\tau_\text{rot}$.
For P{\'e}clet numbers, $\text{Pe}\gtrsim 3.4$, the ISF displays
damped oscillations for wavenumbers that start to resolve the motion on the scale
of the persistence length. At length scales $\bar{D}k^2\tau_\text{diff}\gtrsim 1$
short time diffusion takes over again, see Fig.~\ref{fig:corr}~(b). Inserting the definition of
$\tau_\text{diff}$, one infers that this regime corresponds to length scales $ka \gtrsim\text{Pe}$
where the swimmer moves only a fraction of its size $a$.
In particular, for high P{\'e}clet numbers $\text{Pe}\gtrsim 12$ the short time diffusion
is no longer resolved for the wavenumbers shown in Fig.~\ref{fig:corr}~(c).
For infinite P{\'e}clet number, the translational diffusion is negligible and the ISF
oscillates for wavenumbers resolving the persistence length, Fig.~\ref{fig:corr}~(d).
The physics of these oscillations can be rationalized easily by inspecting the general
expression of the ISF, Eq.~(\ref{eq:sinc}). For wavenumbers such that the rotational and translational diffusion can be ignored,
the trajectories can be approximated by purely persistent motion $|\Delta \vec{r}(t)|=vt$ and
there the ISF follows $F(k,t) =\sin(vkt)/vkt,$
as has been discussed already in Ref. \cite{Berne:1976}.
For infinite P{\'e}clet number the sinc function serves as a good
approximation for wavenumbers $kL\gtrsim 20$.
It is also interesting to ask how the oscillations emerge mathematically from the general
solution in terms of eigenfunctions, Eq.~(\ref{eq:solISF}). Naively, one expects that the ISF is a sum of
relaxing exponentials only, in particular, they should decay monotonically. Yet, the
operator in Eq.~(\ref{eq:Pslm}) for the eigenvalue problem is non-Hermitian, since $R=-\imath kL$ is not real,
such that the eigenvalues can become complex. Indeed one can show (see section Methods), for example $\text{Pe}=\infty$, that
at $|R|=1.9$ the two lowest real eigenvalues merge and bifurcate to a pair of complex conjugates.
Further bifurcations for larger eigenvalues occur at even larger $|R|$. For large P{\'e}clet numbers
the scenario is qualitatively similar, whereas for small $\text{Pe}$ the eigenvalues remain real and no
oscillations in the ISF emerge. Since the eigenvalues depend non-analytically on $|R|=kL$, there
is a finite radius of convergence for the expansion of the ISF in powers of $k$ set by the
first bifurcation point. In particular, the oscillations cannot be obtained by extending the series expansion,
Eq.~(\ref{eq:expansionSinc}), in terms of the moments to arbitrary order.
\section*{Summary and Conclusion}
We have determined exact analytic expressions for the intermediate
scattering function (ISF) of an anisotropic active Brownian particle
in terms of an expansion of eigenfunctions. The solution is validated and
exemplified by stochastic simulations.
Interestingly, the ISF displays a regime
with oscillatory behavior in striking contrast to passive motion in equilibrium
systems. These oscillations are rationalized in terms of bifurcations of the eigenvalue problem
and reflect the directed swimming motion of the active particles.
In addition to the mean-square displacement, we have analyzed the non-Gaussian
parameter and identified a characteristic maximum for positive anisotropies and
large P{\'e}clet numbers.
The non-Gaussian parameter has been derived before for two-dimensional
isotropic swimmers~\cite{Sevilla:2014,Sevilla:2015} by a truncated
mode expansion of the Fokker-Planck equation. Yet, for isotropic diffusion
the non-Gaussian parameter remains negative for all times, in contrast to
experimental observations~\cite{Zheng:2013}. The mode expansion also yields
approximate expressions for the ISF which in
principle also display oscillations in time for the two-dimensional case.
In differential dynamic microscopy experiments for dilute suspensions of \emph{E.\! coli} bacteria in three dimensions
an oscillatory behavior for the ISF has
been observed and analyzed approximately in terms of pure persistent swimming
motion~\cite{Poon:2016}. Our results predict that these oscillations fade out for large
as well as small wavenumbers which should in principle be also measurable
in the set-up. The motility parameters then can be extracted from the measured ISF relying on different
wavenumbers.
The dynamics on small length scales is dominated by translational diffusion,
at intermediate ones by the swimming motion, and, finally at large length scales
by the rotational diffusion.
Furthermore the spatio-temporal information obtained from the ISF allows to
discriminate quantitatively the dynamics of different swimming behaviors,
whereas the mean-square displacement of several models such as simple run-and-tumble
motion~\cite{Martens:2012} is hardly distinguishable to that of an active Brownian particle.
The analytic solution for the active Brownian swimmer derived here
should serve as a reference for more complex swimming behavior.
For example, \emph{E.\! coli} bacteria display a distribution of swimming
velocities, which can be accounted for directly by post-averaging our results
for the ISF. Similarly, the swimming velocity may fluctuate itself~\cite{Romanczuk:2012}
leading to a further smearing of the oscillations in the ISF. Furthermore, the rotational diffusion for
bacteria should be complemented by a run-and-tumble motion~\cite{Berg:1972} as observed by particle
tracking. Species-specific propulsion mechanisms, such
as circular motion of the algae \emph{Chlamydomonas reinhardtii} \cite{Poon:2016}, can be accounted for
by introducing a torque in the Fokker-Planck equation. Our solution strategy can be adapted also
to two-dimensional systems, for instance for the movement of
Janus particles~\cite{Zheng:2013} confined between two glass plates or for the
circular motion of \emph{E.\! coli} bacteria close to surfaces~\cite{Lauga:2006}.
\section*{Methods}
\subsection*{Expansion of the eigenfunctions in powers of the wavenumber}
The starting point of the expansion are the reference solutions
$\text{Ps}_\ell^0(0,0,\eta)\equiv |\ell \rangle :=\text{P}_\ell(\eta)\sqrt{(2\ell+1)/2}$ for
$\ell \in \mathbb{N}_0$ of the eigenvalue problem, Eq.~(\ref{eq:Pslm}), for parameters $R=c^2=0$. By standard perturbation theory one derives
to the desired order $\mathcal{O}=\mathcal{O}(R^3,c^2R,c^4)$
\begin{align}
\text{Ps}_\ell^0(R,c,\eta) &= |\ell\rangle -R\biggl[|\ell-1\rangle \frac{\langle \ell-1|\eta|\ell\rangle}{\Delta A_\ell^{\ell-1}}
+|\ell+1\rangle\frac{\langle \ell+1|\eta|\ell\rangle}{\Delta A_\ell^{\ell+1}}\biggr]
+R^2\biggl[-\frac{1}{2}|\ell\rangle\left(\frac{|\langle\ell-1|\eta|\ell\rangle|^2}{(\Delta A_\ell^{\ell-1})^2}+
\frac{|\langle \ell+1|\eta|\ell\rangle|^2}{(\Delta A_\ell^{\ell+1})^2}\right)\label{eq:Psexpansion}\\
& \ \ \ \ \ + |\ell-2\rangle \frac{\langle\ell-2|\eta|\ell-1\rangle\langle\ell-1|\eta|\ell\rangle}{\Delta A_\ell^{\ell-2}\Delta A_\ell^{\ell-1}}
+|\ell+2\rangle \frac{\langle \ell+2|\eta|\ell+1\rangle\langle\ell+1|\eta|\ell\rangle}{\Delta A_\ell^{\ell+2}\Delta A_\ell^{\ell+1}}\biggr]\notag\\
& \ \ \ \ \ +c^2\biggl[|\ell-2\rangle\frac{\langle \ell-2|\eta^2|\ell\rangle}{\Delta A_\ell^{\ell-2}}
+|\ell+2\rangle \frac{\langle\ell+2|\eta^2|\ell\rangle}{\Delta A_\ell^{\ell+2}}\biggr]+\mathcal{O},\notag
\end{align}
with corresponding eigenvalues
\begin{align}
A^0_{\ell}(R,c) &= \ell(\ell+1)+R^2\left[\frac{|\langle\ell-1|\eta|\ell\rangle|^2}{\Delta A_\ell^{\ell-1}}
+\frac{|\langle \ell+1|\eta|\ell\rangle|^2}{\Delta A_\ell^{\ell+1}}\right] +c^2\langle \ell|\eta^2|\ell\rangle+\mathcal{O}.
\end{align}
Here $|\ell\rangle=0$ for $\ell<0$,
the difference of unperturbed eigenvalues is denoted by $\Delta A_\ell^j = A^0_{\ell}(0,0)-A^0_{j}(0,0)$,
and, the matrix elements of the perturbation
\begin{align}
\langle n| \eta^j|\ell\rangle &= \sqrt{(2n+1)(2\ell+1)}\int_{-1}^1 \diff \eta \ \text{P}_n(\eta) \eta^j\text{P}_\ell(\eta)/2
\end{align}
for $j=1,2$ can be evaluated using the properties of the Legendre polynomials.
\subsection*{Numerical evaluation of the ISF}
For the ISF we need the eigenvalues $\text{A}^0_{\ell}$ and the integrals over the
eigenfunctions $\text{Ps}_\ell^0$, Eq.~(\ref{eq:solISF}). We expand these in terms of the Legendre polynomials~\cite{Yan:2009}
$\text{Ps}_{\ell}^0(c,R,\eta) = \sum\nolimits_{j=0}^\infty d^{0\ell}_j |j\rangle$.
Then the integrals in Eq.~(\ref{eq:solISF}) can be performed and the intermediate scattering function of the anisotropic active Brownian particle reads
\begin{align}
F(k,t) &= e^{-D_\perp k^2 t}\sum_{\ell=0}^\infty [d_0^{0\ell}]^2e^{-D_\text{rot}A^0_{\ell}t}, \label{eq:Fcoef}
\end{align}
Inserting the expansion
into Eq.~(\ref{eq:Pslm})
and projecting onto $\langle n|$ leads to the matrix eigenvalue
problem
\begin{align}
\sum_j[\langle n| c^2\eta^2-R\eta|j\rangle +n(n+1)\delta_{jn}]d_j^{0\ell} &= A^0_{\ell}d_n^{0\ell}.\label{eq:matrix}
\end{align}
\begin{figure}[H]
\centering
\includegraphics[width=\linewidth, keepaspectratio]{Fig-4-Franosch}
\caption{Real (a) and imaginary (b) part of the eigenvalues $A^0_{1}(R,0)$ to $A^0_{4}(R,0)$ for
vanishing translational diffusion ($\text{Pe} =\infty$).\label{fig:bifEV}}
\end{figure}
Since the matrix elements are non-vanishing for $j=n-2,...,n+2$ only,
it is in fact a band matrix with two diagonals on each side. Then the
normalized eigenvectors $\vec{d}^{0\ell}= (d_0^{0\ell},d_1^{0\ell},d_2^{0\ell},...)^{\text{T}}$
and eigenvalues $A^0_{\ell}$ can be efficiently determined numerically.
In practice we truncate the matrix in Eq.~(\ref{eq:matrix}) to sufficiently high order
such that the normalization at time $t=0$ for the ISF, Eq.~(\ref{eq:Fcoef}), is achieved.
Since the generalized spheroidal wave equation is not
Hermitian, the corresponding eigenvalues can become complex.
In fact for $\text{Pe}=\infty$ ($c=0$), the two lowest eigenvalues
merge at $|R|=kL=1.9$ and a bifurcation to two complex conjugates
occurs, see Fig.~\ref{fig:bifEV}. In contrast for small P{\`e}clet number $\text{Pe}=1.1$
the eigenvalues remain real for all wavenumbers.
\begin{acknowledgements}
We acknowledge helpful discussions with Felix H\"ofling at the initial
state of this project. This work has been supported by Deutsche
Forschungsgemeinschaft (DFG) via the contract No. FR1418/5-1 and by the Austrian
Science Fund (FWF): P~28687-N27.
\end{acknowledgements}
\section*{Author contributions statement}
Author contributions: T.F. conceived the project. S.L. designed
the simulation algorithm. C.K. implemented the theory and performed simulations.
C.K. and T.F. interpreted the data and wrote the paper. All authors discussed the results and
commented on the manuscript.
\section*{Additional information}
\subsection*{Competing financial interests:} The authors declare no competing financial interests.
| {
"redpajama_set_name": "RedPajamaArXiv"
} | 5,173 |
/**
* duct front-end core
*/
(function ($) {
// Create modal alert windows (using Foundation reveal)
function modal_alert(bodytext, bodytag, headertext) {
var modal_body;
if (headertext) {
$('#modal-header').empty().text(headertext);
}
if (bodytext) {
modal_body = (bodytag) ? $('<' + bodytag + ' />') : $('<p />');
$('#modal-body').empty().append(
modal_body.addClass('modal-error-text').text(bodytext)
);
}
$('#modal-ok-box').show();
$('#modal-dynamic').foundation('reveal', 'open');
}
function modal_prompt(prompt, bodytag, headertext) {
var modal_body;
if (headertext) {
$('#modal-header').empty().html(headertext);
}
if (prompt) {
modal_body = (bodytag) ? $('<' + bodytag + ' />') : $('<p />');
$('#modal-body').empty().append(
modal_body
.addClass('modal-error-text')
.append(prompt)
);
}
$('#modal-dynamic').foundation('reveal', 'open');
}
function Duct(socket) {
this.socket = socket;
}
// Get all active votes
Duct.prototype.sync = function () {
this.socket.emit('sync');
};
Duct.prototype.intake = function () {
var self = this;
this.socket.on('synced', function (res) {
console.log(res);
});
this.socket.on('contract-created', function (res) {
if (res && res.address) {
$('#contract-address-input').val(res.address);
if (res.functions) {
var html = "<select>";
for (var i = 0, len = res.functions.length; i < len; ++i) {
html += "<option>" + res.functions[i] + "</option>";
}
html += "</select>";
$('#functions-input-label').show();
$('#functions-input').html(html);
}
} else {
console.log("Error creating contract:", res);
}
});
this.socket.on('contract-output', function (res) {
if (res && res.reputation && res.outcomes && res.reporter_bonus) {
$('#status-receiver').slideUp('slow');
var reputation = "Reputation: " + JSON.stringify(res.reputation, null, 3) + "\n";
var output = $('<pre />').text(reputation + outcomes + reporter_bonus);
$('#main-receiver').append(output).show();
} else {
$('#status-receiver').html("<h4>Contract execution unsuccessful</h4>");
}
});
return this;
};
Duct.prototype.exhaust = function () {
var self = this;
$('#modal-ok-button').click(function (event) {
event.preventDefault();
$('#modal-ok-box').hide();
$('#modal-dynamic').foundation('reveal', 'close');
});
$('#create-contract').click(function (event) {
event.preventDefault();
var prompt = '<hr /><div class="row centered">' +
'<form action="#" method="POST" id="create-contract-form">' +
'<textarea id="contract-source" />' +
'<hr />' +
'<button type="submit" class="button expand" '+
'id="create-contract-button">Create</button>' +
'</form></div>';
modal_prompt(prompt, "h2", "Create contract");
$('#create-contract-form').submit(function (event) {
event.preventDefault();
var source = $('#contract-source').val();
if (source == "") {
source = "def main(a,b):\n"+
" return(a^b)";
}
self.socket.emit('create-contract', {
source: source,
gas_input: 70000000
});
this.reset();
$('#modal-dynamic').foundation('reveal', 'close');
});
});
$('form#compile-contract').submit(function (event) {
event.preventDefault();
// c305c901078781c232a2a521c2af7980f8385ee9
});
$('form#run-contract-form').submit(function (event) {
event.preventDefault();
$('#status-receiver').html('<h4>Running contract...</h4>').show();
var data = {
contract_address: $('#contract-address-input').val(),
gas_input: parseInt($('#gas-input').val()),
function_name: $('#function-name-input').val()
};
self.socket.emit('run-contract', data);
this.reset();
});
return self;
};
$(document).ready(function () {
var socket_url, duct;
socket_url = window.location.protocol + '//' + document.domain +
':' + location.port + '/socket.io/';
socket = io.connect(socket_url);
socket.on('connect', function () {
duct = new Duct(socket);
duct.intake().exhaust();
})
});
})(jQuery);
| {
"redpajama_set_name": "RedPajamaGithub"
} | 8,789 |
{"url":"https:\/\/tex.stackexchange.com\/questions\/136749\/super-and-subscripts-with-declarepaireddelimiter\/136767","text":"# Super and subscripts with \\DeclarePairedDelimiter\n\nI defined a macro for the supremum norm by\n\n\\DeclarePairedDelimiter{\\supnorm}{\\|}{\\|_\\infty}\n\n\nwhich works nicely. However, as soon as I want to place a superscript like\n\n\\supnorm{f}^2\n\n\nthings look funny, as the 2 appears after the infty-sign. However, I would like to look things more like what you get by\n\n\\|f\\|_\\infty^2\n\n\nOf course, one can achieve this by using\n\n\\newcommand{\\supnorm}[1]{\\|#1\\|_\\infty}\n\n\nbut then one looses the nice functionality of \\DeclarePairedDelimiter, which I really started to like a lot. Any ideas how one can combine the benefits of both?\n\n## Update:\n\nNow with all the options provided by \\DeclarePairedDelimiter! See the new definition (without using xparse):\n\n\\DeclarePairedDelimiter{\\norm}{\\lVert}{\\rVert}\n\\makeatletter\n\\newcommand{\\@supnormstar}[1]{\\norm*{#1}_\\infty}\n\\newcommand{\\@supnormnostar}[2][]{\\norm[#1]{#2}_\\infty}\n\\newcommand{\\supnorm}{\\@ifstar\\@supnormstar\\@supnormnostar}\n\\makeatother\n\n\nFor example, the following code:\n\n$\\supnorm{\\frac{\\sqrt5-1}{2}}^2$\n$\\supnorm*{\\frac{\\sqrt5-1}{2}}^2$\n$\\supnorm[\\big]{\\frac{\\sqrt5-1}{2}}^2$\n\n\nwill give you:\n\n\u2022 Bad idea, you loose the options provides by \\DeclarePairedDelimiter, and always scaling fences is a bad idea. \u2013\u00a0daleif Oct 7 '13 at 8:08\n\u2022 @daleif: good idea, answer've been updated \u2013\u00a0Francis Oct 7 '13 at 8:51\n\nYou need to get the \\infty outside of the grouping. And that will involve redoing the options:\n\n\\documentclass[a4paper]{memoir}\n\\usepackage{mathtools,xparse}\n\\DeclarePairedDelimiter{\\supnormX}{\\lVert}{\\rVert}\n\\DeclareDocumentCommand\\supnorm{ s o m }{%\n\\IfBooleanTF{#1}{% starred\n\\supnormX*{#3}_\\infty\n}{% not starred\n\\IfNoValueTF{#2}{% no []\n\\supnormX{#3}_\\infty\n}{% data in []\n\\supnormX[#2]{#3}_\\infty\n}\n}\n}\n\\begin{document}\n$\\supnorm{A}^2$\n\\end{document}\n\n\u2022 you're listed as one of the maintainers of mathtools. may i suggest that generalizing the definition of \\DeclarePairedDelimiter there would be a worthy project. \u2013\u00a0barbara beeton Oct 7 '13 at 13:31\n\u2022 Hmm, I'm a bit reluctant to make it depend on xparse and also, why only subscripts..... I'll think about it. \u2013\u00a0daleif Oct 7 '13 at 17:05\n\u2022 hmmm. i didn't mean to imply subscripts only -- certainly superscripts should be dealt with, and i'm not sure whether there are other variations that might be relevant. and i understand why you might want to steer clear of xparse at this moment (and don't disagree). thinking is good. thanks. \u2013\u00a0barbara beeton Oct 7 '13 at 17:42\n\nHere's a different implementation of \\DeclarePairedDelimiter for which I use a different name.\n\n\\documentclass[a4paper]{memoir}\n\\usepackage{amsmath}\n\\usepackage{xparse}\n\n\\ExplSyntaxOn\n\\DeclareDocumentCommand{\\xDeclarePairedDelimiter}{mmmO{}}\n{\n\\NewDocumentCommand{#1}{sO{}m}\n{\n\\IfBooleanTF{##1}\n{\n\\egreg_paired_delimiter_expand:nnnn {#2}{#3}{##3}{#4}\n}\n{\n\\egreg_paired_delimiter_fixed:nnnnn {##2}{#2}{#3}{##3}{#4}\n}\n}\n}\n\\cs_new_protected:Npn \\egreg_paired_delimiter_expand:nnnn #1 #2 #3 #4\n{% Fix the spacing issue with \\left and \\right (D. Arsenau, P. Stephani and H. Oberdiek)\n\\mathopen{}\n\\mathclose\\c_group_begin_token\n\\left#1\n#3\n\\group_insert_after:N \\c_group_end_token\n\\right#2\n\\tl_if_empty:nF {#4} { \\c_math_subscript_token {#4} }\n}\n\\cs_new_protected:Npn \\egreg_paired_delimiter_fixed:nnnnn #1 #2 #3 #4 #5\n{\n\\mathopen{#1#2}#4\\mathclose{#1#3}\n\\tl_if_empty:nF {#5} { \\c_math_subscript_token {#5} }\n}\n\\ExplSyntaxOff\n\n%% the final optional argument to \\xDeclarePairedDelimiter\n%% is a subscript to the right fence\n\\xDeclarePairedDelimiter{\\supnormX}{\\lVert}{\\rVert}[\\infty]\n\n\\begin{document}\n$\\supnormX{A}^2\\quad \\supnormX[\\big]{A}^2\\quad \\supnormX[\\Big]{A}^2\\quad \\supnormX[\\bigg]{A}^2\\quad \\supnormX[\\Bigg]{A}^2\\quad \\supnormX*{\\frac{A}{2}}^2$\n\\end{document}\n\n\nSince daleif has shown interest in extending this idea, I add a way to define the macros with a key-value interface. The main (internal) functions remain the same.\n\n\\documentclass[a4paper]{memoir}\n\\usepackage{amsmath}\n\\usepackage{xparse}\n\n\\ExplSyntaxOn\n\\DeclareDocumentCommand{\\KDeclarePairedDelimiter}{mm}\n{\n\\__egreg_delimiter_clear_keys: % reset to the default\n\\keys_set:nn { egreg\/delimiters } { #2 }\n\\use:x % we want to expand the values of the token variables set with the keys\n{\n\\exp_not:n {\\NewDocumentCommand{#1}{sO{}m} }\n{\n\\exp_not:n { \\IfBooleanTF{##1} }\n{\n\\exp_not:N \\egreg_paired_delimiter_expand:nnnn\n{ \\exp_not:V \\l_egreg_delimiter_left_tl }\n{ \\exp_not:V \\l_egreg_delimiter_right_tl }\n{ \\exp_not:n { ##3 } }\n{ \\exp_not:V \\l_egreg_delimiter_subscript_tl }\n}\n{\n\\exp_not:N \\egreg_paired_delimiter_fixed:nnnnn\n{ \\exp_not:n { ##2 } }\n{ \\exp_not:V \\l_egreg_delimiter_left_tl }\n{ \\exp_not:V \\l_egreg_delimiter_right_tl }\n{ \\exp_not:n { ##3 } }\n{ \\exp_not:V \\l_egreg_delimiter_subscript_tl }\n}\n}\n}\n}\n\n\\keys_define:nn { egreg\/delimiters }\n{\nleft .tl_set:N = \\l_egreg_delimiter_left_tl,\nright .tl_set:N = \\l_egreg_delimiter_right_tl,\nsubscript .tl_set:N = \\l_egreg_delimiter_subscript_tl,\n}\n\n\\cs_new_protected:Npn \\__egreg_delimiter_clear_keys:\n{\n\\keys_set:nn { egreg\/delimiters } { left=.,right=.,subscript={} }\n}\n\n\\cs_new_protected:Npn \\egreg_paired_delimiter_expand:nnnn #1 #2 #3 #4\n{% Fix the spacing issue with \\left and \\right (D. Arsenau, P. Stephani and H. Oberdiek)\n\\mathopen{}\n\\mathclose\\c_group_begin_token\n\\left#1\n#3\n\\group_insert_after:N \\c_group_end_token\n\\right#2\n\\tl_if_empty:nF {#4} { \\c_math_subscript_token {#4} }\n}\n\\cs_new_protected:Npn \\egreg_paired_delimiter_fixed:nnnnn #1 #2 #3 #4 #5\n{\n\\mathopen{#1#2}#4\\mathclose{#1#3}\n\\tl_if_empty:nF {#5} { \\c_math_subscript_token {#5} }\n}\n\\ExplSyntaxOff\n\n\\KDeclarePairedDelimiter{\\supnormX}{\nleft=\\lVert,\nright=\\rVert,\nsubscript=\\infty\n}\n\n\\begin{document}\n$\\supnormX{A}^2\\quad \\supnormX[\\big]{A}^2\\quad \\supnormX[\\Big]{A}^2\\quad \\supnormX[\\bigg]{A}^2\\quad \\supnormX[\\Bigg]{A}^2\\quad \\supnormX*{\\frac{A}{2}}^2$\n\\end{document}\n\n\u2022 You missed mathtools build in fix to the \\left...\\right spacing. \u2013\u00a0daleif Oct 14 '13 at 13:11\n\u2022 @daleif Just load mleftright and use \\mleft and \\mright. ;-) Seriously, I find your approach in mathtools too complicated. \u2013\u00a0egreg Oct 14 '13 at 13:49\n\u2022 Then again you rely on many external packages. \u2013\u00a0daleif Oct 14 '13 at 15:50\n\u2022 @daleif Fixing it with the same trick as mleftright is quite easy. I added it. \u2013\u00a0egreg Oct 14 '13 at 15:58\n\u2022 Isn't that what I do in mathtools? \u2013\u00a0daleif Oct 14 '13 at 19:06","date":"2019-10-15 08:36:26","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\": 0.893034040927887, \"perplexity\": 5972.855664361382}, \"config\": {\"markdown_headings\": true, \"markdown_code\": true, \"boilerplate_config\": {\"ratio_threshold\": 0.18, \"absolute_threshold\": 10, \"end_threshold\": 15, \"enable\": true}, \"remove_buttons\": true, \"remove_image_figures\": true, \"remove_link_clusters\": true, \"table_config\": {\"min_rows\": 2, \"min_cols\": 3, \"format\": \"plain\"}, \"remove_chinese\": true, \"remove_edit_buttons\": true, \"extract_latex\": true}, \"warc_path\": \"s3:\/\/commoncrawl\/crawl-data\/CC-MAIN-2019-43\/segments\/1570986657949.34\/warc\/CC-MAIN-20191015082202-20191015105702-00296.warc.gz\"}"} | null | null |
"Treehouse of Horror IX" är fjärde avsnittet av säsong 10. Det sändes på Fox den 25 oktober 1998. Detta är det nionde avsnittet i serien Treehouse of Horror och består av tre delar. I "Hell Toupée" får Homer en hårtransplantation och i "Terror of Tiny Toon" hamnar Bart och Lisa i Itchy & Scratchy och i "Starship Poopers", Marge visar det sig att Maggies far är Kang.
"Treehouse of Horror IX" skrevs av Donick Cary, Larry Doyle och David S. Cohen, samt regisserades av Steven Dean Moore. "Terror of the Tiny Toon" inkluderar en verklighetssekvens med Regis Philbin och Kathie Lee Gifford. Jerry Springer och Ed McMahon gästskådespelar som sig själv medan Robert Englund gör rösten för Freddy Krueger. Under 1999 blev Alf Clausen nominerad till en Primetime Emmy Award för "Outstanding Music Composition for a Series", för hans arbete med avsnittet.
Handling
Hell Toupée
Snake arresteras efter han rökt på Kwik-E-Mart och Wiggum förklarar för honom att det var hans tredje förseelse vilket leder till avrättning. Innan han förs bort berättar han att Apu, Moe och Bart är vittnen och Snake svär på att han ska döda dem. Snake avrättas i en elektrisk stol i TV-programmet World's Deadliest Executions, som leds av Ed McMahon. Hans kropp doneras därefter till den behövande Barney för hans lever, och Homer för en hårtransplantation. Sedan Dr. Nick transplanterat Snakes hår till Homers huvud börjar håret ta över Homers hjärna och Snake börjar kontrollera hans kropp. Homer mördar därefter Apu och Moe. Bart inser då att han också kommer dö snart men Homer lovar att skydda honom. Senare är Homer ensam med Bart i hans rum och då gör han ett mordförsök på Bart. Familjen inser vad som hänt och Homer lyckas återta kontrollen över sig själv, då han river bort håret. Håret börjar då självt attackera Bart och Homer börjar slå Bart i ansiktet. Polisen dyker upp för att gripa Homer, men förstår snabbt att håret är den skyldige. Håret försöker då fly genom fönstret men blir nerskjutet.
The Terror of Tiny Toon
Marge förbjuder Bart och Lisa att titta på Itchy & Scratchy under Halloween och tar bort batteriet ur fjärrkontrollen. Bart använder då en bit plutonium som han hittar i Homers verktygslåda som batteri. De börjar titta på Itchy and Scratchy men efter lite bråk mellan de båda transporteras de in i TV:n där de ser verklighetens Itchy och Scratchy fäktas. När de upptäcker att de skrattar åt dem så bestämmer de sig för att hämnas och Bart och Lisa blir jagade genom en tecknad värld. Homer börjar samtidigt titta på TV:n men tycker att programmet efter ett tag är tråkigt och byter till Live with Regis and Kathie Lee men han byter sen tillbaka till den tecknade filmen. I båda världarna fortsätter jakten. När Lisa upptäcker att Homer tittar på TV ber hon honom att mata ut dem, vilket han till slut gör, men lite för sent. Barts kropp har blivit uppäten så det bara är ett skelett kvar. Itchy och Scratchy kommer också ut till riktiga världen men är ganska små, så de blir behandlade som husdjur.
Starship Poopers
Marge berättar att Maggie har fått sin första tand. Senare tappar hon sina ben och ut kommer tentakler. Via sin napp kontaktar hon sedan Kang & Kodos. De kommer till familjen Simpson för att hämta Maggie. Marge berättar då att Kang är Maggies far och att hon blev bortrövad och besprutad med laser som gjorde henne gravid. De bestämmer sig för att lösa tvisten om vem som ska ta hand om Maggie på The Jerry Springer Show, men han kan inte lösa det och programledaren Jerry Springer dör. Kang och Kodos hotar att döda alla politiker i Washington om de inte får Maggie. Familjen tror inte på det, så de går hem med Maggie som börjat tala med mansröst.
Produktion
Till skillnad från förra Treehouse of Horror avsnitt gjordes varje del av olika författare. "Hell Toupee" av Donick Cary, "Terror of Tiny Toon" av Larry Doyle. "Starship Poopers" av David S. Cohen. I början av texten kallas Cohen för "David 'Watch Futurama' Cohen" som en referens till att han börjar på den serien. "Terror of the Tiny Toon" inkluderade en verklighetsdel med Regis Philbin och Kathie Lee Gifford från Live with Regis and Kathie Lee. Den delen regisserades av Donick Cary, men resten av rollfigurerna är tecknade. Filmningen tog längre tid än planerat, så en nyhetssändning fick köras från en annan studio. Jerry Springer medverkade som sig själv. Den delen spelades in av Julie Thacker. Stora delar av animation i "Hell Toupée" gjordes av Chris Clements. Moes död var mer skrämmande förut men ändrades efter order av Mike Scully. Animatörerna uppskattade att arbeta med "Terror of Tiny Toon". I "Starship Poopers" visar det sig i en sekvens att Springfield ligger i Louisiana. Dock har man i andra avsnitt påpekat andra platser. Maggies replik i slutet gjordes av Harry Shearer.
Kulturella referenser
Poochie från "The Itchy & Scratchy & Poochie Show" medverkar i "Terror of the Tiny Toon". Soffskämtet innehåller Freddy Krueger från Terror på Elm Street och Jason Voorhees från Fredagen den 13:e. Freddys röst görs av Robert Englund, TV-programmen Live with Regis and Kathie Lee och The Jerry Springer Show ingår också. "Starship Poopers"-titel är en parodi på Starship Troopers.
Mottagande
Avsnitt fick en Nielsen rating på 8,6 och sågs av 8,5 miljoner hushåll och det femte mest sedda programmet på Fox under veckan. Enligt 'I Can't Believe It's a Bigger and Better Updated Unofficial Simpsons Guide anser Warren Martyn och Adrian Wood att "The Terror of Tiny Toon" är den bästa delen medan "Starship Poopers" först blir roligt när Springer börjar medverka. Colin Jacobson på DVD Movie Guide anser att "Hell Toupée" är bäst då de ändra två är förutsägbara. Kay McFadden på The Seattle Times anser att den inte är lika bra som Treehouse of Horror VII men älskar delen med Itchy och Scratchy. Under 2008, blev "Starship Poopers" den tionde bästa Treehouse of Horror-delen IGN. Under 1999 fick Alf Clausen en Primetime Emmy Award nomering för ""Outstanding Music Composition for a Series".
Källor
Externa länkar
"Treehouse of Horror IX" avsnittskapsel på The Simpsons Archive.
"Treehouse of Horror IX på The Simpsons.com
"Treehouse of Horror IX" på TV.com
Avsnitt av Simpsons säsong 10
TV-avsnitt 1998
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\section{Introduction}
The magnetohydrodynamic (MHD) Rankine-Hugoniot relations possess six shock solutions; fast, slow and four types of intermediate shocks, which satisfy entropy condition.
Arguments based upon the theory of strictly hyperbolic systems of conservation laws led to the so-called evolutionary conditions (Lax 1957), and all types of the intermediate shocks are non-evolutionary in the ideal MHD system (Landau \& Lifshitz 1960; Jeffery \& Taniuti 1964; Kantrowiz \& Petschek 1966; Polovin \& Demutskii 1990; and references therein).
These conditions effectively ruled out the intermediate shocks in nature, since they have no neighboring solution corresopnding to small perturbations.
Contrary to this belief, series of numerical experiments in the dissipative MHD system by Wu (1987, 1988, 1990) showed that at least some of the intermediate shocks are admissible and can be formed through nonlinear steeping from continuous waves.
Furthermore, Chao et al. (1993) reported the detection of an interplanetary intermediate shock in the Voyager 1 data.
The dissipative steady solutions that correspond to the fast shock and the slow shock are coplanar (i.e., the velocity field and the magnetic field are in the same plane everywhere).
On the other hand, four types of the dissipative steady solutions of the intermediate shocks can have non-coplanar structure inside their thin but a finite thickness front, and the non-coplanar component magnetic flux is limited by a maximum value, which is proportional to the dissipation coefficients.
These properties were first pointed out by Wu (1990).
The interactions between the intermediate shocks and the Alfv\'en waves were studied by many authors through nonlinear dissipative MHD simulations (Wu 1988; Wu \& Kennel 1992; Markovskii \& Skorokhodov 2000; Falle \& Komissarov 2001).
If classical evolutionary condition is effective even in the dissipative system, the intermediate shocks should instantly disappear.
However, their results do not show such evolution.
Wu (1988) and Wu \& Kennel (1992) showed that the intermediate shock survives a finite time after the interaction with the non-linear Alfv\'en wave whose transverse magnetic field is rotated.
Note that such interaction makes the intermediate shock non-coplanar even in the outsides of the shock structure.
Therefore, they conjectured that there exists a new class of time-dependent intermediate shocks, which do not obey the MHD Rankine-Hugoniot relations, since they violate the coplanarity between the upstream and the downstream, and they are the neighboring states of the intermediate shocks.
Wu (1988) also reported that the intermediate shock remains stable as a result of the interaction with the Alfv\'en wave.
Therefore, the intermediate shocks seem to be evolutionary, but whether the intermediate shock evolves into other shocks and waves or not depends on the nature of the perturbations.
Falle \& Komissarov (2001) showed that the lifetime of the intermediate shock becomes short in a noisy circumstance.
The reason is as follows.
The magnetic flux inside the structure of the intermediate shock is rotated as a result of the interactions with the Alfv\'en waves from the downstream.
Then the intermediate shock eventually brakes up, because of the existence of maximum value of the non-coplanar component magnetic flux that the shock can manage.
Thus, they cautioned that the intermediate shocks tend to survive for a long time in the numerical simulation, because the numerical diffusion makes the maximum value of the non-coplanar magnetic flux large.
The reanalysis of the evolutionary conditions for the intermediate shocks in the dissipative MHD system are done by Hada (1994) and Markovskii (1998a, b).
Hada (1994) reported that there are additional wave modes, which are originated in the dissipations, and the number of outgoing waves from the shock front is lager than the case of the ideal MHD system.
He concluded that the intermediate shocks are evolutionary, but the set of equations are under-determined due to the excessive dissipative modes.
As a result, he introduced the minimum dissipation principle in order to uniquely define solution.
Previous authors who studied the evolutionary conditions in the dissipative MHD system did not take account the continuous structure of the shock front.
Their analyses are based on the linear perturbation theory of the discontinuities or weak solutions.
However, in the dissipative system, we have to treat the unperturbed shocks as a continuous transition layer, and have to solve differential equations instead of the conservation laws in the ideal system.
In this paper, we examine the evolutionary conditions for the continuous MHD shock waves.
\S 2 provides basic equations that describe perturbation of the shock structure.
In \S 3, we formulate the evolutionary conditions for the continuous shock waves.
We adapt our formulation to the various MHD shocks in \S 4 and \S 5.
In \S 6, we summarize our result and discuss their implication.
\section{Basic Equations for Linear Analysis of Dissipative MHD System}
The one-dimensional, dissipative MHD equations are described as follows:
\begin{equation} \label{basic1}
\frac{\partial \vec{u}}{\partial t}+\frac{\partial \vec{f}}{\partial x}=\frac{\partial \vec{d}}{\partial x},
\end{equation}
\begin{equation}
\vec{u}=\left(
\begin{array}{c}
\rho \\ \rho\,v_{x} \\ \rho\,\vec{v}_{t} \\ \vec{B}_{t} \\ e \\
\end{array} \right),\,
\vec{f}=\left(
\begin{array}{c}
\rho\,v_{x} \\ \rho\,v_{x}^{2}+P-B_{x}^{2} \\ \rho\,v_{x}\,\vec{v}_{t}-B_{x}\,\vec{B}_{t} \\ v_{x}\,\vec{B}_{t}-\vec{v}_{t}\,B_{x} \\ (e+P)\,v_{x}-B_{x}\,(\vec{v}\cdot\vec{B}),
\end{array} \right),
\end{equation}
\begin{equation}
\vec{d}=\left(
\begin{array}{c}
0 \\ \left(\frac{4}{3}\nu+\mu\right)\frac{\partial v_{x}}{\partial x} \\
\nu\,\frac{\partial\vec{v}_{t}}{\partial x} \\
\eta\,\frac{\partial\vec{B}_{t}}{\partial x} \\
\left(\frac{\nu}{3}+\mu\right)v_{x}\frac{\partial v_{x}}{\partial x}+\nu\,\vec{v}\cdot \frac{\partial\vec{v}}{\partial x}+\eta\vec{B}_{t}\cdot \frac{\partial\vec{B}_{t}}{\partial x}+\kappa\frac{\partial}{\partial x}\big( \frac{p}{\rho} \big) \\
\end{array} \right),
\end{equation}
\begin{equation} \label{basic4}
e=\frac{1}{2}\rho\,v^{2}+\frac{p}{\gamma -1}+\frac{1}{2}B^{2},\,\,P=p+\frac{1}{2}B^{2},
\end{equation}
where the subscript $x$ denote the $x$-component and the subscript $t$ denote the $y$ and $z$-component.
We use the unit such that the factor $4\pi$ does not appear.
$\nu$ and $\mu$ are the shear and bulk viscosity coefficients, $\eta$ is the electric resistivity, and $\kappa$ is the heat conduction coefficient.
In addition to (\ref{basic1}), we impose $\boldsymbol{\nabla}\cdot \boldsymbol{B}=0$.
This leads $B_{x}$ to a constant in the one-dimensional case.
In the following, we consider the steady shock solution of equations (\ref{basic1})-(\ref{basic4}) as an unperturbed state.
In this case, without loss of generality, we can choose the coordinate system such that the upstream is $x=-\infty$, the downstream is $x=+\infty$ (i.e., $v_{x,0}>0$ everywhere), and the unperturbed velocity and magnetic fields are in the $x-y$ plane at $x=\pm \infty$ (coplanarity).
We also choose the shock rest frame, and assume that the shock structure is around $x=0$.
We call the steady state type 1 if $v_{x}\ge c_{f}$; type 2 if $c_{f}\ge v_{x}\ge c_{i}$; type 3 if $c_{i}\ge v_{x}\ge c_{s}$; and type 4 if $c_{s}\ge v_{x}$, where $c_{f},\,c_{s}$ and $c_{i}$ are the fast, slow and Alfv\'en (intermediate) speed, respectively.
Steady shock solutions of equations (\ref{basic1})-(\ref{basic4}) are well studied (see, e.g. Wu 1990).
There are six types of the shock solutions, $1\rightarrow 2$ fast shock, $3\rightarrow 4$ slow shock and $1\rightarrow 3$, $1\rightarrow 4$, $2\rightarrow 3$ and $2\rightarrow 4$ intermediate shocks, where the numbers before and after the arrow represent the state of ahead and behind the shock front.
Let us consider the small perturbation of the steady shock structure.
We assume that the perturbation of the physical variable $g(x,t)$ takes the following form:
\begin{equation}
g(x,t)=g_{0}(x)+\delta g(x)\,e^{-i\,\omega t}\,,
\end{equation}
where the subscript $0$ denotes the unperturbed variable.
Linearizing the equation (\ref{basic1}), we obtain the perturbed shock equations
\begin{eqnarray}
\frac{d}{dx}\delta\rho&=&\frac{1}{v_{x,0}}\big( i\,\omega\,\delta\rho-\frac{dv_{x,0}}{dx}\delta\rho-\rho_{0}\frac{d\delta v_{x}}{dx}-\frac{d\rho_{0}}{dx}\delta v_{x} \big), \label{EOC}\\
\Big( \frac{4}{3}\nu+\mu \Big) \frac{d^{2}}{dx^{2}}\delta v_{x} &=& \frac{d}{dx}\big( v_{x,0}^{2}\,\delta\rho+2\,\rho_{0}\,v_{x,0}\,\delta v_{x}+\delta p+B_{y,0}\,\delta B_{y}+B_{z,0}\,\delta B_{z} \big)
\nonumber\\&& -i\,\omega\left(\rho_{0}\,\delta v_{x}+v_{x,0}\,\delta \rho \right), \label{EOMx}\\
\nu\,\frac{d^{2}}{dx^{2}}\delta v_{y} &=& \frac{d}{dx}\big( v_{x,0}\,v_{y,0}\,\delta\rho+\rho_{0}\,v_{y,0}\,\delta v_{x}+\rho_{0}\,v_{x,0}\,\delta v_{y}-B_{x,0}\,\delta B_{y} \big)
\nonumber\\&& -i\,\omega\left(\rho_{0}\,\delta v_{y}+v_{y,0}\,\delta \rho \right), \label{EOMy}\\
\nu\,\frac{d^{2}}{dx^{2}}\delta v_{z} &=& \frac{d}{dx}\big( v_{x,0}\,v_{z,0}\,\delta\rho+\rho_{0}\,v_{z,0}\,\delta v_{x}+\rho_{0}\,v_{x,0}\,\delta v_{z}-B_{x,0}\,\delta B_{z} \big)
\nonumber\\&& -i\,\omega\left(\rho_{0}\,\delta v_{z}+v_{z,0}\,\delta \rho \right), \label{EOMz}\\
\eta\,\frac{d^{2}}{dx^{2}}\delta B_{y} &=& \frac{d}{dx}\big( B_{y,0}\,\delta v_{x}+v_{x,0}\,\delta B_{y}-B_{x,0}\,\delta v_{y} \big)-i\,\omega\,\delta B_{y}, \label{IEy}\\
\eta\,\frac{d^{2}}{dx^{2}}\delta B_{z} &=& \frac{d}{dx}\big( B_{z,0}\,\delta v_{x}+v_{x,0}\,\delta B_{z}-B_{x,0}\,\delta v_{z} \big)-i\,\omega\,\delta B_{z}, \label{IEz}\\
\frac{\kappa}{\rho_{0}} \frac{d^{2}}{dx^{2}}\delta p &=& \kappa\,\Big( \frac{2}{\rho_{0}^{2}}\,\frac{d\rho_{0}}{dx}\,\frac{d\delta p}{dx}-\frac{d^{2}\rho_{0}}{dx^{2}}\,\frac{\delta p}{\rho_{0}^{2}} \Big)+
\kappa\,\frac{d^{2}}{dx^{2}}\Big( \frac{p_{0}}{\rho_{0}^{2}}\,\delta \rho \Big)
\nonumber\\&&-\Big( \frac{4}{3}\nu+\mu \Big) \frac{d}{dx}\Big( v_{x,0}\,\frac{d\delta v_{x}}{dx}+\frac{dv_{x,0}}{dx}\,\delta v_{x} \Big)
\nonumber\\&&-\nu\,\frac{d}{dx}\,\Big( v_{y,0}\frac{d\delta v_{y}}{dx}+\frac{dv_{y,0}}{dx}\,\delta v_{y}+v_{z,0}\,\frac{d\delta v_{z}}{dx}+\frac{dv_{z,0}}{dx}\,\delta v_{z} \Big)
\nonumber\\&&-\eta\frac{d}{dx}\Big( B_{y,0}\frac{d\delta B_{y}}{dx}+\frac{dB_{y,0}}{dx}\delta B_{y}+B_{z,0}\frac{d\delta B_{z}}{dx}+\frac{dB_{z,0}}{dx}\delta B_{z} \Big)
\nonumber\\&&+\frac{d}{dx}\Big\{ \big( \delta e+\delta p+B_{y,0}\delta B_{y}+B_{z,0}\delta B_{z} \big)v_{x,0}+\big( e_{0}+p_{0}
\nonumber\\&&+\frac{1}{2}B_{0}^{2} \big)\delta v_{x} \nonumber-\big( \delta v_{x}\,B_{x,0}+\delta v_{y}\,B_{y,0}+\delta v_{z}\,B_{z,0}+v_{y}\,\delta B_{y,0}
\nonumber\\&&+v_{z}\,\delta B_{z,0}+ \big)B_{x} \Big\}-i\,\omega\,\delta e \,.\label{EE}
\end{eqnarray}
Equation (\ref{EOC}) is the first-order ordinary differential equation with respect to $\delta \rho$, and equations (\ref{EOMx})-(\ref{EE}) are the second-order differential equation with respect to $\delta v_{x},\,\delta v_{y},\,\delta v_{z},\,\delta B_{y},\,\delta B_{z},\,$ and $\delta p$ respectively.
Therefore, we have totally the 13th order differential equations.
We denote the order of the perturbed equations as $N=13$.
If we choose the coplanar intermediate shock solution ($v_{z,0}=B_{z,0}=0$ everywhere) as an unperturbed state, equations (\ref{EOC})-(\ref{EE}) are separated into the two sets of equations.
The first set is equations (\ref{EOC})-(\ref{EOMy}),(\ref{IEy}), and (\ref{EE}), which relate the $x$ and $y$- components of the perturbations, and they are totally the $9$th order differential equations ($N_{xy}=9$).
The other set is equations (\ref{EOMz}) and (\ref{IEz}), which relate the $z$-components of the perturbations, and they are totally the $4$th order differential equations ($N_{z}=4$).
\section{Evolutionary Conditions for the Continuous Shock Waves}
In this section, we formulate the evolutionary conditions for the continuous MHD shock waves.
We divide the space into three regions.
Two of them are the regions far away from the shock front, in which the unperturbed physical variables can be regarded as constants.
We call such regions in the upstream and the downstream as region $\cal{U}$ and $\cal{D}$, respectively.
The region between the regions $\cal{U}$ and $\cal{D}$ is the transition region where the unperturbed physical variables change continuously.
We call it region $\cal{T}$.
The situation is schematically illustrated in Figure \ref{fig1}.
\begin{center}
\begin{figure}
\includegraphics[width=10cm]{fig1.eps}
\caption{Schematic of region partition.
}
\label{fig1}
\end{figure}
\end{center}
As shown in the previous section, perturbed shock equations are totally $N$th order ordinary differential equations.
Thus, in order to determine a solution of these equations in the region $\cal{T}$, we need $N$ boundary conditions at the edges of the region $\cal{T}$.
In other words, in the region $\cal{T}$, there are $N$ degree of freedom in order to determine a solution of these equations.
In the regions $\cal{U}$ and $\cal{D}$, unperturbed physical variables are regarded as constants, and we can obtain the asymptotic solutions of the perturbed shock equations.
Omitting the spatial derivative of the zeroth order variable, and Fourier transforming equations (\ref{EOC})-(\ref{EE}) in space ($\partial/\partial x\sim i\,k$), one can obtain the characteristic equation for the asymptotic waves.
The determinant of the characteristic matrix provides the characteristic equation.
\begin{eqnarray}
D_{fse}\,D_{i}&=&0 \,, \label{disp} \\
D_{fse}&=&c_{i}^{2}( \gamma\,\Omega\,K\,V' + 3\,i\,c_{a}^{2}K' ) \nonumber \\&&+
3\,V\,\{ i\,c_{a}^{2}\,K'\,R + \gamma\,\Omega\,K\,(c_{A}^{2}-c_{i}^{2}+R\,V'/3) \} \,, \label{dispfse} \\
D_{i}&=&\left( \Omega+i\,\eta\,k^{2} \right)
\left( \Omega+i\,\nu\,k^{2} \right) -c_{i}^{2}\,k^{2}\,, \label{dispi}
\end{eqnarray}
where
\begin{eqnarray}
\Omega &=& \omega - k\,v_{x,0}\,,\\
c_{a}^{2} &=& \gamma\,p_{0}/\rho_{0}\,,\\
c_{i}^{2} &=& B_{x,0}^{2}/\rho_{0}\,,\\
c_{A}^{2} &=& (B_{x,0}^{2}+B_{y,0}^{2})/\rho_{0}\,,\\
V &=& k^{2}\nu/\rho-i\,\Omega\,,\\
V' &=& k^{2}(3\,\mu-4\,\nu)/\rho-3\,i\,\Omega\,,\\
R &=& \eta-i\,\Omega/k^{2}\,,\\
K &=& (\gamma-1)\,\kappa/\rho-i\,\Omega/k^{2}\,,\\
K'&=& k^{2}(\gamma-1)\,\kappa/\rho-i\,\gamma\,\Omega\,,
\end{eqnarray}
where the zoroth order variables are evaluated in the region $\cal{U}$ and $\cal{D}$, and we use the fact that the non-coplanar component of the unperturbed variables vanish ($B_{z,0}=v_{z,0}=0$) in the asymptotic regions owing to the coplanarity.
The solutions of the characteristic equation $D_{fse}=0$ for $\omega$ as a function of $k$ provide the dispersion relations of the fast, slow, and entropy waves in the uniform dissipative medium, and the solutions of $D_{i}=0$ provide the dispersion relation of the Alfv\'en waves.
\textbf{Definition of Mode }
As shown below, we use the asymptotic solutions in regions $\cal{U}$ and $\cal{D}$ as a boundary conditions of the differential equations (\ref{EOC})-(\ref{EE}).
These conditions are expressed as the superpositions of the independent asymptotic solutions.
The degree of freedom of the spatial behavior of the perturbation for given $\omega$ gives the number of independent asymptotic solutions.
Thus, they are obtained by solving characteristic equation (\ref{disp}) for $k$ as a function of given $\omega$.
In this paper, we call it ``mode" or ``asymptotic mode".
Note that it does not mean solution of characteristic equation for $\omega$ as a function of $k$.
Hada (1994) studied the solutions of $D_{i}=0$ in the limit of small dissipation coefficients.
He found that in addition to the solutions, which correspond to the Alfv\'en modes, there are additional dissipative modes.
Details of the solutions are discussed in the next section.
Let us consider evolutionary condition.
If we steadily throw a small amplitude incident (ingoing) wave whose frequency is $\omega$ toward the shock from the region $\cal{U}$ or $\cal{D}$, then the shock front is perturbed and emit the waves with the same frequency.
Let $m$ $(=m_{\cal{U}}+m_{\cal{D}})$ denote the number of resulting asymptotic modes in the region $\cal{U}$ $(m_{\cal{U}})$ and $\cal{D}$ $(m_{\cal{D}})$ without the incident wave, i.e. the number of modes that can be emitted or raised at the shock front.
In the regions $\cal{U}$ and $\cal{D}$, the solutions of perturbed shock equations should be expressed as the superpositions of the $m$ asymptotic modes and one incident wave.
These asymptotic solutions are determined by $m$ parameters, i.e. amplitudes of the $m$ asymptotic modes.
Note that the amplitude of the incident wave is determined.
Thus, in the case that
\begin{equation}
N=m\,, \label{ECD}
\end{equation}
we can obtain the solution of the perturbed shock equations (\ref{EOC})-(\ref{EE}) in the region $\cal{T}$, which is smoothly connected to the asymptotic solutions at the edges of the region $\cal{T}$, for arbitrary incident wave by choosing $m$ amplitudes of the asymptotic modes.
If $m$ is less than $N$, we cannot obtain solution.
If $m$ is greater than $N$, we can obtain solution, but we need additional conditions or constraints in order to uniquely define the solution.
Therefore, the condition (\ref{ECD}) corresponds to the evolutionary condition in the dissipative system.
If unperturbed shock structure is coplanar, the evolutionary condition (\ref{ECD}) can be divided into the following two conditions
\begin{equation}
N_{xy} = m_{xy}\,, \label{FSECD}
\end{equation}
\begin{equation}
N_{z} = m_{z}\,, \label{ICD}
\end{equation}
where $m_{xy}$ and $m_{z}$ are the numbers of the asymptotic modes of $D_{fse}=0$ and $D_{i}=0$, respectively.
We call the condition (\ref{FSECD}) evolutionary condition for the $x$ and $y$-components, and condition (\ref{ICD}) evolutionary condition for the $z$-components.
Of course, evolutionary shock must satisfy both conditions.
\section{Evolutionary Condition for the $z$-components}
In this section we show that, contrary to the ideal system, all types of intermediate shocks satisfy evolutionary condition for the $z$-components $N_{z}=m_{z}$.
The solutions of $D_{i}=0$ for $k$ under the assumption of small dissipation coefficients, derived by Hada (1994), are
\begin{eqnarray}
k^{(\pm)}&=&\frac{\omega}{v_{x,0}\pm c_{i}}\,+\,\mathcal{O}(\eta,\,\nu)\,, \label{Alfpm} \\
k^{(\pm d)}&=&\frac{-i}{2\,\eta\,\nu}\,\Big( (\eta+\nu)\,v_{x,0}\pm \Big\{(\eta+\nu)^{2}v_{x,0}^{2}-4\,\eta\,\nu\,(v_{x,0}^{2}-c_{i}^{2})\Big\}^{\frac{1}{2}}\Big)\,+\,\mathcal{O}(\eta^{0},\,\nu^{0})\,. \label{Dpm}
\end{eqnarray}
The $(+)$ and $(-)$ modes are the usual Alfv\'en waves, which propagate parallel and anti-parallel to the $x$-axis in the fluid rest frame.
The $(\pm d)$ modes, with no counterparts in the ideal system, are so called the ``dissipative modes".
The dissipative modes do not propagate, because $k^{(\pm d)}$ do not have real part in their primary terms, and they are localized around the shock front.
We exclude diverging dissipative modes as unphysical asymptotic solutions.
Thus, they are physical if
\begin{equation}
\mbox{Im}[\,k^{(\pm d)}\,]>0 \,\mbox{ in the region } \cal{U}, \mbox{ or}
\end{equation}
\begin{equation}
\mbox{Im}[\,k^{(\pm d)}\,]<0 \,\mbox{ in the region } \cal{D}.
\end{equation}
If the flow speed is super-Alfv\'enic in the shock rest frame, then from equation (\ref{Dpm}), the imaginary part of $k^{(\pm d)}$ are less than zero.
Therefore there are two dissipative modes in the region $\cal{U}$, whereas no dissipative mode in the region $\cal{D}$. On the other hand, if the flow speed is sub-Alfv\'enic, the imaginary part of $k^{(+d)}$ is greater than zero and that of $k^{(-d)}$ is less than zero, thus there is one dissipative mode in the region $\cal{U}$ and $\cal{D}$.
Let us consider the evolutionary condition for the $z$-components.
In the case of the fast shock, the flow speeds both in the upstream and the downstream are super-Alfv\'enic.
Therefore, the $k^{(\pm)}$ modes propagate to the downstream in the region $\cal{D}$, and the $k^{(\pm d)}$ modes are in the region $\cal{U}$.
In the same way, the flow speeds of the slow shock are sub-Alfv\'enic in the both sides. There are $k^{(-)}$ and $k^{(+d)}$ modes in the region $\cal{U}$, and $k^{(+)}$ and $k^{(-d)}$ modes in the region $\cal{D}$.
In the case of the intermediate shocks, the upstream flow speed is super-Alfv\'enic and the downstream flow speed is sub-Alfv\'enic.
There are $k^{(\pm d)}$ modes in the region $\cal{U}$, and $k^{(+)}$ and $k^{(-d)}$ modes in the region $\cal{D}$.
The situations are schematically illustrated in Figure \ref{fig2}.
In all cases, the MHD shocks satisfy the evolutionary condition for the $z$-components, i.e. $N_{z}=m_{z}=4$.
\begin{center}
\begin{figure}
\includegraphics[width=10cm]{fig2.eps}
\caption{Illustration of the outgoing ordinary MHD modes and the localized dissipative modes, where the bars represent the region $\cal{T}$, the arrows represent the propagation of the asymptotic modes, and the triangles represent the localized dissipative modes.
The ($\pm$) sign before the Alfv\'en modes denote the propagation direction in the fluid rest frame, and
$c_{i}=(B_{x,0}^{2}/\rho_{0})^{1/2}$ is the Alfv\'en (intermediate) speed.
}
\label{fig2}
\end{figure}
\end{center}
\section{Evolutionary Conditions in the Resistive but Inviscid MHD System}
Let us consider the case where there is only resistivity in the dissipation.
The one-dimensional basic equation of the weakly ionized gas, a main ingredient of the interstellar medium, can be written as a resistive but inviscid MHD equation under the strong coupling (one-fluid) approximation.
Furthermore, Wu (1987, 1990) showed by using one-dimensional simulation in this system that the intermediate shocks are formed through nonlinear steeping from simple waves.
Thus, it is meaningful to analyze the evolutionary condition in this system.
We apply our formalism developed in \S 3 to the resistive but inviscid MHD shocks.
In this system the perturbed basic equations (\ref{EOC})-(\ref{EE}) become first order ordinary differential equations with respect to $\delta \rho,\,\delta p,\,\vec{\delta v}$, and second order one with respect to $\vec{\delta B}_{t}$.
Thus, we have totally 9-th order ($N=9$) differential equations $(N_{xy}=6,\,N_{z}=3$ in the case of the coplanar shocks$)$.
The number of asymptotic modes is determined by the solutions of the resistive version of equation (\ref{disp}).
\begin{eqnarray}
D_{fse}&=&\Omega\,\{\,c_{i}^{2}\,c_{a}^{2}\,k^{4}-i\,\eta\,c_{a}^{2}\,\Omega\,k^{4}-(c_{A}^{2}+c_{a}^{2})\Omega^{2}\,k^{2}+i\,\eta\,\Omega^{3}k^{2}+\Omega^{4}\,\} \label{disprfse}\\
D_{i}&=&\Omega\left( \Omega+i\,\eta\,k^{2} \right)-c_{i,0}^{2}\,k^{2}\,. \label{dispri}
\end{eqnarray}
Equation (\ref{dispri}), which contains three asymptotic modes, was studied by Hada (1994).
Two of them corresponds to two Alfv\'en waves, and third one corresponds to the dissipative mode.
The dissipative mode solution under the assumption of small resistivity is
\begin{equation}
k^{(d1)}=i\,\frac{c_{i}^{2}-v_{x,0}^{2}}{\eta\,v_{x,0}}\,+\,\mathcal{O}(\eta^{0},\,\nu^{0})\,, \label{dissmi}
\end{equation}
which is physical in the upstream for the super-Alfv\'enic state (state 1 and 2), and physical in the downstream for the sub-Alfv\'enic state (state 3 and 4).
Equation (\ref{disprfse}) contains six asymptotic modes.
Five of which correspond to two fast waves, two slow waves and one entropy wave, and the last one corresponds to the dissipative mode.
The dissipative solution under the assumption of small resistivity is
\begin{equation}
k^{(d2)}=i\,\frac{c_{a}^{2}\,c_{i}^{2}-(c_{A}^{2}+c_{a}^{2})\,v_{x,0}^{2}+v_{x,0}^{4}}{\eta\,v_{x,0}\,(c_{a}^{2}-v_{x,0}^{2})}\,+\,\mathcal{O}(\eta^{0},\,\nu^{0})\,. \label{dissmfse}
\end{equation}
In the case of the MHD system with resistivity without viscosity, steady shock structure whose flow velocity is supersonic ($v_{x}>c_{a}$) ahead of the shock front and subsonic ($v_{x}<c_{a}$) behind the shock front is impossible (see, Wu 1990).
Thus, it is possible that the shock structure for $1\rightarrow2$, $1\rightarrow3$, and $2\rightarrow3$ whose flow speeds are supersonic everywhere in the shock rest frame, and $2\rightarrow3$, $2\rightarrow4$, and $3\rightarrow4$ whose flow speeds are subsonic everywhere in the shock rest frame.
The denominator of the righthand side of equation (\ref{dissmfse}) is less than zero in the supersonic state, and larger than zero in the subsonic state.
The numerator is larger than zero in the state 1 (super-fast) and 4 (sub-slow), and less than zero in the state 2 and 3 (sub-fast and super-slow).
Therefore, in the supersonic case, the dissipative (d2) mode is physical in the region $\cal{U}$ for the state 1, and in the region $\cal{D}$ for the state 2 and 3.
In the subsonic case, it is physical in the $\cal{U}$ for the state 2 and 3, and in the $\cal{D}$ for the state 4.
The number of asymptotic modes in asymptotic regions is $m=9\,\,(m_{xy}=6,\,m_{z}=3$ in the case of non-coplanar shock$)$ for all shocks.
We illustrate the situations for the shocks whose upstream velocity is supersonic in Figure \ref{fig3}, and for the shocks whose upstream velocity is subsonic in Figure \ref{fig4}.
Therefore, all the shocks, which are allowed in the MHD system with resistivity without viscosity, satisfy the evolutionary condition $N=m=9\,\,($or $N_{xy}=m_{xy}=6,\,N_{z}=m_{z}=3$).
\begin{figure}
\includegraphics[width=10cm]{fig3.eps}
\caption{
Illustration of the outgoing ordinary MHD modes and the localized dissipative modes for the shocks whose upstream velocity is supersonic, where the bars represent the region $\cal{T}$, the arrows represent the propagation of the asymptotic modes, and the triangles represent the localized dissipative modes.
The ($\pm$) sign before the ordinary MHD modes denote the propagation direction in the fluid rest frame.
}
\label{fig3}
\end{figure}
\begin{figure}
\includegraphics[width=10cm]{fig4.eps}
\caption{
Illustration of the outgoing ordinary MHD modes and the localized dissipative modes for the shocks whose upstream velocity is subsonic, where the bars represent the region $\cal{T}$, the arrows represent the propagation of the asymptotic modes, and the triangles represent the localized dissipative modes.
The ($\pm$) sign before the ordinary MHD modes denote the propagation direction in the fluid rest frame.
}
\label{fig4}
\end{figure}
\section{Summary and Discussion}
We have studied evolutionary conditions for the continuous MHD shock waves in the dissipative system.
We have shown that all types of MHD shocks, even the intermediate shocks, satisfy the evolutionary condition for the $z$-components, which is not satisfied in the ideal MHD system.
Especially in the resistive system, all types of MHD shocks, which are allowed in the resistive system, satisfy the evolutionary condition.
Therefore, the intermediate shocks have neighboring solution, and they can survive interactions with arbitrary small amplitude waves.
The difference of the evolutionary conditions between the ideal and the dissipative system is that, in the dissipative system, there are dissipative perturbation modes, which do not propagate and are localized around the shock front, and they provide degree of freedom, which are lost in the ideal system.
This difference is first pointed out by Hada (1994).
Hada (1994) tried to connect the perturbation by using the Rankine-Hugoniot relations.
As a result, set of equations become under-determined due to the excessive dissipative modes.
In order to uniquely define solution, Hada (1994) introduced the minimum dissipation principle.
In this paper, however, we have shown that perturbations in the asymptotic regions (far away from the shock front) can be uniquely connected by solving differential equations, instead of the conservation laws in the ideal or hyperbolic system, because the shock is not a weak solution but a continuous solution in the dissipative system.
Note that the solution of the differential equations also satisfies the usual conservation laws in the asymptotic regions.
However, the idea of the minimum dissipation principle and use of the perturbed Rankine-Hugoniot relations may be important, when we analyze the linear stability of the intermediate shocks.
In order to analyze the linear stability by using our formulation, we have to solve the differential equation.
This seems difficult, because the equations are stiff, and there need huge amount of calculations to search parameter space.
If the minimum dissipation principle can sort out important dissipative modes from other dissipative modes in order to the solution be unique, the analysis is easily done, because the solution is obtained analytically.
Further study is needed to discover such technique.
Our approach is similar to that of Wu (1988) and Wu \& Kennel (1992).
They studied the interaction between the intermediate shock and the non-linear Alfv\'en wave whose transverse magnetic field is rotated.
They showed by using non-linear simulations that there exists a new class of time-dependent intermediate shocks, since they violate the coplanarity even without the shock structure (i.e., they do not obey the MHD Rankine-Hugoniot relations between the upstream and the downstream), which possibly be a neighboring state of the intermediate shocks.
On the other hand, our approach treat the interaction between the intermediate shock and small amplitude (linear) MHD wave, and showed that the intermediate shocks have neighboring solutions which describe the time evolutions of the perturbed intermediate shocks in the linear regime.
The intermediate shock which has maximum non-coplanar magnetic flux inside the shock structure break up into other shocks and waves as a result of the interaction with the Alfv\'en wave (Markovskii \& Skorokhodov 2000 and Falle \& Komissarov 2001).
Such an unstable phenomenon would be analyzed possibly in terms of the linear stability analysis.
Our formulation is also expected to be used for such analysis.
The existence of the intermediate shocks has been argued mainly in the interplanetary system (Chao et al. 1993; Chao 1995).
However, in the partially ionized interstellar medium, it is known that there are C-type MHD shocks, which have broad shock structure owing to friction between neutral and ionized gases.
Because of the lifetime of the intermediate shock seems to be proportional to the width of the front, the intermediate shocks can be long-lived in the interstellar medium.
The quantitative discussion on this issue remains to be done.
\section*{Acknowledgements}
This work is supported by the Grant-in-Aid for the 21st Century COE "Center for Diversity and Universality in Physics" from the Ministry of Education, Culture, Sports, Science and Technology (MEXT) of Japan.
SI is supported by the Grant-in-Aid (No.15740118, 16077202, 18540238) from MEXT of Japan.
| {
"redpajama_set_name": "RedPajamaArXiv"
} | 5,715 |
Preheat oven to 350 degrees. Combine cream in heavy saucepan with bourbon, cinnamon, salt and honey. Simmer on low heat for 20 minutes.
Put egg yolks in metal bowl. Using a whisk, slowly add cream mixture, brown sugar, granulated sugar and raisins.
Add chopped croissants and smash together. Let bread soak in mixture. Line two soup cups with a small amount of whole butter. Add soaked croissants to soup cups. Pack tight to remove air bubbles. Place cups in small baking tin. Add warm water to within one-quarter inch of the top of the cups. Cover with aluminum foil.
Cook for about 30 minutes until golden brown on top. Use a butter knife and carefully remove and plate the bread pudding. For added appeal, plate the dish surrounded with dollops of caramel sauce and topped with cinnamon whipped cream, spun sugar and a mint sprig. | {
"redpajama_set_name": "RedPajamaC4"
} | 23 |
\section{Introduction}
The leverage effect is one of the well-established phenomena of the financial economics. Historically, \cite{Black1976} discusses a possible relationship between returns and changes in volatility of stocks. The argumentation is based on changes in earnings, where decreasing expected earnings of the company push the price down and in turn it decreases the market value of the company which drives the leverage (ratio between debt and equity) up. Negative relationship between returns and volatility is thus referred to as `the leverage effect'. However, in the modern, high-speed, markets where the market prices of assets are driven by many more forces than simple expected earnings, such an explanation of the effect serves as just a little more than an anecdote. The leverage effect can be simply understood as a negative relationship between returns and volatility which are driven by opposite forces. When negative news reaches the market, volatility of the corresponding asset usually increases because of an uncertain future development. Contrarily, the negative news drives the prices down forming a negative return. The leverage effect thus seems as a natural connection of the two characteristics (returns and volatility) of the traded assets.
The leverage effect is usually tightly connected, and sometimes even interchanged, with a notion of the asymmetric volatility. The standard asymmetric volatility is characterized by a lower volatility connected to a bull (growing) market and a higher volatility connected to a bear (declining) market. The definition and interconnection between the two effects -- the leverage effect and the asymmetric volatility -- is thus very close and sometimes hard to distinguish between. Nonetheless, most authors agree on several characteristics of the relationship between returns and volatility -- returns and volatility are negatively correlated, the correlation is quite weak yet still persists over quite long time (with slowly decaying cross-correlations), and the causality goes from returns to volatility and not vice versa \citep{Pagan1996,Bouchaud2001,Bouchaud2001a,Bollerslev2006}.
Here we analyze the leverage effect in the future contracts of energy commodities, namely WTI and Brent crude oils, natural gas and heating oil. We try to provide a coherent treatment of the leverage effect starting from the long-term memory characteristics of volatility and its potential non-stationarity, then moving to the estimation of the correlation between returns and volatility under borderline (non-)stationary and a typical seasonality of futures contracts, and finally checking the slow decay of the cross-correlation function characteristic for long-range cross-correlated processes. We find that the leverage effect in its purest form (significant negative correlation between returns and volatility) is found for two out of four studied commodities. However, the level of correlation is very low -- lower than levels standardly reported for stocks and stock indices. Moreover, we show that the cross-correlations are not identified as hyperbolically decaying, i.e. there are no long-range cross-correlations between returns and volatility of the studied commodities. An important aspect of our analysis stems in not assuming anything about the relationship between returns and volatility which distinguishes our study from the other studies which are majorly built around assuming some kind of asymmetric volatility model, i.e. the leverage effect and asymmetric volatility are assumed ex ante to be frequently found ex post.
The paper is structured as follows. In Section 2, we provide a literature review of recent studies on the leverage effect and asymmetric volatility on energy markets. Section 3 introduces the most important methodological aspects of our work -- volatility estimation, long-term memory and its tests and estimators, estimation of correlations under borderline (non-)stationarity and seasonality, and long-range cross-correlations testing. Section 4 presents the analyzed dataset and results. Section 5 concludes.
\section{Literature review}
In this section, we review recent literature on the topic of leverage effect and asymmetric volatility in energy commodities in chronological order.
\cite{Fan2008} examine WTI and Brent crude oil prices with various specifications of the generalized autoregressive conditional heteroskedasticity (GARCH) models for purposes of risk management. They find significant two-way spillover effect between both crude oil markets as well as asymmetric leverage effect in the WTI returns but not in the Brent returns. Interestingly, the uncovered leverage effect implies that positive shocks have much higher impact on the future dynamics of the series than the negative ones which is opposite to the leverage effect found in stocks and it can be thus treated as an inverse leverage effect.
\cite{Zhang2008} study an interrelation between the US dollar exchange rates and crude oil prices with a special focus on spillover effects which they separate into three -- mean spillover, volatility spillover and risk spillover. Apart from a significant long-term cointegration relationship, the authors find significant volatility asymmetry. In a similar way to the previous reference, they find the inverse leverage effect which they attribute mainly the non-renewable property of oil and very different roles and behavior of suppliers and demanders of the commodity.
\cite{Aloui2009} examine the relationship between crude oil and stock markets utilizing a two regime Markov switching exponential GARCH model. They show that the volatility clustering and the leverage effect can be significantly reduced by allowing for the regime switching. Transition between regimes is mainly connected to economic recessions together with stock markets behavior. \cite{Agnolucci2009} compares predictive powers of GARCH-type and implied volatility models on the WTI future contract. Apart from showing that the GARCH-type models outperform the implied volatility models, the author also finds no leverage effect for the WTI contract. \cite{Cheong2009} then focuses on both WTI and Brent crude oil markets and applies GARCH specification. The author finds that the WTI volatility is more persistent than the one of the Brent crude oil. Even though the leverage effect is found for the Brent market and not for the WTI market, the out-of-sample forecasting exercise provides an evidence that a reduced GARCH model with no asymmetric volatility outperforms the others.
\cite{Wei2010} study both the WTI and Brent futures and compare a wide portfolio of GARCH-type models. Focusing on the performance of 1-day, 5-day and 20-day forecasting, they find that no single model is a clear winner in the horse race of testing. However, the authors favor the non-linear specifications of GARCh which can control for long-term memory as well as asymmetry. Similarly to the previous studies, the results on asymmetry are mixed for the two markets. Even though the asymmetry is found for a strong majority of specifications for the Brent market, the WTI shows mixed evidence.
\cite{Chang2010} focus on the relationship between crude oil and biofuels. Specifically, they are interested in the dynamics of volatility (using the exponential GARCH model) conditional on various phases of the market with respect to the crude oil prices. A significant asymmetric volatility reaction is found only for the soybean futures during the high oil prices. Other futures show no significant asymmetry. \cite{Du2011} examine the linkage between the crude oil volatility and agricultural commodity markets using the stochastic volatility approach in the Bayesian framework. The authors show that speculation, scalping and petroleum investors form important aspects of the volatility formation. In the model, they find a weak leverage effect between instantaneous volatility and prices.
\cite{Reboredo2011} inspects the crude oil dependence structure with various copula functions. He shows that the correlation structure is similar during both bear and bull markets and further states that the crude oil market is strongly globalized. For the favorited model of the marginals -- exponential GARCH -- the volatility asymmetry is found for all studied crude oil series. The same methodology is then applied in \cite{Reboredo2012} where the relationship between oil price and exchange rates is examined. In general, the connection between the oil and exchange rate markets is reported to be very weak. The evidence of volatility asymmetry is mixed as well. \cite{Wu2012} propose a copula-based GARCH model and use it to model dependence between crude oil and the US dollar. In their specification, the leverage effect is not significant for either of the studied futures.
\cite{Chang2012} employs a combined regime switching exponential GARCH model with Student-$t$ distributed error terms to model crude oil futures returns. The model is able to capture the main stylized facts of the crude oil futures. Importantly, the model combines both the regime switching and asymmetric volatility to capture nonlinear dependencies between returns, volatility and higher moments. In accordance to other works, no leverage effect is found for the WTI futures.
\cite{Ji2012} analyze the effect of crude oil volatility spillovers on non-energy commodities. After controlling for exchange rates, the authors utilize a bivariate exponential GARCH model with time-varying correlation structure. They show that the crude oil plays a core role in the commodities structure as its volatility spills over to other, non-energy, markets as well. The strength of these spillovers even increases after the 2008 financial crisis. Volatility asymmetry is studied as a difference in reaction to bad and good news. The authors find the effect to be significant for majority of the studied pairs.
\cite{Nomikos2012} investigate dynamics of eight energy spot markets on NYMEX. The authors combine a mean-reverting and a spike model with GARCH-type time-varying volatility focusing on risk management issues as well as their forecasting performance. The leverage effect is found for WTI, heating oil and heating oil-WTI crack spread, and the inverse leverage effect is uncovered for gasoline, natural gas, propane and gasoline-WTI crack spread.
Copulas are further utilized by \cite{Tong2013} who study tail dependence between crude oil and refined petroleum markets. Positive dependence is found in both tails so that the markets tend to move together in both bear and bull periods. Asymmetry in tail dependence is found between crude and heating oils, and between crude oil and jet fuel. Interestingly, the upper tail dependence is stronger than in the lower tail for the pre-crisis period. The authors report that the leverage effect, which is found in its standard form, is much stronger for the post-crisis period.
\cite{Salisu2013} study the WTI and Brent crude oil with respect to the structural breaks while controlling for potential volatility asymmetry. Persistence as well as asymmetry of volatility is reported even after controlling for two structural breaks (Iraqi/Kuwait conflict and the financial crisis of 2008) identified for both oil markets. The authors stress that neither of the effects should be studied separately and the constructed models should consider each of structural breaks, volatility persistence and asymmetry.
And \cite{Chkili2014} examine crude oil, natural gas, gold and silver markets using various linear and nonlinear GARCH-type specifications. The nonlinear specifications are found to fare better in a sense of in-sample and out-of-sample performance as well as risk management issues under the Basel II regulations. The direction and significance of the leverage effect are found to be strongly dependent on the model choice.
\section{Methodology}
Studying leverage effect stems primarily on the analysis of the relationship between returns and volatility of the series. As such, this is connected with several issues. Firstly, the volatility itself needs to be extracted from the series. Secondly, the volatility is standardly considered as a long-term memory process. Thirdly, not only is the volatility process long-term dependent but it is usually on the edge of stationarity, i.e. its fractional integration parameter $d \approx 0.5$ and it is thus somewhere between a stationary short-term memory process with $d=0$ and a unit root process with $d=1$. In this section, we introduce methodology and instruments to deal with such specifics.
\subsection{Volatility estimation}
In majority of the leverage effect and asymmetric volatility studies covered in the Literature review, the volatility process has been estimated as a part of the complete model under various assumptions and restrictions. In turn, the volatility series and its characteristics are strongly dependent on the model choice and specifications. For our purposes, the leverage effect emerges from the model only if we assume correlation between the returns and volatility processes. However, if the effect is in reality not present, it can simply occur to be significant during the estimation procedure due to the model misspecification. In our study, we bypass this issue by estimating the volatility outside the returns model.
Historically, the volatility and variance series were estimated simply as a squared or absolute returns of the series. In a sense, the GARCH-type models are built in the same logic. However, these simple measures turn out to be very poor estimators of the true volatility \citep{Chou2010}. Range-based estimators of volatility turn out to be much more efficient and precise than the absolute and squared errors and they stay close to the most efficient realized variance family measures\footnote{We do not opt for the realized variance family measures due to their need of high-frequency data. Moreover, our study is a study of the relationship between returns and volatility, not of finding the best measure of volatility. The range-based estimators are in turn a very fitting compromise as these need only daily open, close, high and low prices which are freely available for practically all publicly traded assets.}.
From several possibilities, we select the Garman--Klass estimator (GKE) as a highly efficient estimator of daily variance. The estimator is defined as
\begin{equation}
\label{GK}
\widehat{\sigma^2_{GK,t}}=\frac{(\log(H_t/L_t))^2}{2}-(2\log2-1)(\log(C_t/O_t))^2
\end{equation}
where $H_t$ and $L_t$ are daily highs and lows, respectively, and $C_t$ and $O_t$ are daily closing and opening prices, respectively \citep{Garman1980}. As the estimator does not take the overnight volatility into consideration, we further work with the open-close returns, i.e. $r_t=\log(C_t)-\log(O_t)$.
\subsection{Long-term memory}
Long-term memory (long memory, long-range dependence) is connected to specific features of the series in both time and frequency domains. In the time domain, the long-term memory process has asymptotically power-law decaying autocorrelation function $\rho(k)$ with lag $k$ such that $\rho(k) \propto k^{2H-2}$ for $k \rightarrow +\infty$. In the frequency domain, the long-term memory process has divergent at origin spectrum $f(\lambda)$ with frequency $\lambda$ such that $f(\lambda)\propto \lambda^{1-2H}$ for $\lambda \rightarrow 0+$. In both definitions, the Hurst exponent $H$ plays a crucial role. For stationary series, $H$ is standardly bounded between 0 and 1 so that $0 \le H <1$. No long-term memory is connected to $H=0.5$, positive long-term autocorrelations are found for $H>0.5$ and negative ones for $H<0.5$. The Hurst exponent is connected to the fractional differencing parameter $d$ in a strict way -- $H=d+0.5$ \citep{Beran1994}.
Hurst exponent is crucial for our further analysis. However, before estimating the exponent itself, we need to test the series for actually being long-range dependent. It has been shown that the estimators of Hurst exponent might report values different from 0.5, and thus hinting long-term memory, even if the series are not long-range dependent \citep{Taqqu1995,Taqqu1996,Teverovsky1999,Lennartz2009,Barunik2010,Kristoufek2010a,Kristoufek2012,Zhou2012}. To deal with this matter, we firstly test for the presence of long-range dependence in the series before estimating the Hurst exponent. We opt for two tests -- modified rescaled range and rescaled variance.
The modified rescaled range test \citep{Lo1991} is an adjusted version of the traditional rescaled range test \citep{Hurst1951} controlling for short-term memory of the series. The testing statistic $V$ is defined as
\begin{equation}
\label{MRS}
V_{T}=\frac{(R/S)_{T}}{\sqrt{T}}
\end{equation}
where the range $R$ is defined as a difference between the maximum and the minimum of the profile (cumulative demeaned original series), $S$ is the standard deviation of the series and $T$ is the time series length. Here $(R/S)_T$ is the rescaled range of the series of length $T$. To control for the potential short-term memory bias (strong short-term memory might be mistaken for the long-term memory), the standard deviation $S$ is used in its heteroskedasticity and autocorrelation consistent (HAC) version. For these purposes, we utilize the following specification which is later used in the bivariate setting and the rescaled covariance test as well:
\begin{equation}
\label{eq:HAC_CC}
\widehat{s_{xy,q}}=\sum_{k=-q}^q{\left(1-\frac{|k|}{q+1}\right)\widehat{\gamma_{xy}}(k)}
\end{equation}
where $\widehat{\gamma_{xy}}(k)$ is a sample cross-covariance at lag $k$, $q$ is a number of lags taken into consideration and the cross-covariances are weighted with the Barlett-kernel weights. For the purposes of the modified rescaled range, we set $S \equiv \widehat{s_{xx,q}}$ as the autocovariance function is symmetric. We follow the suggestion of \cite{Lo1991} and use lag $q$ according to the following formula for the optimal lag:
\begin{equation}
\label{eq2}
q^{\ast}=\left\lfloor\left(\frac{3T}{2}\right)^{\frac{1}{3}}\left(\frac{2\widehat{|\rho(1)|}}{1-\widehat{\rho(1)}^2}\right)^{\frac{2}{3}}\right\rfloor
\end{equation}
where $\widehat{\rho(1)}$ is a sample first order autocorrelation and $\lfloor \rfloor$ is the lower integer operator. Under the null hypothesis of no long-range dependence, the statistic is distributed as
\begin{equation}
F_V(x)=1+2\sum_{k=1}^{\infty}(1-4k^2x^2)e^{-2(kx)^2}.
\end{equation}
As an alternative to the modified rescaled range test, \cite{Giraitis2003} propose the rescaled variance test which simply substitutes the range in Eq. \ref{MRS} by variance of the profile. The testing statistic $M$ is then defined as
\begin{equation*}
M_T=\frac{var(X)}{TS^2},
\end{equation*}
where $X$ is the profile of the original series and the standard deviation $S$ is defined in the same way as for the modified rescaled range test. \cite{Giraitis2003} show that the rescaled variance test has better properties than the modified rescaled range which is further supported by \cite{Lee1996} and \cite{Lee1997}. Under the null hypothesis of no long-term memory, the statistic is distributed as
\begin{equation}
F_M(x)=1+2\sum^\infty_{k=1}{(-1)^ke^{-2k^2\pi^2x}}.
\end{equation}
For the estimation of the Hurst exponent itself, we utilize two frequency domain estimators -- the local Whittle estimator and the GPH estimator. We opt for the frequency domain estimators as these have well defined asymptotic properties and are well suited even for non-stationary or boundary series which turns out to be the case for the analysis we present.
\cite{Robinson1995a} proposes the local Whittle estimator as a semi-parametric maximum likelihood estimator using the likelihood of \cite{Kunsch1987} while focusing only on a part of spectrum near the origin. As an estimator of the spectrum $f(\lambda)$, the periodogram $I(\lambda)$ is utilized. For the time series of length $T$, and setting $m \le T/2$ and $\lambda_j=2\pi j/T$, the Hurst exponent is estimated as
\begin{equation}
\label{eq:LWX}
\widehat{H}=\arg \min R(H),
\end{equation}
where
\begin{equation}
\label{eq:LWX_R}
R(H)=\log\left(\frac{1}{m}\sum_{j=1}^m{\lambda_j^{2H-1}I(\lambda_j)}\right)-\frac{2H-1}{m}\sum_{j=1}^m{\log \lambda_j}.
\end{equation}
\cite{Geweke1983} introduce an estimator based on a full functional specification of the underlying process as the fractional Gaussian noise, which is labeled as the GPH estimator after the authors. The assumption of the underlying process is connected to a specific spectral density which is in turn utilized in the regression estimation of the following equation:
\begin{equation}
\label{GPH}
\log I(\lambda_j) \propto -(H-0.5)\log [4\sin^2(\lambda_j/2)].
\end{equation}
Both estimators are consistent and asymptotically normal. To avoid bias due to short-term memory, we estimate both the local Whittle and GPH estimators only on parts of the estimated periodogram that are close to the origin (short-term memory is present at high frequencies and thus far from the origin). Specifically, we use $m=T^{0.6}$.
\subsection{Correlation coefficient for non-stationary series}
As the leverage effect can be seen as a correlation between returns and volatility, a need for efficient estimators of correlation between potentially non-stationary series is high. Recently, two methods have been proposed in the literature -- detrended cross-correlation coefficient \citep{Zebende2011} and detrending moving-average cross-correlation coefficient \citep{Kristoufek2014a}.
\cite{Zebende2011} proposes the detrended cross-correlation coefficient as a combination of the detrended cross-correlation analysis (DCCA) \citep{Podobnik2008} and the detrended fluctuation analysis (DFA) \citep{Peng1993,Peng1994,Kantelhardt2002}. The detrended cross-correlation coefficient $\rho_{DCCA}(s)$, which measures the correlation even between non-stationary as well as seasonal series, is defined as
\begin{equation}
\rho_{DCCA}(s)=\frac{F^2_{DCCA}(s)}{F_{DFA,x}(s)F_{DFA,y}(s)},
\label{rho}
\end{equation}
where $F^2_{DCCA}(s)$ is a detrended covariance between profiles of the two series based on a window of size $s$, and $F^2_{DFA,x}$ and $F^2_{DFA,y}$ are detrended variances of profiles of the separate series, respectively, for a window size $s$. For more technical details about the methods, please refer to \cite{Kantelhardt2002}, \cite{Podobnik2008} and \cite{Kristoufek2014}. In words, the method is based on calculating the correlation coefficient between series detrended by a linear trend while the detrending is performed in each window of length $s$.
\cite{Kristoufek2014a} introduces the detrending moving-average cross-correlation coefficient as an alternative to the above mentioned coefficient. The method connects the detrending moving average (DMA) procedure \citep{Vandewalle1998,Alessio2002} and detrending moving-average cross-correlation analysis (DMCA) \citep{Arianos2009,He2011a}. The detrending moving-average cross-correlation coefficient $\rho_{DMCA}(\lambda)$ is defined as
\begin{equation}
\rho_{DMCA}(\lambda)=\frac{F_{DMCA}^2(\lambda)}{F_{x,DMA}(\lambda)F_{y,DMA}(\lambda)},
\end{equation}
where $F^2_{DMCA}(\lambda)$, $F^2_{DMA,x}(\lambda)$ and $F^2_{DMA,y}(\lambda)$ are, similarly to the DCCA-based coefficient, detrended covariance between profiles of the two studied series and detrended variances of the separate series, respectively, with a moving average parameter $\lambda$. Contrary to the previous DCCA-based method, the DMCA variant does not require box-splitting but estimates the correlation from the profile series detrended simply by the moving average of length $\lambda$. \cite{Carbone2003} show that the centered moving average outperforms the backward and forward ones so that we apply the centered one in our analysis. For more detailed description of the procedures, please refer to \cite{Alessio2002}, \cite{Arianos2009} and \cite{Kristoufek2014a}.
\subsection{Rescaled covariance test}
Motivated by the rescaled variance test for the univariate series, \cite{Kristoufek2013} proposes the rescaled covariance test which is able to distinguish between long-term and short-term memory between two series. In a similar way as for the univariate series, the long-term memory can be generalized to the bivariate setting so that the long-range cross-correlated (cross-persistent) processes are characterized by asymptotically power-law decaying cross-correlation function and divergent at origin cross-spectrum. By applying the test to the relationship between returns and volatility, we can comment on possible power-law cross-correlated relationship between the two series which is usually connected to the leverage effect \citep{Cont2001}.
The testing statistic for the rescaled covariance test is defined as
\begin{equation}
\label{eq:RC}
M_{xy,T}(q)=q^{\widehat{H_x}+\widehat{H_y}-1}\frac{\widehat{\text{Cov}}(X_T,Y_T)}{T\widehat{s_{xy,q}}},
\end{equation}
where $\widehat{s_{xy,q}}$ is the HAC-estimator of the covariance of the studied series defined in Eq. \ref{eq:HAC_CC}, $\widehat{\text{Cov}}(X_T,Y_T)$ is the estimated covariance between profiles of the series, and $\widehat{H_x}$ and $\widehat{H_y}$ are estimated Hurst exponents for the separate processes. For the estimated Hurst exponents, we use the average of the local Whittle and GPH estimators if the process is found to be long-range dependent. Otherwise, we set the corresponding exponent equal to 0.5. Critical and $p$-values for the test are obtained from the moving-block bootstrap methodology. For more details, please refer to \cite{Kristoufek2013}.
\section{Data and results}
We analyze front futures contracts of Brent crude oil, WTI (West Texas Intermediate) crude oil, heating oil and natural gas between 1.1.2000 and 30.6.2013. As we are interested in the leverage effect, we focus on returns and volatility of the future prices. In Figs. \ref{fig1} and \ref{fig2}, we present returns and volatility based on the Garman-Klass estimator given in Eq. \ref{GK}. From the returns charts, we observe that these behave very similarly to the standard financial returns with volatility clustering and non-Gaussian distribution. However, we also notice, mainly for the natural gas, that returns undergo certain seasonal pattern which is connected to the rolling of the front and back futures contracts. This is dealt with by utilizing detrended cross-correlation and detrending moving-average cross-correlation coefficients which are constructed for such seasonalities. Volatility dynamics again reminds of standard volatility of other financial assets with evident persistence, which is dealt with later on. Again, the natural gas series stands out with more frequent volatility jumps and more erratic behavior.
In Tab. \ref{tab1}, we summarize standard descriptive statistics and tests. All returns series follow quite standard characteristics such as excess volatility, negative skewness (apart from natural gas in this case), non-Gaussian distribution and asymptotic stationarity. For each series, we also find significant autocorrelations. Later, we test whether these can be treated as the long-term ones or not. Apart from the returns and volatility, which we examine in its logarithmic form, we focus on the standardized returns as well. Note that the returns standardized by their volatility are usually close to being normally distributed and in general, they are more suitable for statistical analysis. From this point onward, we focus solely on the relationship between standardized returns and logarithmic volatility so that if returns and volatility are referred to, we work with the transformed series. Standardized returns are all approximately symmetric and do not exceed kurtosis of the normal distribution. Moreover, the autocorrelations have been filtered out by standardizing for three out of four series. For the volatility, we strongly reject normality of the distribution and we find very strong autocorrelations. Moreover, we reject both unit root and stationary behavior of the series. This leads us to an inspection of potential long-term memory in the analyzed series.
In Tab. \ref{tab2}, we show results for the modified rescaled range and the rescaled variance tests. Optimal lag has been chosen according to Eq. \ref{eq2}. We find that neither of the returns series are long-range autocorrelated, even though the testing statistics for natural gas are close to the critical levels. As expected, long-term memory is identified for all volatility series even after controlling for rather high number of lags (between 15 and 20). The results of the long-term memory tests thus give expected results -- no long-term memory for the returns and statistically significant long-term memory for the volatility series.
Based on the previous tests, we take that the returns series are not long-term dependent so that their Hurst exponent is equal to 0.5, which is later used in the rescaled covariance test. For the volatility series, we estimate the Hurst exponent $H$ using the local Whittle and GPH estimators. The estimates are summarized in Tab. \ref{tab3}. We observe that both estimators give similar results -- the Hurst exponent for volatility for all four studied series is estimated around $H=1$. Based on the reported standard errors, we cannot distinguish whether the Hurst exponents are below or above the unity value. Therefore, we cannot easily decide whether the volatility series are stationary long-range dependent or non-stationary long-range dependent but still mean reverting. Nevertheless, this does not discredit any of the following instruments and tests.
As the volatility series are long-term correlated, we need to apply correlation measures which are able to deal with such series. \cite{Kristoufek2014} shows that the standard correlation coefficient is not able to do so. We thus apply the detrended cross-correlation coefficient and detrending moving-average cross-correlation measures which are not only able to work under long-term memory and even non-stationarity but they can also filter out well-defined trends. In the case of the studied futures, the rolling period of a trading month is well-established so that we can set $s=\lambda=20$ and the methods filter the seasonality away. Tab. \ref{tab4} reports the estimated correlation coefficients between returns and volatility of each studied commodity. We find that both crude oils are partially driven by the standard leverage effect connected to negative correlation between returns and volatility. For heating oil, the estimated correlation is also negative but not statistically significant\footnote{$p$-values are constructed using 10,000 series generated using Fourier randomization, which ensures that the autocorrelation structure remains untouched but the cross-correlations are shuffled away.} at 1\% level. Natural gas is then characterized by the inverse leverage effect, i.e. the positive correlation between returns and volatility. Note that even though some of the correlations are found to be statistically significant, the levels are rather weak compared to standardly reported ones for stocks or stock indices.
Tab. \ref{tab5} then summarizes the results of the rescaled covariance test which test possible long-range cross-correlations. We use the same number of lags as for the univariate volatility tests in Tab. \ref{tab3}. Based on the reported $p$-values, we find no sings of long-range dependence in the bivariate setting. This is tightly connected to rather weak correlations found above. Even though the series might be correlated, creating the leverage or the inverse leverage effects, the influence is not strong enough to translate into a long-term connection.
\section{Conclusion}
In this paper, we propose a comprehensive treatment of the leverage effect, focusing on energy commodities futures, namely Brent and WTI crude oils, natural gas and heating oil. After estimating the volatility process without assuming any specific form of its behavior, we find the volatility to be long-term dependent with the Hurst exponent on a verge of stationarity and non-stationarity. Bypassing this using by using the detrended cross-correlation and the detrending moving-average cross-correlation coefficients, we find the standard leverage effect for both crude oils. For heating oil, the effect is not statistically significant, and for natural gas, we find the inverse leverage effect. This points out a need for initial testing for the presence of the leverage effect before constructing any specific models to avoid inefficient estimation or even biased results. Finally, we also show that none of the effects between returns and volatility is detected as the long-term cross-correlated one. The dynamics of the crude oil futures, as ones of the most traded ones, is thus closer to the one of stocks and stock indices whereas the less popular heating oil and natural gas somewhat deviate from the standard behavior. These findings can be further utilized to enhance forecasting models and mainly in the risk management and portfolio diversification.
\section*{Acknowledgements}
The research leading to these results has received funding from the European Union's Seventh Framework Programme (FP7/2007-2013) under grant agreement No. FP7-SSH-612955 (FinMaP). Support from the Czech Science Foundation under projects No. P402/11/0948 and No. 14-11402P is also gratefully acknowledged.
\section*{References}
\bibliographystyle{chicago}
| {
"redpajama_set_name": "RedPajamaArXiv"
} | 1,488 |
/** \file
* \brief Implementation of stricmp
*
* \author Peter 'png' Hille <peter@das-system-networks.de>
*/
#include "config.h"
#include <ctype.h>
#include <string.h>
#include <assert.h>
/** \brief Compare Strings without Case Sensitivity
*
* stricmp compares s1 and s2 without sensitivity to case. All alphabetic
* characters in the two arguments s1 and s2 are converted to lowercase
* before the comparision.
*
* The function operates on null-ended strings. The string arguments to the
* function are expected to contain a null character (\0) at the end of the
* string.
*
* \param s1 Pointer to a \0-terminated C string
* \param s2 Pointer to a \0-terminated C string
*
* \return A value indicating the relationship between the two strings similar
* to the one returned by strcmp(3)
*
* \see http://publib.boulder.ibm.com/infocenter/iseries/v7r1m0/index.jsp?topic=%2Frtref%2Fstricmp.htm
* \see strcasecmp
* \see strnicmp
*
* \note In debug builds the two pointer arguments are checked for NULL
* values using assert(3)
*
* \attention This really shouldn't be used in new projects because it doesn't
* do any bounds checking on the underlying memory areas!
*/
int stricmp(const char* s1, const char* s2) {
assert(s1 != NULL);
assert(s2 != NULL);
#ifdef HAVE_STRCASECMP
return strcasecmp(s1, s2)
#else
while (tolower((unsigned char) *s1) == tolower((unsigned char) *s2)) {
if (*s1 == '\0')
return 0;
s1++; s2++;
}
return (int) tolower((unsigned char) *s1) -
(int) tolower((unsigned char) *s2);
#endif /* !HAVE_STRCASECMP */
}
| {
"redpajama_set_name": "RedPajamaGithub"
} | 367 |
Chytridiomycety (Chytridiomycota) je oddělení primitivních hub, zpravidla označované za nejprimitivnější houby vůbec. Jedná se pravděpodobně o přirozenou, monofyletickou skupinu, pokud se z chytridiomycet vyloučí taxony (oddělení) Neocallimastigomycota a Blastocladiomycota.
Popis
Chytridiomycety mohou mít několik typů stélek, ty nejprimitivnější jsou jen váčkovitá sporangia bez přídatných orgánů, u jiných skupin se však přidávají různě složité rhizoidy. Nejsložitějším typem jsou druhy s tzv. cenocytickým myceliem, které připomíná podhoubí hub. Buňky jsou obvykle mnohojaderné. Buněčná stěna typická pro chytridiomycety je chitin, beta-1,3-glukan a beta-1,6-glukan; celulóza údajně u jednoho zástupce.
Množí se nepohlavně produkcí spor v rozmanitých sporangiích, ale bylo u nich nalezeno i mnoho druhů pohlavního rozmnožování. Zoospory se totiž mohou za určitých podmínek chovat jako gamety a fúzovat spolu za vzniku nového jedince. Jindy jsou však gamety již plně odlišené na základě velikosti či barvy; nejvyšším stupněm vývoje je zřejmě oogamie, kdy samičí gameta (oosféra) zůstává na svém místě v gametangiu a je oplodněna pohyblivými samčími gametami.
Životní styl
Tato skupina je zřejmě primárně vodní, ale vyskytuje se ve velkém měřítku i v půdě. Samostatnou kapitolou jsou Neocallimastigomycota, dnes svébytné oddělení, které žijí v bachoru mnohých býložravých kopytníků.
Nachází se mezi nimi řada parazitických a saprofytických druhů. Parazitovat mohou na jiných chytridiomycetách, cévnatých rostlinách nebo i na hmyzu. Ke známým parazitickým chytridiomycetám patří mimo jiné rakovinec bramborový (Synchytrium endobioticum) parazitující na bramborách, lahvičkovka (Olpidium), případně Batrachochytrium dendrobatidis napadající žáby.
Odkazy
Reference
Externí odkazy
Houby | {
"redpajama_set_name": "RedPajamaWikipedia"
} | 9,733 |
\section{Introduction}
\label{intro}
Coronal mass ejections (CMEs) are large eruptions of magnetized plasma from the solar corona into the heliosphere, and the main driver of geomagnetic storms.
There exists little doubt that CMEs consist of magnetic flux ropes \citep[FRs; e.g.,][]{Chen2017}, although the time of their formation has been debated.
In recent years,
however, there is growing evidence for the existence of FRs before many eruptions
\citep[e.g.,][]{Canou2009, Green2009, Zhang2012, Patsourakos2013, Howard2014, Chintzoglou2015}.
This justifies the use of FR configurations as initial condition for the numerical modeling of CMEs, for both idealized configurations \citep[e.g.,][]{Amari2000, Fan2005, Aulanier2010, Torok2011} and configurations that are constructed using observed magnetograms \citep[e.g.,][]{Manchester2008, Linker2016b}.
Modeling pre-eruptive FRs for observed cases is challenging, since direct measurements of the coronal magnetic field are difficult, so typically the morphology and magnetic structure of the FR can be inferred only indirectly from, e.g., observed filament shapes or the location of flare arcades or dimmings \citep[e.g.,][]{Palmerio2017}.
Multiple trial and error attempts may be required to create a stable magnetic equilibrium that satisfactorily matches the observations.
In contrast, a less rigorous approach based on out-of-equilibrium FRs for initializing CMEs is much easier to apply \citep[e.g.,][]{Liu2008, Lugaz2009, Loesch2011}.
However, there is no guarantee that the resulting CME model will be sufficiently accurate, especially for complex pre-eruptive configurations.
One way to produce equilibrium FR configurations is by mimicking the slow formation of pre-eruptive FRs using a boundary-driven MHD evolution \citep[e.g.,][]{Lionello2002, Bisi2010, Zuccarello2012, Jiang2016}.
This requires the development of photospheric boundary conditions that will lead to the formation of an FR, subject to the constraints of the observed photospheric field.
This approach is non-trivial, computationally expensive, and has no simple means to control the shape and stability of the FR.
Another way to produce such configurations is via non-linear force-free field \citep[NLFFF, e.g.,][]{Schrijver2008} extrapolations.
This method requires the use of observed vector magnetic data.
However, these data are measured at the photospheric level where the magnetic field is not force-free,
and must be ``pre-processed'' to be compatible with the force-free reconstruction \citep{Wiegelmann2006}.
In reality, the magnetic field becomes force-free in a thin layer where the photosphere turns into the chromosphere.
Vector magnetic field observations also suffer from noise and disambiguation issues, making constructing NLFFF-models of FRs that match observable properties a rather non-trivial problem as well.
An alternative approach is the FR insertion method \citep{vanBall2004,Su2011,Savcheva2012}, which uses observations of filaments, loops, etc., to directly constrain the field model.
For this technique, an attempt to find an equilibrium configuration is made in two steps.
First, a field-free cavity following the inferred FR shape is prepared in the related potential-field configuration and filled with axial and azimuthal magnetic fluxes.
Second, the resulting configuration is subjected to a magnetofrictional relaxation to find a force-free equilibrium.
The procedure is repeated by varying the inserted magnetic fluxes and/or cavity until an equilibrium with desirable properties is obtained.
Each initial configuration is highly out of equilibrium, since it is constructed without balancing the magnetic forces in the cavity.
As a result, the relaxation of the configuration can significantly change the inserted fluxes and perturb the ambient structure around the cavity. This makes the properties of the modeled force-free equilibrium difficult to control, and many parameter iterations may be required to reach the desired result.
In contrast, the equilibrium conditions are addressed in the FR embedding method by \citet{Titov2014}, which employs
\fb{a modified version of the model by \citet{Titov1999}, called henceforth TDm model.
}
This method uses the ambient potential field to estimate the axial and azimuthal magnetic fluxes, as well as geometric parameters of the FR.
The estimation \fb{follows from the requirement that
the ambient field superimposed on the FR field must compensate the field component due to FR curvature, yielding an approximately force-free configuration}.
This configuration is then relaxed via a line-tied MHD evolution toward an equilibrium whose parameters are close to the estimated ones---a main advantage of the method.
Unfortunately, since the TDm FR has a toroidal-arc shape, it is difficult to model configurations that reside above a highly elongated or curved polarity inversion line (PIL).
Complex FR shapes can be obtained by merging several FRs \citep[e.g.,][]{Linker2016b}, but this is a rather laborious procedure.
The purpose of the present work is to remove this geometric limitation by generalizing the method for FRs of arbitrary shape.
\section{Regularized Biot-Savart laws}
We propose a general mathematical form that defines at a given point ${\pmb x}$ the magnetic field ${\pmb B}_{\mathrm{FR}}$ of a thin FR with axis path $\cal C$, arc length $l$, radius-vector $ {\pmb R}(l)$, tangential unit vector ${\pmb R}^{\prime} ={\rm d} {\pmb R}/{\rm d}l$, and cross-sectional radius $a(l)$
(Figure~\ref{f:FR_3d}) as follows:
\begin{figure}[ht!]
\centering
\resizebox{0.45\textwidth}{!}{
\includegraphics{fig1}}
\caption{FR with a circular cross-section of radius $a(l)$ and coronal and subphotospheric axis paths $\mathcal C$ and ${\mathcal C}^{*}$, respectively, defined by a radius-vector $ {\pmb R}(l)$, where $l$ is the path arc length.
\label{f:FR_3d}}
\end{figure}
\begin{eqnarray}
{\pmb B}_{\mathrm{FR}} &=& \nabla \times {\pmb A}_{I} + \nabla \times {\pmb A}_{F} \, ,
\label{BFR} \\
{\pmb A}_{I}({\pmb x}) &=& \frac{\mu I}{4\pi} \int_{\cal C\, \cup\,C^{*}} K_{I}(r) \; {\pmb R}^{\prime}(l) \; \frac{ {\rm d}l }{ a(l) } \, ,
\label{AI} \\
{\pmb A}_{F}({\pmb x}) &=& \frac{F}{4\pi} \int_{\cal C\, \cup\,C^{*}} K_{F}(r) \; {\pmb R}^{\prime}(l) \times {\pmb r} \; \frac{ {\rm d}l }{ a(l) ^2 } \, ,
\label{AF}
\end{eqnarray}
where ${\pmb r} \equiv {\pmb r}(l) = \left( {\pmb x} -{\pmb R}(l) \right) / a(l) $.
The field $ {\pmb A}_{I}({\pmb x})$ and its curl represent the axial vector potential and the azimuthal magnetic field, respectively, generated by a net current $I$.
The field $ {\pmb A}_{F}({\pmb x})$ and its curl represent the azimuthal vector potential and the axial magnetic field, respectively, generated by a net flux $F$.
Both integrals are taken over the coronal path ${\cal C}$ and its subphotospheric counterpart ${\cal C}^{*}$ that closes the current/flux circuit.
The path ${\cal C}^{*}$ can be chosen, for example, as a mirror image of ${\cal C}$ to keep the photospheric normal magnetic field outside the FR unchanged, but other choices of ${\cal C}^{*}$ are possible as well.
Externally, a thin FR manifests itself as a thread carrying an axial current $I$ and axial flux $F$,
which means that Equations (\ref{AI}) and (\ref{AF}) should asymptotically coincide with the classical Biot-Savart laws whose kernels are given by $K_{I}(r) = 1/r$ and $K_{F}(r) = 1/r^{3}$ \citep{Jackson1962}.
We take the latter as exact expressions for our kernels outside the FR and extend them to the FR interior by resolving singularities at $r=0$ and making the interior field approximately force-free.
We do this by simply identifying the kernels of straight and curved force-free FRs of a circular cross-section,
which is allowable if local curvature radii ${\mathcal R}_{\mathrm c}(l)$ of the FR axis are large enough, i.e., ${\mathcal R}_{\mathrm c}(l) \gg a(l) $.
We call these kernels the regularized Biot-Savart laws ( ${ \rm _{\rm R\!}BS_{\rm L} }$ ) kernels.
In addition, we assume hereafter, for simplicity, that $a(l)\equiv a = \mbox{const}$ along modeled FRs.
This assumption is justified by the successful applications of our ${ \rm _{\rm R\!}BS_{\rm L} }$ \ method to configurations with coherent FR structures (see Section \ref{s:exmpls}).
Let us use $a$ as a length unit for the distances in our consideration,
and let $\rho$ be the distance from $\pmb x$ to the axis of a straight cylindrical FR.
Then, the arc length of the axis is $l=\sqrt{r^2-\rho^2}$.
Changing the integration variable from $l$ to $r$ in Equation (\ref{AI}) and taking the FR cylinder of length $2L$, we obtain
\begin{eqnarray}
{\pmb A}^{L}_{I}(\rho) = \frac{\mu I \skew{3}\hat{\pmb l}}{4\pi}\; 2 \int_{\rho}^{\sqrt{L^2+\rho^2}} \frac{r\,K_{I}(r)}{\sqrt{r^2-\rho^2}} \; {\mathrm d}r \, .
\label{AIro}
\end{eqnarray}
As expected, this integral diverges logarithmically in the limit of $L \rightarrow \infty$.
However, it can be ``renormalized'' by subtracting the constant ${\pmb A}^{L}_{I}(1)$, such that it becomes convergent in this limit to yield
\begin{eqnarray}
2\int_{\rho}^{1} \frac{r\,K_{I}(r) \; {\mathrm d}r}{\sqrt{r^2-\rho^2}}
- 2\ln\left(1+\sqrt{1-\rho^2} \right)
= A_{\mathrm{ax}}(\rho) \, .
\nonumber \\
\label{eqKI}
\end{eqnarray}
The integration result is equated here to the axial component, $A_{\mathrm{ax}}(\rho)$, of the vector potential that is normalized by $\mu I/(4\pi)$ \fb{and generally determined up to an arbitrary additive constant}.
\fb{In our $A_{\mathrm{ax}}(\rho)$, however, this constant must be fixed by the condition $A_{\mathrm{ax}}(1)=0$ because of the used ``renormalization''.}
The integral of Equation (\ref{AF}) is convergent for the cylindrical FR and straightforwardly reduces to
\begin{eqnarray}
2\,\rho \int_{\rho}^{1} \frac{r\,K_{F}(r) \; {\mathrm d}r}{\sqrt{r^2-\rho^2}}
+ \frac{2\,\rho}{1+\sqrt{1-\rho^2}}
= A_{\mathrm{az}}(\rho) \, ,
\nonumber \\
\label{eqKF}
\end{eqnarray}
where the integration result is equated to the azimuthal component, $A_{\mathrm{az}}(\rho)$, of the vector potential normalized by $F/(4\pi a)$.
\fb{This normalization automatically implies that $A_{\mathrm{az}}(1)=2$.}
Equations (\ref{eqKI}) and (\ref{eqKF}) are related to the Abel integral equation that can be solved analytically \citep[see p. 531 in][]{Polyanin2008}.
Using this fact, we have found general solutions of these equations in terms of the following quadratures:
\begin{eqnarray}
K_{I}(r) &=& \frac{2 \arcsin r}{\pi r}
- \frac{1}{\pi} \frac{\mathrm d\ }{{\mathrm d}r} \left[ r
\int_{r}^{1} \frac{ A_{\mathrm{ax}}(\rho) \; {\mathrm d}\rho }{ \rho \sqrt{\rho^2 - r^2} } \right] \, ,
\nonumber \\
\label{KIgen}\\
K_{F}(r) &=& \frac{2}{\pi r^{2}} \left(\frac{\arcsin r}{r}-\sqrt{1-r^2}\right)
\nonumber \\
&-& \frac{1}{\pi} \frac{\mathrm d\ }{{\mathrm d}r} \left[ r
\int_{r}^{1} \frac{ \left( A_{\mathrm{az}}(\rho) - 2 \rho \right)\; {\mathrm d}\rho }{ \rho^2 \sqrt{\rho^2 - r^2} } \right]\, .
\label{KFgen}
\end{eqnarray}
They determine the ${ \rm _{\rm R\!}BS_{\rm L} }$ \ kernels via given axial and azimuthal components of the vector potential of a cylindrical FR that locally approximates a curved thin FR of any shape.
To make such a curved FR approximately force-free, let us take $A_{\mathrm{ax}}(\rho)$ and $A_{\mathrm{az}}(\rho)$ corresponding to a straight force-free FR.
These components are determined from a force-free equation, for which the profile $j_{\mathrm {ax}}(\rho)$ of the axial current density can be chosen freely.
We take the components derived for a FR with a parabolic $j_{\mathrm{ax}}(\rho)$-profile ($\rho \in [0,\: 1]$) and vanishing axial field at $\rho = 1$ \citep[cf. Equations (63)--(69) in][]{Titov2014}:
\begin{eqnarray}
j_{\mathrm{ax}}(\rho) &=& 2\left( 1-\rho^2 \right) \, ,
\label{jaxpb} \\
A_{\mathrm{ax}}(\rho) &=& \frac{1}{2} \left( 1 - \rho^2 \right) \left( 3 - \rho^2 \right) \, ,
\label{Aaxpb}\\
A_{\mathrm{az}}(\rho) &=& \frac{2\rho}{3\sqrt{3}} \left( 5-2\rho^2 \right)^{3/2} \, ,
\label{Aazpb}\\
F &=& \frac{\pm 3}{5\sqrt{2}} \mu I a,
\label{Fpb}
\end{eqnarray}
where $j_{\mathrm{ax}}(\rho)$ is normalized by $I/(\pi a^2)$.
Integrating Equations (\ref{KIgen}) and (\ref{KFgen}) for this case, we obtain the corresponding ${ \rm _{\rm R\!}BS_{\rm L} }$ \ kernels at $0\le r \le 1$:
\begin{eqnarray}
K_{I}(r) &=& \frac{2}{\pi}
\left( \frac{\arcsin r}{r} + \frac{5-2\, r^2}{3} \sqrt{1-r^2}
\right) \, ,
\label{KI} \\
K_{F}(r) &=& \frac{2}{\pi r^{2}} \left(\frac{\arcsin r}{r}-\sqrt{1-r^2}\right)
+ \frac{2}{\pi} \sqrt{1-r^2}
\nonumber \\
&+&\frac{5-2\, r^2}{2 \sqrt{6}}
\left[ 1 - \frac{2}{\pi} \arcsin\left( \frac{1+2\,r^2}{5-2\,r^2} \right)
\right]
\,.
\nonumber \\
\label{KF}
\end{eqnarray}
\begin{figure}[ht!]
\centering
\resizebox{0.45\textwidth}{!}{
\includegraphics{fig2}}
\caption{The ${ \rm _{\rm R\!}BS_{\rm L} }$ \ kernels for an FR with a parabolic $j_{\mathrm{ax}}(\rho)$-profile and axial magnetic field vanishing outside the FR;
the inner ($0\le r \le 1$) and outer ($r>1$) solutions smoothly conjugate to each other.
\label{f:KIKF}}
\end{figure}
Figure \ref{f:KIKF} shows that they are smoothly conjugate to the classical Biot-Savart kernels outside the FR, as required.
Equations (\ref{AI}) and (\ref{AF}) together with the found ${ \rm _{\rm R\!}BS_{\rm L} }$ \ kernels reduce the computation of $ {\pmb A}_{\mathrm{FR}} \equiv \ {\pmb A}_{I} + {\pmb A}_{F} $ to the calculation of two line integrals.
If one computes $ {\pmb A}_{\mathrm{FR}} $ on a numerical grid,
the key advantage of our method is that the ${ \rm _{\rm R\!}BS_{\rm L} }$ \ integration path is the same for all grid points.
\section{Illustrative Examples}
\label{s:exmpls}
We have implemented our method for magnetic configurations defined on numerical grids under different assumptions on closing the axis path $\mathcal C$ by $\mathcal C^{*}$.
\fb{Below we present several tests of the method to verify its capacity to construct approximately force-free FR configurations.
To assess how close these configurations are to equilibrium, we relax them by using our spherical MHD code, MAS \citep{Lionello2009}, in zero-$\beta$ mode \citep{Mikic2013a}, and then visually compare the initial and final field-line structures.
}
\subsection{Test Case 1: TDm Model}
\label{s:t1}
Our first test is intended to reproduce a
TDm configuration that includes a toroidal FR with a parabolic $j_{\rm ax}(\rho)$-profile given by Equation (\ref{jaxpb}).
\begin{figure}[ht!]
\centering
\resizebox{0.45\textwidth}{!}{
\includegraphics{fig3}}
\caption{Reproducing the TDm model (test case 1): the ${ \rm _{\rm R\!}BS_{\rm L} }$ \ configuration before (a) and after (b) zero-$\beta$ MHD relaxation.
The field lines in both configurations have the same footpoints and colors; the boundary radial field $B_{r0}$ is colored in blue ($B_{r0}<0$) and red ($B_{r0}>0$).
\label{f:TDm}}
\end{figure}
The FR is embedded in an idealized bipolar background field ${\pmb B}_{q}$, which is modeled by two fictitious point sources of strength $|\pm q|=100\:{\mbox{T\:Mm}^{2}}$ placed at a depth of $50\: \mbox{Mm}$ below the photospheric boundary and at a distance of $150\:\mbox{Mm}$ from each other.
The axis path $\mathcal C$ follows an iso-contour, $B_{q} = {\mathrm{const}}$, of a circular-arc shape in the vertical plane of symmetry of the configuration, and is closed by a subphotospheric arc $\mathcal C^{*}$ to form a circle of radius $\mathcal R_{\mathrm c} = 110\: \mbox{Mm}$.
The torus minor radius is set to $a=45\:\mbox{Mm}$, which together with the chosen iso-contour determines the parameters $I$ \citep[Equation (7) in][]{Titov2014} and $F$ (Equation (\ref{Fpb})).
We first compared the initial ${ \rm _{\rm R\!}BS_{\rm L} }$ \ configuration to the equivalent TDm version using the same fluxes, geometry, and parabolic current profile.
We found that their vector potentials differ by less than 2\%,
indicating that this choice for the ${ \rm _{\rm R\!}BS_{\rm L} }$ \ kernels indeed matches the TDm formulation for circular FRs.
We then compare the initial FR configuration (Figure \ref{f:TDm}(a)) to one produced after a line-tied zero-$\beta$ MHD relaxation (Figure \ref{f:TDm}(b)).
In spite of a relatively large FR curvature, $a/{\mathcal R_{\mathrm c}}= 0.41$, the relaxation yields a numerical force-free field that is almost identical to the initial ${ \rm _{\rm R\!}BS_{\rm L} }$ \ configuration, demonstrating that the force-freeness property extends nicely from straight to curved FRs.
\subsection{Test Case 2: Sigmoidal Configuration of the 2009 February 13 CME Event}
\label{s:t2}
Our second test is designed to benchmark the method for simple yet realistic magnetic configurations.
For this purpose, we choose the 2009 February 13 CME event where the pre-eruptive magnetic field
had a characteristic sigmoidal structure above the polarity inversion line (PIL) of the source region \citep{Miklenic2011}.
We do not intend to perfectly reproduce this structure, or preserve the radial component of the photospheric field, $B_{r0}$, obtained from observations.
Rather, our purpose is to check whether the ${ \rm _{\rm R\!}BS_{\rm L} }$ \ method can produce a similar sigmoidal structure by simply superimposing ${\pmb B}_{\mathrm{FR}}$ and the potential field ${\pmb B}_{\mathrm{P}}$ extrapolated from the $B_{r0}$-map.
\begin{figure*}[ht!]
\centering
\resizebox{0.95\textwidth}{!}{
\includegraphics{fig4}}
\caption{Steps of modeling the pre-eruptive configuration of the 2009 February 13 CME event (test case 2): (a) top view of the chosen FR-axis path (green line) in the corona with the grayscaled photospheric $B_{r0}$-map (full magnetic field; black: $B_{r0}<0$; white: $B_{r0}>0$) and the $j_{r0}$-map in blue ($j_{r0}<0$) and red ($j_{r0}>0$); (b) side view of the FR-axis path (thick white line) and field lines for the FR-field only, with the corresponding semi-transparent blue-red $B_{r0}$-map.
The ${ \rm _{\rm R\!}BS_{\rm L} }$ \ configuration before (c) and after (d) zero-$\beta$ MHD relaxation; the $B_{r0}$-map is colored in blue and red.
\label{f:2009Feb13_1}}
\end{figure*}
We first choose an S-shaped axis path $\mathcal{C}$ above the PIL of the $B_{r0}$-map (Figure \ref{f:2009Feb13_1}(a)), as suggested by the observations.
This path is closed by a subphotospheric path $\mathcal{C}^{*}$
that mirrors the path $\mathcal{C}$ about the local horizontal plane passing through the footpoints of $\mathcal{C}$. The superposition of ${\pmb B}_{\mathrm{FR}}$ and ${\pmb B}_{\mathrm{P}}$ will naturally modify the $B_{r0}$-map inside the FR footprints due to the axial flux (Figure \ref{f:2009Feb13_1}(b)), but such a path mirroring causes most of the radial component of the azimuthal FR field to vanish at the photosphere. Stripes of weak $B_{r0}$ remain because of the small spherical curvature of the boundary, but this could, in principle, be eliminated by a small adjustment of $\mathcal{C}^{*}$.
The equilibrium axial current $I$ is estimated in two steps.
First, for some current $I_{0}$ and a middle point of the axis-path ${\pmb R}^{*}$, we calculate the potential field
${\pmb B}^{*}_{\mathrm{P}} \equiv {\pmb B}_{\mathrm{P}}({\pmb R}^{*})$
and the azimuthal field ${\pmb B}^{*}_{I_{0}} \equiv {\pmb B}_{I_{0}}({\pmb R}^{*}) \equiv \left . \nabla \times {\pmb A}_{I_{0}} \right|_{{\pmb x}= {\pmb R}^{*} }$.
Since $|{\pmb B}^{*}_{\mathrm{P}} + c \, {\pmb B}^{*}_{I_{0}}|$ as a function of $c$ has a minimum value at $c = c_{0} \equiv - \left( {\pmb B}^{*}_{\mathrm{P}} {\pmb \cdot} {\pmb B}^{*}_{I_{0}} \right) / { B}^{*2}_{I_{0}}$, we obtain the desired estimate for the equilibrium current $I = c_{0}\, I_{0}$.
Figure \ref{f:2009Feb13_1}(c) shows that
the resulting field ${\pmb B}_{\mathrm{FR}} + {\pmb B}_{\mathrm{P}}$
indeed contains a sigmoidal FR.
The sigmoid expands during line-tied MHD relaxation to form a stable FR of a more pronounced S-shape (Figure \ref{f:2009Feb13_1}(d)).
By changing the coefficient $c$ in the linear superposition $c {\pmb B}_{\mathrm{FR}} + {\pmb B}_{\mathrm{P}}$, one can easily generate a family of solutions with sigmoidal FRs that carry different axial current $I$ and flux $F$.
This is very important for parameter studies, in which, e.g., the critical parameters for the onset of eruption are needed to be found.
\subsection{Test Case 3: Pre-eruptive Configuration of the 2011 October 1 CME Event}
\label{s:t3}
Our third test explores how well the method works for more complex configurations, such as the one that produced the 2011 October 1 CME \citep[Figure \ref{f:2011Oct1}(a);][]{Temmer2017}.
This configuration had a PIL that separated a strong (negative) sunspot and a weak dispersed (positive) flux concentration. The strong inhomogeneity of the ambient magnetic field near the PIL poses a serious challenge to embedding a force-free FR into this region.
\begin{figure}[ht!]
\centering
\resizebox{0.45\textwidth}{!}{
\includegraphics{fig5}}
\caption{Modeling the pre-eruptive configuration of the 2011 October 1 CME event (test case 3): (a) FR-like structure suggested by the AIA 131\:\AA\ image superimposed on the photospheric blue-red $B_{r0}$-map; the ${ \rm _{\rm R\!}BS_{\rm L} }$ \ configuration before (b) and after (c) zero-$\beta$ MHD relaxation; orange and green field lines start near the FR footprints, purple field lines show a magnetic arcade enclosing the FR, whose axis is shown by a red line; field lines are drawn by tracking the motion of selected fluid elements (balls) in time; the same elements are used in (b) and (c).
\label{f:2011Oct1}}
\end{figure}
We first construct the FR-axis path by using SDO/AIA 131\:\AA\ observations, which show a bright, curved, elongated feature over the PIL prior to the eruption (Figure \ref{f:2011Oct1}(a)). This projection does not constrain the height, so we chose a height that was slightly larger than the suggested width of the feature. To improve the match between the observed structure and the modeled pre-eruptive FR, we also explored different types of closures for the FR-axis path.
Figure \ref{f:2011Oct1}(b) presents our best solution. In contrast to test case 2, the observed $B_{r0}$-distribution is preserved using a technique similar to \citet{vanBall2004}. We do this by removing the non-vanishing radial component of the ${ \rm _{\rm R\!}BS_{\rm L} }$ \ field at the photosphere from the original $B_{r0}$-distribution and calculating the corresponding potential field.
Superimposing this and FR fields ensures that the radial field at the boundary exactly matches $B_{r0}$.
The coronal axis path $\mathcal C$ is fine-tuned to minimize a residual Lorentz force along the embedded FR,
by repeating small perpendicular displacements of $\mathcal C$ in the directions that yield the strongest decrease of this force.
Figures \ref{f:2011Oct1}(b) and \ref{f:2011Oct1}(c) show that the configuration with the fine-tuned FR is similar to the force-free equilibrium reached in the line-tied MHD relaxation.
\section{Summary}
\label{sum}
We have developed a new method for constructing force-free FRs embedded into potential magnetic fields.
Our method allows one to use an arbitrary FR axis shape and to estimate the equilibrium parameters from the background field, making it generally applicable and computationally efficient.
The FR field is expressed in terms of the axial and azimuthal vector potentials defined by the ${ \rm _{\rm R\!}BS_{\rm L} }$ s for a given FR axis, total axial current and axial flux.
The axis shape is determined by following the PIL of an eruption's source region, using observed magnetograms, and by using observations of this region.
The height variation along the axis and other FR parameters are estimated via potential field extrapolation.
The FR-axis shape can be iteratively adjusted to minimize the Lorentz force along the FR, after which the configuration is subjected to line-tied MHD relaxation toward a numerical equilibrium.
We successfully tested our method for the TDm model \citep{Titov2014} and the pre-eruption configurations of the 2009 February 13 and 2011 October 1 CME events.
Our tests demonstrate that the ${ \rm _{\rm R\!}BS_{\rm L} }$ \ method is a very flexible and efficient way to construct coherent flux-rope structures of non-trivial geometry.
We envision that this method will be particularly useful for theoretical studies of FRs with complex geometries, and for initializing data-constrained simulations of solar flares and CMEs.
We are currently extending this method for modeling FRs with variable cross-sections, which will further increase its flexibility and allow one to initialize interplanetary CME simulations as well.
\acknowledgments
This research was supported by NASA's HSR, LWS, and HGI programs,
NSF grants AGS-1560411 and AGS-1135432,
and AFOSR contract FA9550-15-C-0001.
Computational resources were provided by NSF's XSEDE and NASA's NAS.
\bibliographystyle{apj}
| {
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<li><a href="http://ChandlerHCSmith.github.io/unit1_projects/week_3/w3_10_technical_blog.html">JavaScript, all the Rage Among Web Developers</a></li>
<li><a href="http://ChandlerHCSmith.github.io/unit2_projects/week_4/w4_08_technical_blog.html">Enumerable Methods</a></li>
<li><a href="http://ChandlerHCSmith.github.io/unit2_projects/week_5/x_w5_07_technical_blog_2.html">A Popular Example of Classes</a></li>
<li><a href="http://ChandlerHCSmith.github.io/unit2_projects/week_6/u2_w6_06_technical_blog.html">Functional vs. Object-Oriented Programming</a></li>
<li><a href="http://ChandlerHCSmith.github.io/unit3_projects/week_7/u3_w7_05_technical_blog.html">SQL databases or NoSQL databases</a></li>
<li><a href="http://ChandlerHCSmith.github.io/unit3_projects/week_8/u3_w8_05_technical_blog.html">Static vs. Dynamic Typed Language</a></li>
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<div class="blogs">
<p>Cultural</p>
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<ul>
<li><a href="http://ChandlerHCSmith.github.io/unit1_projects/week1_cultural_blog.html">Values</a></li>
<li><a href="http://ChandlerHCSmith.github.io/unit1_projects/week_3/w3_11_cultural_blog.html">Thinking Style and Learning</a></li>
<li><a href="http://ChandlerHCSmith.github.io/unit2_projects/week_4/w4_09_cultural_blog.html">The Compass of Shame</a></li>
<li><a href="http://ChandlerHCSmith.github.io/unit2_projects/week_5/w5_08_cultural_blog.html">Issues in the Tech World: Embracing Complexity</a></li>
<li><a href="http://ChandlerHCSmith.github.io/unit2_projects/week_6/u2_w6_07_cultural_blog.html">Stereotype Threat</a></li>
<li><a href="http://ChandlerHCSmith.github.io/unit3_projects/week_7/u3_w7_06_cultural_blog.html">Being the Best Fork of Me Merged into DBC</a></li>
<li><a href="http://ChandlerHCSmith.github.io/unit3_projects/week_8/u3_w8_06_cultural_blog.html">The Pairing Experience</a></li>
<li><a href="http://ChandlerHCSmith.github.io/unit3_projects/week_9/u3_w9_06_cultural_blog.html">Thinking Style and Learning Revisited</a></li>
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<p id="body_paragraphs">Here's to us Boots!</p>
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<p>2014, Adventure Sundays SSR *</p>
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| {
"redpajama_set_name": "RedPajamaGithub"
} | 6,727 |
ALSO BY DAVID MARK
_The Dark Winter_
Published by the Penguin Group
Penguin Group (USA) Inc., 375 Hudson Street,
New York, New York 10014, USA
USA • Canada • UK • Ireland • Australia • New Zealand • India • South Africa • China
Penguin Books Ltd, Registered Offices: 80 Strand, London WC2R 0RL, England
For more information about the Penguin Group visit penguin.com
Copyright © 2013 by David Mark
All rights reserved. No part of this book may be reproduced, scanned, or distributed in any printed or electronic form without permission. Please do not participate in or encourage piracy of copyrighted materials in violation of the author's rights. Purchase only authorized editions. Published simultaneously in Canada
Library of Congress Cataloging-in-Publication Data
Mark, David John, date.
Original skin / David Mark.
p. cm.
ISBN 978-1-101-62111-0
1. Murder—Investigation—Fiction. 2. Police—England—Fiction. 3. Hull (England)—Fiction. 4. Mystery fiction. 5. Suspense fiction. I. Title.
PR6113.A7527O75 2013 2013001237
823'.92—dc23
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, businesses, companies, events, or locales is entirely coincidental.
For Nikki—like everything else
Contents
_Also by David Mark_
_Title Page_
_Copyright_
_Dedication_
_Epigraph_
Prologue
Chapter 1
Chapter 2
Chapter 3
Chapter 4
Chapter 5
Chapter 6
Chapter 7
Chapter 8
Chapter 9
Chapter 10
Chapter 11
Chapter 12
Chapter 13
Chapter 14
Chapter 15
Chapter 16
Chapter 17
Chapter 18
Chapter 19
Chapter 20
Chapter 21
Chapter 22
Chapter 23
Chapter 24
Chapter 25
Chapter 26
Chapter 27
Chapter 28
Chapter 29
Chapter 30
Chapter 31
Chapter 32
Chapter 33
Chapter 34
Chapter 35
Chapter 36
_Acknowledgments_
But I say, anyone who even looks at a woman with lust in his eye has already committed adultery with her in his heart. So if your eye—even if it is your good eye—causes you to lust, gouge it out and throw it away. It is better for you to lose one part of your body than for your whole body to be thrown into hell.
Matthew 5:28–30
Nymphomaniac: a woman as obsessed with sex as an average man.
Mignon McLaughlin, _The Neurotic's Notebook_ , 1960
PROLOGUE
SHOULD HAVE HOOVERED, he thinks, picking a piece of fluff from his tongue. _Should have made it pretty._
He feels a pressure in his lower back.
_Should have had a piss, too_.
He pushes himself up, raising his body from the floor, a mermaid ascending in a crash of spray, and attempts to brush the crumbs and cat hairs from his shiny chest.
_All this bloody oil_ , he thinks. _So slippy. So slick. Going to be like wrestling a dolphin . . ._
The alarm on his phone bleeps. It is gone ten. His visitor is later than he had intended to allow.
_Big girl's blouse_ , he calls himself, and then, in his father's voice: "Fucking poof."
The boy has been here some time. He is feeling uncomfortable. The wrong kind of dirty. Desire is starting to fade.
He wonders if there is a word to describe this opposite of ardor: the dissipation of lust; the moment when passion loosens its noose.
He is beginning to feel a little silly. A little undignified.
He tries to think of a better way to describe the sensation. He likes words. Likes to be thought of as articulate. Uses the apostrophe in the right place when promising to fulfill any lover's desire. Takes an effort with his poetry.
_Shabby._
He is suddenly aware of the shabbiness of this picture. Here, in his cheap, second-floor flat, naked on his cheap carpet, shooing away his cat when she appears at his bedroom door and fixes him with an expression of sneering superiority.
"Five more minutes," he says again, and wonders if this will be another letdown. Whether he will have wasted time and expectation on another coward.
His back and shoulders are beginning to burn in the glare of the three-bar heater. It's an odd feeling. The rest of him is shivering and goose-pimpled. He turns himself over, suppressing a giggle as he thinks of himself as a chicken on a rotisserie.
"Spit-roasted," he says to himself, and laughs into his bare arm.
His face is now in the glare. It's too hot. He turns back again, concerned that he will look red and sweaty. He raises a hand to pick more crumbs and fluff from his face.
The lad is in his mid-twenties, tall and thin. His face, beginning to carry the imprint of the dusty carpet that covers the entirety of his one-bedroom flat, is split by fleshy lips and a too-large nose. He is not attractive, but there are benefits to his company.
"I'm accommodating," he says into the carpet, his mouth and forearm making a pocket of cigarette breath, and wriggles, willing himself back into character.
He is naked. Starfished, facedown on the floor of his living room. There is not much room for his gangly frame. He has had to push back the charity-shop two-seater sofa and throw the old takeaway pizza boxes into his bedroom to be able to suitably accommodate his visitor.
"Five more minutes," he says again, reluctant to accept that tonight's fantasy will remain just that.
He reaches out for his mobile phone, tucked inside one of his battered white sneakers. No new messages.
He reads the recent ones.
_Oh, yes._
Feels the excitement build afresh. Has to reposition himself to accommodate the growing hardness between his legs.
Begins to feel the hunger. A languid luxury easing itself into his movements.
_Time to walk like a panther_. He giggles.
Hard as nails. Pretty as a picture.
_You should charge, boy. You're a fucking treat._
Like a fleetingly sober drunk gulping whiskey, the returning rush of sexuality alters his perceptions. He begins to feel better about the picture he presents. Remembers kind words and grateful embraces. Preens a little as he imagines the picture he presents to the open door. He knows his back and buttocks to be a breathtaking display; the ink that crawls up to his shoulders worth the agony that he screamed into the tattooist's table.
He will make his visitor happy.
There is a sudden creak on the stairs.
He smiles, and his breath comes out in a tremble.
_Here we go._
He arches his back. Presents himself for inspection. Raises his face to ensure the belt, coiled snakelike, is where he left it.
"Is this what you wanted?" he asks, throaty and sensual.
There is silence for a moment. The floorboards creak.
Then he feels the familiar weight on his back. The sensation of being pinned beneath another human being. The excitement of welcome helplessness that comes with giving yourself to another.
In the periphery of his vision, the belt is scooped up in a gloved hand. He closes his eyes, eager to play.
"Am I your fantasy?" he asks again.
The reply, when it finally comes, is hissed into his ear: a tumbled rush of excited words.
"To die for."
There is a sudden, biting, flesh-ripping sensation, as though his Adam's apple is being forced up into his skull.
"Her name!"
Spittle hisses from between his ghoulishly parted lips, frothing on his chin, into the dust and crumbs. His eyes bubbling, popping, like microwaved soup . . .
In an instant, his faculties are at once dulled and frenzied, his thoughts twisted and squeezed.
_Too tight, too hard, too much; fantasy becoming fear._
The words again . . .
"Your friend. Pink blossoms. The laughing girl."
There is only confusion and hurt, a sensation of becoming somehow less; of reducing, melting, puddling into nothing . . .
"The girl. Laughing at me . . ."
Darkness closes in as his oily fingers and skinny legs drum on the dusty floor.
An instant of clarity. A sudden heartbeat of understanding. What this is for. Why he is dying. Why the life is leaving his body and the poetry leaving his soul. What they want. What he must do . . .
The voice again, wet in his ear.
Anger. Venom.
"The one who looked and laughed . . ."
A knee now, hard in his spine; his back arching, teeth bringing blood to his thin lips, blood thundering in his ears . . .
He wants to plead. Wants to beg for his life. Wants this to stop. Wants to live. To write and create. To fuck and dance.
"Name. Her fucking name."
He knows now. Knows these will be his last words. Knows that all the warnings were for nothing. He's going to die, and his final act in this life will be one of betrayal.
The cord loosens for the slightest of moments. The strong hands readjust their grip.
The boy takes a gulp of air. Tries to swallow it. Manages only to hiss, before the cord cuts back under his jawbone and an explosion of sweet-smelling blood flowers and flows from his eyes.
"Suzie . . ."
Her name at once an act of treachery and a dying invocation.
"THEY WEREN'T here when I went to bed at midnight. Bold as brass when I got up at six a.m." The man waves an arm, despairingly. "I mean, when did they turn up?"
Detective Constable Helen Tremberg shrugs her shoulders. "Between midnight and six, I'm guessing."
"But they made no noise! And now listen! It's bedlam. How did they not wake anybody up?"
Tremberg has nothing to offer. "Perhaps they're ninjas."
The man fixes her with a look. He's in his late thirties and dressed for an office job. He has graying black hair and utterly style-free glasses. Something about his manner suggests to Tremberg that he is on a low-risk pension plan, and has a tendency to examine the contents of his handkerchief after blowing his nose. She fancies that after his second glass of wine, his sentences begin to start with the words "I'm not a racist, but . . ."
He saw the travelers from his bathroom window as he was brushing his teeth. Saw, in his words, "the sheer pandemonium" and rang 999. He was not the first person on the leafy street overlooking the football field to do so, but he is the only one who has decided to get in Tremberg's face about the situation.
Until half an hour ago Tremberg had been looking forward to today. She has been pretty much deskbound since her return to work, unable to take part in even vaguely interesting operations until she completed her chats with the force psychologist and had her doctor sign the last of the seemingly endless forms promising that the slash wound to her hand has left no permanent damage. Tonight, all being well, she's allowed back to the sharp end of policing, watching her boss, Trish Pharaoh, slap cuffs on the wrists of a gangland soldier and close down a drugs operation. She wants to be involved. Needs it. Has to show willing and prove she hasn't lost her bottle. Wants to demonstrate to anybody who doubts it that nearly getting her throat cut by a serial killer has been laughed off and dealt with "old-school"—voided from her system with vodka and a good cry.
"When will they be gone?" the man is asking her. "What are you going to do? This is a nice neighborhood. We pay our taxes. I've nothing against them, but there are places. There are sites! What are you going to do?"
Tremberg doesn't offer an answer. She has none. She does not want to talk to this man. She wants to get to work. She doesn't want to be leaning against the goalposts of the playing fields that stitch the affluent villages of Anlaby and Willerby together. She feels like a goalkeeper watching a match take place at the opposite end of the pitch.
"Should have stayed in the car," she says to herself, and looks past the man to where the caravans are parked up, not far from the halfway line of the adjacent rugby pitch. Drinks in the pandemonium.
Six caravans, four off-road cars, a Mercedes and three horse boxes, at least two generators, and, as far as she can see, a portable toilet. They are arranged in a loose semicircle around three floral-print sofas and a sun lounger on which a rapidly multiplying number of traveler women and children are sitting drinking tea, talking to uniformed officers, and occasionally shouting at the schoolchildren and bored motorists who have got out of their motionless cars to watch the commotion through the park railings.
Like most of East Yorkshire, Tremberg is stuck here. Her car is a few streets back, snarled up in the bimonthly gridlock caused by a local transport infrastructure with the breaking strain of a Kit Kat.
Bored, with nothing to do but look at the dark, gloomy sky through the dusty glass of her Citroën, she had switched on the radio in the hope of finding something soothing. She was two minutes into "California Dreamin'," and idly wondering why it appeared to be the only song owned by Radio Humberside, when the traffic report cut in. Half a dozen horses loose on Anlaby Road, and travelers causing uproar on the playing fields by the embankment. She'd had little option but to get out of the car and see if she could lend a hand.
"Are you going to shoot the horses?"
Tremberg gives the man her attention. "Pardon?"
"The police! Will you shoot the horses?"
"Not personally," says Tremberg, close to losing her patience. "The Animal Control Unit is on its way. They're stuck in traffic, too. We're doing our best. I could go get one of the bastards in a headlock if you keep hold of its legs . . ."
Ken Cullen, the thin, bearded, uniformed inspector currently in charge of trying to bring some degree of order to the scene, overhears the dangerous note in the detective's voice and hurries over.
"I'm sorry, sir, we're doing everything we can. If you could just return to your house for the moment and allow us to deal with this . . ."
Tremberg turns away as somebody better equipped to tolerate wankers sends the busybody on his way. The inspector fixes her with a bright smile as he spins back to her.
"Bet you wish you'd never stopped to help, eh?"
"Nothing better to do, Ken. Stuck here with every bugger else. Thought I'd see if I could assist, but this really isn't my cup of tea."
"Dunno, Helen. You've got the physique for crowd control!"
Tremberg shares a laugh with her old uniformed sergeant recently risen to inspector, who has moved, like her, across the water from Grimsby.
"I was pleased to hear you're on the mend," he says, and means it. "All better now?"
Tremberg flicks a V sign at him. "Lost none of my dexterity," she says, smiling.
Cullen gives her a quick once-over. Takes in the thin sports poncho she wears over a sensible pin-striped trouser suit and white blouse. Her hair is cut in a neat bob and she wears no makeup or jewelry. He knows from quiz nights and good-bye parties that she scrubs up well and has extraordinary legs when she hitches her skirt up, but Tremberg is deliberately sexless when on duty. Many female detectives have adopted her approach, appalled by any suggestion they have used their femininity to gain favor, but in so doing have opened themselves up to suggestions of lesbianism. Tremberg frequently wishes she could possess the carefree, fuck-you attitude of Trish Pharaoh, who wears what she wants and doesn't give a damn whether people think she is after dicks or dykes.
For a while the pair of them grumble about the local council closing off the rat runs and giving commuters nowhere to go if the main arteries in and out of town are snarled up. They agree that the local authority is staffed with do-gooders and morons and that the new chair of the Police Authority will no doubt balls it up even more.
Their pleasantly English moan is turning toward the gray skies and the cost of petrol when a young WPC approaches. She looks harassed and windblown in her muddy yellow raincoat.
"We've got all but one of them, sir," she says, in a voice that suggests she has struggled to avoid using a more vulgar term. "Sergeant Parker and Dan managed to box them in. They're in the car park in the Beech Tree. Can't get out. Another bloke with a Land Rover blocked the gap. The owners are trying to get them roped now. It's chaos, sir. Poor Mickey's ripped his trousers trying to pull one back by the hair. The mane. Whatever. Half of Anlaby's covered in horse shit. And the bloody pikey kids aren't helping, singing bloody 'Rawhide' . . ."
Tremberg has had to hide her face as she pictures the local bobbies desperately trying to round up the escaped animals, clapping and hollering and trying to stop the nags from eating the herbaceous borders of anybody important.
"And the last one?" asks Cullen, pulling on his peaked cap.
"It's a real nasty shit. Pikey said it was a stallion who smelled a mare in season. Put a dent in half a dozen cars so far. Seems to particularly hate Audis."
"And the animal team?"
The WPC snorts, herself momentarily horselike. "Having a very helpful meeting in the back of their unit. Lots of flicking through guidelines and phoning vets. I'm not expecting much in the way of action. I'm backing the big fella."
This last she says with a genuine smile.
"Big fella?"
She turns herself to Tremberg. Smiles in a way that the detective is starting to recognize. "Scottish bloke from your unit. The one who . . ."
"McAvoy?" Tremberg's eyebrows shoot up and she looks around as if he may be watching.
"Yeah. One of the lads gave him a ring. Said he knew about animals. Farmer's boy, or something, isn't he? Just turned up a minute ago. Don't know where he parked his car but I think he ran here."
"And what's he doing?"
The officer takes off her hat and gives an appreciative little shake of her head.
"About to start playing tug-of-war with a horse."
• • •
DETECTIVE SERGEANT Aector McAvoy spent his first months in plain clothes taking the title literally. He all but camouflaged himself in khaki-colored trousers, hiking boots, and cheap, mushroom-hued shirts, tearing them fresh from polythene packets every Monday. The disguise never worked. At six foot five inches, and with red hair, freckles, and a Highlander mustache, he is always the most noticeable man in the room.
It was his young wife, Roisin, who put a stop to his attempts to blend in. She told him that, as a good-looking big bastard, he owed it to himself not to dress like "a fecking Bible-selling eejit." Roisin has a way with words.
Despite his objections, he had let her style him like a child playing with a dolly. Under her guidance, and blushing at every alteration to his wardrobe, McAvoy had become known within the force as much for his smart suits and cashmere coat, for his leather satchel and cuff links, as for his detective skills and scars.
Now, flat on his back, staring up at the swollen clouds, with mud and stallion spit on his lapels and horse shit streaking one leg of his dark blue suit, he wishes he were back in khaki.
McAvoy tries to ignore the cheers of the onlookers and climbs back to his feet.
"Right, you bugger . . ."
He had been on his way to the Police Authority meeting when the call came through. One of the constables tasked with corralling the escaped animals had lost his temper after being dragged into the side of a bottle-bin by one of the mares, and had decided it was time for some specialist help. The officer had worked with McAvoy only once, up on the Orchard Park estate. They had been tasked with guarding the door to a crime scene until the forensics van turned up, and had not been made welcome by the locals. He and McAvoy had tolerated the abuse and even the first few bottles and cans, but when the snarling pit bull had been let loose with instructions to see them off, it had been McAvoy who stood his ground while the junior officer tried to persuade a brick wall to absorb him. The giant Scotsman had dropped to his knees and met the dog face-on, turning his head and opening his eyes wide, showing his broad, flat palms to the creature and flattening himself to the cracked pavement, submissive and unthreatening. The dog had stopped as if running into glass, and was on its back having its tummy tickled by McAvoy's great rough hands by the time backup appeared and the crowd were chased away. The young PC had taken McAvoy's number, having the foresight to realize that such a man was worth knowing. Today he had figured the big man was worth a call.
McAvoy, who would have agreed to a head-butting contest with an escaped antelope if it meant taking his mind off the impending Police Authority meeting, had been only too glad to dump his car and sprint to the scene.
He limbers up. Stretches his arms and cracks his neck from side to side. There are a few hoots from the watching motorists, and from the corner of his eye, McAvoy is appalled to see that many of those watching are recording the footage on their camera phones.
"Just shoot it," comes a voice from somewhere in the hubbub. It is a suggestion met with murmurs of approval by some.
"Can't you tranquilize it?"
"I've got a tenner on the big man!"
McAvoy tries to ignore the voices, but the laughter and groans that rang out when he was knocked flat by the charging stallion have turned his cheeks the color of crushed cranberries.
"You shoot that horse, I'll fecking have your eyes."
The voice, its accent unmistakable, causes a momentary silence, and McAvoy turns. The man who has spoken stands to his left, leaning against the bonnet of a blue Volvo. The car's owner has adopted the peculiarly English expedient of pretending he cannot see the large, daunting traveler who is pressing his buttocks into the bonnet of his car.
The gypsy is squat and balding, with a round face and shiny cheeks. Despite the cold and gathering clouds, his arms are bare. His flabby gut and torso are not flattered by the white sleeveless T-shirt or too-blue jeans.
"Yours?" asks McAvoy, with a nod toward the horse.
The man answers with a shrug, but the length of rope in his hand suggests he had been about to try and reclaim his property before he saw McAvoy take the burden upon himself.
"In season?"
The man nods again. "Horny as a Cornishman, first day out the mine."
"Bloody hell."
He'd nearly had him moments ago. The stallion had only been a few feet away, tearing some daffodils from a grass verge of one of the side streets leading off the busy thoroughfare. McAvoy's soft voice and gentle movements had allowed him closer to the animal than anybody else had managed since this unexpected carnival had begun, but as the beast swished its head back and forth, one of the passersby had loudly shouted encouragement, and the burst of noise had spooked it, sending McAvoy, and his expensive clothes, into the dirt.
"Got a name?"
"Me or the horse, sir, me or the horse?"
"The horse."
"Fecked if I know. Try Buttercup."
Slowly, taking care to keep his feet steady on the tarmac, McAvoy moves toward where the animal now stands. Wild-eyed, muddy, and sweat-streaked, it has moved into the garden of one of the nice detached properties set back from the road. Its occupants are staring out of the large double-glazed front windows. With no car in the driveway and the horse showing no apparent interest in their magnolia trees, they are enjoying the show.
"Easy, fella," breathes McAvoy, as he spreads his arms and moves toward the open driveway. "Trust me."
He knows what will happen if he fails. Vets will try and get near with a tranquilizer. They will fail, going in mob-handed and merely scaring the animal. Then some well-intentioned farmer will turn up with a tame horse in the hope of attracting the stallion to within range. The stallion will get overexcited. Damage cars. Damage itself. Eventually a marksman will be called and the horse will be hit with as many bullets as it takes to get the city moving again. McAvoy doesn't want that to happen. The call from the young PC had informed him that the horse had escaped from land where travelers had set up home. In his experience, travelers love their animals, and this one, though gray and with shaggy forelocks that put him in mind of traveled boots, looks like it has been looked after as well as worked hard.
"Easy, boy. Easy."
McAvoy closes the gap. Raises his hand, palm out, and whispers, soft hushes and gentle songs, in the animal's ear. It whinnies. Begins to pull away. McAvoy tilts his head. Exudes both the size and the gentleness that so define him; locks brown eyes with the confused, frightened animal . . .
The horse barely shies as he slips the rope around its neck. He carries on singing. Whispering. Crooning the only traveler song he can remember and wishing he had the same soft voice that his bride uses when she softly hums it into his neck.
This time the cheer from the crowd has little effect on the horse. It allows itself to be led out of the driveway, its unshod hooves making a pleasing clip-clop on the pavement.
McAvoy looks up and sees smiling faces. His cheeks burn and he struggles to keep his face impassive as the motorists give him a little round of applause, delighted to know they will soon be in fifth gear and hurrying toward jobs they hate, to tell the story of this morning's fun and games.
"Good job, sir. Good job."
The traveler has detached himself from the crowd. Unasked, he crosses to the far side of the animal and gently takes it by the ear, leaning in to nuzzle the animal's neck and call it a "great eejit."
McAvoy enjoys the display of affection. The man knows animals. Loves horses. Can't be bad.
Together, they wind their way through the cars and toward the playing fields. Three uniformed officers are leaning, exhausted, against the bonnets of two parked patrol cars. They look ragged and worn out. They nod their thanks as McAvoy passes by. The young constable who called him raises a fist of triumph and leans in to say something to a colleague. There is a burst of laughter and, instinctively, McAvoy presumes himself to have been the butt of the joke.
"We'll tie 'em up, sir," says the traveler. "We thought the fence went right round. Gave me a fright when I saw them gone, so it did."
McAvoy, getting his breath back, looks over the horse's wiry mane at the man. "It's not a campsite, sir. It's a football pitch. You know you can't camp here."
"Ah, would yer not show a little leeway?" the traveler asks, fixing bright blue eyes on McAvoy and suddenly exuding a twinkly, impish charm. "We've had a bit of a barney, me and one of the families up there. Not welcome. Just a night or two, put it to bed, make friends again."
McAvoy isn't really listening. This isn't his call. He's just going with it for now. He was asked to round up an escaped horse and has done so. The excitement is over. Now he has to try and make himself presentable enough for a meeting with the new-look Humberside Police Authority, and try to explain to the new chairman why his unit should be preserved, and exactly why the violent crime statistics are on the rise. It is a prospect that has kept him awake as efficiently as his three-month-old daughter, and its sudden reemergence at the forefront of his mind brings a wave of nausea to his stomach.
A gust of wind brings with it the scent of frying bacon and hand-rolled cigarettes. He raises his head, eager for a breath of cleansing fresh air. Opens his eyes. Stares into a sky the color of a black eye, rain just seconds away.
They approach the semicircle of caravans. There is a whoop that McAvoy traces to one of the women sitting on the sofas outside the nearest caravan. She is in her forties, with curly blond-brown hair, and is wearing a white tracksuit two sizes too small.
"Ah, yer a good lad," she shouts as they get nearer. She puts down her mug of tea and levers her small, curvy frame off the sofa. "Knew it was all reet, didn't I?"
She shouts this last at the two teenage girls who sit on the opposite sofa, each in pink nighties under gray hooded tops. One is perhaps a year older than the other, but both have sleek black hair cut in the same side parting, and wear an equal amount of hooped gold at their throats and earlobes.
McAvoy hands the rope to the man, who gives a genuine bow of thanks. "You're a good man, sir. A good man. Scotsman, ye'll be, yes?"
McAvoy nods. "Western Highlands."
"No kilt?" he asks, with a grin.
"I get enough funny looks."
The traveler laughs louder than the joke deserves. Claps McAvoy on his broad forearm. "By Christ, but you're a big one."
McAvoy's blush threatens to return to his cheeks, so he just gives a nod. Returns to business. "Keep him tied up. Buttercup. It's not fair."
"Aye, sir. Aye."
McAvoy looks around him. At sofas, the generators, and toilets. At the faces emerging from behind spotless net curtains at the windows of the caravans, as interested in what is happening on their doorsteps as the faces behind the glass in the four-bedroom detached properties that ring the fields.
He can't help but picture his wife. She lived like this when they first met. Wasn't much older than the girls on the sofa; her eyes just as distrustful, her world just as small . . .
"McAvoy!"
He turns to see Helen Tremberg and Inspector Ken Cullen walking swiftly across from the adjacent football pitch. He gives a wave, not quite sure whether he is to be treated as a hero or interfering fool.
"McAvoy, is it? Is that what she said?"
There is something in the way the old traveler repeats his name. Something that tells McAvoy he is known.
He gets no chance to press the man. The clouds that have been slung low, like damp laundry, finally split. Rain thunders down. Tremberg, not given to squealing, emits a shriek and stops short, pulling up the hood of her jacket. The travelers emit a cacophony of swearing, and McAvoy's new friend barks orders in an accent so thick it could be a different language. Half a dozen young men appear from inside caravans, and the sofas are quickly dragged under tarpaulins and windows pulled fast shut.
"Christ," says Tremberg, beginning a swift retreat to her vehicle. "They really are ninjas!"
McAvoy doesn't follow her. He's standing, arms wide, letting the downpour soak him to the skin. He knows that he will be tried and tested at this morning's meeting. Knows it will be a painful experience. And knows, too, that he will make life slightly easier for himself if he turns up merely damp, rather than covered in manure.
9:31 A.M. HIGH STREET, OLD TOWN.
BLADES OF RAIN, scything down from a pewter sky. A narrow row of handsome old mercantile palaces. Of insurance brokers and solicitors, art galleries and museums. Detective Sergeant Aector McAvoy, running through the rain—committee papers clutched inside his sodden jacket, rain splashing from his lips and nose.
Up the steps, feet slipping on the mosaic which serves as the welcome mat to the Police Authority headquarters, a rose picked out in red and white tiles, beneath an archway of expensive wood and glass.
He flashes his warrant card at the security guard on the desk, and then bounds up the stairs three at a time.
Assistant Chief Constable Everett is waiting outside the meeting room. He is immaculately turned out, his blue uniform crisp and freshly laundered.
"Good Lord, Sergeant!"
Everett looks aghast at the sight before him. Aector McAvoy has come to serve as his pet symbol of the modern face of Humberside Police. Educated, polite, supremely computer literate, and respectful of every new guideline dreamed up by the powers that be, he has served the assistant chief at endless committee meetings and public engagements.
"Look at the state of you! I needed you at your best, man!"
Strictly speaking, there is no need for a detective sergeant ever to appear before the Police Authority, but ACC Everett is expecting difficult questions from the authority's new chairman, and has managed to ensure that McAvoy is there to answer them. He is pinning his hopes on McAvoy's absorbing the worst of the barrage.
"I'm sorry, sir," gasps McAvoy, trying to catch his breath. "There was a horse . . ."
Everett, a thin-faced and ratty-looking man who managed to rise to the second top job in the force without appearing to be any good at anything, grabs McAvoy's coat and forcibly strips it off his shoulders. Before McAvoy can protest, Everett is pulling out a comb from his back pocket and reaching up to comb his junior officer's hair.
McAvoy backs off. Takes the comb.
"Thank you, sir."
He does what he can. Slicks back his hair and wipes the moisture from his mustache with finger and thumb. Catches his breath. Fastens his suit jacket and secures his tie inside it. Wrings out his cuffs, and straightens the creases with his palm.
Follows Everett into the meeting room.
More than a dozen men and women sit around a number of tables arranged in a vague U shape. The surfaces are covered in jugs of water and empty glasses, notepads, and official-looking papers. At the back of the room, a large pink-and-blue painting of a Manhattan skyline covers one wall. It was a gift to the authority from a previous chairman, and nobody has been impolite enough to take the monstrosity down.
"Bloody hell, Everett, are you making your officers swim here?"
The booming Yorkshire voice emanates from the large, bearded man at the head of the table.
Everett gives a false little laugh as he and McAvoy take a seat at the nearest empty desk. "Sorry, Mr. Chairman, Sergeant McAvoy was called away to deal with an important development in the case you have shown such an interest in."
Tressider waves his hand, dismissively. He looks at McAvoy.
"Important development, eh? You sure you weren't helping a bunch of gypsies round up their horses?"
McAvoy colors instantly. Can feel steam rising from his damp hair.
"Oh, bugger, can't say gypsies, can I?" Tressider turns to the secretary, busy jotting down the minutes of the meeting. "Scribble that out, would you, love?"
The other members of the authority exchange glances, but nobody says a word. Tressider dominates the room. He has a remarkable presence, and enough personality to dominate any environment, even without the benefits of his broad frame and deep Yorkshire accent.
He's a man on the up, is Peter Tressider. One to watch, with coattails worth riding.
Now in his mid-fifties, he was already famous as a businessman before taking his first steps into public office on the Conservative ticket a couple of decades back. His family runs a timberyard, distribution company, and a couple of property agencies, as well as investing heavily in various safe-bet start-up companies. He was elected onto the East Riding Council in 1997, and was moved up to the authority's cabinet soon after, holding high-profile positions on committees responsible for crime prevention, education, and social inclusion.
Tressider was popular in the press from the off, making headlines for his plain speaking, his witty comebacks, and his anti-bullshit stance. He was censured several times for swearing during committee meetings, and reveled in the public perception of him as a proper Yorkshireman who says what he likes, and likes what he bloody well says.
He was elected to the Humberside Police Authority a couple of years back, and set about making it his own. The role suited him. In 2005 Tressider was among the councillors who refused to enforce the home secretary's order that the then chief constable be suspended after Humberside Police's record keeping was found to be dangerously flawed. He gained favor in many quarters for telling the politician to "keep his nose out" of local affairs. Always good for a sound bite, the local papers are having a ball imagining the fun he will have if he is chosen by the Conservatives to stand as an MP at the next general election.
"Anyway, glad you could make it, Sergeant. We have things to discuss."
Tressider pulls his papers toward him and peers at an agenda item. Then he raises his eyes and examines McAvoy more closely. "The _Yorkshire Post_ says you're the face of sexy policing."
There is a titter from the other committee members, who all turn their attention to McAvoy.
"I'm sorry, Mr. Chairman?"
"Here," he says, and locates a photocopied piece of paper among the documents on his desk.
McAvoy recognizes the piece. Feels his heart sink.
Tressider clears his throat, theatrically.
"'Some might say they represent the "sexy" side of detective work. They're the men and women who delve into the very heart of the most high-profile murder cases, using skills and expertise that will eventually jail killers and make the streets a safer place.'"
Tressider looks up. Smiles.
Continues: "'It is a role that spawns images of fictional detectives like Morse, Rebus, and Thorn.'"
Moves his finger along the page before him, enunciating every word.
"'But in a humble room next to the canteen at Courtland Road Police Station on Hull's Orchard Park yesterday, the scene was a far cry from a TV detective show.
"'The _Yorkshire Post_ had been invited to meet a team formed last year with a Home Office grant, which is helping to change the way major incidents are investigated, both locally and nationally. They are the Serious and Organized Crime Unit—the force's murder squad. The team at Courtland Road represents one strand of a hundred-strong pool of civilian and police officers on both banks of the River Humber, which investigates all suspicious deaths and other serious crimes. Many of the civilians are themselves retired police officers, who sift through the mountains of information that pour into the Major Incident Room. They are using decades of experience that the force is reluctant to lose through retirement.'"
Tressider stops. Gives McAvoy a grin.
Reads on.
"'Detective Superintendent Patricia Pharaoh, senior investigating officer, said, "We're trying something a little different, and the assessment so far is that it's working. The volume of documents alone requires such careful and meticulous flow to the right people and our processes are very rigid. It's hard to quantify the successes in the past few months but we know this squad is making a difference. We hope that, even in the face of budget constraints, people realize how important this unit is."'"
There is muttering from some of the other committee members.
"Nicely done," says Tressider with a nod. "There's more, by the way. Shall I?"
McAvoy says nothing. Wonders if he will get the blame for the newspaper article as well as everything else.
Tressider continues. "'Even though the force now uses the Holmes computer system, much of an investigation is still paper-based. The Serious and Organized Crime Unit receive all the information. Every item goes to a receiver, who reads it and decides how it will be dealt with. The indexers then put the information into the system before it is read by a dedicated document reader. This person rereads every document that comes in and decides on any other work that needs to be done. The action manager then allocates work to action teams, based on whether the work is high, medium, or low in conjunction with policy. This all then comes back to the office manager, who gives the final signature on all actions and is responsible for the Major Incident Room running as it should.'"
Tressider stops. Raises his head and gives a mock yawn. "I, for one, was bloody enthralled."
McAvoy looks up, changes his mind, and turns his attention to ensuring his cuffs are the right length. The material squelches between his fingers.
"Sexy, Sergeant?" Tressider gives him a mock once-over. "I'm not sure I can judge. You might be the wife's type!"
He turns to his vice-chair: a gray-haired and nervous woman in a twinset and pearls. "What you reckon, Noreen? Sexy policing?"
The lady gives an embarrassed giggle, which seems to somehow disappoint Tressider. It's clear how the big man became chairman with such ease. Clear, too, what an asset he will be to his party if they give him the nod and jockey him to Westminster the way so many are predicting.
"Good publicity, anyway," says Tressider, picking at his teeth with a large finger. "We'll be looking at your unit in time, Sergeant. Looking at budget usage across the board. But I don't mind headlines like these. Don't mind at all."
McAvoy looks at his papers. Tries to unfold them and finds they are too soggy to come apart. "The reporter made the request for access through the official channels," he says. "I was just there on the day . . ."
Tressider waves him into silence with his paw of a right hand. Sits forward in his chair.
"To business," he says, and there is a general murmur from the assembled committee members.
They represent the great, the good, and the interfering bastards from the local community. The authority consists of seventeen members. Half are elected councillors from the area's four councils, and the others are independents. They are the top bosses. The men and women who make the big decisions and appoint the top brass. And there's not a copper among them.
"Detective Sergeant McAvoy is here to address your particular questions about the increase in violent crime, Mr. Chairman."
Tressider fixes Everett with a withering look. "I believe it was your presence I requested about that, Everett."
Everett squirms. "McAvoy is a key member of the team currently investigating that particular issue, and . . ."
Tressider nods. Turns his attention to McAvoy.
"Vietnamese, I'm told," he says brusquely. "Always been a bugger for the cannabis, ain't they? But it seems to be getting nasty. Stop me if I'm wrong."
McAvoy takes a breath. Wonders where to start.
For the past five years the local cannabis market has been run by Vietnamese gangs, setting up farms in disused warehouses and abandoned buildings, quietly cultivating their crop and then selling on through a network of dealers. Things ran smoothly. The people who got hurt had usually rocked the boat, and Humberside Police paid little attention to the cultivation of a drug they expected to be legalized within the next parliamentary term.
Then a year or so back the Drugs Squad began to hear rumors that, on this coast at least, the Vietnamese were being outmuscled and outgunned. Somebody else was moving in, and their methods of persuasion were not pretty.
A few months ago two Asian men were found unconscious on the shingle at Hessle Foreshore. Their faces showed marks of sustained beating, but it was the injuries to the rest of the two men's bodies that caused the paramedics to gasp.
Naked, fetal, their hands had been nailed to their knees.
Strips of flesh on their torsos and backs had been melted to the color and consistency of burned jam.
There was every indication that a nail gun had been used to drive in their restraints, and a heated paint-stripping tool used to inflict the damage.
The men were alive purely because the message their attackers wished to send was made more potent by their mutilation.
Neither spoke a word of English, but their eyes told a story in a universal language.
A couple of months later a terraced house in the west of the city was burned to the ground—the occupants still inside. The smell that billowed out from the smashed windows put firefighters and officers in mind of a community barbecue. Half of the neighborhood got high on the fumes as a massive amount of fresh-picked cannabis went up in smoke. It could not quite mask the reek of burning flesh.
Despite the protestations of Detective Superintendent Adrian Russell on the Drugs Squad, a decision was taken to make the investigation into the assaults the priority, and Trish Pharaoh was given command.
Nobody has any doubts that the victims were involved in cannabis production. Their clothes had shown traces of marijuana, of fertilizer—even the broad of sparkling mineral water known among the experts to produce a flowering harvest.
She got little from the victims at first, but by pulling in a few favors and suggesting she could assist with their pleas to be allowed to stay in the United Kingdom rather than be returned to Vietnam, she managed to get descriptions of the men who had hurt them. They spoke of big white men. Men who had been giving them orders ever since they smashed down the door to one of their marijuana farms and pressed a mobile phone to their foreman's ear. Their gang leader was relinquishing authority for their operation. The crop, and the workers, were now somebody else's property. They were to cooperate. Work hard. Their families would be taken care of.
The man's transgression was never truly explained. They upset somebody. Did something wrong. Said the wrong thing, perhaps. Made a call they should not have made. They fell foul of their new bosses. And they paid the price.
Little was yet known about these new players on the drugs scene, but the next set of crime statistics was an embarrassment to the top brass. The number of incidents of cannabis possession was up 17 percent in twelve months. More than that, violent crime was on the rise. It wasn't the street dealers who were taking the beatings. It was the people with backroom growing operations. People who grew enough to supply themselves and their friends. They were the ones being beaten down in the street. Beaten beyond recognition. Rendered too afraid or too unintelligible to talk.
Tressider is sufficiently concerned to demand answers. And Everett has none to give.
Stammering at first, and then warming to his theme, McAvoy outlines the situation as best he can. Tells the committee that it is not merely a matter of insufficient resources. It is a case that the new drugs operation is, in no uncertain terms, "very, very good."
"Bloody cannabis," says Tressider. "Should just legalize it. Get it over with. Going to happen, isn't it? Backward and forward this country. Can't have a smoke in a pub but you can drink a liter of supermarket cider for two pounds fifty! And all this nail-gun business! By Christ but that's vicious."
"We've tried to find examples of similar techniques used nationally, but we're having no success, sir. These people seem to have appeared out of nowhere. They took over, and now they're having their way . . ."
"But cannabis? Why not cocaine? Ecstasy? Heroin, even?"
McAvoy feels a vibration in his pocket, and discreetly retrieves his mobile phone. He has to fight to keep the smile from his face.
"We've made a significant breakthrough, sir," he says firmly. "An informant of Detective Superintendent Pharaoh has supplied us with the location of the current bulk of the cannabis operation. We're hopeful a raid will be imminent, and that the perpetrators of the foreshore attacks will be present."
Tressider holds his gaze for a fraction of a second longer than McAvoy is comfortable with. He is not sure what the chairman is thinking, or whether he is about to be praised or bawled out.
"It's a relief to find some bugger who knows what he's doing," says Tressider at length. "Sounds like you've got a busy day ahead of you. We won't detain you further."
McAvoy begins to stand.
"Actually, a piss would be nice. Shall we call a break?"
Amid mutterings of both consternation and agreement, the committee members stand. McAvoy gathers his things.
"A shambles," says Everett under his breath. "Bloody shambles."
McAvoy presumes the remark to be directed at himself. Chooses not to hear it.
Squeezing through the throng of bodies and careful not to touch anybody with his damp clothes, he makes his way out of the room and down the stairs. He can feel a fizz of excitement building inside him. Pharaoh has made progress. Leanne has an address. And within the hour they could have everything they need to kick in some doors and slap on handcuffs.
He emerges back onto the High Street to find that the rain has paused for breath. The cold wind grabs his soaking clothes and instantly brings goose pimples to his skin. He shivers. Looks at his watch and tries to decide what to do for the next hour. He has some time to kill before meeting Pharaoh, a five minutes' walk away in one of the quieter pockets of the city center, and were he to drive back to the office he would only have to turn around and come back again. He looks around.
Next door to the Police Authority stands the Hull and East Riding Museum. He has been here plenty of times with Roisin and Fin, but Lilah is probably still too young to appreciate the giant woolly mammoth that stands in the entrance, or the siege gun commissioned by Henry VIII, which was dug up by archaeologists excavating the city walls and placed on display alongside other exhibits from the city's colorful past.
His feet take him past the entrance and down to the water's edge. The River Hull gives the city its name, and scythes into the city center, then onward into the dark, muddy waters of the Humber. He stares down at the dirty water. At the feet of thick mud, which sit like so much chocolate mousse against the brick and timber walls of the footpath upon which he now stands.
To his left is the _Arctic Corsair_ , an old-fashioned sidewinder trawler transformed into a floating museum by well-meaning types keen to ensure that everybody get a chance to experience the hell of life on board a distant-water fishing vessel.
Idle, directionless, he walks along the towpath by the river. Looks up at the busy divided highway overhead. Past the overpass, to where the curious, curving pyramid structure of the city's aquarium sits, incongruously modern and shiny, on the muddy spit of land called Sammy's Point.
The rain begins to fall again. He wonders for a time whether he should huddle under the bridge until he dries out. Perhaps phone Roisin, or call Helen Tremberg to see if anything has occurred that requires his attention.
Realizing he has thought himself into inertia, he retreats from the downpour and leans against one of the concrete columns that support the overpass. Closes his eyes. Wonders for a time whether he should have responded to ACC Everett's muttered criticisms, or whether he was right to keep his mouth shut.
He looks back the way he has come. Back at the city where he has spent most of his career so far. Where he has risked his life, and captured men and women who have claimed the lives of others. It is a city he cannot love, and yet he feels an affection for it. A closeness. Feels a bond with this city at the end of the motorway, which grew to prosperity on the back of an industry which killed its men, only to slump into listlessness and decay when it disappeared.
At the back of the Police Authority building he can make out the shapes of two stick men. Two silhouettes, picked out against the white paint of the _Corsair_ and the gray of the sky.
He wonders if they are committee members. Whether they are councillors having a shifty smoke, or laughing at the great, hulking sergeant who had turned up damp, but still somehow seemed to persuade Tressider that the sun shone out of his arse.
McAvoy begins walking back. He makes no attempt to protect himself from the rain. He is too soaked to see the point.
Lost in thoughts, adrift in a not-unpleasant daydream, he does not see the two figures depart. He finds himself back at the riverside quicker than he had expected. Gives a last look at the water. Indulges himself in a smile as he looks at the wheels of the supermarket trolley sticking out of the mud bank. The bottles and mattress springs that litter its surface. The mobile phone, sitting on the thick and cloying surface like a tooth left in the frosting of a chocolate cake . . .
He moves to the water's edge. Crouches down.
The mud stops perhaps ten feet below him. Slopes down to water six feet below that.
From this angle, the phone looks relatively new. He wonders if it has slipped from somebody's pocket. Whether it has been kicked accidentally over the side, amid the chaos and frenzy of the rain.
McAvoy screws up his eyes. He's surprised the phone hasn't yet slipped beneath the surface. Whether it is his duty as a policeman to try and recover such obviously valuable property.
Leading down from the footpath, nailed into the river wall, is a metal ladder; its surface slick and grimy, mud-soaked and treacherous.
_Is it worth it, Aector? Seriously?_
He looks at his watch.
_It could belong to one of the committee members. Could be important._
Screws up his eyes.
_You could fix it, if it's broken. Would be a challenge for you._
Lifts one gigantic leg over the side.
_Just see if you can reach it . . ._
Begins to climb down.
10:46 A.M. EIGHTY MILES WEST.
A LIGHT DRIZZLE falling softly on gray, uneven pavements, on plywood shop fronts and untaxed cars.
"Shit-tard bollocking fuckcunts!"
Harry Tattershall is a magnificent and venomous swearer: doing things with words that other people would require a snooker ball and a football sock to achieve. Were he able to do the same with the non-vernacular, he would be poet laureate.
"Twat-box cock cunt!"
He picks up the bundle of dropped keys from the damp, dirty curb. Bangs his head on the wing mirror of his old-style Saab as he rights himself.
"Fucking wank-titting monkey pisser!"
He rubs a hand over his forehead and pushes the raindrops back through his thick, gray hair, then takes off his cola-bottle glasses and smears the moisture and fingerprints into a new pattern, before replacing them on his broken nose. He shivers, wishing he'd thought to pull on more than tracksuit pants and a lumberjack shirt before slamming the door closed at his housing association flat. He is a short, fleshy-limbed man in his late fifties, who does not enjoy the cigarette that is habitually hanging from his lower lip. He just keeps it there to light the next one.
Harry exults in his job title of general manager of the private members' club, but on days like these he can't help but feel like little more than the caretaker. Were it in his power to appoint somebody to the role of watchman he would do so in a flash, but the owners grumble if he so much as changes to fresh from long-life milk, and in his words are "tighter than a ladybird's chuff."
The blue light is flashing on the burglar alarm, but there is no sound. They disabled the bell months ago to keep the neighbors sweet. This is not a nice spot, a mile to the east of central Huddersfield, on the corner of a run-down row of pizza shops and budget hairdressing salons. Despite the less than beautiful location, the club has still faced plenty of problems from protesters and busybodies. Its license is dependent on the council's not having a good reason to shut them down, so keeping the locals happy is paramount.
Harry scrabbles through the many keys on his chain and finds the big one that opens the closed front door. He does not even think to try the handle. He has no doubt the alarm has gone off for no good reason: the same way it always seems to when he gets himself settled in front of a new blue movie with a pot of tea and a packet of HobNobs.
The big blue-painted door swings open, and he stands for a second in the drafty, unpainted, breeze-blocked cubicle where, on weeknights from seven p.m., consenting adults stand in their lace and PVC finery, sliding a ten-pound note and their membership card through a hatch in the interior door, and waiting to be let in for an evening of no-strings coupling, tripling, and, on one memorable occasion, human-centipeding.
He unlocks the inner door and steps into the dark of the downstairs bar. It's red-painted, with brass wall lamps and silhouettes of naked women stenciled artistically around the room. The floor is black lacquer, and the booths and bar stools are covered in imitation crushed velvet that, as Harry knows too well, does not wipe 100 percent clean.
With quick, practiced steps, he crosses to the bar and switches on the downstairs lights. It takes a moment for the bulbs to kick in, and there is a brief flickering before the room is illuminated.
At once Harry knows something is wrong. The computer behind the bar is whirring. It's an old machine and the internal fan is dust-clogged, so it habitually makes a noise like a helicopter in distress. The motor is spinning now. The monitor may be switched off, but recent use of the computer itself is betrayed by a green light winking beneath the bar.
Harry switches the monitor back on. Wiggles the mouse. Screws up his eyes as the database of members' names and addresses gradually comes into focus on the screen.
"Fuckbollocking titshits."
He says this under his breath, resignedly, already knowing that his day has just been ruined. They've had break-ins before, of course. He's turned up at work to find an entire week's worth of booze nabbed from the storeroom, and the fancy leopard-print throw from the circular bed in the viewing room had lasted only a week before it found its way into the depth of a voluminous handbag. But this is the first time the computer has been targeted. He doubts very much any intruder would have deemed the machine itself worth the bother of carrying, but there are bits and pieces stored on its hard drive that he knows, with sudden crystal-clear hindsight, he should have protected better.
"Shit."
He surprises himself with the simplicity of the statement. Pulls up a stool and begins tapping at the keyboard. He would never call himself a computer expert, but he knows how to build a database and surf for porn. He also knows how to transfer footage from the CCTV camera in the swing room to his own personal file.
Harry spins away from the keyboard, grabs a half-pint glass from beneath the bar, and holds it to the vodka optic, gathering up a healthy double measure. He opens the beer fridge and removes a bottle of Holsten. Takes a swig of vodka, then dilutes the burn with the lager. He's not worried yet, but his mind is racing. He wonders if he will be blamed. How they got in. How they got out . . .
It occurs to Harry that he has not yet checked the rest of the building. There is another bar downstairs, with a dance floor, pole, and large flat-screen TV where they show pornos to get clients in the mood. Upstairs there are five private bedrooms with doors that lock, and three where the policy is very much open door.
In the old-school boozer where Harry used to work, there was always a rounders bat next to the till. He wishes he were there now. But there hasn't been any trouble in the two years he has run the place. The members are like-minded and friendly. They know the rules and play the game. They take no for an answer and leave when asked. Harry likes working here. With two students running the bar and a bouncer on the door for three hours on Friday and Saturday nights, the place works like a dream. At the last count they had more than a thousand members, and there can be upwards of fifty people who make this their regular Friday-night outing, turning their backs on regular pubs and clubs for an evening where they can be who the fuck they like in the company of people who don't judge, and are grateful for the attention.
Harry's mind whirs with the same grinding difficulty as the computer fan. He tries to imagine who would break in, and why they would go straight to the machine. He has had plenty of time to get to know the clientele over the past couple of years, lounging at the end of the bar with a mug of coffee, nodding appreciatively at the lads and ladies in their eclectic wardrobes. He could write a book on the sights he has seen. The people. The pervs. The giant, hairy Asian man in the dog collar and leotard. The big one in the gold mask who made a noise like a heifer in distress when he reached orgasm. The woman in her seventies who had to be helped out of the love swing when her hip came out. The fat lass in the pirate costume who cried rape when one of the four men she was fucking tried to put it in her arse . . .
He taps the keyboard again, unsure what to do. Wonders about the potential consequences of inaction. Suppose he confronts a burglar? Suppose they have seen the footage on his private files? He can't afford to be blackmailed. Would simply have to admit culpability to the bosses and look for another job. He doubts there would be charges. But who would target him? He shakes his head and downs his vodka. Perhaps somebody is looking for information. Perhaps a member wants to find out more about somebody who caught his eye or pissed him off.
Perhaps they want to delete their own information. He knows from experience that members are notoriously shy about giving real names and real addresses, so doubts anybody would think it worthwhile to even try and get a name or phone number for another member.
His train of thought is derailed by an unmistakable creak from upstairs.
He closes his eyes, takes a breath, then picks up the nearly empty bottle of Holsten and upends it; a dribble of beer running over the blurry blue ink of his tattooed forearm.
He pulls open the door to the stairway and peers into the gloom. The cord-carpeted stairs disappear into darkness halfway up, and there is nothing appealing about climbing them. Harry pauses, already half decided.
There is another creak.
"Fuck."
He puts his foot on the bottom stair. Puts a hand on the banister and pulls himself up, trying to stand on the corners of each step so as not to make a sound.
As he reaches the top step, the last dribble of beer runs over his wrist. The sudden coldness makes him jump and he lets out a small exclamation, which he follows with a curse.
Harry knows he has given himself away. Whoever is waiting in the dark can fuck-buggering stay there.
He turns. Begins to creep back down the stairs.
This time the sound is unmistakable. Running feet. Sudden movement. Coming closer.
Harry looks up.
_Crack._
Opens his mouth to let rip with a stream of invective, but finds himself wordless. His tongue has been crushed to pulp between his back teeth; a reflex reaction to the hammer blow that has struck him just above the left temple.
Movement. Bone-jarring impact. Thuds and cracks.
Harry finds himself upside down. Right way up again. Feels his old limbs twisted into unnatural directions as they jar with the brick and stair.
Darkness.
Now red clouds.
A sensation of friction at his back and pressure at his wrists.
Now he is looking at the ceiling from an angle he has never seen it before. Now there is dusty, cheap carpet by his face. _How did that happen? Why am I at the bottom of the stairs?_
He blinks. The effort pains him. It seems to awaken other senses.
Agony grabs him. Twists him in its fist.
He looks up. Sees a face. Halfway familiar; attractive and cold.
A voice. Soft, in his ear.
"Her real name. It's not here. Just 'Blossoms.' I already know that."
The voice sounds as if it were underwater. Harry hears an echo. Feels dampness on his skin.
"I never really thought it would be here. I knew you wouldn't check. Just a name and a number. And the number isn't real."
Harry wants to speak. Wants to ask for help. An ambulance.
Harry manages a croak.
"I'm sorry. I'm getting desperate. I had to try. I don't even know if it's her. He said 'Suzie,' but he could have lied . . ."
He croaks again. Tastes blood. Blood and vodka.
"It keeps getting worse. It could have been simple. Now look where we are. There will be more, I know it. I've just made it worse. He'll be so angry . . ."
Harry knows what he wants to say. Can feel the words lining up in his mind. Wants to say that, whatever this is about, he will never speak of it. Wants to say that he can feel himself dying and cannot stand it. Wants to know where his glasses are, and whether they can be fixed.
"I thought your neck was broken. I think it is. I don't know. I could have walked away if your neck was broken. Now it has to be an accident."
Harry tries to move. Realizes he cannot feel his limbs. That it only hurts on one side of his body. On the other, he can feel nothing.
"I'm so sorry."
He lies broken. His limbs broken branches, his back shattered glass. He is on his back, wedged in the doorway. His positioning tells the story of his death. Of a man who slipped climbing the stairs, and who could not put out the flames . . .
His neck is twisted gruesomely to the left, so Harry does not see the cigarette butt that a gloved hand grinds into his vodka-soaked T-shirt. Cannot move his arms to flick it away. Can only watch, eyeballs climbing out of his skull, as it begins to smolder.
He sees his killer walking to the back door, the same hammer in hand that was used to force the lock and crack his skull.
Pain now. Heat. Smoke and flame.
He gulps hard, trying to clear his mouth; to speak.
Swallows clotting blood. Begins to choke.
Coughs and pukes, choking on blood and sick, as the flames take hold of his ragged clothes and spread to the floor.
He is dead before he has to endure the stench of his own cooking skin.
SUZIE'S POSTURE implies prayer. She is bent forward, elbows on her knees, palms clasped fast, both thumbs pressed hard enough into her forehead to make grooves. Her lips move soundlessly, as though begging forgiveness or benefaction.
Her thoughts are far from divine.
She is lost in memory. Consumed by a recollection that has surfaced unbidden.
For a moment, she is entering the red room, with its glitter ball and its velvet sheets. She is gazing upon naked forms. Is recoiling, spluttering in nervous laughter, drunk and giddy enough to change the mood. She is staring into a mask, leering and lascivious, incongruous atop a fleshy body that makes no concealment of its desire.
She is controlling herself now. Saying yes. Letting go. Feeling a warm, familiar hand in her own. Accepting permission like a blessing.
She is on her back, weight upon her. Light making shadows of a grunting, thrusting face, given over to pleasure that could just as well be pain . . .
She shakes it away. Forces the memory back. Pushes her features into a smile. Hides it. Hides her feelings, even from herself . . .
Suzie is twenty-six years old. Petite. A little fleshier than she would like to be around the middle. Kooky, her bosses call her, when clients remark on her multicolored nails and chunky, homemade jewelry. Today she's dressed in a short black skirt over footless leggings, a long-sleeved white top, and flip-flops. The fleecy Disney scarf around her neck covers the top end of a tattoo that her bosses at the law firm have deemed unsuitable for exposure. Her elbow-length lace gloves were considered more off-putting than the butterflies they obscured, so she has taken to wearing fluorescent wristbands. She expects to be asked to remove them as soon as one of the senior partners plucks up the courage. Her shoulder-length hair is dyed a color somewhere between copper and autumn, and today is held back from an unremarkable but pretty face by a pink band. Tiny hummingbirds dangle from the lobes of her multipierced ears.
She is fun to look at.
She makes people smile.
The bells of St. Mary's Church inform her it's one p.m., although she does not need their help. She has always just reached this stage in her lunch when the hour chimes. She fears she's becoming a creature of habit.
Suzie wonders why there are not more people here. It's a pretty spot, and she finds herself surprised on a daily basis to have it to herself. She's five minutes from work and a stone's throw from the relative bustle of the Old Town end of the city center, but in the three months she has been eating her packed lunch here she's had to share this lovely little courtyard garden only a handful of times.
She's in the only green square to be found in the Museums Quarter, hidden away at the center of this pocket of gorgeous old buildings and cobbled streets, constructed two centuries before in the angle between the Rivers Hull and Humber. Here, between Wilberforce House and the Streetlife Museum, she has found a place of near sanctuary. Here, protected by red brick and sloping archways, she feels delightfully invisible, set back under the protective branches of a tree she has come to think of as her own.
The spitting rain picks up its pace. The larger drops make a pleasing noise on the tree's burgundy leaves. She spots one leaf bulging under the weight of collected droplets and reaches out with her left leg so that, when it spills the cold water, it will trickle onto her bare toes. The sensation, when it comes, is exhilarating.
Suzie takes the iPhone from the pocket of her bag. It was an extravagant purchase, forcing her to live for a month on sausage rolls and biscuits from the office tin as a consequence of using her food budget for its acquisition.
She logs on to Facebook. Two pokes from old school friends and a new post from her mum.
A song thrush has fluttered damply down to the nearest flower bed. Suzie looks on the bench for a crumb to give it. She finds one in her scarf and chucks it to the bird, who ignores it and flies away.
"Marmite. Either love it or hate it . . . ," she says under her breath.
She opens her e-mail account. Ignores the messages from the various websites that send her discount codes for music downloads and vouchers for chain restaurants.
"What we got . . . ?"
Two messages.
She finds herself smiling. A tickle of excitement flits between her stomach and chest.
"Still going strong . . ."
He sent one midmorning, and another five minutes before she came out for lunch. A query about whether she touched herself when she woke up, and a one-line missive informing her that he is "so damn hard" at the thought of her.
"Sweet," says Suzie, hitting REPLY.
She got talking to "Dom" last night, halfheartedly at first, distracted by the vampire movie she was watching on the laptop, then later with an enjoyable intensity.
His advert on the website had been straight to the point. "Dominant male seeks under-30 playmate. Must be up for anything. Are you game? Put your body in my control. Be my rag doll." He had put a little _x_ at the end of the posting. She liked that.
"Hey there," she'd written in reply. "Saw your ad. Think we could have fun. Am twenty-six and Ok looking. Have played this game before. Love to be dominated and test myself to the limit. Am I your sort of girl?"
Dom had replied within a minute. Told her he was "aching" to know more. Said he "yearned" for the taste of her. Was "consumed by a need to lick the tears from her face." His words had a lyrical quality that Suzie approved of. Suzie likes words. Completed a year of an English-literature degree before her fiancé's job moved them to Hull and they had decided that his new bumper wages made her continuing on the course a waste of time for both of them. When they split up not long after, she took solace in few things, but words were among them. She enrolled herself on a creative-writing course. Met the skinny, giggly, lovingly absurd little peacock who would become her best friend.
Suzie enjoyed last night's chat with Dom. He seemed genuine. She has been playing these games for a couple of years now and knows that, nine times out of ten, the blokes begging her to fulfill their every fantasy when they're texting each other's brains out will chicken out before meeting up. She has had text sex with countless online finds, but only a handful have had the bottle to say hello in the flesh, and fewer still have been able to deliver on their promises.
"I want you to make me cry."
Suzie presses SEND. Waits a minute. Hopes for an immediate response.
This is the thrill of it. For her, it is not the sex itself. It is the game of it all. The naughtiness. The apprehension and excitement that make her shiver and wriggle as she checks her screen time and again, waiting for a new message, like a wartime bride awaiting a love letter.
Are you a big brave girl? You want to show me what you've got?
Suzie grins as she reads the message and takes another sip of her juice before replying. She had half expected last night's flurry of messages to be a one-off. She is used to the swift curtailment of her cybersex: all too often the result of a spouse coming home early or knocking on the bathroom door.
"I am yours to command."
She stares at the screen for a moment, and when no answer is immediately forthcoming, she opens up one of the Internet pages stored in her FAVORITES section. She looks at the latest tattoo designs and wonders whether she would suit the small posy of dandelion seeds highlighted as the "tattoo of the day." She isn't sure. Her tattoos are all designs she has created herself, though it is the lilies and pink cherry blossoms that wind from the backs of her thighs to the nape of her neck of which she is most proud. She and her friend had gone on the same day: he to be adorned with peacock feathers, she to become a Chinese garden. The results were stunning. The tattooist couldn't stop smiling. Took their pictures from every angle and asked if they would mind him using the images in his promotional material. They had preened and agreed, loving their own prettiness.
Want to see what you can do.
The message flashes up in the corner of the screen. She wrinkles her nose in disappointment. She has a limited amount of time. Wants him to send something not just suggestive, but filthy and obscene.
"Anything."
The memory of the day of blissful agony in the tattoo parlor brings her down. Such thoughts always do. It is six months since she lost her best friend. Half a year since the boy with whom she giggled and cried and gossiped and played wrapped a cord around his neck and hanged himself in the kitchen of the flat she had one day planned to share.
What you doing tonight?
Simon used to keep her safe. They played these games together. Best friends. True friends. He keeping her safe from herself, and providing a reassuring closeness as she indulged in the liaisons that helped her feel alive. She giving him reasons to feel loved and needed; an escape from the dark thoughts that made him seek out punishment and abuse, threatening to pull him under . . .
You promise you've got tattoos?
Suzie sighs, excitement dissipating. "Pink blossoms all over my back. Butterflies on my wrists. A zip on the back of my thigh. All begging for your tongue to trace."
There is no reply. Suzie wonders if this is where it will end. She will not be disappointed. This is the game.
Her phone beeps.
Tonight. Want to see your blossoms. Want to see you get nasty.
Suzie gives a little grin, crossing and uncrossing her legs as she allows herself to imagine that this one may actually happen.
She has no time to reply before the phone beeps again.
Come alone.
• • •
HALF A MILE AWAY, raining twice as hard . . .
Trish Pharaoh looks her sergeant up and down. Then back up and farther up. She places her takeaway cup of coffee between her knees. Reaches forward. Takes his tie in both hands, and wrings it out as if she were throttling an eel.
"Road-testing a new antiperspirant?" she asks sweetly. "It's not working."
McAvoy presses his lips together. Smiles a little, unsure what facial expression to pull, and eventually lets his features settle into a mask of embarrassed gormlessness. It is a countenance he has grown used to wearing in his boss's company.
Pharaoh lets go of his tie and shakes the water off her hand. Wraps both palms around the polystyrene cup. Points at the rain, which billows wavelike across the deserted square. "You did this," she says accusingly.
McAvoy sniffs. "It's coming in off the sea . . . ," he begins defensively.
"Hush now."
She turns away from him. Sips her coffee.
"I didn't get you one," she says, gesturing at her drink without any hint of apology. "Figured you would file a report about attempted bribery or sexual harassment."
McAvoy nods solemnly.
"Oh, bloody hell, Aector, you are as much fun as paper cuts."
McAvoy apologizes. Hangs his head.
They are standing under the awnings of a jewelry shop in Trinity Square. The gray slabs of the piazza have been washed, then varnished, by the downpour, and the great wooden doors of the city's biggest church, a hundred yards from where they stand, have been soaked to a rich chocolate brown. McAvoy gives the church only the briefest of glances. He cuts this thought dead before he begins to question how much rain it would take to wash away the blood that was spilled within Holy Trinity's embrace just a few months ago . . .
"Were they bastards?" asks Pharaoh, finishing her drink and pausing for a moment until the bells of St. Mary's, half a mile away, finish chiming the hour. "The authority? This new bloke as much of a bully as they say he is?"
McAvoy still hasn't made up his mind. "He's in your face," he says, thoughtfully. "Big man. Big personality. Very well informed."
Pharaoh looks at him, expecting more.
"He's clued up on what we're up to. The unit. Seems to read the reports and retain the info."
"That's the last thing we need," says Pharaoh, throwing her cup in one of the bins that dot the square.
"He wants real progress on the drugs, guv. Wants arrests. Busts. A bit of action is what he said."
Pharaoh rolls her eyes. "He wants to be an MP, Aector. He wants some good publicity so he can bugger off to Westminster."
McAvoy says nothing. He puts his hands in his pockets. Feels the outline of the mud-caked phone. Presses his fingers over the keypad. Pictures himself sitting at the kitchen table at home, delicately taking the machine to pieces with fragile tools held in too-large hands. Wonders again what possessed him to pick it up, and whether he has any damn right to root around inside.
"Wish I'd brought a brolly," muses Pharaoh, watching the rain as it scythes down into the square. She looks at McAvoy. "We wouldn't fit you under it, though, would we? You'd have to hold it. Be my slave for a bit, eh?"
He looks away before she can see him blush. Tells himself that she just teases him for fun, and not for meanness. Reminds himself how many times she has stood up for him. Comforted him. Risked her career to back him up.
"Come on, then," she says, when it becomes clear he will not respond. "Let's get wet."
Pharaoh pushes herself off from the wall. Early forties, curvy, and habitually dressed in biker boots, a knee-length dress, and a cropped leather jacket, she does not look much like the head of Humberside Police Serious and Organized Crime Unit. But she's damn good at a job she inherited under difficult circumstances, and she marshals the egos and neuroses of her team like an inspirational primary school teacher.
"She really didn't want to meet somewhere neutral?" asks McAvoy, squinting into the rain. "She wanted us to come to the house?"
Pharaoh shrugs. "I gave her the option. She said to come to her place. I warned her, if you were wondering. Said I didn't advise it."
McAvoy nods. "She knows what she's doing, I suppose."
This time it is Pharaoh who remains silent.
They turn off Trinity Square and walk in silence until they reach the damp cobbles of Dagger Lane. It's only a minute from the Old Town and a quick sprint across the busy divided highway from the bobbing pleasure craft and empty pubs of the marina.
"One at the end," says Pharaoh, nodding at the row of redbrick terraced houses that occupy this old street, the origins of its intriguing name lost to history.
"And she's sure?" asks McAvoy.
"Sounded it."
Pharaoh leans on the bell outside the slim, nondescript terrace. Turns to McAvoy.
"Smarten yourself up, man. You know she fancies you."
"Guv, I . . ."
The door swings open.
Leanne Marvell is forty-one years old, and though she no longer works as a bouncer or competes in the bodybuilding contests that first tempted her into trying steroids, she remains a powerfully built and imposing physical specimen. Though she is not particularly tall, she has a masculine physique, and while her muscles are not as clearly defined as they are in the photographs that McAvoy has seen from her weight-lifting days, she still looks like she could beat him in an arm wrestle.
Her large nose is the only wrong note in a relatively pretty face, which creases into a smile when she sees McAvoy on her doorstep.
"Aector," she says, looking past Pharaoh, "I wasn't expecting you as well."
Self-consciously, Leanne begins to straighten her gray tracksuit trousers, and the belly that sticks out from beneath her workout top miraculously disappears as she breathes in and holds it.
"Let us in, Leanne," says Pharaoh, rolling her eyes. "And don't feel obliged to say his name in Gaelic. It should be bloody Eichann, if you're being picky. I read up on these things. Nobody else is called Aector. It's just him being bloody awkward."
Leanne beckons them into the hallway. Presses herself against McAvoy's damp body as she pushes the door closed.
McAvoy begins to speak. Begins to outline the origins of his name, and the compromise his Gaelic-speaking father and English-speaking mother came to when they chose to name their second son. But he decides to close his mouth instead.
"You'll have to excuse the mess . . ."
Leanne opens the door and ushers the two officers into a joyless and compact living room. It contains a floral two-seater sofa, a cheap coffee table covered in pouches of tobacco and rolling papers, and a huge flat-screen TV. The old-fashioned stone fireplace that is set into the far wall contains no fire: just two wires gaffer-taped to the stone. The walls are papered in a swirl of peaches and pinks, and the only picture that stares down at them is hanging askew. It shows a younger, fitter Leanne, flexing in a purple bikini and fake tan, collecting an award from a man with a shaved head and too many teeth.
"Shaun's not expected?" asks Pharaoh, taking off her coat and hanging it over the back of the sofa, then reaching into her handbag for a hairbrush, which she uses to slick back her hair.
"Not for hours," says Leanne to McAvoy. "You taking yours off, Sergeant?"
"I'm fine," says McAvoy, refusing to catch Pharaoh's eye.
"Sit down, Leanne. Tell us what we're doing."
Leanne perches on the edge of the coffee table. She reaches under the sofa and pulls out a formidable-looking dumbbell. Begins to perform curls with her right arm. If the effort pains her, she does not show it.
"Tonight," says Leanne, looking down at the dirty white sneakers on her feet and the dirtier carpet beneath. "I promise. It's going to be there tonight."
"You sure?"
"I read his phone. He was passed out. I've been reading it all the time. I feel like I'm spying on him."
"You are, love."
"I know, but I don't like the feeling."
"He doesn't know? He's got no idea?"
"He trusts me."
"And you're sure? Really sure you want to go down this road?"
"I've got no choice."
Pharaoh nods. Leanne has already made her decision. She made it months ago while leaning, wet-faced, against the wall of Hull Royal Infirmary, with blood on her clothes and Trish Pharaoh's cigarette at her lips.
Leanne has fallen far since the days she represented her country in weight-lifting championships and landed rosettes and trophies for her bodybuilding. She's one of the Old Town's more colorful characters. Sober, she's caring, thoughtful, and considerate. A good friend. A decent neighbor. Drunk, she's a demon. She's a ferocious ball of anger, who lost her two kids to social services and her job to her criminal record. She has convictions for dealing, possession, wounding, and only escaped a charge of attempted murder when an ex-boyfriend refused to press charges.
McAvoy has read and reread her file, and always found it difficult to reconcile the flirty, friendly woman with the photos of the damage she has caused when the steroids in her bloodstream exploded into rage.
It was temper that brought her to Pharaoh's attention. The night that the two Vietnamese drugs farmers were found at Hessle Foreshore, Leanne was in Accident and Emergency with her boyfriend, Shaun, handcuffed to two different police officers, having been arrested for attacking her partner with a corkscrew. She had managed to get the weapon halfway into his ear, and twice into his chest, before he managed to wriggle free by braining her with a brass ashtray. Quite what they had been arguing about they had been unable to tell the uniformed officers who broke their door down and carted them off to hospital. But it had clearly been important.
As they were being dragged into reception at Hull Royal, Pharaoh was standing at the nearby coffee bar, listening as one of the junior doctors gave her his appraisal of the condition of the two Vietnamese men. She had been scowling into her latte, wincing at the calmness with which he described the nail-gun wounds to the victims' hands and knees, to the burns on their backs and torsos. A paint stripper, he had speculated. Turned the skin to jelly . . .
The doctor had recommended both victims be taken immediately to a specialist unit in Wakefield, where their wounds could be better treated. Pharaoh had acquiesced. Made arrangements. Had the two men wheeled down from the ward, cuffed to the sides of the hospital bed. There was an ambulance waiting outside for them. A police escort, too. Pharaoh had been taking no risks.
And then one of the Vietnamese men spotted Shaun. He was wrestling with two of the constables, trying to get his hands free, desperate to be allowed to speak to Leanne. He was shouting that he loved her. That he would kill anybody who came between them. That he forgave her the fact he was bleeding from his ear and his heart.
Then Shaun stopped. Fell utterly silent. The sudden cessation of noise was more potent than a shout. Heads turned, including Pharaoh's. And she saw the way Shaun was staring at the two men in her care.
The color had drained from his face. The officers holding his arms found the strength had gone out of him, and wrestled him to the floor.
And both of the drugs farmers let fly with a stream of impassioned invective, a gibberish that meant nothing to anybody in the great open lobby, but which told Trish Pharaoh that her victims knew this man, and knew him well.
With the victims safely transported to Wakefield, Pharaoh played a hunch and insisted upon Shaun and the woman he was brought in with being kept apart.
She got their names. Pulled both their records. Acquainted herself with their criminal pasts. Shaun's rap sheet was petty. He had never done more than a week on remand. Had convictions for drugs possession and public violence. She had been more impressed with Leanne's. She had done serious time, and was looking at more.
Pharaoh had found her in a private room, cuffed to a constable, a doctor stitching up the wound on the back of her head and asking that she do her best to stop crying, as it was making it difficult to keep the stitches small.
"He's going to leave me, I know it," said Leanne through the sobs, talking to nobody and barely registering Pharaoh's presence. "He's too young for me. He's got his whole life. He doesn't need this. I'm dragging him down . . ."
Pharaoh had asked the constable to slip the cuffs off. Asked the doctor if he was done. And then she had led Leanne Marvell outside and pressed a cigarette to her lips.
Vulnerable, scared, and doped on a cocktail of painkillers and steroids, Leanne had been perfectly primed for careful questioning. And Pharaoh had obliged. Told her that two Vietnamese drugs farmers had been found tortured and mutilated at Hessle Foreshore, and that they had identified Shaun. Pharaoh had been careful to keep it vague. Had left most of the work to Leanne's imagination. And she had thanked her lucky stars that McAvoy was not there to tell her off.
Despite her formidable appearance and the time she had spent inside, Leanne crumbled. Told her what she knew and begged her to help keep Shaun out of prison.
She had promised to help Pharaoh however she could.
Now, safely registered as a police informant and fully briefed on what will happen if she lies, Leanne is about to earn her pay.
McAvoy, who is legally bound to appear at all meetings between Pharaoh and any of her registered snouts, has an affection for Leanne. She is almost schizophrenic in the change that comes over her when in drink, but here, now, she seems a good person, trying to do her best by her man.
"You can't ever tell him," she says, though she has already had assurances on this point. "You have to say you lost the evidence or something. He can't be the only one to go down for this."
Pharaoh puts a hand on her knee. Offers her a cigarette and then lights it for her. "It's all taken care of, Leanne. We'll look after you."
Over the course of several interviews, it has become clear that Shaun is a relatively minor player in the hierarchy of the gang which has taken over the cannabis supply. His job has been that of a glorified deliveryman, overseeing the movement of crops from one factory to another, and transporting the remaining handful of the Vietnamese workforce around the various properties, where they have been reduced to virtual prisoners. He knows nothing about the muscle side of the business. Doesn't know where the orders come from. Even in drink has confided little in Leanne about his employers, save that they are white and scary as hell.
"I'm not a snitch," says Leanne, and it is a mantra she has repeated endlessly in their meetings. "I know he's been a bad lad. But he wouldn't do that. He's not a violent person, not really. I don't know why they're pinning it on him . . ."
Pharaoh coughs, trying to move the conversation on. She knows McAvoy disapproves of the fact that she is letting Leanne think her boyfriend is in the frame over the torture of his two associates. In truth, he is not a suspect. Through an interpreter, the victims had given only the sketchiest of details about their attackers, but they made it clear that the men who hurt them were higher up the food chain than the man who drove the van. The descriptions they had given were sketchy. Big. White. Well built. Acting under the instruction of a smaller man, who seemed to be enjoying it all far too much . . .
"Do you think we could have a fresh start?" asks Leanne suddenly, putting her dumbbell down to concentrate on her cigarette. She looks at McAvoy. "Do you think you can start again?"
McAvoy tries his best to summon up an encouraging smile. Tries not to let his eyes linger on her paltry possessions, or the signs of frailty and abuse that are starting to creep into her physique.
"We'll take care of you," he says. "I promise."
The words somehow seal it.
Leanne nods.
"The warehouse next to the Lord Line building," she says. "St. Andrew's Quay. Where they used to fish from. Near the memorial."
McAvoy holds Leanne's gaze as Pharaoh begins dialing a number in her mobile phone. He pictures the location. The darkness. The nearness of the Humber and its cold depths.
Sees, in his mind, an area that has witnessed death enough times to make the waters run red.
6:24 P.M. THE CAR PARK AT PETER PANG'S.
RED GLASS LANTERNS clink and sway, disappear, and then reemerge from the shadow of the pagodalike roof.
McAvoy looks. Listens. _Sees._
The sound of waves slapping wood and stone beyond the gray seawall; the broad, brown Humber fading into cloud and drizzle.
The morning's storms have not blown themselves out, but instead hang heavy and threatening in a headstone-colored sky. The river, swollen by the cloudburst, slaps against the rotting timbers of St. Andrew's Dock. Dead flowers and plastic memorial cards skitter and tumble on the wind. Flowers are often left here. This dock was home to the Hull fishing fleet. It is the last glimpse of home that thousands of dead trawlermen ever saw.
On Pharaoh's orders, McAvoy has switched off his phone, but after two hours in this cramped vehicle with nothing to look at but car bonnets and brick, he needs to do something to keep himself alert.
The phone bleeps into life at the push of his thumb. Two hands shake on the blurry, liquid crystal screen. A moment later it vibrates to alert him to three new text messages. One is from Roisin, telling him she loves him and will be wearing nothing but the red leather jacket he bought her for Christmas when he gets home. The other two are from Pharaoh, telling him first that she is BORED, and second that she needs a pee. He presses his lips together to stop himself from laughing.
"Lemon chicken," says DC Andy Daniells, sniffing the air. "Maybe prawns in oyster sauce."
"I'm sorry?"
"Black bean, definitely. Not satay."
McAvoy drags his eyes from the distant bulk of the warehouse, looks across at his colleague.
Daniells, who had told him within the first ten seconds of shaking hands that the double L in his surname was originally Scandinavian and not Welsh, is new to the unit. He's an affable, likable lad in his late twenties, with a bald head and a healthy, ruddy complexion. In the month since Daniells moved across from regular CID, McAvoy has only ever seen him in one outfit. He had clearly decided in his youth that he would never look better than in rumpled navy chinos, a pale blue shirt, and a striped red tie, and had decided to stick with it.
The windscreen wipers squeak inelegantly across the glass of the Corsa, smearing the drizzle into streaks. McAvoy winds down the window, reaches out, and uses the cuff of his jacket to try to make the glass better equipped for surveillance.
"You think this place has got owt to do with it?" Daniells asks, nodding in the direction of the restaurant.
McAvoy, pleased to be back on more familiar ground, gives a shake of his head. "No, we've spoken to the owner. Clean as a whistle. Making a mint and wouldn't want to risk it. Did you know John Prescott's a regular here? Once got in trouble for parking in a disabled bay. Was in the papers . . ."
"Prescott. Deputy prime minister, wasn't he?" asks Daniells, without any hint of embarrassment.
McAvoy pauses for a moment, wondering whether he should instruct the detective on the importance of sound political and local knowledge, but decides that the cheerful, chatty young man will probably pick it up as he goes along. He's only lived on this coast for a year or so, and his Midlands accent remains strong.
"Yeah, he was Blair's number two."
"Must have done a lot for this city, then . . ."
"Yes, you'd think."
They sit in silence for a moment, and McAvoy, who has never felt comfortable in one-on-one situations with colleagues, begins to feel self-conscious. He goes back to his notes, shuffles through the papers in his lap, and checks his watch again.
"Late," says Daniells, lifting his left arm from the steering wheel and showing McAvoy his cheap watch. "She said six."
McAvoy bristles. Can't help himself. "She?"
"Pharaoh. She said six."
McAvoy's mouth becomes a tight line. "Do you mean Detective Superintendent Pharaoh?"
"Yeah," says Daniells, not detecting the warning note in McAvoy's voice. He laughs suddenly, at a memory. "Did you see her trying to get the stab vest on? Could put that on YouTube . . ."
"I beg your pardon, Constable?"
This time, Daniells spots the danger. "Wouldn't want to mess with her, though," he says hurriedly. "Great boss."
"Yes. She is."
He stares out of the window across the gloomy car park.
Spots the rear tire of the surveillance van. McAvoy tries to picture the scene inside: Trish Pharoah, Helen Tremberg, Ben Neilsen, and half a dozen uniformed officers, all sitting cramped and anxious in the half-light, extendable batons greased and palmed, jumping with each crackle of the radio . . .
"We've got movement."
The voice on the radio belongs to Detective Chief Inspector Colin Ray, the second in command of the unit. He's a gangly, goggle-eyed, rat-faced man with a fondness for pin-striped suits. Pushing fifty, and with a greenish pallor to his skin, he is at once feared, respected, and reviled. In the event of impending apocalypse and the collapse of the rule of law, he would find himself getting punched in the face by a lot of colleagues.
McAvoy tries to heighten his senses. Hopes Daniells will do the same.
A black Land Rover glides into the car park, its tires making an expensive-sounding swish on the wet tarmac.
Daniells appears to be about to duck his head below the steering wheel, but a warning hand from his sergeant holds him steady. _No sudden movements_ , suggests McAvoy with his eyes. Nothing to alert the occupant.
"Is it our guy?"
This time the voice is Pharaoh's.
"Too dark. Can't say."
"Fuck."
McAvoy can hear the frustration in his boss's voice.
"This them, d'you think?"
Daniells's voice sounds excited and nervous. McAvoy wonders how many of these operations the young officer has been a part of.
"We'll just have to wait."
McAvoy wishes he were in the van with Pharaoh; able to give her the kind of encouraging smile that tells her he believes in her and that this will come good.
"Steady now," comes Pharaoh's voice.
The Land Rover still has not moved. It remains at a stop, diagonally opposite where McAvoy and Daniells sit. If they are lucky, two burly men will get out and walk across the five hundred yards of wasteland between here and the disused warehouse. Once they are inside, Pharaoh will give the signal, and her team will move in to arrest everybody inside. McAvoy is here in case anybody slips the net: ready to block off the road if someone flees in a vehicle. DCI Ray and DI Shaz Archer are hopefully shivering as they keep watch on top of the giant furniture store that marks the end of the retail park that the road winds through on its way down to this washed-out, run-down location. At the far side of the warehouse, two patrol cars from the Operational Support Unit are parked up behind a wall of containers, ready to block off the escape of anybody who makes it into the storage area of the still-working dock.
Pharaoh's voice: "Keep it together, children . . ."
Seconds tick by.
Minutes.
"Hell of a place, isn't it?" says Daniells broodingly, staring through the glass at the brick building opposite. "All those fishermen . . ."
"Trawlermen," McAvoy mutters under his breath. "Fishermen stand on a bank with a rod. Trawlermen risk their lives in seas harder than you can imagine."
"I'm just saying . . ."
Daniells does not get a chance to say anything more. In a shriek of rubber, the Land Rover roars out of the parking space.
DCI Ray's voice on the radio . . .
"Fucking hell . . ."
The vehicle tears out of the car park, but instead of turning left back onto the road through, it spins right, barreling across the area of wasteland and rubble between Pang's and the nearest tumbledown warehouse.
". . . what's he doing?"
McAvoy feels a fist close around his esophagus. He grabs the radio, but in his haste it slips from his hand and into the footwell. He grabs for it, papers falling from his lap, scrabbling desperately until his fingers close around its bulk.
"Guv, get out of there, it's a setup . . ."
McAvoy doesn't know why, but he is flinging open the car door. He could have instructed Daniells to drive. He will never know why he did not.
He has run only a half-dozen steps when he sees the light. Sees the flame emerge from the dark glass of the Land Rover. Sees it flicker and bounce as the vehicle smashes its way over the ragged landscape. Sees a figure climb halfway out the window of the moving vehicle and draw back its hand . . .
The Land Rover spins 180 degrees and barely slows as it approaches the small outbuilding where McAvoy had seen the telltale smudge of a police van's back tire.
His shout of warning dies in his throat. The light is momentarily airborne, arcing upward, bright against the dark sky, before it tumbles down, down . . . and smashes against the double doors at the back of the police van, stuffed to the gills with police officers: sudden prisoners in a vehicle clothed in flames.
HOME. The back end of the Kingswood estate, a twenty-minute drive from the center of Hull and near enough to the East Riding villages on one side to compensate for the nearness of Europe's biggest council estate on the other.
It's a computer simulation, this place: a sprawl of Identikit houses and lawns the size of bath towels; of used cars bought on finance; and square living rooms costumed with hand-me-down sideboards and January-sale sofas and first-day-at-school photos.
Here, on the curve of one nondescript cul-de-sac, all white paint and bare brick, a rusty blue Peugeot with two wheels on the curb, tasteful ivory curtains and the slightest scent of baking . . .
Roisin McAvoy, pressing her head to her husband's bare chest, absentmindedly tracing the ridged outlines of one of his many scars with her dainty, red-painted fingernails.
McAvoy barely registers her touch upon his dead skin. He can still smell flames. Twenty minutes in the shower scrubbing his face and hair with Roisin's homemade rosemary-and-mint shampoo has not removed the acrid tang of petrol and smoke that clings to his skin like damp linen.
"Another?"
Roisin removes herself from his embrace and nods at her husband's mug, held limp and lopsided between finger and thumb. The marshmallows have melted together and formed a rather pretty roof over the inch-deep sludge of hot chocolate.
"Aector? Another?"
"Not yet," he says, and doesn't know why. "It was lovely."
"It's the cinnamon," she says brightly. "Aphrodisiac, y'know."
McAvoy does know. They've had this conversation before. Roisin knows this, too, but in the past, such chats have led to tickles and fun, so he is pleased she is trying to steer him toward that goal once again, even if he has no energy for the helping of "adult time" she has clearly been craving all day.
"You sure you didn't bump your head, darling?"
"I was nowhere near, Roisin. Didn't even get warm on the flames."
Nobody was badly hurt in the blast. Ben Neilsen had tripped jumping from the van and cut his hand. One of the uniformed constables who had used too much hairspray before suiting up for the operation had found herself looking momentarily angelic when the flames took hold, but Pharaoh had had the presence of mind to push her headfirst into a puddle, and she had escaped without significant injuries.
The operation had not gone well. The four-by-four had managed to lose the patrol car somewhere in the maze of old buildings down by the docks. The helicopter, when it had finally turned up, couldn't pick up the trail. And when Pharaoh and her remaining team had burst through the sagging wooden doorway of the ramshackle warehouse, hoping to salvage the evening by at least seizing a few tons of marijuana, the place had been deserted. The long tables that lined the cold, dark space were covered in dirt and leaf, indicators that the building had indeed been used for cultivation of drugs, but whoever had used the place was long gone. Leanne has not answered her phone, and the uniformed officers dispatched to her house said it was empty and unlit.
"She'll be fine," says Roisin softly. "Pharaoh. She's a big girl."
McAvoy looks at his wife, trying to read her expression. She has not yet met his boss. Despite being married to a policeman, she is not comfortable in the presence of the law. She knows that Pharaoh means a lot to her husband and that there is no risk of him straying, but McAvoy has lately detected an edge in his bride's voice whenever Pharaoh comes up in conversation.
"Briefing in the morning," says McAvoy. "Debriefing, really. See what we can salvage from tonight. I'll go and try Leanne again first thing. I'm sure she wouldn't be involved in any setup. She's not a bad person. She's just, you know . . . it's a mess . . ."
"You'll sort it, Aector. Don't worry."
They are in the kitchen, leaning against the work surfaces. Roisin has just finished the dishes. McAvoy, at her insistence, has not been allowed to help. The arrangement is in part due to her claims that men should not worry about housework, and partly because he has a habit of dropping things and making a mess.
"Oh, I got a call from an old friend today," says Roisin suddenly. "Can get us one of those Toyotas, the four-wheel-drive ones. Two grand and only three years old . . ."
McAvoy winces. Colors instantly. Wishes she had not brought this up. He does not know how to respond to her mentions of "friends" and "contacts"—least of all since this morning's embarrassments with the travelers. He does not believe that any such car will have been procured legitimately. Fears it may even be stolen. He is ashamed of his thoughts and what they say about his prejudices, even toward the person he loves more than any other.
"We'll see," says McAvoy. "The insurance could still pay out."
Roisin barks out a derisory laugh. The McAvoys are locked in a battle with their insurers. Their minivan had been reduced to a burned-out shell a week before Christmas, driven into a brick building by a killer who perished in the resulting blast. McAvoy had escaped with only minor burns. Those wounds have been a picnic compared to the resulting insurance headache. The company claims he is not covered for a "work-related" accident. Refuses to pay up. They have passed him between a dozen different departments; all apparently peopled by twelve-year-olds who keep laughing when they read his description of what caused the accident.
A sudden, halfhearted cry from upstairs causes Roisin to close her eyes in frustration. She is looking tired. Lilah has been difficult all day, grizzling and sobbing, refusing to feed.
"I'll go," says McAvoy, but Roisin waves a hand at him, insisting he go sit down. He does not want to, fearing he will fall asleep as soon as he closes his eyes. She brushes past him, too tired to notice him put out an arm for a cuddle.
McAvoy stands alone in the kitchen for a while. Looks in the bread bin and the biscuit barrel. Eats a couple of peanut butter cookies and takes a swig of milk from the carton in the fridge to swill the crumbs from his teeth. He looks for some kind of chore. Spots his coat over the back of the small kitchen table, and picks it up to go and hang it in the cupboard under the stairs. As he does so, Roisin appears at the top of the staircase. Lilah is red-faced and wet-eyed in her arms.
"I'm throwing those trousers away," she says, nodding at the laundry basket by the bathroom. "Horrible."
She reaches down and picks up something from the floor. "Oh, this was in the pocket."
She throws him the mobile phone.
McAvoy had almost forgotten it. He colors as he looks at it.
"Fancy model, that," says Roisin, mid-yawn. "You going to try and get it working?"
McAvoy runs his tongue around his mouth. Opens his mouth to justify his interest, and realizes Roisin does not need him to. Just nods and enjoys her smile.
• • •
AN HOUR LATER.
_An Irish voice, made snappy by tiredness._
"He fecking is."
Roisin McAvoy is pronouncing that the man on the television is an arsehole.
McAvoy looks up, wondering whom his wife is talking about. He has been lost in concentration, safe in focused hard work. He takes off his reading glasses and lets his eyes focus on the giant flat-screen TV that stands in the corner of the room. He gives a shudder. It's the Thunderbird. Mr. Popple-head. Wanchorman. _That Arsehole_ , to give him his full title. A Hull institution, he has somehow been elevated to the status of a local legend without appearing to have a single fan. He is a slight, creepy, weaselly-looking chap with a head too big for his slim frame and a mustache that has been shaved bootlace-thin and skin that has been sunbed-tanned to the color of damp sand. To McAvoy he always appears to be trying to remember whether he has left the gas on. How he got the gig presenting the local news has been open to speculation for some time, but there are suggestions it involved a complicated ritual and the sacrifice of a goat.
"Oh, God, turn him over," he says, wondering how he has managed to blank out the man's voice until now.
"Can't," she says. "Help!"
Roisin is feeding Lilah, one breast flopping over the top of her nightie, poking out from the folds of her leopard-print dressing gown. "The buttons are over there," she says in mock desperation, nodding at the remote control. It sits taunting her at the other end of the sofa. "I'm stuck."
McAvoy takes the hint. He has a tea tray on his knees, and the mud-caked mobile phone and an assortment of screwdrivers, cotton buds, and brushes laid out on the arm of the chair. He moves them all to one side and stands, padding barefoot to Roisin's side. He retrieves the buttons and hands them to her. She takes them gratefully, but does not yet change the channel.
"How's it going?" she asks, nodding at his tools.
McAvoy pulls a face. "I don't know. It's almost clean. I've got an adaptor and I can charge it through the laptop. The battery from my old Nokia should fit it if that one's fried. SIM's clean, so maybe. I don't know. Wish I'd never found it."
Roisin laughs. "No, you don't."
McAvoy returns to his chair, and Roisin, careful not to dislodge Lilah, fumbles with the controls. Before she can change the channel, Wanchorman introduces a story about changes to the makeup of the Police Authority.
"How did it go?" asks Roisin, remembering.
"It went," says McAvoy. "The new chairman has some interesting ideas. He could go far."
"Sounds like you would like to throw him there."
McAvoy shakes his head. "I can't make my mind up. I guess it makes no difference what I think."
Roisin laughs. "You don't mean that, either."
McAvoy pokes his tongue out at her and turns his attention back to the broken phone, tuning himself out again as his wife makes herself comfortable and settles into her soap opera. He vaguely remembers that he has a cup of tea on the go, but figures that wherever he left it while bathing Fin and telling him his story, it will be too cold to bother about retrieving.
Ten minutes later, satisfied that the phone is as clean as he can make it, he disappears through to the kitchen and out of the back door to the shed. It stands on the nine-slab patio, next to the sandbox and mini-trampoline, and its mingled scent of sawdust and poster paints, linseed oil and solder, reminds him of his father. He has to cling to such links. The two do not speak.
McAvoy's tools are neatly arranged on the wall, each piece of kit outlined in black marker so he can know instantly when something is not in its proper place. He pulls open a plastic drawer and roots through the collection of wires and leads. He has a habit of collecting random things too interesting to be thrown away, and a testament to the hardship of his youth.
He picks up a handful of wires and carries them back to the living room, stopping on his way to retrieve his laptop from where it is charging in the kitchen. Were his hands not so full he would scoop out another handful of lemon meringue pie from the foil tray that sits next to the microwave, but before he can consider sticking his face in the dessert, Roisin's voice cuts through from the living room.
"Leave it. You've had two slices."
He comes back to the living room, his head bowed: busted.
"I wasn't going to have any more . . ."
"Fibber." She raises an eyebrow, catlike. "Am I not feeding you?"
McAvoy looks down at his barrel torso, his chunky thighs and calves, bulging against his cutoff denim shorts and rugby shirt as if he were halfway through a metamorphosis into the Hulk.
"It's soooo good . . . ," he says, a child demanding more cake.
"I'll make another one at the weekend. You can't have everything you want all the time."
The way she says it is enough to make them both laugh without need for a reply.
Some time later, after some gentle cursing and a skewered thumb, McAvoy has managed to create a makeshift adaptor out of an old phone cable and is plugging the phone into his laptop.
"Here we go," he says and holds down the ON switch on the keypad.
Roisin, who is yawning and trying to keep her eyes open for the final credits of her program, can barely find the strength to pretend she is interested. "Working?" she asks as she shifts Lilah into a more comfortable position on her lap.
McAvoy is too engrossed in fiddling with the laptop to reply. He has not used this software before, downloaded from a specialist site dealing in data retrieval and recommended by a colleague in the Technical Support Unit.
"Aector?" His name ends in a slur.
McAvoy looks up. Roisin is starting to doze off, sliding into a half-seated, half-lying-down position, her legs drawn up childlike beneath her. McAvoy carefully moves the computer to the side and crosses to her, taking Lilah from her unresisting grasp. His daughter wriggles and grimaces a little, letting out a tiny cry of disapproval at being disturbed, but McAvoy presses her to his chest and shushes her back to the lightest of sleeps. He slides back into the armchair and watches the screen as the phone's memory is transferred to his desktop.
"Look what Daddy did . . ."
The flickering screen reflects on his daughter's face, turning her apricot cheeks and smooth, almond-colored brow into a shimmering collage of images, words, numbers, names . . .
Lilah wakes again. Reaches up and grabs her father's ear. She holds it, as if deciding whether there is anything to be gained by giving it a yank, and then lets go as she feels the backs of his knuckles stroking her jaw.
McAvoy props his daughter up so she can see the screen.
"I think this might have worked," he says softly in her ear, as if sharing a secret. She looks at the screen wide-eyed, puzzled but fascinated. McAvoy smiles, starts to read. "What have we got?"
He clamps a hand over Lilah's eyes. The movement is unexpected and Lilah gives a gasp of fright that turns into the motorbike rev that signals her intention to cry.
On the sofa, Roisin sits bolt upright. She sees her husband with his hand over their daughter's eyes, blushing furiously and signaling at the laptop with frantic nods of his head.
"Jaysus . . ."
Yawning, exhausted, too tired to sugarcoat, Roisin rolls onto the floor and crosses to him on her knees. She pulls Lilah from his grasp and holds her close, managing to croak a few words of song. With some rocking and a few soft shushes, Lilah settles, and Roisin achily maneuvers herself upright.
"I'll take her up," she says, and there is more honey on her tongue than before. She meets her husband's eyes and manages a bone-weary wink. It's her apology for her sharpness, and McAvoy, never truly convinced of the source from which she draws her love for him, wishes she did not feel compelled to give it.
When the door closes, he looks back at the laptop screen. At the handful of legible messages he can make out among a fog of scrambled numbers, letters, and computer code. The blush is getting redder. He feels the need to lock the door and pull the curtains tight.
"Bloody hell . . ."
A minute later Roisin slips back into the room. Her eyes find his and she raises her arms wide, indicating that she is all ears.
"The phone . . . ," says McAvoy.
"You got it working? Well done."
"Yeah, but . . ." He stops. Pulls an impish face.
"What?"
"'I want to take you inside me. Want to arch my back like a yawning cat, pushing back against your hardness, your manhood so deep inside me that it feels as if I am breathing for you . . .'"
"Fecking hell!"
Her tiredness momentarily forgotten, she all but runs across the room and throws herself over the arm of the chair and onto his lap, knocking loose the lead that connects the phone to the laptop. McAvoy doesn't care. This is fun.
"Is there more?" she asks, looking at the laptop.
McAvoy raises a hand to point at the screen and then stops himself. His wife, bright, witty, beautiful, and gifted, had a traveler's upbringing. Her schooling was sporadic and disjointed. She is not a comfortable reader, despite the patience with which he has helped her develop a love of words. Instead, he picks another phrase at random and reads it to her.
"'I am yours to abuse. I am a toy for your pleasure, a piece of meat to be pounded, clay to be molded—a waiting receptacle for your frustrations and rage . . .'"
Roisin giggles and presses herself against him. They are two teenagers reading a friend's diary; naughty, wrong, and loving it.
"'Want your breath against me, the cord biting into my skin . . .'"
"She's good," says Roisin appreciatively. "Bet he bloody loved it."
"'Want my mind to sculpt your face; your identity to remain the desperate fantasy that first brought your tongue to my shoulders, your hand to my cock . . .'"
McAvoy stops short, and Roisin catches her breath. She gives a snort.
"It's two blokes?"
McAvoy catches himself pulling a face, and a guilty blush thunders from his brow to his neck. His liberal self-loathing grabs a handful of his guts.
"Well, there's nothing wrong with . . ."
Roisin is giggling. "You were loving it," she teases.
"So were you," he protests, and then accepts there is no way to escape this with any dignity, so just starts laughing and buries his face in her chest.
"Did it get you going?" she asks seductively, trying to get a hand inside his shirt.
"No!" Then, sheepishly, "A bit."
"Me too," she says, and presses her face to his.
"SLUTTY," he'd texted, when pressed for a preference on how she should dress. "A dirty girl."
Suzie hadn't really known how to interpret the instruction, but figured it didn't include her Disney scarf or Care Bears rucksack.
Still, she has enjoyed playing dress-up, and her reflection pleased her when she looked in the mirror on the back of the wardrobe door. She has managed to find an outfit in her explosion of clothes that, to her at least, qualifies her as vaguely whorish.
She is shivering in a short blue dress and a secondhand leather jacket that reaches to her bare knees. Her hair is tied back and her makeup is thick enough to ensure there will be no facial damage in the event of a sudden fall.
The high heels her new playmate had insisted upon are on the passenger seat of the Fiat Panda. The stiletto points kept getting caught in the mat when she pressed the accelerator, so had been whipped off at the last set of traffic lights. She is now driving barefoot, unsure whether or not she likes this sensation of damp dirt and metal on the soles of her feet.
It is a miserable night. The rain is a damp net stretched across the black road. It does not seem to fall, but instead hangs, ghostlike and bone-soakingly omnipresent, in the chill, oil-dark air.
Suzie wishes Simon were here. She can picture him with no effort of will; can see him now, smoking a roll-up in the passenger seat and telling her she looks beautiful.
Such a wish is nothing new. Suzie's yearning for his return has become almost a prayer. But tonight it is more through some vague sense of unease over her safety than her usual eagerness to giggle and chat with her best friend.
It's almost nine p.m. This is her third visit to this location, but the first time she has driven here alone.
She remembers Simon's message when she first told him she had heard there was a popular spot for couples and singles on the coast road up to Bridlington.
"Coniston rest stop—where dreams are made."
Ten miles from the city center, between two midsize villages, a little side road has become, in certain circles at least, notorious. Though she does not particularly like the word, it has made the papers as a "dogging spot." Here singles meet, and couples put on participatory shows for the handful of guys who like to spend their spare time sitting in their cars in the dark: each hoping the next set of headlights in the rearview mirror represents a blow job rather than the police.
"What are you doing? Seriously, Suze?"
She asks herself the question as she slowly maneuvers the tiny, battered car into the isolated pitch-dark of the entrance to the rest stop.
It is at least a mile from the nearest house.
There is a nervousness, an excitement, in Suzie's stomach and thighs, but to call the feeling arousal would be inaccurate. In truth she does not do this for the sex. Not really. It is perhaps just to prove herself alive. It is to be somebody who does not just fantasize, but who makes things happen. She does it because she thinks it is weak to deny oneself excitement.
In her years with her fiancé, sex was simplistic and routine. Life was okay. Middle-of-the-road. Safe. When her heart was broken, Suzie lost herself. Did things she could never have previously imagined. Found reserves of lust and rage in equal measure, and made mistakes that catapulted her into a new way of being. She engaged in one-night stands and office flings: sweaty unions in nightclub toilets and in the backs of cars. She read and watched erotica. Bought herself toys with which to pleasure herself when she could not find a partner. Made it clear when starting conversations that she was not just a tease. That she was willing to play.
One such rendezvous introduced her to an attractive older man, who spotted in her a hunger for the unknown. He had introduced her to the websites and forums where like-minded people were able to enjoy grown-up fun. And she had thrown herself into the life. Had quickly come to view ordinary sex as somehow lukewarm and insipid in comparison. Had so grown to love the sordid nastiness of these couplings and triplings that she found herself turning down nights out with potential boyfriends in exchange for late-night assignations with strangers.
Simon was the only friend who knew about it all. Something had happened, shortly after they met, that bonded them together in a friendship without judgment. Both were free to be themselves, whatever that might be. They joined in each other's games and laughed about their adventures. She could not talk to her other mates about such things. Could not stand to be judged or, worse, analyzed. Would not want to hear their aghast musings on what hole in her heart or bump in her brain forced her to subject herself to such abuses and degradation. She does not really want to think about any of that. Just knows that it makes her feel as if she were living life in color after so many years in black-and-white.
"Wish you were here, Si. What am I bloody doing?"
There are two cars in the rest stop. A large estate car is parked up to Suzie's right in the shadow of the mound of shingle and earth that blocks the area from view and gives it such appeal. Its lights, and engine, are on.
In the distance she can spot the shape of another car. It is dark and bulky, lights off, its occupant obscured.
Suzie has been halfheartedly listening to the radio. There has been some sort of accident down at St. Andrew's Quay. A police van has been petrol-bombed and two officers have been taken to hospital. She wonders if it was the speed-camera van and rather hopes that it was.
She takes a deep breath. Parks up on the opposite side of the rest stop to the estate car. Wonders who she is about to fuck.
In the beam of her headlights she can make out that the driver is quite tall. From this remove she guesses he is middle-aged, but cannot be sure. In truth, it doesn't really matter.
She closes her eyes and tries to calm herself. She has done more devilish things than this. She has played more daring games. But in the past Simon had been there to hold her hand.
"God, I miss you."
The first few weeks without him she had had no appetite for such things. She didn't log on to any of the websites that used to bring her such fun. Didn't send a filthy message or put a single kiss at the end of an e-mail. But as grief became bearable, so desire began to return. There were tears when she attended her first swinger party without him, but they had not flown so freely as to inhibit her. The night had gone well. She had enjoyed herself. Had made new friends. Had promised to return for the next gathering. Had even told today's playmate how much she hoped he would join her.
The phone on the passenger seat beeps and Suzie jumps. She picks it up and reads the message.
"Go and make him happy."
The thrill of it all brings goose pimples to her skin. She reaches across for her high heels and slips her cold feet inside them, noting how her fingers tremble as she fastens the buckle. With a quick glance at her reflection in the too-dark mirror, she steps from the car.
A gust of wind pulls at the tails of her leather jacket, and her legs feel unsteady as she totters across the tarmac on her high heels, closing the distance between herself and the vehicle in only a few strides.
The man in the car watches her approach. His head almost reaches the roof of the vehicle. He has a thin, pinched face and rimless glasses. He is dressed in a nice suit, with his tie unloosened almost to the middle of his chest. He is red-faced, and a sheen of sweat is visible on his thinning scalp. As he winds the window down, Suzie is hit by the smell of booze. Bending forward to talk through the glass, she sees the man already has his trousers undone.
"Want to play?"
The line sounds silly and false as she says it, but she can think of nothing better.
The man looks taken aback, and Suzie wonders if he had genuinely expected to find sex here tonight, or had just driven here to see if the rumors were true.
"What you got in mind?"
His voice is slurred, but whether through drink or nerves she cannot say.
"It's cold out here," says Suzie, trying to sound sexy.
"Do you want to get in?"
Suzie remembers her instructions. Wonders if her new friend is watching. Whether he is sitting in the distant car, smiling as she fulfills his fantasy without ever having seen his face.
"You can join me out here. The bonnet of your car looks soooo comfy."
The man fumbles with the car door. He steps from the vehicle, and a half bottle of Jack Daniel's falls onto the road. The man kicks it under his car and stands up straight. He has to reach out to steady himself, and his eyes slide halfway shut.
He is a good foot taller than Suzie, and twice the age.
She looks up at him. Decides they will not kiss.
She wonders if this is turning the watcher on. She is feeling only the slightest frisson of arousal, but that is to do with the sensation of being commanded, being watched, rather than by any desire to have sex with this man.
She goes straight to work. Reaches out and squeezes his groin. He moans and she wonders how long he has been turning himself on, here, alone, in the dark.
"Can I lick it? Lick you? Down there?"
She does not want him to, and tonight's architect had not commanded her to accept any such pleasures.
She shakes her head. "Do me. Now."
Suzie walks as sexily as she can to the front of the car. It is warm and throbbing as she lays herself upon it, face-first, listening to the hum of the engine. Without a word she pulls up the hem of her dress. The cold night air and faint mist of rain feel wonderful on her bare skin.
A moment later, he is behind her, pressing his still-clothed hardness against the backs of her thighs.
She wishes she had her phone in her hand. Wishes she could text him to ask if he is enjoying the show.
She hears the rustle of trousers falling to the wet ground. Feels rough and inexpert fingers between her legs, and then a hand in her hair.
Suzie presses her face onto the wet metal of the car. Feels him fumbling, trying to find the way inside . . .
"Get it over with," she mumbles into the back of her hand.
The sound of a car.
Big, powerful engine roaring into life. Fat, expensive tires on wet tarmac. The sudden scream of a foot stamping on gas.
Suzie turns around. Stares past the grunting, thrusting man. Her eyes widen. It is a sensation of genuine terror.
The other car is screaming toward them, mere feet away and getting faster.
The noise she makes is a strangled squawk. It is an unnatural sound, gargled in her throat.
Desperately, she pushes back against the man, who pins her to the bonnet of his car. Hears him grunt and stagger as he tries to hold her where she lies.
"Get off me!"
Suzie knows she is about to die. Wonders if this is how Simon felt as he gave himself up to the noose.
And then she is squirming, shrieking, slipping out of his grasp: the roar of the car engine drowning out her shouts to "Move!"
She slips free. Throws herself into the dirt at the side of the road.
Turns, just in time to see the four-by-four crush the man, half turning, against the bonnet of his own car in a crash of metal and flesh.
He is pinned between the cars, legs and buttocks still bare, shirttails comically parted like stage curtains to reveal a dying erection.
Suzie cannot make a sound. Her throat has squeezed shut. Her eyes will not close. She stares, unable to yank her gaze away from the man's gulping, gasping mouth, opening, as if with the dying gasps of a fish, as his head falls forward onto the bonnet of the vehicle, which pins him where he stands.
Beneath where Suzie lies, semi-sprawled, the ground is cold. Wet. Her knees are bleeding where she landed on stone. Her mouth is open as if in mimicry of the dying man.
Finally, she is able to raise her dirty hands to her face. To momentarily block it out. To stop her memory from absorbing any more.
She looks up again only when she hears the larger vehicle move. She watches as the four-by-four reverses, pauses, and then turns in a semicircle. It does not pause again. The sound of a boot stamping on the gas rings in Suzie's ears.
A moment later, she is alone, sitting in a ditch at the side of a rest stop, watching a stranger slide to the ground as if made of damp paper; his legs a ruined mess of skin, blood, and bone.
She forces herself to move. Pulls down her dress as if suddenly terrified of being seen. Moves, in jerky increments, to where the man lies.
"I'm sorry," she says, though the sounds do not come out.
She staggers back to her own car. Fumbles with the door. Cries. Tries a dozen times to get the key in the ignition. Breaks a heel as she presses the accelerator to the floor.
She has driven five miles with trembling hands before it occurs to her to call 999.
It is another two before she can find a phone box.
She is nearly home before she has the presence of mind to go back and wipe her prints from the receiver.
LILAH IS WHIMPERING. Thrashing. Kicking fat little limbs the color of uncooked sausages. Turning her cheeks into cold, slapped flesh with the power of her sobs.
"Please, baby girl. Please . . ."
McAvoy's giant hand is splayed upon his daughter's heaving belly, trying to soothe her with his gently massaging fingers.
He leans over the cot. Fills her world with his face. Tries to saturate his eyes with truth, to wordlessly convey to his frightened, agitated child that she has nothing to fear. That Daddy is here. That she need never be afraid or lonely or sad . . .
He scoops her up. Holds her to his chest. Strokes the soft down that covers her warm crown. Shushes her, his stubbled cheek against her soft, untainted skin.
Gradually Lilah settles. One of her tiny hands finds McAvoy's lower lip, and she grips it territorially as she begins to drift back into sleep.
Content to let her keep whatever part of his face she wants, McAvoy leans back against the wall and stares through the glass. Takes in the symmetry and newness, the bland homogeny of the estate.
Allows himself a brief moment of memory. Recalls the gloss of condensation. The smell of smoldering turf. The chill stone floor of the family croft. That view: across the heather and peat of the undulating fields down to the glassy black waters of Loch Ewe . . .
He shakes it away. Concentrates on now. On Hull. Its sky and its streets.
McAvoy has never had cause to use the word in conversation, but he fancies the color of the morning sky, as it bleeds from the orange-tinged black of night to the cloud-covered gloom of day, it could be labeled _isabelline_. It is a word he read in a book as a child, and its cheeky definition ensured it would lodge in his head forever. The word lends itself to the gray-and-yellow parchment hue reputed to be the color of the underwear worn by Isabella, archduchess of Austria, at the end of a three-year siege of her castle home.
It is a word that always makes his nose wrinkle, but it seems strangely appropriate for this damp and ghastly morning.
McAvoy checks his watch. It's just gone six a.m.
He listens for any other sounds inside the house, but there is silence save Lilah's gentle snuffling against his chest. Roisin and Fin remain asleep. He has a moment to himself.
Soundlessly, he crosses back to the cot and tenderly lays Lilah back down. Moving on tiptoes, he leaves the room and closes it behind him, conscious even as he does so how foolish he must look; a man of his size tiptoeing like a burglar, clad only in boxer shorts and suffused with the scent of smoke and too little sleep.
He retrieves his mobile phone from the pocket of the trousers which lie outside his bedroom door, alongside the tie, socks, and underpants he has already picked out for the day. He is used to leaving at strange hours. Does not like to wake his bride by dressing in the bedroom.
McAvoy pads downstairs, checking the messages on his answering service.
He enters the kitchen. Pours himself a glass of milk and adds a squirt of strawberry syrup, then downs it in a gulp.
Within moments he is heading back upstairs. Pulling himself into his clothes and replaying the message that he wishes to God he had picked up when it was left for him at two a.m.
"McAvoy. This is Desk Sergeant Pulis from Queens Gardens. Your request just crossed my desk. I'm sorry, this didn't ring any bells before now. Shaun Unwin, yes? You're looking for him or Leanne Marvell, I understand. Shaun's been with us. In the cells. He's due to be released first thing, but I'll hold him if I hear back from you . . ."
• • •
TWENTY MINUTES LATER McAvoy is walking briskly across Queens Gardens. The skies have not yet unleashed the lake they hold in their bellies, but the air is damp and the morning gray. He is grateful for the long woolen coat he took the time to pick up off the back of the sofa before silently slipping out of the house. He had pushed the car, hand brake off, for two whole streets, before turning the ignition. He wants his family to sleep soundly.
He follows the paved walkway through the neatly tended landscape of duck ponds and grass and up the stairs to the glass-and-concrete frontage of Queens Gardens Police Station.
The sergeant behind the glass raises his eyebrows as the detective walks in, and swivels his eyes to look up at the clock behind him.
"My, you're up with the lark."
"Shaun Unwin," says McAvoy, crossing to the desk. "Has he been released?"
The sergeant, whose name McAvoy recalls as having a military connection, opens a plastic folder and runs a finger along a list of names.
"Released at four forty a.m.," he says. "Pulis told him he could have breakfast before he went home but he was itching to get going."
McAvoy closes his eyes.
Remembers the sergeant's name.
"I gave instructions, Sergeant Uxbridge. It was essential I speak to him . . ."
The sergeant bristles. "I wasn't on shift, mate. Was it a paper request? Only they sometimes get misfiled, see. Now, when it's on the computer, it should flash up to tell you not to let any bugger go if somebody still wants them, but even then it's a hit-and-miss business . . ."
Exasperated, McAvoy turns away. Runs his tongue around his mouth and rasps his hand over his unshaved face.
His mind fills with the snippets of information he was able to piece together on the drive over, in between phone calls to Pharaoh, that had left his head ringing.
Shaun Unwin had been arrested for disorderly conduct at 3:15 p.m. the previous day, even as Pharaoh and her team sat planning the raid at St. Andrew's Quay. He had been knocking back drinks in the Mission. Sparked up a cigarette and refused to put it out. Swung a punch at the barman and smashed his forearm into the Plexiglas frontage of the jukebox. Made a prick of himself, and told the owners that if they didn't like it, they should call the cops.
He didn't run when the police turned up. Seemed to give himself up without any of his usual aggression.
The constable who made the arrest said he could get nothing out of Unwin. Had got no reply when he, like so many others, tried to find Leanne Marvell to inform her of her partner's arrest.
McAvoy closes his eyes. Last night's bust was doomed to failure from the start. Leanne had told her boyfriend that she had told the police. He had gone and got himself banged up, and whether intentional or not, that news would have rung alarm bells with the gang who paid him. Calls would have been made. The cannabis relocated. And then some bastards in a Land Rover dispatched to deliver a flaming warning to the coppers who had thought they were dealing with the usual class of scum . . .
His phone rings. Wincing in advance, he answers as quickly as he can.
"Guv?"
"I already know," says Pharaoh, shouting above the noise of her sports car on the noisy road that leads from her home across the water up to the Humber Bridge. "Fucking idiots. Have you tried the house? He's just thick enough to go back there."
"No, guv. I came straight to Queens Gardens . . ."
"Right. Well, fucking run. Why do these people think they can think? If he wanted to be out of the way, Leanne could have asked us. We could have planned it another way. He could have had nothing to do with any of it. To be sitting in the cells while we were sitting waiting for him—what does he think his bosses were going to think?"
The doors swing open as McAvoy walks back out into the cold. The rain is still holding off, and his feet are steady on the slick pavements as he jogs back across the gardens and over Parliament Street, down onto Whitefriargate, with its shuttered chain stores and its full gutters stuffed with dead leaves, empty bottles, and polystyrene takeaway cartons.
He makes his way across Trinity Square and onto Dagger Lane.
Answers his phone as it vibrates against his thigh.
"Well? Anything? Shaun?" A pause. A note of real concern. "Leanne?"
The street is deserted. The light from the streetlamps shows up the haze of moisture in the gray air, and McAvoy instinctively shivers as he looks at his coat and sees that somehow, despite the absence of rain, he is soaked through.
A voice in his ear: "McAvoy?"
"Nearly there, guv."
"She'll be okay. You've seen her. She's hard. It's not her that told. They just put it together themselves . . ."
They both attempt to persuade themselves into happier, more positive thoughts. They fail.
"Not a sound, guv. He wouldn't come here, though, and we've been trying Leanne all night . . ."
McAvoy stops.
Swears.
"Aector?"
The door to Leanne's terraced house is an inch ajar.
He closes his eyes for a moment.
"The door's open, guv."
"Fuck, Aector. Right, I'm on my way. Call for uniform immediately."
McAvoy eyes the doorway. Reaches out a hand and touches the wet wood. Pushes it open and steps inside.
"Aector, I'm not far off the bridge. I can be there in twenty-five minutes maximum. Don't you even think about going in there."
McAvoy nods, steps back.
Then he smells it. Smells the soft, earthy scent of suffering: of tears and pain. It is an infusion in the air, a whisper of a taste. It catches in his nostrils and stuffs its fingers down his throat.
"Guv, there's somebody inside."
McAvoy says no more. Ends the call and then switches off his phone. Moves, as if trying not to wake a child, back within the embrace of the house.
His feet make no noise as he takes the stairs. He moves slowly, but takes the steps three at a time so as to cut down on the likelihood of one creaking.
He sniffs: a great stag checking the morning air for predators. For prey.
He finds himself moving toward what he presumes to be the bedroom. The door, white-painted and featureless, has been pushed to but not fully closed. He inches toward it. Pulls the extendable baton from his pocket, and then puts it back. He has never swung the weapon. Has seen what it can do. Does not want to add his name to the list of officers who have found themselves disciplined or guilt-ridden after allowing their adrenaline to overtake them while armed with something so deadly.
He pushes open the door.
Shaun Unwin has been tie-wrapped by the ankles to a hard-backed chair. He is naked. His hands are palms down upon his knees, a gory mimicry of a well-disciplined schoolchild.
The room smells of blood. Of lighter fuel. Of burning flesh.
The skin on Shaun's torso has been melted down to bone.
His feet sit, unmoving, in a puddle of blood that runs down from where the nails have been driven through the backs of his hands and deep into his kneecaps.
His head lolls forward: lifeless.
McAvoy crosses the room. Lifts Shaun's head. Recoils as he stares into the slack-jawed ruination of the man's mouth. At the stumps of broken teeth. The blue-black blood. The perforations in his gore-lacquered cheeks.
Shaun's mouth has been filled with a fuel-soaked rag and then set on fire. His tongue is melted black.
McAvoy, fighting his instincts, reaches out a hand and presses his fingers to Shaun's neck.
Moves back to the wall and retrieves his phone.
Pharaoh answers before he can speak.
"He's dead, isn't he, Shaun. I bet the fucking idiot walked straight in the front door."
"They hurt him, guv," says McAvoy, softly. "Must have worked on him for a time. I can't see Leanne. Fuck, what a mess . . ."
A sound behind him makes him stop short.
Shaun would have been home by around five a.m. It's just after seven a.m. now. It would have taken time to do this. Could they still . . .
This time the noise is unmistakable. The bang of wood on brick, and then feet on cobbles.
McAvoy sprints across to the window. Peers left and then right, frantically searching for the source of the sudden sounds.
He catches a glimpse of three figures. A flash of black leather and bristled, porcine skin. Of broad backs and raised collars. A flash of auburn. An insinuation, in the chaos of the picture, of a smaller, more delicate form, quicker than the others, a blur of color and a flash of white.
And they are gone.
McAvoy finds himself alone in a missing informant's flat. Finds himself sinking to his knees, bringing himself level with the ruined body of a man tortured to death for allowing his woman to open her mouth.
"Nobody here," says McAvoy, into the phone, and the words seem to make his tongue swell—make bile rise in his mouth.
He stops himself. Bites back the lies.
"Guv, I'm so sorry . . ."
HOME AGAIN. Tired and guilty, aching and sick.
_It's not your fault. They were playing with bad people. It happened. Leanne could still be okay . . ._
He has heard lots of soothing words in the past few hours, but none has helped him feel any better or cleansed his senses of the stench of Shaun's skin.
Pharaoh has taken over. A murder investigation has been launched, but the top brass have yet to decide whether it is to be folded into Pharaoh's existing investigation, or handed over to a separate CID team. McAvoy believes any attempts to remove it from Pharaoh's grasp would be madness, but knows, too, that his opinion counts for nothing. He's just the cop who found the body. The cop who has spent all day giving statements and having his clothes bagged by forensics officers because he went into the flat without a white suit on and contaminated the crime scene.
He shakes his head, hating everything. Wishing he had listened a little harder. That he had run faster. Caught even one of them. They have nothing to go on. His description is even weaker than that given by the Vietnamese farmers who suffered the same injuries months before. The initial reports on the nails driven into Shaun's knees suggest they came from the same weapon as that used in the first attack, and the doctor's initial impression is that Shaun endured an hour of abuse before his heart gave out.
He has never been as grateful to leave the station. Never wanted to hold Roisin more.
She is upstairs now. Changing Lilah. Pleased to have her husband home early and hoping his presence in the house will allow her a few hours of proper sleep.
McAvoy should be enjoying it, too. Should be up there, making them all giggle. Should maybe be getting his boots on and wandering around to pick up Fin from school. Should be reveling in the look on his son's face, the pleasure and pride at having the biggest dad in the playground.
With no instructions to follow or any other ideas about where to look for Leanne, McAvoy had decided to have one last little look at the contents of the mobile phone he had fished out of the mud of the River Hull. He entertained a hope that by looking at it again he would satisfy his curiosity and be able to sling the damn thing away. Would be able to get focused. Get busy. Make amends.
He plugs the phone into his laptop. Begins to play.
Opening the contacts box, he scrolls through the dots and numbers, whorls and compressed digits. He squints as he tries to make out something intelligible. Mc? MC2? Me?
McAvoy gets up and grabs a piece of paper from the pad by the landline and writes down the half-dozen variations that the numbers may be making up. He crosses back to where his laptop is plugged in, and sits down in his armchair, his computer's battery pleasantly warm on his bare legs.
He logs on. Types in the first number that could vaguely fit with the jumble of numbers. Finds nothing but a string of serial numbers for a courier firm. Tries the next: 07969 . . .
Bingo.
There are three hits. The phone number is linked to a trio of sites.
McAvoy clicks on the first.
"Black cat, three years old, lovely temperament, missing from Anlaby area since last Sunday. If found, please call . . ."
McAvoy, hoping the animal turned up, clicks on the second link.
"New line-dancing club. All ages and abilities welcome. Experienced instructors and fun atmosphere. Every Wednesday at St. Mark's Church Hall, Anlaby Common. Call Simon, 07969 . . ."
McAvoy nods. He is building a picture. Starting to care.
The third link takes him to playmatez.co.uk. He stares at the white screen, its gaudy purple banner; thumbnail pictures of women in fishnets, and men showing off bare torsos, exposed genitals.
_The Number 1 Hook-up Site on the Web! Swing When You're Winning!_
McAvoy turns from the screen. Looks at the door. Prepares an explanation in case Roisin walks in.
Turns his attention back to the laptop, unsure whether he is prying or being a policeman.
He scrolls down until he finds the phone number.
FILL ME UP. MAKE ME YOUR SLAVE. YOUNG, SLIM, OH-SO-EAGER MALE SEEKING DOMINANT MAN. ANLABY AREA. Call 07969 . . .
"Has somebody hurt you?"
His words are said under his breath, but they are laden with the weight of a growing unease.
McAvoy copies the posting. Creates a file on his desktop and saves the link and the words. Does the same with the lost-pet forum and the line-dance club. Wonders why this matters. Why he needs to know. Why he doesn't just put the phone in the bin and agree that it's none of his business unless a crime has been committed. Wonders just how he has convinced himself, with such certainty, that this warrants his time.
"You want to help me?"
The voice floats down the stairs with none of its usual music. Roisin is growing more tired and irritable. She told him earlier that it has been three days since she spoke to another adult. That she had found herself humming the theme tune to _Wibbly Pig_ while walking back from taking Fin to school. That she had to make a conscious effort of will not to ask for the cake in the shape of the "moo-cow" when popping to the bakery last weekend. She is craving stimulation. Needing adult time. Needing to be a young woman rather than a mum.
McAvoy runs his tongue around his mouth to make sure there are no biscuit crumbs to give him away. Gives a slight nod. Makes up his mind.
"I've got an idea . . ."
He hopes she'll squeal when he tells her that this evening they are starting a line-dance class in Anlaby.
• • •
"YOU HAVE TO HOLD ME CLOSER . . ."
The dancer smells of red wine and garlic bread, microwave lasagna, and menthol cigarettes. She's angling her pretty face upward, eyes heavy-lidded and sweat moistening her face at the temples. She is in her mid-twenties, and has clearly done this before. She is grinding her toes into the hardwood floor and lifting her red dress above pink young knees to show firm calves and red-painted toenails. Her arms are shooting out with such ferocity that McAvoy wonders whether she is being operated by remote control. She is even managing to hum along to the music, which, to McAvoy's ear, would sound the same backward.
He tries to ignore her nearness and warmth. Concentrates on his footwork. Counts in his head. Holds her hips as if she were made of glass. Tries to remember whether the hold he is about to place her in is called a hammerlock or a full nelson, and wonders whether the "Suzy-Q" she is performing will lead to osteoporosis in later life.
"One, two, three . . ."
He squints over her shoulder at where his wife is having the time of her life in the arms of a seventy-year-old man wearing yellow corduroy trousers and a designer shirt. His hands are on her buttocks. He appears to be mentally testing a cantaloupe for firmness.
McAvoy and his wife came dressed for country and western. They found it was salsa night.
"Yes, it used to be Wednesdays, but we changed it," said the nice middle-aged woman at the door. "Salsa's more fun. Great for the youngsters. Beginners welcome. Only five pounds each. Refreshments at halftime. And Mike used to be a county champion . . ."
Roisin had squealed and begged him to give it a go. Told him they could still go line dancing another night if that was what he had set his heart on. Said it was a shame to waste the babysitter, and that he might love it.
He is not loving it. Salsa merely gives him indigestion.
"It's in the hips," says Mike, rotating his own in a manner that, if performed outside the confines of the church hall, could see him locked up for indecency. "Excellent. Yes, grind it. Grind it!"
Mike is shouting this last at McAvoy's current partner, and she obeys, putting enough twist into her movements that he wonders whether her high heel will remain screwed to the floor when they separate and move on to the next person in the circle.
"That's it, my lovely. It's about sex!"
McAvoy looks as though he has been running in the rain. He is soaked through with sweat, his white shirt clinging to his skin and his jeans uncomfortably damp. His face is bright red with embarrassment and exertion, and exposed in its entirety due to Roisin's decision to slick his hair back from his face with her hand when she spotted him beginning to drip on his partners.
". . . and rest."
The sound of drums and Spanish guitar crashes to a stop, and the dozen people in the circle give a little cheer and clap for one another.
McAvoy is breathing like a hot bullmastiff, and can barely even muster a polite smile when his partner squeezes him on his sodden arm.
"It takes some getting used to," she says sympathetically. "I took to it straightaway, but some people can take longer."
"There are fish on dry land who dance better than me," says McAvoy, gasping, bending over and placing his hands on his thighs as if he had just run a marathon. He feels her pat him on his broad back.
"Don't give up on it. You've got rhythm."
He straightens up. Manages a little laugh. "Just not the same one as everybody else."
The girl extends her hand. McAvoy wipes his own on his jeans and takes hers in his palm. "Mel," she says.
"Aector," he replies. It feels odd that he has introduced himself by anything other than his rank. He wonders why he has done so. Wonders if he is subconsciously reminding himself that he is not here as a policeman. He is not here on official business. That he's just a nosy bugger, lying to his wife . . .
"Aector, did you see me?"
McAvoy turns as Roisin excitedly bounds up to him. "You were great," he says instinctively.
"I know! This is awesome, Aector."
"This is Mel," he says, by way of explanation for the attractive, sweating woman at his side. "I'm turning her feet into flippers. I don't think she sees me as a potential rosette winner."
Roisin seems to notice her husband's dance partner for the first time. She looks her up and down. Red dress. Hair tied back into a ponytail and tethered with a silk red rose.
"We thought it was line dancing tonight," she says brightly, gesturing at her own red-and-white gingham blouse, tied above her belly button, denim shorts, and fawn, knee-length leather boots. "This is so much better."
"The line-dancing club's changed nights," says Mel.
"So they said."
"There aren't many people go to it now anyway," she says, and she shifts the direction of her conversation from McAvoy to his wife. "And they're all ancient. I went to it a couple of times when it was half decent. Was a real giggle. These days they'll be lucky to get enough people to make an actual line. Not like it was."
"People got bored with it, did they?" asks McAvoy.
Mel shakes her head. "Different tutor," she explains. "Boring lady took over and all the people who used to come for the giggle packed it in."
"The giggle?"
"Simon," she says, and instantly breaks into a smile. "He and his aunt used to run it. Was more of a cabaret night. Was such a laugh."
"Has he gone to another club?" asks Roisin. "We could maybe go there . . ."
Mel shakes her head. "No," she said. She looks away. "It's sad."
McAvoy pinches the sweat from his nose. Forces himself not to push it. Lets the two girls talk. Listens and takes notes in his head.
"We didn't know he was so unhappy," says Mel.
"Quit, did he?" asks Roisin.
"Killed himself," says Mel, matter-of-factly. "Put a rope around his neck and hanged himself in his flat."
McAvoy sniffs.
Blinks once.
"Poor lad," says Roisin.
McAvoy nods. Tries to sound cool. "What was his name again?"
Mel pulls out her phone. Scrolls through. "Simon," she says sadly, and holds up the screen to reveal a grainy picture of a tall, thin, sweaty, and smiling young man. "Simon Appleyard."
McAvoy looks at the phone number displayed across the young man's image. He blinks once, like a camera taking a picture. Files away digits he already knows.
11:13 A.M. COURTLAND ROAD POLICE STATION ON THE ORCHARD PARK ESTATE. A PRETTY NAME FOR A SHITHOLE.
USED TO BE a decent area, this. Still is, in places. Still a few home owners who give a damn and scrub their front step and pick up the crisps packets and empty beer bottles that roll onto their well-tended front lawns. Still people who give a damn, and who believe that once the empty high-rises are torn down and the druggies move on, this community of tiny terraces and low-rent flats will be an address to brag about.
For the time being they're grateful for the nearness of the cop shop.
The Major Incident Team operates from the second floor: a cramped warehouse of grimy computer terminals and coffee-streaked desks; of overstuffed files, bulging in-trays, and mucky cups. A room joyless as a cell, decorated with crime-prevention posters, with overlapping memos and codes of practice, all spreading out from a streaky whiteboard where the names of active cases are scrawled illegibly in red marker pen.
Broken blinds fail to blot out the rain and the view. Strip lights buzz overhead and turn this oppressively quiet room the color of gone-off milk.
McAvoy looks up from his borrowed desk.
Biker boots thudding on the thin blue carpet.
A waft of Issey Miyake perfume strong enough to catch the back of his throat.
Bangles jangling as if their wearer were rattling a tambourine.
Hair swishing sensuously against the collar of her leather jacket . . .
McAvoy feels disloyal for even thinking it, but Trish Pharaoh does not have a natural gift for covert surveillance. The other senses announce her presence long before the eyes take her in.
"Aector, my boy. Mummy needs a hug."
She plonks her backside down on his desk, creasing the computer printouts he is carefully going through with a ruler and highlighter. She leans forward and puts her head on his shoulder, then proceeds to trundle it back and forth. "I hate them all," she says.
McAvoy looks around. There are three civilian support staff sitting at nearby desks, but there are no other police officers in the room. He lets himself smile.
"They being mean, guv? The brass?"
"They are being wankers, Aector."
"Wasn't it you that told me not to expect too much of people? That when an idiot is an idiot, it should not arouse surprise?"
Pharaoh removes her head from his shoulder and pulls a face. "Did I say that? I don't think I said 'idiots.'"
"You said 'tossers,' I think."
"Yeah, it's coming back."
Pharaoh has spent the morning with the head of CID, Detective Chief Superintendent Andrew Davey. His underlings call him "the accountant," though they occasionally drop a vowel. In truth, he's a decent enough career officer in his late forties whose life seems to involve nothing but form filling, committee reports, and a desperate and futile succession of spreadsheets designed to keep the holiday schedules from clashing. He does not tend to interfere in the running of the various CID teams. A small-framed, smartly dressed man with chronic indigestion and glasses that leave grooves in the sides of his long nose, he looks to McAvoy like a man who needs a good cry.
"How did it go?"
Pharaoh rolls her eyes. Her lashes momentarily stick together, and she pulls them apart with chewed fingernails that, though bitten to the quick, have been painted red.
"I've got a 'watching brief,' whatever that means. Shaun's murder's going to regular CID, but under my supervision, though they made it clear they wouldn't trust me to supervise the boiling of a kettle at the moment. They seem to think Leanne's at the root of it, but you and I both know that's bollocks. Davey made it plain that they think she set us up, and Shaun, too, but that's just the way she is. You know that. I know that. She's either so bloody frightened she's gone to ground, or they've got her, too . . ."
McAvoy accidentally meets her eyes and quickly looks away.
"We'll catch them," he says. "Nail Gun and Blow Torch. They can't just . . ."
"They sound like a tag team, Aector. Or really shit superheroes."
"And the third man," McAvoy carries on. "It doesn't feel right. None of it does. They've done these things to send a sign. We need to send one back. You will get them, guv."
Pharaoh smiles. "We will," she says. "Well, somebody will. I won't. I'm being shunted sideways a little. Out of harm's way for a bit. They're asking me to look at some of the 'peripherals' of the case, which has to be one of my favorite phrases of the day."
McAvoy closes his eyes. Shakes his head.
"Colin Ray?"
Pharaoh smiles ruefully. "He's taking over as lead. Taking a fresh look at what we've done so far. I've got Daniells and a list of errands. Ray's the fresh pair of eyes this case needs, apparently."
"You fought it, though," says McAvoy, appalled. "I know you fought it."
Pharaoh holds up a hand and extends the index finger. "I think I left a nail in his desk."
McAvoy doesn't know what to say, so just stares at the carpet. Eventually Pharaoh gives a sigh and then straightens herself up. "Come on," she says brightly. "It won't have much to do with you. You shouldn't be so wary of Ray anyway. He's a good copper, he's just a twat. You've got bigger things to worry about, like writing a report saying I'm ace and the MIT is rubbish and that they should give me more resources and money, and a daily bottle of Zinfandel."
McAvoy rubs a hand over his face as he gives in to a grin. "I'll do my best."
"I'm not even going to ask you how it's looking," she says, clearly asking. "We're expensive, aren't we? The unit? You know we'll be the first to go in the budget cuts, no matter what the new chairman is saying. And we've had a couple of high-profile fuckups these last few days."
"You weren't to blame," he says and means.
She looks through the glass at the rain-lashed car park. Manages a smile.
"Thanks, Aector, but it wasn't my finest hour. Shouldn't have committed to the raid without being one hundred percent. I fought my corner, mind. Told them the pressure on us for results is going to lead to these balls-ups. They don't get it, though. They're too far removed. They don't know much more than they read in the papers, and, according to the tabloids, Hull's going to hell."
"It's never been paradise," says McAvoy, trying to make her smile. "They exaggerate. That's what they do. It's a power struggle over a drug that will be legal in a few years. People are flexing their muscles. Somebody's trying to prove they're a big man, and people are getting hurt."
"We could have had them," says Pharaoh despairingly. "Could have wrapped it up."
"You don't really think these thugs are in charge, do you? They're just muscle. We catch them, there's still whoever is giving them their orders to worry about. And we know they must be serious. The Vietnamese don't play nicely. Whoever's taken over their operation must be one heck of a player."
"That's not for us to think about," says Pharaoh moodily. "They may just be the hired thugs, but they're the ones we want. They're torturers. Now they're murderers. In the public's opinion, they're the ones we want off the streets—not the ones who have a fortune in the bank from farming bloody cannabis. It's not a drugs operation anymore. Not really. It all has to be accounted for, Aector. Has to be on the right fucking spreadsheet . . ."
McAvoy nods. Realizes just what a balancing act it is to chase criminals without offending the other specialist units. Some of this investigation should be in the hands of the Drugs Squad. But Pharaoh, for now at least, is keeping their hands off it—much to the dismay of Detective Superintendent Adrian Russell, who has made his displeasure known. He's good at displeasure. He causes a lot of it.
"They could have killed us," she says. "They could have wedged the doors shut and burned us to death."
"Don't think like that, guv."
"I'm not being morbid, Aector," she says. "I'm confused. These bastards don't think twice about nailing people's hands to their kneecaps, and they get a chance to cook a van full of cops and don't take it?"
McAvoy considers. "Maybe they didn't want that level of interest. Maybe it was a warning. There would have been uproar if anybody had been badly hurt."
Pharaoh shrugs again. "We've got a few leads to follow up, that's the main thing. The car park at Peter Pang's picked up their registration plate. Reported stolen from a high-class car showroom in Doncaster the day of the raid."
"Donny?"
"Yeah, apparently people in South Yorkshire can afford fifty-thousand-pound cars. Who'd have thought?"
"Anything else?"
"Yeah, there's a partial fingerprint recovered from the glass bottle they threw at us. Belongs to a bloke with a record long as your arm. Your arm, not mine. GBH. Embezzlement. Did years for armed robbery. Real piece of work."
"Name?"
"Alan Rourke."
"Doesn't ring a bell . . ."
"Bad sort," she says. "Connected to some real villains . . . Aector, look sharp!"
She jumps off his desk and stands up straight, dragging him to his feet by his collar and bodily spinning him to the door. Peter Tressider, the chairman of the Police Authority, has just entered the room and is waving a hand in the general direction of Detective Chief Superintendent Davey, who is trailing behind him with a look of uncontrolled panic on his face.
Tressider looks around the room, completely ignoring whatever it is Davey is trying to say in his ear. He spots McAvoy, and his mouth opens in pleasant surprise. He crosses the room with his arms outstretched, and for a second McAvoy fears he is about to be taken into a bear hug.
"Sergeant," he says warmly, and pumps McAvoy's hand with a vast, fleshy paw. "Good to see you again. Heard about last night's excitement. Fun and games, eh? My word, I know we gave you a grilling, but we didn't expect you to go and risk your life over statistics. Still, sometimes you have to rush in headlong, eh? I think I read that in a book about the samurai, now I think on it. Excellent read. I'll lend you it. You ever read _The Art of War_? Fascinating stuff. Think I'll give it another once-over if I do make it to Westminster, eh? No shortage of bloody enemies there!"
McAvoy has to stop himself from physically recoiling in the face of the big, bearded politician's enthusiasm. "It was a difficult operation, sir, but there are plenty of positives to take . . ."
Tressider waves a hand. It appears to be a habit of his. McAvoy wonders what political commentators will make of it should he get to the House of Commons. Whether they will applaud his earthy brusqueness or dismiss him as an impatient dinosaur.
The chairman turns to Pharaoh. "You must be the boss, yes? Pharaoh?"
Pharaoh smiles. Takes his hand. Manages not to wince as her palm is squeezed. "Afraid so, sir."
"Delighted you're still with us. Delighted! No shortage of people who would have taken a month off for stress, and yet here you are! Back at work and ready to lead. Impressive. Inspiring!"
Pharaoh gives a little half-laugh, unsure how to deal with this onslaught of optimism.
"I just want to catch the bastards," she says, deciding to just be herself. "Hope the powers that be let me do that."
Tressider gives a nod of understanding. Taps his nose with a plump finger. "We never spoke," he says, winking. "This conversation never happened. But don't you worry. I like your style."
There is a moment's silence. "Can we help you with something, sir?"
Tressider gives them both a warm smile. "No, no, was just here for another meeting and thought I would show my face. Wanted to check you were all fit and healthy and raring to go. I hope I can trust you to keep me informed, and you can trust me to keep my nose out, yes?"
Both officers smile, and he shakes their hands again, even more vigorously than before. McAvoy glances over his shoulder at where Detective Chief Superintendent Davey is a picture of bewildered misery. As he looks back, he sees that the chairman's eyes have swiveled toward McAvoy's computer screen.
"Interesting?" he asks, nodding at the screen. "A lead?"
McAvoy finds himself doing an odd thing with his mouth. Licks his lips. Twitches. Colors instantly. Remembers why he never plays poker.
"Just something I thought was worth checking out . . ."
"Show me."
He clicks on the story he had tried to cover up when he sensed Pharaoh's approach. The _Hull Daily Mail_ article on the death of Simon Appleyard. Pharaoh, as in the dark as Tressider, reads halfway down and turns to McAvoy. He meets her eye purely through fear that, were he to look anywhere else, his view may take in her cleavage.
"Pet project?" she whispers.
McAvoy opens his mouth. Closes it again. Hangs his head. "It's just something that doesn't feel right."
He had found the story during a halfhearted Google search on his midmorning break. It made him sad. The telephone's owner has an identity now. McAvoy has been reading the words of a real person. A loving, gregarious, confused young man who wrapped a cord around his neck and squeezed his own life away.
"The way he writes . . ."
McAvoy struggles to put it into words. Cannot explain why, from the outset, he has felt such unease.
"Writes?" asks Tressider.
"There's something about his death that troubles me," he says, and he feels sweat prickling on his forehead. Can picture two men standing by the River Hull in the pouring rain. Can see the phone sitting on the mud. Can see himself slithering down the dock wall to pick up the device that has led him here. Wonders what the fuck he was thinking and how badly this will end.
"Follow your gut, my boy," says Tressider, and he appears to lose interest. "You can't go wrong if you do that. Anyway, it's been a pleasure. Mrs. Pharaoh, I look forward to our next meeting, and, Aector—did I get that right?—I'm delighted to hear you're unharmed. Do keep me posted."
He turns away. Gathers up Davey the way a tornado snatches cattle, and bangs out of the door like a benign storm.
"I'm thinking about hurting you," says Pharaoh eventually. "I'm not going to do it, but there will be a part of me this afternoon that will regret not punching you in the head, I hope you realize that."
McAvoy looks down. Keeps quiet.
"Were you going to tell me?" she asks.
"There was nothing to tell. Not really."
She gives a frustrated sigh. "Like we're not busy enough."
"I can do it on my own time."
"You haven't got any time, Aector. You're up to your eyes in drug dealers and babies. It was only yesterday that bugger told us he wanted violent crime stats down inside a quarter. And now you want to turn a suicide into a murder investigation?"
"It just needs a bit of a dig, guv . . ."
Pharaoh throws her hands up. Looks at the closed door, as if Tressider were still there. She shrugs. Appears to reach a decision.
"Fuck it. I'm only on peripherals. And the boss said to follow your gut. Give it a dig. And if you balls-up the crime statistics, it was all Colin Ray's fault. Deal?"
McAvoy smiles.
"Deal."
• • •
THE ELEMENTS have made Hull a city of gargoyles. McAvoy has never seen so many faces locked in grimaces. Attractive office workers snarl into the driving wind. Shoppers popping to Marks & Spencer to pick up their evening microwave meal gurn angrily at the rain. The entire Old Town seems to be wincing.
It is just gone lunchtime. Whitefriargate, here on the periphery of the city center's nucleus, should be bustling with shoppers. Instead, the weather has forced everybody indoors. McAvoy has the broad, attractive shopping street to himself. He's one of the few people who bother to look up. Lets his eyes roam past the first floor and enjoys the architecture: the handsome old mercantile palaces that lead down to the Museums Quarter and the waterfront. Enjoys the ornate frontage of the bank on the corner of Parliament Street. Lets himself daydream a little. To imagine this street when Hull was living its best days rather than remembering them.
He's glad he walked. Likes to feel the city beneath his feet. Wishes the destination were farther away.
The files from the Simon Appleyard case have not yet been electronically input, and the paper copies are still at the Coroner's Court. He is enjoying the walk. Looking forward to an hour or two in a quiet room, immersing himself in the final moments of a prematurely ended life.
His booted feet leave large prints on cobbles that seem to have been dyed the color of varnished clay by the incessant rain.
The smell of spit-roast chicken assails him from the butcher's shop. He realizes he has not eaten since yesterday. Makes a mental note to admit this to Roisin when he gets home, so as not to be accused of hiding things from her, and then wonders if it would be kinder to conceal it so she does not feel pressurized into making him something as soon as he walks in. Wonders if she will think that he is trying to make her feel guilty. Whether it would not just be easier to tell her that last night they went to the dance class so he could find out why somebody had buried a mobile phone.
He wonders if this is how other people feel. Wishes, just once, he had a clue how to live.
He continues to salsa in his mind as he makes his way to the attractively named Land of Green Ginger. The little side street is home to two pubs, a legal office, a beauty salon, and a courtroom, though not all of these facilities are mentioned by the leasing agents who try to flog the seemingly endless apartment developments springing up in this part of the Old Town.
He prepares his speech in his head. Wonders if they could help him. Just needs a quick favor. McAvoy knows a couple of the ladies at the Coroner's Court and feels instantly embarrassed when he acknowledges that they will probably cooperate for no other reason than they like him.
He can feel the wind and rain doing him good. Breathes deeply. Enjoys the scent of distant sea spray and motor oil. Inhales the greasy aromas of the butcher's, the sandwich shops. Sucks in a lungful of the ever-present cloud of cigarette smoke that hangs outside the dark blue door of the amusements as the punters spend their slot-machine winnings on tobacco and five-for-a-pound lighters.
He wonders if he loves this city or wishes it dead.
The phone in his inside pocket rings and he ducks into the doorway of a trendy new clothes shop to take the call.
"DS McAvoy, Serious and Organized."
"Sergeant Arthurs, Silly and Slapdash."
McAvoy moves to one side to allow two teenage girls access to the shop. Despite the weather, they have bare legs beneath short, pleated skirts, and their hooded jumpers are soaked through. He wants to know whether they should be in school. Whether they are okay. How old they are. What they want. Whether they're safe . . .
"Thanks for calling me back, Sergeant," he says, forcing himself to turn his eyes back to the street. "It's about an incident you attended last November."
"Yep, you said in your message," Arthurs says brightly. McAvoy has not met the uniformed officer, but from his voice fancies he is in early middle age and almost certainly a dad.
"Simon Appleyard."
"I'm sorry, can you give me a little more . . ."
"The hanging at Springfield Court. You attended."
"Oh, right," says Sergeant Arthurs, recollection dawning. "Yes, sad one, that. The landlord found him, I think. Inquest was just a few weeks back. Open verdict, wasn't it?"
McAvoy nods, and realizes the other man cannot see it. "Yes," he says. "What can you tell me?"
"Well, it's not that exciting," says the other policeman. "The landlord needed to get in to read the meter. Couldn't get any answer. Let himself in, and there he was. Dead in the kitchen. Slumped forward on his knees. I think the rope was tied to a knife rack on the wall. He'd been there a few days. Me and Shelley Dalston attended. WPC Dalston. You know her?"
"No," he says, not wanting to be sidetracked. "I'm on my way to look at your report right now, actually. Can you give me the abridged version?"
Sergeant Arthurs gives a little laugh. "The highlights? Well, he was naked, there's a start for you. Oh, and he was covered in baby oil. Proper covered in the stuff. What else do you want to know? There was no note, I can tell you that much."
McAvoy looks up at the sky. The clouds remind him of curbside snow; bulging and dirty white.
"What's in the forensics report?"
"Well, the whole thing went up to CID after we did our bit, and they gave it about five minutes of their valuable time. When the inquest was coming up, I asked a couple of questions on what had come of it all. Pathologist said cause of death was strangulation. Earned her money that day, eh?"
Across the road, a smartly dressed man appears from the glass-fronted doorway of a legal office. He pushes open a large gold umbrella that almost knocks over a middle-aged woman who is struggling with something heavy in an Argos bag. McAvoy wills the man to apologize. To admit his mistake and help her. To be good. Watches the man walk away.
"Was there anything on his computer, do you know? Was it sent to the tech unit?"
"No computer as far as I can recall," says Arthurs. "That's almost a story in itself these days. I don't think he had much in the way of family, though. I seem to recall he was into dancing, if that's any good to you. What exactly is it you're looking into?"
McAvoy had hoped the question would not be asked, but was prepared for it. "Another police service has been in touch," he lies. "CID in Berkshire have got two apparent suicides. There's a chance they were logging on to a website where people could share tips on how to bump yourself off."
"And they think our lad might have done that? No, like I say, no laptop."
"No phone?"
"Not that we could see. His next of kin lives a few miles away, so it was another force that broke the news. His auntie came to ID him. Lovely lady . . ."
"Did you ask her about his phone?"
There is a pause, as if the other man were thinking. "I'm not . . . oh, hang on, yes. Yes, when we picked her up she said she had been ringing him for the few days before he was found. Hadn't picked up. Hadn't even texted back . . . Yeah, I guess that means he must have had a phone."
"Did you put that in your report? Did CID know that?"
"It will have been in my pocketbook, that's for sure. But, no, I think I'd written up the incident report before I spoke to his auntie."
McAvoy falls silent.
"The other lads gay as well, were they?" asks Sergeant Arthurs, his tone jocular.
"I'm sorry?"
"The other suicides. Gay as well?"
"How do you know he was gay?"
The man laughs. "I'm not being a twat, mate. He was gay, that's all. You could kind of tell."
McAvoy feels himself get hot inside his damp clothes. "And how does one 'just tell'?"
"Well, I don't know many straight lads with peacock feathers tattooed on their backs, do you?"
McAvoy is silent. Swallows. "Well, no."
"Anyway, it said so in the paper, didn't it? At the inquest."
McAvoy scratches his head. Watches the lady with the heavy shopping readjust herself, leaning the burden on a metal post.
She puts her weight on a loose paving slab and dirty water splashes up her leg. Her face twitches. Tears begin to fall.
He asks his next question straight out. "You think it was suicide?"
Sergeant Arthurs blows out a noise that suggests deep thought. "I think so," he says eventually. "I didn't at first. Thought he was into that autoerotic stuff. Maybe a game went wrong. But, no, that was a lonely life. Was just a shabby existence. I think he bumped himself off."
McAvoy thanks the other man for his time. Is about to hang up, already hoping the actual case file will be more useful.
"Oh, hang on," says Arthurs. "There was one thing surprised me when I flicked through the file. There was no mention of the bruise."
McAvoy stops. "What bruise?"
"On his back," says Arthurs. "In among all the peacock feathers and the bloody baby oil."
ELECTRIC FIRE, lit to the third bar.
Glowing red: hot against his cheek.
No other light in this stuffy, airless room.
McAvoy, squinting, struggling to see the words he is scribbling in his notebook with a pen that tears holes in the damp pages.
"Do you think somebody killed him?"
The question comes from nowhere, and is asked in a voice that sounds like an inquiry into whether he would like another piece of cake.
McAvoy doesn't raise his head. He doesn't know what facial expression to pull. He does not know the answer. Does he think Simon Appleyard was killed? Her question forces him to consider his thoughts. He realizes he has been behaving as such from the start. Acknowledges that, in his heart, he already feels he is hunting a killer.
Wonders why.
He is a procedural, methodical detective, given to only occasional flashes of instinct and hunch. He has nothing on which to base his feeling that a life has been taken before its time.
"I think there are questions to be answered," he says, and hopes she will leave it at that. He is habitually beset by feelings of guilt and uncertainty, unsure of who or how to be. Here, now, in this tiny two-bedroom terraced house with its unmowed front lawn and unfashionable wallpaper, its impersonal prints and halfhearted tidiness, he feels he does not deserve to be treated so warmly. He fears that he is, to all extents and purposes, trying to make her cry. He needs her to harbor fears and doubts. Wants her to tell him to dig and claw and kick until he gets her answers. Needs to feel that he is prying into her nephew's death for more noble reasons than his own macabre curiosity.
He looks at her. Nods to show the tea is lovely, and commits her image to memory.
Carrie Ford was probably very pretty twenty years and five thousand cheeseburgers ago. Beneath a hefty layer of fat, McAvoy can make out what was once a willowy, elegant frame. Her green eyes and quick smile are anachronisms in a doughy, makeup-free face that sits atop a careless, dumpy frame.
She is dressed in her supermarket uniform. White polyester dress and green tabard, concealed beneath a plus-sized denim jacket. She looks her age. She looks as though six months of grief have carved a lifetime of wrinkles into her skin.
She had been on her way to work when he knocked on the frosted-glass door.
"Thought you were the taxi," she says again, as she pours McAvoy a second cup of tea. Then, for the third time, "Work can wait, eh?"
She is a nice woman. She lives alone in this two-bedroom semi-detached, a short drive to work and ten minutes from where her nephew died bug-eyed and helpless on a blue cord carpet.
"That's our Simon," she says, waving in the direction of the mantelpiece.
There are half a dozen birthday cards obscuring photo frames and ornaments, and McAvoy cannot see whom she means. She crosses to the fireplace. Retrieves a picture in a cheap frame. Hands it to McAvoy with a smile.
"That's from the class. Our class. The line dancing. See the smile on his face? That's when he was happiest. Performing. Helping people learn. Getting other people excited."
In the poor light, McAvoy has to angle the picture as though trying to cast a shadow on the ceiling with the glass. Squinting, he looks at the picture of the lean, dark-haired lad. He is smiling broadly for the camera and the image, though far from flattering, is a happy one. Simon's fringe is damp with exertion and flopping over his eyes, and there is a sheen of sweat across his neck and chest where it slopes into a sleeveless sweatshirt, slashed provocatively at the collar. McAvoy angles the picture again. Sees his own face in the reflection. Hurriedly tilts it back.
"You had no indication?" he asks. "Never felt he was depressed?"
Mrs. Ford sits down on the high-backed armchair.
McAvoy, on the far corner of the matching two-seater sofa, wishes he had taken her up on her offer to hang up his coat. He is too hot, his damp clothes beginning to steam in the heat from the three-bar electric fire she had turned on instinctively as she led him into the tiny living room.
"He had his ups and downs," she says, and looks at the picture in McAvoy's hands. Wordlessly, he hands it over to her, and is touched by the tenderness with which she looks at the photograph.
"There was mention at the inquest of difficulties with his father. With his sexuality . . ."
Mrs. Ford pulls a face. Holds the photograph to her chest. "Only person who had difficulties with his sexuality was his dad," she says. "My brother, before you ask. Always was an arsehole."
"He didn't have much to do with Simon as a child?"
"Neither of them did. Mum or Dad."
"That can't have been easy," he says. "Nobody to talk to . . ."
Mrs. Ford waves her hands, dismissively. "Simon knew what he was from being a kid," she says. "Never bothered him. He was outgoing, you know? Full of life. He never hid who or what he was. And his dad had no right to even comment."
"But he did comment, yes?"
She sighs. Looks again at the birthday cards on the mantelpiece. "They're from my class," she says. "Line dancing. Not many left now. Not the same without Simon. He was the attraction, I know that. So funny. Would have been a great DJ. Was him that got the crowds in. They didn't come to learn how to line dance."
"I hear he was quite a dancer, too."
"Simon could do anything," she says, and her voice sounds far away.
"Everything I've heard about him suggests he was a lovely lad," says McAvoy. Such lines often help.
"I raised him, you know," she says, appearing to snap back into the present. "His mum wanted nothing to do with him, and his dad, well, he was bloody hopeless. He lived with me most of his childhood. Never had kids of my own. Never married, though don't be thinking I'm some spinster. There have been fellas. Simon was the only constant thing in my life."
McAvoy realizes he is talking to a woman who was all but a mother to the dead man. Tries to understand how that must feel.
"He grew up here?"
"No, love. I only moved in here a few years back. Always lived in Anlaby, though. Him and me have lived in nigh-on every flat in the village. Gypsies, I think we are, though we never move far . . ."
McAvoy sits back in the chair and lets her talk. Tries to craft a person from her memories.
"His dad would come back now and then," she says scornfully. "He'd ring, when he remembered. Would sometimes send a few quid. But there was no closeness there. His dad wanted a lad he could take to the football and down to the pub. Simon wanted to dance and write poetry. It was awful, watching him realize that everything about him was grotesque in the eyes of his own father. He was thirteen the first time his dad called him a poof. Can you imagine?"
McAvoy closes his eyes. Takes a sip of his cold tea.
"At the inquest there was talk of some text messages? An argument with his dad?"
Mrs. Ford puts her hands together in her lap and her leg begins to jiggle. She is either nervous or trying not to let her emotions get the better of her.
"Who hasn't done that? Who hasn't had a couple of drinks and sent some text messages telling the world you're pissed off? That's life. It is these days, any road."
McAvoy nods. "He texted a lot?"
"Fiend for it," laughs Mrs. Ford. "Hundreds of them a day if you were daft enough to reply more than once. Him and Suzie almost starved themselves to death trying to get enough cash to buy one of those fancy phones. He was still saving when it happened. Silly lad. Should have spent it before he did it, don't you think? Go out on a high."
McAvoy makes a show of looking through his notebook. "Suzie?"
Mrs. Ford pinches the bridge of her nose, as if suddenly beset by sinus pain. She looks down at her name badge. Reads it, as if looking for proof of who she is.
"Thick as thieves, them two." She smiles. "Wish I could tell you her last name. Awful, isn't it, not knowing, but you don't ask these days, do you? Lovely girl. Mad as a box of frogs, of course. The clothes! Looks like she's in fancy dress half the time, but would do anything for my Simon. Inseparable, they were." She stops. Appears to be struggling with something. "I don't think they were bad influences on each other or anything. Just brought out the devil in each other. Suzie split up with her fella just before she and Simon became friends. Simon was a good listener. They had some silly idea for getting back at her ex. Made some silly mistakes. But they learned from them . . ."
McAvoy has had time for only the quickest of glances through the case file and has found no mention of any female friend. "School pal?"
"No, they met on a writing course," she says, with a touch of pride. "That's what he would have liked to do, I think. Be a writer. Or a poet, in a different time. He had such a lovely gift with words. Even when he was texting he'd try and make it sound pretty."
Placing his teacup down on the floor beside the sofa, McAvoy leans forward. "Mrs. Ford, it doesn't sound as though you think for a moment that he took his own life."
She grimaces, and then breathes out a sigh that appears to have come from the heart. "What do any of us really know about anybody else?" she asks. "No, I wouldn't have thought of Simon as being that sort of person. He didn't smile all day like a clown, but he enjoyed enough of life to make the bad stuff okay. That's what it's about, isn't it? Life."
McAvoy looks away so he doesn't have to answer.
"Mrs. Ford, I need to ask you about Simon's personal life."
"His sex life?" She smiles, warm and friendly. "We'll both end up blushing."
"He was promiscuous?"
"He was young."
"He had a lot of sexual partners?"
"He enjoyed himself."
McAvoy looks at his notebook. "Mrs. Ford, I need a little bit more. We have evidence that suggests Simon had made contact with a new sexual partner some time before he died."
She shrugs. Looks at the photograph in her hands. "Sergeant, I was his auntie. More of a mum than his real one. We had fun together and giggled more than most, but he didn't think to ring me every time some new bloke rolled off him."
McAvoy instinctively pulls a face. She spots it. He tries to cover it up with a cough. Wants her to know it was her unnecessary crudeness and not the act that she described which caused him to shudder.
"So you didn't meet any boyfriends?"
"I met 'friends,'" she says, smoothing down nonexistent creases in her uniform. "Sometimes he'd get picked up from the church hall by a lad, or he'd tell me he'd been to see a play or for a drink or something and mention some bloke or other, but I didn't like to pry."
"You indicated he was promiscuous . . . ," he says cautiously.
She sits forward in her chair. It appears she is about to stand up and put the photo back on the mantelpiece, but she seems to decide against it and stays where she is.
"It's just things that Suzie and he said to each other," she explains, waving a hand, vaguely. "Maybe they were teasing each other, I don't know. They were like two kids. He'd make her blush, telling me to ask her what she'd been up to some night or another. She'd crack jokes about Simon being too worn out to give his all at some classes. Just the usual." She glances at her watch. "He wasn't shy, I know that much."
McAvoy wonders if there is any more to be gained from this. Whether there ever was. He looks at his notes.
"He had tattoos, I'm told . . ."
"Oh, yes," she says brightly. "Goodness, they were lovely. Got his first when he was just turned sixteen. Some lyric from a band he liked. Got a real taste for it after that."
"I'm told his back was a work of art."
She smiles. "He'd have loved to hear you say that. They were in a magazine, you know. An advert in that glossy mag. I saw it in a doctor's waiting room, not long back. Simon would have been over the moon if he knew. I wasn't sure about it when he told me. You not being from round here, you'll not know, but peacock feathers are awful bad luck."
"They are where I come from as well."
"Really? I thought it was a Hull thing."
"No, I think it's the same everywhere."
Mrs. Ford sticks out her lower lip as though mulling over whether this matters. Decides it doesn't. "Either way, I wasn't sure about them, but he was mad keen. He was going through his more, erm . . ." She searches for the word, before finishing with "flamboyant phase."
"And this was?"
"Not more than a year ago. He and Suzie got their tattoos done the same day. They were both walking like they had sunburn for a few days afterward, but when he showed me, I thought it looked lovely."
"Why peacock feathers?"
"It was something he'd read, I think," she says, looking at her watch again with a sudden expression that suggests work might not be able to wait much longer. "Seemed to like peacocks for a while. It suited him. Just a phase, I suppose."
McAvoy drums his fingers on his notepad. Taps his pen between his teeth. Wonders if he has learned anything. Turns his face away from the three-bar heater and presses the backs of his hands to his too-warm cheeks.
"Do you think somebody may have killed him?"
He asks her the question outright, as she had asked him.
She does not answer for a time. Just looks at Simon's picture and strokes the glass.
"I don't know," she says eventually. "In my heart of hearts, I suppose I'm frightened he died doing something a bit . . ."
"Yes?"
". . . a bit mucky. I've read what people are into these days. I hope it wasn't that."
He wants her to say it. Wants her to fear he was killed. Wants to turn his sense of disquiet into a sterile, official, evidence-led investigation, and needs her pain to make it so.
"Mrs. Ford, do you have suspicions about third-party involvement?"
McAvoy is aware he's leaning forward, perhaps pushing too hard.
She meets his eyes.
"I'm not sure if I want to know," she says, dropping her head. "I can cope with him dead. It hurts, but I know how to grieve. If I thought somebody had hurt him . . ."
She looks back up.
Her eyes are wet but there is no danger of tears.
"'Dead' sounds so much more peaceful than 'killed,' don't you think?"
McAvoy doesn't answer.
He is certain he will soon destroy her peace.
• • •
SUZIE IS SITTING at the kitchen table in the flat she rents above the florist on Anlaby Road. A fine rain mists the single-glazed window. The raffia blind and homemade curtains billow inward as the breeze finds the cracks in the paintwork around the frame.
It is getting dark beyond the glass. Dark enough to switch the light on, should she find the inclination. The gaudy yellow of the streetlights lends the view from the window an industrial, unnatural quality.
Sweet tea, that's what they swear by in the books. Good for shock. Calms your nerves. Better than Prozac, apparently. "Lying tossers," she says, under her breath. Suzie has drunk pints of the stuff since last night. Has put in enough sugar to give an elephant diabetes. And yet she still can't hold the mug steady as she raises it to her lips. She looks again at the _Hull Daily Mail_ website. Three paragraphs and a phone number. Headline in blue, text in white.
MAN HURT IN EAST YORKSHIRE REST STOP
A 44-year-old man is in intensive care after being involved in a suspected hit-and-run at an East Yorkshire beauty spot.
The man, visiting the area on business and said to be from West Yorkshire, was found by motorists at Coniston rest stop on the road to Bridlington late on Tuesday night.
Detectives are keen to talk to the person who made a 999 call from a nearby telephone box shortly after the incident. Anyone with information should call Humberside Police on 0845 6060222, or Crimestoppers, anonymously, on 0800 555 111.
It doesn't seem enough, somehow, but Suzie is pleased there is no more. Were it on the front page, she thinks she would probably buckle. She would call the number. She would admit it was her. She would tell them how she gets her kicks.
She stares at the laptop screen for an age. Reads the story over and over. Eventually the screensaver comes on. Herself and Simon, dressed for a seventies night at the Silhouette nightclub. Orange Afros, silver flares, and huge collars. Big smiles.
Her head drops to her hands. She cried all she needed to when she got home last night and only lasted an hour at work this morning before telling them she felt sick and had to go home. She cried her last when she read the _Hull Daily Mail_ story the first time. She has no tears left now.
She pushes the tea away. Crosses to the worktop and takes a packet of chocolate biscuits from a cupboard. Tears it open and stuffs two into her mouth as she leans back against the draining board. Wishes he were here.
She makes her way back to the table. She does not know what to do.
Flicks, distractedly, to the other stories on the site. There is a special article by the paper's crime reporter on the future of Humberside Police. It predicts swingeing cuts and redundancies. Suzie does not know what the word _swingeing_ means, but it doesn't sound good. She reads a quote from the new chairman of the authority. He promises to secure all the funding he can during his tenure at the top, but warns that there are tough times ahead. The article says he is tipped to be the next MP for Haltemprice and Howden. Suzie wonders if people ever read this kind of shit for fun.
She opens another story. Skims through the details of a nasty-sounding attack on a man down Hessle Road. Anonymous thirty-three-year-old beaten unconscious on his own doorstep. Neighbors claiming that he had been growing marijuana to help him deal with the pain of a motorcycle accident. Witnesses sought. Descriptions given of two heavyset white men and a smaller, younger man, who left the scene in a four-by-four. Said to be the latest in a worrying explosion of drug-related violent incidents, and has been linked to the discovery of a body in the city's Dagger Lane . . .
Suzie loses interest. Turns away from the screen.
Next to the laptop, her mobile sits dead and lifeless. She has not had the courage to turn it on.
_Was it him?_
The question is killing her. Was the man who insisted she perform for his pleasure merely setting her up? Was that his fantasy? To see her crushed beneath the weight of a half-naked stranger amid the sound of grinding metal and frothing blood? Or was it a mere accident? Was the man even there when it happened? Could he have seen something? Could he be a witness? Worse still, she thinks, what if he has already contacted the authorities and told them about her?
She looks at the door, imagining two uniformed policemen banging on it with angry fists, leading her away in handcuffs, her secrets pored over by all.
Suzie takes another bite of biscuit. Looks at the story one more time. At the newspaper's masthead. She remembers how the paper had dealt with Simon's death.
For no other reason than to change the picture on the screen, she types the name Simon Appleyard into the newspaper's search function.
Two stories.
One offering a name and age of the young man found dead in an Anlaby flat two days before.
The other from the inquest; held three weeks ago and too painful to attend.
A Hull woman has paid tribute to her "gregarious and loving" nephew, who took his own life after sinking into a depression over his sexuality.
Hull Coroner Martin Duffy heard yesterday how 24-year-old Simon Appleyard was found dead at his Springfield Court flat in late November last year.
He had tied a length of rope around his neck and to a wall-mounted knife rack in his kitchen. He had been dead for four days when his landlord found his body.
Simon, who worked for a car valeting service in Wincolmlee, had won awards at a national level for line dancing and ran a club in Anlaby.
His aunt, Carrie Ford, of Saffron Close in Willerby, told the inquest: "He had times when he was down, like we all do, but it came as a shock to all who knew and loved him that he would be so unhappy. He had so much to live for. He was always so full of life and happiness. He was such a character, really outgoing and gregarious, with such a loving nature. I don't know what I'll do without him around to make me laugh."
The court heard Mr. Appleyard had been treated for depression as a teenager and had twice been prescribed antidepressants. As a teenager, he had had difficulty accepting his sexuality and clashed with his parents when he came out as gay at the age of nineteen.
Although no note was found, text messages shown to the court by Mr. Appleyard's father show he was feeling unloved and suicidal in the weeks before his death.
Kevin Appleyard, who traveled to the inquest from his home in Derbyshire, said, "He said he was a disappointment. Said we were better off without him. Said I wouldn't accept who he was. I just wish I could hold him one more time . . ."
Suzie turns away from the screen, as disgusted now as when she had first read the stream of lies.
She feels sick that Simon's dad is even mentioned in the article. Disgusted that the man who battered and bullied her friend throughout his childhood would dare to suggest he had ever even hugged his son. Wonders whether, before he gave it to the coroner, Kevin Appleyard had the foresight to change his son's name in his phone to "Simon" from "poof."
An e-mail pops up in the corner of the screen. Instinctively she clicks on it.
Still on for Saturday? Am already getting in the swing of things? Xx
The message is from a couple she knows as J & J. The male half of the relationship is an unattractive blond lad with Leeds United tattoos and poor spelling. The female J is a dumpy brunette with pierced nipples and glasses, who in Suzie's mind at least always seems to be wearing Band-Aids on her heels. They met at a party perhaps a year before. They were quite funny and lived near enough to Simon and her to make them worth being friendly to. The strategy had saved them a fortune in fuel.
She wonders if she should ignore the message. Concentrate on her worries and woes. Manages to hold out for a full ten seconds before deciding she has nothing better to do than drop them a line back in return.
Hey you. Not sure about this time. Not feeling too good. Think I may bring the mood down. XX
She presses SEND.
Gives it a minute, drumming her fingers on the table. Has another biscuit. Starts to reread the _Hull Daily Mail_ report of the hit-and-run, then gets bored with herself and closes it down. She stares at the screen for a while as if debating, then clicks on one of the FAVORITES on her Internet browser. It takes her to the Xanadu homepage.
As ever, she feels a rush of excitement. In the picture at the center of the screen is a group of middle-aged men and women. They have blurred faces, thoroughly average bodies, and are unashamedly nude. With wide grins, they are all giving a thumbs-up to the camera.
Behind them, the flat, green fields of Lincolnshire make the image look like a postcard. It just needs a slogan.
"We're fat and getting plenty," she had once suggested to Simon.
Or, pointing at the thumbnail image of a sixty-one-year-old woman in a love swing and gimp mask, "Does my bum look big in this?"
Go on, Blossoms. It's not a party without you. XX
Suzie can't help but smile. There's a birthday bash at Xanadu this Saturday night. The owner, a tax inspector with matronly boobs and a liking for nipple clamps, is turning fifty. Christine and her husband, Big Dunc, have run Xanadu for eight years. As swinging clubs go, it's pretty plush. Set in a dozen acres, two miles from the nearest house, it's Lincolnshire's best-kept secret. Christine and Big Dunc hold three get-togethers a week. One for couples, one for singles, and one where anything goes.
She flicks through the website, pulling faces and wondering what else she will do this Saturday if she decides not to go along for her slice of cake and free glass of champagne. She's lonely. The thought makes her close her eyes. She has never really acknowledged it before. Since Simon, she has been lost.
Suzie licks her lips. Prepares to type.
A thought occurs. Had she told him? Had she mentioned the party and the location? She has no way of checking without switching on her phone and scrolling through her messages, and she does not want to do that, so decides that she did not.
_But last night . . ._
It comes back again. The sound of the cars colliding. The crunch of bone. The splat of body hitting wet concrete.
_Don't go, Suzie_ , she tells herself. It's not the same anymore.
Would it be so bad? Were she ever asked to justify herself, Suzie would try and explain that there is a big difference between a swinging party and an orgy. The last one she attended at Christine's place, she had spent most of the evening in the kitchen, helping the host make snacks and flicking through a copy of _Good Food_. She spent an hour nattering over a glass of wine with a nice woman from Dorset, talking about the best ways to iron a plastic bra.
Her fingers twitch above the keyboard. The story of the hit-and-run is open at the bottom of her screen. The report of Simon's inquest to the right. But the page that holds her attention offers freedom and fun; a Saturday night of being young and hedonistic. A night of escaping from all the crushing reality.
_You nearly got killed last night_ , she tells herself.
The reply, in her mind, comes in Simon's voice.
_All the more reason to live._
She nods. Makes up her mind.
What time are you leaving? XX
"IS IT TWO TEES or two els?" asks Daniells, his tongue wedged in his lower lip.
Pharaoh turns her head, slowly, a snarl already forming on her face.
"What?"
"Rottweiler," he says, gesturing at his open pad with his Biro. "Double _t_ or double _l?_ "
For a moment, she doesn't answer. Then she leans across, plucks the pen from between his forefinger and thumb, and with as much force as she can muster in the cramped confines of the little sports car, throws it at the side of his head. It bounces off and lands in the footwell.
Daniells pulls a face. Closes his pad and puts it back in his inside pocket. Then he mimes zipping and locking his mouth.
Pharaoh turns back to the window, where the larger of the two guard dogs continues to snarl, occasionally leaping up with an extra riot of barking to leave spit, steam, and paw prints on the glass.
She would like to tell herself it had been a strategic and orderly retreat, but in truth she and Daniells had run for the car like Olympic sprinters when the dogs appeared and bared their fangs. It had been the right thing to do. The dogs had lowered their heads and torn after them in a gnashing fury, and Daniells had lost the tail end of his coat when he slammed the passenger door.
"This is why we should have guns," says Daniells, who has lost none of his cheer and appears to still be enjoying himself, despite very nearly having his leg chewed off by an angry Rottweiler.
"What?"
"Guns," he repeats.
She turns back to him. "You'd shoot the dogs?"
He makes a gun of forefinger and thumb and mimes shooting the muscled, black-and-brown beast that is licking his window. He purses his lips and blows on the tip of his index finger, as if it were smoking. "Would be nice to have the option," he says.
Pharaoh says nothing for a moment. Just stares. It's a look that makes him give an odd, nervous half grin. "Get out," she says.
"Pardon, guv?"
"You're safer out there than in here."
They sit in silence, broken only by the sound of the growling dogs and the thunder of rain on the glass.
"We could go," says Daniells. "Come back with the dog squad."
Pharaoh has been thinking the same thing, but the fact that Daniells has offered advice makes her bloody-minded. "We're going nowhere," she says.
The two-seater is parked on the driveway of Alan Rourke's bungalow. It's an attractive three-bedroom property with imposing double-glazed bay windows framed by luxuriously gaudy drapes of crushed velvet and lace. The doorbell, when they pressed the buzzer, rang to the tune of "Are You Lonesome Tonight?" They had been remarking on Rourke having a nice, albeit tasteless, pad when the two devil dogs emerged from the rear of the property and chased them back to the car.
"He must be able to hear them," says Daniells. "And his neighbors."
It's a quiet cul-de-sac. A dozen detached properties with neatly tended gardens and leylandii trees; Mercedes Estates parked on patterned redbrick driveways and well-tended hanging baskets next to uPVC double doors.
"They'll be used to it," says Pharaoh, pulling down the vanity mirror and checking her mascara and lipstick.
She had not really given the animals much thought when she read through Rourke's file. The last contact the police had had with him was over a complaint that his two dogs had attacked a toddler who had been playing with a ball in his grandmother's garden. The file offered nothing more on the incident except to say the complaint had been withdrawn. Pharaoh could only hope that the incentive to do so was financial rather than fear.
"Come on," she says, more to herself than to her subordinate.
She stares at the door, as if willing it to open.
"Do you think he knows we're police?" asks Daniells.
Pharaoh gestures expansively, a display that takes in the tiny two-seater sports car and their plain clothes. "I wanted the model with the flashing light but couldn't afford that and the CD changer," she says.
She blows out a sigh. Eases down the electric window a fraction.
"Mr. Rourke," she shouts. "This is Detective Superintendent . . ."
Her words are lost in a cacophony of snarling and barking. Yellow teeth and frothing spit crash against the side of the car.
"For fuck's sake."
She puts her head in her hands. Wishes McAvoy were here. Despite his having the social skills of an excitable five-year-old, she rates DC Daniells as an officer. He is enthusiastic and eager. She has spent her working life surrounded by jaded cynics who can convince themselves that anything they can't be bothered to do is not worth doing. Daniells happily throws himself into the most skull-crushingly tedious tasks and is never less than grateful for the opportunity. She had been feeling well disposed toward him when she asked him to tag along for her chat with Rourke. He had called in a favor and managed to speed up the forensic results yielded by the shards of glass found at the scene of the petrol bombing. He had been so excited, she almost offered to pull in at the service station to get him some sweeties as a tip.
She has to bite down on an involuntary smile as she imagines McAvoy, folded up and cramped in the passenger seat. She wonders what he would do. Presumes he would never have run from the dogs in the first place. Would have hummed them to sleep or banged their heads together. She wishes she had seen him talking to the escaped stallion. Can understand fully how he calmed the animal with his big eyes and soft words. Has experienced his almost otherworldly tenderness and empathy. She finds herself oddly jealous of the horse. It got the best of him. She wonders if the horse realizes just what an annoying bugger her brilliant and hopeless sergeant can be.
"Nice place, isn't it?"
They are in one of the handsome villages that lie to the west of the city, Hull's version of a commuter belt and not the sort of place she would associate with a man like Rourke.
She reaches back onto the parcel shelf and grabs the printout of his record.
The mug shot shows a surly, stubble-cheeked man in his mid-fifties scowling out from beneath a shock of coiffed black hair turning gray at the temples. He is sporting a luxurious mustache she would associate with a Breton onion seller, and his eyebrows could use a trim.
"Seven years," she says, running her finger down his long list of offenses and prison terms. "That's a stretch."
"Was that the GBH?"
"No, that was the armed robbery. Only got two years for the GBH. Provocation, apparently."
"I'll bet."
"He's been busy," she says, and as she reads, she finds herself growing more hopeful that she may have struck lucky. That the man she has come to interview is involved with the drugs gang she is so desperate to take down.
"Hang on . . ."
From the back of the property two men are emerging. One is unmistakably Rourke. He is dressed in black jeans, white sneakers, and an oversize Fred Perry T-shirt that is too tight around strong, tattooed biceps and an impressive, overhanging gut. With him is a younger man in bright green tracksuit trousers and an expensive leather jacket worn over a vest. He is thin and pinch-faced, and his hair, though recently shaved, is growing back ginger. He has a cigarette clamped between his lips, and it rises skyward as he sneers angrily at the two occupants of the car.
"Mr. Rourke, I'm Detective Superintendent . . ."
Her voice sets the dogs barking again, and the younger man laughs as she hurriedly pulls her mouth back from the slightly open window.
Rourke removes a tiny, unlit roll-up from his mouth. "Ben. Dara. Cut."
Both dogs stop their barking. They pad, obediently, to his side. He fondles their heads without taking his eyes from Pharaoh. Both dogs lick their muzzles happily. They could be different animals.
"Coppers?" Rourke asks the question.
Pharaoh nods.
Rourke gives a jerk of his head to indicate they are at liberty to get out of the vehicle. Pharaoh takes a deep breath and does so. As she hauls herself free of the low, cramped vehicle, her skirt rides up to reveal a large slice of attractive thigh. She pushes her hair back from her face and straightens her jacket, straightening her back with a movement that accentuates her figure.
Rourke does not change his expression, but the younger lad gives a leer.
"They should be locked up," says Pharaoh, nodding at the dogs as she walks down the driveway and comes to a halt near enough to Rourke to encroach on his personal space. He smells of coffee, nicotine, hair gel, and horses.
"They wouldn't hurt a fly," he says, and his accent contains a tinge of Irish.
"I couldn't care less about flies," she says. "It's sinking their teeth in my arse that I'm not keen on."
"That's a pity," says Rourke, and the younger man laughs louder than the joke deserves. "You look like the sort who'd enjoy it."
Daniells has arrived at Pharaoh's side. He gives them all a big smile. "Are they safe to stroke?" he asks brightly, seemingly having forgotten his desire of moments ago to put a bullet in their heads.
"They don't like coppers," says Rourke. "So, yeah, stroke away."
Daniells moves forward with his hand out, but Pharaoh pulls him back.
Rourke smiles. "He a bit soft upstairs?" he asks, jerking his head in the direction of the young, open-faced detective.
"He thinks the best of people," she says, locking eyes with Rourke. "We're chalk and cheese."
Rourke shrugs. He pushes his damp hair back from his face with a dirty, cigarette-stained hand. "What do you want?"
"Getting out of the rain would be a start," says Pharaoh, looking up at the gray sky.
"Place is a pigsty," says Rourke. "We're grand here."
Pharaoh doesn't push it. "Who's your friend?"
The younger man takes his eyes off her cleavage and raises them to her face. "I'm what you've been waiting for all your life," he says.
If he means the words to be ironic, his face does not betray it.
"Really?" Pharaoh's voice oozes seductive menace. "You don't look like a lottery winner."
Rourke gives a grin at that. His mood seems to soften. "RJ here is doing a bit of work for me," he says.
"Work? Heard you were allergic."
Rourke gestures at his house. "I'm doing okay."
"Bank robberies pay well, I see."
"All in the past, sister. I'm a good boy now."
When he smiles, there is something endearing about Rourke. Although she has skimmed through his record, Pharaoh has to fight to remind herself that he is a violent criminal and not some lovable rogue.
"Tuesday night," she says. "Where were you?"
Rourke lifts his unlit, tiny roll-up to his mouth. Sucks on the end while thinking. "Here," he says. "Like as not."
"All evening?"
"I'll have been drinking," he says. "Can't remember much after _EastEnders_. But, aye, I'll have been here."
"Can anybody confirm that? Your wife, perhaps? Your little friend here?"
"Doubt it," he says, and his blue eyes twinkle with something Pharaoh can only think of as "charm." "I ain't seen the bitch in nigh on three years. And Ro here has business of his own of an evening. Treats the place like a hotel . . ."
"So you live alone?"
Rourke nudges the younger man with a meaty elbow and gives him a playful smile. "I entertain the occasional lady caller," he says, affecting an upper-class English accent.
Pharaoh nods. "Do you want to know why I'm asking?"
"Not really," says Rourke. "But I don't really want you here, either, and that's happened, so chances are you're going to tell me."
"Your fingerprint was found on a shard of glass," she says.
"And?"
"Glass from a bottle that was filled with petrol, set alight, and thrown at a police van I happened to be sitting in."
If Rourke is concerned, he does not show it. He pulls a face and smooths down his mustache. "I'm a bit old for petrol-bombing coppers. Young enough to do other stuff to them, mind."
Pharaoh's patience snaps. "Mr. Rourke, try and imagine what's going on in my head right now."
"Pretty picture?"
"It involves me ringing for a vanload of uniformed officers who will trample all over your front garden, kick in your front door, tranquilize your dogs, lead you out of here in handcuffs, and throw you in a cell. Then we'll have this conversation again, and you'll wish you'd just helped me out when you had the chance. So here's your chance. Why do you think your print would be on that bottle?"
Rourke looks down at his dogs. Gives them a stroke and plays with their ears.
"Where did this happen?" he asks at last.
"Down by the Lord Line building on St. Andrew's Quay. We were involved in a surveillance operation."
"Surveilling what, love?"
"We believe a nearby warehouse was being used as a cannabis farm."
Rourke scoffs. "Cannabis? Who gives a crap?"
"I agree," says Pharaoh. "Couldn't give a chuff, to be honest. But the people who run it are very nasty people, and they hurt somebody nice, so we'd like them to go to prison."
Rourke nods. "Fair enough."
"So all you can tell me is that you were here on Tuesday night and you've no idea how your fingerprint ended up on the glass bottle thrown at the van." Pharaoh smooths down the front of her jacket. "Bit shit," she says.
"The rain looks pretty on your tits."
The younger man is staring at Pharaoh's chest. She gives an incredulous little laugh. "Sorry, son?"
"Pretty," he repeats, and raises his head to look her in the eye. "Bet they don't look so bouncy when you take your bra off, you old bitch."
Pharaoh opens her mouth, but before she can speak, Rourke has slammed a meaty hand into the younger man's chest and pushed him backward. He spins to face him, grabbing him by the lapels and pulling his face close. "You can't help it, can you?" he spits.
_"Maraigh!"_ shouts the younger man, his feet scrabbling on the drive. _"Maraigh!"_
Neither Rourke nor the two detectives have any time to react. The command, screamed in Gaelic, means nothing to Pharaoh or Daniells, but the Rottweilers respond to it as if a bell had been rung.
Time seems to slow down.
To her right, Daniells is reaching into his pocket, trying to extricate his extendable baton from the confines of his battered coat.
Rourke is turning back to her, his eyes widening as his mouth drops open.
The young man is staggering backward. Turning. Preparing to run.
In a heartbeat, the two animals transform into snarling, ravenous killers. Barking, jaws snapping, they turn on the strangers.
Jaws open, spit drooling from finger-long teeth, they leap.
Pharaoh's arms fly to her face but her eyes do not close in time to spare her the image.
Her vision fills with black-brown fur. Fangs. Pink tongue and yellow eyes.
_As she falls, she knows with cold certainty that the word means "Kill."_
__
"HOPE THE DOGS have had their shots," says Colin Ray, leaning across the table to take a chip from Helen Tremberg's plate and managing to drop a blob of ketchup on his mucky pin-striped suit.
"Yeah, would hate for them to get sick," says Shaz Archer, taking a sip from her can of Diet Coke. She then gives a bark of a laugh. It sounds like the braying of an upper-class old man, and not a petite young woman in an expensive dress and patterned tights.
Neither McAvoy, Tremberg, nor Ben Neilsen join in.
"Cheer up, you fuckers," says Ray, reaching for another chip and laughing when Tremberg pulls her plate away from him. "She's all right."
They are sitting in the canteen at Courtland Road Police Station. The news is flickering soundlessly on the TV in the corner of the room, and two uniformed officers are playing pool on the table near the serving hatch, where a fifty-year-old woman in a tabard and hat is red-facedly offering a choice between cottage pie, lasagna, chips, or going hungry to a couple of visiting software salesmen in gray suits and name tags.
Detective Chief Superintendent Davey has just finished stumbling his way through the emergency briefing. Ray and Archer are now entirely in charge of the investigation. The cannabis factories, Rourke, and the attack on their senior officer have all been rolled into one. Ray and Archer are on their way to interview the onetime armed robber, who is busy going mental in the cells and threatening retribution against anybody who refuses to tell him what is happening to his two dogs. The rest of the Major Incident Team are circling, here to help, and ready to take over, if asked . . .
Pharaoh is in a hospital bed with bites to her chest, throat, and hands. Daniells, fresh stitches in his palms, is painfully typing up his witness report on a borrowed desk in the MIT suite.
The teenager with the ginger hair, who gave the order to kill, has not been seen. Every patrol car in the division is out looking for him.
"New boy says she'd be dead if Rourke hadn't called them off," says Ray chattily. "Did the damage in about five seconds."
"Long enough for the lad to leg it," says Archer.
"We'll have an ID by morning, I promise you," says Tremberg through a mouthful of chips. She is appalled at what has happened to their boss, but has an appetite that cannot be slowed by inner turmoil.
"We promise you, too," says Ray, blowing her a kiss. "Paddy will be singing in five minutes, I guarantee it."
"He's not Irish," says McAvoy, with his eyes shut.
Ray pulls a face. "Name like Rourke?"
"Irish descent. Born in South Yorkshire. Traveler family. It's all in his file. You should read it."
There is silence for a moment. "Gyppo, is he?"
Tremberg shoots McAvoy a look. He is staring at the ceiling, his face nearly gray. She knows about his wife's traveler roots and knows, too, of his sensitivity about such subjects. He had arrived at the briefing red-faced and panting for breath, clearly having run all the way from whatever he had been up to when the call came over the radio that Pharaoh had been hurt and that her team were required immediately at Courtland Road. As he heard what had happened, she saw something in his face that was at once bewilderment and despair. She has not yet seen him angry, but has no doubt the feeling is in there somewhere, and that Colin Ray would do well to stop talking.
"He's from a traveler family," says McAvoy again deliberately.
"Gyppo, that's what I said," says Ray, and he and Archer share a laugh.
The two are inseparable. There was a time, when this unit was first being formed, that DCI Ray was expected to get the top job, together with his hard-faced but extraordinarily attractive protégée as number two. Pharaoh had got the gig instead, and the older man had not taken it well—and even less so when he was asked to be her deputy.
"Daniells gave us a good description," says Ben Neilsen, piping up. "Skinny lad, shaved ginger hair, little shit . . . can't be hard to find."
"He says it came from nowhere," adds Tremberg. "The attack, I mean. Rourke was answering the boss's questions. Wasn't exactly friendly, but nothing to worry about. The lad said something about the boss's boobs, and Rourke gave him a clip. Then the lad shouted for the dogs to attack. Whether he wanted them to go for Rourke or the boss, Daniells couldn't say."
"They would never go for their master," says McAvoy quietly. "Not in a million years."
"So why set them on the boss?"
"Maybe he wanted to prove he had the balls."
McAvoy rubs a hand over his face and looks at his watch. He is trying to stay in control of his emotions. He is furious not to have been asked to take over the investigation, but is also well aware that he has no right to expect it and that to even entertain such a hope is somehow to suggest he sees his boss's injuries as an opportunity. He holds himself still.
"Where were you, anyway?" asks Ray. He is twisting in his chair to watch the two lads play pool and does not turn as he asks the question. "Thought you were pen-pushing for Everett or something. Was there a really interesting spreadsheet needed your attention somewhere?"
McAvoy finds himself coloring. "I was taking a look at an old case," he says.
"Cold case?" asks Ray, turning back to the table and giving a hand gesture that suggests he does not think much of the standard of pool playing. "That's Operation Fox, not us."
"It's more recent than their remit. Just something that deserved some proper attention."
Archer leans forward. McAvoy notices that the lacy design of her bra is visible through the white silk of her blouse, and turns away quickly. She is well practiced in using her looks for effect and results, and seems to positively purr when she spots his discomfort.
"Picking and choosing now, are we?" she asks.
"Beg your pardon?"
"Forget it."
Archer drains her can of drink and then stands up, pulling her coat from the back of the hard plastic chair and pulling it on. Colin Ray pulls himself upright, too, and brushes the remnants of his sausage roll from his front. His pin-striped suit was probably expensive when he bought it, but is stained on the lapels and wrinkled at the crotch.
"We'll be off," he says, taking one last chip and winking at Tremberg. "Call us if there's any improvement."
"Yeah, we'll be worrying about those dogs all night."
The pair are laughing as they push their way out of the swing door, so don't hear Tremberg say, "Wankers."
Neilsen, Tremberg, and McAvoy sit in silence. "What's the case?" asks Neilsen eventually.
McAvoy looks at him. Tall and good-looking, he's from a Hessle Road fishing family who don't quite know whether to be proud or ashamed that their youngest is a policeman.
"Young lad," he says, after a pause. "Found a few months back hanging in his kitchen. Barely any investigation. I've got his old phone. It's full of messages from some sexual partner that there's no trace of in the report. They stop suddenly. It feels wrong. There's more to it."
"Forensics?"
McAvoy pauses. Appears to make a decision. Reaches into his bag and pulls out the report and his notes. He pulls himself closer to the table.
"There's no doubt he died from strangulation," he says, reading from his pad. "Pathologist said the rope around his neck was definitely the one that cut off the blood to his brain. Fibers embedded in the skin, she said. He'd been sexually active at some point in the previous twenty-four hours. Had been anally penetrated. No DNA recovered. He was covered in baby oil, apparently. Had eaten a microwave tagliatelle and a Penguin biscuit around two hours before his death. Drank a glass of orange squash."
"And then decided he'd had enough? Those microwave meals are awful, like . . ."
"The file went up to CID and they gave it about thirty seconds. The coroner recorded an open verdict because there was no note, but there's been no investigation. The inventory of his flat's contents make no mention of his mobile, even though he was on it nearly all the time."
"But you've got it?" asks Ben curiously.
_I don't know. Maybe. If he kept his own number in his phone._
"That doesn't matter," he says, waving his hand. "Look, I've spoken to his aunt today. She says there's no reason to think he would kill himself."
Tremberg and Neilsen turn to face each other. Something passes between them, and Tremberg is elected spokeswoman.
"Do you not think it would be better to keep our heads down?"
"I'm sorry?"
"It's just after the cock-up the other night, and now this with Pharaoh, do we really want to be saying CID missed something or didn't give a shit? Do we need to have a murder on the books when we haven't got a suspect? What's that going to do to the crime statistics?"
McAvoy looks at her with what looks like disappointment. He seems almost heartbroken.
"I really don't think that matters," he says, and leaves it at that.
They sit for another few minutes. McAvoy tells them no more about Simon. He has made up his mind. Pharaoh's injuries are sitting in his guts like a snowball, but in her absence he cannot help but see opportunity. His line manager told him to take a look. And she's not around to be contradicted.
They say cheerful good-byes and pull up their collars as they run across the darkening, rain-lashed car park. McAvoy throws open the door of his car and throws himself inside. Turns on the engine just in time for the seven p.m. headlines. A police officer has been hurt in a dog attack in Anlaby. Detectives have renewed their appeal for witnesses to a petrol-bombing incident which took place on St. Andrew's Quay and which is the latest violent incident being linked to an escalation in drugs-related violence . . .
He watches Tremberg's car pull out into traffic. Gives a wave, obscured by the pelting rain, as Neilsen's Suzuki Swift follows. He gives it thirty seconds. Switches off the engine. Steps out of the car and runs back across the car park.
He pulls the telephone from his pocket and holds it in a warm, damp palm as he makes his way to the Technical Support Unit. As he knocks on the white double door and tells them that Trish Pharaoh insisted that they get the results asap, he hopes they put his blush down to his sprint in bad weather, rather than shame at his lies.
COLIN RAY holds the smoke in his lungs and feels his eyes begin to water. Feels the tickle in his chest.
_Hold it, Col, hold it . . ._
He's made it up the stairs without needing to breathe out.
Six, seven, eight steps down the off-green corridor.
Eyes streaming, chest thumping . . .
Turning the handle and entering into Interview Suite B.
He breathes out. Fills the cold, damp room with the smell of cigarettes.
"Thought you deserved a treat," he says, and then gives in to a fit of hacking coughs.
Police interview rooms have been no-smoking zones since 2007. Smokers are at the mercy of their investigators when it comes to a nicotine fix. Colin Ray is not feeling merciful. Left the interview to pop outside for a cigarette, and declined Alan Rourke's request to accompany him.
"Should give them up, lad," says Ray, scraping the chair back from the desk and sitting down forcefully, wiping his eyes and then his nose with the heel of his hand. "They're doing you no good. You look like shit."
Rourke looks up. Shrugs. "Must be like holding up a mirror."
Ray gives a smile. He's enjoyed the past hour of verbal jousting with this hard, unshakable traveler. Rourke has given nothing away. Declined the offer of a solicitor with a wave of his hand, and launched into a variety of "No comments." He looks thoroughly unconcerned. Has the appearance of a man who will sit there forever rather than genuinely help the police with their inquiries.
The door opens again and Shaz Archer walks in on unfeasibly high heels. She's been to change her clothes, having failed to get Rourke's attention in her previous outfit. She's wearing fashionable patterned tights, an expensive mid-length skirt, and a floaty polka-dot top over a black vest. She looks stylish, sexy, and not at all like any of her colleagues. She's the opposite of Helen Tremberg. She emphasizes her sexuality and is happy to give suspects a glimpse down her top if it means they start talking to say thank you. So far Rourke doesn't seem to give a damn.
"Looking hot, Shaz," says Ray, pursing his lips approvingly.
"I was freezing. You could hang your hat on my nips."
"I don't wear a hat."
"Wasn't talking to you, Col."
Across the table, Rourke gives an appreciative, knowing smile. He doesn't bite.
"Smells like an ashtray in here," says Archer, lowering herself into her seat and crossing her legs with a sensual, shushing sound of nylon on silk.
"Spray your scent, love. Give us a treat."
Archer reaches into her handbag and pulls out a bottle of perfume. Gives it a squirt. Sprays some more on her wrists and then elongates her neck to dab it beneath her ears. She does the whole performance sexily, but Rourke pays no attention. Just carries on staring at the wall. Only turns to her when the smell hits his nostrils.
"Smells like a brothel now," he says.
"That's Chanel," says Archer tartly.
"Expensive brothel," he says in reply, and gives an outlandish yawn.
Ray nods to his colleague. She swivels in her chair and turns on the tape recorder.
"It is 8:09 p.m. Detective Chief Inspector Colin Ray and Detective Inspector Sharon Archer interviewing suspect Alan Rourke. Now, Mr. Rourke, where were we?"
Rourke rocks back on his chair. "We were here, love. Having a ball."
"Indeed," says Ray, sucking his cheek and scratching at something unpleasant-looking on the lapel of his dirty pin-striped suit. "We were talking about your fingerprint being found on the bottle that was thrown at a police van. We were talking about your dogs attacking two police officers. We were sitting here, all ears, waiting for you to open your mouth and treat us to some of that fucking gibberish you pikey bastards speak."
With the tape now recording every word, Ray should really be more careful what he says, but if he is concerned about repercussions, he hides it well.
Rourke says nothing. Gives a wry smile.
" _Go n-ithe an cat thu, is go n-ithe an diabhal an cat."_
Ray and Archer look at each other. "Come again?"
"May the cat eat you, and may the devil eat the cat."
Ray scratches his face with his dirty yellow fingernails. Pushes his greasy hair back from his face. Chuckles noiselessly.
"That the family motto?"
"We're a dog family."
"Yeah, we noticed. So did Pharaoh."
Rourke nods, looks down. Sighs. "She okay?"
"Still waiting to hear more," lies Ray. "We're fearing the worst."
Rourke stays silent. "I stayed with her," he says at last. "Could have gone, couldn't I? I locked the dogs up. Called you. Held a towel to her throat . . ."
"You're all heart," says Ray, pushing himself back from the table.
There is silence in the room for a time, Ray and Rourke eyeing each other up. Ray had begun the interview, presuming it would be a matter of time before Rourke gave something up. Here, now, looking across the table into the retired armed robber's eyes, he is beginning to doubt whether he will ever give in.
"You've done long stretches, Alan," he says, changing tack. "You don't want to see another prison cell. We just need some answers. Some information. Let's start with the boy. Our missing teen. What's your connection?"
Rourke turns away again. "No comment."
"It doesn't sound convincing when you say it, Alan. You know you want to comment."
"Honest to God, no comment."
Archer reaches into her handbag and retrieves a piece of chewing gum, which she pops into her mouth. She holds the packet for both Ray and Rourke. Rourke accepts. "Cheers, love."
She smiles, friendly and warm. "You understand how seriously we're taking this," she says, leaning forward. "Two incidents, Alan. The petrol bomb and a dog attack, both placing the lives of respected police officers in real danger. And you linked to both. You must know that this isn't going to go away. I understand completely that you have a code. You don't like the police. But I don't think you're the sort of man who would deliberately set fire to a vanload of cops. And I know it was the lad who gave the order for the dogs to attack . . ."
Anger flashes in Rourke's face. He curses in Gaelic. Apologizes. Nods.
"They are your dogs, though, Alan. You'll be the one crying when they're put down."
For the first time, Rourke's eyes show emotion. He bites down on his lip.
"We have some sway with all this, Alan. It's not a done deal. The dogs are being well looked after. They're safe with our specialist dogs unit. Having a little holiday. But they want to go home, and so do you. Just give us something to think about. Tell us why your fingerprint was on that bottle. Just a story, Alan. Something we can look into and discount you from our inquiries."
Rourke yawns. Chews his gum. Looks up at the ceiling, as if the most interesting story he had ever read is written up there.
Ray loses his temper. "You're going to give me something, lad. You're going to fill in the gaps for me one way or another."
Rourke turns his attention to the senior officer. Gives a rueful shake of his head, as if considering a puppy who has once again failed to control its bladder. "Always comes to that, doesn't it. You're all as bad as each other. Fucking thugs, all of you. All my life I've had you lot looking over my shoulder. Always comes down to the same thing. I've done my time, sir. Moved on. I've not been in trouble in a long time. And still I get you on my doorstep. I told the guy last month, you can threaten me all you want, I don't have anything to say to you . . ."
Ray sits forward suddenly. "Last month?"
"Round face. Smart suit. One of your top dogs."
Ray turns to Archer. Tells her to stay quiet without opening his mouth. Switches his attention back to Rourke.
"You were questioned recently?"
There is nothing on the database to indicate Rourke has had any dealings with the police in a long time.
"Don't know if it was questioned," he muses thoughtfully. "Given a talking to, more like."
"In connection with . . ."
Rourke shuts up again. "No comment."
Ray slams his hand down on the desk. "Which officer?"
Rourke appears to consider the implications of not giving away this snippet of information. "Russell," he says at last.
Archer's body language gives away the fact that this is significant, and Rourke's eyebrows shoot up.
"Did he not put that on your little machine? Hardly a surprise. Would take some balls, that. Though I tell you what, sir, it takes some balls to threaten a man when his Rottweilers are by his side. He turned as green as these bloody walls. Don't think he really made his point the way he wanted to."
Ray slumps back in his chair. Presses his lips together. Wonders whether the traveler is telling the truth. Adrian Russell is head of the Drugs Squad: the last surviving member of the corrupt team that had morphed into the Serious and Organized a year ago. He's also Colin Ray's friend.
"Did he talk to you about drugs?"
"No comment."
"Robbery?"
"Ask him."
"You will fucking talk to me . . ."
Rourke smiles, all teeth and eyes.
"No I fucking won't."
• • •
11:41 P.M., MORPETH STREET, HULL.
Flickering streetlights and pouring rain. A terrace of student bedsits and low-income flats, where every second window shows a poster for a club night, and where the small front yards are home to mountains of ripped garbage bags, pizza boxes, and broken furniture. Where different styles of music hum and blare from open windows, and the shimmering color of giant flat-screen TVs flicker from curtainless front rooms.
Nineteen-year-old Georgie-Lee Suthers sits on the front step of one of the better-looking houses on the street. She is smoking a cigarette and playing with her phone.
She is dressed as a dead bride: her charity-shop dress powdered with talc. There is a livid slash of color at her throat. Panda eyes look out through a ghoulish white face, and her legs in ripped fishnets are a canvas for an advancing army of ink spiders.
The rain is ruining her makeup, but the Mateus Rosé and shots of rum have pitched her past caring. She looks at her phone. Hopes there will be a message. An apology. Advance warning of a busload of guests.
Georgie-Lee bites down on the filter tip of her cigarette and pouts. "Thanks a lot."
It was supposed to be a party, but nobody could, in good conscience, call it anything other than a gathering.
Georgie-Lee has tried for the past three hours to make her housemate's birthday something special, but not even the cramped confines of their two-bedroom flat can make the paltry dozen guests seem anything other than a disappointment.
She had worked hard at making the night a success. Arranged for friends to take Jen shopping while she set to work blowing up balloons and arranging chocolate crispy cakes and half-frozen sausage rolls on the coffee table. Even made a playlist for the iPod from their shared party tunes. Had a nice glass of wine and danced around as she covered the flat in fake cobwebs and skeleton silhouettes. Threw a bag of fake spiders around and drew pupils on Ping-Pong balls to drop in the "witches' brew."
With only half an hour to go before Jen was due back, Georgie-Lee realized that she should have made people confirm their attendance. The thirty or so university friends who said they would definitely try to be there have let her down. Jen has made all the right noises, of course, but even when she went and changed into the skimpy vampire costume that Georgie-Lee had picked out for her, it was clear she was neither in the mood for this nor pleased that nobody else was, either.
Georgie-Lee plays with her phone. Wonders whether she should update her Facebook status to read "Thinking of getting friends who actually give a shit!"
She won't, of course. Georgie-Lee cannot bring herself to be mean to people. She doesn't like conflict or an atmosphere. Instead, she logs on to the site and tells anybody who cares to read it that she is having "the BEST time ever!!"
She rubs out her cigarette on the wall and thinks about going back upstairs. When she came down for some air, a mad professor and a werewolf were going through her DVD collection and demanding she apologize for the absence of any samurai movies. She does not particularly want to go and jolly everybody along and pretend the party is something it is not. She wants them all to go, now. Wants to give Jen her present, then watch a horror film with the lights off.
Georgie-Lee looks up. Tries to catch some rain on her face, but the awning of the doorway means she is relatively dry, even if the cold air makes her shiver. She looks down her street, wondering if she should knock on doors and ask anybody who looks even vaguely interesting if they would like to attend. She wonders whether the two lads who live at number 57 are home, sitting watching DVDs in their downstairs flat, but her view is blocked by the presence of a large four-by-four, parked directly in front of the low wall that marks the border of her own property. It's an expensive-looking vehicle, but it has been in the wars. The front bumper is crumpled and the headlight cracked. She wonders whom it belongs to. It seems out of place.
She scrolls through her Facebook posts. Comments on a friend's photo. Wonders whether she can be bothered to change her relationship status again, and decides she'll leave it for now. She and her boyfriend break up and get back together every week or so, and the frequency of the changes is becoming embarrassing.
She flicks over to Hull Ink. Stefan has appointments free tomorrow. Devon is booked up all day, but he has uploaded some pictures of his old black-and-white stuff.
Indulgently, Georgie-Lee flicks through the various galleries, comparing the images on display. She finds herself wondering, again, who would go to the trouble of having Eeyore or Winnie-the-Pooh tattooed on themselves. Questions why somebody would walk into a tattoo parlor and ask to be branded for life with the lyrics from a Coldplay tune or the picture of a dead granddad. None of it is pretty, and to Georgie-Lee, prettiness is paramount. Her own ink makes her wriggle with pleasure, and to see it on the website, paraded for the masses, still gives her a genuine thrill.
She expands the picture. Looks at her own bare back. A solid brown branch, spreading from the top of her right buttock to the nape of her neck; curving, like a stream, into slender twigs and delicate flowers; a collage of overlying blossoms, against a shimmering lake of lilies.
The design is not really her own. Legally, she supposes, she should not even have it upon her skin. But she had subtly changed the design and added her own little motifs, and Stefan had enjoyed the opportunity to perfect the image that he had inked on another girl some months before.
She looks through the various posts that accompany the image, secretly hoping that somebody will have added a compliment, but the four "like" icons and one "love this" from an old school friend, have not increased in number.
It had hurt less than she thought, and despite predictions that it would take four sittings, Stefan had managed to finish in two. She watched him in the mirror as he worked, his face locked in concentration, the pirate ships on his forearms moving up and down as he drew, as if on gentle seas.
For a time she had planned to make her first tattoo a fairy castle; all daisies and scampering imps. But then she saw the ads. The glossy half-page in the _Journal_. The boy with the peacock feathers and the girl with the blossoms. She had booked herself in almost immediately, brandishing a copy of the mag and insisting that the tattooist who had created the vision do the same for her. He had told her about copyright laws and artistic rights, and she agreed to some tweaks to the design. Added a lily pad, and a tiny hummingbird on an upper branch. Asked if they accepted credit cards, and then stripped herself to the waist.
"You coming back up?"
She looks up. A head is poking out of their second-floor window, masked by a cheap polyester Afro and a Michael Jackson _Thriller_ jacket.
"Won't be a sec," says Georgie-Lee, and the head withdraws.
She takes a breath. Practices her smile. She is incapable of being the gloomy one in the room. Needs, always, to be jollying along and perking up.
She pulls herself up to a standing position and slips her phone into the elastic of her elbow-length black mesh gloves.
She turns away from the street, stopped before she has taken two paces by the sound of a car door slamming. It is close. Close enough to suggest that perhaps another visitor has arrived, more partygoers out to salvage the evening.
She traces the sound to the stranger emerging from the four-by-four. Gives an accepting nod, and spins away from the road.
"Suzie."
Georgie-Lee wonders if she has misheard. On instinct, she turns in the direction of the noise.
In a moment she is crashing backward; strong, powerful limbs bearing her to the ground.
A forearm, pushing up beneath her chin, forcing her head back onto the cold, wet tiles. Another hand, tearing at her wig, rubbing away the white powder, the black eyes.
She squirms. Tries to squeal.
And now she is being flipped. Rolled onto her belly. Feels as though she is being clawed open. Hands, nails, tearing at her dress.
Sudden cool air on bare skin; fingers ripping back the flimsy threads, wrenching her bra strap upward so that the underwire digs in beneath her breasts . . .
"Help. Please . . ."
Her voice a gulp; a breath; an animal noise cut short by the sudden feeling of fists in her hair.
She reaches back. Squirms. Wriggles. Fights for her life.
Wet lips, next to her ear.
"I had to be sure. I'm sorry."
Paving stones rushing toward her face.
_Blackness._
And nothing.
"THEY DON'T MATCH."
Roisin waves her hand in the general direction of McAvoy's feet.
"What don't?"
"Your shoes. One's a sneaker. One's a boot."
He looks at his footwear. Nods. "Yeah."
She turns back to the sink. Fills the kettle. Thirty seconds later she notices it is overflowing and manages to turn the tap off.
"What was I doing?"
"Tea, I think. Or were you sterilizing?"
"It will come to me."
Between them they got around four hours of sleep last night. Lilah had a fever. She screamed until her face was the color of cherry tomatoes. Clenched her fists so tightly that she scored half-moon scars in her tiny palms. Brought both her parents to tears of impotence and exhaustion. Finally passed out through a mild overdose of Calpol around four a.m., lay stiff as a board on a pillow in Daddy's lap.
"You can't go in," says Roisin. "Not in that state, Aector."
She is wearing her nightie and flip-flops, and is soaking wet. She tried to put her leather jacket on a dozen times before she walked Fin to school, but couldn't seem to find the arm holes, and ended up taking him while still dressed for bed. She received sympathetic looks from other parents, familiar with the skull-crushing dishevelment that comes with being mum to a three-month-old baby.
"I can't call in sick for tiredness."
"Work from home."
"Roisin . . ."
"Nnnn."
They are two zombies, communicating in slurred grunts, gestures, and half-finished sentences.
"You need to sleep."
"I got some sleep."
"You got about five minutes, and even then you were sitting up straight. Look at you."
McAvoy pulls himself out of the hard-backed kitchen chair and lifts up the toaster. It's silver and polished to a gleam. He examines his face in the reflection. Unshaven. Dark circles beneath his eyes. A bruise starting to form on the orbit of his eye from having spent too long resting his head on the heel of his hand. He notices his top button is undone. Fixes it. Straightens his red tie and checks the front of his black shirt and light gray suit for signs of porridge or baby spit. Finds nothing that cannot be remedied with a damp towel.
"It was like this with Fin. She'll be better tonight."
Roisin nods. She is too tired to argue.
"You never got me told," she says, lifting the kettle and wondering why it is so full. "She okay? Pharaoh?"
McAvoy is engrossed in sifting through the cupboard under the stairs in search of his matching boot. After a few moments he spots it in his left hand.
"She'll be okay. Sore, but okay."
"That's good."
"Do you have any of that ointment left?"
"No."
Roisin sounds quite final in her pronouncement. She is a gifted herbalist. There are few plants, trees, roots, berries, or leaves that she cannot mix to create a poultice or pill. She would normally volunteer to make a fresh batch, her innate need to help and heal overpowering all other concerns. This morning she is too tired. She is snappy and nauseated and wants to fall asleep on her husband's chest and not wake up until both of her children are old enough to vote.
McAvoy does not get the opportunity to ask her if she will make something to help with the pain of Pharaoh's injuries. His phone rings.
"McAvoy," he says. "Something or other."
"Sergeant?"
"Yes," he says, wearily, running a hand through his hair. "Just about."
"It's Dan from Tech Support. Got something for you."
Drowsily, McAvoy pictures the other man. Dan looks barely old enough to have left university: small, wiry, and with fashionably shaggy mid-length hair. He is usually dressed in a band T-shirt beneath his white coat and wears sneakers with suit trousers.
"Sergeant?"
McAvoy gives himself a little shake. He stands upright and screws his eyes up tightly, trying to bring himself around.
"Yes. Sorry. Terrible night with the little one. I'm listening."
"Right. Anyway, we did it as a rush job. I'm trying to build up some comp time, so I was here overnight. Planning a Glastonbury trip but used my holidays . . ."
McAvoy bites his tongue, which is difficult mid-yawn. He resists the urge to tell Dan to hurry it along, all too aware of his own propensity for rattling on about things he presumes other people will be interested in. Eventually the technician gets around to the phone.
"Yeah, well, like you said, it's knackered," he says with an almost audible shrug. "You did well to recover what you did. We've sent away the remaining soil for analysis . . ."
"There's no need for that, I know where it was found . . ."
"It's procedure," says Dan.
"Right."
"Took the SIM card out again and loaded it into another phone, but like you saw, it's just this mad stream of numbers and fragments. I did have more luck with the phone itself . . ."
"Yes?"
"Well, not all of the contacts were stored on the memory card. A couple were saved to the phone itself. Don't know why people do that, but sometimes they just hit the wrong button."
McAvoy feels himself waking up. He pictures a chill breeze blowing through his skull, scattering the drowsiness and refreshing his thoughts.
"Well?"
"I've e-mailed you the numbers. Only a couple, and there's no way of working out which of the contacts they belonged to, but we've got some complete digits."
McAvoy pulls on his boots as he talks. Unconsciously unfastens then refastens his tie.
"Have you checked who they belong to?"
Dan laughs. "You did say it was Pharaoh wanted this doing, yeah?"
"Why?"
"I'm very eager to please, Sergeant," says the younger man, in a way that McAvoy would have thought of as fun and relatively charming were he not so tired.
"Did you find anything, Dan?"
"Yeah," he says, and there is a note of petulance in his voice. "It's on your e-mail. Let me know if you need anything else."
Dan rings off.
"Bloody hell," says McAvoy to himself, pulling his laptop from his bag and logging on. He opens his e-mail account and brings up the tech report.
"Interesting?" asks Roisin. She is holding a drinkable yogurt and sitting on the floor, leaning back against the fridge.
"Could be," says McAvoy, reading the two short paragraphs of detail. "Got two full numbers in the memory. Dan's requested user details from the service provider, but that could take some time. Ran the digits through a search engine and found them on a procurement committee report. Hull City Council."
"That's nice," says Roisin.
McAvoy runs his tongue around his mouth.
"Council," he says thoughtfully.
The number is listed as being among a batch of phones ordered by the authority's procurement division for officers and elected representatives. McAvoy is not sure how he feels about this development. It could be utterly inconsequential. A telephone number listed as a contact in the phone of a dead man? Does it matter? He finds himself pondering how many hundreds of people have his own phone number stored. Whether he would feel angry were he contacted as a result.
Procedurally, he should not even be looking at the tech report. Dan's work will have to be paid for out of Pharaoh's budget. McAvoy is not officially conducting a murder investigation, and no senior officer besides Pharaoh is even aware of what he is doing. But to make his previous deceit valid, he needs to bend the rules again. He feels the realization settle within him. Feels his mouth begin to salivate, a prelude to sickness. Feels his skin prickle, as if the hairs on his body were being brushed the wrong way. Accepts this as both sacrifice and payment.
McAvoy pulls up the telephone number for Hull City Council. The Google search also brings up a page full of negative headlines. The authority has been at the bottom of just about every league table for as long as McAvoy has lived in the area. It has been slammed by every inspector to have found his way into the wood-paneled corridors of the guildhall.
He dials the number and listens to the automatic prompts.
_For inquiries about rubbish collections, press three. To apply for a council tax rebate, press four . . ._
_"_ If you want to find out why a phone number was stored in the phone of a dead lad, press six," whispers McAvoy to Roisin. She does not reply. Her eyes are closed and she is spilling yogurt on her feet.
"Hull City Council," comes a female voice, when he requests the "any other inquiries" option.
"Procurement, please," he says.
A moment later he is on hold, listening to something monotonously classical. He hums along, but finds his eyes closing, so opens them wide.
"Jacquie Carrington," comes a high, bright voice.
"Hi," says McAvoy suddenly unsure. "My name is Detective Sergeant Aector McAvoy. I'm ringing from Humberside Police . . ."
"Yes?" She doesn't sound disinterested or particularly bothered.
"I'm trying to find out which council official was given a particular telephone."
"The councillors' mobile numbers are all online," she begins.
"Yes, well, I've rung the number and it's dead, but I believe it was among a batch ordered by the authority in July of last year?"
There is no reply for a moment.
"I think I may have to speak to my line manager about that," she says, and it sounds as though she is reading from a script.
"There are obviously ways we can request this information formally," says McAvoy quickly. "I just thought perhaps you could save us some time. Did you say it was Miss Carrington?"
The lady repeats herself. Tells McAvoy she will speak to her line manager and call him back. Takes his name, number, and thanks him for his time. Promises she will not be long.
McAvoy hangs up and puffs out his cheeks. He turns and looks at Roisin and feels his heart swell at how adorable she looks, snoring softly on the floor with her damp clothes clinging to her goose-pimpled skin. Smiling to himself, he crosses to her and scoops her up with the ease he carries their children.
He carries her to the living room and sits her down. She is flopping like a rag doll and makes no protest as he peels off her wet clothes. He lays her down and returns to the kitchen to retrieve a blanket from the tumble dryer. Comes back and covers her up. Presses his lips to her cheek, and whispers, "Love you," in her ear. Fancies he sees the faintest of smiles.
His phone rings again. Roisin opens her eyes at the sudden noise and he frantically shushes her, answering the call as he dashes out of the living room and closes the door behind him.
"McAvoy."
"Detective Sergeant McAvoy?" The voice is nasal. Young.
"Yes. Is that Procurement . . . ?"
"Sergeant McAvoy, my name is Ed Cocker. I wondered if you would be free to talk about one or two matters of a politically sensitive nature."
McAvoy colors. "How did you . . . ?"
"Sergeant, I'm led to believe you are looking into the activities of Councillor Stephen/Steve Hepburn. Can I ask you the nature of the investigation?"
"I'm sorry, I'm not sure . . ."
"A source at the guildhall has informed me that his telephone number has come up in connection with an investigation into an ongoing case. I also have some interest in the councillor. Perhaps we could share some information."
McAvoy is silent for a moment. Wonders just what he is doing and, more important, what he will do next, if not this.
"Perhaps we could."
• • •
THEY ARE IN THE GREEN BRICKS, a pub named after the emerald tiles that adorn the building's frontage. It offers a decent view of the bobbing pleasure craft that moor at Hull Marina, and the city center is only beyond the traffic-clogged main road. It should be a bustling area even in the face of today's harsh gales. Instead, it is an open grave. McAvoy has seen old black-and-white photos of the area in its prime. The fruit market, with its hurly-burly of trade and activity, its carriages and wagons, its chain-smoking men in their dirty jeans and overalls, treading clementines and too-ripe bananas beneath tires and rain boots. Profitable chaos. Trade. Life.
He has seen, too, images of the nearby waterfront. Has read of how the waters, now calmed and framed by the walls of the marina, once rushed haphazardly into the open estuary and provided a living for the captains of many small vessels who would take passengers and produce to New Holland on the opposite bank. Two minutes away is Victoria Pier; the main terminal for ferries to Lincolnshire. McAvoy has heard the stories. Enjoyed the tales of passengers and livestock sharing space on cramped vessels, spending uncomfortable nights stuck on a sandbank midway into the crossing, claimed by the treacherous tides.
He stares out through the glass. Tries to picture it. Tries to give the area new potential. Fresh life. Today most of the units carry TO LET signs, or are home to hairdressing salons where bored stylists read magazines and paint their nails.
To his right, just visible through the dusty, rain-lashed windows, is the stern of the Spurn Lightship, a smudge of black against the gray sky. It is one of Hull's most distinctive landmarks and seems to embody the city and its fortunes. For almost fifty years it sat five miles off the East Yorkshire coast, a floating lighthouse that served as a navigation marker for the thousands of vessels that steamed in and out of the Humber each year, its towering acetylene light visible for more than ten miles against the ink-black waters and skies. It was retired in the mid-seventies, around the same time the city's trawlermen were being told to fuck off home, and has since been turned into a floating museum. Today day-trippers and bored locals are the only ones to clamber inside and try to imagine how it must have felt to live and work in its claustrophobic embrace. It sits at the edge of the marina, black and joyless. As uninviting as its reality.
"Did you order food?" asks Ed Cocker, raising his hand and manfully trying not to wince as it is taken in McAvoy's great paw.
The man, whose business cards declare him a "political consultant," is as tall as McAvoy, but only half the size. He is skeletally thin, with cadaverous cheekbones and sunken eyes. His flesh is stretched so tightly over his skull that McAvoy wonders whether his shaving cuts ever nick bone, and his dark gray, old-fashioned suit covers legs that put McAvoy in mind of a stilt walker. He is perhaps thirty-five years old, and if he is earning a good living, he is not spending the proceeds on his appearance.
McAvoy shakes his head. "I'm not stopping long."
Ed nods. Sips from his bottle of lager. Reaches down and picks up a sheaf of printed pages from the seat next to him. "You've been involved in some very high-profile cases," he says, respectfully flicking through the papers.
"What do you mean by that?" asks McAvoy, coloring. Instinctively, childishly, he presses his cola glass to his cheek.
Cocker brushes past the question. "Gets its fair share of big cases, Hull. Bad business last year with that poor girl in the church."
McAvoy realizes his leg is jiggling.
"New problems, too, I hear. Some new outfit getting in the faces of the Vietnamese? You don't want to go upsetting them, do you? Crazy. What is it about this city?"
McAvoy breathes out. He is on safer ground here.
"There's a thesis to be written on that," says McAvoy. "A sociopolitical doctorate."
"On why it's a shithole?"
"It's not a shithole," says McAvoy, and his words surprise him. "The fishing industry died. Nobody had any work. And the Germans bombed the hell out of the place in the war. No investment. Culturally, a historic lack of impetus on education. And from a geographical perspective, it has a sense of isolation. It's the last stop on the line. It has to deal with more than most. That leads to high crime . . ."
Cocker is listening. He appears to be taking it in.
McAvoy stops. Wonders if he should shut up. Wonders how to explain this city to a stranger.
He turns to stare through the dirty windows. A teenage couple are wincing into the wind and rain, trudging past the glass with their arms folded and faces set in grim determination, their blue jeans made black by the downpour. They are not holding hands. Not talking. Just making their way in resigned silence. McAvoy thinks it would be easier to answer Cocker's question were he just to point at them and tell the southerner to take a look.
"It could be so much," he says, turning back to Cocker, "this place. This city. Used to be. You know that. Biggest fishing port in the world."
Cocker pulls a face. "Doubt it made the workingman a millionaire."
"No," says McAvoy thoughtfully. "But it was something. Something to cling to. An identity. That's what it lacks now. Something to be."
"You got any suggestions?" asks Cocker through a smile.
"I leave that to the politicians," he says, turning away. "I have to hope the people who get paid more than me know more than me."
They catch each other's eye and smile, though for different reasons.
"Anyway," says Cocker.
"Councillor Hepburn," says McAvoy. "You have an interest."
"In him? Not very much. In who he's friends with? Yes, quite a lot."
McAvoy decides to stop dancing around the subject.
"Why are you here, Mr. Cocker? What's your job?"
Cocker gives a nod, as if making a decision. Shrugs. "Your new boss. Peter Tressider. Chairman of the Police Authority."
McAvoy says nothing. Waits for more.
"You must have heard the rumors that the party is interested in him. He could be set for great things."
"He may run at the next election, you mean."
Cocker nods. "And if that goes well . . ."
"People have plans?"
"Indeed."
They sit in silence, both eyeing each other up. McAvoy speaks first.
"And you're seeing if there are any skeletons in his closet."
"In a manner of speaking. I'm a political consultant. I dig. I find out whether we should be worried down the line."
In his mind, McAvoy quickly runs through the many political scandals he has flicked through in the tabloids these past few years. He pulls a face. "There always seems plenty to be embarrassed about in politics."
Cocker gives a grin. "I can't be everywhere."
"And what is it about Councillor Tressider that worries you?"
"Councillor Hepburn."
"I don't understand."
"No. You don't."
Cocker reaches into the inside pocket of his jacket. He pulls out a crumpled roll of paper, covered in photocopied newspaper articles and scribbled notes.
"Let me read you something," says Cocker, clearing his throat. "If you don't mind, that is . . ."
"If it's important," says McAvoy.
"Important? Possibly. Interesting, certainly."
McAvoy waits. Wonders when the other man will make his point.
Cocker reads the words on the page. "'He's the politician who has made a fortune from the pink pound—and who swings to the left, the right, and straight down the middle . . .'"
McAvoy closes his eyes. "Classy. Where's it from?"
Cocker stops. "Political website. One of many. They'd tone it down for a broadsheet feature but the few times he's been in the national headlines, the tone hasn't been far away from this tripe."
McAvoy nods at the papers. "Carry on."
"'Stephen/Steve Hepburn, forty-seven, is the flamboyant, colorful, and rabble-rousing independent councillor and gay-bar owner who is shaking things up in the guildhall in Hull. He's also the man who saw a hole and decided to fill it—and who has yet to hit a bum note on his rise to power. A local boy who was involved in the music scene during the Manchester explosion of the early nineties, he managed bars in London for a time before coming home to Hull a decade back. Hepburn purchased a run-down gay bar not far from the city center and planned to make it the biggest, boldest, and campest club in Britain. Apparently the idea for Slammers came to him in the night. Hepburn faced fierce objection from locals and various civic dignitaries, but despite recommendations that they block the proposals, Hull Council's planning committee allowed Hepburn to proceed with the application. Yes, they bent over and took it. Rumors have since abounded that the authority feared accusations of homophobia, and that Hepburn played on those fears during the consultation process. The high-profile case turned out to be the making of Hepburn, who gave several interviews on radio and TV in which he came across as charming, bold, determined—and very funny. He had the presenter in tears of laughter as he plowed into the objectors and the different members of the committee: mimicking their mannerisms and questioning their motives in a speech that was full of double entendres. The opening night of his new club saw several big names from the music scene put in an appearance, and high-profile gay rights activists applauded his victory—bringing him to national attention. Slammers has gone on to become a hugely popular venue, attracting clubbers from across the country and reveling in a reputation for controlled hedonism where people can gleefully dance on the bones of Hull Council's cock-up . . .'"
McAvoy stops him, holding up a hand. "This is vile."
Cocker spreads his hands. Takes a drink. "This is politics."
"It's not. It's—"
"'During the dispute, Hepburn clearly tasted something he liked,'" continues Cocker. "'He swallowed the plaudits, and so much more. When the next local elections came around, Hepburn put himself forward as a man with something big to offer. Due largely to a low turnout and the fact that the sitting Labour councillor didn't bother to try to drum up any support, Hepburn was elected to the council. When the ruling Liberal Democrats of the time needed an extra vote to get through a key part of their manifesto, they persuaded Hepburn to join them in a loose coalition that gave him a position on the authority's cabinet. He has since been sticking it to the cabinet on every occasion, making political allies along the way, impressed with his silver tongue . . .'"
"Stop."
"Horrible, isn't it? But people read it."
McAvoy screws up his face. Tries to remember who he is and what he's doing.
"And what has this to do with Peter Tressider?" he asks.
"I'm getting to that. Look, known about Hepburn for ages," he says. "His name has come up once or twice. He's a playful man. Every bit as flamboyant as he pretends. That's not what worries us. It's the shadier side. The money. There are questions over where it came from. It wasn't cheap, building that club."
"Shady?"
"Loan from a criminal associate, perhaps," says Cocker, speculatively. "Or perhaps somebody with dough to spare who might not like being linked to that sort of place."
Realization dawns. "You've been through Tressider's financial records, haven't you?"
Cocker does not look away. "That's basic. That's the first job on the list."
"Hepburn's name came up?"
"In a roundabout way."
McAvoy makes no attempt to hide his contempt. "Has anybody done anything wrong?"
Cocker puts his hands flat on the table. Looks away. "That's not the point."
McAvoy waits for more. "Tressider's a businessman. What are you getting at? What does this matter?"
Cocker loses patience. "Look, in this business, rumor can kill you. A whiff of impropriety, you're out on your ear. It doesn't matter how good you are, or even if you've done nothing wrong. It's what people think. And the party has got wind of whispers about a business relationship between our new paragon of virtue and a gay troublemaker, and it's up to me to see if that is something we can swallow."
"You don't care about the truth?"
"I care about the appearance of truth."
McAvoy has to remind himself to breathe. He wants to see if he could fit the man into the empty lager bottle. Wants to pummel this walking embodiment of all he despises about the world.
"And this is a job? A real way to make a living?"
"A good living," says Cocker unashamedly. "And I'm worth every penny."
McAvoy scowls as he puts together the chain of events that have brought him here.
"You slipping cash to some of the officers at Hull City Council? Asking to be told about anything involving Hepburn?"
Cocker grins. "That would be frowned upon, I imagine."
"Mr. Cocker?"
"I have sources."
McAvoy nods.
They sit in silence for a minute. Cocker looks at his watch and then in the direction of the pub kitchen. From the state of him, he could have been waiting for this meal since the mid-seventies.
"There will be eyes on this place soon," he says, gesturing out the window at the dismal, half-empty marina. "The national media will be taking an interest in Tressider if he gets the nod. Right sort of man. Successful. Straightforward. Beautiful wife. Right background. Could go far."
"If you let him."
"Yes."
Silence falls. The sound of glasses being stacked and plates laid on sticky, varnished tables occasionally rises above the relentless patter of the rain on the glass.
McAvoy runs his tongue around the inside of his mouth and wonders if he has missed something. Whether the past half hour has been worth his time.
"We're not investigating Councillor Hepburn," says McAvoy eventually. "I don't think you should be, either."
"What were you ringing about this morning?" asks Cocker, appearing not to register the firmness of McAvoy's gaze.
"It's nothing. We're trying to find out why a certain telephone number was in a certain telephone . . ."
Cocker sits forward, like a jockey planning on giving a horse a few extra kicks toward the finishing post. He can clearly smell a story.
"Hepburn's phone, you mean?" he asks, all smiles.
"I can't tell you that," replies McAvoy, willing himself not to blush.
"The number you gave my contact," Cocker muses to himself. "That was Hepburn's phone. Only had it a month, then reported it lost. Got a new one from his own pocket."
McAvoy looks away before his face betrays him. "And?"
"And what?" says Cocker. He is not in the least bit deferential in his manner. He is talking to McAvoy as if they were mates. McAvoy bridles a little.
"I think we're done," says McAvoy, and begins to stand.
"Are we going to mention it?" asks Cocker. "The thing we both know?"
McAvoy sits back down. He had not wanted this conversation to reach this stage. Yes, he knows about Hepburn's record. Flicked through it as he sat in the car outside the pub. He knows that, as a twenty-something, Hepburn was arrested for the alleged rape of a teenage boy. Knows, too, that the evidence was circumstantial and that the case collapsed well before trial.
"He hasn't been convicted of a thing," says McAvoy. "He hasn't done anything wrong. You're out to get him just because you don't like what he stands for."
"What does he stand for?" asks Cocker, incredulous.
McAvoy colors. "Alternative lifestyles" is the best he can do, and he is embarrassed by the pomposity of his tone.
Cocker does not disguise his laugh. He shuffles his papers and looks as if he were about to read some more. "You want to hear how the papers reported the case?"
This time McAvoy does stand. "If you try and pay off any more council officers I'll arrest you," says McAvoy, walking away.
"I'll be in touch," shouts Cocker at his departing back. McAvoy pulls open the double doors to the pub and all but throws himself out into the gusting wind and rain. It takes all of his willpower to say the word "tosser" only in his head.
GOING FOR the three-course lunch today. Bag of peanuts, packet of crisps, and a pickled egg. All swilling, like croutons, in the red wine Colin Ray has been pouring down his neck for twenty minutes.
It's 1:24 p.m.
He and Detective Superintendent Adrian Russell are sitting by the hearth in the George, representing two-fifths of today's clientele. Warming themselves on the open fire. Ray's got his back to it. There's steam rising from his damp clothes, as though from compost.
It's an unashamedly old-fashioned pub, this. Dark. The smell of cigarette smoke still lingers even now. It has atmosphere. Style. It's a proper boozer, all fingerprints and greasy brass. All leather-studded seats and dust-caked lightbulbs.
Ray looks around. Breathes it in. Wishes he could smoke. Closes his eyes and plays his game. Tries to bring the scene to life in his memory. To paint the picture in his mind and then compare it to the reality. To see how much he can remember.
Hardwood floor, darkened and scuffed.
Mahogany walls, almost black.
Thick, frosted windows.
Old newspaper articles on the wall. Dartboard, more holes than cork. Drawing of something obscene on the blackboard by its side. Warm. Friendly. Comforting. Like crawling inside a hamster's cheek and lighting a fire . . .
Ray opens his eyes. He has every detail memorized. Could draw this place, if asked.
It sits at the bottom of the Old Town, on a deathly quiet cobbled street that carries the most unlikely of names. Land of Green Ginger, the street signs declare. A narrow road which was named either for the profitable trade in exotic spices that brought money to the area centuries back, or in honor of a Dutch joiner who had a yard here once upon a time. Nobody is really sure, but the street name is an interesting enough discussion to knock back and forth over a few drinks.
"Should have come to us in the first place," says Russell. "Drugs bust. The foreshore. You could have had the collar, but when it comes to a bust, that's our territory. That's what we do."
Colin Ray is uncharacteristically diplomatic in his response. He knows that now is not the time to criticize Pharaoh. She's just had her throat torn open by a pair of Rottweilers, which buys her some goodwill among the troops. Has decided to be sympathetic, at least until Russell lays his cards on the table.
"Order came down from on high, Aidy. Don't think she wanted it in the first place. And you've had plenty of results this year, if the _Hull Daily Mail_ is anything to go by."
Russell gives a begrudging nod. "Yeah, we've had some wins."
"We're going to put it all back together," says Ray. "Shaz and me."
Russell takes a swallow of his second pint of bitter. "You're a lucky boy, having Shaz to play with."
"She's lucky to play with me," says Ray as he sips his wine. He drinks lager in the evening.
Russell waits for a juicier tidbit, but when none is forthcoming he picks up his drink again. "Should have come to me in the first place," he says again. "It should never have come to this."
Ray nods his assent. Has the sense to keep his mouth shut. Just picks his back teeth with his tongue and wonders if it would be taking the piss to claim the senior officer's drinks back on expenses. He lets it play out. Doesn't push. Keeps his temper.
"It wasn't that I wouldn't have shared," says Russell slightly petulantly. "Just nobody asked. We're supposed to be the fucking experts . . ."
The boundaries that separate the roles and responsibilities of the different units within Humberside CID may as well be written in water. For every specialist section in the service, there is another team that feels better placed to do the job. For every case that goes to a particular unit, there are half a dozen disgruntled officers who believe the crime comes under their own remit. The Drugs Squad is unsure how to feel about the Serious and Organized team. Their duties frequently overlap, but rarely to anybody's satisfaction. The balls-up during the failed cannabis factory raid at St. Andrew's Quay has given the officers on the Drugs Squad a reason to feel a little better about themselves. They felt they should have been given responsibility for the raid—rather than being reduced to a peripheral role. And their boss, Adrian Russell, has made no secret of the fact. The only thing he has made a secret of is how much he could have helped, had he so chosen.
Despite his rank, Russell is neither liked nor trusted by the majority of the CID team. He is one of the few members of the old CID to have survived the internal inquiry into the corruption so endemic under Doug Roper, Pharaoh's predecessor. Nobody really understands how he scraped through it all—or landed a promotion and a cushy number running a headline-grabbing team. The consensus is that he has friends in low places.
Ray considers him. Russell is in uniform today. Despite being one of the most senior plainclothes officers, he has been called to HQ for a meeting with some local dignitaries and been told to wear his best blues. He and Ray met a decade or so back, when both were unwilling participants on a "community assurance" training course. They were so vocal in their contempt during the two-day seminar that they might as well have stood up and pissed on the course tutor. They had hit it off. Found enough in common to form a halfhearted friendship based on drinking, football, and in-depth discussions about breasts. It was Russell who suggested Ray get on board when the top brass announced the formation of the Serious and Organized Crime Unit. Ray has sometimes wondered whether it was Russell's recommendation that cost him the top job and instead reduced him to Pharaoh's understudy.
"If she'd only asked . . . ," says Russell again, and looks away. Once upon a time the action would have caused a ripple in his fleshy face. These days he's leaner and a damn sight more presentable. When Ray first met him, his gut was spilling over his beige suit and there was a sheen of sickly-looking perspiration on a jowly face that sloped upward to a shock of bristly gray hair. He hasn't been much to look at, but a routine medical had given Russell enough reasons to clean himself up a little, and he had started hitting the gym. He's still a large specimen, but there is now more muscle under his shirt than fat.
Ray finds himself drawing crosses on the hardwood floor with the sole of his shoe. He wants to tell his friend to get on with it. To shit or get off the pot. To give him what he's come for.
"If it had been you running the show, Col . . ."
Ray gives an understanding nod. Condones his senior colleague for being deliberately obstructive. For holding back. For not telling Pharaoh what he's about to spill to one of the good old boys . . .
"Just a steer, Aidy," says Ray, turning away at the sound of the front door opening. A man in a suit pokes his head in. Takes a look at the virtually empty bar and then withdraws. The door bangs again. "We all know it should have gone to you. But it came to us, and Pharaoh played it her way. Maybe she didn't know how much you know. I won't make that mistake, mate. Straight to the horse's mouth."
Russell knows he is being flattered, but the smile on his lips does not suggest that he minds. He takes a longer swallow of beer, and then leans forward, bringing himself closer to Ray.
"I've skimmed the operational reports," he says, and then sneers. "She didn't know half of what she was dealing with. They were never going to be there. This snout she had? This lass who told her where to raid? She's bottom rung, mate. And if she had any sense, she'll have told her bosses what she told Pharaoh the second she had the chance. They'll have cleared out about thirty seconds after she told the brass she had located the latest farm. They're too well connected. We only raid what they can afford to lose. There's a system here, and she's messing with it."
Ray wrinkles his brow. His eyebrows meet in the middle and he blows air into one cheek, as if he has a toothache.
"There's a deal in place? An arrangement?"
Such things are not unknown. During his career he has worked with plenty of senior officers who view their criminal targets as little more than professional associates. He has known officers who have turned a blind eye to wholesale criminality in exchange for being allowed to nab some headline-grabbing, midlevel dealer.
Russell waves the suggestion away. "It's not like that. Not like it used to be. You and I both know the Vietnamese have been looking after cannabis for years. It's almost their national dish. Nobody else even bothers. That's what they do. It's like Colombians and cocaine. Some people have just got a gift for it."
"So who's making waves?"
Russell sighs. "Pharaoh knew a bit of this, but she's only scratched the surface. You think those blokes who got nail-gunned were the only ones? Man, the shit we've heard! We don't know much about where they've come from or how far their ambitions stretch, but there's a new outfit which has got the Vietnamese running scared."
"The Vietnamese don't scare."
"Maybe it's not fear," says Russell irritably, gesturing with his pint glass. "Maybe it's more pragmatic than that. We know these new lads have muscle, but maybe they have money, too. The Vietnamese farmers don't make enough money to even pay the electricity bill on the places where they set up shop. They've got bosses. Paymasters. Maybe our new boys have bought the local labor and are backing it up with a few demonstrations."
Ray raises his eyebrows, expecting more. "That's a lot of maybes, Aidy. You're in charge of the Drugs Squad, mate. Come on, now."
Russell bristles. "You seen the last quarterly reports? You want to know how many raids we've successfully carried out these past few months? It's almost daily. The scale of the operation is enormous. Somebody with a bit of vision has realized that most of the force don't give a damn about cannabis, and they've taken advantage of that to make some serious money."
"Where's your intel coming from?" asks Ray. "The raids? Your targets?"
"Some we get from street dealers who've got an ear to the ground. Deals with people trying to shave a few months off sentence."
"And the bigger raids? The ones that make the papers?" Ray fancies he already knows the answer.
Russell looks into the bottom of his empty glass, but Ray makes no move to fill it. The senior officer sighs. "Anonymous tips," he says. "Good ones. Straight through to the mobile phones of me or a couple of my lads. They're never on the blower more than a few seconds. Just give us an address and a time. We hit the place and the cameras start flashing. Nice picture opportunity, and the chief constable is happy."
Ray sips his wine. Decides it hasn't taken the edge off what he's feeling. He finishes it in a gulp.
"They're pacifying you," he says. "Either that or taking out the competition."
Russell shrugs. Looks at his glass again. Ray turns to the bar and nods to the young, skinny lad in a black Ramones T-shirt, who is fiddling with his mobile phone behind the bar. Signals for two more drinks.
Russell doesn't speak again until he is wiping beer from his upper lip. "It's all part of the business these days," he says. "You know as well as I do we can't get drugs out of our lives. We can't get them off the streets. We can't get them out of bloody prisons. It's about showing that we're trying. And we are trying, Col. But with the resources I'm putting into cannabis raids, the smack heads and coke dealers are having a ball. And how do I know the lads who've taken over the Vietnamese workforce aren't looking after the harder stuff, too? I know bugger all about them except they're hard as nails, very good, and very well informed."
Ray purses his lips. Holds his tongue until the barman has brought him his change. Pockets it and then places his palms flat on the table, as if taking part in a séance. He appears to be thinking.
"Alan Rourke," he says. "I don't take him for some criminal mastermind."
Russell gives a smile. "He's hard, I'll tell you that much. Nearly as hard as his old running partner. The stories I could tell you about him and Giuseppe Noye—"
Ray waves a hand. "Tell me the story I want to hear. Why is Rourke's fingerprint on the bottle that smashed against a police van outside a bloody cannabis factory?"
Russell squeezes one hand with another. "The travelers aren't much different from the Vietnamese," he says, rapping the table with his knuckles to emphasize the words. "They do their thing. They stay in their own community. They fracture some laws and they cause us headaches. That's always been the way. But we live in a multicultural society, Col. Sometimes they branch out."
"And Rourke has branched out? He's connected to this?"
Russell shrugs. "He's a known commodity. He's a tough guy with respect and backup. You've seen the witness reports. It was white guys. Big white guys. What's to say it's not the gypsies who've moved up in the world?"
Ray considers it. Thinks of Rourke: brimming with confidence and utterly unafraid as he kept his silence across the interview table, with a look in his eyes that suggested he would rather bite his tongue off than give up his secrets.
"Where is this coming from, Aidy? I wouldn't even have known you knew the fella if he hadn't mentioned your name. He only did that to see if I'd bite. To show he knew more than me. Made me look a right cunt. Why did you pay him a visit?"
Russell takes another swallow of his drink. Says nothing.
Ray gives a nod of understanding. "You got another call, didn't you? On the mobile. Same voice that tips you the wink on which factories to raid is now telling you who to put the frighteners on. Bloody hell, Aidy, they may as well have you on the payroll. What did they tell you?"
Russell looks down at the numbers on his arm. The flashes on his uniform that remind him he is a very senior officer. "Just told me that Alan Rourke was worth a look. That if we had a bit of a lean on him, we might get a few more phone calls and a few more raids. He's got some lad staying with him. The voice on the line said that we should leave him alone. The teenager. That we should remind Rourke that he's not the boy's father, and that the lad has friends."
Ray pushes himself back from the table. He is always pissed off. Always aggressive. Can only hold himself back for so long. It's taking an effort now.
"You may as well be on the payroll."
Russell reddens. "How can I be working for somebody when I don't know who they are? It was the right decision. A professional decision. I passed on a message, marked his card, and the phone calls kept coming."
Ray isn't listening. "The lad," he says. "It's the same one. The one who set the dogs on Pharaoh. Yes?"
Russell nods. Takes a swallow of beer. "Soon as I saw the description and Rourke's name I decided to call you. You just called me first. I would have rung, Col. Now you're looking after this, we'll get somewhere."
Ray rocks back, the chair on two legs. "How did Rourke react when you went to see him? What did he have to say?"
"Did all but laugh in my face," he says. "Didn't scare. Didn't flinch."
"And the lad? He was there?"
"Turned up as we were leaving. Climbing out of some flash bloody four-by-four as if he was a Lottery winner or Wayne fucking Rooney."
Ray drains his glass. His gaze meets Russell's. The senior officer looks away first.
"I know why you didn't tell Pharaoh what's going on with the Vietnamese," says Ray softly. "It's because you didn't have much to tell without making yourself look like a bloody gopher for some drug dealers. I even get why you went to see Rourke. What I don't get, Aidy, is how you can't see what's going on."
Russell narrows his eyes. "Go on, then, smart-arse."
"The lad's the bloody player. The teenager. He's the one who's valuable to your friends at the end of the phone. Rourke's up to his eyes in all this, but he's overstepping his boundaries. It's the lad who's connected. It's his gyppo connections that are causing your friends the headache."
Russell seems unconvinced. "You're a fucking long way from proving any of that."
"I just need to know it, not prove it," says Ray, putting all four chair legs flat on the floor.
"What's your move?" asks Russell, looking at his watch and clearly deciding that four pints is enough. "The lad's top priority, yes?"
"He set the dogs on Pharaoh. So we want him. But we want to know what's in his head, too. It's all linked. I'm getting a fucking headache here . . ."
Russell gives a smile, and any tension that existed between them seems to evaporate. Both are old-school. Ray would never even consider grassing another officer up. But he does at least recognize that he suddenly has a useful ace up his sleeve.
"The next time you get a call from your friends . . ."
Russell's face drops back into a scowl.
"Go on."
"I want in on the raid. I want to talk to whoever you pull."
Russell seems to think about it. Gives a quick nod. "It could be sooner than you think. It's been a few days . . ."
"Whenever. I'm not planning on getting much sleep."
"Where are you heading next?"
Ray gives a little smile, pulling his phone from his pocket. He punches in a number with long, yellow-stained fingers.
"Shaz? Roll off whoever you're on and get yourself in the car. We're going on our holidays, love. Off to visit a few lovely little caravan parks."
From the other end of the line comes a barrage of confused questioning. Colin Ray shushes her. "The ginger lad. He's run to what he knows. He's back with his people. And we're going to go and make him unwelcome."
4:14 P.M. HA'PENNY BRIDGE WAY, ON THE VICTORIA DOCK ESTATE.
MCAVOY, resting his head on the steering wheel, listening to his wife cry. Her sobs sound in tandem with the rain that beats on the glass.
"Come home," she gasps. "Please."
McAvoy looks out through the waterfall of rain that runs down the windscreen. The air is slate gray, the sky reaching all the way down to the deepening puddles and burst drains that are starting to obscure the roads and pavements.
"One stop, Roisin," he says again. "One more stop and then I'll come and take over."
"She won't stop crying," she says again, and the desperation in her stabs into his chest like an icicle.
"An hour," he says, closing his eyes.
"She's screaming," begs Roisin. "I can't . . ."
She hangs up. She has never hung up on him before. He begins to call her back. His phone rings again. "Roisin . . ."
"No, lad. DCI Ray. Where the bloody hell are you?"
McAvoy winds the window down a crack. Lets some air into the vehicle. Watches the reeds sway in the tatty duck pond that separates the two modern apartment blocks that loom over the patchwork of semi-detached properties.
"I've been—"
"Don't care," says Ray. "Anyway, Tremberg's been bending my ear about getting in touch with you, so I am, because it might make her shut the fuck up. Got her well trained, that one, ain't you? We spoke to Alan Rourke. Gypsy bastard barely said a word. No ID on the lad who set the dogs on Her Highness, but Shaz is in with Rourke again now and is promising to stop his dogs getting the chop if he's a bit more helpful. We've got fuck all on the Vietnamese or the petrol bombing. Nowt that would interest you, anyway. Reckon we should be checking the England cricket team from the way they threw that thing. Bloody awful bowling action. Anyway, Ben Neilsen's over in Doncaster rounding up CCTV from when the Land Rover was nicked. Turns out they've got other vehicles missing, too. Merc, Audi, a Lexus or two. In Doncaster! You feeling in the loop now? Right. Fuck off. Bye."
McAvoy is too tired to subject the exchange to analysis. He just nods to himself. Eventually closes his phone.
Beyond the glass, the storm is getting worse. The news headlines on Radio Humberside were starting to sound a little hysterical when McAvoy tuned in on the drive here. It is only a few years since Hull suffered near-biblical floods. There are still people living in caravans in their front gardens. Every time it rains the city holds its breath. The weather was top story, ahead of an appeal for witnesses following a nasty attack on Morpeth Street last night. A nineteen-year-old girl is in hospital with severe head injuries after being set upon by an unknown assailant around midnight. Friends have described her as a happy-go-lucky girl who would do anything for anyone. A spokesman for Humberside Police said it was too soon to speculate whether the incident could be linked to the escalation of violent crime in the city, rumored to be linked to the drugs trade.
It had made McAvoy grimace. Made him wonder, too, why he was chasing so hard after answers about Simon Appleyard when the living were being mercilessly persecuted.
With some difficulty, McAvoy hauls himself from the car. He is far too big for its cramped confines. Feels as though he should cut a hole in the roof to poke his head through. Worries he will pull the door off every time he grips the handle.
The wind and rain refresh him briefly. He looks up at the slate sky and allows the downpour to soak his face. Slicks back his hair and licks the collected droplets from his lips.
The flat he is after is on the ground floor, offering a decent view of the dense mess of reeds that all but obscure the water of the large rectangular pond. McAvoy and Roisin had considered getting a place on this estate when they first moved to the city. It is only a ten-minute walk from the center and was designed with upwardly mobile families in mind. The houses are small but neatly built and well looked after, but the sense of inner-city community that the developers had been aiming for when they called it an "urban village" has never truly materialized. Many of the properties are let to young flat-sharers by distant landlords, and there are too many FOR SALE and TO LET signs to suggest it is a place where people are desperate to stay. It is beginning to look tired, not least because of the moonscape that many of the streets are beginning to resemble, thanks to the widely predicted subsidence. A former working dock, it was landfilled to make way for the housing development. There are fears that the whole estate is beginning to sink.
McAvoy takes care on the damp timbers of the footbridge. Looks for signs of life in the duck pond. Wonders if there are any ducks hiding in there or if they, too, have left the estate for more comfortable accommodation in the East Riding villages.
He checks his notepad to confirm the address. Realizes he is already standing outside the doors to the apartment block and cannot put this off any longer. He rings the bell.
Seconds go by. Water drips down the back of his neck.
A burst of static from the intercom.
"Hello?"
"Councillor Hepburn," says McAvoy, louder than he had intended. "I'm a policeman. Could I have some of your time?"
There is another pause.
"Come in."
The door buzzes and McAvoy pushes it open. He finds himself in a wide, gray-carpeted, buttermilk-painted atrium. There is a set of stairs at the far end, and three brown wooden doors set in the remaining walls.
Number 29 is opening.
He recognizes the man in the doorway from the newspaper and TV appearances. He is in his late forties, with dyed blond hair swept back from a long face that nature has made unremarkable, but vanity has colored. The tuft of hair beneath his lower lip is dyed peroxide blond, and his sideburns are razored to a neat, almost devilish point. He has two rings in his left ear, and McAvoy doubts that his eyebrows are naturally as jet-black as they appear.
He is smiling broadly, a politician's grin. He is wearing a purple V-neck sweater and loudly checked trousers that are more Rupert the Bear than Harris tweed. He is in relatively good shape, but the shape of sagging pectorals can still be made out through the material of his sweater.
"Plainclothes, eh?" Hepburn asks warmly. "Intriguing."
McAvoy looks down at himself, standing in a puddle, all but raining on the floor.
"Still drizzling?" asks Hepburn.
McAvoy forces a smile.
"Come in," Hepburn says. "I'll get you a towel."
He steps back inside the apartment and holds the door open. McAvoy wonders if he should offer to remove his boots, but remembers the fuss he had getting his shoes on this morning, and very much doubts that his socks match.
He follows the councillor inside and down a short corridor decorated with black-and-white prints.
He is led into a large living room, painted terra-cotta and designed the vaguely Javanese style. The blinds are raffia, and the prints on the wall are of elephants and traditional fishing skiffs, pots of spice and gentlemen's club antique maps. There is an expensive-looking rug on the cream-carpeted floor, and the red chesterfield sofa gives the place the feel of a British Empire hotel gone slightly to seed. The two-seater table at one end of the room supports a huge vase of lilies. The other end of the long room is given over to a large flat-screen TV.
"Paula, can you bring me a towel, love?"
Hepburn shouts this last as he throws himself down on the sofa. There is a laptop computer on the middle cushion, and a mobile phone on the floor.
"What can I do for you?"
McAvoy is about to speak when a woman appears in the doorway. She is around the same age as Hepburn, and almost as imposing a physical specimen as McAvoy. She is pushing six feet tall, and broad across the shoulders. Her hair is a collage of different shades of blond and is cut in a choppy, neck-length bob that looks to McAvoy's inexpert gaze as if it was expensive. She is wearing a white blouse and cropped trousers with wedged high heels. She hands McAvoy a fluffy yellow towel, which he takes gratefully, and uses to dry his face and hair.
When he has finished, he does his best to smooth down his curls, and is grateful that he is facing away from the mirror that dominates one wall.
"Paula," says Hepburn to the woman in the doorway, "this is . . ." He pulls a quizzical face. "Did you tell me?"
"Detective Sergeant McAvoy," he says, and is embarrassed by the squeak in his voice.
"McAvoy," says Hepburn thoughtfully. Snaps his fingers, as if placing the name. "Indeed. This is Paula."
"How do you do," says McAvoy, extending his hand.
Paula gives a curt nod. Raises an eyebrow at Hepburn.
"Coffee," she says, and it does not sound like a question. She turns her back. Leaves them to it.
"So," says Hepburn again, "what can I do for you?"
McAvoy realizes he has been pressing his lips together. Takes a breath.
"Councillor, I probably shouldn't be here, but today I received information that suggests you are the target of some kind of journalistic investigation designed to discredit you."
He stops. Hepburn widens his eyes, and the playful smile on his face seems to grow.
"Really? Do tell."
"I had reason to speak to a reporter from a national newspaper about another matter. He informed me they are planning a story looking into some criminal connections in your past."
Hepburn gives a whistle.
"Anything else?"
"There is some suggestion that your nightclub has been financed by drug money."
Hepburn is openly laughing. McAvoy feels sick.
"Councillor Hepburn?"
The other man pulls himself off the couch. Stands up straight and gives a wide grin. "And you're bringing this to me because . . ."
McAvoy allows himself to look baffled. The answer should be obvious. "Because that's not right."
Hepburn pulls himself together. "But you don't know me," he says, looking straight at McAvoy. "People get shit written about them all the time. I got elected by people who either wanted a drink at midnight or liked the idea of pissing off Labour. Seriously, Sergeant, another story about me being a bad boy is not going to destroy me. Might even be good publicity. Come on, what's the ulterior motive?"
McAvoy feels his cheeks flush. He had not expected to have his integrity questioned. Is appalled to have been seen through so easily. Fears what it says about him that he could so easily be identified as a liar.
"Come on, Sergeant," says Hepburn.
McAvoy meets Hepburn's gaze. There is a fierce intelligence in his blue eyes. He remembers his TV appearances. His quick wits and sharp tongue. Realizes he was wrong to blunder in here so ill prepared and cack-handed.
"Simon Appleyard," he blurts, and then has to all but wrestle with himself to stop his right hand coming up to his face in a childish show of regret.
Hepburn narrows his eyes. "And who might that be?"
"Simon Appleyard was found hanged at his home last year. We are looking again at the circumstances surrounding his death. I'm going through the numbers in his telephone. Your number is among those that were stored."
Hepburn shrugs, and it is not an unfriendly gesture. "I'm sorry, but I really don't think I know the name. I'm a public figure. I run a club. I change phones quite a lot . . ."
"This is the telephone you were given by the city council."
"Ah," says Hepburn, with a grin. "Right. Had that about a fortnight, then it went missing. Probably nicked from the club. Felt a right fool reporting it missing. Been using my personal one ever since. Got two SIM cards in it. Really snazzy . . ."
"And you don't know the name Simon Appleyard? He was in his mid-twenties. Tall. Ran a line-dance class . . ."
Hepburn shakes his head.
McAvoy plows on.
". . . peacock feathers tattooed all over his back . . ."
For a fraction of a second Hepburn's smile seems to die at the eyes. Then it is back. Wide. Charming. Naughty.
"Leave me your card, Sergeant," he says, still being friendly. "I'll have a think. Get in touch with you."
Paula reappears at the door. She is not holding coffee cups. She is not smiling.
McAvoy looks at Hepburn, whose raised eyebrows represent a friendlier version of Paula's hard stare. He is being invited to leave.
"My card," says McAvoy, handing the councillor a damp rectangle of blurred ink. "I'm sorry to have bothered you. I just thought you should know about the reporter . . ."
Hepburn is nodding as he gets to his feet. Makes a show of shaking the officer's hand. Is only a pace behind him as he corrals him out of the door, past the bulk of his unfriendly companion, and into the hall.
"If I hear anything more . . ."
"Thank you, Sergeant," says Hepburn.
The door shuts behind him and McAvoy finds himself back in the lobby.
He feels his cheeks burning, but this time it is with temper instead of shame. The way she looked at him. The playful little smirk on Hepburn's face. He had felt like a teenager caught out in an untruth. Had been made to feel a fool.
He would have taken such feelings as penance, were it not for that moment, that flicker of recognition, that caused the councillor's grin to lock in fleeting falsehood.
McAvoy came here in the hope of unearthing something to vindicate his instincts. Hoped to find a whiff of something that meant he was not wasting his time. For a moment, here in the heat of tired irritation and embarrassment, he feels he may have found it.
The rain is less furious as McAvoy lets himself out of the doors and into an ankle-deep puddle. He barely pays it any heed. He pulls his phone from his pocket. Calls Roisin.
"Darling," he says, when she answers on the eighth ring. "I'm on my way home. I'm so sorry."
They talk for five minutes. Her apologizing. Telling him she understands. Him begging forgiveness for his remoteness. His uselessness. Telling her, excitedly, about his five minutes in the home of Councillor Hepburn, and his suspicions that the man knows more than he is willing to admit. That there is a case here. A real investigation.
He is still talking when his phone beeps, and he tells Roisin he will have to go. He will see her soon.
"Sergeant McAvoy," he says. "Serious and Organized."
"Sergeant. This is Assistant Chief Constable Everett. I want you in my office right away. There has been a complaint that you are harassing a senior member of Hull Council."
The color drains from McAvoy's face. He closes his eyes.
He can already hear the tears in Roisin's voice.
• • •
THE MAN in the tan leather jacket is losing his temper. Suzie is no expert in body language, but from the shape of his shoulders and his white-knuckled grip on the counter, she senses an imminent explosion of anger.
"Am I speaking a different language here?"
Suzie, sweating despite her damp clothes and beginning to feel a little feverish, shares his pain.
She looks at her own cashier. Tries her sweetest smile. Hopes her exasperated grin will find a kindred spirit. Gets nothing in response. The lady behind the glass is younger than her, but has the sour face and unmoving expression of a lifelong doctor's receptionist.
"It was eighteen p," says Suzie, again. "Eighteen p! That's how much I was overdrawn by. You charged me for the letter you sent to tell me that, and then charged me for three days of unauthorized overdraft use. I would have cleared the eighteen p, but the charge put me fifteen pounds into the red, and then the extra charges . . ."
She stops. It is the fourth time she has tried to persuade the cashier behind the glass that she is being treated unfairly. She can feel the back of her neck getting hot and prickly. The injustice of it is making her words catch in her throat. Misery sits in her stomach like a snowball.
"It's my money," says the man in the leather jacket, and his voice has increased in volume. "You look after it for me. That's your job."
The neighboring cashier is equally intractable. "Without a passport or driving license, we can only give you five thousand pounds."
"But it's my money!" he shouts.
Both debates have been going on for some time, and the lengthening queue in the bank is watching the two exchanges with a mixture of impatience and interest.
Suzie stands in silence, shaking her head and trying to think of another collection of words that might make the cashier change her mind. She fears that her eyes are on the verge of filling with tears.
"It's an issue of your security," says the adjacent cashier.
"The charges are all explained on the website," says Suzie's tyrant.
The man in the leather jacket is looking around, as if for an ally who can help explain to him the workings of this insane and alien world. He looks at Suzie. Their eyes meet. He is a handsome enough man, pushing forty, and his clothes are casual but expensive. His face softens a little as he takes in her red face and sodden hair, damp clothes, and watery eyes.
"Can you believe this?"
Suzie shakes her head. Turns to the cashier.
"I'm having a bad time," she says softly. "It was eighteen p. And now it's pushing a hundred pounds. Just with charges. Can I give you the eighteen p? Or a token gesture, or something. I can't get back into credit until payday . . ."
"The charges are all explained on our website."
Tears come. Unbidden, Suzie realizes her eyes have overflown.
Salt water runs down her powdered cheeks and her shoulders start to shake.
"I'm sorry," says the woman behind the glass, with the same expression she has worn since Suzie reached the front of the queue and asked for a little leniency.
"I'm on my lunch," sobs Suzie, as if this might make a difference. "I normally sit in a little garden . . ."
She cries openly into her hand. She hates the pathetic picture she knows she must be presenting. Hates being so feeble. Wants to turn and run, to hide until somebody finds her and promises that it will all be better soon.
She has still not had the courage to turn on her telephone. Has heard nothing more about the man she left to die.
"We own you anyway," says the man in the leather jacket, turning his attention from his own cashier to Suzie's. "The taxpayers. You belong to us."
He looks around at the queue behind him, as if trying to drum up support for a revolution. He gives a sigh as he takes in the collection of damp shoppers and office workers, shivering in wet clothes and waiting for their turn to go and shout impotently at the staff.
"The rules are there on the website."
Suzie stiffens as the man moves closer. He ducks down to place his face in her line of vision. He is looking into her eyes. It is a caring stare, devoid of malice or threat. He wants to see if she's okay.
"They won't listen," he says to her quietly. "Charges, is it?"
Suzie tries to smile. She feels wretchedly miserable.
"How much to get back in credit?" he asks, leaning in close enough for his words to tickle her wet earlobe and neck.
"Nearly ninety pounds."
The man gives a nod. He puts a hand in his pocket and takes out a roll of notes. "Take it," he says, handing her five twenty-pound notes.
"What? No . . . !"
Suzie's chest constricts. She begins to protest. To tell him that it's not his problem. That he has problems of his own.
The notes have found their way into her hand. A damp smile has made its way to her face.
"Please," says the man softly. "Let me."
"I don't—"
"I don't want these bastards taking another penny off anyone. Please."
Suzie, flustered and unsure, turns to the cashier, who is struggling to keep the grimace off her face.
"Here," she says, through a face of tears and snot. "I'd like to make a deposit."
The man does not return to his own cashier. He leans against the counter, and looks at Suzie with amused affection.
He looks her up and down. Likes the view.
"Does that buy your phone number?" he asks.
Suzie freezes. Gives a girlish, embarrassed smile that Simon would have mocked her for.
"I know," says the man, raising his hands. "It spoils the selfless gesture."
"I'm not sure . . . ," she begins.
"Take mine," he says, and writes a scrawl of digits on the back of a deposit slip. "No pressure."
Suzie looks up again and finds herself blushing.
"I don't know if I'll call," she says, picking up the number.
"I'll be hoping," he says, and turns away.
"Bye," she says, shocked and embarrassed.
"Bye, Susan," he says, and is gone.
Suzie stays at the counter for a few seconds, wishing she had somebody to share this odd moment with. She wonders whom she will tell about the handsome man who came to her rescue. Whether she will log on to Facebook and tell her friends. Thinks of pulling out her phone.
Freezes, as paranoia strikes.
Her name. He knew her name!
She turns from the counter and pushes through the crowd, down the half-dozen steps and through the double doors onto Whitefriargate. She looks this way and that, squinting in the rain, trying to make out the shape of the leather-jacketed man.
Cautiously, taking care in her flip-flops on the cobbles, she splashes through a puddle and runs up the street.
"Hey," she shouts, and finds herself giving a peculiar little laugh at the indignity of the scene. "Hey!"
Up ahead, outside the store where she and Simon had bought the _Twilight_ box set and then argued over custody, she spots him. She half falls into his back, clumsily grabbing his shoulder.
He turns. Surprised at first, then pleased.
"How did you know my name?" she asks, breathlessly. "You said, 'Susan.'"
The man rubs a hand over his face and screws up his eyebrows.
"What?"
"My name. You knew my name."
He looks around him, almost as if searching for a hidden camera crew. When he finds none, he looks at her and closes one eye as he speaks, as if wincing at the words.
"It's on your bank card," he says, gently but confused. "What's wrong?"
Suzie breathes out, hard. Fresh tears prick her eyes.
"I'm sorry," she says, looking down at the ground.
The man stands there for a moment.
Suzie is a statue. Looking down at wet cobbles and her own soaking, dirty toes.
Then she feels his arms around her.
Her shoulders shake and she weeps against his chest, clinging to a stranger in the pouring rain.
IT IS PUSHING eleven p.m. when McAvoy opens his front door.
_Home_ , he thinks gratefully. _Thank you._
He is so tired he can barely lift his feet. Too drained to notice that the rain has stopped or to comment on the brightness of the near-full moon, which hangs like a disk of crumpled parchment in a blue-black, cloudless sky.
Too exhausted to remark upon Roisin's absence. She is normally here, smiling in the doorway, waiting for him. Waiting to kiss him home and slide herself into his embrace.
"Roisin?"
He finds her in the darkened living room, curled up on the sofa. Finlay is wrapped around her, face-to-face, snoring softly into her open mouth. He is wearing a woolen hat, pulled down over his ears. McAvoy takes it as a sign that his eldest child has grown tired of his sister's cries.
"I'm so sorry," he whispers, and hopes that it will cover his multitude of sins.
Quietly he heads upstairs, avoiding the creaking steps.
Lilah is lying spread-eagled in his bed, prisoner in a rectangle of pillows. She is healthily pink-faced and her sleep looks a lovely and peaceful thing. He wants to kiss her. To smell her head. To say sorry for not being what she needs. He persuades himself not to wake her. Tiptoes back downstairs.
Roisin is disentangling herself from Fin. She looks up as he appears at the door.
"Hi." She smiles drowsily. "What time is it?"
"Too late," says McAvoy, crossing to her. "I'm so sorry."
He bends down and crushes her in an embrace.
"Aector, easy . . ."
He is holding her too tightly. Lets go. Tips her face upward with his index finger and stares into her eyes. Again. "I'm so sorry."
Her smile, though tired, is warm and genuine. She kisses him.
He tastes the sleep in her mouth. Tastes the black currant juice she and Fin have shared. The tang of hand-rolled cigarettes.
The last few hours have been torture, made worse by the cold agony of separation.
Everett swallowed his story about wanting to warn Councillor Hepburn about the newspaper investigation. McAvoy had even been commended by the tall, ferrety man for his diplomacy and foresight, and had managed to keep his mouth shut about Simon Appleyard. He had been starting to let himself think he could still make it home in time to bathe the little ones when he had been asked to cast his expert eye over a speech Everett was due to present. It took hours. McAvoy has many times rued the day he first put together an expenditure report for a committee briefing. It had been coherent, simple to follow, and correctly spelled. In Everett's eyes, it had marked him out as a borderline genius and the go-to guy for any job that required somebody who doesn't move his lips when he reads.
"How was it?" Roisin asks quietly, leading him into the kitchen so as not to wake Fin. "You a naughty boy?"
McAvoy manages a little laugh. "I think Hepburn has friends in high places," he says.
"Let's hope they're going to jump," she replies, setting to work making him a sandwich with fresh bread and homemade jam.
"I can't have been out of there thirty seconds before he made a call," he says, taking a swig of the glass of milk she hands him. "He was okay when I was there."
"Arsehole."
She hands him his sandwich. Watches him take a bite. Seems pleased with his grunt of appreciation.
McAvoy notes she is wearing the same clothes she had on this morning.
"I could bathe you," he says through his dinner. "Candles. Wash your hair. Shave your legs. Paint your nails."
Roisin grins. "Sounds lovely," she says. "But let's just go to bed. I've got a surprise for you."
McAvoy wonders if he has the strength to keep pace with whatever surprise she has planned. He is about to suggest they just hold each other when she gives a bright smile. "Wait here," she says, and runs from the kitchen.
Puzzled, McAvoy finishes his sandwich. Drains his drink. Takes a chocolate biscuit from the tin by the microwave and polishes it off in a bite.
Like a deflating bouncy castle, he folds himself into the kitchen chair and drops his head to the table. He closes his eyes. Treats himself to a moment devoid of thought.
It hits him then. Just how reckless he has been. How disloyal and vain. He has been pursuing proof of his own instincts. He has been trying to vindicate a feeling. While he has been trying to prove that he can sense a crime the same way his father can smell the nearness of snow, a real investigation has gone tits up, and the only colleague who truly believes in him has been attacked by dogs.
_Simon Appleyard._
He decides it's time to make the case more official. He will approach one of the detective superintendents in regular CID tomorrow. Tell him there is a case to be looked into. Take the withering looks and jaded sighs and simply insist that the investigation is carried out, and properly.
"Don't be cross."
Roisin is standing in the doorway. She is smiling and has changed into a silkier nightdress. Her hair is piled high on top of her head, exposing her dark, scented neck.
McAvoy blinks a few times, muzzy-headed. He smiles as he takes her in.
She holds out her hands.
On her left palm is a mobile phone.
"You got him one, did you . . . ?" begins McAvoy, then stops, his smile freezing, as a picture surfaces in his addled memory.
"I'm sorry I was so mean," she says, and walks toward him, waiting for her hug.
McAvoy's mouth falls open and the color bleeds from his face.
His wife is holding an unfamiliar mobile phone.
He doesn't know whether it is instinct, or simply the hopeful, helpful look on his wife's face, but he knows at once it belongs to Councillor Hepburn.
• • •
"THAT'S HIM," says Suzie, pointing through the railings. "Trevor, say hello."
Beside her, in the dark, she can hear Anthony smiling. It is an odd feeling. She can sense him staring. Grinning at the side of her face. He has been looking at her with affectionate bemusement much of the evening, and now appears to be enjoying the note of sleepy drunkenness that has entered her voice.
"That's where I sit," she adds, pointing at the bench in the courtyard garden. "Every day. Me and Trevor, setting the world to rights. I do most of the talking but he's a great listener."
Anthony scratches his stubbly chin and gives her an encouraging smile.
"He's a lovely tree," he says, and then has to stifle a little grin. He has never really imagined having to use such a phrase, and wonders what his mates would think of this strange, colorful girl. He finds himself hoping they will find out.
Their date has gone relatively well. Suzie called him from the work phone midafternoon to apologize for being weird and to reassure him that she wasn't mental. He had laughed and insisted she could only make it up to him if she met him for a drink.
It is now just gone eleven, and they have the Old Town to themselves. The endless rain seems to have swept the city clean, and there are no raised voices or passing cars to break the perfect silence that exists here in this darkened pocket of Hull.
Suzie is wearing a long blue dress onto which she has embroidered a large felt heron. She is wearing a beret and her earrings are owls in cages. It took her a long time to get ready. She was excited and scared, and wished she had somebody standing behind her telling her she looked nice, would have a good time, and that there was very little chance of having to jump out of the way of a speeding four-by-four while mid-fuck.
The alcohol in her system coupled with the bracing night air is making her feel teary and tired. She is overemotional. Confused. She has talked endlessly. Managed to keep the conversation away from one-night stands and casual sex without really knowing why. She wonders if she is ashamed. Or simply cautious about driving away this nice man by revealing who and what she really is.
She had been pleased he wanted to meet Trevor.
"I tried to persuade myself he was Simon," she says suddenly. "But he couldn't be, could he? Trevor's been here for years. Simon hasn't been dead long. What do you think? Could it have sucked up his soul?"
As she asks the question, she leans her forehead on the damp brick. She closes her eyes. She has drunk too much, eaten nothing, and feels truly intoxicated on the newness of this evening. She has enjoyed talking. Letting her mouth run away. Unburdening herself. She feels somehow free tonight. Anthony is nice. He seems to find her interesting.
Anthony puts an arm around her shoulders and gently pulls her back from the wall. He bends down a little to better look into her eyes.
"I'm sure he's happy, wherever he is."
Anthony has not followed her stories perfectly. Suzie is not the most linear of narrators. He understands that her best friend died some months ago and that, since then, she has felt isolated and alone. He is not sure how to ask more without prying, or what he would do with the answers.
"Do you think?"
He nods as solemnly as he can.
"You're nice."
She wonders if this is what dates are usually like. Her life has not been like this. She was with her first boyfriend from childhood, and segued into promiscuity at the relationship's end. She has never been romanced. Tonight, sharing a couple of bottles of wine in the attractive Russian vodka bar at the bottom of Whitefriargate, has felt pleasantly bizarre. She feels more nervous, here and now, than during her countless trips to sex clubs. In that environment, she has never found herself timid or unsure. Each patron came with one goal in mind.
Here, on a regular date, with a nice man who wants to know more about her, she feels twitchy and confused. She doesn't understand what he wants.
"I'm nice?" he asks, pretending to be offended. "Just what every man wants to hear."
Suzie smiles. She is feeling tired. "Nice in good ways, I mean. You wanted to see my tree . . ."
"It's a grand tree."
"He."
"He's a grand tree."
Drunkenly, impulsively, she leans forward and kisses him. She catches him just below the lips, and presses too hard, hurting both of their faces.
"I'm sorry," she says as she pulls back.
"Don't be," he replies, laughing and rubbing his lips.
They look at each other, awkwardly, for a moment. Anthony is thirty-nine years old and, up close, it is clear he shaves his head because he is balding anyway. He is wearing the same tan leather jacket he had on in the bank, and smells faintly of some kind of aftershave balm. He is an attractive man, and had looked embarrassed while telling her that he makes his living hiring out play equipment for children's parties, and renting out mobile discos. He has two children from a failed marriage, and lives alone in an apartment on Victoria Dock. They are walking distance from his home.
"I'm sorry I've prattled on," says Suzie, suddenly unsure what to say next.
"I like the way you talk. It's soothing."
Suzie looks at him again and wonders what to do. If this were a club night, she would simply take him by the hand and lead him to a private room. She is willing to have sex with him. But this feels different. She would quite like to kiss him, too. Wants to know how it would feel to have his arms around her and her head on his chest.
"I have more wine at home," he says with a slight smile. "It's not far . . ."
Suzie looks down at her feet. She is wearing her flip-flops and is standing in a puddle. It feels nice. If she wriggles her toes, she can feel grit on the soles of her feet.
The sensation feels familiar.
She is suddenly back on the rest stop at Coniston. She is being fucked over the bonnet of a car by a stranger, while somebody slams his foot down on the accelerator . . .
She is at the sex club. She is on the floor; one man inside her, three more waiting their turn. Simon, leaning against the wall with a rolled-up cigarette, talking to a handsome man with gray hair and a flamboyant shirt, open to the waist.
She is crying into her phone, unable to hear Simon's auntie's words of condolence as she tries to digest the news that her best friend is dead and had never trusted her enough to share his pain . . .
Suzie flops back against the wall. Her eyes fill with tears. She does not know what she wants or who to be. She just knows she misses her friend and that her life has felt empty and lonely ever since he hanged himself in his kitchen.
"Nobody understands," she murmurs.
She needs to feel alive. She needs to close her heart and open her legs. She does not need love, she tells herself. Does not need to be held, or kissed, or praised, or romanced. She needs to take her pleasures and please those who want her, and she needs to close herself down to all the anguish that threatens to spill in and out of her if not controlled.
"Suzie?"
"I'm sorry," she says, with her eyes closed. "I'm not ready for a relationship."
Anthony's face flashes with confusion. "I didn't think I was offering one," he says, and when he realizes how harsh that sounds, adds, "I just offered a drink."
Suzie cannot really hear his words. The blood is rushing in her head, and she is feeling dizzy. Sick, suddenly. Her whole sense of self has become a finger painting; all intermingled swirls of contradiction and insecurity.
Anthony is a nice man. She has enjoyed his company. He is funny and charming and seems to care. And she knows he could do so much better. She knows that she is not right for him. Not built for satisfying the heart.
"Just do what you want," she says dozily, and turns her back on him, pulling up the hem of her dress and then stumbling to one knee.
She lies there, face against the brick, tears rolling down her face.
Anthony looks down at her, bare leg and thigh exposed, dirt and brick dust streaking her pale skin; the tail end of a tattoo upon her flesh.
For the briefest of moments desire fills him. The sight of bare, youthful skin thrills him. And then it is gone, replaced by pity. More than that. Tenderness. Affection.
He sits with her and strokes her hair until the taxi arrives. When he tells it to take her to her own home, and not to his, it is with only a hint of regret.
He hopes there will be other times with this odd, kooky, pretty girl, who talks to trees and mourns dead friends and draws peacocks on beer mats with tears in her eyes.
In her drunken sleep Suzie knows there will not.
• • •
11:18 P.M. The Kingswood Estate. The kitchen of a cookie-cutter house made wedding-cake white by three days of rain. Aector and Roisin McAvoy, angry and disappointed with each other for the first time in their lives.
Even as they argue, their voices are whispers. Their row does not get loud enough to wake the baby. Their tempers do not supersede practicalities.
"I'm a policeman! This is theft. It's burglary. You robbed a member of the public . . ."
"You said he was a suspect! You said . . ."
"But I don't tell you things so you can go and do this! I tell you because I've got nobody to talk to . . ."
"I wanted to help. I'd been so grumpy and you've been working all these hours, and I thought if you caught somebody maybe you could come home . . ."
"But I'm a policeman!"
Roisin's face is flushed. McAvoy cannot tell if she is angry at him for being so pathetically moralistic, or for not just saying thank you and giving her a kiss.
He is trying to control his panic. He feels as though the back door could be kicked off its hinges by internal affairs officers at any moment. He tries to picture himself as anything other than a policeman. Wonders whether he will be kept in a special wing in prison to save him from the other inmates . . .
"I'll have it taken back," she says sadly, and McAvoy realizes she is upset purely because she had tried to do something nice and he does not like his present.
Even now, his heart thundering in his chest, his hands trembling, he cannot maintain his anger at her. He crosses to her. Pulls her in close. Feels her resist and then acquiesce. She looks up at him.
"I'll phone my pal," she says. "He'll drop it off."
McAvoy gives a nod. Tries to calm himself. He wants to ask her who took it. Wants the name and address of this criminal whom his wife seems to have no compunction about contacting.
"I don't understand," he says softly, moving away. "I never ask, Ro. Never ask you what you do to make money or the people you know. I'd never want to hear the answers. But I wouldn't know who to ring to get a house broken into, and I'm a policeman . . ."
Roisin shrugs and moves herself onto one of the kitchen chairs. When she raises her head, she is looking at him as though she is talking to a child. He is a decade her senior and has spent his adult life chasing killers, but sometimes he thinks she sees him as extraordinarily naive. He suddenly questions whether she stays with him not because he is her big, strong protector, but because she pities him as an unworldly innocent.
"Everybody knows somebody who can do that kind of shite," she says. "It's just a phone. He took it from his pocket . . ."
"He?"
"My pal. It was supposed to be a nice surprise."
McAvoy cannot help but smile. He pinches the bridge of his nose and looks again at his wife. She has put the phone on the kitchen table. It sits there invitingly.
He bends down and kisses the top of Roisin's head. Turns her face and kisses her pouting mouth.
"Next time could you just make me a lemon meringue?"
Roisin giggles.
"I knew you'd be a bit mad," she says, still grinning. "But go on, admit it, you're pleased."
McAvoy pretends to look indignant, then gives in.
"I can't look at it," he says, and wishes he were somebody else.
"Oh, for God's sake, Aector," she says, exasperated, and picks up the phone. She starts pressing buttons and pulling faces. "Ooh. Wow. Wait until you see this."
Laughing despite himself, he takes the phone from her hand. "Go on up," he says, nodding in the direction of the stairs. "Five minutes, I'll join you."
She raises an eyebrow. "I put this on for you," she says, gesturing at her short nightie. She puts her bare legs and feet on the kitchen table. "Don't let me fall asleep."
McAvoy blows her a kiss as she stands, and feels a wonderful warmth inside as she flashes him her bum on the way out of the door. He loves her enough to go to jail. Would die for her. Would rather break the law than let her think he doesn't appreciate her present to him.
He takes a breath. Locks the back door and moves to the living room. Checks on Fin, still snoring peacefully on the sofa. Removes his coat and sits down in his armchair. Closes his eyes, as if in prayer, then turns his attention to the phone.
It is an HTC Wildfire, its touch screen as awkward as all others McAvoy has tried to get to grips with. His big fingers prod at the surface, and he navigates his way to the phone's message facility.
He reads through the last few texts Hepburn has received. There is one unopened; obviously sent since the phone was taken, sent from a contact named Gwen. He flicks through those that have already been opened. Reminders about a meeting the following day from somebody named Carl. A query from Tim about whether he can run an Artois Cidre promotion at the club. A thread of half a dozen messages for P.
McAvoy reads backward, from apology to insult.
I'm sorry. So grumpy. Not your fault. Too much pressure, sometimes. Talk soon. Xx
You don't give a damn about anybody but yourself.
I have so much wrapped up in you. I couldn't just stand by. What are friends for? Xx
You've got enough on. I wasn't trying to make you cross.
You don't understand. You break my heart when you are like this. I hate you.
McAvoy carries on scrolling through the in-box. The messages were sent between five and six p.m. He flicks to the SENT ITEMS and finds only two messages in return.
Wish I could be what you need. Xx
I give a damn about you. Xxxx
He opens the phone's diary function. Leafs through Hepburn's schedule. Council meetings. Officer reports. Interview with _Mixmag_.
Then:
Plmtz, 8pm, Saturday. Birthday bash.
He closes his eyes again. Wonders if it is the right thing to stop now. Asks himself, in all seriousness, whether Hepburn deserves to have his personal details pored over like this when he has done, so far, nothing wrong.
He opens his photo files. Flicks, quickly, through hundreds of party-night pictures; all neon lights and shadows; sweating shapes and amplifiers, half-empty glasses and screaming girls.
Opens another file. Pictures from a holiday. Hepburn, in a blue Speedo, on a sun lounger with something exotic and fruity. Two young men, grinning for the camera: shirtless and tanned. A figure on a Jet Ski, far out on a blue sea . . .
McAvoy cannot help it. He opens the file marked FUN.
He does not look for long. The images are clear in their content.
Naked male flesh. Hard cocks and bare buttocks, wet mouths and body hair.
Men making love.
He recognizes Hepburn. All smiles. A man having fun.
Opens another file. More of the same. So much skin.
As he watches, a message flashes up on the screen.
He can't help it. He opens it.
Needed you tonight. You said you would call. X
The message is from a contact called MC. McAvoy jots down the number. Picks up his laptop. Enters it in Google with one hand, while keying Simon Appleyard's number into the mobile phone with the other.
There are no hits on Hepburn's phone. Nothing that links him to the dead man.
He looks at the laptop. Closes his eyes.
MC is Hull City Councillor Mark Cabourne. Vice-chairman of the Planning Committee, and member of the Police Authority. Portfolio holder for health and equalities and executive member of the Yorkshire Regional Flood Defence Committee. A face. A name. A former council planner turned politician.
He tries to make sense of this. Tries to diminish the impact of his string of semi-discoveries and fresh questions. Two colleagues texting? So what? Perhaps the kiss is an accident. Perhaps the message is non-sexual. Perhaps none of it is any of his fucking business.
He switches off the phone. Makes a decision.
Achily, exhaustedly, he pulls himself out of his chair. His wife is waiting for him to thank her properly. Tomorrow is a new day. The rain has blown itself out and none of his crimes so far cannot be remedied.
He feels a sudden tremble against his chest. Feels like crying as he pulls out his phone.
"McAvoy? It's Helen. Helen Tremberg. I just bumped into Shaz Archer. Detective Inspector Archer, whatever. Colin Ray's been following a lead. Reckons the lad we're after is with the gypsies on the playing fields. There's a link to Rourke. He's been gone ages. He didn't take backup. She's going up there after him. I think you should come . . ."
MIDNIGHT. DEAD ON.
MCAVOY RUNS across the wet grass, mud splashing his trousers and lapping at his boots, acid belching into his throat.
He can hear dogs barking. Raised voices. The crackle of throaty laughter.
He squints ahead at the semicircle of caravans. There are black figures etched against the darkness, the lights spilling from the curtained windows and the melee of parked cars making the picture dance and flicker.
Tremberg filled him in on the way here, shouting into the phone he held guiltily to his ear as the little car whined its way to seventy miles per hour on the divided highway.
It was pure luck that Colin Ray had made the connection between Alan Rourke and the makeshift traveler camp that had caused McAvoy so much embarrassment on the first day the rains came. Ray had been taking a break from the endless "No comment" interview and had used it to go through the report of Rourke's known associates that one of the civilian workers had left on his desk. He had been looking at a printout of an armed robber by the name of Daragh Fitzroyce when Helen Tremberg came back to the office and recognized the mug shot as that of the leader of the gypsy camp causing chaos up at Anlaby. "Buttercup's owner," she had told the confused DCI, smiling, only to find out that the tall, gray-haired, scowling detective inspector had not heard about the incident or, as far as she could tell, the invention of the Internet. She had filled him in on McAvoy's horse wrangling of a couple of days before.
Ray didn't believe in coincidence. Reread Rourke's own file.
Suddenly convinced of some gypsy conspiracy, he retrieved Shaz Archer from the interview room, told her where he was headed, and went off to the playing fields. That was some time ago, and he has not yet checked in.
"McAvoy!"
He spins his head. Helen Tremberg is running toward him out of the darkness, having parked on a nearby side street and waited for his arrival. He recognizes her shape and instantly feels bad for it, while noting that she, too, must have identified him from his mass.
"Nice timing," she says as she gets close enough for him to make her out. The front of her pin-striped suit is covered in mud and her face is flushed, though she is not out of breath.
"Have you called for uniform backup?" he asks.
Tremberg shakes her head. "What if he's fine? He'll go bloody spare. He's our superior, not the other way around. He can go in guns blazing if he wants."
McAvoy can see her point, but knows that procedurally they are in serious trouble. "What if he's not fine?"
"That would be a mixed blessing," she says, meeting his eye. "I'm sorry for calling you out. I just thought . . ."
"I know. It's okay."
They stand in silence. "They wouldn't hurt a policeman . . . ," he begins.
"Let's see," Tremberg says, and nods in the direction of the caravans.
Sighing, unsure, hating himself, McAvoy nods, and together they walk briskly toward the lights.
The dogs bark more loudly as they get nearer, and moments later figures are emerging from caravans and cars, lifting themselves from the sofas that sit in the center around what looks, at this remove, like a selection of patio heaters.
"Police," shouts Tremberg at the approaching crowd.
"Serious and . . ."
McAvoy falls silent.
Half a dozen men are approaching the two officers. Their faces are angry and territorial. McAvoy recognizes one of them from the other day and tries a half smile, but gets nothing but anger in return.
"We've done nottin," says one man, over shouts and protests from the others.
"We're not fucking going anywhere, I'll tell you that!"
"Leave us be, ya guard bastard."
"Jaysus, look at the size of the fucker."
McAvoy raises his hands, as if trying to pacify an angry dog. He pushes through the crowd into the center of the clearing.
"Detective Chief Inspector Ray! Detective Chief Inspector Ray!"
He wonders what he will do next. Whether he will start opening doors and checking under caravans. Wonders what the fuck is the point of thinking of himself as an asset.
"Christ, it's the Highlander!"
On the sofa, next to the patio heater, Colin Ray and Shaz Archer are sitting as if in their living room. Ray is drinking from a bottle of Newcastle Brown Ale. Archer is sipping tea from a mug. Opposite them, on a leather recliner, lounges the man McAvoy now knows as Daragh Fitzroyce. He is drinking from a glass bottle of fresh orange juice, and welcomes McAvoy and Tremberg with big smiles.
"Mr. McAvoy," he says warmly. "Buttercup's been missing you!"
There is laughter from the rest of the group, who assemble like Roman senators to watch what will follow.
"What are you two after?" asks Ray angrily. "And how do you know McAvoy?"
Fitzroyce grins, mischievously. "Just for his cowboy skills, Mr. Ray, just for his cowboy skills."
Ray nods, suddenly remembering the connection. McAvoy breathes out, relieved beyond explanation that his wife's name has not been mentioned. He tries to catch Fitzroyce's eye to see if this omission has been deliberate, but the camp leader has turned his attention back to the two people on his sofa.
"If we'd known there would have been this many of you, I'd have got the wife cooking," he says. "Does the best cottage pie. Can do a Sunday dinner on a two-ring hob. Wonderful woman."
On the doorstep of the largest caravan, the woman who was sitting with Fitzroyce the day the horses escaped is smoking a cigarette and raising a mug of tea, as if saluting her husband's words.
"Can I get you a drink? Beer? Cider? Got some crème de menthe if you're partial."
McAvoy stands still, unsure how to steer the situation. He is suddenly conscious of how he looks. The mud on his clothes, the redness of his face.
"By Christ, you're wearing some funny stuff for jogging," says Fitzroyce, and there is laughter from the crowd.
He turns to Ray. "Am I going to have to say all this shite again?"
Ray glowers at the two newcomers, and Shaz Archer gives a condescending shake of her head. "You've said nowt as it is, Fitzroyce."
The other man grins warmly. He turns to McAvoy. "Your boss man reckons I'm hiding me a dangerous fugitive," he says. "Reckons a lad I used to do a bit of naughtiness with has told him I'm looking after somebody you want. Ginger little scrub-headed bastard, by the sounds of it. Set another man's dog on a woman? Wrong. Just wrong. If them dogs are destroyed . . ."
He shakes his head, tailing off.
"I told your man here I ain't seen Big Al in bloody years. Your man says he lives around here. That's grand. Pint would be nice sometime. But I haven't seen him, and I don't know nothing about this lad you want. I've got enough problems . . ."
McAvoy is watching the crowd. There must be thirty people here now. He tries to put his finger on what is bothering him, besides the hell of it all.
Fitzroyce beats out a little rhythm on his thighs and finishes his juice. He looks as though, were they not already outside, he would like to politely show them all the door.
"You want a few more of the bottles?" he asks Ray, gesturing at the ale. "Got plenty. Take them away with you. Make sure it's not a wasted trip . . ."
"Kids in bed?" asks McAvoy suddenly, looking at Fitzroyce. He realizes what is wrong. He has been to many traveler sites and has never seen one without children, even at this hour. Indeed, he can only see a couple of women. The rest are aged between their mid-teens and mid-fifties.
The moment of concern that ripples across Fitzroyce's face is soon replaced with a smile. "Tucked up warm," he says. "Been cold. It's late."
"Must be tucked in cupboards, if all these lads are sleeping here, too," says McAvoy, gesturing.
As he looks around, he takes in the various vehicles parked around the camp. There is only one that looks expensive, a black Lexus. He squints. It has been here some time.
"We're good at making room," says Fitzroyce, though his eyes flick to where his wife sits.
Colin Ray notices the gesture and glances at McAvoy. He levers himself out of the chair and motions to Shaz Archer to do the same.
"Nice motor," says McAvoy, pointing at the Lexus. "Yours?"
"I wish," laughs Fitzroyce. "Belongs to a pal. Letting me have a play."
"Insured?"
McAvoy says it playfully—two mates having a laugh. Fitzroyce gives a grin. "Of course, sir, of course."
McAvoy nods. Scans the crowd again. Reaches into his jacket and finds the clunky radio he had the presence of mind to pick up on his way out the front door. Flicks it on and fills the sudden silence with static. "Control, this is Sergeant Aector McAvoy. I need a PNC check on a vehicle . . ."
"Oh, now, Mr. McAvoy, there's no need . . ."
"Close the lips, fella," says Ray, holding up a hand. "Let the officer do his job."
Fitzroyce looks again at his caravan, then back into the crowd.
McAvoy moves around as he talks into the radio. Changes his position.
"You," he says, and points at the crowd. "Name?"
The man in front of him is tall and heavily built with a shaved head. He is wearing a black T-shirt over large muscles, and his forearms are covered in amateurish prison tattoos.
He looks McAvoy up and down. Gives a little snort of laughter, full of contempt.
"Fuck this . . ."
McAvoy's feet slip from under him as the man pushes him in the chest and muscles past him. As he falls, he hears a thud from the caravan. He turns and the air leaves his body as he hits the ground. He hears shouts. A scream. Glances up and sees Fitzroyce's wife sprawled on the floor. A scrawny ginger lad is struggling to get past her.
Through a mass of bodies, he sees the man who had pushed him over sprinting for the Lexus. He hauls himself up, begins to run, but a shout behind him pulls him up short.
He turns back. In front of him stands a ginger teen. He is unmistakably the weaselly little bastard who set the dogs on Trish Pharaoh.
Behind him, Colin Ray is on his knees, holding a hand to his head and struggling to gain his feet.
McAvoy blocks the lad's way. "Don't you fucking try it . . ."
The lad, clad in a white vest and tracksuit trousers, suddenly swings wildly with an object in his left hand. It is a large crucifix, and McAvoy only just manages to dodge backward as it arcs up toward his chin.
Helen Tremberg, trying to grab him from behind, is not so lucky. He chops down with the crucifix, and there is a sickening thud as it cracks into her kneecap. She goes down, roaring.
Triumphant, furious, the teen turns back to McAvoy. "Out of my way, ya jock shite . . ."
He swings the wooden crucifix like it's a hatchet, and McAvoy has to fight to keep his feet as he steps backward. He looks up and sees Fitzroyce tending to his wife. Sees Ray and Tremberg on the floor. Sees Shaz Archer disappearing into the darkness, sprinting after the disappearing lights of the Lexus, which roars, wetly, into the distance . . .
"Come on, then!"
The ginger lad is screaming in his face, spittle hissing from bared teeth. He lunges, the crucifix hacking down as if chopping wood. McAvoy sees the blow coming and dodges backward, his gait that of a boxer, his hands becoming fists. He throws out a right hand, and snaps it back just before it collides with the young lad's face.
The lad, furious to know he could have been so easily knocked out, snarls again and begins to turn away.
McAvoy lunges forward. He barrels into the young lad with all of his weight, a rugby player to his core, and plants him in the dirt.
"Backup urgently required . . . ," he begins, fumbling again for his radio.
"We'll kill you," screams the lad, squirming underneath him. "We'll fucking kill you all . . ."
THERE IS milky breath in McAvoy's face.
He kisses Lilah on the eyebrow and she snuggles in closer. He looks up and locks eyes with Fin. His son is lying on his tummy at the bottom of the bed, laid out across McAvoy's legs. He is reading a photocopy of McAvoy's report into the mess at the gypsy camp. There are pages all over the bed.
"What's a crucifix?" asks Fin quietly.
McAvoy smiles at him. "Tell you later," he whispers. "Put that down now, son. You go play."
Fin does as he's told. Shuffles off the bed and heads to his room, where moments later McAvoy can hear the sound of a five-year-old boy telling one of his toys not to mess with Detective Sergeant Finlay McAvoy, and then pretending to fight a bad man.
McAvoy snuggles into the blankets. He looks at the clock by the bed and realizes he has been asleep for nearly thirteen hours. It feels good. He is warm and contented: happier than he has been in a while.
He has no right to feel this way, of course. Yesterday was a catastrophe. The well-muscled man in the Lexus got away. A search of the database showed it had been stolen from a car showroom in Doncaster.
The lad with the ginger hair has been named by Fitzroyce as Ronan Gill, though the traveler is giving no further information, and volunteered that much only through gritted teeth. Ronan is sixteen years old, a minor in the eyes of the law, who can be interviewed only in the presence of an appropriate adult. He is not cooperating. Has not stopped screaming and swearing yet, forcing every attempt at interview to be abandoned. Grabbed the left breast of one of the well-meaning volunteers who had agreed to sit in on the interview in the absence of a parent or guardian.
A psychiatric consultation had been ordered by the force's medical examiner, but so far they have not been able to find anyone who is available. As a result, time is ticking on, and they have no answers about who the man in the Lexus was, why Ronan set the dogs on Trish, or why Alan Rourke's fingerprint was on the petrol bomb.
Rourke was released last night, having stuck to his "No comment" answers throughout, breaking his silence only to thank Shaz Archer with all his heart when she revealed his dogs were still okay, and were being cared for at a nearby shelter until a decision could be made as to whether they were to be destroyed.
The plus side of having such a busy day of form filling, paperwork, and desk-bound inquiry was that McAvoy was able to get home at a sensible hour. He had come home to cottage pie and a four-pack of bitter. Drank one. Watched his bride polish off the other three. Had told his children stories.
He hears a faint buzzing sound. Wonders if there is any chance of reaching his mobile phone. Manages to disentangle himself and climbs out of bed. Finds his trousers on the bedroom floor and curses, silently, as the call ends.
He looks at the number. Tremberg. He pulls on a pair of rugby shorts and a hooded top, and pads downstairs, intent on making breakfast before doing anything that might upset Roisin on his day off.
Entering the kitchen, he has the vaguest of memories. Last night. Just after nine p.m. Giggling, here, by the sink, as she held aloft a rolling pin like a club. Roisin telling him that her contact couldn't be trusted to put Hepburn's phone back, and suggesting they smash it up instead. Him, knowing that it was the right thing to do, but unable to acquiesce.
He picks up the phone from where it lies on the counter. Switches it on. Fills the kettle as he lets the phone pick up messages and calls. Looks again at the screen. A dozen texts and seventeen missed calls.
He would like to give the phone to the tech unit. Wants them to go through it and make it evidence. To make it clinical and somehow policemanlike. At the moment it is still prying.
Halfheartedly, lips pursed, he glances through the messages.
More, from Mark Cabourne.
ARE YOU IGNORING ME?!
He's rung again! What is it he wants? Please. Xx
Have I done something?
Why are you being like this? I need you. I need this. Please text. Xx
McAvoy rubs his face, the peace of sleep evaporating. He cannot help himself. He cannot stop now.
He makes a mug of tea and opens the back door. The day is clear, bright, and blue skied, and the cold air feels good on his bare legs. He sips his tea, and winces, as if it were too hot. It is not. The grimace is the result of making up his mind.
He dials a number. Waits only three rings.
"Councillor Cabourne? This is Detective Sergeant Aector McAvoy . . ."
• • •
THE NOISES coming from the cell seem to be a mixture of English, Gaelic, and Demon. Spits and shouts, screams and cries, all made unintelligible by the fury with which they froth from Ronan Gill's mouth.
"And he thinks he's angry now . . . ," says Colin Ray grimly to himself, as he passes the custody desk and makes his way to cell four.
He takes a breath. His ribs ache. There is mud caking his suit. He has a headache where his teeth slammed together, and he can taste blood. And he's feeling pretty good.
From within the cell come another series of crude screams and threats.
"He'll have you. All of you. And him! Fucking cunt! He'll take you down. All of you!"
A noise behind Ray makes him turn. He is surprised to see the tall, imposing shape of Helen Tremberg. It strikes him as odd to have a female companion other than Shaz Archer. Shaz has gone home to get changed, but Tremberg is less concerned about the dirt on her clothes and has made little effort to sponge her knees or face clean. She wants to be here. To be involved. To see what happens next.
Ray seems about to tell her to piss off. To ask why she's here and not busy putting antiseptic on McAvoy's grazes, telling him what a big brave soldier he is.
He loses interest in insulting her. Just gives her a shrug, as if to warn her that he's about to do things his way and it's up to her whether she stays or goes.
"Where's my fucking solicitor? I'm not speaking. Not a fucking word. You know how much my brief costs? He'll have you all. All your jobs . . ."
Ronan Gill has learned nothing from his guardian in terms of keeping his mouth shut. None of Alan Rourke's stoic silence has rubbed off on the teen. He has been like this since the uniformed officers dumped him on the cell floor and began stripping him of his clothes. The sergeant in charge, who finished the job with a bleeding lip and bruised knuckles, said that putting him in the paper suit was like trying to put a lobster in a rubber glove, though Ray doesn't know who did his research.
"I'll have you all . . . !"
Ray bangs his palm on the metal door.
"Shut the fuck up, son. Back it up."
The warning prompts another burst of Gaelic. Ray finds himself smiling back at Tremberg, who pulls a face. It is the warmest moment that has ever passed between the two.
"I need to talk to you, lad. I can come in with a dozen uniformed officers and we can do this in a way that hurts."
There is silence for a moment, then Ronan's voice, thick with rage. "I'm bleeding! They assaulted me. That's assault. When my brief gets here . . ."
"Easy now," says Ray, and reaches up to open the viewing flap in the metal door. A moment later a comet of spit shoots through the gap, and Ray thanks experience for not having been in the way.
"Feel better now?"
"Fuck you!"
"I can stay out here if you like. We can talk from here. You obviously enjoy your privacy."
There is another stream of spit.
"You're gonna dehydrate, son."
Quickly, deftly, Ray glances through the viewing window. Ronan is bouncing on the balls of his feet, fists at his side, face crimson, like a baby with wind. He has torn the paper suit to shreds, which hang off him as if he has burst out of them from within. The mattress from the bunk is propped against the far wall with fist-shaped dints at its center. The toilet pipe is leaking water, as if it has been booted again and again.
"I've been going through your things, Ronan," says Ray. "You're going to have some explaining to do."
The silence from the cell is longer this time. Without looking, Ray beckons Tremberg closer. He reaches into his pocket and hands her a slick, expensive mobile phone and a handful of bits of scrap paper. She takes them without asking why. _Read them,_ he mouths at her.
"You break my phone and I'll break you," says Ronan, though his voice has taken on a slightly more whiny tone.
"I'd have thought you'd have wanted me to break it," says Ray. "Thought you'd have wanted it smashed."
"There's nothing in it," says Ronan, but there is a note of uncertainty there now.
Ray smiles at Tremberg as she looks up from going through the ragged scraps. She looks puzzled. Doesn't seem to know whether it would make her look like an idiot to admit she doesn't know what she's looking at.
Ronan had fought like a tiger to keep the phone. He was brought into the custody suite with an officer holding each limb, screaming and roaring, and any attempt to book him in properly would have ended in somebody's blood. He should have been asked his name, age, and address, and been given a list of items that were in his possession at the time of arrest. Instead, he had been dragged to the cell, forcibly stripped, and the contents of his pockets stuffed into a carrier bag, to be given to Colin Ray as soon as he arrived.
Ray's fears that any messages in the phone's history would be in Gaelic were unfounded. He had made sense of it all pretty swiftly. It was clear that Ronan did not use the gadget for personal reasons. There are no messages from girlfriends or mates in the in- or out-boxes. It's all business.
"It's not even my fucking phone," shouts Ronan.
Ray grins, and in the lurid half-light of the corridor it's a ghoulish thing.
"You gonna stand back so I can open the door, son? Gonna play nice and let me in for a little chat?"
"I ain't speaking until my brief gets here. I told you."
"We'll do the interview, Ronan. We'll go do it all properly, you'll see. Be super-official and very polite. You'll follow your brief's advice and keep your trap shut. I can see it all now. Don't worry, we'll follow procedure. I just wanted to have a little parley—two grown-ups together. But we can leave it. Don't fret. You have a nice time smashing your cell up and tearing your clothes to confetti. We'll talk later."
"Fuck you," comes the reply.
"Like autopilot, isn't it?" says Ray to Tremberg. "Pavlov's dogs. They hear me speak and start salivating swear words."
Tremberg looks up from where she is scrolling through the half-dozen messages in the phone's history. She can't make much sense of what she sees. Letters. Numbers. The occasional smiley face. It seems more gibberish than code.
Ray takes the phone from her hand. Holds it up. Looks at the most recent message.
He reads aloud. "'Eleven. H-four. Nine. Agreed. Two crew. Four-oh-one. Transfer H-six.'"
There is silence in the corridor.
"I just sunk your battleship," says Ray through a grim smile.
"I don't know what that fucking means."
"No," says Ray. "Neither did I at first. Code of some kind, I reckon, because I'm smart like that. And I reckon that if I spent the rest of my life trying to break it I'd only get a headache. Thing is, son, I don't need to, do I?"
There is silence from the cell.
"You took that illegally . . ."
Despite the pain in his ribs, Ray starts laughing. "Always hide behind the law, these fuckers," he says to Tremberg.
"You don't know what you're doing," says Ronan, and there is desperation in his voice.
Ray takes the handful of paper scraps from Tremberg. Holds one up; a lined page scrawled with a Biro.
"H4—Division Road," he says clearly. "Nine equals movement of crops. Nips know. Pick up by Lee. Four hundred one plants. Transfer—New Bridge Road. Before weekend."
There is silence in the corridor.
"I'm surprised you remember much from your school days, Ronan, but it's nice to see you show your workings out."
Ray slams the viewing window shut to muffle the screams and threats and pounding fists that rattle against it. He gives Tremberg a nod and walks past her, favoring his left side.
"Sir?"
Ray turns back. "Stupid prick couldn't remember the code. Wrote it down in English for us and shoved it in his pocket. Little shit thought he was untouchable."
Tremberg throws her hands up. "I don't understand."
"He's running the Vietnamese crew for somebody. He's moving the crop before the weekend. The farmers know to expect him and a crew of two and take it all to the next house on the list."
In the poor light, it takes Tremberg considerable effort to show her shock and skepticism. "He's just a little thug."
"They all started out like that, love," he says, and for the first time he does not sound as though his every word is spraying bile. "Ronan was on the ladder."
"Was?"
Ray rubs a hand over his unshaven face. "I don't think they'll fast-track him once we raid Division Road."
• • •
EVEN IF he were not already a vaguely familiar face, McAvoy would still recognize the councillor as he enters the diner. There is an air of dread and panic about him; a cloud of anxiety that dampens his face and slicks down his hair.
The man scans the room. Takes in the monochrome baseball and Rat Pack photos on the wall, the black-and-white tiles on the floor, the expensive flowers by the till, and the open grill at the back of the room, where white-suited chefs toss pancakes and grill bacon.
McAvoy waves a hand. Beckons him over.
"Detective?" he asks, approaching. He lifts his hand to shake, drops it to his side and then lifts it again.
McAvoy begins to stand. Smiles through a mouthful of breakfast. Realizes that, even in this half crouch, he towers over the other man, and is quick to sit down again so as not to be instantly intimidating.
Cabourne slides into the seat opposite him. He is full of nervous energy. Drumming his hands on the table. Playing with the saltcellar. Jiggling his legs.
"That your partner?" Cabourne asks the question with what is intended as a little laugh, but it comes out as a strangled, high-pitched giggle. He is nodding at Lilah, fast asleep in a car seat at McAvoy's side.
"Saturday parenting duty," says McAvoy. "You got children?"
Cabourne looks away.
McAvoy already knows that his brunch companion is a father. A married man. Home owner and former council officer turned politician. Fourteen years on the local authority. A member of the Police Authority and face on more committees than he could name. This is an important man, and he looks like a child summoned to the headmaster.
"Nice here," says Cabourne distractedly. "Chain, is it."
McAvoy nods. Approves. Wishes they would switch back on the Italian jazz they had been playing when he arrived.
He and Cabourne are among only a handful of customers in this imitation-American diner. It sits between the hamburger joint and the fried-chicken chain that constitute a major part of the "retail and leisure" end of the Kingswood estate.
Roisin has taken Fin to see a Disney film at the nearby cinema. There is talk of slush puppies and bowling afterward. It could yet be a nice family day within walking distance of home. There has been no need to tell Roisin that his offer to take Lilah for breakfast is not entirely selfless. He is not sure how he would have arranged things if Cabourne had not agreed to meet him here.
As it happened, Cabourne had been only too willing to help—happy to meet the detective whenever and wherever he wanted, and not once asking what it was about.
"Can I get you something?"
McAvoy passes the brunch menu across the table. He takes a sip of his chocolate milk shake, and skewers another pancake with his fork, teaming it with a half rasher of bacon and enough maple syrup to fossilize a woodpecker.
"Erm, coffee would be nice. And water, please. I'll get them . . ."
Cabourne plunges his hand into his pocket and tries to retrieve some change. As he does so he seems to get his sweaty palms stuck, and as he wrenches his hand free, change spills onto the hardwood floor.
"Shit!"
A waiter in black trousers and shirt comes to help as McAvoy levers himself out of the booth and starts retrieving coins. The councillor just sits there, arms folded, looking down at the black lacquer of the table, seemingly unsure what to do or say.
"Coffee," says McAvoy to the waiter, as they both deposit a handful of change in front of Cabourne. "And water, please. Tap." As McAvoy slides back into his seat, Cabourne gives him a grateful smile. "I've always been clumsy," he says. McAvoy looks him up and down. He is around six foot. Late forties to early fifties. Gray hair swept back from a thin face, made stern and bookishly intelligent by rimless glasses. He is dressed in a thick mauve shirt and chinos, and his only adornments are a simple gold wedding ring on his left hand and a thin silver chain at his throat. To McAvoy, he has the air of a foreign football manager. He looks like he can afford his own breakfast.
"I appreciate your coming," says McAvoy, pushing his plate away. "As I explained, we are at the very earliest stage of an investigation and I am talking to you purely out of courtesy . . ."
Cabourne holds up one hand. He closes his eyes. Takes off his glasses and rubs his eyes.
"I think I already know," he says quietly.
They sit silently as the waiter leaves the coffee and water on the table.
"Councillor?"
Cabourne sips his water. Puts the glass down. Lifts it and gulps some more.
"I didn't know it was illegal," he says.
McAvoy sits in silence, content to let things play out.
Cabourne's eyes are darting, flitting from booth to booth, table to table, although whether for familiar faces or a way out, McAvoy cannot say.
"Why don't you get it off your chest?"
The older man seems to sag. It is as if he has been punctured. When he looks up again, McAvoy has removed Lilah from her car seat and is sitting her, floppily, on his knee. Deep down, he knows he is using his daughter as a prop: putting the councillor at ease by making this a chat between fathers rather than an interview with a policeman, but to acknowledge it would be an admission of manipulation, and that is an admission he does not want to make.
"Hepburn's ignoring me now," Cabourne says. "I think he's more scared than he's letting on. That's Steve, though. Always the same."
McAvoy strokes his daughter's cheek with the back of his knuckle. Dips his finger in the dregs of maple syrup and lets her lick it, while nuzzling her head with his nose.
"Councillor, I know you want to tell me something. You'll feel better. You're not under caution. This is just a chat."
Cabourne seems to galvanize his resolve. Gives a nod.
"He's left me so many messages. This Ed Cocker. Some sort of political fixer. I don't know what he wants me to say."
McAvoy gives an encouraging nod.
"Some people get sports cars or motorbikes when they hit middle age. I did this."
"This?"
Cabourne looks suspicious suddenly. "Can I see your warrant card?"
McAvoy raises his eyebrows. Pulls out his card from his shirt pocket and slides it across the table. Cabourne studies it. Nods.
"This Ed Cocker. He won't take no for an answer."
McAvoy sighs. "What's the question, Councillor?"
"He says Hepburn's the story, but he's not, is he? Not when he finds out."
McAvoy runs his tongue over his lips and strains his brain. Thinks of the desperation in Cabourne's messages to Hepburn's stolen phone. The kisses. Looks now at the father of three, sweating and panicking in the seat opposite him.
"Councillor, your personal life is your own. Whom you have relationships with is not police business."
Cabourne sags again. "It's not a relationship," he says. "It was just one of those things."
_Something that you wish would continue_ , thinks McAvoy. "And the journalist from the _Hull Daily Mail_ knows about it?"
"I don't know. Steve would never tell, no matter how much he likes the limelight. And I haven't told anybody. But we've made mistakes. And I've hardly been discreet."
McAvoy raises his hands to stop the councillor's flow. Takes a breath.
"Councillor, I'm obliged to inform you that I am here to talk to you about the circumstances surrounding the death of a young man named Simon Appleyard. Simon died in November last year. Hanged himself. There are reasons to consider looking again at his death. Your name has come up in connection with the investigation."
"Oh, God!" The councillor collapses in on himself, his face red, his mouth open. "I knew," he says, hugging his arms. "I knew."
McAvoy does not know how to respond, so simply kisses his daughter and waits for Cabourne to meet his eye.
"Do you know Simon Appleyard?"
"I don't know," replies Cabourne angrily. "Fuck!"
McAvoy gestures in the direction of his daughter. "Don't swear."
Cabourne, rubbing his face, apologizes. He sips more water. Has his face in his glass when McAvoy slides a picture of Simon across the table. It is a photocopy of the image that the dead man's aunt had given him.
Cabourne shrugs. Looks away.
"Do you know Simon?"
Cabourne forces himself to study the photo. "It was dark."
"When?"
"Every time!"
They look at each other, each trying to gauge what the other knows.
"You have been meeting men for sex," whispers McAvoy, conscious of Lilah's nearness. "Am I right?"
Cabourne sips his coffee. Meets the detective's eye. "Men, not boys."
"Simon was twenty-five."
"I didn't mean that. I mean it's not illegal."
"No, it's not."
Cabourne breaks first. "I love my wife," he says, suddenly pitiful. "I think I do, anyway. We've been together so long. It was just—"
"One of those things?"
"Exactly. I've never cheated with a woman. Not really."
"Not really?"
"Only when they've been there, too."
"Where?"
"The parties!" he says exasperated. "The clubs. The dates."
McAvoy puts Lilah back in her seat. Scratches his head. Lets the pieces drift together.
"Sex parties. That is what Ed Cocker is investigating?"
"It must be!"
"Parties that Councillor Hepburn organizes?"
"He doesn't organize them," says Cabourne defensively. "Why would he? He can have what he wants. Take what he wants."
Cabourne sniffs. McAvoy passes him one of Lilah's wipes, which he takes gratefully and uses to clean his face.
"Councillor Cabourne, my brain is starting to hurt. What is it you're frightened of?"
Cabourne looks up, blinking.
"Playmatez," he says, under his breath. "It was just to try it out. I'd always had this fantasy . . ."
McAvoy nods, keeping his eyes impassive. Non-judgmental. "You went on a website, yes? A dating site?"
"I wanted to try it. Everybody was there for the same thing. It was free. I must have been drunk when I signed up. Just put a bit about myself and what I liked. Didn't even think about it at first. Linked it to my private e-mail."
"And?"
"And I got loads of responses. Men and women! I didn't even say I wanted girls, and there they were, turned on just at the thought. I e-mailed a few of the lads back. Said I was a novice. Didn't know what to do or what I wanted. Said discretion was everything . . ."
"You met?"
Cabourne finishes his coffee and looks away. "Cheap hotel near Goole," he says. "A married man, out for the same thing as me."
"And?"
Cabourne shrugs, all pride lost. "I wanted more. Met more."
"When did this begin?"
"A year ago, maybe. No more."
"Simon," says McAvoy, nodding again at the picture. "Did you ever meet Simon?"
Cabourne picks up the picture again. "No," he says at length. "I'm sorry. No. This is the dead man?"
"Did you ever read this post?"
McAvoy slides a piece of paper across to Cabourne. The councillor's lips twitch as he reads the words on the page. It is Simon's posting on the Playmatez website. An invitation to fill him up and a phone number.
"It rings a bell," he begins, noncommittal.
"Did you respond to that posting?"
"Possibly," he says, with a shrug that is far from uncaring. "I replied to so many."
McAvoy looks around him. There are balloons on a table to his left, already laid out for a party later in the day. Beyond the wooden blinds the rain is holding off temporarily. Shoppers and diners walking past the glass are coatless. Some have bare arms. He wants to be in that sunshine now. Not here, where the clouds are gathering and the air smells of rain.
"Councillor Hepburn," says McAvoy, deliberately vague, "explain how that happened."
Cabourne closes his eyes. He pulls his phone from his shirt pocket and looks nervously at the screen, as if checking for messages. "We had friends in common," he says, and appears to be watching footage of the night in the cinema of his memory. He is almost smiling.
"Friends?"
"I let my mouth run away with me. Told a guy my real name. What I did for a living. I don't know why. Just trying to impress."
"And he knew Hepburn?"
"Everybody knows him."
"And?"
"And even though I begged him to forget what I said, it wasn't long before I got a text from Steve telling me he knew I'd been a bad boy."
"That must have been a difficult moment."
"Horrendous. I panicked. Told him I had no idea what he was talking about."
McAvoy reaches across and takes a sip of Cabourne's water. He can think of no other way to show the other man that he is not disgusted. That he does not feel repulsed by these admissions and is still a safe listening ear.
"He didn't believe you?"
"We were only colleagues, not friends," explains Cabourne. "We'd had a few rows in council meetings. Been in the same bars after meetings. I'm Labour, he's an independent, but he wasn't exactly a political enemy. Nor was he a great mate. Just a guy I kind of knew. A guy famous for his lifestyle choices, who now knew everything about me."
"What did you do?"
"I didn't have to do very much," he mutters with a half smile. "Hepburn didn't make a big deal of it. After those couple of texts he was just his usual self. Said hello when we passed on the stairs, gave me a grilling at committee. Usual stuff. I didn't see anybody else for a while. Then one day he just asked me, out of the blue, if I fancied a drink. He was casual about it. Just said it one day as we were coming out of committee. I panicked. But I said yes."
"And?"
"And we talked. He didn't try and put pressure on me to admit what I'd been doing, but I just blurted it out. Told him it all. He just listened. Let me be myself."
Cabourne purses his lips. Distractedly brushes at the front of his shirt. Looks at his phone and puts it down again.
"You had an affair?"
Cabourne shakes his head. "We just became friends."
McAvoy looks skeptical. "Friends?"
"He made my life more interesting. He knows everyone. Has been living the right kind of life for an age."
"The right kind of life?"
"Fun," he says bombastically. "Alive."
"You went to parties together? Sex parties?"
"Nothing around here," says Cabourne, as if trying to prove he has not been a complete fool. "We'd go to London. Manchester. There's one in Blackpool . . ."
"All men?"
"All sorts."
They stop talking. McAvoy stares hard at the other man. He is trying to decide how he feels about him. He wonders if Cabourne has done anything wrong. What "wrong" even means.
"You really don't know Simon?" asks McAvoy at length.
"I could have e-mails from him," Cabourne says, trying to be helpful. "So much of this stuff happens online. Most times it leads to nothing. Some people leave their mobile numbers on the site but I could never do that. Too risky. I could check . . ."
McAvoy waves him into silence. "Dial this number," he says, flipping open his notebook and showing Simon's digits to the councillor. "Dial it and show me your phone."
Obediently, like a child, Cabourne does as he is bid. The councillor punches the final digit, and waits for it to ring. Before the warning message flashes up to tell him the number is unavailable, the phone does the hard work for him. The number is linked to a contact called "Peacock."
Cabourne's mouth drops open. "Him?"
McAvoy looks at the other man with an expression that says he does not appreciate being lied to.
"I swear I just took the number down," he says desperately. "I've contacted so many people on there. I just kept the numbers when they gave them. Look, look . . ."
Cabourne is turning the phone around, scrolling through the contacts. Names flash by.
"Paul T," he says, pointing. "That's for 'throat.' He said he liked having his neck squeezed. And there, look. Vampire. He said he was into biting. They're just for me to help remember who is who . . ."
"And Peacock?"
"I think he said he had tattoos." Cabourne stops, memory dawning. "He e-mailed me," he says, eyes wide. "There was a line of poetry on the bottom of his message. Something he said he liked. Peacocks and lilies."
McAvoy drops his head to his hands. He has more questions than answers.
Suddenly he looks up. "He e-mailed you? Not texted?"
"Definitely."
McAvoy begins rummaging through his papers. He is trying to find a mention anywhere in the various reports that suggests Simon owned a computer.
"I'm an idiot . . . ," mumbles McAvoy.
"I'm sorry?"
"Were the e-mails from a smartphone?"
"I'm sorry, I don't, I don't think so . . ."
McAvoy stops. He realizes the man in front of him is guilty of betrayal. Of confusion. Of weakness and lust. But he does not see a criminal.
"Keep your head down, Councillor Cabourne," he says, sliding himself out of the booth and picking up Lilah's car seat. "Ed Cocker isn't after you. He's after somebody much bigger."
Cabourne looks up at him, unsure whether to give in to the magical sense of relief that threatens to flood him.
"I'll check my old e-mail account," he begins. "I'll do anything I can to help you."
McAvoy nods. "Yes. You will."
11:47 A.M.
A BLUE twelve-year-old Vauxhall Frontera, steamed up and idling on the double yellow lines that edge this quiet side street off old Hessle Road.
Four cops inside—damply smoldering, jittery with unused adrenaline.
There's a brightly lit takeaway to their left. It's all glass and white paint, cartoon characters, and gaudy lettering. The relentless rain jewels the large, dirty windows and turns the skinny, fifty-something woman behind the counter into a fragmented caricature of herself: mechanical, joyless, shaking spice into paper bags full of chips.
There's a barbershop to their right. Black gloss—bought in bulk and applied too thickly, collecting in rivulets in the gaps between the bricks.
Shutters down today. Down most days.
Helen Tremberg sits in the back of the unmarked car. A sergeant from the Drugs Squad stares out of the window beside her, watching the raindrops dribble haphazardly down the glass. He hasn't spoken since giving her a grunt of acknowledgment as she slid into the back of the car and wiped the rain from her face with a warm palm. He smells faintly of stale beer and wet dog.
DCI Ray is sitting in the front, passenger side, sucking the chocolate off a Twix.
At the wheel is Detective Superintendent Adrian Russell. Everything about his manner suggests he is in a foul mood. He is moving chewing gum around his mouth, but the look upon his face is more in keeping with that of a man trying not to acknowledge the gone-off oyster under his tongue.
There is silence inside the vehicle, save the drumming of rain on the roof, and the occasional swish of damp tires as cars pass by on Hessle Road.
Tremberg feels uncomfortable. Out of place. Unwelcome. She has never worked with Russell or his underlings, and has no bond with her DCI. She is here because the opportunity arose. Here because she is an ambitious officer who wants to be there when a high-profile raid goes down. Here because Shaz Archer can't be rustled up, and because with Pharaoh out of the picture and McAvoy out on a limb, she is feeling lost. There has been no fanfare to welcome her return. No hugs or tears. She came back after risking her life to catch a killer, and was very nearly on fire before the end of her first shift.
"Should have parked ourselves in Rayner's," says Ray chattily, throatily, while jabbing a thumb over his shoulder at the legendary pub across the street. "Could have bought you a Babycham and a packet of peanuts."
Ray angles the rearview mirror until he can see into the back. A bite of Twix moves around in his mouth as he talks.
"Never been in," says Tremberg, turning in her seat to stare across at the building on the corner. "Doesn't look welcoming. They do scampi in a basket?"
"Proper pub," says Ray. "I read up on it when I moved to this shitty city. Hessle Road was already on its arse by then but, fuck, that place had character."
It is not the first time Tremberg has heard about this boozer, or its place at the very heart of the old fishing community. This is where the trawlermen drank on their three days home, their refuge after six weeks risking their lives in distant waters. It is where scores were settled and where tensions erupted into bloody violence. Where feuds ended in bloodshed or in forgiveness. Where men tried to dilute the ocean in their veins with pint after fucking pint. It was a hard man's watering hole. A place of mourning and of celebration. A place that numbered countless dead among its regulars, and where it was said that the ghosts of recently dead trawlermen would call in for a drink before sailing on to purgatory.
"What's it like in there now?" asks Tremberg, for something to say.
Ray shrugs. "Only been in there once. Decent pint. Few old boys with a story to tell. Bit sad, really, when you think what it was. What all this was . . ."
Ray stops talking as he realizes he is sounding soppy. He gestures at the run-down side street beyond the glass. Waves an arm halfheartedly at the cut-price furniture shops and the empty greasy-spoon cafés.
"It was probably all shit in the good old days, too," he says, by way of antidote to his display of nostalgia. "Fifty years from now Hull folk will reckon life nowadays was fucking peachy."
Silence again.
Adrian Russell, chewing his gum.
The sergeant beside her stifling a burp and then blowing out the faint smell of last night's beef Madras . . .
Tremberg wondering if she should text McAvoy. Tell him what Ray has arranged. Ask him if he knows why the fuck the detective superintendent seems to have ceded operational authority to his junior officer, and appears to be swilling sick around his gob.
They all jump as Russell's phone rings, the riff from Gary Numan's "Cars."
A look passes between Ray and the detective superintendent.
Russell closes his eyes. Answers in little more than a whisper.
"Russell. Yes. Yes, as a matter of courtesy . . . No. Well, obviously. I do appreciate that. No. It's not my call. There are limits, you understand . . . I'm not sure that would be wise . . . No, I realize that. Different breed, you might say. Of course I understand the benefits. Yes. If you're sure . . ."
Russell hands the phone to Colin Ray.
Ray is all smiles.
"Detective Chief Inspector Colin Ray. Very Serious and Vaguely Organized."
He puts the phone onto speaker. Seems to take pleasure in the other officer's shiver of discomfort.
The car is filled with a stranger's voice: tinny and robotic.
"Mr. Ray, I'm sorry we have not had a chance to be properly introduced before now. I would have made it my business to do so, but I was unaware of your existence till today."
The voice is almost accentless. The enunciation clear but giving nothing away.
"That's okay, son, I don't know much about you, either. Know you're going to have a bad day, though."
Ray's words seem not to register with the speaker.
"In the past hour I have remedied my aberration. I have acquainted myself with several of your personal details. Allow me to express my sadness that such an experienced officer should find himself so poorly remunerated at such a time in his career. You have given up so much for this job, and you are rewarded with a childless existence, and more ex-wives than a man can afford. To be only a few years from retirement, and still to be an underling . . . it saddens me. A man of your experience should be better rewarded."
In the mirror Tremberg watches Ray's face for any glimmer of discomfort. Sees none.
"Aye, you're right there," he says, as if chatting to an old friend. "I'm surrounded by fucking ingrates and incompetents. I'm sure you know the feeling. That's what you get for working with Chinks and pikeys. You should put your hand in your pocket, son. Bring in some lads who can think and tie their shoes at the same time."
For an instant there is no reply. Then the voice continues, as though Ray has not spoken.
"The house on Division Road is not expecting you, Chief Inspector. The details of my arrangement with your colleague were clearly miscommunicated."
Russell reaches out to take the phone, mumbling words of protest. Ray raises his arm and splays his fingers. Keeps the phone beyond the other man's reach, until Russell sinks back into his seat.
"Like I said, son, bad day for you."
"I have experienced bad days before. What happens today will be of significance to you, but of little or no consequence to me and the people I represent."
"And yet you took the time to ring . . ."
"If inconvenience can be avoided, I believe it to be worth the gesture."
"You're not going to avoid this inconvenience, boy. One of your little helpers took a swing at me with a fucking crucifix. That doesn't buy you much in the way of goodwill."
Ray catches Tremberg's eye. Winks. He seems to be enjoying this.
"Some of my associates are spirited individuals," says the man. "They have unique character traits and skill sets that we attempt to harness. I am not one to stand in the way of youthful exuberance."
Ray laughs. "That what you call it when you nail somebody's hands to their knees? When you petrol-bomb a police van? You're no fucking big shot, whoever the fuck you are. You run a few cannabis factories. You've scared a few Chinks. You think you'll make my memoirs when I retire?"
Now it is the other man who emits a chuckle. "I presume that you are recording this conversation, Chief Inspector, so I will refrain from unburdening myself with regards to my regret for recent incidents. But to presume my associates are limited to such matters represents a degree of shortsightedness that they will find amusing."
"Did you actually want something, lad? Only I've got a drug den to raid and a couple of fucking Chinks to arrest."
The man does not speak for several seconds.
Finally he gives a little sigh.
"Your colleagues," he says. "The large gentleman who looks like he should be carrying a claymore. The lady in the biker boots and breasts. Tell them not to feel guilty. They had a job to do. Miss Marvell was big enough to make her own decision. And do tell Detective Superintendent Russell that I will be in touch."
The call is terminated. The speaker begins to emit slow beeps, like a life-support machine.
Ray looks at the side of Russell's face for a spell. Looks as though he is about to spit.
"Sir?" Tremberg is the first to speak. "Do you think that bloke runs this lot, then? That he's the boss? He didn't sound right. Didn't sound like just some drugs thug . . ."
Ray picks his teeth for a spell.
Says nothing.
Finally picks up the radio from between his legs.
"Go."
A dozen car lengths ahead, the double doors swing open at the back of a white van. Half a dozen uniformed officers emerge, fast and furious.
Farther up the street, four plainclothes Drugs Squad detectives step into the rain.
As one they descend upon a deceptively large town house halfway up the street.
Tremberg opens her own door. Puts her left foot down in a puddle. Pulls her extendable baton from the pocket of her raincoat. Listens, above the footsteps and the resurgent rain, for the sound of the police dogs as they pour into the property's backyard, straining at the leashes of their handlers . . .
Watches a burly officer muscling his way to the front of the pack.
He hefts the Enforcer, the rubber-ended metal battering ram that can deliver three tons of kinetic energy in a single swing.
Brings it forward: expert and practiced.
The wooden door at the front of the house is smashed back off its hinges.
She hears shouts. Warnings. Watches the officers streaming forward—a blur of color and rain—as they surge through the busted door.
Colin Ray puts out a hand.
"No point being in there first, love. Being last out, that's what you want. Slapping the cuffs on and watching as the bastards take their last look."
Tremberg looks at him. At the rain running down his sallow, unhealthy-looking face. At his stained teeth and sodden, stained pin-striped suit. Wonders whether, if he could just be a bit less of a cunt, she could learn a lot from this man.
More shouts. A roar, full of frightened energy.
"Fuck! Fuck!"
One of the detectives emerges from the property. He is breathing hard. Puts out a hand to steady himself against the redbrick wall.
Tremberg follows Ray as he walks briskly up to the house.
"Well?"
The officer is around Tremberg's own age. Fleshy cheeked and earnest, all supermarket suit and inoffensive haircut.
"Fucking forest up there," he says, wheezing. "Got one lad. The other did a bunk out the back."
The radio in Ray's hand crackles. The dog unit has cornered an Asian-looking gentleman in the backyard.
"Job well done, then," says Ray, about to step into the property.
The constable shakes his head. Something is wrong.
"There's a woman up there, sir. Big girl. There was a report, couple of days ago, a misper . . . missing person . . . I think it's her . . . fuck, sir, what they've done . . ."
Tremberg steps inside the house. Pushes past the throng of uniformed officers who line the hallway and staircase, uncomfortable in their damp raincoats, and makes her way up the stairs.
The carpet beneath her feet is patterned with swirls, and her head spins as she pushes open the doors to room after room set up for the cultivation of the finest-quality marijuana. Here blocks of resin, stacked like house bricks, set up for collection. There sacks of leaf, dried out and also ready for collection, sitting like bags of Christmas presents against white-painted walls.
She follows the sound of foreign shouts. Of brutal curses and angry threats, frothing on a tongue bitten bloody by gnashing teeth.
Sees a young, dark-haired Vietnamese man, in vest and shorts, writhing on the ground, tie-wrap cuffs behind his back, an officer on his legs and another pinning his shoulders.
Looks past him. Past the detective leaning against the door frame of a bedroom wrapped in plastic sheeting and hemmed with snaking wires.
Takes in, briefly, the plants in their varying stages of growth: some flowering, verdant and glossy, beneath yellow hydroponic lights.
"In here."
Tremberg approaches. Looks inside.
The woman is alive, but barely. She lies on her side, hair plastered to her face, an officer's uniformed jacket covering her naked, fetal form.
"We've called an ambulance. We didn't like to move her."
Tremberg crosses to the woman's side. Gently pulls back the coat.
The heads of the nails scarcely protrude an inch from the putrefying entry wounds in the back of her hands. The tips are buried three inches into her kneecaps. Blood has run down her legs to her ankles and blackened her feet. She was sitting up when this was done to her, before being thrown down here for more blood to trickle and congeal upon the lumpy, linoleum floor.
Her bare breasts appear, at first glance, to be covered in a matted, sticky hair.
Tremberg peers closer.
Sees the horror of the mutilated flesh. The blackening and burning of her skin.
Tremberg, face gray, turns back to the door. Colin Ray is standing there, smile gone.
"Pharaoh's snout," says Tremberg through bile. "McAvoy's admirer."
Ray scowls. Turns away.
Tremberg brushes Leanne's hair back from her face. Feels the big, well-muscled woman shiver and pull away. Her eyes flicker open and closed. Her lips move. Tremberg has to place her ear next to her mouth to make out what she is saying.
"Shaun—is he okay? Shaun? They wouldn't tell me. They kept asking me where he was, then laughing when I said I didn't know."
Tremberg, despite herself, feels tears prick at her eyes.
She wonders who will have to tell this tortured, broken creature that the man she has been protecting is already dead. That she has been mere practice, and sport.
THE REPORTER is in her thirties and plain as a cheese sandwich. She has brown hair, glasses, and her waterproof coat betrays no flair or sponsor. She's BBC to her bones.
Helen Tremberg tries not to let the sauce from her bacon sandwich drip as she stands in the canteen and watches the bulletin.
The reporter is being lashed by a heavy, gusting rain, and winces slightly as she talks to the camera.
"I'm here on Division Road, just off Hessle Road in the west of the city, where residents were this morning witness to the latest in a series of citywide raids by Humberside Police. We're told that this morning, in an operation involving the force's helicopter and a dozen officers, Drugs Squad operatives smashed their way into the property you see behind me and recovered hundreds of cannabis plants, along with equipment used for their cultivation. There are reports that one of the suspects removed from the house was transferred immediately to a medical facility, though where they sustained their injuries remains unknown.
"I'm joined here by Detective Superintendent Adrian Russell, who oversaw the hugely successful operation."
Tremberg takes a bite of her sandwich and watches as the senior officer enters shot. He has pulled on a coat and made an attempt to slick back his hair, but the unhealthy pallor of his skin and fretting of his hands betray his discomfort.
Tremberg finishes her lunch in two bites as the reporter asks a series of bland questions, to which Russell gives anemic answers.
She tries to pay proper attention. Focuses in on what he is saying.
"It's too early to say at this stage whether this setup has anything to do with a larger organization, but this is clearly an important result. These drugs would have a street value of hundreds of thousands of pounds. We found seedlings and plants in thirteen rooms in this abandoned house, as well as a complex setup. Corridors between the rooms were snaked with electric wires and pipes to vent the smell of the drugs out of the building. The energy to heat the equipment came from a generator that had been custom-built to hide the noise. The front of the property appears totally derelict—"
He is interrupted by the reporter, asking the only question that matters.
"And the two men you arrested?"
Russell looks as though he wants to be sick. "I can only tell you that a fifteen-year-old youth and a thirty-year-old man, both believed to be Vietnamese in origin, have been arrested and are currently being questioned by senior detectives."
Tremberg smiles to herself. Wipes her face with a napkin. She likes being called a senior detective.
Throwing the napkin in the bin, she pushes through the swinging canteen doors and heads for the interview room. She was grateful when they took a break for a midafternoon lunch. She was starting to worry that the vein in Colin Ray's head was about to pop. He is truly struggling with the concept of people not really being able to speak English. Seems about to reach across the desk and do some serious harm.
As she nears the interview suite, one of the doors bangs open and Colin Ray stomps out, furious.
"Fucking Chinks!" he screams at nobody in particular, and then glowers at Tremberg when he sees her. "They understood Ronan easy enough, and he sounds like he's drowning half the time. And they don't understand me? They can say 'solicitor' well enough, lying bastards. Where you been, anyway? Fucking part-timer . . ."
Tremberg bows her head as she is bawled out, and suddenly feels an extraordinary rush of affection for McAvoy and Pharaoh. She wishes to high heaven they were here. Wishes they were running this. She has seen Colin Ray get results today. Seen him, somehow, twist people inside out. And yet it only added to the acid in his gut and the distaste on his face. There is something vile within him. A genuine, bona fide malevolence. She realizes he is dangerous. That if he were not so damned obsessive about catching crooks he would be one.
"Is the translator on her way?" asks Tremberg at last.
Ray spits on the linoleum floor of the corridor. "Hours away. And the assistant chief constable is sniffing around. Talking about procedure."
This morning's brief sensation of victory is souring. The two Vietnamese farmers are saying nothing. If they speak English, they are hiding it well.
Ray stares into space for a while.
"Ronan's picture," she says. "Anything?"
Tremberg had stayed with the older man in the interview room while Ray worked on the younger one. She has not yet heard how it went, though from Ray's face she can guess.
"Knows him, course he does," says Ray viciously. "Eyes like bloody saucers when I showed him. Then it was all this Vietnamese shit and plenty of 'No, no, no.' Same with the picture of Shaun Unwin. And the two other Chinks from the foreshore. Christ, you'd think they'd want to help their mates. Don't they know what they're looking at? Even if they didn't do the harm to Pharaoh's informant, they've been busy growing weed while she lay there rotting and begging for help."
Ray slams a fist into his palm. "They're not going to talk, are they?"
Tremberg doesn't answer.
"Neither's Rourke. Or Ronan. His brief's got him to shut his trap. Shaz can't get a word out of him."
They stand in the corridor, and for a moment neither knows what to do.
Within seconds of each other their phones begin to ring.
They turn away. Ray to Archer. Tremberg to McAvoy.
"Hello, Helen. Are you okay? I heard about the raid. What's happening? I thought you were going home last night. I would have come with you. Were you with Ray? And Leanne, she's okay, yes? Does Pharaoh know? Are you okay?"
Tremberg gives in to a smile.
This is the most appreciated she has felt all day.
• • •
"COULD YOU BUTTER these, please, Suze?"
The middle-aged lady nods at a tray of bread rolls. The gesture comes as a relief to Suzie. The lady is wearing a plastic apron over what appears to be a corset paired with school socks, and Suzie had momentarily feared she was going to be asked to do something unusual with a tub of margarine.
"Don't go mad," she says as Suzie sets to work. "Just a scraping."
This is the aspect of swinging and wild sex that Suzie finds most pleasantly surreal. Underneath the costumes and the impromptu blow jobs, these gatherings are little different from a normal house party. Although most of today's guests will spend the day naked, the owners are putting on a finger buffet, and so far everybody who has arrived has brought a bottle, a plate of homemade cakes, or a card for the birthday girl.
It is four p.m. on a bright but cold Saturday afternoon. Suzie has gravitated toward the large, old-fashioned kitchen of the white-painted farmhouse that stands in a dozen acres of private fields and woodland. She is dressed in a short denim dress, thigh boots, and a Venetian mask, which sits on her head as she sips from a plastic beaker of lemonade and helps the host and her best friend make snacks.
"Throw a few cherry tomatoes on the tray," says the woman. She shakes a bottle of homemade salad dressing with enough force to send her lopsided breasts jiggling. "Make it look pretty."
Suzie is pleased she came. She is not planning on staying all evening, and has no hopes or ambitions for how the party will play out, but she is enjoying the feeling of relaxed escapism that always settles upon her when she finds herself in the company of people who, to some degree at least, understand her.
"I wish I'd brought a cake or something," says Suzie as she gaily drops cherry tomatoes onto limp-looking ham sandwiches. "It was just a last-minute thing."
"Don't you worry," says Christine. "Just nice to see you."
Suzie turns. Adjusts her glasses so she can slide the mask back onto her face. Smiles at the hostess. "Are you having a nice birthday?"
"Ask me again when a few more turn up." Christine laughs. She had greeted Suzie with a big, full-breasted cuddle and a kiss on both cheeks.
"Are you expecting many?" asks Suzie, taking another drink. "Maybe the weather will put people off."
Christine looks out through the thick glass. The sky is a rich blue, but the trees that bound the paddock are shaking in a chill breeze.
"We'll see," she says. "Got a party with the family tomorrow anyway. This is just a normal club night, even if I do get a few extra presents."
"I like your outfit," says Suzie.
"Took some getting into," says Christine. "It's not real leather. You have to cover yourself in talc to get it on, and when it comes off, you can still see the shape of it on your skin. Hopefully be too dark for anybody to notice by then."
Outside, Suzie hears the sound of a car pulling up on the gravel driveway. She heads to the back door and steps into the cold air; her boots precarious on the uneven cobbles.
On the patio, four or five couples are lying, in various states of undress, on deck chairs and sun loungers. When she came here with Simon, Suzie had thought it funny that wicker chairs were also laid out. They had laughed uncontrollably when she had nudged him and pointed at the back and buttocks of a sixty-year-old man who had recently vacated one armchair. "Crinkle cut," she'd said.
There are smiles all around as she is noticed on the patio. None of the people who have turned up so far are particularly attractive, but all have enjoyed dressing up. Unfortunately the cold weather and muddy fields have rather spoiled their ensembles. On one striped deck chair, a woman in her early thirties is wearing a waterproof parka over a crotchless body stocking, while her fifty-year-old partner is holding his lighter as if it were a portable heater, cupping his hands around the flame as he shivers in denim shorts and a tight-fitting T-shirt.
On the other side of the patio, two couples are chatting animatedly about the rising cost of fuel. The tall, dark-haired man who arrived with a younger, bespectacled girl in a PVC catsuit and red knitted cardigan is complaining that it cost him eight pounds more to fill up the car than it did the last time they made the drive over here from their home in Morecambe Bay. The younger, stockier man he is talking to looks genuinely interested. He is complaining that he just spent twenty thousand pounds on a new car, but that it has no cup holder, and beeps at him when he doesn't wear his seat belt. It is a pleasant conversation, and neither of the men seems to mind that the younger chap is wearing an unfastened white dressing gown and wellies.
Later they will pair up and team up. They will drink and smoke and giggle and splash in the hot tub that sits at the bottom of the far field, next to a cheap imitation Hawaiian bar.
Those brave enough will stride naked to the small stream with its halfhearted waterfall that bisects the apple orchard half a mile from the house, where Suzie and Simon once sat and smoked a joint with a gay couple from Leeds.
"Everybody, this is Jarod and Melissa. Say hello."
Big Dunc, the home owner and husband of the birthday girl, Christine, is introducing two newcomers to the rest of the group. Jarod is no more than twenty-five years old. He has short blond hair and an unremarkably pleasing face. He is wearing a black muscle vest that shows a slim but well-defined physique, and looks happy, if slightly ill at ease.
The lady is older. Larger. Expensive and imposing. Black hair, cut short. She could be his mum, were it not for the fact she is holding his hand.
"Just the one single," says Christine, smiling and pointing at Suzie. "Plenty of couples to pick from soon enough. We're really pleased you could join us. Now can I get you a drink?"
Suzie takes a sip from her glass as the newcomers look at her. Jarod smiles. Melissa does, too, but it comes a moment later and is not so wide.
"Where do we put our stuff?" asks Jarod of the lady in the waxed jacket. He gestures at his sleeping roll and overnight bag.
"Big Dunc will sort all that," she says. "Just leave it there for now. You can trust everybody. There's nobody comes up unless they're here for this, so there's never any thefts."
Jarod smiles a thank-you. The woman in the parka, who Suzie seems to think might well be called Karen, gives the man a once-over with her eyes. She looks at her partner and they share a grin.
"First time here?" she asks the newcomers.
"Yeah, thought we'd try," says Jarod. "Game for anything, us."
"Couple are you? Or just a swinging couple?"
Melissa turns to her. "We're just here to play," she says, and there is something in her voice that suggests no further questions are welcome. Those present respect her wishes. Such gatherings are based on trust. All participants in these parties have told some lie or another about where they are going. Some are with playmates they met on the Internet. Others lead completely separate lives with other partners and spouses, only coming together with their "swinging partner" for such parties and club nights as these. And others are here with their husbands and wives, keeping their relationships fresh and exciting by fucking strangers, and terrified at the prospect of their kids finding out what Mummy and Daddy were up to when they went away for the weekend.
Suzie feels a bit of a spare part. She had not felt in the mood to be chauffeured by J & J, and so had driven here on her own. If they turn up later, she will apologize and if need be make it up to them. She is resisting the urge to drink alcohol so she can drive herself home if and when she feels like it, but is beginning to feel an eagerness to claim a glass of wine.
She has not switched on her phone yet. Does not know if Anthony has called. Has vague memories of him putting her in a taxi and sending her home, but the thought of switching on her phone and sifting through the last four days of messages and voice mails fills her with dread. She's thinking of him though. Remembering the puzzled little smile with which he listened to her ramblings. The tenderness with which he had held her in the street as she wept in his arms. The way he stood his corner in the bank and saved her with his white-knight generosity.
"Here."
Suzie has been staring across the flat, green fields and trying to work out if she can see Lincoln Minster in the distance, and is startled when Melissa places a bottle of beer in her hand.
"I'm sorry," she says nervously, taking the drink. "Miles away."
Melissa looks at her. There is an intensity to her gaze. "I like your mask," she says. "That would have been a good idea. Do people usually wear them?"
"Only if they want to," says Suzie, absentmindedly sipping the beer that she had not asked for but is grateful to receive. "It's not about secrecy. Everybody knows everybody."
"Yes?"
Suzie thinks about it. "Well, no. I guess everybody trusts everybody."
"These people know you?"
"They know my face."
Melissa gives her first real smile. "I bet they know more than that."
Suzie takes another drink, and points with the bottle at where Jarod is talking to another couple of newcomers about the difficulty he had in getting this location to show up on the GPS. "Jarod, was it? Interesting name."
Melissa shrugs, as if to suggest that not much about Jarod interests her at this moment. Suzie feels vaguely uncomfortable under the older, larger woman's stare. She has played this game before, of course. She has experimented time and again. She did not think she was averse to doing so again tonight, but at present there are no stirrings of desire within her. She is just enjoying looking across the fields and not really existing for a while. The events of the week are a mound of cold coins in her gut. She feels weighted down and toxic. She fancies she can taste blood when she swallows. She is existing in moments of exhilaration and numbness, unwilling to let any of her thoughts develop into questions. She knows she cannot ignore what happened. Knows that she left a man to die. Knows, too, that she feels somehow fearful for her own safety. But she cannot distinguish this feeling from the loneliness and solitude that have been constant since Simon died. More than anything, her thoughts keep returning to Anthony. It has been a long time since she had these feelings. Is feeling the lovely terror of wondering if somebody likes her . . .
"You've polished that off," says Melissa, pointing to Suzie's empty bottle. "I'll get you a proper drink."
Suzie lifts her mask, then drops it again. She likes being half hidden like this. She readjusts her dress. Exposes the lilies inked on her skin.
"Hi," comes a voice, close enough to her ear to goose-pimple her skin.
She turns. Sees Jarod staring into her eyes, his own a piercing green.
"Beautiful ink," he says, tracing a hand over the design. His touch makes her tremble.
"Thank you." Her voice catches. In her throat.
A half smile on the young man's face; his eyes on her tattooed skin.
"I feel like I've been looking for you."
• • •
NIGHTTIME. A shapeless landscape in northern Lincolnshire; green fields and neatly tended apple trees. Two figures laughing: stick drawings etched in tar.
"Are they okay with this?"
"Of course," says Suzie, laughing. "They're okay with everything."
This is a pleasant drunkenness. Suzie does not feel sick, and the dizziness is that of a carousel rather than a fairground waltzer. She feels light. Not content, but happy enough with this sensation of giddiness.
"You cold?"
"I'll live."
The night sky is the color of bruised fruit, but remains cloudless, and though the air is cold and close, the wind has dropped.
Both Suzie and Jarod are wearing dressing gowns over naked skin. Until a few moments ago they had been drinking wine in the hot tub with a married couple who had driven up from Reading, and a large Asian man with an extreme amount of body hair whom nobody seemed to know.
Suzie has been drinking for seven hours. She has long since given up the notion of going home. Here, intoxicated, giggly, excited, she can see nothing to rush home for. Cannot bear the thought of the empty flat. Shudders at the thought of sitting at her kitchen table, trying to think of something wholesome to do, before giving in and searching dating sites and porn channels for something that will divert her attention from the fact that somebody tried to kill her, and that her best friend took his own life . . .
"Down here," she says, holding open an old wooden gate and pointing to the six stepping-stones that lead to the river.
"Pretty," says Jarod, touching her hip with his palm. He takes the lead and follows the sound of tumbling water.
"Anybody there?"
He and Suzie pull expectant faces as they listen for answers, then giggle at the silliness of it. Suzie feels her insides warming. Enjoys herself, throwing herself into silly games with this young, attractive, playful man. Imagines, for the smallest of moments, that the past few months have not happened. That she is giggling with Simon and that death has not touched her life.
"Is it deep?"
The stream is at its widest point here, beneath the miniature waterfall. It is perhaps six feet across. The riverbed is silt and stone, and sandbanks slope upward to soft, damp grass.
"Up to your waist," says Suzie, cautiously tiptoeing to the water's edge. She is cold—the water from the hot tub turned icy cold on her flesh during their walk across the fields.
"I can't believe we're doing this," says Jarod, and laughs. He leans in and gives Suzie a light kiss on the cheek. It is friendly and not sexual. They may have been naked in the hot tub together, but there has been no suggestion so far of anything happening besides giggles and laughter. They have enjoyed each other's company. They are the youngest people here. Have laughed themselves drunk at each other's gently barbed comments about the other guests. Have talked football and music, stayed away from anything that matters.
"Do you think she saw us?" asks Jarod, peering into the darkness. "She's like a bloodhound."
Melissa, the lady he came with, has not been a popular party guest. She has barely taken her eyes off Jarod or Suzie all day, and anybody who has approached her with an offer of finding a private room or a place to get to know each other better has been rewarded with an icy stare. Suzie does not want to know the dynamics of her new friend's relationship with the older lady, but fancies it is not destined for marriage and kids.
"Ooh, it's freezing!" Suzie has dipped a toe in the water. She winces. Takes her glasses off and lays them on the bank. She pulls up the hem of her borrowed dressing gown and steps, ankle deep, into the water.
"I'm game if you are," says Jarod. He doesn't look particularly game. In truth, he suddenly looks cold and reluctant.
"It was your idea," says Suzie, and her laugh rings out, the only sound besides the tumbling water.
"How did we end up here?" asks Jarod thoughtfully. He appears to be trying to distract Suzie from making him good on his skinny-dipping promise.
"You said you wanted a plunge pool. You said you were too hot in the hot tub. Which you would be. That's its job . . ."
"No, here," he says, casting an arm around. "What did you say you were? Twenty-six? I'm twenty-two. They're all, like, old."
Suzie frowns at him. "They're just people having fun," she says. "You're not going to get Angelina Jolie at a place like this." She pauses. "You might, actually. She seems into all sorts."
"I didn't expect it to be like this."
Suzie pouts. "You're not having fun?"
Jarod waves in the direction of the house. "We're not a couple," he says a little drowsily. "We've done it a few times. Met her on the Internet and it turns out she lives near me. I don't fancy her or anything. I don't even know how we ended up in bed."
Suzie is shivering now, up to her knees in the water, not really listening.
"This is her fantasy," he says. "She says she wants to see me do it to somebody else."
Suzie shrugs. "She doesn't seem like she wants to."
Jarod nods enthusiastically. "I'm not really called Jarod, by the way. I'm Luke. I just liked the name Jarod."
Suzie smiles. "I'm really called Suzie. Some people call me Blossoms."
"It suits you."
"Thanks. Jarod is a good name. You're more a Jarod than a Luke."
They smile at each other, half drunk, half happy, here, knee-deep in a silted-up stream.
"Fancy getting soaked?" asks Jarod, looking at the water.
Suzie is not sure now. She knows it will be exhilarating to plunge into the water, but it suddenly seems too cold. Too dark, even. Her thoughts turn to Simon before she can stop them. To the last time she threw herself into this water, hand in hand with her best friend.
"Next time," she says, and begins to inch her way back to the bank.
Above the sound of the falling water she hears voices. She looks up the slight slope to see a naked couple and the Asian man in a giant bath towel appear at the top of the stepping-stones.
"Hi," shouts Jarod, to alert the newcomers. "Water's lovely."
The trio of fellow bathers wave and laugh. "Is it freezing?" comes a woman's voice.
"Too cold for us," says Jarod.
They pass one another, awkwardly, wet and naked, on the stepping-stones. Suzie gets a whiff of beer and marijuana. The fat Asian man gives her a smile that is guileless and innocent. She wonders if he has turned up here by mistake.
Suzie and Jarod begin to walk back toward the house. They are barefoot and the wet grass feels nice on their feet. Behind them, they can hear fading shrieks of alarm and excitement as the three bathers enter the pool.
"Do you think Melissa is making friends?" asks Suzie quietly as they pass under the low-hanging branches of an apple tree. She lets the leaves play through her fingers.
"Doubt it," says Jarod, with a laugh. "Here, did you—?"
He does not get to finish his sentence.
Suzie turns at a sudden movement in time to see Jarod falling to his knees. He is crumpling as if demolished from beneath. Even in this darkness, she can see the sudden explosion of crimson that colors his expressionless face as he folds in on himself.
Suzie begins to shriek, but finds no words. She spins, her world chaos and movement, darkness and noise, and then there is a hand in her hair and she is being pushed to the ground.
Her face is in the grass, her mouth full of dirt. There is pressure on her back, now. Strong arms upon her shoulders, a fist in her hair.
She feels a frenzied tugging at her clothing and, for a moment, she knows what will happen. Knows she is to be raped. Knows that without Simon to protect her, her fears are coming true . . .
She is yanked back and down again as the dressing-gown belt is tugged free. Suzie tries to throw elbows backward, to claw at the pressure upon her, but she is suddenly aware of her weakness, her glasses pressed painfully into her face, the sudden taste of blood in her mouth as she mashes her teeth on her tongue.
Now the belt is free. Her bare stomach and breasts are pressed into the grass. There is more dirt on her tongue.
A hard yank, her hair tearing at the roots, and now the belt is around her throat: a hissing sound fighting the blood in her ears as her neck is squeezed shut.
_Simon. Please. Simon . . ._
_"_ What the fuck?"
A chorus of shouts. Sudden protests.
"Who . . . ? Get off, you bastard."
The pressure suddenly loosens. She can breathe. She can breathe!
"Come here, you fucker . . ."
"Stop!"
Suzie: coughing up blood and earth, gasping for breath, trying to turn herself. To see who did this to her. To see who it is that is trying to end her life.
Tears in her eyes. Blood streaking her face.
Suddenly feeling lighter than air. Flying. Rising high: a half-drunk rapture.
Being picked up in the arms of a fat Asian man. Her face pressed into a wet, hairy chest. Heart thudding, masking the sound of running footsteps, and distant shouts . . .
SUNDAY, MIDMORNING.
A LEG OF LAMB roasting in the oven and the smell of garlicky meat and fat filling this small two-bedroom house.
McAvoy looks at his wife. She is wearing a purple velour tracksuit top and shorts. She has taken her makeup off, and her dark, tanned skin looks kissably soft in the half-light of the bedroom, illuminated only by the ghost-shaped lamp that sits on Fin's chest of drawers.
"You happy, darling?"
Roisin gives her husband a huge grin. Then playfully shouts, "Catch," and pretends to throw him their daughter. He adopts a rugby player's stance, and they share a laugh together over his instinctive response.
"Are we going to watch the film now?" asks Fin.
The lad had been upstairs, playing with his toys, when he had asked if his sister could come and join him. Roisin had taken Lilah up and told him he had to play nicely and not let her near the toys that could come to bits. Ten minutes later Fin had shouted for his parents and told them his sister had given a noise that was a definite laugh. His parents had needed proof, and set about putting on a comedy routine. Lilah had not responded to silly voices or Roisin's jumping jacks, but had started showing signs of mirth when McAvoy plucked his wife out of the air and threw her on the bed.
"Sure, Fin, we'll put it on. You finished playing?"
McAvoy is interrupted by the sound of a Shakira song. Roisin fumbles in her cleavage for her phone, and puts Lilah on her hip as she speaks.
She rolls her eyes at McAvoy as she asks who it is.
Her smile fades. She stops looking at her husband. Turns away from him.
"Daddy, can we—?"
McAvoy shushes his son. Crosses to his wife and turns her to face him.
"But that's mad," his wife is saying. "It's not an honor thing now. How can it be? He'll never say yes. He's a policeman. No, that's . . ."
McAvoy is rubbing his wife's forearm. Trying to get answers. He has a feeling between his guts and chest, an uneasiness. A queasy feeling of foreboding.
"Tell him no," says Roisin. "No."
She hangs up the phone. Turns to McAvoy. Her face is pale. The dark lines beneath her eyes, invisible when she was laughing just moments ago, seem suddenly to have deepened to a bruise.
"Fin, can you watch your sister for five minutes? There's a good lad."
Roisin's voice has a slight tremble. Its tone is gray.
She settles Lilah back on her play mat and takes McAvoy's hand as she leads him from the room and into their own bedroom. She switches on the bedroom light and sits down on the bed, looking up at him with wide eyes.
"Did you hurt Ronan?"
McAvoy, the nervousness inside him threatening to make his hands tremble, is too bewildered to answer. He tries to predict what he will be told. Cannot think fast enough.
"There's a new campsite at the playing fields in Anlaby," she says. "Some of the lads from Cottingham have set up there."
McAvoy spreads his hands, eager to find out how much he needs to worry. "Yeah, I was there a few days ago, there was an escaped horse, I told you . . ."
"You were there a couple of nights ago. You arrested Ronan."
McAvoy frowns. An image of the ginger lad fills his mind. Sees himself, pinning him to the dirt and wrenching his hands behind his back. Hears, again, the hissed threats. "Do you know him? He's the one who set the dogs on Trish."
Roisin waves the question away. "I think we were once at a wedding together. That's not the thing." She stops. "Aector, do you know who his godfather is?"
McAvoy's mind is struggling to keep up. "What? No."
"Look, Aector, people know who you are. They know you're the big ginger copper that Roisin Byrne ran off with and got herself married to. They know your name."
"What does that matter?"
McAvoy's voice betrays his feelings. They have not had to discuss such things in many years. His wife's past and heritage are things they have both long since assimilated into their union. They have been a couple since she was seventeen. Their first meeting was on a campsite just outside Carlisle. She was a girl, giggly and raven-haired, entertained but not enthralled by the giant, young, uniformed policeman who blushed so furiously as he spoke to the men on the site about a spate of petty thefts. It was only later that their passing knowledge of each other was cemented. Bonded by fire. Turned into something deep and unyielding in a moment of violence that left McAvoy with blood on his hands, and a weeping girl in his arms: she rescued from her attackers by luck, providence, and a giant man with flame-red hair and furious righteousness in his eyes.
"Aector, Ronan's godfather has heard about what you did. Ronan's called him somehow. Told him you beat him up. Tied his hands and battered him."
"That's insane," splutters McAvoy. "I would never . . ."
"It doesn't matter," she says, her eyes pricking with tears. "He believes it. And he wants a straightener."
McAvoy opens his mouth. Pulls a face. He breathes out, relieved that the problem is no bigger than the ones he is already facing.
"A straightener? I'm a policeman! You told them that, yeah?" He pauses. Furrows his brow. "Who was that on the phone?"
Roisin looks at her phone distractedly, as if it doesn't matter. "Just somebody giving us warning."
"Friendly or unfriendly?" asks McAvoy, and there is an edge to his voice now.
"Aector, there are still people who care for me. I'm not dead to everyone."
McAvoy sees the flash of temper in her cheeks and sits next to her on the bed. He puts an arm around her slim, toned shoulders. "I didn't mean that," he says.
He knows how much she has sacrificed to become his wife.
Knows that her mother and father can barely bring themselves to acknowledge that their youngest daughter has married a policeman, in a simple registry office ceremony. Her two brothers deny her existence. Roisin was brought up believing in family above all else. He knows that part of her soul was fractured the day she told her parents that she had fallen in love with the policeman who had twice arrested her dad.
"Aector, his godfather is Noye."
McAvoy searches her face, waiting for more information. None comes.
"Noye?"
"Giuseppe Noye. Pepe."
McAvoy stands again. There is a half-full glass of water on the bedside table, and he takes a sip, swilling it around his mouth until it is warm.
"I'm a policeman, Roisin. We don't have fights. We deal with dangerous people all the time."
Roisin stands now, coming close to her husband. There is genuine fear in her expression.
"He won't care about that," she says. "It's a traveler thing. An honor thing. Ronan's told him you hurt him, and that's that. The uniform won't matter."
McAvoy sighs. He could do without this. "Roisin, seriously, he can't expect me to go and have a bare-knuckle fight . . ."
"He does! That's what he's demanding."
"Well, he hasn't demanded anything of me."
"This is how it works, Aector," she says patiently, as if explaining to a child. "The word gets out. A message gets to you. A time and place is arranged. You meet and you fight. And you keep going until one of you gives up."
"Dead?"
"No, not dead. There are rules. There's a ref. He keeps it from getting—"
"Deadly?"
"Yeah. But people get hurt. Really hurt. And they get hurt by Giuseppe Noye."
McAvoy finishes the glass of water. Sits back down and pulls Roisin to his knee. In truth, he is not overly concerned. He is sad that his wife is upset, and knows that he will probably have to deal with this situation at some point, but in terms of what he has to deal with at present, he will not be giving Giuseppe Noye much thought. He mentally puts a circle around the name. Makes a note to check him out, and cross-reference for any links to Vietnamese drugs gangs.
"I can look after myself," says McAvoy. "This is what I do."
Roisin does not seem pacified. "Would you fight him, Aector? If you had to? For honor?"
McAvoy looks at her. He realizes he has been wrong. Her fear is not that Noye will hurt him. It is that he will not fight.
"There's no honor in this," he says coldly. "I'd die for what I thought is right. But this? Is that what you think I am?"
Roisin drops her face to her hands. "I don't know what I want. Sometimes I feel like a stranger. The way things are, the way you all behave."
"Who's 'you all'?"
They sit in silence. For a moment, McAvoy entertains the notion of agreeing. Of standing his ground and taking his bruises from a bare-knuckle fighter. He laughs under his breath. Reaches out and strokes his wife's hair.
"I'll be whatever you want me to be, Roisin. I'd die to make you smile."
She shakes her head. "I don't want that. I don't even want you to fight. I want you to be you. To be good and brave and caring. But then I see my mam's face and how she would sneer if one of her boys said no to a straightener and I don't know who to be myself."
McAvoy pulls her close. Holds her. They were married when she was so young. Her life was among the travelers, and she took to his world without a backward glance. There are times they both feel they married somebody from a different age.
He tries to make her smile.
"Lilah was awesome, wasn't she?"
With an effort of will, Roisin manages to let herself be steered into more pleasant thoughts.
"She's got my laugh, not yours."
"That's a relief," says McAvoy. "She'd scare people."
Fin appears in the doorway. He is scowling and clearly ready to watch the film.
"Go on down with Mammy," says McAvoy. He eases Roisin into a standing position. "I'm going to make a call or two, then I'll be down, too."
She looks at her husband. Ruffles his hair and bends forward to stroke the rasping stubble on his cheeks. "You're my hero."
The family head downstairs, leaving McAvoy alone in the bedroom. He picks the laptop up from where it has been charging by the bed, and places it on his knees as he shuffles back against the headboard. The machine had run out of power when they were looking at holiday destinations in bed the night before. The picture is frozen on an image of a lake in Sweden. It is the view from the remote log cabin he hopes to be able to afford to take his family to for a week or so in the winter. Whether they make the trip or stay at home will depend on whether the insurance company pays out for the minivan. He is not getting his hopes up.
He logs on to his work e-mail, using his remote access code and password. Checks his messages. Nothing from the tech unit yet, and a brief line of thanks from ACC Everett for rewriting his speech. It had gone well.
Pursing his lips, unsure whether he is simply inviting more worry, he accesses the Police National Computer. He enters the name Giuseppe Noye and breathes out through a tight mouth as the screen is filled with the criminal activities of the forty-eight-year-old repeat offender. He scans the various crimes. Armed robbery. Wounding. Receipt of stolen goods. He has served for different lengthy sentences. Was released from a stretch only last September and has not kept any of his parole meetings. A warrant for his arrest is currently active.
McAvoy brings up the mug shot. Maximizes the image until it fills the screen. Looks into the face of a thickset, bovine man with close-cropped hair and piggy eyes, his jowls and jaw covered in gray stubble. McAvoy checks his height. Six feet, two inches. He gives a little nod.
"Okay," he breathes.
He is about to close the screen when it occurs to him to check Noye's associates. He does not know whether he expects to find Ronan's name, or Roisin's.
Scrolling down, he looks for familiar names. Stops at Alan Rourke. The pair did an armed robbery together in 1993. Held up a post office in a village just outside Leicester. It had been a straightforward raid: lots of noise and shouting and a shotgun shoved in the postmistress's face. They would have got away had Noye not realized, on his way out of the door, that he had used the name Al when shouting instructions at his partner. Despite Rourke's protestations, he had climbed out of the getaway car to go back in and silence the witnesses. The decision was costly. Rourke and Noye were still arguing on the pavement over whether or not to add murder to their list of crimes when the police turned up. The chase was a short one. Rourke crashed their stolen Toyota, and both men were sent down. They served seven years of a twelve-year sentence.
McAvoy jots down a couple of notes. Closes his eyes, aware he is about to be shouted at, then picks up his mobile. Calls Colin Ray.
"What do you want?" The voice is tired and grumpy.
"It's about Alan Rourke," says McAvoy, determined simply to say what he has to, and then get off the phone. "One of his associates. A Giuseppe Noye. He's worth checking out."
There is silence at the other end of the phone. McAvoy wonders where the other man is. Realizes he knows precious little about his life. Knows only that he is twice divorced and lives in an apartment somewhere in the city center. He tries to picture his life. Finds it hard to imagine the older man without Shaz Archer in his shadow. A thought crosses his mind. He wonders if there is anything more to their relationship than the master-and-protégée dynamic. Realizes that many of his colleagues must have questioned it before him. Wonders, briefly, whether such rumors would ever circulate about his own bond with Trish Pharaoh.
"It's Sunday morning, lad. I'm busy."
"Oh, yes?" McAvoy tries to sound chatty. Can't help but be curious.
"Picking up the lads, as it happens. Football match."
"Yes? Who's playing?"
"We are, you daft bastard. Bridlington away."
McAvoy vaguely recalls some conversation he had with Colin Ray when he first joined the unit. Remembers that the older man coaches one of the divisional police football teams. Remembers, too, the detective chief inspector's expression when he told him he was a rugby and boxing man, and did not follow football.
"Are you driving?" McAvoy is about to offer to call him back when it is safe to take the call.
"What do you fucking want?"
McAvoy feels the blush. Wishes he could talk to people with some degree of comfort or aplomb.
"One of Alan Rourke's past associates. He's a real villain. A Giuseppe Noye. He's also the godfather of Ronan."
A pause at the other end of the line. "Noye?"
"Yes. Armed robber." He thinks for a second about whether to reveal more. Realizes he must. "Traveler."
Ray gives a bark of a laugh. "You don't say."
McAvoy falls silent. "I thought it might be worth checking out, that's all."
He has done his best to maintain an interest in the Rourke investigation, but knows only that the old armed robber kept his trap shut during the interview. Gave "No comments" all the way. Young Ronan gave only slightly more. Lost his temper, shouted and screamed his way through questioning. Neither Ray nor Archer had managed to get a useful word from either of them, and though they made a fuss when both were given bail, they had expected little else. Ronan gave his address as Rourke's place, and the older man was put down on paper as being his current guardian. Social services went away happy. And Ronan fucked off the second he walked out of the door.
"I know the name Noye," says Ray quietly, appearing to be struggling with a memory. "Fighter, isn't he? Bare-knuckle stuff."
McAvoy isn't sure how to respond. Starts Googling Noye's name for something to distract himself. "A boxer? I don't think . . ."
"Traveler fighting," says Ray. "Bare-knuckle stuff. I think he's part of that crowd."
McAvoy finds a link to the gypsy's name. Clicks it. Feels himself closing down inside as he presses play on a video showing Noye stripped to the waist, knuckles taped, pounding right hand after right hand into the ribs of a younger man while a crowd of lads form a rough circle around the fight. A muscular man in a white T-shirt tries to separate them. To keep some kind of order. He is struggling.
"That's illegal."
"Piss off, lad," says Ray. "Everything fun is illegal. And the gyppos have been doing this shit for centuries. Straighteners, they call them. Honor fights. Big business now. They're arranged like pro fights. Big crowds. And the DVD sales are massive."
"I'm watching him fight now," says McAvoy. "How can I be watching an illegal fight? I just clicked one button . . ."
Ray gives a joyless little chuckle. "I'd love to see the world like you do, lad. Fucking hell."
McAvoy pauses the video, just as the camera zooms in on Noye's snarling, blood-spattered face. His bound knuckles, too, are caked in red.
"The interview," says McAvoy. "Ronan."
Ray laughs again. "Fuck all so far," he says. "Had to sedate the little bastard. Every time he went in his cell he lost it. Started bouncing himself off the walls. Not happy with you."
"Me?"
"He's feeling a bit miffed that you put him down like a sack of shit."
McAvoy isn't sure whether to preen or be humble. "I'm a policeman."
Ray says nothing for a moment. Then, as if it hurts him to say it, adds, "You did good, by the way. Taking him down. I lost my feet. Little shit got me right in the jaw. Landed on a rib. Hurts like hell . . ."
McAvoy knows that if he were to speak, he would spoil the moment, so simply nods. "Any news on Pharaoh?" he manages.
"Back tomorrow, so she reckons," says Ray, equally glad to have had the subject changed. "She could have strung this out for months, silly cow. Obviously needs to come and make sure we can still wipe our arses."
McAvoy lets the other man talk. He is wondering what Pharaoh's return means. Whether he has done enough wrong to get more than a telling-off. Whether he will be able to get the report back from the tech unit in time to present her with evidence of the need for a genuine murder inquiry. Whether he should just do what he's been told. Wonders, for a moment, why he is not trying this hard to catch the two shaven-headed thugs who have outmuscled the Vietnamese and caused a spike in the violent crime statistics.
"Enjoy the match, sir," says McAvoy. "Hope you win."
"Enjoy whatever it is you fucking do," says Ray, and ends the call.
McAvoy stares for a moment longer into the eyes of Giuseppe Noye. Shakes his fears away. Calls the tech unit and asks for Dan.
"Sergeant," says the young man when he comes to the phone. "All good?"
"I was rather hoping to have your report this morning," says McAvoy. "Superintendent Pharaoh did specify that it was very urgent."
"I know she did," says Dan. "I was up till three for her. She's worth an all-nighter, don't you think? That's why I sent her the report."
McAvoy closes his eyes. "You sent it to Superintendent Pharaoh?"
"Yes. And?"
"I asked for it to come to me, I said!" He sounds exasperated. Childish.
"Does it make any difference? I wanted her to know how much effort we'd gone to . . ."
McAvoy is spared the effort of trying to formulate a sentence by the sound of the doorbell. He takes the phone away from his ear and listens to the muttered conversation from downstairs. A moment later there is the sound of footsteps on the stairs.
"I'll call you back," he says into the phone.
He looks up, ready to smile for his wife, as the door opens. His face freezes, locked in place, color falling into his shirt.
In the doorway of his bedroom, gauze strapped to her throat, hands bandaged, dressed in jeans, a too-tight vest, and a leather jacket, stands Trish Pharaoh. Her eyebrows are raised almost to her hairline.
"Guv, I . . ."
He is suddenly aware that he is wearing nothing but shorts. That he has a laptop balanced on his lap. He shuts the screen like a guilty teenager looking at porn.
Pharaoh raises a fist full of computer printouts.
"Time for a chat?" she asks.
Her voice could shatter steel.
"Your missus is a looker," says Pharaoh, leaning against the bedroom wall and making no attempt to look away as McAvoy pulls on a hooded sweatshirt and smooths down his hair with the palm of his hand. "Not what I pictured."
McAvoy wonders what his wife made of his boss. What she will make of him, when she is gone.
"Thank you," he says, distractedly. He waves an arm vaguely to indicate her injuries. "How are you?"
Pharaoh's expression does not change. She continues to watch him with wide-eyed detachment. "Sore. Getting attacked by Rottweilers will do that to you."
"I wish I'd been there . . ."
"I know you do."
McAvoy stops. Stands, next to the bed, and meets her gaze. "I was worried," he says.
Pharaoh softens. For a moment, she is an indulgent mother accepting a thank-you fridge drawing from a naughty toddler. "Daniells stepped up. Kicked one of them right in the nuts."
"He got hurt, too."
"Poor lamb."
"He did good."
"He's not you," she says, shrugging, and what she means by it is left unsaid.
McAvoy cannot help himself. He points at the papers she clutches. "Tech report?" he asks, wincing.
"Yes, e-mailed to me at three a.m., together with a little note from some computer geek who wanted me to know how hard he had worked on this, along with all the info I requested and assigned to my budget."
McAvoy rubs his face. Realizes he is biting the webbing between forefinger and thumb.
"It's the case I told you about, guv," he says. "You suggested I have a look."
Pharaoh runs her tongue around the inside of her lower lip. Despite her injuries, she is wearing makeup. He wonders whether she applied it herself. Why she bothered, if only to come and shout at her sergeant.
"Does the expression 'bane of my life' mean anything to you, Aector?"
McAvoy grabs the question like a lifeline. " _Bane_ is an old English word for 'murderer.' That morphed into meaning 'something that causes death.' That's where you get the name for poisonous plants from, like wolfsbane and henbane . . ."
"No, Aector. _You_ are the bane of my life. I spend a lot of time deciding whether or not to stab you in the head."
McAvoy stops talking.
She looks at him hard. Gives the room a quick once-over. Lets her gaze linger, for the tiniest fraction of a second, on the leopard-print silk nightdress that hangs from the foot of the bed.
"I hesitate to ask this of your big brain, but is there a landlubber word for a Jonah?"
"Are you really asking?"
"No," she says. "But I am implying that you seem like a magnet for shit."
McAvoy looks at the papers in her hand as she gesticulates. Is desperate to unroll them and read the hidden words.
"I thought it was important."
Pharaoh smiles, rolling her eyes. "It is important. You were right. You're nearly always right. Doesn't mean I don't think about hurting you."
McAvoy's skin prickles. He does not know how to comport his face, so stands still, looking expectant. "I was right?" He is hesitant to ask which of his half-formed theories and vague gut instincts have been vindicated.
"Right about Simon Appleyard," says Pharaoh, crossing the room and sitting down, unasked, on his bed. He stands up, in turn. He has to fight the urge to turn scarlet at the intimacy of the moment. His boss, here, in his bedroom. Family downstairs. Words to stroke his ego in her hand.
"He was murdered?" asks McAvoy, instinctively, leaning back against the wall in the position she has just vacated.
Pharaoh shrugs. "You're right that he could have been. I've spoken to the pathologist again. She e-mailed me through images of the body and the postmortem exam. Pretty boy, wasn't he?"
"His back, you mean? The tattoos?"
"Yeah, lovely work. Will have to show you mine someday. Anyway, I can see why she didn't see it, but she'll be getting a bollocking at some stage."
"See what?"
"The bruise," says Pharaoh, rustling through her papers.
McAvoy crosses back to her. Sits next to her on the bed. Catches a hint of her perfume. Notices that she is wearing open-toed shoes instead of her usual biker boots, and that her feet are not as pretty as his wife's. Wonders why he is even thinking about such things.
"You can see it here," she says, holding up a color printout.
McAvoy looks upon the photograph of the boy whose death has so troubled him. Simon is laid out naked on a steel table. The clinical aluminum and white of the mortuary frame the exotic colors of his body. Makes his slim frame appear almost skeletal. McAvoy stares into the mass of ink. Squints, between the eyes of the tattooed peacock feather, at the slight blur of discoloration.
"Sergeant Arthurs told me about that," he says. "Said he was surprised the pathologist missed it."
"Nothing surprises me," says Pharaoh. She hands him more pictures. Simon, kneeling forward, slumped and lifeless, his skin a mottled red and blue. A rope trailing from his throat, tongue hanging forward from between open lips, a black slug.
"He had been there some time," recalls McAvoy.
"Heater was on the whole time, too. Decomposition started quickly. She'll have a good reason for not seeing it."
"If it's anything at all," warns McAvoy.
"True," says his boss. "But it looks like a footprint to me."
"Or a knee," he says, looking again at the image.
They look at each other, close as lovers on the edge of the bed. McAvoy looks away first.
"Why did you contact the pathologist, guv?" he asks.
"Boredom?" She laughs. Then her face turns serious. "No, Aector, I trust your instincts. You've made your usual balls-up of going about it all, but there's something here."
McAvoy is torn between feeling flattered and insulted. Tries to ignore both feelings and just ends up jiggling his leg. His mind is trying to work out how much she knows. Whether she is already further ahead than him.
"Dan's report?"
"I'm pleased it came to me first. You'd have had a heart attack. But you're right. I think we made a right cock-up looking into it."
"I've spoken to his aunt," says McAvoy. "She doesn't know what she thinks. Doesn't know if she wants to know. But she says he was living life to the full, if you get my meaning. And had a friend who went everywhere with him. I haven't started tracking her down yet."
"What else?"
McAvoy looks skyward. Realizes he has no real justification for sharing more, but does not want to hold anything back. "Two city councillors," he says, at last. "Cabourne and Hepburn. They're connected. They're lovers."
"Lovers! Christ, the way you talk. They're both blokes, yes?"
"Yes. Hepburn's the one who . . ."
"Yes, I know him. Character, yes? Some shady stuff in his background, but nothing he hides." She waggles her tongue thoughtfully. "And have they got a connection to Simon?"
"Cabourne has been meeting men for sex—using the same dating site that Simon posted his details on. Simon's phone number is on there."
"And does Cabourne remember Simon?"
"He thinks they shared some messages but nothing ever happened."
"But?"
McAvoy shrugs. "Hepburn knows more than he's letting on. And I think Simon might have met somebody on that site who didn't want his secret getting out."
"Cabourne?"
He considers it. "I don't know."
They sit in silence.
"Anyway, turns out Dan's more of a technical genius than you are. He's got plenty more info off it than you managed."
She hands him another sheaf of papers, made almost illegible by the amount of creases.
"What's this?" He looks at the images. Turns it around. Widens his eyes.
"Yep," says Pharaoh with a smile. "That appears to be a picture of our dead man having a little bit of fun with himself, though quite why Dan thought I'd want that to be the picture I looked at over my breakfast is anybody's guess."
"He sent these?"
McAvoy looks at the images. They are unmistakably of Simon Appleyard, naked and pleasuring himself.
"Bloody hell."
"Yes. And they were sent as picture messages at about nine p.m. on the day Simon was last seen."
"That's . . ."
"Yes, about an hour before the pathologist reckons Simon died."
"So he was sending this kind of stuff at nine p.m. and then hanging himself at ten?"
"Doesn't mean he didn't bump himself off," says Pharaoh, taking the pictures back. "Just means there is a hell of a lot to explore here."
"What else?"
"Some more poetry. Messages he sent . . ."
McAvoy takes the report. Reads Simon's words out loud: "'You move inside me as a puppeteer. Take ownership of my body. Fold me into your vision of desire . . .'"
"You say the nicest things. Look at what he was getting in return. These were in the in-box."
McAvoy turns the page. "'Am going to hurt you. Take you. Make you my bitch . . .'"
"They were my wedding vows."
"'Will scratch my mark on you, tear open the ink on your skin . . .'"
"Yep, I do."
McAvoy stops. "So he knew he had tattoos? Had they met before? Or did he send him pictures of his back, too?"
Pharaoh sighs. "What we can get is in there. Simon's poetry, and this other person wanting to hurt him and dominate him."
"Is it just a game?"
Pharaoh raises her eyebrows. "I'm not the expert," she says. "I know people go online a hell of a lot looking for sex, and I know people have fantasies they want to come true and those that they don't. Anyway, this is all just maybes. I'm not here on a Sunday for maybes. You haven't got to the interesting bit yet."
McAvoy turns to the last page of the sheaf. Reads the words underlined in red. The words Simon Appleyard received the night he died.
Want you on your belly when I arrive. Naked. Body waiting for my touch. Hold the rope in your hand. Leave the door unlocked. Show me your ink as I arrive, then let me take possession of you. Let me make you feel pretty . . .
McAvoy looks up. "Fuck."
Pharaoh smiles. "Yes indeed."
THE WATER tastes of early mornings. Of last night's booze.
Dirt.
Grass.
Blood.
"Thanks." She grimaces. Her throat is full of cold stones. "Lovely."
She shuffles herself into a more comfortable sitting position. Watches the sunlight stream in through the conservatory glass. Dazedly soaks up the view. The flat green landscape and the swaying trees, the painted-on symmetry of the distant apple trees and the blueness of the clear sky.
"Still sore?"
Suzie winces again as she finishes the drink. "Will be okay when it opens up a bit. Christ, I sound like Louis Armstrong."
She is in the large, L-shaped living room of a remote Lincolnshire farmhouse that, three nights a week, becomes a sex club. This morning it is just a home, and she is an injured guest, convalescing, wrapped up in a blanket on the sofa and with her hair stuck up on one side.
"Did you have bad dreams?"
Suzie gives a shrug. "I don't remember," she says. "Maybe. Doesn't matter if you don't remember, does it?"
Christine is up and dressed. She looks comfortable in old jeans and a rugby shirt. Big Dunc is doing something onerous out on the shingled drive. She can hear the scrape of a rake on the pebbles.
"You must be hungry," says Christine. "Yogurt? Fruit?"
Suzie screws up her face. "I'm going to get off this morning. I'll stop at a McDonald's on the way home. I'm fine."
"Suzie, you can stay as long as you want."
"Honestly," she says. "I need to go."
Christine seems unsure. Suzie can understand her feelings. Here, on the sofa in the living room, she can be watched. She can be gently spoken to and nursed. She can be persuaded of the benefits of chalking Saturday night down to experience, and keeping her bloody mouth shut.
"Really," says Suzie, stretching. "You've been good to me. I'm okay to go."
Christine still looks worried, but she forces her lips into a tight smile. "I'll make you a sandwich for the journey," she says, and takes the empty glass from Suzie's hand before heading back to the kitchen.
Suzie rummages around in her thigh boot and finds her watch. It's just gone lunchtime. A McDonald's breakfast is out of the question.
She fell asleep just after three, just as the last vehicle left the grounds, and as news filtered back from the hospital that Jarod had a fractured skull but was going to be okay. He'd been smashed across the head with a branch. It has been disposed of. Suzie did not have much to say about that, or anything else. She was on the sofa, knees curled up beneath her, sucking an ice cube, some of Christine's hand cream turning the redness of her rope-burned neck into something ghoulishly shiny.
Last night is coming back to her in stages. Nobody had wanted to call the police. There had even been dissenting voices when Big Dunc said he was going to take Jarod to hospital. Were it not for the fact that she could barely swallow and that her heart was still racing, as if to justify its reprieve, she would have found the discussions of the previous evening comical. Even in her dazed, drunken, semi-throttled state, she could feel a bizarre giggle building inside herself as she took in the scene. A score of men and women—some in white dressing gowns, like Greek philosophers, and others with towels around their waist. One man entirely naked, sitting on the edge of a wicker chair with his shrunken manhood sitting on his balls like a hat. Jarod laid out on the patio in a mess of mud and blood. Angry voices and fist-shaking accusations.
The man who carried her back to the house said his name was Matt. He was a chartered accountant from Bradford, and spoke with a thick West Yorkshire accent. He did not let her go until she was ready. Held her in his arms as if comforting a child. Placed one large hand over her left ear and pressed the other ear to his chest, while the argument raged about what had happened and what should be done.
Suzie does not judge the others for wanting to keep their secrets. Few would be proud to have their names and addresses taken by detectives in connection with an attempted murder at a sex party. Fewer still would want their lives pored over by sniggering police officers, or their wives and partners to be questioned over their movements and bedtime habits.
Suzie's attacker had not been found. Whoever it was disappeared into the shadows with barely a sound; the shouts of pursuers roused the rutting couples who were enjoying the party, and resulted in a hastily convened, babbling argument in the conservatory about what was for the best.
"My life will be over! This can't get out. There'll be interviews. Police. The papers. My wife!"
"It's about right and wrong. Somebody's attacked him. He could die!"
"No, it's too important. Secrecy, remember. That's what the website says. Discretion assured."
"This changes things. It's life and death. It could be one of us."
"He probably slipped. She might have had too much to drink. It might even be her who did it."
Like a tennis umpire, she watched the debate go back and forth. At length, they had convinced Suzie that her attacker was probably a local teenager. Big Dunc revealed that their website had received a few e-mails from youngsters who had heard there was a sex club upon their doorstep, promising to pay a visit next time there was a party. Suzie had merely nodded. Kept her own counsel. Swallowed painful mouthfuls of blood and picked the dirt from between her teeth.
Here, now, she knows. Knows full well that she has been running from her own thoughts. Knows that Simon did not hang himself. That she never truly believed that he did. Simply refused to let her fears take her to a conclusion that terrified her. She knows, more than anything, that whoever killed him is now after her.
Filling herself with a deep, painful breath, she rummages through her handbag and finds her phone. It has been switched off for days. She half expects her hands to tremble as she turns it on, but is surprised to find that she is in control. She feels detached somehow. Not numb, but somehow separated from what she is doing. She did not feel her soul leave her body as her attacker strangled her, but now she almost feels as if she were looking down on herself from above.
Suzie disentangles her legs from the quilt. She is still wearing the dressing gown. Somebody has brought her clothes from the side of the hot tub, but she is in no mood to dress as yesterday. She stuffs them in her handbag after removing a long blue dress, spotted with snowflakes and with an owl on the left breast. She pulls it on and wriggles her feet into her boots as the phone begins beeping. Spewing out messages and missed calls.
Closing her eyes, preparing herself, she moves to the conservatory door and slides it open. Takes a lungful of cool, fresh, air. Plugs back into her life against a soundscape of chattering birds and scraping gravel.
The message she seeks was sent ten minutes after the car plowed into the man she had been ordered to fuck.
So sorry. Can't make it. Will you still go play for me and tell me how he touched you? Wish I could see what a dirty girl you are. Xx
Suzie swallows again. Sneers, ever so slightly, and makes fists with her hands.
His next message was sent early the following morning.
Were you a bad girl last night? Xx
Then:
You've gone quiet on me. Did you pussy out on me? Are you a tease?
There is a hiatus of a few hours. Then, angrily:
Knew you would be just like the others. Knew you were all talk.
She scrolls on. Finds his next missive.
Sounds like you had a lucky escape. Bad accident at the rest stop. Lucky girl. x
Finally:
May make an appearance at the party you mentioned. Lincolnshire, you said. Googled it and sounds a ball. Would I be welcome? x
Suzie stares out across the fields. Watches a fat, purple-throated pigeon walk delicately along the wooden fence. Squints, and wonders if the brown creature she can see near the hedgerow is a rabbit or somebody's discarded UGG boot from the night before.
She sifts through her other messages. Nothing from the police about the accident, but plenty of inquiries from work, from friends, about where she is and what she's doing. A Facebook alert from her mum.
Her eyes close, almost involuntarily. The sensation of dislocation is dissipating. She is coming back to herself, guided by the pain in her throat and the cold emptiness in her gut. She is unsure, right now, how she feels about herself. She knows that she has let Simon down by accepting his suicide without question. Believes herself to have cheapened their memory by giving in to fear, and never demanding a less palatable truth.
It was the shabbiness of his life that Simon hated. The smallness of it. The inability to shine as brightly or as brilliantly as he wanted to. But such miseries would not claim his life. No, he was killed by somebody he wanted to make happy, and Suzie wants to cry at the thought.
"Somebody wants to kill me, Si," she says, under her breath. "Somebody who killed you."
She opens eyes that threaten to fill with tears. "I'm so sorry."
"Beg your pardon, love?" Suzie turns. Christine has entered the conservatory with a fat ham sandwich and a mug of tea. "Made you a little something," she says, putting it on the stout table that has been cleared and wiped down sometime between the party and today.
"Oh, I'm sorry, I don't think . . ."
"You need to eat," she says. She puts an arm out and gives Suzie a squeeze. "Lovely dress," she says.
Suzie can barely find the strength to smile. She wants to run suddenly. Wants to get away from these old people, with their sagging flesh and foul-tasting skin and their desperation to touch her and make themselves feel alive. She hates herself now. Hates seeing herself as this link to vitality. This young sacrifice being fawned and pored over, tongued and tasted, by men and women fleeing the grave. She feels disgusted with herself. Here, now, she feels the wrong kind of dirty. Feels the wrong sort of whore.
"I have to go," she says, bundling out of the conservatory door, spilling yesterday's knickers from her bag as she fumbles for her car keys.
"Dunc!" Christine is shouting her husband's name. "Dunc, she's going . . ."
The big man appears suddenly from behind a white-painted outbuilding. He is all smiles.
"You off, sweetie? There's no rush. I'll give you a lift later . . ."
Suzie can't think of anything to say. She just pushes past him. Runs to where her crappy blue car is parked on a patch of grass. Pulls open the door and climbs inside, willing the engine to work. Now her hands tremble as she turns the key, and she laughs with relief as it bursts into life.
She turns the car in a ragged semicircle, scattering the neatly raked gravel, and puts her foot down. She feels alive suddenly. And so very scared of death.
The trees and hedgerows fill her windows as the car bumps and jerks down the rutted path. She is barely looking at the road. Instead, she fiddles with her phone. Scrolls through her numbers. Finds the number she could have called six months ago.
Rings the auntie of her dead friend.
Doesn't even manage a hello.
"Simon was murdered," she says.
And in this moment, a sluice gate opens inside her. The tears finally come.
"ARE YOU NOT GOING TO EAT IT?"
McAvoy holds the plastic Tupperware box on his lap as if it were a ticking bomb.
"Maybe on the way back."
"It'll get cold."
"It's nice cold. It's fine."
"If you'd done as I'd said, you could have sat down to dinner with the rest of us."
"I wasn't that long. I just wanted a quick shave . . ."
Pharaoh has a little smile on her face during the exchange. She is not deliberately trying to embarrass him but is enjoying having a little play with his shyness. Likes these married-couple chats that they too rarely have the opportunity to indulge in.
"She wasn't cross, was she?" he asks, his eyes closed, like a toddler trying to make himself invisible.
"Aector, I don't think she could be cross with you if she found you nuts deep in a squirrel."
McAvoy is grateful he is already looking out of the window. This way he does not have to disguise his blush or his smile.
Pharaoh stayed for lunch. Sat down to roast lamb and minted peas and potatoes while McAvoy was upstairs showering, shaving, and slipping into a brown tweed-effect three-piece suit. He is not wearing a tie. It is his concession to working on a Sunday.
"There's a lot to hate about your wife," says Pharaoh chattily as she turns the little sports car onto Anlaby Road and slams down the clutch with her bare left foot.
"I beg your pardon?"
"Gorgeous. Slim. Lovely. I should hate her."
McAvoy, who had swiveled his head toward his boss, looks away again. "Oh."
"Seriously, Aector. If that lamb knew how well his body would be treated, he would have handed himself in at the abattoir. I have never had gravy like that. Could I take a slice home for a sandwich?"
McAvoy holds the lunch box a little tighter. Of all the crimson hues that his cheeks have taken this past week, none was as deep as that which exploded in his face when his wife handed him a Tupperware box with his roast dinner inside it, kissed him on the cheek, and told him to have fun.
"We had a nice chat," says Pharaoh devilishly. "Can't believe we've let it go this long without ever properly meeting."
McAvoy splutters a little. "Yeah, well, next pub quiz or something . . ."
"No, I owe you a dinner now. We'll have you over to ours."
He does not know how to reply. Has never been able to picture his boss's home life, or felt comfortable enough to ask about it. He knows only that she has teenage children, and a wheelchair-bound husband, though whether his condition is from injury or disease, he has never ascertained.
"What's your specialty?" he asks her, by way of conversation.
"The wine list," she says, her eyes on the road. "Up here?"
McAvoy gives a nod. "Yep. Second right, then pull in."
In the past two hours Simon Appleyard's death has become a murder. They are taking the first tentative steps toward telling the top brass in CID that the violent-crime statistics for last year are going to show another unlawful killing. They have only briefly mentioned the discovery of Leanne Marvell. Neither wants to address it. Neither wants to question whether they could, or should, have done more. They will visit her, together, when this is done. Help her somehow. Make it better . . .
"These are all right," says Pharaoh, pulling into a space and looking up at the properties. "Small, but neat enough."
"Three hundred fifty pounds a month, or thereabouts," says McAvoy. "One-bedroom, but properly looked after. All the same landlord."
They disentangle themselves from Pharaoh's two-seater sports car. It is as close-fitting around McAvoy's bulky frame as a tailored suit, but Pharaoh adores the vehicle and wears it far more comfortably than her sergeant.
"Going to be a nice one," says Pharaoh, looking up at the blue sky. "Any more rain on the way, Farmer Boy?"
McAvoy smiles. Gives a sniff of the air. "Maybe a bit of drizzle tomorrow."
"And what perfume am I wearing."
He inhales again. "Issey Miyake. And roast lamb."
McAvoy straightens his clothes and checks that his notebook is open on a fresh, dated page.
They stand for a moment on the pavement outside the little apartment block. Springfield Court in Anlaby. A nice enough neighborhood with decent schools and a couple of twenty-four-hour supermarkets. Flats for young couples saving for a deposit on a starter home, and for singletons content to rent.
"Still nothing from the landlord?" asks Pharaoh.
McAvoy checks his phone and shakes his head. He has left a message for the property owner, but has heard nothing back. A quick search of the electoral register indicates that Simon's old flat is now occupied by a Mr. Paul Essex, though how precise the information is, they cannot say.
"Flat two B, yeah?"
"Or not."
"What?"
" _Hamlet_. Forget it."
Pharaoh rings the bell. They stand for a moment on the step, staring at the white paint. Out of habit, Pharaoh tries the door handle. It doesn't move.
"In bed or out, you reckon?"
McAvoy considers the crisp blueness of the day, and doubts that the door will be opening anytime soon.
"Try the neighbor," says Pharaoh, stepping back from the door to peer up at the second-floor window. Net curtains obscure the glass.
McAvoy moves around to the other side of the building. Rings the next bell. Stands for a time, tapping his foot, then rings again.
"Luck of the Irish, you, ain't you?" says Pharaoh snappily. She sighs. Cocks her head. "Can you hear that?"
McAvoy listens. He can hear guitar music coming from inside the ground-floor flat. He struggles to place it. It sounds Spanish. Classical. " _'_ Asturias,' _"_ he says, nodding.
"What?"
"The piece."
"Aector, you're a fucking idiot."
Pharaoh pushes him aside and hammers on the door with her fist. The music stops. She lifts the letterbox. "Police," she yells, then looks at McAvoy. "Sort of."
A moment later the door is answered by a lad in his early to mid-twenties. He is short and thin with curly red hair and a pale, freckly complexion. He is wearing a faded black shirt and tight jeans with sneakers, and he holds a battered guitar in his right hand.
"Seriously?" he asks, halfway between a curse and a sigh. "Does this have to be now? I was caught up in the moment."
"That's okay," says Pharaoh. "I get like that when I'm Hoovering."
She puts her foot on the step and muscles him back into the cramped hall, which is littered with discarded sneakers and a mountain of takeaway leaflets.
"We just need a moment," says McAvoy by way of explanation. He follows Pharaoh down the hall, the bewildered lad following with much huffing and protestation. They arrive in a small, busy living area that does not appear to have been decorated to the occupant's own tastes. The carpet is a gray cord, and the wallpaper is patterned with floral swirls. A pink two-seater sofa covered in sheet music and draped in drying T-shirts is pushed back against one wall, and the open-plan kitchen to their right is home to a mountain of dishes and polystyrene chips cartons.
"If you don't like the mess, you can tidy before you leave," he begins, aggressively.
"It's okay, sweetheart, I'm not your mum." Pharaoh looks around her. Decides she will not sit down. It's not overly dirty here, but she feels like she is in a teenager's bedroom, and is wary about sitting on anything that will not rub out with a wet cloth.
"What's this about?"
"Your neighbor," says Pharaoh, turning to him, with a bright smile. "Bit on the dead side."
The boy looks strangely relieved. Gives a shrug. "Lad from two B?"
"Simon," says McAvoy. He feels claustrophobic in the little room. Is grateful the electric heater is not switched on.
"You know him?" asks Pharaoh.
The lad sits down on the sofa. Holds his guitar comfortably in his lap. His expression is unreadable, but not unfamiliar. McAvoy has seen it too many times. He has interviewed too many youngsters who truly do not give a damn about anybody but themselves. He has looked into the eyes of too many people who truly do not give a damn.
"How long have you lived here, Mr. . . . ?"
"Woodmansey," he says. "Darren. Been here just under a year now."
"So Simon was your neighbor."
Woodmansey gives a grunt. Sighs, like a teenager being asked if he has done his homework. "Haven't we done this? Back when it happened?"
"You've spoken to a detective before?" asks McAvoy.
"Dunno about detective. He was a copper. Fella in uniform. Told him I barely knew him."
"You knew his name was Simon?"
Darren looks up, mulling this over with exaggerated wariness. "Yeah, I think so. I mean, I know now, what with the inquest and everything. But yeah, he introduced himself."
"What was your opinion of Mr. Appleyard?" asks Pharaoh, crossing to the window and looking out at the neatly tended grass verges and hedges that border the little group of properties. She watches a pigeon pecking at a discarded piece of takeaway fried chicken. Wonders if it qualifies as cannibalism. She turns back. "Did you get on?"
Darren smiles a little and plucks at the guitar strings. "Can't say I had much of an opinion of him at all," he says. "Not in a bad way, like. He was just there. I was just here. Y'know? People come and go. Who fucking cares?"
"Ever go in his home?" asks McAvoy, watching the young lad and trying not to let himself imagine putting him over his knees. He suddenly remembers he hasn't showed him his warrant card, and does so now, if only to remind himself who he is and what he does.
"Round for dinner, you mean?" asks Darren, with an attempt at a laugh.
"Round for whatever, sweetheart," says Pharaoh. "It's like this one, isn't it? Same layout?"
Darren shrugs. Appears to think. "I helped him carry a mini-oven up there about a month after I moved in. Was struggling with it when I got home, so I gave him a hand."
"Neighborly of you," says Pharaoh.
Darren shrugs again. It is an irritating habit—a display of affected nonchalance that McAvoy considers inappropriate. "He spotted me. Asked. I couldn't get away."
"And his flat? Layout like this, yes?"
"Far as I can recall," Darren says, and strums a complicated-looking chord.
McAvoy nods. Moves into the kitchen. Looks at the knife rack screwed into the wall next to the drainer. According to the crime-scene photos and the incident report, it was a mooring just like this that Simon Appleyard hanged himself from. Tied a belt around his neck, the other end here, and leaned forward until he was dead.
McAvoy taps the plasterboard wall with his knuckles. Catches Pharaoh's eye. She nods, and reaches into her pocket for her purse, retrieving a twenty-pound note and wordlessly handing it to the guitarist. "Sorry," she says.
McAvoy grabs the knife rack and pulls. There is barely a moment's resistance before the screws are wrenched free and the object clatters onto the draining board: knives and ladles spilling noisily over the linoleum floor.
"What are you doing?" asks Darren, angry and shocked. "This place isn't mine. It's rented."
"Bit of grout and a deeper screw, and it will be okay," says McAvoy, distractedly. "It can take a bit of weight."
"But not a body?" asks Pharaoh.
"No."
McAvoy and Pharaoh hold each other's gaze. McAvoy takes over.
"We're trying to find out a little more about Simon's death," he says, crossing back to the seating area and deliberately looming large over the small, seated young man. "There is evidence to suggest he might have been murdered."
Darren looks from one officer to the other. He appears genuinely bewildered.
"What? Who?"
"That's what we're trying to find out. We're led to believe that Simon was promiscuous. Do you remember seeing many other people calling at his home?"
Darren puts the guitar down. "I'm not the Neighborhood Watch," he says, masking his discomfort with a mild attempt at aggression.
"Mr. Woodmansey . . ."
"Yeah," he snaps. "Yes," he corrects himself, looking up at the towering police officer. "There were knocks on the door sometimes. And I'd see the odd soul come and go, like."
"Do you remember faces? Dates?"
"Of course not," he says, incredulous. "Just blokes."
"So you were you aware that Simon was a homosexual."
Darren snorts. "Er, just a bit," he says, sarcastically. "He couldn't have been any gayer, really. Not that I'm bothered, like. Whatever, y'know."
"And you didn't mind him entertaining people on your doorstep?"
"I might have had something to say about it on my doorstep," says Darren, attempting to stand up and then sitting back down again when he realizes he is only as tall as McAvoy's chest. "I don't give a shit what he did in his bedroom. What's this got to do with me?"
Pharaoh moves beside McAvoy. "Were you sad to hear he had died?"
Darren looks appalled by the question. He leans over the arm of the sofa and picks up a glass ashtray that looks like it has been stolen from a pub. He pulls a cigarette from the packet on the floor and lights it, taking a breathless drag. "Whatever. Tough break, man."
"And you didn't suspect foul play?"
"I didn't think about that side of it."
Pharaoh pulls a face. "A man dies next door and you don't think about it?"
"I thought it was a shame he had died," says Darren, examining the end of his cigarette. "He seemed a nice enough fella. But, y'know . . ."
McAvoy's face is impassive. "I know what?"
"People's lifestyles," he says, grasping around for a way to explain himself. "You don't know what they get up to behind closed doors, do you? For all we know, he was into all that autoerotic stuff . . ."
"Oh, so you do have one theory," says Pharaoh acidly. "Any others?"
"I don't mean that," he says, uncomfortable in the ferocity of her sudden glare. "I mean, what am I allowed to think, these days? I vote Liberal. I mean, I would do. If I voted. And if the Liberals were still liberal. I don't mind what people do. I used to be in a band with a gay bassist."
There is silence for a moment. McAvoy looks at the young man and wonders how many years there are between them. How different their views of the world. Wonders how it would feel to look out on the world with such little interest.
"I know that song," begins Darren, as a Curtis Mayfield tune blares out from Pharaoh's handbag.
Pharaoh waves a hand at him, shushing him while reaching for her phone. "Did you think he could have been murdered?" she asks.
Darren turns to McAvoy. Seems to think for a moment. He shrugs. "Maybe for a minute. I don't know."
McAvoy sighs. "The time he died. Not the day, I know you can't remember. Just vaguely around the time. Did he have any visitors that you can recall?"
"There were always people coming and going."
McAvoy arches his back and his chest muscles strain against his shirt. He is getting tired of nobody giving a damn. "Mr. Woodmansey, I appreciate that it is a Sunday afternoon and that this was not what you were expecting when you answered the door . . ."
"Bollocks to it," says Pharaoh. Her phone stops vibrating before she can take the call. "McAvoy, leave the lad alone."
She turns on the youngster, suddenly a mum, furious with her son. "Did you, or did you bloody not see or hear anything that I might find even vaguely fucking interesting?"
The young man backs himself into the sofa as though retreating from punches. He looks desperately up at McAvoy, and then appears to start thinking hard.
"He had a friend," he says. "A lass. Bit odd-looking. Smiley. She picked him up sometimes . . ."
"Anything else? Ages of his visitors? Anything?"
Darren gives up. Looks at the end of his cigarette. "I guess they didn't look gay."
McAvoy's shoulders sag. "What does gay look like?"
"Like Simon! These were blokes. Like, just blokes."
For a moment there is silence. Wordlessly, Pharaoh takes back the twenty-pound note. "You haven't done anything wrong," she says quietly as McAvoy stomps from the room. "I just don't like you."
She follows McAroy down the corridor and finds her sergeant leaning against the brick wall, shaking his head slightly and looking cross.
"Nobody cares," he says. "People die next door and their neighbors just think it's none of their business."
"He helped him carry his oven," she points out.
"It's hardly the same."
"McAvoy, people don't want to think about it. That lad could barely afford decent fags. He sure as hell couldn't afford to move. He's not going to want to let himself think his neighbor's been killed."
"You were as hard on him as I was," he says defensively.
"I'd have happily been harder." She smiles. "I got bitten by a dog this week. I'm not in a happy place."
McAvoy leans his head back. Closes his eyes. "I'm not sure if I should ever have started this. I feel like a bloody amateur . . ."
Pharaoh is about to offer some words of comfort when her phone begins to ring again. She answers, and listens for a moment.
"Right," she says into the mobile. "Send me the number." She takes the phone from her ear. Looks at the screen. "Got it. You sure? Right."
She hangs up. McAvoy looks at her expectantly.
"Dan's managed to unlock some of the mobile's call history," she says. "Knows his stuff, that lad, even if the kisses on his e-mails are in capitals. And the specialist lab he sent it off to reckons there are two different types of dirt present in the phone's insides. Silt and sand, as you would expect, but also mud. Seeds that have no business in a tidal river. It looks like it was buried twice."
McAvoy says nothing. Takes it in. "The call history," he says at last. "Tell me."
"It called a taxi firm the day Simon died. Made no calls afterward. And it belonged to Simon."
Pharaoh is already dialing the taxi firm. Introduces herself and asks for the manager. Explains what she needs. Uses the right amount of sweetness and snarl. Hangs up and motions for McAvoy to follow her back to the car. She leans against the bonnet, waiting for another call, and breathes out with a whistle.
"Exciting, police work, isn't it?"
McAvoy, despite himself, manages a little grin. He watches as she unlocks the car, removes his Tupperware box from the passenger seat, and starts eating the cold roast lamb and gravy. "If you want this, you're going to have to fight me for it," she says, licking her fingers. "And be warned—I bite."
Her phone rings. Between mouthfuls, she answers. McAvoy hands her his notepad and pen. She scrawls down the address and says thanks.
"The taxi company says according to their records they received two calls at that time, on that day. One was a cab between the Empress in town and the Tiger in Cottingham," she says, one half of her mouth curling up and the other still chewing a roast potato. "The other was a pickup from Morrison's going to Beck Lane. Welton."
"Near the Dale?"
"Near enough."
"Address?"
She nods. Swallows. Pulls her police radio from her handbag and contacts the control room.
"This is Trish Pharaoh," she says. "I need you to check an address for me. Beck Lane, Welton. Thanks."
They stand in silence. It feels like waiting for a diagnosis.
McAvoy frowns. "He wouldn't get a taxi home, would he? You don't kill somebody and then call a cab . . ."
Pharaoh shrugs. "Morrison's is a minute from here. Could have bumped Simon off and walked it. Ordered a cab, dumped the phone when they got near home. Would never have expected anybody to find out. Nothing sinister in getting a cab home with your shopping."
"Did the passenger have shopping? Did we get a description?"
"Driver's in Marbella, apparently. They're trying to rustle him up on the mobile."
Moments pass. McAvoy, for something to do, plucks a leaf from a privet hedge and folds it into quarters. Pharaoh presses her knuckles into her forehead.
Both police officers jump as her radio crackles. "This is control, guv. We've got the details you asked for. That property belongs to a Peter Tressider. Councillor, it says here . . ."
Pharaoh and McAvoy stare at each other. After a moment, Pharaoh switches off the radio.
"It could be nothing to do with all this," says McAvoy instinctively, but even as he speaks, his mind is soaring back to the riverbank: to the two stick figures in the distance, and the phone, winking up at him, from the mud.
"No," says his boss, quietly. "But."
"Yeah. But."
Pharaoh drops her head to the car bonnet. Wonders if her injuries are sufficiently well healed to remove her bandages.
She wants to look her best when she goes to question the new chairman of the Police Authority in connection with murder.
ON SUNNY DAYS all roads lead to the Country Park Inn. It sits no more than a few hundred yards from where the Humber Bridge stitches Yorkshire to Lincolnshire, and offers the best view in the county of the towering road and its metal harp strings, albeit from virtually underneath. _Mole's-eye view_ , McAvoy had said when he brought Roisin here. She had been good enough to laugh.
The tables and chairs on the patio area at the front are constantly occupied, families and friends sipping iced cider and flicking cigarette butts onto the shingle beach that leads down to the coffee-colored waters. Across the water is another strip of mud leading up to Barton. There's a wildlife sanctuary over there that McAvoy has yet to get around to visiting. An art gallery that was once the longest tiled building in Europe, and which used to be a rope-making factory before it fell to ruin. McAvoy read once that they made the ropes for Hillary's conquest of Everest. It is a snippet of information that refuses to leave his brain.
_Could be another country_ , thinks McAvoy, staring across the water. North Lincolnshire remains somehow "over there."
The river that separates the two counties is the same stretch of mud and swirling currents that, centuries earlier, halted the Romans in their march north. Today it is still a barrier. "Humberside" was reviled on both sides when the government tried to create a new county that included towns north and south of the water.
_Yellow-bellies_. That's what the Yorkshiremen call people from Lincolnshire. _Miserable, tight-arsed bastards_ is the rather less poetic riposte from across the water.
There had been rejoicing all around when the boundary lines were put right. Hull became Yorkshire again. Humberside Police have yet to change its name to something more popular. It still polices both banks.
McAvoy likes it here. So do plenty of others. Although the wind still whips in cold from the east, the glimpse of blue sky has been enough to persuade the county's drinkers that they should be outside, and there are perhaps fifty people thronging the outside of the pub, wrapped up warm and holding glasses and bottles, as McAvoy and Pharaoh cross the car park.
"Bloody mad," says Pharaoh, nodding at a girl in her late teens who appears to have dressed for a tropical beach, and who is turning a shade of blue that matches her bottle of WKD.
"Inside, yes?"
"Too right."
They enter the large, brightly lit bar. On the walls are posters advertising tribute acts and local singers. They sit uncomfortably next to mass-produced abstracts on canvas and blackboards advertising the daily restaurant specials.
Pharaoh orders herself a double vodka with lemonade and lime, and McAvoy decides that half a pint of bitter would be a welcome anesthetic. They take their drinks to a corner booth circled by glass, and sit opposite each other.
"Cheers."
They clink glasses.
"Here's to following your nose."
McAvoy looks down, ashamed to be the victim of the sarcastic toast.
Eventually Pharaoh gives a snort of laughter, then shakes her head. "They don't like me anyway."
"Who?"
"Top brass."
"Oh." McAvoy looks out of the windows. Watches the green channel marker bobbing on the swollen waters of the estuary. "I'm sure they respect you."
She shrugs. "I don't know. No black-and-whites, are there? They like it when I catch villains. Don't like it when I don't."
"Your cleanup record is top-drawer," points out McAvoy, gulping half of his drink and then surreptitiously spitting some of it back in the glass. He has no more change in his pocket. He has to make his drink last.
"My predecessor's cleanup was nigh on a hundred percent," she says.
McAvoy bites both lips. Any mention of Doug Roper makes his scars hurt. "It was all lies," he says.
"Yeah." She pouts. "Wish I could tell some."
They drink their drinks. Watch the channel marker. Think their thoughts.
"You going to tell me what's happening in there?" asks Pharaoh, making a gun of her forefinger and thumb and pointing it at her sergeant's forehead.
McAvoy rubs his hands together. Wonders if he should just keep his trap shut. Realizes that he can't.
"It's a sex thing," he says, looking away, and realizing as he does so just how feeble and prudish it makes him look. "Simon Appleyard has been meeting men off the Internet for sex. One of those men killed him. They set him up to be lying on the floor of his living room when they arrived. Even made him pick his own noose. Strangled him to death. Made it look like suicide. Took his phone and his laptop. Dumped them."
Pharaoh is nodding thoughtfully. She clinks the ice in her glass. Runs a short, jeweled finger around the rim.
"Were they killing Simon specifically, or just anyone they could get their hands on?"
McAvoy picks up a beer mat. Starts spinning it on its end as he thinks. "They wanted to kill Simon," he says, and hearing it out loud makes it seem more real. "He knew something. Saw something . . ."
"Based on what?"
McAvoy gestures with his hands, casting around as if trying to pluck the right answer from the air.
"So far, three councillors' names have come up in connection with this. I wish one of them hadn't, but it has. Fuck, just saying that makes me need to pee. Why did it have to be Tressider's place? Anyway, fucking hell. So . . . yeah . . . three councillors. And that's just with my tiny bit of digging. None of them would want their secrets coming out."
"Hepburn got himself elected by playing the gay card," says Pharaoh, warming to her role of devil's advocate. "He wouldn't care."
"And Cabourne looks like he wouldn't say boo to a goose."
"So."
They meet each other's eyes.
"Tressider," says McAvoy.
"Bloody hell," says Pharaoh.
They sit in silence. McAvoy's mind turns to the day by the river. Himself, damp to the bone, the smell of horses on his hands, and two stick figures talking by the water. Could one have been Tressider? Could the big, bearded future MP really have come straight from the Police Authority meeting and then thrown Simon Appleyard's phone into the mud of the River Hull?
Pharaoh finishes her drink. Looks at McAvoy's. "Drink it, you girl."
He does so. She goes to the bar and comes back with the same again for both of them.
"So if he's just got himself a high-profile job and he's on the fast track to Westminster . . ."
"Then maybe he has reasons to make sure all the skeletons in his closet are very much dead."
They look at each other. "We're being harsh on him," says Pharaoh at last. "He's a big lovable bugger, after all. Likes you. All we have is the fact that somebody took a cab from Morrison's to his house on the day Simon Appleyard was murdered. Called the cab on a mobile that was very likely used to set up the murder of a practicing homosexual . . ."
McAvoy finishes his first drink and starts on the next. "It's not exactly damning."
"No. Nor does it look good."
After a moment, Pharaoh starts tapping her fingers on her teeth. "Show me the website."
"Guv?"
"This Playmatez thingy he was on. Where all these people seem to meet to get their kicks. Cabourne was on it, yeah? And I bet Hepburn's no stranger. Let's see how it all works."
McAvoy looks around him. The pub is almost deserted. He pulls his laptop from his bag and opens it up. Searches for the website.
"Come round this side," says Pharaoh. "We're not playing Battleship."
McAvoy slips in beside his boss on the other side of the table and spins the laptop to face him. The desktop wallpaper is a photo of Roisin, laughing, as a baby Fin closes his pudgy fingers around one of her gold necklaces.
"You take that?"
He nods.
"Pretty. She looks young."
"Eighteen," he says.
"And you were?"
"Twenty-seven," he says, not looking at her.
"Girls mature quicker than boys," says Pharaoh, looking at the side of his head as if willing him to glance in her direction.
He says nothing. Intends saying nothing more.
"Go on, then," says Pharaoh, sighing, and pointing at the screen. "Show me some bummers."
He finds the website. Blue background with toned bodies and lustful glances superimposed.
"Women, too?" asks Pharaoh, looking at the images.
"You click your preferences," says McAvoy. "Tell them whether you want a man, a woman, or you're not particular."
"And they say romance is dead. Show me something."
McAvoy gives her a quick guide around the site. Shows her how to create a personal profile and where to post messages.
"So if you're feeling randy you just pop on here, tell the world you're after a bit, and then you get an e-mail from somebody who wants to pummel you from behind, yeah?"
McAvoy bites down on his embarrassed smile. "Something like that. There are different categories of membership. You can just log on and post a message, like you can on Craigslist or Gumtree or any of those . . ."
"I've heard of them."
"Or you can become a member, create a profile, and then it's more like a dating site."
"Or?"
"Or you can become a gold member, pay a membership fee, and have access to even more stuff."
"Like what?"
"Well, you get to see all the other member profiles. Pictures as well. And you can message them direct. That might be what happened with Simon."
"But Simon's phone number was on here. That's how you found him."
"It's against the rules, but it's not run like some big multinational company. It will have an administrator who checks for things like pictures of kids or threats or anything, but it's easy for things to slip through."
"So how do we see Simon's profile?"
"We'd need to know his user name."
"And we don't?"
"No."
"Could we try a process of elimination? Look for members between twenty and thirty, local to this area, certain body type, into certain things, tattoos and whatnot?"
McAvoy smiles, pleased that she has picked it up so quickly. "There are thousands of members. It's still a needle in a haystack. If we had even a part of his name we could narrow it down . . ."
He stops. Closes his eyes. One picture flashes into his mind whenever he thinks of Simon. Of his inked skin and flamboyance. His love of words.
Types, slowly, into the website's search facility.
P-E-A-C-O-C-K.
Four matches are revealed. Each has a user name that contains the name of the bird. Only one belongs to Simon. The image that accompanies his profile is a close-up shot of a hard, firm torso. The other three profiles are illustrated with erect penises.
"Distinctive," says Pharaoh, peering at Simon's picture. "Skinny as a rake. What did he have to say for himself?"
McAvoy brings up the member details of Peacock1990. The information is scarce.
Young, slim, tattooed male, seeks dominant master. Want to be hurt and controlled. Non-smokers preferred.
"Non-smokers?" laughs Pharaoh. "Fucking hell."
McAvoy looks at the section of the profile that details sexual preferences. All relate to being controlled and dominated.
"What's that?" asks Pharaoh, as he scrolls down the page. She puts her finger on the screen. Reads aloud. "'Remember that the most beautiful things in the world are the most useless: peacocks and lilies, for instance.'" She pauses. "That's nice."
"That's his profile signature. He must have been on the forums. You can give yourself a signature. Some line from a film or something so people know it's you. That's his." McAvoy squeezes his fist with the palm of his other hand. Thinks for a moment. "That must be what his auntie was talking about. Poetry. Some line that meant the world to him. That must be what he put on the messages he exchanged with Cabourne."
"So it was definitely him?"
"As definite as it can be."
"Can we go on the forums, then?"
McAvoy looks at her. He is suddenly aware that his cheeks are no more flushed than usual, and wonders what to make of it. He is not embarrassed. He is looking at a sex site, crushed against his boss in the corner of a secluded pub, and he feels more like a policeman than he has in days.
"You need to be a member," he says. "You have to pay."
She shrugs. "Pay."
"I haven't got my credit card . . ."
"Oh, Aector." She pulls her purse from her handbag. It's designer and looks expensive, but when she opens it, it's filled with receipts and battered business cards. "Here," she says, handing him a Visa card. "Shall we do you or me?"
Now the blush comes. McAvoy's face turns scarlet.
"Christ! Fine, we'll make somebody up. Relax."
For the next fifteen minutes they enjoy themselves crafting a sub-dominant twenty-something pretty boy with big muscles, tattoos, and, at Pharaoh's giggled insistence, ginger hair. They choose not to upload a picture, and tick the same preferences boxes as Simon. They give themselves the user name ruffstuff69, which McAvoy hopes is in reference to his boss's date of birth rather than anything else. Moments later, Pharaoh's phone buzzes. An e-mail has arrived, activating her account.
"Nice one," she says, smiling. "Go on, then. Show me."
McAvoy navigates them onto the discussion forums as Pharaoh brushes past and heads back to the bar for more fuel. In her absence he checks his phone. He has a missed call from a withheld number, and an "I love you soooo much xxxx" from Roisin.
"What's he got to say for himself?" asks Pharaoh, sitting back down. "He leave any messages saying he was bummed then strangled to death by three local politicians?"
McAvoy takes another sip of beer. He can feel it doing him good. Types Simon's user name into the forum to see what he has posted on. Pulls a face when he gets nothing back.
"Not very chatty," says Pharaoh.
"I'll try some of his areas of interest."
They try line dancing. Hull. Anlaby. Dominance.
All bring up discussion threads but none that Simon contributed to.
McAvoy drops his head to the table and gives a moan as Pharaoh takes over on typing duty.
"The spelling is shocking. I suppose it can't be easy to care when you've only got one hand to spare."
McAvoy listens as his boss murmurs ideas. Feels the vibration in his forehead as her fingers thunder on the keys. "Hey, Aector, I've got a message. Wahey, somebody loves me!"
He looks up. In the corner of the screen is an icon indicating the arrival of new mail.
"Open it," he instructs.
Pharaoh reads: "'No picture? You tease. Bet you're pretty in the flesh. Want to meet?'"
McAvoy shrugs. "Nothing."
"Dunno," muses Pharaoh. "Sounds quite nice."
"Click that one," says McAvoy suddenly, looking at a discussion title. "Go on."
Pharaoh does as she is told. The discussion is titled "All talk, no action—left me lying."
"Scroll down. Click that. There."
They read it together. It is little more than a chat between two members, with occasional comments from observers. The first posting was made in August of last year: a missive from a member called Adams71 furious at having been led on by a potential partner, only to be left wanting. A reply, from RedKen1960, details a similar experience.
"'Was so embarrassing,'" reads Pharaoh, from the screen. "'Days of texts, getting me so horny and hard, I did everything he wanted, and he just left me there.'"
"'Ditto, mate,'" reads McAvoy. "'Feel sick thinking about it. Just took a look at me and left. I thought it was part of the game.'"
"He's online now," says Pharaoh suddenly. "Look."
A red icon is flashing on the screen. RedKen1960 is logged on.
"Let me," says McAvoy, grabbing the laptop. Quickly, he types: "Hey you. Read about your problems with that no-show tease. What happened? xx."
They say nothing for a moment. Just stare at the screen. Tap their fingers on the table. Inhale, in unison, ahead of the dejected sigh that will come if there is no reply.
"There," says Pharaoh, first.
She clicks the message icon. Opens it up. Reads aloud.
Still fuming about it! Met some teases but that was just cruel. Even thought about reporting him, but he'd closed down his profile by the time I came back on the site to give him a telling off. Spent the day e-mailing me, getting me so horny, all these things he wanted to do to me. So kinky. Wanted me naked, waiting for him when he got there. Was just going to take me without a word. Even asked me to get a belt so he could tie my hands. Did everything he wanted, he came in the room, and then just sodded off without touching me. Soooo embarrassing. Anyway, was his loss. Am over it now. Wot about you? See we're into the same things. You got any playmates you can recommend?
Pharaoh pushes herself back from the table so she can look at McAvoy without his face going blurry through nearness. As expected, he has his eyes shut.
"I can think and keep my eyes open at the same time, y'know," she says sweetly. "Multitasking, they call it."
His eyelids flick open. He sees her staring at his face and looks away. When he finds the courage to return her gaze, she is looking back at the computer screen.
"He was looking for Simon," says McAvoy quietly. "He wanted to see if they had tattoos. When they didn't, he left. When he found the right man, he killed him."
Pharaoh sucks in her cheeks. Blows out. Crosses her legs, then lifts one by the ankle and angles it across the other. The material of her clingy dress shows the shape of her thighs, and McAvoy has to fight the impulse to commit the image to memory.
"Am I replying?"
Pharaoh nods. "Tell him to see if he can find the original user name of the person he got in touch with. We need the messages, too. If we do take this further, we'll need it all to give to the website administrator."
"If?" asks McAvoy.
Pharaoh nods, openly. "Yeah, if. At the moment this is just Aector McAvoy's intellectual exercise. It's not a murder investigation. It's you and your boss knocking theories about and trying to decide whether there's enough here to take it further."
McAvoy widens his eyes. Shows his frustration. He feels as if he is running over breaking ice. "I thought you agreed with me."
Pharaoh smiles indulgently. Puts her hand on his knee, as if he were an angry teenager refusing to accept her advice. "I do agree with you. I agree CID didn't look into this when they should have and I agree there is a bloody good chance Simon Appleyard was murdered. But the lad has been cremated. The only evidence we have are some knackered mobile phones and some theories. I've got to think about the likelihood of a conviction. If not, there's just another unsolved murder on the books."
McAvoy turns his face to her. He is flushed and prickling with sweat across his back and shoulders. "So what does that make us? If we're more concerned with numbers than justice, then who holds it all together? What are we here for?"
He has raised his voice more than he intended, and Pharaoh's face turns angry. "Don't count me in with the number crunchers, boyo. When somebody does something wrong, I want them caught and I want them punished. When somebody has been hurt, they deserve to know that there has been some kind of payback."
"And when somebody has been murdered?"
"We catch who did it," she says, then adds, "if we can."
They sit in silence, looking past each other, unsure of whether to make up or take the argument further.
"What next?" asks McAvoy.
Pharaoh shrugs. "Next, we take a step back. We see what else Dan can find on the phone laptop. We wait for a description of the taxi passenger. We try and find out why they took a cab. We learn more about Simon. Then we have a think."
McAvoy nods sullenly. He can see the sense in the suggestion.
"What if he's trying to hurt somebody else?" he asks.
"You said it yourself, he was after Simon. He's got him."
McAvoy cannot meet her gaze, so turns back to the computer screen. Starts flicking through Simon's details again. Looks at the FRIENDS section of the site.
"You think any of those are Suzie?" he asks Pharaoh, highlighting some of the female contacts that Simon has listed on his page. "Should I e-mail them? See how they knew him?"
Pharaoh nods. "Good idea," she says.
"It's a whole world we don't know about," says McAvoy thoughtfully, as he plows through the endless profiles. "People must be so lonely."
Pharaoh looks at him as if he's from outer space. "Not everybody has what you have," she says at last. "People need excitement. Some people drink. Some smoke. They gamble. They meet strangers for sex. They put themselves in the hands of a sadist because it makes their heart beat faster. Life's so tame sometimes, Aector. People just need badness sometimes."
McAvoy wishes he had something else to drink. "I just can't imagine spending my evenings having sex in the back of a car with a stranger."
"You wouldn't fit in a car. You'd need to go dogging in a van." McAvoy takes no notice of her words. He just hears "dogging" and has a moment's flash of inspiration. He clicks out of the website and finds a search engine. Types "dogging, East Yorkshire" into Google. "Good job your missus doesn't check your search history," says Pharaoh.
Moments later, he is on a website called swingingheaven.co.uk. He scrolls through dozens of postings left by members with names like luvbstolik and trev69, until he sees one that mentions East Yorkshire. Opens up the discussion thread and finds a score of messages mentioning the A46, Coniston rest stop.
He goes back to Google. Types in the road name. Is taken straight to a story on the _Hull Daily Mail_ website.
MAN HURT AT EAST YORKSHIRE REST STOP
A 44-year-old man is in intensive care after being involved in a suspected hit-and-run at an East Yorkshire beauty spot.
The man, visiting the area on business and said to be from West Yorkshire, was found by motorists at Coniston rest stop on the road to Bridlington late on Tuesday night.
Detectives are keen to talk to the person who made a 999 call from a nearby telephone box shortly after the incident. Anyone with information should call Humberside Police on 0845 6060222, or Crimestoppers, anonymously, on 0800 555 111.
As he turns to Pharaoh, her sigh is powerful enough to tickle his damp fringe. "Guv?"
She pulls out her phone. Rings through to control. Asks which uniformed officer dealt with the incident and whether he is working today. As she waits for an answer, she mouths "I hate you" at McAvoy, who scowls and then gives a nervous laugh.
"Really? I think I know him, yeah. Radio through and ask him to ring me on this number. Thanks." She hangs up. Turns to McAvoy. "It's gone up to Tony Laws at Bridlington. Control are asking him to get in touch."
"Why don't we know about this?" asks McAvoy.
"We're just a little unit," says Pharaoh. "We look after very specific crimes. You remember the regular CID workload, Aector. You can't keep track of it. And nobody knows you and I are doing this. We're supposed to be finding out which bastards nailed lots of people's hands to their knees. They probably don't think we have time for dogging."
Her phone rings. She answers politely.
"Tony, hi. Yeah. No, I know. I won't keep you. Forget all that ma'am stuff. Guv will be fine. Or Trish, once you've bought me a drink. Look, Coniston rest stop, I'm told somebody got a bit carried away . . ."
McAvoy listens as his boss learns more in five minutes of charm and chat than he has in days of solo grandstanding and analysis.
She hangs up, having made a new friend.
"Right," she says as he looks at her expectantly. "Victim was one David Stoneleigh. Letting agent from Morley. It's near the Ikea roundabout, before you ask. Leeds way. Over here looking to link up with another letting firm, or at least that's the story. Tony Laws reckons he came all this way to meet somebody up the rest stop. Apparently it's endless up there. They ignore most of it. Do the occasional sweep of the area but tend to turn a blind eye. Anyway, they got a call last week from a phone box in the next village. Female voice, told ambulance to get to the rest stop. Somebody badly hurt. Police were alerted automatically. Patrol car arrived. Found this bugger flat on the ground, pants around his ankles, legs smashed in and hips broken. Death's door. Got him to hospital and he was unconscious for two days. Operated on Friday and he's lost his spleen, but he's conscious again. Not talking very much. Probably shitting his pants trying to think of something to tell the wife. She's used to it, mind. He was cautioned for curb crawling in Bradford in 2003."
McAvoy digests it all. "Nasty business. But I don't see any connection."
"No, neither did I. Was about to go back to being sensible and doing this cautiously. Then he gave me the other news."
"Yes?"
"They've fingerprinted the bonnet of his car. His own prints, and another set."
McAvoy looks at her expectantly.
"Susan Devlin. Twenty-six. Arrested two years ago for an attack on her partner. Criminal damage. Attacked her boyfriend when he was tied up. Sex thing."
McAvoy tries to link the information, but cannot put it together. Pharaoh is smiling.
"Received a suspended sentence when it went to magistrates' court. So did her co-accused."
"Co-accused?"
Pharaoh grins. "Simon Appleyard."
7:17 P.M. WELTON.
COUNCILLOR Peter Tressider's big white house: screened by leylandii trees and tall black railings, so as to be almost invisible from this wide, quiet street.
Trish Pharaoh, pulling in to the driveway in her two-seater sports car, sucking two extra-strong mints and smoking a black cigarette.
She looks up at the property. Gives a grudging nod. It seems tailor-made for an aspiring politician. It suggests wealth without pretentiousness, success without pomposity. Pharaoh would use the word _tasteful_ , if asked.
She steps out of the car. Checks her reflection in the window. Ensures there are no errant herbs between her teeth, and then grinds her cigarette out with the heel of her boot. She's been home. Changed into a lemon-yellow blouse and black skirt. Put a scarf around her neck and brushed her hair. Slipped into her biker boots and pulled on a suit jacket, which she has since discarded and thrown on the passenger seat. It was a long drive, just to make herself presentable. Sixty-mile round trip, over the bridge and back. But she's pleased she made the effort. Feels less self-conscious about her bandaged cuts and scars now that she is dressed for her day job.
A slight pause. A breath and a moment of darkness, hiding behind her eyelids. Then up to the front door. Two taps with the brass knocker, followed by a ring of the bell.
Five seconds. Ten.
She tries the handle. Nothing. Listens for sounds from inside the property. Fancies she can hear activity somewhere past the glass conservatory that marks the western boundary of the long, brick-built property.
Pharaoh crunches over the gravel and onto the deep green grass. Is silent as she moves to the back of the house. Pushes open a wooden side gate and emerges in a long, well-tended garden. A raised patio area gives way to a hundred yards of landscaped lawn. At its center is a Chinese-style pagoda, overlooking a large, teardrop-shaped pond. On a raised platform above, water shoots from an ornamental fountain to splash merrily across polished, colored rocks.
Peter Tressider is sitting with his feet in the pond. He is wearing a white short-sleeved shirt with a jumper folded like a cape about his neck. His trousers are rolled up and he is reading from a sheaf of A4 papers while sipping beer from a can.
"Councillor Tressider, sir?"
He looks up, eyebrows knitting together, as Pharaoh crosses the grass. He's a burly, square-shouldered chap with a dark, thick beard that looks as though it would regrow within the hour if shaven.
"No, no, this is my private residence, I'm afraid I . . ."
He starts getting up, pulling a pale, fleshy right foot out of the water and bracing his hands on his thigh to lever himself into a standing position. He recognizes her as she gets closer.
Gives a show of surprise.
"Pharaoh, isn't it? Aector's mate?"
She nods, happily accepting the description. "Yes, sir. I'm so sorry to intrude . . ."
He waves a hand. Lowers himself back down.
She stands at the water's edge, watching her reflection being distorted by the water falling from the fountain. Catches sight of a large, orange-and-white carp moving slowly in the depths of the pond.
"You're welcome to have a dip," he says warmly, pointing at the pond. "Wonderfully refreshing once you get used to the cold. I used to go for a dip every New Year's Day at Bridlington, y'know. Very bracing. Don't think the heart would stand it now. Will settle for getting chilly to the ankles."
Pharaoh notices a wooden fold-up chair in the gazebo and brings it down to the side of the pond. She erects it and sits down carefully.
"You okay with that chair?" he asks. "I notice you've hurt yourself."
"Bit of a scrap with a couple of dogs," she says matter-of-factly. "They came off worse."
Tressider frowns. "Are you the officer involved in the gypsy case?" He catches himself. Looks around, feigning guilt. "I can't say that, can I? Gypsy? What's the politically correct term for them? It was you, though—yes? A suspect set his dogs on you and another officer? Am I right in thinking it's all linked to the drugs business? Yes, yes. Goodness, how you keep it all in one head I'll never know. You're having quite a time, aren't you? All this just to keep the spreadsheets looking pretty. It's a world gone mad. Can't wait to change it! Glad you're back on your feet." He stops. Looks suspicious. "This isn't about compensation, is it?"
Pharaoh pinches her nose and sits forward in her chair. It's nice here, with the tumbling water and the lowering sky. She looks back up at the house. There is a lot of glass and expensive-looking pleated curtains. She fancies that from the balcony you would be able to see down to the Humber from the second floor.
"It's actually quite a delicate matter, sir," she says, conspiratorially. "I'm sorry for intruding and turning up here unannounced, but I was keen to be as discreet as possible."
A half smile plays at the corner of Tressider's mouth. He takes a sip from his can of beer. "Now I am intrigued," he says, and stifles a burp. "Pardon me. Goodness, my insides are disintegrating. Can you overdose on antacids? I've taken about twenty today."
"I used to suffer," says Pharaoh companionably. "Too much white wine. Doctor put me on pills that made me feel like I was full of polystyrene. Decided just to live with it. Friend of mine's wife knocked up an herbal potion for me, actually. Don't know what was in it. Tastes of cardamom and wet dog, but it does the job when you're struggling."
"Sounds like a good friend to have," says Tressider. "Could use something similar myself. I'm ninety percent bile."
Pharaoh tries to steer the conversation back where she had intended it to go. "Councillor . . ."
"Peter, please."
"Councillor, I'm looking into a case from last year. Some questions have been raised. I'm talking about the death of Simon Appleyard, a man in his twenties who was found strangled in his flat in Anlaby last November."
Tressider looks at her, open-faced, awaiting more. "I know that name. Aector was having a ponder, wasn't he? On his computer screen when I popped in at Courtland Road? Small world, eh? Right, yes, well, what else?"
"The coroner recorded an open verdict because there was no suicide note. But evidence has since come to light that suggests Mr. Appleyard may have been murdered."
She has Tressider's full attention, but his expression still shows nothing more incriminating than intrigue.
"Councillor Tressider, this is not a proper investigation yet. We're just taking a look. And out of courtesy I wanted to tell you face-to-face."
Tressider wrinkles his brow, confused. "Well, I know I asked you to keep me in the loop, but I trust CID to investigate cases as they see fit. You don't need to worry about the authority peering over your shoulder . . ."
Pharaoh looks down into the deep, dark water. "Councillor, I'm not talking to you in your role as chairman of the authority. I'm here to ask you some questions about your own knowledge of the case."
There is silence for a moment. Tressider's brow is so creased as to be almost knotted.
"I'm sorry, am I somehow a suspect in all this?"
His voice is quiet. There is no menace. Just a genuine inquiry. He looks confused. Bewildered. Lost.
"Councillor, we have evidence that suggests you took a taxi on fourteen November last year. It took you from Morrison's in Anlaby to your own front door. The mobile phone that called for that taxi belonged, as far as we can tell, to Simon Appleyard."
Tressider's face pales. "I don't have a bloody clue what you're talking about," he says, and throws himself angrily backward from the water—hauling himself into a standing position.
Pharaoh stands. "Councillor, I wanted to talk to you here, privately like this, so we could clear up any misunderstandings. As I said, this is not an investigation. Not at this stage."
Tressider is windmilling his arms now. Looking around him as though expecting more enemies to jump out of the bushes.
"You've made a big mistake here. A big bloody mistake. Is this the best they've got?" He steps close to Pharaoh, face right in hers. "Do you think I'm a bloody idiot?"
Pharaoh holds her ground. Her heart is beating hard, but she is careful to remain calm. Professional.
"Councillor Tressider, did you take that taxi? Did you know Simon Appleyard? He was a practicing homosexual. Was involved in online dating. We believe he was known to one or two of your colleagues over on Hull Council."
Tressider turns away. Drops his head to his palm. Appears to be tugging at his hair.
"I'm not having this," he says when he spins back. "I've got nothing to hide. I've only been chairman five bloody minutes. The selection process for the next candidate doesn't start until next year. Who's so bloody scared of me they have to resort to this? I told him and I'll tell you, I don't even know if I want the nomination."
"Told who, Councillor?" asks Pharaoh, reaching out to put a gentle hand on his arm and not letting go when he tries to shake her off.
"That slimy bastard. Cocker, or whatever. Upset Paula. Made me look a prize berk."
"I don't understand."
"Cocker," he says again, angrily. Then he screws his eyes closed and throws himself back down to the grass, thrusting his feet back in the water.
"Cocker is the political fixer, yes? Guy who checks for skeletons in the closet of party members?"
Tressider rustles around in his top pocket. Pulls out a couple of receipts and then a business card. Hands it to Pharaoh. She takes it and looks at the logo, and Ed Cocker's name and job title. "What did he want?"
Tressider casts around with his hands. Picks up his empty beer can and tries to find a drop of comfort. He looks exhausted suddenly.
"Stephen bloody Hepburn," he says, and it almost pains him to say the words. "Cocker seems to think he's a story. Could ruin my chances at the election. Not the authority one, that's a done deal. The real election. If I stand. If they let me. If my heart doesn't pack it in first. The git turned up here last Saturday . . ."
"Hepburn?"
"Cocker. Knocked on my door, bold as brass. Told Paula he wanted to speak to me. She said I wasn't home. So he started on her. Asked her if she knew Hepburn. Whether she had any knowledge of his business dealings. Whether she knew that I had invested significantly in his club . . ."
"The gay bar? In Hull."
"It's bollocks," he says dejectedly. "I never invested in any bloody gay club. I loaned a business associate some money to assist with the marketing of a new club he was buying into. It happened to be Hepburn's."
"Much money?"
"Fifteen thousand pounds. A pittance, really."
"Who was this friend?"
"That's not important. It's all there in this paper trail Cocker says he's got. It's enough to bugger things up for me. Enough to give the party the jitters about me. Cocker's the guy who will see if there's enough there to be scared of."
Pharaoh pulls a face. "Councillor, I don't think that's a story. Not these days. I don't think anybody would care."
Tressider looks up at her. "He upset Paula. I called him when she told me. Tried to be polite, but I lost my temper. Told him to leave us alone. Said I had done nowt to be ashamed of and they either wanted me or they didn't. But he's on good money to do this stuff. Has a job to do, so he says. A report to write. Reckons there are enough positives about me to make me worth digging a little deeper into . . ." He pauses. "Flattered me, I guess. I mellowed a bit. Said I didn't like his methods but that I was listening."
"How did he take that?"
Tressider looks down into the pond. Raises his feet and looks at his toes, as though confirming he is real.
"He seemed confused, I suppose. Said he wanted to explain properly. Wanted to meet up."
"And?"
"And then he started asking questions about my family life. Even about Paula. Told me it was common practice to compile reports like these—about prospects and their partners. I lost my temper. I put the phone down. Tried to forget about it. And now you're on my bloody doorstep."
Pharaoh feels suddenly sorry for the man. She cannot explain it, but there is something about the tenderness with which he describes his wife that she connects with. She squats down next to him.
"You're going to have to get used to people prying, Councillor. If you're going to be an MP. If you're this rising star . . ."
Tressider snorts. "I'm fifty-six," he says. "I go to the toilet three times a night. I'm not on the bloody rise, love. I'm a decent councillor. I'm a good businessman. I could be a good MP, and I promise you I'll be a good chairman. But I don't know how much of it I actually want."
They sit in silence for a spell.
"Simon Appleyard," says Pharaoh at last.
Tressider looks away. Turns back to face her with his eyes still closed. "I don't know anything about that, love. I don't know the name. I haven't taken a taxi in bloody ages. I don't shop at Morrison's. I've got one mobile phone and it's in my pocket. You can look if you want. I've had the same number for years."
He fumbles in his trouser pockets, and his wallet falls to the ground. Pharaoh picks it up. It has fallen open, and the front flap shows a picture of a smiling blond middle-aged woman holding a glass of wine and with candlelight catching in her blue eyes.
"She's stunning," says Pharaoh, though in truth the woman is little more than well groomed.
Tressider looks at the picture. "He really upset her," he says, almost to himself.
"Is she home?" asks Pharaoh. "We could ask her if she wants to make a harassment complaint . . ."
Tressider blusters. Brushes it away. "She's a tough lass. She can take it. We breed them hardy. She's from Lancashire, to begin with, but I don't hold that against her. One of ours now."
Pharaoh considers him for a moment. Wonders how far to push it. Whether all she has succeeded in doing is alerting him that they are investigating a murder that he may have his fingerprints all over.
"I'm sorry to have troubled you, Councillor," she says at last. "You can imagine how difficult it was to know how to proceed . . ."
Tressider nods, lips thin, eyes glassy and dark. "You have a job to do," he says. "I appreciate your being so discreet."
Pharaoh stays crouching for a moment longer. Then she extends her hand. Shakes the one that is offered in return, and while doing so, scoops up one of the receipts that has fluttered to the damp grass. She slips it into her boot.
She turns and begins walking across the damp grass.
"Simon," he says suddenly. Pharaoh spins.
"Pardon, sir?"
"The boy," he says. "Did he suffer much?"
Pharaoh considers it. Looks up. Clouds are rolling in. Against the darkening sky they turn the heavens into a muscled back.
"I think he always suffered," she says. "But his death was no relief. It was murder."
From this remove, she cannot see the councillor's expression. But she can tell that his head has dropped, and his feet, in the water, are still.
9:43 P.M. TRANBY RISE, ANLABY.
A POLICE VAN, swaying erratically past nice middle-class houses and neat lawns.
Two angry Rottweilers making a racket in the back. Two animals in the front, hungry for blood . . .
"Shut the fuck up!"
Colin Ray twists in his seat and instantly regrets it. Pain grips his ribs; a bony handful of flesh and bone. He winces, then covers it up. Curses. Hopes Tanner didn't see.
"Fucking gyppo," through gritted teeth.
He hopes the pain is muscular, left over from his tussle with Ronan. He likes pain to be the result of something tangible. An impact or collision. He can understand the notion of cause and effect. Illness perplexes him. He is disquieted by syndromes. He wants his rib to be broken because that would explain why it hurts so much. The alternative diagnosis involves his heart, and he does not believe there is good news to be found in that line of inquiry.
"You okay, boss?"
"Little git definitely potted a rib. Thought it would have worn off by now."
"You should get yourself on sick. Have a few months. Bit of compo."
"And who'd look after you lot, eh?"
Ray looks across at his traveling companion. Malcolm Tanner is a sergeant in the dog section of Humberside Police. He is a round-faced and affable man, with thinning brown hair and a tendency to swallow his top lip with his lower one when smiling. The habit makes him look a little like a sock puppet, and as such, he answers to Socko around his football buddies. He's a better man than his presence here suggests. He has drunk too much, and recklessness has made him willing and cruel.
Ray considers his friend and for once he is grateful that he is not in the company of Shaz Archer. She's busy tonight. Up to no good with one of her pretty boys. He'll want details from her in the morning, and she'll be willing to oblige. He'll be glad to have her back by his side. Tremberg was happy enough to get stuck in, but if he needed some feminine wiles to get the Vietnamese to talk, he'd have been better off slipping into a dress than asking that fucking brontosaurus to act sexy.
Tanner's good company, even if he's not much to look at. For tonight's adventure he has changed back into his uniform, but the collar of his goalkeeper jersey is poking out above his white shirt, and his knees are grass-stained and muddy beneath his creased navy-blue trousers.
They are sitting together in the front of a dog van, two streets from the home of Alan Rourke. They have the radio up high in a bid to drown out the Rottweilers' incessant barking. The noise of the animals is muffled by the panel that closes off the driving area from the back, but the dogs are in a fury and the noise cannot be completely eclipsed.
Ray is almost grateful for the din. It keeps him angry. Keeps him looking forward to the moment, mere minutes from now, when he can put the gun barrel against the first dog's skull, pull the trigger, and watch a lying gypsy bastard cry.
He looks down at the object in his lap. Enjoys its shape and heft. Its sleekness. Its clarity of purpose.
"It's called a captive-bolt stunner," Tanner had said beerily as he reached under one of the panels in the back of the van and pulled out an object wrapped in a burlap bag. "Most humane thing there is."
Ray had looked the man in his eyes to see if he was taking the piss. "Why you got one of these, lad? You're meant to be a fucking animal lover."
"That's why," he'd said, removing the gun from the bag. "You know the places we get called to. You seen animals screaming, boss? Did you know animals can scream? Sometimes you can't wait for the vet. Just can't listen. They'd have my warrant card if they knew, but I'm not the only one. Quick blast with this, it's over."
"And you can do that, can you?" Ray had asked. "These dogs aren't dying, son. They just belong to a cunt who needs to talk."
Tanner had laughed off the suggestion he would not be up for whatever was required.
"They went for a copper, boss. And, besides, it's you who'll be pulling the trigger, if it comes to it."
Ray feels the stun gun's weight in his hand. He has absolutely no doubt about his willingness to make good on the threats he is about to make. Can feel bile and venom rising up his chest as they get nearer to the target. Can already see Rourke's face in his mind's eye, pleading for his dogs' lives and giving them chapter and verse . . .
They move off, quickly, slewing right as Tanner pulls onto one of the quiet side streets and narrowly misses a parked Mercedes.
"Fucking Italians," says Ray.
"German, aren't they? Mercs."
"Dunno." Ray considers it. Tries to remember whom he is mad at. "Make good cars."
Both men are too drunk to be driving. A couple of hours ago, furious at the command from on high that both Ronan and his uncle be released due to lack of evidence and the insinuation that Ray had broken plenty of rules in dealing with the younger prisoner, this had all seemed a superb idea. They had sunk half a dozen pints of dry cider apiece as they celebrated their team's 5–2 victory over Bridlington. The rest of the lads had called it quits after a pint or two, sloping off home to watch a period drama with the missus or pick up a curry and a six-pack ahead of a night in front of a DVD. Ray and Tanner had shown no such compunction. Ray has nobody to go home to. Dad of three, Tanner merely doesn't want to go home.
If asked, neither man would be able to decide accurately which of them had taken credit for their current course of action. The idea was born around teatime, in a pub called the Coach and Horses on the road back from Bridlington. It's only a short drive from an area known to be popular with swingers and doggers looking to get their kicks, and where an out-of-town businessman was nearly crushed to death while cruising for sex a few nights ago.
Alan Rourke's Rottweilers are due to be returned to him tomorrow. His solicitor presented an emergency petition to the city magistrates, who ruled there was insufficient reason to have the animals destroyed. Rourke's brief said the dogs had never harmed anybody before and were only defending their owner. What's more, they had been responding to an order to kill given by a third party. The magistrates had taken mere moments to rule that the dogs be returned to their owner from the police-approved kennels where they were being held.
Ray had told the story to his goalkeeper over their celebratory ciders. Some time later they decided to take the dogs. They drank more alcohol. Talked about gypsy bastards and ginger cunts. And then Tanner had told him about the little tool he kept in the back of the van in case of emergencies. And Ray had risen from the pub table like a monster, teeth clamped and finger already twitching to caress the trigger.
The van pulls in to Tranby Rise. Behind thick curtains, tasteful lighting and TV screens glow. This middle-class street of bungalows and wind chimes smells of roast-beef dinners and family get-togethers. It is a place for families who all have the same surname. Colin Ray does not like the fact that it is home to Alan Rourke.
"That one. Like a bloody cartoon house, isn't it?"
They park on the road, blocking the driveway and crushing two well-tended bushes that bloom beside the neat lawn.
The dogs, perhaps sensing themselves near home, double their frenzied barking. Listening to their angry, frothing cries, Ray wonders that they were able to get the dogs in the back of the van without losing important body parts. He had marveled at the way Tanner had corralled the snarling animals into the specialist vehicle, using only a long pole with a slipknot noose, and some well-placed swear words.
Ray steps down from the vehicle. Arches his back and winces again at a second stab of pain.
"Tasteless bastard," he mutters, looking at the large bungalow and the two large Honda four-by-fours parked on the redbrick drive.
"Bet he's got chandeliers," says Tanner, appearing at his side. "They always bloody do."
A light comes on beyond the frosted-glass door of Alan Rourke's home. The door swings inward. Rourke is silhouetted on his step, a can of beer in his hand, wearing only tracksuit trousers and leather slip-on shoes.
"Them my dogs?" he asks, advancing down the drive. "Jesus, but you've got them worked up. That you, Mr. Ray?"
They have left the vehicle lights on, and the glare of the headlights means Ray and his friend are hard to see. Rourke raises an arm as he approaches and squints his eyes.
"Mr. Ray? Jesus, I didn't expect personal service, sir. My brief said to just go pick them up tomorrow meself. My, you're a grand fella, so you are."
Ray runs his tongue around the inside of his mouth. He feels angry and sick. It is a feeling he is used to. He suffers from stomach ulcers that would be enough of a reason for retirement. He sometimes feels as though his insides are decaying. When he is drunk and melancholy, the gases that belch up into his throat are rank with the taste of corruption. Of the grave.
"Couldn't expect you to put yourself out, Mr. Rourke," says Ray, sneering. "That's what we're here for, lad. To serve people like you."
Rourke stands in front of them, hand veiling his eyes. He looks from one to the other with a half smile on his face that fades a touch when it is not returned. Both men are looking at him coldly. His excitement at being reunited with his dogs begins to fade.
"You want to reverse into the drive so you can let the animals straight in the back?" he asks chattily. "May be easiest, eh? They'll be overexcited, and we don't want to wake this snooty bunch up, eh?"
His attempt at making the two men warm to him gets nowhere, so he shrugs. Returns to the sullen unhelpfulness he exhibited throughout his interviews.
"The lad wrapped up warm, is he?" asks Ray.
"Ah, Ronan will be out with his pals, sir," says Rourke. "I'm not his jailer. He'll be home soon enough, and pleased to see my dogs back safe and sound."
Ray hopes that Rourke can smell the beer on their breath. Hopes he can tell how they feel about him. The stun gun is in his pocket, cumbersome but reassuring.
He turns to Tanner. "Nice night for it, eh, Tanner? Would love to be out for a wander with my pals. Having a drink or two. Packet of fags. Fingering some tart round the back of the skips. Christ, he's living the life, eh? Must be great coming to stay with Uncle Alan."
"Uncle?" asks Tanner, as if they have prepared the exchange.
"Oh, not his real one. Friend of the family, like. Isn't that right?"
Rourke spits. Shrugs. Has heard enough. Wants his dogs.
"He's got an uncle, though, hasn't he? Godfather, or whatever these godless bastards call them."
Rourke's jaw tightens. He sips from his can of beer, then throws it into his garden.
"Scary bastard, from what I've heard," says Tanner quietly.
"Aye, he is that. Big man in Ronan's world, though. Big name."
"What was it again, boss?"
Ray cocks his head. Looks skyward. Appears to be thinking. "Italian sounding, I reckon. Can't bring it to mind. You want to help me out, Mr. Rourke?"
Rourke considers the pair. Looks back up to the warmth of his own front door.
"You got any more you want to get off your chest, or can I have my dogs?"
Ray gives a tight-lipped smile. "That's the thing, son. That's the thing."
Rourke considers the detective. Looks closely at the fifty-year-old man in his disheveled black raincoat over soft cords and golfing jumper. Looks again at the face that has snarled at him across an interview-room table time and again these past days. There is nothing new about the distaste and contempt he sees in the policeman's eyes, but tonight, away from the police station and accompanied by the sound of enraged barking, it is an undisguised malevolence.
"The magistrates—"
Ray laughs. "You hiding behind the law now, boy? You bomb a police van. You set your dogs on an officer. You spend days making me look a prick . . ."
"Sir, I told you what I knew, and it was nothing . . ."
Ray is shaking his head now, getting angrier. He does not know what he truly expected to happen when they arrived.
"You made me look a prick, lad. But that's going to change."
"Give me my dogs." Rourke's voice is rising.
"I'm going to appeal to your better nature."
"My dogs, sir."
"I'm going to ask you the same questions I've asked you all fucking week . . ."
"Ask what you like!"
"And if you don't tell me what I want to know, I'm going to kill your fucking dogs and throw them in the river. And if anybody asks what happened to them, we'll say it was gypsies."
Rourke's face twitches. He shows teeth. Pushes his hair back from his face. "You okay, girls?" Shouts this last at the side of the van, and is rewarded with a cacophony of barking.
Ray has had enough. He pulls the gun from his pocket and Rourke instantly backs away.
"Don't you worry," says Ray through a grimace. "It's not what you think it is. I'm not going to put a bullet in your knee, though God knows I'd fucking love to. No, this is for your little darlings. You seen one before?"
Rourke is shifting his weight from one foot to the other, looking in turn at the officers and the weapon in Ray's hand.
"Abattoir gun," Rourke says, his teeth locked.
"Give the man a prize," says Ray, his voice high and unhinged. "I think they call them a stunner. They fire a metal bolt several inches into the brain. Render an animal unconscious in a heartbeat, to give you a bit of time to enjoy slitting their throat."
"They're illegal."
"I give a fuck?"
"You wouldn't fucking dare."
Ray strokes the gun as if it's a pet. "I hope you don't tell me, to be honest. I hope I get to look you in the eye while I run a straight blade across your darlings' windpipes."
Beside him, Tanner shifts. This is ugly. This is more than the game he was expecting. There is something about Ray's posture, his stance, that is more terrifying than the weapon in his hand.
"Suppose I'm telling the truth?" says Rourke breathlessly. "Suppose I know nothing?"
Ray spits. Hawks up something vile from his chest and launches it like a bullet. "Get the back doors open, Tanner. This selfish prick isn't going to help his doggies."
Rourke stares into the officer's eyes. Tries defiance. "You wouldn't do it," he says. "Not really."
Ray takes a step toward him. His eyes are only an inch or two from the traveler's. He says nothing. Just lets Rourke make up his own mind about whether Ray has the balls to make good on his promise.
"You sick fuck. You sick, sick fucker," says Rourke desperately, looking to Tanner in the hope that the younger man, at least, is bluffing. "Please, officers. I can't. This isn't right. It's not right . . ."
"Open the van doors, son."
Ray's voice is cold now. Almost a whisper. He is no longer expecting answers from Rourke. So he is going to kill his dogs.
For a moment Tanner hesitates. The cold night air is cleansing him of the alcohol that has got him this far. He looks at Colin Ray and realizes what he is doing. Realizes that Ray never expected the man to talk. That he has been brought here to commit murder.
"Just tell him," says Tanner, suddenly beseeching. "He'll do it. Look at him. He'll fucking kill them both."
Rourke's gaze flits between the two of them. For endless hours this man sat in cold cells and colder interview rooms, refusing to give more than a "No comment" or a "Fuck you." Here, now, he is crumbling. He seems to be getting smaller under the weight of his indecision. He seems to be trying to decide whether to take a swing or run away. Whether to close his lips or spill his guts.
"Open the fucking doors, Tanner . . ."
"Noye," says Rourke, and the name erupts from his lips like air from a popped balloon. "Giuseppe Noye. Ronan's godfather."
Ray nods. Says nothing more. The look on his face is somewhere between fury and disappointment. The gathering wind plays with the tails of his coat. Takes some of the redness out of his face. He wants to shiver suddenly. Wonders if he is ill or in pain. Takes a cigarette from his pocket, lights it, and hands another to Rourke, who lights it with an expensive Zippo and inhales deeply.
"Talk, boy. You've got to find a lot of words in the next two minutes or I swear I'm going to—"
"We did time," says Rourke, gabbling. "Pepe and me. He's an important man. Not somebody to piss off. A friend."
Ray takes a drag on his cigarette. "I'm not gripped with excitement, lad."
"Pepe's done a lot of time. Last stretch was a long one. He made some new contacts. Saw a new line of business. Saw an opportunity."
"Contacts?"
Rourke blows out a cloud of smoke. "Asians," he says quietly. "Vietnamese."
Ray spits. "Bollocks. Your lot don't work with that lot. And they don't work with outsiders, neither."
"It's all changed, sir," says Rourke, staring at the end of his cigarette as if looking for answers in the glowing tip. "Vietnamese may look after some things, but the people who give them their orders are people Pepe has no problems working with. Never used to, anyways."
"Spit it out."
"Pepe's nephew dotes on him. Ronan. Wanted to be like him his whole life. Wanted to impress him . . ."
"And?"
"And Pepe threw some work his way. Asked him to look into this new opportunity for him. He did. Showed a bit of heart. Balls, even. Pepe said he could be the man for these new opportunities."
"Are we talking in code, Rourke?"
"I'm giving you what I can," he says, bunching his fists.
"Ronan got in over his head?"
"He doesn't think so. He thinks he's the big man. Ronan's gone off the rails. These new people Pepe set him up with, they're bad news. Filled his head with big ideas."
"These are the people who run the drugs operation?" asks Ray. "The cannabis factories?"
Rourke closes his eyes. "They're big. Bigger than us. Than Pepe. All I know is, Ronan got caught up with people that weren't good for him. And so Pepe asked me to try and get him out of it. Keep him under my wing. Look out for him."
"And you were willing to do that? Take this nutter in?"
"When Pepe asks, you say yes. You don't upset him."
"And Ronan didn't want to leave his new mates behind?"
"He wouldn't come. Had to get Pepe to reach out to him direct and tell him that he'd gone too far. That he had to come back with me. Ronan agreed in the end. Did as he was asked. But his new mates didn't give a shit about what Pepe wanted. They said Ronan was part of the operation now. They were going to set fire to the whole bloody campsite. I didn't know which fucking way to turn. It all got out of hand . . ."
Ray pulls a face. Rubs the stun gun absentmindedly over his sore ribs. "Doesn't sound like it was ever in hand. We got him easy enough . . ."
Rourke grinds out his cigarette with the palm of his hand. "Copper will pay for that, I promise you. You warn him."
"Who?"
"Big guy, so Ronan says. Ginger, Scots fella. Fucking giant, according to Ronan."
Ray looks confused. "McAvoy?"
"Aye. Noye's taking it personally. He can't touch the lads who are driving Ronan astray, but he can bloody sort this."
Ray turns to Tanner. Gives a tiny shake of his head. Screws up his face, trying to make sense of it.
"So Pepe tells this kid to go play with villains, then decides they're too naughty and wants him to come home? Why didn't he sort it himself?"
"He doesn't want the connection ruined," says Rourke, as if eager to get every last word out of himself while he still can. "But he wants Ronan out of there."
"And Ronan's enjoying himself too much?"
Rourke looks down. "He's running wild. I can't control him. He's giving orders and people are following them. He's just a boy and these fuckers are doing what he says. Had one of his heavies hold some Chink woman's hand in hot oil. Melted it down to the bone. Burned down a house on Bransholme where somebody said there was a little cannabis operation. He's living in his head. He's out of it. Threatening us. Threatening fucking coppers. Got his uncle involved now . . ."
"Did you throw the petrol bomb, Rourke?"
Rourke stops talking. Looks away.
"I read his phone when he was having a shower. He'd got a message from his contacts. Told him the warehouse was being watched. Said he wanted a message sent to the coppers outside. I took it on. Called some friends. Cleared the warehouse for him."
"And the petrol bomb?"
Rourke nods. "Me and a pal." He looks up, voice quickening. "You must know we never wanted to cook anybody. We threw it as wide as we could. Just wanted to show it had been done. Then Ronan wasn't in trouble and nobody was hurt. You tell me—what was I supposed to do? Pepe asks me to keep the boy safe, and next thing I'm in the middle of all this shit . . ."
Rourke stops. Closes his eyes. Looks tired and old. Looks like a man who has been pulled in too many directions.
"How did your print get on the bottle?"
"I wore gloves, but the bottle we used . . ."
"Yeah?"
"One of my own," he says, shaking his head. "I'm a thick bastard."
"The car you drove?"
"Ronan's. Been driving round like a rock star."
Ray scratches his face thoughtfully. "Bet Ronan wasn't pleased."
"Threatened me with his uncle. Threatened me with his new pals."
Ray gives a tiny nod. "And Noye?"
"He knows I've done what I can."
"The drug contacts?"
Rourke shrugs. "Who fucking knows?"
They stand and consider each other for a time.
Broken, humbled, Rourke cannot even raise his head as he asks whether he can have his dogs.
Ray looks at him hard. Turns to Tanner. "Give him the fucking things."
He leans back against the side of the van. Listens as the barking increases in volume, and has to suppress a smile when he sees the swarthy, unbreakable traveler down on his knees, weeping into the necks of two excitable, slobbering dogs.
Rourke catches his eye.
"Would you have done it? Pulled the trigger?"
Ray gives his first real smile.
"I still might."
SUZIE WISHES she had an addiction.
Wishes that drink or cigarettes or sticking a fucking needle in her veins brought her some vestige of comfort and relief. She has nothing. No chemical crutch. Doesn't know how to soothe herself.
"Breathe, Suzie, breathe . . ."
She halfheartedly punches the passenger seat—a backhand slap that would have made Simon giggle were he there to receive it.
"It's all bollocks."
She sneers as she says it. Drops her head. Realizes that her images of peace are all clichés. Knows that were she to shut her eyes and take deep, cleansing breaths, she would feel no more at ease than she does now: wide-eyed, teary, staring into the gathering darkness with her fingers wrapped around the steering wheel like ivy around a tree trunk.
It is pushing ten p.m. She does not know why she has driven here, to Coniston rest stop. Doesn't know what she hopes to achieve. To see. To feel. It brings her no comfort, but nor did she expect any.
Her day has been a haze. She drove home after fleeing Big Dunc's house, but found her flat as cold and distant as an unloving partner. Wraithlike, she drifted from room to room, looking for familiarity. For warmth. For something that would trigger happier memories or more encouraging thoughts. She found nothing. Took her laptop to the pub and spent money she does not have ordering clothes she does not need. Drank two vodka-and-Cokes, then threw up in the toilets and headed to the park. Sat reading a book and picking at a shop-bought sandwich: absorbing nothing and tasting less.
It is only here, now, that she is finding herself again able to connect with her own thoughts. Here, now, she thinks in her own voice.
She fiddles with the radio. Something pop has been playing and it seems inappropriate. She fiddles with the dial and finds a classical station. Gives herself over to a melancholy cello concerto that seems a more fitting accompaniment to the gloomy, cloud-shrouded sunset she watches through the dirty glass.
"Miss you, Si," she says again. She has been saying it a lot. Not chatting to him. That would be odd. Just acknowledging his memory, his presence. The fact that she failed him. Let him be murdered and did not care enough to make a fuss.
Later, should she be given the chance, she will find a place for her guilt. She will never excuse herself, but she will accept that she went mad for a time when her friend died. Became a half-thing. Existed emptily. Closed herself off to thoughts she could not stomach and to fears she dared not acknowledge. She will tell herself that she was young, naive, and that the notion of murder, or deliberate death, had never filtered into her bouncy, silly life. But here, now, she hates herself for not demanding answers when Simon died. For letting it seem she did not care.
"See you soon, Si," she says softly.
She senses there is a certain truth in the statement. She is not entertaining thoughts of self-harm, but twice this week somebody has tried to kill her. She cannot swallow without pain. Cannot close her eyes without seeing the man crushed against his own car. Keeps remembering the sound of crunching bone and squelching blood, the snippets of vile audio collected as she tumbled down the grass verge, knickers trailing from one leg, dirt and grit on her face.
She looks at her phone. Wishes she had somebody to counsel her. Somebody she could ask whether it is a good idea to text the man who tried to kill her, and ask him whether he murdered Simon, too.
She is parked away from the main shadow of the rest stop. The car's lights are off, but there is enough daylight remaining for her not to worry, or care, about another vehicle slamming into her. She has been passed by two vehicles already. Saw another half-dozen when she drove through half an hour back.
"What are you going to do?"
She asks the question of herself. Pulls down the sun visor and looks at herself in the vanity mirror. Asks it again and looks at the shape her mouth makes. Looks into her own eyes. Takes off her glasses and wipes away the steam that her hot, wet eyes have created on the lenses. Wipes her eyes with the heels of her hands. Sniffs noisily and pulls a face at the rather revolting sound.
Suzie has never felt so alone. So lost. She cannot help but consider the shabbiness of her life. To think of her debts. Her tiny flat with its charity-shop and hand-me-down furniture. Her stalling career. Poor CV. Her one, failed, relationship.
"You're not even pretty," she says out loud, and suddenly feels angry about it. "Ugly bitch," she snaps venomously at herself. "Fuck you!"
She slams the visor back against the car roof. Takes a breath, but as she blows it out, her throat hurts and she starts coughing. She begins to cry again, then loses her temper with her frailty and locks her teeth. Angrily, she turns the car key. Stamps down on the accelerator as she wrestles the old Fiat into gear. Throws the car forward, disappearing into the darkness behind the mound of grass and trees that shields the rest stop from the road, and makes the whole area so appealing.
Four cars are parked up, two on either side of the road. A mass of figures congregates around the windows of a large family sedan. As Suzie approaches, heads turn. She pulls into the curb close by. Feels, perhaps, a frisson of terror. Remembers, fleetingly, the first time she came here. Remembers watching from the safe remove of her car as a husband made love to his wife in the back of a hatchback and a man stood at the window, touching himself, as the pair put on their show. It had all been a novelty then. She had been fresh out of love, eager to live, to experience. To be bad. It had struck her as odd at first. Unusual that men and women should cluster around a cheap car and watch other people fuck. She has seen so much since that she finds it unusual some people don't.
Impulsively, recklessly, she steps out of the car. She is wearing flip-flops and a long dress under a baggy jumper. She has on a scarf to hide the ligature marks. Looks okay. Would be considered a prize were she to let anybody touch her this evening. She wonders whether she will. Whether she'll watch and smile or let somebody enjoy her.
There are three men standing at the driver's side of the sedan. A couple, wrapped up in each other, are on the far side, watching through the passenger window. With the car doors open, the interior light is switched on, and gives off enough light for Suzie to understand the nature of the scene taking place within.
Heads turn her way as she gets closer to the car. She smiles instinctively. Closes her eyes tightly as she passes the spot where, just a few days ago, a man was nearly killed in her stead.
"Evening."
A male voice. Local. Friendly. Middle-aged.
Suzie nods a hello to the nearest of the group. Feels eyes upon her. Fixes her gaze on the shapes moving in the car. Stops. Looks inside. Locks eyes with the man in the passenger seat. He's young. Maybe eighteen or nineteen. He's wearing a knockoff designer T-shirt and his tracksuit bottoms are pooled around his ankles, sitting atop his dirty white sneakers. He is not unattractive. A little ratty and disheveled, perhaps, but he is well muscled and his features handsome.
The shock of hair bobbing up and down in his lap belongs to a large, bulky woman whose clothes suggest middle age. She has taken off her seat belt to better go about her work, but Suzie knows from experience that the gear stick will be hurting her breastbone. Squinting, she notes that there is a label sticking out the back of the woman's jumper. It declares her a size sixteen and a discount shopper.
"You on your own?" asks the same voice.
Suzie looks at its owner. Mid- to late forties. Quite tall. Decent-sized gut pushing at cord trousers. Turtleneck and check suit jacket. Graying hair and two days' beard on his oval, fleshy face. He has nice eyes.
"You think I need a bodyguard?" she asks.
The man smiles. "Been at it for a while," he whispers, nodding at the car. "Thought they'd be done by now. When I was his age I was a two-minute man. Opposite problem now." He looks her up and down. "You wanting to play or just watch? My car's nice and warm . . ."
"We'll see," says Suzie, and realizes she has no clue why she is here or what she wants. The knowledge is almost freeing.
Suzie cannot help but want to see more. She gently pushes past the man with the nice eyes and bends down to better see what is happening. She is not aroused. Just curious. Eager to witness something that is worth opening her eyes for.
"You!"
Suzie's head whips left. Searches the source of the enraged exclamation.
The back of the car. Broad back and shoulders against the glass. Skintight leggings and a floaty silky top. Multicolored bob and rage in her eyes. Melissa. Jarod's friend. The lady from the swingers' party who didn't smile, and who ended the evening with her partner being rushed to hospital.
Suzie's mouth falls open. She starts to say hello but a sudden fear has taken hold of her. She backs away from the window but pushes against the firm, unyielding bulk of the man with nice eyes.
She sees the woman in the front seat stop her work. Turn her head. She looks familiar. She, too, was at the party. She is in her late thirties. Plain. Pinch-faced. Spent some time in the swing being pleasured by Big Dunc and a vending-machine stockist from Selby.
The back door of the car is swinging open. Melissa levers herself out. Angrily pushes aside the young couple in her way and comes round the back of the vehicle.
"Little slag," she's saying, face contorted. "Jarod's going to be a vegetable 'cause of you. One night was all I wanted. One night with him. And you make eyes and he's off and then he's getting a brick around his fucking skull and you're playing the victim and it's all 'poor me, poor me.'"
Her words come out in an angry spit. Her lips froth. Teeth are bared. Suzie doesn't know whether to turn and run or stand her ground and defend herself from the woman's lies.
"I'm sorry, nothing happened with us, we were just having a moment . . ."
"You're a little tease," says Melissa, her face in Suzie's. "You wave your bits in people's faces, then leave them begging."
"That's not true . . ."
"Little teases, all of you. I've met people like you. The ones on the Internet who say they'll turn up. The ones who go to parties and don't join in . . ."
"I do join in . . ."
The blow comes from nowhere. It is not a slap. It is a right hand delivered with a closed fist, and it knocks Suzie backward and to her knees. She is dizzy. Sick. The same dirt and gravel in her mouth she has tasted for days.
A boot now. A foot to her ribs, tipping her to her side. Pain explodes inside her. She begins to vomit, but is still gasping for breath and begins to choke, her swollen throat closing.
Now there are fingers in her hair. She is being dragged upright. She hears voices. Protestations. Sees the shape of men moving away. Hurried footsteps, running for cars. Lights, engines.
Her face slams against metal. The cold of the car bonnet. Spit and blood pooling in the corner of her mouth as her face is pushed painfully down . . .
"I'll show you little bitches . . ."
Spit on her face. Another punch to the back of her head.
And now she can feel the cold wind on the back of her thighs. Can feel her dress being pushed upward and hands clawing at her skin.
She knows now. Knows what she came for and what she will get.
Then there is a new voice. Loud. Firm. Clear.
She hears Melissa spit and swear. The throaty laugh that follows her suggestion the speaker should wait his turn.
And then the pressure is released. There are no hands upon her. There is no pressure on her face or breeze upon her thighs.
There is a shrieking. More angry threats, growing more frenzied and yet more distant.
Suzie slides down the bonnet of the car. Collapses on the gravel, a mess of twisted limbs and pain.
Her eyes close. None of it seems to matter anymore. All the noise. The pain. The threats. She feels somehow free of it. Light. Feels a warmth and closeness around her that she has not felt in an age.
Soft, rough hands upon her face. A giant, tender palm upon her cheek.
She opens her eyes. Lets the features swim into place.
Lets the background roar of her ears tune to the frequency of the gentle, probing voice.
Catches, among the other words, a name.
McAvoy. Detective Sergeant Aector McAvoy. Don't worry, I'm here, you're safe . . .
• • •
THE COFFEE is bitter and tastes faintly of soup and orange squash. More than anything, it tastes of the flask it has been carried in. Suzie drinks it gratefully. It warms her. Fills her with a flavor that is blessedly removed from the blood and spit of her mouth.
"Better?"
Suzie nods, cupping both hands around the metal mug. She lets the steam rise beneath her nostrils. Inhales deeply.
"Cute," she says, and forces a smile.
McAvoy is holding Lilah against his hip. The baby is wide awake. Trying to reach her daddy's ear with one tiny, grasping fist.
"Thanks," says McAvoy, kissing his daughter on the head and shushing her softly. "All babies are cute, though, aren't they?"
Suzie appears to think about it. "I prefer bunnies."
"Yes? You must be a city girl. Grow up in the country, you don't feel the same. Bloody pests."
"Wouldn't trust anybody who didn't like bunnies."
McAvoy smiles. "Didn't say I didn't like them. Said they were pests."
Suzie sips her coffee again. "You don't seem like a policeman," she says quietly.
"It's the outfit, isn't it?" he says acceptingly. "I know, I know."
He is dressed in pajama trousers and a rugby shirt underneath a waterproof jacket he managed to find in the back of the car. When he took Lilah for a drive to settle her before bed, he had not admitted his intentions to himself. Even went to the trouble of feigning surprise when the car took him to Suzie's flat. Told himself then it would be dereliction of duty not to follow when he saw her climbing into the Fiat and slamming the door.
He has not yet phoned Roisin to tell her why he has been out so long. Will tell her tomorrow that he did not want to wake her. That he and Lilah fell asleep somewhere pretty and had a pleasant night.
Suzie pulls a face as she looks up at the hulking form of McAvoy, and then at his car. She is sitting in the passenger seat, legs outside, door open. She cannot imagine him squeezing himself into the tiny vehicle.
"This really your car? Bet you don't need roll bars."
McAvoy laughs. Takes the cup from her and sips from it thoughtfully. "Used to have a minivan. It blew up."
"You have all the luck."
"Luck of the Irish, my friend says."
"But you're Scottish. You sound it, anyway."
"That's the joke."
"Oh."
In silence they pass the drink between them. It feels strangely intimate. They feel oddly comfortable in each other's nearness.
"You were following me?" asks Suzie, struggling to digest the information McAvoy had first imparted as he scooped her from the ground in arms that could have lifted the car, too.
McAvoy nods. "You know what I want to ask you about . . ."
Suzie nods. Bites her lip. She looks up at the sky, where the moon is partially obscured by clouds that put her in mind of burned popcorn.
"It was me," she says acceptingly. "The other night. Here. The man."
McAvoy pauses. "I know that. We have your prints."
She frowns. "I wiped the receiver."
McAvoy looks away. "We got them from the bonnet of the car."
Suzie pulls a face, embarrassed. "Oh."
"What happened?"
"Don't know."
"Suzie . . ."
"Car came from nowhere," she says flatly, finishing the coffee. "We were playing. Then the car hit him. I got out of the way just in time."
"And you left him there."
"I phoned for an ambulance."
"But you didn't call the police?"
"No."
Suzie has tried to hold McAvoy's gaze, but gives now. Looks at the moon.
McAvoy considers her for a moment. There is dried blood on her chin. One arm is protectively tucked against her bruised ribs. She had refused to go to hospital in the police car or the ambulance that both arrived within minutes of McAvoy's call. A unit from the Holderness Policing Team has taken Melissa to Priory Road Police Station, where tomorrow she will most likely be charged with anything from sexual assault to attempted murder. The young man who had been receiving the passenger-seat BJ needed the ambulance. He had attacked McAvoy while he was trying to restrain Melissa. McAvoy had shrugged him off. The shrug had been enough to cut him down and snap a tendon in his ankle. McAvoy has not yet had time to worry about the repercussions, or to hate himself properly for how much he longed to slam Melissa's head on the bonnet of his car as she struggled in his grasp and he fumbled for his radio.
"You're not what I was expecting," he says at length.
Suzie looks up at him and there is a flicker of warmth in her eyes. "Did you think I'd be in a corset and clear heels?"
"I should be so lucky," he says, and, unable to help himself, wipes the dried blood from her chin.
"Thank you," she says, and then impulsively, desperately, grabs his hand with hers. She holds it against her cheek. Closes her eyes. Allows herself to feel a moment of safety. Of solace. Builds comfort and secure into the sensation of this touch.
"Simon Appleyard," McAvoy says as he slowly withdraws his hand. "Your friend. He was murdered."
Suzie nods.
"You knew?" he asks.
"I think I always knew." She seems to consider it. Shivers and moves back a little into the car. "Maybe I didn't."
"But you're convinced now?"
She pulls down her scarf. Shows him the ligature mark. "I'm next," she says.
McAvoy crouches down. Examines the marks. When he looks at her again, his face is only inches from hers. Her reflection swims in his eyes, and in this mirror the girl who looks back is lovably pretty.
"Last night," she says. "Party. Somebody tried to strangle me. Hurt somebody else."
McAvoy's face changes. He begins rooting in pockets for bits of paper. Pencil. "I'll need the details."
She shrugs it all away. "I'm tired," she says, and the statement is completely accurate.
McAvoy seems to realize that he is missing an opportunity to crowbar some proper police procedure into his investigation. The notion is tantalizing. He feels like a maverick, a private investigator hiding in the police force who is very much at odds with the man he has always been.
"I can take you to hospital now. We can talk on the way."
"No, not yet. Let me enjoy the breeze."
McAvoy narrows his eyes. Tries to understand this girl. To better appreciate her reluctance to go to hospital. To talk to the uniformed officers. To leave his side. It occurs to him what may be going through her mind. "Suzie, you're not going to get arrested. You're not going to prison."
"I have a record. Simon and me. We had this idea. Watched a film where this girl tattooed a man who was mean to her. We decided to do it. He was still texting me, my ex. Treated me like shit, but still thought I'd come running for sex. I don't know whose idea it was to persuade him to be tied up. But he liked it. And then, when he was under me, I don't know. I had the blade. He was crying. Squealing. I felt sick. Made a couple of scratches and then ran. I was frightened to untie him. I went and got Simon. He came back with me and untied him. He went for Simon, and Simon fought back. We ran. It was a mess. It wasn't like we planned."
"He called the police?"
She nods. "They took his side."
McAvoy looks again at this plumpish, bizarrely dressed girl, and finds it hard to reconcile the details on the charge sheet with the person in front of him.
"He must have really hurt you," he says at last. "Your ex. For you to get into all . . . this."
"I needed to be something more than I was. Needed to be more than this timid, downtrodden little girl."
"There are other ways. You know, you would both still be in prison if the victim had given evidence, don't you?"
"And maybe Simon would be alive."
McAvoy nods. Sighs. Lowers himself to the ground and sits cross-legged: Lilah across his knees.
"You look like one of those Scottish kings," says Suzie, sniffing. "Like you should have one of those big swords and be on a throne of skulls."
"Did Scottish kings have thrones of skulls?"
"I would."
McAvoy laughs. Shakes his head.
"Who do you think killed him? Who is trying to kill you?"
"I just know he can spell."
McAvoy stiffens. "Pardon?"
Suzie hands over her mobile phone. Shows him how to navigate to the string of messages. Tells him how her admirer came into her life and what he has cost her.
"You did this? That's why you were here?"
McAvoy is reading the man's instructions. Trying to work out why she would debase herself to please a stranger.
"It's a game."
He looks her in the eye and tries to let concern instead of contempt fill his face. "It's not."
"No," she says, holding his gaze. "I know."
For a time he thinks about lecturing her. Telling her that her lifestyle killed her friend. But he is not convinced of the truth of the accusation, and already likes her too much to hurt her this way.
"He knew you would be at the party?"
"Yes."
"And you've had no more messages? Just his temper tantrum?"
"I've been thinking about texting him . . ."
McAvoy clamps his lips together. Considers. "Not yet," he says. "We need to know more."
"What do you actually know?" asks Suzie, and though there is no accusation in her voice, McAvoy implants his own.
"I think Simon might have been killed because he saw someone, or met someone, who does not want it known how they spend their spare time."
Suzie nods. "Some people are like that. They're ashamed."
"Are you?"
She considers it. "I didn't think I was. I thought I was empowered. I'm not sure."
"Simon was your protector, yes? He kept you safe while you played?"
"He was my everything."
McAvoy nods. Readjusts Lilah and puts his smallest finger in her mouth as he thinks. "The parties you attend. The people you met. Do you have any kind of record? Any diary? Any way of identifying people?"
Suzie shakes her head. "People take care. It's all false names and personal e-mail addresses and pay-as-you-go mobiles. People go to the other side of the country to shag a stranger so the wife doesn't find out. It's not like dating."
McAvoy stands. The movement seems to pain him. He hands Lilah to Suzie, who takes her without thinking, and passes her back without comment.
"If I showed you some photographs, would you be able to remember whether they are people you may have come across in your private life? Whether they are people Simon knew?"
Suzie nods. "Tonight?"
McAvoy shakes his head. "Tomorrow. I have to go home. Put the little one to bed. Explain to the wife why I've been up at a dogging spot . . ."
He colors as he says it, afraid he has offended her. Suzie merely smiles.
"Not a nice word, is it? I don't even like dogs. Puppies, yes. But puppying sounds wrong."
McAvoy realizes that when he gets in the car and starts the engine, he will be taking this young, confused girl to hospital and pretty much dumping her. Knows she will be getting a taxi back to her cold and lonely flat. Is not sure he can allow that.
"How are your ribs?"
"Sore."
"My wife's a healer."
"A doctor?"
"No. She's just good at making people feel better."
Suzie grins. Realizes she does not want to leave this man's side. "You're a good pair."
MONDAY, MIDMORNING.
A GRASSY AREA set back from a quiet country road, shielded by high hedges and cherry blossoms.
They sit side by side. Arms resting on the damp, bowed wood of the sagging picnic table.
A storm in the air.
Pharaoh sucks an inch off her black cigarette. Sips at her takeaway coffee and is angered to discover the polystyrene cup is empty. She reaches across for McAvoy's bottle of lemonade. Takes a swig and grimaces.
"Did you get crisps and candy, too? Packet of cola cubes and a caramel shortbread? It's amazing you've got teeth."
McAvoy takes back the lemonade. Puts it down on the bench beside him, away from her reach. Tries one more time to get an answer he can work with.
"Guv, do you think he could have killed Simon?"
Pharaoh throws her cigarette butt onto the ground, then stares at its glowing tip. "Why did I do that? There were three drags left."
"Guv . . ."
"Oh, for God's sake, Aector, yes. Okay? Yes, he could have killed him. Happy? Anybody could have killed anyone. People act oddly. For instance, I've got this giant fucking idiot of a detective sergeant who works for me. He let a suspect in a murder case sleep at his house last night. Then he invited me for a picnic."
McAvoy allows himself the tiniest of glances at her chest. It is not the shade of crimson that it goes when she is truly cross, nor is there a sheen of perspiration at her temples or on her upper lip, so he knows her temper is not as intense as she is making out.
"She had nowhere to go. She was hurt. She's got nothing . . ."
Pharaoh looks at him so intensely that he has to turn away. For a moment it feels as though she is reading the back of his skull.
Finally she rubs her hands through her hair and gives a stretch that comes with no accompanying yawn. "You know the reason we don't carry guns in this country, Aector? It's because, if we did, I'd shoot you."
"Dead?" he asks, as though this will make a difference.
"No," she says, thinking about it. "Just sore. I'd maybe just hit you with it."
He smiles. "Thanks, guv."
She smiles at him, maternal again. Seems about to beckon him close for a cuddle.
"How is she?"
"Sore. Achy ribs. Nasty bruise."
"Not her. Roisin."
McAvoy pulls a face. "Okay, I think. Says she's okay. Was making her breakfast when I left."
"Anything nice?"
"Scrambled eggs with smoked salmon. Fresh chives."
"What kind of toast?"
"Not sure. I can ring if you like . . ."
Pharaoh gives in to laughter. "Fucking hell."
McAvoy cannot work out which of his indiscretions he should feel most ashamed of. For a long time he wondered whether other people were given a handbook in childhood outlining what is acceptable and what is not. He is never truly sure.
"She likes helping people."
She looks at him. Pulls a face. "Yeah, I'd imagine she's big on lost causes."
Pharaoh wishes she hadn't said it as soon as the words are out of her mouth. Curses when she sees the impact on his face. Pain and uncertainty pass across his face like a ripple on a still pond. He absorbs it and then it is gone.
"He was okay with you?" McAvoy asks, his voice catching. "Tressider."
"Professional. Decent guy, really. Inasmuch."
"Yeah. Inasmuch."
McAvoy is staring at the damp grass. Watching a cherry blossom that has become trapped in the wooden supports of the picnic table. Wants to free it so that it can join its friends and dance on the breeze, but fears more contempt.
"This is something now, isn't it?" he asks quietly. "It really is a murder."
Pharaoh seems about to argue, but loses enthusiasm. She nods.
"Are we going to take it to the top brass? Start the investigation? Do things properly?"
Pharaoh shrugs. Sips again at her empty coffee. Reaches into her handbag and retrieves another black cigarette, which she holds but does not light.
"It's my first day back, Aector," she says to the side of his face. "As far as the brass know, my team is looking into the drugs. The petrol bombing. Alan Rourke and the ginger runt. That's what we're doing. And we're not doing it particularly well. Simon Appleyard doesn't figure on anybody's radar."
"One phone call," he says, looking at her. "We make it happen."
Pharaoh looks up at the sky. There is still some blue up there, but the dark, rain-swollen clouds are rolling back. Their undersides hang low, as if waiting to be sliced open with a blade. They seem oppressive. Ominous. They could just as well be as full of black eels as rain.
"It's not me you have to convince," she says. "I can't say I'm relishing telling people that Peter Tressider needs to be formally interviewed in connection with a murder inquiry, but I'm willing to do it. What I need before we do is something more than a few concidences, some intuition, and a big leap of faith."
"Suzie," says McAvoy, reaching down and freeing the cherry blossom, then letting it go on the next gust of wind. "She'll tell them what has been happening to her."
"And she's a reliable witness, is she? A swinger with a record."
"She was attacked."
"She was at a dogging pit and a sex party."
"She's a victim."
"She's a tart."
"That's not fair. All she's been through . . ."
Pharaoh throws the cup down on the wooden table. It bounces and rolls onto the grass. "These aren't my words, Aector. They're what I'll be hearing."
They sit in silence for a time. McAvoy has much to say but cannot find the right order for his words. Sits wondering instead whether Roisin is angry with him. Pharaoh, in her turn, lets her mind drift to less confusing thoughts. Looks at her black patterned tights and wonders whether she should have shaved her legs before putting them on. Whether she will have a bruise in her armpits from the underwire of her ill-fitting bra. Whether she should have eaten the brownie McAvoy bought her, or gone for a piece of fruit instead . . .
"Ray didn't seem pleased to see me," she says resignedly. "Didn't get much from the interviews, did he? But there was a look in his eye."
"If he's got something, he has to tell you. That's the chain of command."
"Oh, aye," she says, sarcastically. "We're all about procedure."
The blue sky darkens a shade. The bellies of the clouds sag farther. They watch a sparrow flutter down to a neighboring picnic table and peck at a discarded bottle top before flying away.
"Pretty here," says Pharaoh. "Suzie would like it."
"I'm sorry?"
"Secluded," she says, by way of explanation. "You can get up to all sorts."
McAvoy half turns to her, but realizes he is already mid-blush, so stops and continues staring ahead.
"You know what we should do," he says quietly. He takes a breath. Lets the color bleed from his face. Turns, at last, to meet her gaze. "You know it will work."
Pharaoh places the cigarette in her mouth and flicks the filter with her tongue. It waggles in her mouth as she thinks.
"She'd be putting herself in harm's way."
"I'd be there."
"Would she do it?"
"I think so."
Their nods are imperceptible, their acquiescence unspoken. They simply accept the truth of what must happen.
"We need more," says Pharaoh at last. "If it comes to trial, we'll need to demonstrate we had just cause . . ."
McAvoy reaches into his pocket. Pulls out his phone. "If it's not him . . ."
"I'll be delighted," says Pharaoh. "That's the worst bit of all of this. If we've just been off the reservation but come back with a killer, we're on easy street. If we come back with the chairman of the Police Authority in handcuffs, we're making the wrong kind of headlines."
In the open air, better able to pick up the signal, McAvoy's phone rings. He mouths, "Excuse me," and takes the call.
Quietly, discreetly, one of the civilian support workers has been jockeying the database. At McAvoy's request, she has been working through the log, putting together a list of all cars reported lost, stolen, or abandoned within a five-mile radius of Simon Appleyard's flat between November of last year and March of this. Pharaoh called McAvoy's suggestion an "informed hunch" but green-lighted his use of resources.
He has been awake most of the night, Lilah dozing contentedly on his chest and Roisin's warm back and buttocks against his side. He listened for a while to Suzie's soft crying, and wondered whether he should take her a blanket or a glass of warm milk before tiredness robbed him of enthusiasm for the trip downstairs. Instead, he thought about the taxi ride. About why a killer would take a cab from a murder. Why he would order one to pick him up from a brightly lit, bustling supermarket. He tried to put himself in the killer's shoes. Manipulative. Intelligent. Cunning. He would have driven to the scene, no question. Perhaps parked a couple of streets away from Simon's home, just to be safe. But why the taxi? Why not drive away? The answer hit him as he considered his current car. Its tendency to stall in second gear. Its leaking radiator and shot A/C. Suzie's car, too, had groaned painfully as it took the sharp left at the entrance to his estate.
Cars, he thought.
Bloody unreliable things.
He utters some platitudes and thanks into the phone. Holds up a finger to delay Pharaoh. Offers heartfelt gratitude. Hangs up and then looks at his screen as the report comes through.
"A blue Honda CR-V. Left on Mortimer Close, two minutes from Simon's place. Was blocking the entrance to somebody's driveway. Looked like it had been abandoned. Community Support officer attended. Vehicle was registered to a timber company at South Cave."
"Owned by?"
"Registered in the name of Paula Tressider. Executive shareholder for her husband's firm."
There is little celebration in his voice.
"Tressider was told?"
"Company was. Tow truck came the next day."
"Any follow-up?"
"No need. Job done."
They look at each other.
"His car broke down," says Pharaoh. "He went to kill Simon and his car broke down. Walked himself to the nearest busy place and called a cab on Simon's phone."
"Why his own address? Why home?"
"Why not? He planned on dumping the phone. Simon was always going to be a suicide."
McAvoy flares his nostrils in temper. "He barely considered us," he snaps. "Didn't even try. Chairman of the bloody Police Authority and he knew we were too crap and lazy to give a damn."
Pharaoh finds herself nodding. "You care," she says quietly. "Me too, though don't tell anyone."
"We've got enough now, surely," says McAvoy. "I'll do all the spade and legwork, you know that. We can fill in the gaps once we've got the cuffs on."
Pharaoh is about to speak when the sound of an approaching car silences her. She hears the tick-tock of an indicator, and then an old-school Volvo pulls into the picnic area. It is dirty and mud-splattered, and its driver takes little care as he pulls in, deliberately, between Pharaoh's sports car and McAvoy's little hatchback.
"Looker, isn't he?" says Pharaoh, sniffily, as Ed Cocker climbs out of the car.
The political fixer is tieless, in a gray suit with a dark blue shirt. He smiles broadly as he approaches, notebook sticking out of the pocket of his suit jacket.
"Very cloak-and-dagger, Sergeant McAvoy," he says, smiling. "You couldn't have just asked me to meet you in a pub?"
"We like the fresh air," says Pharaoh, making no move to stand up, and surreptitiously tugging McAvoy back to a seated position as he begins to stand. "Like to be able to see who's lurking in the bushes."
"And you must be Trish Pharaoh," he says, extending his hand.
"I'm Detective Superintendent Trish Pharaoh, yes," she says, and steeples her fingers under her chin.
"Heard about the petrol bomb. And the dog bites."
"Been a rough week," she says.
"Sounds it. You could go for compensation, you know. You would have a lot of public sympathy. Senior female officer? Could be a mint. Cracking story . . ."
"I'm fine."
"Well, if you decide you want to tell your side," he says, and puts a business card on the table in front of her. "I have a lot of friends in the media. People who owe me favors. Take the card . . ."
"I've got one," she says, flicking it onto the grass. "You gave one to Peter Tressider. And Stephen Hepburn. Mark Cabourne. My colleague here, too, but for different reasons. You're very open about what you're doing here. Won't be long until somebody on one of the local papers hears about you snooping. Or is that what you want now? Are you trying to discredit him? Do you want Tressider or don't you?"
Cocker looks from one to the other. Spreads his hands in surrender.
"We're keeping our options open," he says smoothly. "He could be a find for us or an embarrassment. We all have the capacity to be both."
"This is what politics comes down to, is it?" Pharaoh looks like she'd like to spit.
Cocker makes no attempt to charm either of them. He has clearly been here before. "Is this the bit where you suggest I fuck off before I make things difficult for powerful people?"
Pharaoh snorts. "Which powerful people? A bunch of councillors? People don't care. People don't give a damn. For God's sake, our MPs can do what they like and we at least know what some of them look like. You think people will be shocked that a councillor has a bit of a dodgy past? Seriously? Smells like bullshit to me."
Cocker opens his mouth, then closes it again. Unasked, he slides onto the opposite bench, then gives an accepting nod.
"Hepburn's not the story," he says. "Not really. There is a story there, though. Some way down the line. He's a liar. Bloody fraud, I can tell you that. Got himself into a powerful position playing the gay card and, I can tell you, that particular card isn't all pink. I haven't always done this, you know. I was a lobbyist for a while. I've just got a knack for sniffing out trouble. And there's trouble here that could embarrass the people who pay my wages . . ."
"You thought Tressider and Hepburn might be having a fling?"
Cocker shrugs. "It happens. More than you might think."
"And what did you find out?"
Cocker looks skyward, almost reluctant to admit his hunches have failed to play out. "Hepburn's a bit of a lad. Some dodgy connections, but nothing that powerful people the world over haven't got in their backgrounds. I know he had a fling with another city councillor. Cabourne or someone. I know he likes a bit of variety in his life."
"Variety?"
"I got hold of a police report. From a while back. He got a bit of a telling-off from a patrol unit who caught him nuts deep in trouble in the public toilets at Fraisthorpe. You know the caravan park on the road to Bridlington? Arse-end of nowhere? Police caught him in there getting up to no good."
"Was he arrested?"
"Got a telling-off, but no official caution."
Pharaoh and McAvoy look at each other. "Do you know who the other person was?"
Cocker smiles, broadly. "Officers were sympathetic. All boys together. Didn't make her give her name."
McAvoy's eyes narrow. "Her?"
"Told you," says Cocker. "That pink card he plays is raspberry ripple. He likes a bit of both."
McAvoy begins to speak, but a sudden pressure on his wrist indicates Pharaoh wants him to be quiet.
"Did you confront Hepburn with this knowledge?"
Cocker nods. "He's not who we're interested in. But I thought he might be honest with me. Tell me if there really was anything to worry about. He didn't like being asked. Said he didn't think anybody would give a damn. Said he is a single man who enjoys himself."
Pharaoh smiles. "So it was just a fishing trip when you visited Tressider?"
"Wanted to see if I could shake something loose. He lost his temper with me. Went crackers about me approaching his wife. She wasn't happy, neither."
"So what will you be writing?" asks McAvoy. "Your report. Will you recommend him?"
Cocker pulls a face. "Probably. There's nothing we couldn't manage. I think he'll be a decent MP," says Cocker grudgingly. "I'm sure the decision will come down from on high, not to mention the money he put Hepburn's way. Doesn't look great, and if that's his only indiscretion, would be a shame to lose him for the party. He's our sort of person. No kids, but they're a strong couple. Respectable. Bit panicky but they'll get used to that."
"Panicky?"
Cocker waves it away as if it didn't matter. "You can't go ranting and roaring at the people you need on your side. You need some dignity. Screaming, he was. We like them best in photo-shoot mode. Like in that mag you lot have got up here. There was a piece in it about them. 'At Home with the Tressiders' sort of thing."
"The _Journal?_ "
"Yeah, our sort of people." Cocker stops. Rummages in his suit pocket and, frowning, runs back to the car. He comes back with a copy of the mag. Hands it to McAvoy.
"You may as well keep it," he says. "Angle's all changed."
Pharaoh curls her lip. "We're done," she says.
"Yeah? That was it? Thought I was going to get pummeled."
"Don't count it out," she says, then waves sweetly to tell him to be on his way. Moments later the Volvo is pulling out and roaring off.
"Weasel," says Pharaoh, and turns to McAvoy, who is opening the mag and leafing through. The weak sunlight bounces off the glossy pages, and he has to hold it straight out in front of him before the images come into focus. He looks at Peter and Paula Tressider. Blinks. Looks again. He swallows hard.
"She was at Hepburn's house," he says quietly, and scratches his eyelid. "Offered me a towel."
Pharaoh takes the mag. "Perhaps they're friends," she begins cautiously.
"Perhaps."
They look at the pictures for another full minute. McAvoy lets his eyes scan the text.
Paula, 53, runs two boutique shops in Beverley, but is also a director of two of her husband's companies. She admits to finding the idea of being a politician's wife very daunting, but says she will be at her husband's side throughout his journey to Westminster if he is selected, as many predict, next year.
Pharaoh takes over. Reads Paula's quotes out loud. "'That's a wife's role—to support her husband. We have always been a strong unit. We have not been blessed with children but we don't feel our lives are incomplete. We've always had this feeling that it's us against the world. It will be hard to let people in.'"
She and McAvoy realize their legs are jiggling. Their breathing has slowed. They inhale and exhale almost as one.
"It's a lovely house inside," says Pharaoh, to break the silence.
He flicks through the mag again. Skims a feature on an up-and-coming polo player, and a six-page spread about an organic deli opening in North Ferriby. Scans the adverts. Butchers. Bakers. Bloody ornamental candlestick makers.
Turns to the back page. An advert for a jeweler's. A posh hair-dresser's, at Kirk Ella. A tattoo and henna parlor on Newland Avenue . . .
Squints his eyes against the glare on the glossy page. Knows, even before the image swims into focus, that the picture will be of a skinny young man with peacock feathers on his back, and a fleshy girl with blossoms and lilies upon her shoulder.
Closes his eyes as he passes the magazine across to Pharaoh. "Same edition," he says, under his breath. "Same edition outlining their future, they got a glimpse of their past."
HALF AN HOUR LATER. NEWLAND AVENUE, HULL.
BAKERS, butchers, charity shops, and a couple of decent restaurants and wine bars.
A busy street, where asylum seekers, students, and daytime drinkers sit elbow to elbow with besuited businessmen and schmoozing city councillors, smoking everything from roll-ups to fat cigars at metal tables and chairs.
It's the city's melting pot: a beacon of multiculturalism and a place where most things can be acquired, be it a secondhand Kappa tracksuit, a bag of weed, or a tenner's worth of rotisserie chickens.
McAvoy parks down the side street by Planet Coffee. The place is doing a brisk trade. Young office workers reading the papers over soup bowls of latte. Students sharing a muffin at low tables and sifting through purses full of bus tickets to find enough change for the jukebox.
"That one," says Pharaoh needlessly, as she lets herself out of the car.
Hull Ink occupies a corner plot on the opposite side of the road. The sign declares it to be an award winner, though it is clearly not for interior design. The large glass window is papered to halfway up with black-and-white designs, while the glass door is covered in a double-page spread from the _Hull Daily Mail_ , showing one of the tattoo artists hard at work scrawling something indistinct into the back of one of the paper's less dull feature writers.
"Me?" asks Pharaoh, as they cross the road.
"Guv?"
"The talking. Me?"
McAvoy isn't sure what to say. Doesn't know whether he should tell his senior officer to leave it to him. "We'll just play it by ear."
McAvoy pushes open the door into a large, cream-painted room. The floor is a chessboard of tiles and the walls are a collage of different designs. Behind the till sits a large, heavily inked woman aged around thirty. There are piercings in her eyebrows and through her septum. She is dressed in black, and though her arms are uncovered, they could not be called bare. No flesh tones can be seen. She is patterned magnificently, a tapestry of intermingled designs and glorious color.
She smiles brightly. For a goth, she seems a happy soul.
"Nice," says Pharaoh enthusiastically. She crosses straight to the girl by the till and begins cooing over her arms.
"You like?" asks the girl, her Hull accent strong.
"Love it! Was it all one design or has it just evolved?"
"Bit of both," says the girl. "Got a treble clef and then some stars. Then a band around my bicep. Was becoming a bit chaotic, so we started joining them up. Took a while . . ."
"I bet! Gorgeous. Would love more myself but work would frown. Would love something pretty, though. A bird, maybe. Something free."
McAvoy turns. Tries not to wonder where she hides her ink. Forces himself back to flicking through the different designs that hang in plastic folders from a newspaper rack.
"That what you're after, is it?" asks the girl. "We've only got Devon in today. Stefan has Mondays off. We've got an appointment in about an hour."
"Sorry, love, it's actually more business than pleasure," says Pharaoh, still in the same bright tone. "I'm Trish Pharaoh. Detective Superintendent, if you're asking. That big lump is Detective Sergeant McAvoy. I know. To tattoo him you'd need a javelin and a pot of emulsion, wouldn't you?"
"I'm just the receptionist," says the girl, though she doesn't seem worried. "Devon's upstairs with a client . . ."
"Sounds dubious, when you say it like that," laughs Trish, who has turned her attention to the magazines on the counter. "That's nice," she says, pointing at a little pixie wrapped around a bluebell, which snakes down the back of a pale-skinned model. "It's lilies I'm interested in. Peacocks, too. I saw your advert . . ."
McAvoy crosses to the counter. "In the back of the _Journal_ ," he says. "A boy with peacock feathers and a girl with lilies and blossoms."
The girl nods enthusiastically. "Got a good response from that. Stefan's work, though it was their own designs. That's caused a few problems, actually. People who liked the advert can't have the same design. Copyright laws, you see. I mean, it's not like the courts can repossess it if we do breach copyright, but it's not really the done thing."
"Do you know the couple? The boy and girl?"
"I remember them. He was quite flamboyant. She was fun."
"What records do you keep? If somebody got in touch with you and wanted to know about the models . . . ?"
The receptionist pulls a face. "We wouldn't tell. We take a phone number and a name to reserve the appointment but that's it, really. Of course, if you go online, you can see who our regulars are, and our best work goes on there . . ."
McAvoy stops her by holding up a hand. "Has anybody else shown an interest in these particular images?"
The girl looks up, as if trying to see into her own mind. "Young girl brought in a copy of the mag. Wanted the same as the model. We told her about copyright but she made a few tweaks, and that was okay. Wasn't that long since. Go on Facebook—I think it's on there . . ."
McAvoy and Pharaoh exchange a glance. He turns away from the desk, reaching into his pocket for his phone.
"Can you show me?" asks Pharaoh.
The girl, eager to please, retrieves a laptop from under the desk. Opens it up and flicks over to the shop's page.
"There's no real order to it," she says, flicking through a collage of beautifully inked skin. "Black-and-white stuff. Flowers. The rest is all just crammed in."
"Would the peacocks and lilies be on here?"
"I don't think so. Stefan wouldn't use somebody else's designs on here, I don't think."
"But the advert was okay? In the magazine?"
"Who reads magazines? It was just nice work. There was no strategy . . . aah, there we go. Georgie-Lee. Beautiful, isn't it?" She turns the screen so Pharaoh can see. McAvoy views it over her shoulder. Bare skin patterned with branches and blossoms, lily pads, and petals.
"Click on the girl."
The receptionist does as asked.
A moment later, all three are reading page after page of messages, all posted on the wall of a nineteen-year-old girl who, according to friends, is in their prayers.
"'Can't believe it,'" reads Pharaoh slowly. "'You don't deserve this.'"
"'Hope they kill whoever did this to you,'" reads McAvoy.
"Is that what this is all about?" asks the receptionist. "What happened? Is she okay?"
Pharaoh scribbles down a name on the front cover of one of the tattoo mags. Rips it haphazardly. Stuffs it in her bag.
"Take the whole thing if you like," says the girl, but Pharaoh has already turned her back. She is muttering in the big man's ear.
"Do you need to see Stefan?" asks the girl as the detectives walk quickly across the tiles.
The door bangs behind them.
• • •
PHARAOH IS SITTING in the passenger seat, phone to her face, finger in her other ear, frustration in her eyes as she tells McAvoy to shut up and let her listen.
"And she's home already, yeah? Morpeth Street? That's two minutes, isn't it?"
McAvoy fidgets. Debates turning on the ignition. Decides not to.
Pharaoh hangs up. Raises a hand, then looks at her phone.
"Coming through now," she mutters.
McAvoy looks at her expectantly.
"DC Jensen took the statement," she says. "Queens Gardens CID. Would have come to us in time."
"Guv?"
"Serious and Organized. That's what it smelled like. That, or just some nutter on his way home. Was on its way to our in-box anyway."
Her eyes narrow as she reads the statement Georgie-Lee gave yesterday morning from her hospital bed, as she waited for a nurse to remove the cannula from her wrist and to check that the stitches in her forehead weren't likely to burst open before she made it to the taxi.
"Friend's birthday party . . . outside for a smoke and a breath of fresh air . . . saw a vehicle, big thing, four-by-four, damage to front end . . . heard a door slam."
Pharaoh stops.
"The attacker said, 'Suzie.'"
McAvoy closes his eyes. Turns his head away. Looks out of the window, where a man in running gear is sipping a hot chocolate and smoking a cigarette while chatting up two students. He finds himself shaking his head. Wonders if there will be exhilaration later. He feels none now.
"Pushed to the ground . . . choked her . . . hands in her hair . . . turned her over . . . thought she was going to be raped . . ."
"Jesus," says McAvoy softly.
"Ripped her dress. Tore it nearly to shreds."
"Looking for her ink," says McAvoy needlessly.
"Smashed her head into the ground, after . . ."
Pharaoh pauses.
"Guv?"
"They said sorry."
McAvoy licks his teeth. Turns the key in the ignition only to have something to do with his hands.
"Description?"
"Yes. Sketchy, but yes."
"And she's home?"
"Yes. Two minutes."
McAvoy pulls out. Makes the brief drive through a network of back roads and one-way streets.
"That one," says Pharaoh, and he pulls to a stop.
"You have it?" he asks, and Pharaoh pulls the magazine from her bag.
They look at each other. They are suddenly bone-tired. Cold. Drained by the endlessness of what they do.
The front door is answered by an attractive girl in tracksuit bottoms and a skimpy top. She announces herself as Jen and tells them excitedly she is Georgie-Lee's best friend. Tells them, too, that she has been looking after her. Keeping her comfortable. Keeping her spirits up, because that is what she always does for everybody else. Tells them, a manic note in her voice, that Georgie-Lee had just organized the "best birthday party ever" when the attack happened, and that she doubts she will ever get over the shock of finding her there, bleeding and unconscious on the front step.
"Georgie-Lee! Visitors!"
She opens a white-painted bedroom door. A young girl, childlike in pajamas and bandages, is sitting up in bed, reading a magazine and sipping hot black currant tea. The room is decorated with band posters and dream catchers, tattooed rockers, and colorful unicorns. She smiles as her friend enters the room, then freezes as McAvoy and Pharaoh enter behind her.
"Are you police?" she asks, instinctively pulling up the quilt. "Have you got them?"
Pharaoh sits down on her bed.
Does not speak. Just holds the young girl's gaze, and tries to tell her that everything will be okay.
After a moment she pulls the magazine from her pocket. Leafs through. Flinches at the sound of glossy papers catching on the quilt and the tearing of quality paper.
Lays the mag down on the bed, open at the picture of politician and bride.
She does not have to ask Georgie-Lee the question. The shiver of fear and vile memory that ripples across her face is enough.
• • •
SUZIE IS UPSIDE DOWN. The blood is pooling in her head and there is a thunderous rushing sound in her ears, broken only by the ceaseless shrieking that fills the living room.
"Give it up. You won't last . . ."
She does as she is bid. Tumbles right way up and gives in to fits of breathless noise.
She and Roisin are having a headstand competition on the sofa, and Lilah is laughing so hard, there is a risk of losing at least one eye.
"You sure this isn't hurting?" asks Roisin, as they right themselves and pull faces at the giggling baby.
"Hurts anyway," Suzie says with a shrug. "May as well be upside down."
Roisin nods and seems to think about the sentence. "I like that. We should have that on a T-shirt."
"Or a pair of knickers."
They giggle like two old friends.
Roisin and Suzie have shared little in the way of their stories. They have not probed each other's secrets. Roisin knows only that the kooky young woman in her house is somebody her husband wants to keep safe. She does not doubt him. Suzie strikes her as somebody the world requires.
Suzie, meanwhile, thinks Roisin may be the best person ever. She wants to tell her. Blurts it out, as the petite, dark-haired gypsy girl straightens her hair in the big mirror above the fire.
"You're so lovely," she says. "I can't believe I'm giggling. Being silly."
"We're a silly family," says Roisin, blowing a raspberry on Lilah's tummy.
"I'm not sure your husband is silly," says Suzie, carefully. "He seems quite serious."
Roisin smiles warmly, as if looking at a picture nobody else can see. "He's serious about some things. He's a big eejit a lot of the time."
"And this is okay?" Suzie asks.
"Nice to have company. Nice to be silly."
Suzie looks at herself in the mirror. She is wearing one of McAvoy's shirts, and a pair of Roisin's leggings. She has no makeup on, though Roisin has promised to remedy this after lunch. She feels odd, looking at this unfamiliar reflection. This plain face and unremarkable hair. Comfy clothes and covered-up tattoos.
"Do you think I'd look good with my hair black, like yours?" she asks.
Roisin considers it. "Might be a bit severe for you. You've got a warm face. Need a warm color. Your hair suits you."
Suzie smiles. "Thanks."
Roisin picks Lilah out of her playpen. Pretends to bite her tummy. "Want a hold?"
Suzie shakes her head. "I'll just stick to performing. I'm clumsy."
"Aector is, too. Should see him trying to get his change ready to pay the toll at the bridge. He can't manage five-p pieces. Hands are too big. Gets in a right state."
The way she says it is not critical. Roisin seems to think her husband's clumsiness is every bit as laudable as his strength.
"He must make you feel very safe," Suzie says, and instantly wonders if she has overstepped an invisible boundary.
Roisin looks at her quizzically. Grins. "There's no world without him."
They enjoy a moment, two new friends together. As they stand here in front of the mirror, fixing their hair and praising each other, the first fresh handfuls of rain start to beat against the glass. They cross to the window, amazed by the thunderous onslaught.
"It's gone so dark," says Suzie, marveling at the sudden gloom beyond the glass. "Could be nighttime."
"Going to be a good summer," predicts Roisin. "Crappy spring means warm summer."
"That true?"
Roisin shrugs. "No. If we say it enough, though, it will be eventually."
As they talk, there is a screech of tires and a fountain of spray as a car, traveling too fast, is flung around the turn-in to the little close. It is followed by another, uncomfortably close, and both scream to a halt on the curb opposite.
"Aector?"
Both women watch as the doors are flung open. McAvoy clambers from the driving seat of the hatchback in the lead. A middle-aged, busty woman in leather boots and a too-tight V-neck jumper wrenches open the driver's door of the little two-seater sports car. From this remove, Suzie thinks her bra looks painfully tight.
"His boss," says Roisin, by way of explanation. "Likes lamb."
"Yeah?" asks Suzie, confused. "Bunnies, personally."
The door swings open and McAvoy, red-faced, bursts into the living room, knocking a picture off the wall. Pharaoh is just behind him.
"These two," says McAvoy, pulling out a mag and throwing it open at a picture of a smiling, fifty-something couple in a posh, expensive-looking house.
Suzie looks to Roisin. Glances at Pharaoh, whose eyes are wide and face unreadable.
"Look," says McAvoy. "Do you know these two?"
Suzie takes the mag. Looks up into McAvoy's face and gives a nod.
The big detective spins away from her, hands in his hair. Throws a look at his boss. Goes and stands in front of the window with his hands on the sill, collecting himself. Roisin, wordlessly, slips to his side.
"You're sure," asks Pharaoh.
"It was a place over in West Yorkshire," says Suzie, and when she hears Lilah's little squeal, drops her voice, as if embarrassed. "Private members' club."
"A sex club?"
"Yes."
"And?"
"Simon had made friends with somebody online. Said we should try it out."
"When was this?"
Suzie sucks her lip. "Not even a year, I don't think. I'd only had the tattoos done a wee while. Simon, too." She stops, nods excitedly as she remembers. "Yeah, that was his grand unveiling. Couldn't wait to show them off. The tattooist was really pleased. Said he was going to use them in his adverts. Was going upmarket . . ."
Pharaoh moves them both to the sofa. Sits them down. "Were there many couples there? Would there be more witnesses?"
She frowns. "People don't like to talk. They give false names. You're pretty safe there."
"What happened?" asks McAvoy, crossing from the window. "What did you do?"
Suzie looks at each of their intense faces. "We played. Him with me. Then him with Simon. Then all four of us. He was nice. Simon said he was amazing. She was a bit of a cold fish. Liked my tattoos, though . . ."
"This couple," says McAvoy, pointing at the page. "You're sure."
Suzie's mouth drops open, horrified she may have misled them. "Not him," she says hurriedly. "I don't know him. Just her. She had this mask. She was a big woman. Like a man, with boobs. She was wearing this silly mask when we went in. I think she'd been to posher parties than ours. We were a bit of a comedown, but she liked roughing it. She took the mask off soon after. Was really into it. Into me. There was another guy, too. Just joined in. It's a bit embarrassing talking about all this . . ."
Pharaoh spins in her seat. Locks eyes with McAvoy. He pulls out his phone and quickly finds his way to the Hull Council website. Finds the right picture. Crouches down and shows it to Suzie.
"Him, yes? He was the other man? Stephen Hepburn."
She nods. "Yeah. Friendly guy. Funny. Simon liked him. Are they not a couple, then? Who's the guy with the beard in the mag? That her husband?"
Pharaoh gives a laugh. "That's Peter Tressider. Chairman of the Police Authority. Future MP."
Suzie looks at McAvoy, not understanding.
"And he killed Simon?"
McAvoy shakes his head.
"No," he says, rubbing his head with a large, clumsy hand. "She did."
THE MASK sits on the dressing table in the master bedroom, propped against the gilt-edged frame of the expensive oval mirror and surrounded by vintage perfume bottles, which flicker in the soft light of the large church candles that burn behind the four-poster bed.
Paula remembers the mask's purchase. A little shop filled with grinning faces, laughing gargoyles, down a Venetian side street near the grand hotel where she and her new husband were honeymooning.
"Do you like it?" he'd asked, already reaching for his wallet.
She didn't need to answer. She was mesmerized. Lost in the sightless eyes of the gold-and-crimson face she yearned to pull over her own.
A _bauta_ mask, the seller had said. Worn in the eighteenth century by men and women keen to disguise their identities at the gaming tables.
She reaches for it now. Strokes the glossy paint. Touches its nose and its detailed jaw with the back of her knuckles.
Paula has never felt more alive than when looking out through its eyes.
This is the face she wears when she lets herself play. At parties. In hotel rooms. Letting herself be free.
It was only naughtiness at first. Just a chance to feel sexy with a man or two. It became an addiction. And then more than that.
She stares at the mask again.
The colors are entrancing. Traditionally, it should be painted in plain black or white, but the harlequin pattern of luxurious red and gold catches the light better. It is an exquisite work, a gorgeous example of its type. Tied with ribbon at the back, it covers the whole face, but the square jawline points upward, allowing the wearer to eat and drink without its removal.
From behind this magnificent veil, Paula has experienced pleasure and pain in equal and exquisite measure. She has tongued and tasted, felt and fucked. She has given in to every instinct and desire. And she has never had to look at her face in the mirror.
Of course, her identity had not mattered at the start. She had been a successful man's wife, but the risk of having sex with strangers was no greater for her than for anybody else.
Then his political career took off.
She began having her photo taken. She began to become recognizable.
And they started to talk about Peter becoming an MP.
She had trusted to good fortune at first. Told herself that anybody who recognized her from her tawdry couplings would have a vested interest in keeping it to themselves. But she could not stop herself from remembering. Could not help but think back to all the nights when she had risked everything in the pursuit of faceless sex.
Alone among the many indiscretions troubling her was the night they slummed it. When she and Hepburn found a couple of playmates online and decided to take a risk.
During their Internet chats, the couple mentioned the private members' place in Huddersfield. Told her and Hepburn all about the love swing. The chains. It had sounded deliciously seedy. Wonderfully down-market. Instantly arousing in its griminess.
They decided to take the risk. Convinced themselves eighty miles was far enough from home.
They had let their fantasies take shape and worked themselves up. Given false names and paid their membership. Had a drink with the foulmouthed old bastard who ran the place and then headed upstairs to one of the private rooms.
Paula had worn the mask. Been waiting in a private room, spread-eagle on the bed, when Simon and Suzie walked in.
Suzie had laughed. Taken a look at the tall, broad-shouldered, middle-aged woman on the bed in her hook-nosed Venetian mask, and giggled.
So Paula had taken it off. She wanted the girl as soon as she saw her. Wanted to touch her warm, young skin. Wanted to trace her tongue against the blossoms on her back. She hadn't wanted the evening to dissolve into silliness. She'd taken off the mask and pulled Suzie between her thighs. And the party had begun.
After a time, when her need for pleasure had outweighed all else, she had instructed Hepburn to open the door. To let in the first man he saw. She opened her legs, and allowed herself to be entered by a stranger. His name was Connor Brannick, and the few seconds he spent inside her would eventually cost him his life.
Here, now, Paula drops her head to her hands. She can hear her husband mowing the lawn in the back garden. Wishes she were out there, too. Perhaps sitting on a blanket. Drinking wine. She can't go out there now. Can't even look at the fish pond.
She knows that this cannot go on. That soon her husband will find the time to investigate properly the continued death of his expensive carp. Will drain the pond. Will find Connor's body, stones wedged into his motorbike leathers, decomposing on the plastic bottom of the deep pool . . .
Her husband does not know she has killed. But he knows something has changed. Knows that she is lying. Has asked, more than he should have done, when the motorcycle in his garage will be going back to the "friend" for whom she claims to be looking after it.
She knows, too, that the big Scottish sergeant is getting closer. That Hepburn's phone call to his superiors has done nothing but convince him there is something to investigate.
Knows that, far more than her husband, it is her lover, Stephen Hepburn, who is closest to making the accusation. To asking her whether she has killed two people, and is trying, so damn hard, to do it again.
She holds the mask to her face. Stares out through its eyes. Smells the stale sweat and shivers as she remembers the moment she first closed it over her countenance.
Her phone bleeps.
Behind the mask, she gives the faintest of smiles.
• • •
THEY DIDN'T THINK she was going to reply at first. Sat for hours watching Suzie's phone and waiting for a message back. It came around six p.m., as Pharaoh and McAvoy were sitting at the laptop in his kitchen, eating ham-and-mustard sandwiches and filling in the gaps in their murderer's life.
Paula Tressider was born in 1959. Nice middle-class Manchester family. Two sisters. Arty mum and businessman dad. Started university and met the man who would be her first husband. Played at being a political activist but seems to have been more about the outfits than the cause. Married the history student at twenty-two and divorced him a year later. Took a job in a boutique in Leeds. Became the manager. Met Peter Tressider. Married in 1989. Became the good wife. Started appearing on the boards of his various businesses. Turning up on his arm at political functions. Moved to East Yorkshire and opened two fashion houses. Gave talks to the Women's Institute about the importance of a stable family unit. Started wearing twinsets and pearls. Joined the board of governors at a local school. Joined the Conservative Party. Became a pillar of the community and the cardboard cutout of a politician's wife.
"She must have been terrified," says McAvoy, softly.
"Don't start that," says Pharaoh between mouthfuls. "You go soft on me, I'll kick your teeth in."
"Sorry, guv. Just, I thought we were after somebody who was doing this for kicks. She just wanted her secrets to stay hidden."
Pharaoh shrugs. "She got her kicks a different way. Killed to cover it up."
"You think he knows?"
"Her husband? No. If he does, he turned a blind eye."
McAvoy looks again at the website. Paula Tressider, pictured in hundreds of pounds' worth of designer gear, wearing a forced smile for the camera, shaking hands with the prime minister at a Conservative fund-raiser a year before.
"When she saw their tattoos in the magazine . . ."
"Yeah."
The phone buzzes. They each take a breath, before McAvoy reads the text aloud.
"'Think it's time we finished our game. You're on.' _"_
His smile contains no mirth. Just a relief that his own message, carefully constructed with Suzie's help, has been received and accepted.
Roisin gives Suzie a cuddle as they head out of the door. She does not know what is happening, but her new friend seems trembly and scared.
"He'll take care of you," she says, gesturing at her husband.
"I know."
McAvoy bends down and gives Lilah a tickle. Bumps fists with Fin, who is sitting in front of the TV, eating pasta and pesto with sliced-up hot dogs.
"They called," whispers Roisin in his ear as he stands.
He turns to her. Looks quizzical.
"Noye," she says. "The new campsite. Anlaby. He wants you there."
McAvoy's face contorts. He wonders if any more burdens will be laid upon his broad shoulders tonight.
"I'm a policeman."
She makes sure he is looking straight at her as she replies, "You're a man."
He does not speak again. Just quietly closes the door as he leaves. Does not turn up his collar or lower his head as he walks through the pounding rain. Opens the car door and climbs inside. Starts the engine and finds something soothing on a classical station.
Watches the lights come on.
Checks his radio and gives a nod.
Suzie leads the way, her tiny Fiat at the head of this three-car convoy. She squints through the rain and the gathering gloom, wincing at the distorted headlights of the cars in front and behind, aware that the only reason she is not shaking her legs is because she does not want to stall the car.
On the passenger seat, the phone Pharaoh gave her bleeps. She reads the message. It has been forwarded from her own phone.
Need to Taste Your Skin. Don't be Late.
She closes her eyes for as long as she dares while driving. Instinctively looks across to the passenger seat. Wonders if she can really feel Simon's presence or just wants to.
The journey takes more than an hour in the slow-moving traffic. Twice she fears she has lost McAvoy and Pharaoh, but whenever she prepares to park up and wait for them, her phone flashes to tell her they can see her. That she is not alone.
It is easier when she hits the motorway. She sticks at a steady seventy in the inside lane. Tries to find comfort in the sound of the wet tires on the road. Concentrates on her breathing. Half wishes she had let Roisin petition her husband into being allowed to come, too.
She has been to this hotel before. It sits off the motorway, three miles from Goole. She sat in the car park for two hours while Simon entertained a man he'd met on the website. She had done some drawing and eaten a McChicken sandwich. Simon had enjoyed his afternoon. Said the man was grateful and kind.
Suzie parks. She wants to look at the other cars in the dark, wet car park. Wants to see if her murderer is already there. Does not let herself. Climbs out of the vehicle and, straight-backed, face upturned, walks through the puddles and into the hotel.
"Can I help you?"
The man at reception is younger than she is. He looks bored, and his shirt is too big for his skinny frame.
"I have a room booked."
She gives her name. Tries to keep calm as he fiddles with the machine and then finally hands her a key. He looks her over, as if appraising livestock. Even has the temerity to nod.
"Second floor," he says.
She takes the stairs. Cannot bear the thought that the lift may be mirrored. Does not want to see herself.
Balling her fists, clenching her jaw, she finds the room. Slides the pass card into the lock and pushes open the door. Switches on the light and looks round the dark, characterless room, her heart thudding painfully against her broken ribs.
Another message on her phone, this time from McAvoy.
Be strong. I'm here.
She undresses. Peels off her borrowed shirt and leggings. Tries to rub the creases out of her imperfect skin. Takes the length of cord from her handbag and wedges the door open with a flip-flop.
Slowly, as if every moment pains her, and each breath is a countdown, she moves to the bed. Lies facedown and naked. Feels the cool blankets against her warm skin. Grips the phone tight. Texts her killer.
I'm ready.
Time slows. Suzie does not know how long she has been here. Her mind drifts. She could not say with any certainty that she has not fallen asleep in the time she has been lying here, in this beige room, with its white sheets and thin mattress.
Just knows that this was how Simon died. And that by lying here, like this, she is helping to catch a killer.
There is no prelude to the attack. She hears nothing. No creak of floorboard or clever threat.
One moment she is lying facedown on the hotel bed. The next there is a pressure upon her back, and the cord she has draped so invitingly across her buttocks is tight around her neck.
She gasps. Fights. Thrashes like an animal. But the weight upon her shoulders is too great. The hands too rough. It is the same weight that pinned her to the grass two nights ago and was the last thing her friend felt as he died. It hurts.
Her mouth opens. The tendons in her neck feel, for a moment, to be snapping like a fistful of twigs.
And then it is gone. She is facedown on the bed. Her face is on the pillow. The tears upon her cheeks are soaking into the mattress. Warm, tender hands are upon her.
She turns. Manages to wriggle onto her back. Whips her head this way and that. Looks at the devastation of the room. The smashed TV. The spilled kettle and cups. The door that hangs from only one hinge.
"Suzie."
Shirt torn, bleeding from the nose, McAvoy is standing in the doorway. He gives her a bone-weary smile.
"You okay?"
"Did you get her?"
"Are you okay, Suzie?"
She nods. Breathes, deep and slow. "Please . . ."
"We've got them both, Suzie. It's a mess . . ."
JULY, IT WAS. Evening. A Sunday. Halfway through some costume drama on BBC One. A bottle of wine already drained and gravy-streaked dinner plates daringly abandoned on the coffee table.
That was when Paula Tressider took the call that made her a killer.
Four and a half rings, then a weary hello: warm plastic receiver against fleshy cheek.
TV on pause and a shared look of exasperation . . .
Five whole seconds of silence, then a male voice. She didn't recognize it at first. Had not heard it say very much the only time they met. Just a few grunts and a thank-you.
Clipped West Yorkshire tones . . .
"You might not remember me. I remember you. Huddersfield, it was. Some enchanted evening. Does your sweetheart know how you spend your evenings?"
Thirty seconds more.
No words, save his breathing.
"I think you have the wrong number . . ."
The laugh. The snuffling, nasal sniggering.
"No, I've got you. Was a surprise, like. Didn't think somebody in your position would be in the phone book. Then again, I didn't think somebody in your position would do the things I saw you do . . ."
Cold fear in a churning belly.
White roses blossoming on red flesh.
"I'm sorry, would there be a better time for you to ring to discuss this more fully? Perhaps if you left me your number, I could contact you at a more convenient time . . . ?"
No laughter this time. Just ice in his voice.
"I'll call you. I'll call again, and again, and then I'll call somebody else. I'll tell. I'm sorry to be doing this. I really am."
A moment's consideration. Eyes closed, hiding from it all. Memories folding inward over everything else, like petals at dusk.
"Tomorrow. Call me tomorrow."
A wordless nod.
_Click._
A week without food. Of hands trembling and broken sleep. Needing a piss every thirty bloody seconds. Throwing up the wine that brought such feeble relief. Snapping at every gentle inquiry. Swearing at every question over health and happiness . . .
He called back, of course. The other man.
Midday, it was. Friday. Sweetheart at the shops.
Alone in the house. Glass knocking against the receiver; clutched in shaking hands.
"You ready to talk now?"
A nod. Then a deeper, more assertive reply.
"I've done nothing to be ashamed of."
His smile.
"No? That's fine, then. I'd probably get plenty of money from the papers. I'm only calling you out of courtesy."
"That's what you want, is it? Money? What makes you think I'd pay? Or that I could?"
Scorn, then. A note of uncertainty? A diversion from the script . . .
"You've got more than me. Everybody's got more than me."
Resisting feebly. Trying to talk him round.
"How do you know it's a secret? People who know me already know what I like . . ."
"Bollocks. I remember. You told me a dozen bloody times what a risk you were taking. You were slumming it. Roughing it. Playing away, too close to home. You don't want this coming out. I read the papers. I know what they're saying about your future. You're a big deal. And I saw you on top of that pretty little girl with the blossoms, loving every minute of it. Even me, when you let me join in . . ."
Tears. A coughing fit that became puke.
"How much do you want . . . ?"
_Click._
Life became timeless. Hours became a shapeless, colorless mass.
Days, nights, all spreading out from that one moment, in the early hours, when her mind was made up.
Broken by tiredness, racked by fear.
Acquiescing to will.
Weighing payment with risk.
Nodding in the dark. Eyes fixed on the ceiling. Tears on cheeks.
Paula Tressider acknowledged what must be done. Decided they all had to die.
• • •
HERE, NOW, in the interview room at Courtland Road Police Station, with the rain thundering down outside and their breaths forming into ragged strips on the cold, damp air, Paula Tressider sobs.
McAvoy prompts gently. Nudges along her confession. Tries to pretend they already know what she is giving up so freely . . .
"I said I would pay," she says, her voice muffled as she drops her face into her fleshy palms. "Told him to come."
McAvoy leans forward. "You need to say his name for the tape."
"Connor," she says, choking on the word. "Connor Brannick."
It means nothing to him. He tries not to show it. Feels the breeze as Pharaoh leaves the room, name scrawled on her palm in ballpoint.
Paula is too caught up to notice the sudden absence. Just keeps talking. Sniffing. Wiping away tears with the heel of her hand.
"He came on his motorbike. End of summer. Hot day, I remember that. I could barely keep my hands still. It was real then. Him on the front drive, in his helmet and leathers, asking me if he could put the bike in the garage so the sap from the trees didn't drip on the paintwork . . ."
"Go on."
"He was different from how he'd been on the phone. Embarrassed, even. When he took his helmet off, he looked like he was about to cry. Was talking and talking. Said the house was lovely—that his wife would like it. He seemed sorry to be asking me for money. Tried to justify himself by telling me he was struggling. Said he would never do this if he could just get work."
"Go on, Mrs. Tressider."
"He was talking so fast. Just gabbling on. He was as nervous as I was. I told him to come through to the back garden, be a bit more discreet. He followed. Saw the pond. Started saying how he could fit lights in it. Saying what a good electrician he was. Would look lovely lit from underneath. I was hardly listening. Managed to tell him the money was in the gazebo. I went to get it."
"And then what, Paula?"
"I came back with the hammer."
Pharaoh reenters the room. Slides a warm piece of paper in front of McAvoy.
PARTNER OF MISSING MAN SAYS SHE HAS NOT GIVEN UP HOPE
The common-law wife of a Morley electrician missing for almost eight months has made a renewed appeal for him to come home.
43-year-old Connor Brannick vanished last September. He told his partner, 39-year-old Gwen Simmons, that he was going to price up a new job of work, but never returned home.
Ms. Simmons, the mother of his four-year-old son, Andrew, waited several days before contacting police, as she said it was not unlike him to go away for several days at a time for work.
But as time went on she began to worry, and calls to his mobile phone went unanswered.
Today she told the _Huddersfield Examiner_ that after he vanished, she discovered that he had been hiding major financial problems.
She said, "I just wish he'd spoken to me about it. I know everybody's saying that he's done himself in, or just run off and left me to it, but I have to cling to the hope that he's okay and will come home.
"Our son keeps asking where Daddy is. I need him here. It was never about the money. I wish he'd told me how deep a mess we were in. I don't know what to do next or where to turn. I just want him home."
Mr. Brannick's motorcycle, which he was riding when he left the family home, is also missing. Anybody with information should call West Yorkshire Police . . .
McAvoy looks up. "His body?"
Paula raises her head long enough to glare at him, then the flicker of defiance is gone. She looks away. "In the pond."
"You smashed his skull in?"
Paula nods.
"For the tape, please?"
"Yes."
For a moment there is silence in the room. Then McAvoy says the name that has brought them here. "Simon Appleyard."
Paula turns to the gray-suited solicitor who sits to her left, and who has done nothing but polish his glasses on his tie since she told him to shut up and let her speak.
"The magazine."
McAvoy nods. "The _Journal_. The advert."
"They were both there. Him and her. The boy Stephen found for us and the girl he brought. Their tattoos, mocking me, like she did that night. The night she made me take the mask off . . ."
McAvoy licks his teeth. "Mrs. Tressider, do you really believe that either Simon Appleyard or Suzie would ever have tried to blackmail you? Do you think that even if you became the prime minister's wife, they would have any notion of who you were, or try and use that to their advantage? Not everybody is like that."
For the first time, Paula meets his gaze. "Don't tell me about people. I know what people are. I know what's under the skin. It's not pretty. It's base and it's desperate, and it takes what it wants . . ."
It is Trish Pharaoh who stops her short, slamming her palm down on the desk.
"Did you kill Simon Appleyard?"
She holds Pharaoh's gaze. "Yes."
"And you are responsible for the attack on Georgie-Lee Suthers? On the boy at the swingers party? Repeated attacks on Suzie Devlin?"
"Yes."
Pharaoh breathes out. Looks the burly, disheveled politician's wife up and down. "It's always the quiet ones."
AS HE WALKS across the car park in the teeming rain, tired to his bones, aching to his soul, McAvoy considers desire. Wonders at the nature of lust. Pictures Simon Appleyard, naked and holding his own noose, waiting for the stranger who would kill him where he lay. Considers Suzie: still visiting darkened rest stops and opening herself for strangers, even as the bruises burned on her skin.
Thinks of Paula Tressider.
They'd always gone farther afield, she'd told them in her interview. She and Stephen. Had crossed the country to find playmates. She thought Huddersfield was too close to home. Only eighty miles away. Too risky. But she'd been excited. Liberated. Daring. Had thrown herself into the evening and had fun with the young couple with the tattoos. And then the magazine had arrived. The one she'd been so proud of. The photo shoot of her beautiful home. The picture of her and Peter. Her fingers locked around her husband's. Every inch the politician's wife. And there, mocking her, in the adverts at the back: skin she had tasted and which had touched her own. She didn't know when she'd decided to commit murder. Just knew that she had to make sure they could never talk. Knew only that the boy was into being dominated and liked certain websites. Knew, more than anything, he liked to please. Liked words. Liked peacocks. Could be found, and could be persuaded, to contribute to his own death.
"Aector."
He turns, one hand on the door handle, rain slicking his hair to his face, soaking his already sodden clothes.
Pharaoh runs across the puddle-filled car park. She has her jacket over her head.
"Suzie," she says, and it is a question.
"She's okay," he says. "Roisin's making a fuss of her. Sounds like they're having a sleepover."
"She's at your house?"
"No. Roisin's at hers."
"The kids."
"There, too."
"And where are you going?"
McAvoy looks at her for a while. Watches the rain run down her face. Sees the black mascara pool in her eyes. Sees himself on her pupils.
"What was all this for?" he asks, raising his voice above the noise of the rain.
Pharaoh gives him an encouraging smile. "We've got a confession. We've solved a murder."
"Nobody knew it was a murder."
"Does that matter? You knew."
"Why did I do this?" he asks, and she cannot tell if it is rain or tears that spills down his face.
" _'_ Cause you're one of the good ones."
He shakes his head. "I don't feel any better for it."
"Is that what you thought would happen?"
"I don't know. I feel worse."
"Oh, Aector, it's not your fault. You're the one that did things right. She'd have killed Suzie, you know that. Tressider had a weak heart. He kept it secret. Thought it would ruin his political chances. You heard her. That's why she turned to Hepburn."
The affair had started a little more than a year ago. Her husband had introduced them at a civic function. Mentioned they had done a little business together, and then left them to share a quiet corner and glasses of tasteless white wine. Hepburn had been vibrant, larger than life. Exciting. Flirty. She had thought he was gay until he asked to smell her perfume and then licked her jawline all the way to her mouth. Had found herself a secret life she did not know she wanted, and to which she became addicted.
For a moment McAvoy and Pharaoh could do nothing but look at each other, trying to see a better kind of sense in the other's approach to crime. They turn as they hear a car door slam, a soft, mechanical sound barely audible over the rain.
They peer over at the side street that leads to the front entrance of the police station. Dressed in a T-shirt that clings to his skin and a pair of nondescript trousers, they almost do not recognize Stephen Hepburn. His shoulders are slumped. He is looking back over his shoulder at the car he has parked haphazardly on a curb.
He pauses now. Standing by the barrier that blocks the entrance to the staff car park.
"News travels fast," says Pharaoh.
McAvoy is already moving toward the distant figure. He does not bow his head in the face of the gale. Takes the cold and the rain upon his face without complaint or evasion.
Hepburn sees him coming. Straightens himself. Pulls the damp material from his skin. Pushes a hand through his hair and then drops his arms to his side. They hang there, awkward and limp.
"Is it true?"
Hepburn's voice is a tremble.
McAvoy stands in front of him, saying nothing. Looks the smaller man up and down. Tries to place this ragged, unremarkable man at the scene of so many stories. Imagines him, for the briefest of moments, rutting with Paula Tressider on the Welton hillside. Imagines him at his keyboard, persuading sexed-up strangers to meet them in hotel rooms and car parks. Remembers the arrogant, cocksure man who curled his lip in halfhearted contempt when McAvoy told him he was investigating a murder.
"Please," says Hepburn. "Paula. Did she kill the boy? The one from the party?"
McAvoy looks him up and down. Stares into eyes filled with bewilderment and tears.
"You really didn't know," he says, and it is not a question. "You just didn't give a damn, did you?"
Hepburn opens his mouth to speak.
McAvoy silences him with a shake of his head.
"She didn't just kill _him_ , Councillor. The man who joined you that night in Huddersfield. She did him in, too. She killed anybody who might tell about the time she took her mask off."
A stream of water falls from the end of Hepburn's nose.
"I didn't know," he begins, before protestation gives way to self-preservation. "I had nothing to do with it . . ."
McAvoy works his jaw in a slow circle, then clenches his teeth. "You won't be charged," he says quietly. "I don't know what we could make stick. I don't know if there's even a charge for what you've done. I don't even know how I feel about you. I don't know if you've done anything wrong. I just know you've got away with something and I hope you never forget that."
Hepburn looks up into McAvoy's brown eyes. Sees himself mirrored, fuzzy and indistinct. A dark, shadowy thing, blurred by the storm.
"I just wanted to play."
McAvoy walks away. Is grateful that the rain running into his mouth tastes so foul. It gives him a legitimate reason to spit.
"All of it," says McAvoy quietly, as he returns to his car to find Pharaoh waiting, soaked to the skin. "Trying to run Suzie over. Attacking her at the party. Smashing the lad over the head and trying to throttle her. She did all of it just to make sure nobody told."
"I think she got a taste for it," says Pharaoh, choosing not to ask him what he said to Hepburn. "I think she started pushing the boundaries. Maybe this became another game. I don't believe that shit about burying the past."
"What's going to happen, d'you think?" He asks the question cautiously, as if walking on breaking ice. "Politically, I mean."
Pharaoh purses her lips. The bandage on her neck is sodden with rain and she reaches up to smooth it back down. "I think we'll be okay. We did it with kid gloves, didn't we? Kept it off the books. Had a look and then got a result."
"We should go and see him," says McAvoy. "Now. He'll want to talk."
Pharaoh holds his gaze. "You think he knew?"
McAvoy nods. "I think he threw the phone where I would bloody find it. I think I was cheaper than hiring a private detective."
11:14 P.M. HULL ROYAL INFIRMARY.
PETER TRESSIDER is sitting up, limply, in his hospital bed, wearing borrowed surgical scrubs. He appears shrunken. Diminished. Small. There is a clump of hair missing from the side of his head, and red skin glares from beneath the coarse hair at his throat.
As he enters the private room, McAvoy is put in mind of a skinned bear. The image flashes through his mind unbidden. His thoughts are filled with raw, pink flesh and bloodied fur. He sees the man in the hospital bed as a beast, hunted, wounded, mutilated inside and out.
"Councillor."
Tressider opens his eyes. Looks at his visitors. At Aector McAvoy, soaked to the skin and expressionless in his gaze. At Trish Pharaoh behind him, makeup on her cheeks, rain on her chest.
"I won't be hearing that again," he says softly.
Pharaoh closes the door behind them. Sits down on a hard-backed plastic chair. McAvoy doesn't move. Just holds Tressider's stare.
"You followed her," Pharaoh says at length. "Tonight."
Tressider swallows. Looks away.
McAvoy steps to the side, putting himself back inTressider's eye line.
"How long have you known?"
Tressider lifts himself up a little. Rubs his hands in his beard. Lets his fingers fall to his throat.
"You nearly broke my windpipe," he says, and coughs as he does so. "When you dragged me off. You're stronger than you look. And you look bloody strong. Scalped me, too. Pulled half my bloody hair out by the roots."
McAvoy doesn't speak. There is silence in the room, save the rustle of Pharaoh pushing her hair back from her face, and recrossing her legs.
"How long have you known, Councillor? We can do this properly, if you'd prefer. Take you to the station. You can call your brief. There may be photographers at the station, though . . ."
Weakly, as if he is past caring about such things, Tressider waves his hand. "You think I care about all that? You think I ever cared?"
McAvoy says nothing. Waits for the other man to fill the silence.
Tressider screws up his eyes. Talks to the image that is playing in his imagination.
"You're asking if I knew," he says, licking his lips. "I'm asking myself, too."
McAvoy leans forward. Peers into the councillor's face like a pathologist examining a corpse.
"Speak to me, sir." He says it softly. "So we understand her. So we understand Paula. So we understand the woman you loved."
Tressider's eyes lock on McAvoy's. Both see the other, reflected in their pupils.
He breathes out, and there is a sickliness to the sound. A weariness. An approach of the grave.
At length he reaches to his bedside and takes a sip of water. Savors it.
"I knew when we got married that she was full of life," he says, staring up at the ceiling with its gray tiles and garish lights. "I knew she had fire in her. Did I know she was playing around? Not at first, no. I didn't think that way. We were happy. Whatever we had, it worked. She seemed to love me, I know that much. Seemed to enjoy our lives . . ."
"But you began to suspect?"
Eyes still closed, Tressider nods. "She got a second mobile phone. The first one we got through the business. Claimed the tax back on her business calls, you see. All aboveboard. But when you live with somebody, you can't hide everything, can you? I saw it in her handbag. Knew she had hidden it from me. You can't help but think the worst, can you?"
"Did you confront her?"
Tressider swallows painfully. Shakes his head. "I don't know if I wanted the truth. Not really. Not then."
"What happened, Councillor?"
Tressider opens his eyes. There are no tears, but his face is pale and drawn, his lips gray. He is a pencil sketch of himself.
"I tried to stop thinking about it," he said. "Told myself that whatever we had, it worked. Tried to be a modern man, I guess. When she got pally with Steve Hepburn, I told her to go for it. To enjoy herself. I thought having a flamboyant, gay friend like that would appeal to whatever part of her I wasn't satisfying. They hit it off. She even suggested we put some money into one of his businesses . . ."
"The phone, Councillor. Simon Appleyard."
Tressider smoothes down the front of his pajamas. Presses his lips tight together.
"She got a phone call. A few months ago. We were sitting at home and she answered a call on our home phone. Came back white as a sheet. Wouldn't speak. Wouldn't tell me anything. I tried to cheer her up, but I knew something was wrong."
"The blackmailer," says McAvoy, turning to Pharaoh. "Connor."
"She was weird for days. Told me she was just feeling under the weather. Told me to leave it to her and to concentrate on work. On council work. The authority. Getting into the good books with the party . . ."
Tressider's bottom lip shakes. He bites it, willing himself to be strong.
"What is going to happen to her?" he asks.
McAvoy rubs his hands through his hair. Picks the damp material of his trousers from his legs.
"She's going to be charged with murder, Councillor Tressider."
Tressider swallows again. Says nothing for a full ten seconds.
"The pond," he says at last. "Connor's in the pond."
McAvoy turns to Pharaoh. Back to the man in the bed.
"You put him there?"
He gives a shake of his head. "I found him there. Staring up at me. Eyes like headlights . . ."
McAvoy needs to move. Has held himself still too long. He crosses to the window and stares, through his own reflection, at the lights of the city. At the yellows and blues that flicker and glare in the darkness and the rain.
"You were a member of the Police Authority and didn't think to call the police?"
He feels Tressider's eyes upon him. Refuses to turn.
"I knew," he says flatly. "Knew what she had done."
Pharaoh clears her throat. "You didn't confront her?"
"I wanted to," says Tressider, and his voice is almost a wail. "But she seemed so happy suddenly. Gleeful. Bouncing, almost. I kept telling myself we'd just have a few days like that, and then we would talk. But days became weeks. We were happy."
McAvoy turns from the window, face red. "There was a dead man in your pool, Councillor! You must have needed to know."
Tressider wipes his nose with the back of his hand. "I tried to forget . . ."
"And then the happiness stopped," says McAvoy. "She started behaving strangely again."
"It was that damn magazine," he spits. "She was so bloody proud of us. Kept flicking through it. Loved the thought of what we were going to become. And then she changed. Became cold. Stopped talking . . ."
"The phone," says Pharaoh. "Where did you find it?"
Tressider turns to her. Tries to scowl, but lacks the strength. "I followed her," he says, breaking eye contact. "One Sunday, a month before Christmas. She'd come home in a taxi. Said the car had broken down and needed towing near Anlaby. She had no reason to be there. What was she doing? I couldn't stand it. She said she needed to clear her head and went out again almost as soon as she got in."
"Where did she go?"
"The dale," says Tressider, lost in memory. "Top of Welton. Pretty place."
McAvoy nods. He knows the area. Steep-sided and tree-lined, and scented with bluebells and cow parsley, fresh air and dirt.
"I saw her bury something," Tressider says. "Pulling up clumps of dirt with her bare hands. She was crying. I wanted to hold her . . ."
"But you wanted to know what she was doing."
Tressider falls silent.
"You dug it up," says Pharaoh.
"Not at first," says Tressider, as if that's important. "I tried to stop myself. Tried to tell myself that it was all over. Waited for her to get happy again, like before. But she didn't. She was colder than ever. Always on the computer, out at all hours."
"You went back."
Tressider nods. "I dug up what she had buried. Dug up the phone."
"It was broken," says McAvoy. "You couldn't make it work. You couldn't get answers."
"I tried," says Tressider, and his hands make fists around the bedclothes. "But I didn't know what to do . . ."
"You asked Hepburn," says McAvoy. "That day. The first meeting of the Police Authority. You tried to get answers. Wanted to know what he knew . . ."
"He told me they had been having an affair," he says, and snatches away a tear. "He didn't hide it. Said he wasn't as gay as people thought. Said he was sorry. Said he hoped we could all move on."
Pharaoh pulls herself out of the chair. Crosses to his bedside.
"And you decided that you could," she says icily. "You decided you could live with the body in the pool. You could forgive her whatever she had done. And you threw the thing in the river."
Tressider turns away from her. Stares at McAvoy.
"I saw you," he says softly. "Had this glimpse of what I thought the police should be. I suppose I trusted in fate . . ."
McAvoy scoffs openly. Sneers with contempt. "Did you want me to find it?" he asks, making fists. "Did you leave it for me? Was I your fucking errand boy?"
Tressider looks down at himself. Gives a half laugh as he takes in the sight he presents.
"I don't know."
McAvoy spins back to the window. Presses his head against the cool glass.
"Tonight," he says, and his breath fogs the pane. "You followed her to the hotel."
In the reflection he sees Tressider nod.
"You read her phone."
Another nod.
"You thought she was meeting another man."
Softly: "Yes."
McAvoy licks his lips, lets his eyelids close. He is suddenly exhausted.
"You're done," says Pharaoh behind him, and though she addresses her comments to the councillor, it is McAvoy who feels the sting of her words.
His thoughts turn to love. To utter, blinding devotion. To Roisin. He asks himself how much he could forgive. How much he could tolerate. How much pain he would endure to make her love him, and never leave.
He turns away from the window. Stares into Tressider's eyes.
"She loved you best," he says softly. "All the things she did were to protect you from people finding out that she had strayed. She wanted you to get all you ever wanted."
Pharaoh looks at him quizzically, taken aback by the sudden gesture of compassion.
"Yeah," she says scornfully. "She was all fucking heart."
McAvoy holds Tressider's gaze. Breathes out slowly and pulls open the door. Gives a nod to the uniformed constable in the corridor, and stomps, damply, down the hall. He takes the stairs, two at a time. Crashes across the reception area and bursts into the storm beyond the glass.
"McAvoy!"
He doesn't turn, but the sound of Pharaoh's boots on the linoleum is unmistakable, and she pulls him back by the arm.
"You don't want the arrest?"
McAvoy twitches his mouth into a ghost of a smile.
"I don't know what I want."
Pharaoh opens her mouth. Her tongue flicks out and glosses her full, red lips. She puts a hand on his arm and squeezes, never taking her eyes from his.
"You did good, Aector."
McAvoy looks away. Shrugs. Begins to walk away.
"Where are you going?"
He answers with one word.
"Roisin."
He doesn't hear her move. Can picture her standing there, watching him get smaller.
Wonders what she will read into his answer.
Whether she knows he is on his way to a fistfight with a gangster.
11:18 P.M. ANLABY PLAYING FIELD.
COLIN RAY is pressed against the damp brick of the changing rooms, tucked into the blackest pocket of shadow that he can find. He is soaked to the bone. His suit is clinging to his gangly frame and every few seconds he shivers, sending a fresh mist of rain from the brim of his borrowed black baseball cap.
"Anything?"
Shaz Archer's voice comes from between locked teeth. She is behind him, better concealed in the doorway of the outbuilding.
"Still just talking."
The travelers are gone. The caravans and the horses, the furniture and four-by-fours disappeared sometime this afternoon. Anybody who saw them go is keeping quiet about it.
They are not why the two police officers are here. They are after the men who sit in the nearby Lexus, parked up on a patch of rutted, rain-lashed gravel a hundred yards away from where they shiver in the sodden clothes and wait.
Ray is in a foul mood. Already news has filtered through that McAvoy and Pharaoh have brought in a murderer he did not even know they were seeking. Already there is a chance that the collar he makes tonight will not be the most eye-catching of the day.
"Col, are we sure . . . ?"
Ray holds up a hand to shush her. A car is pulling up, nosing in past the wooden fence and coming to a halt around thirty yards from the Lexus.
"Bloody gangsters are on hard times these days," muses Ray, squinting through the rain, trying to focus on the figure climbing out of the little hatchback.
He feels Archer beside him, unable to keep herself quiet.
"Is that . . . ?"
Ray nods. "McAvoy."
They watch silently as the bulky Scotsman walks sure-footedly across the car park to the Lexus. See him tap on the blackened glass.
"Col, what's he doing?"
McAvoy is stepping back. Taking off his coat. Folding it up and laying it on the roof of the big posh car.
"Oh, Christ, that's what he meant . . ."
Suddenly Ray remembers Alan Rourke's words. What he said about Noye's need for respect. His intention to do harm to the copper who hurt his godson.
"Is he on his own?"
Ray doesn't answer. Just watches as the doors open on the Lexus. Watches four men climb out.
"Your eyes are better than mine," he whispers, and grabs Archer by the pocket of her soaking denim jacket. Pulls her close. "Tell me."
"Ronan," she says softly. "Noye."
"Fucking hell."
Wordlessly, they watch McAvoy back up. Watch one figure break off from the advancing quartet. Take the lead.
Ray raises the radio. "Are you getting this?"
"Sir."
"Hold positions."
Through the veil of rain, the lead figure becomes Giuseppe Noye. Becomes a thickset, burly, middle-aged man in jacket and jeans.
He is talking to McAvoy. Leaning in. Face-to-face. Pressing a finger in the bigger man's chest.
"Col, he's going to get himself killed . . ."
Ray is not being malicious in his stalling. It is pragmatism that keeps him here in the dark. He sees an opportunity for an arrest. Sees a distraction better than any he could have planned.
"Sir."
He looks down in irritation as the radio crackles in the dark. He raises his head again.
"Let them play . . ."
• • •
"THIS ISN'T YOUR FIGHT, NOYE. He's lying to you."
McAvoy says it again. Shouts it loud enough for Ronan to hear. Sees the ginger teen flick a V sign, bookended by two leather-jacketed thugs.
"You're going to get broken, copper. Broken."
McAvoy tries to keep his feet as the smaller man comes forward, swinging brisk, powerful blows at his head. He absorbs them with his forearms, jabs, and moves away, fending off this bare-knuckle fighter with speed and agility. He tries to make this a boxing match. Something vaguely noble. Remembers his bouts at university; all head guards, mouth guards, vests, and padded shorts. The experience helps him little now, on this patch of rutted concrete, lit by strips of moon, and fighting a man whose hand wraps are stained almost black from the blood he has spilled in countless similar bouts.
"Close in now, lads, close in."
McAvoy doesn't know the referee. He's a short, slightly built man in his middle years, who is exposing little of his face between the collar of his sheepskin coat and the peak of his flat cap. He had given them a brief rundown of what passed for the rules, and told Noye he wanted none of his usual bollocks. Had asked McAvoy whether he had anybody to stand beside him, and given a shake of his head in response to McAvoy's.
"I'll leave you bleeding, boy," says Noye. "I'll leave you crying for your ma."
Noye's words are whispered promises, softly snarled as McAvoy tries to gather him in and hold him, to test his strength and sap his energy.
"He's lying to you," says McAvoy in Noye's ear, as a short left-handed blow thuds into his ribs. "Your godson. He's a liar. He's turned his back on all of you. And now you're fighting his battles . . ."
Another blow connects with his body, and this one hurts him. He winces and Noye scents victory. He swings a hard right hand and catches McAvoy behind the ear. Follows up with a blow to his chin, thumped home with the heel of his hand.
McAvoy's vision blurs. He hears high-pitched song, then static.
He is down to one knee. Raising a hand. Trying to block the blows that rain down upon him.
The referee pulls Noye back before he can deliver a boot to his fallen opponent. There are rules here. A code. No kicking. No punching when on the ground. No biting, unless the opponent is trying to rip your tonsils out. Everything else is fair game.
McAvoy pulls himself up, groggy, disoriented. Strong arms push him forward into another flurry of punches. He brings his hands up. Takes the impact on his forearms. Tries to grab the smaller man as if they are on the ropes of a boxing ring.
Hard, thudding right hands pound into his ribs. The air leaves his body. The fight leaves his legs . . .
• • •
COLIN RAY lifts the radio. Prepares to give the order to move in.
McAvoy is still upright. Refusing to go down. Refusing to do much more than make himself a target.
"Fight, you jock bastard," says Ray under his breath. "Take his fucking head off."
• • •
NOYE BACKS AWAY, looking at the other men, as if unable to understand. The orders he receives in their glances and nods are unmistakable. _Finish it._
He moves back in, arms by his sides, preparing to swing upward from the floor at McAvoy's exposed jaw.
McAvoy sees it coming. Sees the scarred, cracked knuckles coming straight up to smash beneath his chin.
He lashes out. A straight right. His fist crunches into Noye's.
It is the gypsy who yelps, a high, effeminate squeal, like a pained cat.
And now McAvoy is moving forward. He is moving as a boxer, feet balanced, hands raised.
He throws a left that snaps Noye's head backward. Another that staggers him. Hurls a right that would have taken his head off had he not pulled it at the very last instant . . .
• • •
"GO ON, SON . . ."
Ray watches, openmouthed, as McAvoy hurls himself forward and bodily picks up Noye by the waist. Charges across the car park with the other man in his embrace and slams him into the side of the Lexus with enough force to buckle the doors.
"Go, go, go . . ."
Ray has seen enough. Enjoyed every fucking second of it.
As Ray and Archer run across the car park, they see the three other figures fall upon McAvoy. Begin thumping elbows, fists, knees into his big broad back as the unconscious Noye slithers to the ground.
Sirens now. Flashing blue lights and Colin Ray's shouts.
McAvoy, swinging wildly, taking hold of the nearest head and slamming it into the car. Planting a meaty right on the side of a slick, shaved skull.
Chaos.
The two figures that remain upright seem to freeze.
Then McAvoy drops to one knee. And Ronan runs.
"You all right, son?" bellows Ray, above the rain, as he approaches his fallen colleague. Around him, uniformed officers are jumping out of police cars. To his right, Shaz Archer is slipping cuffs on a black-jacketed, shaven-headed man who is lying groggily in a puddle.
McAvoy looks up at him from under a swelling eye. "Sir?"
Ray gives a relieved little burst of laughter. Turns from him and takes over from a uniformed officer who is cuffing the other, larger leather-jacketed man. Giuseppe Noye is being tended to by two officers. In the distance, two constables in fluorescent raincoats are disappearing into the darkness, sprinting after Ronan's vanishing form.
McAvoy takes an offered hand. Hauls himself upright.
Looks around dazedly. At the reassuring sight of men in uniform and villains in cuffs.
"Sir, I'm not sure . . ."
Colin Ray returns to his side. "Noye," he says, nodding at the man on the ground, groaning and clutching his ribs. "You were right, son. Alan Rourke gave him up. Tipped us the wink that he was coming up here on business tonight. We figured it was the Vietnamese . . ."
"Sir?"
"His godson, Ronan. He's working for the new outfit that's outmuscled the Chinks. He's got his own little crew. He's the one who's been giving our crime statistics the battering."
McAvoy presses a hand to his head, trying to take it all in. "The other two?" he asks, gesturing at the other two men who are being manhandled into the back of squad cars.
"Muscle for the new outfit. Ronan's thugs. Nice little sideline in stolen cars before they started putting ladies' hands in pans of boiling oil."
"That was these two?"
"According to Rourke."
"And he'll give evidence?"
"Nope. But Noye will."
McAvoy screws up his eyes. "What?"
"His godson. Ronan. Little shit's gone off the rails. Noye will see the benefits of getting him away from his new friends."
McAvoy seems about to fall to his knees. He steadies himself. Rubs the rainwater from his face and winces as he touches his bruised face.
"He won't give evidence, sir."
Ray smiles and puts a hand on his back. "I've got ways and means, son. Noye's a proud man. He won't like finding out what these two bruisers have been doing to his godson."
"What?"
"He's got quite the temper, has Pepe. And when he finds out his new business partners have been abusing his blue-eyed boy . . ."
McAvoy looks into the long, ratty face of the older man. "Have they, sir?"
"We're not dealing with a genius here, son. We're dealing with a very bad man."
They regard each other for a time. Standing in the rain. Soaked to the bone. McAvoy's blood on both their hands.
"Heard you caught a killer," says Ray eventually.
"She's confessed, yes."
For a time it is just them and the rain. The sound of three men coming around from painful injuries to find themselves in cuffs.
"You really came here to fight?" asks Ray softly.
McAvoy allows himself the ghost of a smile. "I hoped I could talk him out of it."
"Didn't work?"
"Apparently I'm not all that persuasive."
Ray shakes his head. Grins. Looks around him and gives a grudging nod at a fine night's work. Runs, painfully, across to Shaz Archer, and pretends he is McAvoy. They giggle as he pretends to pick her up and slam her into the Lexus.
McAvoy stands alone. Closes his eyes and waits for the thumping dizziness to cease.
Lifts his face and lets the rain wash him clean. Sniffs hard, but the only blood he can smell is his own.
Finally, he crosses to the Lexus and retrieves his coat. It is soaked through, but he pulls it on anyway. Removes his phone from the pocket and looks at the message on the screen.
We love you so much. xx
He holds the phone in his hand for a time. Caresses it, as if it is all that keeps him upright.
Smiles to himself, as he realizes that it is.
ACKNOWLEDGMENTS
Much affection and gratitude go the way of David, Phoebe, and Eliza at Blue Rider for their support, their encouragement, and their Herculean levels of tolerance.
Many thanks to the Quercus team who make my books better—Jon, Richard, Lucy, and Ron.
Gratitude, as ever, goes the way of superagent Oil Munson. Cheers for putting up with me.
And here's me raising a glass to Rob L and Babs for stopping me being too, well, me.
I'd also like to give a manly punch on the shoulder to my boy, George, and some form of stomach raspberry to my daughter, Elora. Seriously, I'm nowt without you.
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\section{Introduction}
\label{s:intro}
In the last decade, much effort has been spent towards the development of sound localisation systems. Classical approaches include estimation of the \glspl{tdoa} between microphone pairs using the \gls{gcc}~\cite{Knapp1976, brandstein1997practical}, beamformer based models such as SRP-PHAT~\cite{dibiase2000high}, and spectral estimation-based methods such as the multiple signal classification algorithm (MUSIC) \cite{schmidt1986multiple}.
More recently, localisation systems based on \glspl{dnn} have shown promising performance. In \cite{sun2018indoor}, probabilistic neural networks were used to estimate the \gls{doa} in an indoor environment using \gls{gcc}-based features. A similar scenario was studied in \cite{vesperini2018localizing} which used a \gls{cnn} to predict speaker coordinates. Binaural cues are employed in \cite{MaEtAl2017dnn}, where the \gls{ccf} was used as features in a \gls{dnn} to estimate the azimuth of a sound source with simulated head movement. \gls{cnn} architectures were also used in \cite{chakrabarty2017multi, adavanne2018sound} using frequency-domain features such as the phase or the magnitude of the signal.
All of the approaches so far are based on hand-crafted features explicitly extracted from the waveform. Such a feature extraction process may lead to a loss of information which can affect the performance. Human listeners, on the other hand, are able to use waveforms from just two ears to reliably determine the location of a sound source~\cite{Blauert97}. It is well known that this ability is largely based on both binaural cues, such as the \gls{itd} and the \gls{ild}, and monaural spectral cues created by direction-dependent filtering of the outer ears. However, it is less clear how these cues are seamlessly combined and processed by the auditory cortex for sound localisation~\cite{GrotheEtAl2010}.
Recently, much effort has been spent in the development of end-to-end systems for many audio applications. For example, a model for end-to-end \gls{asr} is proposed in \cite{sainath2017multichannel}, which combines localisation, beamforming, acoustic modelling and speech enhancement in a unified \gls{dnn}. In audio generation, several end-to-end methods were proposed to directly generate waveforms from text~\cite{van2016wavenet, mehri2016samplernn}.
This paper proposes a novel end-to-end approach for sound localisation, referred to as \textit{WaveLoc}.
Instead of an explicit feature extraction stage, the proposed approach uses a \gls{cnn} with a cascade of convolutional layers to implicitly extract features directly from the raw waveform for sound localisation. One of the key stages in the network is the frequency analysis, and two different approaches are investigated. The first approach is auditory-inspired and uses a convolutional layer based on the gammatone filterbank~\cite{PattersonEtAl1992}. The gammatone filter is a widely-used model of auditory frequency analysis, with bandwidths set to reproduce human critical bandwidths~\cite{WangBrown2006}. In the second model, we adopt a standard convolutional layer which is intended to learn how to perform frequency analysis along with the training process of the entire network. After frequency analysis, further convolutional layers with 2-D kernels operates directly on the signals from both ears to extract features that are similar to the binaural cues used by the auditory system. The extracted features are finally concatenated and used as input to a \gls{dnn} with fully connected layers, in order to map them to the corresponding source azimuth.
Our evaluation shows that the proposed WaveLoc systems are able to accurately estimate the azimuth of a sound source in the anechoic condition. However, the performance of the data-driven WaveLoc approach is poor in reverberant conditions when trained only on anechoic signals. This leads to a detailed investigation of the benefits of \gls{mct}, following which we are able to demonstrate robust performance of the wave-based approaches across a range of challenging reverberant conditions.
The rest of the paper is organised as follows. Section~\ref{s:system} presents the proposed end-to-end sound localisation framework, with a focus on two waveform-based approaches that differ in the frequency analysis stage. The experiment setups are described in Section~\ref{s:eval} and results are presented with discussions in Section~\ref{s:results}. Finally, Section~\ref{s:conc} concludes the paper and makes suggestions for future work.
\begin{figure*}[!h]
\centerline{\includegraphics[width=.9\textwidth]{cnn_waveloc_gtf}}
\caption{The proposed end-to-end WaveLoc-GTF system using convolutional neural networks for binaural sound localisation. }
\label{f:system}
\end{figure*}
\section{System Description}
\label{s:system}
\subsection{Overview}
\label{ssec:overview}
The proposed end-to-end sound localisation approach is illustrated in Fig.~\ref{f:system}. The convolutional neural network can be broadly divided into three stages: (i) a frequency analysis stage that takes the framed binaural ear signals as input, (ii) a feature extraction stage with a cascade of convolutional layers to extract suitable features for sound localisation, and (iii) a sound localisation stage based on several dense layers to perform sound localisation as a classification task.
The raw waveforms of the left and right ear signals, as indicated by `L' and `R' in Fig.~\ref{f:system}, are directly used as inputs to the proposed \glspl{cnn}. The ear signals are sampled at 16\,kHz and framed with 20\,ms window size with 10\,ms overlap. In each frame the left and right channels are stacked together to form an input matrix of size $2\times320$.
It is well established that the auditory system performs a frequency analysis that divides the ear signal into frequency bands, and then does analysis on the fine time signal in each band~\cite{ehret1978stiffness, Yost2013}. Such processing has been shown to improve the robustness when exploited in a binaural sound localisation system, particularly in reverberant environments~\cite{MaEtAl2018loc}. To simulate this operation, the first stage of the \gls{cnn} performs a frequency analysis which filters the ear signals in the time domain with convolutional kernels.
Two frequency analysis strategies were investigated in this study. In the first system, named \textit{WaveLoc-GTF}, the frequency analysis is performed by a convolution layer which is broadly based on a gammatone filterbank~\cite{PattersonEtAl1992}. As shown in Fig.~\ref{f:system}, the frequency analysis layer consists of a number of frequency channels. The following layers in each frequency channel elaborate upon the frequency analysis output, in order to extract frequency-dependent features. The second system, named \textit{WaveLoc-CONV} imposes no constraint on frequency analysis. Instead, a convolutional layer with 1-D convolutional kernels is exploited to analyse frequency, with parameters learned from the data as part of the network training process.
In both systems, the frequency analysis is followed by a layer of 2-D convolutional kernels to extract features based on correlations of the left and the right channels. In WaveLoc-GTF these kernels are applied separately for each output of the gammatone filters, while in WaveLoc-CONV they are applied to the single frequency analysis layer. The correlation-based features are closely related to \gls{itd} and \gls{ild} cues, which are further elaborated by another convolutional layer with 1-D kernels in order to search for specific patterns that are related to the localisation task. Finally, the features produced by the convolutional layers are flattened and concatenated, before being passed to two dense layers. A softmax activation function is used in the output layer in order to perform sound localisation as a classification task.
\subsection{WaveLoc-GTF}
\label{ssec:waveloc-gtf}
Fig.~\ref{f:system} illustrates the first proposed \gls{cnn}: WaveLoc-GTF. As discussed, the frequency analysis is performed by a gammatone filter bank, which consists of 32 filters spanning between 70 and 7000\,Hz with peak gain set to 0\,dB. These filters are directly coded into \textit{non-trainable} \gls{cnn} kernels of size $1\times320$, with a linear activation function. The gammatone impulse response is given by:
\begin{equation}
w[t] = at^{n-1}\cos(2{\pi}ft+\phi)e^{-2\pi bt}
\end{equation}
where $t$ is time, $a$ is the amplitude, $f$ is the centre frequency, $\phi$ is the phase of the carrier, $n$ is the filter's order, and $b$ is the filter's bandwidth. In order to perform a time convolution, each filter is flipped in time so that the kernel operation is defined as:
\begin{equation}
y[t] = \sum_{ m = -M}^{M} x[m] w[t - m]
\end{equation}
where $x$ is the input signal, $w$ the weights of the filter, $t$ is the index of the actual value and $M$ is the filter length.
In each frequency band, the resulting feature maps share the same dimensions ($2\times320$) of the input matrix. A normalisation layer is then applied which looks for the maximum absolute value across all the gammatone channels before dividing them by this value. Hence, the output feature values range between [-1,1], which are further processed with $1\times2$ max pooling.
A separate stack of two further convolutional layers processes each normalised channel, searching for specific patterns related to localisation. The first convolutional layer has 2-D kernels of size $2\times18$ and the second layer has a set of 1-D kernels of size $1\times6$. Both convolutional layers are followed by $1\times4$ max pooling and employ \textit{ReLU} activation. Finally, the processed channels are concatenated. and fed into two fully connected dense layers.
Each dense layer consists of 1024 hidden units with \textit{ReLU} activation and a dropout rate of 0.5.
The output layer consists of 37 nodes corresponding to the 37 azimuth classes, with \textit{softmax} activation.
\subsection{WaveLoc-CONV}
The neural architecture of the second system, WaveLoc-CONV, employs a single convolutional layer dedicated to frequency analysis. Its key difference from WaveLoc-GTF is that the frequency analysis of this model is learnt during the training process together with other parameters of the network. A convolutional layer with 64 1-D kernels of shape $1\times256$ is employed as time domain filters for frequency analysis. It is reasonable to expect that the shape of a convolutional kernel directly trained on a raw waveform will be similar to all the sinusoidal components that form the waveform itself. In other words, the convolutional kernels are characterised by a set of sinusoidal functions, which lead to a particular frequency response of the kernel itself~\cite{sainath2017multichannel}.
The convolutional layer is followed by $1\times2$ max pooling with a linear activation function applied. As in WaveLoc-GTF, two more convolutional layers are employed to search for features suitable for localisation. However, instead of acting separately for each channel as in WaveLoc-GTF, they now jointly process all the output of the frequency analysis stage. The first of the two layers uses 64 2-D kernels of size $2\times18$ to look for correlations between the left and right channels. The second uses 64 1-D kernels of size $1\times6$. Both layers use the ReLu activation function and are followed by $1\times4$ max pooling. Finally, the outputs are flattened and fed into a two fully-connected hidden layers with 1024 units each. The output layer uses softmax activation with 37 neurons.
The hyperparameters or both end-to-end architectures are chosen based on an optimisation process using a development dataset.
\section{Evaluation}
\label{s:eval}
\subsection{Binaural simulation}
Binaural signals were simulated by convolving speech recordings with the Surrey \gls{brir} database \cite{HummersoneMasonBrookes10}. The Surrey \glspl{brir} were captured using a Cortex \gls{hats} in both anechoic and reverberant rooms. A total of 37 azimuth angles were used, ranging from [-90\degree, 90\degree] in steps of 5\degree, where 0\degree \space is located exactly in front of the head.
Four reverberant rooms were employed, denoted A--D. The reverberation time (T$_{60}$) and \gls{drr} of each room are listed in \tableref{t:rooms}.
\begin{table}[thb]
\caption{Room characteristics of the Surrey \gls{brir} database~\cite{HummersoneMasonBrookes10}.}
\label{t:rooms}
\vspace{1mm}
\centering
\begin{tabular}{@{} l c c c c @{}}
\hline\hline
& Room A & Room B & Room C & Room D \\
\hline
$\mathrm{T}_{60}$ (s) & 0.32 & 0.47 & 0.68 & 0.89 \\
$\mathrm{DRR}$ ($\deci\bel$) & 6.09 & 5.31 & 8.82 & 6.12 \\
\hline\hline
\end{tabular}
\end{table}
Speech signals belonging to the DARPA TIMIT database \cite{Gar1993} were convolved with each \glspl{brir}. The initial and final frames of each speech utterance were truncated if silence was present. The training dataset was obtained by randomly selecting 24 sentences per azimuth from the TIMIT training subset, while another 6 sentences composed the validation dataset. 15 more sentences per azimuth were selected from the TIMIT test subset to create the test dataset.
\subsection{Experimental setup}
For training the \textit{Adam} optimiser with a learning rate of $1\mathrm{e}{-3}$ and a batch size of 128 samples was employed. The training process lasted for 50 epochs, but early stopping was applied if no improvement was observed on the validation set for more than 5 epochs. A decreasing learning rate was employed to improve training, being multiplied by 0.2 if no lower error was achieved after 2 epochs.
The networks were trained in two acoustic room conditions: (i) using anechoic signals only for training; (ii) multiconditional training, in which the networks were trained using data from all the reverberant rooms apart from the one used for test.
The evaluation results are reported based on chunks. Each chunk is 250\,ms long (25 frames). The prediction made for each frame in a chunk is averaged to report a single azimuth location for the chunk. Chunk-based evaluation was adopted in order to avoid the issue that a speech signal typically includes short pauses where there is no directional sound source. The accuracy of the models was finally measured in terms of \gls{rmse} given in degrees.
\subsection{Baseline system}
The baseline system is a state-of-the-art \gls{dnn}-based localisation system using GCC-PHAT features as inputs \cite{vesperini2018localizing,xiao2015learning}. GCC-PHAT features are computed as the inverse transform of the frequency domain cross-correlation of two audio signals captured by a microphone pair.
The binaural signals sampled at 16\,kHz are framed at 20\,ms, with 10\,ms overlap. Since a distance of 18\,cm occurs between the two microphones, the first 37 values are selected from the inverse transform. Unit variance and zero mean normalization is then applied.
The baseline network consists of an input layer, two hidden layers of 1024 units each and an output layer of 37 classes. Dropout equal to 0.5 is applied after the two hidden layers. Softmax is selected as the activation function for the output layer, while a sigmoid activation function is used for the hidden units. All the hyperparameters were optimised using the development dataset.
\section{Results and Discussion}
\label{s:results}
\subsection{Anechoic training}
\label{ssec:anec_train}
\tableref{t:anechoic_train} shows results using systems trained in the anechoic condition. The best overall performance is achieved by the baseline GCC system. The proposed WaveLoc-GTF performed slightly worse compared to the baseline, while the localisation errors for WaveLoc-CONV were considerably larger across all reverberant conditions.
\begin{table}[!h]
\center
\caption{Localisation \gls{rmse} results in degrees for the models trained in anechoic environment.}
\label{t:anechoic_train}
\vspace{1mm}
\begin{tabular}{@{} l c c c c c @{}}
\hline \hline
Room & Anechoic & A & B & C & D \\ \hline
Baseline & 0.1\degree & \textbf{2.6\degree} & \textbf{9.3\degree} & 2.6\degree & \textbf{10.1\degree} \\
WaveLoc-GTF & \textbf{0\degree} & 9.1\degree & 10.7\degree & \textbf{1.6\degree} & 10.5\degree \\
WaveLoc-CONV & \textbf{0\degree} & 37.7\degree & 41.8\degree & 37.3\degree & 44.4\degree \\
\hline\hline
\end{tabular}
\end{table}
\begin{figure}[!h]
\center
\includegraphics[width=0.9\columnwidth]{spec_WAVEFORM_BASELINE_1_train_Surrey_Anechoic_CLEAN.pdf}
\caption{Log-power spectra of the kernels in the first convolutional layer of WaveLoc-CONV when trained in the anechoic environment.}
\label{f:spec_train_anec}
\end{figure}
It appears that the WaveLoc-CONV system has a tendency for overfitting compared to the other two systems. \figref{f:spec_train_anec} shows the log-power spectra of all the 64 kernels in the first convolutional layer in WaveLoc-CONV. It is clear that the kernels, when trained in the anechoic condition, act largely as a set of band pass filters, mostly enhancing the frequency bands between 300--600\,Hz and between 2300--2800\,Hz. It is widely known that binaural features such as \glspl{itd} are more reliable in the low frequency region below 1600\,Hz while others such as \glspl{ild} become more robust in the high frequency region above 1600\,Hz~\cite{Blauert97}. It is possible that the network extracts related binaural features which are most effective in these two bands for sound localisation in the anechoic condition. Such behaviour, however, failed to generalise to unseen reverberant conditions as these frequency bands could become unreliable due to reverberation.
The WaveLoc-GTF model, on the other hand, performs frequency analysis with the gammatone filterbank layer which forces the system to exploit all frequency bands and thus extract the most effective localisation features in each band.
\subsection{Multiconditional training}
\label{ssec:mct_train}
It has been shown in the past that \gls{mct} can mitigate overfitting and increase the robustness of sound localisation in reverberant conditions~\cite{MaEtAl2017dnn, MayvandeParKohlrausch11a}. This can be done by adding either diffuse noise or reverberation to the training signals. In this study, a reverberant training approach was adopted as our preliminary experiments showed it to be more effective. Specifically, the anechoic training dataset was supplemented with reverberant versions by convolving it with various \glspl{brir}.
For each one of the four reverberant room under evaluation, all the remaining three were included for \gls{mct}.
\begin{table}[thb]
\center
\caption{Localisation \gls{rmse} results in degrees using \gls{mct}.}
\label{t:train_mct_diffuse_room}
\vspace{1mm}
\begin{tabular}{@{} l c c c c @{}}
\hline \hline
Room & A & B & C & D \\ \hline
Baseline & 2.7\degree & 3.3\degree & 3.1\degree & 5.2\degree \\
WaveLoc-GTF & \textbf{1.5\degree} & 3.0\degree & 1.7\degree & 3.5\degree \\
WaveLoc-CONV & 1.7\degree & \textbf{2.3\degree} & \textbf{1.4\degree} & \textbf{2.4\degree} \\
\hline \hline
\end{tabular}
\end{table}
\tableref{t:train_mct_diffuse_room} lists the results of all the models. The anechoic condition was excluded in this study, as all the models performed well even without \gls{mct}. All the models benefitted from \gls{mct}, especially the proposed WaveLoc models. The best overall performance in reverberant conditions is achieved by the WaveLoc-CONV model, which has an average localisation \gls{rmse} less than 3\degree compared to over 30\degree without \gls{mct}.
To investigate the effect of \gls{mct} on the convolutional kernels, we again plot the log-power spectra of all the 64 kernels in the first convolutional layers of the WaveLoc-CONV model. Only the plot for Room D is shown in \figref{f:spec_train_mct_rooms_NO_D} but plots for all the other rooms are similar.
It can be seen that the first convolutional layer is now composed of a set of distributed bandpass filters emphasising mainly the 1500-4000\,Hz range, with some kernels stretching up to 6--7\,kHz.
The low frequencies below 1500\,Hz are less exploited by the WaveLoc-CONV model. It is interesting to notice that the data-driven model learns to use more high frequency cues in a reverberant environment, which suggests \gls{ild} become more useful than \gls{itd}. It is reasonable to expect that the \gls{itd} is more affected by reverberation, while the \gls{ild}, created by the head shadowing effect mainly for frequencies higher than 1600\,Hz, is more robust to reverberation. Indeed, psychophysical cue-trading studies find that human listeners give \gls{ild} more weight than \gls{itd} when localising sounds in reverberant conditions \cite{nguyen2014}.
\begin{figure}[t]
\center
\includegraphics[width=0.9\columnwidth]{spec_WAVEFORM_BASELINE_MCT_ROOMS_NO_D_train_Surrey_Anechoic_CLEAN.pdf}
\caption{Log-power spectra of the kernels in the first convolutional layer of WaveLoc-CONV when trained using \gls{mct}.}
\label{f:spec_train_mct_rooms_NO_D}
\end{figure}
\section{Conclusions}
\label{s:conc}
This paper described a new approach for localising a sound source directly from the waveform, by proposing two novel end-to-end \gls{cnn} systems. Machine localisation systems typically employ hand-crafted features, such as the \gls{itd} and \gls{ild}. Such explicit feature extraction may limit the model performance since it implies a lossy transformation of the input signals. Instead, the proposed end-to-end approach employs a cascade of convolutional layers to extract features directly from the waveform, that are suitable for localisation in reverberant environments. When \gls{mct} was used across reverberant conditions, both end-to-end systems outperformed a state-of-the-art \gls{dnn} system using conventional features.
Two \gls{cnn}-based systems were introduced. The first system, WaveLoc-GTF, is inspired by the auditory system and employs a convolutional layer that is largely based on a gammatone filterbank. The second system, WaveLoc-CONV, employs a data-driven approach, where a convolutional layer with trainable 1-D kernels is dedicated for frequency analysis. Although the gammatone filterbank is in some sense more `principled', since it approximates the filtering characteristics of the human auditory system, it does not work as well as a system that is trained (i.e., finds its own filters) across a number of reverberation conditions. One reason for this is that the system may elect to emphasise frequency regions during training that provide more robust cues to localisation.
Indeed, we found that when \gls{mct} was used, the WaveLoc-CONV model was better able to exploit features in the high frequency regions above 2\,kHz, which tend to be less corrupted by reverberation. This mirrors findings from human perception suggesting that \gls{ild} (which is primarily available at high frequencies) is more robust than \gls{itd} when reverberation is present.
Future work will focus on improving the ability of end-to-end systems to generalise to unseen room conditions and multiple sources. Another possible direction is to combine sound identification with sound localisation within an end-to-end system~\cite{CakirVirtanen2018}. Finally, we plan to conduct `psychophysical' studies on trained networks in order to fully understand their underlying mechanisms, e.g. by using the cue trading protocol described in \cite{nguyen2014}.
\vfill\pagebreak
\bibliographystyle{IEEEbib}
| {
"redpajama_set_name": "RedPajamaArXiv"
} | 2,310 |
WARNING: THIS BLOG CONTAINS BODYCOUNT. HIGH RISK OF SPOILERS. ENTER IF YOU DARE.
TV Terror: The Naughty List (2021)
The Naughty List (American Horror Stories Episode 4/Season 1 (2021)
Rating: ***
Starring: Kevin McHale, Nico Greetham and Dyllon Burnside
It's Christmas in July and Santa has come to town once more to spread some holiday fear in this deservingly gruesome entry from the horror anthology series American Horror Stories, a spin-off of Ryan Murphy's American Horror Story.
Taking a bloody stab at internet fame, a group of three dude bro influencers and their more timid tech guy-slash-camera man, collectively known as the 'Bro House", decided one December day to try something a little different from their usual posts of internet challenges, pranks and other tomfooleries; camping out near a bridge infamous for suicide jumpers, the gang records a man's demise and posts it online with little hesitation, hoping for millions of new subscribers. The result though, as many would've guessed, is a ton backlash from their soon-to-be former fans and sponsors, leading to the Bro House boys to try everything they can to salvage their channel and get their increasingly declining follower count to spike back up.
One of these sad attempts happen to be raiding a mall Santa's workshop where they harass staff and visiting families, unsuspectingly leading to a nasty surprise for them later: as it turns out, the Santa working there (played by the ever intimidating yet badass Danny Trejo) is actually a serial killer who snuffs out real mall Santas to take their place and you can take a good wild guess who just ended up in this murderous Sick Nick's naughty list...
How well you would take in this mini-movie episode depends on how much toxic behavior and dumbassery your stomach can handle before the carnage because it is going to take a while to get there; for a good sum of the story, we're treated to three douchebags flexing their perceived invincibility as social media influencers and their belief that they're above any form of decency and morality just because they have a website and a fanbase. Admittedly, this is both frustrating and satisfying to watch as these characters' monumental stupidity to even understand the gravity of their actions meant seeing them struggle to grasp and accept the situation they caused on their own, henceforth further digging their own graves deeper not just as internet stars, but also literally when they finally cross the wrong person. I love the fact they're suffering from their choices, but I cannot help but think this could have been written and implemented better, perhaps with a more compelling set of characters to really give that gut a good ole' punch rather than going for the easy route of Paul Logan caricatures with even less functioning braincells.
In a way, this does make the resulting massacre all that more satisfying given how much these character made themselves so deserving of being targeted by a holiday-themed slasher. I love the fact Danny Trejo's slaughtering Santa has little to go by in terms of solid origin and depth, as it kinda gives him this mysterious air of simply existing there to do harm in what he sees as his deed as Saint Nick. The kills are bloody at best, the most creative among them being a nasty way out involving Christmas lights and a pool, but with it happening around the last third of an episode with a near 40 minute run time, I did wish it was a tad more longer so we could further engross ourselves to the unfolding horror.
In a nutshell, American Horror Stories' The Naughty List (2021) embodies the worst and best elements of a simple slasher story, focusing first on a slowburn rabbit hole of internet toxicity and cancel culture before getting on the gory goods of candy cane colored crossbows and human gullet-decorated Christmas trees. It's not the grandest example of a holiday slasher story done right (heck, it barely even felt like its Christmas throughout the episode), but if you can survive your way through annoying characters without a hitch just to see that sweet, sweet mayhem for all that trouble, I say go for it and catch this!
Bodycount:
1 male jumps off a bridge
1 male found inside a box dismembered into pieces
1 male had his head twisted
1 male tangled in Christmas lights and thrown into a pool, electrocuted
1 male shot to death with a crossbow
1 male shot on the leg with a crossbow and set on fire with gasoline, later succumbs to his injuries in a pool
Posted by Kaijinu at 4:36 PM No comments:
Labels: 2020s, Franchises, revenge, reviews, Santa gone wild, slasher, The Holidays, TV Terror
Happy 10th Birthday, StickyRed!
Ten years.
Ten years of all things hacking and stabbing. Ten years of filling up this compendium of bodycounts from slasher movies and its kin. I never really thought I could make it this far!
It kinda feels like it was just yesterday when I finally decided to start writing about my favorite horror sub-genre after visiting sites like Hysteria-Lives, Slasherpool and Retroslashers. (Dang, who remembers the last two?) I wanted an outlet and this site gave me that opportunity to do just that so, thank you.
Thank you for those who dropped by to see what this little man has offer in insight regarding your favorite and/or new slasher titles out there. (I sure do hope I didn't spoil any movies considering the bodycounting I do here! haha) Thank you for tolerating my grammar, which I swear I check and re-check, over and over. Thank you for commenting and sharing a few suggestions for me to watch and read. (You know who you are, you good people!)
Thank you, to all the little friends I made here. Knowing you take a time off your day to see what I write about simply puts a smile on my face. (What can I say? I'm a man of simple pleasures!)
So ten years has passed since I started this blog. Here's to another ten years and more! So long as evil walks on two legs and there are final girls and boys to stop them, I will power on to see what this fine subgenre has to offer! So here's to StickyRed: Bodycount Compendium! So now, dear readers...
Posted by Kaijinu at 10:49 PM 3 comments:
Labels: Badly Drawn masterpieces of mine, Random Stuff
The Witch Forever Lives: Fear Street Part Three: 1666 (2021)
Fear Street Part Three: 1666 (2021)
Rating: ****
Starring: Kiana Madeira, Ashley Zukerman and Gillian Jacobs
Previously on Fear Street; After learning through a fellow Shadyside massacre survivor's own harrowing account back in 1978 that they can end the curse afflicting their town by reuniting a dead witch's severed hand with the rest of her remains, siblings Deena and Josh find the witch's hand and head forth to her suspiciously misplaced grave. But when Deena bleeds unto the remains, however, she begins to see things in the past through the witch's eyes...
It's the early 17th century, at a settlement called Union; Sarah Fier is just like any other youth around the community, spending her days helping around with her family's chores and enjoying whatever free time she has left with her friends. But when one night of festivities under the full moon had her confessing her love for her friend Sarah Miller and got intimate, it somehow kicked off a curse that spoiled the town's harvests, poisoned the water and possessed the settlement's pastor Cyrus Miller into murdering a dozen children. As the town's grievance boils into blaming rage, they set their eyes on hunting down Sarah Fier and Hannah Miller as many claim to have seen the two practicing witchcraft.
Now on the run and blamed for the mysterious plague, Sarah Fier looks for a way to save not only herself, but her beloved Hannah, this including striking a deal with the Devil himself. But not everything is what it appears and Sarah, along with our present day Shadysiders, will find out the horrible truth about the darkness befalling them...
Much like the first two Fear Street movies, 1666 subverts expectations by practically handling a story that is its own rather than a homage to a popular slasher plot; running around nearly two hours, the film is split into (further) two parts, with the first being a folk horror story taking place during the fire-and-brimstone era of colonial America. This is where the trilogy bring forth its most intriguing and strongest plot point as it plays with the idea of forbidden romance, paranoia and religious fanaticism through a twisty mystery as a mean to flesh out a workable overarching story even more. The unexpected turns within these parts work fittingly well with the overall tone of the story as a whole, keeping us on our toes as we watch Fier and Miller (both played by the two main casts from the previous two films, a nice touch that gave their characters a bit of legacy) figure out what's causing the misfortunes, all the while surviving an extremist mob on a murderous witch hunt. Through its gloomy camera work and suppressive score, this half is simply evocative and intense down to the expected yet still shocking and surprisingly bittersweet end that leads us back to the present.
Without giving away the big reveal, the second half of 1666 follows up on where the events of 1994 left off so far, now with a stronger sense of urgency and tighter atmosphere as a surprise villain is made aware that their secretive exploits have been compromised and is now willing to do anything to make sure it stays hidden. The whole second act just ties neatly to everything that been happening, made better with the fact that it brought back the spunky energy of the first movie which was noticeably absent from the second. This means more creative Scooby-Doo friendly plans to stop yet another small army of stabby, masked and not-so-masked villains (I swear, these killers better get their own movies!), ending with a satisfying finale that tug heartstrings and a last shot implying some possible mayhem in the future. It perfectly balances out the bleak and gloomy first half, a cleverly entertaining way to wraparound the entirety of the story.
With the story straying away from the typical hack-and-stab film plot, this leaves Fear Street 1666 a bit of an oddball within the interconnected slasher trilogy with its lack of sizeable onscreen kill count or recognizable teen horror tropes. It does earn points here, however, as a general horror flick with its bold moves of having the first set of slayings involve children (done offscreen and it's only for a single scene, but the shown aftermath of the carnage is still unnerving) and Sarah Fier's own hanging being effectively distressing considering the twist. It further makes up for its lack of solid slasher build with a very engaging story and character development and this is something I personally see as not only this movie's winning factor, but the entire trilogy's; there's respect for the slasher tropes we all know and love here, addressing it before giving it a spin of its own. This fresh take on the subgenre is exactly the kind of welcome creative response we need and is why each film works.
A strong last chapter to a trilogy of slasher throwbacks and then some, Fear Street 1666 is as good as it can get when it comes to a fun mystery, intense thrills and ghoulish cavalcades of masked killers and traitorous spellcasters. It certainly has its little flaws (I find it kinda hard to buy that puritan teens sneaks out for parties, drinking the fruits of the land and take berries as drugs but, eh!), but for all the good it has to offer, I can honestly say this is a one gratifying entry for a of a kind slasher event!
12 children found murdered, eyes plucked out with a horseshoe hook pick
1 male stabbed in the side with a pitchfork
1 female had her throat cut with a knife
1 female hanged
1 male stabbed through the neck with a knife
1 male knifed in the gut
1 male knifed in the eye
Posted by Kaijinu at 10:33 PM No comments:
Labels: 2020s, band of killers, child in peril, Franchises, masked slasher, murder mystery, offscreen massacre, revenge, reviews, slasher, supernatural, Twisted Twist
No Talking. No Texting. No Breathing: Al Morir La Matinée (2020)
Al Morir La Matinée (Uruguay/Argentina, 2020) (AKA "Red Screening", "The Last Matinee", "Bloody Matinee")
Starring: Ricardo Islas, Luciana Grasso and Franco Duran
Sometimes you don't have to flex a high concept to make a fun horror movie. Sometimes you could just work it out with some guy with a bag full of tools, a fairly sizable bodycount and set of gnarly kills, all taking place in a old school movie theater.
It's a rainy day in 1993 and young projectionist Ana (Luciana Grasso) is taking over her emphysema-stricken father's shift at the Cine Opera downtown theater, which is mostly empty that day. The few patrons who are staying, though, are treating themselves with a showing of Frankenstein, The Monster (With clips taken from Frankenstein: Day of the Beast (2011)) all the while going on with their own shenanigans of hooking up with girls, giving their dates a hand and, for one kid, sneaking in to brave a movie that might be too scary for him. Unbeknown to them all, a gloved maniac is staying in the very same theater and he has set his eyes on killing them all.
While it isn't new to have a slasher film set in a movie theater seeing we have titles like The Meateater (1979), Movie House Massacre (1984), Anguish (1987), and Matinee (1989) already, as well as a few horror titles opening with a murder taking place in crowded theaters such as He Knows You're Alone (1980) and Scream 2 (1997), none of them really took the sleek and gory approach as good as Al Morir La Matinée (2020). Basically, the film's direction is to have the hack and stab elements stripped down to its very core, a throwback of sorts with a simple premise and somewhat okay characters being themselves while a killer picks them off one by one. It never strays away from this approach. No twists or curveballs. Just have a psycho stalk the cinema and murder away anyone he can grab hold on to.
This, in turn, means that the characters can be lacking a bit of depth or personality, but the interactions between themselves and others do have their subtle, cheeky and cheesy moments, thus keeping their scenes far from being boring whenever the film focuses on them. It also helps that the group isn't that big so the focus isn't all over the place, as well as the fact that the plot is paced nicely between the horror and non-horror scenes, giving us enough time to take a quick breather before our killer decides who to stalk and stab next.
And speaking of which, Al Morir La Matinée (2020) will undoubtedly make a gorehound's day as not only is it very bloody, but there's also a strong Italian giallo influence in its cinematography so expect outrageous murder angles, evocative tinted lighting and stylized brutality that would make the likes of Argento or Bava proud. (Best of which being a smoker's demise when the killer went for their throat!) What further works here is that there's a sense of tension and creepiness to the killer's massacre, utilizing the nearly empty cinema setting to give our maniac all the darkened halls and rooms to sneak and prowl around, as well as use the screams from the movie playing to drown out the sound of his weapons and distract unsuspecting moviegoers from the murder happening just a few rows next to them. There's definitely a reason behind all of these methodic killings, but who this killer is or how did he ended up this way is something the movie's keeping to itself and I'm frankly okay with that.
Smoothened with a beatastic synth soundtrack and a strong finale involving a trio of potential survivors, Al Morir La Matinée (2020) is a real winner in my eyes as an old school-style slasher done right. It doesn't beat around the bush with too many layered side-plots and its brutal enough to keep a horror fanatic excited, all the makings of a bonafide slasher experience in my book! So whenever you find yourself a chance to see this, I guarantee you a keeper here!
1 elderly male brained with a hammer
1 male had his throat cut with a knife
1 male and 1 female had their heads skewered together with a broken long iron hook
1 male had his face bashed against the floor, stabbed with a knife
1 female had her chest and eye stabbed with a knife
1 male knifed to death
1 female had her head repeatedly crushed with a projector's lamphouse lid
1 male ran through with a broken long iron hook
Posted by Kaijinu at 1:44 AM No comments:
Labels: 2020s, cannibalism, child in peril, europe, foreign, Giallo, gore, influenced, reviews, slasher
Bad Things Always Happen To Shadysiders: Fear Street Part Two: 1978 (2021)
Fear Street Part Two: 1978 (2021)
Rating: ***1/2
Starring: Sadie Sink, Emily Rudd and Ryan Simpkins
Previously in Fear Street; in the cursed town of Shadyside, teen siblings Deena and Josh Johnson, along with Deena's girlfriend Sam Fraser, found a way to stop a trio of resurrected killers after seemingly disturbing a fabled witch's grave, losing their dear friends in the process. Unbeknowst to them, the curse is far from done as Sam gets possessed with a murderous frenzy. Desperate for a solution, Deena and Josh turn to the only person who can help them now, a fellow Shadyside resident named C. Berman who dealt with the witch's curse before.
As the siblings manage to restrain Sam and travel to Berman's house, much to the latter's uneasiness and reluctance to get involved with the curse again, they try convincing her to at least share what she knows. Giving in eventually, Berman recounts her own tragedy during that one night at Camp Nightwing.
It was 1978, when the rivalry between the prosperous town of Sunnyvale and the downbeaten Shadyside gets put to the test once more through the camp's annual color war. Shadyside sisters Ziggy and Cindy Berman, polar opposites of one another personality-wise, will see themselves caught in an alarming situation through their own encounters with the camp nurse, Mrs. Lane, whose own daughter Ruby Lane became one of Shadyside's spree killers. Ziggy, the rebellious one, notices Lane's distressed ramblings about the nature of her daughter's murder spree whilst getting her wounds treated, while Cindy, the uptight one, gets a far more harrowing experience when Lane attacked her and her boyfriend, Tommy Slater, after claiming she saw the boy's name on a wall.
Fortunately, Lane gets dealt with before any further damage can be done, prompting the campers to bring up the supposed Shadyside curse and Sarah Fier, the supposed witch responsible for damning the land. Being the ever rational person, Cindy doesn't buy any of this and goes snooping around the nurse's office with Tommy, looking for proof that she may just be high on something during the attack. Crashing in to join them is Cindy's former friend Alice and her boyfriend Arnie, and they find not only a bottle of unlabeled drugs, but also Lane's journal full of notes and details about Sarah Fier, her deal with the devil and the curse itself.
As Cindy and her group goes deeper into the woods looking for Fier's house to understand Lane's obsession over the witch, Ziggy deals with a group of Sunnyvale bullies and gets an unexpected help from a fellow Sunnyvale teen, a counselor named Nick Goode who, more or less, is smitten with her. It would have been a typical Summer night of misadventures for the two groups, but little do they know that the curse is real and it has taken another soul that evening. A soul damned to commit murder under the name of Sarah Fier...
Clocking for almost two hours, about an approximate two-thirds of Fear Street 1978 have us watching more heart-to-heart moments between the doomed teens as they discuss their troubles and outlooks in life through the eyes of troubled youth, as well as dive deeper into the hints and details as to who and what Sarah Fier could be and how the curse works. This means that, although influenced by backwoods slashers, particularly the Friday The 13th and Sleepaway Camp movies, the film subverts expectations by primarily focusing more of its narrative on the characters and the mythology behind the witch's curse, leaving the slasher elements to run its course in the background for the killcount and, frankly, very gruesome shock value. This can be a relative drawback as the approach may not sit well for those expecting full on backwoods-set hacking and stabbing through and through, especially with its comparably slower pace than the first entry and the abundance of lore talk (most of it taking place inside a cave, mind you), but the movie does make up for these with its stellar performances, inspired scripting and genuinely bleak atmosphere.
Much of the development stems from the two lead actresses Sadie Sink and Emily Rudd as Ziggy and Cindy Berman respectively, sisters who hadn't seen each other's desperation for love and acceptance for quite some time, now thrown in an otherworldly massacre that's both human and supernatural. The writing for these two and how they were acted simply flesh out the uneasy and tense history between them, giving the girls time to develop and open up with their troubles to further establish themselves as individuals and sincerely grow as family. Sad to say, as it is established early into the film, one of them will bite the big one and the impact of the demise is still no short of tragic.
When Fear Street 1978 does show its slasher side, it is mostly what you would expect from a backwoods slasher set-up featuring a mad axeman, with people getting chopped left and right with explicitly strong gore. The shock factor here, though, is that almost half of the victims were preteen children and albeit it's mostly offcamera, the fact that this massacre included young kids is pretty ballsy. It undoubtedly shows the kind of serious mood they're setting for, at the price of the camp and snark which does make the first movie memorable, sadly. The supernatural-tainted finale does also call for some fair word about it for how dark it was despite, again, knowing the outcome.
Acting as a solid midway between a planned trilogy, as well as a competent standalone campsite slasher, Fear Street 1978 is a treat for backwoods slasher enthusiasts who doesn't mind a fair bit of teen drama and witchy threats tagging along the bodycount. It may lack the vigor of the first film, yes, but I'll give it points for attempting something fresh out of the typical backwoods teen horror and for being a consistent narrative within a bigger story.
1 male repeatedly axed on the face
1 boy hacked to death with an axe
1 female hacked to death with an axe
2 boys and 2 girls axed to death offscreen
1 male decapitated with an axe
1 female hacked on the chest with an axe
Labels: 2020s, backwoods, band of killers, child in peril, Franchises, masked slasher, reviews, Sacred Bloodshed, slasher, supernatural
Where Your Worst Nightmares Live: Fear Street Part One: 1994 (2021)
Fear Street Part One: 1994 (2021)
Starring: Kiana Madeira, Olivia Scott Welch and Benjamin Flores Jr.
Like many horror-inclined kid growing up in the late 90s and the early 2000s, R.L. Stine's Goosebumps books were a staple reading for me during them past lazy weekend afternoons, that was until I elevated my taste for horror novels by reading my first Stephen King work. (Cujo, for those who are curious) From that point, I kinda associated R.L. Stine with kiddie horror for a while and never really bothered with the author until I started looking for slasher novels during college and rediscovered Stine through his other literary series, Fear Street, about teens living in a town called Shadyside and their deadly encounters with the paranormal, the murderous, or sometimes both!
In this first entry of a planned three-parter movie event inspired by the books, we start off in a small town mall during its closing hours and one "Shadysider" teen named Heather Watkins, clocking off from her bookstore cashier job. A figure in a robed skeleton costume suddenly shows up and starts a deadly cat-and-mouse chase with her, hunting knife at hand and a couple of other unfortunate folks littered outside already murdered. "Skull Mask" here eventually catches up on Heather after an intense stalk-and-stab act, knifing her to death just as she uncovers that Skull Mask is actually her friend Ryan Torres.
The killer then gets himself snuffed out with a bullet to the head when a responding sheriff finally makes it there, thus ending another Shadyside massacre...
As the following morning comes, news of the spree killing have the entire town talking once again, among them being the students of the local high school who some believe this to be the work of a witch who cursed the town centuries ago, leading to murders like this being an unwanted norm for Shadyside. Young Deena Johnson doesn't believe in this, however, nor does she have the time to; still bitter about a recent break-up, she just wants to get the day done and over with so she can quit the school band and go back home to sulk. Her fun junkie friends Kate and Simon, though, convince her to stay and join them in a memorial service over at the neighboring town of Sunnyvale, whose football team they'll be fighting in a later game.
The problem, of course, is that Deena's ex-girlfriend, Sam, just happens to be now living in Sunnyvale and attending the same school opposing Shadyside's. The meet-up gets as awkward as it can get when Deena starts going off on Sam for breaking her heart, made worse as a fight breaks out during the service when a Sunnyvale teen made some very unsavory remarks against Shadyside.
It all escalates for the worse when a group of Sunnyvale punks, along with Sam as an unwilling participant, decided to harass the school bus Deena and her friends are at while heading back to town. Their attempt to retaliate ends in a car crash, which lands Sam in the hospital and her boyfriend threatening Deena that he'll get her back for what happened.
And wouldn't you know it, at the night after the incident, someone in a Skull Mask get-up begins stalking Deena and the gang, lurking around and breaking into houses, seemingly preying on something or someone. Could this be, perhaps, a pissed-off boyfriend's sick and twisted attempt to get even? Or is there something far more dangerous and deadlier at play here? Something involving a witch's curse and a town's dark legacy as America's murder capital?
With its fast pace, fun characters and captivating lore, not only does Fear Street 1994 captures the feel and tone of the book series it is adapted from, but it also twists a refreshingly creative take on throwback slashers that simply delivers; rather than being a straightforward bodycount mystery, it delved into other elements such as witchcraft horror and a bit of survivalist thrills as the middle run of the plot becomes a chase flick with not only one, but three slashers on a hunt and its up for our small troubled group to think up plans to stop them, these being the best bits of the film.
The key point that makes this direction work so well is that it gave us a chance to know our characters a bit more and see their interaction with one another despite not altogether meeting eye-to-eye at the beginning. They start out as typical slasher victim labels of junkies, nerds, cheerleaders and an obviously marked final girl, but their development during the plot's downtimes made them closer and I genuinely love the writing done for them, may it be for laughs or feels, so much so that I worry the outcome of their predicament. And, seeing this is a slasher, they eventually have to chance a gruesome fate and when it happens, it's purely cathartic.
The supernatural theme, thankfully, didn't overly complicate matters as it remains pretty easy to follow and is used to greatly fair effect here through the slashers; without spoiling a lot, the three killers are basically henchmen working under the influence of a curse and it is through this power that keeps them ticking and going. What I love about these killers is that they're pretty diverse in their theme and the lore behind them do make me wish Fear Street 1994 isn't going to just contain itself within three films as the mythology behind the curse hinted more slashers at work from the past and we don't often get slasher concepts taking place during periods of times away from the 90s, 80s, or 70s. (We got a brief look of a 1950s milkman slasher and a deformed boy on a bashing spree in 1920s. How often do we get that?!)
When it comes to the kills, Fear Street 1994 remains generous with the bloodletting, though a good bulk of the numbers are from offscreen murders and quick carnal flashbacks. Whenever they do decide to show some serious onscreen carnage, its mostly practical with one memorable murder involving a supermarket's electric bread slicer that I doubt will leave everybody's minds as soon as the film ends. (Eat your heart out, Intruder (1989)!) Other highlights include evident homages to classic slasher scenes such as Scream (1996)'s opening act of murdering a big star like Maya Hawke of the Stranger Things fame, as well as Kate's babysitting fiasco as a possible throwback to John Carpenter's Halloween (1978).
It's a pretty strong start for Fear Street's trilogy of supernatural slasher terror and I, for one, cannot wait to see where this entry's cliffhanger ending will lead us to in terms of lore, twists and, hopefully, bodycounting. A strong love letter to slashers of old with a witty witchy twist, whether you love the books or slasher in general, I highly recommend not missing out on this unexpectedly entertaining chiller, or be damned by the witch's curse forever!
1 female repeatedly knifed, bled to death
1 male shot on the head
6 victims seen and mentioned murdered
1 male ran through with a knife
1 female found with a cut throat
1 male knifed in the throat
1 female had her throat cut with a razor (flashback)
1 female seen knifed to death (flashback)
1 boy seen with his head being bashed with a bat (flashback)
1 victim seen being drowned (flashback)
1 female had her head ran through an electric bread slicer
1 male axed on the head
Posted by Kaijinu at 10:22 AM 3 comments:
Labels: 2020s, band of killers, campus corpses, female psychos, Franchises, influenced, masked slasher, reviews, slasher, supernatural
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Kaijinu
I'm a Filipino Nerd with a penchant for all things weird, messy and overly theatrical. Loves to draw, write, and read at a highschool level. Has a thing for slashers, monsters, comic books, Doctor Who and collecting knick-knacks such as a certain line of toys based on a 2010 reboot of an 80s cartoon about talking, rainbow colored ponies.
No Killers were harmed in the making of this blog. Their victims, however, aren't so lucky. Awesome Inc. theme. Powered by Blogger. | {
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Q: Does my Arduino-based device need FCC certification? I've built a small consumer device that contains an Arduino Nano. It's coupled to a custom daughter board that allows it to pulse a 12V electromagnet at about 1 hertz as well as inteface to some sensors. It does not intentionally produce any RF emissions like wifi or bluetooth.
I'd like to sell my device in the US, and I'm trying to determine what certification I need to legally sell it. From what I've read about FCC certification, including similar questions here, it's needed by nearly all electronic devices that oscillate above 9 kHz.
So, if I understand this correctly, my custom daughter board wouldn't require FCC certification? The Arduino Nano contains a clock that oscillates at 16 MHz, but I believe it already has FCC certification. Does my composite device constitute something that needs to be re-certified by an FCC approved testing lab? I'm not sure how much I'll be able to sell the device for, and don't expect to make much money, so if I can avoid wasting $10,000 on worthless certification for an unintentional emitter, I'd like to do so.
I'm not sure if this is an appropriate question for this site. If it's not, where could I find an answer to this? I've checked the FCC's website, but aside from vague FAQs, I can't find any way to contact anyone with a clue. I've seen some test labs offering to give me a quote to answer this question, but since they have nothing to gain by telling me "no don't bother paying us thousands to test your device", I'm hesitant to trust a response from them.
A: Code of Federal Regulations, Title 47, Part 15 (47 CFR 15) of Federal Communications Commission (FCC) rules and regulations regarding unlicensed transmissions. Nearly every electronics devices sold inside the United States radiates unintentional emissions, and must be reviewed to comply with Part 15 before it can be advertised or sold in the US market.
The verification procedure requires that tests be performed on the device to be
authorized. These tests measure the levels of radio frequency energy that are radiated
by the device into the open air or conducted by the device onto the power lines. After
these tests are performed, a report must be produced showing the test procedure, the
test results, and some additional information about the device including design
drawings. The specific information that must be included in a verification report is
detailed in Part 2 of the FCC Rules.
Sections 2.951
through 2.957
Once the report is completed, the manufacturer (or importer for an imported device)
is required to keep a copy of it on file as evidence that the device meets the technical
standards in Part 15. The manufacturer (importer) must be able to produce this report
on short notice should the FCC ever request it.
There is no filing
with the FCC required for verified equipment.
There are a number of exemptions outlined but not limited to the following:
EXEMPT "Test equipment" includes devices used for
maintenance, research, evaluation, simulation and other analytical or
scientific applications in areas such as industrial plants, public
utilities, hospitals, universities, laboratories, automotive service
centers and electronic repair shops.
Suggestions
Something that pulses once a second could easily go under the radar of scanning Sweep tests. So you ought to minimize the unintended radiation with twisted pairs, CM chokes and or snubbers so that it cannot be hear on an AM radio 30m away on a weak channel.
A: You are confusing certification and emissions requirements. Only intentional radiators need to be certified. From your description, your device is not a intentional radiator.
However, you are still obligated to ensure it does not radiate excessively. The limits are defined in part 15 of the FCC rules.
How you determine for yourself and ensure that the device does not radiate more than allowed is up to you. The FCC doesn't go looking at the millions of devices that are unintentional radiators and test them for compliance. However, your competitors might. If they find your device radiates illegally, they can file a complaint with the FCC.
The worst case is if some communication got interfered with, the FCC investigates, and finds one of your devices causing the problem. Then it gets serious fast.
Large resellers may require a recognized lab to certify that your device radiates legally, or they won't carry it.
All that said, for a little guy selling a few 100 gizmos a year off some web site, there is very little chance anyone is going to check whether the device radiates within the limits. If you follow best practices, like a good overall grounding strategy, filtering of external wires, etc, chances are very low your device will radiate enough past the limits for anyone to notice or to care.
As Dirty Harry would say: "you've got to ask yourself one question: 'Do I feel lucky?' Well, do ya, punk?"
A: You will require an emissions certification for your product. The certification process would include your complete product in it's enclosure and would have to be tested with any normal accessories like an AC adapter attached. It is also typical that representative cables must be plugged into all interface connectors that would normally be used during operation of the device.
If your device is operated off the mains power line you have to undergo additional certifications for conducted emissions and immunity to certain applied disturbances like electrical spikes and surges.
There also requirements for your product to be tested for immunity to static discharge. Safety testing may also be required depending on the product category and customer /user location.
The fines and potential liability claims can be considerable if you do not follow the rules and would most certainly be far in excess the cost of having certification done.
A: You'll want to look at this answer that talks about when testing is compulsory:
https://electronics.stackexchange.com/a/16938/39344
You have an "unintentional radiator" regulated under Part 15, Subpart B. Your 12V electromagnet can be a much bigger source of problems that you might imagine, depending on how sharp the signal leading to it is.
Your testing need not be expensive: get multiple quotes, educate yourself before calling, make it clear you're an easy quick pass.
You can "get away with it" for a while, if your volume and ambitions are low. But it's not all that hard to do it right.
A: If it makes any sense for your device, perhaps you can consider your daughterboard a 'subassembly'. In other words, if it makes any sense for you to market the daughterboard apart from the Arduino itself, that daughterboard could be considered a 'subassembly' and thus exempt from FCC authorization:
You may be asking yourself how companies such as Sparkfun, a business
based on selling electronics kits and wireless development kit
companies continue to sell large numbers of non FCC authorized kits,
with seeming impunity. For Sparkfun, the rules that apply in most
cases relate to "subassemblies". This just means that Sparkfun's
customers will most likely use the products to build larger products
containing a number of subassemblies. For example, that may include an
Arduino™ processor board along with several sensors or peripherals and
an LCD. The user may even put all of these parts into an enclosure. If
the user sells this product containing multiple subassembly parts in
an enclosure, for all intents and purposes they are now a
"manufacturer" and their equipment is subject to the normal FCC
authorization procedures.
EMC FastPass
There is also an "Exempted" product list, that perhaps your product might fall under:
*
*A digital device utilized exclusively in any transportation vehicle including motor vehicles and aircraft.
*A digital device used exclusively as an electronic control or power system utilized by a public utility or in an industrial plant. The term public utility includes equipment only to the extent that it is in a dedicated building or large room owned or leased by the utility and does not extend to equipment installed in a subscriber's facility.
*A digital device used exclusively as industrial, commercial, or medical test equipment.
*A digital device utilized exclusively in an appliance, e.g., microwave oven, dishwasher, clothes dryer, air conditioner (central or window), etc.
*Specialized medical digital devices (generally used at the direction of or under the supervision of a licensed health care practitioner) whether used in a patient's home or a health care facility. Non-specialized medical devices, i.e., devices marketed through retail channels for use by the general public, are not exempted. This exemption also does not apply to digital devices used for record keeping or any purpose not directly connected with medical treatment.
*Digital devices that have a power consumption not exceeding 6 nW.
Joystick controllers or similar devices, such as a mouse, used with digital devices but which contain only non-digital circuitry or a simple circuit to convert the signal to the format required (e.g., an integrated circuit for analog to digital conversion) are viewed as passive add-on devices, not themselves directly subject to the technical standards or the equipment authorization requirements.
*Digital devices in which both the highest frequency generated and the highest frequency used are less than 1.705 MHz and which do not operate from the AC power lines or contain provisions for operation while connected to the AC power lines.
*Digital devices that include, or make provision for the use of, battery eliminators, AC adaptors or battery chargers which permit operation while charging or that connect to the AC power lines indirectly, obtaining their power through another device which is connected to the AC power lines, do not fall under this exemption.
The FCC And Open Source Hardware
A: Define it as OEM device instead of Consumer device. In such case, it's the end user that should put it in an adequate shielded case and perform the FCC tests if he intends to sell the device. If it's just for his personal use, it's considered evaluation. It's surely worth checking if such backdoors exist in the law.
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\section{Introduction}
Heterogeneous graphs (HGs) model compositions of different types of entities and relations and naturally emerge in various real-world applications. Such examples include bibliographic networks, social networks, and recommendation systems. Encoding this high-dimensional, non-Euclidean, and heterogeneous information to a low-dimensional Euclidean embedding space facilitates various data related tasks that benefit from operating on Euclidean spaces, such as classification, clustering, and link prediction. Such an encoding could be done, for example, by using traditional machine learning techniques, which have been proven to be powerful when learning embeddings (a.k.a. representations). However, the graph nature (non-Euclidean structure) and heterogeneity of HGs pose great challenges to their representation learning. Moreover, embedding HGs based on self-supervision signals allows the re-usability of the representations for various tasks and fine-tuning on specific tasks with few labeled data.
When dealing with homogeneous graphs, different self-supervised graph embedding methods have been proposed for learning representations. Early approaches rely on random walk techniques (e.g., DeepWalk~\cite{perozzi2014deepwalk}) and they mainly preserve the input structure, i.e., nodes that are close in the input should have similar representations. The limitation of these methods is that they are not inherently designed to handle additional feature data of nodes, which are present in most real-world graphs. As a result, graph neural networks (GNNs) have been proposed~\cite{kipf2016semi}, which are deep learning methods designed for handling both the graph structure and the node attributes. GNNs mainly estimate node representations through a recursive neighborhood aggregation scheme. Since GNN already capture the structural information, recent graph embedding methods learn the representations to encode different scales of the data (such as clusters of nodes, e.g., GIC~\cite{mavromatis2020graph}), which conveys higher order information.
A natural approach of HG representation learning is to extend the existing homogeneous-designed methods such as they handle the heterogeneity of the data. A common way is to model different compositions of HG relations as different meta-paths~\cite{sun2011pathsim}, which then, can be treated as homogeneous graphs by disregarding intermediate relations. Representations of each meta-path can be encoded and optimized by any self-supervised method for homogeneous graphs (e.g., random walk on each meta-path~\cite{dong2017metapath2vec}, or a GNN encoder with GIC self-supervision). Oftentimes, in order to capture the full semantics of the data, it may be necessary to combine and fuse information across various meta-paths. This is mostly achieved with heterogeneous graph neural networks (HGNNs), which aggregate the different meta-path representations to a compact representation (e.g., HAN~\cite{wang2019han}). In this case, the supervision is directly optimizing the compact representation, which is used for downstream tasks.
\textbf{Contribution}. In this paper, we advocate that relying solely on homogeneous-designed self-supervision signals is not sufficient for learning good representations, since they do not help different meta-paths to learn from each other. We propose a heterogeneous-designed method, HeMI, that allows information exchange and knowledge discovery across meta-paths. Specifically, per-meta-path node representations are aggregated together to a fused representation and these representations are jointly optimized so that their mutual information is maximized. As a result, the fused representations encode information that is mutually helpful to all meta-paths and integrate globally shared semantics to each meta-path representation.
We evaluate HeMI on four standard datasets using node classification, node clustering and link prediction as the downstream tasks. Experiments show that HeMI outperforms both unsupervised and semi-supervised competing approaches in all tasks. HeMi's improvement over the \emph{best} competing method is over 1\%, 2\%, and 3\% in the majority of the datasets for node classification, node clustering and link prediction, respectively. HeMI is also powerful as an augmented loss objective providing better performance by 2\% and 10\% for node classification and link prediction, respectively.
\section{Preliminary}
\begin{figure}[t]
\centering
\includestandalone[width=\textwidth]{figs/example}
\caption{An illustrative example of a heterogeneous graph (citation network) with three types of nodes and three types of edges. }
\label{fig:example}
\end{figure}
\textbf{Heterogeneous graph}~\cite{sun2013mining}. A heterogeneous graph (HG), denoted as $\mathcal{G} = (\mathcal{V}, \mathcal{E})$, consists of an object set $\mathcal{V}$ and a link set $\mathcal{E}$. It is used to represent a network with
multiple types of nodes $\mathcal{A}$ and links $\mathcal{R}$. A heterogeneous graph is associated with a node type
mapping function $\chi_\mathcal{V}: \mathcal{V} \xrightarrow{} \mathcal{A}$ and a link type mapping function
$\chi_\mathcal{E}: \mathcal{E} \xrightarrow{} \mathcal{R}$. $\mathcal{A}$ and $\mathcal{R}$ denote the sets of predefined node types and
link types, where $|\mathcal{A}| + |\mathcal{R}| > 2$. The type of a link automatically defines the types of nodes on its two ends.
Fig.~\ref{fig:example}(a) shows an example of a HG where $\mathcal{V}$ consists of \{paper, keyword, author\} nodes and $\mathcal{E}$ consists of \{cites, contains, writes\} links. Here, $\chi_\mathcal{V}$ corresponds to the shape used for illustration of each node type (circle, rectangular, and diamond) and $\chi_\mathcal{E}$ corresponds to the color used for illustration of each link type (green, blue, and red).
\textbf{Meta-path}~\cite{sun2011pathsim}. A meta-path $P$ is defined as a path in the form $A_1 \xrightarrow{R_1} A_2 \xrightarrow{R_2} \cdots \xrightarrow{R_l} A_l$ (abbreviated as $A_1A_2 \cdots A_l$), which describes a composite relation $R = R_1 \circ R_2 \cdots \circ R_l $ between node types $A_1$ and $A_l$, where $\circ$ denotes the composition operator on relations.
In Fig.~\ref{fig:example}(c), example meta-paths include
\begin{itemize}
\item Paper $\xrightarrow[]{\text{cites}}$ Paper (PP)
\item Paper $\xrightarrow[]{\text{contains}}$ Keyword $\xrightarrow[]{\text{contained}}$ Paper (PKP)
\item Paper $\xrightarrow[]{\text{written}}$ Author $\xrightarrow[]{\text{writes}}$ Paper (PAP).
\end{itemize}
Different meta-paths reveal different semantics of the data. Note that we could obtain more meta-paths, such as PAPK, PKPPA, etc.
\textbf{Meta-path based neighbor}.
Given a meta-path $P$ whose starting vertices are of type $A_1$ and ending vertices are of type $A_l$, the meta-path based neighbors $\mathcal{N}^P_v$ of $v \in A_1$ are defined as the set of $A_l$ nodes that connect with $v$ via meta-path $P$ (following the right type of relations).
Note that the node's neighbors include itself.
Fig.~\ref{fig:example}(d) shows the meta-path based neighbors of the upper paper node via meta-path PP.
\textbf{Meta-path based graph}. Given a meta-path $P$ whose starting vertices are of type $A_1$ and ending vertices are of type $A_l$, the meta-path based graph $\mathcal{G}^P$ is a graph constructed from all node pairs $v \in A_1$ and $u \in A_l$ that connect via meta-path $P$. Meta-path based graphs can exploit different aspects of structure information in heterogeneous graph.
\textbf{Heterogeneous graph representations}. Given a
heterogeneous graph $\mathcal{G} = (\mathcal{V}, \mathcal{E})$, with node attribute matrix $\mathbf{X} \in \mathbb{R}^{|\mathcal{V}| \times d_{\text{in}}}$, heterogeneous graph embedding is the task of learning the $d$-dimensional node representations $\mathbf{z}_v \in \mathbb{R}^d$ for all $v \in \mathcal{V}$. It is desired that the representations are able to capture rich structural and semantic information involved in $\mathcal{G}$. The learned (low-dimensional) representations can be applied to downstream graph-related tasks such as node classification, node clustering, and link prediction. We focus on learning the representations of specific types of nodes in this paper, i.e., $\mathbf{Z} \in \mathbb{R}^{|\mathcal{V}_t| \times d}$, where $\mathcal{V}_t$ is the set of nodes of the target types.
Here, we denote vectors by bold lower-case letters and matrices by bold upper-case letters. We also use the terms \emph{representation} and \emph{embedding} interchangeably.
\section{Related Work}\label{sec:rel}
\textbf{Homogeneous graph representation learning}.
Early graph representation learning approaches, such as DeepWalk (DW)~\cite{perozzi2014deepwalk}, optimize the embeddings so that nodes that tend to co-occur on short random walks over the graph have similar embeddings. Following the paradigm of DeepWalk, node2vec (N2V)~\cite{grover2016node2vec} and LINE~\cite{tang2015line} are more generalized versions of DeepWalk with biased random walks and additional proximity measures, respectively. Since these methods are not inherently designed to handle node attributes, GNNs have been proposed to encode both the structural and semantic information of the graph and its attributes. Widely used GNNs include GCN~\cite{kipf2016semi}, which at each layer learns to convolve features (or embeddings) of 1-hop neighbors, and GAT~\cite{velickovic2018graph}, which substitutes the statically normalized convolution operation of GCN with an attention mechanism. Leveraging GNNs to handle the structural properties of the graph, recent self-supervised methods do not rely on structure-based proximities, but optimize the representations based on higher order similarities. DGI~\cite{velickovic2018deep} encodes information that is shared among all nodes of the graph, while GIC~\cite{mavromatis2020graph} extends DGI to encode additional shared information within clusters of nodes. In a similar manner, MVGRL~\cite{hassani2020contrastive} uses data augmentation techniques to obtain different views of the graph and to encode the information that is shared among these views.
\noindent
\textbf{Heterogeneous graph representation learning}. Inspired by homogeneous graph representation learning, most heterogeneous based approaches handle and encode the heterogeneity of the graph by operating on meta-paths, which can also be treated as homogeneous graphs. For example, metapath2vec (M2V)~\cite{dong2017metapath2vec} and ESim~\cite{shang2016meta} are random walk methods applied on a single meta-path and multiple meta-paths, respectively. Unlike ESim, that cannot learn the importance of each meta-path, HERec~\cite{shi2018herec} utilizes a fusion algorithm to better combine information from different meta-paths. The fusion strategy inspired the development of HGNNs to better handle the heterogeneity of the graph compared to conventional GNNs. For example, HAN~\cite{wang2019han} uses a GAT-like encoder on each meta-path and learns to aggregate the information from these meta-paths with an additional attention mechanism. This strategy can be applied to any homogeneous self-supervised method to handle the heterogeneity of the graph; for example, we later extend GIC to heterogeneous GIC (HGIC). HDGI~\cite{ren2019hdgi} acts in similar manner to obtain compact representations, which are optimized with the DGI objective. DMGI~\cite{park2020dmgi} enhances the DGI objective with a consensus regularization so that meta-path representations are similar to each other. Different from most methods that treat each meta-path as a homogeneous graph, MAGNN~\cite{fu2020magnn} extends HAN by considering both the meta-path based neighborhood and the nodes along the meta-path, so that all the meta-path information is preserved for learning. Finally, since most heterogeneous graphs can be treated as knowledge graphs and vice versa, many learning methods on knowledge graphs, e.g., RGCN~\cite{schlichtkrull2018rgcn} and RotatE~\cite{sun2019rotate}, can often be applied to heterogeneous graph learning. Applications of HGNNs range from citations networks~\cite{wang2019han} and recommendation systems~\cite{shi2018herec} to biological networks~\cite{shui2020hmgnn} and drug interactions~\cite{ioannidis2020panrep}.
\section{Methodology}
\subsection{Hypothesis and motivation}
We assume that the HG $\mathcal{G}$ is given as distinct meta-path graphs $\mathcal{G}^{P_1}, \dots, \mathcal{G}^{P_M}$, which can be regarded as different views (multi-views) -- each describing a different perspective.
We also assume that all meta-paths contain useful information that needs to be encoded (the meta-paths could be provided by experts or data engineers) and our methodology is postulated on the following:
First, specific knowledge of one view can be exploited by other views in order to better describe the underlying data and unveil hidden semantics. Thus, \emph{meta-path fusion}, i.e., information exchange, among views seems necessary in order to obtain good representations.
Second, good representations are the ones that simultaneously benefit all meta-paths. Thus, \emph{meta-path consensus} among the views is needed in order to decide which information is globally helpful and to prevent over-emphasizing or neglecting information associated with specific meta-paths.
For the example in Fig.~\ref{fig:example}, suppose we wish to make recommendations based on a particular paper. Then, a natural approach is to regard simultaneously which papers it cites, which keywords it contains, and by whom authors it is written (meta-path fusion). Since not all citations, keywords or authors are equally important (e.g., out-dated prior work, keywords co-used for multiple domains, co-authors of different disciplines), a good representation of the paper is the one that preserves the information that best describes its overall contribution (meta-path consensus). By computing such representations, good recommendations include papers with similar scientific contributions.
\begin{figure}[t]
\centering
\includestandalone[width=0.7\textwidth]{figs/framework}
\caption{HeMI framework. The representations for node $v$ for meta-path $P_1$, $\mathbf{z}^{P_1}_v$, and for meta-path $P_2$, $\mathbf{z}^{P_2}_v$, are computed with encoders $f_1$ and $f_2$, respectively. The representations are aggregated with a function $g$ to the fused representation $\mathbf{z}_v$. In addition, nodes' representation in $P_1$ and $P_2$ are summarized to $\mathbf{s}^{P_1}$ and $\mathbf{s}^{P_2}$, respectively. All $\mathbf{z}^{P_1}_v, \mathbf{s}^{P_1}, \mathbf{z}^{P_2}_v$, and $\mathbf{s}^{P_2}$, along with $\mathbf{z}_v$, are optimized by maximizing their consensus, i.e., mutual information, with $\mathbf{z}_v$. Different MI estimators, $\mathcal{I}_{\psi_1}, \mathcal{I}_{\phi_1}, \mathcal{I}_{\psi_2},$ and $\mathcal{I}_{\phi_2}$, are employed for each consensus case.}
\label{fig:framework}
\end{figure}
\subsection{A mutual information maximization solution}
In this subsection, we present our proposed framework (a graphical illustration is given in Fig.~\ref{fig:framework}).
For each meta-path graph $\mathcal{G}^{P_j}$ we capture its structure and attributes by employing a GNN/HGNN encoder $f_j$ to obtain representations for each node $v$; i.e., $\mathbf{z}^{P_j}_v = f_j(\mathcal{G}^{P_j}_v)$.
In order to achieve meta-path information fusion, we use an aggregation function $g: \mathbb{R}^{|\mathcal{V}_t| \times d} \times \dots \times \mathbb{R}^{|\mathcal{V}_t| \times d} \xrightarrow{} \mathbb{R}^{|\mathcal{V}_t| \times d}$ which combines and compresses the representations $\mathbf{z}^{P_j}_v$ together to $\mathbf{z}_v$ such as $\mathbf{z}_v = g(\mathbf{z}^{P_1}_v, \dots, \mathbf{z}^{P_M}_v)$ (for simplicity, we assume the output dimensionality is the same).
The representations are optimized by focusing on achieving meta-path consensus. However, instead of achieving consensus between all meta-path combinations (which exponentially grow to the number of the meta-paths), we use $\mathbf{z}_v$ to globally guide the consensus between the meta-paths. In order to measure the consensus, we use the mutual information (MI) $\mathcal{I}$ between $\mathbf{z}_v$ and each one of the meta-path specific representations; i.e., $\mathcal{I}(\mathbf{z}^{P_j}_v; \mathbf{z}_v)$. We maximize the consensus between all $\mathbf{z}^{P_j}_v$ by maximizing the MI of each $\mathbf{z}^{P_j}_v$ with $\mathbf{z}_v$ to achieve global optimization, such as
\begin{equation}
\max \; \sum_j \mathcal{I}(\mathbf{z}^{P_j}_v; \mathbf{z}_v).
\end{equation}
The MI measure has several advantages. If the encoders $f_j$ are injective~\cite{xu2018powerful}, we have $\mathcal{I}(\mathcal{G}^{P_j}_v; \mathbf{z}_v) = \mathcal{I}(\mathbf{z}^{P_j}_v; \mathbf{z}_v)$, and thus can achieve MI maximization with the actual input; here, $\mathcal{G}^{P_j}_v$ is a subgraph centered around node $v$. Moreover, if a meta-path contains a lot of noise, this does not increase the MI and thus, is not preferred to be encoded; in contrast to other measures, such as Euclidean distances, which are affected by noisy instances.
Since mutual information is in general intractable, it is often estimated by a parametrized function $\mathcal{I}_\psi$, such as a deep neural network. The estimated MI is a lower bound of the true MI, which is maximized during optimization in the course of maximizing the true MI. The MI estimation becomes
\begin{equation}
\sum_j\mathcal{I}(\mathbf{z}^{P_j}_v,\mathbf{z}_v) \geq \sum_j\mathcal{I}_{\psi_j}(\mathbf{z}^{P_j}_v;\mathbf{z}_v).
\label{eq:lb1}
\end{equation}
In self-supervised settings for graphs, it is common to maximize the MI between representations and coarse-grain parts of the graph, such as the full graph representation~\cite{velickovic2018deep} or representations of clusters of nodes~\cite{mavromatis2020graph}. This facilitates to propagate higher order properties to the representations. We follow a similar strategy and compute the summary of each meta-path graph as $\mathbf{s}^{P_j} = c(\mathbf{Z}^{P_j})$, where $c: \mathbb{R}^{|\mathcal{V}_t| \times d} \xrightarrow{ } \mathbb{R}^{1 \times d}$. Due to the data processing inequality of MI and then by using MI estimators, we have
\begin{equation}
\mathcal{I}(\mathbf{z}^{P_j}_v; \mathbf{z}_v) \geq \mathcal{I}(\mathbf{s}^{P_j}; \mathbf{z}_v) \geq \mathcal{I}_{\phi_j}(\mathbf{s}^{P_j};\mathbf{z}_v).
\label{eq:lb2}
\end{equation}
Eq.~\eqref{eq:lb1} and Eq.~\eqref{eq:lb2} are two lower bounds of the desired MI $\sum_j \mathcal{I}(\mathbf{z}^{P_j}_v; \mathbf{z}_v)$, which both can be maximized.
By multiplying each bound with $\lambda$ and $(1-\lambda)$, respectively, and adding them together, we obtain the final objective
\begin{equation}
\max \sum_j \lambda \mathcal{I}_{\psi_j}(\mathbf{z}_v^{P_j},\mathbf{z}_v) + (1-\lambda) \mathcal{I}_{\phi_j}(\mathbf{s}^{P_j},\mathbf{z}_v),
\label{eq:obj}
\end{equation}
where $\lambda \in \mathbb{R}$ controls the relative importance of each objective term.
Our proposed method is designed to compute embeddings that optimize Eq.~\eqref{eq:obj}.
Since our method relies on \underline{He}terogeneous graph \underline{MI} maximization, we name our method \underline{HeMI}.
\subsection{Implementation}
\textbf{MI estimators}. In order to estimate the MI we use the Jensen-Shannon MI estimator. The estimator's objective maximizes the expected $\log$-ratio of the samples from the joint distribution (positive examples) and the product of marginal distributions (negative examples). It acts as a binary cross-entropy (BCE) loss between positive and negative examples with an aid of a function $T_\psi: \mathbb{R}^d \times \mathbb{R}^d \xrightarrow{} \mathbb{R}$ with learnable parameters $\psi$, which assigns higher scores to the positive than the negative examples. The objective for each $v \in \mathcal{V}_t$ is
\begin{equation}
\min \; \lambda \mathcal{L}_f + (1-\lambda) \mathcal{L}_c,
\label{eq:loss}
\end{equation}
where
\begin{align*}
\mathcal{L}_f = & \frac{1}{M} \sum_j \Big( \log T_{\psi_j}(\mathbf{z}^{P_j}_v, \mathbf{z}_v) + \log \big(1 - T_{\psi_j}(\mathbf{z}^{P_j}_v, \tilde{\mathbf{z}}_v) \big) \Big), \numberthis
\label{eq:fine}
\\
\mathcal{L}_c = & \frac{1}{M} \sum_j \Big( \log T_{\phi_j}(\mathbf{s}^{P_j}, \mathbf{z}_v) + \log \big(1 - T_{\phi_j}(\mathbf{s}^{P_j}, \tilde{\mathbf{z}}_v) \big) \Big). \numberthis
\label{eq:coarse}
\end{align*}
Positive examples are pairings of $(\mathbf{z}^{P_j}_v, \mathbf{z}_v)$ and $(\mathbf{s}^{P_j}, \mathbf{z}_v)$, and negatives are pairings of $(\mathbf{z}^{P_j}_v, \tilde{\mathbf{z}}_v)$ and $(\mathbf{s}^{P_j}, \tilde{\mathbf{z}}_v)$. We obtain $\tilde{\mathbf{z}}_v$ as the output of our model with a corrupted input, which in our case is the original graph with row-shuffled features (for node $v$ we use the attributes of a randomly selected node). As functions $T_{\psi_j}, T_{\phi_j}$, we use a bilinear scoring function, followed by a logistic sigmoid nonlinearity $\sigma(\cdot)$, which converts scores into probabilities, as $T_{\psi_j}(\mathbf{z}^{P_j}_v, \mathbf{z}_v) = \sigma(\mathbf{z}^{P_j}_v \mathbf{W}_{\psi_j} \mathbf{z}_v)$ and $T_{\phi_j}(\mathbf{z}^{P_j}_v, \mathbf{z}_v) = \sigma(\mathbf{s}^{P_j} \mathbf{W}_{\phi_j} \mathbf{z}_v)$, where $\mathbf{W}_{\psi_j}, \mathbf{W}_{\phi_j} \in \mathbb{R}^{d \times d}$ are learnable weights. Finally, the summary of each meta-path is computed as $\mathbf{s}^{P_j} = \sigma \big( \frac{1}{|\mathcal{V}_t|} \sum_v \mathbf{z}^{P_j}_v \big)$.
\noindent
\textbf{Information fusion}. Recall that in order to fuse information among meta-paths, we use an aggregation function $g$ as $\mathbf{z}_v= g(\mathbf{z}^{P_1}_v, \dots,\mathbf{z}^{P_M}_v)$. We employ an attention based function $g$, which is common in the literature~\cite{wang2019han,fu2020magnn} to learn the importance of each meta-path. Specifically, we first compute
\begin{equation}
\mathbf{e}^{P_j} = \frac{1}{|\mathcal{V}_t|} \sum_v \mathbf{q}^T ( \mathbf{W}_{\text{sem}} \mathbf{z}^{P_M}_v + \mathbf{b} ),
\end{equation}
where $\mathbf{W}_{\text{sem}} \in \mathbb{R}^{d_m \times d}$, $\mathbf{b} \in \mathbb{R}^{d_m}$ are learnable parameters and $\mathbf{q} \in \mathbb{R}^{d_m}$ is a parametrized attention vector. The importance of each meta-path $\beta^{P_j}$ is obtained by normalizing $\mathbf{e}^{P_j}$ with a softmax function and the representation $\mathbf{z}_v$ is the weighted average of each meta-path graph representation as
\begin{equation}
\mathbf{z}_v = \sum_j \beta^{P_j} \mathbf{z}^{P_j}_v, \text{ with } \beta^{P_j} = \frac{\exp(\mathbf{e}^{P_j})}{\sum_j \exp(\mathbf{e}^{P_j})}.
\end{equation}
\noindent
\textbf{Encoders}. In order to compute $\mathbf{z}^{P_j}_v$ we can employ both GNN and HGNN encoders. When using GNN encoders, each meta-path is reduced to an adjacency matrix $\mathbf{A}^{P_j} \in \mathbb{R}^{|\mathcal{V}_t| \times |\mathcal{V}_t|}$ with $\mathbf{A}^{P_j}_{vu} = 1$ if nodes $v$ and $u$ connect through meta-path $P_j$, and $\mathbf{A}^{P_j}_{vu} = 0$ otherwise. For example if we use a GCN~\cite{kipf2016semi} encoder, $\mathbf{Z}^{p_j}$ is computed as
\begin{equation}
\mathbf{Z}^{p_j} = \sigma( {\mathbf{D}^{P_j}}^{-0.5} \bar{\mathbf{A}}^{P_j}{\mathbf{D}^{P_j}}^{-0.5} \mathbf{X}_t \mathbf{W}^{P_j}_t ),
\end{equation}
where $\bar{\mathbf{A}}^{P_j} = \mathbf{A}^{P_j} + \mathbf{I}_t$ with $\mathbf{I}_t$ the identity matrix, $\mathbf{D}^{P_j}$ is the diagonal node degree matrix of $\bar{\mathbf{A}}^{P_j}$, $\mathbf{W}^{P_j}_t \in \mathbb{R}^{d_{\mathcal{V}_t} \times d}$ is a learnable matrix, and $\sigma$ is a nonlinear activatios such as PReLU.
HGNN encoders, such as MAGNN~\cite{fu2020magnn}, offer the advantage that they also encode intermediate nodes (of a different type) between two nodes of the same type for each meta-path (intra-meta-path information). Node attributes of different node types may have different dimensions, so first, they are projected to a space with same dimensions as $ \mathbf{X}'_t =\mathbf{X}_t \mathbf{W}_t $, where $\mathbf{W}_t \in \mathbb{R}^{d_{t} \times d'}$.
Let $P(v,u)$ be a meta-path instance connecting the target node $v$ and the meta-path-based neighbor $u \in \mathcal{N}^P_v$, the intermediate nodes of $P(v,u)$ are defined as $\{m^{P(v,u)}\} = P(v,u) \text{\textbackslash} \{u,v\}$. An intra-meta-path encoder $g_{P(v,u)}$ transforms all the node features along a meta-path into a single vector, for example at the first layer, as
\begin{equation}
\mathbf{h}_{P(v,u)} = g_{P(v,u)}\Big( \mathbf{x}'_v, \mathbf{x}'_u, \big\{\mathbf{x}'_k, \forall k \in \{m^{P(v,u)}\}\big\}\Big),
\label{eq:aggr}
\end{equation}
where $\mathbf{h}_{P(v,u)} \in \mathbb{R}^{d'}$. Various useful $g_{P(v,u)}$ meta-path encoders can be found in~\cite{fu2020magnn}, and the final representation can be obtained by any aggregator (mean, attention, sum, etc.); e.g., the mean aggregator computes $\mathbf{h}^{P}_{v} = \frac{1}{\mathcal{N}^P_v} \sum_{u \in \mathcal{N}^P_v} \mathbf{h}_{P(v,u)}$, followed usually by a nonlinear activation, such as PReLU or ELU.
\section{Experimental Methodology}
In this section, we describe the experimental configuration and evaluation of our and competing methods.
We found out that competing methods treat the same dataset in various different ways to report their performance. In order to show that our method is efficient under different configurations, we employ two different evaluation protocols (EP) as proposed in~\cite{ren2019hdgi} (EP1) and in~\cite{fu2020magnn} (EP2).
\subsection{Datasets}
Four widely used heterogeneous graph datasets from different domains are adapted to evaluate the performance of HeMI. Specifically, we use the ACM, DBLP, IMDB, and Last.fm datasets\footnote{\url{https://dl.acm.org/}, \url{https://dblp.uni-trier.de/}, \url{https://www.imdb.com/}, \url{https://www.last.fm/}}, of which statistics are summarized in Table~\ref{tb:data} and their detailed descriptions in the Supplementary Material. DBLP and IMDB have different configuration for EP1 and EP2. For IMDB, we extract additional labels from the raw data in Section~\ref{sec:hess}, as explained in the Supplementary Material. For nodes without attributes, we assign them one-hot identity vectors.
\addtolength{\tabcolsep}{2pt}
\begin{table}[t]
\centering
\caption{Datasets.}
\label{tb:data}
\resizebox{\textwidth}{!}{
\begin{tabular}{cllccc}
\toprule
Dataset & Node-type: \#Nodes & Edge-type: \#Edges & \#Features & Labels & Meta-paths\\
\midrule
\multirow{3}{*}{ACM} & Paper (P): 3,025 & Paper-Author: 9,744 & 1,870 & P: \textit{Database}, & PAP\\
& Author (A): 5,835& Paper-Subject: 3,025 & & \textit{Wireless Communication}, & PSP \\
& Subject (S): 56 & & & \textit{Data Mining} & \\
\midrule
\multirow{5}{*}{DBLP} & Author (A): 4,057 & Author-Paper: 19,645 & 334 & A: \textit{Database}, & APA\\
& Paper (P): 14,328& Paper-Conference: 14,328 & & \textit{Data Mining}, & APCPA \\
& Conference (C): 20 & Paper-Term-EP1: 88,420 & & \textit{Machine Learning}, & APTPA\\
& Term (T)-EP1: 8,789 & Paper-Term-EP2: 85,810 & &\textit{Information Retrieval} & \\
& Term (T)-EP2: 7,723 & & & & \\
\midrule
\multirow{4}{*}{IMDB-EP1} & Movie (M): 4,275 & Movie-Actor: 12,838 & 6,344 & M: \textit{Action}, & MAM\\
& Director (P): 2,082& Movie-Director: 4,280 & & \textit{Comedy}, & MDM \\
& Actor (A): 5,431 & Movie-Keyword: 20,529 & & \textit{Drama} & MKM\\
& Keyword (K): 7,313 & & & & \\
\midrule
\multirow{3}{*}{IMDB-EP2} & Movie (M): 4,278 & Movie-Actor: 12,838 & 1,232 & M: \textit{Action}, & MDM, MAM\\
& Director (P): 2,081 & Movie-Director: 4,278 & & \textit{Comedy}, & DMD, DMAMD \\
& Actor (A): 5,257 & & & \textit{Drama} & AMA, AMDMA\\
\midrule
\multirow{3}{*}{Last.fm} & User (U): 1,892 & User-User: 12,717 & - & & UU, UAU\\
& Artist (A): 17,632& User-Artist: 92,834 & identity & - &UATAU, AUA \\
& Tag (T): 1,088 & Artist-Tag: 23,253 & features & & AUUA, ATA\\
\bottomrule
\end{tabular}
}
\end{table}
\addtolength{\tabcolsep}{-2pt}
\subsection{Tasks}
The node-related tasks for evaluation are node classification, node clustering, and link prediction.
\begin{itemize}
\item \textbf{Node classification}. In the node classification task, the unsupervised learned representations are fed to a downstream classifier, while the supervised methods output the classification result as end-to-end models. The classification is repeated 10 times, and average Macro-F1 and Micro-F1 scores are reported.
In EP1, the training set sizes (i.e., given labels for training) is 20\% or 80\% of full datasets, and the validation set and test set sizes are fixed at 10\% of full datasets. The downstream classifier is logistic regression classifier which is optimized using the Adam~\cite{kingma2014adam} with a learning rate of~0.001 and with an early stopping on the validation accuracy.
In EP2, the training, validation, and testing sets are of 400 (9.35\%), 400 (9.35\%), and 3478 (81.30\%) nodes respectively in IMDB and 400 (9.86\%), 400 (9.86\%), and 3257 (80.28\%) nodes respectively in DBLP. The downstream classifier is a linear SVM classifier with default parameters, which only uses the test node representations (i.e., 3478
nodes for IMDb and 3257 nodes for DBLP). For training, the test nodes are re-divided into training and testing proportions, namely 20\%, 40\%, 60\% and 80\%.
\end{itemize}
\begin{itemize}
\item \textbf{Node clustering}. In the node clustering task, the $K$-Means algorithm is used to conduct the clustering based on the learned representations. The number of clusters $K$ is set as to the number of target node classes. Unsupervised methods do not use any label during learning representations, while supervised methods do. The clustering process is repeated 10 times and the average normalized mutual information (NMI) and adjusted rand index (ARI) evaluation metrics are reported (computed for all nodes in EP1 and computed for the test nodes in EP2).
\end{itemize}
\begin{itemize}
\item \textbf{Link prediction}. In link prediction, the goal is to correctly predict whether two nodes connect or do not connect. Usually, some positive and negative edges of the original graph are masked and used to test the predictions. Given two node representations $\mathbf{z}_u$ and $\mathbf{z}_v$ generated by the trained model, the probability that $u$ and $v$ link together is calculated as $p(u,v) = \sigma( {\mathbf{z}_u}^T \mathbf{z}_v)$, where $\sigma(\cdot)$ is the sigmoid function. The probability should be close to 1 for positive edges and close to 0 for negative edges. The evaluation is performed by reporting the mean area under the ROC curve (AUC) and average precision (AP) scores after 5 runs.
For EP1, we mask 50\% (45\% for testing and 5\% for validation) of edges (and we obtain an equal number of negative edges) of each meta-path and use the computed representations to predict them. Heterogeneous methods use the same representation for all meta-path link predictions, while homogeneous use the per-meta-path representation for each meta-path.
For EP2 and Last.fm, the graph is divided into batches with size of 64. To make the task more challenging, we train the model on the subgraphs of the obtained batches, and evaluate to an equal number of completely unseen subgraphs of other batches. The number of batches experimented is 1, 10, 50.
\end{itemize}
\begin{itemize}
\item \textbf{Loss augmentation with HeMI}. We also evaluate the effectiveness of HeMI by employing it as an additional loss function to end-to-end trained models. The final representations of the models are obtained by $\mathbf{h}_v = \mathbf{W}_{\text{task}} \mathbf{z}_v$, where $\mathbf{W}_{\text{task}}$ are learnable weights with proper dimensions. Then, $\mathbf{h}_v$ is optimized based on a task specific loss.
In semi-supervised node classification, we minimize the cross-entropy over all labeled nodes between the ground-truth labels and the predicted labels as
\begin{equation}
\mathcal{L}_{\text{nc}} = - \sum_{v \in \mathcal{V}_Y} \mathbf{Y}_v \ln \mathbf{H}_v,
\end{equation}
where $\mathcal{V}_Y$ is the set of nodes that have labels, and $\mathbf{Y}_v$ and $\mathbf{H}_v$ are the labels and the predicted label probabilities of the labeled nodes, respectively.
In link prediction, we minimize the binary cross-entropy loss function between positive and negative edges as
\begin{equation}
\mathcal{L}_{\text{lp}} = - \sum_{ \{(v,u)\}} \log \sigma( {\mathbf{h}_v}^T \mathbf{h}_u) - \sum_{\{(v,u')\}} \log \sigma( -{\mathbf{h}_v}^T \mathbf{h}_{u'}),
\end{equation}
where $\sigma(\cdot)$ is the sigmoid function, $\{(v,u)\}$ is the set of observed (positive) node pairs, $\{(v,u')\}$ is the set of negative node pairs sampled from all unobserved node pairs (the complement of $\{(v,u)\}$).
In both cases, HeMI is used to optimize the intermediate representations $\mathbf{z}_v$ based on Eq.~\eqref{eq:loss}.
\end{itemize}
\subsection{Reproducibility}
For competing methods, we report their performance by reusing the reported results of~\cite{ren2019hdgi,fu2020magnn}. Whenever we need to implement a method by our own, we do it based on their original implementations and proposed configurations\footnote{\url{https://github.com/cmavro/Graph-InfoClust-GIC}}, and for a fair comparison, we use the same encoders for these methods as explained below. All competing approaches are described in Section~\ref{sec:rel}.
For HeMI, we set representations' dimensions to $d=256$ and the semantic attention dimensions to $d_m=16$, in all experiments; we only set $d=64$ in link prediction tasks. In EP1 experiments, we use 1-layer GCN encoders, and for IMDB dataset only, we use the same GCN encoder for all views. In EP2 experiments, we use either a neighbor-based MAGNN encoder ($\text{MAGNN}_{\text{nb}}$) or a RotatE~\cite{sun2019rotate} MAGNN encoder ($\text{MAGNN}_{\text{rot}}$) with layers selected from \{1,2\} and we use a mean aggregator in Eq.~\eqref{eq:aggr}. HeMI is optimized with Adam, with a learning rate of 0.001, for epochs selected from \{1000,2000\} with an early stopping of 50 patience epochs on the training loss. The hyper-parameter $\lambda$, which controls the relative importance of each objective term, is selected from \{0, .25, .5, .75, 1\}, with a parameter study provided in the Supplementary Material.
We implement HeMI by modifying HDGI original code\footnote{\url{https://github.com/YuxiangRen/Heterogeneous-Deep-Graph-Infomax}} (which is implemented with PyTorch~\cite{paszke2017automatic}) for EP1, and by modifying MAGNN original code\footnote{\url{https://github.com/cynricfu/MAGNN}} (which is implemented with PyTorch and Deep Graph Library~\cite{wang2019dgl}) for EP2. All experiments were performed on a Nvidia Geforce RTX-2070 GPU on a i5-8400 CPU and 32GB RAM machine.
\addtolength{\tabcolsep}{2pt}
\begin{table}[t]
\centering
\caption{Node classification and clustering performance for EP1.}
\label{tb:hyp-cl}
\resizebox{0.7\textwidth}{!}{
\begin{threeparttable}
\begin{tabular}{lc|cc|cc|cc}
\toprule
& Fusion/ & \multicolumn{2}{c}{ACM} & \multicolumn{2}{c}{DBLP} & \multicolumn{2}{c}{IMDB} \\
& Consensus& Classif. & Clust. & Classif. & Clust. & Classif. & Clust.\\
\midrule
DGI+concat & No/No & 93.28 & 67.31 & 93.21 & \textbf{74.97} & 59.54 & 1.61 \\
GIC+concat & No/No & 93.18 & 67.92 & \textbf{93.28} & 73.33 & 62.11 & 5.95 \\
MVGRL & No/Yes & 92.19 & 66.75 & 92.40 & 72.79 & 60.37 & 2.07 \\
HDGI & Yes/No & 92.27 & 71.83 & 91.75 & 69.25 & 58.93 & 1.87 \\
HGIC & Yes/No & 93.58 & 69.32 & 92.57 & 69.80 & 57.45 & 8.87 \\
DMGI & Yes/Yes & 92.55 & 67.48 & 80.96 & 56.07 & 58.88 & 2.18 \\
\textbf{HeMI} & Yes/Yes & \textbf{93.97} & \textbf{73.45} & 93.06 & 69.98 & \textbf{63.65} & \textbf{12.23}\\
\bottomrule
\end{tabular}
\begin{tablenotes}
\item The scores reported are Micro-F1 with 20\% train nodes and NMI, for classification (Classif.) and clustering (Clust.), respectively.
\end{tablenotes}
\end{threeparttable}
}
\end{table}
\addtolength{\tabcolsep}{-2pt}
\addtolength{\tabcolsep}{2pt}
\begin{table}[t]
\centering
\caption{Link prediction performance for EP1.}
\label{tb:hyp-lp}
\resizebox{\textwidth}{!}{
\begin{threeparttable}
\begin{tabular}{lc|ccc|cccc}
\toprule
& & \multicolumn{3}{c}{ACM} & \multicolumn{4}{c}{DBLP} \\
& Fusion/ & PAP & PSP & avg. & APA & APTPA & APCPA & avg. \\
&Consensus & AUC/AP & AUC/AP & AUC/AP. & AUC/AP & AUC/AP &AUC/AP &AUC/AP \\
\midrule
DGI+concat & No/No & 63.26/62.34 & 51.27/54.38 & 57.26/58.36 & 73.79/72.10 & 61.30/62.61 & 72.00/76.50 & 69.03/70.41\\
GIC+concat & No/No & \underline{84.38/79.63} & 44.53/56.76 & 64.46/68.20 & 76.25/74.99 & \underline{73.39/74.59} & 80.24/82.57 & 76.63/77.38\\
MVGRL & No/Yes & 49.67/54.58 & 47.33/58.53 & 48.50/56.55 & 67.81/68.85 & 54.30/54.12 & 66.13/69.07 & 62.75/64.02 \\
HDGI & Yes/No & 71.50/72.15 & 62.05/58.28 & 66.77/65.21 & 81.51/81.94 & 69.00/70.55 & 80.12/80.01 & 76.88/77.50 \\
HGIC & Yes/No & 73.22/73.11 & 60.77/59.48 & 67.00/66.29 & 87.35/85.00 & 69.42/68.96 & 92.94/92.27 & 83.24/82.08 \\
DMGI & Yes/Yes & 69.71/62.44 & 66.23/60.17 & 67.97/61.31 & 88.96/88.94 & 68.11/69.83 & 67.84/67.20 & 74.98/75.32 \\
\textbf{HeMI} & Yes/Yes & 74.69/74.64 & \underline{70.77/64.92} & \textbf{72.87/69.24} & \underline{89.83/87.54} & 69.60/67.37 & \underline{96.54/96.74} & \textbf{85.32/83.88 } \\
\bottomrule
\end{tabular}
\begin{tablenotes}
\item We report scores for each meta-path graph (PAP and PSP in ACM, and APA, APTPA, and APCPA in DBLP) and overall scores of all meta-paths (avg.).
\end{tablenotes}
\end{threeparttable}
}
\end{table}
\addtolength{\tabcolsep}{-2pt}
\section{Results}
In this section, we wish to answer the following research questions (RQ):
\begin{itemize}
\item \textit{RQ1}. Is meta-path fusion and consensus important when learning HG representations?
\item \textit{RQ2}. How does the proposed method perform compared to other methods?
\item \textit{RQ3}. Can the proposed objective be used as an augmented loss function for better representation learning?
\end{itemize}
\subsection{Results using the first evaluation protocol (EP1)}\label{sec:hyp}
\textit{(RQ1\&RQ2)}. Table~\ref{tb:hyp-cl} shows that HeMI, which relies on both meta-path fusion and meta-path consensus, outperforms all other methods in ACM and IMDB. An exception is the DBLP dataset, since the labels (conference types) are directly obtained by the APCPA meta-path and thus, fusion does not convey additional information; here, DMGI performs the worst due to its strict consensus regularization. Moreover, for node classification, methods that simply concatenate the per-meta-path representations (DGI and GIC) can outperform their fusion-based counterparts (HDGI and HGIC) because the downstream classifier is powerful enough to extract hidden interactions, in the same way fusion does. However, this does not happen for node clustering since fusion-based methods compute more compact representations (with the exception of DBLP).
Table~\ref{tb:hyp-lp} shows that HeMI outperforms (avg. column) all other methods for link prediction tasks, indicating that is able to encode useful information from all meta-paths simultaneously. Note that fusion-based methods do not perform as well as they appear to overestimate information from specific meta-paths, e.g., HGIC and HDGI overestimate PAP in ACM. Although, non-fusion methods, like GIC, outperform other methods on some meta-paths, e.g., on PAP in ACM and APTPA in DBLP, they cannot generalize as well for all meta-paths, since information exchange is necessary. Finally, other consensus-based methods either lose the per-meta-path useful information for link prediction (MVGRL) or are not as powerful as HeMI (DMGI).
We also release full results for node classification, node clustering, and link prediction in the Supplementary Material.
\addtolength{\tabcolsep}{2pt}
\begin{table}[t]
\small
\centering
\caption{Node classification results for EP2.}
\label{tb:classif2}
\resizebox{\textwidth}{!}{
\begin{tabular}{ccc|cccc|ccccccc}
\toprule
Dataset & Metric& Train& \multicolumn{4}{c|}{Supervised} & \multicolumn{7}{c}{Unsupervised}\\
& & & GCN & GAT & HAN & MAGNN & LINE & N2V & ESim & M2V & HERec & HGIC & \textbf{HeMI}\\
\midrule
\multirow{8}{*}{IMDb} & \multirow{4}{*}{Macro-F1}& 20\% & 52.73 & 53.64 &56.19 & 59.35 & 44.04 & 49.00 & 48.37 & 46.05 & 45.61 & 57.93 & \underline{\textbf{59.40}}$\pm$ 0.59 \\
& & 40\% & 53.67 &55.50 &56.15& 60.27 & 45.45 &50.63& 50.09 &47.57 &46.80 &59.06 & \underline{\textbf{60.72}}$\pm$ 0.92\\
& & 60\% & 54.24& 56.46& 57.29& 60.66 & 47.09& 51.65& 51.45& 48.17 &46.84 & 59.53 & \underline{\textbf{61.51}}$\pm$ 0.90 \\
& & 80\% & 54.77& 57.43 &58.51 &61.44 & 47.49& 51.49& 51.37& 49.99 &47.73 & 59.57 & \underline{\textbf{62.87}}$\pm$ 1.31\\
\cline{2-14}
& \multirow{4}{*}{Micro-F1}& 20\% & 52.80& 53.64 &56.32& \underline{59.60} & 45.21& 49.94 &49.32& 47.22& 46.23 & 58.29 & \textbf{59.47}$\pm$ 0.58\\
& & 40\% & 53.76 &55.56 &57.32 & 60.50 & 46.92& 51.77& 51.21 &48.17& 47.89 & 59.32 & \underline{\textbf{60.76}}$\pm$ 0.87 \\
& & 60\% & 54.23& 56.47& 58.42& 60.88 & 48.35& 52.79& 52.53& 49.87& 48.19 & 59.81 & \underline{\textbf{61.51}}$\pm$ 0.92 \\
& & 80\% & 54.63& 57.40 &59.24 &61.53 & 48.98& 52.72 &52.54 &50.50& 49.11 & 59.84 & \underline{\textbf{62.88}}$\pm$ 1.29 \\
\midrule
\multirow{8}{*}{DBLP} & \multirow{4}{*}{Macro-F1}& 20\% & 88.00& 91.05& 91.69& 93.13 &87.16 &86.70 &90.68 &88.47& 90.82 & 92.66 & \textbf{93.50}$\pm$ 0.44 \\
& & 40\% & 89.00 &91.24& 91.96 &93.23 & 88.85& 88.07& 91.61& 89.91& 91.44& 92.85 & \textbf{93.67}$\pm$ 0.38\\
& & 60\% & 89.43& 91.42 &92.14 &93.57& 88.93& 88.69 &91.84& 90.50 &92.08 & 92.96 & \underline{\textbf{93.80}}$\pm$ 0.46 \\
& & 80\% & 89.98 &91.73 &92.50 &94.10 & 89.51 &88.93& 92.27 &90.86& 92.25 & 93.02 & \underline{\textbf{94.13}}$\pm$ 0.73\\
\cline{2-14}
& \multirow{4}{*}{Micro-F1}& 20\% & 88.51& 91.61& 92.33& 93.61 & 87.68& 87.21& 91.21& 89.02& 91.49& 93.33 & \underline{\textbf{94.01}}$\pm$ 0.42\\
& & 40\% & 89.22 &91.77 &92.57& 93.68 & 89.25& 88.51& 92.05& 90.36& 92.05 & 93.39 & \underline{\textbf{94.17}}$\pm$ 0.36 \\
& & 60\% & 89.57 &91.97& 92.72& 93.99 & 89.34& 89.09& 92.28& 90.94 &92.66 & 93.43 & \underline{\textbf{94.30}}$\pm$ 0.41 \\
& & 80\% & 90.33& 92.24 &93.23 &94.47 & 89.96 &89.37 &92.68 &91.31 &92.78 & 93.50& \underline{\textbf{94.59}}$\pm$ 0.67 \\
\bottomrule
\end{tabular}
}
\end{table}
\addtolength{\tabcolsep}{-2pt}
\addtolength{\tabcolsep}{2pt}
\begin{table}[t]
\small
\centering
\caption{Node clustering results for EP2.}
\label{tb:clust2}
\resizebox{0.8\textwidth}{!}{
\begin{tabular}{cc|cccc|ccccccc}
\toprule
Dataset & Metric & \multicolumn{4}{c|}{Supervised} & \multicolumn{7}{c}{Unsupervised}\\
& & GCN & GAT & HAN & MAGNN & LINE & N2V & ESim & M2V & HERec & HGIC & \textbf{HeMI}\\
\midrule
\multirow{2}{*}{IMDb}& NMI &7.46 &7.84 &10.79 &\underline{15.58}& 1.13& 5.22& 1.07& 0.89 &0.39 & 7.2 & \textbf{9.74}\\
&ARI &7.69& 8.87 &11.11 &\underline{16.74} &1.20 &6.02& 1.01& 0.22& 0.11& 7.3 & \textbf{10.04}\\
\midrule
\multirow{2}{*}{DBLP}& NMI &73.45& 70.73& 77.49 &\underline{80.81} & 71.02 &77.01& 68.33& 74.18& 69.03& 70.51 & \textbf{77.18}\\
&ARI &77.50 &76.04 &82.95& \underline{85.54}&76.52 &81.37& 72.22 &78.11 &72.45& 76.19 & \textbf{81.71}\\
\bottomrule
\end{tabular}
}
\end{table}
\addtolength{\tabcolsep}{-2pt}
\subsection{Results using the second evaluation protocol (EP2)}
\textit{(RQ2)}. Tables~\ref{tb:classif2}~and~\ref{tb:clust2} show the node classification and clustering performance, respectively, for EP2. As it can be seen, HeMI outperforms both semi-supervised and unsupervised methods for node classification. Specifically, HeMI outperforms the best performing unsupervised method by up to 3.3\% in IMDb and by up to 1.1\% in DBLP dataset. HeMI achieves slightly better than MAGNN (about 1\%) and outperforms other semi-supervised methods by more than 4\% and 1.3\% in IMDb and DBLP, respectively. For node clustering, HeMI performs better than other unsupervised HGNN/GNN methods in both datasets. However, it is not able to outperform semi-supervised methods HAN and MAGNN since they have already seen the labels during training. Here, we only report HGIC from methods of Section~\ref{sec:hyp}, since we found out that it performs the best with a MAGNN encoder and is a generalized version of HDGI.
\addtolength{\tabcolsep}{2pt}
\begin{table}[t]
\caption{Link prediction and HeMI as objectives for Last.fm.}
\label{tb:hemilp}
\small
\centering
\resizebox{0.7\textwidth}{!}{
\begin{tabular}{l|cc|cc|cc}
\toprule
Train/Test & \multicolumn{2}{c}{1 batch} & \multicolumn{2}{c}{10 batches} & \multicolumn{2}{c}{50 batches}\\
Metric & AUC & AP & AUC & AP & AUC & AP \\
\midrule
\textbf{Supervision} & & & & & & \\
Link Prediction & 65.65 & 56.50 & 67.95 & 56.85 & 87.46 & 83.74 \\
Link Prediction + HeMI ($\lambda=0$) & \textbf{78.56} & \textbf{66.96} & 71.49 & 60.12 & 88.28 & 83.80 \\
Link Prediction + HeMI ($\lambda=1$) & 73.17 & 63.92 & \textbf{75.52} & \textbf{65.83} & \textbf{94.37} & \textbf{95.16} \\
\bottomrule
\end{tabular}
}
\end{table}
\addtolength{\tabcolsep}{-2pt}
\addtolength{\tabcolsep}{2pt}
\begin{table}[t]
\caption{Node classification and HeMI as objectives for IMDB-EP2 with multiple labels.}
\label{tb:heminc}
\small
\centering
\resizebox{0.7\textwidth}{!}{
\begin{threeparttable}
\begin{tabular}{l|cc|cc|cc}
\toprule
Labels & \multicolumn{2}{c}{\textit{Genre}} & \multicolumn{2}{c}{\textit{Rating}} & \multicolumn{2}{c}{\textit{Budget}} \\
\midrule
& Classif. & Clust. & Classif. & Clust. & Classif. & Clust.\\
\midrule
\textbf{Supervision} & & & & & & \\
\textit{Genre} & 61.53/59.60 & 15.58 & 38.89/37.99 & 0.52 & 44.08/42.33 & 2.50\\
\textit{Genre}+HeMI & 62.62/\textbf{61.70} & \textbf{16.63} & 42.41/40.73 & 1.34 & 48.86/47.48 & 2.87\\
\cline{2-7}
\textit{Rating}& 47.42/44.88 & 1.09 & \textbf{44.58}/41.95 & 2.75 & 45.47/43.32 & 0.06\\
\textit{Rating}+HeMI& 49.43/45.81 & 1.17 & 43.14/\textbf{42.69} & \textbf{3.26} & 46.76/45.12 & 1.95\\
\cline{2-7}
\textit{Budget}& 51.03/47.58 & 2.80 & 38.62/37.04 & 1.82 & 47.27/45.90 & 2.97\\
\textit{Budget}+HeMI& 51.32/48.45 & 3.97 & 39.95/37.41 & 2.47 & 49.79/47.15 & 4.43\\
\cline{2-7}
HeMI& \textbf{62.88}/59.47 & 9.74 & 43.91/41.05 & 1.04 & \textbf{54.21}/\textbf{51.14} & \textbf{7.29}\\
\bottomrule
\end{tabular}
\begin{tablenotes}
\item The scores reported are averaged the Micro-F1 with 80\%/20\% train nodes and NMI, for classification (Classif.) and clustering (Clust.), respectively.
\end{tablenotes}
\end{threeparttable}
}
\end{table}
\addtolength{\tabcolsep}{-2pt}
\subsection{Loss augmentation with HeMI}\label{sec:hess}
\textit{(RQ3)}. Table~\ref{tb:hemilp} shows that HeMI benefits the MAGNN encoder when used together for link prediction. Specifically, when using 1 batch for training and 1 for testing, the fine-grain consensus ($\lambda=0$) leads to superior performance by more than 10\% compared to only using the link prediction loss. Here, the coarse-grain consensus ($\lambda=1$) does not work as well, since there are few nodes in the batch. By increasing the batches, the coarse-grain consensus' power is also increased and outperforms the link-prediction-only supervision by 8\%, in average.
Table~\ref{tb:heminc} shows that HeMI preserves useful information that can be used for various node classification tasks, compared to a supervised MAGNN encoder. When using \textit{Genre} labels with HeMI self-supervision, it outperforms the label-only supervision by more than 2.5\%, in average. \textit{Genre} labels depend more on the plot of a movie (node attributes) and HeMI gives extra importance on the features by using fake attributes.
\textit{Budget} label depends usually on the cast of a movie, e.g., director and actor, and consequently on the information MDM and MAM meta-paths provide. HeMI is designed to preserve this information and thus, leads to superior performance by more than 1\%, in average. \textit{Rating} labels are more challenging to predict, and thus, using labels during training facilitates the learning. Finally, HeMI as a self-supervision alone outperforms its semi-supervised counterpart for all labels and tasks, which indicates that it preserves all the information useful for the classification tasks.
\section{Conclusion}
In this paper, we have proposed HeMI for learning representations from heterogeneous graphs. HeMI relies on meta-path information fusion and consensus to optimize the representations. Experiments show that HeMI outperforms competing approaches in various node-related tasks, such as node classification, node clustering, and link prediction.
\subsubsection*{Acknowledgements}
This work was supported in part by NSF (1447788, 1704074, 1757916, 1834251), Army Research Office (W911NF1810344), Intel Corp, and the Digital Technology Center at the University of Minnesota. Access to research and computing facilities was provided by the Digital Technology Center and the Minnesota Supercomputing Institute.
\bibliographystyle{splncs04}
| {
"redpajama_set_name": "RedPajamaArXiv"
} | 4,884 |
ProMaster 71 series CNC point to point machining centers from Holz-Her available in three, four, and 5 axis configurations.
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Padawan.sublime
===============
**No plans for Windows support**
Padawan.sublime is a Sublime Text 3 plugin for [padawan.php server
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https://github.com/mkusher/padawan.php#how-to-use)
2. Install this plugin via [Package control](https://packagecontrol.io)
or clone this repo to `/path/to/your/sublime-text-3/Packages/`
# Demo video
Click the image below to watch a short video on what
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| {
"redpajama_set_name": "RedPajamaGithub"
} | 9,610 |
\section{Introduction}
\label{intro}
It has long been known that hadrons containing a single
charm or bottom quark exhibit approximate heavy quark spin
and flavour symmetries and can be described to a reasonable approximation
by an idealised system in which the heavy quark is taken
to be infinitely massive~\cite{isgur}. In this limit the heavy quark
symmetries become exact and the system consists of
light degrees of freedom bound to a static colour point source.
The static approximation can be systematically improved using a
power-counting rule in $\Lambda_{\rm{QCD}}/{\rm{m}}_{\rm{Q}}$, which
forms the basis for heavy quark effective theory (HQET).
For hadrons containing
a bottom quark this approach is particularly suitable. In this
case one expects leading-order corrections to be already quite small, at the
level of ten percent.
On the lattice, mass-dependent errors which arise from the discretisation
of the Dirac operator can limit the accuracy of simulations of heavy-light
systems. In principle, one can avoid such errors by using the HQET lagrangian to
describe the heavy quark dynamics. An alternative approach is to perform a number
of simulations both at unphysically light heavy quark masses using the lattice Dirac operator
and in the static limit and
to interpolate the results to the physical heavy quark mass guided by the predictions of HQET.
Therefore, precise calculations in the static limit are of considerable interest to the
lattice community.
In addition, little is known experimentally of the excited state spectrum of heavy-light hadrons
and accurate simulations of their simpler static-light counterparts will have considerable
phenomenological impact.
However, traditionally, simulations of static-light systems have been extremely noisy.
Ultimately, this stems from the fact that the use of conventional point-to-all propagators for the light
quarks restricts interpolating operators for the static-light hadron to a single
spatial lattice site.
This problem could be overcome if it were possible to compute
all elements of the light quark propagator.
This would allow
source and sink operators for the static-light correlators to be placed at every spatial lattice site, yielding
a dramatic increase in statistics. Indeed, simulations of static-light mesons are the natural
testing ground for all-to-all techniques since they require a single light quark propagator per configuration while
the static quark propagator is easy to evaluate. It is therefore not surprising that a number of groups
have attempted to tackle this particular problem~\cite{green,burch}.
In this contribution, we describe a preliminary study of the excited state spectrum of static-light mesons
using the all-to-all propagator technique introduced in Ref.~\cite{mise}.
In this approach, the important contribution of the low-lying eigenmodes of the
Dirac operator to the fermion propagator is computed exactly. A noisy estimate
is then used to correct for the contribution of the remaining eigenmodes.
This estimate is obtained by inverting the Dirac operator on a set of $\rm{Z}_{4}$ noise
sources which have been partitioned in some subset of quarkfield indices. This partitioning
or `dilution' yields a substantial reduction in the variance of the stochastic estimate.
This study has been performed on a 3+1 anisotropic lattice, where the temporal lattice spacing
$a_{t}$ is much finer than the spatial lattice spacing $a_{s}$.
The fine temporal lattice spacing proves particularly useful when determining a plateau in the effective
masses of the higher excited states.
The correlators have been computed on $\rm{N}_{\rm{f}}=2$
background configurations.
\section{Interpolating operators}
\label{sec:1}
Due to heavy quark spin symmetry, states which lie in the same
hyperfine multiplet are degenerate and in the continuum it is
conventional to label these multiplets by the angular momentum
and parity of the light degrees of freedom~\cite{isgur2}. For example, in the static
limit the pseudoscalar and vector correspond to $ {J_{\ell}}^{P_{\ell}} = {\frac{1} {2}}^{-}$.
At finite lattice spacing, it makes sense to classify states according to the
irreducible representations (irreps) of the octahedral point group $\rm{O}_{h}$~\cite{basak1}. This 48 element
group is the direct product of the 24 proper spatial rotations allowed by the lattice with the group consisting
of the identity and the space inversion operator. The relationship between these representations
and continuum quantum numbers is not difficult to determine. Restricting the continuum irreps
to the elements of the lattice symmetry group generates representations for $\rm{O}_{\rm{h}}$
which are in general reducible and identification of the constituent irreps is a straight-forward exercise in group theory.
Here, we take advantage of the degeneracies which arise from the heavy quark spin symmetry
and construct interpolating operators specifically for the light degrees of freedom.
Therefore, our operators transform according to the double-valued or spinorial representations of
${\rm{O}}_{\rm{h}}$.
There are 6 such representations
\begin{eqnarray}
{\rm{G}}_{1 \rm{u}}, {\rm{G}}_{1 \rm{g}}, {\rm{G}}_{2 \rm{u}}, {\rm{G}}_{2 \rm{g}}, {\rm{H}}_{\rm{u}},
{\rm{H}}_{\rm{g}}.
\end{eqnarray}
where the subscripts $u$ and $g$ denote representations which
are odd and even under spatial inversion respectively.
\begin{table}
\begin{center}
\begin{tabular}{|c|c|c|}
\hline
Lattice irrep & Dimension & ${J}$ \\
\hline
${\rm{G}}_{1}$ & 2 & $\frac{1} {2}, \frac{7} {2}$... \\
\hline
${\rm{G}}_{2}$ & 2 & $\frac{5} {2}, \frac{7} {2}$... \\
\hline
H & 4 & $ \frac{3} {2}, \frac{5} {2}, \frac{7} {2} $.. \\
\hline
\end{tabular}
\end{center}
\caption{Irreducible representations of the group of 24 proper spatial
lattice rotations with low-lying angular momentum content.}
\label{tab:J}
\end{table}
Table~\ref{tab:J} lists the irreps of the subgroup of proper rotations and
the angular momenta of their lower-lying constituent states.
From this table we see that states with the same angular momentum but with different ${{J}}_{z}$
may be scattered across the lattice irreps.
For example, states which transform according to the rows of the 6-dimensional
spin $\frac{5} {2}$ representation are divided between the ${\rm{G}}_{2}$ and H irreps.
In a numerical study we should therefore expect to observe energy levels in these irreps which are
degenerate, up to lattice artifacts, corresponding to the ${J}_{\ell} = \frac{5} {2}$ states.
In this study, we were particularly interested in orbital excitations of the static-light meson.
There is a single S-wave state with ${J}_{\ell}^{{P}_{\ell}}$ quantum numbers ${\frac{1} {2}}^{-}$,
the P-wave doublet is ${\frac{1} {2} }^{+}, {\frac{3} {2}}^{+}$ and the D-wave states are labelled
${\frac{3} {2}}^{-}, {\frac{5} {2}}^{-}$. Of these, only the S-wave and the ${\frac{1} {2}}^{+}$ P-wave
which lie in the $\rm{G}_{1 u}$ and $ \rm{G}_{1 g}$ irreps can be accessed using local operators, while all
other irreps require the use of extended interpolating operators. It is worth noting that when one uses
all-to-all light quark propagators the construction
of extended interpolating fields requires no additional fermion matrix inversions and the extra
computational cost incurred is minimal.
In order to determine accurately the excited state spectrum it is important to choose operators
which couple strongly to the states of interest and we tested a number of operators across the
lattice representations.
Further details of our choice of interpolating operators will be presented in Ref.~\cite{todhcai}.
\section{Simulation details}
The lattice Dirac and gauge actions used in this study
have previously been described in detail in Refs.~\cite{quarkaction,gaugeaction}.
Both are specifically formulated for 3+1 anisotropic lattices.
The Wilson-like fermion action is accurate to $\mathcal{O}(a_{t}, a_{s}^{3})$ and employs
stout-smeared spatial links to reduce radiative corrections.
The gauge action incorporates tree-level Symanzik and tadpole improvement and is accurate to
$\mathcal{O} ( a_{t}^{2}, a_{s}^{4}, \alpha_{s} a_{s}^{2} )$.
The anisotropic lattice breaks hypercubic symmetry and introduces two new parameters:
the bare quark and gauge anisotropies, denoted $\xi_{q}^{0}$ and $\xi_{g}^{0}$, which must be tuned such that the measured anisotropy, $a_{s}/a_{t}$, assumes a consistent value. In dynamical QCD the tuning procedure is quite complicated and is
detailed in Ref.~\cite{morrin}. In this study we use bare anisotropy values taken from that paper
of $\xi_{q}^{0} = 7.52$ and $\xi_{g}^{0} = 8.06$ which yields a measured anisotropy of 6 with a
three percent error.
We employ the standard Eichten-Hill action~\cite{eichten} for the static quark and the corresponding
static quark propagator is a product of unsmeared temporal links.
Measurements have been performed on 249 background configurations on an $8^{3} \times 80$ lattice
with a spatial lattice spacing of about $0.2\rm{fm}$.
The mass of the sea quarks and the light valence quark was found to be
approximately the strange quark mass. For the light quark propagator, we
computed the contribution of 100 low-lying eigenvectors of the
Dirac operator. Two independent $\rm{Z}_{4}$ noise sources which were
diluted in time and colour were used in the stochastic estimate.
To extract excited state energies we applied the standard variational
analysis to correlation matrices which were obtained by applying 5 or 6
levels of Jacobi smearing to the light quark fields. This was facilitated
by the use of all-to-all propagators which meant that no additional
inversions were required to smear the quark fields.
\section{Results}
Fig.~\ref{all} shows effective masses for the lowest-lying states across a number of the
lattice irreps. In this case the lowest effective mass corresponds to the S-wave ${\frac{1} {2}}^{-}$ state
and the higher-lying states are orbital excitations. The highest-lying states lie in D-wave channels.
Clear plateaux are evident in each channel.
In addition, we obtain strong signals for a number states
in each of the lattice irreps. Fig.~\ref{swave} shows the ground state and the
first two excited states
in the $\rm{G}_{1 u}$ (S-wave) irrep.
These excited states shown here
are almost certainly radial excitations in the S-wave channel.
Since we are working in the static limit the energies determined
by fitting to the effective masses contain an unphysical shift.
This shift cancels in energy differences, which are therefore physically
meaningful. Fig.~\ref{splitting} plots the energy differences between the lowest-lying
state in the $\rm{G}_{1 u}$ representation (i.e. the S-wave multiplet) and a number of
excited states. We have not presented these preliminary results in physical units
because we have performed our simulation on quite a small spatial volume which
might distort the energies of the orbitally and radially excited
states.
The main point here is the precision to which we can compute the spectrum.
Nevertheless, it is interesting to note the position of the lowest-lying states in the
$\rm{H}_{u}$ and $\rm{G}_{2 u}$ irreps. Naively, one expects the lightest state in the $\rm{H}_{u}$ representation
to correspond to the ${\frac{3} {2}}^{-}$ D-wave while the ${\frac{5} {2}}^{-}$ ought to be the lightest
state in the $\rm{G}_{2 u}$ irrep. However, there appears to be no significant splitting between
these states which may indicate that the ${\frac{5} {2}}^{-}$ multiplet
is lighter than the ${\frac{3} {2}}^{-}$. Such a scenario has been predicted
by quark models~\cite{schnitzer2,isgur3}.
\begin{figure}
\resizebox{0.48\textwidth}{!}{%
\includegraphics*{all.eps}
}
\caption{Effective masses for the lowest-lying states in 5 lattice irreps. These include P-wave and D-wave excitations.}
\label{all}
\end{figure}
\begin{figure}
\resizebox{0.48\textwidth}{!}{
\includegraphics*{swave.eps}
}
\caption{Effective masses for the lowest-lying states in the $G_{1 u}$ (S-wave) irrep.}
\label{swave}
\end{figure}
\begin{figure}
\resizebox{0.48\textwidth}{!}{
\includegraphics*{splitting.eps}
}
\caption{Mass differences between the ground state S-wave multiplet
and higher lying states.}
\label{splitting}
\end{figure}
\section{Conclusions and outlook}
We have presented preliminary results for the spectrum
of static-light mesons in $\rm{N}_{\rm{f}} = 2$ QCD. We have been
able to compute energy differences between the S-wave
multiplet and a number of orbital and radial excitations.
The use of all-to-all propagators allows us to minimise the
statistical uncertainty in our measurements and facilitates the
use of spatially-extended interpolating operators and the
variational approach.
The results presented here have allowed
us to make some tentative remarks about the nature of the spectrum;
however, we have not yet made a serious
attempt to identify the continuum quantum numbers of the observed states.
More recently we have
performed similar simulations using larger operator bases on
two different spatial volumes. These runs
should allow us to assess finite volume effects and identify
multi-particle states.
We are currently analysing the results of these
simulations and we intend to present our findings in the near future~\cite{todhcai}.
| {
"redpajama_set_name": "RedPajamaArXiv"
} | 9,969 |
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<h2 id="treebank-statistics-ud_gothic-proiel-relations-nummod">Treebank Statistics: UD_Gothic-PROIEL: Relations: <code class="language-plaintext highlighter-rouge">nummod</code></h2>
<p>This relation is universal.</p>
<p>195 nodes (0%) are attached to their parents as <code class="language-plaintext highlighter-rouge">nummod</code>.</p>
<p>149 instances of <code class="language-plaintext highlighter-rouge">nummod</code> (76%) are right-to-left (child precedes parent).
Average distance between parent and child is 1.14358974358974.</p>
<p>The following 7 pairs of parts of speech are connected with <code class="language-plaintext highlighter-rouge">nummod</code>: <tt><a href="got_proiel-pos-NOUN.html">NOUN</a></tt>-<tt><a href="got_proiel-pos-NUM.html">NUM</a></tt> (162; 83% instances), <tt><a href="got_proiel-pos-NUM.html">NUM</a></tt>-<tt><a href="got_proiel-pos-NUM.html">NUM</a></tt> (20; 10% instances), <tt><a href="got_proiel-pos-ADJ.html">ADJ</a></tt>-<tt><a href="got_proiel-pos-NUM.html">NUM</a></tt> (8; 4% instances), <tt><a href="got_proiel-pos-PRON.html">PRON</a></tt>-<tt><a href="got_proiel-pos-NUM.html">NUM</a></tt> (2; 1% instances), <tt><a href="got_proiel-pos-ADV.html">ADV</a></tt>-<tt><a href="got_proiel-pos-NUM.html">NUM</a></tt> (1; 1% instances), <tt><a href="got_proiel-pos-PROPN.html">PROPN</a></tt>-<tt><a href="got_proiel-pos-NUM.html">NUM</a></tt> (1; 1% instances), <tt><a href="got_proiel-pos-VERB.html">VERB</a></tt>-<tt><a href="got_proiel-pos-NUM.html">NUM</a></tt> (1; 1% instances).</p>
<pre><code class="language-conllu"># visual-style 7 bgColor:blue
# visual-style 7 fgColor:white
# visual-style 6 bgColor:blue
# visual-style 6 fgColor:white
# visual-style 6 7 nummod color:blue
1 jah jah CCONJ C- _ 8 cc _ Ref=MATT_5.41
2 jabai jabai SCONJ G- _ 5 mark _ Ref=MATT_5.41
3 ƕas ƕas ADJ Px Case=Nom|Gender=Masc|Number=Sing 5 nsubj _ Ref=MATT_5.41
4 þuk þu PRON Pp Case=Acc|Gender=Masc|Number=Sing|Person=2|PronType=Prs 5 obj _ Ref=MATT_5.41
5 ananauþjai ana-nauþjan VERB V- Mood=Opt|Number=Sing|Person=3|Tense=Pres|VerbForm=Fin|Voice=Act 8 advcl _ Ref=MATT_5.41
6 rasta rasta NOUN Nb Case=Acc|Gender=Fem|Number=Sing 5 obl _ Ref=MATT_5.41
7 aina ains NUM Ma Case=Acc|Gender=Fem|Number=Sing 6 nummod _ Ref=MATT_5.41
8 gaggais gaggan VERB V- Mood=Opt|Number=Sing|Person=2|Tense=Pres|VerbForm=Fin|Voice=Act 0 root _ Ref=MATT_5.41
9 miþ miþ ADP R- _ 10 case _ Ref=MATT_5.41
10 imma is PRON Pp Case=Dat|Gender=Masc|Number=Sing|Person=3|PronType=Prs 8 obl _ Ref=MATT_5.41
11 twos twai NUM Ma Case=Acc|Gender=Fem|Number=Plur 8 obl _ Ref=MATT_5.41
</code></pre>
<pre><code class="language-conllu"># visual-style 5 bgColor:blue
# visual-style 5 fgColor:white
# visual-style 6 bgColor:blue
# visual-style 6 fgColor:white
# visual-style 6 5 nummod color:blue
1 wesun wisan AUX V- Mood=Ind|Number=Plur|Person=3|Tense=Past|VerbForm=Fin|Voice=Act 6 cop _ LId=1|Ref=MARK_5.13
2 uþ -uh ADV Df _ 6 advmod _ Ref=MARK_5.13
3 þan þan ADV Df _ 6 discourse _ Ref=MARK_5.13
4 swe swe ADV Df _ 6 advmod _ Ref=MARK_5.13
5 twos twai NUM Ma Case=Nom|Gender=Fem|Number=Plur 6 nummod _ Ref=MARK_5.13
6 þusundjos þūsundi NUM Ma Case=Nom|Gender=Fem|Number=Plur 0 root _ Ref=MARK_5.13
7 jah jah CCONJ C- _ 6 cc _ Ref=MARK_5.13
8 afƕapnodedun af-ƕapnan VERB V- Mood=Ind|Number=Plur|Person=3|Tense=Past|VerbForm=Fin|Voice=Act 6 conj _ Ref=MARK_5.13
9 in in ADP R- _ 10 case _ Ref=MARK_5.13
10 marein marei NOUN Nb Case=Acc|Gender=Fem|Number=Sing 8 obl _ Ref=MARK_5.13
</code></pre>
<pre><code class="language-conllu"># visual-style 2 bgColor:blue
# visual-style 2 fgColor:white
# visual-style 1 bgColor:blue
# visual-style 1 fgColor:white
# visual-style 1 2 nummod color:blue
1 þat sa ADJ Pd Case=Acc|Gender=Neut|Number=Sing 3 obj _ Ref=JOHN_9.25
2 ain ains NUM Ma Case=Acc|Gender=Neut|Number=Sing 1 nummod _ Ref=JOHN_9.25
3 wait witan VERB V- Mood=Ind|Number=Sing|Person=1|Tense=Pres|VerbForm=Fin|Voice=Act 0 root _ LId=1|Ref=JOHN_9.25
4 ei ei SCONJ G- _ 5 mark _ Ref=JOHN_9.25
5 blinds blinds ADJ A- Case=Nom|Degree=Pos|Gender=Masc|Number=Sing|Strength=Strong 1 appos _ Ref=JOHN_9.25
6 was wisan AUX V- Mood=Ind|Number=Sing|Person=1|Tense=Past|VerbForm=Fin|Voice=Act 5 cop _ LId=1|Ref=JOHN_9.25
7 iþ iþ CCONJ C- _ 5 cc _ Ref=JOHN_9.25
8 nu nu ADV Df _ 9 advmod _ Ref=JOHN_9.25
9 saiƕa saiƕan VERB V- Mood=Ind|Number=Sing|Person=1|Tense=Pres|VerbForm=Fin|Voice=Act 5 conj _ Ref=JOHN_9.25
</code></pre>
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\section{Introduction}
distinctive feature of real-time systems is to be subject to severe
time constraints that arise from critical interactions between the
system and its environment. Since reasoning about real-time systems is
difficult, it is important to be able to apply formal validation
techniques early during the development process and to define formally
the requirements that need to be checked.
In this work, we follow a classical approach to model checking: (1) we
use a high-level language to describe a model of the system; (2) we
use a logical-based formalism to express requirements on the system;
and (3) the verification consists in compiling the system's model and
requirements into a low-level model for which we have the appropriate
theory and the convenient tooling. We propose a new treatment for this
traditional approach. In particular, for point (2), we focus on a
dense real-time model and we use \emph{real-time patterns} for the
specification of the system instead of timed extensions of temporal
logic. Our patterns can be interpreted as a real-time extension to the
specification patterns of~\citet{ppsfsv1999}. Time patterns can be
used to express constraints on the timing as well as the order of
events, such as the compliance to deadline or minimum time bounds on
the delay between events. Concerning verification, point (3), we work
with Time Transition Systems (see Sect.~\ref{sec:time-trans-syst}), an
extension of Time Petri Nets with data variables and priorities.
Our first contribution is to propose a decidable verification method
for checking real-time patterns on Time Transition Systems (TTS). The
method is based on the use of observers and model-checking techniques
in order to transform the verification of patterns into the
verification of simpler LTL formula.
Our observers are proved correct and non-intrusive, meaning that
they compute the correct answer and have no impact on the system
under observation. This is why we say our approach is verified. The
formal framework we have defined is not only useful for proving the validity
of formal results but also to check the soundness of optimisation
in the implementation.
Our second contribution is to provide a reference implementation for
these timed patterns. The complete framework defined in this paper has
been integrated into a verification tool chain for
Fiacre~\citep{filfmvte2008}, a high-level modelling language that can
be compiled to TTS. Fiacre can be used as input language for two
verification toolboxes: TINA, the TIme Petri Net Analyzer
tool set~\citep{tina}, and CADP~\citep{cadp}. In our tool chain
(described in Fig.~\ref{fig:intro}) a Fiacre specification is combined
with patterns and compiled into a TTS model using the Frac compiler (the Fiacre
language compiler).
Then the model can be checked using the TINA toolbox. This is not a
toy example. Indeed, Fiacre is the intermediate language used for
model verification in Topcased~\citep{ttptosfcasd2006}, an Eclipse
based toolkit for critical systems, where it is used as the target of
model transformation engines from various languages, such as SDL, BPEL
or AADL~\citep{aadl2fcr}. Therefore, through the connection with
Fiacre, we can check timed patterns on many different modelling
languages.
\begin{figure*}
\centering
\begin{tabular}[c]{l}
\includegraphics[height=6cm]{image.pdf}
\end{tabular}
\caption{The global verification tool chain}
\label{fig:intro}
\end{figure*}
Due to space limitations, we only give a partial descriptions of our
timed patterns and give only part of our theoretical results. A
complete catalogue of timed specification patterns is given
in~\cite{FRP11}, while the complete formal framework is defined in a
long version of this paper~\citep{VRTS11}.
For the purpose of this work, we focus on a simple \emph{deadline
pattern}, named \pop{leadsto}, and define different classes of
observers that can be used to check this pattern. We define observers
for the \pop{leadsto} patterns that are based on the monitoring of
places or transitions. In addition to these two traditional kind of
observers, we propose a class of TTS observers that monitor data
modifications. The goal is to choose the most efficient observer in
practice. We give some experimental results on the impact of the
choice of observer on the size of the state graphs that need to be
generated---that is on the space complexity of our verification
method---and on the verification time. The goal of this particular
study is not to define a method for automatically generating an
observer from a pattern. Instead, we define a set of possible
observers that are compared in order to choose the best one in
practice.
\subsubsection*{Outline}
The paper is organised as follows. We start by introducing our formal
framework in Sect.~\ref{sec:time-trans-syst}. This section is useful
to define the notion of composition and non-interference for our
observers. In Sect.~\ref{sec:prop-extens-}
and~\ref{sec:real-time-properties}, we describe a subset of our
real-time specification patterns and the verification framework. We
describe the implementation of our tool chain and give some
experimental results on the use of the \pop{leadsto} pattern in
Sect.~\ref{sec:experimental-results}. We conclude with a review of the
related work, an outline of our contributions and some perspectives on
future work.
\section{Formal framework}
\label{sec:time-trans-syst}
We define some formal notations that are used in the remainder of this
paper. In our approach, the observers and the systems are presented as
Time Transition System (TTS), an extension of Time Petri Nets (TPN)
\citetext{see e.g.~\citealp{merlin}} with data variables and
priorities. Our formal framework is based on the work
of~\citet{OCTPN}, where the authors define formally the composition of
two TPN. Their presentation has been extended to the full TTS model
in~\citet{VRTS11}.
The notion of composition is important in our work since we use TTS
models for both the system and the observer and, for verification, we
use TTS composition to graft the system with the observer.
This section is organised as follows: first, we introduce informally
a TTS example. Then, we give a formal definition
of TPN following the presentation of~\citet{OCTPN}, which is then extended
to TTS. The semantics of TTS is defined using sets of
timed traces. Finally, we define the composition of two TTS.
\subsection{Informal Presentation of the TTS Model}
\label{sub:introduction}
We introduce next a graphical syntax of TTS using a simple
example that models the behaviour of a mouse button with
double-clicking, as pictured in Fig.~\ref{fig/dble-TTS}.
The behaviour, in this case, is to emit the event
\lbl{double} if there are more than two \lbl{click} events in
\emph{strictly less} than one unit of time (u.t.).
\begin{figure}
\centerline{
\begin{tikzpicture}[node distance=4.5ex and 7.5ex, label distance=-0.5ex]
\node[place, label=below:$s_0$, tokens=1] (s0) {} ;
\node[vtransition, right=of s0, label=above:click] (t1) {}
edge[pre] (s0) ;
\node[place, label=below:$s_1$, right=of t1] (s1) {}
edge[pre] (t1) ;
\node[vtransition, right=of s1, label=below:{$[1;1]$}, label=above:{$\tau$}] (t2) {}
edge[pre] (s1) ;
\node[place, right=of t2, label=below:$s_2$] (s2) {}
edge[pre] (t2) ;
\node[vtransition, below=6ex of s1, label=above:double,
label=below:\begin{tabular}{c}\preact{dbl == true}\\\postact{dbl := false}\end{tabular}] (t3) {}
edge[pre, in=240, out=0] (s2)
edge[post, in=-60, out=180] (s0) ;
\node[transition, above=3.5ex of s1, label=left:click, label=above:\postact{dbl := true}] (t4) {}
edge[readarc] (s1) ;
\node[transition, above=7.5ex of t4, label=below:single,
label=above:\begin{tabular}{c}\postact{dbl := false}\\\preact{dbl == false}\end{tabular}] (t5) {}
edge[pre, in=120, out=0] (s2)
edge[post, in=60, out=180] (s0) ;
\draw[prio] (t2) -- (t4) ;
\end{tikzpicture}}
\caption{The double-click example in TTS}
\label{fig/dble-TTS}
\end{figure}
Ignoring at first side conditions and side effects (the \pop{pre} and
\pop{act} expressions inside dotted rectangles), the TTS in
Fig.~\ref{fig/dble-TTS} can be viewed as a TPN with one token in place
$s_0$ as its initial marking. From this ``state'', a \lbl{click}
transition may occur and move the token from $s_0$ to $s_1$. With this
marking, the internal transition $\tau$ is enabled and will fire after
exactly one unit of time, since the token in $s_1$ is not consumed by any other transition.
Meanwhile, the transition labeled \lbl{click} may fire one or more
times without removing the token from $s_1$, as indicated by the
\emph{read arc} (arcs ending with a black dot). After exactly one unit
of time, because of the priority arc (a
dashed arrow between transitions),
the \lbl{click} transition is disabled until
the token moves from $s_1$ to $s_2$.
Data is managed within the \pop{act} and \pop{pre} expressions that
may be associated to each transition. These expressions may refer to
a fixed set of variables that form the \emph{store} of the TTS. Assume
$t$ is a transition with guards \pop{act}$_t$ and \pop{pre}$_t$. In
comparison with a TPN, a transition $t$ in a TTS is enabled if there
is both: (1) enough tokens in the places of its pre-condition; and (2)
the predicate \pop{pre}$_t$ is true. With respect to the firing of
$t$, the main difference is that we modify the store by executing the
action guard \pop{act}$_t$. For example, when the token reaches the
place $s_2$ in the TTS of Fig.~\ref{fig/dble-TTS}, we use the value of
the variable \lbl{dbl} to test whether we should signal a double click
or not.
\subsection{Labeled Time Petri Nets and Time Transition Systems}
\label{sub:TPN-def}
Labeled Time Petri Nets (or TPN) extend Time Petri
Nets~\citep{merlin} with an action alphabet and a function labelling
the transitions with those actions.
Notation~: Let $I^+$ be the set of nonempty real intervals with non
negative rational endpoints. For $i \in I^+$ , the symbol
$\mathop{\downarrow}\kern -0.4ex i$ denotes the left end-point of the interval $i$ and
$\mathop{\uparrow}\kern -0.4ex i$ its right end-point, if $i$ is bounded, or $\infty$
otherwise. We use $\mathbb{N}$ to denote the set of non negative integers.
\begin{definition}\label{def:TPN} A labeled Time Petri Net (or TPN)
is a 8-tuple $(P, T, B, F, M_{0}, I_{s}, \sum, L)$ in which:
\begin{itemize}
\item $P$ is a finite set of places $p_{i}$;
\item $T$ is a finite set of transitions $t_{i}$;
\item $B$ is the backward incidence function\\
\begin{tabular}{c}
$B:T \rightarrow P \rightarrow \mathbb{N};$\\
\end{tabular}
\item $F$ is the forward incidence function\\
\begin{tabular}{c}
$F:T \rightarrow P \rightarrow \mathbb{N};$
\end{tabular}
\item $M_{0}$ is the initial marking function \\
\begin{tabular}{c}
$M_{0}:P \rightarrow \mathbb{N};$
\end{tabular}
\item $I_{s}$ is a function called the static interval function\\
\begin{tabular}{c}
$I_{s}:T \rightarrow I^+;$
\end{tabular}\\
Function $I_{s}$ associates a temporal interval $I_{s}(t) \in I^+$ with
every transition of the system. $\mathop{\downarrow}\kern -0.4ex I_{s}(t)$ and $\mathop{\uparrow}\kern -0.4ex I_{s}(t)$
are called the static earliest and latest firing times of
$t$,
respectively.
Assuming that a transition t became enabled at time $\tau$,
then
$t$ cannot fire before $(\tau + \mathop{\downarrow}\kern -0.4ex I_{s}(t))$
and no later than $(\tau + \mathop{\uparrow}\kern -0.4ex I_{s}(t))$ unless
disabled by firing some other transition.
\item $\sum$ is a finite set of actions, or labels, not containing
the silent action $\varepsilon$;
\item $L : T \rightarrow \sum \cup \{\varepsilon\}$ is a transition
labelling function.
\end{itemize}
A marking is a function $M : P \rightarrow \mathbb{N}$ that records the
current (dynamic) value of the places in the net, as transitions are
fired. The transition $t \in T$ is enabled by $M$ iff $(M \geqslant
B(t))$. The dynamic interval function $I : T \rightarrow I^+$ is a
mapping from transitions to time intervals. The dynamic interval
function is used to record the current timing constraints associated
to each transition, as time passes.
\end{definition}
A transition $t$ can fire from $(M, I)$ if $t$ is enabled at $M$ and
instantly fireable, that is $0 \in I(t)$. In the target state, the
transitions that remained enabled while $t$ is fired ($t$ excluded)
keep their time interval, the intervals of the others (newly enabled)
transitions are set to their respective static intervals. Together
with those ``discrete'' transitions, a time Petri Net adds the ability
to model the flowing of time. A transition of amount $d$
(i.e. taking $d$ time units) is possible iff $d$ is less than $\mathop{\uparrow}\kern -0.4ex
I(t)$ for all enabled transitions $t$.
The definition of TTS is a natural extension of TPN that takes
variables and priorities into account. Details are presented
in~\citet{VRTS11}.
\begin{definition}[Timed traces]
\label{def/timed-trace}
A timed trace $\s$ is a possibly infinite sequence of events
$t \in T$ and duration $d$ with $d \in \real^{+}$.
Formally, $\s$ is a partial mapping from $\mathbb{N}$ to $\dotT
= T \cup \set{d \mid d \in \real^{+}}$ such that
$\s(i)$ is defined whenever $\s(j)$ is defined and $i \leq j$.
The domain of $\s$ is written $\mathop{\mathsf{dom}}\s$. If $\mathop{\mathsf{dom}}\s$ is finite, the
\emph{duration} of $\s$, denoted $\Delta(\s)$, is the sum of the
delays in $\s$, that is $\sum_{i \mid \s(i) \in \real^{+}} \s(i)$.
\end{definition}
The semantics of a TPN (resp. TTS) is the set of its timed traces.
(see details in~\citet{VRTS11}).
\subsection{Composition of TTS and Timed Traces }
\label{sub:semantics}
We study the composition of two TTS and consider the relation between traces of
the composed system and traces of both components. This operation is
particularly significant in the context of this work, since both the system and the
observer are TTS and we use composition to graft the latter to the former.
In particular, we are interested in conditions ensuring that the
behaviour of the observer does not interfere with the behaviour of the
observed system.
The ``parallel composition'' of labeled Petri nets is a fundamental
operation that is used to model large systems by incrementally
combining smaller nets. Basically, the composition of two labeled TPN
$N_1$ and $N_2$ is a labeled net $N \smash[t]{{\overset{\text{def}}{=}}} (N_1 \comp N_2)$ such that:
the places of $N$ is the cartesian product of the places of $N_1$ and
$N_2$, and the transitions of $N$ is the fusion of the transitions in
$N_1$ and $N_2$ that have the same label. A formal definition for the
composition of two TPN is given in~\citet{OCTPN}. Composition of TTS
is basically the same~\citep{VRTS11}, with the noticeable restriction
that transitions which have priority over other transitions may not be
synchronised across components. This is required to ensure the
compositionality theorem, which we introduce below.
In the same way, we can define the composition of timed traces as an
operation that builds a timed trace $\s_1 \comp \s_2$ from two traces
$\s_1$ and $\s_2$. The trace $\s_1 \comp \s_2$ is obtained by merging
the events with the same labels. This operation is well-defined for
pairs of \emph{composable traces}. Let $N_1$ (resp. $N_2$) be a TPN,
and $\s_1$ (resp. $\s_2$) one of its traces. We say that $\s_1$ and
$\s_2$ are \emph{composable} iff $\mathop{\mathsf{dom}}{\s_1} = \mathop{\mathsf{dom}}{\s_2}$, and for
all $i \in \mathop{\mathsf{dom}}{\s_1}$, (1) $\s_1(i) = d \wedge d \in \real^{+} \imply
\s_2(i) = d$, and (2) $\s_1(i) = t \wedge t \in T \imply L(\s_1(i)) =
L(\s_2(i))$.
The compositionality theorem states that the behaviour of the composed
system (expressed as a set of timed traces) is a subset of the
behaviour of both components. In other terms, composing a system with
an observer cannot generate new behaviour.
\begin{theorem}[Compositionality]
\label{prop/composition}
Let $N_1$ and $N_2$ be two TTS and $N = N_1 \comp N_2$ be their
composition. Then, for every timed trace $\s$ of $N$, there exist two
timed traces, $\s_1$ and $\s_2$, such that: (1) $\s_i$ is a trace of
$N_i$ for $i \in 1..2$ and (2) $\s = \s_1 \comp \s_2$.
\end{theorem}
In the compositionality theorem, the trace $\s_1$ (resp. $\s_2$) is
obtained from $\s$ by ``erasing'' all transitions of $N_2$
(resp. $N_1$). Due to lack of space, we omit the proof here and
invite the reader to consult~\citet{VRTS11}.
\section{Real-Time Specification Patterns}
\label{sec:prop-extens-}
We have defined in~\citet{FRP11} a set of specification patterns
that can express constraints on the delays between the occurrences of
two events or on the duration of a given condition. In our context, the event of a model can be: a transition that
is fired; the system entering or leaving a state; a change in the
value of variables; \dots The advantage of
proposing predefined patterns is to provide a simple formalism to
non-experts for expressing properties that can be directly checked
with our verification tool chain. Our patterns can be viewed as a
real-time extension of Dwyer's~\citeyearpar{ppsfsv1999} specification
patterns. In his seminal work, Dwyer shows through a study of 500
specification examples that 80\% of the temporal requirements can be
covered by a small number of ``pattern formulas''. We follow a similar
philosophy and define a list of patterns that takes into account
timing constraints. At the syntactic level, this is mostly obtained by
extending Dwyer's patterns with two kind of \emph{timing modifiers}:
(1) $P$ \pop{within} $I$, which states that the delay between two
events declared in the pattern $P$ must fit in the time interval $I$;
and (2) $P$ \pop{lasting} $D$, which states that the condition
defined by $P$ must hold for at least duration $D$. For example, we
define a pattern {\pop{Present} $A$ \pop{after} $B$ \pop{within}
$]0, 4]$} to express that the event $A$ must occur within 4 unit of
time of the first occurrence of event $B$, if any, and not
simultaneously with it. Although seemingly innocuous, the addition of
these two modifiers has a great impact on the semantics of patterns
and on the verification techniques that are involved.
We describe our patterns using a hierarchical classification borrowed
from~\citet{ppsfsv1999}, with patterns arranged in categories such as
universality, absence, response, etc. In the following, we give some
examples of \textit{absence} and \textit{response} patterns based on
the TTS example of Fig.~\ref{fig/dble-TTS}. Each of these patterns can
be checked using our tool chain. A complete catalogue of patterns, with
their formal definition, is given in~\citet{FRP11}. In this section,
we focus on the ``response pattern with delay'', to give an example of
how patterns can be formally defined and to explain our different
classes of observers.
\subsection{Absence Pattern with Delay}
This category of patterns is used to specify delays within which
activities must not occur. A typical pattern in this category is:
\[\tag{absent} \pop{absent}
E_2 \pop{after} E_1 \pop{for interval} [d_1; d_2]~,
\]
which asserts that a transition (labeled with) $E_2$ cannot occur
between $d_1$ and $d_2$ units of time after the first occurrence of a
transition $E_1$. An example of use for this pattern would be the
requirement that we cannot have two double clicks in less than~$2$
units of time (u.t.), that is: \pop{absent} \code{double} \pop{after}
\code{double} \pop{for interval} $[0; 2]$. (This property is not true
for our example in Fig.~\ref{fig/dble-TTS}.) A more contrived example
is to require that if there are no single clicks in the first $10$
u.t. of an execution then there should be no double clicks at
all. This requirement can be expressed using the composition of two
absence patterns using the implication operator and the reserved
transition \code{init} (that identifies the start of the system):
\[
\begin{array}{l}
\big(\pop{absent} \code{single} \pop{after} \code{init}
\pop{for interval} [0;10]\big)\\
\quad \Rightarrow \big(\pop{absent} \code{double}
\pop{after} \code{init} \pop{for interval} [0;\infty[ \big)~.
\end{array}
\]
\subsection{Response Pattern with Delay}
This category of patterns is used to express that some (triggering)
event must always be followed by a given (response) event within a
fixed delay of time. The typical example of response pattern states
that every occurrence of a transition labeled with $E_1$ must be
followed by an occurrence of a transition labeled with $E_2$ within a
time interval $I$. (We consider the first occurrence of $E_2$ after
$E_1$.)
\[\tag{leadsto} E_1 \pop{leadsto} E_2 \pop{within} I~.\]
For example, using a disjunction between transition labels, we can
bound the time between a \code{click} and a mouse event with the
pattern: \code{click} \pop{leadsto} $(\code{single} \vee
\code{double})$ \pop{within} $[0,1]$.
\subsection{Other Examples of Patterns}
To give a feel of the expressiveness of our patterns, we briefly
describe some other examples. For each pattern, we give just a textual
definition. In each example, $E_1$, $E_2$ and $E_3$ refer to events in
the system and $d_1$ (resp. $d_2$) stand for the left end-point (resp.
right end-point) of the time interval $I$.\\
\fichepattern
{\code{Present} $E_1$ \code{after} $E_2$ \code{within} $I$}
{Predicate $E_1$ must hold
between $d_1$ and $d_2$ u.t after the first occurrence
of $E_2$. The pattern is also satisfied if $E_2$ never holds.
}
\fichepattern
{\code{Present first} $E_1$ \code{before} $E_2$ \code{within} $I$}
{The first occurrence of $E_1$ should be between $d_1$ and $d_2$
u.t. before the first occurrence of~$E_2$. The pattern also holds if
$E_2$ never occurs.}
\fichepattern
{\code{Present} $E_1$ \code{lasting} $D$}
{Starting from the first occurrence when the predicate $E_1$ holds, it
remains true for at least duration $D$. This pattern makes sense
only if $E_1$ is a predicate on states (that is, on the marking or
store); since transitions are instantaneous, they have no
duration.}
\fichepattern
{\code{Absent} $E_1$ \code{before} $E_2$ \code{for duration} $D$}
{No $E_1$ can occur less than $D$ u.t. before the first occurrence of~$E_2$. The pattern holds if there are no occurrence of $E_2$.
}
\fichepattern
{$E_1$ \code{leadsto} first $E_2$ \code{within} $I$ \code{before} $E_3$}
{Before the first occurrence of $E_3$, each occurrence of $E_1$ is
followed by an occurrence of $E_2$ which occurs both before $E_3$, and in the time
interval $I$ after $E_1$. The pattern holds if $E_3$ never occurs.}
\fichepattern
{$E_1$ \code{leadsto first} $E_2$ \code{within} $I$ \code{after} $E_3$}
{Same than with the pattern ``$E_1$ \pop{leadsto first} $E_2$ \pop{within} $I$''
but only considering occurrences of $E_1$ after the first $E_3$.
}
\subsection{Interpretation of Patterns}
\label{sec:interpr-patt}
We can use different formalisms to define the semantics of
patterns. In this work, we focus on a denotational interpretation,
based on first-order formulas over timed traces (with equality and
trace composition). We illustrate our approach using the pattern $E_1
\pop{leadsto} E_2 \pop{within} I$.
For the ``denotational'' definition, we say that the pattern $E_1
\pop{leadsto} E_2 \pop{within} I$ is true for a TTS $N$ if and only
if, for every timed-trace $\s$ of $N$, we have:
\[
\forall \s_1, \s_2 \ .\ (\s = \s_1 E_1 \s_2) \Rightarrow \left
(\begin{array}[c]{@{}l@{}}
\exists \s_3,\s_4 \ .\ \s_2 = \s_3 E_2 \s_4\\
\quad \wedge \Delta(\s_3)
\in I \wedge E_2 \notin \s_3
\end{array} \right )
\]
where $\Delta(\s_3)$ is the sum of all the duration in $\s_3$. The
denotational approach is very convenient for a ``tool developer'' (for
instance to prove the soundness of an observer implementing a pattern)
since it is self-contained.
For another example, the denotational definition for the pattern
$\pop{absent} E_2 \pop{after} E_1 \pop{for interval} I$ is given by
the following condition on the traces $\s$ of a system:
\[
\begin{array}[c]{l@{}l}
\forall \s_1, \s_2, \s_3 \ .\ & (\s = \s_1 E_1 \s_2 E_2 \s_3)\\
& \ \wedge (E_1 \notin \s_1) \Rightarrow (\Delta(\s_2) \notin I)
\end{array}
\]
On our complete catalogue of patterns~\citep{FRP11}, we provide an
alternative (equivalent) semantics for patterns based on MTL, a timed
extension of linear temporal logic \citetext{see e.g.~\citealp{MITL}
for a definition of the logic}. For instance, for the \code{leadsto}
pattern, the equivalent MTL formula is $\ltlall\big(E_1 \Rightarrow
((\neg E_2) \mathrel{\mathbf{{U}}}_I E_2)\big)$, which reads like a LTL formula enriched
by a time constraint on the until modality $\mathrel{\mathbf{{U}}}$.
\section{Patterns Verification}
\label{sec:real-time-properties}
We define different types of observers at the TTS level that can be
used for the verification of patterns. It is important to note that we
do not give an automatic method to generate observers. Rather, we
define a set of observers for each patterns and, after selecting the
``most efficient one'', we prove that it is correct (see the
discussion in Sect.~\ref{sec:experimental-results}). We make use of
the whole expressiveness of the TTS model to build observers:
synchronous or asynchronous rendez-vous (through places and
transitions); shared memory (through data variables); and
priorities. We believe that an automatic method for generating the
observer, while doable, will be detrimental for the performance of our
approach. Moreover, when compared to a ``temporal logic'' approach, we
are in a more favorable situation because we only have to deal with a
finite number of patterns.
\subsection{Observers for the Leadsto Pattern}
\label{sec:observ-leadsto-patt}
We focus on the example of the \code{leadsto} pattern. We assume that
some events of the system are labeled with $E_1$ and some others with
$E_2$. We give three examples of observers for the
pattern: $E_1$ \code{leadsto} $E_2$ \code{within}
$[0,\mathit{max}[$. The first observer monitors transitions and uses a
single place; the second observer monitors places; the third observer
monitors shared, boolean variables injected into the system (by means
of composition). We define our TTS observers using a classical
graphical notation for Petri Nets, where arcs with a black circle
denote \emph{read arcs}, while arcs with a white circle are
\emph{inhibitor arcs}. (These extra categories of arcs can be defined
in TTS and are supported in our tool chain.) The use of a \emph{data
observer} is quite new in the context of TTS systems. The results of
our experiments seem to show that, in practice, this is the best
choice to implement an observer.
\subsubsection{Transition Observer}
The observer $O_t$, see Fig.~\ref{fig:dummy-1}, uses a place,
\lbl{obs}, to record the time since the last transition $E_1$
occurred. The place \lbl{obs} in $O_t$ is emptied if a transition
labeled $E_2$ is fired, otherwise the transition \lbl{error} is fired
after $\mathit{max}$ unit of time. The priority arc (dashed arrow)
between \lbl{error} and $E_2$ is used to observe the transition
\lbl{error} even in the case where a transition $E_2$ occurs exactly
$\mathit{max}$ u.t. after the place \lbl{obs} was filled.
By definition of the TTS composition operator, the composition of the
observer $O_t$ with the system $N$ duplicates each transitions in $N$
that is labeled $E_1$: one copy can fire if \lbl{obs} is empty---as a
result of the inhibitor arc---while the other can fire only if the
place is full. As a consequence, in the TTS $N \comp O_t$, the
transition \lbl{error} can fire if and only if the place \lbl{obs}
stays full---there has been an instance of $E_1$ but not of
$E_2$---for a duration of $\mathit{max}$. Then, to prove that $N$
satisfies the \code{leadsto} pattern, it is enough to check that the
system $N \comp O_t$ cannot fire the transition \lbl{error}. This can
be done by checking the LTL formula $\ltlall (\neg \code{error})$ on
the system $N \comp O_t$.
The observer $O_t$ given in Fig.~\ref{fig:dummy-1} is
\emph{deterministic} and will ``react'' to the first occurrence of
$E_2$ that miss a deadline. It is also possible to define a
non-deterministic observer, such that some occurrences of $E_1$ or
$E_2$ may be disregarded. This approach is safe since model-checking
performs an exhaustive exploration of the states of the system; it
considers all possible scenarios. This non-deterministic behaviour is
quite close to the treatment obtained when compiling an (untimed) LTL
formula ``equivalent'' to the \code{leadsto} pattern, namely $\ltlall
(E_1 \Rightarrow \ltlexist E_2)$, into a Büchi
automaton~\citep{FBAT}. We have implemented the deterministic and
non-deterministic observers and compared them taking in account their
impact on the size of the state graphs that need to be generated and
on the verification time. Experiments have shown that the
deterministic observer is more efficient, which underlines the benefit
of singling out the best possible observer and looking for specific
optimisation.
\begin{figure}[t]
\begin{center}
\begin{tikzpicture} [node distance=4.5ex and 7.5ex, label distance=-0.5ex]
\node[transition, label=above:$E_1$] (t1) {};
\node[transition, right=of t1, label=below:error, label=above:{$[\mathit{max},\mathit{max}]$}] (t2) {};
\node[transition, right=of t2 , label=above:$E_2$] (t3) {};
\node[place, below=of t2, label=below:obs] (event1) {}
edge[post] (t2);
\draw[-stealth] (t1) to[in=125,out=-15] (event1);
\draw[o-] (t1) to[in=150,out=-50] (event1);
\draw[o-] (t3) -- (event1);
\node[transition, below=of t1, label=below:$E_1$] (t4) {}
edge[readarc] (event1);
\node[transition, below=of t3, label=below:$E_2$] (t5) {}
edge[pre] (event1);
\draw[prio] (t2) -- (t5) ;
\end{tikzpicture}
\caption{Transition Observer: $O_t$}\label{fig:dummy-1}
\end{center}
\end{figure}
\begin{figure}
\begin{center}
\begin{tikzpicture} [node distance=7.5ex and 10.5ex, label distance=-0.5ex]
\node[transition, label=below:{\postact{flag := true}}, label=above:$E_1$] (t1) {};
\node[transition, below=of t1 , label=above:error,
label=below:{\begin{tabular}{c}\preact{flag ==
true}\\\end{tabular}}, label=right:{$[max,max]$}] (t2) {};
\node[transition, label=below:{\postact{flag := false}}, below=of t2, label=above:$E_2$] (t3) {};
\draw[prio] (t2) -- (t3) ;
\end{tikzpicture}
\caption{Data Observer: $O_d$}\label{fig:dummy-3}
\end{center}
\end{figure}
\subsubsection{Data Observer}
We define the data observer $O_d$ in Fig.~\ref{fig:dummy-3}. The data
observer has a transition \lbl{error} conditioned by the value of a
boolean variable, \lbl{flag}, that ``takes the role'' of the place
\lbl{obs} in $O_t$ (every boolean variable is considered to be
initially set to false). Indeed, \lbl{flag} is true between an
occurrence of $E_1$ and the following transition $E_2$. Therefore,
like in the previous case, to check if a system $N$ satisfies the
pattern, it is enough to check the reachability of the event
\lbl{error}. Notice that the whole state of the data observer is
encoded in its store, since the underlying net has no place.
\subsubsection{Place Observer}
We define the place observer $O_p$ in Fig.~\ref{fig:dummy-2}. In this
section, to simplify the presentation, we assume that the events $E_1$
and $E_2$ are associated to the system entering some given states
$S_1$ and $S_2$. (But we can easily adapt this net to observe events
associated to transitions in the system.) We also rely on a
composition operator that composes TTS through their places instead of
their transitions~\citep{OCTPN} and that is available in our tool
chain. In $O_p$, we use a transition labeled $\tau_1$ whenever a
token is placed in $S_1$ and a transition $\tau_2$ for observing that
the system is in state $S_2$ (we assume that the labels $\tau_1$ and
$\tau_2$ are fresh---private to the observer---and should not be
composed with the observed systems). The remaining component of $O_p$
is just like the transition observer. We consider both a place and a
transition observer since, depending on the kind of events that are
monitored, one variant may be more efficient than the other.
\begin{figure}[h]
\begin{minipage}[b]{0.85\linewidth}\centering
\begin{tikzpicture} [node distance=4.5ex and 7.5ex, label distance=-0.5ex]
\node[place, label=below:$S_1$] (event1) {} ;
\node[transition, right=of event1, label=right:{$[0,0]$}, label=above:$\tau_1$] (t1) {}
edge[readarc] (event1);
\node[place, below=of t1, label=left:obs] (beginobserver) {};
\node[vtransition, right=of beginobserver , label=below:error, label=above:{$[max,max]$}] (t2) {}
edge[pre] (beginobserver) ;
\node[vtransition, below=of beginobserver, label=right:{$[0,0]$}, label=below:$\tau_2$] (t3) {}
edge[pre] (beginobserver);
\draw[-stealth] (t1) to[in=50,out=-50] (beginobserver);
\draw[o-] (t1) -- (beginobserver);
\node[place, left=of t3, label=below:$S_2$] (event2) {};
\node[vtransition, below=of beginobserver] (t3) {}
edge[readarc] (event2);
\draw[prio] (t2) -- (t3) ;
\end{tikzpicture}
\caption{Place Observer: $O_p$}\label{fig:dummy-2}
\end{minipage}
\end{figure}
\subsection{Proving Innocuousness and Soundness of Observers}
\label{sec:prov-innoc-observ}
The goal of this section is to show how to prove that an observer for
a pattern is correct. We demonstrate our approach on the particular
examples of observers for the pattern $E_1 \pop{leadsto} E_2
\pop{within} [0, \mathrm{max}[$, given in the previous section.
We say that an observer $O$ for this pattern is \emph{sound} if it can
``detect'' the traces of a system $N$ that do not hold for the
pattern. More formally, if there is a trace $\s$ of $N$ such that: $\s
= \s_1 E_1 \s_2 E_2 \s_3$ with $\Delta(\s_2) \geq \mathrm{max}$ and
$E_2 \notin \s_2$, then there should be a trace $\s'$ in $N \comp O$
such that $\lbl{error} \in \s'$. (The condition on the trace $\s$
directly follows from the denotational definition of the pattern, see
Sect.~\ref{sec:interpr-patt}) On the opposite, the observer is
\emph{correct} if it can detect that a system satisfies a pattern: if
for all trace $\s'$ of $N \comp O$ we have $\lbl{error} \notin \s'$
then for all trace $\s$ of $N$ the pattern holds.
From our compositionality theorem, see Sect.~\ref{sub:semantics}, we
have that every trace $\s'$ of $N \comp O$ can be defined as the
composition $\s \comp \s_o$ of a trace $\s$ of the system $N$ with a
trace $\s_o$ of the observer $O$. Therefore, to prove that an observer
is correct, it is enough to prove that the pattern does not hold for a
trace $\s_o$ in $O$ iff $\lbl{error} \in s_o$. Indeed, if there is a
trace $\s$ in $N$ that does not hold for the pattern, then we obtain a
trace $\s \comp \s_o$ in $N \comp O$ that does not hold either.
We can use our formal framework to prove the soundness of an observer
(work is currently under way to mechanise these proofs using the Coq
interactive theorem prover). Correctness proofs are more complicated,
since they require to reason on the traces of a system composed with
the observer to figure out the behaviour of the system
alone. Therefore, instead of proving that an observer is correct, we
prove a stronger assumption, that is that observers should be
\emph{innocuous}. A net is said to be \emph{innocuous} if it cannot
interfere with a system placed in parallel. More formally, the TTS $O$
is innocuous if for all TTS $N$ and for all trace $\s$ in $N$ there
exists a trace $\s_o$ in $O$ such that $\s \comp \s_o$ is a trace in
$N \comp O$. Innocuousness means that the observer cannot restrict the
behaviour of another system. This is particularly useful in our case
since, with innocuous observer, any trace $\s$ of the observed
system $N$ is preserved in the composed system $N \comp O$: the
observer does not obstruct the behaviour of the system (see
Lemma~\ref{lem/non-intrusive} below).
Instead of proving that observers are non-intrusive in a case by case
basis, we can give a set of sufficient conditions for an observer
$O$ to be innocuous. These conditions are met by the three
observers given in Fig.~\ref{fig:dummy-1}--\ref{fig:dummy-2}.
Given a TTS $N$, we say that a transition $t$ of the observer is
\emph{synchronised} when there exists a labeled transition $t'$ of $N$
such that $L(t) = L(t')$ (and the label $L(t)$ is not $\epsilon$). We
write $\T_{sync}$ the set of synchronised transitions of the observer and
$L_{sync}$ the labels of the synchronised transitions. The transitions
in $\T_{sync}$ are the transitions used by the observer to probe the
system. In the examples defined in the previous section, the only
synchronised transitions are the ones labeled $E_1$ and $E_2$ in the
data ($O_d$) and transition ($O_t$) observers.
We define $\T_{imm}$ as the set of transitions of the observer whose static
time interval is $[0, 0]$. By construction, no transiton in $\T_{sync}$ can also be part of
$\T_{imm}$.
\begin{lemma}
\label{lem/non-intrusive}
Assume $O$ satisfies the following three conditions:
\begin{itemize}
\item all synchronised transitions have a trivial static time interval
and no priority (that is, for every $t$ in $\T_{sync}$, $I_s^t = [0 ;
+\infty[$ and $t$ has no priority over another transition in $O$);
\item from any state of the observer, and for every label $l \in
L_{sync}$, there is at least one transition $t$ in $O$ with label
$l$ that can fire immediately;
\item from any state of the observer, there is no infinite sequence of
transitions in $\T_{imm}$.
\end{itemize}
then, for all timed trace $\s$ in $N$ there exists a timed trace $\s
\comp \s_o$ in $N \comp O$ such that $\s_o$ is a trace of $O$.
\end{lemma}
The proof of Lemma~\ref{lem/non-intrusive} can be found
in~\citet{VRTS11}. A few comments on these conditions. The first
condition is necessary for defining the composition of two TTS (see
Sect.~\ref{sub:semantics}). The second condition ensures that the
observer cannot delay the firing of a synchronised transition ``for a
non-zero time''. Assume $s$ is a state of the observer $O$ and
$\s$ a finite trace of $O$ starting from state $s$. We define
$O(s, \s)$ to be the (necessarily unique) state reached by the
observer after trace $\s$ has been executed. From the second
condition, in every reachable state $s$ of $O$, and for every label
$l$ in $L(\T_{sync})$, there exists a (possibly empty) finite trace $\s$
not containing transitions in $\T_{sync}$ such that the duration of $\s$
is 0 and there exists $t \in \T_{sync}$ with $L(t) = l$, which is fireable
in state $O(s, \s)$. Note also that the observer cannot involve
other synchronised transitions while reaching a state where $l$ is
firable, since this would abusively constrain the behaviour of the main
system $N$, not to mention deadlock issues. This condition is true for
the observer $O_t$ in Fig.~\ref{fig:dummy-1} since, at any time,
exactly one of the two transitions labeled $E_1$ (resp. $E_2$) can
fire.
\section{Experimental Results}
\label{sec:experimental-results}
Our verification framework has been integrated into a prototype
extension of \emph{frac}, the Fiacre compiler for the TINA toolbox.
This extension supports the addition of real time patterns and
automatically compose a system with the necessary observers. (Software
and examples are available at \url{http://homepages.laas.fr/~nabid}.)
In case the system does not meet its specification, we obtain a
counter-example that can be converted into a timed sequence of events
exhibiting a problematic scenario. This sequence can be played back
using two programs provided in the TINA tool set, \emph{nd} and
\emph{play}. The first program is a graphical animator for Time Petri
Net, while the latter is an interactive (text-based) animator for the
full TTS model.
We define the \emph{empirical complexity} of an observer as its impact
on the augmentation of the state space size of the observed
system. For a system $S$, we define $\mathit{size}(S)$ as the size (in
bytes) of the \emph{State Class Graph} (SCG)~\citep{tina} of $S$
generated by our verification tools. In TINA, we use SCG as an
abstraction of the state space of a TTS. State class graphs exhibit
good properties: an SCG preserves the set of discrete traces---and
therefore preserves the validation of LTL properties---and the SCG of
$S$ is finite if the Petri Net associated with $S$ is bounded and if
the set of values generated from $S$ is finite. We cannot use the
``plain'' labeled transition system associated to $S$ to define the
size of $S$; indeed, this transition graph maybe infinite since we
work with a dense time model and we have to take into account the
passing of time.
The size of $S$ is a good indicator of the memory footprint and the
computation time needed for model-checking the system $S$: the time
and space complexity of the model-checking problem is proportional to
$\mathit{size}(S)$. Building on this definition, we say that the
complexity of an observer $O$ applied to the system $S$, denoted
$C_O(S)$, is the quotient between the size of $(S \comp O)$ and the
size of $S$.
\begin{figure*}[htb]
\centering
\includegraphics[height=6cm]{unverifiedcase.pdf} \hfill
\includegraphics[height=6cm]{verifiedcase.pdf}
\caption{Compared complexity of the data and place observers (in
percentage of system size growth)---average time for invalid
properties (right) and valid properties (left).}
\label{fig:example}
\end{figure*}
We resort to an empirical measure for the complexity since we cannot
give an analytical definition of $C_O$ outside of the simplest
cases. However, we can give some simple bounds on the function
$C_O$. First of all, since our observers should be non-intrusive, we can show that the SCG of $S$ is
a sub graph of the SCG of $S \comp O$, and therefore $C_O(S) \geq
1$. Also, in the case of the \code{leadsto} pattern, the transitions
and places-based observers add exactly one place to the net associated
to $S$. In this case, we can show that the complexity of these two
observers is always less than~$2$; we can at most double the size of
the system. We can prove a similar upper bound for the \code{leadsto}
observer based on data. While the three observers have the same
(theoretical) worst-case complexity, our experiments have shown that
one approach was superior to the others. We are not aware of previous
work on using experimental criteria to select the best observer for a
real-time property. In the context of ``untimed properties'', this
approach may be compared to the problem of optimising the generation
of Büchi Automata from LTL formulas, see e.g.~\citet{FBAT}.
We have used our prototype compiler to experiment with different
implementations for the observers. The goal is to find the most
efficient observer ``in practice'', that is the observer with the
lowest complexity. To this end, we have compared the complexity of
different implementations on a fixed set of representative examples
and for a specific set of properties (we consider both valid and
invalid properties). The results for the \code{leadsto} pattern are
displayed in Fig.~\ref{fig:example}. For the experiments used in this
paper, we use three examples of systems selected because they exhibit
very different features (size of the state space, amount of
concurrency and symmetry in the system, \dots):
\begin{itemize}
\item TRAIN is a model of a train gate controller. The example models
a system responsible for controlling the barriers protecting a
railroad crossing gate. When a train approaches, the barrier must be
lowered and then raised after the train's departure. The valid
property, for the TRAIN example, states that the delay between
raising and lowering a barrier does not exceed 100 unit of time. For
the invalid property, we use the same requirement, but shortening
the delay to 75.
\item APOTA is an industrial use case that models the dynamic
architecture for a network protocol in charge of data communications
between an air plane and ground stations~\citep{FVAMFT}. This
example has been obtained using an translation from AADL to
Fiacre. In this case, timing constraints arise from timeouts between
requests and periods of the tasks involved in the protocol
implementation. The property, in this case, is related to the
worst-case execution time for the main application task.
\item CAR is a system modelling an automated rail car system taken
from~\citet{tap}. The system is composed of four terminals connected
by rail tracks in a cyclic network. Several rail cars, operated from
a central control center, are available to transport passengers
between terminals. When a car approaches its destination, it sends a
request to the terminal to signal its arrival. Passengers in the
terminal can then book a travel in the car. The valid property, for
the CAR example, states that a passenger arriving in a terminal,
must have a car ready to transport him within 15\, unit of time. For
the invalid property, we use the same requirement, but shortening
the delay to 2 unit of time.
\end{itemize}
In Fig.~\ref{fig:example}, we compare the growth in the state space
size---that is the value of $C_o(S)$---for the place and data
observers defined in Sect.~\ref{sec:observ-leadsto-patt} and our three
running examples. We do not consider the transition observer in these
results since the events used in the requirements are all related to a
system entering a state (and therefore our benchmark favor the place
observer over the transition observer). We use one chart to display
the result for patterns that are invalid and another for valid
patterns.
In Fig.~\ref{fig:example2} (page~\pageref{fig:example2}), we give
results on the total verification time for the APOTA example.
The
value displayed in the table refer to the time spent generating the
complete state space of the system and verifying the property. The row
SYSTEM gives the time needed for exploring the complete state space of
the system (without adding any observer) while ``VALID'' and
``INVALID'' refer to the state space of the system synchronised with
data observer and state observer in the case of valid and invalid
property respectively.
In our experiments, we have consistently observed that the observer
based on data is the best choice; it is the observer giving the
minimal execution time in almost all the cases and that seldom gives
the worst result.
We can explain the efficiency of the data observer by the fact that it
adds less transitions than the state observer; which means that it
adds less intermediary states to the state space of $(N \comp O)$.
\begin{figure}[htb]
\centering
\begin{tabular}[c]{|p{0.15\textwidth}|p{0.1\textwidth}|p{0.1\textwidth}|}
\hline
\centering
Example & State observer & Data observer \\
\hline
SYSTEM & \hfill 2.861 & \hfill 2.861\\
\hline
VALID & \hfill 11.662 & \hfill 10.652\\
\hline
INVALID & \hfill 11.611 & \hfill 10.179\\
\hline
\end{tabular}
\caption{Total verification time for APOTA (in seconds)}
\label{fig:example2}
\end{figure}
\section{Related Work}
Two broad approaches coexist for the definition and verification of
real-time properties: (1) real-time extensions of temporal logic~\citep{H98}; and
(2) observer-based approaches, such as the Context Description Languages
(CDL) of Dhaussy et al.~\citep{ECOVFM} or approaches based on timed
automata~\citep{MITL,MCRTTA,TPRTTA}.
Obviously, the logic-based approach provides most of the theoretically
well-founded body of works, such as complexity results for different
fragments of real-time temporal logics~\citep{H98}: Temporal logic with
clock constraints (TPTL); Metric Temporal Logic---with or without
interval constrained operators---; Event Clock Logic; etc. The
algebraic nature of logic-based approaches make them expressive and
enable an accurate formal semantics. However, it may be impossible to
express all the necessary requirements inside the same logic fragment
if we ask for an efficient model-checking algorithm (with polynomial
time complexity). For example, Uppaal~\citep{behrmann04atutorial} chose
a restricted fragment of TCTL with clock variables, while Kronos
provide a more expressive framework, but at the cost of a much higher
complexity. As a consequence, selecting this approach requires to
develop model-checkers for each interesting fragment of these
logics---and a way to choose the right tool for every
requirement---which may be impractical.
Pattern-based approaches propose a user-friendly syntax that
facilitates their adoption by non-experts. However, in the real-time
case, most of these approaches lack in theory or use inappropriate
definitions. One of our goal is to reverse this situation. In the
seminal work of~\citet{ppsfsv1999}, patterns are defined by
translation to formal frameworks, such as LTL and CTL. There is no
need to provide a verification approach, in this case, since efficient
model-checkers are available for these logics. This work on patterns
has been extended to the real-time case. For example,~\citet{RSP} extends the patterns language with time constraints
and give a mapping from timed pattern to TCTL and MTL, but they do not
study the decidability of the verification method (the
implementability of their approach). Another related work
is~\citep{PTPS}, where the authors define observers based on Timed
Automata for each pattern. However, they do not provide a formal
framework for proving the correctness or the innocuousness of their
observers and they have not integrated their approach inside a
model-checking tool chain.
Concerning observer-based approaches, \citet{TPRTTA,MCRTTA} use test automata to check
properties of reactive systems. The goal is to identify properties on
timed automata for which model checking can be reduced to reachability
checking. In this framework, verification is limited to safety and
bounded liveness properties. In the context of Time Petri Net, a
similar approach has been experimented by~\citet{TCVMBTPN}, but they propose a less general model for
observers and consider only two verification techniques over four
kinds of time constraints. \citet{APTABI} propose a method
to verify the correctness of their approach formally. However,
they do not prove formally all their invariants (patterns in
our case).
\section{Contributions and Perspectives}
\label{sec:contr-persp}
In contrast to these related works, we make the following
contributions. We reduce the problem of checking real-time properties
to the problem of checking LTL properties on the composition of the
system with an observer. We define also a real-time patterns language
based on the work of~\citet{ppsfsv1999} and inspired from real-case
studies. To choose the best way to verify a pattern, we defined, for
each pattern, a set of non-intrusive observers. We are based on a
formal framework to verify the correctness of an observer, whether it
can interfere with the behaviour of the system under observation.
Our approach has been integrated into a complete verification
tool chain for the Fiacre modelling language and can therefore be used
in conjunction with Topcased~\citep{ttptosfcasd2006}. We give several experimental results
based on the use of this tool chain in
Sect.~\ref{sec:experimental-results}. The fact that we implemented our
approach has influenced our definition of the observers. Indeed,
another contribution of our work is the use of a pragmatic approach
for comparing the effectiveness of different observers for the same
property. Our experimental results seem to show that data observers
look promising.
We are following several directions for future work. A first goal is
to define a new low-level language for observers---adapted from the
TTS model---equipped with more powerful optimisation techniques and
with easier soundness proofs. On the theoretical side, we are
currently looking into the use of mechanised theorem proving
techniques to support the validation of observers. On the experimental
side, we need to define an improved method to select the best
observer. For instance, we would like to provide a tool for the
``syntax-directed selection'' of observers that would choose (and even
adapt) the right observers based on a structural analysis of the
target system.
| {
"redpajama_set_name": "RedPajamaArXiv"
} | 36 |
\section{Introduction}
The purpose of this contribution to the debate on \lq\lq Quantum
and emergent gravity\rq\rq is fourfold.
First of all, we would like to introduce the group field theory
(GFT) formalism \cite{iogft, iogft2,laurentgft}, that has recently
attracted interest in the general area of non-perturbative quantum
gravity, and is currently mainly used in the context of Loop
Quantum Gravity \cite{LQG}. We will describe the general features
of the formalism, at both kinematical and dynamical level, and
provide an interpretation for them.
Second, we would like to portrait a picture of group field
theories as a common framework and a unifying language for several
approaches to quantum gravity, in particular loop quantum gravity
and simplicial quantum gravity (i.e. quantum Regge calculus and
dynamical triangulations), by sketching how the basic ingredients
of these various approaches can be identified within the GFT
setting. We will argue that the pictures of quantum spacetime,
developed in the various approaches, are compatible and can help
completing each other, while acquiring a new interpretation within
the GFT framework. GFTs can then represent a suitable context in
which all these different approaches can inform, cross-fertilize
and improve each other with the achieved results and insights into
the nature of quantum geometry, and with the tools they have
developed to study it. In doing so, of course, we will discuss why
we think is useful to move from the contexts provided by each of
these quantum gravity approaches to the GFT one.
Third, we want to stress the need to devote our research efforts
to tackle the issue of the continuum approximation of the quantum
discrete structures that these various approaches identify as the
fundamental building blocks of spacetime. Only if we are able to
show convincingly that a good continuum description of spacetime,
with its dynamics governed by (some modified version of) General
Relativity, emerges naturally from the formulation of quantum
gravity we favor, we will have a truly convincing argument for
believing this formulation. This is of course well-known by
researchers working in non-perturbative quantum gravity, and in
particular in the approaches we have just mentioned: loop quantum
gravity (and spin foam models), quantum Regge calculus and
(causal) dynamical triangulations. Indeed, many techniques and
strategies have been developed, within these various approaches,
to solve the continuum (and semi-classical) riddle, and many
results already obtained. We will briefly discuss, and try to
re-phrase, them in the GFT language. This will allow us to both
understand them as providing insights about different regimes and
features of the same type of models, and clarify in which sense
they do not represent the most convenient or natural way to
approach the continuum problem from a GFT point of view.
Last, we will argue that group field theories offer new and
powerful tools to tackle the problem of the continuum in quantum
gravity, together with a new perspective on the whole issue, that
could prove decisive for settling it, at the same time developing
further and going beyond the insights obtained from the other
approaches mentioned above. The suggestion will basically be that
we could try to view spacetime as a (peculiar indeed) condensed
matter system, with the GFT representing the microscopic
description of its \lq\lq atoms\rq\rq, and providing the starting
point for studying both the statistical mechanics and the
effective dynamics of large number of them, which we will
tentatively identify with continuum physics. In particular, group
field theories can offer the context and the tools to realize
explicitly the intriguing idea of spacetime as a condensate of
fundamental building blocks and of continuum geometry as an
emergent concept. We will then put forward a proposal for this GFT
condensate, suggest some concrete research directions (some of
which currently pursued), and offer some speculation on how a
continuum spacetime and General Relativity can emerge in this
scheme, again making use (also) of the condensed matter analogy.
Given its aims, this article will contain a limited amount of
technicalities, only those needed to introduce the main GFT idea
and general formalism, and only references to and brief
discussions of the many results obtained both in the GFT context
and in the context of the other approaches to quantum gravity we
will mention. At the same time, it may contain a more than average
amount of speculations, especially in its last part, when we will
try to forecast where the new perspective we are advocating may
lead to. We will hopefully compensate for this by trying to be as
precise as possible in presenting the main ideas, motivations and
arguments behind this perspective, and to convince the reader that
this may be an intriguing and reasonable picture of what recent
results in quantum gravity research are pointing to.
\section{The group field theory formalism}
We now proceed to introduce the main features of the GFT
formalism. We refer to the literature, in particular the reviews
\cite{iogft,iogft2,laurentgft}, for a more complete and detailed
treatment and a more extensive list of references.
\subsection{Kinematics: the fundamental building blocks of quantum space}
We start from a field taken to be a
$\mathbb{C}$-valued function of D group elements, for a generic
group $G$, one for each of the D boundary (D-2)-faces of the
(D-1)-simplex that the field $\phi$ represents: $$\phi(g_1,g_2,...,g_D):
G^{\times D}\rightarrow \mathbb{C}.$$
In models (aiming at) describing D-dimensional quantum gravity,
this field is interpreted as a second quantized (D-1)-simplex,
with (D-2)-faces of the same labelled by group theoretic data,
interpreted as (pre-)geometric elementary quantities, or discrete
quantum gravity variables. Equivalently, the same data can be
associated to the links of a topologically dual graph, and the
field is then seen as the second quantization of a spin network
functional \cite{LQG}. This means that GFTs can be seen
equivalently as a second quantized formulation of spin network
dynamics or as a field theory {\it of} simplicial geometry. We can
identify the ordering of the arguments of the field with a choice
of orientation for the (D-1)-simplex it represents, and we require
invariance of the field under even permutations $\sigma$ of its
arguments and trade odd permutations with complex conjugation of
the field. Other symmetry properties can also be considered. An
additional symmetry that is usually imposed on the field is the
invariance under diagonal action of the group $G$ on the D
arguments of the field: $\phi(g_1,...,g_D)=\phi(g_1g,...,g_Dg)$;
but this is again model-dependent, of course, and in the models of
\cite{generalised,iotimgft}, for example, only invariance under a
certain proper subgroup is imposed. This is the simplicial
counterpart of the Lorentz gauge invariance of continuum and
discrete first order gravity actions, and it has also the
geometric interpretation, at the simplicial level, of requiring
the D faces of a (D-1)-simplex to close.
A momentum representation for the field and its dynamics is
obtained by harmonic analysis on the group manifold $G$. The field
can be expanded in modes as:
$$\phi(g_i)=\sum_{J_i,\Lambda,k_i}\phi^{J_i\Lambda}_{k_i}\left( \prod_iD^{J_i}_{k_il_i}(g_i)\right) C^{J_1..J_D\Lambda}_{l_1..l_D}, $$ with the $J$'s
labelling representations of $G$, the $k$'s vector indices in the
representation spaces, and the $C$'s being intertwiners of the
group $G$. We have labelled an orthonormal basis of intertwiners
by an extra parameter $\Lambda$ (depending on the group chosen and
on the dimension D, this may actually be a shorthand notation for
{\it a set} of parameters). That this decomposition is possible is
not guaranteed in general, but it is in fact true for all the
known quantum gravity GFT models, which are based on the Lorentz
group or on extensions of it. The proper geometric interpretation
of the field variables can be identified by looking at the Feynman
amplitudes for the GFT at hand, that either have the form of
discrete path integrals for some gravity action
\cite{generalised,iotimgft} or can be derived from one
\cite{iogft,iogft2,laurentgft}. This interpretation depends of
course on the specific model considered. However, generally
speaking, the group variables are seen to represent parallel
transport of a (gravity) connection along elementary paths dual to
the (D-2)-faces, and the representations $J$ are usually put in
correspondence with the volumes of the same (D-2)-faces.
\begin{figure}[t]
\includegraphics[width=15.4cm, height=3cm]{field-triangle-vertex.eps}
\caption{For the $D=3$ case, the association of a field with a
2-simplex, or equivalently its dual vertex, and of its arguments
with the 1-faces of it, or equivalently with the links incident to
the vertex, together with the labelling by group-theoretic
variables.}
\end{figure}
Just as one identifies a single field with a single (D-1)-simplex,
a simplicial space built out of $N$ such (D-1)-simplices is
described by a suitable polynomial in the field variables, with
constraints among the group or representation data, implementing
the fact that some of their (D-2)-faces are identified. For
example, a state describing two (D-1)-simplices glued along one
common (D-2)-face would be represented by: $\phi^{J_1
J_2..J_D\Lambda}_{k_1 k_2...k_D} \phi^{\tilde{J}_1
J_2...\tilde{J}_D\tilde{\Lambda}}_{\tilde{k}_1 k_2...\tilde{k}_D}$,
where the gluing is along the face labelled by the representation
$J_2$, and effected by the contraction of the corresponding vector
indices (of course, states corresponding to disjoint
(D-1)-simplices are also allowed).
\begin{figure}
\includegraphics[width=12.8cm, height=3cm]{trianglesglued.eps}
\caption{A \lq 2-particle state\rq (again, in the D=3 example)}
\end{figure}
We see that states of the theory are then labelled, in momentum
space, by {\it spin networks} based on the group $G$ \cite{LQG}.
GFT observables are given \cite{laurentgft} by gauge invariant
functionals of the GFT field, and can be constructed in momentum
space using again spin networks according to the formula:
$$ O_{\Psi=(\gamma, j_e,i_v)}(\phi)=\left(\prod_{(ij)}\int dg_{ij}dg_{ji}\right) \Psi_{(\gamma, j_e,i_v)}(g_{ij}g_{ji}^{-1})\prod_i \phi(g_{ij}),$$
where $\Psi_{(\gamma, j_e,i_v)}(g)$ identifies a spin network
functional \cite{LQG} for the spin network labelled by a graph
$\gamma$ with representations $j_e$ associated to its edges and
intertwiners $i_v$ associated to its vertices, and $g_{ij}$ are
group elements associated to the edges $(ij)$ of $\gamma$ that
meet at the vertex $i$.
Thus, {\bf group field theories describe a quantum space in terms
of fundamental building blocks, the quanta of the GFT field, that
acquire then the status of \lq\lq atoms of space\rq\rq in this
setting, and that can be represented both as spin network vertices
or as elementary (D-1)-simplices. A generic quantum state will be
a \lq\lq many-particle\rq\rq configuration for these quanta,
representing some extended discrete structure (a larger spin
network or a larger (D-1)-triangulation)} characterized by both
the \lq\lq particle number\rq\rq and by additional symmetries or
constraints imposed, specifying how the fundamental building
blocks are glued together. This picture can be made more precise
and a Fock space characterization of the GFT state space (and thus
of quantum space, in this framework) can be obtained after
Hamiltonian analysis of specific GFT models \cite{iojimmy}.
\subsection{Dynamics: the interaction and evolution of the atoms of space}
On the basis of the above kinematical structure, one aims at
defining a field theory for describing the interaction of
fundamental atoms of space, and in which {\bf a typical
interaction process will be characterized by a D-dimensional
simplicial complex. In the dual picture, the same will be
represented as a spin foam (labelled 2-complex).} This is the
straightforward generalization of the way in which 2d discretized
surfaces emerge from the interaction of matrices (graphically,
segments)\cite{mm}, or ordinary Feynman graphs emerge from the
interaction of point particles. A {\it discrete} spacetime emerge
then from the theory as a virtual construct, a possible
interaction process among the GFT quanta.
In order for this to be realized, the classical field action in
group field theories has to be chosen appropriately. In this
choice lies the main peculiarity of GFTs with respect to ordinary
field theories. This action, in configuration space, has the
general structure:
\begin{eqnarray} \hspace{-0.4cm} S_D(\phi, \lambda)= \frac{1}{2}\left(\prod_{i=1}^D\int
dg_id\tilde{g}_i\right)
\phi(g_i)\mathcal{K}(g_i\tilde{g}_i^{-1})\phi(\tilde{g}_i)
+
\frac{\lambda}{(D+1)!}\left(\prod_{i\neq j =1}^{D+1}\int dg_{ij}\right)
\phi(g_{1j})...\phi(g_{D+1 j})\,\mathcal{V}(g_{ij}g_{ji}^{-1})\;\;, \label{eq:action}
\end{eqnarray}
and it is of course the choice of kinetic and interaction
functions $\mathcal{K}$ and $\mathcal{V}$ that define the specific
model considered. Obviously, the same action can be written in
momentum space after harmonic decomposition on the group manifold.
The interaction term describes the interaction of D+1
(D-1)-simplices to form a D-simplex (\lq a fundamental virtual
spacetime event\rq) by gluing along their (D-2)-faces (arguments
of the fields), that are {\it pairwise} linked by the interaction
vertex. The nature of this interaction is specified by the choice
of function $\mathcal{V}$. The kinetic term involves two fields
each representing a given (D-1)-simplex seen from one of the two
D-simplices (interaction vertices) sharing it, so that the choice
of kinetic functions $\mathcal{K}$ specifies how the information
and therefore the geometric degrees of freedom corresponding to
their D (D-2)-faces are propagated from one vertex of interaction
to another. One can consider generalizations of the above
combinatorial structure, corresponding to the gluing of
(D-1)-simplices to form different sorts of D-dimensional complexes
(e.g. hypercubes etc).
Some examples of GFT actions are: 1) those corresponding to the
kinetic and vertex functions:
\begin{equation} \mathcal{K}(g_i,\tilde{g}_i) = \prod_{i=1}^{D}
\delta(g_i\tilde{g}_i^{-1}),\;\;\;\;\;\;\mathcal{V}(g_{ij},g_{ji})
= \prod_{i<j=1}^{D+1}\delta(g_{ij}g_{ji}^{-1}), \label{BF} \end{equation}
which produce a perturbative quantum dynamics that can be related
to topological BF theories in any dimension, for internal gauge
group $G$; 2) models in which suitably defined additional
constraints on the same BF-type kinetic and/or vertex terms are
imposed, and which aim at representing the GFT equivalent of the
constraint reducing BF theory to gravity in a Plebanski-like
formulation of the same \cite{DP-F-K-R,P-R,iolaurentkirill}; 3)
extended models based on more than Lorentz group variables and
characterized by a proper differential operator playing the role
of kinetic term, one example of which is the class of models in
\cite{iotimgft}, using a complex field on $(G\times X)^D$, with
$G$ being the Lorentz group and $X$ a metric space isomorphic to
the Lie algebra of $G$, and based on the kinetic and vertex terms:
\begin{eqnarray}
\hspace{-0.5cm} \mathcal{K}(g_i,x_i, \tilde{g}_i,\tilde{x}_i) = \,
\prod_i\,\left( \triangle_{i} + \square_i \right)\delta(g_i
\tilde{g}_{i}^{-1})\delta(x_i - \tilde{x}_{i}^{-1}) \;\;\;\;\;\;
\mathcal{V}(g_{ij},x_{ij}) =\,\prod_{i\neq j}^{} \delta(g_{ij}
g_{ji}^{-1})\delta(x_{ij} - x_{ji}) \label{EF}
\end{eqnarray}
where $g_i\in G$, $x_i\in X$, $\triangle$ is the Laplace-Beltrami
on $X$ and $\square$ is the Laplace-Beltrami on $G$; these last
models produce Feynman amplitudes with the interpretation of
simplicial path integrals for 1st order gravity actions
\cite{iotimgft}.
\ \
Let us now turn to the quantum dynamics. Most of the research in
this area has concerned the perturbative aspects of this dynamics
around the no-particle state, the complete vacuum, and the main
guide for model building have been, up to now, only the properties
of the resulting Feynman amplitudes:
$$ Z\,=\,\int
\mathcal{D}\phi\,e^{iS[\phi]}\,=\,\sum_{\Gamma}\,\frac{\lambda^{N_v(\Gamma)}}{sym[\Gamma]}\,Z(\Gamma),
$$
where $N_v$ is the number of interaction vertices $v$ in the
Feynman diagram $\Gamma$, $sym[\Gamma]$ is the number of
automorphisms of $\Gamma$ and $Z(\Gamma)$ the corresponding
Feynman amplitude. Each edge of the Feynman graph is made of $D$
strands, one for each argument of the field and each one is then
re-routed at the interaction vertex, with the combinatorial
structure of an $D$-simplex, following the pairing of field
arguments in the vertex operator.
\begin{figure}[here]
\setlength{\unitlength}{1cm}
\begin{minipage}[t]{3.5cm}
\hspace{-0.3cm}\includegraphics[width=3.5cm,
height=2.5cm]{propagator3d.eps}
\end{minipage}
\hspace{0.3cm}
\begin{minipage}[t]{5.5cm
\includegraphics[width=3.5cm, height=2.5cm]{vertex3d.eps}
\end{minipage}
\hspace{0.1cm}
\begin{minipage}[t]{5cm
\includegraphics[width=7cm, height=3cm]{propvertaction.eps}
\end{minipage}
\caption{The basic building blocks of the GFT Feynman diagrams
(for $D=3$).}
\end{figure}
Each strand in an edge of the Feynman diagram goes through several
vertices, coming back where it started, for closed Feynman
diagrams, and therefore identifies a 2-cell (for open graphs, it
may end up on the boundary, but still identifies a 2-cell). Each
Feynman diagram $\Gamma$ is then a collection of 2-cells, edges
and vertices, i.e. a 2-complex, that, because of the chosen
combinatorics for the arguments of the field in the action, is
topologically dual to a D-dimensional simplicial complex. Notice
that the resulting 2-cells can be glued (i.e. can share edges) in
all sorts of ways, forming for example \lq\lq bubbles\rq\rq, i.e.
closed 3-cells.
No restriction on the topology of the diagram/complex is imposed,
a priori, in the construction, so the resulting
complexes/triangulations can have arbitrary topology. Each of them
corresponds to a particular {\it scattering
process} of the fundamental building blocks of space,
i.e. (D-1)-simplices/spin network vertices. Each line of
propagation, made as we said out of D strands, is labelled, on top
of the group/representation data, by a permutation of $(1,..,D)$,
representing the labelling of the field variables, and all these
data are summed over in the construction of the Feynman expansion.
The sum over permutations affects directly the combinatorics of
the allowed gluings of vertices with propagators\cite{DP-P}.
\begin{figure}[here]
\includegraphics[width=12.5cm, height=3cm]{constructionFD3d.eps}
\caption{The gluing of vertices of interaction through
propagators, again in the D=3 example. The rectangles represent
the additional integrations imposing gauge invariance under the
action of $G$, while the ellipses represent the implicit sum over
permutations of the (labels of the) strands to be glued.}
\end{figure}
As said, each strand in a propagation line carries a field
variable, e.g. a group element in configuration space or a
representation label in momentum space. After the closure of the
strand to form a 2-cell in a closed diagram, the same
representation label ends up being associated to this 2-cell.
Therefore in momentum space each Feynman graph is given by a spin
foam (a 2-complex with faces labelled by representation
variables), and each Feynman amplitude (a complex function of the
representation labels, obtained by contracting vertex amplitudes
with propagator functions) by a so-called spin foam model
\cite{SF} (in the models \cite{generalised,iotimgft} the labelling
of the spin foam 2-complex is slightly more involved). The inverse
is also true: any local spin foam model can be obtained from a GFT
perturbative expansion \cite{mikecarlo,laurentgft}. The sum over
Feynman graphs gives then a sum over spin foams, and equivalently
a sum over triangulations, augmented by a sum over algebraic data
(group elements or representations) with a geometric
interpretation, assigned to each triangulation. This perturbative
expansion of the partition function also allows for a perturbative
evaluation of expectation values of GFT observables, as in
ordinary QFT. In particular, the transition amplitude (probability
amplitude for a certain scattering process) between certain
boundary data represented by two spin networks, of arbitrary
combinatorial complexity, can be expressed as the expectation
value of the field operators having the same combinatorial
structure of the two spin networks \cite{laurentgft, iogft}.
$$
\langle \Psi_1\mid\Psi_2\rangle = \int
\mathcal{D}\phi\,O_{\Psi_1}\,O_{\Psi_2}\,e^{iS(\phi)} =
\sum_{\Gamma/\partial\Gamma=\gamma_{\Psi_1}\cup\gamma_{\Psi_2}}\,\frac{\lambda^N}{sym[\Gamma]}\,Z(\Gamma)
$$
where the sum involves only 2-complexes (spin foams) with boundary
given by the two spin networks chosen.
\ \
The above perturbative expansion involves thus two types of sums:
one is the sum over geometric data (group elements or
representations of $G$) entering the definition of the Feynman
amplitudes as the GFT analogue of the integral over momenta or
positions of usual QFT; the other is the overall sum over Feynman
diagrams. We stress again that, in absence of additional
restrictions being imposed on the GFT, the last sum includes a sum
over all triangulations for a given topology and a sum over all
topologies.
\subsection{A peculiar quantum field theory (still, a proper field theory!)}
In the end, {\bf GFTs are a peculiar type of quantum field
theories}, defined on specifically chosen group manifolds. The
main reasons why they are rather peculiar, from a purely
field-theoretic perspective, are:
\begin{itemize}
\item the way in which field arguments are paired in the
interaction term, which makes them a sort of {\it combinatorially
non-local field theories};
\item the resulting combinatorial structure of Feynman diagrams,
given, as we discussed by fat graphs dual to simplicial complexes,
but also presenting no true vertex of interaction, in the usual
QFT sense of simultaneous identification of more than two
configuration variables, and constituted only by \lq loops \rq
(closed lines of propagation of the individual field arguments)
and \lq bubbles\rq (3-cells bounded by several such loops);
\item the fact that all the arguments of the field are naturally
treated on equal footing; if a specific time parameter can be
identified among the group coordinates, still there would be one
such parameter for each argument of the field, thus D in total,
leading to a sort of \lq multi-time dynamics\rq; in the
Hamiltonian analysis of GFTs \cite{iojimmy}, this implies the need
for a polysymplectic canonical formulation and has several
interesting consequences;
\item the fact that, for GFTs characterized by kinetic functions
formed by differential operators, there is then naturally one such
operator for each argument of the field, and a product structure
of the full kinetic term, reproducing again this independent
propagation of field arguments, but also producing technical
complications.
\end{itemize}
However, as for the rest, we have an almost ordinary field theory,
in that we can rely on a fixed background metric structure, given
by the invariant Killing-Cartan metric on the group manifold (or
extensions of it), a fixed topology, given again by the topology
of the group manifold, the usual splitting between kinetic
(quadratic) and interaction (higher order) term in the action, and
the usual conjugate pictures of configuration and momentum space.
This allows us to use all usual QFT techniques and language in the
analysis of GFTs, and thus of quantum gravity, even though we
remain in a background independent (in the physical sense of \lq
spacetime independent\rq) context. The importance of this, in a
non-perturbative quantum gravity framework, should not be
underestimated, we think, and it is at the roots of the strategy
we will propose later on to tackle the issue of the continuum and
semi-classical approximation.
\section{Group field theory as a common framework for discrete quantum gravity}
{\bf GFTs can potentially represent a common framework for
different current approaches to quantum gravity}, in particular
canonical loop quantum gravity\cite{LQG} and simplicial quantum
gravity formalisms, namely quantum Regge calculus \cite{QRC} and
(causal) dynamical triangulations \cite{CDT}, because the same
mathematical structures that characterize these approaches also
enter necessarily and in very similar fashion in the GFT
framework. {\bf We believe in the need to learn from all of them
in order to solve the remaining challenges towards a complete
theory of quantum gravity, and the GFT formalism may be the most
suitable framework in which the many lessons we can draw from all
of them can be brought together and to fruition.}
\subsection{Convergence of formalisms, structures and languages}
Historically, GFTs can be understood as being born as a
generalisation of matrix models \cite{mm} for 2-dimensional
quantum gravity. This generalisation is obtained in two steps: 1)
by passing to generic tensors, instead of matrices, as fundamental
variables, thus obtaining a generating functional for the sum over
D-dimensional simplicial complexes that was the essence of the
dynamical triangulations approach to quantum
gravity\cite{gross,ambjorn}; 2) adding group structure defining
geometric degrees of freedom. The last step is what turns a tensor
model into a proper field theory. In fact, the first example of a
GFT was the group-theoretic generalisation of 3d tensor models
proposed by Boulatov \cite{boulatov}, corresponding to the $D=3$
and $G=SU(2)$ case of (\ref{BF}). Already at this initial stage,
group field theories allowed a direct contact between simplicial
quantum gravity and what we now call spin foam models \cite{SF},
as the Boulatov model produces Feynman amplitudes given by the
so-called Ponzano-Regge spin foam model. As we have discussed
above, we now know that this is just one example of a very general
result \cite{mikecarlo}: the equivalence between (local) spin foam
models and GFT Feynman amplitudes. In turn, spin foam models
\cite{SF} have been a very active area of quantum gravity research
in the past ten years, for two main reasons. First, one obtains a
spin foam model when considering a path integral quantization of
discrete gravity formulated as a gauge theory. Second, spin foams
arise naturally when considering the dynamics of the kinematical
quantum states of geometry as identified by canonical loop quantum
gravity \cite{LQG}. Indeed, from the LQG perspective, spin foams
represents the histories of spin networks and are thus the crucial
ingredient of any path integral or covariant formulation of the
quantum gravity dynamics in LQG. From both the simplicial and
canonical perspective, a sum over spin foams/triangulations,
weighted by appropriate amplitudes, is a crucial ingredient in
defining the dynamics of the gravitational field: in simplicial
quantum gravity because such sum can compensate the truncation of
geometric degrees of freedom that the restriction to a given
lattice imposes; in LQG, because a complete path integral
formulation of the dynamics needs, in general, a sum over all the
histories between given spin network states. At present, group
field theories are the only known tool to define uniquely and
completely such sum over spin foams. Now let us give a closer look
at how the various ingredients of these various approaches, that
all have historically contributed, with hindsight, to the
development of the group field theory formalism, can be identified
and re-interpreted within the formalism itself.
\subsection{Loop quantum gravity and group field theory}
We have mentioned already the first and most basic link between
the group field theory formalism and loop quantum gravity: {\bf
boundary states of generic GFTs are spin networks}, i.e. what has
been identified by the canonical loop quantization programme as
the kinematical quantum states of geometry \cite{LQG}. {\bf The
GFT field itself}, as we have seen, {\bf is interpreted as the
result of a 2nd quantization of a spin network wave function.}
This correspondence can be made more precise, and one can in fact
show \cite{ioeteraSN} that a generic spin network wave function
can be re-expressed as a direct analogue of a multi-particle wave
function, with the particle degrees of freedom being associated to
the spin network vertices; a standard second quantization
procedure applied to these multi-particle wave functions, then,
leads to a field defined on the same group manifold from which
spin network data are taken, and that can be straightforwardly
identified with the GFT field. {\bf GFTs therefore define possible
dynamics for these quantum states of geometry, in a 2nd quantized
formulation,} and in a way that identifies the basic dynamical
degrees of freedom as those associated to the vertices of the spin
networks themselves, that in turn have been shown in LQG to
correspond to elementary chunks of space volume. From these
kinematical considerations, it immediately follows that any
quantum operator that can be defined in the 1st quantized LQG
setting has a 2nd quantized GFT counterpart, that can be, at least
in principle, identified. More importantly, this suggests that the
LQG {\it dynamics} can be embedded and studied within the GFT
setting. There are two equivalent ways in which this can be done.
First of all, as in any QFT, the GFT classical action should
encode the full 1st quantized dynamics, and the classical
equations of motion should correspond to the full dynamical
equations of the 1st quantized wave function. Solving the GFT
classical equations, then, means identifying non-trivial quantum
gravity wave functions satisfying {\bf all} the quantum gravity
constraints, an important and still unachieved goal of canonical
loop quantum gravity, except in some simplified situations. The
same classical equations of motion can be solved, implicitly, also
at the level of the perturbative Feynman expansion: one could
consider the restriction of the GFT perturbative expansion given
above to {\it tree level}, for given boundary spin network
observables \cite{laurentgft}. This is the GFT definition of the
canonical inner product between two spin network states. The
definition is well posed, because at tree level every single
amplitude $Z(\Gamma)$ is finite whatever the model considered due
to the absence of infinite summation (unless it presents
divergences at specific values of the configuration/momentum
variables). Moreover, it possesses all the properties one expects
from a canonical inner product \cite{laurentgft}. This means that
the physical Hilbert space for canonical spin network states can
be constructed starting from the above definition of the inner
product. This shows a concrete example of how the dynamics of spin
network states can be encoded covariantly in a sum over spin
foams, in the same sense in which the dynamics of canonical
gravity in ADM variables can be formulated, in principle, as a
covariant path integral over geometries (see \cite{alex} for more
details on this perspective on spin foam models).
There are of course many open issues regarding the exact
connection between the LQG and the GFT frameworks. One concerns,
for example, the role of spatial topology change. Its status
within LQG is not obvious at present: on the one hand, LQG being
the result of a canonical quantization on a globally hyperbolic
manifold, one would expect spatial topology change to be ruled out
almost by definition; on the other hand, the resulting quantum
states of geometry unavoidably describe also quantum spatial
geometries with degeneracy points, and thus seem to admit the
possibility of branchings of space at those points). In GFT, as we
have seen, non-trivial topologies appear in perturbative expansion
as soon as one goes beyond tree level, and there is no known
mechanism to either suppress or avoid them. Another open issue is
the interpretation, from the quantum gravity point of view in
general, and within LQG in particular, of the GFT coupling
constant; for some proposals on this, we refer to the literature
\cite{iogft}. One more unsettled point is whether one should
expect a direct link between the GFT and the LQG dynamics, i.e.
between the GFT action and the LQG Hamiltonian constraint already
at the level of the, supposedly, microscopic definition of the GFT
itself, or at the level of some macroscopic, effective QFT action
defined starting from the microscopic GFT dynamics. After all, the
Hamiltonian constraint operator of LQG is obtained by a direct
quantization of continuum (and possibly effective) General
Relativistic dynamics, and while one can be lucky enough to
capture some kinematical properties of the microscopic description
of a system, in general one should not expect to capture the exact
microscopic {\it dynamics} of the same starting from some
effective macroscopic description \cite{volovik}, although it is
certainly a possibility. More specific open issues concern the
exact choice of the gauge group, which is usually the full Lorentz
group in GFTs and the $SU(2)$ subgroup in LQG, the need for the
GFT restriction on the valence of spin network vertices, etc
However, while it is clear that much more work is needed to
explore and settle these issues, their presence does not spoil or
modify drastically, we think, the above general picture of the
GFT-LQG relation, and most importantly, all these issues can be
tackled {\it within the GFT formalism itself}.
\subsection{Simplicial quantum gravity and group field theory}
The GFT Feynman diagrams, as we have seen, identify simplicial
complexes to which the GFT assigns geometric data, weighted by
quantum amplitudes that can be related to path integrals for
simplicial gravity on the given complex, and indeed share the same
interpretation. These Feynman diagrams/simplicial complexes are
summed over to define the GFT partition function in perturbative
expansion, and thus the full dynamics.
$$ Z\,=\,\int \mathcal{D}\phi\,e^{iS[\phi]}\,=\,\sum_{\Gamma}\,\frac{\lambda^{N_\Gamma}}{sym[\Gamma]}\sum_{\{J_i\}}\,A_\Gamma(J_i). $$
The relation between GFTs and traditional approaches to discrete
quantum gravity is therefore clear, at least in its general
features. {\bf For given Feynman diagram, and thus fixing a single
triangulation as a discrete model of spacetime, the GFT provides a
quantization of gravity in the spirit and language of quantum
Regge calculus}, by an assignment of geometric data that are (more
or less direct) analogues of the edge lengths used there, and
summing over all such possible assignments. The full amplitude
weighting such assignments, i.e. the specific function of the
geometric data to be used, is specified uniquely b the specific
GFT model one is considering. Schematically:
$$ Z\,=\,\int \mathcal{D}\phi\,e^{iS[\phi]} \curvearrowright \,
Z_{QRC}\,=\,\sum_{\{J_i\}}\,A_\Gamma(J_i) \approx \lq\lq \int
\mathcal{D}g \,e^{iS_{GR}(g)} " $$
{\bf If, instead of fixing the triangulation, i.e. considering a
specific GFT Feynman diagram, one freezes GFT field degrees of
freedom (thus fixing the geometric data) to some constant value,
the same GFT provides a definition of the dynamics of quantum
geometry} via a sum over triangulations weighted by purely
combinatorial amplitudes, i.e. functions of the combinatorics of
the simplicial complexes only. This is a definition of quantum
gravity {\bf in the same spirit and language of the dynamical
triangulations approach.} Schematically:
$$ Z\,=\,\int
\mathcal{D}\phi\,e^{iS[\phi]}\curvearrowright
Z_{DT}\,=\,\sum_{\Gamma}\,\frac{1}{sym[\Gamma]}\,A_\Gamma
(\lambda)\approx \lq\lq \int Dg\, e^{i S_{GR}(g)} " $$
The quantum amplitudes weighting histories of the gravitational
field are given, in both quantum Regge calculus and dynamical
triangulations, by the exponential of the Regge action for
discrete gravity, while in most spin foam models the connection
between the quantum amplitudes and the Regge action is clear only
in a particular regime and even there it is rather involved
\cite{SF}. However, such relation is much clearer in the recent
GFTs of \cite{iotimgft}, whose amplitudes have indeed the form of
simplicial gravity path integrals, with clearly identified
classical simplicial gravity actions. GFTs can then be said to
incorporate both traditional simplicial quantum gravity
approaches, and to do so in a nice complementary way. We do not
know, however, is they also do it correctly or whether, by doing
so, they extend the definition of both beyond what is useful or
needed. Much more work is needed, for example, to study in
greater detail the (classical and quantum) simplicial geometry
corresponding, for given triangulation, to the known GFTs. And
much more work is needed in order to understand what is the QFT
meaning of many of the configurations, e.g. those corresponding to
non-trivial spacetime topologies, or the non-manifold-like ones,
appearing in the perturbative GFT sum over triangulations; how one
could gain control over them is an open, important issue. Also, in
the modern {\it causal} dynamical triangulations approach, the
nice result (that we are going to discuss in the following)
concerning the continuum limit of the sum over triangulations seem
to depend on specific {\it causality} restrictions on the class of
triangulations summed over \cite{CDT}; whether and how one can
understand and implement such restrictions from a field theory
perspective and within the GFT setting is presently unclear. At
the same time, there is hope that the sum over triangulations may
provide a more powerful alternative to the refinement procedure of
Regge calculus to lift the restriction to a fixed simplicial
complex, and that the additional field-theoretic data and
associated gauge symmetries and non-perturbative information of
GFTs can be useful not only because they provide the theory with a
well-identified space of states etc, but also for gaining control
of the sum over triangulations of the dynamical triangulations
approach \cite{GFTopenmath}. To summarize, even given the present
limited level of understanding of GFTs, it is clear that they
represent a unification and a generalisation, that can perhaps
turn out to be useful in the future, of both quantum Regge
calculus and dynamical triangulations, together with a radical
change of perspective on them: GFTs define the 2nd quantized
description of the dynamics of fundamental simplicial building
blocks of space, and simplicial quantum gravity path integrals
arise in a perturbative definition of this dynamics around the
vacuum, either when considering single virtual interaction
processes, i.e. single Feynman diagrams (quantum Regge calculus),
or the full perturbative Feynman sum restricted to its purely
combinatorial properties (dynamical triangulations).
\subsection{Cui prodest?}
So far so good. All this may be interesting and indeed it is
intriguing to speculate of a unifying framework for all discrete
quantum gravity approaches, that encompasses loop quantum gravity
structures as well as simplicial quantum gravity ones. But is it
useful? Can it be helpful in solving any of the outstanding open
problems that these various approaches face? Does it really offer
a new perspective on them and on quantum gravity in general?
In fact, we believe it does offer such a new perspective and that
because of this it can be very useful in helping to solve some of
the current open problems, also by providing new technical tools
for doing so. We have mentioned already some of the possibilities,
e.g. the issue of the dynamics and of the definition of the
physical inner product in LQG, or a possible grasp on non-trivial
topologies in dynamical triangulations. However, what we have
mainly in mind is the issue of the continuum limit, because it is
here that the change in perspective offered by GFTs can be most
relevant. We are going to discuss this issue at length in the
following. Here, we limit ourselves to sketch very briefly what
this change in perspective amounts to and what new tools it
suggests and provides.
The change in perspective, with respect to all the other
approaches we have mentioned, stems from the following
consideration: {\bf all of them, spin foam models, quantum Regge
calculus, dynamical triangulations, arise in perturbative
expansion around the \lq no-particle fundamental vacuum\rq, as
Feynman amplitudes or Feynman diagrams sums.} This means two main
things: 1) that, {\bf from the point of view of GFTs, the
discretization of spacetime used by all of these approaches in
describing the dynamics of geometry, and encoded in a 2-complex
(spin foams) or in a simplicial complex (simplicial quantum
gravity) is not a regularization of the theory (gravity, here) in
the usual lattice gauge theory sense, but corresponds to
describing the physics of \lq few-particles\rq} (be them spin
network vertices or simplices) and virtual processes, with no
individual meaning themselves, except in very limited and specific
approximations; 2) that, at the same time, {\bf the GFT formalism
is in principle suited for going beyond this regime and describe
the many-particle as well as the non-perturbative physics} of the
same system, that is, unless -all- of these approaches are wrong,
quantum spacetime.
Together with a change in perspective, luckily, comes therefore
the possibility of using new mathematical tools and physical
ideas, provided as well by the GFT formalism. This, as said, is a
2nd quantization of the same basic kinematical (space) structures
used in the other approaches, and we know very well how
advantageous it is to have at one's disposal a 2nd quantized and
field-theoretic framework for studying the dynamics of a physical
system described in terms of \lq particle-like objects\rq. A 2nd
quantized, field theory description allows: to overcome the
supreme impracticability of solving the 1st quantized equations of
motions involving many particles (here, very complex spin networks
or extended triangulations), to deal in an easier way with the
symmetries and statistics of the fundamental quanta, to have full
control on quantum (e.g. self-energy) effects. Most importantly, a
field theory description is the best way of: studying the
properties of systems with many degrees of freedom (and, again,
gravity in general, and complex spin networks or extended
triangulations, are certainly examples of such systems);
connecting microscopic many-particle physics and macroscopic,
collective dynamics of the same, its statistical mechanics and the
corresponding thermodynamical quantities. We are going to expand
on this point in the following.
\section{Building up a coherent picture of quantum spacetime}
Once we have seen how (the basic ingredients of) different
discrete approaches to quantum gravity are incorporated within the
group field theory formalism, we can take a fresh look at the many
important results obtained in them, regarding the classical and
quantum nature of gravity and spacetime, and try to re-interpret
them in the GFT language and framework. We are going to be rather
brief, and possibly superficial, in our attempt to summarize in a
few key points what we have learned during many years of quantum
gravity research in such diverse directions, due to space (and
time) constraints, as well as our limited knowledge. We apologize
for this. This exercise has two purposes. 1) It may help in
acquiring a new understanding of the insights the different
approaches provide, and in analyzing their mutual compatibility,
and possibly also suggests ways in which what we have learned from
one approach can contribute to solving presently open problems of
another or common to all. 2) It is needed in order to check
whether a single coherent picture of quantum gravity, patching
together all these various insights and results, is possible,
within the GFT setting. If it turns out that, indeed, it is
possible, then we believe it would be arguably the best thing to
use it and develop it further.
\subsection{Insights from loop quantum gravity and spin foam models}
So, what have we learned about quantum gravity from loop quantum
gravity \cite{LQG} and spin foam models \cite{SF}? We have learned
first of all that the kinematical degrees of freedom of quantum
space can be captured and encoded in discrete, purely
combinatorial and algebraic structures, spin network states:
graphs labelled by group representations. And this applies as well
to kinematical semi-classical states approximating continuum
geometries. Of this space of states we have strong mathematical
results concerning inner products, kinematical observables,
functional properties and much more \cite{LQG}. Moreover, although
all this has been discovered by a direct canonical quantization of
continuum classical General Relativity with Einstein-Hilbert
action, we now understand this result as a very generic feature of
any description of geometry based on: 1) diffeomorphism invariance
and background independence, requiring a purely relational
description of space, hence the purely combinatorial substratum;
2) a formulation of geometry in terms of connections (and local
reference frames), i.e. a gauge-theory-like formulation of
gravity, hence the use of group elements and representations to
encode gravitational degrees of freedom. These are purely
kinematical considerations, referring solely to the way
information about space and its geometry can be encoded, to a
\lq\lq possible backbone\rq\rq of any theory of quantum gravity,
and thus may well hold regardless of specific dynamical details,
e.g. choices of action, additional symmetry requirements,
spacetime dimension, etc. Similar considerations apply to the
dynamics of space, that can as well be represented in purely
combinatorial and algebraic terms. We have learned this already
from the quantization of the Hamiltonian constraint in LQG
\cite{LQG}, but this is all the more evident in the spin foam
description \cite{SF} of the dynamics of quantum space. As we have
seen, we have again purely combinatorial structures (2-complexes)
labelled by purely algebraic data (group representations and
elements) to represent possible histories of geometry, at the
quantum level. And again, this general features follow naturally
from the requirements of background independence and from a
description of gravity as a gauge theory, either imported from the
canonical formulation, or implemented in some discrete
re-formulation of lagrangian quantum gravity, or somewhat implicit
in categorical quantizations of geometry, which are the main ways
in which a spin foam formalism arises \cite{SF}. Recent results
have confirmed that a spin foam formulation indeed is capable of
describing key properties of the dynamics of quantum gravity, both
in 3 and 4 dimensions, including matter coupling and graviton
propagation, at least in the approximation in which the relevant
spacetime geometry information can be encoded in discrete
structures. For these results, we refer to the literature (see,
e.g. \cite{SF,laurentlibro,graviton}). And also the canonical LQG
formulation of the dynamics has been shown to provide very
interesting physical insights on quantum geometry, at least in the
symmetry reduced context of Loop Quantum Cosmology \cite{LQC}.
From the overview of the GFT formalism that we have given earlier,
it should be clear that all these insights are are not only
compatible but also already fully incorporated in the GFT
framework. In this context, they imply the following: 1) that {\bf
GFT quantum multi-particle states encode correctly quantum
geometric degrees of freedom in a very precise sense, at least at
a kinematical level, and satisfy the requirements of background
independence}; 2) that {\bf GFTs are also able to describe the
corresponding multi-particle dynamics, at least in the
approximation in which the whole perturbative series needs not be
re-summed or high order Feynman diagrams can be neglected.} In
particular, the results on the coupling of matter Feynman diagrams
to spin foams \cite{laurentlibro} show how natural it is to treat
matter Feynman diagrams on the same footing as spin foams, i.e.
GFT Feynman diagrams, which is also confirmed by the corresponding
GFT formulation of the same gravity+matter models
\cite{iojimmymatter}. And the nice results on graviton propagator
in LQG/spin foams \cite{graviton}, using as well and in a crucial
way GFT techniques, seem to us to indicate that GFTs (as LQG and
spin foam models) permit first of all to re-formulate perturbative
gravity questions in a fully background independent language
(which it is we believe the greatest achievement, so far, of this
line of work), and also that GFT perturbative particle dynamics
can in fact reproduce general relativistic semi-classical dynamics
in the (semi-classical, large distance and close to flat)
approximation in which discrete gravity is directly applicable:
GFT few particle physics, and where, in particular, GFT Feynman
amplitudes reduce to semi-classical quantum Regge calculus, which
indeed is at the heart of these results \cite{graviton}, together
with LQG semi-classical kinematical states.
\subsection{Insights from quantum Regge calculus}
Let us then turn then our attention to what we have instead
learned up to now from (quantum) Regge calculus, referring to the
literature for more details \cite{QRC,hamber}. The main lesson, we
believe, is at the classical level: Regge calculus represents a
beautiful and faithful discretization of classical geometry and of
its dynamics. It has been shown, in fact, that classical Regge
calculus reproduces General Relativity in the continuum
approximation in at least two main ways: 1) the Regge action
approximates well the Einstein-Hilbert action (and the
correspondence generalises to higher-derivatives extensions of the
same) in the sense of measures, and 2) solutions of the linearised
Regge equations converge to analytic solutions to the linearised
Einstein's equations, when some appropriate conditions are met.
Even more confidence in the correctness of the Regge
discretization of classical geometry stems from the possibility of
identifying characteristic symmetries of continuum gravity in the
simplicial setting, including diffeomorphisms, when appropriately
defined, as well as the related discrete Bianchi identities (but
see, on this, \cite{renatediscrete}. In the GFT language, this can
be re-phrased by saying that, {\bf for GFT models that possess
Feynman amplitudes of the form of simplicial path integrals for
(some version of) the Regge action, or in the approximation in
which such form is obtained, there is evidence that the \lq\lq
classical dynamics\rq\rq of the GFT particles can correctly
reproduce relevant features of classical gravity, including
symmetries, and better and better the more GFT particles we
consider.} This already hints at the relation between continuum
geometry and the thermodynamic limit in GFTs (large number of
particles), on which we will say more in the following.
At the quantum level, the results are also interesting \cite{QRC}.
In particular, in the semi-classical, large scale, and flat
approximation, quantum Regge calculus reproduces very well the
graviton propagator and thus Newton's law, plus quantum
corrections, even for simple triangulations. It is quite natural
to expect this to be the case also in GFTs with a simplicial path
integral form of the Feynman amplitudes, and indeed the mentioned
results on the spin foam propagator of the lattice graviton seem
to confirm it, while at the same time confirming the correctness
of the choice of boundary states operated in that context. Many
other results concern matter coupling, quantum cosmology, etc. As
for the definition of the full gravitational path integral in
quantum Regge calculus, the situation is more controversial, and
much debate in particular has focused on the issue of the quantum
measure to be used \cite{QRC}. More precisely, the object of
interest is the continuum limit of the discrete path integral
defined by Regge calculus, on which there are interesting but not
fully conclusive results \cite{hamber}, and about which we will
say more in the next section. As explained above, this discrete
path integral is nothing more (for specific GFT models, or in
special limits of the same) than the GFT Feynman amplitude for a
particular interaction process of GFT quanta.
\subsection{Insights from matrix models and dynamical triangulations}
In matrix models for 2d quantum gravity and in their
higher-dimensional extensions, i.e. tensor models, as well as in
the strictly related dynamical triangulations (DT) approach
\cite{mm,CDT}, the goal is to obtain a consistent and computable
definition of the gravitational path integral, i.e. of the sum
over geometries for given spacetime topology, with some results
being obtained also on the limited extensions of the same to
non-trivial topologies. As such, the classical simplicial geometry
is of limited interest, and indeed it cannot be fully captured by
the approach due to the truncation of the geometric degrees of
freedom associated to the individual lattices. The classical {\it
continuum} geometry, on the other hand, is possibly reproduced to
the extent in which the DT partition function reproduces the
gravitational continuum path integral. {\bf In GFT terms} this is
easily understood, as it this means that, {\bf once one has frozen
the field degrees of freedom, the classical particle dynamics}
(classical simplicial gravity) {\bf cannot be reproduced in a
satisfactory manner, but at the same time the continuum field
dynamics} (continuum quantum gravity) {\bf could still in
principle be reproduced, at least to the extent in which the
truncated sum over Feynman diagrams, restricted to its
combinatorial properties, reproduces properties of the full field
partition function.} Therefore, all the many results obtained in
this approach refer to the continuum approximation of the discrete
gravitational path integral defined as a sum over triangulations,
and we defer their discussion to the next section. Here we limit
ourselves to notice that work in matrix models and dynamical
triangulations has resulted in an immense amount of results and
available tools, both analytical and numerical, in an almost
complete understanding of 2d quantum gravity with a nice
discrete-continuum correspondence, in both Riemannian and
Lorentzian cases, and in important results obtained recently for
higher dimensions concerning this discrete-continuum
correspondence, in the Lorentzian context of so-called {\it
causal} dynamical triangulations \cite{CDT}.
\section{The problem of the continuum: current strategies from a GFT perspective}
{\bf Given our favorite formulation of quantum gravity, using
discrete structures of some sort to describe spacetime and to
encode quantum geometric degrees of freedom, does it reproduce, in
some controlled and well defined approximation, a smooth
spacetime, and is the quantum dynamics of spacetime geometry
effectively approximated, in the same regime, by continuum General
Relativity, possibly modified by quantum effects?} This is the
problem of the continuum in quantum gravity, for how we see it.
And this is, in our opinion, {\it the} outstanding unsolved issue
that all the current approaches to quantum gravity, and certainly
the ones we have mentioned, loop quantum gravity and spin foam
models, quantum Regge calculus, dynamical triangulations, have to
tackle hard and solve, to be considered successful. The same, of
course, is true for group field theory. The importance of
obtaining a satisfactory understanding of this issue cannot be
overstated, we believe, as it would amount to showing that our
favorite formalism, whatever it is, does indeed provide at least
{\it one possible} quantum theory of gravity. In absence of such
result the connection with gravity would remain a (more or less
plausible) hypothesis, and, as stressed, for example, in
\cite{renate}, any interpretation of the discrete expressions one
has in terms of quantum spacetime structures can be taken only as
a suggestion, {\it before} a physically correct continuum
approximation to them has been found. {\bf The group field theory
formalism, in the perspective we are proposing, can offer new
tools to solve this issue} to each of the different approaches it
(potentially) subsumes, and at the same time capitalise on their
results and insights. However, we believe that {\bf it also calls
for a change in perspective and for a consequent new strategy.} We
will be arguing in this direction in the next section; here we
would like first to briefly overview the strategies currently
adopted within the other approaches, all of course sensible and
potentially successful, and then \lq\lq translate\rq\rq them in
the GFT language, since this translation will make clear why a
change in perspective and strategy is naturally suggested.
\subsection{The loop quantum gravity/spin foam strategy}
Research on the semi-classical and continuum approximation in loop
quantum gravity and spin foam models has been mainly carried out,
at least in the 4-dimensional setting, in the canonical
formulation and is mainly confined to the kinematical
setting\footnote{The exceptions, that may come to mind, are the
many results in Loop Quantum Cosmology \cite{LQC}, and the recent
progress on the spin foam calculation of the lattice graviton
propagator. However, the first apply to symmetry reduced
situations, where it is possible to encode {\it all} the (finite
number) degrees of freedom of the continuum theory in the discrete
spin network structures. The second is limited to perturbative
physics {\it around a semi-classical space geometry}, first of
all, but, more important, remains confined at the level of
(justified) discrete approximations and large scale information,
thus not really addressing the issue of the continuum in this
framework.} The starting point of the LQG/SF strategy (using
$SU(2)$ spin networks and related observables) for recovering
continuum physics is the construction of appropriate kinematical
quantum states of space which approximate continuum space
geometries in some sense. The first type of such
semi-classical/almost continuum states are the so-called \lq\lq
weaves\rq\rq \cite{weave,weavestat}. These are defined by a
(directed) graph embedded in a reference compact space $\Sigma$,
the links are dressed with holonomies of an $SU(2)$ connection in
the representation $j=1/2$, with appropriate intertwiners
labelling the vertices of the graph. This graph is taken to be a
huge collection of loops in $\Sigma$, uniformly distributed with
respect to some classical 3-geometry $h_{ab}$. The mean spacing
between the loops (akin to a sort of lattice size for the graph)
is of the order of the Planck length $l_P$. This means that the
number of loops is approximately $N=(\frac{L}{l_P})^3$ where $L$
is the distance scale corresponding to the volume region one is
interested in approximating, measured in the reference metric
$h_{ab}$. The observables considered as a probe of the
semi-classicality of our quantum state are areas of surfaces in
$\Sigma$ and 3-volumes of regions contained in it. The nice result
is that for large enough volume regions, the areas and volumes as
computed quantum mechanically on the weave state are very close to
the ones measured in the classical continuum metric $h_{ab}$, and
with very small uncertainties. In this sense, one can say that the
quantum state considered has a good continuum and semi-classical
approximation \cite{weave}. This type of construction can be
extended to consider random weaves and averages over ensembles of
graphs, using statistical techniques \cite{weavestat}. A different
type of improvement of this construction is to change test
observables \cite{coherent}, using for this scope the basic
canonical pair of variables of loop quantum gravity, i.e. triad
and holonomy operators. The resulting quantum states are then
semi-classical {\it coherent states} providing expectation values
for both of them that are close to the classical continuum values,
as well as minimizing the uncertainties of both, in an appropriate
sense. The resulting quantum states are then an even more
satisfactory approximation of continuum 3-geometry, and many nice
results can be proven for them \cite{coherent} (overcompleteness,
Ehrenfest properties, etc). Notice however that we are still
confined at the kinematical level, while what we are really
interested in reproducing, starting from out quantum gravity
formalism, is the continuum {\it dynamics} and the {\it spacetime
continuum}. The way to do this test in the Hamiltonian/canonical
setting would be to study the action of the Hamiltonian
constraint of the theory on these weave or coherent states. This
is extremely complicated, due also to the intrinsic complications
involved in the very definition of the Hamiltonian constraint
operator, and has not been done, to the best of our knowledge.
More work has been devoted recently to the spin foam formulation
of the dynamics, so maybe one would want to use these weave or
coherent states in that context. It has not been done, yet. The
general idea however would be to use the above
semi-classical/almost continuum states as boundary states for an
appropriate spin foam model and compute the quantum gravity
analogue of 2-point functions between two of them; the spin foam
amplitudes would impose the quantum dynamics and the result should
then be compared with continuum path integral
calculations\footnote{One would also have to compute observables
other than 2-point functions, but this does not alter our
argument.}. The calculation could be done for fixed spin foam
2-complex, but most likely should involve a sum over spin foams,
that could then be truncated because of some physical
requirements. One way to define such sum would be through the
corresponding GFT formulation, with the GFT here used only as a
auxiliary tool, devoid of physical meaning, for generating the sum
over 2-complexes. All this is possible and sensible. However,
notice the orders of magnitude that would be involved, generally
speaking, in such calculations: if we aim at reproducing continuum
physics over a scale of, say, $L=10^{-19}cm$ (the distance scale
of a quark), we would need boundary states, in our spin foam
calculations, that are weaves with about $N=10^{42}$ loops, or,
which is arguably the same, spin networks with a similar number of
vertices. The complexity of the spin foam complex would go
accordingly. It is not obvious that such a calculation would be
doable, and at the very least we are lead to look for some
alternative, more efficient procedure.
Let us look at the GFT translation of the same procedure, taking
the GFT formalism to be physically meaningful in itself and not
just a mathematical tool, and see how the above sounds like. In
the GFT language, interpreted {\it realistically}, the procedure
would then be the following: 1) consider a carefully chosen
multi-particle state of a given GFT (a quantum field theory)
corresponding to a wave function satisfying some carefully
specified conditions (with respect to your favorite choice of test
observables); this state should contain about $N=10^{42}$ GFT
quanta, say; more precisely you should consider two of these
states, one per boundary in a typical \lq\lq scattering\rq\rq
process; 2) construct the corresponding field observable and
insert it in the GFT partition function; 3) expand the GFT
partition function in perturbative expansion around the vacuum
state (i.e. the state with no GFT quanta), i.e. in Feynman
diagrams; these Feynman diagrams give all the possible virtual
interaction processes of the $10^{42}$ initial and final
particles, including all quantum loops, self-interactions etc.;
even for the simplest diagrams (e.g. tree level and next to tree
level), their complexity will be of the same order of and scale
with the complexity of the boundary states; 4) compute the
transition amplitude in this Feynman expansion, maybe truncating
the expansion to some given order in the GFT coupling constant
(notice that the needed order would be necessarily extremely
high).
{\bf The strategy is not wrong, in any sense, but it definitely
does not look like what one would naturally do to study the
physics of such hugely populated multi-particle state in a field
theory context.} The basic point is that, {\bf when we choose as
our system of interest a hugely populated particle state, we put
ourselves immediately in the situation in which the vacuum
no-particle state and its physics is not relevant, the Feynman
diagrams of the individual particles are not relevant, in a sense
the microscopic dynamics itself is not relevant
anymore}\footnote{One can of course be more optimistic and hope
that a smooth continuum spacetime arise, and a general
relativistic description of it, holds already, say, for distances
100 times the Planck length; this will make the number of needed
particles $N=10^6$. The numbers are then vastly different but the
result is the same: for this number of quanta, the direct solution
of the corresponding microscopic dynamical equation for wave
functions or the study of their dynamics via Feynman expansion
around the vacuum are at best unpractical and possibly even
conceptually mistaken. If such a lucky situation occurs, it would
simply mean that already at the order of $10^6$ particles, we are
free to take the limit $N\rightarrow\infty$.}. {\bf In any case,
the Feynman diagrammatics and the individual particle picture is
not the most convenient language to describe the relevant physics
of these states.} We are lead to look for an alternative.
\subsection{The quantum Regge calculus strategy}
In quantum Regge calculus\cite{QRC,hamber}, the theory is defined
by the Euclideanized (or statistical) discrete gravity path
integral on a fixed lattice (most often hypercubic, then
subdivided into simplices) $T$ (thus also for fixed topology,
usually the sphere or the torus): \begin{equation} Z_T=\prod_e\int
\mathcal{D}l_e\, e^{- S_{Regge}(l_e)}, \end{equation} where $e$ labels the
edges of the lattice, $l_e$ are the corresponding edge lenghts,
which are the fundamental variables, integrated over with some
measure $\mathcal{D}l_e$, and the most studied version of the
discrete action, in 4d, is the Regge one augmented by quadratic
higher derivative terms (a discretization of the Riemann tensor
squared):
\begin{equation} S_{Regge}(l_e) = \sum_{t} \left( \lambda V_t(l_e) - k A_t(l_e)
\epsilon_t(l_e)+ a \frac{(A_t(l_e)
\epsilon_t(l_e))^2}{V_t(l_e)}\right), \end{equation} where the sum runs over
the triangles of the 4d simplicial complex, $A_t$ are their areas,
$\epsilon_t$ the associated deficit angle (discrete curvature),
and $V_t$ is the contribution of the given triangle to the total
4-volume of the lattice \cite{QRC,hamber, renatediscrete}. The
partition function is then a function of the coupling constants
$\lambda$ (cosmological constant), $k$ (inverse of Newton
constant), and $a$. The integration over the edge lengths is
usually cut-off both in the IR and UV, to ensure convergence.
Studying the continuum approximation of this theory means studying
the above partition function and appropriate geometric observables
(average curvature, average square curvature, etc) for very large
simplicial lattices (often at fixed total 4-volume) in a scaling
limit, while removing the cut-offs, as a function of the coupling
constants. The aim is to show that in a region of the parameter
space the above reproduces continuum spacetimes and continuum
geometric observables, thus representing a good (regularised and
computable) substitute of the formal continuum gravity path
integral. As said, this analysis has been done exclusively for
statistical path integrals over euclidean geometries, and mainly
numerically. The main results are the evidence for a two-phase
structure: for a certain $k_c$ the average curvature vanishes; for
$k> k_c$ (small $G_N$) the simplicial complex degenerates into a
crumpled phase incompatible with a smooth geometry with simplices
of very small volumes and large curvature; for $k<k_c$ there is
instead evidence for a smooth phase, depending also on the value
of $a$ and $\lambda$, with small (and negative) curvature. See
\cite{hamber} for more details. There is evidence for a second
order nature of the phase transition \cite{hamber}, which is what
one needs in order to have long-range correlations, but this
evidence does not seem to be considered fully conclusive by the
community (see, e.g. \cite{renatediscrete}). The result seems to
be rather generic, i.e. not too strongly dependent on the specific
measure $\mathcal{D}l$ chosen or on the specific topology or
lattice structure chosen, even if for irregular lattices the phase
structure is more involved (more critical points) and singular
structures seem to appear (spikes) and the choice of measure
becomes more important. In the end, we cannot yet conclude whether
this approach reproduces continuum physics or not, but we
definitely have gained lots of insights in the properties of
similar discrete gravity path integrals, and many tools to analyze
them have been developed.
Once more, this does not look like the most natural procedure to
adopt to study the continuum approximation of the same structures
when embedded and re-interpreted in a GFT context. The discrete
gravity path integral on a fixed lattice, in fact, amounts to the
evaluation of a single GFT Feynman amplitude for a given
interaction process of the GFT quanta, and all the lattice
prescriptions used in quantum Regge calculus require a Feynman
diagram with about $10^3-10^4$ vertices of interaction (numerical
simulations have been performed with up to $16^4\sim 6 \times
10^5$ lattice size, and the continuum approximation is expected to
be only improved going to larger lattices). {\bf Such a huge
Feynman diagram computation would indeed capture some information
of the many-particle physics of the corresponding GFT, which is
again suggested to be the regime corresponding to continuum
gravity, but the truncation to a single Feynman diagram is most
likely not consistent within the GFT setting. Moreover, just as in
LQG, it seems that to study the many-particle dynamics of the
theory at the level of perturbative expansion around the vacuum is
definitely not the most convenient thing to do.}
\subsection{The dynamical triangulations strategy}
In the traditional euclidean triangulations programme, the theory
is defined by the partition function:
\begin{equation} Z=\sum_{T}\frac{1}{C_T}\,e^{- S_{Regge}(l, k, \lambda)}, \end{equation}
i.e. by a sum over {\it equilateral triangulations} $T$, at fixed
topology (usually the spherical one), with fixed edge length $l$
(which is interpreted as a cut-off), weighted by a symmetry factor
(the automorphism group of the triangulation, $C_T$, and a
euclideanized exponential of the same Regge action usually limited
to a cosmological and a curvature term, thus in the end a function
of the combinatorics of the triangulation only and of the
parameters $l$,$k$,$\lambda$. The continuum approximation involves
again evaluating explicitly this sum $Z(\lambda,l,k)$ or, more
precisely, its Legendre transform $Z(N_4,l,k)$ which corresponds
to work for fixed number of 4-simplices $N_4$ and thus with fixed
4-volume $V \sim l^4 N_4$. Having done this, one is interested in
the thermodynamic limit $N_4\rightarrow \infty$, $l\rightarrow 0$,
$V \sim$ constant. Simplifying a bit, the resulting phase
structure was found to be given again by two phases separated by a
critical value of $k$, $k_c$, depending ont he volume $N_4$. For
$k<k_c$ we have a crumpled phase characterised by small curvature,
high graph connectivity and very large Hausdorff dimension. For
$k>k_c$ one finds an elongated phase with large and positive
curvature and an effective branched-polymer geometry with
effective Hausdorff dimension equal to 2. One could still hope
that a continuum theory is defined at the transition point, if the
transition was second order, but further analysis (again not fully
conclusive) suggested that the transition is instead first order.
For more details and further references, see
\cite{renatediscrete}.
The situation changes drastically in the modern form of this
approach, the so-called {\it causal dynamical triangulations}, to
the point that one can even make the case
\cite{CDT,renate,renatediscrete} for the {\it origin} of the
troubles encountered in the euclidean dynamical triangulations, as
well as, to some extent, in euclidean quantum Regge calculus, in
finding a good continuum approximation to be the dominance of
pathological configurations such as baby universes and other types
of singular geometries. These configurations are basically
unavoidable in the euclidean setting. They are not so, however, in
a Lorentzian one, where instead one can indeed identify conditions
on the triangulations summed over that rule out them from the
start (i.e. by construction). This is what is achieved in the
causal dynamical triangulations approach. Here the basic
ingredients for the construction and definition of the
triangulations summed over are (see \cite{CDT} for more details):
1) a local light cone structure, i.e. a differentiation between
spacelike and timelike edges (which have a relative
proportionality factor $\Delta$ for their values, on top of the
difference in the sign of their square); 2) the existence of a
global discrete time function; 3) no spatial topology change
allowed with respect to this \lq time\rq structure. The
triangulations are then weighted by a complex exponential of the
same Regge action but now for Lorentzian simplicial geometries.
The results are striking \cite{CDT}. There are now three phases:
a) for large $k$ a phase characterised by 3-dimensional slices of
a branched-polymer type, so not a 4d smooth geometry, once more;
b) for small $k$ and small asymmetry parameter $\Delta$ a phase
with crumpled 3-dimensional slices, similar to the euclidean
setting; so, again, not a smooth 4d geometry; c) for sufficiently
small $k$ and sufficiently large $\Delta$, a stable, extended
4-geometry, with Hausdorff dimension equal approximately to 4 and
a global shape of spacetime related to a simple minisuperspace
model of gravity, similar to those used in quantum cosmology. This
is strong and exciting evidence for a smooth geometry and thus a
continuum limit, even though several features of the model itself
and of the resulting dynamics of geometry are yet to be
understood, such as whether the results are robust with respect to
limited extensions of the ensemble of triangulations considered,
how much exactly of the full dynamics of general relativity is
recovered in the continuum approximation, whether there is a way
to generate analytically the above sum over triangulations, that
is at present constructed algorithmically and only studied
numerically, etc.
How does the GFT translation of the above sound like? In the GFT
language, the above corresponds to the following: 1) consider a
specific GFT model, producing Feynman amplitudes with appropriate
exponential form (either real or complex) for a discrete gravity
action (with field theoretic data interpreted as either euclidean
or lorentizian discrete geometries); 2) fix all the field
theoretic data, e.g. the momenta of the GFT field to some constant
value, giving then equilateral triangulations dual to the GFT
Feynman diagrams (producing the parameters $l$ and, in the causal
case, $\Delta$); this corresponds to restricting to a specific
momentum regime for the GFT particles, i.e. to particles all
having the same momentum; 3) restrict the perturbative sum over
Feynman diagrams to only those diagrams of some given topology
(and further restrictions have to be in place to recover the
causal restrictions of \cite{CDT}); finally, perform the {\it
whole} sum computing in this way the corresponding restricted
sector of the theory partition function, and appropriate
observables. Once more, we see that {\bf one necessarily needs to
study Feynman diagrams or arbitrary combinatorial complexity, and
involving huge numbers of GFT quanta, supporting further the idea
that continuum physics corresponds to the many-particle physics of
the theory.} Importantly, the work on dynamical triangulations
provide lots of technical tools for studying it. {\bf The CDT
results,} moreover, {\bf seem to indicate that, at least in that
regime of the GFT, some continuum physics can indeed be captured
in satisfactory form by this procedure}, which is exciting indeed.
However, once more the above procedure seems not so convincing
from the GFT perspective (that of course one is free not to take):
first of all it is well possible that the DT and the CDT
restrictions at the level of GFTs are not consistent from a field
theory perspective. Here we are not so much concerned by the
restriction on the momenta, which may well simply correspond to a
particular sector of the GFT, and thus to a reasonable
approximation of the full theory. Rather, what may be more
problematic is the restriction to fixed topology and, in the CDT
case, to fixed slicing structures of the diagrams summed over. We
have a too poor understanding of the GFTs themselves
\cite{GFTopenmath} to specify what these restrictions amount to,
from a purely field theory perspective. For example, we may run
into problems in asuming these restrictions, if they do not
clearly amount to a classical limit, say, as for example the
large-N planar limit of matrix models, and still involve removing
the GFT analogue of quantum loops or the like. Modulo these
remarks, {\bf it is clear that the (C)DT restriction does indeed
amount to extract at least some non-perturbative information far
beyond the physics of the GFT vacuum state, i.e. the few particle
physics, so it is a sensible thing to do even within the GFT
setting; in practice, in fact, amounts to solving the theory
(computing the partition function) at least in a restricted
sector, which may well turn out to be the one in which continuum
gravitational physics lies.}
{\bf However}, the same doubt put forward concerning the other
approaches applies: {\bf how convenient is it to study the
non-perturbative many-particle physics, and the corresponding
vacuum state and its dynamics, using what remains a perturbative
expansion around the no-particle vacuum, encoding the the
many-particle dynamics in hugely complicated Feynman diagrams, and
then re-summing all of them?} Again, the GFT perspective calls for
the use of different tools and for a change in strategy.
\subsection{Lessons and further motivations}
Let us summarize briefly the outcome of this sketchy overview.
First of all, {\bf {\it all} these strategies and approaches {\it
do} teach us something about GFTs}, when embedded in it. Second,
{\bf {\it all} of them suggest or strongly support the view that
continuum physics corresponds to the many-particle sector of the
GFT formalism}, and most likely involve collective and
non-perturbative effects. Third, {\bf their translation in the GFT
language suggests that maybe, although we have been indeed
studying the relevant sector of the (GFT) theory, we have not used
the most convenient set of tools} and language for doing so.
Fourth, luckily enough, {\bf the GFT formalism potentially
provides us with all the non-perturbative, field-theoretic tools
and concepts for trying out a different strategy.} As we had
stressed, in fact, we know from condensed matter physics and
statistical physics that field theory and 2nd quantization
language are the most convenient ones to study many-particle
physics, the corresponding phases, collective behaviour, etc.
\section{Quantum spacetime as a condensed matter system}
Let us now give some more specific suggestions for what this GFT
perspective seems to imply. We will put forward an hypothesis for
the continuum phase of a GFT, i.e. the phase or regime of the
theory in which we expect continuum gravitational physics to be
reproduce, and some general hints at what the strategy to check
this hypothesis could be. The general idea for the above will be
to take GFTs seriously for what they (formally) seem to be, at
least as a working hypothesis, and consider them as the
microscopic description of a very peculiar condensed matter
system, which is quantum space. In other words, {\bf we will
consider the GFT quanta}, that can be pictured, as we have seen,
as spin network vertices or (D-1)-simplices, {\bf as the true
\lq\lq atoms of quantum space\rq\rq , its fundamental hypothetical
constituents, and the GFT formalism as the microscopic
(fundamental?) formulation of their quantum dynamics}, thus
described in terms of a peculiar (non-local, etc) quantum field
theory, but a quantum field theory nonetheless. Then, we will
broaden the discourse a little and try to summarise some of the
general insights that, once we have taken this standpoint, come to
us from condensed matter physics (and from condensed matter analog
gravity models).
\subsection{If GFT is its microscopic description, what is the
continuum and how to get there?}
We have seen that all current approaches seem to suggest that
continuum gravitational physics is obtained in what is, in the GFT
language, the (very) many-particles sector of the theory. This
perfectly match the working hypothesis of GFTs as the microscopic
description of the atoms of space. In other words, we most likely
need a {\bf very large number} of them to constitute a region of
space that can be governed effectively by continuum gravitational
physics, and be described by a continuum space to start with.
Moreover, from results in these approaches, as well as from
general physical intuition and, again, from the perspective of
spacetime as a \lq\lq material\rq\rq of some sort, made of (GFT)
constituents, we would expect these many constituents making up
the continuum to be {\bf very small}, in the appropriate sense,
probably of order of the Planck length (volume). In GFT terms,
generally speaking, this translates into the {\it GFT quanta to be
in a low momentum regime}. On top of this, we expect the quantum
constituents of space to be {\bf governed, in their continuum
phase/regime, by collective dynamical laws}, not anymore by the
microscopic individual dynamics, simply because otherwise we would
have noticed already the \lq\lq true\rq\rq atomic nature of space.
Finally, whatever the exact phase looks like, whatever the
symmetries characterizing it are, and whatever the effective
dynamics governing it is, we expect our condensed matter system,
i.e. quantum spacetime, modelled by our favorite GFT, to be {\bf
very close to equilibrium}. In other words, we expect a continuum
description of spacetime to prove itself correct, and not only
possible, when close to an {\it equilibrium} and stable
vacuum/phase of the (GFT) system, at least to scales close to the
sector of the physical world that has already probed (by us).
Again, this is simply because otherwise we would have most likely
already noticed a failure of the continuum description of
spacetime. From this perspective, the breakdown of general
relativistic theories of geometry in cosmological situations or in
black hole physics can be {\it speculated} to be a sign of a phase
transition occurring in the (fundamental?) GFT system.
Notice that none of the above implies that the continuum
approximations goes necessarily in hand with the semi-classical
approximation, which may be needed later on to simplify/extract a
specific dynamics or for capturing some relevant features of the
system in the regime we are interested in, but as far as the above
reasoning is concerned, the possibility of {\bf a continuum
description may even be {\it the result of a purely quantum
property of the system}}. We will give later on an explicit
proposal for this.
So, {\bf a continuum space is a very large number of very small
GFT quanta very close to equilibrium, i.e. very close to some yet
to be determined many-particle vacuum, to be described
collectively and whose dynamics is to be given by continuum
larger-scale equations}. This seems (to us, at least) just a
description of a fluid (whether gaseous or liquid or what else, is
to be determined by hard, technical, future work), close to
equilibrium, governed indeed by hydrodynamical equations.
The picture that seems to come out of the above reasoning, then,
and more indirectly (we admit that) from work in the various
approaches to discrete quantum gravity we have discussed is that
of {\bf {\it quantum spacetime as a (quantum) fluid of GFT
particles, governed microscopically by the GFT partition function,
but macroscopically by a suitably identified GFT effective
hydrodynamics.}}
As we had stressed, this is at present just a suggestion, of
course, given the little we understand GFTs themselves and the
(basically nihil) amount of work that has been devoted up to now
to develop and test it. But we find it a very intriguing and, most
important, convincing one. It immediately implies one thing: at
least for a while, at least from the GFT standpoint, and only if
we intend to tackle the issue of the continuum approximation and
its effective dynamics, it may be convenient to partially forget
about spin foams and even the simplicial gravity description of
the GFT system, and focus our attention on other aspects of the
formalism. This is simply because, as we have stressed, the
perturbative formulation of GFTs, which is where the spin foam and
the simplicial gravity descriptions appear, is very useful for the
physical interpretation of the system, of its quanta and field
theoretic data (indeed, we have relied exclusively on it for all
of the above reasoning), but it is {\it technically} useful for
describing the system in its few-particle regime. If we are
interested in describing the many-particle behaviour of the same
system we should move away from the no-particle vacuum.
In its stead, we need to develop first and then use a {\bf {\it
statistical group field theory}} formalism for identifying first
and then select the different phases of the theory, i.e. the
possible equilibrium configurations in which the system may find
itself, hoping that some of the GFT models we have or we will
construct for the scope allow for the existence of at least one
with the properties that allow for a continuum geometric
description. Second, we need to obtain an {\bf {\it effective
field theory or hydrodynamic description}}, coming from the
fundamental GFT, for describing the dynamics close to the
different phases, and probably tied to each particular phase under
consideration. We will speculate more, but also try to be more
specific, about how both may look like in the next section.
\subsection{What can quantum gravity learn from condensed matter theory and analogue gravity models}
The idea of spacetime as a condensed matter system in general, and
as a fluid in particular, and of GR as an hydrodynamic effective
description of it, is of course not new and has been advocated
many times, and very convincingly, in the past
\cite{jacobson,hu,volovik,laughlin,padmadaran}, and is both
motivated by and an inspiration for the many condensed matter
analog gravity models \cite{analog}. What is new here is only the
argument that it is the very research in non-perturbative quantum
gravity carried out to date, and the many results obtained in the
many approaches it is split into, that points in this direction.
Also, what is new here is the hypothesis that GFTs can represent:
1) the framework in which these many approaches to quantum gravity
and their insights can be seen as part of a single coherent
formalism and physical picture of spacetime; 2) also because of
this, a solid and motivated formalism to be used to realise
concretely, in mathematical and physical terms, the suggested idea
of spacetime as a condensed matter system of a peculiar type, and
a concrete, if tentative only, description of its microscopic
structure. This description, moreover, as we stressed repeatedly,
uses a field theory language that may facilitate the application
in this context of traditional condensed matter ideas and tools
(probably suitably adapted).
This is probably the main contribution that GFTs can provide
researchers working in condensed matter analog gravity models:
{\it a concrete formalism and system} on which to apply their
insights, if they are interested in unravelling the true
microscopic structure of quantum spacetime, and not only in
finding out more about its effective continuum description, once
interpreted as a condensed matter system, or in using the same
gravitational analogy and the general relativistic tools to
discover more interesting properties of the usual condensed matter
systems (Bose-Einstein condensates, etc)\footnote{Needless to say,
both things are definitely worthwhile and of fundamental
significance; simply, they are not quantum gravity issues.}.
In other words, it is often stated in the analog gravity
literature and in the condensed-matter-but-interested-in-gravity
community that \cite{hu,volovik,analog}: 1) quantum gravity is not
so much about quantizing general relativity in a strict sense, but
rather about identifying the microscopic constituents of space and
provide a tentative description of their microscopic dynamics; 2)
we do not know what this microscopic structure and dynamics is; 3)
the current top-down approaches to quantum gravity are so
different and so complicated that no coherent picture and no
clear indication about the fundamental structure of space is
provided by them, that could serve directly for the application of
the insights coming from condensed matter theory. What we have
argued is the following. The first thing is true, and {\bf it is
the very same approaches to non-perturbative quantum gravity that
have lead (in a rather tortuous way) to the GFT formalism which
itself is {\it not} a quantization of classical GR} (just look at
the GFT action). The second statement is true in a sense, {\bf we
do not have a clear and complete picture of the spacetime
microscopics}, but false or at least overly pessimistic in
another: {\bf we have several candidates for this microscopic
structure, and one, the GFT formalism, that seem to encompass many
of them}. The third statement is false: {\bf the respective
pictures that at least some of these approaches to quantum
gravity} (those we have discussed) {\bf provide are not only
compatible and coherently build up a tentative picture of quantum
spacetime} (the one encoded in the GFT formalism), {\bf but also
one that allows for a rather direct application of condensed
matter concepts, formalisms and techniques for understanding the
microscopic-macroscopic and discrete-continuum transition.}
What quantum gravity, and in particular the GFT approach, can
learn from condensed matter (CDM) physics and from condensed
matter analog gravity models is much more.
Concretely, the main help that condensed matter techniques can
provide stems from the fact that {\bf in that context}, as
stressed in \cite{analog}, {\bf the transition from discrete
microscopic physics and continuum macroscopic one is well
understood} conceptually and there are many theoretical tools that
can be applied to its analysis and study. As we have seen, this is
the main open problem of the discrete quantum gravity approaches
have to solve, even after they have provided a tentative
description of microscopic spacetime. This holds for GFTs as well,
and its field theory setting makes the application of CDM
techniques even more straightforward. More generally, taking a CDM
perspective means also a conceptual shift with respect to what we
expect from our theory, and how we approach our physical
challenges. We list here only some of the CDM wisdom (for more,
see \cite{hu,volovik,laughlin,anderson,analog}), that is useful
for approaching our quantum gravity problems, in our opinion. We
should not expect a rigorous, deductive path from the microscopic
dynamics to the macroscopic one, and even the kinematics (relevant
variables, symmetries, etc) at the macroscopic scale or in the \lq
continuum phase\rq , thus in the hydrodynamic regime, can be very
loosely related to the one of the corresponding microscopic
theory. In other words, even the relation between microscopic
variables and collective ones is often less that direct, and the
specific form of the microscopic QFT for your atoms is often not
at all similar to the macroscopic effective QFT for the resulting
fluid. In particular, many of the small details of the microscopic
theory become irrelevant at the hydrodynamic effective level. This
is governed mainly by general macroscopic symmetries and
associated conservation laws, that should acquire thus a
fundamental importance in our model building. It is not
reasonable, in light of the above, and at least if one is first of
all interested in showing that a continuum approximation exists,
to demand necessarily for exact treatments or to look for exact
solutions of microscopic dynamics, because this exactness will
almost inevitably end up being irrelevant at a different (larger)
scale. All this should apply to our future treatment of the GFT
formalism, in our attempt to use it to obtain the correct
macroscopic effective continuum description. Nothing revolutionary
here, of course, but things that are worth keeping in mind in
quantum gravity research, and in particular when one sees
spacetime as a condensed matter system, because they are often
neglected (by us, at least). Also, we are warned that experimental
is absolutely crucial for guiding model building and for guessing
what are the relevant features in the hydrodynamic regime, thus at
the effective level. The recent development of quantum gravity
phenomenology \cite{giovanni} it therefore of extreme importance,
also in this condensed matter interpretation.
At the same time, as stressed very nicely in \cite{anderson} (see
also \cite{laughlin}): \lq\lq The behavior of large and complex
aggregates of elementary particles, it turns out, is not to be
understood in terms of a simple extrapolation of the properties of
a few particles. Instead, at each level of complexity entirely new
properties appear\rq\rq . This is a warning but also an
encouragement because it implies richness and potential fun in
unravelling it.
\section{Guessing the future: several research directions, an hypothesis and some speculations}
The above discussion has been very general, serving only the
purpose of sketching what are further inputs to the GFT
perspective on the continuum we are advocating, again, as a
working hypothesis. Now we will try to be a bit more specific
about how one can develop further and what may come out of this
condensed matter perspective, in concrete terms, in the GFT
framework. We will put forward one specific proposal for what can
be the phase of the GFT, i.e. the relevant vacuum for the GFT
multi-particle physics, where continuum geometry and its dynamics
could be reproduced, and then explore, tentatively, some
possibilities for the dynamics of the theory in this phase, and
how it can relate to known formulations of classical continuum
gravity.
It should be clear that, given our present understanding of the
GFT formalism, any guess in this direction can be only partially
based on known results, but rather speculative. The study of the
GFTs in their own right, treated as peculiar but {\it bona fide}
field theories, is in its infancy and only the first basic steps
have been or are being taken \cite{iojimmy}. Nevertheless, they
already provide some hints of what may come next, and we are going
to build upon these hints in the following.
Before we do so, let us mention three other directions of work
that, in the perspective we are advocating, are certainly relevant
(see also \cite{GFTopenmath} for a more detailed discussion). One
if the development and use of renormalization group techniques.
The renormalization group is in fact one of the most powerful
tools we have in field theory and in condensed matter physics to
explore the structure and behaviour of our system at different
scales. It is indeed applied routinely in condensed matter for
investigating phase structures, which is exactly what we have
argued we have to do in our GFTs. In particular, we believe that
it would be very important, and of great direct relevance for
solving the problem of the continuum, to develop the formalism of
the Wilsonian Exact Renormalization Group for group field
theories, with the construction of the effective action and the
analysis of the corresponding flow, for specific GFT models. This
would not only prove the consistency of the given models
(renormalisability, etc) but also suggest what is the relevant
form of the theory (action) at the scales we expect to be related
to continuum physics. A second one is the study of classical
solutions of the GFT equations. Of course, they encode
non-perturbative information about the system, and thus are also
relevant for the continuum phase. This work has started
\cite{elainstantons}. However, we would like also to stress that,
from a condensed matter point of view, it may be even more
important to construct {\it approximate} solutions to the GFT
dynamics, tailored to the multi-particle situation. The third, and
maybe most important, is the analysis of the GFT classical
symmetries, to be done both at the lagrangian and hamiltonian
level \cite{iojimmy}; this is because, as stressed, macroscopic
behaviour and hydrodynamics in particular are likely determined
more by these symmetries, or their broken version, than by the
exact microscopic GFT dynamics.
\subsection{Geometrogenesis using GFTs}
Our proposed general scheme for the emergence of continuum
geometry from the dynamics of the GFT quanta can be seen as a
particular possible implementation of the {\it geometrogenesis}
idea.
This is the catchy name given in \cite{ltf} to a conjectured phase
transition of a combinatorial and algebraic model of quantum space
described by a a labelled graph, much alike spin networks, between
a high-temperature \lq pre-geometric phase\rq in which space has
the form of a complete graph, and thus no notion of locality or
geometry (e.g. distance), to a \lq geometric phase\rq in which the
graph acquires a more regular, local structure, where geometric
data can be identified. Furthermore, the data labelling the graph
then allow for the emergence of matter degrees of freedom, having
the role of qausi-particle moving on the resulting regular
lattice, in the same way as the model of topological order studied
by Wen et al \cite{wen} does, in terms of string condensation.
Now, the details of the model do not concern us here. We just want
to note the similarity with the idea we are proposing for the
emergence of the continuum in GFTs. The basic quantum states of
the GFTs, as we have seen, are characterized by labelled
combinatorial structures as well, of the spin network type (or,
dually, of a simplicial type). It seems to us that because of
this, any phase transition in a GFT setting will be described by a
transition from some irregularly structured and labelled graph or
from an ensemble of such graphs to a more regular and ordered one
at lower temperatures, in the same spirit as the model of
\cite{ltf}. Further, we are suggesting that after the ground state
has been identified its own effective dynamics will be described,
if the scenario we are suggesting is correct, by an effective
continuum field theory with a geometric interpretation, and in
principle derivable (but not necessarily deducible) from the
microscopic GFT. Both the hamiltonian function driving the
transition, and thus the selection of the ground state lattice,
and the effective hamiltonian governing the dynamics of
quasi-particles around the resulting ground state, the two main
ingredients of the model in \cite{ltf}, can in principle be
derived from any given choice of GFT action, whose dynamical
content is indeed the same, after appropriate simplifications. If
our understanding is correct, then, the model of \cite{ltf} can be
interpreted as an effective simplified GFT Hamiltonian, and
similar models can be constructed and inspired by the GFT
formalism as well. Conversely, we believe that more work in the
direction opened by the model \cite{ltf} will be of importance
also for the research programme we are suggesting, in that it will
amount to explore models that may indeed capture relevant features
of GFT phase transitions and vacua as well.
\subsection{Spacetime as a condensate: a GFT realisation?}
\subsubsection{Continuum space as a GFT Bose-Einstein condensate}
Our tentative proposal for a relevant vacuum of a GFT model in
which a continuum approximation could be expected, i.e. a
continuum and geometric phase of the model, is a simple one: a
Bose-Einstein condensate. Again, here it is not so much important
the idea in itself, because the similarities between continuum
spacetime and condensates have been noticed long ago and a similar
possibility has been advocated by several authors, and very
convincingly \cite{hu,volovik} and the effective (and emergent)
spacetime character of real Bose-Einstein condensates (those
stored in laboratories) is the basis of many condensed matter
analog gravity models \cite{analog}. What is important here is the
fact that the concrete realization of this scenario within a
specific microscopic model of quantum spacetime, i.e. a GFT model,
seems to us not only possible, but within reach. Of course, such
scenario involves first of all the development of a statistical
group field theory formalism, the identification of the GFT
analogues of relevant thermodynamical quantities, and more, and,
as we have noticed above, even basic steps in the analysis of GFTs
apart from their Feynman amplitudes have been taken only recently
\cite{iojimmy}. We will now sketch, also based on these initial
results, how thermodynamical quantities in a GFT setting could be
defined and then how the possibility of a Bose-Einstein condensate
of GFT quanta could be realized, including some likely features of
the resulting vacuum state.
GFT thermodynamic quantities \cite{iostat} will have to be defined
in a formal way, letting ourselves be guided, at first, only by
the field theory look of the GFT formalism, and only in a second
stage one should try to match the definition of each of them with
a corresponding physical interpretation. In turn, this physical
interpretation will have to rely almost exclusively on the
(pre-)geometric interpretation that the GFT variables have in the
context of the Feynman expansion, i.e. in the context of
simplicial gravity. This can be done more easily in an Hamiltonian
setting, and in the same context we will give now a sketch of a
possible concrete definition of Hamiltonian (thus of a GFT \lq\lq
energy\rq\rq) and temperature, while for other quantities we can
only offer guesses, at this point, although reasonable ones, we
hope.
Consider a GFT action like (we restrict here to the free theory,
which sffices for our present purposes)\cite{generalised}:
$$ S=\left( \prod_i \int_G dg_i
\int_\mathbb{R} ds_i\right) \,
\phi^\dagger(g_1,s_1;...;g_D,s_D)\prod_i\,\left( i\partial_{s_{i}}
+ \square_i \right)\phi(g_1,s_1;...;g_D,s_D) + h.c.$$ with $g_i\in
G$, $s_i\in\mathbb{R}$, $\square$ being the Laplace-Beltrami
operator on $G$, for generic group $G$ (Riemannian or Lorentzian).
The kinetic term has the structure of a product of differential
operators, each acting independently on one of the D (sets of)
arguments of the field. Each of them is a Schroedinger-like
operator with \lq \lq Hamiltonian\rq\rq $\square$. This suggests
that one should consider the variables $s_i$ as \lq\lq time\rq\rq
variables, to be used in a GFT generalization of the usual
time+space splitting of the configuration space coordinates, with
the group elements treated instead as \lq\lq space\rq\rq. This
implies that we have a field theory with D \lq\lq times\rq\rq, all
to be treated on equal footing. The approach chosen in
\cite{iojimmy} is to use the DeDonder-Weyl generalized Hamiltonian
mechanics, as developed at both the classical and quantum level as
a {\it polysymplectic (or polymomentum) mechanics} by Kanatchikov
\cite{kanatchikov}, as a starting point and to adapt it to the
peculiar GFT setting.
The general idea is the following \cite{iojimmy}. One starts from
a \lq\lq covariant\rq\rq definition of momenta, hamiltonian
density, Poisson brackets, etc treating all \lq\lq time
variables\rq\rq on equal footing at first, i.e. when defining
densities. Then one defines 'scalar' quantities referring to each
\lq time direction\rq (to be turned into operators at the quantum
level), including a set of D Hamiltonians, by integration over
appropriate hypersurfaces in $(G\times\mathbb{R})^{\times D}$, so
that each Hamiltonian refers to a single time direction, but at
the same time all time directions are treated equally but
independently. A similar procedure is adopted for other canonical
quantities, e.g. Poisson brackets, scalar products etc.
Let us sketch one example of such procedure, for the case $D=2$,
referring to \cite{iojimmy} for more details. We start from the
naive phase space $(\phi,\phi^\dagger,\pi_\phi^i=\frac{\delta
L}{\delta\partial_{s_i}\phi},\pi_{\phi^\dagger}^i=\frac{\delta
L}{\delta\partial_{s_i}\phi^\dagger})$, with the product structure
of the kinetic term resulting in a peculiar expression for the
momenta, e.g. $\pi_\phi^1 = (-i\partial_2
+\square_2)\phi^\dagger$, and define the DeDonder-Weyl Hamiltonian
density (summation over repeated indices understood):
$$ \mathcal{H}_{DW} = \pi_{\phi}^i
\partial_{s_i}\phi + \pi_{\phi^\dagger}^i
\partial_{s_i}\phi^\dagger - L = 2 \pi_{\phi^\dagger}^1 \pi_\phi^2 + i
\pi_\phi^1 \square_1 \phi + i \pi_\phi^2 \square_2 \phi + h.c. .$$
One then proceeds to re-write it as a sum of two contributions,
each uniquely associated to a single time parameter:
$\mathcal{H}_{DW} = \mathcal{H}_1 + \mathcal{H}_2$ , with
$\mathcal{H}_i = \pi_{\phi^\dagger}^1 \pi_\phi^2 + i \pi_\phi^i
\square_i \phi + h.c$. The Hamiltonians governing the \lq time
evolution\rq\ with respect to the different time directions
identified by each variable $s_i$ are then defined by integration
over independent hypersurfaces, each orthogonal to a different
time direction, e.g. $H_1 = \int ds_2 dg_i \mathcal{H}_1$. Each
$H_i$ results in being independent of time $s_i$.
One can then proceed, after suitable decomposition in modes of
fields and momenta, the definition of (a GFT-adapted version of)
the covariant Poisson brackets, etc, to the canonical quantization
of the theory, with the definition of a Fock structure on the
space of states. We refer once more to \cite{iojimmy} for the
results of this analysis.
From the above results, it is easy to guess how the notion of GFT
temperature may be defined, because it simply involves following
the usual QFT procedure. One could repeat the analysis above but
now requiring periodicity of the fields in the $s_i$ variables,
with period $\beta$, and would then be left with a partition
function in hamiltonian form:
$$
Z= \int \mathcal{D}\phi \mathcal{D}\phi^*\; e^{i\sum_i \int ds_i
H_i(\phi,\phi^*) }
$$
with the integration over $s_i$ restricted to the interval
$(0,\beta)$, and thus obtaining, after Wick rotation in the same
$s_i$ variables:
$$
Z= \int \mathcal{D}\phi \mathcal{D}\phi^*\; e^{-\beta \sum_i
H_i(\phi,\phi^*) } =\int \mathcal{D}\phi \mathcal{D}\phi^*\;
e^{-\beta H_{tot}(\phi,\phi^*) }
$$
with $\beta=\frac{1}{kT}$ defining the GFT temperature. The notion
of temperature, then, may be defined, and indeed the corresponding
quantity will play the role of a temperature at least at the
formal level. However, its physical interpretation will have to be
studied with care (even its dimensions may not be those of a
temperature). In other words, just as the variables $s_i$ played
the role of time in the formalism, and could be treated formally
as such in a consistent way, but still do not have the geometric
interpretation of time variables on any physical spacetime, not
even at the simplicial level, similarly the GFT temperature $T$
may be found to correspond, say, at the simplicial level, to a
geometric quantity that a priori has no similar interpretation,
even though the GFT sees it indeed as a temperature parameter. An
even clearer example is the notion of energy in the above simple
GFT. The hamiltonian in each \lq time direction\rq is given by
$\square_i$ acting on the group manifold $G$ for the i-th field
argument, and corresponding to a particular set of field modes
solutions of the GFT equations of motion. In momentum space, i.e.
in representation space, it is given simply by the Casimir of the
group $G$, and for compact groups (Riemannian models) it will have
a discrete spectrum with minimal eigenvalue $0$. Thus we see that
the group representations $J$ correspond to the \lq\lq
energy\rq\rq of the GFT. However, their geometric interpretation
(at least at the simplicial level) is that of (D-2)-volumes, i.e.
distances, areas etc according to the dimension chosen. This is
the type of procedure we were envisaging above for defining
thermodynamical GFT quantities: be guided first by the field
theory formalism, then look for a geometric interpretation. As a
further example, as the GFTs are field theories on the group
manifold $G^{\times D}$, its is (the normalisation chosen for)
this group manifold and any eventual cut-off in the group
integrals that will provide a definition of GFT \lq\lq
volume\rq\rq in which the GFT quanta could be confined. From this
quantities, and the partition function itself, one can proceed to
define other thermodynamical quantities, standard statistical
ensembles etc.
What is most relevant for us here is that within the same type of
formulation, a straightforward proof of Bose-Einstein condensation
seems possible, at least for the free theory, and in the case in
which indeed the GFT quanta are bosons (which is not obvious
\cite{iojimmy}). Indeed, one expect to be able to even adapt to
the peculiar GFT setting the standard (textbook) derivation of the
Bose distribution and proceed as usual. In the model sketched
above, in fact, one expects that for fixed number of particles
(GFT quanta) and at low temperature $T$, the system will reach its
ground state represented by (almost) all the GFT quanta condensed
into the same state $J=0$. Again, according to the simplicial
geometry emerging from GFTs in perturbative expansion, this means
having all (D-2)-volumes being of Planck size. Work on this is
currently in progress \cite{iobec}.
The interpretation of this {\bf vacuum state} is exciting, we
think. It corresponds to {\bf a free gas of spin network vertices
or of (D-1)-simplices that has condensed in momentum space, i.e. a
Bose-Einstein condensate of spin network vertices/simplices;
geometrically, a Bose-Einstein condensate of the fundamental
building blocks of quantum space all of Planck size}.
This also resembles, in general terms, the heuristic picture of a
\lq\lq semi-classical state\rq\rq in LQG, with two differences: no
embedding is needed for its definition, and it is selected \lq\lq
dynamically\rq\rq , in a GFT statistical setting.
From a more general perspective, there are many reasons why a
condensed phase of this kind would be a very attractive
possibility, in our opinion, for the vacuum relevant for the
continuum limit. We have mentioned the first: it is realisable in
concrete terms, and not just an hypothesis. Still at the practical
level: the theory of Bose-Einstein condensates is vast and lots is
known about them (see for example \cite{bose}), so in principle
many tools from the condensed matter theory of BEC systems can be
imported in the GFT setting to study the property of this new
phase. At the theoretical and conceptual level it is also very
attractive: it is {\bf a purely quantum phenomenon}, thus a
realisation of the possibility we anticipated that the emergence
of a continuum spacetime from GFT structures could be considered
indeed a quantum effect; it is {\bf rather generic} \cite{bose},
being robust to the presence of interactions, even strong ones, if
they are repulsive, but surviving (when dealt with much care) also
small attractive ones; it gives rise to a pletora of emergent
phenomena \cite{hu,volovik,analog}; as we will discuss in slightly
more details in the following, {\bf the approximate collective
motion of the condensate admits (in mean field theory
approximation) a description in term of a classical (better, 1st
quantized) equation}, the Gross-Pitaevskii equation; {\bf
condensate atoms move as a whole, so that small purely quantum
effects can be amplified}, and one can speculate the same to
happen for this quantum gravity condensate, thus leading (we are
speculating!) to observables quantum gravity effects or, more
likely, to the possibility that large scale properties of
spacetime (e.g. features of GR) that we are accustomed to, can be
understood as originating from purely quantum features of this GFT
vacuum.
To summarise, we are proposing the possibility that {\bf GFT will
produce geometrogenesis in the form of a condensation of the GFT
particles in momentum space accompanied by the approach to
equilibrium of the system} (otherwise, no hydrodynamic description
is possible).
Let us close this section with a comment, that will be relevant
for the following guesses at the effective dynamics of the
condensate. GFT quanta (think of them now as open spin network
vertices) are labelled by both representations of $G$ and by
corresponding vector indices in the representation spaces. It may
happen (and indeed is what we would expect because of symmetry
considerations at the level of the GFT action) that the GFT
hamiltonian, and thus the energy of the vacuum state does not
depend on these additional parameters. Now, suppose that the
condensation is not complete, so that the vacuum state is actually
a mixture of spin net vertices with $J=0$ and $J=1/2$, for
$G=SU(2)$, or in general of lowest eigenvalue (which has also a
single value for the vector indices) and next to lowest eigenvalue
for the energy. Alternatively, suppose that the lowest eigenvalue
is forbidden by some symmetry or by the quantum measure; or, more
generally, the lowest allowed eigenvalue (for some group $G$ and
choice of GFT action) may have a representation space of dimension
bigger than 1. What this means is that we do not necessarily
expect the condensation to lead to a unique vacuum state, even in
the $T\rightarrow 0$ limit. Instead, it may lead us to any of the
quantum states corresponding to $N$ spin network vertices for the
lowest allowed representation parameters and some given choice for
their vector indices. Now in particular, one can consider all
linear combinations of such states, obtained by contracting in all
possible ways the spin network vertices along their open links
labelled by the vector indices. Each of these possible
contractions, which is equivalent to a gluing of the dual
(D-1)-simplices, corresponds to a possible choice of the topology
of the corresponding quantum space, formed by the same spin
networks/simplices. Of course each possible choice also
corresponds to a different effective condensate wave function
\cite{bose}, that then carries a dependence on the resulting
topology of quantum space. If on the other hand, the GFT dynamics
or some additional symmetry consideration will select a specific
contraction of the vector indices or the absence of any such
contraction, once more this will amount to selecting one specific
space topology for our quantum space in this phase.
\subsubsection{Effective dynamics of spacetime from GFT}
Let us move to discuss how we could try to extract and study the
effective dynamics, actually the hydrodynamics, of the GFT
condensate. In discussing this issue, once more the present status
of the field will force us to remain at the level of arguments,
guesses, speculations. Again, we hope the reader will find them
interesting.
Generally speaking, the effective collective dynamics will depend
heavily on the phase the system is in, i.e. on the vacuum selected
by the GFT microscopic dynamics. At this stage, even to guess it
is impossible. However, we can try to forecast some general
features and ask ourselves very general questions about it.
We are assuming here that a sort of Bose-Einstein condensate has
formed, that the system is at equilibrium or very close to it,
that we have made one specific choice of vacuum state, obtaining a
specific effective vacuum wave function \cite{bose}, or
equivalently a classical field (the order parameter).
It is possible that a clever redefinition of the field variables
will bring us collective variables with a direct geometric
interpretation, say connection field or a metric, so that we
could hope that the effective hydrodynamics for these collective
variables is given directly by some extended gravity theory.
However, we find this possibility very unlikely, for the following
reasons:
\begin{itemize}
\item while the effective topology of the physical quantum space
is probably determined by the vacuum (following the comments at
the end of the previous section), nothing seems to select for us
the effective topology of {\it spacetime}; in general, we should
expect an effective theory in which spatial topology change and
non-trivial spacetime topologies are included;
\item in analog gravity models \cite{analog}, the effective
spacetime that quasi-particles see may be very different from the
original spacetime on which the microscopic field theory is
defined, in both geometry and topology, but the spacetime on which
the {\it hydrodynamics} is defined is very close to the one one
started from;
\item in particular, the GFT we have started from has the
interpretation of a discrete 3rd quantized formulation of gravity
and indeed, at least in perturbative expansion, produces discrete
virtual spacetimes of arbitrary topology, and moreover it was a
theory on an internal group manifold and not a physical continuum
spacetime; we expect neither the \lq\lq formal level of
quantization\rq\rq nor the nature of the manifold on which the
effective field is defined to change with respect to the original
microscopic (group) field theory.
\end{itemize}
For all the above reasons, and some others, we expect the
effective GFT dynamics for the chosen condensate vacuum to be not
directly of the form of an extended gravitational theory on a
fixed spacetime, but rather of the form of a continuum 3rd
quantized field theory of gravity, i.e. of a quantum field theory
on a continuum superspace (space of continuum geometries). This
type of gravitational theories have not been much studied, beyond
the original definition \cite{giddingsstrominger,mcguigan}, but
are supposed to have the general action (schematically):
\begin{equation} S = \int_\mathcal{S} \mathcal{D}X \Psi^*(X)
\mathcal{H}(X)\Psi(X)\, + \Lambda \int \Psi^{n}(X) V(X) \end{equation}
where $\Psi(X)$ is a scalar field on the superspace $\mathcal{S}$,
i.e. the space of all space geometries (not spacetime) for given
space topology $\Sigma$, and $X$ are then coordinates on this
space, i.e. some geometric variables (3-metrics, connections,
etc); the (non-local) interaction term $V(X)$ generates, in
perturbative expansion spatial topology changing processes
(producing disconnected universes) while the free kinetic term is
given by a canonical Hamiltonian constraint $\mathcal{H}$. Notice
that the superspace $\mathcal{S}$ is a metric space itself
\cite{dewitt}.
As we have said, for our GFT condensate, we expect the effective
field, call it $\Psi$ as well, to be determined by the vacuum
state, from which would most likely inherit also the choice of
space topology $\Sigma$ and the topological and metric properties
of the effective superspace $\mathcal{S}$, that will depend on the
space topology chosen. In turn, as we have said, the properties of
the vacuum state depend on the original choice of GFT field and of
group manifold $G^{\times D}$. We then expect the emergent
superspace to be some sort of group manifold, with an exact
structure determined by the topology of space we have selected
with the vacuum, and thus again parametrised by group elements or,
equivalently by a (gravity) connection.
To summarise, we would probably obtain, as our effective GFT
hydrodynamics of quantum space, 1st order versions of the old
quantum field theories on superspace. Nothing is known (to the
best of our knowledge) about how these may look like, and a
detailed analysis of such possible field theories (involving the
metric structure of a 1s order superspace, first of all) is called
for.
In general, then, our effective GFT hydrodynamics, in the GFT
analogue of the mean field approximation, will be a continuum
field theory of the form:
\begin{equation} S = \int_\mathcal{S} \mathcal{D}X \Psi^*(X)
\mathcal{K}(X)\Psi(X)\, + \int V(\Psi,\Psi^*) \end{equation}
for some kinetic term $\mathcal{K}$ and higher order (non-local)
interaction $V(\Psi,\Psi^*)$.
The corresponding equations of motion with be our hydrodynamics
equations, non-linear equations for the field/wave function $\Psi$
that will represent the GFT analogue of the Gross-Pitaevskii
equation for Bose-Einstein condensates \cite{bose}. Notice that
the above field theory can be easily recast in a more customary
hydrodynamic form by redefining the basic variables to $\Psi(X) =
\sqrt{\rho(X)} e^{i \theta(X)}$ where $\rho(X)$ is the condensate
density and $v(X) = \nabla\theta(X)$ is the condensate velocity
field.
Let us now see how the link with continuum GR (in some extended
form, probably) can be investigated. The type of gravity theory we
would have obtained will be encoded, and hopefully fully
specified, by the quadratic term in the above action, that would
give the effective Hamiltonian constraint of the corresponding
canonical theory. Notice that all of the above (and of the
following) is at the level of {\it classical} effective theories.
We then would have to extract the quadratic part of the action,
here represented by $\mathcal{K}$. However, it is clear that the
split of the above action, and more generally the very form of the
effective hydrodynamics action depends strongly on the specific
mean field ansatz one has chosen to obtain it\footnote{As they
say, mean field theory, and in general the procedure of
constructing effective dynamics for collective variables, is a
complicated art.}. Anyway, assuming that, in some approximation,
we have got up to here, we could then compare the kinetic term
$\mathcal{K}$, which would be in general a differential operator
on an effective 1st order superspace $\mathcal{S}$, and thus
depending on connection variables and their conjugate variables,
with the classical Hamiltonian constraints of various canonical
1st order formulations of gravity for space topology $\Sigma$, or
re-interpret it as such, and study in this way what type of
effective gravity theory our GFT reproduces in this phase, i.e.
for this choice of condensate vacuum state\footnote{In principle
it would be also possible to extract the corresponding lagrangian
form for the same gravity theory and even the corresponding
continuum path integral, i.e. the 2-point function for the
corresponding free field theory on superspace. Obviously this
would have only a formal meaning, and limited applicability, just
as the formal quantization of hydrodynamics has, and in any case
will not resembles at all the original GFT we started from, just
as the quantization of hydrodynamics for ordinary quantum fluids
does not reproduce at all the underlying microscopic atomic theory
\cite{volovik}.}.
Another possibility, that we mention en passant, comes from the
interpretation of classical gravity as a single particle theory on
superspace \cite{greensite}. In our case, the continuum superspace
is effective and corresponds to the effective manifold on which
our GFT condensate lives. The procedure for identifying classical
gravity in our hydrodynamic field theory on superspace is
consistent with this interpretation. But what if classical gravity
is a {\it \lq\lq quasi-particle\rq\rq} of the above theory on
superspace, and not a particle? Then the effective superspace it
would live in would not be given by $\mathcal{S}$, but by a space
with an effective geometry function of $\rho(X)$ and $v(X)$
\cite{analog}. We are not going to expand on this, but it is clear
that in this case the body of knowledge developed in condensed
matter analog gravity models \cite{analog} would become even more
directly relevant.
It is clear that the realm of possibilities for the structure of
the vacuum and even more for the way to extract effective dynamics
for it, and to find our what back to classical gravity, is
enormous. This is true even if one accepts the idea of the correct
vacuum being represented by a condensate of the type we suggested.
And there are for sure many other plausible hypothesis that can be
made at this stage. Again, condensed matter physics wisdom
suggests to be cautious because condensed matter systems are rich,
and always richer than we imagine. {\bf We simply wanted to
suggest
{\it one possible path from the microscopic discrete to the
macroscopic continuum}}: microscopic GFT $\rightarrow$ condensate
$\rightarrow$ condensate hydrodynamics $\rightarrow$ effective
continuum 3rd QGR $\rightarrow$ approximate free theory
$\rightarrow$ classical (extended) GR.
This probably means we have been un-cautious enough already.
\section{Conclusions}
We have presented a brief introduction to the group field theory
formalism for quantum gravity. We have then argued that GFTs may
provide a common framework for several other discrete approaches
to quantum gravity (loop quantum gravity, quantum Regge calculus,
dynamical triangulations), and shown how the connection with these
other approaches can be understood. Having done so, we have tried
to sketch the elements of a single coherent picture of quantum
spacetime, incorporating the insights and results achieved in all
these different approaches, as seen from a GFT standpoint. We have
tried to argue that the GFT formalism offers also a new
perspective on the same structures.
We have then stressed the importance of solving the open problem
of the continuum approximation of the discrete structures
representing spacetime at the quantum level in these quantum
gravity models, including GFTs, and overviewed the strategies
adopted in loop and simplicial approaches to do so, and the
results obtained. At the same time, we have translated these
strategies in the GFT language, showing that the GFT formalism
would suggest a different one instead, and then sketched what we
believe is a new GFT perspective on the continuum problem in
quantum gravity. This amounts to consider quantum spacetime as a
condensed matter system and the GFT as the microscopic quantum
field theory for its fundamental constituents. We have finally
outlined a GFT strategy from tackling the problem of the emergence
of the continuum, put forward an hypothesis for the relevant GFT
phase, a Bose-Einstein condensate, and sketched a (rather
speculative, at present) programme for realizing this idea and
connecting GFT microscopics to continuum gravity and GR, obtained
from the effective hydrodynamics of the GFT condensate.
\medskip
We hope that, in spite of necessary conciseness of the first part
of this contribution, and of the speculative nature of much of the
second, we have managed to elicit interest for the ideas presented
and for this, we believe, very exciting area of fundamental
theoretical physics that is non-perturbative quantum gravity. The
hope is also that the reader will then join the efforts of
researchers working in this area, and contribute to turning the
present speculations into solid results, in the conviction that
most of the many impressive results already obtained in this
fascinating field have been just tentative suggestions or
speculations at an earlier stage.
\section{Acknowledgements}
I would like to thank J. Ambjorn, A. Ashtekar, F. Dowker, L.
Garay, B. Hu, J. Henson, S. Liberati, L-K. Lim, R. Loll, F.
Markopoulou, P. Massignan, P. Machado, C. Rovelli, L. Smolin, M.
Visser, G. Volovik, R. Williams, for insightful and useful
discussions, comments and criticisms. I would also like to thank
the organizers of the Conference \lq\lq From Quantum to Emergent
Gravity: Theory and Experiments\rq\rq, and especially F. Girelli,
for the invitation to participate and for a very interesting and
stimulating conference.
| {
"redpajama_set_name": "RedPajamaArXiv"
} | 1,149 |
{"url":"http:\/\/clandiw.it\/ervj\/names-with-double-letters-at-the-end.html","text":"Use of corporate letterhead. The sound is [ah], as is \u0905\u0924\u0903 [atah] meaning therefore. When adding suffixes to one-syllable words, it's helpful to follow the CVC rule. With a formal typed letter, this is possible by including a carbon copy notation at the end of your message. Found 107 words that end in zz. And don't worry, I totally got the job. Forums: Word Games, Palindromes Email this Topic \u2022 Print this Page. In the following example all file names ending with *. Examples: rabbit, manner, dagger, banner, drummer. Block Format. \u0b9c\u06e9\u06de\u06e9\u0b9c WELCOME TO COOL LETTERS NOTE: Some may look weird when used in-game. May 28, 2019 @ 10:29. Words formed from any letters in ll , plus an optional blank or existing letter List all words starting with ll , words containing ll or words ending with ll. Use bulleted lists for items that are in no required order. D E Shaw had $3 million invested in the company at the end of the quarter. Enharmonic Interval - Two notes that differ in name only. You can probably see why Mc and Mac names typically contain a second capital letter. -- The filesize list only goes to 48 pts but you can type in larger sizes-- up to 127 pts. \u201d \u201cE\u201d is the letter which most commonly occurs third in a word, and is the third most common second letter in a word. For instance, use personal pronouns, such as \"you,\" \"we,\" and \"I. A change is as good as a rest. To count the number of cells that end with specific text, you can use the COUNTIF function. This lesson is a quick reminder and is not meant to be all-inclusive or definitive. See Rules 1b and 1c of Apostrophes for more discussion. Create a folder as the working directory, and set that in the Start In field in your R shortcut. Lowercase Letters Symbol Command Symbol Command Symbol Command Symbol Command \\alpha \\beta \\gamma \\delta \\epsilon \\varepsilon \\zeta \\eta \\theta \\vartheta \\iota \\kappa \\lambda \\mu u \\xi \\pi \\varpi \\rho \\varrho \\sigma \\varsigma \\tau \\upsilon \\phi \\varphi \\chi \\psi \\omega. Savage: XR is a new patch for Savage, created by the Newerth. I shared that letter with all of you, just as I had shared the rest of our Elf on the Shelf antics. Put the egg in my bag. Mississippi has three sets of doubled letters, two double s pairs, and one double p pair. Write a simple message in the body of the email to let the hiring manager know you\u2019ve attached. This translates into your \"finishing\" number. 94% Words that end with double letters Answers, Solutions, Tips and Walkthroughs. To simplify matters, we\u2019re demonstrating the block format on this page, one of the two most common formats. Actually \u201ct\u201d begins the most words, followed by \u201co. 94% by SCIMOB. Example of format when not using a letterhead: Part 2. ), and write directly to the individual. There are 7 countries whose names end in the letter R. 1 updated on August 2017. Double vowels are always long: maan ('moon') - slaap ('sleep') But single vowels can be long or short: hel ('hell') - hele ('whole') Diphthongs (AU EI EU IJ OE OU UI) are always long. Combine, we can reliably concatenate a file name with the temporary path received. E y ending baby names and what they mean, with 334 results. As a journalist for 40 years, I have had the unfortunate duty from time to time to own up to a mistake I made and to write a correction about a fact I got wrong\u2014a wrong date, a misspelled name. Note: some of the answers may differ from device you use or from the version of the game you have on your phone. So do a lot of proper nouns ( Mr. Words that end with the letter Y end with the long e and long i vowel sounds, not the \/yuh\/ consonant sound. Energico - A symbol in sheet music a direction to play energetically. The Cricut Design Space defaults to letter spacing of 1. If name is a string, then name[0] represents the first character in the string, name[1] represents the second character in the string name, and so on. The most common Y names are Henry at number 16 and. Page 1: family, Day, Thursday, Tuesday, thirty, ay, happy, sixty, country. Do you know why we spell some of these with a single letter before the -le and some with double letters? title \/ little, idle \/ middle,. The four-syllable Gentleman Jack, of Cedar Grove, N. \"(227) 1 The letter, as a \"symbol,\" is thus the central subject of the book; Hester's story itself only corresponds to the \"small roll of dingy paper\" (32) that provides an insight into the elusive meaning of the letter. Sample request letters; Business letter format. Goodbye, Norma Jean. That is, of the 2290 words, 1745 have the ending sound of the twelve consonants. Write a simple message in the body of the email to let the hiring manager know you\u2019ve attached. Enharmonic Interval - Two notes that differ in name only. Names with ss, nn, tt, ff, dd and mm, the double letter is repeated twice without a bounce. You may envision your dog having a name with an ending sound of the long \"e\", which can be spelled with a \"y\" as the last letter of the name. names without pronunciations are excluded from results * is a wildcard that will match zero or more letters in the pronunciation. At the bottom of the last page of a business letter, end notations may show who typed the letter, whether any materials are enclosed with the letter, and who is receiving a copy of the letter. Ancient parenthetical words. Consider the examples below: 1 The word \u201cdrop\u201d becomes dropped or dropping. I went on a nine month quest for inspiration to find unique names for my first three, and if you could please help me, I would SO appreciate it. There are many names out there that contain double-letters within them. A suffix is a group of letters added at the end of a word to change its meaning. Variables in a computer program are analogous to \"Buckets\" or \"Envelopes\" where information can be maintained and referenced. Skip one line between paragraphs. The jury is still out on how to form the possessive for s -final names. Explication : A complete and detailed analysis of a work of literature, often word-by-word and line-by-line. Mississippi has three sets of doubled letters, two double s pairs, and one double p pair. Chicago and MLA spec\u00adify one\u2014de\u00adbate ended\u2014but the pop\u00adu\u00adlar ar\u00adgu\u00adments in sup. The ending starts with a consonant and the last letter in sad is not doubled. The acute accent was first used in the polytonic orthography of Ancient Greek, where it indicated a syllable with a high pitch. Place the parenthetical citation at the end of the block, after the end punctuation. \u201cWords that end with \u201cd\u201d and words that end with \u201cg\u201d). one-dollar bill has been marked as a replacement note. Other languages have other marks, such as French using guillemets, \u00ab \u00bb as quotation marks. Generally, the best method is to use the traditional full block format with all the lines of the letter starting at the left. Also try our list of Words that start with f, and words that contain f. On the other hand, in a word like beyond there is an obstacle to the breath which can be. Consider the examples below: 1 The word \"drop\" becomes dropped or dropping. This is the doubling rule. 5, or double spacing. Following is the complete list of seven letter (7 letters) words starting with S and ending in S for domain names and scrabble with meaning. These double letters are: - ll, - ff, - ss, and - zz. Proofread the letter. Why do I need a website for my business?. Also note. She swallows another sword. An insurance representative lets himself be talked by a seductive housewife into a murder\/insurance fraud scheme that arouses the suspicion of an insurance investigator. Enjambment (or enjambement): A line having no end punctuation but running over to the next line. Game available on iPhone, iPod, iPad, Kindle and Android. The al ending shows the presence of the -CHO group. one-dollar bill has been marked as a replacement note. The letter Y can be used to represent different sounds in different words, and can therefore fit either definition. A list of words that end with Ee. List of suffixes ending in \"ing\". The basic marking when combined with any price. , 9), or underscores. First look at the base word. Rule: To show singular possession of a name ending in ch , add \u2019s on the end of the name. How could I do it?. Hello, I am new to powershell, but I was wondering how I could get it to remove some text at the end of a file that I do not want. The chart above shows the Hebrew alphabet with the name of the letter, and the approximate equivalent sound in English (and the numerical value). The ch sound: At the beginning of a word, use ch. To draw a horizontal line (or horizontal rule), set the x-slope to 1 and the y-slope to zero. Consider another word, this time a place name near Hilo: Puainako. Four Letter Words ending in O are often very useful for word games like Scrabble and Words with Friends. Others also add another s. Skip down one line space and type Signer 1's position or title, tab over and do the same for the Signer 2 and Signer 3 so their titles are directly under. NOTE- THERE ARE A FEW NAMES WITHIN THIS LIST THAT ARE ALSO USED BY SOME NON-JEWS. Initials included at the bottom of a business letter are called typist\u2019s initials. My TRUE TALE for today is a bit unique, because it involves me writing a letter to my son, whom I re-connected with in 2013 after being estranged from him. The figure-eight knot is also known as the savoy knot or the Flemish knot. Step 18 \u2013 FF, LL, and SS. HIs royal poem baits. Many thanks to our Face Book fans for their help in supplying words!. Last updated: December 25, 2019. 79% gain for the first quarter of 2020. This page lists all the 4 letter words that end with 'e'. And: An \"L\" instructors the C# compiler that 10000 is a long type. Use the Japanese numeral system for vertical letters. See also \"more\". Example 1: Write the structural formula for propanal. For example, I said \"Must End With 's'\" and \"Must Contain This Sequence: 'm'\" You could switch those letters around, use them in the \"Must Begin With\" option, etc. After First _ Letters And Before Last _ Letters Specifies the minimum number of characters at the beginning or end of a word that can be broken by a hyphen. example: *lee will match names which end with the sound lee (s) will match exactly one syllable in the pronunciation. 12) is reserved to the implementation for any use. For normal text ( not markup), there are no special characters except < and &: just make sure your XML Declaration refers to the correct encoding scheme for the language and\/or writing system you want to use, and that your computer correctly stores the. Alphabet: Italian words are made up of the same 26 letters as employed by English, although the letters j, k, w, x and y are considered foreign and are only used in import words. In the generic form of the formula (above) txt represents the text that cells should end with, and \"*\" is a wildcard matching any number of characters. For multiple senders, include each name on a separate line. Share : Feel free to propose us other boy names, and we will add them in the database. Words\/names with double letters BB, LL, RR, KK: the hand should \u201cbounce\u201d (just a small bounce; nothing too exaggerated, please) with each iteration of the doubled letters, as in \u201cBobby\u201d, \u201cBilly\u201d, \u201cCarrie\u201d, and \u201cNikki\u201d. At the end discuss any letter names that proved tricky to remember. Or type in-between sizes (like the very useful 11 pts). Knot names have evolved over time and there are many conflicting or confusing naming issues. The more fashionable birth names in this compilation are Alani (#347), Kailani (#389), Leilani (#120), Naomi (#64) and Remi (#177), while Ansari (TOP 6%) and Pisani (7%) are conventional -i surnames. The key is an AutoText entry called AddressLayout in English (see Non-English Word for the correct name in other languages). In other words, each letter within the blend is pronounced individually, but quickly, so they \"blend\" together. Murray compares her letters to Laurence Whistler\u2019s 1964 war memoir The Initials in the Heart, but that book\u2019s heroine, Jill Furse, is a far rarer, less limited soul than Eileen. My TRUE TALE for today is a bit unique, because it involves me writing a letter to my son, whom I re-connected with in 2013 after being estranged from him. How many can you name? Using the same names as the JetPunk Countries of the World quiz. But none of us has a name that matches the direction we face,\u201d said the man facing north. 0 Points - Blank tile. The most common double letter is L, with LL accounting for 0. The four-syllable Gentleman Jack, of Cedar Grove, N. I do it to to give a text a more 'sing-song' or extended quality. The end of the story finds Winston at the Chestnut Tree Caf\u00e9, sitting by a chess board and drinking gin. And their type is, in fact, a null-terminated array of characters. Click on the audio button to hear how the Spanish word is pronounced. Angelina Jolie and Brad Pitt chose a few 'X' names (Pax, Maddox, Knox) and so do the Simpson sisters. You can also remove words once they have been added to your account. They should clearly summarize the information that was provided in the assessment part of the letter. one-dollar bill has been marked as a replacement note. It tells the first vowel to say its name. The typist's initials, in lowercase letters, follow the initials of the author, in capital letters, and a colon or a front-slash ( LCP:ecb or LCP\/ecb ). example: (s)(s)ra will match names which have two syllables and then the sound rah (c) will match. pl silent letters in english words and its orgins. Alphabet: Italian words are made up of the same 26 letters as employed by English, although the letters j, k, w, x and y are considered foreign and are only used in import words. Baby names with the letter 'X' seem to be the hot way to go for new parents. The bad thing about testing against EOF is that if the file is not in the right format (e. The block names supported by Pattern are the valid block names accepted and defined by UnicodeBlock. Sentences begin with a capital letter and end with a period. 35 bonus points are awarded whenever a player uses all 7 tiles on their rack in a single turn. Press the space bar once and type the person's name. A String is a sequence of symbols or digits. If you want to find trivia questions about what is the most popular finishing double in professional darts? or are hosting a trivia night, this is the place to find the answer. If name is a string, then name[0] represents the first character in the string, name[1] represents the second character in the string name, and so on. Say the name of each of the letters of the alphabet (in a muddled up order) and ask children to write the small case letter and upper case letter. Prefix is a letter or a group of letters that appears at the beginning of a word and changes the word's original meaning. 24-30 says:. names without pronunciations are excluded from results * is a wildcard that will match zero or more letters in the pronunciation. Son of a gun! Classic names with a solid 'son' ending -- Dawson, Jackson, Jefferson, Emerson, Addison, Kason, Orson and more. About the \"My name\" from previous lesson, I think that it should be written (FADY) in English because in Arabic it ends with the letter \"Ya' \"which is a \"Y\" in English, but if the name was to be written in Arabic with a \"Kasra\" at the end instead of the letter \"Ya' \" then it will be \"Fadi. 5, or double spacing. CC: Jarrod Curtis or. Food List | Eating A to Z \u2013 I want A to Z food name list. List all words ending with ll sorted by length or by how common the words are. is a holding company that gives ambitious projects the resources, freedom, and focus to make their ideas happen \u2014 and will be the parent company of Google, Nest, and other ventures. Knowing the name of the person shows that you have taken the initiative to learn more about the company. Most popular job search locations: West Cheshire and Chester. Review Sample Business Letters: Check out a few business letter examples before composing your letter and then. In the following example all file names ending with *. To show plural possession, simply put an apostrophe after the s. Within the double X two very important symbol letters appear, the W and X. Write your first and last name, followed by a comma and the correct abbreviation. The complimentary close begins with a capital letter and ends with a comma. Kids activity games, worksheets and lesson plans for Primary and Junior High School students in United States. It is best not to divide a word this way. **A root word is the original word in its root form without any prefixes or suffixes attached e. List of 3,963 words that have double vowels. It is best if you avoid the use of special characters altogether and stick exclusively to the letters A-Z, numbers 0-9, and the characters _-+#$ when naming lists. Recommendation Letter. Double the final consonant before adding an ending that begins with a vowel when the last syllable of the word is accented and that syllable ends in a single vowel followed by a. Larger letters or names: irregularities in letter or name size within the signature draw attention to certain parts of it, stressing their importance to the writer. The letters \"kst\" are in the word, \"in\" is at the beginning, with \"and\" at the end. Then find the names in the word search. Kids activity games, worksheets and lesson plans for Primary and Junior High School students in United States. Also try our list of Words that start with f, and words that contain f. It is used both to indicate a change in vowel quantity as well as quality and that the stress should be on this, normally unstressed, syllable. 31 Script data double escape end state. Tab over and type the name of Signer 2; tab over again and type the name of Signer 3. In fact, I once received a Christmas card from far-away family that was simply signed \"The Jones,\" because of how. I've seen this in on-line games. A basic division is between free-standing or single-family detached homes and various types of attached or multi-family residential dwellings. It is the shortest English word with three consecutive double-letters. If the input is \"abcxyz\", it matches. Others also add another s. , with violin family instruments, a note. Write your first and last name, followed by a comma and the correct abbreviation. Civili-sation. The Cricut Design Space defaults to letter spacing of 1. The finished fabric is completely reversible, lending itself to colorwork, contrasting fibers, or simply a knit-faced fabric that is twice as warm!. Tes has the largest selection of academic, education, teaching and support positions for the world's largest network of teachers and teaching professionals. txt has an extension of. At the bottom of the last page of a business letter, end notations may show who typed the letter, whether any materials are enclosed with the letter, and who is receiving a copy of the letter. The carbon in that group counts as one of the chain. This is tested by calling the member function eof(). The first pic is of food mostly white theres a rectangle of something white with a red top and dollops of white stuff topped with shaved turnip and little purple flowers and the second pic is a man in front of a class writing on a white board, the third pic is a man teeing off, the last pic is man on a computer with a lady. My children's names are Corinne, Weston, and Keely. 4 (four) letter words starting with E. According to \"The Encyclopedia of Business Letters, Faxes and E-Mail\", the carbon copy method of addressing multiple people is the preferred style for business communication. The hard \u05d1\u05bc is pronounced \"b\". Spelling can be tough, especially in the English language. Ancient parenthetical words. ) Download and Installation (Windows). \u00adAll modern United States currency contains either a 10- or 11-digit serial number in order to make each bill unique. A relative few, in more modern times, have made the jump directly as loanwords. For start and end dates, use either full years (eg 1998-2000) or the first three letters of the month followed by the last two digits of the year (eg Jun 98-Sep 00) In the right-hand column, list the name of the school or university on one line, followed by further details (the course name or the number of exam subjects passed) on the next line. Analysis of 9,481 English works (3. The carbon in that group counts as one of the chain. Double Letters at End. (68) and so on. \" followed by a colon. Alphabet supports and develops companies applying technology to the world\u2019s biggest challenges. If you have any additions, changes, corrections, or suggestions, or if you would like to reference this site or the information herein, please feel free to let us know. The beginning and the end of k and g rest on the line of writing. The - ff ending is a bit less common, and - zz is not a very common word ending. \u0b9c\u06e9\u06de\u06e9\u0b9c WELCOME TO COOL LETTERS NOTE: Some may look weird when used in-game. Word lists with a letter at position \u2026 Click to choose the letter. Personal names around the world Intended audience: HTML content authors (using editors or scripting), script developers (PHP, JSP, etc. Portrait layout. Word lists containing a sequence of letters. In Word, go to the \"Page Layout\" menu and select \"size. Corrections, suggestions, and new documentation should be posted to the Forum. The most common double letter is L, with LL accounting for 0. With a formal typed letter, this is possible by including a carbon copy notation at the end of your message. They should clearly summarize the information that was provided in the assessment part of the letter. The complications of \u201cw\u201d are doublefold because of its name, \u2018double u\u2019 and its shape, \u2018double v\u2019. Skip down one line space and type Signer 1's position or title, tab over and do the same for the Signer 2 and Signer 3 so their titles are directly under. Hebrew language. The U makes a little splash of saliva when I put it down. This website offers you more than 7300 baby boy names, you can sort them by letter (names that start\/end with something) or by origin (country). Many thanks to our Face Book fans for their help in supplying words!. For example, at, cat, hat, and fat are a family of words with the \"at\" sound and letter combination in common. 4 Letter Words Ending with t: 5 Letter Words Ending with t: 5 Letter Words Ending with t: 6 Letter Words Ending with t: 6 Letter Words Ending with t. example: (s)(s)ra will match names which have two syllables and then the sound rah (c) will match. Does it start with a vowel? If you can say yes to these two questions, double the last letter in the. The Martinis are coming to dinner. Unlike most other DNA repair and DNA recombination pathways, nonhomologous DNA end joining (NHEJ) in prokaryotes and eukaryotes evolved along themes of mechanistic flexibility, enzyme multifunctionality, and iterative processing in order to achieve repair a diverse range of substrate DNA ends at double-strand breaks (DSBs) (1-3). If the word ending with S is plural, add an apostrophe at the end to make it possessive: the aardvarks' route. Functional Group Names: The ending of the name as a suffix tells the type of compound or functional group present. How do you know when you should put a double letter at the end of a word? Flossy words are words with short vowel sounds that end in a doublet. Like any other data type, strings in TestComplete are represented as OLE-compatible variants. We have undertaken the difficult task and created the following list of over 85 animals that end with letter E. 1 A list is a sequence Like a string, a list is a sequence of values. English words from Latin ending in xious include anxious, noxious, and obnoxious. In the while loop, we keep on reading usernameand score until we hit the end of the file. \u201cThere is no real ending. Reply Thu 8 Jun, 2017 12:06 pm I wrote a quick BASIC program to analyse a word list I downloaded from the. The educational health content on What To Expect is reviewed by our medical review board and team of experts to be up-to-date and in line with the latest evidence-based medical information and accepted health guidelines, including the medically reviewed What to Expect books by Heidi Murkoff. Long and Short Vowels. To be on the safe side -simply avoid doing it. Three Letter Baby Names - 3 Letters. One benefit of using double quotes is that the string can have a single quote character in it. R and l rest on the line of writing as a saucer would rest on a table. With this function, you not only can know how many letters or numbers in the string of the cell, but also know the orders of the letters and numbers. For example: C sharp and D flat. To show possession with a singular noun, add an apostrophe plus the letter s. Angelina Jolie and Brad Pitt chose a few 'X' names (Pax, Maddox, Knox) and so do the Simpson sisters. Addressing it Dear Sirs and Madams: What is the proper notation to indicate to whom the letter was sent? Do I list them at the bottom, like a CC, and if so, what is the method, abbreviation, etc? Thnx Tom. With a formal typed letter, this is possible by including a carbon copy notation at the end of your message. A business letter is a formal letter with six parts: The heading contains the return address with the date on the last line. Many great quality resources are available that explain the different types of cryptic crossword clues and I suggest you read as many as you can. English is a living language so rules do change over time, but I have always used Ross' instead of Ross's. Also try our list of Words that start with f, and words that contain f. Several types of verbs may be included such as linking verbs. Ever missed the end to a film and wondered how does the movie end? Then this is the website for you! Movie Spoilers, Film Endings and we have a bit of fun too!. Double check the spelling of the person's name before you send off the letter. How do you know when you should put a double letter at the end of a word? Flossy words are words with short vowel sounds that end in a doublet. Names with ss, nn, tt, ff, dd and mm, the double letter is repeated twice without a bounce. 4 (four) letter words starting with E. f, l, and s tiles, Word Cards 101-110. This is a list of house types. But that's not all: *** We also double up some words that are more than one syllable when the last syllable is stressed. Format Your Business Letter to Make It More Readable: Leave 1-inch margins and a double-space between paragraphs. Capitalize proper nouns\u2014and adjectives derived from proper. com is a participant in the Amazon Services LLC Associates Program, an affiliate advertising program designed to provide a means for sites to earn advertising fees by advertising and linking to Amazon. Here Dr is an abbreviation for the word Doctor. The letters B, C, F, K, L, Q, and R are usually pronounced at the end of a word. Footnotes indicated by lower-case letters. X is a Christogram, a symbol representing Christ as in Xmas, derived from the Greek letter chi that begins the word \u201cchristos. Acknowledgements should be brief, and should not include thanks to anonymous referees and editors, inessential words, or effusive. Some variable types. com \u2022 Contact NamePlayground. If a word ends with a \"kasra\" tashkeel, then the tanween is indicated by writing two \"kasra\"s (one above the other); same with the \"dhamma\", you write two dhammas, one beside the other. To show possession with a singular noun, add an apostrophe plus the letter s. Corrections, suggestions, and new documentation should be posted to the Forum. Short A Vowel Sound. When Adding a Second \"PS\" at the End of a Letter, It's \"PPS\", Not \"PSS\" I'm subscribed to dozens of email newletters from the top names in the email business, and I see PSS probably twice as often as PPS. A generic name is a 'collective name. Your customizable and curated collection of the best in trusted news plus coverage of sports, entertainment, money, weather, travel, health and lifestyle, combined with Outlook\/Hotmail, Facebook. Baby names with the letter 'X' seem to be the hot way to go for new parents. If you typed the letter yourself, omit the typist initials. If you're sending the letter to one address, try to include all names. A diamond is forever. The values in list are called elements or sometimes items. The chart above shows the Hebrew alphabet with the name of the letter, and the approximate equivalent sound in English (and the numerical value). is for others to add it to the attorney's name when writing to a practicing attorney (e. This is the doubling rule. The letters \"kst\" are in the word, \"in\" is at the beginning, with \"and\" at the end. ) Answer: (a \u222a b)\u2217(aa \u222a bb) (e) The language {w \u2208 \u03a3\u2217 | w does not end in a double letter}. Tile Values. ) Another example is to select a style: use Styles\/Alt+F8, press the letter \"N\", then type a letter or two to speed down the list. In the example, cell F4 contains this formula: =COUNTIF( B4:B11,\"*r\") How the formula works. Explication : A complete and detailed analysis of a work of literature, often word-by-word and line-by-line. Savage: XR is a new patch for Savage, created by the Newerth. You then include the name and address of the person to whom you are sending the letter. Do NOT include your name in this section \u2014 when selecting this style, it simply looks better to sign off with your name at the end of the letter. I went on a nine month quest for inspiration to find unique names for my first three, and if you could please help me, I would SO appreciate it. Note: The three letter baby names below were found searching for three underscores which represent three of any letter. Understanding the different types of word-play used in cryptic crosswords is crucial to your success. The regexprep function returns the updated text in newStr. Writing is therapeutic! My Last Letter to My Son. If I say \u201cSomething you eat with\u2026. End your cover letter header by inserting the date of writing before moving on. Double-barrelled names have seen a recent growth in popularity in the UK as parents get more creative with the challenge of naming their little one. CatherineTheSoSo. ) Answer: (a \u222a b)\u2217(aa \u222a bb) (e) The language {w \u2208 \u03a3\u2217 | w does not end in a double letter}. This list will help you to find the top scoring words to beat the opponent. Punctuation \u2014 whose shapes can\u2019t be adapted \u2014 fares particularly badly. Civili-sation. I do like names like Rocco and Viggo, but I'm not in love. For example: You take the word \"Example\" and using the first two letters \"EX\" you then create = Extra. The most common consonant in English is \"t. A change is as good as a rest. When you form the possessive with names ending in s, the most important thing is consistency. Street sales for the newspaper were extraordinary that day; the edition sold out in a remarkably short time. The four-syllable Gentleman Jack, of Cedar Grove, N. The notes occupy the same position. Learn, teach, and study with Course Hero. Coffee has double f's; so does its key ingredient for us non-morning people: Steven Soderbergh award winning film or Los Angeles gridlock: A runny nose may cause a person to do this tiny snort, to others' annoyance: You can shuffle off to this city which is famous for Wings and Bills. If someone can read the double letter you're definitely golden. Recommendation Letter. A business letter is a formal letter with six parts: The heading contains the return address with the date on the last line. Its full name has 189,819 letters. \u201cThere is no real ending. Teach kids about phonics with this set of printable worksheets that focus on the recognition of different consonant sounds at the end of words. Heartspring commented on the list double-letter-words. 10-letter words ending with EE ATTENTION! Please see our Crossword & Codeword , Words With Friends or Scrabble word helpers if that's what you're looking for. Others also add another s. A business letter is a formal way of communication and that is why it requires a special format. is lesson will teach three new phonograms and that letters f, l, and s may be doubled at the end of a word. Beginning, Middle, & Ending Sounds. Before you start, ask children to listen out for any letters that are vowels and to put a star next to the vowel letters. In this video lesson Rochelle Barlow teaches how to fingerspell double letters in ASL. Example: Harry Birch\u2019s house. Dividing Words at End of Line Hyphens are used to divide words at the end of a line when the word cannot fit on the remainder of the line. That may mean using a different method to sign its double letters, or combing methods. [Plain language] has a bad name among some lawyers. Cap Height - The height of capital letters from the baseline to the top of caps, most accurately measured on a character with a flat bottom (E, H, I, etc. example: *lee will match names which end with the sound lee (s) will match exactly one syllable in the pronunciation. \u201cThere is no real ending. A business letter is to be composed on the company\u2019s letterhead, with margins of 1 to 1. A diamond in the rough. Each of these strokes begins and ends on the same plane. Appearance and Sound of Hangul The Consonants. An example is that many English Surnames have their origin in Latin, but then at one time Latin was the primary language of communication through out many parts of the known world. The bad thing about using eof() is that if the file is not in the right format (e. -ey names are used more often as masculine names. For instance, use personal pronouns, such as \"you,\" \"we,\" and \"I. The root name tells the number of carbons in the longest continuous chain. The more frequent route of Arabic words into English was in previous eras, often traveling. The most common consonant in English is \"t. Rule 1: Words ending with a Consonant-Vowel-Consonant Pattern (Review Consonants and Vowels) One-syllable words: ED = If the word ends in a CVC pattern, it gets a double consonant + ED. ET The journalists at BuzzFeed News are proud to bring you. 01 Introduction. When a name ends in \u201cs,\u201d you would normally show possession by adding the apostrophe-s ending. The text should be left-justified and double-spaced between paragraphs. If the person has a first name that could be the name of a man or a woman, use his or her full name in the salutation, for example \"Dear Terry Smith. Alphabet: Italian words are made up of the same 26 letters as employed by English, although the letters j, k, w, x and y are considered foreign and are only used in import words. Words from Arabic have come into English in two different ways. Abbreviations must be clearly distinguished from contractions. Other common double-letter bigrams are SS, EE, OO, and TT. Depending on the position of a letter in a word, the number of letters in the word, and a letter's relationship to the letters around it, that letter can have its [\u2026]. Visit our Articles & News Page to read other FYTI Articles. A set of hand shapes used to spell words is know as a \"manual alphabet. example: (s)(s)ra will match names which have two syllables and then the sound rah (c) will match. You will see that Spanish words are pronounced as they are written. For example, at, cat, hat, and fat are a family of words with the \"at\" sound and letter combination in common. First, we have to determine the number of syllables in the word. The way I learned it is xx\ud83d\ude18 stands for kisses And xoxo stands for hugs & kisses BUT that might be meant in a non-literal way\u2026 it may not actually imply a desire to hug & kiss, it could just be an expression of affection\u2026 but maybe in a caring kind. Ten-digit serial numbers were on all bills until the \"new style\" came out in 1996. For some reason, I hate her more. The XML FAQ \u2014 Frequently-Asked Questions about the Extensible Markup Language. I ending baby names and what they mean, with 578 results. They also can be used to indicate irony and introduce an unfamiliar term or nickname. If you know the name, you can jump to it much quicker-- type the first letters of the name, and it will jump to the closest match. Spelling can be tough, especially in the English language. Of the 7,180 baby names in our database the following baby names have exactly 3 letters. If a word with a certain vowel in it says the name of the vowel, then that vowel is making a \u201clong\u201d sound. The first character must be an English letter (A, B, C,. It is not necessary to type a return address. A phone number and email address below your name can make it that much easier for the hiring manager to get in touch with you. At the simplest level of scientific classification, each plant has a name made up of two parts, a generic (or genus) name and a specific name or epithet. Angelina Jolie and Brad Pitt chose a few 'X' names (Pax, Maddox, Knox) and so do the Simpson sisters. 6% of all bigrams. Blank tiles will have no point values. In 2018, their total usage was 2. All children develop as individuals. Our suggestion is that you update to the latest version of the game. It seems to only be for words above 3 characters. 1) True \/False: When typing in your source code into the computer, you must be very careful since most of your C++ instructions, header files, and variable names are case sensitive. Rule: To show singular possession of a name ending in ch , add \u2019s on the end of the name. and I want to remove the underscore and the last 3 digits, so the (_001,_010,_150, etc). Quiz by SnipeLikeYoda. 292% with 38 -ah names listed among the top 1000. You can use these Five letter words for finding good domain names, while playing scrabble or in research. square brackets. Double Consonants: When b, d, g, m, n, or p appear after a short vowel in a word with two syllables, double the consonant. If the person has a first name that could be the name of a man or a woman, use his or her full name in the salutation, for example \"Dear Terry Smith. In other words, a variable has a name, a type and stores a value. Reply With Quote. There are 14 two-letter words ending with E: AE BE DE TE WE YE. A proposed name is rebuttably presumed to imply an affiliation with, or a subsidiary relationship to, a business entity possessing an existing name if the proposed name is the same or deceptively similar to the existing name except for the addition or absence of the word \"of\" followed by a geographic designation at the end of the name or. If you write a one page press release, at the bottom of the copy add three pound signs (###), the number thirty (-30-), or the word \":end\": in capital letters (END). 2circles 3d a accent acronym aigreen almostred ambigram american-typewriter ampersand animal apostrophe arc arrow ball balloon beakorange bean bear beehive beige bell benettongreen bird bold bowtie brackets c caliper canvas car cartoon cat cc chain checkerboard circle circles circumflex claw clock contour crest crimson crocodile cross crosshair. Larger letters or names: irregularities in letter or name size within the signature draw attention to certain parts of it, stressing their importance to the writer. Learning Letter Sounds. The name is needed to uniquely identify each variable, so as to assign a value to the variable (e. all dishes chinese. I have debated with liberals all over the world about many geopolitical things and one thing I have noticed is at the end of every debate EVER if you don't agree with them they name call you and scream at you and don't want to talk to you ever again and or treat you differently while opening the conversation with saying they are very \"tolerant and open minded\". Remember the uppercase letter at the beginning and the period (. A memo has no indentations; it is single-spaced, with double spaces between heading and paragraph and among paragraphs. Consider the examples below: 1 The word \u201cdrop\u201d becomes dropped or dropping. A dictionary of sewing terms and terminology you may use on your sewing journey. We're looking for suggestions for names with double letters that also have a good nickname. Some variable types. The sound is [ah], as is \u0905\u0924\u0903 [atah] meaning therefore. This regex matches any numeric substring (of digits 0 to 9) of the input. example: *lee will match names which end with the sound lee (s) will match exactly one syllable in the pronunciation. Example of format when not using a letterhead: Part 2. Ah yes, the double-s ending. Use the Japanese numeral system for vertical letters. Catch a little wanderlust with ideas, tips and pointers to help you get inspired on how you're going to plan your next getaway. The Y doesn't appear until Emily at number 12. Coffee has double f's; so does its key ingredient for us non-morning people: Steven Soderbergh award winning film or Los Angeles gridlock: A runny nose may cause a person to do this tiny snort, to others' annoyance: You can shuffle off to this city which is famous for Wings and Bills. By sorting the diagonal elements, you can find the double-letter combinations that appear most frequently in the corpus. Double-space all of the quote. Ten-digit serial numbers were on all bills until the \"new style\" came out in 1996. The spelling curriculum for kindergarten should cover kindergarten spelling words start with basic two letter words, or three letter consonant-vowel-consonant words, and become more complex. One of the advantages to using Microsoft Exchange or Microsoft Outlook is the ability to use information from the Address Book in Microsoft Word documents. This is usually because they don't understand enough about it to judge it properly. Two dots are used to the right of letters. With these files, your students will practice reading and writing words with the short a vowel sounds. By Albert Samaha and Katie J. Used by over 70,000 teachers & 1 million students at home and school. As a journalist for 40 years, I have had the unfortunate duty from time to time to own up to a mistake I made and to write a correction about a fact I got wrong\u2014a wrong date, a misspelled name. Often there is a line skipped between the address and the date. Add length, consonants, vowels, syllables, origin, spelling and more. Here's how to end a cover letter: End your cover letter on a high note. Don\u2019t assume gender or marital status (Mr. Alphabet supports and develops companies applying technology to the world\u2019s biggest challenges. Now the earth was formless and empty, darkness was over the surface of the deep, and the Spirit of God was hovering over the waters. Capitalize the first word of a document and the first word after a period. If we had a 4th and it was a girl I would like to have a name with a double letter since all the girls in the family have a double letter in their name. Some companies require them so that they know who actually typed the letter versus who composed it, in order to determine who is responsible for typos, misspellings, and other mistakes that took place when the letter was produced. How many can you name? Using the same names as the JetPunk Countries of the World quiz. Back in the day, it was common to differentiate people with the same name by also calling them by the names of their fathers, which is how this sort of surname started to become popular. cc: Jarrod Curtis. Counterscientific 2). A proposed name is rebuttably presumed to imply an affiliation with, or a subsidiary relationship to, a business entity possessing an existing name if the proposed name is the same or deceptively similar to the existing name except for the addition or absence of the word \"of\" followed by a geographic designation at the end of the name or. The ch sound: At the beginning of a word, use ch. =A1&\"A\" or whatever letter you want to Add. You can search english words that ending with or starting with Very usefull for lettergames addicts or song writers. Why do I need a website for my business?. But the Chicago Manual of Style says that forming the possessive with names ending in s is just like forming the possessive with names that don\u2019t end in s: add an apostrophe-s (\u2019s). The root name tells the number of carbons in the longest continuous chain. My professor wants it to say Y for another student and N to quit at the end. -ey names are used more often as masculine names. The acute accent was first used in the polytonic orthography of Ancient Greek, where it indicated a syllable with a high pitch. Son of a gun! Classic names with a solid 'son' ending -- Dawson, Jackson, Jefferson, Emerson, Addison, Kason, Orson and more. A phone number and email address below your name can make it that much easier for the hiring manager to get in touch with you. For others, do a Google search for list of words that begin and end with same letter. This list will help you to find the top scoring words to beat the opponent. What works for a fundraising letter also works for a thank you letter. Words with an \/i\u02d0\/ sound or an \/\u0430\u026a\/ sound. The YoLinux portal covers topics from desktop to servers and from developers to users. Such letters are written for official purposes to authorities, dignitaries, colleagues, seniors, etc and not to personal contacts, friends or family. Typically, formal people increase the size of the surname, for example. Lesson 6: Letter Group 2 Meg has a bad leg. If you typed the letter yourself, omit the typist initials. We search a large Scrabble dictionary for words ending with the letter or word you enter, and generate all words ending with Ee (words with the suffix ee). Blends are consonants whose \"sounds blends together\". A vowel sound is considered long when that vowel is read as its name, like the letter A in the words \u201caim\u201d and \u201cenable. Four Letter Words ending in O are often very useful for word games like Scrabble and Words with Friends. I know, it's awkwardly worded but otherwise it wouldn't be a riddle. and who knows what others. In statistical publications containing tables with numerous footnotes, numbers are normally used instead of letters. Only include your street address, city, state, and zip code. Second, the files are imported one-by-one using a for loop where the original names are assigned to the generated data frames with the assign function. Spelling the long vowel sound \/a\/ a-e, ai, ei, ay. You can add an appropriate title such as \"Ms. in the column I tried find and replace and I'm obviously doing it. Capitalize the first word of a document and the first word after a period. Links may lead off this site. -ey names are used more often as masculine names. 30 Script data double escaped less-than sign state; 12. example: (s)(s)ra will match names which have two syllables and then the sound rah (c) will match. I hope she has lousy letters. Wordbrain Themes, Words With Friends, Scrabble, 4Pics1Word, Word Cookies cheats, answers, and more. End the recommendation letter with a couple sentences at most. These tips are presented in three parts- how to organize the Letterhead and Opening at the top of your business letter, the Body, and finally the Closing at the bottom. A line has no ends ! Change the position of points A and B. Son of a gun! Classic names with a solid 'son' ending -- Dawson, Jackson, Jefferson, Emerson, Addison, Kason, Orson and more. If it was. The game ends when one player plays every tile in his rack, and there are no tiles remaining to draw from. Your enthusiasm suggests that you\u2019re organized and vigorous, but adding a note of urgency by inviting an employer to call you may motivate the employer to prioritize the interview and move up the timetable for a meeting. Subsequent characters can be letters, numeric digits (0, 1,. One of the advantages to using Microsoft Exchange or Microsoft Outlook is the ability to use information from the Address Book in Microsoft Word documents. The name, Myles, always ends in \"s\" even though it is singular. Last updated: December 25, 2019. A dictionary of sewing terms and terminology you may use on your sewing journey. Discover and collect the things you love, and buy it all in one place!. Asprey, Plain Language for Lawyers 11 (2d ed. long-toed tropical freshwater wading bird. Many thanks to our Face Book fans for their help in supplying words!. Below are the point values for each letter that is used in a Scrabble game. Walsh becomes Walshes, and Malkovich becomes Malkoviches. The rarely used plural of consensus is consensuses , but some words from Latin that end in us have a plural that ends in a long i sound (\\\u012b\\) and is spelled with i. thefreedictionary. (a-a-a-a-a) and maybe just including company name + clothing in the page heading\/elsewhere on the landing page. A bunch of fives. is in the U. The rules for splitting words at the end of the line in the English language are quite complicated, and in many cases rather subjective. A String is a sequence of symbols or digits. Does it have One, One, One (one syllable ending in one consonant after one vowel)? Then look at the ending. The name is needed to uniquely identify each variable, so as to assign a value to the variable (e. Usually, try and split it in the middle of the word. Knot names have evolved over time and there are many conflicting or confusing naming issues. Additional resources. Quiz and answer stats >> Enter country here. Wordbyletter purpose a crosswords solver. The 'CC' notation usually includes names of people to whom you distribute copies, sometimes you could include their addresses as well. \" When the last three letters of a one-syllable word follow the CVC pattern, the last consonant should be doubled when adding the ending. On behalf of all the children, staff and Governors, I\u2019d like to warmly welcome you to the Woodlands Primary School website. 5, or double spacing. Here is a list of the most commonly used Spanish words similar to English words starting with the letter N. Plural words that don't end with S, such as \u201c children,\u201d do take an apostrophe-S at the end for possession. Rule: To show singular possession of a name ending in ch , add \u2019s on the end of the name. Tips, stories, tournaments, and fun!. But the Chicago Manual of Style says that forming the possessive with names ending in s is just like forming the possessive with names that don\u2019t end in s: add an apostrophe-s (\u2019s). By \u201cname\u201d we mean, the name of the actual letter. com \u2022 Contact NamePlayground. After the concluding line, you should write the farewell expressions which sound apt to end a business letter. Double-space all of the quote. A phone number and email address below your name can make it that much easier for the hiring manager to get in touch with you. Adding \"CC\" at the end of a letter is easily done. There are over 100 less boy double letter names in the Top 1000 for 2012 than girl ones, with 138 before combining spellings (116 after). The Letter Sorting Word Generator helps you to make words from letters. \" There are many different manual alphabets throughout the world. They are to help understanding and the correct reading. 0 Points - Blank tile. Whether you have a son or daughter, my advice to you is to be honest and bare your soul. I see Jed. I ending baby names and what they mean, with 578 results. 843% of baby girls being given -ah names. First, conisder that a double-t ending is almost always preceded by a vowel. Welcome to Starting Out with C++: From Control Structures through Objects, 7th edition. \"Wanna Learn\" wrote in message. Tip: Unless previously defined, the length of the variable is set to 41 characters. Answer: \u03b5 \u222a a \u222a b \u222a (a \u222a b)\u2217(ab \u222a ba) (f) The language {w \u2208 \u03a3\u2217 | w contains exactly one double letter}. Want to learn some words with double letters? Read on. For examples, If the input is \"abc123xyz\", it matches substring \"123\". Reply With Quote. The first pic is of food mostly white theres a rectangle of something white with a red top and dollops of white stuff topped with shaved turnip and little purple flowers and the second pic is a man in front of a class writing on a white board, the third pic is a man teeing off, the last pic is man on a computer with a lady. In the following example all file names ending with *. This hyphen is invisible, unless the word gets split at the end of a line. HOW TO READ MUSIC NOTES (QUICK-LEARN CHEAT SHEETS), Page 8 Phrase \u2013 a complete musical thought or a musical sentence. Some are used to notate pitch, tempo, metre, duration and articulation of a note or a passage of music. Savage: XR is a new patch for Savage, created by the Newerth. Different letters in the game will have various point values and this will depend on how rare the letter is and how difficult it may be to lay that letter. R and l rest on the line of writing as a saucer would rest on a table. For instance, what if you wanted to make all vowels upper case:. To show possession with a singular noun, add an apostrophe plus the letter s. One tell-tale sign of a self-published book is tiny, tiny margins!. The al ending shows the presence of the -CHO group. - should not be used when in contact with a hyphen or apostrophe. \u201cWords that end with \u201cd\u201d and words that end with \u201cg\u201d). Word Finder by WordTips gives you a list of words ordered by their word game points of your choice. Tom Hanks has sent a heartfelt letter and a Corona brand typewriter to an Australian boy who wrote to him about being bullied over his name, Corona. Scientific Names - naming the plant - rules - type specimens Cultivar Names; Common Names; Scientific names. Here is a list of the most commonly used Spanish words similar to English words starting with the letter N. The - ll and - ss endings are very common. If she doesn't like the person she will respond with single lettered words. 28 Script data double escaped dash state; 12.","date":"2020-05-30 21:38:12","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\": 0.1724419891834259, \"perplexity\": 2159.7393808343963}, \"config\": {\"markdown_headings\": true, \"markdown_code\": true, \"boilerplate_config\": {\"ratio_threshold\": 0.18, \"absolute_threshold\": 10, \"end_threshold\": 15, \"enable\": true}, \"remove_buttons\": true, \"remove_image_figures\": true, \"remove_link_clusters\": true, \"table_config\": {\"min_rows\": 2, \"min_cols\": 3, \"format\": \"plain\"}, \"remove_chinese\": true, \"remove_edit_buttons\": true, \"extract_latex\": true}, \"warc_path\": \"s3:\/\/commoncrawl\/crawl-data\/CC-MAIN-2020-24\/segments\/1590347410352.47\/warc\/CC-MAIN-20200530200643-20200530230643-00509.warc.gz\"}"} | null | null |
Als opaker (aus lat. opacus = schattig/dunkel) Datentyp bezeichnet man auf dem Gebiet der Informatik einen Datentyp, dessen physikalische Darstellung (Repräsentation) entweder unbekannt oder irrelevant ist. Die Datenstruktur eines opaken Datentyps ist nicht alleine auf einen Grenzbereich definiert. Die konkrete Darstellungsart bleibt für den Benutzer undurchsichtig (verborgen) und die sichtbare Umsetzung ist unvollständig.
Opake Datentypen, also undurchsichtige Datentypen, werden häufig benutzt, um abstrakte Datentypen zu implementieren.
Anwendungen
Typische Beispiele für diese undurchsichtigen Datentypen sind etwa die umfassenden Ressourcen eines Betriebssystems, welche mit Hilfe von Softwareanwendungen dem Benutzer zur Verfügung gestellt werden. Dabei wird der Datentyp vor dem Anwender versteckt, da er nur für das Betriebssystem von Bedeutung ist. Es besteht ebenso die Möglichkeit, diesen Datentyp betriebssystemseitig zu ändern, ohne dass der Quellcode der Applikationsprogramme angepasst werden muss.
Es handelt sich dabei etwa um das Portable Operating System Interface (POSIX), eine Anwendungsprogrammschnittstelle, welche eine Schnittstelle zwischen Anwendungssoftware und dem Betriebssystem darstellt.
Definition in modularen Programmiersprachen
Opake Datentypen werden auch in Modulare Programmiersprachen wie Modula-2, eine Fortentwicklung von Pascal, verwendet. Eine Realisierung bzw. Umsetzung erfolgt dabei mit Hilfe sogenannter Module. Alle separat vom Hauptprogramm übersetzten Teile werden in zwei Dateien aufgespalten, ein Definitionsmodul (Definition Module) sowie ein Implementationsmodul (Implementation Module).
Im Definitionsmodul werden lediglich der Typ-Name und die Schnittstelle angegeben. Die Typangabe selbst wird nicht angeführt, womit die Struktur des Datentyps verborgen bleibt. Diese muss daher im Implementationsmodul beschrieben werden. Somit handelt es sich um einen opaken (undurchsichtigen) Datentyp.
Im Gegensatz dazu steht ein Datentyp, dessen Darstellung sichtbar wird. Dieser Datentyp wird wiederum als "transparent" bezeichnet.
Verwendung opaker Datentypen (Beispiel)
var stack: TStack; // (1)if stack.Count() > 0 then // (2)
stack.Pop();
else
...
end;
Beschreibung
Bei einem opaken Datentyp ist die Strukturbeschreibung der zugehörigen opaken Datenobjekte an anderer Stelle definiert und dem Verwender des Datentyps nicht zugänglich. Er verwendet nur den Namen des opaken Datentyps TStack, um opake Datenobjekte zu deklarieren (1). Der Zugriff erfolgt nun nicht dadurch, dass man Objekte direkt verändert oder liest, indem man die Kenntnis ihrer Detailstruktur benutzt, sondern ausschließlich Zugriffsoperationen benutzt (2).
Verwendung transparenter Datentypen (Beispiel)
type TSpace = array[1..n] of TItem; // (3)var space: TSpace; // (4)if ... then
space[index] := x;
else
...
end; // (5)
Beschreibung
Ein normaler Datentyp wird durch eine Typendeklaration eingeführt (3), die den Namen des Typs festlegt, in diesem Beispiel TSpace, und die in der anschließenden Typendefinition seine Struktur angibt, hier ein n-elementiges Feld irgendeines Datentyps. Diese Struktur wird als offengelegt betrachtet, weshalb ein normaler Datentyp auch transparenter Datentyp genannt wird. Mit Hilfe des Typbezeichners der die Strukturbeschreibung repräsentiert, kann nun in einer Datenobjektdeklaration (4) ein Datenobjekt dieses Typs deklariert werden. Da dessen Detailstruktur bekannt ist, kann bei der Manipulation dieses Datenobjekts darauf Bezug genommen werden (5).
Siehe auch
Abstrakter Datentyp
Einzelnachweise
Programmiersprachelement
Datenstruktur | {
"redpajama_set_name": "RedPajamaWikipedia"
} | 4,604 |
import os
import sys
# Add the project to the path
sys.path.insert(0, os.path.dirname(os.path.dirname(__file__)))
PROJECT_MODULE_PATH = os.path.dirname(os.path.dirname(os.path.realpath(__file__)))
PROJECT_APPS_PATH = os.path.join(PROJECT_MODULE_PATH, 'ehb_service/apps')
PROJECT_ROOT, PROJECT_MODULE_NAME = os.path.split(PROJECT_MODULE_PATH)
# setup the environment
os.environ.setdefault('DJANGO_SETTINGS_MODULE', 'ehb_service.conf.settings')
os.environ.setdefault('PYTHON_EGG_CACHE', '/tmp')
sys.path.insert(0, PROJECT_MODULE_PATH)
sys.path.insert(0, PROJECT_APPS_PATH)
from django.core.management import execute_from_command_line
execute_from_command_line()
| {
"redpajama_set_name": "RedPajamaGithub"
} | 2,344 |
Theodore Lane Sampley (July 17, 1946 – May 12, 2009) was an American Vietnam War veteran and activist. He primarily advocated for those servicemembers still considered missing in action or prisoners of war (POW-MIA) as of the end of hostilities in 1975. A staunch political conservative, he also ran for local political office several times. He is credited with the research that identified Air Force Lt. Michael Blassie as the Vietnam fatality buried at the Tomb of the Unknown Soldier, and for his role in organizing the annual Rolling Thunder motorcycle event in Washington. In Kinston, North Carolina, where he lived for much of his adult life, he was known for his local civic activism, most notably his effort to build a replica of the Confederate ironclad CSS Neuse, the only full-size replica of a Confederate ironclad, in the city's downtown.
A native of Wilmington, North Carolina, he enlisted in the Army in 1963. Two years later, he was deployed to Vietnam with the 173rd Airborne Brigade, where he did a year's tour of duty as a combat infantryman. Afterwards he became a Green Beret and served another tour leading and training a Civilian Irregular Defense Group along the Cambodian border, earning four Bronze Stars, an Army Commendation Medal and the Vietnamese Cross of Gallantry. After returning to Fort Bragg to train other Special Forces soldiers for duty in Vietnam, he left the Army in 1973 with the rank of staff sergeant.
Following his honorable discharge, Sampley worked in journalism and then settled in Kinston, North Carolina, where he opened a craft store selling ceramics, an art he had learned from local artisans in his off-duty time while stationed on Okinawa at the beginning of his military career. In the early 1980s he began his activism, after learning that not all the POWs and MIAs in Vietnam at the end of the war had been accounted for, joining groups demanding that the U.S. put pressure on the Vietnamese government. He started and published U.S. Veterans Dispatch, a newspaper primarily devoted to the issue.
Sampley soon became known as an outspoken activist for his cause, using confrontational tactics similar to those used by antiwar protestors. He was particularly hostile to Senators John Kerry and John McCain, both of whom had served in Vietnam and were members of the 1993 Senate select committee which found that no POWs or MIAs remained alive in Southeast Asia. McCain himself, whom Sampley frequently accused of having been brainwashed by the Vietnamese during his years as a POW in the Hanoi Hilton, said Sampley was "one of the most despicable people I have ever had the misfortune to encounter"; Sampley was convicted of assault after a fight with McCain's chief of staff. He was criticized further for using the POW-MIA cause for his aggrandizement and personal enrichment; sculptor Frederick Hart successfully sued Sampley for unpaid royalties over his unauthorized use of Hart's The Three Soldiers on T-shirts he sold near the Vietnam Veterans Memorial on the National Mall.
1946–1973: Early life and military career
Theodore Lane Sampley was born on July 17, 1946, in Wilmington, North Carolina. After growing up on a tobacco farm there, he enlisted in the Army in 1963, aged 17. Following basic training, Sampley went through advanced infantry training and then Airborne School. The next year he was assigned to the 173rd Airborne Brigade, then stationed on the Japanese island of Okinawa. While there, when off-duty, he visited local potters and began to study ceramics.
In 1965, the 173rd was deployed to Vietnam. Sampley served a year's tour of duty as a combat infantryman, recalling later that he had known very little about the growing conflict at that time. He was then promoted to sergeant and spent a second tour commanding a B-36 MIKE Force unit of indigenous minority population along the Cambodian border, part of the Civilian Irregular Defense Group program. In that capacity he earned four Bronze Stars, the Army Commendation Medal and the Vietnamese Cross of Gallantry.
Sampley wrote in his online biography that he was one of a few Americans sent to train at the British Army's Jungle Warfare School in Johor Bahru, Malaysia. The two-month course was taught by instructors from the Australian and New Zealand armies as well as the British. He and the other Americans wore British uniforms so the British Army could better keep it secret that they were training Americans there.
Returning to the U.S., Sampley became a Green Beret assigned to first the 3d, then the 6th Special Forces Group. He trained in, and trained others, in military subjects from guerilla warfare to High Altitude Low Opening parachute jumping. He also learned to speak Arabic and Japanese fluently.
Part of the Special Forces curriculum was training in how to be a prisoner of war (POW), an issue Sampley said later he did not know much about until then. During the early 1970s, he began volunteering with Americans Who Care, a group formed in nearby Fayetteville to raise awareness about the issue. When the war ended, with US POWs released and returning home, Sampley believed, like many other Americans, that they had all been accounted for. Having attained the rank of staff sergeant, he was honorably discharged in 1973.
1973–1983: Post-military career
After returning to civilian life, Sampley recalled, "I just kind of withdrew back into myself, like a lot of vets did." He worked in journalism, both for a local weekly newspaper and a television station. His interest in pottery, first piqued during his time on Okinawa, returned, and after building his own kiln decided to try making and selling his own pottery. He started his business, The Potter's Wheel, and within two years had produced and sold 90,000 pieces, some of which were featured in a 1980 Country Living pictorial on rural potters in North Carolina.
Sampley also became active in local politics. He served on the New Hanover County Republican committee. In 1976, 1978 and 1980 he ran unsuccessfully for county commissioner.
Sampley's political efforts were not without controversy. During his 1976 campaign, the county's sheriff sued him for slander. Three years later, he tried to have the sheriff arrested. The county's Republican chairman would later refer to him as "a millstone around our necks".
In 1982, during a relative's court case, Sampley got into an altercation in the courthouse. He was arrested, and ultimately convicted of assaulting a law enforcement officer. Years later, he expressed regret for the incident. "Some of those guys were my friends", he said.
1983–2009: POW/MIA activism
In 1982, Sampley was one of the many Vietnam veterans who went to Washington for the dedication ceremonies of the Vietnam Veterans Memorial on the National Mall, a memorial whose design he had criticized. At the ceremonies he heard many other veterans express doubt that all the POWs and other servicemembers officially considered missing in action from Vietnam-era combat operations in Southeast Asia had been accounted for as the government had claimed since 1975, particularly those they had known personally. Sampley concluded from those discussions that the government knew more than it was publicly disclosing, and decided to get the answers.
"When I came back from the war, I had trouble with authority figures", he recalled in 2001. "I heard people talking about how it was time to get over the war, I thought, How? When you've seen human beings all around you reduced to rotting flesh, you can't just flip a switch and turn things off."
After returning home, he resumed his activism on behalf of prisoners of war and those servicemembers still listed as missing in action (POW/MIA) in Vietnam. He, along with some families of the POW/MIAs unaccounted for at the end of the war, believed not only that the Vietnamese government knew more than it had publicly acknowledged about the fate of some of those men, but that some had survived the end of the war and were even still alive in captivity. The National League of POW/MIA Families (NLF), an organization founded during the war by Sybil Stockdale, wife of James Stockdale, the highest-ranking U.S. Navy POW during the war, took on a public role lobbying on this issue, and Sampley joined those efforts.
Sampley took a highly visible public role in the movement, leading demonstrations and speaking to the media. He also started his own newspaper, U.S. Veterans News and Report (later U.S. Veterans Dispatch), to publicize the issue; it would later be described as "required reading for the MIA hardcore." Many of his protests had elements of civil disobedience—he often had the daughters of MIA servicemen chain themselves to the gates of the White House or other government buildings, and sometimes threw fake blood on security officers, especially the uniformed Secret Service officers at the White House.
Other protests Sampley organized were more confrontational. He had protestors in bamboo cages, similar to those in which some POWs were exhibited publicly during their time in Vietnam, placed on the front lawn of Donald Regan, then chief of staff to President Ronald Reagan. He also led another group to the house of Defense Secretary Frank Carlucci during a heavy snowstorm, where they blocked his driveway with 1,800 care packages intended for POWs they believed were still alive and being held in Laos.
The effort to draw attention to those prisoners believed to be held in Laos took Sampley and other American activists to that country in 1988. They traveled to its border with Thailand along the Mekong River and surreptitiously attempted to distribute dollar bills stamped with the reward offer, some floated in the river, offering a US$2.4 million reward for information about possible U.S. POWs allegedly still held in the country. While two members were captured and detained by Laotian authorities for six weeks, Sampley, who called Laos a "black hole" for missing Americans, was only briefly detained when he returned to Thailand illegally.
Even allies in the POW/MIA effort found themselves targeted by Sampley's protests for what he considered to be a willingness to compromise. He organized a "bounty hunt" in which participants were encouraged to target NLF head Ann Mills Griffiths and other organization officials with water balloons, cream pies and rotten tomatoes if they saw them. At another time he led a group of protestors to occupy the NLF office in the American Legion's headquarters. While Sampley later described the effort as peaceful, Griffiths recalled that he threatened to kill her before being handcuffed and led out of the building.
Sampley later acknowledged his tactics were inspired by those used in late 1960s political protests. "I took my training in guerrilla warfare and I turned it around on the U.S. government," he told the Phoenix New Times in 1999. "I started thinking of as many types of tricks as I could pull to disrupt the system ... The idea was ... to disrupt the process, to cause the government to have to talk about the POW issue."
The Last Firebase
In 1984 Homecoming II, a POW/MIA information project named after Operation Homecoming, the original return of POWs after the war, was founded in Kansas as a local effort. The founders began compiling information on POW/MIA cases, from biographies to reports of their possible survival in captivity, and began lobbying the U.S. and Vietnamese governments. Within a year, they realized they had compiled and made public more information about the cases than the Pentagon had given a congressional task force investigating the issue.
The following year the group established a booth on the Mall near the Lincoln Memorial where they initially kept a vigil to raise awareness of the POW/MIA issue and sold merchandise to support it. They also supported demonstrations in the area by veterans, including one who locked himself in a bamboo cage and began a hunger strike. It was named The Last Firebase, after the fire support bases established by U.S. artillery units in Vietnam to support infantry operations against the Vietcong.
In 1989 the founder of Homecoming II stepped down due to family issues and asked Sampley to take over. He led the organization until its dissolution in 1993. The archives the group had compiled were transferred to the newly created The Last Firebase Veterans Archive Project. A year later Sampley would use information from those files to identify the Vietnam Unknown Soldier as missing Air Force Lt. Michael Blassie.
Rolling Thunder
In 1987 Ray Manzo, a Marine veteran of the war, visited the memorial in Washington. After stopping at a booth near the memorial run by one of Sampley's organizations, he learned about the POW/MIA issue. As a Marine trained to leave no man behind, the idea that living POWs might still be in Southeast Asia disturbed him, and he resolved to do something about it. He joined forces with Sampley and two other veterans to organize Rolling Thunder, a motorcycle ride from the Pentagon parking lot to the memorial named after a bombing campaign during the war, to show that veterans still cared about their missing comrades. The first run, held the following year, attracted 2,500 riders, a number that has grown with the years. It has since been held every year on the Sunday of Memorial Day weekend, and has become one of the capital's best-attended annual events.
The organization established to support the annual runs has also lobbied for passage of laws supportive of the POW/MIA issue. In 1993 Congress passed the Missing Service Personnel Act, which requires that the Defense Department have substantial evidence that a missing servicemember was killed in action before listing them as such. Two years later the Postal Service issued a stamp with the POW/MIA flag on it, following Rolling Thunder's lobbying.
Identification of Vietnam Unknown Soldier
In 1984 Sampley had gone to Washington again for another Vietnam War-related ceremony, the interment of apparently unidentified remains from that war in the Tomb of the Unknown Soldier. A decade later he published an article in his newspaper asserting that the remains could, in fact, be identified. Items found with the remains in 1972, he alleged, indicated that the fallen serviceman was Air Force Lt. Michael Blassie, whose A-37 Dragonfly jet fighter had been shot down in the area five months before six bones found during the Battle of An Lộc. No other U.S. MIA within of where the remains were found, Sampley wrote, would have had the life raft, parachute, holster and identity card found with the body.
Sampley contacted Blassie's family after publishing his article. Their inquiries found that the South Vietnamese Army patrol which found the bones had also found his ID card and a wallet with a picture of his family and relayed that information to U.S. forces in the area. However the Army lab charged with identifying remains found it unlikely that the remains were Blassie's based on identification techniques later found to be questionable; the ID and wallet had been lost or stolen during their transfer there. In 1980 they were classified as unknown; the remains were later buried as the Vietnam Unknown Soldier, despite not meeting selection criteria calling for 80% of the body, in the wake of political pressure from the president and Congress.
In 1997 Vince Gonzales, a junior CBS News correspondent, read Sampley's article and began replicating the research with requests for documents under the Freedom of Information Act. He came to the same conclusion as Sampley, and the following year CBS reported that the remains were likely those of Blassie. The remains were later exhumed and DNA was found to be a match to Blassie's family; they were reburied near his home and the crypt in which he had lain for 14 years at Arlington was deliberately left empty to symbolize the fallen from Vietnam still not returned home. Sampley told CNN the whole process "was at the very best premature and at worst a politically expedient attempt to further close the books on the POW/MIA issue".
Controversies
Some of Sampley's activities on behalf of POW/MIA servicemembers led to controversy and, in one case, another assault conviction.
Allegations of fabrication and backlash
In 1988 Vietnam returned the remains of Navy Cmdr. Edwin B. Tucker, who had been listed as MIA and presumed dead after his plane was shot down over the country in 1973. Shortly after they were buried, Sampley held a news conference in Norfolk, Virginia, home to the Navy's Atlantic Fleet, where Tucker had been based during his service. Sampley claimed that instead of having been just discovered by the Vietnamese, Tucker's remains had in fact been kept on public display during the intervening years, after he had been beaten severely by the villagers where he parachuted into. In order to get them back, Sampley claimed, the Tucker family had been forbidden to say this. Tucker's son publicly denied this the next day and questioned why anyone would keep an unpreserved human corpse on display under glass for that long.
According to Susan Katz Keating, a former reporter for the conservative Washington Times who went from believing completely in the possibility of living POWs to considering it a hoax, Sampley told a similar story of a downed American pilot killed by natives of the country he had parachuted into early in the 1991 Persian Gulf War. She recalled him calling her about Robert Wetzel, another Navy pilot shot down over Iraq, saying that he was carved up by natives who found his nametag and assumed him to be Jewish, then distributed portions of his body as souvenirs and was telling her despite the Pentagon's attempts to keep the story secret; he had nevertheless told Wetzel's family. Wetzel was later released unharmed; Keating recalled that Sampley later told her he had never completely believed the story but told it anyway to keep attention on the plight of POWs still believed to be in Southeast Asia. Later, Sampley responded that he had indeed found a story reporting that an American pilot had been beaten and killed by a Baghdad mob after being shot down and distributed it in good faith; he never told the Wetzels about it.
In 1992 one of Sampley's protests resulted in the NLF publicly distancing itself from his actions. He had organized a group to disrupt a speech by President George H. W. Bush to the organization's annual assembly. NLF officials were able to prevent Sampley himself from attending the event, and Secret Service officers arrested him for trespassing before Bush appeared. But his group began heckling the president with chants of "No more lies!". After they refused Bush's requests to let him finish, he finally yelled "Would you please shut up and sit down!" at them; the incident made national news. The NLF's leadership was fearful that this would damage their relationship with the administration, and Griffiths not only personally apologized to Bush in a letter but took out newspaper ads with the same message.
The Three Soldiers copyright infringement lawsuit
In the late 1980s an organization Sampley founded, The Last Firebase (TLF), began operating a booth near the Vietnam Veterans Memorial, where it sold T-shirts and other memorabilia to support the POW/MIA cause. This brought them into conflict with other veterans' groups, such as the NLF and the Veterans of Foreign Wars. Not only did they object to commercial activity so close to what they considered to be a solemn site of reflection and remembrance, they pointed out that TLF and two other organizations that were using the adjacent space for similar purposes were operating under National Park Service (NPS) permits that were intended for public gatherings on federal parkland, such as the 24-hour vigils for POW/MIA servicemembers that they had started out doing.
Jan Scruggs, president of the Vietnam Veterans Memorial Foundation (VVMF), who had spearheaded the effort to erect the memorial, was especially incensed. "[They're] a blight on what's supposed to be one of the most beautiful places in the country," he told the Washington City Paper in 1991, pointing to a photo of the trash the booths left behind. Scruggs' opposition was particularly antagonistic to Sampley and other POW/MIA activists who volunteered at the booths since Scruggs was highly skeptical of claims that any American military personnel otherwise unaccounted for in Southeast Asia, having suggested that believing they were was akin to believing that UFOs were real.
Scruggs had also angered the veterans at the booths by saying that they perpetuated stereotypes of Vietnam veterans as disgruntled and alienated from society. He noted that he himself, like many other veterans, had gone onto a professional career, in his case in law, after his return from Vietnam and efforts to build the memorial. "I suppose some of them are down there having a good time", he said in 2001. "It's better than working at Wal-Mart."
The NPS was reluctant to take action against TLF and the other organizations operating the booths, since they were in compliance with the terms of their permits, which allow merchandise sales. In 1991 Scruggs and sculptor Frederick Hart, whose The Three Soldiers was added to the memorial shortly after its completion to make those it memorializes less abstract, discovered TLF was selling T-shirts with the sculpture on it, without having licensed the image from the sculptor and the VVMF, who jointly owned the copyright, and filed suit against Sampley and TLF for infringement.
Sampley refused multiple offers from Scruggs to settle the case out of court, offering to let him and TLF license the image. He also rebuffed similar efforts by Tom Burch, another outspoken POW/MIA activist, to work out a settlement. Instead, Sampley attacked Hart in his newspaper, noting that by the sculptor's own recollection he had been gassed during antiwar demonstrations in the late 1960s and asserting that Hart had made a large amount of money from others' licensing fees (in fact, he did not keep any of the money he received).
Scruggs came in for some criticism as well. Sampley noted that under the legislation which authorized the memorial, the fund Scruggs started to pay for it should have spent its remaining money and dissolved itself in 1984 after the memorial was completed and opened to the public. Instead, Sampley wrote, Scruggs had turned it into the VVMF and made it permanent, on the argument that some of the memorial's granite panels had already cracked and needed repair.
Sampley noted that money that had been raised from the public to pay for those repairs was instead being spent on high-priced Washington lawyers in the lawsuit. He argued in court that since the statue had been placed on public land and was maintained at public expense, it was a "national symbol" that could not be copyrighted. Nevertheless, he offered to settle the case by having Homecoming II pay for the panel repairs.
In 1993 the federal court hearing the case held for Hart and Scruggs. They were awarded almost $360,000 in unpaid royalties; as part of the judgment they were entitled to seize Sampley's house and business. Records he had filed with the court showed that he had made more money than the judgment from T-shirt sales.
Sampley had formed one company, Red Hawk, to make the T-shirts, which it then sold to Homecoming II, a non-profit POW/MIA organization started by others in the 1980s whose founders had asked Sampley to take control of after a few years. Sampley himself claimed to have almost no money, but Red Hawk grossed almost $2 million over three years (Sampley claimed later that Scruggs's lawyers had deliberately exaggerated that number by including revenues from his for-profit businesses). At the same time, Keating wrote in her book, the Homecoming II volunteers who manned the booths received nothing beyond free lodging at a house owned by the organization.
Before Hart and Scruggs could collect, however, Sampley, who had vowed never to pay "homage" to Scruggs, shut down both Red Hawk and Homecoming II, transferring their assets to new companies and organizations. By the time collection efforts began, there was nothing to collect. He never paid, and Scruggs and Hart eventually stopped trying to collect.
Sampley did comply with the aspect of the court's ruling enjoining him from further sales of T-shirts with the statue on them. In 1995, as a result of the court's ruling, the NPS banned all organizations with permits to operate booths on the Mall from selling T-shirts, save the company that operates the guest services kiosks; two years later the D.C. Circuit Court of Appeals upheld the ban as not infringing on First Amendment rights. Sampley later attributed the whole affair to his own stubbornness. "It was the biggest mistake of my life", he told the City Paper in 2001.
Scruggs later told Keating that two months before Sampley's death, he ran into him at an event at the memorial, and the two had a pleasant and civil conversation over their mutual heart problems; he recalled that Sampley looked weak. Sampley later donated $5,000 to the foundation for an educational center. Although Sampley never said so, Scruggs believes it was an attempt to reconcile.
Attacks on John Kerry and John McCain
In 1991, the Senate convened a select committee to examine the POW/MIA issue, chaired by John Kerry and thus known as the Kerry Committee. It began holding hearings during the next Congress. Sampley and other activists were dubious about whether the committee, whose membership included all the sitting senators who were Vietnam-era combat veterans, was really committed to fully investigating the issue. They believed that its true purpose was to resolve the issue by concluding that all POW/MIAs had been accounted for and none were still alive in order to clear a major obstacle to normalizing relations with Vietnam.
The committee did indeed reach a conclusion that if there were any remaining living POWs, there were not many and all the other cases that could be accounted for had been, prompting criticism from one of its members, Bob Smith of New Hampshire, who had introduced the resolution that created the committee. Kerry and John McCain, ranking Republican on the committee and himself a former POW, both rejected claims that the committee had covered up evidence contradictory to that conclusion, or allowed federal officials to lie under oath to the committee. Early in the process, Sampley had called for Kerry to resign after what he alleged was witness tampering by committee staff who reported to him; in response, he claimed, the committee began investigating him and his business activities related to POW/MIA activism; when Keating published her book the next years, with an entire chapter devoted to him, Sampley accused her of doing so at the behest of the Defense Intelligence Agency, which he believed had been trying to discredit him and other activists who believed there were still living POWs.
McCain in particular had drawn the activists' ire early in the hearings. He had stated publicly that most of them were "not zealots in a good cause. They are criminals and some of the most craven, most cynical and most despicable human beings to ever run a scam." McCain also publicly embraced former North Vietnamese Army colonel Bùi Tín while reducing one POW/MIA family member to tears with harsh questioning.
Fight with McCain aide
As the hearings wound down, in December 1992, Sampley wrote an article about McCain for his newspaper, calling him a "Manchurian candidate", beholden to the Vietnamese and depicting him on the cover with a queen of diamonds in the background, alluding to the film and novel from which the term came. While in Washington, he began leaving copies of his newspaper in every senator's office. Although he intended to skip McCain's, he said later that he had been in a hurry and did not look closely at the name on the door, and thus left one in the Arizona senator's.
McCain's chief of staff, Mark Salter, confronted him when he entered and ordered him to leave, which Sampley said he was already doing when Salter followed him into the hall because, according to Salter, Sampley had told him he had something to tell him. In Sampley's account, that was an invitation to a fistfight outside. Salter, by his own later admission, touched Sampley first, pushing him from behind—after, Sampley wrote, following him down the hall; Salter says he merely tapped Sampley's shoulder. In response, Sampley "decked" him.
Capitol police intervened and arrested Sampley after, he claimed, Salter misrepresented the situation. After two days in jail, he was released. Salter asked him if he would agree to stay away from the senator and his staff, but Sampley refused, so he pressed the assault charge. The judge at trial took Salter's word, and sentenced Sampley to time served and 180 days' probation. He also issued a restraining order similarly barring Sampley from contact with McCain or his staff.
McCain and Kerry presidential campaigns
In 2000 McCain sought the Republican presidential nomination. Sampley and other veterans still angry with the senator over his 1993 performance formed Vietnam Veterans Against John McCain. On his newspaper's website, Sampley posted a page of links to articles by himself and others, going back to 1997, criticizing McCain's service as a POW. In 1993 during the hearings, Sampley claimed, McCain along with Rep. Pete Peterson, another former Vietnam POW who became the first postwar U.S. ambassador to Vietnam, had privately beseeched the Vietnamese government to never make its files on American POWs public. Sampley suggested that during his confinement, McCain had collaborated with the Vietnamese at one point in exchange for medical treatment, and wanted to prevent that from becoming public knowledge.
Four years later, Kerry, a Navy veteran of the war who had called Sampley a "stupid ass" in response to his attacks on McCain, won the Democratic nomination for president. Sampley helped organize Vietnam Veterans Against John Kerry and disseminated material disparaging the Massachusetts senator's antiwar activism as unpatriotic and a betrayal of his fellow veterans. McCain came to Kerry's defense, saying that Sampley was "the most despicable person I have ever had the misfortune to encounter." Sampley called the remark "unbecoming" a senator and later addressed it more specifically: "What does that say about his relationship with the Vietnamese prison guards whom he claims brutally tortured him daily?"
In 2008, McCain again ran for president and this time won his party's nomination, although he lost to Democrat Barack Obama in the general election. Once again, Sampley organized efforts to oppose him and distributed his accusations of collaboration, this time in the form of Vetting John McCain, a self-published book. He was joined by other veterans who had supported the effort before and former Republican congressmen Bill Hendon and John LeBoutillier, who had strongly supported the cause of POW/MIA activists during brief House terms in the early 1980s. "He took away the only leverage we had for getting those soldiers back", Sampley told the New York Daily News, referring to McCain's role in normalizing relations with Vietnam in the 1990s. "Why? He was paying back the Vietnamese for keeping quiet about him."
In his 2002 memoir Worth the Fighting For, McCain's opinion of Sampley was mutual:
McCain wrote that he later decided to take Sampley's accusations in stride, referring to himself as the "Manchurian Candidate" in speeches "probably more often than Sampley has repeated the accusation." It was repeated and disseminated among McCain's critics after Sampley's death, especially during the administration of Donald Trump, whom McCain had criticized frequently until his own death in 2018.
CSS Neuse replica and local activism
Sampley was also involved in efforts to revitalize downtown Kinston, where he owned several other businesses besides his pottery shop. In 1991 the Lenoir County Chamber of Commerce recognized his efforts. The following year the Raleigh News & Observer honored Sampley as a "Tar Heel of the Week".
In the late 1990s, he grew disgusted with the sight of one overgrown vacant lot on a major intersection. He proposed to a group of Kinstonians he assembled at a local cafe that they organize to build a life-size replica of the CSS Neuse, an early ironclad of the Confederate States Navy whose remains are listed on the National Register of Historic Places, and put it on the lot as a tourist attraction.
Listeners were doubtful, but two weeks later, at their next gathering, Sampley was accompanied by Alton Stapleford, a retired master boat builder. He explained how it could be done, and in 2002 the Neuse II Foundation was established and construction began. Local volunteers spent the next several months putting timbers into place; the replica is the only full-size replica of a Confederate ironclad. It opened to tourists in 2009.
In the 1990s the state's Department of Natural and Cultural Resources had built the first structure to house the remaining hull of the original Neuse on another lot in downtown Kinston. Sampley and other local historians believed that site possibly also contained the true grave of North Carolina's first governor, Richard Caswell, who had also done the original land survey of Kinston. Sampley announced a contest to find that site, with the state to judge the winner; however officials at the department were angry that he did so without informing them. Sampley was unperturbed: "It's a question the state should have answered a long time ago", he wrote.
In 2004 Sampley and a fellow activist pressured both the Lenoir County commissioners and the Kinston city council to pass a resolution stating that God was the foundation of American government and that the Founding Fathers had never intended for the modern degree of separation of church and state. It was modeled on a resolution that had recently been passed in Greene County, Tennessee. While it passed the city council unanimously, many members said they felt they had been "coerced" into doing so by Sampley's tactics. "I feel like I have a very strong belief in God.", said one. "But I don't like getting stuff like this crammed down my throat."
Personal life and death
Sampley was married twice. He had two children by his first wife, Kiku Uehara, who was from Okinawa; the couple later divorced. Both Kiku and her daughter with Sampley predeceased him.
Later he married Robin Owen, daughter of Army officer Robert Owen, who went missing in Laos in 1968. They had a son together before divorcing in the 1990s. Sampley continued to wear the POW/MIA bracelet with his former father-in-law's name on it afterwards.
Sampley died on May 12, 2009, at the Veterans Affairs Medical Center in Durham, North Carolina, of complications from heart surgery. He was interred at Dyson Cemetery in Ivanhoe, with full military honors.
Notes
References
External links
1946 births
2009 deaths
United States Army personnel of the Vietnam War
Members of the United States Army Special Forces
Military personnel from North Carolina
American ceramists
Artists from North Carolina
Vietnam War POW/MIA activists
Activists from North Carolina
American people convicted of assault
20th-century American newspaper publishers (people)
North Carolina Republicans
People from Wilmington, North Carolina
People from Kinston, North Carolina
United States Army non-commissioned officers | {
"redpajama_set_name": "RedPajamaWikipedia"
} | 7,610 |
\section{Introduction}
\IEEEPARstart{N}{etwork} robustness has various meanings in different scenarios for different concerns. In this article, it refers to the ability of a network to sustain its normal functionality when a fraction of the network fail to work due to attacks. Today, malicious attacks widely exist in many engineering and technological facilities and processes, which degrade or even destroy certain network functions, typically through destructing the network structural connectivity thereby disabling the network to continue its functioning. It is therefore crucial to strengthen the network robustness against such attacks and failures\cite{Barabasi2016NS,Newman2010N,Cohen2010Book,Chen2014Book,Callaway2000PRL,Holme2002PRE,Shargel2003PRL,Bashan2013NP,Iyer2013PO,Yuan2015PRE,Dey2019PNAS}. The study of network robustness generally includes measuring and evaluation, attacking and defending, as well as topological optimization\cite{Ellens2013arXiv,Chan2016DMKD,Freitas2022TKDE}. The concerned damage caused by attacks and failures is typically the degeneration or destruction of network functions, such as connectivity\cite{Schneider2011PNAS,Ellens2013arXiv,Freitas2022TKDE}, controllability\cite{Liu2011N,Yuan2013NC,Xiang2019CSM}, data transmission and communication abilities\cite{Lu2019CL,Mburano2021ISNCC}, and so on. Among these functions, network connectivity is fundamental and essential to support other functions, although good connectivity does not necessarily guarantee good performance of a certain function on the network. In this regard, the subject of network connectivity, controllability, and communication robustness is of fundamental and practical importance, which has been extensively investigated with applications to, for example, the fields of nervous systems\cite{Yan2017NAT}, wireless sensor networks\cite{Qiu2019TN}, power grids\cite{Chen2017TCASII}, and transportation networks\cite{Yang2018TITS,Cai2020SSCI}, among many others. This survey article focuses on measuring the network structural robustness with respect to network functions, in particular the network connectivity, controllability and communication ability against destructive attacks. This survey only discusses the robustness of single-layer networks with static connections, since the structural robustness is not the main issue for networks with dynamic and temporal connections.
Measuring is the first step in analyzing and optimizing the network robustness. There are quite many network robustness measures. In this paper, robustness measures are categorized into two classes according to whether attack simulations are needed for the measurement, namely, the \textit{a priori} measures that do not require attack simulations and the \textit{a posteriori} measures that require so.
\textit{A priori} measures are generally quantified by certain indicative network features that can be calculated without performing attack simulations, including: 1) topological measures, e.g., binary connectivity\cite{Diestel2017GT}, efficiency\cite{Latora2007NJP}, betweenness centrality\cite{Freeman1977Soc}, and clustering coefficient\cite{Watts1998N}; 2) adjacency matrix-based spectral measures, e.g., spectral radius\cite{Van2011PRE}, spectral gap\cite{Malliaros2012ICDM}, natural connectivity\cite{Wu2010CPL}; and 3) Laplacian matrix-based spectral measures, e.g., algebraic connectivity\cite{Fiedler1973CMJ}, effective resistance\cite{Klein1993JMC}, and the number of spanning trees\cite{Butler2008Book}. \textit{A priori} measures require only one-time calculation and usually have lower time and computational complexities comparing to \textit{a posteriori} measures\cite{Freitas2022TKDE,Chan2016DMKD}.
\textit{A posteriori} measures, on the other hand, are quantified by the sequence of values that record the concerned functionality of the remaining network after a sequence of node- or edge-attacks, typically removal attacks. The ratios of largest connected components (LCC)\cite{Schneider2011PNAS}, driver nodes (DN)\cite{Liu2011N,Yuan2013NC} and communicable node pairs (CNP)\cite{Lu2019CL,Mburano2021ISNCC} are the most widely-used measures for the connectivity, controllability, and communication ability, respectively. In turn, the robustness of connectivity, controllability, and communication ability is quantified by a sequence of values that record the corresponding remaining measures after a sequence of node- or edge-attacks, respectively. A network is considered to be more robust if it can maintain higher values of the fractions of nodes in LCC and CNP, but lower fractions of DN, throughout the attack process.
\begin{figure*}[htbp]
\centering
\includegraphics[width=.8\linewidth]{fig/mes_cmp.pdf}
\caption{General framework for \textit{a priori} and \textit{a posteriori} measures of network robustness. \textit{A priori} measures perform one-time calculations to evaluate the network robustness, where \textit{PM} represents an \textit{a priori} measure; while \textit{a posteriori} measures require an iterative process to gain the robustness.}\label{fig:mes_cmp}
\end{figure*}
\textit{A priori} measures are easy-to-access and predictive; while \textit{a posteriori} measures are iteratively calculated after each of the sequence of (simulated) attacks, which are usually time-consuming especially for large-scale networks. However, the predictive \textit{a priori} measures have limited scopes of applications\cite{Yamashita2019COMPSAC}. Moreover, the \textit{a posteriori} measures are effective when the attack process is terminated by a specific criterion, whereas the \textit{a priori} measures do not consider the stopping criteria. Therefore, the time-consuming but precise \textit{a posteriori} measures remain to be the main approach for real-world applications today.
The general framework of \textit{a priori} and \textit{a posteriori} measures is shown in Fig. \ref{fig:mes_cmp}, which shows that \textit{a priori} measures evaluate the network robustness in a straightforward one-time process; while \textit{a posteriori} measures require an iterative process until the stopping criterion is met. It is clear that \textit{a posteriori} measures could have different options on the configuration of stopping criteria and attack strategies, while this is invalid for \textit{a priori} measures.
With desirable robustness measure(s) chosen and used as the objective(s) to optimize, network robustness can be enhanced by model design\cite{Sha2013PCS,Yan2016SR,Lou2018TCASI,Hayashi2018NS,Lou2019R}, edge addition\cite{Beygelzimer2005PA,Freitas2022TKDE}, or edge rewiring\cite{Wu2011PRE,Zeng2012PRE,Louzada2013JCN,Schneider2013SR,Liang2015CPL,Chan2016DMKD,Lou2021TCASII,Wang2020TEVC,Wang2021TEVC}. In so doing, whether or not a modification of the network structure can enhance the robustness has to be evaluated, usually by using \textit{a posteriori} robustness measures that usually requires attack simulations.
Other than attack simulations, network robustness can also be estimated using both analytical and computational methods without iterative calculation. Analytical approximations are applicable when the \textit{a prior} knowledge of the concerned network is available and the attack strategy can be well modeled\cite{Sun2019ICSRS,Sun2021TNSM,Lou2023IJCAS}, e.g., random attacks. In contrast, computational methods are generally data-driven and thus applicable to any attack methods with or without a specific pattern\cite{Lou2022TCYB,Lou2022TNNLS,Lou2023NN}.
In the literature, some general survey papers emphasizing more on \textit{a priori} measures of network robustness are available\cite{Freitas2022TKDE,Ellens2013arXiv}, but there does not seem to be any that specifically emphasizes on the \textit{a posteriori} measures. To fill the gap, this article presents a survey of the \textit{a posteriori} measures of network robustness, including definitions, computation, applications, and optimization. The main contributions of this survey are summarized as follows:
\begin{itemize}
\item[1)] The \textit{a posteriori} robustness measures are summarized and compared, from the perspectives of network functionality, attack strategies, robustness performance prediction, and structural optimization.
\item[2)] A threshold of network destruction is proposed, which suggests a more practical robustness measure of the functionality, especially when a network has been severely destructed.
\item[3)] Both \textit{a posteriori} and \textit{a priori} robustness measures are experimentally compared on a series of directed and undirected network examples. It is found from simulations that \textit{a posteriori} measures have broader applicability.
\item[4)] Some possible future research directions with respect to network robustness are suggested.
\end{itemize}
The remainder of this article is organized as follows.
Section \ref{sec:pst} reviews the \textit{a posteriori} measures of network robustness, from the perspectives of network functionality, malicious attacks, robustness performance prediction, and optimization.
Section \ref{sec:des} introduces a threshold of network destruction.
Section \ref{sec:cmp} experimentally compares \textit{a posteriori} and \textit{a priori} measures.
Section \ref{sec:ftw} presents some prospective research directions with respect to \textit{a posteriori} robustness measures of complex networks.
Section \ref{sec:end} concludes the survey.
\section{\textit{A Posteriori} Measures for Network Robustness}\label{sec:pst}
Network robustness can be defined differently with different practical meanings in graph theory, control systems, communication networks, biological structures, transportation frameworks, etc.\cite{Barabasi2004NRG,Kitano2004NRG,Boccaletti2006PR,Liu2017FCS,Logins2021TKDE,Logins2020WWW}.
On the one hand, it is possible to consider different robustness measures for the same network. On the other hand, the same measure might be applied to different scenarios, for example to both power grids\cite{Cuadra2015ENG} and food webs\cite{Bellingeri2013TE}, where the main concern is the remaining largest connected components after suffering attacks, and the attacks can be physical or cyber destruction to power stations in power grids or extinction of species in food webs.
Here, the focus is on the ability of a network to sustain its specific function(s) when a fraction of the network fail to work due to attacks. Random failures and malicious attacks\cite{Wandelt2022RESS} occur on nodes or edges, or both, in the form of removal or malfunctioning, under different conditions. In implementation, the consequence of attacking a node could be either removal or malfunctioning (without removal), while that of attacking an edge is only edge removal in typical cases. When a node is attacked and removed, all of its connected edges will also be removed; while under edge-attacks, no nodes will be removed.
The remainder of this section is organized as follows. Subsection \ref{sub:fun} reviews the \textit{a posteriori} robustness measures from three commonly concerned network functions: connectivity, controllability, and communication ability. Some extensions of these measures will be discussed in Subsection \ref{sub:othmes}. Various attack strategies and robustness estimation methods are reviewed in Subsection \ref{sub:atk} and Subsection \ref{sub:pre}, respectively. Finally, Subsection \ref{sub:opt} presents some robustness optimization technicians based on \textit{a posteriori} measures.
\subsection{Network Functions}\label{sub:fun}
\textit{A posteriori} measures iteratively calculate specifically concerned network function(s) after each occurrence of attacks. The general form of \textit{a posteriori} robustness measures is as follows:
\begin{equation}\label{eq:r}
R=\frac{1}{K}\sum_{i=1}^{K}{w_i\cdot f(i)}\,,
\end{equation}
where $f(i)$ represents the residual functionality of the remaining network after a number (or proportion) of $i$ objects (either nodes or edges) have been attacked; $K$ represents the total number of attacks; $w_i$ represents the weight of $f(i)$ in calculating $R$. When different network functions are concerned, such as the connectivity, controllability or communication ability, to be discussed below, $f(\cdot)$ will be specified accordingly. The weighting parameter $w$ is usually considered as a parameter for normalization, such that the robustness performances of different-sized networks can be compared. However, these weights also shift the importance of the attacks in the attack sequence, which is often overlooked.
\begin{figure}[htbp]
\centering
\includegraphics[width=.5\linewidth]{fig/functionality.pdf}
\caption{[color online] Widely-used \textit{a priori} and \textit{a posteriori} robustness measures of the three network functions. }\label{fig:functionality}
\end{figure}
Figure \ref{fig:functionality} shows the widely-used \textit{a priori} and \textit{a posteriori} robustness measures of the three network functions. The details of \textit{a posteriori} measures are summarized in the following.
\subsubsection{Connectivity Robustness}
The connectivity of an undirected network means that there is at least one path between any pair of nodes. For a directed network, it is strongly connected if there is at least one directed path from any node to an other node, while it is weakly connected if its underlying undirected network is connected.
LCC is the most commonly-used \textit{a posteriori} measure for connectivity robustness. Under a sequence of node-malfunctioning failures or removals, the connectivity robustness is evaluated by calculating the remaining LCC after each attack\cite{Schneider2011PNAS}, formulated as follows:
\begin{equation}\label{eq:lc}
R_1=\frac{1}{N}\sum_{i=0}^{N-1}{n_L(i)}=\frac{1}{N}\sum_{i=0}^{N-1}{\frac{N_L(i)}{N}}\,,
\end{equation}
where $n_L(i)$ and $N_L(i)$ represent the proportion and the number of nodes in the remaining LCC after a total number of $i$ nodes have been attacked. Specifically, $f(i)=N_L(i)$ measures the remaining connectivity and $w_i=1/N$ is the uniform weight, assuming that the malfunctioned nodes are still counted as a part of the $N$-node networks.
In contrast, if the attacked nodes are removed from the network, its connectivity robustness is calculated by
\begin{equation}\label{eq:lc2}
R_2=\frac{1}{N}\sum_{i=0}^{N-1}{\frac{N_L(i)}{N-i}}\,,
\end{equation}
where $w_i=1/(N-i)$ means that the $i$th attacked nodes have been removed from the network. Compared to Eq. (\ref{eq:lc}), $w_i=1/(N-i)$ assigns higher weights to the later attack stages as $i$ increases. Different weighting parameters $w_i$ also change the range of robustness measure, where $R_1\in[0,0.5]$ but $R_2\in[0,1]$.
The measure shown in Eq. (\ref{eq:lc}) for network robustness under node-attacks can be extended to edge-attacks\cite{Zeng2012PRE}, as follows:
\begin{equation}\label{eq:lce}
R_1^{e}=\frac{1}{M+1}\sum_{i=0}^{M}{n^{e}_L(i)}=\frac{1}{M+1}\sum_{i=0}^{M}{\frac{N^{e}_L(i)}{N}}\,,
\end{equation}
with the superscript $e$ indicating edge-attacks, where the denominator remains $N$ under the assumption that the number of nodes is unchanged during edge-attacks.
When the values $n_L(i)$ or $n^{e}_L(i)$ are plotted, a curve is obtained, which is called the connectivity curve. A higher $R_1$, $R_2$, or $R^{e}_1$ value indicates an overall better connectivity robustness against attacks.
\subsubsection{Controllability Robustness}
Controllability robustness reflects how well a networked system is in maintaining its controllable state. Consider a general linear time-invariant (LTI) networked system, $\dot{{\bf x}}={\bf A}{\bf x}+{\bf B}{\bf u}$, where ${\bf{x}}\in\mathbb{R}^N$ and ${\bf{u}}\in\mathbb{R}^b$ are the state vector and control input, respectively, and ${\bf A}\in\mathbb{R}^{N\times N}$ and ${\bf B}\in\mathbb{R}^{N\times b}$ are constant matrices of compatible dimensions. Conceptually, this LTI system is state controllable if and only if there exists a control input $\bf{u}$ that can drive the system state $\bf{x}$ from any initial state to any target state in the state space within finite time. A commonly-used criterion is that the LTI system is state controllable if and only if the controllability matrix ${\bf C}=[{\bf B}\ {\bf AB}\ {\bf A}^2{\bf B}\ \cdots {\bf A}^{N-1}{\bf B}]$ has a full row-rank\cite{Chen1998Book}. The concept of structural controllability is a slight generalization of the state controllability, to deal with two parameterized matrices ${\bf A}$ and ${\bf B}$, in which the parameters characterize the structure of the underlying system in the sense that if there are specific parameter values ensuring the system to be state controllable then the system is structurally controllable.
When considering a network of many LTI systems, the node system with control input is called a driver node (DN). Network controllability is investigated from two aspects: 1) to gain the full control of the entire dynamical system, one aims to determine how many and which nodes to control\cite{Liu2011N,Yuan2013NC}; 2) for each single node, the aim is to determine the dimension of its controllable subspace\cite{Hosoe1980TAC,Liu2012PO}.
Define the density of DNs by $n_D=\frac{N_D}{N}$, where $N_D$ is the minimum number of DNs needed to retain a full control of the network, which can be calculated using either the minimum inputs theorem (MIT)\cite{Liu2011N} for directed networks or the exact controllability theorem (ECT)\cite{Yuan2013NC} for both directed and undirected networks, defined as follows:
\begin{equation}\label{eq:2nd}
N_D=
\begin{cases}
\text{max}\{1, N-|E^*|\},& \text{using MIT\cite{Liu2011N}},\\
\text{max}\{1, N-\text{rank}(A)\},& \text{using ECT\cite{Yuan2013NC}},\\
\end{cases}
\end{equation}
where $|E^*|$ represents the number of edges in the maximum matching $E^*$, which is a basic concept in classical graph theory\cite{Liu2011N}. Under node-attacks, the controllability robustness is measured by
\begin{equation}\label{eq:nd}
R_3=\frac{1}{N}\sum_{i=0}^{N-1}{n_D(i)}=\frac{1}{N}\sum_{i=0}^{N-1}{\frac{N_D(i)}{N'}}\,,
\end{equation}
where $n_D(i)$ and $N_D(i)$ represent the density and number of DNs needed to retain the network controllability after a total of $i$ nodes have been attacked; $N'$ can be set to either $N'=N-i$ or $N'=N$, depending on specific preference, namely whether or not an attacked node still belongs to the network depends on the situation under consideration. Usually, attacked nodes are assumed to be malfunctioned (but still in the system) in connectivity robustness measures, but will be removed from the network in controllability robustness measures.
Similarly, controllability robustness under edge-attacks is measured by
\begin{equation}\label{eq:nde}
R^{e}_3=\frac{1}{M+1}\sum_{i=0}^{M}{n^{e}_D(i)}=\frac{1}{M+1}\sum_{i=0}^{M}{\frac{N^{e}_D(i)}{N}}\,,
\end{equation}
where $M$ is the number of edges in the network. When the values $n_D(i)$ or $n^{e}_D(i)$ are plotted, a curve is obtained, which is called the controllability curve. A lower $R_3$ or $R^{e}_3$ value represents a more robust controllability against attacks.
Different from considering the density of DNs, the control centrality measures the control ability of a single node in a directed network\cite{Liu2012PO}, defined by $c^{(j)}_c=C^{(j)}_c/N$, where $C^{(j)}_c=\text{rank}_g({\bf C}^{(j)})$ is the generic dimension of the controllable subspace of node $j$ that can be calculated according to the Hosoe theorem\cite{Hosoe1980TAC}; ${\bf C}$ represents the controllability matrix. Under this measure, the greater the $c^{(j)}_c$ value is, the more ``powerful'' the node $j$ is as a DN.
The expected robust control centrality (ERCC)\cite{Usman2020MSC,Usman2019CCC} is a control centrality-based robustness measure for node-attacks, defined as follows:
\begin{equation}\label{eq:ercc}
R_{4}^{(j)}(i)=E[C_c^{(j)}(i)]\,,
\end{equation}
where $C_c^{(j)}(i)$ represents the control centrality of node $j$ after a total number of $i$ nodes have been attacked; $E[\cdot]$ is the statistical expectation. The generic robust control centrality (GRCC)\cite{Usman2020MSC,Usman2019CCC} is a generalization of ERCC, defined as follows:
\begin{equation}\label{eq:grcc}
R_{4}^{e,(j)}(\{e\})=E[C_c^{(j)}(\{e\})]\,,
\end{equation}
where $C_c^{(j)}(\{e\})$ represents the control centrality of node $j$ after a set of edges $\{e\}$ have been removed, under either node- or edge-attacks. Both ERCC and GRCC measure the significance of a single node in controlling part of the system, under random node- and edge-attacks, respectively.
The reachability-based controllability robustness\cite{Parekh2014ICSIT,Sun2021TNSM} is also a control centrality-based robustness measure. Given a fixed number of $H$ controllers that can be pinned anywhere (``free control'' mode), the controllability robustness is calculated by
\begin{equation}\label{eq:freec}
R_5=\frac{1}{N}\sum_{i=0}^{N-1}{\sum_{j=1}^{H}} {\frac{c_c(j)}{N'}}\,,
\end{equation}
where $\sum_{j=1}^{H}{c_c(j)}$ represents the dimension of the controllable subspace by the given $H$ DNs. During the attack, these DNs can be freely set in the remaining network, as long as the control centrality is maximized. Again, $N'=N-i$ or $N$, depending on the specific situation under consideration.
In the case that the given external controllers are fixedly pinned at a set of given nodes (``fixed control'' mode)\cite{Sun2021TNSM}, the controllability robustness is also measured using Eq. (\ref{eq:freec}), where however $\sum_{j=1}^{H}{c_c(j)}$ counts the dimension of the controllable subspace by the given $H$ fixed controllers.
\subsubsection{Communication Robustness}
Different from the \textit{a priori} measures of general connectivity robustness, which are either spectral measures or topological features, the \textit{a priori} measures of communication robustness are more comprehensive. For example, the $r$-robustness\cite{LeBlanc2013JSAC,Zhao2014CDC,Zhang2015TCNS} based on reachability, and the comprehensive measure proposed in\cite{Morales2018CL} consisting of three indices, including edge betweenness centrality, number of edge cut-sets, and node Wiener impact\cite{Wiener1947JACS}. Nevertheless, the \textit{a posteriori} measures for connectivity robustness remain useful for measuring communication robustness\cite{Qiu2017TN}.
The CNP-based robustness measure is a widely-used \textit{a posteriori} measure for communication robustness, defined as follows\cite{Lu2019CL}:
\begin{equation}\label{eq:com}
R_6=\frac{1}{N}\sum_{i=0}^{N-1}{\sum_{j=1}^{\Gamma(i)}} {\frac{{S_j\choose 2}}{{N\choose 2}}}\,,
\end{equation}
where $\Gamma(i)$ represents the number of connected components in the remaining network after a total of $i$ nodes have been attacked; $S_j$ represents the number of nodes in the $j$th connected component; ${S_j\choose 2}$ represents the number of communicable node pairs, while ${N\choose 2}$ is the number of all possible node pairs. When ${S_j\choose 2}={N\choose 2}$, the network is fully connected, thus each pair of nodes are communicable; while for the networks that are not fully connected, the number of communicable node pairs should be less than the all possible node pairs, namely ${S_j\choose 2}<{N\choose 2}$.
The following simplified communication robustness\cite{Mburano2021ISNCC} provides a simpler CNP-based measure:
\begin{equation}\label{eq:scom}
R_7 =\frac{1}{N}\sum_{i=0}^{N-1}{\sum_{j=1}^{\Gamma(i)}} {\frac{{S_j^2}}{{N^2}}} \,,
\end{equation}
which ignores the non-dominant terms in Eq. (\ref{eq:com}) but keeps only the dominant ones. The computation complexities of both measures are the same, $O(NM)$.
When the CNP values are plotted, a curve is obtained, which is called the communication curve. Apparently, higher values of $R_6$ or $R_7$ represent better communication robustness against attacks.
\subsection{Variants of Robustness Measures}\label{sub:othmes}
Based on the fundamental \textit{a posteriori} robustness measures presented in Subsection \ref{sub:fun}, several variants have been developed with different concerns.
\subsubsection{Rank-based Measure}
Before being attacked, the initial proportions of LCC for all connected networks are the same, namely $n_L(0)=1$. In contrast, the initially required proportion of DNs to fully control a network varies from case to case. This inequality of initial controllability may influence the measurement of robustness.The rank-based controllability measure offers an alternative to diminish this influence, which is defined by
\begin{equation}\label{eq:rank}
R_8=\frac{1}{N}\sum_{i=0}^{N-1}{r_D(i)}\,,
\end{equation}
where $r_D(i)$ is the rank of the controllability matrix after a total of $i$ nodes have been attacked. Lower ranks are assigned to the networks that possess better controllability.
Figure \ref{fig:eg_rank_measure} shows an illustrative example, where net1 requires a larger initial proportion of DNs than net2. The controllability curve of net1 is flatter than that of net2. Under two different measures, $R_3$ returns that net2 has better controllability robustness than net1, but $R_8$ returns that they have same performance. Clearly, $R_8$ diminishes the influence of the initial states.
\begin{figure}[htbp]
\centering
\includegraphics[width=0.35\linewidth]{fig/eg_rank_measure.pdf}
\caption{[color online] Example of two different controllability robustness measures. $R_3$ and $R_8$ are calculated using Eqs. (\ref{eq:nd}) and (\ref{eq:rank}), respectively. }\label{fig:eg_rank_measure}
\end{figure}
\subsubsection{Combinatorial Measure}
Although connectivity robustness has a certain positive correlation with controllability robustness and communication robustness, they actually have very different measures and objectives. In general, good connectivity is the prerequisite for good controllability and communication ability, but the former does not guarantee the latter in general.
Considering connectivity robustness and controllability robustness together, adjustment is necessary since better robustness means maximization Eq. (\ref{eq:lc}) but minimization Eq. (\ref{eq:nd}). To unify them (e.g., both being maximization), a combinatorial measure can be defined using either the opposite of $n_D(i)$\cite{Xiao2014CPB}, as follows:
\begin{equation}\label{eq:xiao}
R_{9}=\frac{1}{N}\sum_{i=0}^{N-1}(1-n_D(i)) \,,
\end{equation}
or the reciprocal of $n_D(i)$\cite{Wang2018TNSE}, as follows:
\begin{equation}\label{eq:wang_tnse}
R_{10} =\frac{1}{N}\sum_{i=0}^{N-1}\frac{n_L(i)}{n_D(i)}\,.
\end{equation}
Maximizing $R_9$ is equivalent to minimizing $R_3$, while maximizing $R_{10}$ is equivalent to either maximizing $R_1$, or minimizing $R_3$, or maximizing $R_1$ and minimizing $R_3$ together.
\subsubsection{Averaged Measure}
All the above-mentioned \textit{a posteriori} robustness measures, except for ERCC and GRCC\cite{Usman2020MSC,Usman2019CCC}, are calculated based on a specific attack sequence, namely, each robustness value is one-to-one corresponding to a specific attack sequence. If network robustness is required to be measured by a number of repeated simulations, or several different attack sequences are required to be considered, then the averaged robustness measure can be applied, which is defined as follows:
\begin{equation}\label{eq:rpt}
R_{11}=\frac{1}{P\cdot Q}\sum_{p=1}^{P}\sum_{q=1}^{Q}R_{p,q}\,,
\end{equation}
where $R_{p,q}$ represents the network robustness measured under the $p$-th repeated simulation using the $q$-th attack strategy; $P$ is the number of repeated attack simulations; $Q$ is the number of different attack strategies. After averaging, a robustness value will not be corresponding to a specific attack strategy or sequence.
\subsubsection{Other Measures}
When cascading failure-based attacks are considered, the robustness measure can be slightly modified, as follows:
\begin{equation}\label{eq:tang}
R_{12}=\frac{1}{N}\sum_{h=1}^{H}f(h)\,,
\end{equation}
where $H$ is the required number of attacks to achieve the attack task, for example, a significant destruction of functionality\cite{Pu2012PA,Nie2014PO,Tang2016SR,Chen2017PA,Hou2019ICISBDE}. Here, $H\leq N$ implies that it is not always necessary to attack all nodes in order to destroy the network functions.
When the community structure is concerned, the community robustness can also be calculated using Eq. (\ref{eq:r}), where $f(i)$ could be either the community integrity that counts the number of remaining nodes in the community\cite{Ma2013PRE}, or the normalized mutual information\cite{Wang2017JSTAT}.
It is noted that this survey paper focuses on reviewing the robustness measures of the networks with static connection, whereas the networks with dynamic and temporal connections are not discussed. This is because the robustness measures of dynamic and temporal networks have very different characteristics and applications. For example, the robustness of a temporal network is measured by calculating the relative loss of efficiency caused by attacks\cite{Scellato2011TMC}, as follows:
\begin{equation}\label{eq:temporal}
R_{13}=1-\frac{\Delta{\epsilon}}{\epsilon_0}\,,
\end{equation}
where $\epsilon_0$ represents the global efficiency of the temporal network within a given time window, and $\Delta{\epsilon}$ represents the efficiency loss caused by attacks. Although it may be regarded as an \textit{a posteriori} measure, it has a different form from Eq. (\ref{eq:r}) that performs iterative attack-and-evaluation operations.
\subsection{Attack Strategies}\label{sub:atk}
From the attacker's perspective, searching for the most destructive attacking sequence is a desirable task, which can also help the defender in considering how to design a best possible network topology with the strongest robustness. Therefore, attack strategy is also a focal topic in the study of network robustness.
For a given network, \textit{a priori} measures return a single deterministic value about the network robustness, which will not change when different attack strategies or different numbers (rounds) of attacks are applied. In contrast, \textit{a posteriori} measures are able to reflect different robustness performances when attack strategies (or attack sequences) vary. The issue of network robustness within different contexts has been extensively investigated, with many edge- and node-attack strategies proposed to destruct the network functions, regarding the connectivity, controllability, communication ability, and so on.
Random attacks remove or malfunction randomly-selected objects (nodes or edges), while targeted attacks aim at attacking deliberately-selected objects, for example, the highest-degree node or the largest-betweenness edge. Given an importance measure $g$ for either nodes or edges, targeted attacks perform sequential attacks to object $j$, with $\argmax g$, meaning that object $j$ is the most important according to measure $g$.
\subsubsection{Degree- and Betweenness-based Attack Strategies}\label{sub:dbc}
For targeted attacks, it is assumed that the targeted object is more important than the others in maintaining the network functionality. The most frequently-used measures of importance are the degree centrality and betweenness centrality, for both nodes and edges. In fact, the maximum degree-based targeted attack (MDTA) and maximum betweenness-based targeted attack (MBTA) are the most widely-used strategies.
To integrate multiple importance measures into one, weights and probabilities may be considered:
\begin{equation}\label{eq:pj}
p_j=\sum_i \alpha_i\times\frac{g_{i,j}}{\sum_{j=1}^{K}{g_{i,j}}}\,,
\end{equation}
where $p_j$ is the probability of attacking object $j$; $\alpha_i$ is the weight for importance measure $g_i$; $g_{i,j}$ is the importance measure $g_i$ for object $j$. For example, $p_j=\alpha_1\times\frac{k_j}{\sum_{j=1}^{K}{k_j}} + \alpha_2\times\frac{b_j}{\sum_{j=1}^{K}{b_j}}$ represents a combination of degree and betweenness, where $k_j$ and $b_j$ are the degree and the betweenness of node $j$; weights $\alpha_1$ and $\alpha_2$ adjust the distributions of different features, which can be set manually\cite{Nie2015PA}, or with $\alpha_2$ being replaced by $1-\alpha_1$\cite{Gao2018PA}.
Similarly, three parameters can be used\cite{Hao2020PA} to control the weights of degree, betweenness and harmonic closeness, respectively. Attacking the highest-betweenness node inside the LCC makes MBTA more destructive in the later stages of the attack process\cite{Nguyen2019PA}. These measures have also been used in some strategies to attack interdependent networks\cite{Huang2011PRE,Dong2012PRE,Gao2018PA,Cui2018PA,Hao2020PA}, networks of networks\cite{Dong2013PRE,Liu2015CSF}, and weighted networks\cite{Bellingeri2018PA}.
Both MDTA and MBTA are not only destructive to connectivity robustness, but also effectively degrade other network functions such as controllability and communication ability\cite{Pu2012PA,Nie2014PO,Chen2019TCASII,Lu2019CL}.
\subsubsection{Topology-based Attack Strategies}
Beside degree and betweenness, commonly-used measures of importance include closeness\cite{Borgatti2005SN}, Katz centrality\cite{Katz1953PSY}, neighborhood similarity\cite{Ruan2017APS}, branch weighting\cite{Simon2017MOE}, structural holes\cite{Yang2020SYM}, and so on. However, ranking the importance of nodes or edges is practically intractable for large-scale networks, since most measures cannot guarantee that removing the targeted object will globally and consistently cause the greatest damage to the network.
The hierarchical structure of a directed network enables the random upstream (or downstream) attack to the network controllability, which results in a more destructive attack strategy than random attacks\cite{Liu2012PO}. The module-based attack strategy\cite{daCunha2015PO,Shai2015PRE} aims at attacking the nodes with inter-community edges that are crucial to maintain the connectivity among communities. Practically, the removal costs for different nodes are not the same, so attack strategies could also be designed to minimize the total costs\cite{Ren2019PNAS}.
Given an $N$-node network, which is subject to node-attacks, there are $N!$ possible attack sequences in total. Thus, it is quite possible to have different or even opposite conclusions for network robustness depending on some topological issues. For example, it is observed that homogeneous networks are more robust than heterogeneous networks against random attacks, MDTA, and MBTA\cite{Lu2016PO}. Also, when the attack strategy aims at removing the three-level tree structures (including random, maximum- and minimum-degree nodes)\cite{Hao2016PA}, homogeneous networks are more robust than heterogeneous networks. However, if one aims at removing approximately the longest simple path from a network, then homogeneous networks are more vulnerable than heterogeneous networks\cite{Pu2015PA}. Moreover, for networks with special topological features, the efficiencies of different attack strategies are also different; for example, MDTA causes greater damages to local-world networks\cite{Li2003PA} with larger local-world sizes, while networks with smaller local-world sizes show better robustness regarding both connectivity and controllability\cite{Sun2016PLA}.
\subsubsection{Damage-based Attack Strategies}\label{subsub:dmg}
The concept of ``damage''\cite{Wang2014PA} in network connectivity helps to evaluate and guide attacks. The damage of a specific node is quantified by the change of the LCC size, before and after attacking the node. Therefore, it is natural that an efficient greedy attack strategy can be formed by sequentially attacking the node whose removal or malfunctioning will cause the greatest damage to the network\cite{Wang2014PA}. With damage as the importance measure, the most destructive node-removal sequence can be searched by solving a combinatorial optimization problem, using genetic algorithm\cite{Lin2022SC}, memetic algorithm\cite{Yang2018PA}, or other advanced optimization tools.
Different from the damage of connectivity, the damage of controllability is defined based on the categorization of edges or nodes. An edge or node is critical if and only if its removal increases the number of needed DNs; otherwise, it is non-critical\cite{Liu2011N,Sun2019ICSRS,Lou2021CNSNS}. The damage of controllability helps to form effective attack strategies, where critical edges or nodes will be removed with the highest priority\cite{Sun2019ICSRS,Lou2021CNSNS}.
Damage-based attack strategies are intuitive and the maximal destruction is guaranteed for every single attack. However, they have two clear disadvantages: 1) the maximal destruction of a series of continuous attacks cannot be guaranteed; 2) the computational cost of calculating the damage is not negligible.
\subsubsection{Computational Intelligence-based Attack Strategies}
Searching for a desirable attack sequence from the large number of possible choices is an NP-hard combinatorial optimization problem\cite{Karp1972CCC,Braunstein2016PNAS}. Evolutionary algorithms have been applied to dealing with this problem, such as genetic algorithms\cite{Zhang2016CEC}, artificial bee colony algorithm\cite{Lozano2017IS}, Tabu search algorithm\cite{Deng2016PA,Qi2018CHAOS}, and other metaheuristic algorithms\cite{Ventresca2012COR,Li2020ESA}. Candidate attack sequences referred to as individuals form a population, which are evolved towards the optimal destruction of networks.
Moreover, machine learning techniques have been increasingly used to explore optimal attack strategies on large-scale networks. Ensemble learning is employed to estimate node importance, where node damage (see Subsection \ref{subsub:dmg}) is used for training the model, such that nodes with higher damages can be identified, thus an efficient attack strategy can be designed\cite{Li2019ISBSCI}.
The minimal set of critical nodes is identified using graph attention networks\cite{Velivckovic2017arXiv}, which is then used to effectively disintegrate a complex network\cite{Grassia2021NC}. Such an attack strategy can be successful based on deep reinforcement learning\cite{Fan2020NMI}. A sequential attack process can also be modeled by a Markov decision process, whereas deep reinforcement learning\cite{Mnih2015N,Sutton2018Book} can be used to find optimal attack sequences\cite{Yan2016TIFS,Tian2021CSF,Yan2022TNSE}.
Recently, a combination of convolutional neural networks (CNN) and graph neural networks (GNN)\cite{Kipf2016arXiv,Hamilton2017NIPS,Hamilton2020Book} has been used for measuring the node importance in virus spreading models\cite{Zhang2022Neur}.
The computational intelligence-based attack strategies require a non-negligible or even substantial amount of computational cost in the stages of robustness evaluation and model training. The difference is that evolutionary algorithm-based strategies aim at finding the most destructive attack sequence for the given networks, while machine learning-based strategies also pursue the generalizability to unknown data, for which greater computational cost is needed in the training stage.
\subsection{Robustness Performance Prediction}\label{sub:pre}
Evaluating \textit{a posteriori} measures by attack simulations is generally very time-consuming. In case that the exact robustness values are not required, approximated values can be estimated by either analytical or computational methods. In so doing, the time complexity is either constant for analytical methods\cite{Lou2023IJCAS} or increasing significantly slower than that of attack simulations for computational methods\cite{Lou2022TCYB}.
\subsubsection{Analytical Approximation}
Analytical approximations require full knowledge of the network structure and the applied attack strategy that can be well-modeled\cite{Sun2019ICSRS,Cai2021TSMC}, such as random attacks. Given the network adjacency matrix, the controllability configuration and critical edges can be found, so that the controllability curve under random edge-attacks can be approximated based on the uniformly-random decreasing process of critical edges\cite{Sun2019ICSRS}. This analytical method is applicable to approximating the controllability robustness\cite{Sun2019ICSRS}, as shown in Eq. (\ref{eq:nde}), and the reachability-based controllability robustness\cite{Sun2021TNSM}, as shown in Eq. (\ref{eq:freec}), under random edge-attacks.
For random-graph networks, based on the fact that the generation mechanism is essentially the same as the random edge-removal process from a fully-connected network in a reverse manner, a precise approximation can be designed. The given random-graph networks are classified as either ``dense'', ``median'', or ``sparse''. Then, the hybrid approximation method uses different prior knowledge to approximate the controllability curves\cite{Lou2023IJCAS}.
In comparison, the approach of \cite{Lou2023IJCAS} performs significantly better in predicting the controllability curves of random-graph networks under random edge-attacks; while its disadvantage is clear that it is applicable only to the above-mentioned scenario.
\subsubsection{Machine Learning-based Prediction}
Machine learning algorithms, such as linear regression, random forest, and neural networks, have been successfully applied to predicting the number of DNs under random or targeted edge-attack, such that the controllability curves can be fitted\cite{Dhiman2021MLN}.
During the network robustness optimization processes, calculating the exact robustness values may not be required for every generation. Therefore, fast estimation can be used to improve the efficiency. For example, in Refs.\cite{Wang2020TEVC,Wang2021TEVC}, three algorithms, including radial basis function\cite{Hardy1971JGR}, inverse distance weighting\cite{Zhou2005CEC} and least-squares\cite{Shepard1968ACMNC}, form a surrogate ensemble for estimating connectivity robustness values; attack simulations are intermittently performed for obtaining real robustness values, which are used for simultaneously evaluation and updating the surrogates. The computational time of optimization can be significantly reduced by using such a surrogate ensemble\cite{Wang2020TEVC,Wang2021TEVC}.
The CNN-based prediction approach treats complex network data as gray-scale images\cite{Lou2022TCYB}, thereby fast approximating the robustness performance against different attacks in an end-to-end manner. Prior knowledge is useful for pre-processing and filtering, which is utilized to build an improved predictor\cite{Lou2022TNNLS}, showing lower prediction errors. A limitation of this straightforward approach is that it cannot deal with the situation where the network size is significantly different from the input size of the CNN\cite{Wu2022IJCNN}. Graph representation learning\cite{Kipf2016arXiv,Hamilton2017NIPS,Hamilton2020Book}, which is specifically designed for processing graph data, provides a solution to overcome this problem\cite{Lou2022TCYB2}. In graph representation learning, the raw complex network data ($N\times N$) are compressed and unified to lower-dimensional representations ($W\times U$, with $W<N$ and $U\ll N$); thus, not only the input size problem is solved but also the topological features can be better extracted and utilized.
\subsection{Robustness Optimization}\label{sub:opt}
It is crucial to understand the relationship between the network structure and its functionality robustness. Generally, dense homogeneous networks have better robustness than spare heterogeneous ones, regarding the network connectivity, controllability, and communication ability. However, it is also possible that well-designed heterogeneous networks have better robustness than homogeneous networks\cite{Yan2016SR}. For general heterogeneous networks, it is known that onion-like structures that possess higher assortativity coefficients are robust against attacks\cite{Herrmann2011JSTAT,Schneider2011PNAS,Wu2011PRE,Tanizawa2012PRE,Hayashi2018SR}. Given suitable robustness measure(s) as the objective(s), network robustness can be optimized using evolutionary algorithms\cite{Gunasekara2018MOO,Liu2019ECCN}.
Figure \ref{fig:ea} shows a general flowchart of using evolutionary algorithms for network robustness optimization. Rewiring is the most widely-used strategy to perform disturbances onto the network structure, while in some specific applications adding edges is more cost-effective. Constraints such as degree preservation for all nodes guarantee some given prerequisites. After one or several edge rewiring operations, whether the disturbance enhances the robustness has to be evaluated by using either \textit{a priori} or \textit{a posteriori} measures. Here, within the focus of this survey, only the latter is discussed.
\begin{figure}[htbp]
\centering
\includegraphics[width=.5\linewidth]{fig/ea.pdf}
\caption{Flowchart of network robustness optimization using evolutionary algorithms. }\label{fig:ea}
\end{figure}
\begin{table*}[htbp]
\centering \caption{Summary of using heuristic algorithms to optimize network robustness.}
\begin{tabular}{|l|l|l|l|l|}
\hline
\multicolumn{1}{|c|}{Work} & \multicolumn{1}{c|}{Measure} & \multicolumn{1}{c|}{Algorithm} & \multicolumn{1}{c|}{Constraint} & \multicolumn{1}{c|}{\begin{tabular}[c]{@{}l@{}}Attack\\Object\end{tabular}} \\ \hline
Schneider, et al.\cite{Schneider2011PNAS} & Eq. (\ref{eq:lc}) & random rewiring & \begin{tabular}[c]{@{}l@{}}degree preservation\\ for each node\end{tabular} & node \\ \hline
Herrmann, et al.\cite{Herrmann2011JSTAT} & Eq. (\ref{eq:lc}) & Monte Carlo-based algorithm & \begin{tabular}[c]{@{}l@{}}degree distribution\\ preservation\end{tabular} & node \\ \hline
Buesser, et. al.\cite{Buesser2011ICANCA} & Eq. (\ref{eq:lc}) & simulated annealing & \begin{tabular}[c]{@{}l@{}}degree preservation\\ for each node\end{tabular} & node \\ \hline
Zeng and Liu\cite{Zeng2012PRE} & Eq. (\ref{eq:lc}) & hybrid greedy algorithm & \begin{tabular}[c]{@{}l@{}}degree preservation\\ for each node\end{tabular} & node and edge \\ \hline
\begin{tabular}[c]{@{}l@{}}Peixoto and\\Bornholdt\cite{Peixoto2012PRL}\end{tabular} & Eq. (\ref{eq:lc}) & \begin{tabular}[c]{@{}l@{}}BFGS\cite{Fletcher2013Book} and\\ other swarm-based algorithms\end{tabular} & \begin{tabular}[c]{@{}l@{}} average degree\\preservation\end{tabular} & node \\ \hline
Cao, et al.\cite{Cao2013CSF} & Eq. (\ref{eq:lc}) & strategies of adding edges & N/A & node \\ \hline
Zhou and Liu\cite{Zhou2014PA} & Eq. (\ref{eq:lc}) & memetic algorithm & \begin{tabular}[c]{@{}l@{}}degree preservation\\ for each node\end{tabular} & node \\ \hline
Xiao, et al.\cite{Xiao2014CPB} & Eq. (\ref{eq:xiao}) & dynamic optimization & \begin{tabular}[c]{@{}l@{}}degree preservation\\ of each node\end{tabular} & node \\ \hline
Bai, et al.\cite{Bai2015CPL} & Eq. (\ref{eq:lc}) & hill-climbing search & \begin{tabular}[c]{@{}l@{}}degree preservation\\ for each node\end{tabular} & node \\ \hline
Yang, et al.\cite{Yang2015PO} & Eq. (\ref{eq:lc}) & \begin{tabular}[c]{@{}l@{}}greedy bypass\\rewiring algorithm\end{tabular} & \begin{tabular}[c]{@{}l@{}}preserving both the\\ degree distribution and\\ community structure\end{tabular} & node \\ \hline
Tang, et al.\cite{Tang2015EPL} & Eq. (\ref{eq:tang}) & memetic algorithm & \begin{tabular}[c]{@{}l@{}}degree preservation\\ for each node\end{tabular} & node \\ \hline
Ma, et al.\cite{Ma2016PA} & Eq. (\ref{eq:lc}) and Eq. (\ref{eq:lce}) & edge-replenishment strategy & \begin{tabular}[c]{@{}l@{}}keep the total numbers\\ of nodes and edges\end{tabular} & node and edge \\ \hline
Sun, et al.\cite{Sun2016PA} & Eq. (\ref{eq:lc}) & tabu search & \begin{tabular}[c]{@{}l@{}}degree preservation\\ for each node\end{tabular} & node \\ \hline
Tang, et al.\cite{Tang2016SR} & Eq. (\ref{eq:tang}) & memetic algorithm & \begin{tabular}[c]{@{}l@{}}degree preservation\\ for each node\end{tabular} & node \\ \hline
Park and Hahn\cite{Park2016PRE} & Eq. (\ref{eq:lc}) & \begin{tabular}[c]{@{}l@{}}greedy bypass\\ rewiring algorithm\end{tabular} & N/A & node \\ \hline
Wang and Liu\cite{Wang2017SR} & \begin{tabular}[c]{@{}l@{}}2 objectives: \\ Eq. (\ref{eq:lc}) and cooperation\\ (fraction of cooperators)\end{tabular} & \begin{tabular}[c]{@{}l@{}}multi-objective\\ evolutionary algorithm\end{tabular} & \begin{tabular}[c]{@{}l@{}}degree preservation\\ for each node\end{tabular} & node \\ \hline
Wang and Liu\cite{Wang2017CEC} & community integrity\cite{Ma2013PRE} & genetic algorithm & \begin{tabular}[c]{@{}l@{}}degree distribution \\ preservation\end{tabular} & node \\ \hline
Wang, et al.\cite{Wang2017JSTAT} & \begin{tabular}[c]{@{}l@{}}normalized mutual \\ information\cite{Wang2017JSTAT}\end{tabular} & simulated annealing & \begin{tabular}[c]{@{}l@{}}degree distribution \\ preservation\end{tabular}& node \\ \hline
Wang and Liu\cite{Wang2018TNSE} & \begin{tabular}[c]{@{}l@{}}2 objectives:\\ Eq. (\ref{eq:wang_tnse}) and cooperation\\ (fraction of cooperators)\end{tabular} & \begin{tabular}[c]{@{}l@{}}multi-objective\\ evolutionary algorithm\end{tabular} & \begin{tabular}[c]{@{}l@{}}degree preservation\\ for each node\end{tabular} & node \\ \hline
Rong and Liu\cite{Rong2018PA} & Eq. (\ref{eq:lc}) & heuristic algorithm & \begin{tabular}[c]{@{}l@{}}degree preservation\\ for each node\end{tabular} & node \\ \hline
Gunasekara, et al.\cite{Gunasekara2018MOO} &
\begin{tabular}[c]{@{}l@{}}Eq. (\ref{eq:lc}) and two\\spectral measures\end{tabular}
& \begin{tabular}[c]{@{}l@{}}multi-objective \\ evolutionary algorithm\end{tabular} & \begin{tabular}[c]{@{}l@{}}N/A\end{tabular} & node \\ \hline
Liu, et al.\cite{Liu2019TEVC} & Eq. (\ref{eq:lc}) & evolutionary algorithm & N/A & node \\ \hline
Liu, et al.\cite{Liu2019ECCN} &
\begin{tabular}[c]{@{}l@{}}Eq. (\ref{eq:lc}) and Eq. (\ref{eq:lce})\end{tabular}
& \begin{tabular}[c]{@{}l@{}}multi-objective \\ evolutionary algorithm\end{tabular} & \begin{tabular}[c]{@{}l@{}}degree preservation\\ for each node\end{tabular} & node and edge \\ \hline
Qiu, et al.\cite{Qiu2019TN} & Eq. (\ref{eq:lc}) & multi-population co-evolution & \begin{tabular}[c]{@{}l@{}}degree preservation\\ for each node\end{tabular} & node \\ \hline
Cai, et al.\cite{Cai2020SSCI} & \begin{tabular}[c]{@{}l@{}}2 objectives:\\ 1) maximize algebraic\\ connectivity\\ 2) minimize the\\ number of removed edges\end{tabular} & \begin{tabular}[c]{@{}l@{}}NSGA-II, NSGA-III,\\ and MODPSO\end{tabular} & N/A & edge \\ \hline
Wang, et al.\cite{Wang2020TEVC} & Eq. (\ref{eq:lc})* & \begin{tabular}[c]{@{}l@{}}surrogate-assisted\\ evolutionary algorithm\end{tabular} & \begin{tabular}[c]{@{}l@{}}degree preservation\\ for each node\end{tabular} & node \\ \hline
Wang, et al.\cite{Wang2021TEVC} & Eq. (\ref{eq:lc})* & \begin{tabular}[c]{@{}l@{}}surrogate-assisted\\ multi-objective \\ evolutionary algorithm\end{tabular} & \begin{tabular}[c]{@{}l@{}}degree preservation\\ for each node\end{tabular} & node and edge \\ \hline
Lou, et al.\cite{Lou2020TCASI} & Eq. (\ref{eq:nd}) & \begin{tabular}[c]{@{}l@{}}random edge rectification\end{tabular} & \begin{tabular}[c]{@{}l@{}} average degree\\preservation\end{tabular} & node and edge \\ \hline
Lou, et al.\cite{Lou2021TCASII} & Eq. (\ref{eq:nd}) & \begin{tabular}[c]{@{}l@{}}random rewiring\end{tabular} & \begin{tabular}[c]{@{}l@{}} average degree\\preservation; and\\ underlying-topology\\preservation \end{tabular} & node \\ \hline
\multicolumn{5}{l}{* with the assistance of surrogates and assortativity}
\end{tabular}\label{tab:ea}
\end{table*}
Table \ref{tab:ea} summarizes 29 methods on network robustness optimization. The most common scenario of these methods is to use evolutionary algorithms to optimize network robustness under node-attacks measured by Eq. (\ref{eq:lc}), where the degree (or both in- and out-degrees) for each node remains unchanged during the optimization process. Extensions of this common scenario include considering different measures, using advanced algorithms, imposing different constraints of topology disturbances, and targeting different objects. Since a single measure sometimes cannot fully reflect the network robustness\cite{Liu2017FCS,Mburano2021ISNCC}, multiple robustness measures are usually considered, which are simultaneously optimized by using multi-objective optimization algorithms\cite{Gunasekara2018MOO,Liu2019ECCN,Wang2021TEVC}.
\section{Measuring Network Destruction}\label{sec:des}
It is observed that \textit{a posteriori} measures not only have intuitively clear meanings for a network function, as discussed in Subsection \ref{sub:fun}, but also have clearer descriptions about the network robustness, as introduced in Section \ref{sec:cmp}. One significant disadvantage of \textit{a posteriori} measures, however, is that their calculations are generally time-consuming. Nevertheless, in some applications, this can be (partially) solved by using analytical and computational methods, as discussed in Subsection \ref{sub:pre}.
Here, another common shortcoming of \textit{a posteriori} measures is addressed. The calculation of most \textit{a posteriori} measures is based on the entire process from attacking the first object to ending the attack at the last object. Practically, if a network has been severely destructed or malfunctioned, measuring its functionality will have no meaning. Also, complete disconnection of networks may not always be important in many applications. for example, when cascading failures are concerned, as shown in Eq. (\ref{eq:tang}), complete disconnection is unnecessary to attempt. Therefore, it is not always necessary to attack all nodes or edges for measuring the network robustness.
Clearly, determining when to stop attacking or whether a network is severely destructed or malfunctioned is application-dependent. For example, the robustness of food webs is widely measured by $R50$, which is the proportion of species (nodes) that has to be removed to cause the extinction of $50\%$ of the species in the food web\cite{Dunne2004MEPS,Curtsdotter2011BAE,Bellingeri2013TE}.
Different from the Molloy--Reed criterion\cite{Molloy1995RSA}, which states that a network will lose its giant component if $\langle k^2\rangle/\langle k\rangle >2$ is reached, here a new measure of network destruction is proposed, based on the change of the number of connected components (NCC)\cite{Uehara1999TR}. Specifically, for the \textit{a posteriori} measure $f(\cdot)$ that considers the network destruction, Eq. (\ref{eq:r}) can be rewritten as
\begin{equation}\label{eq:rt}
R_{14}=\frac{1}{T+1}\sum_{i=0}^{T}{w_i\cdot f(i)}\,,
\end{equation}
where $T$ ($T<N$) is the counted number of removed objects before the threshold of ``severe destruction'' is reached. Here, the threshold integer $T$ separates the attack process into two parts: before $T$ is reached, the network is considered as normal; after $T$ is reached, the network is deemed breakdown.
Network robustness will be measured only before this threshold is reached. In the literature about node-attacks, $T$ is set to be $0.5N$ in\cite{Fan2020NMI}, $0.05N$ in\cite{Lordan2019RESS}, or less than $0.2N$ in\cite{Lu2016PO}. All are user-defined fixed integers.
In this paper, instead of setting $T$ to be a fixed value, the network destruction is considered from the perspective of NCC, which changes non-monotonically during the attack process. In general, there is a clear turning point in the curve of NCC. For a connected network, its initial NCC is 1. The value of NCC increases as nodes and edges are being attacked. During the targeted attack process, the isolated nodes (generated due to attacks) will never be removed until there are only isolated nodes left in the residual network, since connected nodes are always targets if they exist. Therefore, the turning point of the NCC tendency curve can be used as the indicator of severe network destruction, namely, when this turning point appears, it means that there are only isolated nodes left in the residual network. This indicator of destruction is studied from a general perspective but not for a specific application. Note that removing isolated nodes is possible in any step of random attacks; hence, it is not suitable to use this turning point as the indicator of destruction.
The number of DN is non-decreasing and the numbers of LCC and CNP are non-increasing. Moreover, stagnation of DN, LCC, and CNP may occur frequently; therefore, it is difficult if not impossible to determine the network destruction using the changes of DN, LCC, or CNP.
Let $D(i)$ denote the NCC values after a total of $i$ nodes have been attacked, where $D(i)\in[1,N]$ and $i\in[0,N-1]$.
The turning point of $D(i)$ is calculated by
\begin{equation}\label{eq:t}
T=\underset{i}{\argmax}~{D(i)}\,.
\end{equation}
In attack simulations, $T$ can be determined when $D(i)<D(i-1)$ is successively detected for $\floor{pN}$ times. Then, $T=i-\floor{pN}$, where $p$ is a small decimal. It is empirically observed that the determination of $T$ is insensitive to the change of $p$. Set $p=0.05$ in the simulation, which means that when $D(i)<D(i-1)$ is successively detected for $\floor{0.05N}$ times, one has $T=i-\floor{0.05N}$.
Equation (\ref{eq:t}) can also be applied to edge-attacks. Since NCC will not decrease under edge-attacks, $T=i-\floor{pN}$ can be determined when $D(i)=D(i-1)$ is successively detected for $\floor{pN}$ times.
Figures \ref{fig:ndeg} and \ref{fig:ebet} show the attack simulation results under node MDTA and edge MBTA, respectively.
Here, a total of 10 synthetic network models are simulated, including the Erd{\"{o}}s--R{\'e}nyi (ER) random-graph\cite{Erdos1964RG}, Newman--Watts small-world (SW-NW)\cite{Newman1999PLA}, Watts--Strogatz small-world (SW-WS)\cite{Watts1998N}, random triangle (RT)\cite{Chen2019TCASII}, random hexagon (RH)\cite{Lou2022ACCESS}, extremely homogeneous (EH)\cite{Lou2020TCASI}, Barab{\'a}si--Albert (BA) scale-free\cite{Barabasi1999SCI,Barabasi2009SCI}, generic scale-free (SF)\cite{Goh2001PRL}, onion-like generic scale-free (OS)\cite{Schneider2011PNAS}, \textit{q}-snapback (QS)\cite{Lou2018TCASI,Wu2022TCNS} networks. In all simulations, the network size is set to $N=1000$ with $\langle k\rangle=10$. Four network functions are measured, that is, controllability robustness, connectivity robustness, communication robustness, and the number of connected components. The resultant values are normalized, so they are all in $[0,1]$.
\begin{figure*}[htbp]
\centering
\includegraphics[width=\linewidth]{fig/ndeg.pdf}
\caption{[color online] Network robustness in terms of the number of connected components ($n_{NCC}$), controllability ($n_D$), connectivity ($n_L$), and communication ability ($n_{CM}$), together with the threshold of destruction $T$. Here, $\delta$ represents the portion of nodes removed from the network. Node MDTA is implemented.}\label{fig:ndeg}
\end{figure*}
\begin{figure*}[htbp]
\centering
\includegraphics[width=\linewidth]{fig/ebet.pdf}
\caption{[color online][color online] Network robustness in terms of the number of connected components ($n_{NCC}$), controllability ($n_D$), connectivity ($n_L$), and communication ability ($n_{CM}$), together with the threshold of destruction $T$. Here, $\delta$ represents the portion of edges removed from the network. Edge MBTA is implemented. }\label{fig:ebet}
\end{figure*}
As can be seen from Fig. \ref{fig:ndeg}, all the vertical green lines well match the turning points of the controllability curves (blue dashed curves). In contrast, for the connectivity and communication curves (brown and black dashed curves), the vertical green lines appear consistently later than the turning points of these curves.
This means that, under node MDTA, Eq. (\ref{eq:t}) suggests a good threshold for the network destruction in terms of controllability robustness, but not for connectivity and communication robustness. In contrast, as shown in Fig. \ref{fig:ebet}, the vertical green lines match the turning points of the connectivity and communication curves better than the controllability curves. This implies that, under edge MBTA, Eq. (\ref{eq:t}) suggests a good threshold for connectivity and communication robustness, but not for the controllability robustness. This means that there is no turning point in the controllability curve under edge MBTA; the turning point only appears under node MDTA.
\begin{table*}[htbp]
\centering \caption{Comparison of robustness performance under two measure schemes: complete disconnection (CD) and threshold-based disconnection (TD). Numbers in parentheses represent the corresponding ranks of robustness. }
\begin{tabular}{|cc|llll|llll|}
\hline
\multicolumn{2}{|c|}{\multirow{2}{*}{}} & \multicolumn{4}{c|}{Node MDTA} & \multicolumn{4}{c|}{Edge MBTA} \\ \cline{3-10}
\multicolumn{2}{|c|}{} & \multicolumn{1}{c|}{\begin{tabular}[c]{@{}l@{}}Controllability\\Robustness\end{tabular}} & \multicolumn{1}{c|}{\begin{tabular}[c]{@{}l@{}}Connectivity\\Robustness\end{tabular}} & \multicolumn{1}{c|}{\begin{tabular}[c]{@{}l@{}}Communication\\Robustness\end{tabular}} & \multicolumn{1}{c|}{$T$} & \multicolumn{1}{c|}{\begin{tabular}[c]{@{}l@{}}Controllability\\Robustness\end{tabular}} & \multicolumn{1}{c|}{\begin{tabular}[c]{@{}l@{}}Connectivity\\Robustness\end{tabular}} & \multicolumn{1}{c|}{\begin{tabular}[c]{@{}l@{}}Communication\\Robustness\end{tabular}} & \multicolumn{1}{c|}{$T$} \\ \hline
\multicolumn{1}{|c|}{ER} & \begin{tabular}[c]{@{}c@{}}CD\\TD\end{tabular} & \multicolumn{1}{l|}{\begin{tabular}[c]{@{}l@{}}0.247 (2)\\0.155 (3)\end{tabular}} & \multicolumn{1}{l|}{\begin{tabular}[c]{@{}l@{}}0.476 (3)\\0.534 (6.5)\end{tabular}} & \multicolumn{1}{l|}{\begin{tabular}[c]{@{}l@{}}0.331 (3.5)\\0.372 (6.5)\end{tabular}} & 891 & \multicolumn{1}{l|}{\begin{tabular}[c]{@{}l@{}}0.390 (6)\\0.286 (4)\end{tabular}} & \multicolumn{1}{l|}{\begin{tabular}[c]{@{}l@{}}0.835 (4.5)\\0.991 (2)\end{tabular}} & \multicolumn{1}{l|}{\begin{tabular}[c]{@{}l@{}}0.806 (4.5)\\0.983 (2)\end{tabular}} & 381 \\ \hline
\multicolumn{1}{|c|}{\begin{tabular}[c]{@{}c@{}}SW-\\NW\end{tabular}} & \begin{tabular}[c]{@{}c@{}}CD\\TD\end{tabular} & \multicolumn{1}{l|}{\begin{tabular}[c]{@{}l@{}}0.248 (3)\\0.146 (2)\end{tabular}} & \multicolumn{1}{l|}{\begin{tabular}[c]{@{}l@{}}0.474 (4)\\0.538 (1.5)\end{tabular}} & \multicolumn{1}{l|}{\begin{tabular}[c]{@{}l@{}}0.331 (3.5)\\0.376 (4)\end{tabular}} & 881 & \multicolumn{1}{l|}{\begin{tabular}[c]{@{}l@{}}0.361 (3)\\0.269 (3)\end{tabular}} & \multicolumn{1}{l|}{\begin{tabular}[c]{@{}l@{}}0.834 (6.5)\\0.985 (5)\end{tabular}} & \multicolumn{1}{l|}{\begin{tabular}[c]{@{}l@{}}0.808 (3)\\0.971 (5)\end{tabular}} & 406 \\ \hline
\multicolumn{1}{|c|}{\begin{tabular}[c]{@{}c@{}}SW-\\WS\end{tabular}} & \begin{tabular}[c]{@{}c@{}}CD\\TD\end{tabular} & \multicolumn{1}{l|}{\begin{tabular}[c]{@{}l@{}}0.259 (4)\\0.159 (4)\end{tabular}} & \multicolumn{1}{l|}{\begin{tabular}[c]{@{}l@{}}0.473 (5)\\0.537 (3.5)\end{tabular}} & \multicolumn{1}{l|}{\begin{tabular}[c]{@{}l@{}}0.330 (5)\\0.375 (5)\end{tabular}} & 881 & \multicolumn{1}{l|}{\begin{tabular}[c]{@{}l@{}}0.372 (4)\\0.234 (1.5)\end{tabular}} & \multicolumn{1}{l|}{\begin{tabular}[c]{@{}l@{}}0.834 (6.5)\\0.998 (1)\end{tabular}} & \multicolumn{1}{l|}{\begin{tabular}[c]{@{}l@{}}0.806 (4.5)\\0.995 (1)\end{tabular}} & 347 \\ \hline
\multicolumn{1}{|c|}{RT} & \begin{tabular}[c]{@{}c@{}}CD\\TD\end{tabular} & \multicolumn{1}{l|}{\begin{tabular}[c]{@{}l@{}}0.302 (7)\\0.190 (6)\end{tabular}} & \multicolumn{1}{l|}{\begin{tabular}[c]{@{}l@{}}0.459 (7)\\0.533 (8)\end{tabular}} & \multicolumn{1}{l|}{\begin{tabular}[c]{@{}l@{}}0.326 (7)\\0.378 (3)\end{tabular}} & 861 & \multicolumn{1}{l|}{\begin{tabular}[c]{@{}l@{}}0.376 (5)\\0.302 (6)\end{tabular}} & \multicolumn{1}{l|}{\begin{tabular}[c]{@{}l@{}}0.830 (8)\\0.964 (6)\end{tabular}} & \multicolumn{1}{l|}{\begin{tabular}[c]{@{}l@{}}0.801 (6.5)\\0.936 (6)\end{tabular}} & 426 \\ \hline
\multicolumn{1}{|c|}{RH} & \begin{tabular}[c]{@{}c@{}}CD\\TD\end{tabular} & \multicolumn{1}{l|}{\begin{tabular}[c]{@{}l@{}}0.292 (6)\\0.178 (5)\end{tabular}} & \multicolumn{1}{l|}{\begin{tabular}[c]{@{}l@{}}0.464 (6)\\0.538 (1.5)\end{tabular}} & \multicolumn{1}{l|}{\begin{tabular}[c]{@{}l@{}}0.327 (6)\\0.379 (2)\end{tabular}} & 861 & \multicolumn{1}{l|}{\begin{tabular}[c]{@{}l@{}}0.344 (2)\\0.294 (5)\end{tabular}} & \multicolumn{1}{l|}{\begin{tabular}[c]{@{}l@{}}0.836 (2.5)\\0.918 (8)\end{tabular}} & \multicolumn{1}{l|}{\begin{tabular}[c]{@{}l@{}}0.812 (2)\\0.892 (8)\end{tabular}} & 455 \\ \hline
\multicolumn{1}{|c|}{EH} & \begin{tabular}[c]{@{}c@{}}CD\\TD\end{tabular} & \multicolumn{1}{l|}{\begin{tabular}[c]{@{}l@{}}0.227 (1)\\0.132 (1)\end{tabular}} & \multicolumn{1}{l|}{\begin{tabular}[c]{@{}l@{}}0.478 (2)\\0.536 (5)\end{tabular}} & \multicolumn{1}{l|}{\begin{tabular}[c]{@{}l@{}}0.332 (2)\\0.372 (6.5)\end{tabular}} & 891 & \multicolumn{1}{l|}{\begin{tabular}[c]{@{}l@{}}0.327 (1)\\0.234 (1.5)\end{tabular}} & \multicolumn{1}{l|}{\begin{tabular}[c]{@{}l@{}}0.836 (2.5)\\0.986 (4)\end{tabular}} & \multicolumn{1}{l|}{\begin{tabular}[c]{@{}l@{}}0.813 (1)\\0.975 (4)\end{tabular}} & 411 \\ \hline
\multicolumn{1}{|c|}{BA} & \begin{tabular}[c]{@{}c@{}}CD\\TD\end{tabular} & \multicolumn{1}{l|}{\begin{tabular}[c]{@{}l@{}}0.443 (8)\\0.288 (8)\end{tabular}} & \multicolumn{1}{l|}{\begin{tabular}[c]{@{}l@{}}0.418 (8)\\0.534 (6.5)\end{tabular}} & \multicolumn{1}{l|}{\begin{tabular}[c]{@{}l@{}}0.307 (8)\\0.393 (1)\end{tabular}} & 782 & \multicolumn{1}{l|}{\begin{tabular}[c]{@{}l@{}}0.424 (7)\\0.372 (8)\end{tabular}} & \multicolumn{1}{l|}{\begin{tabular}[c]{@{}l@{}}0.841 (1)\\0.936 (7)\end{tabular}} & \multicolumn{1}{l|}{\begin{tabular}[c]{@{}l@{}}0.800 (8)\\0.896 (7)\end{tabular}} & 446 \\ \hline
\multicolumn{1}{|c|}{SF} & \begin{tabular}[c]{@{}c@{}}CD\\TD\end{tabular} & \multicolumn{1}{l|}{\begin{tabular}[c]{@{}l@{}}0.783 (10)\\0.601 (10)\end{tabular}} & \multicolumn{1}{l|}{\begin{tabular}[c]{@{}l@{}}0.205 (10)\\0.376 (10)\end{tabular}} & \multicolumn{1}{l|}{\begin{tabular}[c]{@{}l@{}}0.149 (10)\\0.273 (10)\end{tabular}} & 545 & \multicolumn{1}{l|}{\begin{tabular}[c]{@{}l@{}}0.632 (10)\\0.610 (10)\end{tabular}} & \multicolumn{1}{l|}{\begin{tabular}[c]{@{}l@{}}0.778 (10)\\0.829 (10)\end{tabular}} & \multicolumn{1}{l|}{\begin{tabular}[c]{@{}l@{}}0.669 (10)\\0.718 (10)\end{tabular}} & 465 \\ \hline
\multicolumn{1}{|c|}{SO} & \begin{tabular}[c]{@{}c@{}}CD\\TD\end{tabular} & \multicolumn{1}{l|}{\begin{tabular}[c]{@{}l@{}}0.768 (9)\\0.589 (9)\end{tabular}} & \multicolumn{1}{l|}{\begin{tabular}[c]{@{}l@{}}0.215 (9)\\0.380 (9)\end{tabular}} & \multicolumn{1}{l|}{\begin{tabular}[c]{@{}l@{}}0.156 (9)\\0.276 (9)\end{tabular}} & 564 & \multicolumn{1}{l|}{\begin{tabular}[c]{@{}l@{}}0.615 (9)\\0.588 (9)\end{tabular}} & \multicolumn{1}{l|}{\begin{tabular}[c]{@{}l@{}}0.781 (9)\\0.839 (9)\end{tabular}} & \multicolumn{1}{l|}{\begin{tabular}[c]{@{}l@{}}0.676 (9)\\0.733 (9)\end{tabular}} & 460 \\ \hline
\multicolumn{1}{|c|}{QS} & \begin{tabular}[c]{@{}c@{}}CD\\TD\end{tabular} & \multicolumn{1}{l|}{\begin{tabular}[c]{@{}l@{}}0.286 (5)\\0.207 (7)\end{tabular}} & \multicolumn{1}{l|}{\begin{tabular}[c]{@{}l@{}}0.484 (1)\\0.537 (3.5)\end{tabular}} & \multicolumn{1}{l|}{\begin{tabular}[c]{@{}l@{}}0.333 (1)\\0.369 (8)\end{tabular}} & 901 & \multicolumn{1}{l|}{\begin{tabular}[c]{@{}l@{}}0.448 (8)\\0.361 (7)\end{tabular}} & \multicolumn{1}{l|}{\begin{tabular}[c]{@{}l@{}}0.835 (4.5)\\0.989 (3)\end{tabular}} & \multicolumn{1}{l|}{\begin{tabular}[c]{@{}l@{}}0.801 (6.5)\\0.978 (3)\end{tabular}} & 381 \\ \hline
\end{tabular}\label{tab:t}
\end{table*}
Table \ref{tab:t} shows the comparison of robustness performance under two measure schemes, namely the complete disconnection (CD) scheme as described by Eq. (\ref{eq:r}) and the threshold-based disconnection (TD) scheme as described by Eq. (\ref{eq:rt}). In the table, the numbers in parentheses represent the corresponding ranks of robustness. It is clear that, under different schemes, the robustness performance can be measured very differently. The TD robustness measures are recommended (or even necessary) to use for the following reasons: 1) the resultant ranks are unique, such that the robustness measures can be distinguished for different networks; 2) the TD measures require much fewer numbers of attacks to measure the robustness and thus requires less computational time.
\section{Comparison between \textit{A Priori} and \textit{A Posteriori} Measures} \label{sec:cmp}
Now, experimental results on \textit{a priori} and \textit{a posteriori} measures are compared under 3 different node-attack strategies, namely exhaustive attack (EXA)\cite{Lou2020TCASI}, MDTA, and MBTA.
EXA averages the robustness values of a given $N$-node network over all the $N!$ possible node-attack sequences. Note that any intentional attack (e.g., MDTA, MBTA) is a particular case of the exhaustive attacks. Since the sample size of $N!$ becomes enormous as $N$ increases, only $N=4$ is tested here, which well serves for the purpose of demonstration. Figs. \ref{fig:dir4} and \ref{fig:undir4} show the topologies of the 4-node directed and undirected networks, respectively.
\begin{figure*}[htbp]
\centering
\includegraphics[width=\linewidth]{fig/dir4.pdf}
\caption{Twelve four-node directed networks: (A) fully-connected (FUL); (B) weak fully-connected (WKF); (C) loop (LOP); (D) ring-shaped non-loop (RIN); (E) cactus (CTS); (F) star-shaped out (SSO); (G) star-shaped in (SSI); (H) star-shaped random (SSR); (I) directed chain (DCH); (L) chain-shaped (UCH); (K) disconnected (DIS); and (L) isolated (ISO). }\label{fig:dir4}
\end{figure*}
\begin{figure*}[htbp]
\centering
\includegraphics[width=\linewidth]{fig/undir4.pdf}
\caption{Six four-node undirected networks: (A) fully-connected (FUL); (B) loop (LOP); (C) star-shaped (STR); (D) cactus (CTS); (E) undirected chain (CHA); (F) isolated (ISO). }\label{fig:undir4}
\end{figure*}
Four \textit{a posteriori} measures are simulated together, namely the connectivity robustness measured by Eq. (\ref{eq:lc}), controllability robustness measured by Eq. (\ref{eq:nd}), communication robustness measured by Eq. (\ref{eq:scom}), and connectivity robustness measured by NCC, which is defined as follows:
\begin{equation}\label{eq:ncc}
R_{15}=\frac{1}{N}\sum_{i=0}^{N-1}{N_{NCC}(i)}\,,
\end{equation}
where $N_{NCC}(i)$ represents the number of connected components after a total of $i$ nodes have been attacked.
\subsection{A Priori Measures}\label{sub:pri}
\textit{A priori} measures are quantified by specific network features that can be calculated without performing attack simulations. \textit{A priori} measures require only one-time calculation and they usually have lower time and computational complexities compared to \textit{a posteriori} measures\cite{Chan2016DMKD,Freitas2022TKDE}.
Four topological \textit{a priori} measures are compared with \textit{a posteriori} measures for both directed and undirected networks, namely efficiency (EFF)\cite{Latora2001PRL}, node betweenness (NB)\cite{Freeman1977Soc}, edge betweenness (EB)\cite{Freeman1977Soc}, and clustering coefficient (CC)\cite{Watts1998N}.
Spectral measures are widely used to measure network connectivity robustness for undirected networks\cite{Chan2016DMKD}. Here, 3 adjacency matrix-based spectral measures, namely spectral radius (AS-SR)\cite{Van2011PRE}, spectral gap (AS-SG)\cite{Malliaros2012ICDM} and natural connectivity (AS-NC)\cite{Wu2010CPL} measures, and 3 Laplacian matrix-based spectral measures, namely algebraic connectivity (LS-AC)\cite{Fiedler1973CMJ}, number of spanning trees (LS-NS)\cite{Butler2008Book}, and effective resistance (LS-ER)\cite{Klein1993JMC}, are compared with \textit{a posteriori} measures for measuring undirected networks.
\subsection{Attack Simulations}
\begin{table*}[htbp]
\centering \caption{Robustness ranks of the 12 directed 4-node networks, using 4 \textit{a posteriori} measures and 4 \textit{a priori} measures. }
\begin{tabular}{|ccl|c|c|c|c|c|c|c|c|c|c|c|c|}
\hline
\multicolumn{3}{|c|}{\begin{tabular}[c]{@{}c@{}}Directed 4-Node\\ Networks\end{tabular}} & FUL & WKF & LOP & RIN & CTS & SSO & SSI & SSR & DCH & UCH & DIS & ISO \\\hline
\multicolumn{1}{|c|}{\multirow{12}{*}{\rotatebox[origin=c]{90}{\textit{A Posteriori} Measures}}} & \multicolumn{1}{c|}{\multirow{4}{*}{EXA}} & $R_1$ (Eq.(\ref{eq:lc})) & 1.5 & 1.5 & 3.5 & 3.5 & 5 & 8 & 8 & 8 & 8 & 8 & 11 & 12 \\\cline{3-15}
\multicolumn{1}{|c|}{} & \multicolumn{1}{c|}{} & $R_{15}$ (Eq.(\ref{eq:ncc})) & 1.5 & 1.5 & 3.5 & 3.5 & 5 & 8 & 8 & 8 & 8 & 8 & 11 & 12 \\\cline{3-15}
\multicolumn{1}{|c|}{} & \multicolumn{1}{c|}{} & $R_3$ (Eq.(\ref{eq:nd})) & 1.5 & 1.5 & 3 & 4.5 & 4.5 & 10.5 & 10.5 & 8 & 6 & 9 & 7 & 12 \\\cline{3-15}
\multicolumn{1}{|c|}{} & \multicolumn{1}{c|}{} & $R_7$ (Eq.(\ref{eq:scom})) & 1.5 & 1.5 & 3.5 & 3.5 & 5 & 7 & 7 & 7 & 9.5 & 9.5 & 11 & 12 \\\cline{2-15}
\multicolumn{1}{|c|}{} & \multicolumn{1}{c|}{\multirow{4}{*}{MDTA}} & $R_1$ (Eq.(\ref{eq:lc})) & 3.5 & 3.5 & 7 & 3.5 & 3.5 & 10 & 3.5 & 10 & 3.5 & 8 & 10 & 12 \\\cline{3-15}
\multicolumn{1}{|c|}{} & \multicolumn{1}{c|}{} & $R_{15}$ (Eq.(\ref{eq:ncc})) & 3.5 & 3.5 & 7 & 3.5 & 3.5 & 10 & 3.5 & 10 & 3.5 & 8 & 10 & 12 \\\cline{3-15}
\multicolumn{1}{|c|}{} & \multicolumn{1}{c|}{} & $R_3$ (Eq.(\ref{eq:nd})) & 2.5 & 2.5 & 6 & 2.5 & 5 & 11 & 7.5 & 10 & 2.5 & 9 & 7.5 & 12 \\\cline{3-15}
\multicolumn{1}{|c|}{} & \multicolumn{1}{c|}{} & $R_7$ (Eq.(\ref{eq:scom})) & 3.5 & 3.5 & 7 & 3.5 & 3.5 & 9.5 & 3.5 & 9.5 & 3.5 & 8 & 11 & 12 \\\cline{2-15}
\multicolumn{1}{|c|}{} & \multicolumn{1}{c|}{\multirow{4}{*}{MBTA}} & $R_1$ (Eq.(\ref{eq:lc})) & 2.5 & 2.5 & 6.5 & 2.5 & 6.5 & 6.5 & 2.5 & 10.5 & 9 & 6.5 & 10.5 & 12 \\\cline{3-15}
\multicolumn{1}{|c|}{} & \multicolumn{1}{c|}{} & $R_{15}$ (Eq.(\ref{eq:ncc})) & 2.5 & 2.5 & 6.5 & 2.5 & 6.5 & 6.5 & 2.5 & 10.5 & 9 & 6.5 & 10.5 & 12 \\\cline{3-15}
\multicolumn{1}{|c|}{} & \multicolumn{1}{c|}{} & $R_3$ (Eq.(\ref{eq:nd})) & 1.5 & 1.5 & 5 & 3.5 & 3.5 & 10 & 7 & 11 & 7 & 9 & 7 & 12 \\\cline{3-15}
\multicolumn{1}{|c|}{} & \multicolumn{1}{c|}{} & $R_7$ (Eq.(\ref{eq:scom})) & 2.5 & 2.5 & 6 & 2.5 & 8 & 6 & 2.5 & 10 & 9 & 6 & 11 & 12 \\\hline
\multicolumn{2}{|c|}{\multirow{4}{*}{\textit{A Priori} Measures}} & EFF & 1 & 2 & 3 & 5 & 4 & 10 & 10 & 8 & 7 & 10 & 6 & 12 \\\cline{3-15}
\multicolumn{2}{|c|}{} & NB & 2.5 & 10 & 11 & 5.5 & 9 & 2.5 & 2.5 & 5.5 & 8 & 2.5 & 7 & NA\\\cline{3-15}
\multicolumn{2}{|c|}{} & EB & 2.5 & 8 & 11 & 5 & 10 & 2.5 & 2.5 & 6 & 9 & 2.5 & 7 & NA\\\cline{3-15}
\multicolumn{2}{|c|}{} & CC & 1.5 & 1.5 & 8.5 & 8.5 & 4 & 8.5 & 8.5 & 8.5 & 8.5 & 8.5 & 3 & 8.5 \\\hline
\end{tabular}\label{tab:dir}
\end{table*}
\begin{table}[htbp]
\centering \caption{Robustness ranks of the 6 undirected 4-node networks, using 4 \textit{a posteriori} measures and 10 \textit{a priori} measures. }
\begin{tabular}{|ccl|c|c|c|c|c|c|}
\hline
\multicolumn{3}{|c|}{\begin{tabular}[c]{@{}c@{}}Undirected 4-Node\\ Networks\end{tabular}} & FUL & LOP & STR & CTS & CHA & ISO \\\hline
\multicolumn{1}{|c|}{\multirow{12}{*}{\rotatebox[origin=c]{90}{\textit{A Posteriori} Measures}}} & \multicolumn{1}{c|}{\multirow{4}{*}{\rotatebox[origin=c]{90}{EXA}}} & $R_1$ (Eq.(\ref{eq:lc})) & 1 & 2 & 4.5 & 3 & 4.5 & 6 \\\cline{3-9}
\multicolumn{1}{|c|}{} & \multicolumn{1}{c|}{} & $R_{15}$ (Eq.(\ref{eq:ncc})) & 1 & 2 & 4.5 & 3 & 4.5 & 6 \\\cline{3-9}
\multicolumn{1}{|c|}{} & \multicolumn{1}{c|}{} & $R_3$ (Eq.(\ref{eq:nd})) & 1 & 4 & 5 & 2 & 3 & 6 \\\cline{3-9}
\multicolumn{1}{|c|}{} & \multicolumn{1}{c|}{} & $R_7$ (Eq.(\ref{eq:scom})) & 1 & 2 & 4 & 3 & 5 & 6 \\\cline{2-9}
\multicolumn{1}{|c|}{} & \multicolumn{1}{c|}{\multirow{4}{*}{\rotatebox[origin=c]{90}{MDTA}}} & $R_1$ (Eq.(\ref{eq:lc})) & 1 & 2 & 5 & 3.5 & 3.5 & 6 \\\cline{3-9}
\multicolumn{1}{|c|}{} & \multicolumn{1}{c|}{} & $R_{15}$ (Eq.(\ref{eq:ncc})) & 1 & 2 & 5 & 3.5 & 3.5 & 6 \\\cline{3-9}
\multicolumn{1}{|c|}{} & \multicolumn{1}{c|}{} & $R_3$ (Eq.(\ref{eq:nd})) & 1 & 4 & 5 & 2.5 & 2.5 & 6 \\\cline{3-9}
\multicolumn{1}{|c|}{} & \multicolumn{1}{c|}{} & $R_7$ (Eq.(\ref{eq:scom})) & 1 & 2 & 5 & 3.5 & 3.5 & 6 \\\cline{2-9}
\multicolumn{1}{|c|}{} & \multicolumn{1}{c|}{\multirow{4}{*}{\rotatebox[origin=c]{90}{MBTA}}} & $R_1$ (Eq.(\ref{eq:lc})) & 1 & 2.5 & 5 & 2.5 & 4 & 6 \\\cline{3-9}
\multicolumn{1}{|c|}{} & \multicolumn{1}{c|}{} & $R_{15}$ (Eq.(\ref{eq:ncc})) & 1 & 2.5 & 5 & 2.5 & 4 & 6 \\\cline{3-9}
\multicolumn{1}{|c|}{} & \multicolumn{1}{c|}{} & $R_3$ (Eq.(\ref{eq:nd})) & 1.5 & 4 & 5 & 1.5 & 3 & 6 \\\cline{3-9}
\multicolumn{1}{|c|}{} & \multicolumn{1}{c|}{} & $R_7$ (Eq.(\ref{eq:scom})) & 1 & 2 & 5 & 3 & 4 & 6 \\\hline
\multicolumn{2}{|c|}{\multirow{10}{*}{\rotatebox[origin=c]{90}{\textit{A Priori} Measures}}} & EFF & 1 & 2.5 & 4 & 2.5 & 5 & 6 \\\cline{3-9}
\multicolumn{2}{|c|}{} & NB & 1 & 2.5 & 4 & 2.5 & 5 & NA\\\cline{3-9}
\multicolumn{2}{|c|}{} & EB & 1 & 2.5 & 4 & 2.5 & 5 & NA\\\cline{3-9}
\multicolumn{2}{|c|}{} & CC & 1 & 4.5 & 4.5 & 2 & 4.5 & 4.5 \\\cline{3-9}
\multicolumn{2}{|c|}{} & AS-SR & 1 & 3 & 4 & 2 & 5 & 6 \\\cline{3-9}
\multicolumn{2}{|c|}{} & AS-SG & 1 & 2 & 4 & 3 & 5 & 6 \\\cline{3-9}
\multicolumn{2}{|c|}{} & AS-NC & 1 & 3 & 4 & 2 & 5 & 6 \\\cline{3-9}
\multicolumn{2}{|c|}{} & LS-AC & 1 & 2 & 3.5 & 3.5 & 5 & 6 \\\cline{3-9}
\multicolumn{2}{|c|}{} & LS-NS & 1 & 2.5 & 4.5 & 2.5 & 4.5 & 6 \\\cline{3-9}
\multicolumn{2}{|c|}{} & LS-ER & 1 & 2 & 4 & 3 & 5 & NA\\\hline
\end{tabular}\label{tab:und}
\end{table}
Tables \ref{tab:dir} and \ref{tab:und} show the robustness ranks of different 4-node networks, where equal robustness share the same ranks. Intensive simulations show the following general results:
1) \textit{A posteriori} measures return different values when different attack strategies are applied, while \textit{a priori} measures return a unique value for the network under different attacks. This suggests that \textit{a priori} measures are not capable of distinguishing the network robustness under different attacks; while \textit{a posteriori} measures are capable of capturing even tiny robustness difference of a network under different attacks.
2) Most measures can reflect the basic fact that fully-connected networks possesses the best robustness, followed by connected networks, then disconnected networks, and lastly isolated networks, which possess the worst robustness. This means that both \textit{a posteriori} and \textit{a priori} measures can generally reflect the robustness of networks, regardless of the attack strategies.
3) For the 12 directed networks, no measure can distinguish all of them with different robustness values; whereas for the 6 undirected networks, the measures described by Eq. (\ref{eq:nd}) under EXA, Eq. (\ref{eq:scom}) under EXA and MBTA, as well as the 3 adjacency matrix-based spectral measures, all return 6 distinguished robustness values. This suggests that the robustness measures for directed networks remain a challenging research topic for future investigation. It is also intuitively clear that different directed networks may possess the same robustness performance, which are impossible to distinguish.
4) The two connectivity robustness measures, $R_1$ in Eq. (\ref{eq:lc}) and $R_{15}$ in Eq. (\ref{eq:ncc}), return identical robustness values (ranks) in all the cases for both directed and undirected networks, indicating that these two measures are highly-correlated. It is worth mentioning that $R_1$ has been widely used for robustness measure, while $R_{15}$ is rarely used. As can be seen from Section \ref{sec:des}, more useful information can be dug out from NCC, which may be underestimated as compared to LCC.
5) CC cannot provide distinguished robustness values for different networks in many cases; while NB, EB, and LS-ER cannot measure the isolated networks. In contrast, the \textit{a posteriori} measures are able to return different robustness values for all networks. This suggests a wider applicability of the \textit{a posteriori} measures.
The advantages of \textit{a posteriori} measures, which are summarized from the experimental comparisons shown in Tables \ref{tab:dir} and \ref{tab:und}, are as follows:
1) \textit{A posteriori} measures have intuitively clear meanings for each network function and each attack strategy, while \textit{a priori} measures always return a unique robustness value for a given network.
2) \textit{A posteriori} measures can provide more robustness information about the given networks. For example, although the 3 adjacency-spectral measures can return distinguished robustness values for different undirected networks, as shown in Table \ref{tab:und}, they disagree with the assertion that LOP is more robustness than CTS. In contrast, \textit{a posteriori} measures tell much more useful information: LOP is more robustness than CTS with respect to $R_1$, $R_{15}$, and $R_7$, but CTS is more robustness than LOP with respect to $R_3$ under EXA and MDTA; two networks are equally robust under MBTA with respect to $R_1$ and $R_{15}$.
3) It is clear that the spectral measures provide better indicators than the topological measures, but the former can only be applied to undirected networks. Moreover, NB, EB, and LS-ER cannot even be calculated for isolated networks. In contrast, \textit{a posteriori} measures can be applied to any network, and provide better indicators to network robustness, thus have a wider applicability.
\section{Prospective Research Directions}\label{sec:ftw}
Some prospective research directions are summarized from four aspects: 1) exploring better weighting methods and termination criteria for Eq. (\ref{eq:r}); 2) designing more efficient and precise analytical and computational estimation methods; 3) performing more efficient robustness optimization; and 4) exploring more real-world applications.
\subsection{Weighting the Attacks}
The currently widely-used \textit{a posteriori} robustness measures assign unique weights for each single attack in the attack sequence, assuming equal contributions of all the remaining network functionalities to the calculation of the overall robustness. As shown in Eq. (\ref{eq:r}), when $w_i=1/N$, it means that $f(i)$ and $f(j)$ ($j\neq i$) are equally important to the overall network robustness. Although the importance of attacking nodes $i$ and $j$ can be partially reflected by the different values of $f(i)$ and $f(j)$, this is clearly insufficient in many scenarios.
Practically, the removal or malfunction of some nodes will cause greater damages than other nodes. If such important nodes are attacked at the very beginning, the robustness measure should be different from the scenario that these important nodes can be protected until the later stages. The robustness measure can be delicately adjusted for different applications, by setting proper configuration of $w_i$. Meanwhile, if important nodes can be attacked at the beginning, more credits should be assigned to the attack strategy; otherwise, it means that the attack strategy is less efficient.
Moreover, the network robustness under different attack strategies can be weighted, depending on the specific situation and concern. Weighting values can be added into Eq. (\ref{eq:rpt}), where its current form implies uniform weights for different attack strategies. In practice, if the probabilities of a network suffering different attacks are different, then it is meaningful to impose different weights to them.
Possible realistic weighting methodologies include decaying weights, importance-based weights, adaptive weights, etc.
\subsection{Termination Criteria}
A realistic threshold of destruction is introduced in Section \ref{sec:des}, which gives an alternative threshold to the conventional settings, such as the Molloy--Reed criterion\cite{Molloy1995RSA} and the fixed-proportion threshold. However, there still lacks a systematic investigation on the determination of the time when a networked system is deemed breakdown thereby the attack process can be terminated. The destruction of networks can be investigated from the perspectives of topological structures, network functions, or both.
To determine proper termination criteria, analytical and theoretical studies can be carried out, for example, further development of the Molloy--Reed criterion\cite{Molloy1995RSA}, percolation theory\cite{Li2021PR}, and so on. Empirical studies such as the realistic threshold introduced in Section \ref{sec:des} can also be further investigated. Moreover, machine learning techniques may be utilized for solving this problem more effectively from a data-scientific perspective, based on both real-world networks and synthetic models. For example, given real-world data of network destruction as training data, machine learning can be used to estimate whether a given network is considered breakdown, or when it would be breakdown, under attacks.
\subsection{Robustness Estimation}\label{sub:est}
It is important to precisely and cost-efficiently approximate various robustness of large-scale networks.
The existing analytical approximations are applicable only to very limited specific issues of complex networks, e.g., controllability robustness under random or critical edge-attacks\cite{Sun2019ICSRS,Sun2021TNSM,Lou2023IJCAS}. Considering Eq. (\ref{eq:nd}) as the controllability robustness measure, attacking a single node (or edge) may either increase the number of DN by 1, or it does not change the number of DN at all. Thus, the maximum damage to the network controllability is limited. In contrast, when Eq. (\ref{eq:lc}) is used to study the connectivity robustness, the range of damages caused by each attack to LCC could vary from 0 to $N-1$, namely with all possibilities. Therefore, predicting the connectivity robustness is much more uncertain and challenging than predicting the controllability robustness, either analytically or computationally\cite{Lou2021TNSE}.
In this direction, if the pattern of malicious attacks can be well modeled using mathematics and statistics tools, then analytical approximation methods are recommended; but if there is no such a pattern (neither random not specifically targeted), then analytical methods are inapplicable while computational techniques are effective.
A comprehensive investigation of analytical approximation to robustness is needed, where some potential research topics include: 1) modeling more intrinsic attacks other than random or degree-based attacks; 2) exploring the relationships between the topological features and the robustness performance, where if direct relationships cannot be revealed then indirected relationships may be explored, for example some critical points (e.g., turning points) of the robustness curve might be estimated using topological features, so that a robustness curve can be fitted based on these critical points. As for computational approaches, not only the state-of-the-art machine learning techniques can be developed and applied, but also prior knowledge and theoretical findings can be used to further improve the prediction performances.
\subsection{Robustness Optimization}
Network robustness optimization via topological rewiring is NP-hard\cite{Kempe2003KDD}. The development of evolutionary algorithms helps in effectively resolving this difficult problem.
Robustness optimization for large-scale complex networks is higher-dimensional and computational expensive in general. In this regard, dimension reduction can be archived by applying graph embedding or using GNN\cite{Kipf2016arXiv,Hamilton2017NIPS,Hamilton2020Book}, which not only compress higher-dimensional network data into lower-dimensional representations, but also extract structural features for further processing.
As for the computational expenses in robustness evaluation, surrogate models are advantageous for improving the search efficiency and capability\cite{Wang2020TEVC,Wang2021TEVC}. Robustness estimation techniques, as introduced in Subsection \ref{sub:est}, can provide even better estimation tools than the commonly-used surrogates methods in evolutionary computation.
Since the thriving development of evolutionary computation has provided useful approaches for complex networks to evolve towards more robust structures, the key issue in this research direction is how to substantially reduce the computational cost of robustness evaluation. Although surrogates and easy-to-access indicators (such as assortativity coefficient) have been employed, the runtime of optimization remains high in many real-world applications\cite{Wang2020TEVC,Wang2021TEVC}.
Fast and precise estimation methods, both analytical and computational, can be applied to further reduce the runtime. Since real evaluations of robustness are inevitable, the ratio and arrangement between the real evaluations and the estimations should be investigated, such that the cost-efficiency can be maximized. Moreover, instead of using adjacency matrices as the chromosomes, better network representations may be explored, such that the feasibility of robustness optimization in the lower-dimensional representation domain (other than the higher-dimensional topological domain) can be explored.
\subsection{Real-world Applications}
The study of network robustness not only has been extended to many different network types, including weighted networks\cite{Bellingeri2018PA}, network of networks\cite{Gao2011PRL,Dong2013PRE,Havlin2014EPJST,Liu2015CSF}, interdependent networks\cite{Huang2011PRE,Gao2012PRE,Cui2018PA,Gao2018PA,Zhang2018PO}, and multiplex networks\cite{Min2014PRE,Chen2017TKDE}, but also has been applied to more and more real-world applications, for example land and air transport networks\cite{Zhang2013EPL,Sun2017CJA,Yang2018TITS,Lordan2014TRPE,Cai2020SSCI,Jiao2020TRPD,Li2020TRPA,Lordan2020PO,Zhu2018PA,De2012TRPC}, wireless sensor networks\cite{Qiu2017TN,Qiu2019TN,Hu2020CC}, power grids\cite{Guo2017TPS,Chen2017TCASII,Tu2018TCASII}, Internet of Things\cite{Qiu2022Book}, and so on.
Together with the development of realistic robustness measures, fast and precise robustness estimation, and cost-efficient optimization techniques, it is expected that these findings and the developed techniques can significantly extend and facilitate broader applications of real-world network problems in the near future.
\section{Conclusions}\label{sec:end}
The rapid development of complex networks research demands effective measures on various types of network robustness, especially for practical \textit{a posteriori} measures.
This survey presents a summary and overview of the comprehensive network robustness research development, focusing on the \textit{a posteriori} robustness measures. Specifically, the \textit{a posteriori} robustness measures are reviewed from four perspectives, namely the network functionality, malicious attacks, robustness estimation, and network robustness optimization. Moreover, a practical threshold of network destruction due to attacks is introduced. Network robustness is suggested to be measured only before the threshold of destruction is reached, thereafter the network is deemed breakdown and so further measuring its functionality is not meaningful anymore. Extensive simulations confirm that the proposed threshold is suitable for certain functional robustness under some specific attack strategies. Thereby, further systematic investigation is recommended for determining network destruction with respect to \textit{a posteriori} measures.
Moreover, experimental comparisons of \textit{a posteriori} and \textit{a priori} measures on directed and undirected example networks are performed and analyzed. Compared to \textit{a priori} measures, the advantages of \textit{a posteriori} measures are obvious: 1) \textit{a posteriori} measures have intuitively clear meanings for every network function and attack strategy; 2) \textit{a posteriori} measures provide more useful robustness information; and 3) \textit{a posteriori} measures have wider applicability.
Finally, some prospective research directions with respect to \textit{a posteriori} robustness measures are suggested, including weighting and termination of \textit{a posteriori} measures, analytical and computation-based robustness estimation methods, robustness optimization techniques, and some potential real-world applications.
\bibliographystyle{IEEEtran}
| {
"redpajama_set_name": "RedPajamaArXiv"
} | 8,417 |
package support;
import org.eclipse.jgit.api.errors.GitAPIException;
import org.eclipse.jgit.lib.Repository;
import org.eclipse.jgit.lib.RepositoryBuilder;
import org.eclipse.jgit.revwalk.RevCommit;
import playRepository.GitRepository;
import java.io.BufferedWriter;
import java.io.File;
import java.io.FileWriter;
import java.io.IOException;
public class Git {
public static RevCommit commit(Repository repository, String fileName,
String contents, String commitMessage) throws IOException, GitAPIException {
String wcPath = repository.getWorkTree().getAbsolutePath();
org.eclipse.jgit.api.Git git = new org.eclipse.jgit.api.Git(repository);
BufferedWriter out = new BufferedWriter(new FileWriter(wcPath + "/" + fileName));
out.write(contents);
out.flush();
git.add().addFilepattern(fileName).call();
return git.commit().setMessage(commitMessage).call();
}
public static Repository createRepository(String userName, String projectName, boolean bare) throws IOException {
String wcPath = GitRepository.getRepoPrefix() + userName + "/" + projectName;
String repoPath = wcPath + "/.git";
File repoDir = new File(repoPath);
Repository repository = new RepositoryBuilder().setGitDir(repoDir).build();
repository.create(bare);
return repository;
}
}
| {
"redpajama_set_name": "RedPajamaGithub"
} | 3,381 |
As your guests enter your ceremony you want to set a mood and ambiance for your ceremony as a first impression, just like your wedding invitation. Music sets the tone of your ceremony.
Prelude music prior to your processional songs should start at least 15 minutes prior to your ceremony start time for guests who arrive on time.
Are you having live musicians play, or are you having a disc jockey? Make sure you have a microphone for your officiant, and if you are at an outside location, would make sure you have umbrellas or a canopy to shield the Arizona sun, or heaters for warmth for your guests and musicians/disc jockey. | {
"redpajama_set_name": "RedPajamaC4"
} | 1,155 |
Federal Cases
776 F.3d 94 (2nd Cir. 2015), 13-0627-cv, Stratte-McClure v. Stanley
Docket Nº: 13-0627-cv
Citation: 776 F.3d 94
Opinion Judge: Debra Ann Livingston, Circuit Judge :
Party Name: JOEL STRATTE-MCCLURE, Plaintiff, FJARDE AP-FONDEN, Plaintiff-Appellant, PLAINTIFF STATEBOSTON RETIREMENT SYSTEM, STATE-BOSTON RETIREMENT SYSTEM, Movant-Appellant, -v- MORGAN STANLEY, a Delaware Corporation, JOHN J. MACK, ZOE CRUZ, DAVID SIDWELL, THOMAS COLM KELLEHER, THOMAS v. DAULA, Defendants-Appellees, GARY G. LYNCH, Defendant
Attorney: David Kessler (Andrew L. Zivitz, Kimberly A. Justice, Richard A. Russo, Jr., Joshua A. Materese, on the brief) Kessler Topaz Meltzer & Check, LLP, Radnor, PA, for Plaintiff-Appellant Fjarde AP-Fonden and for the Class. JAVIER BLEICHMAR (Jonathan M. Plasse, Joseph A. Fonti, Wilson M. Meeks, on the...
Judge Panel: Before: CABRANES, WESLEY, and LIVINGSTON.
Case Date: January 12, 2015
Court: United States Courts of Appeals, Court of Appeals for the Second Circuit
776 F.3d 94 (2nd Cir. 2015)
JOEL STRATTE-MCCLURE, Plaintiff, FJARDE AP-FONDEN, Plaintiff-Appellant, PLAINTIFF STATEBOSTON RETIREMENT SYSTEM, STATE-BOSTON RETIREMENT SYSTEM, Movant-Appellant, -v- MORGAN STANLEY, a Delaware Corporation, JOHN J. MACK, ZOE CRUZ, DAVID SIDWELL, THOMAS COLM KELLEHER, THOMAS
DAULA, Defendants-Appellees, GARY G. LYNCH, Defendant
No. 13-0627-cv
United States Court of Appeals, Second Circuit
Argued: December 10, 2013.
Appeal from April 4, 2011 and January 18, 2013 orders of the United States District Court for the Southern District of New York (Batts, J.), granting the Defendants' motions to dismiss. For the reasons stated here and in a summary order issued simultaneously with this opinion, we AFFIRM the order granting the Defendants' motion to dismiss.
David Kessler (Andrew L. Zivitz, Kimberly A. Justice, Richard A. Russo, Jr., Joshua A. Materese, on the brief) Kessler Topaz Meltzer & Check, LLP, Radnor, PA, for Plaintiff-Appellant Fjarde AP-Fonden and for the Class.
JAVIER BLEICHMAR (Jonathan M. Plasse, Joseph A. Fonti, Wilson M. Meeks, on the brief), Labaton Sucharow LLP, New York, NY, for Movant-Appellant State-Boston Retirement System and for the Class.
ROBERT F. WISE, JR. (Charles S. Duggan, Andrew Ditchfield, on the brief), Davis Polk & Wardwell LLP, New York, NY, for Defendants-Appellees.
Before: CABRANES, WESLEY, and LIVINGSTON.
Debra Ann Livingston, Circuit Judge :
Lead Plaintiffs State-Boston Retirement System and Fjarde AP-Fonden brought this putative securities fraud class action on behalf of themselves and other similarly situated investors (" Plaintiffs" ) pursuant to Sections 10(b) and 20(a), 15 U.S.C. § § 78j(b) and 78t(a), of the Securities Exchange Act of 1934. They allege that Morgan Stanley and six of its officers and former officers -- John J. Mack, Zoe Cruz, David Sidwell, Thomas Colm Kelleher, and Thomas Daula (collectively, " Morgan Stanley" or " Defendants" ) -- made material misstatements and omissions between June 20, 2007 and November 19, 2007 (the " class period" ) in an effort to conceal Morgan Stanley's exposure to and losses from the subprime mortgage market. As a result, Plaintiffs claim, they suffered substantial financial loss when Morgan Stanley's stock prices dropped following public disclosure of the truth about Morgan Stanley's positions and losses.
The United States District Court for the Southern District of New York (Batts, J.) dismissed all claims on the pleadings for failure to state a claim, and we affirm. For the reasons stated in this opinion, we conclude that the district court properly dismissed Plaintiffs' claim that Defendants' omission of information purportedly required to be disclosed under Item 303 of Regulation S-K, 17 C.F.R. § 229.303(a)(3)(ii) (" Item 303" ), violated Section 10(b). We also affirm its order dismissing Plaintiffs' other claims in a summary order issued simultaneously with this decision.
This case arises out of a massive proprietary trade executed by Morgan Stanley's Proprietary Trading Group in December 2006. The trade consisted of two components: a $2 billion short position (" Short Position" ) and a $13.5 billion long position (" Long Position" ). In the Short Position, Morgan Stanley purchased credit default swaps (" CDSs" ) on collateralized debt obligations (" CDOs" ) backed by " mezzanine tranches" of subprime residential mortgage-backed securities (" RMBSs" ).2 These CDSs operated like insurance policies -- Morgan Stanley paid annual premiums for the assurance that, if the housing market worsened and the mezzanine RMBS tranches backing its CDOs defaulted or declined in value, it would receive payments. In the Long Position, Morgan Stanley sold CDSs. These CDSs, like those it bought for the Short Position, referenced CDOs backed by mezzanine tranches of subprime RMBSs. But the CDOs referenced by the Long Position were super-senior tranches of CDOs that were higher-rated and lower-risk than the CDOs referenced by the Short Position. Through the Long Position, Morgan Stanley therefore received premium payments for the guarantee that it would pay the CDS purchasers in the event that these lower-risk CDO tranches defaulted or declined in value. Morgan Stanley could use the income from those premiums to finance the Short Position, but would have to make payouts if the CDO tranches referenced by the Long Position suffered defaults -- up to a maximum of $13.5 billion in the event of a 100 percent default in these CDOs. In essence, the company was betting that defaults in the subprime mortgage markets would be significant enough to impair the value of the higher-risk CDO tranches referenced by the Short Position, but not significant enough to impair the value of the lower-risk CDO tranches referenced by the Long Position.
According to the Plaintiffs, " [b]y mid-2006, the biggest housing bubble in U.S. history had popped." J.A. 465. Subprime mortgages issued in 2005 and 2006, like those backing Morgan Stanley's proprietary trade, rapidly began to suffer from delinquencies and defaults. " On February 12, 2007, Morgan [Stanley] economist Richard Berner acknowledged that these '[s]oaring defaults signal that the long-awaited meltdown in subprime mortgage lending is now underway[.]'" J.A. 469. Although Morgan Stanley's Proprietary Trading Group had correctly predicted the direction that the subprime housing market would turn, it apparently underestimated the magnitude of the collapse. The value of Morgan Stanley's swap positions declined substantially over the course of 2007, and Morgan Stanley ultimately lost billions of dollars on the proprietary trade.
Plaintiffs allege that Defendants made numerous material misstatements and omissions from June 20, 2007 through November 19, 2007 to conceal Morgan Stanley's exposure to and losses from this subprime proprietary trade. The second amended complaint identifies two categories of misrepresentations and omissions: (1) misrepresentations and omissions regarding Morgan Stanley's exposure to credit risk related to the U.S. subprime mortgage market arising from its Long Position (the " exposure claim" ), and (2) misrepresentations regarding Morgan Stanley's losses arising from the Long Position (the " valuation claim" ). Plaintiffs allege that these misstatements and omissions fraudulently inflated Morgan Stanley's stock price during the class period and caused them to suffer financially when the market learned the truth about Morgan Stanley's exposure and losses.
A. Exposure Claim
The second amended complaint alleges that Defendants materially misrepresented Morgan Stanley's exposure to the subprime mortgage market. Plaintiffs rely on four statements from Morgan Stanley officers, and one alleged omission. First, on a June 20, 2007 call with market analysts about Morgan Stanley's second quarter earnings, Defendant Sidwell stated that " concerns early in the quarter about whether issues in the sub-prime market were going to spread dissipated." J.A. 498. Second, on that same call, Sidwell responded to a request to characterize Morgan Stanley's position in the mortgage market and to explain the decline in the company's fixed income revenues by stating that Morgan Stanley " really did benefit" from conditions in the subprime market in the first quarter of 2007, and " certainly did not lose money in this business" during the second quarter. J.A. 498, 499. Third, during another earnings call with market analysts on September 19, 2007, Defendant Kelleher stated that Morgan Stanley " remain[ed] exposed to risk exposures through a number of instruments [including] CDOs," without describing the extent of that exposure. J.A. 506-07. And fourth, Kelleher stated in an October 24, 2007 interview with CIBC World Markets analyst Meredith Whitney that he " [did] not see further write-downs to [Morgan Stanley's] carrying values over the near term." J.A. 516. Plaintiffs claim that each of these statements was materially false or misleading.
As pertinent here, Plaintiffs also allege that Defendants made material omissions in their 10-Q filings by failing to disclose the existence of the Long Position, that Morgan Stanley had sustained losses on that position in the second and third quarters of 2007, and that the company was likely to incur additional significant losses on the position in the future. They argue that Item 303 of Regulation S-K and related guidance requires companies to disclose on their 10-Q filings any " known trends, or uncertainties that have had, or might reasonably be expected to have, a[n] . . . unfavorable material effect" on the company's " revenue, operating income or net income." J.A. 465. Plaintiffs claim that " [b]y July 4[, 2007,] at the latest, Defendants knew that the Long Position was reasonably expected to have an unfavorable material effect on revenue." J.A. 482. It is not disputed that Morgan Stanley did not make this Item 303 disclosure on its 10-Q filings in 2007.
B. Valuation Claim
In a separate claim, the second amended complaint alleges that Morgan Stanley overstated its earnings in the third quarter of 2007 because it did not sufficiently write down the value of its Long Position. According to Plaintiffs, the Long Position's
value was " inherently linked" to an index of RMBSs known as the ABX.BBB.06-1 Index (the " ABX Index" ). Thus, when the ABX Index declined by 32.8 percent in the third quarter of 2007, Morgan Stanley should have marked down the value of the Long Position by that same percentage and disclosed the loss in its quarterly statement. Instead of taking that $4.4 billion markdown, however, Morgan Stanley recognized only a $1.9...
milliken v meyer case brief
Hurtado v California
Macpherson v Buick Motor Co
McGrain v Daugherty
Administrative Decisions | {
"redpajama_set_name": "RedPajamaCommonCrawl"
} | 4,692 |
{"url":"https:\/\/www.zbmath.org\/?q=an%3A1143.26002","text":"# zbMATH \u2014 the first resource for mathematics\n\nAn extension problem related to the fractional Laplacian. (English) Zbl\u00a01143.26002\nThe authors relate the fractional Laplacian of a function $$f: \\mathbb{R}^{n}\\rightarrow \\mathbb{R}$$ to solutions $$u:\\mathbb{R}^{n}\\times [0,\\infty )\\rightarrow \\mathbb{R}$$ of the extension problem $\\left\\{ \\begin{matrix} u(x,0)=f(x) \\\\ \\Delta _{x}u+\\frac{a}{y}u_{y}+u_{yy}=0. \\end{matrix} \\right.$ It is shown that $\\lim_{y\\rightarrow 0}y^{a}u_{y}(x,y)=u_{z}(x,0)=-(-\\Delta )^{s}f(x)$ where $$s=\\frac{1-a}{2}$$ and $$z=\\left( \\frac{y}{1-a}\\right) ^{1-a}.$$ This work extends the well-known fact that the operator $$(-\\Delta )^{1\/2}$$ can be obtained from the harmonic extension problem to the upper half space as the operator that maps the Dirichlet boundary condition to the Neumann condition. Therefore, the present work generalizes this characterization to general fractional powers of the Laplacian. This is also done for other integro-differential operators and some properties of these integro-differential equations are derived.\n\n##### MSC:\n 26A33 Fractional derivatives and integrals 35J70 Degenerate elliptic equations\n##### Keywords:\nDegenerate elliptic equations; fractional Laplacian\nFull Text:\n##### References:\n [1] Almgren F. J., Almgren\u2019s Big Regularity Paper (2000) \u00b7 Zbl\u00a00985.49001 [2] Athanasopoulos I., Amer. J. Math. [3] Bogdan K., Studia Math. 123 pp 43\u2013 (1997) [4] DOI: 10.1353\/ajm.1997.0010 \u00b7 Zbl\u00a00878.35039 [5] DOI: 10.1002\/(SICI)1097-0312(199604)49:4<365::AID-CPA3>3.0.CO;2-A \u00b7 Zbl\u00a00854.35032 [6] Fabes E., Ann. Inst. Fourier (Grenoble) 32 pp 151\u2013 (1982) \u00b7 Zbl\u00a00488.35034 [7] DOI: 10.1080\/03605308208820218 \u00b7 Zbl\u00a00498.35042 [8] Fabes , E. B. , Kenig , C. E. , Jerison , D. ( 1983 ). Boundary behavior of solutions to degenerate elliptic equations . In Conference on Harmonic Analysis in Honor of Antoni Zygmund, Vols. I, II. (Chicago, Ill., 1981), Wadsworth Math. Ser. Belmont , CA : Wadsworth , pp. 577 \u2013 589 . [9] Landkof N. S., Foundations of Modern Potential Theory. (1972) \u00b7 Zbl\u00a00253.31001 [10] DOI: 10.1080\/00036818308839425 \u00b7 Zbl\u00a00513.35013\nThis reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.","date":"2021-10-22 11:59:52","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\": 0.7576908469200134, \"perplexity\": 1942.0903453541373}, \"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\/1634323585507.26\/warc\/CC-MAIN-20211022114748-20211022144748-00636.warc.gz\"}"} | null | null |
The Stillwater Dam is a hydroelectric dam on the Stillwater River in Old Town north of downtown Orono in Penobscot County, Maine. As a part of the Penobscot River restoration and the removal of the Great Works and Veazie dams, the Stillwater Dam and the Orono Dam will be upgraded to maintain previous levels of power generation.
References
Dams in Maine
Buildings and structures in Old Town, Maine
Hydroelectric power plants in Maine
Dams completed in 1937 | {
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T.I. is the king of a lot of things. The Southern rapper has established himself as a very reputable artist by dropping a slew of platinum albums. He has showcased his acting prowess in several films, and also a humorous family dad on his VH1 show T.I. and Tiny. Besides all of those things, T.I does reign supreme in is his hat game.
For whatever reason, his uncanny ability to tilt his hat is somewhat of a trick that everyone has tried to replicate. Well, someone dared to come after The King's throne and usurp him from his title recently.
Let's just say, T.I. crushed him with relative ease.
Twitter user @KriegLaFlare tweeted T.I. the following: "I challenge you to an anti gravity hat off Clifford. You have 24 hours to respond or I am the new King @Tip." After garnering thousands of retweets, T.I. got word and of course decided to clap back.
24 hours later, T.I. responded with a photo of him defying the odds once again with a deflating response to silence @KriegLaFlare. "How dare he @KriegLaFlare!Like my son startin to golf today&callin out Tiger.Be gone FROM my presence, mere peasant," he wrote as his caption on Twitter.
Here's a lesson to all you kiddies out there: never try to come at your elders unless your ready to face their wrath. SHARE this story with your friends. | {
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Елка Рачева Добрева е българска езиковедка, семиотик и културолог. Главен асистент (1984), доцент (1989) и професор (2010) в ШУ "Епископ Константин Преславски".
Биография
Родена е на 15 декември 1952 в гр. Шумен. Завършва Класическа филология в Софийския университет (1975), след което е редовна аспирантка по Общо и теоретично езикознание. Доктор по филология с дисертация на тема "Към въпроса за комуникативната функция на глаголните граматически категории" (1982). От 1992 г. работи в Катедрата по български език към Шуменския университет като преподавател по Общо езикознание. Специализирала е в областта на текстолингвистиката и теорията на комуникацията в Института по общо езикознание към Виенския университет (1986 г.) и в Университета в Увърхемптън, Англия (1997). Доктор на филологическите науки с дисертация на тема "Нетолерантност и нулева толерантност в съвременния български печат (Критически лингвосемиотичен анализ)" (2008).
Гост лектор в Бакинския славянски университет в Баку, Азербайджан (2002) и университета "Лудвиг Максимилиян" в Мюнхен, Германия (2005).
Зам.-ректор на Шуменския университет. Декан на Факултета по хуманитарни науки. Зам.-декан на факултета по хуманитарни науки. Ръководител на катедра Журналистика и масови комуникации.
Член на Съюза на учените в България.
Носителка на наградата на Шумен за наука (1988).
Библиография
Монографии
1988 – "Теорията за знака в лингвистиката и литературната наука". София: Наука и изкуство, 1988, 228 с. (в съавт. с Добрин Добрев).
1990 – "Проблеми на изграждането на текста". София: Народна просвета, 1990, 120 с. (в съавт. с Ивелина Савова).
1994 – "Проблеми на изграждането на текста". Шумен, 1994, 202 с. – второ изд. (в съавт. с Ивелина Савова).
1992 – "Справочник на семиотичните термини". Шумен: Глаукс, 1992, 156 с. (в съавт. с Добрин Добрев).
2000 – "Текстолингвистика. Уводен курс". Шумен: Унив. издат. "Епископ Константин Преславски", 2000, 257 с. (в съавт. с Ивелина Савова).
2002 – "Писмените ученически текстове". София: Кръгозор, 2002, 243 с. (в съавт. с Маргарита Георгиева).
2009 – "Толерантност, нетолерантност и нулева толерантност в съвременния български печат (Критически лингвосемиотичен анализ)". Фабер: Велико Търново, 2009, 390 с.
2009 – "Текст & дискурс. Терминологичен справочник". Фабер: Велико Търново, 2009, 275 с. (в съавт. с Ивелина Савова)
2011 – "Аспекти на масмедийната "реалност". Фабер: Велико Търново, 2011, 274 с.
Учебници и учебни помагала
1997 – "Български език". Ополе, 1997 (в съавт. с колектив под ред. на Стефана Димитрова).
2001 – "Тестове по български език и литература. Помагало за кандидатстуденти". Шумен: Унив. издат. "Епископ Константин Преславски", 2001 (в съавт. с Ивелина Савова, Добрин Добрев и Младен Енчев).
2003 – "Български език и литература. Правила, понятия, тестове". Кръгозор: София, 2003 (в съавт. с Ивелина Савова и Добрин Добрев), 255 с.
2004 – "Увод в общото езикознание". Шумен: Унив. издат. "Епископ Константин Преславски", 2004, 243 с., второ изд. Фабер: Велико Търново, 2009, 243 стр.
2007 – "Трансформиращ преразказ". София: Кръгозор, 2007 (в съавт. с Маргарита Георгиева), 100 с.
Източници
Библиография на Елка Добрева до 2002 г., LiterNet, 20 октомври 2002 г.
Книги на Елка Добрева в Националния регистър на издаваните книги в България
Външни препратки
Биография на проф. Елка Добрева на сайта на Шуменския университет
Списък с публикации на проф. Елка Добрева на сайта на Шуменския университет
От и за Елка Добрева в Своден каталог НАБИС - национален каталог на академичните библиотеки в България
Публикации на проф. Елка Добрева на сайта Литернет
Страница на проф. Елка Добрева на сайта на издателство Фабер
Страница на проф. Елка Добрева на сайта на издателство Кръгозор
Български езиковеди
Възпитаници на Софийския университет
Възпитаници на Виенския университет
Преподаватели в Шуменския университет
Родени в Шумен | {
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Nicola Marie Bolger (* 3. März 1993 in Westmead, New South Wales, Australien) ist eine australische ehemalige Fußballnationalspielerin, die im Mittelfeld spielte. Im Januar 2013 gewann sie mit Sydney FC das Grand Final um die australische Meisterschaft. 2012 wurde sie erstmals in der australischen Fußballnationalmannschaft der Frauen eingesetzt.
Werdegang
Vereine
Die in einem Vorort von Sydney geborene Bolger begann 2008 bei Sydney FC. Am Ende der Saison 2008/09 stand ihre Mannschaft auf dem 4. Platz der Tabelle und war damit für die Playoffs qualifiziert. Im Halbfinale am 11. Januar 2009 trafen sie auf Queensland Roar. Da es nach 120 Minuten 1:1 stand, musste das Elfmeterschießen entscheiden. Bolger scheiterte als einzige Schützin und ihre Mannschaft verpasste damit das Grand Final. Besser lief es in der folgenden Saison. Am Ende der Punktspielrunde stand Sydney auf Platz 1 und erreichte dann auch das Grand Final gegen Brisbane Roar. Beim 3:2-Sieg wurde sie aber nicht eingesetzt.
2011 wurde erneut das Finale erreicht und wieder war Brisbane der Gegner. Diesmal wurde sie zwar eingesetzt, aber ihre Mannschaft verlor mit 1:2. Danach wechselte sie zu Newcastle United Jets. Dort hatte sie zwar mehr Einsätze, aber als Tabellenfünfter wurden die Playoffs verpasst. Bolger kehrte danach nach Sydney zurück und erreichte 2013 erneut das Grand Final, diesmal gegen Melbourne Victory. Beim 3:1-Sieg erzielte sie das erste Tor. Ein Jahr später konnte sich Melbourne im Halbfinale revanchieren und Sydney durch ein 3:2 den erneuten Finaleinzug verwehren. Und auch im folgenden Jahr war im Halbfinale Endstation, diesmal gegen Perth Glory. 2016 wurde wieder das Finale erreicht, aber mit 1:4 gegen Melbourne City verloren. 2017 war Perth Glory im Halbfinale wieder stärker. Sie wechselte darauf zum Halbfinalgegner. Als Sechste verpasste Perth aber die Finalrunde. Nach 100 Liga-Spielen beendete sie ihre Karriere.
Nationalmannschaften
Am 27. Juni 2012 wurde sie gegen Neuseeland erstmals in der Nationalmannschaft eingesetzt.
Für die Fußball-Asienmeisterschaft der Frauen 2014, bei der Australien den Titel nicht verteidigen konnte, wurde sie nominiert, aber nur im zweiten Gruppenspiel gegen Jordanien eingesetzt.
Im März 2015 nahm sie mit Australien am Zypern-Cup 2015 teil, wo sie beim 0:3 gegen England eingewechselt wurde und beim 3:0 gegen Finnland in der Startelf stand. Am 12. Mai 2015 wurde sie für den australischen WM-Kader 2015 nominiert. Bei der WM wurde sie aber nicht eingesetzt.
Weblinks
Profil auf der Sydney FC Webseite
Profil auf Football Federation Australia
Einzelnachweise
Fußballnationalspieler (Australien)
Fußballspieler (Newcastle United Jets)
Fußballspieler (Sydney FC)
Teilnehmer an einer Fußball-Weltmeisterschaft (Australien)
Australischer Meister (Fußball)
Australier
Geboren 1993
Frau | {
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Knemodynerus tectus är en stekelart som först beskrevs av Fabricius 1781. Knemodynerus tectus ingår i släktet Knemodynerus och familjen Eumenidae. Inga underarter finns listade i Catalogue of Life.
Källor
Steklar
tectus | {
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Q: BizTalk and SalesForce integration: the query or retrieve function returns unknown type I'm integration a legacy on premise BizTalk Server with SFDC (Salesforce CRM on Demand). I'm using BizTalk 2009 with WCF custom ports. I have imported the enterprise WSDL and successfully used it to create an accounts in SFDC. The problem occurs when I try to use the retrieve (or query) function to get user details, everything is working nicely except when I try to "use" the response message.
Request:
<ns0:retrieve xmlns:ns1="urn:sobject.enterprise.soap.sforce.com" xmlns:ns0="urn:enterprise.soap.sforce.com">
<ns0:fieldList>Name, Email</ns0:fieldList>
<ns0:sObjectType>User</ns0:sObjectType>
<ns0:ids>005900000023xmcAAA</ns0:ids>
</ns0:retrieve>
Receive pipeline is the standard XMLReceive.
Response message:
<retrieveResponse xmlns="urn:enterprise.soap.sforce.com">
<result xsi:type="sf:User" xmlns:xsi="http://www.w3.org/2001/XMLSchema-instance">
<sf:Id xmlns:sf="urn:sobject.enterprise.soap.sforce.com">005900000023xmcAAA</sf:Id>
<sf:Email xmlns:sf="urn:sobject.enterprise.soap.sforce.com">jredwood@charteredaccountants.com.au</sf:Email>
<sf:Name xmlns:sf="urn:sobject.enterprise.soap.sforce.com">Julian Redwood</sf:Name>
</result>
</retrieveResponse>
Error Details:"Unable to read the stream produced by the pipeline.
Details: The value 'sf:User' is invalid according to its schema type 'http://www.w3.org/2001/XMLSchema:QName' - 'sf' is an undeclared namespace. ".
A: Yes, that response is rather messed up.
It declares the a default name space at the root xmlns="urn:enterprise.soap.sforce.com"
It doesn't declare the sf namespace prefix at the root e.g. (xmlns:sf="urn:sobject.enterprise.soap.sforce.com").
And then for user has it as xsi:type="sf:User" where it then doesn't have the sf prefix defined for the result node.
Either it needs to declare it either in the root or at the result node level.
Option 1) If you the ESB Toolkit you could try and use the ESB Add Namespace pipeline component and add the NamspacePrefix = sf and NamspaceBase = urn:enterprise.soap.sforce.com
Option 2) Raise it as in issue with Salesforce as that is not valid.
Or both.
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Where to Preorder:
B & N
"We're messy. We're raw and rough and passionate, and we say the wrong things and stumble over each other, but there is no one I'd rather stumble through the rest of my life with than you. I don't want ordinary. I want mind-blowing."
By day, Allie Greene stays busy with her family diner, and keeping tabs on her teenage daughter. What's really exhausting Allie, however, are the nights. Not that she minds Bash Anderson unbuttoning her naughty desires—if only in her dreams. But what was he doing there at all? He's her best friend, and a father figure to her girl. Talk about awkward. Talk about OMG-heat-and-fireworks that are flipping fifteen years of normal upside down. And now, when Allie needs him as a friend more than ever, logic doesn't stand a chance against his lips and irresistible deep-blue eyes . . .
Sure, Bash has fantasized about Allie, but there's no way he'd act on it. She and her daughter are the closest thing to family he's ever known. With the exception of one drunken moment fifteen years ago, he and Allie have stayed on this side of the line—until that impulsive kiss of hers knocked him on his butt. That's just one hurdle. Not only does Allie need Bash's help to save her diner, but his apiary is in trouble, too. To stir the pot further, they've been roped into vying for the town's Honey King and Queen contest—a sweet event that's making them closer than ever. Something's bound to come undone. Bash just hopes it's not the friendship he's worked so hard to hold on to.
Before and Ever Since review by Cocktails & Books
"Sharla Lovelace is a gifted storyteller. She creates characters that are very down-to-earth, flawed and understand that there isn't a fairy tale happily ever after waiting for them at the end. It's raw, it's gritty and it's not tied up with a pretty bow, which makes it feel so very real. I look forward to reading more from Sharla Lovelace in the future." ~Cocktails & Books
http://sharlalovelace.com/testimonials/before-and-ever-since-review-by-cocktails-books/
Loving The Chase review by Publishers Weekly
"...the excitement of Texas's dramatic weather and Maddi and Zach's dramatic history make [Loving The Chase] an exhilarating and satisfying read." ~Publishers Weekly
http://sharlalovelace.com/testimonials/loving-the-chase-review-by-publishers-weekly/ | {
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# **ANDY STEVES' EUROPE**
**CITY-HOPPING ON A BUDGET**
## **FOREWORD**
When I wrote my first guidebook 35 years ago, it was a practical manual for a generation of independent travelers taking overnight trains, sleeping in hostels, carrying traveler's checks, and picking up mail at American Express offices.
Times have changed, and today my son Andy has written a new kind of guidebook for Millennial travelers with an entirely new range of options to choose from. They jet between hot spots on the cheap (something inconceivable when I was Andy's age). They find accommodations through social networks like Airbnb and couch surfing. And they leverage digital and online tools that would have been a space age dream in my student vagabond days.
Yet while travel styles and tools may change from generation to generation, the fundamentals of travel remain the same. And today, as much as ever, if you equip yourself with good information and expect yourself to travel smart, you can get the most out of two precious resources: your time and your money.
Another key to good travel is the importance of meeting people. I remember when Andy went on his first solo European trip just out of high school, I laid out all the museums and castles and galleries he should be sure to visit. When he came home he triumphantly declared that he had skipped most of those conventional sights and, instead, made friends in each country.
Andy's right: it's connecting with people that carbonates your travel experience, and he's an expert at that. When you meet people from different cultures face to face, you broaden your perspective. And when your new friends take you out of your comfort zone, the world becomes your playground.
In this guidebook, Andy shares lessons gained from a decade of living on the road while organizing his Weekend Student Adventures tours for tens of thousands of young travelers. In the era of globalization, a complete education includes some foreign travel experience. Andy's advice on sightseeing and practicalities will help any young traveler city-hopping on a budget have an economical and unforgettable adventure in Europe.
A closing note to Andy: This is not your father's guidebook. It fills the needs of young travelers in a way that I never could and it does so brilliantly.
Bravo!
**Rick Steves**
## **CONTENTS**
**Index**
**Introduction**
**London**
**Paris**
**Amsterdam**
**Rome**
**Florence**
**Venice**
**Madrid**
**Barcelona**
**Berlin**
**Prague**
**Budapest**
**Dublin**
**Edinburgh**
**Appendix**
**Photo Credits**
## **INTRODUCTION**
**Modern travelers have all the technology in the world at their fingertips. Today, thanks to budget airlines and online resources, backpackers have become precision travelers, popping in and out of major destinations all over Europe at the spur of the moment. We're visiting cities rather than countries, and packing incredible adventures into increasingly shorter time slots. This book will help you make the most of your time and money by zeroing in on each destination's must-sees.**
### **TRAVEL PHILOSOPHY**
Good travel is really all about two things: bringing the right mind-set to embrace the unexpected, and sorting out the practical logistics. The second part of that equation is what the rest of this book is all about—the logistics of getting from A to B in the most time-efficient and cost-effective way, and having a blast while you're there. I'll be with you every step of the way!
But your mind-set is just as important. We live in a complex world, with many different points of view swirling around us in more media formats and screens than ever before. Technology is accelerating progress around the world, and things are changing at an ever-faster rate. It's never been more important to understand and connect with people from other cultures. The lower to the ground you travel, the more you experience each fascinating destination and achieve life-changing experiences.
#### **Connect With the Culture**
If you've never traveled or studied outside the country, being somewhere completely different can be nerve-wracking. People, food, language...even the laws will be different than what you're used to. The resulting anxiety can lead you to trap yourself inside the "American Bubble."
The American Bubble is that safe group of American friends you meet abroad. They listen to the same music as you do, eat the same food, take the same classes, and are therefore quite easy to relate to. Don't get me wrong; I enjoy meeting Americans while traveling. However, if at the end of your time abroad, the only new friends you made are English-speakers, you may need to reevaluate why you came to Europe in the first place.
Don't let the American Bubble prevent you from experiencing Europe to the fullest. Break free and immerse yourself in something new. Dive into a local festival, sample strange food, flirt with that cute French guy (or girl!). You'll make memories that will ultimately make you a better, more worldly human being.
#### **Say Yes**
In my experience, simply saying "yes" to things I haven't tried before is the most direct route to creating unforgettable memories. Of course, some lines should never be crossed, but in general, being a yes man or woman for the day (or night) can be really fun. You have a chance to catch a _fútbol_ match in Madrid? Say yes! Never tried smoked herring? Taste it in Amsterdam. Do as the Berliners do and stay out till sunrise. Expose yourself to new experiences. You won't regret it.
#### **Ask Questions**
Take every opportunity to learn something new about each place you visit. Ask locals where _they_ go on a Saturday night, what their views on European and American politics are, and how they celebrate their holidays. They'll appreciate your interest, and they are sure to have questions for you as well.
When it comes to making conversation, don't ask where someone is from unless you've got a follow-up question ready, or conversation will die shortly afterward. Instead, have a lineup of compelling questions to ask those sitting next to you on the train or at the bar to get people to open up.
#### **Some Suggestions:**
• Where are you headed, and what is there to do there?
• Where else should I travel?
• I really like X. Where should I go to experience that?
#### **Maintain Perspective**
As this book was nearing completion in late 2015; the headlines were dominated by terrorist shootings in Paris. Terrorism is a fact of life in Europe and many other parts of the world, including the U.S., and there's always a chance that another attack will occur in one of the cities covered in this book. If that happens, take a deep breath and try to keep the risk and tragedy of terrorism in perspective. The chance of your being affected directly are astronomically minuscule—you can look it up—and as every European understands, the best way to counter terrorism is by refusing to be terrorized.
### **ABOUT THIS BOOK**
#### **Who is Andy Steves?**
I grew up traveling with my father, travel guru Rick Steves. From infancy through high school, I spent my summers learning the ins and outs of budget travel.
My approach to travel fuses my father's love of culture with all the modern tools available today thanks to technology. Through college, I worked as a tour guide for Rick Steves' Europe Through the Back Door. While studying abroad in Rome, I started organizing trips for my friends each weekend, leading groups of 5-10 fellow students. By the end of the semester, these groups numbered over 30.
Upon returning to the United States, I began formulating ideas for tips and trips designed specfically for budget travelers and study abroad students. That led to **Weekend Student Adventures** (wsaeurope.com), which is now the leading student tour company in Europe. Weekend Student Adventures offers an affordable, local experience to budget travelers in Amsterdam, Paris, Barcelona, Berlin, Budapest, Prague, Rome, Krakow, Edinburgh, Dublin and more!
#### **How to Use this Book**
I've adapted the tried and tested itineraries I developed for Weekend Student Adventures for the purposes of this book. Each time I visit a city, I try to do it all: visit the most blogged-about restaurants, hit the main museums, pop into the city's churches, relax in the best parks, find the best panorama, grab a beer at the hottest pubs and clubs. Once I tackle the big names, I spend the rest of my time getting off the beaten path and writing about it to share with you. Each chapter gives you the nuts and bolts you need to plan a quick trip to a given city, from tips on calibrating your budget to transportation information to a detailed three-day itinerary that lets you experience that city to its fullest.
### **BEFORE YOU GO**
Think of your trip as one big piece of artwork: Start with the outline first, then get down to the details. Make the outline of your trip by nailing down the following constraints:
• Where you want to go
• How much time you have—and how much of that you're willing to spend in transit
• Budget—daily target and total
• What you want to do (what's your priority—sightseeing, music, culture, architecture, or nightlife?)
#### **Decide Where to Go**
If you've got three weeks or less and want to hit a wide range of destinations, like Dublin, London, Edinburgh, Paris, Amsterdam, Berlin, Prague, and Budapest, here's how I would break it down. To split eight cities across 21 days, you'd have a travel day almost every other day, barely giving you a chance to catch your breath. I'd recommend slicing at least two of these destinations from the itinerary. Consider the following factors for each city: inbound and outbound flights, similarity of one city to the next, your own desires and preferences, what exactly you want to do in the destinations, and time and cost to get from one city to the next.
These cities provide a pretty solid range of experiences. Dublin, London, and Edinburgh are of course separated from the continent. While I love Dublin, if it's your first time through Europe, I'd recommend cutting out this city in favor of having a little more time and flexibility.
If you're studying abroad and will be taking multiple trips throughout your school semester, map out when to go based on each destination. Go to the beach cities (like Barcelona) when it's warmer, visit active or pedestrian cities (like Prague) before it gets too cold, and save "museum" cities (like Berlin) for the winter.
#### **Plan Your Route**
Now that I've narrowed down my itinerary to a manageable list, it's time to consider how to get from A to B to C. Consider trains, buses, and flights—which are generally the quickest and best option. If you're on a tight budget, book transportation ahead of time, as prices can climb the closer you get to your travel date. For tips on choosing the best mode of transit, along with step-by-step instructions on booking your transportation between cities, see here.
**Links:** Try skyscanner.com, kayak.com, cheapoair.com, momondo.com for flights; sbb.ch for train travel; eurolines.com, orangeways.com, berlinlinienbus.de, renfe.com, and studentagencybus.com for bus travel; carpooling.co.uk for ride-sharing.
**Apps:** For a list of helpful transportation apps, see here.
#### **Decide Where to Sleep**
Travelers often ask me whether they should book accommodations ahead of time. To decide, weigh the value of accommodation at potentially cheaper prices against flexibility and spontaneity. If you're on a tight budget or traveling during a famous event, book your hostel as far in advance as possible. For more tips on booking accommodations, see here.
If you do book in advance, go the extra mile and find out how to get to your hostel _before_ you leave for your trip. Nothing is worse than arriving in a foreign city with a heavy backpack and not knowing where to go. Print out directions or a map, and know which metro line to take and how much it costs.
**Links:** Use hostelworld.com or airbnb.com. Also check out each site's helpful app.
#### **Calibrate Your Budget**
Know roughly how much your trip is going to cost before jumping in headfirst. How much is an individual meal? How expensive is a night out? What's the admission price for that famous art museum? To help you out with this, I've added city-specific tips on calibrating your budget at the beginning of each chapter.
**Credit cards** are widely accepted across Europe. Visa tends to be more common than Mastercard, and it's difficult to find anyone who takes American Express. For security reasons, the "chip-and-pin" system is ubiquitous in Europe, though swipe cards do still work just about everywhere. It's worth asking your bank to issue a chip-and-pin card ahead of time. **Cash** is often preferred in southern European countries like Italy and Spain. And you may encounter minimums (around €10) set by merchants to use a card.
**Apps:** Use the XE Currency app to keep track of fluctuating conversion rates and the Mint.com app to balance your budget.
#### **Plan for Precision Sightseeing**
Make a list of the top 3-5 things you want to see, and make reservations online to save yourself hours in line. At the beginning of each chapter, you'll find a list of sights that you should reserve in advance. Avoid trying to see too much—travel is a lot more than snapping a selfie in front of a famous monument.
Also consider booking a **local guide.** Doing so costs money, but guides help you get more out of your travels and will reduce the amount of time getting lost.
**IF YOU LIKE...**
**Grand museums:**
Paris, London, Berlin, Rome
**Culture and live performances:**
Dublin, Madrid, Edinburgh
**Outdoor recreation:**
Edinburgh, Barcelona
**Art history:**
Paris, Rome, Venice, Florence, Madrid
**Cuisine:**
Rome, Madrid, Venice
**Festivals:**
Budapest, Venice, Amsterdam
**Cheap thrills:**
Madrid, Budapest, Prague
**GO TO...**
**Euro-trash techno discotecas:**
Berlin, Barcelona, Rome
**Pub life and beer:**
Berlin, Prague, Dublin, London
**Alternative scene:**
Berlin, Budapest, Prague
**Casual and social ambience:**
Paris, Amsterdam, Prague
**Old-world atmosphere:**
Prague, Budapest
**Late, late nights:**
Berlin, Barcelona, Madrid
**LGBT nightlife:**
Amsterdam, Paris, London, Barcelona
**CITY-HOPPING**
Group cities geographically so that you don't waste time and money zigzagging across the continent. Each of these itineraries can also be done in the opposite direction, and you can link them together for a longer trip—for example, Best of the North followed by Best of Britain (Amsterdam → Paris → London →Edinburgh → Dublin). Prices are approximate, and tend to be lower the farther in advance you book.
**Best of the North**
**London** → **Paris** → **Amsterdam**
In as little as one week, you can get a taste of three of Western Europe's major cultural capitals.
• From London's St. Pancras station, take the fast train to Paris (2.5 hours, from €50).
• From Paris, hop a train to Amsterdam (3.5 hours, around €75).
**Best of Eastern Europe**
**Berlin** → **Prague** → **Budapest**
Explore complex history by day and dive into an alternative nightlife at night. This itinerary is great for history buffs and architecture fiends.
• From Berlin, use studentagencybus.com or orangeways.com to book a bus to Prague (4.5 hours, €40).
• From Prague, consider a Wizz Air or Czech Airlines flight to Budapest (1 hour, from €60), or take a train (7 hours, €75).
**Best of Spain**
**Barcelona** → **Madrid**
Whether you love nightlife, tapas, museums, or simply having a good time, Spain's two most important cities have got you covered.
• Fast trains connect Barcelona and Madrid (3 hours, from €50) several times daily. Buses are also an option. Book at renfe.com.
**Best of Italy**
**Rome** → **Florence** → **Venice**
This route is a no-brainer for pizza- and gelato-loving foodies. Add in robust art history and architecture, and you'll see why this trip has become a classic.
• From Rome, take a fast train (1.5 hours, from €25, reservations required) to Florence. Or save money on a slow train (8 hours, from €20). Avoid rush hour for cheaper prices.
• From Florence, take a fast train (2 hours, from €30, reservations required) or a slow train (7.5 hours, €15) to Venice. As with the previous leg of the journey, avoid rush hour for the best price.
**Best of Britain**
**London** → **Edinburgh** → **Dublin**
Head to the capital cities of the British Isles for pubs, live music, and the ease of connecting with friendly, English-speaking locals.
• From London, hop a cheap flight to Edinburgh (1 hour, €60), or take a train (4.5 hours, €110).
• From Edinburgh, numerous airlines run cheap flights to Dublin (1 hour, from €45) multiple times a day. For the best price, avoid rush hour. Many people commute between these cities for business.
### **ON-THE-GROUND TRAVEL TIPS**
#### **Carry a Map**
Getting lost in foreign cities can be fun, but it's always nice to know the way back home. Always carry a map, and take time to look up from it every once in a while to orient yourself. I always take my first morning in town to study up on my favorite sights and mark them on the map. I can then strategically group close-together sights to save commuting time.
#### **Flash Your Student ID**
Some cities (like Madrid) offer free or discounted admission to students with a valid ID. It's always worth checking when booking ahead or buying tickets.
#### **Customize Your Experience**
Whether it's sporting matches, DJs, or architecture, it's worth researching what's on in each city during your visit.
Music, sailing, and cycling races are some of my personal hobbies, so I watch for related events when I travel. If I'm in France in July, I make a point to catch a stage of the Tour de France. On a nice day in May in Berlin, I look up a lake outside of the city and take a friend sailing for an afternoon.
Ready to hit the road? Read on! _Bon voyage, bon viaggio, and gute Fahrt!_
**Andy Steves**
London Maps
London 101
Three Day Itinerary
Top Neighborhoods
Top Sights
Top Eats
Top Nightlife
Top Shopping & Markets
Top Parks & Recreation
Top Tours
Top Hostels
Transportation
Day Trips
Help!
London, a world leader in style, design, art, industry, politics, and pageantry, is the home of modern Western culture. Having rebounded after fighting tooth and nail in World War II, it's now one of the most modern cities on the planet. London proudly offers some of the world's best museums (mostly free!), nightlife that ranges from classic pubs to trend-setting clubs, and probably the best chicken tikka masala outside India. Get ready for a good time, because, as The Clash so famously put it, London's calling.
### **LONDON 101**
London was founded as a far-flung outpost in the Roman Empire nearly 2,000 years ago. The city's strategic location on the Thames put it squarely on the route leading from Britannia to the continent of Europe.
By the Middle Ages, London was a tangle of overcrowded, filthy streets. Disease spread quickly in the dense tenements. In 1350, the bubonic plague rode through on the backs of rats, killing more than 100,000 Londoners in a few short months. Mass graves dating back to these difficult years are often discovered today when city center construction projects break ground.
In spite of the plague, the city continued to grow, but in 1650, tragedy struck again when a three-day blaze now known as the Great Fire of London destroyed nearly 90 percent of the homes in the City of London, displacing tens of thousands of citizens. Numerous plans for rebuilding were submitted but never realized, and many modern streets mirror those of London's haphazard medieval layout.
The Industrial Revolution toward the end of the 18th century transformed London once more. Technologies like the telegraph and industries like the railway propelled wealth and population growth in London as well as Britain's burgeoning global empire. By 1800, London was home to one million inhabitants, becoming the first city to cross that benchmark since the Roman Empire. By this time, British naval prowess had stretched a small island's influence around the globe to the Americas, Africa, and the Far East, concentrating riches never before seen in London.
In 1863, London broke ground on the world's first underground railway, the Underground, aka the Tube. An idea many contemporary skeptics initially laughed at now transports over one billion passengers a year and is the lifeblood of the city, removing a large portion of traffic from the congested streets.
These same tunnels provided shelter when the Germans bombed the city tirelessly during the Battle of Britain in World War II. Winston Churchill conducted the war effort from his own bunker just a few blocks from Westminster Palace. You can visit the Churchill War Rooms today and see them just as they were when the war ended in 1945.
After the war, the city grew even more, as rebuilding efforts focused on reducing density by encouraging residents to move to communities farther from the city center. Former colonies were also allowed open immigration to England, making London one of the most diverse cities in Europe.
Since its early days, the City of London has been the economic capital of England, with prohibitive real estate prices driving many residents out. Each borough beyond the City of London used to be its own independent township, until eventually the borders dissolved and the boroughs combined to become the City of London we know today.
### **PLAN AHEAD**
#### **RESERVATIONS**
Reservations are recommended for the following sights:
**Tower of London** (hrp.org.uk/TowerOfLondon)
**London Eye** (londoneye.com)
#### **LOCAL HAPPENINGS**
##### **Wireless Festival**
Founded in 2005, the three-day Wireless Festival (wirelessfestival.co.uk), held annually in late June and early July, offers world-renowned bands and performers from all over the world. Going down each year at Finsbury Park, the Wireless Festival exchanges corporate sponsors often between banks and tech companies. Headliners from recent years included Justin Timberlake, John Legend, Jay Z, and Rihanna.
##### **Bonfire Night**
"Remember, remember, the 5th of November." This quote was made famous to Americans through the movie _V for Vendetta_ , but it is actually a ditty written about Guy Fawkes Night, now known as Bonfire Night. Fawkes was a Catholic conspirator who organized a group of fellow Catholics to blow up the Houses of Parliament in 1605 (the Gunpowder Plot). Unfortunately for him, his plot was discovered just in time and he was executed for high treason. Today, Brits everywhere celebrate the prevention of this terrorist plot with fireworks, bonfires, and a lot of drinking on November 5. Battersea Park now has a huge bonfire and light show each year.
**KNOW BEFORE YOU GO**
**KEY STATS & FIGURES**
**Currency:**
**British pound ( _£_ ); _£_ 1 = about 1.5 USD**
**Population:**
**8,173,194**
**Language:**
**British English**
**Number of metro lines:**
**11**
**Amount of tea consumed daily in the United Kingdom:**
**120,000,000 cups**
**National dishes:**
**fish-and-chips, chicken tikka masala**
**Key Cockney phrases:**
**Brahms and Liszt = pissed (drunk), tomfoolery = jewelry, rub-a-dub = pub**
**CALIBRATE YOUR BUDGET**
**TYPICAL PRICES FOR:**
**Hostel dorm bed:**
**_£_ 16**
**Two-course dinner and drink:**
**_£_ 14**
**Pint of beer:**
**_£_ 4**
**Daily bicycle rental:**
**from _£_ 2**
**Single Tube pass:**
**_£_ 2.30 with Oyster Card**
**Museums:**
**mostly free!**
**MOVIES TO WATCH**
_The King's Speech, Love Actually, A Clockwork Orange, V for Vendetta, Emma, and A Fish Called Wanda_
**THREE DAY ITINERARY**
A good itinerary of London flanks the south side of the Thames on the first day, and the north side of the Thames on the second day. On your third day, you'll still find plenty to keep you busy.
**DAY 1: WELCOME TO LONDON**
**MORNING**
Spend your morning on the South Bank. First, head out for my favorite walk along the water: **Millennium Mile**. It's an enjoyable three-hour stroll. Start with breakfast from a food stall in the **Borough Market**. Continue on to see **Shakespeare's Globe** and the **Tate Modern** (featuring one of the best contemporary art collections in the world). This stroll will orient you and give you a sense of the scale and layout of the city as it wraps around the Thames River.
**AFTERNOON**
Drop south a couple blocks from the roundabout at Waterloo Station to grab lunch at the best fish-and-chips joint in the city, **Master's Super Fish**.
Head back to the Millennium Mile for a ride in the **London Eye**. Afterward, cross Westminster Bridge to glimpse the famous exterior of **Big Ben** and **Westminster Palace.**
Hop on the Circle Line at the Westminster station over to Bayswater Tube and stop for a bike tour of the Royal Gardens with **Fat Tire Bike Tours** (leaves daily at 15:00 May 15 through September). If you're outside the season, flip today's itinerary to catch their daily morning departure at 11:00. If you prefer to stay seated, take **bus 11** for a ride straight through the heart of the City, connecting Victoria and Liverpool Street stations.
**EVENING**
Get dinner in the eccentric Covent Garden neighborhood. Choose gastropub grub at **Porterhouse** , authentic Southern Mexican at **Wahaca** , or spicy Thai at **Busaba Eathai**. The nightlife and bars around Covent Garden are sure to keep you busy. **Freud** is an excellent hidden little spot for a nightcap. Or if wine is your thing, head to **Gordon's Wine Bar** right next to the Embankment stop for candlelit glasses within exposed-brick cellar walls.
**LATE**
Stay in the West End and explore the back streets toward Soho for the night. Consider catching a West End show: Find last-minute discount tickets in **Leicester Square** at the **TKTS** booth.
**DAY 2: CHECK OUT THE CITY**
**MORNING**
Spend an hour or so at **Westminster Abbey** (check hours beforehand; the abbey can close for special events), where you'll find all the VIPs of British history: royals, philosophers, and artists like Chaucer and Charles Dickens. Next, pop into a **Pret a Manger** for a quick, cheap lunch along Whitehall Road.
**AFTERNOON**
From Westminster Abbey, take a slow stroll north and east along Whitehall Road, taking the rest of the day to make your way over to the Tower of London. After Trafalgar Square, Whitehall Road becomes the Strand, London's and Westminster's main boulevard. You'll pass numerous important landmarks along the way, including the traditional parade grounds of the **Horse Guards Parade** , grand **Trafalgar Square** , the **National Gallery,** and **Nelson's Column**. If these sights pique your fancy, be sure to stop in and check them out!
In two miles, you'll come upon the historic but easy-to-miss **Ye Olde Cheshire Cheese** pub (145 Fleet St) and the massive **St Paul's Cathedral**. At this point, you're also crossing the border between Westminster and the City of London. About a mile after St Paul's, keep an eye out on the right for the **Monument** (the tall, singular column), which commemorates the Great Fire of London; you can climb it for only _£_ 3.
Finally, arrive at the **Tower of London** and pick up a ticket. Time your visit to catch the wonderfully enthralling Beefeater-guided tour (leaving every half hour from the main gate) and of course see the Crown Jewels! All in all, you'll spend a couple hours here.
**EVENING**
Take the half-hour walk north from the river to **Brick Lane** for the world's best chicken tikka masala. Soak in all the hipster culture while you're in **Shoreditch** , one of London's trendiest neighborhoods, offering diverse nightlife venues like **Nightjar,** which does just the trick for me.
**DAY 3: MUSEUMS & SHOPPING**
**MORNING**
Spend the morning in the **British Museum** and take in everything it has to offer. Then find **Goodge Street** for an early lunch. Choices abound for every type of cuisine you could imagine. Continue up to **Oxford Street** and wander toward **Hyde Park** to enjoy London's main shopping zone. Give yourself the "London look" with a stop in **Selfridges.**
Alternatively, if it's Sunday, spend your morning shopping at **Old Spitalfields Market** (open daily, but best on Sunday). The food's great, but better options are nearby at the weekly **Sunday Upmarket** on Brick Lane. After your fill, wander north on **Brick Lane** through more streets of vintage clothing shops, hipster goods galore, and delicious snacks and food stalls.
**AFTERNOON**
If you opted for the department store shopping route, drop south from Selfridges through **Soho** and explore the many music stores, market stands, adult shops, cafés, and eccentric people that crowd the winding neighborhood streets.
Make your way west past Hyde Park into **Harrods,** London's iconic department store. If you still have energy, the nearby **Victoria and Albert Museum** boasts the world's largest collection design.
**EVENING**
Head up to **Camden Market** to explore and check out the alternative scene via the Piccadilly line, transferring to the Northern line. Get ready for a bohemian, eclectic scene with loads of student nights and discount drinks that budget travelers love. Center your efforts around the Camden Market Tube stop and market and north for a few blocks along Chalk Farm road.
### **TOP NEIGHBORHOODS**
The River Thames bisects London west to east, creating a useful navigational aid. The City of London, north of the river, is a mere square mile that was once enclosed within medieval London's walls. Today, **The City,** as it's known, is where you'll find St Paul's Cathedral and the Tower of London, among other sights.
West of The City is London's **West End.** This is _the_ entertainment district, with endless restaurants, bars and clubs, and great shopping on Oxford Street. The West End includes the hip neighborhood of Soho (which is the hub of London's LGBT community), the theater district of Leicester (pronounced "Lester") Square, and Covent Garden, known for good food and fun, casual bars.
**Westminster,** just south of the West End, is the political, royal, and religious hub of Britain. This is where you'll find Buckingham Palace and Westminster Abbey.
South of the river is the **South Bank,** which has famous sights like the London Eye, the Tate Modern museum, and Shakespeare's Globe theater, all conveniently linked by the Millennium Mile riverwalk.
London is big, and there are worthwhile attractions even on its outskirts. Northwest of the City, the **British Museum neighborhood** offers a couple of convenient hostels and awesome budget food options on Goodge Street. **West London** is home to the Victoria and Albert Museum, along with famous Hyde Park. **Shoreditch** (northeast of the City) and **Camden** (about 20 minutes north of the city center on the Tube) are great nightlife districts. These two neighborhoods attract a young, hipster crowd with counterculture bars and shops.
### **TOP SIGHTS**
#### **Tower of London**
The Tower of London is a castle right on the banks of the Thames, with foundations dating back to the 11th century. Over the following centuries, the Tower of London was under constant expansion and served as both a defensive fortress and the palace of kings and queens, up until the Tudors in the 1500s.
Besides its defensive and royal purposes, the Tower of London also served as a high-security prison with a long, bloody history of torture and public executions. The lively **Beefeater-guided tours** that depart every half hour (included with admission) provide an enjoyable history of the complex. Inside the tower itself, an exhibit walks you through the 500-year history of royal armor, medieval architectural plans and relics, and full-size replicas of siege weaponry.
Cap your visit with a viewing of the **Crown Jewels,** located just opposite the grounds from the tower. Seeing the jewels is included with admission and definitely worth the wait. Have your camera ready as you get on the moving sidewalk machine to see them. Taking a Beefeater tour, seeing the museum in the tower, and peeking at the Crown Jewels takes around 2.5 hours total.
£24 online, £25 in person, Tues-Sat 09:00-17:30, Sun-Mon 10:00-17:30, closes one hour earlier Nov-Feb, The City, +44 (0)844 482 7777, hrp.org.uk/TowerOfLondon, Tube: Tower Hill
#### **Tower Bridge**
After visiting the Tower of London, walk down toward the river and you'll find yourself a great photo op of the Tower Bridge. It's one of the main icons of the city, and tourists commonly mistake this bridge for the London Bridge, which is actually the next, less-exciting bridge upriver. For its 120th anniversary in 2014, Tower Bridge was decked out with a glass walkway that visitors can walk (or crawl) across, taking in the bridge and traffic from a new and impressive angle.
£9, daily 10:00-18:00 in summer, 09:30-17:30 in winter, +44 (0)20 7403 3761, towerbridge.org.uk, The City, Tube: Tower Hill
#### **St Paul's Cathedral**
St Paul's Cathedral, originally founded in 604, was rebuilt 1675-1710 after succumbing to the Great Fire of London. (Townsfolk thought that the massive stone cathedral would be a safe refuge, but the structure's wooden scaffolding caught fire, causing great damage inside and out.) The rebuilding was carried out by London's most revered architect of the time, Sir Christopher Wren. Wren drew direct inspiration from St Peter's Basilica in Rome and strove to use all the formulas of the popular neo-Renaissance style to create his finest masterpiece.
During the German blitzkriegs of World War II, St Paul's Cathedral stood tall and remained unharmed. Many took this as a sign that even in the darkest of times, God would watch over the British people and see them through. During this time, St Paul's dome was actually protected by a crew of brave blokes who lived at the church and would run water buckets at the first sign of danger. Thanks to the success of defying the Germans, and the good fortune it took to escape the flames of war, St Paul's Cathedral became a symbol of British national identity.
Take in beauty of overwhelming magnitude at this Anglican cathedral, located in the middle of the city and on the tallest hill in all of London. With its large, white stone construction, St Paul's is impossible to miss. Inside, you'll discover gilded architectural accents and hundreds of intricate golden mosaics. Climb the 271 steps to the top of this architectural marvel for a beautiful panorama.
£16 online, £18 in person, Mon-Sat 08:30-16:30, Sun worshippers only, The City, +44 (0)20 7246 8350, stpauls.co.uk, Tube: St Paul's
#### **The Monument**
The single stone column known simply as the Monument was erected between 1671 and 1666 to commemorate the infamous Great Fire of London, which ravaged the city in 1666. If the column were to fall down toward the east, the end point would mark the exact spot of the start of the fire, in a bakery on Pudding Lane. The column offers a hike of 311 steps and a beautiful panorama of downtown London. A location right in the city of London (a couple blocks from the Tower of London) and an entry price of only _£_ 3 make this sight both convenient and cheap!
£4, daily Apr-Sept 09:30-18:00, daily Oct-Mar 09:30-17:30, Fish Street Hill, The City, +44 (0)20 7626 2717, themonument.info, Tube: Tower Hill
#### **Westminster Palace & Big Ben**
Westminster Palace was designed and built during the 19th century, when the sun never set on the British Empire. This building was an important project for the architect, Sir Charles Barry. Strategically located just outside of England's capital city, the City of London, Westminster Palace is where the wealthy and powerful had to come to get their voices heard in government. Today, the aggressively neo-gothic building that housed the parliament of the most powerful empire of the world is quite a sight to behold. While England's politics have played out in this governmental center since the palace's completion in 1870, England has been ruled from this neighborhood for even longer—numerous wooden-construction palaces have been lost to fire.
The Elizabeth Clock Tower is the one tourists like to call **Big Ben.** While it's the tallest tower at Westminster Palace, Big Ben is actually the name of the 13-ton bell (which you can't see from the street) that has tolled every hour since 1859. The best angle for selfies is from a few paces down Westminster Bridge. The clock face is a full 23 feet wide, and the minute hand moves six feet every five minutes.
**LONDON'S BEST VIEWPOINTS**
Some of London's top sights also offer amazing panoramic city views:
**St Paul's Cathedral**
Climb the 271 steps to the top of this architectural marvel on the tallest hill in all of London and take in a beautiful panorama of the City.
**Tate Modern**
The top floor of this museum doubles as a restaurant with a beautiful view of the City. The restaurant is pricey; the view is not.
**The Shard**
Ride to the top of this London landmark or enjoy a posh cocktail on the 31st floor. Just as at the Eiffel Tower, the best views are seen from halfway up. Otherwise your vantage point is too far removed and the buildings fade into the distance.
**London Eye**
Take in impressive views from this 450-foot-tall Ferris wheel. Go at dusk for shorter lines and beautiful sunset shots.
While it is possible to visit inside Westminster Palace (see parliament.uk/visiting for information), those on a three-day visit to London are happy with a view from the outside.
Free, open when Parliament is in session, Westminster, Tube: Westminster
#### **Buckingham Palace**
You can't leave London without a selfie in front of Buckingham Palace! This is the grand and opulent residence of the royal family, located about a 15-minute walk west of Westminster Abbey. William and Kate shared a kiss on the neoclassical balcony, as did Prince Charles and Lady Diana in years past. Notice the perfect balance of the facade, a classic example of Renaissance-style architectural beauty. Located at the edge of St James's Park, Buckingham Palace contains a total of 775 rooms and has been the official home of British sovereigns since 1837. While today the palace functions as both a residence to the queen and an administrative building, it also serves the grand purpose of being one of the biggest tourist attractions in all of London. Make sure to time your visit during the changing of the guard.
From July through September, it's possible to tour the palace. In this half-mile experience that takes about two hours, you can check out the royal State Rooms (where the royal family entertains guests) and the Royal Mews (the stables), and see dozens of priceless art pieces from greats like Rembrandt, Titian, and Van Dyck in the Queen's Gallery. Tickets for a visit can be purchased all together in the Royal Day Out package (from _£_ 20) or in combinations of the various attractions.
Changing of the guards 11:30 every other day (refer to monthly schedule on website), +44 (0)20 7766 7300, Westminster, check full palace information and purchase tickets at royalcollection.org.uk, changing-the-guard.com/dates-times.html, Tube: Victoria
#### **Westminster Abbey**
With nearly 1,000 years of coronations, London's premier gothic church, complete with pointed arches, stained glass, and flying buttresses, has played host to the major life events of England's royal family. This Anglican church is still quite active: It was the site of Prince Charles's wedding to Lady Diana (1981), Princess Diana's funeral (1997), and Kate and William's wedding (2011). Many of the United Kingdom's most famous thinkers and national figures are entombed here, including Darwin, Chaucer, and Newton, along with 17 kings and queens of England underneath sequoia-like pillars supporting breathtaking arches and structuring ribbing.
£20, Mon-Fri 09:30-16:30, Wed till 19:00, Sat 09:30-14:30, Sun for worship only, 20 Deans Yard, Westminster, +44 (0)20 7222 5152, westminster-abbey.org, Tube: Westminster
#### **Churchill War Rooms**
Can you imagine running a world war in which millions of lives counted on you, coordinating movements of troops by pinning their positions on a wall in a damp, dark bunker deep in the earth? With the help of serious quantities of tobacco and whiskey, that's exactly what Winston Churchill and his military chiefs of staff did for the entirety of World War II, from a series of reinforced concrete rooms located deep beneath the Treasury Building.
Some of these war rooms have been frozen in time to create a museum. Tour each room, seeing the maps, note pads, and communication systems left as they were on VE Day in 1945. You can also walk through a fascinating exhibit on wartime England, which takes you through the daily lives of Londoners during the war, visually presenting everything from ration tickets and propaganda posters to spying technology and what it was like to seek cover from German bombs in the Tube. This is a fascinating museum—especially for WWII buffs—that fills up an enjoyable two hours.
£18, daily 09:30-18:00, Clive Steps, King Charles St, Westminster, +44 (0)20 7930 6961, iwm.org.uk/visits/churchill-war-rooms, Tube: Westminster
#### **Horse Guards Parade**
This graveled parade ground has served as an open space in the heart of the city for events, ceremonies, and practices of the British military and royal family. For the 2012 Olympics, which London hosted, the Horse Guards Parade was transformed into a stage for beach volleyball, and it had a capacity for 15,000 fans surrounding the single court. Today the parade grounds make for a great, quick photo in front of the mounted guards and a chance to see the court through the gates to the back, on Whitehall leading from Big Ben and Westminster Abbey.
Free, guards change daily at 11:00 (10:00 Sun) and dismount at 16:00, Whitehall, Westminster, Tube: Charing Cross
#### **Millennium Mile**
The Millennium Mile is an easy-to-stroll riverwalk on the south side of the Thames. Anchored by the **London Bridge** and the **Shard** on one end, and **Big Ben** and the **London Eye** on the other, it's an extremely pleasant and convenient way to see a number of London's most famous sights. Between those major sights, you'll also pass by the **Borough Market, Shakespeare's Globe** theater, the **Tate Modern** museum, and the **Millennium Bridge**.
Free, always open, South Bank, Tube: London Bridge or Waterloo
#### **The Shard**
London's newest and most visible landmark is the Shard. Named for its distinctive glass design tearing into the London sky, the skyscraper was completed in 2013. Take the elevator to the 32nd floor, where you can enjoy a coffee in the posh restaurant Oblix. The best views are seen from halfway up; otherwise your vantage point is too far removed and the buildings fade into the distance.
Viewing platform £26 in advance/£31 in person adult over 16, £21 in advance/£26 in person student with valid ID; Apr-Oct daily 10:00-22:00 (last entry 21:30); Nov-Mar Sun-Wed 10:00-19:00 (last entry 17:30) and Thurs-Sat 10:00-22:00; 32 London Bridge St, South Bank, +44 (0)333 456 4000, the-shard.com, Tube: London Bridge
#### **Shakespeare's Globe**
While the design of Shakespeare's original theater was innovative, it was not particularly fire safe, and the theater burned down in 1613 after a malfunctioning cannon set the stage on fire. Along with the thatch roof and the stage sets, the architectural plans for this ground-breaking theater went up in flames.
This replica, with the trademark black timber-framed construction and intimate stage, was completed in 1997. It's meant to resemble the original theater as it would have looked in the 17th century. With capacity for about 850 seated nobles and 700 standing peasants, the Globe is an active theater, and viewing a performance here is fascinating. Plays are performed as they would have been in Shakespeare's day, with no voice amplification or artificial lighting. Check showtimes and book online.
It's also possible to **tour the theater** with an excellent guide who speaks about the history of the original building, how plays work in this purpose-built space for live entertainment, and the function of the venue today. Just like Shakespeare's plays, this guided walk appeals to intellectuals as well as potty-humor fans. Tours leave every 30 minutes throughout the day.
Exhibition £6, tour £13.50, price for shows varies, daily 09:00-17:30, Box Office West Piazza, 21 New Globe Walk, South Bank, +44 (0)20 7902 1400, shakespearesglobe.com, Tube: London Bridge
#### **Tate Modern**
This imposing remodeled power plant was opened in 2000 to house London's premier modern art museum, featuring one of the best collections of contemporary art in the world. With stark, dark brick construction and wide open gallery spaces, the Tate is packed with works from an all-star roster of artists. You can seek out Andy Warhol's prints, Roy Lichtenstein's large-scale comics, Donald Judd's sculptures of spatial explorations, and pieces by Cézanne. Plan to spend about an hour and a half here. The bookstore alone can keep you enraptured for hours. The top floor doubles as a restaurant with a beautiful view of the City. The restaurant is pricey; the view is free.
Free, Sun-Thurs 10:00-18:00, Fri-Sat 10:00-22:00, South Bank, +44 (0)20 7887 8888, tate.org.uk, Tube: Blackfriars
#### **Millennium Bridge**
This beautiful, modern, glass-and-steel pedestrian bridge spans the Thames from directly in front of the Tate Modern on the south side of the river to St Paul's Cathedral on the north side of the river, in the City of London proper. The Millennium Bridge opened in June of 2000, only two months late and _£_ 2 million over budget, but the opening day wasn't all applause and champagne. As the first few walked across, the bridge moved, so it became obvious the bridge needed additional support. It was closed back down for two years, and millions more were spent as engineers added support to make the bridge more rigid.
Free, always open, directly between St Paul's Cathedral and the Tate Modern, South Bank, Tube: Blackfriars
#### **London Eye**
Hoisted up in 1999 to celebrate the Millennium, this massive observation wheel— _not_ Ferris wheel, as the staff and everyone else in London will have you know—is 450 feet tall and the United Kingdom's most popular tourist attraction, with over 3.5 million visitors each year. The 32 air-conditioned glass and steel pods, each holding up to 25 people, rotate around on a wheel on a single, cantilevered arm holding the entire contraption out above the Thames at a constant, crawling speed, completing only two revolutions each hour. Guests hop in and out of the moving pods as they skim the boarding area at the bottom. Insider tip: Sunny days wash out all your pictures. Go at dusk for shorter lines and striking sunset shots.
£20 online, £21.50 in person, daily 10:00-sunset, Riverside Bldg, County Hall, Westminster Bridge Rd, South Bank, +44 (0)871 781 3000, londoneye.com, Tube: Waterloo
#### **British Museum**
Opened in 1759, the British Museum was the first national public museum in the world free for all those who want to visit. Showcasing some of the greatest historical treasures the world has ever known, like the Rosetta Stone, the Parthenon Sculptures, and Egyptian artifacts, this well-appointed museum is dedicated to history, art, and culture. It takes about two hours to see the highlights. The museum also offers free 40-minute walking tours in the exhibit of your choice throughout the day. Find the relevant times on the free brochure when you walk in, or check the website.
Free, daily 10:00-17:30, Great Russell St, British Museum Neighborhood, +44 (0)20 7323 8299, britishmuseum.org, Tube: Tottenham Court Road
#### **National Gallery**
The National Gallery is a collection of two-dimensional artwork spanning back from the Dark Ages all the way to impressionism. Enjoy the free entry, and ponder 2,000 of the greatest paintings in existence, from Renaissance bosses like Leonardo and Michelangelo to baroque Caravaggio and impressionist Van Gogh.
The National Gallery is right on **Trafalgar Square** , a hub of London's history and public transportation. In the middle of the square, you'll find **Nelson's Column,** dedicated to the valiant Admiral Nelson, who was killed in battle at Cape Trafalgar in 1805 while leading his navy to a crucial victory over the French and Napoleon's navy.
Free, daily 10:00-18:00, Fri until 21:00, Trafalgar Square, West End, nationalgallery.org.uk, Tube: Charing Cross
#### **West End Plays**
The business of live entertainment dates back hundreds of years, well before the time of Shakespeare. Theaters began popping up in London on the west side of town, or the West End. Today, the district is home to dozens of theaters—large and small—and it's a popular option to catch a show while you're in town. Some plays, like the classics _Les Miserables_ and _Mousetrap,_ have been running for decades, while others, like _Wicked_ and _The Lion King,_ offer a more modern take. Find the **TKTS** booth in Leicester Square for tickets up to half off in the hours and minutes before the show starts.
Theater locations, prices, and show times vary, West End, Tube: Leicester Square, Covent Garden, or Piccadilly Circus
#### **Victoria and Albert Museum**
With a collection of more than 4.5 million design and decorative art pieces from all around the world, this massive museum can overwhelm any visitor in both quantity and quality of artifacts dating from the last five millennia. A stop at the information desk is key to organizing your time. Exhibits span religions, centuries, styles, materials, subjects, geography, and media. Beeline toward the subjects that interest you most, because it's easy to get lost in the 145 galleries, which make you feel like you're walking through a life-size encyclopedia. Don't miss my favorite piece, an original, revolutionary "Job" cigarette poster by Alphonse Mucha.
Free, daily 10:00-17:45, Cromwell Rd, West London, +44 (0)20 7942 2000, vam.ac.uk, Tube: South Kensington
### **TOP EATS**
London has undergone a culinary renaissance in the last couple decades. People used to turn up their noses at the thought of British cuisine, but now visitors can find affordable places to eat throughout town. Heads-up: Sit-down lunches can easily run _£_ 13 ($20), and dinners will can cost past _£_ 19 ($30). My suggestions will help you either save or get the best bang for your buck. There is no need to tip at fast food or takeaway shops. If you're sitting down for a meal, take a look over your bill and assess whether cover and service are included. If not, feel free to round up 10 percent or so, based on the service you received.
As **fish-and-chips** is the national dish of England, it would be a crime not to duck into one of the many local establishments that offer it. Master's Super Fish and FishcoTeque are two of my favorite places in town to sample it.
#### **Master's Super Fish**
Take a short walk down from Waterloo station and get ready for the city's best tempura goodness! Come for the great fish-and-chips, and don't let the gruff service get to you. This is a no-frills bar that feels more like your grandma's living room than a fast-food joint. The food is sure to make up for any hard feelings.
£5, Mon-Sat 04:30-22:30, 191 Waterloo Rd., South Bank, +44 (0)20 7928 6924, Tube: Waterloo
#### **FishcoTeque**
Fish are brought in daily from the Billingsgate market to this street stand restaurant near Waterloo station. The restaurant has recently been renovated to recapture the original 1950s feel it had when it opened. You've got a streamlined menu offering just about anything fried: chips, fish, burgers, and chicken burgers. Eat in or get takeaway to keep your momentum up.
£5.95, daily 11:00-23:00, 79A Waterloo Rd, South Bank, +44 (0)20 7928 1484, Tube: Waterloo
#### **Ping Pong**
Modern and fresh both in decor and food, Ping Pong is a stylish Asian fusion joint with delicious steamed dim sum. Ping Pong does spring rolls and beef dumplings especially well. Be sure to try their signature and seasonal Ping Pong cocktail for a tasty treat.
Meals from £10, Mon-Fri 11:00-24:00, Sat 12:00-24:00, Sun 12:00-22:30, 45 Great Marlborough St, West End, +44 (0)20 7851 6969, pingpongdimsum.com, Tube: Oxford Circus
#### **Dishoom**
This is my favorite spot for Indian food in the center of London. You pay a little more in this casual and refined atmosphere, but the quality of the naan (go for the garlic!), curries (especially the unique mahi tikka), and grills (I love the lamb roti) is superb. Top it all off with a mango lassi, and you'll remember this meal for sure. I like the open interior, which has a modern cafeteria feeling with wooden chairs and large windows. In addition to the Covent Garden location listed, you'll find three other locations throughout town. These guys don't take reservations, so you might wait up to an hour for your food. Trust me—it's worth it!
From £12, Mon-Thurs 08:00-23:00, Fri 08:00-24:00, Sat 09:00-24:00, Sun 09:00-23:00, 12 Upper Saint Martin's Ln, West End, +44 (0)20 7420 9320, dishoom.com, Tube: Covent Garden
#### **The Breakfast Club**
Famous for serving one of the best breakfasts in London, Breakfast Club has lines out the door—sometimes as long as 45 minutes—from opening to closing. Whether you come for the delicious beverages (coffee, cappuccino, or even hot chocolate), the thick-sliced bacon, the pancakes, or the indulgent eggs Benedict, the Breakfast Club may have you coming back every morning of your stay. Slept in? Don't worry, the Breakfast Club serves breakfast until 17:00 daily. Thanks to six locations in town, you're never far from a face full of their cinnamon apple French toast.
Breakfast from £8.50, Mon-Sat 08:00-22:00, Sun 08:00-19:00, 33 D'Arblay St, West End, +44 (0)20 7434 2571, thebreakfastclubcafes.com, Tube: Oxford Circus or Tottenham Court Road
#### **Burger & Lobster**
These guys do two things, and they do them well: burgers and lobster. Actually, they do their drinks and desserts (for example, the Lemon Strawberry and Mint cocktail or the chocolate amaretto crunch) exceedingly well, too. Service is friendly, and the menu is simple: choose burger, lobster, or lobster and salad roll (basically a thickly sliced sandwich) and decide on your sides, such as chips (French fries) or a tasty green salad. Deals are offered frequently, at a much better price than you'd expect from anything with "lobster" in the title. Eating a meal here feels like sitting down to a crawfish boil thanks to the long wooden tables and paper tablecloths.
Burger and lobster from £20, Mon-Wed 12:00-22:30, Thurs-Sat 12:00-23:00, Sun 12:00-22:00, 36-38 Dean St, West End, +44 (0)20 7432 4800, burgerandlobster.com, Tube: Leicester Square or Tottenham Court Road
#### **Busaba Eathai**
Kick-ass curry and delicious pad thai make this one of my favorite places in London. All of Busaba's locations are done up in a mod, dark Asian-fusion theme, with welcoming service that is happy to make recommendations. Their _tom yam goong_ , or spicy and sour prawn soup, will make you feel like you're back in Bangkok. The Songkhla-style red curry is my favorite. In addition to the Soho shop, you'll find many other locations across town.
£8-12, Mon-Thurs 12:00-23:00, Fri-Sat 12:00-23:30, Sun 12:00-22:00, 106-110 Wardour St, West End, +44 (0)20 7255 8686, busaba.com, Tube: Piccadilly Circus
#### **Wahaca**
Wahaca offers some of the best tacos, burritos, grilled steak, and _horchata_ (a beverage) I've had this side of the Atlantic. I love the fresh ingredients and authentic dishes that remind me of my travels south of the border. Expect colorful decorations, fast and cheerful service, and a casual, welcoming atmosphere. Orders are noted on your big paper tablecloth. There are many locations across town.
£11, Mon-Sat 12:00-23:00, Sun 12:00-22:30, 80 Wardour St, West End, +44 (0)20 7734 0195, wahaca.co.uk, Tube: Piccadilly Circus
#### **Chipotle**
I have to mention this fine delicatessen only because London is one of the few places outside of the US that have this chain eatery. It's fast, it's cheap, and—let's be honest with ourselves—it's delicious. You'll find Chipotle outposts throughout the city.
£8, Mon-Sat 11:00-23:00, Sun 11:00-22:00, 114-116 Charing Cross Rd, Trafalgar Square, West End, +44 (0)20 7836 8491, chipotle.com, Tube: Leicester Square
#### **Pret a Manger**
This chain, serving sandwiches that are freshly made daily along with salads and snacks, is a London mainstay—like Starbucks is to Seattle (and just about every other city around the world). Everyone from backpackers to businessmen comes to this functional and clean fast-food chain to enjoy a healthy, cheap lunch. Grab your sandwich from the display rack, order coffee, and pay at the bar. In addition to the Westminster location, you'll find branches of Pret a Manger throughout the city.
£2-5, Mon-Fri 07:30-21:00, Sat 08:00-20:00, Sun 12:00-18:00, 47 Great Peter St, Westminster, +44 (0)20 7932 5401, pret.com, Tube: Westminster or St James's Park
#### **Bubbledogs**
Bubbledogs, located near the Goodge Street eateries, is a novelty experience as much as anything else: Get a gourmet hot dog served in a classic red plastic bowl, and wash it down with a glass of champagne. The irony seems to be lost as soon as you walk in the door, because the staff take both their bubbles and their dogs seriously. The Mac Daddy, a brat topped with piping hot mac-n-cheese, fried onions, and bacon bits, is a favorite of everyone except cardiologists. Add a side of tater tots, and enjoy the experience of dining on what you would've had for lunch in the third grade, yet in completely different surroundings, with aproned servers suggesting which champagne to pair with your main.
For a step up, find the hidden Michelin restaurant in the back, called **Kitchen Table,** for an experimental culinary experience that will take both your taste buds and your wallet for a ride. Saddle up to the bar along with 18 other diners, and enjoy the show for the next several hours as the chef and cooks dish up unforgettable meals. Find more information and make reservations at kitchentablelondon.co.uk.
Dogs from £6, Tues-Thurs 11:30-15:00 and 17:30-22:30, Fri-Sat 11:30-22:30, 70 Charlotte St, British Museum Neighborhood, +44 (0)20 7637 7770, Tube: Goodge Street
#### **Goodge Street Restaurants & Food Stalls**
Goodge Street between Tottenham Court Road and Cleveland Street, along with the streets that surround it, offers your highest density of inexpensive, fresh and fast casual restaurants in all of London. At every lunch hour, this street packs out with London's young professionals, who enjoy the span of choice from gourmet hot dogs at **Bubbledogs,** burritos at **Benito's Hat** (56 Goodge St), pad thai at Thai Metro (38 Charlotte St), Greek salads at **Andreas** (40 Charlotte St), and sushi at **Roka** (37 Charlotte St). You'll discover classic British tea at **Yumchaa** (9 Tottenham St), tapas at **Salt Yard** (54 Goodge St), and delicious pizza at **ICCo** (Italian Coffee Company, 46 Goodge St)—all in the space of just a few blocks. All places listed are casual with fast service and are run like serious businesses: If you're there to eat and have the money to pay, you'll be taken care of in cool and quick fashion. One of the more unique restaurants in the district, **Bubbledogs,** is listed separately.
Goodge St between Tottenham Court Rd and Cleveland St, British Museum Neighborhood, Tube: Goodge Street
#### **Brick Lane Restaurants & Food Stalls**
Brick Lane between Fashion Street and Bruxton Street is home to many immigrants from the former Southern Asian British colonies. On this meandering street and in the surrounding district, you'll find some of the most authentic Indian food in the world. It's a truly intercultural dining experience, from the moment you're approached by restaurant reps to finally stepping inside your selected restaurant. Do a lap up and down the lane and be prepared to barter for the price and inclusions of your meal! Part of the fun of Brick Lane is the unknown—you never know for sure if you're choosing the right restaurant until after your meal. I usually start my lap at **Brick Lane Clipper** (104 Brick Ln) and **Cinnamon** (134 Brick Ln) restaurants. And I avoid **Saffron** (53 Brick Lane) and **Preem & Prithi** (118-122 Brick Lane), where quality and service vary widely.
Shoreditch, Tube: Aldgate East
### **TOP NIGHTLIFE**
London sports one of the hottest nightlife scenes around, with bars, pubs, and clubs to suit just about any interest. When going out, keep some things in mind: The Tube closes at midnight, as do all the pubs in town. Some bars and most clubs stay open late. Always bring valid **photo ID** to get into any bar or club—it puts a damper on the night when you're turned away at the door!
#### **NIGHTLIFE DISTRICTS**
London is a massive city, so it's best to plan your nights out based on which neighborhoods you identify best with. Figure out if you're looking for the bright, easy, central, touristy pubs and bars, the posh cocktail lounges, or the serious nightclubs, and go from there.
##### **The West End**
This West End is always happening on the weekends and has a ton of nightlife joints that span the spectrum from ale houses to throbbing _discotecas_. **Soho,** one of its hippest neighborhoods, offers numerous music venues, off-the-wall bars, and bohemian cafés. Soho is also the hub for the **LGBT** community and has many gay-friendly bars and clubs to choose from.
West End, Tube: Tottenham Court Road, Piccadilly Circus, Leicester Square, or Covent Garden
##### **Shoreditch**
Full of hipsters, artists, and interesting people, this diverse neighborhood on the east side of town feels a bit more off the beaten path. Shoreditch is widely regarded as the trendiest neighborhood in London. You'll find cocktail bars with aproned bartenders who take their craft seriously ( **Nightjar** ), artisanal beer houses ( **Brewdog** ), and fun clubs. The nearby **Brick Lane** area also caters to the trendy hipster set with skinny jeans and beards.
Shoreditch, Tube: Old Street
##### **Camden Town**
Camden is the heart of London's punk and bohemian scene. Revelers flock here for the underground music, as reflected by residents with their many tattoos, piercings, and dyed hair. While it's sliding toward the mainstream, you can still find excellent live music in venues like **Proud Camden.**
Camden, Tube: Camden Town
##### **Soho**
One of the hippest neighborhoods in London, Soho offers numerous music venues, off the wall bars, and bohemian cafes. Soho is also the hub for the LGBT community and has many gay-friendly bars and clubs.
Soho, Tube: Tottenham Court Road
#### **BARS & PUBS**
Spend a few days in London and you'll notice the sheer quantity of pubs—one on nearly on every corner throughout the city. And they all have proud, bold signage featuring names that stretch your imagination, like **Shakespeare's Head** (29 Regent St), **Ye Olde Cheshire Cheese** (145 Fleet St), and the **Hung Drawn and Quartered** (26-27 Great Tower St). King Richard II declared in 1393 that pubs must have signs in front of them to mark their location. So pubs sprang up with names that would differentiate them from all the others in town. The signs featured big, bright illustrations that still hang today, because writing the names of the pubs was mostly useless because of the modest literacy rate at the time.
##### **The Anchor**
If you're looking for the classic London pub, check out this historic one, located right on the banks of the Thames, on the Millennium Mile between the Borough Market and Shakespeare's Globe. Pint-sized exposed-brick rooms with overstuffed red leather chairs and benches with your classic dark wooden decor make this a favorite pit stop for tourists, locals, and even tour guides.
**LGBT LONDON**
The gay scene in London is alive and well. All you need to know is **Soho.** In this tangle of streets in central London, you'll find hotels, shopping, bars, restaurants, cafés, dance bars, clubs, lounges, and theaters that cater to the gay community, especially along Old Compton Street. For drinks and good music, try **G-A-Y** (30 Old Compton St, +44 20 7494 2756) or **SheSoho** (23-25 Old Compton St, she-soho.com). **Heaven** (covers around _£_ 5, coat check _£_ 1, 11 The Arches, +44 (0)20 7930 2020) is the go-to club for a fun night out. They've got a massive dance hall with three bars and a side room for hip-hop—and the drinks are strong! All are welcome for the show. Check the website for events. Find Heaven underneath Charing Cross station.
Pints from £4, daily 11:00-23:00, 34 Park St, South Bank, +44 (0)20 7407 1577, Tube: London Bridge
##### **The London Cocktail Club**
For one helluva classy cocktail bar that expertly blends life's simple pleasures with high-society London, check out the London Cocktail Club. With innovative options like Jägerbombs from an oyster shell, martinis featuring fried bacon, and even a sexy love potion that comes complete with a condom (in a wrapper), your mind is sure to be blown. While the unpretentious bartenders are experts at whipping up bizarre concoctions, they don't forget the staples like old-fashioneds and tasty mojitos. In addition to the Covent Garden location, you'll find branches of the London Cocktail Club at 224a Shaftesbury Avenue, 61 Goodge Street, 29 Sclater Street, 6-7 Great Newport Street, and 4 Great Portland Street.
Drinks along with the experience £8-10, Mon-Sat 16:30-23:30, 6-7 Great Newport St, West End, +44 (0)020 7836 9533, londoncocktailclub.co.uk, Tube: Leicester Square
##### **The Rocket**
This pub, right next to King's Cross Station, is a great place for students and locals to grab a casual drink or hit the dance floor rocking out to Top 40 on weekends. Imbibers flock here for the fun, casual atmosphere. This is the type of place you can't help but stop into when you walk by. Get there early if you're hoping to not wait in line.
Drinks from £4.50, Mon-Thurs and Sun 08:00-24:00, Fri-Sat 08:00-02:00, 120 Euston Rd, British Museum Neighborhood, +44 (0)20 7388 0021, therocketeustonroad.co.uk, Tube: King's Cross
##### **Freud**
With a name like Freud you might expect a bawdier crowd, but this little basement bar is a favorite of the classy hipster set. Come out for some "speed cocktails" (i.e., cocktails you can make quickly) and enjoy the welcoming scene. Instant gratification, eh?
**LONDON ALE TRAIL**
For a mile-long stretch of London's best ale houses, look no further than the London Ale Trail. It wanders through London's oldest streets in Westminster and into the City. Starting near the Holborn Tube stop and wrapping up at Blackfriars, this route is an excellent way to sip the afternoon away without too many steps in between.
**The Princess Louise**
(208 High Holborn): Customers have been knocking back the ales since 1891 in this renovated ale house with dark wood Victorian interior. The classic Brit pub offers no music or TV.
**Cittie of Yorke**
(22 High Holborn) **:** Step back in time in another classic pub serving both ales and hearty fish-and-chips and pies.
**The Knights Templar**
(95 Chancery Ln): With high ceilings, carpeted floors, and large windows, the Knights Templar feels more like a living room than one of London's more famous pubs.
**The Old Bank of England**
(194 Fleet St): Power lunchers come here for the stunning interior, and since it's just around the corner from the high courts, you may well see some big shots enjoying a snack and pint before heading back into their legal proceedings next door. You pay for the sumptuous interior; pints break _£_ 4, and pies top _£_ 9.
**Ye Olde Cheshire Cheese**
(145 Fleet St): This amusingly named venue is easily one of London's most historic pubs.
**The Blackfriar**
(174 Queen Victoria St): I love this pub for its charm and beautiful interior, with intricate mosaics and detailed woodwork. You feel like you're stepping into a chapel that serves beer and fish-and-chips.
Drinks from £5, daily 11:00-23:00, till 01:00 on weekends, 198 Shaftesbury Ave, West End, +44 (0)20 7240 1100, freud.eu, Tube: Tottenham Court Road
##### **Porterhouse**
Brewing its own collection of beers, the Porterhouse is a dependable sports café. Come out to catch just about any game televised, or descend into the dark basement to enjoy your fresh brew in one of the nooks and crannies spread out across several levels in this creatively constructed pub. In good weather, come early to snag a spot on the partly sunny patio. It packs out once the workday is done!
Pints from £4.50, Mon-Fri 12:00-23:00, Sat 12:00-24:00, Sun 12:00-22:30, 21-22 Maiden Ln, West End, +44 (0)20 7379 7917, theporterhouse.ie, Tube: London Charing Cross, Leicester Square, or Covent Garden
##### **Bounce**
Ever spent a night out at a "social Ping Pong club"? Come out to Bounce to experience Europe's first and largest Ping Pong club, featuring the finest tables and paddles around. This place has it all: fun atmosphere, activities like Ping Pong to keep you busy, and delicious pizza upstairs.
Half hour on the table £13, Mon-Thurs 16:00-24:00, Fri-Sun 12:00-01:00, 121 Holborn, The City, +44 (0)20 3657 6525, bouncepingpong.com, Tube: Chancery Lane
#### **SPEAKEASIES**
One of London's hottest current trends is the Prohibition-era cocktail bar in a low-lit lounge setting. Sprinkled across town, they're intentionally hard to find, but if you follow my tips on where to look, you'll be having one of the best old-fashioneds in your memory—or why don't you try a nice gimlet?
##### **Nightjar**
Step back in time to a speakeasy-esque experience at the Nightjar. Look forward to live music, a cozy atmosphere, and excellent Prohibition-era cocktails in this trendy little bar. Due to its popularity, it can be quite hard to get in. Dress the part to better your chances at the door.
Cocktails £10-12, daily 18:00 till late, 129 City Rd, Shoreditch, +44 (0)20 7253 4101, barnightjar.com, Tube: Old Street
##### **Underdog at Brewdog**
For great cocktails and brews, head to Brewdog, a lively bar. Even better, find the canvas sheet at the bottom of the stairs near the main door. Pull it aside and take the stairs down to the clandestine Underdog bar and music venue. Underdog puts on great live music and even DJ sets throughout the week, and the trendy, hipster crowd is less pretentious than what you'll find at similar places elsewhere. After its secret entrance and stairway, the venue is actually much larger than what you'd expect at a "speakeasy," so the party gets that much louder and that much better.
Drinks from £5, Mon-Thurs and Sun 12:00-24:00, Fri-Sat 12:00-01:00, 51-55 Bethnal Green Rd, Shoreditch, +44 (0)20 7729 8476, brewdog.com/bars/uk/shoreditch, Tube: Aldgate East
##### **Mayor of Scaredy Cat Town**
At the back of the Spitalfields location of the Breakfast Club, find a secret bar that serves exquisite cocktails like a Peat-nut Butter Cup. You've never had peanuts with a bourbon and Scotch kick like this! Their beautifully presented Bombay Sapphire is to die for. Enter through the refrigerator door by giving the password ("I need to speak with the mayor") to get into this retro, American-style diner and cocktail lounge. Be sure to read and obey the house rules! A full food menu available as well, in all its diner goodness (think burgers and pork sandwiches).
**ACT LIKE A LOCAL**
**Be Trendy**
Londoners take keeping up to speed seriously, from fashion and food to neighborhoods and living arrangements. Pick up a **_Time Out_** magazine to know what I'm talking about. There is something going on every night of the week, which is part of what makes this town so lively and fun. Year after year, the most popular neighborhoods are constantly shifting to different locations, and lots of effort is put into being "indie" or "underground." As soon as that unique flavor becomes mainstream, the trendsetters set off to somewhere new.
Drinks from £10, Mon-Thurs 17:00-24:00, Fri 15:00-24:00, Sat 12:00-24:00, Sun 12:00- 22:30, 12-16 Artillery Ln, Shoreditch, +44 (0)20 7078 9639, themayorofscaredycattown.com, Tube: Aldgate East
##### **Experimental Cocktail Club**
If you like classics with a twist, this dimly lit, Roaring '20s-era cocktail lounge is for you. Utilizing proprietary syrups and bitters along with fresh ingredients, these guys have rejuvenated your grandma's vodka tonic. Besides updates on your standard vintage cocktails, the ECC has invented an extensive list of new cocktails featuring exotic liquors, top shelf spirits, and fresh ingredients like homemade ginger syrup and habanero bitters. For a refreshing splurge, try the Watermelon Cooler. They're absolutely capable of making the classic lineup of cocktails, too. All drinks are served in the appropriate vessel, whether that's a lowball or martini glass, which adds to the experience.
Innovative cocktails from £9.50, daily 18:00-03:00, 13a Gerrard St, West End, no phone, chinatownecc.com, reservation@chinatownecc.com, Tube: Leicester Square
#### **CLUBS**
##### **Fabric**
Of all the clubs in London, Fabric, with an epic reputation on the electronic music scene, is the one that cannot be missed. Located just north of The City, it's famous for its resident DJs and its "bodysonic" room, which shakes you to your core with sound waves. You'll think that half of London is waiting in the queue with you to get in. Fortunately for you, if you stay at one of my suggested hostels, they can make a reservation for you—and drop the cover down to half price.
Tickets online from £7, Fri-Sun 23:00-05:30, sometimes open later, 77A Charterhouse St, The City, +44 (0)20 7336 8898, fabriclondon.com, Tube: Barbican
##### **Piccadilly Institute**
Explore this labyrinth of rooms, each decorated according to an insane asylum theme, exploring different shades of craziness. I've had great experience with the service and music choice at this venue with five bars and two clubs, which draws a young, stylish crowd. Dress with nice shoes and collars to get past the bouncers.
£5 cover weekdays, £10 cover weekends, daily 17:00-late, cover charges after 21:00, 1 The London Pavilion, West End, +44 (0)20 7287 8008, piccadillyinstitute.com, Tube: Piccadilly Circus
##### **XOYO**
XOYO is a London clubbing classic, well-established on the scene. Any big-time DJ coming through is sure to schedule a gig here. Two floors and top-of-the-line sound systems mean you can rock the night away till the sun comes up. Consider avoiding the long lines by getting on the guest list online or hitting the club up on a Monday or Tuesday. Music is of the indie electronic DJ set.
Cover varies up to £20, Mon, Tues, Thurs, Fri, Sat 20:00-03:00, 32-37 Cowper St, Shoreditch, +44 (0)20 7354 9993, xoyo.co.uk, Tube: Old Street
##### **Ministry of Sound**
A true institution, Ministry of Sound got so popular and so big that they've created their own recording label. Located on the south side of town (about a 15-minute walk south of the River Thames), the Ministry puts their commitment to music and sound above all else, especially to the beats of house and techno. They're clearly onto something, with a packed house every night since the early '90s. Note that this area can get a bit dodgy later at night.
Pricey covers up to £20, Tues and Fri -Sat 22:00-late, 103 Gaunt St, South Bank, +44 (0)20 7740 8600, ministryofsound.com, Tube: Elephant & Castle
##### **Proud Camden**
What was formerly an equestrian hospital is now a massive multipurpose venue hosting everything from private events to drag shows and live music concerts. Former horse stables are now VIP booths, fitting up to 25 parties each. Each has been renovated in a unique style, such as Asian pinup tattoo, the Desert Orchid room, and the gold bedazzled '70s disco room. Enjoy a meal here, or even relax in the sun in their garden and take a dip in their hot tub. When the sun goes down, the party heats up with live music—primarily indie rock bands—taking over the large dance hall. Proud Camden truly is a microcosm of Camden in London, all of the funkiness distilled into one venue.
Check the events listings online ahead of time to see what's going down. The crowd is young, hip, and professional: a well-dressed set that likes to let their hair down.
Show tickets run from £10, drinks from £4, Wed 11:00-01:30, Thurs-Sat 11:00-02:30, The Horse Hospital, The Stables Market, Chalk Farm Rd, Camden, +44 (0)20 7482 3867, reservations through opentable.co.uk, Tube: Camden Town
#### **WINE BARS**
##### **Gordon's Wine Bar**
Dating back to the 1890s, this place feels like a step back to the Middle Ages. That feeling is due mostly to the candlelit brick cellars you're delving into. On nice days, get a bottle and enjoy the shaded outdoor terrace. The food selection goes well with the wines—from finger food and tapas to meat dishes and Sunday roasts, rest assured you can get both your drink and your eat on at Gordon's.
Reasonable bottle prices from £15, daily 11:00-23:00, 47 Villiers St, West End, +44 (0)20 7930 1408, gordonswinebar.com, Tube: Embankment or London Charing Cross
##### **Cork & Bottle**
You may be surprised to find food and wine of this caliber in such a touristy district, just a block off Leicester Square. But at Cork & Bottle, count on an incredible selection of international wines available by the glass or bottle, along with a slew of unforgettable entrées and finger food like Cajun chicken and an impressive ham-and-cheese pie. Cork & Bottle feels like a cozy little slice of France, with hardly a tourist in sight. The servers are happy to walk you through both the wine list and menu.
Sides and salads from £7, mains from £13, daily 11:00-24:00, 44-46 Cranbourn St, West End, +44 (0)20 7734 7807, thecorkandbottle.co.uk, Tube: Leicester Square
#### **PUB CRAWLS**
It is still possible to find a sub- _£_ 3 pint in central London; it just takes some determination. Pub crawls offer guided parties that run nightly in London and can be a good choice if you're looking to save money and meet travelers. The savings you get from discounted drink deals and what you would have spent on club entry will cover the ticket price.
##### **1 Big Night Out**
This pub crawl operates nightly 19:30-21:00 in Leicester Square.
£15, meets nightly at ZOO Bar, 17 Bear Street, +44 20 7836 9995, 1bignightout.com, Tube: Leicester Square
##### **Camden Pub Crawl**
As the name suggests, Camden Pub Crawl operates in Camden. Pub crawls run from 19:30 to 21:30.
£14 weekdays, £16 weekends, save £2 by booking online, meets nightly at Belushi's Camden, 48-50 Camden High Street, undiscoveredlondon.com, info@undiscoveredlondon.com, Tube: Mornington Crescent
##### **London Gone Wild**
London Gone Wild is a pub crawl option in Shoreditch.
£13.50, meets Thurs, Fri and Sat between 20:00-21:00 at The Shoreditch, 145 Shoreditch High St, +44 20 7096 0371, nightsgonewild.com, Tube: Aldgate East
### **TOP SHOPPING & MARKETS**
#### **SHOPPING DISTRICTS**
##### **Oxford Street**
Oxford Street is quite possibly London's single best shopping street and stretches for just about one mile from east to west on the north side of the city. It's here you'll find all the major retailers, along with **Selfridges** (Mon-Sat 09:30-21:00, Sun 11:30-18:15, 400 Oxford St, +44 (0)113 369 8040, Tube: Bond Street), one of London's famous top-end department stores. Be sure to venture southward off of Oxford down **Regents Street** to top off that shopping fix of yours.
West End, Tube: Oxford Circus
#### **MARKETS**
##### **Harrods**
Visit this massive monument to consumerism to get a taste of what it feels like to be a lowly commoner. Back in the day, London was one of the few cities with enough concentrated wealth to support and kick off what has become the modern department store and mall. Everything—from luxury chocolates to designer lingerie and large-than-life stuffed Paddington Bears to posh home furnishings—is sold across seven full floors of hedonism. Harrods is basically your one-stop shop for everything high end that you could ever want.
Mon-Sat 10:00-20:00, Sun 11:30-18:00, 87-135 Brompton Rd, West London, +44 (0)20 7730 1234, Tube: Knightsbridge
##### **Portobello Market**
Every Saturday, one of the world's biggest antiques markets takes over the Notting Hill neighborhood all up and down Portobello Road. If you're actually planning on buying the jewelry, vintage clothing, and other items on offer, be sure to bring enough cash, because the ATMs often run out or have insane lines. Saturday mornings before 10:00 are best, as you've got the selection without the crowds. Be sure to stop into the famous **Hummingbird Bakery** (cupcakes from _£_ 1.50, Mon-Fri 10:00-18:00, Sat 09:00-18:30, Sun 11:00-17:30, 133 Portobello Rd, +44 (0)20 7851 1795) for a cupcake to give you that sugar rush to get you through the day.
Mon-Wed and Fri-Sat 08:00-18:30, Thurs 08:00-13:00, West London, Tube: Notting Hill Gate
##### **Camden Market**
London's original craft market dates back to the 1970s. Today, it's a collection of extremely popular street markets, seeing tens of thousands of visitors each weekend. Explore large open-air clothes vendors and thrift shop-type stalls, food shops, and, my favorite, a large shopping complex in a repurposed barn. Save time for food, and stick around for the nightlife, as this is one of London's most intriguing neighborhoods and the center of the alternative fashion scene. The market, open daily, is busiest on weekends.
Daily 10:00-18:00, Camden, Tube: Camden Town
##### **Borough Market**
South of the river, this food market is _the_ place to challenge your taste buds, with loads of free samples ranging from wild boar to ostrich sandwiches. Conveniently located near the Millennium Mile, Borough Market has permanent shops, restaurants, and cafés built into the ironwork of the awnings, along with farmer stalls that pop up daily to sell fresh goods. Those who just want to nibble can pick up cheeses and meats by the slice. Otherwise, dive into the delicious sandwiches, wraps, and smoothies available throughout the market.
Mon-Wed 10:00-15:00, Thurs 11:00-17:00, Fri 12:00-18:00, Sat 08:00-17:00, 8 Southwark St, South Bank, boroughmarket.org, Tube: London Bridge
##### **Old Spitalfields Market**
This grand covered arts-and-crafts market has been located here for the better part of 400 years. Today, find items like graphic tees, hipster shoulder bags, real leather belts, and hand-made art pieces and frames. In the massive barn-like warehouse are both permanent retail stores and semi-permanent stalls. The market makes for great window shopping and even better photos. Skip the overpriced restaurants inside and opt for food at the **Brick Lane Market** nearby.
Daily 08:00-23:00, 16 Horner Square, Shoreditch, Tube: Liverpool Street
##### **Brick Lane Market & Sunday Upmarket**
Brick Lane Market is a collection of warehouses and open streets where you can wander and discover untold pop-up shops, custom bike stores, and tons of bars and cafés. Sundays, when street performers and crowds create quite a buzz, are the best day to visit. If vintage is your thing, come here for heaven on earth. Find the **Boiler House** (open Sun, 91 Brick Ln) for yet another international food hall.
Near the Boiler House, what was an old brewery is now the **Sunday Upmarket** , an international food extravaganza. Street food stalls sell everything from Greek to Indonesian, Brazilian to Nepalese. Grab your food at each counter, and find an empty spot on a bench to chow down. You'll leave here with a full belly without hurting your wallet. Between courses, stop at the clothing stalls selling hand-made and designer pieces.
Permanent stores and shops are open throughout the week, best day for the food halls is Sun 10:00-17:00, 91 Brick Ln, Shoreditch, +44 (0)20 7770 6028, bricklanemarket.com, Tube: Algate East
### **TOP PARKS & RECREATION**
#### **St James's Park**
St James's Park is right off of Whitehall Street near the Parliament building, conveniently providing a direct path between Buckingham Palace and the Horse Guards Parade. You can walk across this park in barely five minutes. Wander through the gardens and have a sandwich on a bench if you need a break from the hustle and bustle of the city. Enjoy the well-manicured meandering paths wrapping around a beautiful little lake.
Westminster, Tube: St James's Park
#### **Hyde Park**
London's "Central Park" is the largest in the city, spanning more than 350 acres. Highlights include the Diana Memorial Fountain, the large Serpentine Lake, and the Speakers' Corner, a place where Londoners can come and speak their minds loudly and proudly on any subject they wish. Take **paddle boats** ( _£_ 10/30 min, _£_ 12/1 hr) out on the lake, or enjoy a drink at any one of the outdoor cafés that line it. Any time there are important sports matches going on, thousands flock to the park to watch on massive projector screens. Music events and festivals take place frequently throughout the year, too.
West London, Tube: Hyde Park Corner
#### **Battersea Park**
Well off the beaten track for most tourists, this is my favorite park in town because it offers miles of running trails, waterfront views, and a unique and spectacular 100-foot-tall Buddhist Peace Pagoda monument to reflect on. It's also the site of a huge bonfire and light show held every November 5 to commemorate the thwarting of a terrorist plot to blow up the Houses of Parliament in 1605.
South Bank, Tube: Sloane Square
### **TOP TOURS**
#### **Fat Tire Bike Tours**
My favorite way to see the city! Take the **Royal Gardens Bike Tour** and enjoy the musings of a great guide through some of the city's most beautiful scenery. Learn a ton from your funny and informative guide as you cruise through Hyde Park, Kensington Gardens, Green Park, and down the Mall, stopping at each interesting point and photo op along the way. I like how these bike tours avoid street traffic as much as possible, sticking to parks for the most part.
£18, low season Mon and Thurs-Sun 11:00, shoulder season daily 11:00, high season daily 11:00 and 15:00, meets at Queensway Tube Station, West London, fattirebiketours.com/London, Tube: Queensway
#### **Sandeman's New Europe Walking Tours**
Take a free walking tour and see some of London's most popular sights with your own personal guide. Satisfied? Tip your guide for a job well done—their earnings are purely gratuity based. You'll learn about some of the top sights in London over this three-hour tour, including Big Ben, Buckingham Palace, Westminster Abbey, and Churchill War Rooms.
Free (tip based), daily 11:00 and 13:00, meets by Wellington Arch, Westminster, neweuropetours.eu, Tube: Hyde Park Corner
### **TOP HOSTELS**
#### **Clink78 Hostel**
If buildings could talk, this hostel would certainly have some stories to share. Originally a courthouse, it's made up of dorms that used to be prison cells and an Internet café and movie room that used to be courtrooms. And for those of you who need to brush up on your rock 'n' roll history, this hostel is where the famous punk group The Clash went on trial back in 1978. Today, you'll find a large hostel with comfortable beds and a very social atmosphere with an on-site Clash-themed bar and music. Some complain about the noise and sleep quality, but for those looking for a good time out in London, Clink78 is a great choice.
From £10, £3 Wi-Fi, 24-hour reception, laundry, bar, free breakfast, 78 King's Cross Rd, British Museum Neighborhood, +44 (0)20 7183 9400, clinkhostels.com, reservations@clinkhostels.com, Tube: Kings Cross St Pancras
#### **St Christopher's Village, London Bridge**
Get your kicks here if you're looking for your first ever London hangover. London's ultimate party hostel features a basement nightclub, DJs, a bar and restaurant, karaoke nights, a cinema, and a comedy club. Its rooms and bathrooms are clean, and it's next to great sights such as the Tower of London, Tower Bridge, and the famous Tate Modern. Other St Christopher locations in Camden, Greenwich, Hammersmith, and Shepherd's Bush feature similar social scenes and amenities. Check them out and book online at st-christophers.co.uk.
From £15, free Wi-Fi, free breakfast, 24-hour reception, free lockers, 161-165 Borough High St, South Bank, +44 (0)20 7939 9710, st-christophers.co.uk, village@st-christophers.co.uk, Tube: London Bridge
#### **Generator Russell Square**
Generator hostels are dotted all over Europe. All of them seem to offer a reliable standard: clean rooms, affordable prices, and more amenities than you know what to do with. Definitely aimed at the younger generation of travelers, this hostel has its own bar, game room with pool tables, free walking tours, a lounge equipped with multiple flat screens, and daily events like karaoke night and drinking games. It's also just a five-minute walk from Euston, St Pancras, and King's Cross Tube stations and train stations.
From £27, free Wi-Fi, laundry, 24-hour reception, lockers, 37 Tavistock Plaza, British Museum Neighborhood, +44 (0)20 7388 7666, generatorhostels.com, info@generatorhostels.com, Tube: Russell Square
#### **Wombat's**
One of the Wombat chain's newest locations has opened up to rave reviews. Just outside The City on the edge of East London, Wombat's puts you right where you want to be for sightseeing by day and partying by night. It's a five-minute walk from the Tower of London and Tower Bridge, and ten minutes from the Brick Lane food scene and the nightlife in Shoreditch. Count on a welcoming social atmosphere, a helpful staff that provides good recommendations, and delicious breakfast. It's done up in shabby chic decor with a reception desk that's built from recycled lumber. You'll feel cool from check-in till you're downing drinks under brick arches at the on-site bar. Rooms are clean and bright, featuring a plug and reading lamp at each bunk.
Beds from £24, 24-hour reception, on-site bar, hair dryers, adapters, and towels available, pool table, free Wi-Fi, breakfast extra, 7 Dock St, The City, +44 (0)20 7680 7600, wombats-hostels.com, booklondon@wombats.eu, Tube: Tower Hill
#### **Astor Hyde Park Hostel**
The Astor chain is a great option for budget travelers. Set in a 19th-century mansion, this old building has been retrofitted to feature all the modern amenities but retains its authentic wooden soul. Beds are comfortable, the staff is extremely welcoming and helpful, and the location near Hyde Park is spectacular. Simple rooms offer everything from private twins up to 12-bed mixed dorms—bunks are constructed of tubular steel. The Hyde Park branch is my favorite Astor location, but other options include Victoria, Queensway, and near the British Museum.
Bunks from £18, free Wi-Fi, 24-hour reception, laundry facilities, common room with foosball table; towels, hair dryers, and breakfast extra, 191 Queensgate, West London, +44 (0)20 7581 0103, astorhostels.com, hydepark@astorhostels.com, Tube: Gloucester Road
#### **Barmy Badger Backpackers**
The Barmy Badger includes everything in your booking. There are no hidden fees, meaning Wi-Fi, continental breakfast, power sockets next to your bed, kitchen use, and "travel advice" are freely offered and included in the price of your bed. Get extras like towels and adapters for a refundable deposit. The owner's two dogs are thrown in for good measure, too. While the bunks are stacked three high in the six-bed dorms, each cubby sports a comfortable mattress, personal light, and charging port, so you get a solid night's sleep. The place feels a bit like your aunt's town home in the posh Earls Court district of London, complete with a large living room and a nice couch to sink into.
Six-bed dorm £24, privates from £65, 24-hour reception, cable TV, free maps, laundry facilities, free Wi-Fi, outdoor terrace, breakfast included, towels with deposit, 17 Longridge Rd, West London, +44 (0)20 7370 5213, barmybadger.com, barmybadger@hotmail.com, Tube: Earls Court
### **TRANSPORTATION**
#### **GETTING THERE & AWAY**
Although England is an island, international travel into London has never been easier thanks to the city's efficient infrastructure and extensive public transit system.
##### **Plane**
The city of London has five international airports. All have public transportation options into the center.
**London City** (LCY, londoncityairport.com) is the airport closest to downtown London. Hop on the **DLR** (dlrlondon.co.uk), a commuter train that connects to and uses the same tickets as the Tube, and transfer onto the Tube system at Canning Town to connect into the rest of London. Your journey from the airport to the center will cost _£_ 4.80.
From **Heathrow** (LHR, heathrow.com), take the Piccadilly Tube line straight into the center (this takes about an hour, stopping at every station along the way) and connect to your accommodation's Tube stop from there. Single fare runs _£_ 5.70. Other connection options are longer, overpriced, and not worth it.
From **Gatwick** (LGW, gatwickairport.com) take the **Southern Train** (southernrailway.co.uk) service, which runs every 15 minutes, to Victoria station (45 minutes, single fare _£_ 15). You'll be tempted to spring for the **Gatwick Express** (30 minutes, single fares from _£_ 18, gatwickexpress.com), but the Southern Train saves you cash without sacrificing too much time.
**Luton** (LTN, london-luton.co.uk) has connections into London every 15 minutes via **EasyBus** (easybus.com); the 80-minute journey puts you at Victoria station. Or take a shuttle to Luton Airport Parkway, which offers different rail services that can get you into the center. For the cheapest fares, visit **Trainline** (thetrainline.com).
From **Stansted** (STN, stanstedairport.com), your best options are either the **Stansted Express** train ( _£_ 22.50, runs every 15 min, 45-min journey, stanstedexpress.com), which will take you to Liverpool Street station, or **EasyBus** (easybus.com, runs every 20 min, 75-min journey), which will take you to Baker Street station. Visit easybus.com for pricing. Buses can take three hours in rush hour, so be sure to factor that in!
Some budget airlines are also beginning to offer cheaper flights into regional airports well outside of London. Note that flying into these can add considerable time and costs to your travel plans. From London South End, for example, it takes at least 1.5 hours to get to the center.
##### **Train**
There are nine major train stations in London. All are connected to the Tube. Rates from Paris—on fast trains taking just under 2.5 hours—range from about _£_ 42 (€50) and up, depending on student discount and how far in advance you book your ticket. For more information, visit the **Eurostar** website (eurostar.com).
##### **Bus**
There are bus options to London from the rest of the UK as well as Paris, Amsterdam, and Brussels. Rates can be significantly cheaper than your train options, but remember; hours spent on buses can add up quickly. Check out routes and fares at **Eurolines** (eurolines.com), **Megabus** (uk.megabus.com), and **Student Agency** (studentagencybus.com).
##### **Car**
By car, London is 460 kilometers (about 5.25 hours) from Paris and 660 kilometers (about 8 hours) from Edinburgh.
#### **GETTING AROUND**
London is a massive but very walkable city. **Walking** is a great way to explore one block at a time. Keep an eye out for dark blue neighborhood map signs posted around the city near bus stops. These are a great way to get oriented to the neighborhood immediately surrounding you. To give you a sense of scale, the walk from Big Ben to the Tower of London takes about an hour at a relaxed pace.
Prepare to be amazed, and possibly intimidated, by the London **public transportation** system. Comprising 11 metro lines and over 700 bus routes, it's one of the best and most extensive public transport systems in the world. Buy yourself an **Oyster Card** ( _£_ 8, includes a _£_ 5 deposit) at any Tube station to reduce the cost of your bus and Tube tickets by about half.
##### **Tube**
The Underground, also known as the Tube, closes at midnight. (Remember that before getting too deep into your drinking routine!) At the time of writing, 24-hour service was in the works. A single journey costs _£_ 4.80 ( _£_ 2.30 with the Oyster Card). Pick up a free metro map from the desk at any Tube station. Or use the **TubeMap** app, which is just what it sounds like. It also stays up to date, advising you with the latest closures.
##### **Bus**
If you ever find yourself lost or disoriented, each little bus stop has a map that shows you where you are and which buses stop there. Do your homework before going out at night to see which night bus can take you back home. Ask the driver when you exit the bus if you are ever in doubt of your return stop. A single journey costs _£_ 2.20 ( _£_ 1.30 with the Oyster Card). Map out your journey beforehand with the **Citymapper** app.
**Bus 11** between Liverpool Street station and Victoria offers an excellent cross-section of London with an orientating route leading from The City into Westminster and past the royal parks to Buckingham Palace.
##### **Taxi**
The taxi industry in London is highly regulated and quite competitive. It's also quite expensive (about _£_ 8.50/12-min ride). If you need to call for a taxi, use only a licensed **Black Cab** or a taxi service recommended by your hostel. A good number to call if you're unable to hail a taxi from the street is **087 1871 8710,** which gives you access to hundreds of licensed taxis 24 hours a day. Cabbies evaporate when all the bars let out, and they're nearly impossible to wave down. **Uber** is now well-established in London.
##### **Bicycle**
**Santander Cycles** (tfl.gov.uk/modes/cycling/santander-cycles) runs London's public bike-rental system. For _£_ 2, you get access to the bikes. Each ride is free as long as you check it back in to a Santander stand within 30 minutes. If you want to keep the bike longer, each subsequent half hour costs _£_ 2. Find more information online and in the app store.
### **DAY TRIPS**
#### **Leeds Castle**
It's easy to get lost in the urban jungle that is London. Thankfully, a two-hour train ride can zip you from the city center to rural England, where you'll find quite a different setting, completely transforming your perspective with a castle fit for a king. From humble medieval beginnings, this castle was rebuilt and rebuilt to now stand as a proud example of Tudor architecture and defenses. The castle played home to six of England's queens and Henry VIII to boot. Today, the castle is a private museum with well-manicured grounds and a hedge maze to get lost in. It's often deemed England's "loveliest" castle.
The castle and grounds are located in Kent. From London, take the Southeastern Rail from Victoria station. In about 1.25 hours, you'll arrive to Bearsted. From Bearsted, it's a 10-minute taxi ride straight to the castle. The whole trip will cost you about _£_ 40 round-trip, not including entry into the castle. Tour services like **Golden Tours** (goldentours.com) run from London frequently and offer day trips from _£_ 84.
Pricey individual entry at £24, daily 10:30-17:00, +44 (0)1622 765400, leeds-castle.com
#### **Stonehenge**
This world-famous prehistoric rock formation is within reach of a day trip from London! The purpose, meaning, and possible construction methods of this monument—carbon dated to before 3000 BC—continue to evade experts. So we are left to use our own imaginations to try to figure out how and why such massive stones (weighing up to four tons each) were hoisted and placed in perfect alignment, all without machinery, modern engineering, or even nylon ropes.
An early morning trip will get you out to Stonehenge in time to enjoy it for a bit. Numerous day-trip options exist; they are worth it because public transportation options can take up to four hours. **London Toolkit** offers a day trip with a stop in Bath ( _£_ 47, londontoolkit.com). **Premium Tours** has a Stonehenge-only option (from _£_ 35, premiumtours.co.uk). **Viator** (viator.com) has curated options available online.
I recommend booking Stonehenge with a tour that also takes you to **Windsor Castle,** a sprawling Tudor and English gothic-style royal castle with foundations dating back to the 10th century, or **Bath,** an old Roman fort town named after the hot springs that bubble out of the ground in the area. Otherwise, it's quite a long time on the bus to see a stack of large rocks, no?
£14.50, £13 with valid student ID, daily 09:30-19:00, closes at 17:00 in winter, last admission 2 hours before closing time, +44 (0)0370 333 1181
### **HELP!**
#### **Tourist Information Centers**
London has excellent services for visitors. All the information you'll need for your visit can be found online (cityoflondon.gov.uk and visitlondon.com). For many sights, you can even book tickets through these websites. Keep in mind there are shops throughout London that purport to be unbiased tourist resources but in fact make money selling you shows and tickets. This is London's only impartial tourist information center:
City of London Information Centre, St Paul's Churchyard, +44 (0)20 7332 1456
#### **Pickpockets & Scams**
As in the rest of Western European cities, the crime rate in London is pretty low. However, be alert for pickpockets in the larger touristy areas, restaurants, metros, buses, and trains. Teams on scooters have been known to snatch the phone right out of your hand while they drive by. If you need to call for a taxi, you should use only a licensed Black Cab or a taxi service recommended by your hostel.
#### **Emergencies**
In an emergency, dial 999.
#### **Hospital**
St Thomas' Hospital
Westminster Bridge Rd, Lambeth, SE1 7EH
+44 (0)20 7928 9292
#### **US Embassy**
24 Grosvenor Square
+44 (0)20 7499 9000
Paris Maps
Paris 101
Three Day Itinerary
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Help!
Paris is a treasure trove of art, cathedrals, and magnificent architecture. From the Eiffel Tower to Sacre Coeur, from _Mona Lisa's_ coy smile to the gargoyles that guard Notre Dame, Paris is full of refined beauty. But—like a deliciously flaky _pain au chocolat_ —the city's real treat lies beneath its surface. Even if you don't fall in love _in_ Paris, you will fall in love _with_ Paris. Dig deep into the city to discover the people, sights, sounds, tastes, and smells that will make your trip unforgettable.
### **PARIS 101**
Paris's location on a couple islands in the Seine River made the city, founded in 250 BC, a strategic stronghold. By 1200, Paris was one of the most powerful cities in Europe, with a massive castle, cathedrals, trade unions, and all the makings of a major capital. As the city evolved, the French royalty levied untenable taxes on their population to finance wars and opulent palaces like Versailles. Peasants were starving, and a groundswell of anger gave rise to the French Revolution in 1789. Over 10 long years, thousands of nobles were jailed or publicly beheaded by the guillotine.
The revolution was a success in terms of eliminating the ruling class. But it takes more than that. You've got to govern. By 1799, France was struggling to find a leader who could harness their nationalist fervor. A man rose through the ranks (and well beyond his short stature) to become self-proclaimed emperor of France. His name? Napoleon Bonaparte. After conquering much of Western Europe, Napoleon was exiled—twice. He escaped his first exile quickly, suffered tremendous losses at the battle of Waterloo, was exiled again, and lived out his days in disgraced solitude on a small island between Brazil and Africa. But Napoleon did leave his mark on modern society. Our legal and public educational systems and even units of measurement today derive from the developments Napoleon undertook during his reign.
As France entered the modern age, Napoleon III, Napoleon's nephew, was elected in France's first democratic election and reigned nearly 18 years. Napoleon III wanted to beautify his capital city, making it fit for an emperor. The new emperor commissioned his favorite architect and city planner, Georges-Eugène Haussmann, with the immense project of modernizing Paris. Thousands of Parisians were displaced to make room for new wide boulevards and the grand avenues that we see today.
Paris enjoyed a golden age around the turn of the 20th century. An engineer, Gustave Eiffel, completed his tower for the World's Fair, while an eclectic group of outcast artists flocked to the city and developed a new artistic style they called impressionism. The 20th century brought two world wars, but even through the difficult years, Paris continued to attract artists and bohemians. Paris was on its heels after World War II, but thanks to Allied support, the city began the process of reconstruction.
Today, France is liberal and progressive, and the Parisian joie de vivre is as palpable as ever.
### **PLAN AHEAD**
#### **RESERVATIONS**
Reservations are recommended for the following sights:
**Eiffel Tower** (toureiffel.paris/en)
**Louvre** (louvre.fr/en)
#### **PASSES**
##### **Paris Museum Pass**
The **Paris Museum Pass** (en.parismuseumpass.com) gives you free entry and line-skipping privileges at some of Paris's major museums, including the Louvre, the Orsay, and Versailles. Choose from two-day (€42), four-day (€56), and six-day (€69) options. The pass is particularly smart if you plan to visit Versailles, which costs €18 on its own. In peak tourism months, the line-skipping privileges are even more valuable than the savings. Purchase the pass online or at any participating museum. Activate it by writing the date of entry to your first sight.
#### **LOCAL HAPPENINGS**
##### **Tour de France**
The Tour de France, which loops more than 2,000 miles around France each July, is the world's most famous and competitive professional cycling event. Cyclists from all over the world drive themselves to near exhaustion in a three-week-long stage race. The starting point changes yearly (with recent years kicking off in England, Germany, Belgium, and the Netherlands), but the race always ends with eight laps through central Paris, up and down the Champs-Élysées and under the Louvre in a grueling sprint. The festive atmosphere draws tens of thousands of visitors every year. Weathered but exuberant cyclists take a slow victory lap afterward, celebrating the mere feat of completing the world's most punishing race.
##### **Bastille Day**
France's national holiday, Bastille Day (July 14), commemorates the day when rebels stormed the Bastille prison and kicked off the French Revolution. On Bastille Day, a grand parade is accompanied by fireworks, musical performances, dances, large communal meals, and, of course, wine and champagne. Expect the entire country—including sights and museums—to shut down. Don't fret; the parties are much more fun anyway!
**KNOW BEFORE YOU GO**
**KEY STATS & FIGURES**
**Currency:**
**France uses the euro (€); 1 EUR = about 1.06 USD**
**Population:**
**2.25 million**
**Language:**
**French**
**Number of metro lines:**
**14**
**National drink:**
**champagne**
**National dish:**
**crêpe**
**Year of last guillotine use:**
**1977**
**CALIBRATE YOUR BUDGET**
**TYPICAL PRICES FOR:**
**Hostel dorm bed:**
**€24**
**Two-course dinner:**
**€15**
**Pint of beer:**
**€6.50**
**Bicycle rental:**
**€12**
**Single metro pass:**
**€1.70**
**MOVIES TO WATCH**
_Midnight in Paris, Jeux d'Enfants (Love Me if You Dare), Chocolat, Les Misérables, Amélie_
THREE DAY ITINERARY
It's important to be efficient when making your rounds of Paris's top sights. Get ready to walk—a lot—and get comfortable on the metro to explore the best corners of this beautiful city.
**DAY 1: BIENVENUE Á LA GAUCHE**
**MORNING**
Grab breakfast at the hostel or pop into your local _patisserie_ (pastry shop) to try your first heavenly croissant or _pain au chocolat_ (chocolate croissant).
Head into Île de la Cité and spend a few hours exploring Paris's most quintessential sights: the epically gothic **Notre Dame** cathedral, the compelling and thought-provoking **Deportation Memorial** , and the stunningly beautiful royal chapel of **Sainte-Chapelle**.
Catch a free, three-hour Latin Quarter walking tour with **Discover Walks,** meeting at 11:00 (May 1-Oct 15 daily) at the equestrian statue in front of Notre Dame. This casual, tip-based walking tour will take you to the Left Bank, Paris's student and bohemian quarter. You'll wander through the narrow lanes while enjoying a fun narrative bringing the streets to life.
**AFTERNOON**
Your walking tour will take a lunch break at a local bakery. If you're on your own, **Les Delices du Fournil** offers fast and dependable lunch options: A baguette sandwich, dessert crêpe, and drink can all be had for around €5, a steal in the touristy Latin Quarter.
After your tour, hop the metro across town up to **Montmartre** (Metro: Blanche, Pigalle, or Abbesses) to explore the artistic and alternative bohemian neighborhood and make a visit to **Sacre Coeur**. If it's a nice day, grab a beverage and snack to tuck into on the steps, taking in the view and enjoying being serenaded by the buskers around you.
**EVENING**
After freshening up for the night at the hostel, head out to have dinner at **La Varangue** (Metro: Champ de Mars) in the shadow of the Eiffel Tower.
Cap off the evening by heading up the **Eiffel Tower** and taking in the City of Lights at its finest, just as the lights begin to twinkle. Grab a bottle or two of wine afterward to enjoy on **Champ de Mars**. Pick up some cheeses, salami, baguettes, and an extra bottle to share with fellow picnickers and make some friends!
**LATE**
Sufficiently wined, and dressed right, make your way over to **Showcase** (Metro: Invalides), Paris's glamorous river-front club located underneath the Pont Alexandre III, the same bridge that played the backdrop to Adele's "Someone Like You" music video.
**DAY 2: THE ROYAL RIGHT**
**MORNING**
Get a big breakfast to gear up for the **Louvre.** If you've got an EU student card or EU student visa in an American passport, bring it to get free entry! It'll take you about three hours to see the Louvre's destination masterpieces ( _Winged Victory, Venus de Milo,_ the French Crown Jewels, the large-scale paintings, and of course the _Mona Lisa_ ). Be sure to pick up a free museum map as you enter to help navigate the endless halls. Afterward, grab a sandwich in the well-appointed cafeteria at the mezzanine level overlooking the ticket lines underneath the main glass pyramid.
**AFTERNOON**
Hop on the metro, which will zip you uphill to the **Arc de Triomphe** (Metro: Charles de Gaulle-Étoile). Climb the arch for a beautiful panorama of Paris's most exclusive districts. Watch the traffic snake around the busiest intersection in France—and the only roundabout in France where incoming traffic has the right of way.
Spend the rest of the afternoon on the **Champs-Élysées,** shopping till you drop. Get lost wandering all the way down the grandest boulevard in the country.
**EVENING**
Stop back at the hostel to change for the night and to grab some layers. Head over to the Dupleix metro stop near the Eiffel Tower and catch **Fat Tire** 's amazing three-hour evening bike tour at 18:00. Enjoy the leadership of your fearless guide through the streets of Paris, and cap your ride with a cruise on the River Seine toasting to the good life (untold amounts of wine included!).
Because of the awkward start time, it may be a good idea to pick up a baguette sandwich and snacks on your way to enjoy during the tour. Otherwise, the bars you'll visit next also have kitchens for a later dinner.
**LATE**
Hop on the metro line from the nearby La Motte-Picquet-Grenelle metro stop over to Place de la République to kick off a crawl up the **Canal Saint-Martin** to experience Paris's trendy hipster nightlife. The funky **Le Comptoir General** (Metro: Goncourt) is just a couple blocks north along the canal and will be bumping by this time in the evening.
**DAY 3: CHOICES, CHOICES, CHOICES**
If you've got another full day in Paris, consider a day trip to either **Disneyland Paris** or the palace of **Versailles**. Alternatively, enjoy one of Paris's markets and a quiet afternoon at the city's most famous cemetery.
**MORNING**
Head to the trendy **Marché des Enfants Rouge** (Metro: Filles du Calvaire, closed Mon). Go for coffee on arrival, then take a lap to make the difficult choice of where you're going to want to eat. You'll have to choose between Italian, Korean, Latin American, veggie, sandwiches, and of course French street food: crêpes.
**AFTERNOON**
Following the market, walk 30 minutes or take the metro to **Parc des Buttes-Chaumont** , a beautiful manicured park that embodies 19th-century French romanticism, or take the metro to **Père Lachaise Cemetery** (Metro: Père-Lachaise) to view some of the most ornate graves around and to pay your respects to greats like Chopin, Jim Morrison, and Oscar Wilde.
### **TOP NEIGHBORHOODS**
Paris is divided into numbered neighborhoods, or _arrondisements_. Each _arrondisement_ has its own personality, but it's not all that helpful for visitors to understand the intricacies of each one on a short visit.
The River Seine bisects the city. Two small islands, **Île de la Cité** and **Île Saint-Louis** , rest in the heart of it all. These picturesque islands are home to Notre Dame Cathedral, the Deportation Memorial, and Sainte-Chapelle. To the west of these islands is the **Louvre,** on the north side of the Seine.
The north side of the Seine, known as the Right Bank, has historically been the wealthier part of Paris. The Right Bank is home to the grand shopping boulevard **Champs-Élysées,** which sits at its western edge. The chic **Marais** neighborhood, on the eastern edge of the Right Bank, caters mostly to rich yuppies but offers lively shopping and nightlife. The Marais area is also Paris's gay district, and is home to the Pompidou Center, a modern art museum. To the north, you'll find trendy nightlife, several recommended hostels, and the pleasant Parc des Buttes-Chaumont along **Canal Saint-Martin.** Finally, the bohemian district of **Montmartre** hovers just north of the city center. Here, you'll find the Moulin Rouge and the lovely hilltop cathedral of Sacre Coeur, with sweeping views of the city.
The south side of the Seine is known as the Left Bank. Traditionally home to starving artists, students, and philosophers, the Left Bank includes the **Eiffel Tower** and nearby Rue Cler market as well as the touristy **Latin Quarter,** just south of Île de la Cité and Île Saint-Louis. **Odéon,** west of the Latin Quarter, is close to Luxembourg Gardens and offers fun nightlife. In **Montparnasse,** find a couple of good restaurants that are worth the jaunt south of the Latin Quarter and Odéon.
### **TOP SIGHTS**
#### **Eiffel Tower**
It's not until you gaze upon this iconic building in person that you realize that the photos you've seen come nowhere close to doing it justice. Built in 1889 by Gustave Eiffel for the World's Fair, this 1,063-foot tower was the tallest building in an otherwise strictly height-capped Paris, and still offers the best views of the city. The building comprises three different levels. The first two are accessible by an elevator and by stairs, and the third level by elevator only. To save yourself some money, hike up the first two levels and then decide if you want to purchase the extra ticket for the ride to the top. I find the view is actually much better from the first two levels, from which you can see the city's landmarks in detail. At a thousand feet, the detail fades and the views are like looking out of a plane. Be sure to go up the Eiffel Tower, but don't forget to admire it from all parts of the city, up close and from far away. What was initially an eyesore for Parisians is now their dearest icon. After sunset, the tower sparkles with 20,000 computer-controlled strobe lights for five minutes every hour on the hour, drawing a fun crowd of picnicking spectators each night all up and down the Champ de Mars, the open field nearby.
**EIFFEL TOWER STATS**
The Eiffel Tower was built for the World's Fair in 1889. Though it was originally only meant to stand for a few years, this iron monument to industrialization is still breathtaking today. These stats will help you begin to comprehend its scale:
**Time to build:**
Two years, two months, and five days
**Cost to build:**
US$36 million in today's money
**Original purpose:**
Radio tower
**Height:**
1,063 feet
**Mass:**
7,300 tons
**Area of base:**
410 square feet
**Shift potential:**
7 inches at the top due to warming and expansion of iron in the sun
**Metric tons of paint covering the tower:**
60
**Frequency of repainting:**
Every seven years
**Number of lights:**
20,000 bulbs and 336 lamps
**Steps to the top:**
1,665
**Average daily visitors:**
25,000
**Distance traveled by one Eiffel Tower elevator in one year:**
64,001 miles
**Total number of visitors:**
250 million, and counting...
Stairs to 2nd floor €5 for ages 12-24, €7 for 25+, elevator to all levels €14.50 ages 12-24, €17 for 25+; mid-June-early Sept daily 09:00-24:00, rest of the year daily 09:30-23:00; 5 Av Anatole France, Eiffel Tower Neighborhood, +33 (0)8 92 70 12 39, tour-eiffel.fr, Metro: Bir-Hakeim
#### **The Louvre**
What was once the French monarchy's decadent city center palace has been converted to the world's most famous and most visited museum, with 8.5 million visitors annually. Before the revolution, the French royalty enjoyed opulence far beyond what the starving people of France could even imagine. Built on the foundations of an old medieval fort, this Renaissance palace is one of the massively expensive projects that helped bring about the climax of resentment of the have-nots toward the haves. Heads rolled, and today we get to enjoy these residences as a public art museum.
The collection of art is so vast that it is said if you spent 30 seconds in front of each piece, you'd spend the better part of a year in here. Pieces span the course of human history, from prehistoric artifacts to cutting-edge modern art. Clearly, it's best to know what kind of art you're interested in before visiting. Definitely hit the key features, as you don't want to go back home saying you missed the _Mona Lisa, Nike of Samothrace (Winged Victory), The Raft of the Medusa, The Coronation of Napoleon, Psyche Revived by Cupid's Kiss,_ or _Venus de Milo_. All of these popular pieces are highlighted on the free map you can pick up at the information desk directly underneath the largest glass pyramid.
For those of you retracing the facts blended with the liberal artistic freedoms of the popular book _The Da Vinci Code,_ the Louvre hosts numerous stars from the book. Keep your eyes peeled for the meridian rose on the steps leading up to the _Winged Victory_. Stop by Da Vinci's works on display in the Italian Renaissance wing, including the _Virgin of the Rocks, Virgin and Child with Saint Anne,_ and of course, the lady herself, _Mona Lisa_. And don't leave without finding the inverted glass pyramid, where it all ends, just outside the security in the main entrance wing (follow the hallway out past Starbucks).
Buy your ticket in advance online (ticketnet.fr/index) or enter the Louvre via the metro Palais Royale, where you'll pass through a large food court on your way in (great place to refuel after your visit!).
€12, free with EU student card or EU student visa in American passport, free first Sun of the month, open Mon, Thurs, Sat, and Sun 09:00-18:00, Wed and Fri 09:00-21:45, closed Tues, Louvre Neighborhood, +33 (0)1 40 20 50 50, louvre.fr, Metro: Palais Royale
#### **Île de la Cité & Île Saint-Louis**
These two islands, where the city was founded in pre-Roman times, make up the heart of Paris. This tiny patch of real estate in the middle of the Seine River contains several of Paris's most famous sights, but strolling their picturesque lanes is a quintessentially Parisian experience in itself.
**Île de la Cité** is bustling with tourists. On the north side, you'll find the grand **Palais du Justice,** the judicial center of Paris and former prison, where Marie Antoinette was kept before being guillotined in 1793. Next to it is the beautiful stained-glass private royal chapel, **Sainte-Chapelle**. **Hotel-Dieu,** Paris's oldest and most prestigious hospital, sits in the middle of the island. And on the south side, you can't miss the iconic bell towers of **Notre Dame**. The low profile of Paris's **Deportation Memorial** , behind the cathedral, is more discreet. The metro stop, Cité, is worth a mention for the art nouveau design and style of the signage. Fans of the _Bourne_ series will recognize **Pont Neuf** , the bridge on the northernmost point of the island, and the Samaritaine department store (Bourne's meeting point and vantage point), which closed down in 2005 due to safety code issues. The new owners have been navigating French and Paris city bureaucracy ever since.
**Île Saint-Louis** offers a special quiet zone smack dab in the middle of the city. Cross the bridges here and the noises of the otherwise busy streets fade away, and you'll find yourself in one of Paris's most exclusive residential districts, where the rich and famous (including Johnny Depp) keep their second (or sixth) homes. Find local-feeling _boulangeries_ (bread bakeries), and don't miss the **Amorino** gelato shop at 47 Rue Saint-Louis en l'Île.
Free, always open, Île de la Cité and Île Saint-Louis, Metro: Cité
#### **Notre Dame Cathedral**
Notre Dame de Paris (Our Lady of Paris) never ceases to amaze me. One of the world's most famous gothic cathedrals is also one of the most visited churches in the world. Construction began in 1163 and continued in fits and starts for 170 years, whenever there was money and the will to build. The next several hundred years weren't so easy for our lady, who endured acts of vandalism, the looting of all her valuables, and the decapitation of her statues. At one point, Notre Dame even served as a stable for livestock, falling into further disrepair. It was in terrible shape when an obscure author, Victor Hugo, inscribed _The Hunchback of Notre Dame,_ in 1831, almost single-handedly thrusting the cathedral back into the realm of public interest. A series of renovations, including the addition of the prominent pointy spire, has restored the cathedral to its present grandeur.
Inside, observe the beautiful stained glass of the South Rose Window and imagine the countless hours it took to complete such a grand design. Reflect on the fact that it took generations of builders and artisans to complete the church. After exploring the great halls of this cathedral, climb the stairs to the top of the tower and step outside for a magnificent view of the city. You won't meet the fabled hunchback up here, but you will meet all his creepy gargoyle friends, which line the outer ledges of the building. Note that the tower has separate hours.
Free, climb the towers for €8.50, cathedral open daily 08:00-18:45, Sat-Sun till 19:15, tower open daily 10:00-18:30 Apr-Sept, Fri-Sat 10:00-23:00 July-Aug, Oct-Mar 10:00-17:30, 6 Parvis Notre-Dame-Place Jean-Paul II, Île de la Cité, +33 (0)1 42 34 56 10, notredamedeparis.fr, Metro: St-Michel
#### **Sainte-Chapelle**
Dating back to the 13th century, this private chapel was created so the royal family could worship under striking floor-to-ceiling gothic stained-glass windows. The intricate windows tell Bible stories, starting on the left side of the chapel with Genesis and progressing to the right through the New Testament. Thanks to a recent cleaning and restorations, it's never been better—or brighter—to take in the beauty of Sainte-Chapelle.
€8.50 adult, Mar-Oct daily 09:30-18:00, Wed til 21:30 mid-May-mid-Sept, Nov-Feb daily 09:00-17:00, 8 Bd du Palais, Île de la Cité, +33 (0)1 53 40 60 80, sainte-chapelle.monuments-nationaux.fr, Metro: Cité
#### **Deportation Memorial**
Situated on the eastern tip of Île de la Cité, this memorial (full name: Memorial to the Martyrs of the Holocaust) commemorates the 200,000 people who were deported from Paris to concentration camps during the Nazi occupation of World War II. The architect who designed it, Georges-Henri Pingusson, wanted to convey the feeling of claustrophobia, which is achieved the moment you descend the narrow staircase. The noise of the street fades away, and each step takes you deeper into an open-topped chamber. Find the corridor with one crystal for each deported Parisian victim. You can watch the clouds and river pass by above and below your feet, but you feel stuck—imprisoned. It's a moving experience.
Free, Tues-Sun 10:00-19:00 Apr-Sept, Tues-Sun 10:00-17:00 Oct-Mar, Square de l'Île de France, Île de la Cité, +33 (0)1 46 33 87 56, Metro: Cité
#### **Champs-Élysées**
In French, _champs_ means "fields." This entire area used to be the open fields beyond the old walls of medieval Paris. Today, the grazing fields have been replaced by the world's most famous shopping street. You can find everything here, from Bugattis to brioche, Maseratis to McDonald's, and fashion superstores to nightclubs. To save on energy, start at the top, near the Arc de Triomphe, and enjoy your stroll downhill, popping into any store that suits your fancy. While it's highly recommended for shopping during the day, the boulevard and surrounding streets get touristy and a bit seedy at night.
Free, always open, Champs-Élysées Neighborhood, Metro: Charles de Gaulle-Étoile
##### **Arc de Triomphe**
Commissioned by Napoleon and finished in 1833, this 165-foot arch, situated at the top of the Champs-Élysées, honors those who died in the Revolutionary and Napoleonic Wars. When you stand under it, the purpose is clear: It's a pure power move by one of the greatest emperors Europe has ever seen, meant to make you feel small in relation to the size and supremacy of the French Republic.
Underneath the epic nationalist arch, you'll find the eternal flame marking the grave of France's unknown soldier, interred here and guarded constantly since World War I. Climb the 284 stairs for a beautiful panorama of the entire city, where you can look back down the Champs-Élysées and toward the skyscrapers of modern Paris in the opposite direction.
€8, Apr-Sept daily 10:00-23:00, Oct-Mar daily 10:00-22:30, Champs-Élysées Neighborhood, +33 (0)1 55 37 73 77, arcdetriompheparis.com, Metro: Charles de Gaulle-Étoile
#### **Orsay Museum**
The Musée d'Orsay was built as a train station and converted into a museum in 1986 to showcase artwork from 1848 to 1914. The museum is most notable for its impressionist and post-impressionist collections, containing artwork from Monet, Manet, Renoir, and Van Gogh, to name a few. The open floor plan is fun to explore to discover both 2-D and 3-D pieces. The art nouveau furniture upstairs is one of my favorite exhibits around. Find the museum on the Left Bank, a 15-minute walk across the Seine from the Louvre.
€8.50 ages 18-25, €11 over age 25, free first Sun of the month, Tues-Sun 09:30-18:00 (Thurs 09:30-21:00), closed Mon, last tickets sold at 17:00 (21:00 on Thurs), 5 Quai Anatole France, Louvre Neighborhood, +33 (0)1 40 49 48 14, musee-orsay.fr/en, Metro: Solferino
#### **Pigalle**
Located in the bohemian neighborhood of Montmartre, the Pigalle quarter, centering around the Pigalle metro stop and the street Rue Jean-Baptiste Pigalle, is Paris's old red light district. While you won't have any ladies grabbing you off the street, if you look up you may notice that buildings have heavily covered windows. Many of these house Paris's porn studios. The famous **Moulin Rouge** (Red Mill, dinner and show €190, show only €105, 82 Bd de Clichy), attributes its fame to the song and movie of the same name. Today, it's a pricey cabaret show with can-canning peacocks strutting their stuff. The **Erotic Museum** (€8, 72 Bd de Cliché, +33 (0)1 42 58 28 73) is worth a gander, and much cheaper.
Free, always open, Montmartre, Metro: Pigalle or Abbesses
#### **Sacre Coeur**
The church of the Sacred Heart is a beautiful, all-white modern cathedral located at the top of Montmartre. It was built relatively recently (between 1875 and 1919), during a time of religious resurgence. At the end of World War I, Sacre Coeur was dedicated to those who died in the conflict.
Sacre Coeur is one of my favorite churches in Europe, sporting both a beautiful refined facade and spectacular mosaic interior with the added bonus of one of the best views of the city of Paris. When climbing the Eiffel Tower or the Arc de Triomphe, Sacre Coeur is easily recognizable across the entire city, capping Montmartre hill. The steps leading up to the church appeared in numerous scenes of the fanciful drama _Amélie._ The best part is that both the church and views are free!
Free, daily 06:00-22:30, 35 Rue du Chevalier de la Barre, Montmartre, +33 (0)1 53 41 89 00, sacre-coeur-montmartre.com, Metro: Abbesses
#### **Pompidou Center**
The Pompidou's eclectic exterior houses a fabulous collection of modern art on its fourth and fifth floors. The building was an ambitious project by a trio of famous architects from Italy and England. The design puts the guts of the building (like air-conditioning and wiring) on the exterior rather than inside the walls. Inside, the largest modern art museum in Europe takes you through the most comprehensive 20th-century timeline of creative expression, with pieces from cubist Picasso to colorful Warhol and avant-garde Chagall.
The streets surrounding the Pompidou Center are interesting to explore. Local food options abound just to the east in the nearby Jewish district. Otherwise, check out the rooftop restaurant for a midpriced post-museum snack, with sandwiches around €6.
€14 museum entry, free first Sun of the month, Wed-Mon 11:00-22:00, closed Tues, no admissions sold after 20:00, 19 Rue Beaubourg, Marais and Vicinity, +33 (0)1 44 78 12 33, centrepompidou.fr/en, Metro: Rambuteau
**MEANDER THROUGH MONTMARTRE**
Historically, the neighborhood of Montmartre existed outside the old city walls of Paris—and therefore outside the city's authority and tax jurisdiction. This was where you would find prostitutes, cheap alcohol, and strip shows...and, eventually, starving artists and students, who could not afford to live elsewhere. Montmartre has been the home to some of the world's most famous artists, including Van Gogh, Matisse, Degas, and Renoir. There are even tales of cafés accepting sketches from these poor struggling artists as payment for their dinners.
Today, Montmartre is a wonderful neighborhood for a hilly stroll. Starting from the Blanche metro stop (and Pigalle and Moulin Rouge), continue uphill toward Montmartre (Hill of the Martyrs) to reach Sacre Coeur. In the streets leading to the church, you'll pass through some of Paris's most fascinating streets, lined with cafés, restaurants, galleries, and textile shops. If you get off the main streets leading to the church, you'll find a neighborhood where today's daily Parisian life is not overshadowed by the thousands of tourists that pass through every day.
#### **Père Lachaise Cemetery**
Populated with everyone from poets and rock stars to political leaders and revolutionaries, Paris's most famous and most beautiful cemetery is toward the northeast part of the city, northeast of the Marais. This cemetery is even more densely populated than Paris's streets. Memorials, tombs, and sculptures practically trip over each other throughout this peaceful park. Must see graves: Jim Morrison, Georges Rodenbach, Frédéric Chopin, Colette, Oscar Wilde, and Victor Noir. The cemetery is quite large, and graves can be difficult to find without the help of a map. Download one ahead of time, or snap a pic of one of the maps posted at each of the entrances of the cemetery to help navigate.
Free, Mon-Fri 08:00-18:00, Sat 08:30-18:00, Sun 09:00-18:00 (17:30 in winter), 16 Rue du Repos, Marais and Vicinity, +33 (0)1 55 25 82 10, pere-lachaise.com, Metro: Gambetta
### **TOP EATS**
You can't talk about France without talking about its famous cuisine. The tastes of Paris should easily be the highlight of your visit, and my recommendations will fit the bill across all budgets and preferences. Stay away from the touristy areas like the Latin Quarter, where restaurants tend to be overpriced. Menus with a _prix fix_ sign used to be a good value, but now more often than not, these menus hide cover and service charges and I prefer to steer clear. Tipping is not as assumed as it is in the US. Look over your restaurant bill to see if service is already included. If it is, you'll see the words " _service compris._ " Otherwise, round up to about 10 percent if you liked the service.
For a cheap and tasty way to dine, opt for a **crêpe** (rhymes with "prep"). They're ubiquitous throughout the city, with options found on nearly every street corner, with street stands to boot. Crêpes can be had sweet or savory. Nutella-and-banana is a favorite. For lunch, ham-and-cheese is a classic, and I'm a fan of the tuna with a fried egg option.
#### **La Varangue**
Located near the Eiffel Tower in Champ de Mars, La Varangue should really be called Philippe's Restaurant. Philippe is a one-man show who cooks, serves, and entertains in his own funky, Parisian way. If he stops everything and makes you sing happy birthday to a fellow diner, it's best to do what he says. Come to this tiny little restaurant that seats about 18 people and enjoy Philippe's magical duck, beef, escargot, and chicken dishes in a warm and intimate atmosphere. La Varangue will surely be one of your most memorable dining experiences in Paris.
€8-20, daily 12:00-14:00 and 19:30-22:00, often closes down Dec-Jan, 27 Rue Augereau, Eiffel Tower Neighborhood, +33 (0)1 47 05 51 22, Metro: École Militaire
#### **Les Delices du Fournil**
Paris is brimming with _boulangeries_ and _patisseries_ showcasing freshly baked breads and pastries, respectively. Les Delices du Fournil's excellent location in the Latin Quarter is your best budget option for filling ham, tuna, turkey, and vegetarian baguette sandwiches and sweet, fresh crêpes. Ask about their lunch menu for both sandwich and crêpe with a drink. Be sure to avail yourself of the restroom downstairs at this utilitarian café before striking back out onto the streets!
€7, Mon-Wed 06:00-19:30, Fri-Sat 06:00-19:30, Sun 06:30-12:30, 2 Rue Carmes/49 Bd Saint-Germain, +33 (0)1 43 54 14 47, Latin Quarter, Metro: Maubert-Mutualité
#### **Coffee Parisien**
For a deluxe taste of home, head to the Coffee Parisien, a retro American-style diner serving up classics like three-layered club sandwiches, massive bacon cheeseburgers, eggs Benedict, thick pancakes, hash browns, and more. The food and coffee are delicious in this fun, casual eatery, but the service does lag a bit. Make your orders clear, and don't come here if you're tight on time. I like the place mats, which illustrate a history lesson of all past US presidents.
Mains from €8, daily 12:00-24:00, 4 Rue Princesse, Odéon, +33 (0)1 43 54 18 18, coffee-parisien.fr, Metro: Saint-Germain-des-Prés
#### **La Durée**
Welcome to the world headquarters of macarons! Consider a stop at La Durée as part of your Parisian pilgrimage. Stop in for a fancy afternoon tea break and sample some of these delightful meringue cookies, a quintessential French treat. La Durée moved into their fanciful Champs-Élysées location in 1997 and they haven't looked back.
Snacks from €5, daily 07:30-23:00, 75 Av Des Champs-Élysées, Champs-Élysées Neighborhood, +33 (0)1 40 75 08 75, laduree.com, Metro: George V
#### **Angelina**
Step into this gilded (and a bit pretentious) classic luxury café and into a place seemingly obsessed with achieving the world's finest hot chocolate. Add to that pastries, macarons, and fudge-drowned sugar-coated desserts, and Angelina will floor you (and your wallet). If you're hungry, tuck into the poshest salads, omelets, and eggs Benedicts you've ever tried. Find Angelina on the north side of the Tuileries gardens.
Artfully presented starters from €25, tea from €7, Mon-Fri 07:30-19:00, Sat-Sun 08:30-19:30, 226 Rue de Rivoli, Louvre Neighborhood, +33 (0)1 42 60 82 00, Angelina-paris.fr, Metro: Palais Royal-Musée du Louvre
#### **Le Café Marly**
Find this decadent café in the arcade facing the glass pyramids at the Louvre. Breaking just about any budget, it's a welcome glass of white wine after a long day inside the museum. See if you can't convince yourself to splurge on one of their many delectable desserts (from €14). You're paying for the setting and the ambience, and as such are definitely among well-heeled tourists. Reservations are required.
Meals from €30, daily 12:00-23:00, 93 Rue de Rivoli, Louvre Neighborhood, +33 (0)1 49 26 06 60, cafe-marly.com, Metro: Palais Royal-Musée du Louvre
#### **Derrière**
Derrière (which translates to "backside" in English) is a trendy and eclectic spot tucked behind the sister 404 Restaurant in a retrofitted mansion in the Marais. Derriere is a unique dining experience from the minute you open up the artistic menu. Enjoy your meal at the foot of a bed, on a sofa, or even in the kitchen and bathrooms. It's as much about the experience of playing a game of Ping Pong or sneaking through a wardrobe into a hidden room as it is about the food itself—as delicious as it is. The friendly—if French-paced—service is happy to share about their artisanal dishes, like salads made with those multicolored carrots, beef filet, and tuna tartare. The €38 buffet brunch (Sun 12:00-16:00) is a winner. Reservations are required. After dinner consider a drink at the cool 'n' cozy cocktail bar next door, **Andy Wahloo.**
Mains from €30, daily 12:00-01:30, find the unmarked passageway leading to a courtyard at 69 Rue des Gravilliers, Marais and Vicinity, +33 (0)1 44 61 91 95, derriere-resto.com, Metro: Arts et Metiers
#### **Amorino**
If you happen to pass one of these _gelaterie_ , stop in and grab some gelato that—when fresh—rivals even the best in Italy. The best part: Ask for as many flavors as you want, and they'll arrange them into an edible bouquet for you. _Speculoos_ (graham cracker) is my personal favorite. You'll find several other well-located Amorino branches sprinkled around Paris's most popular neighborhoods.
From €3.50, daily 12:00-24:00, 47 Rue Saint-Louis en l'Île, Île Saint-Louis, +33 (0)1 44 07 48 08, amorino.com/en, Metro: Pont Marie
#### **L'Avant Comptoir**
I stumbled upon this gem when I noticed a bunch of locals waiting in line for a spot at the bar. This cozy bar and restaurant pours wine and serves up Spanish-style tapas like _jamón_ (ham) and cheese platters, generous sandwiches on rustic bread, and, my favorite, _pimientos de padrón_ (fried, salted peppers) by the truckload, all presented with Parisian sophistication. Find the menu and bottles available hanging from the ceiling and enjoy the reasonable prices and a fun, local, in-the-know vibe.
€8-20, daily 12:00-23:00, 9 Carrefour de l'Odéon, Odéon, +33 (0)1 44 27 07 50, Metro: Odéon
#### **Heureux Comme Alexandre**
If you're looking for a dinner of fondue, this is my top choice in town. Come for the €16 unlimited salad and potatoes fondue dinner, and stick around for the fun atmosphere and easygoing service. Enjoy the excellent location in the Latin Quarter without the hordes of tourists, leaving you, as the name of the place suggests, "happy as Alexandre." Reservations recommended.
Dinner €16, daily 11:00-24:00, 24 Rue de la Parcheminerie, Latin Quarter, +33 (0)1 43 26 49 66, heureuxcommealexandre.com, Metro: Cluny-La Sorbonne
#### **La Closerie des Lilas**
Enjoy a wonderfully Parisian evening on an unforgettable splurge dinner at this Michelin star restaurant just steps from the Luxembourg Gardens. You'll understand why this place was a favorite of Hemingway, Man Ray, and Sartre as soon as you step through the threshold into this Secret Garden-esque experience. You'll dine on a covered patio under flowers and foliage while being serenaded by the pianist inside. If you can stomach the blow to the wallet, the menu spans fish and shellfish to meats to the classic _steak frites_ (steak and fries). Make reservations well in advance, and be sure to ask if the pianist will be playing.
Mains from €40, soaring past €50, cocktails from €16, daily 12:00-00:30, 171 Bd du Montparnasse, Montparnasse, +33 (0)1 40 51 34 50, closeriedeslilas.fr, Metro: Vavin
#### **Cubana Café**
I prioritize a return to Cubana Café on just about every visit to Paris. Here, you'll get a consistently good meal at a fair price. On the restaurant side, you'll find welcoming staff and a delicious menu complete with shredded pork, beans, and plantains. If it's a cigar you're after, select it with the waiter at the bar and step through the glass doors into the smoking room. This is where you may have the chance to rub elbows with some high rollers in the Paris scene, as well as enjoy some fine Caribbean tobacco with a Havana rum and Coke. Freedom never tasted so good.
Mains from €12, daily 11:00-03:00, 47 Rue Vavin, Montparnasse, +33 (0)1 40 46 80 81, cubanacafe.com, Metro: Vavin
#### **Chez Prune**
The quays along Canal Saint-Martin are packed with people chilling and drinking all night, and Chez Prune is a great place to stop for a cappuccino or excellent cheese and meat plates during the day. This cool artsy bar is popular yet seems to escape the tourist eye, drawing a crowd of hip young Parisian professionals. This nondescript café is located right on the canal, so enjoy your drinks outside on nice, sunny days.
Dishes from €13, daily 08:00-02:00, 36 Rue Beaurepaire, Canal Saint-Martin, +33 (0)1 42 41 30 47, Metro: Gare de l'Est or Jacques Bonsergent
### **TOP NIGHTLIFE**
Parisian nightlife is different than what young Americans may be used to. On an average night out, Parisians may have only a couple of drinks but go through packs of cigarettes. Parisians don't drink to get drunk, unless you somehow find yourself invited to a house party (in which case, bring two bottles of wine for yourself and a third to contribute to the party). Since going out on the town is so expensive, many students prefer to drink with friends in each other's cramped-yet-cozy little apartments to either pregame a night out or spend the entire evening in. By the way, French students take their foosball seriously. So if you've got skills, get yourself to the front of the queue and get ready to school or be schooled—a great way to make friends either way.
Anytime you go out, be sure to bring valid **ID,** as proof of age is required at the door of many bars and clubs. Your best bet is to bring your driver's license and a copy of your passport; leave the real thing at home! It really puts a major damper on the night when you're "bounced" because you left yours at home.
#### **NIGHTLIFE DISTRICTS**
Paris is big enough for you to find exactly what you're looking for, but it does come with the typical uptick of pretentiousness—you are in Paris after all. This city is full of districts with distinct personalities catering to all preferences and tastes. The student neighborhoods of Bastille and Oberkampf and the trendy Canal Saint-Martin tend to be the most easygoing and fun, so that's where I go when I'm in town.
##### **Canal Saint-Martin**
What used to be a backwater of Paris has now emerged as one of the city's trendiest and most exciting districts. The low rents have attracted bohemian types, and cafés and bars have been popping up in the streets running along the canal, leading north from Place de la République, a mile north of Bastille. On a nice day, there's nothing better than going for a late afternoon stroll along this 4.5-kilometer canal, stopping at any of the hipster joints that pique your fancy. Start your canal crawl from the north side of Canal Saint-Martin near several recommended hostels, then work your way south.
Canal Saint-Martin, Metro: Gare de l'Est, Goncourt, Jaures, or Louis Blanc
##### **Oberkampf**
The narrow, unrefined streets that wind through this hilly neighborhood in northern Paris, near Canal Saint-Martin, create a very Parisian atmosphere, topped off with quaint, locally owned cafés. This neighborhood is popular among students, as it offers some of the cheapest food and drink prices you can find in all of Paris. You'll find most of the action between the **Avenue de la République** and **Rue Moret.**
**THE MOST FAMOUS NON-SIGHT IN PARIS**
The Bastille neighborhood is named after the notorious castle-turned-prison. Built in the 14th century, the castle was converted into a prison by the oppressive monarchs in the 1600s. On July 14, 1789, peasant hordes swarmed the Bastille, thinking it contained hundreds of their imprisoned compatriots. The 600 rebels took over the defenses, but found only a small contingent of opposition, along with just seven degenerate prisoners. It was a glorious success to no real point. Shortly afterward, the Bastille was demolished, as it symbolized the oppression of the revolution's ideals: _liberté, égalité, fraternité_ (liberty, equality, fraternity). July 14 is now observed as Bastille Day, an annual holiday.
Though the castle is long gone, you've got to swing through the Bastille neighborhood at some point during your visit. Yes, you may see clueless tourists trying to find the castle, but you'll find only a massive roundabout circling an obelisk (the July Column), with the Opéra Bastille anchoring one side. The canal leading away from the square makes for a wonderful afternoon stroll in good weather, as you check out the boats and munch on fresh baguettes. In the back streets surrounding the grand plaza, you'll find bohemian nightlife, student bars, and a tasteful array of restaurants. Trendy Parisians are now turning their collective nose up at this neighborhood, as it's been well-discovered by tourists, but you can find gems, especially on Rue de Lappe.
Canal Saint-Martin, Metro: Parmentier or Oberkampf
##### **Bastille**
East of the Marais, the Bastille district draws a young crowd. It's ideal for a night of pub crawling, as all the action centers on one street: **Rue de Lappe,** just north and east of the Place de la Bastille. This street satisfies the classy and grungy alike, with everything from sports bars and lounges to small, crowded, boutique-style drinkeries. Rue de Lappe used to be a Parisian secret, with tons of places to score cheap drinks, but prices have aligned themselves with the neighborhood's newfound reputation. It's worth taking a lap to evaluate your many options for happy hour or to find the place just right for you.
Marais and Vicinity, Metro: Bastille
##### **The Marais**
Bordered on the west by Rue du Renard and the Pompidou Center, and to the east by the grand Boulevard de Beaumarchais and the Bastille district, this is one of Paris's most eclectic and fun neighborhoods. Chic and a bit yuppie, the Marais is known as Paris's gay district.
Marais and Vicinity, Metro: Saint-Paul
##### **Odéon**
Once the center of intellectual life, this Left Bank neighborhood was just teeming with philosophers contemplating existentialism, artists toying with surrealism, and musicians exploring the depths of jazz. Now, it's a popular neighborhood with some great restaurants and bars, like **Le Dix Bar,** a cozy sangria bar. The best part is that bars and restaurants center around a single street, **Rue Princesse.** Why is all this goodness so tightly concentrated, you ask? Because it's located just across the street from the Sorbonne, the University of Paris! Expect to see some revelers out.
Odéon, Metro: Odéon
#### **BARS**
##### **Bar Ourcq**
Bar Ourcq is a great place to kick off your evening along Canal Saint-Martin. It feels like your comfortable corner bar where you can get cheap drinks, socialize, enjoy some good music, or take your drinks to go and tip them back along the canal. Organic snacks, €1 sandwiches, and the convenient location make for a great start to the night. From the simply decorated café, snag a chair and enjoy your drink overlooking the canal.
Drinks from €3, Wed-Thurs 15:00-24:00, Fri-Sat 15:00-01:45, Sun 15:00-22:00, 68 Quai de la Loire, Canal Saint Martin, +33 (0)1 42 40 12 26, barourcq.free.fr, Metro: Lumière
##### **Point Éphémère**
Run by a nonprofit agency charged with converting old and abandoned buildings into lively cultural centers for the arts, this place has it all: great food, art studios and galleries, a concert venue, terrace, cheap drinks for the canal, and a cool crowd without the usual Parisian snob. The Point, having taken over an old canal-side industrial workshop, usually has a number of freshly tagged graffiti pieces scrawled across the exterior. They keep their art and live music shows fresh with a constantly updated calendar of events, drawing a cool and eclectic crowd. Staying true to its name, each visit back to the Point offers something new. Check their website for the upcoming events.
Drinks from €4, book concert tickets online, Mon-Sat 12:00-02:00, Sun 12:00-23:00, 200 Quai de Valmy, Canal Saint-Martin, +33 (0)1 40 34 02 48, pointephemere.org, Metro: Jaures or Louis Blanc
##### **Le Comptoir Général**
I love this multifaceted artistic and cultural space set back from the canal in an old warehouse—Paris's best equivalent to the popular Budapest-style ruin bars. Begin exploring the complex to discover a serious coffee bar, excellent cocktail bar (of course), space for designers to sell their concepts from pop-up shops, live music acts, and hanging gardens. The crowd and vibe are really as hipster as it gets, but not in a bad way. Enjoy film screenings and concerts, and check their calendar ahead of time to see if tickets are available for hot upcoming events. Heads-up: The line can be long on weekend nights, but that's not the only time to stop by and explore! While the party can ramp up occasionally on their small dance floor, it's much more a quiet and chill place to enjoy a drink while exploring all the rooms in this open and airy garden bar.
**LGBT PARIS**
The Marais, which overlaps with the Jewish Quarter, is known as Paris' gay district. In this bustling tangle of medieval lanes, you'll find art galleries, design shops, adult stores, high fashion boutiques, florists, cafés, bakeries, and cozy restaurants by day. By night, the fun runs the length of **Rue de la Verrerie,** leading into **Rue de Roi Sicile,** with crowds spilling out onto the surrounding streets. The bears feel at home in the burly **Bear's Den** (6 Rue des Lombards, +33 (0)1 42 71 08 20). If you like beer, stop into **La Caves a Bulles** (45 Rue Quincampoix, +33 (0)1 40 29 03 69), an artisanal brew shop where you get bottles to go and enjoy them on the square in front of the Pompidou Center.
Free entry, donation suggested, daily 11:00-02:00, 80 Quai de Jemmapes, Canal Saint-Martin, +33 (0)1 44 88 24 48, lecomptoirgeneral.com/en, Metro: Goncourt
##### **Le Dix Bar**
Le Dix Bar is a cool sangria bar with a dark-wood interior in one of my favorite neighborhoods, Odéon. Come out with friends and select your size of pitcher. Give it your best guess, then round up, because the sangria the friendly old man pumps out is delicious and refreshing. Enjoy your fruit-infused wine upstairs or head downstairs to join the social crowd of drinkers.
Pitchers from €12, daily 18:00-late, 10 Rue de l'Odéon, Odéon, +33 (0)01 43 26 66 83, le10bar.com, Metro: Odéon
##### **Chez Georges**
For a truly Parisian experience, come to this classic bar with a twist and enjoy some wine with a group of your friends. Or fly solo and sit down at one of the many communal tables to strike up a conversation with your fellow winos. It draws a slightly older crowd. Don't miss the exposed brick arch cellar downstairs, and join the sweaty dancing that kicks off on weekend nights. While it closes at 02:00, it's located near Odéon, which is convenient if you want to keep the night going.
Glasses from €6, Mon-Fri 15:00-late, Sat 12:00-02:00, Sun 19:00-02:00, 11 Rue des Canettes, Odéon, +33 (0)1 43 26 79 15, Metro: Mabillon
##### **Café Oz**
Unapologetically touristy and international, this large Aussie sports bar with an open layout also fits the bill for downtown dancing. Pub crawls kick off here, so it gets packed then empties out each night, and you can count on just about every game being televised to be up on the screens—they've got a thing for rugby here. Come out and enjoy the rowdy student crowd dancing to Top 40 and hip-hop spinning after the matches finish up. In town for a holiday? Check out their frequent theme nights for Halloween, Christmas, and more online.
Drinks from €5, Mon-Fri 17:00-late, Sat-Sun 12:00-late, 18 Rue Saint-Denis, Louvre Neighborhood, +33 (0)1 40 39 00 18, cafe-oz.com, Metro: Châtelet
#### **CLUBS**
Remember to bring ID with birth date on it, and dress to impress to get past the bouncers. They want you inside having a good time, but they're also picky to keep the party at the level of class they want. Most recommended clubs don't open till 23:00 or so.
##### **Showcase**
This posh and cavernous club features an excellent view of the Eiffel Tower from the smoking section from underneath the Pont Alexandre III bridge. Opt for the shots, as the long drinks seem to be watered down. Dress to impress and get past the bouncers, as this colonnaded dance floor is one of the most popular in the city. Get down to the famous resident and visiting DJs spinning electronic and techno and enjoy the energized crowd.
€20 cover, beers and drinks from €8, Fri-Sat 23:00-late, Port des Champs-Élysées underneath Pont Alexandre III, Champs-Élysées Neighborhood, +33 (0)1 45 61 25 43, showcase.fr, Metro: Invalides
##### **YoYo Palais de Tokyo**
If you like to dance and have moved beyond the college-dorm party scene, YoYo is a great choice. This large venue in an old theater has plenty of room to get your groove on. Thankfully, the air-conditioning is turned up so you can keep cool without breaking a sweat. Famous DJs visit often to spin house and techno—it's a good idea to purchase entry ahead of time if there's an upcoming gig you like. Be prepared for long, slow lines outside and plenty of security. But once inside, you'll love it! Pregame on the Champ de Mars with some wine before crossing the river and finding this club tucked underneath the Modern Art Museum of Paris.
Covers from €10, Wed-Sat 23:00-late, 13 Av du Président Wilson, Champs-Élysées Neighborhood, +33 (0)1 84 79 11 70, yoyo-paris.com, Metro: Iena or Alma Marceau
##### **La Machine Du Moulin Rouge**
This is a massive triple-level dance club with DJs in each room spinning a wide range of music, from house and electronic to dubstep and the more generic Top 40. Don't let the location, right in the middle of the city's famed red light district, unnerve you, as the neighborhood is clean and safe. This place really puts on the party—packed with locals, backpackers, and a fun-loving crowd. Famous acts come through often, so check the program ahead of time to see if there are any names you recognize.
Covers and drinks €10, Fri-Sun 23:00-late, 90 Bd de Clichy, Montmartre, +33 (0)1 53 41 88 89, lamachinedumoulinrouge.com/en, Metro: Blanche
##### **Queen**
Champs-Élysées is not the best place to go out, but glitzy Queen is the one exception. It's a popular gay club, featuring costumed male strippers on raised platforms putting on dueling acts throughout the night. Wednesday is ladies night and evens the ratio a bit, but all are welcome any night of the week. Expensive drinks, but shots are passed out to the lucky ones when the managers want to ramp the party up.
€20 cover, daily 23:00-05:00, 102 Av. des Champs-Élysées, Champs-Élysées Neighborhood, +33 (0)1 53 89 08 90, queen.fr, Metro: George V
#### **PUB CRAWLS**
When it comes to planning a night out in Paris, first consider the way Parisians go out: for one glass of wine and a pack (or three) of cigarettes. If you tend to be thirstier, an organized pub crawl may be a great choice for you.
##### **Sandeman's New Europe Pub Crawl**
You know the pub crawl drill: Pay the €25 cover, then get free drinks for an hour at the meeting point. Your itinerary will include about three stops, with a final stop at a club with a free entry. It'll be up to you to get home, but between the opening bell and when "Piano Man" comes on to close the night, you're sure to have a good time with a set of 20-120 new international friends.
€25, Thurs-Sat 20:00, meet at the Bastille metro stop, Marais and Vicinity, newparistours.com, Metro: Bastille
### **TOP SHOPPING & MARKETS**
Stores in Paris close down on Sunday. Stores that do stay open actually pay a fine week after week for keeping their doors open.
#### **SHOPPING DISTRICTS**
##### **Champs-Élysées**
The most famous shopping street in the world and open every day of the week—despite French law requiring shops to close on Sunday—Champs-Élysées is lined with everything from Louis Vuitton to Gap. Spend a day lost in luxury, making sure to stop for a crêpe or a box of macarons to prevent the "drop" from the famous adage of "shop till you drop."
Champs-Élysées Neighborhood, Metro: Charles de Gaulle-Étoile
##### **The Marais**
Paris's gay and Jewish district is also one of its trendiest, with excellent shopping options and delicious restaurants and cafés. You'll find unique boutique stores and the more common high-end designer brands in this neighborhood of winding medieval streets. **Rue de Rivoli** and **Rue Vieille du Temple** are the main intersecting streets in this area, with dozens of lanes turning off them to explore. This neighborhood is a great place to shop on Sunday, as the stores are some of the only ones open in the whole city.
Marais and Vicinity, Metro: Saint Paul
##### **Rue Montorgueil**
One of my favorite pedestrian avenues in Paris, **Rue Montorgueil** boasts pastry shops, flower stalls, bistros, and an open-air market that sells the wide variety of meats, cheeses, and other goodies. You'll also find top-notch nightlife, with a large variety of bars that appeal to every person imaginable. Leading north from the Châtelet-Les Halles metro stop, all the way toward Montmartre, Rue Montorgueil offers a casual scene for those who like to spend an evening on a low-key stroll, stopping for snacks, drinks, and gelato on the way.
Louvre Neighborhood, Metro: Étienne Marcel or Sentier
#### **MARKETS**
Traditionally, life in Paris centered around the goods purchased at the daily market. Markets are a great way to catch a glimpse of daily Parisian life, with grandmas picking out the freshest fruit and children running home with their daily staple of baguettes. While many cute local markets have been swallowed up by large grocery and department stores, some manage to maintain their charm (no matter how many guidebook-toting tourists plod through). These are some of my favorites, but keep your eyes open for other local neighborhood markets while you wander.
##### **Rue Cler**
The market on Rue Cler, a quaint street just east of the Eiffel Tower, is a local market that stubbornly retains its identity. You can see schoolchildren running about, enjoy a crêpe on the street, and pick up some French produce. Though the prices at the restaurants on this street have become a bit outrageous due to its popularity, Rue Cler offers a little slice of picturesque Paris with the cafés, corner markets, and food stands that line these three little blocks.
Eiffel Tower Neighborhood, Metro: École Militaire
##### **Marché des Enfants Rouges**
Marché des Enfants is a permanent, semi-covered food market in an old courtyard that has been around since 1628. It's quite popular among trendy Parisians. Pop in for a quick and tasty lunch, from Japanese cuisine to exotic African dishes. Food stalls let you take your food tray to the communal benches and tables around the market.
Tues-Sun 08:30-13:00 and 16:00-19:30, closed Mon, 39 Rue de Bretagne 3e, Marais and Vicinity, Metro: Filles du Calvaire
#### **SHOPPING CENTERS**
##### **Galeries Lafayette**
Claiming to be one of the oldest shopping malls in the world, this shopping center has an enormous variety of well-known international shops and a spectacular four-level atrium interior under an ornately gilded glass dome. Stop by in December to see their famous 50-foot-tall Christmas tree reaching toward the ceiling.
Mon-Sat 09:30-20:00, 40 Bd Haussmann, Louvre Neighborhood, Metro: Chaussée d'Antin-La Fayette
##### **Forum des Halles**
Located right in downtown Paris, Forum des Halles is a massive shopping center dug deep into the ground and sitting right atop the metro. Very popular for Parisians coming in from the _banlouies_ (the suburbs of Paris), it's a top choice for blue-collar brands like H&M and Jack Jones.
Mon-Sat 10:00-20:00, Louvre Neighborhood, forumdeshalles.com, Metro: Les Halles
### **TOP PARKS & RECREATION**
Paris has numerous city parks to explore. They're a great option for a cheap picnic and to catch a breather from the bustle of the city.
#### **Luxembourg Gardens**
This beautiful 60-acre park is just a few blocks south of the Notre Dame Cathedral, and filled with flower gardens, fountains, ponds, and statues, topped off with an enormous Florentine Renaissance-style palace overlooking it. The manicured lawns and paths make it a favorite among reflective Parisians for an afternoon stroll. Catch a game of _pétanque_ (the French game of field bowling) on the southwest side of the park.
Free, daily 07:00-dusk, Odéon, Metro: Odéon, RER: Luxembourg
#### **The Tuileries**
The Tuileries are Paris's oldest garden. Be sure to walk through this sprawling park as you're making your way from the Louvre to the Champs-Élysées. Take a moment to relax in the reclining park chairs and enjoy watching little French kids sail model boats in the ponds. Fun fact: _Tuile_ means "tile." Paris's tile and ceramics factories were located here until Queen Catherine of the Medici family moved them out in 1564 to make room for her royal gardens, which were meant to be reminiscent of those she would have seen in her home town of Florence.
Free, always open, Louvre Neighborhood, Metro: Tuileries
#### **Champ de Mars**
The Champ de Mars, or "Field of War," stretches southwest from the Eiffel Tower. This large open space was the parade grounds where officers practiced maneuvering troops. Today, it's a favorite spot for Parisians to picnic on nice summer evenings, watching the glittering Eiffel Tower sparkle for five minutes each hour. When there are important events or sports matches here, large Jumbotrons are brought in for the crowds to watch. It's also just about impossible to find a free square inch here on New Year's Eve, as the fireworks display pops off from the Eiffel Tower.
Free, always open, Eiffel Tower Neighborhood, Metro: École Militaire
#### **Parc des Buttes-Chaumont**
This beautiful park in the northeast of Paris features more than a mile of romantic paths and hundreds of trees to make for a wonderful break from the busy city. Typical of 19th-century ingenuity, this ambitious park project with an artificial lake and a peak over 150 feet tall was reclaimed from a garbage dump and stone quarry during the beautification projects under Napoleon III. (This happened during the same era that the boulevards and avenues of Paris were being widened.) I enjoy the rope bridge and short climb to the temple capping the peak for a panoramic view across town. The park is also close to several great hostels and the Canal Saint-Martin nightlife district.
Free, always open, Canal Saint-Martin, Metro: Botzaris
### **TOP TOURS**
Paris is a massive city, and a knowledgeable guide really helps you grasp the complex history and understand better what you're looking at. Fortunately, you've got both walking tours and bike tours to choose from.
#### **Discover Walks**
This walking tour company offers some great itineraries through the heart of Paris. Tours are led by fun and engaging locals who are paid only by your generosity in the form of tips at the end of the tour. Their numerous itineraries, organized by theme and neighborhood, could fill up an entire visit to Paris. I recommend their **Latin Quarter Walk,** which will take you through the oldest neighborhoods of the city, including Île de la Cité and Notre Dame Cathedral.
Free, 11:00 and 14:30 daily, meet at the Charlemagne statue in front of Notre Dame, Île de la Cité, discoverwalks.com/paris-walking-tours, Metro: Cité
#### **Fat Tire Bike & Boat Tour**
Cruising around by bike is my favorite way to experience the City of Lights. Go for the evening tour and cap your bike adventure with an evening boat cruise! Highlights include Notre Dame, the Louvre, a stop for ice cream, and then a ride down the Seine. With free wine included during the boat tour, this is the perfect way to cap an evening.
Student single €28, day+night €42, tours mid-Feb-Nov, daily mid-Mar-Oct, check website for times and off-season dates, 24 Rue Edgar Faure, Eiffel Tower Neighborhood, fattirebiketours.com/paris, Metro: Dupleix
#### **Paris Noir**
Paris Noir (Black Paris) is run by a good friend of mine, Kevi Donat. Kevi leads visitors through the streets of Paris speaking from the perspective of black Parisians, reviewing their oral, musical, and artistic culture. Recognized by French and international media, Kevi does an excellent job of researching and developing new content and delivering these fascinating stories from the turn of the century to today.
Tours on request, blackpariswalks.com
### **TOP HOSTELS**
Paris is woefully lagging in budget accommodations, but those listed are a good value. Use **hostelworld.com** to find the latest hostel ratings, and don't forget to check out **airbnb.com** for private apartment options.
#### **Generator Hostel Paris**
The Generator Hostel chain continues to take the continent by storm, and the new Paris location does not disappoint. The Generator team knows exactly what budget travelers are looking for and does an excellent job of providing a fun, chill place to crash after your long days out on the town. Besides the new, clean rooms, mod decor, ubiquitous free Wi-Fi, and plugs and reading lights for each bed, the location, right in the hot spot of Canal Saint-Martin's trendy scene, is a favorite feature of the hostel. In the opposite direction, the beautiful Parc des Buttes-Chaumont is only a few minutes away. Deluxe penthouse and private rooms are available as well, offering views of the neighborhood.
Beds from €22, 24-hour reception, mini grocer, vending machines, free Wi-Fi, terrace, travel options desk, on-site bar and restaurant, laundry facilities, lockers and adapters for rent, 9-11 Place du Colonel Fabien, Canal Saint-Martin, +33 (0)1 70 98 84 00, generatorhostels.com, Metro: Colonel Fabien
#### **St Christopher's Inn Gare du Nord**
This hostel, which opened its doors in May 2013, lives up to the St Christopher's brand of large hostels with clean rooms, loads of amenities, and a great location on the north side of Paris, just steps from Paris's northern train station and a couple blocks from the trendy, up-and-coming Canal Saint-Martin district. With over 600 beds, this hostel brings in loads of travelers from all over the world, creating a fun and social atmosphere to meet and connect with other globetrotters.
€24, free breakfast, optional upgrade, free Wi-Fi, bar, café, 24-hour reception, free linens, 5 Rue de Dunkerque, Canal Saint-Martin, +33 (0)1 70 08 52 22, st-christophers.co.uk/paris-hostels/gare-du-nord, paris@st-christophers.co.uk, Metro: Gare du Nord
#### **St Christopher's Inn Canal**
This is the original location (and my second choice of the two St Christopher's hostels in town), with a fun vibe and a music bar downstairs featuring acts that even bring in the locals. While it's a half-hour metro ride to sights like the Louvre and the Cathedral of Notre Dame, the comfy beds, helpful yet busy staff, free breakfast, and Wi-Fi make this one of Paris's better budget choices. It's also about a 30-minute walk north of the fun Canal Saint-Martin nightlife district. You can't miss the funky exterior: It looks like a bunch of giant rubber bands wrapped around all six levels of this large hostel overlooking the canal.
€24, daily breakfast, free Wi-Fi, "chill-out room," bag lockers, 24-hour reception, free linens, 159 Rue de Crimée, Canal Saint-Martin, +33 (0)1 40 34 34 40, st-christophers.co.uk/paris-hostels/canal, paris@st-christophers.co.uk, Metro: Crimée
#### **The Loft Boutique Hostel**
The Loft tries hard to be Paris's top boutique hostel and is successful in many ways. Highlights include the fun intimate atmosphere in the common rooms, cleanliness (thanks, cleaning staff!), and the included continental breakfast. The outdoor patio and nightly happy hour make for a nice social vibe. The neighborhood feels a bit bohemian and off the beaten path, away from the major tourist sights of Paris, yet right between the Parc des Buttes-Chaumont and Père Lachaise Cemetery.
Beds from €23, 24-hour reception, outdoor terrace, on-site bar, free Wi-Fi, included breakfast, kitchen and laundry facilities available, 70 Rue Julien Lacroix, Canal Saint-Martin, +33 (0)1 42 02 42 02, theloft-paris.com, Metro: Pyrénées
#### **Trendy Hostel**
While this hostel southeast of the city center (about 20 minutes away on the metro) suffers in location, backpackers love it for just about everything else: comfortable beds, fun staff, great local restaurants, and a long list of amenities. The value is great, but it may feel like you're commuting into town each time you want to sightsee or go out at night.
Beds from €18 with female dorms available, 24-hour reception, free maps and Wi-Fi, laundry facilities available, linens included, breakfast included, 2B Rue Édouard Vasseur, Greater Paris, +33 (0)982 479 024, trendyhostel.com, Metro: Pierre et Marie Curie
#### **BVJ Louvre**
With a title that translates to "Office of Youth Travel," this hostel is painfully institutional, but still worthwhile thanks to the superb location. BVJ Louvre completely misses the mark when it comes to providing what backpackers want these days (i.e., fun, social atmosphere, comfortable mattresses, free Wi-Fi—yes, that's three strikes against BVJ), but you just can't deny how nice it is to roll out of bed straight into the Louvre. If BVJ Louvre is full, you can try their other location, **BVJ Opéra** (1 Rue de la Tour des Dames, +33 (0)1 42 36 88 18, Metro: Trinité d'Estienne d'Orves), offering mostly the same dull experience in another central location.
Beds from €24, towels and breakfast not included, Wi-Fi working sometimes, 20 Rue Jean-Jacques Rousseau, Louvre Neighborhood, +33 (0)1 53 00 90 90, bvjhotel.com, Metro: Louvre-Rivoli
### **TRANSPORTATION**
#### **GETTING THERE & AWAY**
##### **Plane**
There are three major airports in Paris: **Charles de Gaulle** (CDG), **Orly** (ORY), and **Beauvais** (BVA). Find information for all three at aeroportsdeparis.com. All offer quick and reliable transport into the city.
From **Charles de Gaulle,** take Paris's regional train system, the **RER.** After arrival, follow the signs marked RER leading you to the ticketing area. A one-way ticket will cost you €9.10, with trains departing between 05:00 and 00:15 every 15 minutes. Remember; you need your ticket to exit the RER system, so keep it handy. This is your cheapest and fastest option into the city, with opportunities to connect onto Paris's extensive metro system, including service to Gare du Nord (for St Christopher's Hostels), Châtelet-Les Halles (central Paris), and Luxembourg (south side of Paris) stations. By RER and metro, it takes about an hour to get from Charles de Gaulle into the center of Paris.
Another option is the **Roissybus** express bus (€10). Find it by following the signs marked "Roissybus." Purchase a ticket from an RATP (the name of Paris's public transportation system) vendor, which is clearly marked and located right by the bus stop. Roissybus takes about an hour and a half into the city center.
Lastly, you can hail **taxis** outside the terminal for a €45 one-hour trip into the city.
From **Orly,** take the **Orlybus,** which is a direct route into the center of the city. The bus will drop you at Denfert Rochereau metro station, where you can connect to the metro and ride it to your desired stop. Buses depart every 15 minutes, running from 05:35 to 23:00. The journey costs €7.70 one way and will take you about 25 minutes.
You can also use Air France's **shuttle service** (€9), departing every 15 minutes running to Porte d'Orléans, Montparnasse, and Invalides metro stations. It runs from 06:00 to 23:00, with a journey length of 25 minutes. Or, take a **taxi** into the center for about €25.
From **Beauvais** , catch the Beauvais Airport official **shuttle** (€16), which is a non-stop journey from the airport to the Porte Maillot metro stop in Paris. The journey takes about 75 minutes.
##### **Train**
Rates from London—on fast trains taking just under 2.5 hours—range from €50 to €150 and up, depending on student discount and how far in advance you book your ticket. For more information, visit the **Eurostar** website (eurostar.com). Trains to Amsterdam (about €75) run often and take about 3.5 hours.
Six major train stations operate in Paris, so it's important to confirm your departure station when the time comes for you to leave. **Gare du Nord** (in northern Paris) serves connections to London, Amsterdam, and Brussels. **Gare de l'Est** (just east of Gare du Nord) serves Munich, Frankfurt, Luxembourg, and beyond. **Gare de Lyon** (east of the Marais) most likely has your night connections to Nice, Barcelona, and Milan. All train stations are well connected via metro, and it is possible to buy tickets for your departure at any train station for any other station.
##### **Bus**
Paris has numerous bus options for France, Germany, Spain, Italy, and even the UK. For the cheaper price, remember, you're trading time en route and comfort. **Eurolines** (eurolines.fr) is the dominant company. Check their prices and full list of destinations online.
##### **Car**
Thanks to Paris's excellent public transportation system, I advise against renting a car for any visit to the city, and caution those who are considering taking a road trip. It's not that it's dangerous or unpleasant, but trains, planes, and buses will all get you to your destination quicker, in more comfort, and most likely at a cheaper price. When budgeting for your trip, don't forget to account for gas (about $10/gallon), parking, and speeding tickets from pesky automated machines all on top of the rental fee, service, and insurance for the car itself.
#### **GETTING AROUND**
Take advantage of the fantastic public transportation this city offers, which comprises train, bus, and metro lines. The system is integrated: A single **ticket** costs €1.70. A **_carnet_** of 10 tickets (€12.70) saves you about 25 percent, and the 10 rides can be shared between multiple people.
##### **Metro**
Paris's metro system (ratp.fr) is easily one of the best in the world, and is an extremely useful way to get around. The metro operates Sunday-Thursday 05:30-00:30 and Friday-Saturday 05:30-02:00. You'll rarely walk two blocks without passing a metro stop, and all routes are color-coded and numbered, safe, and easy to navigate. Here's how to use it: On a metro map, find the station you're at and the one you want to get to. Follow the lines stemming from both and see where they meet up to figure out if you need to transfer. You can get nearly anywhere in Paris with one or two transfers between different lines. Once you've figured out which line(s) to take, follow signs to the side of the platform with the final destination indicating the direction toward your stop. The **RATP Lite** app offers a full map of the metro. Be aware that **pickpockets** operate throughout Paris's metro system. Keep your valuables tight and safe.
##### **RER**
The RER system, Paris's express regional train, zips straight through the heart of town and to destinations outside the city. It's much speedier than the metro or bus. Keep an eye out for any potential reasons to use it during your stay.
The RER network is lettered A through E. Tickets within the city cost the same as on the metro; remember to hold onto your validated ticket because you need it to exit the system. The RER system also provides fast connections to the airports as well as popular day-trip options outside the city center, like Versailles and Disneyland Paris. Those tickets are more expensive. Numerous easy-to-use ticket kiosks have options in English; you can purchase your tickets with both card and cash.
##### **Bus**
Paris has an extensive bus system that offers great views of the city while you ride. The maps at each bus station are helpful for both bus riders and those lost trying to find their way in the neighborhood. While the skinny maps are oriented tightly around the bus routes in the bus stop signage, pedestrians can orient themselves while checking out the maps from the numerous lines. If you buy your ticket on board, you aren't allowed any transfers. If that's an issue for you, consider buying a _carnet_ of 10 tickets from any metro station and using them in the integrated system, because each ticket gives you 90 minutes worth of travel with as many transfers as you need—much better!
##### **Taxi**
Heads-up: Taxis in Paris have a reputation of scamming tourists. They're a great solution if you're splitting a ride home from the club late at night, but they dry up quickly once all the bars let out. Thankfully, **Uber** works in Paris. Before getting a regular cab, be sure to know your route home. Insist on seeing the meter before you pay. Estimate fare for a cross-town ride from the Eiffel Tower to the recommended hostels near Gare du Nord, for example, should run about €25. If you feel your driver is ripping you off, walk into your hostel and ask for help.
##### **Bicycle**
Paris is home to the world's most extensive public bike system, **Vélib'** (en.velib.paris.fr). To use it, you need a chip and pin debit or credit card to purchase a 24-hour access pass (€1.70). Users have 30 minutes free to return the bike. After that, subsequent hours cost a euro each. The most popular way to get across town for free on longer rides is to simply check in your old bike and pull a new one.
Before checking out a bike, make sure the tires are full of air, the brakes work, and you can move the seat to the right place. A saddle facing backward signals to the maintenance staff that a bike needs service. Now, with a fully functional bike, you're free to cruise the 400 kilometers of bike paths in the city. But remember to play it safe! There are many Vélib' accidents each year, and drivers don't necessarily yield to bike lanes. Helmets are not provided at the rental stations. If you want to bike but want to do it with a knowledgeable guide and without the stress of trying to navigate, consider **Fat Tire Bike Tours** (24 Rue Edgar Faure, fattirebiketours.com/paris).
### **DAY TRIPS**
Thanks to the RER regional rail system, you've got two excellent day trip options just outside of the city, reachable in an hour or two. With an early start (07:30), you should be able to beat the crowds on just about any day.
#### **Versailles**
A visit to Versailles is the perfect way to cap your time in Paris. Constructed under the rule of the Sun King, Louis XIV, this unbelievably extravagant palace, along with its seemingly never-ending gardens, symbolizes absolute monarchy. It's tough to wrap your mind around the sheer scale of this palace, so here are some statistics: 67,000 square feet, 67 staircases, 700 royal rooms, and nearly 2,000 acres of gardens. Tour the palace to see the gilded room where the Sun King slept, the floor-to-ceiling mirrors in the grand Hall of Mirrors (meant to show off royal wealth at a time when hardly anyone could afford a pocket mirror), and much more. After the angry hordes had taken Paris during the revolution, they came out the 24 kilometers to Versailles to drag Marie Antoinette and Louis XVI back to town for a date with the guillotine.
The trip from Paris takes 30 minutes to an hour on the RER-C, depending on the train you catch, and costs €8.20. You will arrive at the Versailles RER train station, with the palace and breathtaking gardens just a few steps away.
€18 for palace and gardens, €15 for palace, daily 09:00-17:30, Place d'Armes, +33 (0)1 30 83 78 00, en.chateauversailles.fr
#### **Disneyland Paris**
Can you believe the wonderful world of Disney is just a 32-kilometer train ride away from Paris's city center? Head to this amusement park to live it up with Mickey and Minnie—speaking in French accents—and all the rides you would find at home. If Belle and Aladdin are your thing, budget a full day to visit Europe's happiest kingdom, making kids of all nationalities smile since 1992. Tickets are also sold to the Walt Disney Studios, a theme park that focuses on recent movies like _Toy Story_. These rides, coasters, and experiences take you on interactive behind-the-scenes trips and are geared toward a slightly older audience.
Ticket prices vary, depending on season and day of week. I recommend purchasing tickets ahead of time online (from about €50 for a single-day, single-park ticket, and €80 for combo tickets for both parks). To get here, take RER A4 (€7.60) toward Marne-la-Vallee Chessy from Châtelet-Les Halles. The last train home leaves shortly after midnight. Trains run every 15 minutes and take about an hour.
Ticket prices vary, daily 09:00-20:00, +33 (0)8 25 30 05 00, disneylandparis.com
### **HELP!**
#### **Tourist Information Centers**
Paris has a couple of tourist offices in town. They are great places to stop in and pick up a free map. Get information online at en.parisinfo.com, or find Paris's official tourism office (open daily 10:00-19:00) and info desk at:
25 Rue des Pyramides
+33 (0)1 49 52 42 63
#### **Pickpockets & Scams**
As with most European cities, violent crime within the city of Paris is extremely low; however, be on the lookout for pickpockets, as they are numerous. Beware if someone approaches you holding an object and asking if you've dropped it, or if a stranger asks you to take a survey or sign a petition. Also, a large percentage of pickpockets are children under the age of 16, so be on your guard if a group starts to crowd around you. Lastly, keep a close watch on handbags or purses, and never set them down and walk away, even if it's just for a second.
#### **Emergencies**
Dial 112 for emergencies, 17 for English-speaking police, and 15 for an ambulance.
#### **Hospital**
Hôpital Américain de Paris
63 Bd Victor Hugo
+33 (0)1 46 41 25 25
#### **US Embassy**
4 Av Gabriel
+33 (0)1 43 12 22 22
Amsterdam Maps
Amsterdam 101
Three Day Itinerary
Top Neighborhoods
Top Sights
Top Eats
Top Coffeeshops
Top Nightlife
Top Shopping & Markets
Top Parks & Recreation
Top Tours
Top Hostels
Transportation
Day Trips
Help!
Amsterdam has a rich history of social inclusion and individual liberty. Spend a few days biking through this canal-laced city, stopping to smell the tulips, sample the goods at the coffeeshops, and tip back a half-pint with the very accepting Dutch. A weekend here will open your eyes to the possibilities of doing things just a little bit differently. So go on—take a taste!
### **AMSTERDAM 101**
The Dutch are a proud lot. It was their ancestors who turned a small, marshy port into the home base for the world's first global corporation. The Dutch East India Company, established in 1602, was the largest company in its time to trade with Asia. It was so successful that private navies were employed to protect its interests halfway around the globe.
The impact of these far-reaching companies made the Dutch language become the first internationally spoken language, and the Dutch guilder was the first internationally traded currency. A small town on the riverbanks of a marshland quickly grew into one of the world's richest cities, sparking a Golden Age of arts, culture, innovation, and a wee bit of gluttony.
Along with its riches, Amsterdam also grew in physical size, but only through the labor-intensive process of driving millions of wooden pilings into the soft marshy surroundings. As you walk through this city, remember that every square inch has been reclaimed from the sea, putting quite a premium on real estate. That's the reason staircases are steep (so as to reduce their footprint) and buildings are built high and essentially stapled together (look for the metal brackets on the older, leaning buildings throughout the city).
In the face of any obstacle, the Dutch have been able to overcome, but not without painful lessons. One such lesson the Dutch learned occurred when speculation over flower bulbs led to the world's first stock market bubble and collapse. "Tulip Mania" saw these plants trading at the price of entire estates, and people were making money hand over fist...until the market realized tulip bulbs just weren't worth that much. Overnight, vast wealth disappeared, shaking the Dutch economy to its core. The small, sea-level country eventually recovered and has now made a triumphant comeback to become the vibrant capital of commerce that it is today.
The Netherlands, as an economic powerhouse, welcomed migrants into its multicultural quilt. This history of integration and tolerance is also reflected in Dutch laws. They've seen few gains in trying to legislate morality. Rather, the business-savvy Dutch have taken a progressive approach in decriminalizing and taxing industries like cannabis and prostitution. Here's the Dutch rule of thumb for determining if an activity should be legal: Does it harm anyone? Is it good for business? Is it discreet?
As long as you answer these correctly, you can safely assume that whatever it is you want to do is legal, or at least decriminalized. This pragmatic approach has done a lot to show how legalization and regulation can undercut gangsters and pimps, and provide needed capital for support and recovery services for those who want out of their respective activities. What better way to experience this fascinating history and rich international culture than in the capital of the Netherlands itself, Amsterdam!
### **PLAN AHEAD**
#### **RESERVATIONS**
Reservations are recommended for the following sights:
**Anne Frank House** (annefrank.org)
**Van Gogh Museum** (vangoghmuseum.nl)
**Rijksmuseum** (rijksmuseum.nl)
**Heineken Experience** (heinekenexperience.com)
#### **LOCAL HAPPENINGS**
##### **Keukenhof**
Keukenhof (€21, 08:00-19:30,Stationsweg 166A, +31 (0)252 465 555, keukenhof.nl) is the most famous tulip field garden of the Netherlands. It's only open for a few weeks a year during bloom (the last week of March through the second week of May), so it's definitely worth penciling this into your itinerary should you be around at the right time. Make your visit efficient by checking it out on your day of arrival or departure, as the only connection is on bus 58 through Schiphol Airport.
##### **King's Day**
April 27 is Amsterdam's biggest celebration of the year, commemorating the king's birthday. The Dutch simply use it as an excuse to 1) sell their junk (the entire city turns into a yard sale) and 2) party hardy with free world-class events and music going on in numerous locations across the entire city. Make your hostel reservations long in advance, and be prepared to pony up a bit more than usual!
##### **Gay Pride Fest**
Every first Saturday in August, the historic world capital of tolerance and acceptance turns into one massive party. Club parties, canal parades, and street festivities draw hundreds of thousands and make this one of the most popular weekends of the year to be in Amsterdam.
**KNOW BEFORE YOU GO**
**KEY STATS & FIGURES**
**Currency:**
**The Netherlands uses the euro (€); 1 EUR = about 1.06 USD**
**Population:**
**820,000**
**Nationalities Represented:**
**177**
**Language:**
**Dutch**
**Bicycles:**
**880,000**
**Bike paths within the city:**
**250 miles**
**Bikes dredged from canals annually:**
**10,000**
**Canals:**
**165**
**Bridges:**
**1,281**
**Houseboats:**
**2,500**
**CALIBRATE YOUR BUDGET**
**TYPICAL PRICES FOR:**
**Hostel dorm bed:**
**€30**
**Two-course dinner and drink:**
**€15**
**Pint of beer:**
**€6**
**Day bike rental:**
**€7**
**Single tram pass:**
**€2.90**
**MOVIES TO WATCH**
_Ocean's Twelve, The Diary of Anne Frank, The Fault in Our Stars_
**VIDEOS TO WATCH**
_Holland: The Original Cool_
THREE DAY ITINERARY
Amsterdam's touristy center is walkable, but consider renting a bike to get around in a jiffy. Make reservations for the Anne Frank House on the second day ahead of time.
**DAY 1: WELCOME TO AMSTERDAM**
**MORNING**
Head to the south end of the heart of Amsterdam for a delicious breakfast at **De Laatste Kruimel.** Come hungry for their delicious pastries and thirsty for some fresh caffeine. Consider picking up a sandwich to go to munch on during your bike tour later. This bustling breakfast joint is just down the way from the University of Amsterdam campus, and a few steps from the floating **Bloemenmarkt** flower market and the **Dampkring** coffeeshop (where the "lost in translation" scene in _Ocean's Twelve_ was filmed).
**AFTERNOON**
Take the better part of the afternoon to get oriented on a bike tour. Rides with **Mike's Bike Tours** kick off daily at 12:00 at Kerkstraat 134 (3 hours), about a 10-minute walk south of De Laatste Kruimel. You'll cover a great deal of ground on the bikes, with plenty of time to enjoy a pit stop at the windmill brewery, **Brouwerij 't Ij.** (Not into bikes? Meet a **Sandeman's New Europe Walking Tours** guide at 11:15 at the National Monument, Dam Square for a three-hour tour.)
After wrapping up back at the shop, consider keeping your bike for your stay at a discounted rate. Walk or hop on tram 2 or 5 south to Museumplein for either the **Van Gogh Museum** or the newly renovated **Rijksmuseum**.
**EVENING**
Take tram 2 or 5 back north to rest up and relax for a bit at your hostel. When you're ready, hop a tram back to Central Amsterdam, exiting at Dam station, to grab a tasty Thai dinner at **Thai Snack Bar**. Then wander the **Red Light District** , where you'll be navigating some high-density raunch. You're likely to run into plenty of window-shopping creeps among the touristy bars and coffeeshops in the RDL, and you'll have no trouble satisfying the munchies with the plentiful food options around.
**LATE**
If you're looking to experience Amsterdam's rollicking nightlife scene, drop south to the **Rembrandtplein** entertainment district, where you're sure to find a loud, crowded, sweaty party—Dutch style.
**DAY 2: ANNE FRANK HOUSE & JORDAAN**
**MORNING**
Snag breakfast at your hostel, or enjoy some classy English fare at **Greenwoods** before you head to the **Anne Frank House,** a 10-minute walk away. Arrive early to avoid the crowds. Reservations made ahead of time are highly recommended!
**AFTERNOON**
Grab lunch just down the street at the **Pancake Bakery** (worth calling ahead for a reservation). Save some room for Amsterdam's best apple pie at **Winkel 43** in the **Noordermarkt** just across the canal (you can thank me later). If it's Saturday or a Monday, spend a while exploring the bustling farmers market, taking every sample that's offered to you. Take this opportunity to wander deeper into the **Jordaan** district as well.
Walk about 30 minutes or use tram 14 (exit Waterlooplein) to loop across town to **Gassan Diamonds** for a free diamond-cutting demonstration and factory tour. On the way, stop at the **Cannabis College** after for an education in all things green.
**EVENING**
Recharge at your hostel for a bit. Then head to dinner at **Kantjil & de Tijger** for an Indonesian meal that's easy on the wallet.
**LATE**
Head down Leidsestraat to Leidseplein, and catch a comedy show at **Boom Chicago**. Dive into any number of watering holes, or dress to impress to get past the bouncers at clubs **Paradiso** or **Melkweg**. To party like a rock star, catch the **Amsterdam's Ultimate Party pub crawl** kicking off each night with a power hour at 20:30 at Candela Bar.
**DAY 3: MUSEUMPLEIN & MORE**
**MORNING**
Starting in Dam Square, stroll down **Kalverstraat,** then **Leidsestraat.** Farther down Leidsestraat, take a lunch to go from the best sandwich spot in the city, **B &B Lunchroom,** and sample local cheeses at **Henri Willig Cheese & More**.
Continue onto **Vondelpark,** a beautiful city park laced with gardens, running paths, and bike trails. Enjoy a snack at **'t Blauwe Theehuis** in the middle of the park or snag a park bench and watch Amsterdam ride by while you munch your sandwich.
**AFTERNOON**
Exit the park through the south side toward Museumplein and pop into either the **Rijksmuseum** or the **Van Gogh Museum** —whichever you didn't visit on day one. Don't forget your photo op at the I Amsterdam sign on the square! (Insider secret: The front of the sign is always crowded with tourists. Snap your shot from the back, and flip it around in Photoshop afterward to have the sign all to yourself.)
After you get your fill of museums, walk about 15 minutes over to Europe's largest daily market, **Albert Cuyp Market.** Cute _bruin cafés_ (brown cafés) line the street, perfect for a coffee or tea. If you're hungry, nearby **Bazar** has delicious ethnic dishes. Drop down to relax in **Sarphatipark** , a block south of Albert Cuyp Market.
You've got a choice: Head to the **Heineken Experience** for a sensational brewery tour, then stick around the **De Pijp** neighborhood, popular among locals for food and nightlife. Or, hop on tram 5 to return to the center. Head to the **City Library** near the train station for a panoramic view of the city at sunset. The **V &D La Place** café on the seventh floor makes for a great early dinner option.
**EVENING**
Take an hour-long evening canal cruise; they leave hourly from the **Lover's Canal Cruises** stand just across from the train station, opposite the Play In casino. Bring a discreet six-pack to sip on while taking in the city from the canals.
**LATE**
Take it easy on **Nieuwmarkt,** enjoying some half-pints with the locals at the cafés opening up onto the square.
### **TOP NEIGHBORHOODS**
Amsterdam fans out south of the River Ij. At its heart, historic **Central Amsterdam** is home to the famous Red Light District and the Bloemenmarkt, as well as restaurants, coffeeshops, nightlife venues, and hostels.
West of Central Amsterdam and Singel canal, **West Amsterdam** contains the Anne Frank House, the Nine Streets shopping district, and the beautiful, residential Jordaan neighborhood. Jordaan is unique in Amsterdam in that canals don't cut through the streets, and it's fun to explore, with cafés and cute little shops.
The ring of canals south of Central Amsterdam is home to the **Leidseplein** entertainment district, including recommended restaurants and coffeeshops. **Southeast Amsterdam** has a couple of noteworthy attractions (Gassan Diamonds, the Jewish Historical Museum), along with the Rembrandtplein nightlife district. The southwestern corner of the city is Amsterdam's **Museum District,** home to the Rijksmuseum and Van Gogh Museum (which cluster around the square called Museumplein) and Vondelpark, Amsterdam's city park. East of the Museum District is the trendy nightlife neighborhood of **De Pijp** and the Heineken Experience.
### **TOP SIGHTS**
There are dozens of fascinating museums in this city to keep you busy for weeks, but on a short three-day visit, these are the ones worth seeing.
#### **Anne Frank House**
The city's most famous historical landmark, this is where Anne Frank and her family hid for more than two years during the Nazi occupation of Amsterdam. It's from this house that she wrote in her diary about the secret annex, growing up, and her personal thoughts. Her diary is now one of the world's most translated books, second only to the Bible. Otto Frank, Anne's father, survived the war, and insisted on keeping the apartment as it was, opting for empty rooms rather than filling them with replica furniture. Today, visitors can walk through the apartment, even passing through the hidden bookcase through which the family entered the secret annex. You'll see photos of the family, their concentration camp registration cards, and the diaries themselves.
€9, Apr-Jun and Sept-Oct daily 09:00-21:00, Sat until 22:00; July-Aug daily 09:00-22:00; Nov-Mar daily 09:00-19:00, Sat until 21:00; Prinsengracht 267, West Amsterdam, +31 (0)20 556 7105, annefrank.org, Tram: Westermarkt
**ACT LIKE A LOCAL**
**Gezellig-what?**
The Dutch have an expression, one word that is used for just about everything that makes you feel all warm and fuzzy on the inside: _gezellig_ (pronounced GHE-zell-igh, with guttural-sounding Gs). For Americans, bigger and newer is just about always better in my opinion. For the Dutch, _gezellig_ is the goal, and it's used to describe just about anything that is warm, inviting, and cozy. Tucked in front of the fire at Christmas is _gezellig_. A good brown pub in Amsterdam is just about always _gezellig_. Use your newfound cultural lingo to strike up a conversation. Where's your favorite _gezellig_ place?
**Make it a half-pint**
Prices for a half-pint here in Amsterdam are usually exactly half the price of a pint. So locals prefer to drink a half-pint at a time for a couple reasons: First, their drink stays cooler. Second, Dutch drinking culture tends to be more social—the Dutch often do-si-do between drinks and cigarettes.
#### **Rijksmuseum**
Art from the last 800 years of Amsterdam and the Netherlands takes you from the city's humble beginnings through masterpieces of the Dutch Golden Age. The museum features artists such as Rembrandt, Goya, Van Gogh, Cuyp, and Vermeer. The museum is presented across three levels. You'll notice the exterior looks remarkably similar to that of the Centraal Station. The same architect, Pierre Cuypers, designed both buildings.
€17.50, daily 09:00-17:00, last entry 30 minutes before closing, Museumstraat 1, Museum District, +31 (0)900 0745, rijksmuseum.nl, Tram: Rijksmuseum
#### **Van Gogh Museum**
Containing all the masterpieces of the most famous and crazed Dutch artist, this museum is wonderfully compact, spreading out over just three floors. Exhibits include Van Gogh's pieces along with those that inspired him and those whom he inspired. Some highlights include: _The Potato Eaters, Sunflowers, The Bedroom,_ and _Wheatfield with Crows,_ the last painting Van Gogh ever did, which some say was a portrayal of his inner sadness and imminent death.
€17, daily 10:00-17:00, Fri until 22:00, Paulus Potterstraat 7, Museum District, +31 (0)20 570 5200, vangoghmuseum.nl/en, Tram: Rijksmuseum or Van Baerlestraat
#### **Red Light District**
Centering around the Old Church (Oude Kerk), this is where Amsterdam grew up from a small fishing village into a world power. The old docks used to line the north side of these streets. So this is where the ladies of the night used to catch sailors on their last night in town or to help lighten their freshly lined pockets upon return. Today, you'll see the sex trade in action as you stroll by nearly nude prostitutes advertising for themselves behind glass windows. You'll also see touristy cafes and restaurants, coffeeshops, and souvenir stores. In recent years, the city council has made a point to reduce the number of prostitute windows in this district, replacing them with art installations and boutiques. There's a lot to keep you busy, but unfortunately, many tourists don't venture beyond this district to experience the more authentic parts of Amsterdam.
Free, always open, Central Amsterdam, Tram: Dam
#### **Cannabis College**
This college in the Red Light District shows you the truth about cannabis and all its potential uses. Pop in for an education in all things green, including information about the plant's medical uses. Ask them where they'd recommend picking up some kush, and they'll bust out a map with their favorite nearby shops. No marijuana is sold here. Pay a few euros to see the college's flowering cannabis garden, a handful of plants in various stages of growth from "babies" to flowering beauties.
**BE SMART: AMSTERDAM'S SMART SHOPS**
Smart shops sell magic legumes or other "herbal mood enhancers." Selling psychedelic mushrooms is now illegal in the European Union, so producers sell truffles to the same "magic" effect. These shops are generally staffed by helpful in-the-know people who make recommendations based on a customer's past experiences, as well as the experience the customer is seeking. If you choose to give this a try, pack something sweet like juice to rescue yourself from a bad trip.
Free, donations welcome, daily 11:00-19:00, Oudezijds Achterburgwal 124, Central Amsterdam, +31 (0)20 423 4420, cannabiscollege.com, Metro: Nieuwmarkt
#### **Heineken Experience**
This is a top priority for many who come to Amsterdam. Their sensational tour features a virtual reality "bottling experience," two half-pints, and long lines if you don't make reservations ahead of time. For a less mass-produced feel, consider visiting the Brouwerij 't IJ instead.
€18, €16 online; Mon-Thurs 10:30-19:30, last entry at 17:00; Fri-Sun 10:30-21:00, last entry at 19:00; longer hours Jul-Aug; Stadhouderskade 78, De Pijp and Vicinity, +31 (0)20 523 9222, heinekenexperience.com, Tram: Stadhouderskade
#### **Gassan Diamonds**
Ever been in the presence of a two-carat diamond? Learn about the craft of diamond cutting from the best with an introductory tour in English through the factory, where you'll see craftspeople going about their business and get an education in the "five C's": cut, color, clarity, carat, and certification. With a free entry and tour, Gassan is a hidden gem, if you know what I mean.
Free guided tour leaves often, daily 09:00-17:00, Nieuwe Uilenburgerstraat 175, Southeast Amsterdam, +31 (0)20 622 5333, gassandiamonds.nl, Tram: Mr. Visserplein
#### **City Library**
The last 150 years has seen a complete transformation of Amsterdam's harbor, starting with the construction of Amsterdam Centraal Station, which was completed in 1889. One of the most recent additions to the docklands has been Amsterdam's City Library, a beautiful 10-story modern steel-and-glass structure just northeast of Central Amsterdam. The top floor offers a magnificent view over the center of Amsterdam. Stop in for lunch and enjoy a wonderfully colorful and affordable spread from a branch of **V &D La Place,** a trendy Dutch self-service cafeteria brand.
Free, daily 10:00-22:00, Oosterdokskade 143, Central Amsterdam, +31 (0)20 523 0900, oba.nl, Tram: Centraal
#### **EXTRA CREDIT**
If you've got the time or the desire, check out these options for more ideas beyond the essentials.
##### **Jewish Historical Museum**
This in-depth museum takes you through the history and synagogues of this resilient community.
€6, daily 13:00-17:00, Nieuwe Amstelstraat 1, Southeast Amsterdam, jhm.nl, Tram: Mr. Visserplein
##### **Bag & Purse Museum**
Come visit this museum, located east of Leidseplein, to enjoy a collection of more than 4,000 different bags, trunks, pouches, and purses, dating all the way back from the Middle Ages to today.
€5, daily 10:00-17:00, Herengracht 573, tassenmuseum.nl, Leidseplein and Vicinity, Tram: Rembrandtplein
##### **Sex Museum**
Classic sex museum featuring erotic art and artifacts from the dawn of time through to today. Kinky, eh?
€5, daily 09:30-23:30, Damrak 18, Central Amsterdam, +31 (0)20 622 8376, sexmuseumamsterdam.nl, Tram: Centraal
##### **Electric Ladyland**
Made famous through Jimi Hendrix's third album, this experimental museum offers a psychedelic experience, as it's filled with trippy, fluorescent mini-displays and artwork.
€5, Tues-Sat 13:00-18:00, Tweede Leliedwarsstraat 5, West Amsterdam, electric-lady-land.com, Tram: Bloemgracht
### **TOP EATS**
Delicious food may or may not be the first thing on your mind when thinking of Amsterdam, but you might be surprised by the options. True Dutch cuisine is typical of other northern European countries: meat and potatoes. Their most famous dish is an extremely filling one, called **_stamppot:_** potatoes mashed together with your choice of veggies and a thick bratwurst-style sausage. **_Bitterballen_** are also a mainstay. Think flash-fried balls of mushy...well, I don't really know. Take it easy on the first bite, though, as they're always steamy on the inside! **Herring** is also a popular lunch dish in this sea-faring culture. Locals stop at any of the dozens of herring stands across the city for a quick lunch of raw herring and veggies.
The **Red Light District** is filled with places to grab a quick bite on the cheap (and satisfy the munchies). Trendy, fresh upscale restaurants can be found around nearly every corner throughout the rest of town. Tipping is not expected, but most locals round up a euro or two. These are some of my favorite budget options.
#### **De Laatste Kruimel**
Find this gem just on the south side of the Red Light District, a stone's throw from the University of Amsterdam. Come in for fresh scones, savory and sweet pies, delicious pastries, and hearty sandwiches to get your day off on the right foot. The name means "The Last Crumb" in Dutch, and you'll know why when you're searching your pastry paper for every last morsel.
Pastries and coffee from €3, daily 08:00-20:00, Sun from 10:00, Langebrugsteeg 4, Central Amsterdam, +31 (0)20 423 0499, delaatstekruimel.nl, Tram: Spui Rokin
#### **Greenwoods**
Need something more than a continental breakfast? Come here for a classy English breakfast complete with tea or coffee. In nice weather, enjoy your eggs and toast at the outdoor seating right next to the canal. In addition to the one on Singel, find another branch at Keizergracht 465.
Breakfasts from €6 and lunches from €8, daily 09:30-18:00, till 19:00 on weekends, Singel 103, Central Amsterdam, +31 (0)20 623 7071, greenwoods.eu, Tram: Nieuwezijds Kolk
#### **Blue Café**
In a glass box on the top floor of the Kalvortoren shopping center, this café offers up close views of Central Amsterdam. Though I love cafés with a view, I come here mostly for the coffee, as the food tends to be a bit overpriced.
Sandwiches from €7.50, Tues-Sat 10:00-18:30, Mon 11:00-18:30, Sun 12:00-18:30, Singel 457, Central Amsterdam, +31 (0)20 427 3901, blue-amsterdam.nl, Tram: Spui (Rokin)
#### **V &D La Place**
This is one of those delicious self-service cafeterias where the employees are well-paid and happy, the food is local and delicious, the ingredients are fresh and healthy, and the building is LEED-certified. Did I mention the free Wi-Fi? What else can you ask for? Pop into one of the several locations in town for an affordable, casual lunch at communal tables. All feature a wide range of culinary choices: fresh cooked fish and meats, curries and stir fry, personal pizzas, salads, fresh-pressed juices, and more. In addition to this branch, located on the seventh floor of the City Library just northeast of Central Amsterdam, find another city center location at Kalverstraat 203 in Central Amsterdam.
€10-15 lunches, daily 09:00-20:00, from 11:00 weekend mornings, City Library, Oosterdokskade 143, Central Amsterdam, laplace.com, Tram: Centraal
#### **Thai Snack Bar**
Find some of the best Thai food you'll ever taste just steps from the Red Light District in Amsterdam's China Town. This cheap eatery is almost always packed with in-the-know locals stopping by for a quick bite or takeout dinner. If the snack bar is full, consider the sister restaurant across the street with the same great menu. I love the friendly (if curt) service, warm atmosphere, fresh pad thai, and the spicy curries.
Entrées €10, 14:00-21:00, Zeedijk 77, Central Amsterdam, +31 (0)20 420 6289, thai-bird.nl, Metro: Nieuwmarkt
#### **Kantjil & de Tijger**
This is Indonesian at its finest. Kantjil has the feel of an Asian fusion restaurant but cranks out Indonesia's most typical delicious dish, _rijsttafel_. Think of it as a culinary experience that comes with a smorgasbord of small bowls of richly spiced rice, meats, and veggies. The poor man's _rijsttafel_ is _nasi rames,_ which includes many of the same great-tasting samplers as its bigger brother. This sit-down restaurant has a cheap, delicious twin takeaway joint next door. At the takeaway shop, pick your size, noodles, and meat to take advantage of one of the best dinner values in town.
Entrées from €15, takeaway €5-7, daily 12:00-23:00, Nieuwezijds Voorburgwal 342, Central Amsterdam, +31 (0)20 620 0994, kantjil.nl/en, Tram: Spui
#### **Winkel 43**
You haven't tried apple pie until you've indulged in this homemade, deep-dish, steaming slice of heaven. Share a piece with a friend because the portions here are generous. (Who am I kidding? Keep it all for yourself!) Winkel also has a decent menu with sandwiches, salads, and appetizers if you don't ruin your appetite on pie, though I always do.
Worlds of steaming crusty pleasure for around €6, daily 08:00-01:00, Noordermarkt 43, West Amsterdam, +31 (0)20 623 0223, winkel43.com, Tram: Marnixplein
#### **Pancake Bakery**
Hands down some of the best pancakes you'll ever have. Choose from a traditional variety or take a walk on the wild side and try one of the bakery's international versions, ranging from Hungarian to Thai to Mexican. Those with a sweet tooth should indulge in the _poffertjes_ , mini pancakes layered in ice cream, chocolate, and powdered sugar. The staff can seem a bit peeved at all the customers that flock here, so don't expect smiling service.
Pancake plates €12-16, daily 09:00-21:30, Prinsengracht 191, West Amsterdam, +31 (0)20 625 1333, pancake.nl, Tram: Westermarkt
#### **B &B Lunchroom**
Stop here along your shopping route for some of the best made-to-order sandwiches you'll ever have, especially considering the price. The options are endless and the staff couldn't be friendlier. Snag a delicious, freshly baked muffin to go for a snack later. In addition to this branch, you'll find other locations around town.
Sandwiches €5, daily 08:00-18:00, Leidsestraat 44, Leidseplein and Vicinity, +31 (0)20 642 1816, onzecatering.nl, Tram: Prinsengracht
#### **Vleminckx Friteshuis**
They do one thing, and they do it well! This place is famous for its _frites_ (French fries). Crowds flock here for a delicious snack in a cone. The line often wraps out down the street, but it moves quickly, and the fries are well worth the wait.
€3, daily 12:00-19:00, Voetboogstraat 33, Central Amsterdam, +31 (0)20 624 6075, vleminckxdesausmeester.nl, Tram: Spui
#### **Febo Fast Food**
Ever bought a burger or fries out of a vending machine? Well here's your chance! Not recommended for the quality of the food, but more so for the prices and the experience. Step into a small shop with floor-to-ceiling cubbies—like the P.O. Box room at your local post office, except that every little door is transparent. Simply drop in a euro or two, turn the knob, and reach in to grab your snack. Burgermeisters are busy on the inside constantly restocking the popular selections. Febo seems to hit the spot for many a late-night reveler. In addition to this branch, you'll find numerous locations around town.
€3, daily 10:30-02:00, till 04:00 on weekends, Damrak 6, Central Amsterdam, +31 (0)20 638 5318, www.febo.nl, Tram: Centraal
#### **Albert Heijn Grocery Stores**
Albert Heijn is to Amsterdam as Starbucks is to Seattle: You've got one on just about every corner. Those on a budget can dive into the aisles of this grocery store for a cheap lunch of anything from pre-made curries to sushi and sandwiches. Don't forget to pick up some of my favorite sweet treats: the _stroopwafel_ (syrup-filled waffle). You'll find numerous locations throughout the city, including one inside the train station.
Daily 08:00-22:00, Nieuwezijds Voorburgwal 226, ah.nl, +31 20 421 8344, Tram: Dam
#### **Henri Willig Cheese & More**
Thanks to numerous locations across town, you can't spend a few days in Amsterdam without stumbling into at least one Henri Willig cheese shop. Sample typical cheeses like Gouda, goat, sheep's, smoked, and even spicy cheeses, all made in the Netherlands. While it may feel a bit mass-produced and commercialized, a round of cheese makes for a great souvenir...if you can get it home without eating it beforehand, that is.
Pound cheese rounds from €10, daily 09:00-19:00, Leidsestraat 52, Leidseplein and Vicinity, +31 (0)20 620 9030, henriwillig.com, Tram: Kaizergracht
#### **'t Blauwe Theehuis**
A welcome respite from the bustling city center, this teahouse with a load of outdoor seating is right in the middle of Vondelpark. Considering the idyllic location, prices aren't bad. Pop in for a recharge that won't hurt the wallet.
Simple sandwiches from €5.50, daily 09:00-18:00, Vondelpark 5, Museum District, +31 (0)20 662 0254, blauwetheehuis.nl, Tram: Van Baerlestraat
#### **Bazar**
This Amsterdam institution is located right in the Albert Cuyp Market. As soon as you step into the Bazar, you forget you're in Amsterdam and feel like you've traveled to another continent. Come out for fresh kebabs, falafel, sandwiches, and other cuisine from the Middle East. The friendly service is happy to explain dishes that are foreign to you.
Breakfasts and lunches from €6, dinners from €12, Mon-Fri 11:00-24:00, Sat-Sun 09:00-24:00, Albert Cuypstraat 182, De Pijp and Vicinity, +31 (0)20 675 0544, bazaramsterdam.com, Tram: Albert Cuypstraat
### **TOP COFFEESHOPS**
The first question I get from Amsterdam-tourists-to-be: "Where are the best coffeeshops?" First off, if you're looking for coffee, you'll want to head to a café. As all Amsterdammers know, a coffeeshop is a place to buy and smoke the city's famous high-grade marijuana. There really isn't one "best" option in town. I look for friendly staff, fair prices, and a welcoming atmosphere. If the coffeeshop checks those boxes, I'll order a mint tea and make myself at home.
#### **Dampkring**
Made famous by the movie _Ocean's Twelve,_ Dampkring is a cozy one-roomed rose-colored coffeeshop with a cat named Bowie in permanent residence. The menu is on the pricey side but features an array of award-winning strains. The edibles are also excellent, but quite strong! Order your greens in the back, and drinks at the bar to the left.
Grams from €10, daily 10:00-01:00, Handboogstraat 29, Central Amsterdam, +31 (0)20 638 0705, dampkring-coffeeshop-amsterdam.nl, Tram: Spui
#### **Easy Times**
Popular among tourists and locals alike for their space cakes, clean and trendy atmosphere, and friendly service, Easy Times is my pick for the Leidseplein district. Go easy on the cakes, though. They're notoriously strong. On nice days, the canal-side outdoor seating cannot be beat. The interior feels a bit more like a mod hookah lounge than dark coffeeshop.
Grams from €8, daily 09:00-01:00, Prinsengracht 476, Leidseplein and Vicinity, +31 (0)20 626 5709, easytimes-amsterdam.com, Tram: Prisengracht
#### **Grey Area**
Grey Area, just west of Central Amsterdam, is a classic in Amsterdam's coffeeshop roster. Pop into this tiny café to stock up, and snag one of the six seats they have in house to roll up and meet fellow tokers. The staff is particularly welcoming, and happy to take beginners under their wing to explain the menu.
**COFFEESHOPS & SMOKING ETIQUETTE**
Amsterdam operates by a discreet code: You buy coffee in cafés, marijuana in coffeeshops, and psychedelic mushrooms in "smart shops." Nothing is legal but rather it's "decriminalized." This is why you'll see no advertisements or signs. Coffeeshops can't even have their own website. But they're not hard to find: Just look for a numbered green and white license in the front window of each shop.
Pop into a few shops on your first day to get a sense for the options out there. You're not obligated to buy anything, but it is frowned upon to consume things that weren't purchased there.
The drink bar and marijuana bar are usually separate. Find the menu at the weed bar, and talk to the staff to get exactly what you're looking for. Take your time and don't be afraid to ask questions—if they're rude, walk out and find the next shop. Coffeeshops on the tourist track from the train station to the Red Light District (and inside the canal rings) with Bob Marley and fluorescent lights in the windows cater to tourists and are likely overpriced.
You can no longer smoke tobacco indoors. Hard drugs like heroin and cocaine are strictly illegal, and possession carries stiff penalties. And remember; green-thumbed scientists have been breeding marijuana for years to make it stronger and stronger. It's quite intense, so don't overdo it. Don't mix marijuana with alcohol, as unpleasant side affects like sweating and nausea can arise. Drink something sweet like juice or soda if you're not feeling well. It's good to stick with your friends and have the hostel address handy if you need to make a quick exit.
By the way, per capita, the Dutch smoke less than half as much as compared to Americans. Interesting, eh? Here are some tips to get you started on your coffeeshop experience.
**First, some terminology:**
• Marijuana = weed.
• Hashish = a small brown brick you rub off into tobacco cigarettes.
• Joint = weed only in a paper wrap.
• Spliff = weed and tobacco (what the Dutch would normally smoke—if they smoked).
• Sativa is the "high" strain (think: high, happy, giggly).
• Indica is the "stoned" strain (think: chill, relaxed, sleepy, heavy).
**Here are some other things that are good to know:**
• Don't pay more than €11/gram. Any more and you're in a touristy zone!
• A good coffeeshop has helpful staff, a welcoming vibe, nice ambience, and comfy seating.
• After purchasing a gram ask for a few papers for free. Purchase 5 grams and you should get a sleeve of papers included.
• Avoid pre-rolled joints. It's much better to purchase a gram or two of loose weed and buddy up with someone in the shop to help you out. People are usually happy to help, and it's a great way to strike up a conversation with new friends.
• You'll usually get a 10-15 percent discount on purchases of 5 grams (which should be plenty for a few friends over a weekend).
• Take it easy on edibles. Don't drink or smoke more after having eaten them, and wait at least 1.5 hours before taking anything else or anything more. Make sure to ask how many doses are in the edible you're purchasing—some have one, some are good to split across four friends. They usually run €5-15.
• Actually looking for just a coffee? Go to a café!
Grams from €9, daily 12:00-20:00, Oude Leliestraat 2, West Amsterdam, +31 (0)20 4204301, greyarea.nl, Tram: Dam
#### **The Bulldog**
I list this place not so much to recommend it, but as a place to pop into to calibrate the kind of scene you're looking for. Here you'll find loud music, a grungy vibe, and touristy prices. It's never far away, thanks to numerous locations in the RLD and Leidseplein.
Grams from €10, daily 09:00-01:00, till 03:00 on weekends, Leidseplein 15, Leidseplein and Vicinity, +31 (0)20 625 9864, thebulldog.com, Tram: Leidseplein
#### **The Grasshopper**
While touristy, it's worth popping into this multi-service establishment, where you'll find a steak house upstairs, a bar on the ground floor, and a coffeeshop downstairs. The views of the Centraal Station from the canal-side seating are hard to beat while enjoying a beer and a fresh roll.
Grams from €8, daily 10:00-01:00, Oudebrugsteeg 16, Central Amsterdam, +31 (0)20 626 1259, thegrasshopper.nl, Tram: Dam
### **TOP NIGHTLIFE**
The drinking culture among the Dutch is a very social one and is done over a few beers in the square. Don't be afraid to strike up a conversation with them. Their English is probably better than yours!
**_Bruin cafés_** ("brown pubs" or "brown cafés") are a mainstay of Dutch and Amsterdam culture. Affectionately so-called because of their dark wood and tobacco-stained interior, these picturesque spots are perfect for an afternoon tea or snack. Each individual café is rather unremarkable, but they do offer a nice place for a break any time of day: coffee in the mornings, light lunches and snacks throughout the day, and half pints towards the evening. **Café van Zuylen** (snacks and coffee from €5, Torensteeg 8, +31 20 639 1055, cafevanzuylen.nl) inside of Singel canal is one supremely typical _bruin café_ with canal-side seating in the summer.
#### **NIGHTLIFE DISTRICTS**
Amsterdam's nightlife can be broken down by neighborhood. Each district has its own personality. Match your preferences to my descriptions and get ready for a wild ride!
##### **Red Light District**
The Red Light District boasts a rainbow of venues, bars, clubs, street food, peep shows, and more. And it's the most dense, most touristy, and most overpriced part of town. While the RLD is worth an evening out, many tourists unfortunately never get beyond this area in the day or night. Notice the window-shopping tourists change as the day turns to night. At night come obnoxious stag parties and creepers doing more than just window-shopping. Sprinkled with numerous police departments, the neighborhood is actually the safest in town, but keep a close hold on your valuables in the busy and crowded streets. Spend an evening wandering through the canals, bridges, and seductive windows. One redeeming venue in this district is the recommended brewery **Brouwerij de Prael.**
Central Amsterdam, Tram: Dam
##### **Nieuwmarkt**
What was once a major gatehouse to enter the city and where Amsterdam's gallows were located is now a popular square lined with bars just on the edge of the Red Light District. You'll find a more subdued, sophisticated vibe with locals sipping half-pints on the east side of the square.
Central Amsterdam, Tram: Dam
##### **Leidseplein**
The best entertainment district in Amsterdam is named after a town 20 miles down the road: Leiden. Farmers would bring their milk, produce it, sell it, and then blow all their money before going home. To this day, this area is known for its music venues, concert halls, comedy shows, and restaurants. It's where tourists and locals go out to club. Prices unfortunately reflect its popularity, and you'll often have to pay to use the bathrooms, even in bars where you're drinking. So don't break that seal! Beyond numerous dance bar options on the square, **Melkweg** and **Paradiso** are music venue institutions that have headliners like Lady Gaga and Beyonce pass through often.
Leidseplein and Vicinity, Tram: Leidseplein
**LGBT AMSTERDAM**
As one of the most inclusive and progressive cities in the world, Amsterdam has long been known to the members of the LGBT community as an excellent place to visit, live, and work. Gay Pride dominates the city each year during the last week of July leading into August. There is no need to specifically seek out gay friendly restaurants and bars in this town, as most options are open to all—and there are many venues with rainbow flags in the windows. You'll find the gay, BDSM and Bear-focused street on Warmoestraat just around the corner from Dam Square to the east. For more information, find the **Pink Point LGBT** information stand on Westermarkt, in the shadow of Westerchurch and just steps from the Anne Frank House, and the **Homomonument** (daily 10:30-18:00, Westermarkt, +31 20 48 1070, facebook.com/PinkPointAmsterdam).
##### **De Pijp**
On the south side of town behind the Heineken brewery, you'll find Amsterdam's best-kept secret, where the locals hang out before heading to Leidseplein for the clubs. This is Amsterdam's trendiest and most up-and-coming district. Most of the action centers around a square, **Marien Heinekenplein,** which is lined with a half dozen social bars.
De Pijp and Vicinity, Tram: Stadhouderskade
##### **Rembrandtplein**
This square, east of Leidseplein, and the surrounding streets tend to be packed with blue-collar Dutch types and tourists in the know. While a bit pricey, the parties get rowdy here on a nightly basis. You'll find cafés and brown pubs, restaurants, bars, and even Starbucks' flagship location in all of Europe right here on this lively square named after one of the most famous painters to emerge from the Dutch Golden Age. A rock star in life, I think Rembrandt would be happy with the scene that takes over his square on a nightly basis.
Southeast Amsterdam, Tram: Rembrandtplein
#### **BREWERIES**
Heineken isn't the only brewery with an Amsterdam connection. Try out these spots as well:
##### **Brouwerij 't IJ**
The city has given tax breaks to any company willing to conduct their business in windmills in order to protect the heritage and keep these monuments to Dutch history alive and well. Thankfully, a local brewery took the city up on the offer, and now you can go in and taste a range of house brews and enjoy a fascinating, quite affordable tour of the hoppy facilities. Brouwerij 't IJ is east of Amsterdam's city center.
Pints from €4.50, pub open daily 14:00-20:00, €4.50 tours Fri-Sun at 15:30 and 16:00 (reservations recommended), Funenkade 7, Greater Amsterdam, +31 (0)20 622 8325, brouwerijhetij.nl, Tram: Hoogte Kadijk
##### **Brouwerij de Prael**
Real estate is so hard to come by in Amsterdam that breweries like this are rare in the center, but this is a wonderful exception to the rule. Step into the Prael brewery and you'll feel at home in the cozy tasting room lined with mugs owned by regulars who frequent this place. Head to the back to see the magic in the making with big fermentation vats, and avail yourself of any one of their 10 house-made brews. Free brewery tours leave on the hour in the afternoons Tuesday-Sunday.
Pints from €5, daily 12:00-01:00, Oudezijds Voorburgwal 30, Central Amsterdam, +31 (0)20 408 4470, deprael.nl, Tram: Centraal
#### **BARS**
Unfortunately, many Dutch brews are often overshadowed by another agricultural production—weed. If artisan beer and unique pubs pique your fancy, follow my **ale trail** (from **In de Wildeman** to **Arendsnest, Café de Prins,** and finally **De Zotte** ) for a low-key night of sipping Dutch brews.
##### **In de Wildeman**
Famous for its wide selection of beers both in bottles and on tap, this distillery-turned-bar is a great place to start an ale crawl, right in the old town. The cozy setting and helpful staff help you whittle down your hundreds of international brew-tastic choices.
Tastings from €2.40, Mon-Sat 12:00-01:00, Kolksteeg 3, Central Amsterdam, +31 (0)20 638 2348, Tram: Nieuwezijds Kolk
##### **Arendsnest**
You may well need guidance by the friendly bartenders to pick your beer at this spot, which proudly serves Dutch beers only. This is a one-stop shop for you to go on a happy hoppy tour of the Netherlands. Get started with the mini tasting of four beers for €6.
Daily 16:00-12:00, later on weekends, Herengracht 90, West Amsterdam, +31 (0)20 4212057, Tram: Nieuwezijds Kolk
##### **Café de Prins**
For your typical little Dutch bar where you can enjoy everything from coffee and snacks to beer and music, Café de Prins checks all the boxes. Across the canal from the Anne Frank House, Café de Prins does a good job of staying under the radar and keeping all its cozy _gezellig_ atmosphere.
Soups and sandwiches from €4, daily 10:00-01:00, Prinsengracht 124, West Amsterdam, +31 (0)20 624 9382, Tram: Westermarkt
##### **De Zotte**
Drop into De Zotte for its impressive lineup of Belgian brews. A constantly rotating menu offers a new experience for each returning visit. Watch out, the Trappists are coming! Hungry by now? They've got some nice little cheese and meat plates to snack on.
Pints start around €4, daily 16:00-01:00, Raamstraat 29, Leidseplein and Vicinity, +31 (0)20 626 8694, Tram: Raamplein
##### **Winston Bar & Venue**
Located in the ground floor of one of my favorite hostels, St Christopher's Winston Hotel, this bar is popular with backpackers who come for the cheap prices, open patio in the back, and smoking lounge. The in-house music venue frequently has rock bands playing. Check the website for the calendar.
Drinks from €4.50, daily 10:00-01:00, till 03:00 on weekends, Warmoesstraat 129, Central Amsterdam, +31 (0)20 623 1380, winston.nl/bar, Tram: Dam
##### **Kingfisher Café**
At this comfortable, modern _bruin café_ , you'll find locals sipping on beers and nibbling away at cheese plates. Pull up a stool to the bar and strike up a conversation with their friendly bartenders. If you're looking for a quiet local's perspective on a night out, this is a great place to start.
Pints from €6, Mon-Thurs 10:00-01:00, Fri-Sat 10:00-03:00, Sun 12:00-01:00, Ferdinand Bolstraat 24, De Pijp and Vicinity, +31 (0)20 671 2395, kingfishercafe.nl, Tram: Stadhouderskade
#### **CLUBS**
##### **Melkweg**
Melkweg is a serious live-music Amsterdam institution that draws crowds by the thousands. Be sure to check the program online, and purchase tickets ahead of time if there are acts that you don't want to miss. The venue is expansive: a massive dance floor faces a stage lined with two observation levels above. International DJs, rock bands, rappers, and pop singers all come here to rock out with the enthusiastic young crowds. Plan on lines and cover.
Covers and show tickets from €7, shows generally start 20:00-23:00, Lijnbaansgracht 234A, Leidseplein, +31 (0)20 531 8181, melkweg.nl, Tram: Leidseplein
##### **Paradiso**
This famous nightlife venue, housed in an old church, has hosted many big names since opening their doors in the 1960s. Their artist list reads like a who's who of the music and entertainment industry: Amy Winehouse, Lenny Kravitz, the Rolling Stones, James Brown, Daft Punk, Lana del Ray, and plenty more. The genres of events span a range of tastes.
Covers and show tickets from €10, the party picks up around 23:00 and goes til late every night of the week, Weteringschans 6-8, Leidseplein, +31 (0)20 626 4521, paradise.nl, Tram: Leidseplein
#### **PUB CRAWLS**
##### **Amsterdam's Ultimate Party**
Take the work out of a night out and join Amsterdam's Ultimate Party pub crawl. You've got two choices: exploring the Red Light District's bars and clubs or hitting the clubbing district of Leidseplein. I'd personally opt for the latter, as it's easy to wander through the RLD and create your own adventure. Show this book for a possible discount.
Pricey at €20, kicks off nightly at 20:30, meets at Candela Bar (Korte Leidsedwarsstraat 85), Leidseplein and Vicinity, +31 (0)20 776 7888, joinultimateparty.com, info@joinultimateparty.com, Tram: Leidseplein
#### **SEX SHOWS**
Nothing is beyond the boundaries in this city, including the option to pay for and witness bored-looking sex performers pound through a series of erotic acrobatics and naughty spectacles.
##### **Casa Rosso**
If you're in the market for a sex show, this is Amsterdam's most popular. Pay €40 for entrance, or €50 for two watered down drinks to be included, and you're free to stay for as long or as little as you want. Six erotic shows with solo or more actors perform their scandalous deeds for rotations of about 10 minutes in length, so after an hour you've basically seen it all. For a cheaper alternative, drop two euros into the peep show at **Sex Palace** (#84) next door for some instant gratification.
€40 entry, daily 19:00-02:00, till 03:00 on weekends, Oudezijds Achterburgwal 106-108, Central Amsterdam, +31 (0)20 627 8954, casarosso.nl, Tram: Dam
#### **COMEDY SHOWS**
##### **Boom Chicago**
Dig comedy shows? This is your place for no-holds-barred English improvisational comedy that artfully blends the quirky Dutch sense of humor with a frank and politically incorrect attitude. Now a mainstay on Amsterdam's cultural scene, Boom Chicago has been putting on critically acclaimed shows since the early '90s.
Seats start at €11, shows run Wed-Sun, doors open at 20:00, show time generally 20:30, Rozengracht 117, West Amsterdam, +31 (0)20 217 0400, boomchicago.nl, Tram: Rozengracht
### **TOP SHOPPING & MARKETS**
#### **SHOPPING DISTRICTS**
##### **Nieuwendijk, Kalverstraat & Leidsestraat**
This long, semi-unbroken chain of streets leads from the Centraal station, through Dam Square and all the way down to Leidseplein and Vondelpark. You could easily spend an afternoon strolling down these streets, which evolve from kitschy and sleazy by the train station to stylish and trendy as you approach Leidseplein. On this mile-long stretch, you'll find fashion boutiques, chocolate shops, grocery stores, souvenir shops, housewares stores, and more.
Central Amsterdam and Leidseplein, Tram: Dam
##### **The Nine Streets**
The Nine Streets neighborhood (theninestreets.com), packed with cute, trendy, and independent shops and eateries, is so called because its three streets (Hartenstraat, Wolvenstraat, and Huidenstraat) cross over three canals (Herengracht, Keizersgracht, and Prinsengracht). It's just south of Anne Frank's House and Westerkerk and between the Singel and Prinsengracht canals.
West Amsterdam, Tram: Westermarkt
#### **MARKETS**
##### **Albert Cuyp Market**
Named after a famous Dutch author, the Albert Cuyp Market is Amsterdam's largest and is a main reason for the De Pijp neighborhood's recent popularity. This market features 300 vendors offering everything from chocolate-glazed waffles, fresh fruit, and fish to sparkly pants and leather goods. It's a cultural grab bag that speaks to the diversity of this city.
Free, daily 09:00-17:00, Albert Cuypstraat, albertcuypmarkt.com, De Pijp and Vicinity, Tram: Albert Cuypstraat
##### **Bloemenmarkt**
At this famous floating tulip market just south of the center, flower stalls are parked permanently on houseboats. Vendors are happy to package your bulbs to make sure you get them through customs on your way home.
Free, Mon-Sat 09:00-14:30, Sun 11:00-17:30, Singel near the Mint Tower, Central Amsterdam, Tram: Koningsplein
##### **Noordermarkt**
Come here on a Saturday or Monday, and this square will be packed with vendors hawking their fresh produce, meats, cheeses, and more. It's not only a food market, though. Go deeper and you'll find antiques, clothes, jewelry, and pretty much anything else you can imagine.
Sat and Mon 09:00-14:00, West Amsterdam, noordermarkt-amsterdam.nl, Tram: Marnixplein
##### **Royal Delft Experience**
Beyond the cliché windmill or phallic souvenirs, Delftware makes a great gift for loved ones back home. Named after a nearby town from where it originates, Delftware is classic blue on white and often depicts typical Dutch scenes. Splurge on hand-painted pieces or find much more affordable, printed versions toward the back of most stores. Find the Royal Delft Experience ceramics shop in the Munt Tower (Mint Tower) just on the south side of the center of town. You can pay to tour the working factory, or just shop.
Experience: Apr-Oct daily 10:00-20:00, Nov-Mar daily 10:00-17:30, €5; shop: Apr-Oct daily 09:30-21:00, Nov-Mar daily 09:30-18:00; Muntplein 12, Central Amsterdam, Tram: Spui
### **TOP PARKS & RECREATION**
Don't leave Amsterdam without exploring at least one of the city's beautiful parks. Each park offers a wonderful respite from the hectic city center.
#### **PARKS**
##### **Vondelpark**
NYC's Central Park was modeled after Amsterdam's largest and most popular one, located south of Leidseplein. On a sunny day, it's hard to beat this picturesque setting for a relaxing picnic or bike ride. You can spend hours exploring the paths and discovering the small lakes that dot this 120-acre park.
Free, always open, Museum District, Tram: Emmastraat
##### **Sarphatipark**
Be sure to stop here while making your trek out to Albert Cuyp Market. This small, beautiful park is skipped by most tourists, so you'll easily find a great spot to picnic and take a load off after a long day of sightseeing. Manicured lawns and fanciful bridges reflect the Romantic movement of the times from the 19th century. The park is located just south of the city center.
Free, always open, Albert Cuypstraat 2, De Pijp and Vicinity, Tram: Albert Cuypstraat
#### **CANAL CRUISES**
##### **Lover's Canal Cruises**
Amsterdam's most budget-friendly canal cruises offer day, evening, and dining cruises. Trips are narrated in multiple languages and often come with an entertaining captain. A canal cruise is a relaxing way to change your perspective on the city and take it in from the angle that it was built for. Find the kiosk just in front of Centraal station.
Cruises from €15.50, Stationsplein 10, Central Amsterdam, +31 (0)20 330 1374, lovers.nl, Tram: Centraal
### **TOP TOURS**
#### **Mike's Bike Tours**
A tour with Mike's Bike Tours is a great way to see central Amsterdam. It includes a pit stop at a local brewery in a windmill. They'll start you off with a brief story of Amsterdam with some beautiful historical maps, then saddle you up on cruisers to hit the town with a trusty guide. They've also got Dutch countryside tours, where you'll see windmills, tulips, and happy Dutch cows all along the way. You'll take a 30-minute break at a local cheese farm, where you can sample some of their delicious Gouda.
€19, 12:00 daily except Dec 13-Jan 4, 3.5-hour tour, Kerkstraat 123, Leidseplein and Vicinity, +31 (0)20 622 7970, mikesbiketoursamsterdam.com, Tram: Keizersgracht
#### **Sandeman's New Europe Walking Tours**
Take a free three-hour walking tour with New Europe to help orient yourself to the city. Your guide will entertain you with Amsterdam's wild history of drugs, prostitution, and Nazi occupation. While your tour group may be a bit large, the guides are skilled at leading you to different parts of the city that you'd never find on your own.
Free (tips required), meets daily at 11:15 and 13:15 at the National Monument, Central Amsterdam, newamsterdamtours.com, Tram: Dam
### **TOP HOSTELS**
Some of my favorite hostels I've ever stayed at are in Amsterdam. Each has its own identity, and you can generally depend on a cool vibe and fun backpacking atmosphere. All of my recommendations are clean and safe, and all have lockers, towels, and free Wi-Fi available. Many have their own drug policies, so be sure to note and respect them. Generally, you can smoke in a designated area, though harder drugs are never allowed on the premises (or anywhere in the city for that matter). While staying in group dorms, always have your valuables locked whether you're in the room or not, just to play it safe. Real estate is at a premium in Amsterdam, so prices for bunks can climb upward of €30, €40, or even €60 during popular events and weekends.
#### **St Christopher's Winston Hotel**
If you're looking to stay in the middle of the action, look no further. Quentin Tarantino camped out here while writing the script for _Pulp Fiction_. Enjoy the fun bar, extensive free breakfast, music venue, smoking lounge, and patio out back, which backs up against the rosy red lights.
Bunks from €35, 24-hour reception, free Wi-Fi, breakfast included, lockers available, full bar, Warmoesstraat 129, Central Amsterdam, +31 (0)20 623 1380, winston.nl, winston@winston.nl, Tram: Dam
#### **Flying Pig Downtown**
The downtown branch of the Flying Pig is in one of the oldest parts of town, just minutes from the Red Light District. It's known for its fun atmosphere, which you can also experience at the Flying Pig Uptown (bunks from 16, same amenities, Vossiusstraat 46/47, +31 (0)20 400 4187, flyingpig.nl, uptown@flyingpig.nl, Tram: Van Baerlestraat), near the mouth of the sprawling Vondelpark, close to the Leidseplein entertainment and party district.
Bunks from €19, 24-hour reception, free Wi-Fi, breakfast included, lockers, café, bar, Nieuwendijk 100, Central Amsterdam, +31 (0)20 420 6822, flyingpig.nl, downtown@flyingpig.nl, Tram: Nieuwezijds Kolk
#### **Bulldog Hotel & Hostel**
Smack-dab in the center, the Bulldog hostel is a great option to get the most out of your stay. It has a great atmosphere, and a fantastic breakfast is included. Amsterdam is sprinkled with coffeeshops and bars of the same brand, though they don't sport the value of the namesake hostel.
Bunks from €27, 24-hour reception, free Wi-Fi, laundry, lockers, breakfast included, Oudezijds Voorburgwal 220, Central Amsterdam, +31 (0)20 620 3822, bulldoghotel.com, info@bulldoghostel.nl, Tram: Dam
#### **Cocomama**
For those who shy away from grungy backpacker hostels, this boutique chain is redefining budget accommodations. While you'll pay a premium for it, you'll find a very different, much more chill vibe. You can relax in the lounge in the common room and sleep easy in comfortable, purpose-built bunks. This hostel east of Leidseplein, run by fun and helpful staff, is non-smoking.
Bunks from €35, 6-bed dorms, free Wi-Fi, 24-hour reception, laundry, lockers, Westeinde 18, Leidseplein and Vicinity, +31 (0)20 627 2454, cocomama.nl, info@cocomama.nl, Tram: Frederiksplein
#### **Ecomama**
Run by the same company as Cocomama, Ecomama, just east of Central Amsterdam, has a similar vibe. It's also non-smoking.
Bunks from €32, 5-, 7-, 8-, and 12-bed dorms and private doubles, free Wi-Fi, 24-hour reception, laundry, lockers, Valkenburgerstraat 124, Central Amsterdam, +31 (0)20 770 9529, ecomamahotel.com, ecomamahotel@gmail.com, Tram: Mr. Visserplein
### **TRANSPORTATION**
#### **GETTING THERE & AWAY**
The excellent public transportation system in Amsterdam makes getting from the train station or airport to your accommodations easy.
##### **Plane**
From **Amsterdam Schiphol Airport** (AMS, schiphol.nl), trains leave for Amsterdam Centraal Station every 10 minutes from platforms one and two, and take about 15 minutes to get to the city center. Grab a snack from the grocery store so you'll have change on hand for the ticket machines. Purchase a ticket (€3.60) there or from the ticket desk. There is an express train called Fyra, but the time saved isn't worth the additional cost. Luggage storage is available in the basement of the airport.
If you're flying into **Eindhoven Airport** (EIN, eindhovenairport.nl), connections are made via bus and take you straight to Amsterdam Centraal Station. Transfers cost €25 each way and take about 90 minutes.
##### **Train**
Trains to Paris (about €75) run daily and take about 3.5 hours. For Berlin, trains cost about €50-60 and take 6.5 hours.
The **Amsterdam Centraal Station** is on the northern border of Central Amsterdam. If your hostel is anywhere in the center, it'll be no more than a half mile away from the train station, so consider walking if you're comfortable with your bags. If your hostel is in the canal rings or farther, opt for the tram. There is a helpful **tourist information center** (look for the blue-and-white VVV logo) just out front of the train station across the tramway. Drop in there to get directions and pick up a map.
##### **Bus**
Buses from other destinations in Europe drop off at **Amsterdam Centraal Station.** Check eurolines.com for routes and prices.
##### **Car**
Amsterdam is about 500 kilometers (5 hours) north of Paris via the A1 highway. It's about 650 kilometers (6.5 hours) west of Berlin via the A2 highway.
#### **GETTING AROUND**
Most of the Dutch speak very good English. Don't be afraid to ask for directions!
Amsterdam is small and walkable. You can get from one side of the historic district to the other in about 45 minutes. I suggest organizing your time by seeing sights close to the center on day one, then renting a bike or mapping out tram routes for the following days to explore the sights and activities farther from the center. The city center bustles with pedestrians, trams, bicycles, mopeds, and cars all jockeying to get to their respective destinations. It's important to keep your wits about you when on the city streets, especially if there are tram tracks around. As my old football coach used to say, "keep your head on a swivel!"
The **Amsterdam Centraal Station** is the main hub for all local transportation. Many trams and buses start their lines here. The public transit system is integrated. The rates are €2.90/hour, €7.50/day, €12.50/2 days, and €17/3 days.
##### **Metro**
While Amsterdam does sport a fast metro, you likely won't need to use it on your visit (though the Nieuwmarkt stop can be useful to access some sights in Central Amsterdam). A new line is supposed to open up in the next few years cutting straight through the center, but it has been delayed by construction and funding difficulties.
##### **Tram**
If walking and biking aren't your thing, trams will do just the trick. Purchase your ticket on board. **Trams 2** and **5** are particularly useful to tourists. They run north to south, connecting Amsterdam Centraal Station with Leidseplein and Museumplein. **Trams 13, 14,** and **17** run east to west from Gassan Diamonds to the Anne Frank House.
##### **Bus**
Amsterdam does have a bus system, but it's much less helpful for visitors than trams.
##### **Bicycle**
If you like to ride bikes, renting one is a must! Being on two wheels allows you to zip across town to all the popular sites in no time. It's by far the most popular form of transportation for locals, and you'll need to dust up on those riding skills to join the Dutch rush. For those who don't like riding bikes, or are rusty, I'd advise against saddling up, as traffic can get overwhelming occasionally. I love **Mike's Bike Tours** (Kerkstraat 123, Leidseplein and Vicinity, +31 (0)20 622 7970, mikesbiketoursamsterdam.com), and they offer competitive rates for a bike rental (€7/day, €5/half-day). For a bike rental option only, **StarBikes Rental** (Daily 8:00-19:00, De Ruijterkade 143, +31 (0)20 620 3215, starbikesrental.com) is the closest to the train station and has bikes that blend in rather than advertise they're rented (€7/day). Exit the station, hang left past Mac Bikes and under the bridge, and follow the buildings to the right, down about 100 yards.
### **DAY TRIPS**
#### **North Amsterdam**
Modern Amsterdam has expanded north beyond the River Ij. What used to be warehouses and industrial yards are now trendy modern studios, offices, and apartments. It's worth a lunchtime visit out to the **Ij Cantine** (lunch from €6.50, daily 9:00-24:00, Mt. Ondinaweg 15-17, +31 (0)20 633 7162, ijkantine.nl) restaurant. As soon as you step off the ferry, you'll feel like you're in Amsterdam's secret little neighborhood. Studios are tucked away in shipping containers and hipster restaurants open out on the small sandy spots, with collapsing benches abounding. Come out here for urban exploring more than anything else. Free ferries take pedestrians and cyclists every 10 minutes from just behind the train station to the NDSM Wharf in less than ten minutes. If you're on bike, continue north and cruise through the quiet Dutch suburbs.
#### **Haarlem**
If you've got an extra day and want to see what small-town Holland looks like, a visit to the quaint Dutch town of Haarlem is a great way to spend an afternoon. In Haarlem's city center, you'll find shopping, cafés, and a beautiful gothic church right on the main square. To get here, head to Amsterdam Centraal Station and look for departures that leave every 10 minutes for Haarlem, which is just a 30-minute commuter train away. The train station drops you on the edge of town, and it's just a 10-minute walk into the center.
### **HELP!**
#### **Tourist Information**
Amsterdam's tourist information offices are good places to stop for information and maps, but they're often crowded.
#### **Pickpockets & Scams**
Always have your wits about you, as pickpockets like to target overwhelmed, bewildered tourists in the busy areas of the city. It's very unlikely you'll fall victim to violent crime during your visit, but if you're high out of your mind, you make an easy target. Stick with your friends, have a map with your hostel circled on it, and you'll have no problem at all!
#### **Emergencies**
In an emergency, dial 112.
#### **Hospital**
VU University Medical Center
De Boelelaan 1118
+31 (0)20 444 4444
#### **US Consulate**
Museumplein 19
1071 DJ Amsterdam
+31 (0)20 575 5309
Rome Maps
Rome 101
Three Day Itinerary
Top Neighborhoods
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Top Eats
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Help!
Rome is the city to which all roads lead, the center of the known world during the Roman Empire, and the capital of Italy today. The city overflows with ancient, medieval, and Renaissance architecture and art—and also comes complete with romantic little side streets, beautiful panoramas, and unforgettable cuisine. You could spend a lifetime here and still struggle to grasp this modern city with ancient foundations. Set aside time for the sights, but also make sure to pause for espresso, partake in the _passeggiata,_ and drink in _la dolce vita._
### **ROME 101**
Legend has it that Rome was founded in 753 BC by a pair of orphans who were cast off by their human mother on the Tiber River. Their little raft landed in a bend of the river downstream, and they were taken in by a she-wolf and her pack. The fledgling city quickly grew into a regional, then continental powerhouse over the next 1,000 years thanks to technological innovations, incredible engineering feats, and a series of ambitious leaders with a well-organized military to back them up. Rome's population grew to over one million inhabitants and the city was the richest and largest the world had ever seen. Rome exerted its influence across an entire continent thanks to technological developments like the highway (many of today's European freeways follow the path of Roman roads), arches, and water delivery systems like aqueducts and plumbing. Today, you'll see evidence of Ancient Rome's advanced technical prowess at sights like the Colosseum, brilliantly designed to allow tens of thousands of spectators to exit the venue in a mere 15 minutes, and the perfectly domed (and often-imitated) Pantheon.
Rome kept control of far-off territories by installing loyal governors enticed by land grants and plush appointments. Rome also let the inhabitants of newly conquered territories retain a good amount of autonomy—and even their pagan religion—as long as they paid their taxes to Rome. The Roman Empire's decline, which began around AD 300, was the result of overextension as well as weak and corrupt leadership. Leaders were struggling to hold together an empire that stretched across 6.5 million square kilometers without technologies like Skype and Snapchat. Can you imagine?
Rome was finally sacked numerous times by barbarian hordes from the north, driving the entire continent into a period of lawlessness and violence called the Dark Ages. In medieval times, Rome was a dank backwater rife with crime and disease—not a place you'd want to hang out. With a population of only 15,000 after its peak of a million, Rome suffered from constant outside attacks and dealt with outbreaks of disease as mosquitos and pests thrived in the warm, humid climate.
The Vatican City State stepped into this vacuum of power to clean up the city, and inspired the entire continent to come out of the Dark Ages by financing artistic and technical innovations. Eventually, the Catholic church began supporting artists, sculptors, and designers to deck out churches in a beauty and elegance meant to showcase the power of Christianity. The Renaissance sparked artistic and architectural developments, leading to the design and building of ever-greater churches like St Peter's Basilica. Since then, Rome has been a hub of visual and culinary arts, with a contagious love of life and rich Roman personalities to match their city's long history.
### **PLAN AHEAD**
#### **RESERVATIONS**
Reservations are required for the **Galleria Borghese** (galleriaborghese.it) and the **papal audience** (papalaudience.org/tickets). When planning, remember that the **Vatican Museums** are closed most Sundays, while the worthwhile (and free) **Porta Portese** market is only open on Sunday.
Reservations are recommended for the following sights:
**Colosseum** (tickitaly.com/tickets/colosseum-tickets.php)
**Vatican Museums** (biglietteriamusei.-vatican.va)
#### **PASSES**
##### **Roma Pass**
The **Roma Pass** (€36, valid for three days upon validation, romapass.it) offers free entry—and line-skipping privileges—to the first two covered museums of your choice, the most cost-effective being the Colosseum and the Galleria Borghese (reservations are still required for the Galleria), followed by discounted rates to a number of other sights. See the website for a complete list of covered sights. (The Vatican Museums are not covered.) The Roma Pass also earns you free use of all public transportation in Rome during those 72 hours. Pick up the pass at all participating sights and don't validate it until you enter your first sight. Tally up the sights you want to see that are included in the pass to determine if it will be a good value for you.
#### **LOCAL HAPPENINGS**
##### **Easter**
Easter, which occurs in March or April, is one of the biggest festivals in Rome, with millions of pilgrims coming in from around the world. If you plan to be in Rome during this time, reserve long in advance, and be prepared to pay more for your stay. Christmas and New Year's are also popular times to visit Rome.
#### **HIGH AND LOW SEASON**
June, July, and September are Rome's uber-busy months of tourism. August is when all of Italy shuts down for vacation. Don't expect to see much going on in August here. You can still experience all the tourist sites of course, but anyone who can afford to heads to the beach for a month.
**KNOW BEFORE YOU GO**
**KEY STATS & FIGURES**
**Currency:**
**Italy uses the euro (€); 1 EUR = about 1.06 USD**
**Population:**
**2.7 million**
**National language:**
**Italian**
**National religion:**
**soccer (football)**
**Number of metro lines:**
**2**
**Kilos of pasta consumed annually per capita:**
**28 kilos (over 50 pounds!)**
**Souvenir of choice:**
**Sexy Wine (try some at Miscellanea), postcards, limoncello liqueur**
**CALIBRATE YOUR BUDGET**
**TYPICAL PRICES FOR:**
**Hostel dorm bed:**
**€20**
**Two-course dinner:**
**€12**
**Glass of wine:**
**€4**
**Metro pass:**
**€1.50**
**MOVIES TO WATCH**
_The Talented Mr. Ripley, Angels & Demons, Eat Pray Love, EuroTrip, Ben Hur, To Rome with Love, Roman Holiday_
THREE DAY ITINERARY
Rome's can't-miss attractions will easily fill three days. Organize your visit chronologically, starting with Ancient Rome on day one and continuing onto Renaissance Rome on day two. Remember that the Vatican Museums are closed on Sunday, so you may need to switch days one and two to avoid missing it, depending on your arrival day.
**DAY 1: ANCIENT & MEDIEVAL ROME**
**MORNING**
Catch breakfast at a neighborhood café near your hostel in Termini. Do a lap around the block and pop into any café that has locals inside. Once fortified, grab a bottle of water and catch metro line B down to the **Colosseum** (two stops to Colosseo). I like to tackle the Colosseum before the heat of the day, so make your reservations early to avoid having to stand in line. Download Rick Steves' free podcast and audio tours, which will walk you through the ancient ruins.
Spend at least an hour exploring the ruins, then head to the back of the Colosseum for a quick snack and coffee before your next Ancient Rome experience: the **Roman Forum**. You'll spend about an hour here as well.
**AFTERNOON**
From the Roman Forum, climb the staircase to Capitoline Hill to look back over the Forum. Continue to the square at the top of the hill, the **Campidoglio** , from which you can look across modern Rome on the other side of the hill. Looking down the steps from the Campidoglio, you'll see the hilltop **Santa Maria in Aracoeli** to your right. Behind that is the **Vittorio Emanuele Monument** , offering both a panorama of the city and an excellent free war museum.
Continue down the stairs and cross the busy street leading to Piazza Venezia and wander through the back streets of Rome toward the **Largo di Torre Argentina**. From the railing tracing the perimeter of the square, look down onto the ruins of three temples dedicated to pagan gods. See any stray cats? You're gazing into Rome's main stray cat sanctuary as well.
Continue to the opposite corner of Largo Argentina toward the Pantheon. After a small block, **Isola del Panino** , my favorite sandwich shop in Rome, will be in an alley on your left. Pop in and order by pointing. Don't forget to say hi to my friend Fabio, who's been cranking out delicious sandwiches for as long as I've been coming to Rome.
Take your sandwich and enjoy it in the shadow of the **Pantheon.** Head inside the open-topped site after and take your time exploring. (If it's Saturday, consider doubling back here later for Mass at 17:00).
Afterward, grab one of my favorite treats in all of Rome: _granita al caffè_ at the **Casa del Caffè Tazza d'Oro** coffee shop on the corner to the right as you leave the Pantheon. If you're in the mood for gelato instead, one of Rome's best _gelaterie,_ **Giolitti,** is barely two blocks north.
Continue north on the **Via del Corso,** a popular shopping street, toward the **Spanish Steps** and take in the scene there. Relax on the steps or destroy your credit card in the city's top designer fashion stores. Whenever you're ready, catch metro line A from the Spanish Steps back to Termini and the hostel to rest.
**EVENING**
Take bus H from Termini into Trastevere (get off at the first stop after the bridge over the river, about 15 minutes) to explore Rome's medieval neighborhood. Enjoy a _passeggiata_ and take part in the Italian tradition of seeing and being seen. You have some of Rome's best choices for dinner in Trastevere: my favorite pizza in town at **Dar Poeta** or typical Roman dishes at the local favorite, **Carlo Menta.**
**LATE**
Party it up with the locals in **Trastevere**. There are tons of great venues, like **Niji Café** (classy cocktails) or **Freni e Frizioni** (social drinks), to choose from. So much of Italian nightlife goes down out in the streets, so get a drink and enjoy _la dolce vita_ (the sweet life), Roman style.
**DAY 2: RENAISSANCE & BAROQUE ROME**
**MORNING**
Do your best to get full from breakfast, and knock back an extra espresso, as you'll be seeing one of the world's largest collections of art, sculptures, and priceless artifacts today at the Vatican Museums!
Catch metro line A to the Ottaviano stop for **Vatican City.** Make reservations online ahead of time to skip the lines. From the metro, follow signs pointing toward Musei Vaticani, walking past the tour hawkers and following the walls and crowds to the entry gate. If you've made reservations ahead of time, continue straight to the entry gate and flash your voucher. Otherwise, be prepared to wait. Once inside, go with the flow through the **Sistine Chapel** and on to **St Peter's Basilica.** Oftentimes, the museum will spit you out a one-way exit, and you'll have to go back through security to enter the basilica. If there's an option to leave from the back right corner of the Sistine Chapel, take it and alight directly in St Peter's Basilica. Otherwise, now's a great time for a lunch break.
**AFTERNOON**
Find my choice for the best pizza by the slice in this touristy district at **Alice Pizza** on Via delle Grazie. Reward yourself with a visit to **Old Bridge** _gelateria_ (ice cream shop) just across the street from the Vatican City walls.
Head back to Piazza San Pietro, the square in front of the massive **St Peter's Basilica,** and get in line for security. While the line may be long, it does move quickly through multiple checkpoints. Once inside, look for Michelangelo's famous **_Pietà_** immediately to the right as you enter. You can go beneath the basilica into the free crypts, or climb the cupola for beautiful panoramic views.
If you're beat from the day of sightseeing, hop on bus 40 or 64 any time to get back to your hostel in Termini. For superwomen and -men, I highly recommend taking the 30-minute walk/climb up **Gianicolo Hill** for another panorama across the entire city from above Trastevere. With your back to the front of St Peter's Basilica, the hill is to your right beyond the buildings. You should be able to make out the trees behind church administrative buildings. This is where the moped-riding lovers of Rome bring their squeeze after the sun goes down to escape their parents' watchful eyes.
I like to spend the afternoon wandering back into the center of Rome from either St Peter's or Gianicolo Hill, exploring the old side streets of Rome. Take the bridge directly in front of Castel Sant'Angelo toward Largo di Torre Argentina to catch the bus back up to Termini for a rest. But between the Tiber River and that bus stop, you'll find all sorts of boutique shops, _gelaterie,_ cafés, and more beautiful baroque and Renaissance churches.
**EVENING**
Hop onto metro line A from Termini to the Piazza di Spagna stop, and enjoy an evening _passeggiata_ past the **Trevi Fountain** and west toward the **Pantheon.** Grab dinner at **Miscellanea,** right outside the Pantheon, to get the party started with four courses of deliciousness including unlimited wine and water. Mention my name and get unlimited wine with dinner. You're welcome!
**LATE**
Depending on the breed of nightlife you're looking for, you've got tons of options in front of you. Party with the Americans? Stroll over to **Campo de' Fiori,** a favorite of the study-abroad students in Rome, with staples like **Sloppy Sam's** and **Drunken Ship.** More likely than not, you'll finish your night with a crowd of new friends heading to the Irish bar open late, **Scholar's Lounge.**
Another night out with Italians? Head to **Piazza Navona,** a short walk west of the Pantheon. Keep walking through Piazza Navona and exit the touristy square on the west side, finding an alleyway called Via di Sant'Agnese in Agone, right next to Tre Scalini. This alley leads to some of my favorite nightlife venues, **Bar del Fico** and **Abbey Theater.** (I've dubbed Via di Sant'Agnese in Agone the "Gateway to Hell," because every time I pass through it, a miserable hangover awaits.)
Time to get your clubbing on? Split a cab with friends and head south to the clubs in **Testaccio.**
**DAY 3: CHOICES, CHOICES, CHOICES**
Shake off that hangover with a _cornetto_ pastry. Then consider a day trip to **Via Appia Antica** or the fascist urban development of **EUR** , or stay closer to the city center and visit the Galleria Borghese and/or Porta Portese market.
**MORNING**
If it's Sunday, leave your valuables at the hostel and take bus H down to Trastevere where you'll find the notorious **Porta Portese** market. It's easy to fill a couple hours strolling the length of the market.
Leaving the market, cross the river and climb the hill up to the **Giardino degli Aranci,** where you'll find the famous viewpoint through the keyhole, showing three sovereign nations in one view.
Wander across the nearby **Circo Massimo,** then north along the river into the **Jewish quarter,** where more restaurants will likely be open—try the kosher gelato!
**LATE AFTERNOON**
Anyone who appreciates art, especially baroque art and sculpture, should make plans for the **Galleria Borghese.** From Termini accommodations it's a 20-minute walk north, or you can take metro line A to the Piazza di Spagna stop. From there, it's about a 25-minute walk northeast and uphill to the museum. If you've made reservations ahead of time, excellent. Otherwise, head over for a chance to get a last-minute entry. Save time before or afterward for a picnic in **Villa Borghese,** the big park that surrounds the Galleria Borghese. You can rent four-seater bikes and pedal around the park without worrying about the crazy traffic!
### **TOP NEIGHBORHOODS**
Rome straddles a bend in the Tiber River. Most of the tourist sites are on the east side of the river, clustered in the Ancient Rome and Pantheon neighborhoods. This area (along with Vatican City) is sightseeing ground zero. **Ancient Rome** comprises the Colosseum and Roman Forum. Northwest is the **Pantheon neighborhood,** including that famous structure along with Trevi Fountain and Largo di Torre Argentina (often shortened to Largo Argentina). The Pantheon neighborhood also offers romantic alleyways and some good (if touristy) options for food and nightlife, especially near Campo de' Fiori and Piazza Navona.
North of the Pantheon neighborhood, **North Rome** contains the Galleria Borghese, Spanish Steps, Piazza del Popolo, and Rome's most expensive shopping. Restaurants are either upscale or tourist traps, so it's better to eat elsewhere.
On the west side of the river is **Vatican City.** A few worthwhile dining options cluster around this holy city-state, but the sights are your top priority. South of Vatican City, and west from Ancient Rome, is **Trastevere** , one of my favorite neighborhoods for its authentic, local feel. You'll find winding cobblestone streets, medieval churches, and quaint squares with ivy climbing up the walls. It's easy to blend in with the locals on their nightly stroll to dinner, then gelato, then the bars.
Back on the east side of the river, East Rome is home to the main train station, **Termini,** which gives its name to the surrounding neighborhood. All recommended hostels are located here. While the streets immediately surrounding the station aren't the most interesting, they've recently cleaned up quite a bit. Find student-oriented nightlife in **San Lorenzo,** a 15-minute walk south and east from Termini station. From Termini, it's an easy metro or bus ride into most of the city center sights.
**Testaccio,** directly south from Circo Massimo, has an excellent neighborhood feel and a daily produce and meat market. You'll find yourself here for one other main reason, though: the club scene.
### **TOP SIGHTS**
#### **Colosseum**
Commissioned in AD 70 by Emperor Vespasian, the Colosseum took a mere 10 years to build. This amphitheater (or double theater) packed in 50,000 jeering Romans of all walks of life and castes. It was segregated, of course, with nobles and senators in the lower levels, merchants and soldiers in the mid levels, and slaves and women in the highest rafters. Entry was free with a numbered ceramic fragment. Emperors and campaigning local politicians sponsored events with exotic animal and gladiator fights, food, VIP seating, and more. Hey, I'd vote for anyone who gave me free tickets to the Seahawks! Thanks to the design, all 50,000 spectators could exit the venue within 15 minutes, something most modern stadiums could only dream of.
Nearly 2,000 years after its construction, the Colosseum is Rome's most popular attraction. Queue along the outer walls, purchase tickets just inside, and continue directly up to the first floor, where you can complete a lap, looking down and imagining the battles that occurred here. Complete your visit with another lap on the ground floor, this time with a close-up view of the network of passageways used by gladiators, cages and cells for animals, and trap door loading areas that would have been covered by a wooden floor and sand.
**ACT LIKE A LOCAL**
**Go Take a Passeggiata!**
Italian culture is all about seeing and being seen—and looking good at all times. There's even a term for it: keeping _la bella figura,_ meaning keeping up appearances no matter what. That's why clothing labels are so prominent here: They know people are watching! If you're going to the gym, you'll bring a change of clothes, as it's not OK for people to see you sweat. If you see someone running outside, they're either a marathoner or not Italian.
And what better way to display your labels and the cute thing on your arm than to stroll through town before and after dinner? This aimless little neighborhood walk is called a _passeggiata,_ a nightly ritual that you've got to join at least once. _Passeggiate_ are also taken just to get out of the house. Roman apartments are small, with the paradoxical result that Italians often have to go out in public to get any sort of privacy from the family. Sunday afternoons are another popular time to stroll the old streets, as many close down to be pedestrian only.
**Breakfast, the Italian Way**
You won't find bacon or hash browns anywhere in Rome! Italian breakfasts are simple, and they're consumed at the corner café. Step in and observe the scene: career baristas who are passionate about coffee serving pastries to regulars who come in five days a week ordering the same thing. Everyone seems like their old friends. Depending on the café, you might pay for your order first, _then_ deliver the receipt to the barista to redeem your coffee. Otherwise, order at the bar and pay before leaving—just watch what the locals do. Remember, sitting down at a table often comes with a small surcharge on each item ordered. The locals often have a single espresso at the bar to wash down a _cornetto,_ a pastry very similar to—but not quite as good as—the French croissant.
**Bread Basics**
It's an American invention to dump olive oil and vinegar onto a side plate and sop it up with bread while you wait for your food. Italians would never do such a thing, as it kills your appetite and fills you up with carbs. Italians use bread only during their meal to _fare la scarpetta,_ an expression meaning "to make the little shoe" on—or scoop up—your leftover pasta sauce.
**It's Pronounced "Broo-sketta"!**
No matter what our friends at the Olive Garden tell you, anytime you see an H following a C in the Italian language, it turns that letter into a hard C. In other words, you pronounce the delicious tomato and garlic toast as broo-SKET-a, not broo-SHET-a. Conversely, _cioccolato,_ Italian for chocolate, starts with the same "ch" sound as our word because it lacks the H after the C.
**Street Smarts**
Traffic is to Rome as Carrot Top is to just about everything he touches: obnoxious and overwhelming. Traffic will never stop unless you boldly step out into the intersection—as long as you're within the zebra stripes!—staring down oncoming drivers. Here's the secret to getting across: Find a local and pair up at the hip (unbeknownst to them), and cross when they do. Obey any traffic lights if they're there. If there is no walking signal, wait for a slight break in the traffic. Wave back at the tourists who are too timid to give it a shot, and once you're an expert, bring them along!
The Colosseum and Roman Forum share a single entry ticket. Consider picking your ticket up at the gate to the Forum if the line at the Colosseum is long. Heads-up: As soon as you arrive to the area, you'll be hounded by promoters offering private tours and ways to skip the line. On days when the line is long, taking an organized tour may be worth it, but be sure to understand what you're getting for your money. Also, take a peek at the line yourself to gauge the value of skipping it. Many tour operators take your money first, then make you wait until they find other customers in order to run a full tour.
€15.50 ticket also covers Roman Forum, free first Sun of the month, daily 08:30 til one hour before sunset, last entry one hour before closing, Ancient Rome, +39 06 3996 7700, get tickets at tickitaly.com, Metro: Colosseo, Bus: Piazza Venezia
#### **Roman Forum**
The Roman Forum was the nerve center of the most impressive empire the world has ever seen. It's in this district that daily life went down, with toga-wearing politicians rushing between votes and making impassioned speeches, shopkeepers selling their wares, and triumphant military commanders parading their way back into town in victory processions lasting days on end as they touted their booty, new slaves, and plunder. Over the centuries, much of the Roman Forum disappeared due to urban cannibalism. (It was much easier to come here and take precut stones for your home rather than go to the quarry and cut them from the ground.) Floods carrying river silt and debris also covered the ruins up over time.
Two thousand years later, it is still possible to come here and see many remnants of Ancient Roman society, including the Temple of the Vestal Virgins, the arch of Septimius Severus, and the well-preserved Roman Senate House. When walking on the original Roman cobblestones, note how much lower in the ground you are than street level. This area was right in the floodplain of the Tiber River until the walls were built in the 20th century to prevent the floods.
€15.50 ticket also covers Colosseum, free first Sun of the month, daily 08:30 til one hour before sunset, last entry one hour before closing, Ancient Rome, +39 06 0608, get tickets at tickitaly.com, Metro: Colosseo
#### **Vatican Museums & Sistine Chapel**
The Vatican is the center of the global Catholic church. Its public museums offer a glimpse into their awe-inspiring collection of worldly riches amassed over the centuries. The hallways are gilded in baroque embellishments and encrusted in priceless Renaissance masterpieces. Michelangelo's Sistine Chapel is quite literally jaw-dropping, as are the Raphael rooms ( _stanze di Rafaello_ ). Reserve your entry well ahead of time for the Vatican Museums to skip the line. Otherwise, beat the lines by getting up early or going late. Allow at least three hours inside to see everything. You might want to bring a bottle of water and snacks to survive. I highly recommend Rick Steves' audio tours as a way to learn about what you're looking at at your own pace.
€16, additional €4 for online reservation, Mon-Sat 09:00-18:00, last entry at 16:00, closed Sun but open last Sun of the month 09:00-14:00 with free entry before 12:30, Vatican City Neighborhood, +39 06 6988 3332, biglietteriamusei.vatican.va, Metro: Ottaviano
#### **St Peter's Basilica**
Catholicism's proudest cathedral is dedicated to the first pope, Peter. It is said that Peter was crucified between chariot races at the racetrack that used to loop around this neighborhood. After he died, Peter's followers took him down and buried him in a clandestine Christian grave in the hill behind the racetrack. After hundreds of years and multiple chapels and churches, the altar today stands about 60 feet above these catacombs, with Bernini's swirly bronze baldachin rising high above. Designed by an ever-changing team of Renaissance master architects, St Peter's was built to an otherworldly scale that makes visitors feel like ants. Take a look high above at the inscriptions that wrap the interior walls. Those letters are six feet tall. And every image you see in this entire church is not a painting, but rather a mosaic. Paintings fade, so 10,000 square meters of intricate mosaics depict past popes, lessons from the Bible, and all characters you'd expect to see in Catholicism's most important cathedral.
As you enter, immediately to the right is Michelangelo's masterpiece, the **_Pietà_.** In elongated and exaggerated features, Jesus' body weighs heavy on Mary's lap. Michelangelo completed this piece when he was only 24 years old, and only returned to sign it when he heard rumors that it was being attributed to someone else. In 1972, a crazed Italian man came at Mary's face and arm with a hammer, knocking chunks off the priceless sculpture. Luckily, the Pietà has been restored with marble taken from the back of the piece (raw marble is like a thumbprint—you'll never find two pieces the same coloring, hue, and pattern). Thanks to our disturbed friend, you'll observe the Pietà from 30 feet away and through 2 inches of bulletproof glass.
Visitors can go beneath the basilica into the free crypts or climb the cupola for a beautiful panorama of the entire city (€5 by stairs, €7 by elevator). Papal audiences with Pope Francis take place every Wednesday at 10:30 (arrival recommended at 08:30, free tickets required, papalaudience.org).
Free, daily Apr-Sept 07:00-19:00, Oct-Mar 07:00-18:30, dome open daily from 08:00, last entry to stairs 17:00 (16:00 Oct-Mar), to elevator 18:00 (17:00 Oct-Mar), Piazza San Pietro, Vatican City Neighborhood, +39 800 038 436, vatican.va, Metro: Ottaviano
#### **Pantheon**
A revolutionary feat of architecture and engineering, this domed temple has been studied and replicated since its creation. What was a one-stop shop to worship all the gods of the Roman Empire ( _pan_ = all, _theon_ = gods), is now an active Catholic church technically called Church of St Mary and the Martyrs. Raphael, the Renaissance master, is buried in the church, along with the first two kings of Italy and one of their wives, Queen Margherita. (She's the one the popular pizza was named after. The three main ingredients—tomatoes, mozzarella, and basil—represent the three colors of the newly united Italy's flag.)
Two wooden structures burned down on this site before Emperor Hadrian commissioned a stone version and completed it around AD 125. At its base, the walls are nearly 30 feet thick, built with heavy brick and stone. As the walls climb higher and arch into the ceiling, the building materials become lighter. The uppermost portion is made of concrete mixed with light volcanic ash. The ceiling is only three feet thick, and is recessed both to add decoration and to reduce mass overhead.
The Pantheon is the world's largest and oldest unreinforced concrete dome. It inspired the designs of some of the most famous domes in Rome and the world, including St Peter's Basilica, the Duomo in Florence, Jefferson's Monticello house, and even the American Capitol building in Washington DC.
Today, the Catholic church is active. You can catch Mass here Saturday at 17:00 and Sunday at 10:30. It's a special experience because the noisy tourists are ushered out and the faithful stay in quiet reflection.
Free, Mon-Sat 08:30-19:30, Sun 09:00-18:00, holidays 09:00-13:00, Pantheon Neighborhood, +39 06 6830 0230, Bus: Largo Argentina
#### **Trevi Fountain**
Rome's greatest fountain is dedicated to the aqueduct that powered it for more than 400 years. You've got to remember that running water was a luxury (and still is sometimes in modern-day Rome!), and the Romans harnessed this technology to drive population growth beyond one million inhabitants. That's something that didn't happen again until industrialized London nearly 2,000 years later.
The fountain itself dates from the 18th century, when Pope Clement XII commissioned the master, Salvi, to renovate it to today's stature. As you gaze upon the fountain, you'll see Poseidon with two horses, one rearing its head and the other docile. These opposing postures represent the two personalities of the sea: violent and calm. On the reliefs high above, you'll see a girl leading the Roman centurions to the headwaters of the aqueduct that powered this fountain and many others in Rome, and the cornucopia beneath represents the bounty of the sea. All in all, this fountain is a masterpiece of symbolism and detail—see how many types of flora and fauna you can spot.
And as you take the scene in, look to the right and find the scraggy chunk of rock sticking up out of nowhere. As noisy construction dragged on far beyond the scheduled completion date, a local shopkeeper was complaining about the annoyance. In the end, Salvi graciously left a permanent block of stone just in front the shop to obscure the view of his fountain as a thank-you note for the trouble caused.
Today, the Trevi Fountain is swarmed by tourists, knickknack sellers, and Italian Casanovas looking for love. Reopened in November of 2015 after 16 months of renovation (some of it funded by Fendi design house), it's more pristine than ever. Tradition says that all who wish to return to Rome must come to the fountain and toss a coin in with their right hand over their left shoulder. See if it works for you!
Free, always open, Pantheon Neighborhood, Metro: Barberini
#### **Piazza Navona**
Piazza Navona, another of the city's famous squares, used to be a track for chariot racing downtown. If you look closely, the elongated outline of the square is rounded at one end, following the shape of the old track from back in the day. Today, Piazza Navona fills up with artists and trinket vendors and is a prime spot for the _passeggiate_ (strolls) that locals take every night. Don't miss the famous chocolate truffles at Tre Scalini and the excellent nightlife in the district just through the alleyway leading toward the river next to the church.
**TREVI TRIVIA**
Thanks to the tradition of tossing coins into the Trevi Fountain to guarantee a return trip back to Italy, the pool in front of the fountain collects thousands of euros daily. (Apparently, many tourists believe the higher the value of coin, the higher the chance the superstition will actually work.) Well, where does all this money go? For a number of years, it just disappeared. No one knew what was happening...until the police caught a homeless man swimming in the fountain.
They caught him once, then again...and again...and again. He hadn't really committed a real crime, so he couldn't be detained for more than 24 hours. Each time he was caught, he was released. This went on until someone had the brilliant idea to create a waterproof fountain Roomba and donate the money to the Italian Red Cross. Now, several million euros collected from the fountain are donated to the Red Cross each year. _Grazie,_ tourists!
Free, always open, Pantheon Neighborhood, Bus: Largo Argentina
#### **Largo di Torre Argentina**
One of Rome's busiest squares for bus connections also looks down onto this open-air archeological dig. While the purposes for each are unknown, you'll see the foundations for three Roman temples, which today are called Temple A, Temple B, and Temple C. Rome used to have a serious feral cat problem in the downtown area, and they were able to solve the problem by relegating all stray cats to this one sunken square. On the southwest corner of the square, descend a short staircase to visit Rome's **Cat Sanctuary** (hours vary). Largo Argentina today is more important to know as a reference point for public transportation than anything else, and has conveniently located shops like a bookstore, _tabacchi_ (tobacco store), and _gelateria_ , as well as many recommended restaurants nearby.
Free, always open, Pantheon Neighborhood, Bus: Largo Argentina
#### **Campidoglio**
After climbing out of the Forum, you'll pass through this beautiful, perfectly balanced Renaissance square at the top of Capitoline Hill. Michelangelo was brought in to complete this half-built square in the 16th century. The Campidoglio exemplifies what Renaissance masters held to be most important about architecture: city planning and order. Michelangelo had his work cut out for him, as the pope wanted a square that connected modern Rome with its ancient past; it had to reference the Roman Forum but look forward into the future. Michelangelo's solution was ambitious, but served to unite one of Rome's most important hilltops by reflecting and updating the existing inharmonious medieval palazzi. He overlaid the square with a geometric pattern, reflecting the advances in math at the time, and squeezed a mirroring palace (the one closer to the Vittorio Emanuele Monument) to make the square symmetrical.
The equestrian statue in the middle of the square is Marcus Aurelius. Most bronze statues were melted down in medieval times for material, and this statue still exists today only because it was thought to be of Emperor Constantine, the one who turned Rome to the Christian faith—so it was protected by the church. Everyone thought the statue's outstretched arm was meant to be blessing the audience, but instead, Marcus Aurelius is commanding his armies.
Free, always open, Ancient Rome, Metro: Colosseo, Bus: Piazza Venezia
#### **Santa Maria in Aracoeli**
As you descend from the Campidoglio down the Renaissance stairway, Santa Maria in Aracoeli is up and to your right. Sitting on the most holy and important hill in Rome, this church rests on ruins dating back to the Roman Empire. In this single church, you can study a wide range of artistic and architectural styles, from Romanesque to gothic and medieval to renaissance and baroque. Thankfully, the church was not destroyed during the construction of the nearby Vittorio Emanuele Monument.
Free, daily 09:00-18:30, Ancient Rome, Metro: Colosseo, Bus: Piazza Venezia
#### **Vittorio Emanuele Monument**
The colossal Vittorio Emanuele Monument (aka Altare della Patria) to Italy's first president is located on Piazza Venezia and caps the head of Via del Corso. From the nicknames locals have given it—the Typewriter, Dentures, Wedding Cake—you can guess their sentiments on this neoclassical feat of architecture. An enthralling free military museum is definitely worth a gander on your way up to the elevators zipping visitors to the roof of the monument. The €7 elevator ride to the top offers the best views of both ancient and modern Rome around.
Free, daily 09:30-18:30, ticket offices close 45 minutes earlier; museum €5, daily 09:30-18:30, Piazza Venezia, Ancient Rome, +39 06 678 0664, Metro: Colosseo, Bus: Piazza Venezia
#### **Circo Massimo**
This was Ancient Rome's largest stadium, seating more than 150,000 Romans at once. The sheer scale of the stadium is mind-blowing. It was built over 2,000 years before the Big House, and it's still substantially bigger! The entertainment that attracted these crowds varied: You could come for an afternoon of chariot racing, singing or musical events, gladiator combat, or athletic demonstrations. Located right in the downtown of Ancient Rome, with the Emperor's Palace overlooking the track, the Circo Massimo was near and dear to the empire's heart.
Today, this empty field and long running track are great for a picnic or to get a few laps in. I was here during the 2006 World Cup when Italy faced France and won. You couldn't imagine the melee of happy Italians blaring air horns, shooting bottle rockets off sideways, and riding their mopeds tandem long into the night.
Free, always open, Ancient Rome, Metro: Circo Massimo
#### **Galleria Borghese**
This Baroque villa houses possibly my favorite museum on the planet, featuring some of the most incredible baroque paintings and sculptures ever completed, like Caravaggio's dramatically lit _David with the Head of Goliath_ and Bernini's spectacular _Apollo and Daphne_. The quality of the exhibition is underrated and missed by the hordes of tourists, but it should be at the top of anyone's list when visiting Rome. Reservations are required; get them at galleriaborghese.it. The Galleria Borghese is about a 15-minute walk from the listed metro stops. From Termini, consider taking bus 910, which will drop you off closer to the museum, or just walk about 20 minutes.
€13, includes mandatory €2 reservation fee, free first Sun of the month, Tues-Sun 09:30-19:00, Piazzale del Museo Borghese 5 (located inside the Villa Borghese park), North Rome, +39 06 841 3979, Metro: Barberini or Flaminio, Bus: Museo Borghese
#### **Spanish Steps**
Another example of Roman baroque exuberance, this staircase was built to connect the church and neighborhood high above with the rest of central Rome. Ironically, the name is a misnomer. The steps were paid for by a French diplomat, not the Spanish. We call them the Spanish Steps because the Spanish envoy to the Vatican is just around the corner near the tall freestanding column in front of the nearby McDonald's, and it rolls off the tongue a little nicer, doesn't it? The Spanish Steps are busy throughout the day now—some of the best shopping in Rome is to be had in the streets leading away from the steps. This neighborhood and the steps specifically have always been a meeting place for lovers, romantics, and poets. All it takes is a stop here yourself, and you'll understand why.
Free, always open, North Rome, Metro: Spagna
#### **Piazza del Popolo**
This piazza is an excellent example of a Renaissance city planning technique called the _tridente_. See the three grand avenues leading south away from the square into Central Rome? The central one is Via del Corso. To the left, Via del Babuino leads straight to the Spanish Steps, and Via di Ripetta leads off to the right toward the old medieval center and the bridge that connects the heart of Rome with Vatican City. The streets didn't naturally happen this straight; it was the urban planning of an ambitious pope in the 17th century—any homes in the way were simply demolished in the interest of beautifying the church's capital city.
Today, Piazza del Popolo (People's Square) is a great spot to people-watch and soak in Rome's rich architecture without the bustle of the bigger sites. Demonstrations often use the square to kick off a procession leading into the center of town. Pop into the churches on the square to view some beautiful baroque art, including a piece by Caravaggio.
Free, always open, North Rome, Metro: Flamino
#### **Jewish Quarter**
Rome has a rich Jewish history dating back hundreds of years. Though the Jewish community even predates that of the Christians, Pope Paul IV relegated the Jews to this district just outside the bend of the River Tiber in a Papal Bull in 1555. This was a deliberate ploy to place the Jewish community right in the path of the river that floods numerous times each year. The Jewish community persevered through these horrid conditions, then enjoyed a short respite upon the unification of modern Italy toward the end of the 19th century, until the Nazi fervor retook the continent by the 1930s.
Today, the Jewish quarter is well worth a stroll. Keep your eyes open to find inlaid stones with Hebrew inscriptions in buildings throughout the neighborhood. While many other parts of the city close down on Sunday, the Jewish quarter is alive and bustling. Don't forget to look down—you'll notice numerous small bronze cobblestones—aka "stumblestones"—sprinkled throughout the area. They're placed in front of homes from which Jewish victims of the Holocaust were deported in 1939 and on. You'll see the name of each victim inscribed with their birth date, occupation, which camp they were deported to, and last date known to be alive. It started here in Rome, but you'll notice these bronze stumblestones throughout Europe in front of buildings where victims of the Holocaust lived.
Wander through this district and be sure to try and find the following: fried artichoke in the Jewish style (think Bloomin' Onion, _stilo Romano_ ), kosher gelato, and Bernini's _Fountain of the Turtles_ (dedicated to the Jewish community). The turtles represent carrying all your earthly belongings on your back until finding a land to call your own.
Free, always open, Pantheon Neighborhood, Bus: Largo Argentina
#### **EXTRA CREDIT**
##### **Churches, Churches, & More Churches**
Historically, the church has been the richest patron around, so churches in Rome often house beautiful works of art. (And Rome has more churches than Seattle has Starbucks outlets.) As you explore the city, pop into any church that piques your curiosity to see what treasures you discover. You'd be surprised what masterpieces you can discover by breaking off and exploring on your own. **Santa Maria Maggiore** (Termini, Bus: Equilino-Cavour, bus 70), with its beautiful ceiling and cavernous space; the science-oriented **Santa Maria degli Angeli e degli Martiri** (Termini, Bus: Repubblica, bus 64 or 70); and **San Bartolomeo dell'Isola** (Isola Tiberina, Bus: Sonnino, bus H), uniquely located on the island in the river, are some of my favorites.
Free, open dawn-dusk
##### **Monti District**
If you're exhausted from marathoning through the Roman Forum and Vatican City, this quiet neighborhood just north of Ancient Rome might be a welcome change of pace. Monti offers a glimpse into Roman life and is a hive for up-and-coming artisan shops. Wander along **Via Leonina,** which runs parallel to the bigger avenue, **Via Cavour,** to find some of Monti's gems. On Via Leonina alone, you'll find numerous excellent restaurants, **Finnegan's Irish Pub** (Via Leonina 66); **Rome's Ice Bar** (Via della Madonna dei Monti 28), where everything inside is made out of ice, including the cups you drink from; and **MercatoMonti** (Via Leonina 46), an awesome boutique designer minimarket where local jewelry and fashion designers come to sell their one-off handmade goods. MercatoMonti is most popular in the evening hours.
Free, always open, Ancient Rome, Metro: Cavour
### **TOP EATS**
The cuisine is easily one of the highlights of any visit to Rome. There are choices to suit just about any budget. Breakfast is a cheap and simple affair: pastry and espresso. Your target price for each should be about a euro. If you're paying more for that, you're likely in a touristy café. For lunch, I love stopping into any little bakery ( _paninoteca_ ) for delicious freshly made sandwiches. Some of my favorite Roman dishes are **_cacio e pepe_** (pasta with cheese and pepper), **_carbonara_** (egg, bacon, pecorino cheese, and pepper over spaghetti), and **_amatriciana_** (tomato-based sauce over buccatini, a type of thick spaghetti with a hole running through it, reminiscent of a pool noodle). When looking at dinner menus, watch out for "false friends" as we called them in my language classes: _Pepperoni_ are bell peppers, _acciughe_ are anchovies (not artichokes), and _margherita_ refers to the tomato, basil, and cheese pizza—not the tequila-laden drink.
And there's an Italian tradition that you just cannot miss: **_aperitivi._** Many bars offer access to an extensive, yet simple buffet included with the price of any drink. You'll find many young professionals heading to their favorite bar after work to take advantage of this incredible deal. The spread often includes things like cold bean and pasta salads, tiny slices of pizza, and tasty focaccia bread. Get your fill for the price of one €6 drink.
Tipping in Italy is not required, but about 10 percent is appreciated. Look at your receipt to determine whether or not _"servizio incluso"_ appears, which means a service charge was included.
#### **Miscellanea**
Frisky Mikki runs this popular lunch and dinner spot located just at the back corner of the Pantheon. Pumping out pasta dishes, meat, salad, and delicious bruschetta, Miscellanea is a favorite of all who come here, from Italian senators and the Swiss Guard to poor study-abroad students and tourists like you. All are attracted to the fun and quick service and the welcoming atmosphere. Immensely popular among students is a menu I designed with these guys a few years ago: For €15 you get bruschetta, pizza, two types of pasta, dolce, water, red and/or white wine, and of course, Mikki's famous Sexy Wine. Be sure to tell him _"ciao bello"_ for me!
**THE ORDER—AND THE ORDERING—OF FOOD**
In Italy, food is like a religion: You do things a certain way because that's the way it has always been done, and who are you to question it?
**_Aperitivo:_** Kick off your meal with a drink to toast with. This can often be a flute of Prosecco or Spumante.
**_Antipasti:_** Begin your feast with cold cuts ( _salumi_ ) and a selection of cheeses. Prosciutto, salami, and savory cheeses are quite typical.
**_Primo:_** This is typically a pasta or otherwise carb-focused course, like a risotto. Don't overload on this course, because you've still got several more on the way!
**_Secondo:_** This is your meat course. Don't ask for chicken pesto pasta because Italians don't mix pasta and meat. It's sacrilege. So enjoy your pasta course, and then dig into something like pork chops, steak, or chicken breast with potatoes and veggies on the side.
**_Insalata:_** While we start with a salad, the Italians finish with it. Expect it to be light and leafy.
**_Formaggi/frutta/dolce:_** What better way to finish your meal than with a selection of cheese or fruit? These will come before any sort of _dolce_ (sweet), which may well be gelato, tiramisu, pannecotta, or cake.
**_Caffè/digestivo:_** To keep you going into the night, cap everything off with a single espresso. At this time, many also have a round of whiskey or a typical grappa, _limoncello_ (lemon liqueur), or _sciacchetrà_ (Italian sweet wine).
Plates and sandwiches from €8, feast for €15, daily 11:00-late, Via Della Palombella 34/35, Pantheon Neighborhood, +39 06 6813 5318, Facebook: Miscellanea Pub, Bus: Largo Argentina
#### **Isola del Panino**
Between Largo Argentina and the Pantheon, you'll find one of the best values in the heart of Rome. This one-man show, run by friendly Fabio, attracts power lunchers and budget backpackers alike. Fabio doesn't speak much English, so order your sandwich fillings by pointing to the fresh ingredients (chicken, beef, ham, turkey, salads, cheeses, other toppings, and sauces) in the glass case. They'll toast the sandwich and wrap it in a way you can eat while on the go; there's no seating here, just a skinny bar on the sidewall.
From €3.50, Mon-Sat 11:00-14:00, Via Monterone 19, Pantheon Neighborhood, +39 06 6830 7769, +39 328 785 8717, Bus: Largo Argentina
#### **Obicà Mozzarella Bar**
I love the experience of eating at Obicà: tour Italy through the prism of fresh mozzarella cheese. Your place mat is a map of Italy that explains where your locally and regionally sourced ingredients are coming from. The restaurant has a mod, sophisticated decor, and I like how the servers are clearly passionate about their ingredients.
Dinners from €12, daily 06:30-02:00, Piazza Campo de' Fiori 16, Pantheon Neighborhood, obica.com, Bus: Largo Argentina
#### **Il Forno**
Not a restaurant, but an excellent spot to drop in and get a bready snack at any time day or night—I love their salty focaccia bread. Il Forno (The Oven) has an excellent location right on the corner of Campo de' Fiori, making it a convenient stop whenever the mood strikes.
Snacks from €1.50, Mon-Sat 07:30-14:30 and 16:40-20:00, Piazza Campo de' Fiori, Pantheon Neighborhood, +39 06 6880 2366, Bus: Largo Argentina
#### **Tre Scalini**
Tre Scalini, with their outdoor seating spilling out onto Piazza Navona, is famous for their to-die-for chocolate-ice cream truffles. Imagine dark, white, and milk chocolate rolled into a ball then dusted with chocolate powder and sprinkles and you're almost there. The only other thing you've got to do is shovel it down! When you're ready for nightlife, some of my favorite joints are just down the alley leading away from the square toward Piazza del Fico.
From €6, daily 12:30-15:50 and 19:00-24:00, Piazza Navona 30, Pantheon Neighborhood +39 06 687 9148, ristorante-3scalini.com, Bus: Largo Argentina
#### **Casa del Café Tazza d'Oro**
This café still offers one of my favorite frozen treats in Rome: their famous _granita al café_ , or coffee granita. Pay at the counter in the back first, _then_ bring your receipt to the front to claim your prize. Personally, I like to ask for more granita, less whipped cream: " _Per favore, piu granita, meno crema."_
From €1.50, Mon-Sat 07:00-20:00, Sun 10:30-19:30, Via degli Orfani 84, Pantheon Neighborhood, +39 06 678 9792, tazzadorocoffeeshop.com, Bus: Largo Argentina
#### **Alice Pizza**
Alice Pizza near the Vatican offers my favorite pizza by the slice. You know it must be good when old nuns elbow you out of the way to order. Head into either of the doors, and get to the front of the line. Don't hesitate when it's your turn to order or you'll miss your chance, and don't be afraid to ask for more _or_ less! The ladies at this place serve delicious pizza, but don't expect any smiles.
From €4, Mon-Sat 11:00-18:00, Via delle Grazie 9, Vatican City Neighborhood, +39 06 687 5746, Metro: Ottaviano
#### **Dar Poeta**
Rome's best pizzeria is tucked away in a small alley in Trastevere. Run by three friends, this place pumps out an incredible menu of wood-fired pizzas for a great price. If you're in a spicy mood, go for the _lingua del fuoco_. Hungry? Ask for _"pizza alto"_ to get a double crust. Be sure to save some room for their famous _calzone cioccolatto_ (chocolate calzone), big enough to split amongst friends... Or don't share and keep it all for yourself. I won't tell.
Pizzas from €8, daily 12:00-15:00, 19:00-late, Vicolo del Bologna 45, Trastevere, +39 06 588 0516, darpoeta.com, Bus: Sonnino/Piazza Belli
#### **Carlo Menta**
Carlo Menta is a Trastevere institution that packs out with locals for fresh pasta, artichokes, and lasagna. If you lived in Rome, this is where you might come for Sunday afternoon brunch with your family and grandparents. I love it for the cheap prices (pizza from €3.50!). Don't expect any niceties from the busy staff, but you can count on solid value even with the €1.50 cover charge. Split the liter of table wine with friends for a reasonable €8.
Dinners from €5, daily 12:00-24:00, Via della Lungarina 101, Trastevere, +39 06 580 3733, Bus: Sonnino/Piazza Belli
#### **Hostaria del Moro (aka Tony's)**
A favorite of the study-abroad students in Rome, this is your go-to feast for €15 on the west side of the Tiber. Tony and the boys have boiled the experience down to a science: fast service, great food—pastas, pizzas, meats, fish, and desserts—at fair prices. Your best bet is to sit down and go with the flow, enjoying whatever comes your way—seriously, Tony will dictate the menu to you, and I always go for whatever is recommended. It's up to you to engage with the waiters. When they're busy, you may need to request the check a few times.
Dinners from €11, daily 19:00-late, Vicolo del Cinque 36, Trastevere, +39 06 580 9165, Bus: Sonnino/Piazza Belli
**ITALIAN COFFEE**
Italians love their espresso almost as much as their Vespas. It can seem a bit intimidating to join in on the tradition, but you've got to do it at least once. Italians consider coffee a digestive, not to be consumed _with_ food, only _after_ food. Expect to pay about €1 per espresso, €2 per cappuccino, and double that if you'd like to sit down rather than take it at the bar.
Here's a breakdown of your options at any normal Roman _caffè:_
**_Caffè:_** Your standard single shot of espresso, consumed in one toss back.
**_Doppio:_** A double espresso.
**_Macchiato:_** Espresso "stained" with just a touch of hot, steamed milk. This is my favorite one to try.
**_Cappuccino:_** Your standard espresso and steamed milk mix, though it's smaller than the Starbucks counterpart you may be used to. _Never_ to be ordered or consumed after the clock strikes noon. Italians believe that milk must not be consumed anytime after breakfast, so keep this in mind while preparing to order.
**_Caffè Americano:_** American-style coffee, as interpreted by Italians, meaning espresso diluted with hot water.
**_Caffè Corretto:_** An espresso "corrected" with a shot of liquor. Italians enjoy mixing coffee with grappa. My favorite is to have it with a splash of Baileys, the perfect blend of liquor and cream.
**_Caffè Ristretto:_** An even denser, more intense shot of espresso.
#### **Fish Market Trastevere**
For fresh seafood at a great price, head to Trastevere's best fast seafood joint, Fish Market. The jovial, welcoming staff squeezes you like sardines into tables packed with locals, handing you a paper menu and pencil. Order à la carte oysters, prawns, fish, salads, and more, checking off all the various samplings you want, then take your order up to the register and pay. With all the _frittura_ (fried food), it can be a heavy meal, but oh so good! Don't forget a nice bottle of white wine to wash down your selections!
Plates from €5, Mon-Fri 19:30-01:00, Sat-Sun 13:00-16:00, Vicolo della Luce 2, Trastevere, +39 366 914 4157, fishmarket-roma.com, Bus: Sonnino/Piazza Belli
#### **Pizzeria del Secolo**
When I'm in the mood for a snack in the Termini neighborhood, this is where I come. Friendly pizza maestros keep the ovens hot all day, creating steaming pizza masterpieces like _caprese_ (Capri style), sausage, potato, and eggplant. Order and pay by weight, and wait as they heat up your selection before chowing down at their high-top tables. This is Italian fast food at its finest.
From €3, daily 11:00-23:00, Via Vicenza 46, Termini, +39 041 0298 0580, Metro: Termini
### **TOP GELATO**
Gelato is not just a treat for tourists. At just about any time of year, you'll see locals of all ages frequenting their favorite _gelateria_ (ice cream shop). Zeroing in on the places where you hear the most Italian is a surefire way to get good gelato in your cone. There are tons of knock-your-socks-off _gelaterie_ across Rome. For tips on finding the best, see here.
#### **Giolitti**
Pay for your order as you step in the door, then take your receipt over to the overflowing supply of chillingly good flavors. Giolitti is located a three-minute walk away from the Pantheon, so can you really _not_ go to this place every day? I don't think so.
From €2.50, daily 07:00-01:30, Via degli Uffici del Vicario 40, Pantheon Neighborhood, +39 06 699 1243, Bus: Largo Argentina
#### **Gelateria del Teatro**
I like this place because you can watch your gelato being made right in front of your eyes. Pop in here and ask what's being made, and what is the freshest off the press. I tend to go for the coffee and pistachio flavors.
From €2, daily 12:00-24:00, Via dei Coronari 65, Pantheon Neighborhood, +39 06 4547 4880, Bus: Largo Argentina
#### **Old Bridge**
This famous _gelateria_ is perfectly situated to fortify you for (or help you recover from) the daunting Vatican experience. There's always a line halfway around the block for Old Bridge, but don't be scared; just roll up your sleeves and dive into the crowd. Get up to the front and place your order, paying as you do so. This business has thankfully opened up another location in Trastevere (Via della Scala 70), making for another perfect post-dinner stop.
From €2.50, Mon-Sat 09:00-02:00, Sun 14:30-02:00, Viale Bastioni di Michelangelo 5, Vatican City Neighborhood, +39 06 4559 9961, gelateriaoldbridge.com, Metro: Ottaviano
#### **Ice Cream Factory**
I was skeptical at first, but was pleasantly surprised that this new entry into the gelato scene adheres wonderfully to the traditional processes. Don't be afraid to ask for a sampler from the friendly staff: " _Posso assaggiare_?" (POSS-so assad-JAH-ray).
From €2, daily 10:00-24:00, Via Palestro 47, Termini, Metro: Termini
### **TOP NIGHTLIFE**
While some Americans are always about the next shot or moving on to the next bar, their Italian counterparts much prefer getting lost in the moment and catching up with friends in the streets. Italian students also work with a rather tighter budget generally—so it's easier to enjoy a beer on the square rather than pay for a mixed drink at the bar. That's why you may notice mostly internationals inside the bars, with the locals hanging out outside.
Unfortunately, creepers abound at nightlife venues. That's not to say that ladies can't have a good time out. But to my surprise, there seem to be plenty of men grabbing ass at the clubs. It's also important to keep an eye on your drink, as roofies are unfortunately not all that rare in the discos in town. There are just as many creepers in Testaccio as there are at your local Phi Delta Theta chapter. Don't let this get in the way of your fun, but do be smart and keep your wits about you.
#### **NIGHTLIFE DISTRICTS**
Nightlife varies by neighborhood in Rome. The cheap student bars tend to bunch together, with the glitzy clubs sprinkled here and there throughout the city. These are my favorite neighborhoods.
##### **Trastevere**
Local students like to drink beers on the steps of **Piazza Trilussa,** located right at the end of the Ponte Sisto bridge crossing the Tiber River. American study-abroad students love to hang out around **G Bar** and **Almalu Shot Bar**. And I like to explore and get lost in Trastevere's windy streets. You're sure to stumble upon some great hole-in-the-wall bars or cafés where you could have experiences absolutely unique to Rome.
Trastevere, Bus: Sonnino/Piazza Belli
##### **Campo de' Fiori**
Overrun with grungy locals and American students, this "field of flowers" morphs from a daytime market to a happening nightspot after the sun goes down. Do a lap of the square to pick out the vibe you like most. Its proximity to Largo Argentina bus stop is great for taking the night bus home or catching a taxi. Hungry later? **Aristocampo** _paninoteca_ (Piazza della Cancelleria 93) makes excellent sandwiches fresh to go from €3.50!
Pantheon Neighborhood, Bus: Largo Argentina
##### **Piazza Navona**
Classy—if touristy—cafés line this square. Just off the square to the west you can find excellent and affordable local pubs, like **Bar del Fico,** offering great drinks and an extensive _aperitivo_ , with no Americans in sight.
Pantheon Neighborhood, Bus: Largo Argentina
##### **San Lorenzo**
With dark streets full of mingling Italians and not a tourist in sight, this is one of the most authentic nightlife districts in Rome. East of Termini, San Lorenzo bars offer €4 drinks and there is _pizza al taglio_ (by the slice) on just about every corner. The action centers on two main streets, **Via Tiburtina** and **Via dei Sardi,** from which you can turn in just about any direction and discover your own tangle of bars and cafés. Some travelers may associate the dark streets and people milling about with a bad vibe, but this is just what makes San Lorenzo that much more authentic. Start your night at the dependable **La Piazetta** (Largo Degli Osci 15-19) for cheap beers, _aperitivo,_ and pizza by the slice.
San Lorenzo, Metro: Termini
##### **Testaccio**
Rome's ancient garbage dump for used clay containers has decomposed into a hill into which some of Rome's trendiest clubs have burrowed. (Look closely, and you'll see the entire hill is made of broken down and stacked ceramics). For the clubs, dress up and play the part, as this is definitely Rome's top neighborhood and draws a posh crowd. I recommend doing a lap around the district and selecting the club that looks best to you—do your best to get a sense of the ratio of ladies and gents inside before you pay, as you'll often find the ratio leaning far to the guys. Ask at your hostel for best transportation connections. You'll probably be cabbing it back at the end of the night. Remember, Uber works in Rome!
**LGBT ROME**
Italy is generally very accepting of the LGBT community, and LGBT travelers will have no trouble experiencing Rome. The streets directly behind the Colosseum, following Via San Giovanni in Laterano, have come to be known as Rome's gay district. Here, you'll find great lunch spots, cafés, restaurants, and bars. **L'Alibi** (€10 cover, Via di Monte Testaccio 44, +39 06 574 3448), in Testaccio, is one of Rome's most popular gay clubs. For more information and listings, head to **Arcigay** , Italy's LGBT association (Via Goito 35b, +39 06 645 011 02, arcigay.it).
Testaccio, Metro: Piramide
#### **BARS**
##### **Niji Café**
Easily my favorite cocktail bar in Rome, this speakeasy-style lounge kicks ass. Sometimes you step into a place staffed by wannabe hipsters. Niji only employs real ones, complete with aprons and, if you're lucky, handlebar mustaches. Rather than fumbling through complex, Prohibition-era drinks, they know them like the back of their hand, and the drinks they throw together are made with tender loving care, all in a refined and cozy faux-reading room setting. Order an old-fashioned and sip on the good life.
Drinks from €7, daily 18:00-02:00, Via dei Vascellari 35, Trastevere, +39 06 581 9520, Bus: Sonnino/Piazza Belli
##### **Almalu Shot Bar**
This teensy tiny little bar back in the alleys of Trastevere is famous for its creative shots. Known for varieties such as the Harry Potter shot and the Blowjob shot, Almalu features drink specials throughout the week and was the innovator that brought the €1 shot night onto the scene. This bar is often standing room only, with customers spilling out and socializing on the nearby streets.
Shots from €1, Mon-Sat 18:00-late, Via Della Scala 77, Trastevere, +39 06 5833 3558, Bus: Sonnino/Piazza Belli
##### **Vendita Libri, Cioccolata e Vino**
I suppose every bar needs a gimmick here in Trastevere. Some pull it off quite well and others struggle, but this spot, the name of which translates to the Book, Chocolate, and Wine Store, takes the cake. Each shot is served in a shot glass made of chocolate. More of a curiosity than anything else because it feels like you're drinking in your local bookstore, this place is worth popping into for a quick one and then moving on.
Shots from €2, Mon-Fri 18:30-02:00, Sun 14:00-02:00, Vicolo del Cinque 11a, Trastevere, +39 065 830 1868, Bus: Sonnino/Piazza Belli
##### **G Bar**
Gilded in ostentatious gold, G Bar draws the crowd that loves rocking out to Lady Gaga and Nicki Minaj. Getting a drink upstairs takes patience, and you'll be lucky if you can breathe downstairs. (This dance bar does not enforce a max capacity.) Nonetheless, this bar is very popular with the study-abroad crowd and local observers. The drinks are cheap and strong!
Shots from €2, daily 15:00-late, Vicolo Dei Cinque 60, Trastevere, +39 347 994 1825, g-bar.it, Bus: Sonnino/Piazza Belli
##### **Q's Rummeria Rum Bar Trastevere**
This place takes sugar cane-derived alcohol seriously. With an entire wall full of more shades of amber than you can shake a sugar stick at, the Rum Bar serves just what you'd think: shots and concoctions made with rum as the base ingredient. Be sure to give the freshly casked honey rum a try! Once you get your sipper, climb the stairs to get lost in their stash of games, like Connect Four and Pick Up Sticks. Don't miss the retro _Playboy_ magazine covers in the stairwell, either.
Drinks from €3, daily 19:30-02:00, Vicolo Moroni 53, Trastevere, +39 331 996 6996, qsrummeria.it, Bus: Sonnino/Piazza Belli
##### **Akbar**
Akbar is Roman shabby chic on steroids. Come out to this bar for coffee in the morning, an extensive _aperitivo_ , or drinks late into the night. I love the funky decoration, unique seating, and cool crowd that packs this place out late every night.
Drinks from €5, Mon-Thurs 08:30-02:00, Fri-Sat 08:30-03:00, Sun 10:00-02:00, Piazza in Piscinula, Trastevere, +39 06 580 0681, Bus: Sonnino/Piazza Belli
##### **Freni e Frizioni**
This trendy _aperitivo_ bar has taken over what used to be a mechanics garage with a name that fits: Brakes and Transmissions. Chow down on their extensive nightly buffet and get stuffed for the price of a drink. In warm weather, the cool crowd spills out onto an intimate little square. Come out to this place and feel like a local.
_Aperitivi_ from €6, daily 06:30-02:00, Via del Poloteama 4/6, Trastevere, +39 064 549 7499, Bus: Sonnino/Piazza Belli
##### **Il Baretto**
If you can find and get into this place, located about a 15-minute walk from the Sonnino/Piazza Belli bus stop, you'll think you've discovered an oasis of Italian trendiness unspoiled by the tourist hordes. It's bordering on pretentious, so dress nice and act the part to enjoy an excellent happy hour and an _aperitivo_ that suits any budget. At sunset, enjoy the outdoor patio overlooking the city with a crowd that smugly knows it is indeed in the know.
Drinks from €8, Mon-Sat 07:00-02:00, Via G. Garibaldi 27, Trastevere +39 06 589 6055, Bus: Sonnino/Piazza Belli
##### **Sloppy Sam's**
With seating spilling out onto Campo de' Fiori, Sloppy Sam's is an excellent choice for unpretentious—if touristy—drinks and people-watching. Frequent specials throughout the week draw budget-oriented boozehounds and students. The bar inside is like any other dark dive bar, but the international bartenders seem to do just fine passing out drinks with a smile.
Drinks from €5, daily 09:00-03:00, Piazza Campo de' Fiori 10, Pantheon Neighborhood, +39 06 6688 02746, sloppysamsrome.com, Bus: Largo Argentina
##### **Drunken Ship**
Drunken Ship is a fave with the expat crew looking to down shots and play some beer pong. Don't come looking for especially cheap drinks or for an authentic local experience. Do come for an enjoyable time getting drunk with fellow international backpackers. Somehow the guys at Drunken Ship have figured out the impossible: convincing cheap students to buy expensive drinks every night of the week!
Drinks from €5.50, daily 15:00-03:00, Piazza Campo de' Fiori 20, Pantheon Neighborhood, +39 06 6830 0535, drunkenship.com, Bus: Largo Argentina
##### **Caffè Peru**
Caffè Peru is possibly the cheapest pregame bar open late in Rome. If the Giordano Bruno statue walked forward, turned left and exited Piazza Campo de' Fiori, turned right after a block and continued north one block, it would arrive at the brightly lit Café Peru. Don't come here for the decor, but this is a great place to enjoy beers with friends in an unassuming, mostly empty bar that no one seems to know about.
Drinks from €3, Mon-Sat 06:30-02:00, Sun 09:00-21:00, Via di Monserrato 46, Pantheon Neighborhood, +39 06 687 9548, Bus: Largo Argentina
##### **Bar del Fico**
Named after the piazza outside, Bar del Fico is a popular and casual spot to grab a drink with friends. It's best known for its all-you-can-eat _aperitivo_ buffet, from which I'm guilty many times over of getting my dinner fill, at the price of my rum and Coke.
Drinks from €5, daily 08:00-02:00, Piazza del Fico 26, Piazza Navona, Pantheon Neighborhood, +39 066 880 8413, bardelfico.com, Bus: Largo Argentina
##### **Abbey Theater**
This Irish pub just around the corner from Piazza Navona is a great spot to catch a game on one of its 14 TVs. It also boasts free Wi-Fi, excellent happy hours, and live Irish music on the weekends! But did you really come all the way to Rome for €12 burgers and Guinness? Who knows, maybe you did!
Drinks from €5.50, daily 12:00-02:00, from 11:00 on weekends, Via del Governo Vecchio 51, Piazza Navona, Pantheon Neighborhood, +39 06 686 1341, abbey-rome.com, Bus: Largo Argentina
##### **Scholar's Lounge**
Located between Largo Argentina and Piazza Venezia, Scholar's is a cornerstone to almost every semester-abroad experience in Rome. They're open late, with karaoke nights twice a week and excellent cover bands on the weekends. Game on? They'll be playing it no matter the hour.
Drinks from €5, daily 11:00-03:30, Via del Plebiscito 101, Pantheon Neighborhood, +39 06 6920 2208, scholarsloungerome.com, Bus: Largo Argentina or Piazza Venezia
##### **Bar Open Baladin**
Beer buffs will notice it's difficult—and expensive—to find a nice cold pint in Rome. IPAs are just about impossible to find in Italy, but Baladin will get you as close as possible with their wide selection of beers both on tap and by the bottle. This bar serves artisan beer with a full menu of massive burgers, from the standard bacon cheeseburger to more creative versions topped with mozzarella and basil mayonnaise. Their hand-cut fries are delicious, too!
Pints from €5.50, daily 12:00-01:00, Via degli Specchi 6, Pantheon Neighborhood, +39 06 683 8989, openbaladinroma.it, Bus: Largo Argentina
#### **CLUBS**
If you're trying to club, Testaccio is the place to go. More than a dozen clubs and bars encircle this mound that rises up near the Piramide metro stop, and your best bet is to dress to the nines, pregame hard (drinking at the clubs is expensive), and split a cab home with friends. Do your best to pick a place close to midnight, and plan on sticking around—once the club goers show up, lines get long!
##### **Akab**
Akab is Testaccio's most famous and longest-running dance club, spinning a wide mix of music from house and dance music to even "Straight Outta Compton" hip-hop and R&B. Check the website for upcoming events that take over this club, which has two floors and a massive dance floor, and be sure to dress well to get through the door.
€10 cover, Thurs-Sat 23:30-05:00, Via di Monte Testaccio 69, Testaccio, +39 06 5725 0585, akabclub.com, Metro: Piramide
##### **L'Alibi**
L'Alibi is a popular gay-and-straight-friendly club featuring drag shows during breaks in the house techno and dance music. Check out the website for details and shows. Recently, Friday has been deemed "Hetero Night," but all are welcome throughout the weekend. Rock out over two dark levels with low-slung arches, plus an open patio and terrace during the hot summer months.
€10 cover, Thurs-Sun 23:30-05:00, Via di Monte Testaccio 44, Testaccio, +39 06 574 3448, lalibi.it, Metro: Piramide
##### **Shari Vari Playhouse**
Shari Vari, a posh and ultra-pretentious nightclub, comes with discerning doormen and velvet ropes to keep you out, as well as sexy bartenders and great DJs to keep the party going once you get in. It's popular among Rome's young professionals and jet-set elite; those who come here can afford the €12 drinks, and they dress like it too. You can stop in here for coffee and breakfast and even _aperitivo_ before the venue transforms into its trendy, swanky house and techno-spinning self. Themed nights go down often. Consider reserving a table with friends ahead of time to skip the fuss at the door—as long as the budget permits.
Cover and drinks from €10, daily 08:00-04:00, Via di Torre Argentina 78, Pantheon Neighborhood, +39 06 680 6936, sharivari.it, Bus: Largo Argentina
#### **PUB CRAWLS**
Since the city enacted an ordinance outlawing pub crawls, these moving parties have taken a direct hit. Recently, they've reincarnated as parties that start in one place then go to their last stop at a club, which renders them technically legal but of questionable value.
##### **Rome's Ultimate Party (aka Spanish Steps Pub Crawl)**
These guys have dominated the Roman party scene for the last decade and have adapted their itinerary to the new laws laid out by the city council. While the route may struggle as far as quantity of stops, the party definitely does not.
€25, free wine, beer, and mixed drinks until 22:00, pizza and drink specials included, +39 06 4544 7204, pubcrawlrome.com
### **TOP SHOPPING & MARKETS**
Many know Italy to be the world capital of fashion and fine living. You can find everything in this city, from €1,200 purses and €5,000 suits all the way down to wares and goods for us mere peasants. Enjoy the window-shopping, find some great deals if you time your visit with the semi-annual sales, or _saldi_ (July-mid-August and January-mid-February), and get even better deals at the outdoor markets.
#### **SHOPPING DISTRICTS**
##### **Via del Corso**
Translating directly to "Way of the Race," Via del Corso was once just that: a horse-racing track. Horse racing here was a popular spectator sport until a noblewoman saw a gruesome accident right underneath her VIP window. After that, the city council outlawed horse racing in Rome. Today, Via del Corso, extending south from Piazza del Popolo toward Piazza Venezia, is a bustling modern avenue with tons of recognizable Italian brands and shops, including Diesel, H&M, and Zara. It's relatively more "blue collar" than the district immediately toward the Spanish Steps.
North Rome, Metro: Flaminio or Spagna, Bus: Piazza Venezia
##### **Shopping Triangle of Death**
Leave Via del Corso toward the Spanish Steps and you'll enter the area I've affectionately dubbed the Shopping Triangle of Death, home to high fashion stores Zegna, Gucci, Prada, Yves Saint Laurent, and Louis Vuitton. The district is near the Piazza di Spagna, with the streets of Via del Corso, Via del Babuino, and Via del Tritone forming the triangle. This is where fashion-conscious Italians come to spend their inheritance. This high-end fashion district is designed for deep pockets...and big-brand window-shopping for the rest of us.
North Rome, Metro: Spagna
#### **MARKETS**
##### **Porta Portese**
This Sunday morning market is fundamental for a true Roman experience. Throughout history, cheap vendors have hung out just outside city walls to avoid paying city taxes. While the walls of Rome have fallen, the tradition survives today. Go early on Sunday morning, and leave everything but a little bit of spending cash: Porta Portese is notorious for pickpockets who prey on bewildered tourists lost in the throngs of shoppers. I would also think twice before biting into one of the pork sandwiches for sale on the street... Just speaking from personal experience.
Sun 06:30-13:30, Trastevere, Bus: Sonnino/Piazza Belli
### **TOP PARKS & RECREATION**
#### **Giardino degli Aranci**
Located on the south side of town, across the river from Trastevere and just north of Testaccio, this picturesque little garden of orange trees faces north and west. It's a quiet and welcome respite from the busy streets of Rome. With Testaccio's daily bustling farmers market nearby, this place makes an excellent picnic spot. Enjoy the view from the square, and before descending back into Rome's screaming traffic, check out **Il Buco di Roma** (The Mouth of Rome) nearby. Il Buco di Roma is a keyhole through which you can peer to see the sovereign Priory of the Knights of Malta in the foreground, Roman buildings in the distance, and the dome of St Peter's far off on the horizon. It's a fascinating visual effect with three independent states in a single view.
To get here, take metro line B to Circo Massimo and head north along the track of the Circo Massimo. From there hang a left and go up the hill after about 300 meters.
Free, 07:00-sunset, Testaccio, Metro: Circo Massimo
#### **Gianicolo Hill**
Gianicolo (jah-KNEE-koh-lo), rising behind Trastevere and south of the Vatican, offers a beautiful panoramic view of the heart of Rome. See if you can't pick out the domed roof of the Pantheon, green Palatine hill, the crest of the Colosseum, and even Termini train station way off in the distance. The impressive building off to your left is the old Palace of Justice. For local teenagers, and even older _mammoni_ , this is the place to sneak away to for a necking session at sunset. This hike is a good way to cap a visit to the Vatican and St Peter's Basilica.
Free, always open, Vatican City Neighborhood, Metro: Ottaviano
**MAMMONI: DECONSTRUCTING THE MAMA'S BOY**
Everyone loves a mama's boy—and in Italy, you've got about 10 million of them. Italian mothers put their sons on pedestals and smother them with love, food, clothes, and just about anything else you can imagine. With life so good, why would you ever move out? Add that to the prohibitive costs of buying an apartment or condo in town, and you've got a dilemma that is facing nearly all men in Italy: Should I stay or should I go?
Though it feels a bit like the plot of _Step Brothers,_ in Rome, it's not unusual for Italian men to live at home until they break the ripe age of 40. Even then, the economic prospects are difficult when it comes to buying property. So you've got half the population with more than enough to live on at their parents' house, but nowhere near enough capital to buy a place. So where does their expendable income go? Food, fashion, and nightlife. Party!
This also introduces a dilemma: Where do you take a hot date? Well, if you get a feeling that Italian men are content with a grope and making out on the dance floor, it's because they really don't have a place to take a date home. You'll notice that viewpoints like the one on Gianicolo Hill are populated with couples who, ironically, go out in public to find a little privacy.
#### **Villa Borghese**
This tree-lined park, just to the north and east of town above Piazza del Popolo, was once a noble family's private gardens. Today, it's Rome's largest public playground, with manicured paths, a full-scale replica of Shakespeare's Globe Theatre, thousands of trees, the Galleria Borghese, and even Rome's zoo. It's a beautiful place for a picnic on a sunny day. Rent fun four-person **pedal carts** (€15/hour) or **bikes** (€6/hour) to explore the far corners of the park.
North Rome, Metro: Barberini
### **TOP TOURS**
If you're on a tight budget, a high-quality tour in Rome may be beyond the means. But for history buffs, a tour guide's inside knowledge is well worth the price.
#### **Walks of Italy**
Walks of Italy connects you with excellent local guides who bring the ancient, confusing, and overwhelming history of Rome to vivid life. I appreciate Walks of Italy's attentive booking service and their passionate guides, who adapt to their audience depending on the balance of heavy history vs comedic relief preferred.
Walking tours starting at €29, museum tours from €49, +39 069 480 4888, walksofitaly.com/rome-tours
#### **Eating Italy Food Tours**
Gastro food tours have taken the world by storm, and Kenny and his team have developed an excellent walking tour through one of Rome's richest neighborhoods, Testaccio. Sample the fruits, sweets, pastries, _salumi_ (cold cuts, including but not limited to salami), cheeses, pastas, coffees, and gelati of daily Roman life, all on this three-hour neighborhood walk. Eating Italy tours kicked off a craze that has been replicated many times over now by companies seeing just how successful these experiences are.
Testaccio Food Tour €75, +39 391 358 3117, eatingeuropetours.com
### **TOP HOSTELS**
Hostels cluster around the Termini train station. This neighborhood has really cleaned up its act, and it gives you the best value for your money by far. What was a seedy set of streets is now welcoming and brighter, with some restaurant options nearby. Find more hostel options at **hostelworld.com.**
#### **The Yellow Hostel**
The Yellow just finished their subterranean micro-club, meaning that it finally lives up to its claim as a party hostel, and it has actually developed a following for its nightly specials. The rooms are clean, and the staff is welcoming and helpful. Private budget rooms and hotel rooms are also available. The on-site café and bar churns out great breakfast sandwiches and coffee to get the day started.
Dorms from €25, privates at €120, free Wi-Fi, iPad rentals, Via Palestro 44, Termini, +39 06 49 382 682, the-yellow.com, Metro: Termini
#### **Alessandro's Palace**
Alessandro's Palace, located near Termini, is fun, safe, clean, and bright, with an in-house bar downstairs, air-conditioning in all rooms, and a new rooftop terrace. Alessandro Palace's sister location, **Alessandro Downtown** (Via Carlo Cattaneo 23) is another solid budget option, located opposite Termini train station.
Dorms from €22, privates at €90, free Wi-Fi, sheets included, showers, towels for rent, Via Vicenza 42, Termini, +39 06 4461 958, hostelsalessandro.com, Metro: Termini
#### **The Beehive Hostel & B&B**
The Beehive has been helping budget travelers feel at home in Rome for years. With recent updates like an organic, on-site vegetarian café and welcoming outdoor garden, it's now more popular than ever. Choose from either a handful of dorm beds or a private room (reasonable €40/night). Check out their website for current rates and more details. If these guys don't have availability, they have a network to fall back on that will get you sorted.
Dorms from €25, privates from €40, check-in 07:00-23:00, check-ins outside this window for a fee, free Wi-Fi, sheets, showers, towels for rent, non-smoking, Via Marghera 8, Termini, +39 064 470 4553, the-beehive.com, Metro: Termini
#### **Ciak Hostel**
This is a smaller, more intimate option near the Termini neighborhood. I like this place as it feels more like home than a big commercial hostel. With only about 35 beds in funkily decorated, comfortable rooms (like a Madonna picture frame set over classic car wallpaper), you'll feel like you're hanging out at a friend's house rather than a hostel. Deluxe private rooms with double beds are worth inquiring about.
Beds from €19, free Wi-Fi, sheets, showers, towels for rent, Viale Manzoni 55, Termini, +39 06 7707 6703, ciakhostel.com, Metro: Termini
#### **M &J Hostel**
This recently renovated hostel is fun, laid-back, and staffed with helpful people at the reception desk. It's in a great location, just steps from Termini station, but consider yourself warned: The building is quite old; the hot water cuts out at times, and the sewage system seems like it needs to be improved. The smell can get unfortunate sometimes.
Dorms from €18, posh privates at €90, free Wi-Fi, bar venue downstairs, Via Solferino 9, Termini, +39 06 446 2802, mejplacehostel.com, Metro: Termini
### **TRANSPORTATION**
Rome is well-connected by all modes of transportation. Just remember that not everything works the way you may think it should. Strikes frequently mess up travelers' plans; be sure to ask at the hostel if any may be coming up. Usually, public transport sectors try not to strike simultaneously, so as to avoid really messing the city up...but sometimes they do. Just do your best to roll with Italy's punches and you'll do just fine.
#### **GETTING THERE & AWAY**
Trains and buses arrive at Rome's main train station, **Termini,** located in the Termini neighborhood east of the city center. From Termini you can catch local buses, metro connections, and taxis to anywhere you need in Rome. All of my recommended hostels are within walking distance of the train station.
##### **Plane**
Rome has two airports: **Da Vinci-Fiumicino** (FCO, adr.it) and **Ciampino** (CIA, adr.it). Both airports take roughly the same amount of time and money to get to and from Termini.
The **Da Vinci Express Train** departs every 30 minutes from Da Vinci-Fiumicino and will get you to Termini in 30 minutes (€15). This is the fastest and most comfortable option connecting to Termini, but it's also more expensive than the buses or slower train.
Both airports are serviced by **Terravision** (terravision.eu) buses, your simplest and cheapest option at €8 single and €13 round-trip. Departures are frequent throughout the day and take about one hour. Check timetables and get more details online. On your departure day, it's worthwhile to go to the Terravision office well ahead of time to ensure you get on the bus you need to go on. On busy days, buses can fill up and you'll be stuck waiting for the next one or shelling out for a taxi.
**Taxis** now charge a flat rate from anywhere within Roman city walls to both airports (and vice versa). The rate to and from FCO is €48. To and from CIA is €30.
##### **Train**
Though most trains arrive at **Termini** station, remember that Rome has other major train stations, including **Trastevere** and **Tiburtina**. Be sure to confirm your station if connecting out of Rome via train. From Florence, Rome is 1.5 hours via the fast train (from €25, reservation required) and 3.5 hours via the slow train (from €20). It's about 3.5 hours (€70) from Venice. You can find overnight options on slow trains as well.
**Pickpockets** take advantage of the commotion at train stations to snag tourists' wallets. Take the time on travel days to arrange your valuables securely into one safe place, either on you or in your bag that you know will not go anywhere. Be wary of distractions staged by pickpocketing teams in an effort to take your attention off your pockets. If you need cash, find the Bancomat (ATM) in the least trafficked area of the station.
##### **Bus**
All buses connect into Termini station, but most of the travel across Italy is done via train. You can find cheap domestic connections through **Eurolines** (eurolines.it) and their affiliates.
##### **Car**
I don't recommend driving in Italy. It comes with just too much headache: traffic, parking, hidden speed trap cameras, etc. To rent a car, you'll need a passport and an international driver's license (purchase for US$20 from AAA before leaving home). Overall costs run high when you factor in everything that you'll be paying for: rental, insurance, gas, highway tolls, and any tickets you accidentally rack up.
#### **GETTING AROUND**
Navigating the streets is your first mission to accomplish, but with frequent _caffé_ and gelato stops, I know you'll do just fine! The **bus, tram, and metro systems** use the same **ticket** (€1.50).
##### **Metro**
Rome's two metro lines intersect under the main train station, Termini. The red line, **line A,** will get you to the Spanish Steps (stop: Piazza di Spagna) and the Vatican (stop: Ottaviano). The blue line, **line B,** zips you over to the Colosseum (stop: Colosseo) and the Testaccio neighborhood farther south (stop: Piramide). There's a fabled line C under construction, and I'm hoping it will be completed at some point during my lifetime.
##### **Bus**
Buses in Rome work well but operate without timetables. Purchase a _biglietto_ (ticket) at any _tabacchi_ (tobacco shop). Once you're on the bus, be sure to validate it in the yellow boxes. Non-validated tickets can earn you a €50 fine. If an inspector catches you without a validated ticket, he or she may insist on taking you to an ATM so you can pay the fine on the spot. Ask for the written ticket instead to ensure these inspectors are legitimate, and whatever you do, don't hand over your passport or ID.
Orient yourself by getting familiar with the main bus points in the city: **Termini** (near most hostels), **Largo Argentina** (Pantheon, historic center, nightlife), and **Piazza Venezia** (Vittorio Emanuele Monument, Via del Corso, Colosseum, and Scholar's Lounge). There are a few major bus routes you might find yourself using a lot. **Bus H** is the express bus from Termini to Trastevere, a neighborhood untouched by the two metro lines. For Trastevere, get off at **Sonnino/Piazza Belli,** just after you cross the river. **Buses 40** and **64** connect Termini and Vatican City, hitting the major Largo Argentina and Piazza Venezia bus stops along the way.
##### **Tram**
Rome has a limited tram network. **Tram 8** connects Piazza Venezia in central Rome with the Trastevere neighborhood. You may see other lines in town. As they don't intersect, riding them is really quite simple. Validate your ticket on the tram just as you would on the bus or metro.
##### **Taxi**
Taxis are a good way to get home at the end of the night and affordable if you split with friends. Make sure that the meter reads Tariffa 1 and that it starts at no more than €5 when you get in. The Tariffa 1 zone covers all of Rome inside the old city walls. As soon as you go outside of that, you'll jump to Tariffa 2. At night, taxis start their meters at €6.50. While higher in initial cost, I find Rome's taxi rates to climb relatively more slowly than taxis in other cities.
##### **Bicycle**
Rome isn't bicycle-friendly at all, so getting around on two wheels is not recommended. There are rental options in Villa Borghese, where you've got a few kilometers of trails without a car in sight. Via Appia Antica, the old Roman highway leading south to Naples and Sicily, is another place that's great to explore by bike, and you can stop at cafés and catacombs along the way.
### **DAY TRIPS**
Touring Rome's major sights will easily fill three days and more. But if you've got more time, there are excellent options for day trips outside the city center.
#### **Via Appia Antica**
The Ancient Appian Way was one of the most important roads leading to Rome during the Roman Empire. It connected all settlements and territories south of Rome to the capital. And this several-kilometer section just outside of the modern city of Rome can still be seen today. Numerous crumbling churches, catacombs, tombs, and ancient ruins are sprinkled along the road, making taking a bike out here to explore very worthwhile. You'll also get a nice glimpse of the Italian countryside.
Allow just about all day for this excursion from the city center. Take bus 118 (leaves often) heading south toward the Via Appia Antica. Get off at Appia Pignatello—Erode Attico and follow signs to the "Via Appia Antica." From there, it's a three-kilometer walk back into the city center, where you can hop on the metro line B at Circo Massimo.
Free, always open, Greater Rome
#### **EUR**
Find Mussolini's best attempt at blending Ancient Roman architecture with stark and imposing fascist architecture just south of Rome at the development called EUR—a massive planned district complete with offices, residential buildings, a church, reflecting pools, and a massive sports stadium. It feels a bit like a college campus, without the life or soul. The EUR was supposed to embody the legacy of Ancient Rome while showing how advanced and perfect Italian culture and life was, but a walk through this inhabited ghost town that didn't quite realize its dream of becoming the New Rome gives you an eerie post-apocalyptic feeling. Accessible by metro, it's easy to spend a few hours exploring these streets. Three metro stops will put you at EUR: EUR Magliana, EUR Palasport, and EUR Fermi.
Free, always open, Greater Rome
### **HELP!**
#### **Tourist Information Centers**
Many tourist information centers purport to be unbiased, but in fact only promote a limited number of tours. Pop in for a quick run-down of the city and a free map, but remember this when being offered tour options.
#### **Pickpockets & Scams**
Ladies should expect plenty of attention from locals. Keep your wits about you, whether you're in a cab or enjoying a night out. Italian men can be persistent and aggressive. It's nothing to worry about, but good to be aware of.
Rome does have a pickpocketing problem. Pickpockets usually work in teams and hang out in the most touristy areas: metros, bus 64, Spanish Steps, Termini train station, Campo de' Fiori, the Colosseum, and even inside the Vatican Museums. Try to identify pickpockets by sight, especially the ones swaddling "babies," as the extra clothing often conceals a sly, quick hand. If there is any commotion around you first make sure your valuables are secure. Oftentimes the teams create a distraction on the bus or in the metro, and while everybody's attention is turned one way, there's a guy making off with purses and wallets in the opposite end of the bus.
#### **Emergencies**
Dial 118 for an ambulance, or 113 for police.
#### **Hospital**
Policlinico Umberto I
Via del Policlinico 155
+39 06 499 71
#### **US Embassy**
Via Vittorio Veneto 119
+39 06 467 41
Florence Maps
Florence 101
Three Day Itinerary
Top Neighborhoods
Top Sights
Top Eats
Top Gelato
Top Nightlife
Top Shopping & Markets
Top Tours
Top Hostels
Transportation
Day Trips
Help!
Florence is the birthplace of the Renaissance. You're sure to catch a whiff of inspiration yourself as you glimpse the city's iconic dome, gaze up at the _David_ statue, and imagine Michelangelo himself walking these regal cobblestone streets. Luckily, this Tuscan city's larger-than-life reputation hasn't tarnished its charm. As a study-abroad hot spot and a key stop on the international backpacking circuit, Florence offers modern attractions, too—from fun student nightlife to cheap and tasty eats and unforgettable gelato.
### **FLORENCE 101**
Founded as an important Roman outpost in AD 59, Florence sat on an important trade route from Rome leading north to the rest of the continent. With the fall of the Roman Empire, all of Europe fell into chaos. The Florence region was one of the first to bounce back economically from the turmoil of the Dark Ages after several hundred years. Traded back and forth between warring rivals, Florence finally emerged independent around AD 1100. The city extended its influence through banking and trade around the subcontinent and beyond as it grew into the medieval age.
By the 14th century, a quarter of Florence's population was supported by the wool and textile industries. With this trade, one family, the Medicis, rose to immense riches and power—it didn't hurt that the Medicis were the bankers to the pope, either. Cosimo de Medici (1389-1464) was the first of the family to hold significant influence over the city from behind the scenes. The wealthy Florentines had the luxury of splurging on the finer things in life, like art and architecture, which helped Cosimo kick off a wave of artistic, literary, and architectural innovation.
Lorenzo de Medici, Cosimo's grandson, was the family's biggest patron of the arts and devoted vast sums to a handful of promising artists. Some of his personal friends (Michelangelo, Leonardo da Vinci, Botticelli) changed the course of history with their contributions thanks to the support of Cosimo. This enlightenment of the human spirit and intellect—known as the Renaissance, or rebirth—ushered in developments at an accelerating rate, slowly wrenching the rest of Europe from the grip of feudalism. During the Renaissance, buildings became taller, windows bigger, and decoration more exuberant. Paintings were brighter and more lifelike. Artists came to be paid in relation to their skill, not as a function of the costs of their raw art materials. Technical developments like sewage disposal allowed for population growth and healthier, longer lives. It may seem like a stretch, but one wealthy family's investment in the arts revolutionized the world and continues to have a lasting impact today.
Florence was annexed as part of the Holy Roman Empire in the 18th century. Population growth and modernization efforts like cutting new boulevards for traffic and commerce continued into the 20th century. In World War II, Florence rested under Nazi occupation for a year before Allied troops chased them out. The departing Nazis destroyed as many river crossings as they could but spared the Ponte Vecchio, thanks to a rare consideration by Hitler of the historic importance of the bridge.
Today, Florence's lifeblood is tourism, tourism, and more tourism. This historic city sees millions of visitors every year, but with my suggestions, you'll be able to find your own little corner of this fascinating Renaissance town.
### **PLAN AHEAD**
#### **RESERVATIONS**
Note that most museums in Florence, including the Uffizi and Accademia, are **closed on Monday.**
Reservations are recommended for the following sights:
**Accademia** (firenzemusei.it)
**Uffizi Gallery** (firenzemusei.it)
#### **PASSES**
##### **Firenze Card**
Serious sightseers should consider the **Firenze Card** (€72, valid for 72 hours, firenzecard.it), which provides free entry (and line-skipping privileges—just go to the head of each line and flash the card) to Florence's major museums. Your three days of exhaustive sightseeing kick off the moment you validate the pass, so be sure to line up your visits within this window. Before purchasing the card, add up the entry fees for all the sights you plan to see, then see how that total compares to the cost of the Firenze Card. The card includes the Uffizi, Accademia, Palazzo Vecchio, Medici Chapels, Bargello Museum, the Duomo (including the Baptistry and dome climb), Pitti Palace, the archaeological site of Fiesole, and over 50 other sights. Purchase the pass or find the full list of covered sights online, at the ticket offices of these museums, or at either of the tourist information centers in town. If you're planning on seeing each of the sights I've listed, this pass will just barely cover the cost—as you see, you've got to be quite busy (and up early) to get your money's worth. The Firenze Card also includes your public transportation for the three days of validity.
##### **Duomo Sights**
The Duomo itself is free, and a single ticket (€15) covers entry to all Duomo-related sights (Baptistry, dome climb, belltower climb, and the museum). The ticket, which you can buy at the ticket office (7 Piazza San Giovanni), is valid for six days after purchase. You must enter all the specified sights within 24 hours after your first entry.
**KNOW BEFORE YOU GO**
**KEY STATS & FIGURES**
**Currency:**
**Italy uses the euro (€); 1 EUR = about 1.06 USD**
**Population:**
**370,000, 1.5 million in the greater Florence area**
**Tourists per year:**
**over 10 million**
**Language:**
**Italian**
**Percentage of citizens killed by the black death plague:**
**50 percent**
**Typical dishes and products:**
**_ribollita_ (Tuscan bean soup), _trippa_ (tripe), _lampredotto_ (tripe sandwich) _, bistecca fiorentina_ (Florentine steak), Ruffino wine (Chianti Classico)**
**Number of steps to climb the Duomo:**
**463**
**Height of Michelangelo's _David_ :**
**14 feet, 3 inches**
**CALIBRATE YOUR BUDGET**
**TYPICAL PRICES FOR:**
**Hostel dorm bed:**
**€16**
**Two-course dinner:**
**€14**
**Pint of beer:**
**€5**
**Basic leather purse:**
**€16**
**Leather jacket:**
**€120 (or more)**
**MOVIES TO WATCH**
_A Room with a View_
THREE DAY ITINERARY
As the de facto capital of the Renaissance, Florence is packed to the rafters with famous museums displaying priceless artwork. Any short visit here is a mad dash mostly on foot to take in top sights. For the Uffizi and Accademia, I highly recommend making reservations well ahead of time to skip lines and avoid hours of waiting. Otherwise, opt for the pricey but effective Firenze Card as another way to skip the lines.
**DAY 1: BENVENUTO A FIRENZE!**
**MORNING**
Grab a hearty breakfast at your hotel before striking out to join the free walking tour. Consider today your chance to get acquainted with the city.
At 11:00, meet up with your fun local guide just in front of Santa Maria Novella for a free two-hour walking tour with **Florence Free Tour** to learn the basics about the history and culture of the city. You'll stroll past the **Duomo,** numerous famous piazze, and the **Uffizi,** and will learn some background info on the Ponte Vecchio.
**AFTERNOON**
Depending on where you end up, try out **Oil Shoppe** or **All'Antico Viniao** for fresh and cheap sandwiches to keep you going.
Continue on toward the iconic **Ponte Vecchio** for a beautiful view of the river. Cross the bridge into the **Oltrarno** neighborhood and stop at the **Piazza Santo Spirito** for an afternoon spritz cocktail.
**LATE AFTERNOON**
Stop to pick up a bottle of wine and maybe a snack to enjoy later, and climb up to the **Piazzale Michelangelo** for an awe-inspiring late afternoon panorama of the entire city of Florence. Bust out your picnic supplies and take in the scene. You can see the Duomo and San Lorenzo, along with the spine of Santa Croce and spire of the Palazzo Vecchio. Selfie with David, anyone? There's a replica of the famous statue up here in this square.
**EVENING**
Hungry? Start walking back the way you came, but hang a right on Via dei Benci and cross the Ponte alle Grazie to my favorite spot in the Santa Croce neighborhood for _aperitivi,_ **Moyo.** Stop in here for a drink and access to an impressive buffet of simple local dishes while listening to the cool grooves from the DJ.
It's still early, so go back to your hostel to rest up, shower, and get ready for a crazy night out!
**LATE**
Return to the Santa Croce neighborhood to kick your night off right at some of my favorite spots, including **Naima** , **The Lion's Fountain** , and **Kikuya** , and go late at **Red Garter**.
**DAY 2: UFFIZI & ACCADEMIA**
**MORNING**
Overachiever? Get in line by 07:30 (yes, AM) at the **Uffizi** to avoid the lines that stretch through the rest of the day. Or better, sleep in and pat yourself on the back for making reservations ahead of time! You'll spend about two hours exploring this world-class museum.
**AFTERNOON**
Walk north about 10 minutes to the Piazza del Duomo for pizza by the slice at **Pizzeria del Duomo** , or stroll down Via dell'Oriuolo to the east of the Duomo to **Caffetteria Oblate** for a tasty lunch at a hidden yet beautiful library. If you saved room, some of my favorite _gelaterie,_ **Grom** and **Perché No!,** are just a couple blocks away.
Spend the rest of the afternoon **climbing the dome** of the Duomo to get a better grasp on the scale of the engineering feat that truly kicked off the Renaissance. Can you imagine construction starting on the church's foundations before the technology existed to build a dome big enough to cap the church?
And if you're on your game, catch your afternoon **Accademia** reservations around 16:00 today and enjoy the one-floor museum to see the original of Michelangelo's masterpiece, _David._ Walk back to the hostel to rest up a bit and put those feet up.
Make sure you've got reservations for dinner if you're picking a popular spot like Buca Mario.
**EVENING**
On the way to dinner, pop into **Casa del Vino** for a sampling with my friend Gianni. Admire the old-school pictures hanging up on the wall as you sample some of Tuscany's best vintages. Then enjoy authentic Florentine fare at one of my favorite restaurants, like **Osteria Zio Gigi's** , **Buca Mario,** or **Osteria Brincello.**
**LATE**
Get classy at **Hotel Cavour's rooftop bar** and enjoy a cocktail with a beautiful view of central Florence. Afterward, join the party at my favorite downtown club, **Club TwentyOne,** just around the corner. Good night!
**DAY 3: DAY TRIP OR SHOPPING**
If you have the whole day, consider a day trip to **Fiesole** or a **Tuscan hill town,** just a short trip away. Otherwise, spend the morning shopping for leather goods and delicious local produce at the **San Lorenzo Market**. More time? Walk over to another one of Florence's world-class museums, the **Bargello** , in the afternoon.
### **TOP NEIGHBORHOODS**
Florence's compact medieval center is called the **Centro Storico.** You can trace the old walls, tight medieval streets, and expansion beyond the walls on any city map. With the Duomo in the middle, the Centro Storico contains most of the sights tourists will want to see on a short visit. Here, you'll find the Accademia, the Uffizi Gallery, and the famous Ponte Vecchio, along with recommended restaurants, hostels, and nightlife venues.
**Santa Croce,** a few blocks east of the Duomo on the eastern edge of the historical center, is one of my favorite neighborhoods. This is where you'll find some of Florence's best nightlife, from trendy _aperitivo_ bars to lounges and student favorites like Kikuya and Red Garter.
Crossing the Arno River, you enter into **Oltrarno,** translating roughly to "the other side of the Arno." This district is quieter, with more upscale restaurants and a fun local scene centered around Piazza Santo Spirito. It's also home to the Pitti Palace and the access point to the Piazzale Michelangelo viewpoint.
### **TOP SIGHTS**
#### **Uffizi Gallery**
This extensive art museum fills out what were the offices of the astronomically wealthy Medici family. Today, you can tour these long hallways and corridors packed with priceless medieval and Renaissance works of art. The Uffizi museum takes you through their collection in chronological order, so you can see the transition from flat, gold-encrusted medieval triptychs to the voluminous and dimensional Michelangelo masterpieces. Find Botticelli's _Birth of Venus,_ as well as paintings by da Vinci, Raphael, and Titian. Lines can be long. Reserve ahead, or go early or late to avoid the crowds.
€12.50, free first Sun of the month, Tues-Sun 08:15-18:35, closed Mon, Piazzale degli Uffizi, Centro Storico, +39 055 975 7007, uffizi.firenze.it
#### **Accademia**
Thankfully occupying only a single floor, this museum is the permanent home of Michelangelo's colossal little squirt, **_David_**. This magnificent example of sculptural _contraposto_ (a stance where the weight is placed on one foot, with the corresponding hip jutting out slightly, adding tension and movement in otherwise static paintings and sculptures) captures the moment just before the underdog slew the giant in the biblical story of David and Goliath. Catch his gaze, and notice exaggerated features like furrowed eyebrows and large, veiny, oversized hands. The posture and embellishments go a long ways in bringing this stone to life. Monitors nearby allow visitors to get a close-up view on a high definition 3-D scan of the sculpture.
As you walk down the hallway from the entryway, toward the larger-than-life sculpture, you're flanked by Michelangelo's "prisoners," pieces that he never quite finished. Michelangelo believed he simply freed shapes from the confines of each block of marble. He could visualize the finished piece inside the stone before he ever took his first whack.
€12.50, additional €4 for reservation, free first Sun of the month, Tues-Sun 08:15-18:50, closed Mon, via Ricasoli 66, Centro Storico, +39 055 215449, accademia.firenze.it
#### **The Duomo**
Many say that the Cattedrale di Santa Maria del Fiore, or Duomo for short, is the building that kicked off the Renaissance. Brunelleschi, an engineer and architect, won the contract to dream up what was to be the greatest church around. He designed the floor plan and began construction on the cathedral before the technology to build a dome big enough to cap it even existed. A perfect example of form and function, the beautiful cathedral now dominates the Florentine skyline. Just about every large modern dome, from the Vatican to the Capitol building in Washington DC, can point to this structure for its inspiration. Even today, the scale and beauty of the Duomo is captivating as you **climb the dome,** moving through the space between the double-domed structure to get to the top. The inside, lighter dome provides the ceiling to the church. The tiled outer dome supports the lantern at the top and protects the church from the elements, a design that greatly diminished the weight of this structure. You can also **climb the belltower** for views over Florence. A heads-up for the claustrophobic: Both stair-stepping hikes gets quite tight and darker the higher up you go.
Back in the day, you were not allowed to enter a Catholic church without having been baptized. As such, the **Baptistry** of the Duomo was separated, and new converts were baptized there so they could then enter the church holy and cleansed. This famous octagonal baptistry sports three sets of large bronze double doors. These outward facing doors were a prime place to introduce the unbaptized and the illiterate to Christianity. The world-famous eastward-facing set recounts 10 stories from the Old Testament. Lorenzo Ghiberti's bronze doors were ground-breaking works of art, vividly depicting three-dimensional scenes on flat bronze panels. The doors you see are replicas. To see the originals, head to the interesting and worthwhile Duomo Museum.
Duomo free, €15 ticket covers dome climb, belltower, Baptistry, and Duomo Museum (you must enter all sights within 24 hours of first entry), Mon-Fri 10:00-17:00, Thu until 16:00 May and Oct, until 16:30 Nov-Apr; Sat 10:00-16:45, Sun 13:30-16:45, Piazza del Duomo, Centro Storico, +39 055 230 2885
#### **Duomo Museum**
If you're anything like your dear author, you love geeking out about how revolutionary structures like the Duomo are built. The Museo dell'Opera del Duomo is located just behind Brunelleschi's dome. This museum showcases Ghiberti's original Baptistry doors, along with historical drawings and images of Florence's main cathedral.
€15 ticket also covers Duomo dome climb, belltower, and Baptistry (must enter all sights within 24 hours after your first entry), Mon-Sat 09:00-19:00, Sun 09:00-13:45, last entry 45 minutes before closing, Piazza del Duomo 9, Centro Storico, +39 055 230 2885, museumflorence.com
#### **Basilica di Santa Croce**
The imposing facade of the main Franciscan church in Florence is a Renaissance veneer over a more modest 14th-century beginning. Famous for its permanently interred residents—Galileo Galilei, Lorenzo Ghiberti, Niccolo Machiavelli, Michelangelo Buonaroti, and Dante are all buried here—the church now costs €6 to go inside, but it's worth it for those really interested in Tuscan-style Renaissance churches. Just behind the church and through a garden, you'll find the **Leather School of Santa Croce** (free, observe students' ongoing work Mon-Fri 10:00-17:30, entrance at Via San Giuseppe 5R, +39 055 244 533, scuoladelcuoio.com), a workshop that carries on the long tradition of leather working.
€6, Mon-Sat 09:30-17:30, Sun 14:00-17:30, Piazza Santa Croce 16, Santa Croce, +39 055 246 6105, santacroceopera.it
#### **Bargello Museum**
Among the great museums of Florence, this one tends to be overlooked, but by no fault of its own. I actually prefer it as the poor man's option to see great sculptures by Renaissance masters, such as Donatello's _David_ , a much more subtle representation of the Bible character than that of his rival Michelangelo. Another one of Donatello's works, the armored _St George_ statue representing the armorers guild, was taken from its original exterior niche at the Orsanmichele church and placed here to keep it out of the elements and for better protection. The museum is housed in a 13-century palace, which has also served as a prison and even police barracks; this is one of the oldest and grandest buildings in Florence—and half of my reason for visiting is the chance to see the inner courtyard, staircase, and large halls.
€7, free first Sun of the month, Tues-Sat 08:15-17:00, 08:15-13:50 if there are no special exhibits, also open on the second and fourth Mon and the first, third, and fifth Sun of each month, Via del Proconsolo 4, Centro Storico, +39 055 238 8606, polomuseale.firenze.it
#### **Palazzo Vecchio**
This is Florence's old fortified Town Hall. It was built like a fortress because of the turmoil enveloping Florence in the 16th century: Both internal politics and external city-state were threatening the city, which made defensive features necessary. In fact, when a physical fight broke out between Florentine politicians in 1567, a chair was thrown from the window, striking the _David_ in the Piazza della Signoria far below. The statue's left arm broke into three pieces, prompting the move of the statue to safe cover in the Accademia museum.
Today, it takes visitors about two hours to wander through a tangle of more than 20 rooms extensively decorated with more Renaissance artwork. Keep an eye out for interesting temporary exhibits, as the museum has a strong rotation of them. Climbing the tower costs extra but proffers excellent downtown Florence views with the Duomo in the frame.
Courtyard free, museum €10, tower climb €10, museum and tower €14, Apr-Sept Fri-Wed 9:00-24:00, Thurs 9:00-14:00, Oct-March Fri-Wed 9:00-19:00, Thurs 9:00-14:00, tower has shorter hours, last entry to either one hour before closing, Piazza della Signoria, Centro Storico, +39 055 276 8325, commune.fi.it
#### **Basilica di San Lorenzo & Medici Chapels**
San Lorenzo is where the noble Medici family went to church, and as such, it's one of the grandest churches in town. The exterior is imposing (yet dwarfed by the Duomo nearby), but the interior displays the true treasure: the Medici family tombs. Just about every important member of the noble family is buried here under grand tombs, giving the viewer no doubt what a big deal these guys actually were.
Basilica €4.50, chapels €8, free on first Sun of the month, basilica: Mon-Sat 10:00-17:30, Sun 13:30-17:30, chapels: Apr-Oct Tues-Sat 08:15-16:50, Nov-March Tues-Sat 08:15-13:50, Piazza di San Lorenzo, Centro Storico, +39 055 238 8602, polomuseale.firenze.it
#### **Orsanmichele Church**
Possibly more famous for its exterior than the interior, this church visually demonstrates just how important trade was in Renaissance Florence. The church auctioned off niches around the facades of the building to the networks of professional guilds in Florence at the time. Wanting to outdo each other, the guilds hired master sculptors to artfully represent their sponsors. See if you can't pick out the niches representing the wood- and stoneworkers, the bankers, the wool producers, the shoemakers, and the doctors. Ghiberti, Donatello, and Brunelleschi all gave their best shot at their respective guilds' patron saints. Donatello's _St George_ (with the large shield), designed and sculpted for the armorers guild, is my personal favorite. Remember that making a statue out of bronze was 10 times more expensive than creating the same thing out of stone. So to display their success, wealthy guilds like the merchants of Florence created theirs from bronze. Most original statues are now distributed around town for permanent protection and have been replaced with copies.
Inside the church, check out the beautiful and ornate tabernacle, off center due to the fact it was a repurposed monastery kitchen before it became a church. Ever-practical Florence used the upstairs as grain storage. Just inside the entrance to the left, you'll see the hole in the ceiling that was used to feed grain down to ground level.
Free, daily 10:00-17:00, Via dell'Arte della Lana, Centro Storico, +39 055 23885
#### **Ponte Vecchio**
Right up there with the Duomo as a key icon of the city, this is the oldest bridge in town and the most important crossing point of the Arno River in Florence. To avoid the common Florentine people, the Medici family constructed a long passageway connecting their private residences in Oltrarno to their offices, the Uffizi. A stretch of this passageway, called the Vasari Corridor, actually runs along the spine of the Ponte Vecchio. Historically, the bridge was populated by butchers and fishmongers. But they were deemed too smelly by the Medici above, so they banned them and replaced them with goldsmiths and jewelers. Ahhh, much better.
Today, the bridge is still full of jewelers, shops, and tourists snapping selfies. My favorite photo op of the bridge itself is from half a block north and west along the river, where you can look back to see the shops stacked haphazardly like Jenga blocks along the length of the bridge.
Free, always open, Centro Storico
#### **Piazzale Michelangelo**
Offering the best viewpoint of the whole city, Piazzale Michelangelo is worth the hike (and sweat) up and out of town, especially if you bring a bottle or two to enjoy at sunset. What's actually a glorified parking lot is capped with a replica of Michelangelo's _David_. Also in the square, you'll sometimes find a company called **TestDriveFirenze** (+39 331 20 55 888, testdrivefirenze.com, info@testdrivefirenze.com), which lets you drive a Ferrari in chunks of 15 minutes for €60. Bucket list, anyone?
Free, always open, Oltrarno
#### **EXTRA CREDIT**
##### **Pitti Palace**
It's impossible to miss the outside of this dominating and imposing palace containing dozens of richly decorated rooms and tens of thousands of pieces of art. If you're big into late-Renaissance artwork, this is a great museum for the likes of Caravaggio, Titian, Rembrandt, and Raphael.
€13, Tues-Sun 08:15-18:50, Piazza de Pitti 1, Oltrarno, +39 055 294 883
### **TOP EATS**
Here you are, in the middle of Tuscany, a region world-famous for its rustic and delicious cuisine! Take every opportunity to indulge in the extra-virgin olive oil, fresh produce, and wine the region is so well-known for.
Like many northern Italian towns, Florence has a culture of **_aperitivi_** , an extensive buffet of pastas and salads all for the price of a glass of wine—or even better, the popular spritz, a refreshing drink made with sparkling Prosecco and a dash of sweet Aperol or bitter Campari. I prefer the Aperol version myself.
Examine your bill before paying to determine whether or not service is built into the final price. Many restaurants include either a mandatory or optional tip, but it's up to you to determine it. Additionally, _coperto_ (a "cover fee" for sitting down at a table) should also be considered before simply rounding up your bill about 10 percent for service.
#### **Pangie's Bistrot**
Pangie's makes great focaccia sandwiches for €5 and also has a full menu of delicious _primi, secondo,_ and platters with bruschetta, meats, and cheese. This casual, welcoming shop is small, with a handful of tables. The sweet owners, Mario and Francesca, are happy to make made-to-order sandwiches to keep you going between visits to museums and sites. They are open for lunch and dinner daily except for the second and third Sundays each month.
Sandwiches from €5, Tues-Sun 12:00-15:30 and 19:00-24:00, Via del Parione 43-45r, Centro Storico, +39 055 295 439, pangies2010@gmail.com
**DINING TIPS**
Keep these factors in mind when dining in touristy Florence:
**_Coperto:_** Many restaurants charge a "cover fee" (usually €1-2) for sitting down at their tables. It's meant to cover the bread brought out to you and will be added to your final bill. You'll find _coperto_ listed in fine print at the bottom of the menus out front of most restaurants. If the charge is any more than €2, keep looking.
**_Servizio:_** Check your bill closely to see whether service is included. If it is, you don't need to tip beyond that. If service was not included in the bill, feel free to round up to add about 10 percent.
**Tap water:** Most restaurants do not offer free tap water. If you order water, they will ask you "Still or sparkling?" and then deliver a nice bottle of water (which you'll pay for) to your table. Be sure to check the price in the menu. Interestingly enough, table wine is often cheaper, and bottles of water can run past €6!
#### **Osteria Brincello**
Osteria Brincello offers what is easily one of the most solid values and authentic experiences in town. Run by a jovial crew of friends and family, Brincello has a limited, constantly refreshing menu of delicious and ultra-typical Tuscan pastas, like tagliatelle with meat sauce, and Florentine steaks. Count on fresh ingredients on every visit. The location of this sit-down and easygoing restaurant is another plus—not far from the train station, and close to many of the hostels in town.
Plates from €12, Fri-Wed 12:00-15:00, 19:00-23:00, Via Nazionale 10, Centro Storico, +39 055 282 645
#### **Caffetteria delle Oblate**
Stop by the Oblate library for a spectacular and unique close-up view of the Duomo while enjoying a budget snack served cafeteria-style on the top floor of a historic library. The open-air loggia in the upper floor of the library is my favorite little getaway in central Florence. The atmosphere and people-watching are endlessly interesting: This is your chance to observe the Italian version of your local library.
Free entry, eats from €4.50, Mon 14:00-19:00, Tues-Sat 09:00-24:00, Sun 11:00-18:00, Via dell'Oriuolo 26, Centro Storico, +39 055 263 9685, lospaziochesperavi.it
#### **Trattoria Nerone Pizzeria**
Imagine an Italian take on the American Buca di Beppo chain, and you've got this casual sit-down pizzeria, Nerone. The restaurant is done up in a gregarious fashion, but the service and pizza are thankfully less obnoxious. As soon as you step in, you're greeted straightaway with the warmth of their firewood stove, cranking out fresh and delicious pizza left and right.
Pizzas from €7, daily 11:30-23:00, Via Faenza 95r, Centro Storico, +39 055 291 217
#### **Mercato Centrale**
Florence's bustling main market is great spot to pick up picnic supplies for a hike up to Piazzale Michelangelo. Drop by vendors selling everything from pigs' feet to fresh veggies and fruit. I love asking " _puo fare un panino_?" ("will you make a sandwich?") and getting a fresh sandwich made right before my eyes. Just pick the places that have bread, meat, and cheese visible and order by pointing. For a quick, casual, and cheap eat, find **Nerbone** inside the market and sharpen your elbows to grab a plate of their delicious Tuscan fare. Pastas, sandwiches, pork meat, and other dishes all go for around €5 each.
Eats from €4, daily 07:00-14:00, Piazza San Lorenzo, Centro Storico
#### **Casa del Vino**
If sampling wine is your thing, I highly recommend stopping in to say " _ciao_ " to Gianni, who's the latest in a family line of sommeliers. This shop has been in the family for over 70 years. Admire the old school photos on the wall, and get an education in local and regional Tuscan wines. Pick up a bottle for the road if you're in the mood!
Tasting platters from €8, Mon-Sat 09:30-20:00, Via dell'Ariento 16r, Centro Storico, +39 055 215 609, casadelvino.it
#### **Pizzeria del Duomo**
Imagine biting into a slice of thick-cut, freshly made pizza while taking in one of the best views of the Renaissance (the Duomo and Baptistry) in the warm Tuscan sun. That's exactly what's offered at my favorite _pizza al taglio_ (pizza by the cut, or slice) spot in Florence! Order your pieces in the display windows, and grab a spot either outside on the square or downstairs in their cellar. To find this place, look for the Pizzeria al Taglio awning at 5 Piazza di San Giovanni behind the Baptistry. You'll enjoy the best pizza and cheapest view around. I'm just praying they don't get spoiled by the tourist popularity.
Slices from €2, daily 11:00-24:00, Piazza di San Giovanni 21r, Centro Storico, +39 055 210 719
#### **Oil Shoppe**
This fast and casual sandwich shop is famous for their generously appointed sandwiches filled with local meats called _salumi_ (prosciutto, ham, salami), veggies (eggplant, tomato), and cheeses (pecorino and mozzarella). It's a favorite among the study-abroad students in Florence. Stop in for a bite to take away or eat in along the bar on the wall. Don't let the name of the place throw you off: Vegetarian and healthy options are plentiful!
Overflowing sandwiches from €3.50, Mon-Sat 10:00-19:00, Via Sant-Egidio 22r, Centro Storico, +39 055 386 0091, facebook.com/theoilshoppe.it
#### **Pane & Toscana**
Another excellent choice for a fast and easy city-center lunch, Pane & Toscana tops their menu with two dozen typical Florentine sandwiches incorporating ingredients like artichokes, mushrooms, smoked tuna, and a wide array of sauces. Select your fillings and have them on either salty focaccia or a healthy wrap. Tourists and locals enjoy the fast service and tasty sandwiches. Try to grab a spot on the bench outside.
Sandwiches €3, Tues-Sun 09:00-22:00, Borgo degli Albizi 31, Centro Storico, +39 345 173 3604
#### **Osteria Zio Gigi's**
Zio Gigi's offers surprising quality and value for a sit-down restaurant this close to the Duomo. I love the comfortable, welcoming atmosphere and the set menu for large portions without the large price tag. Pop in here for simple and hearty Tuscan fare, like pasta with meatballs, salad with prawns, and Florentine steaks. Gigi is a load of fun and will take good care of you for both lunch and dinner.
Dinners from €12, daily 12:00-15:00 and 19:00-23:00, Via Portinari Folco 7r, Centro Storico, +39 055 215 584
#### **Moyo**
I love this place for an _aperitivo_ that will tide you over but still leave room for drinks and gelato later—very important! Step in and grab a spot, and a waitress will take your order for a drink—I always opt for the spritz with Aperol. Then, head to the buffet table—free as long as you have a drink—and enjoy an eclectic yet tasty spread of everything from salad and cheese pizza to hummus, risotto, and French fries. Good food, well-poured drinks, suave mod atmosphere, and a spinning DJ all make this place a favorite of mine.
_Aperitivo_ €7, daily 08:00-02:00, Via dei Benci 23r, Santa Croce, +39 055 247 9738, moyo.it
#### **Buca Mario**
If you like steaks, this is your place to splurge. Buca Mario is possibly Florence's best steakhouse. Wash your meal down with typical Chianti from the region based on recommendations from the suited-up servers. Reservations recommended due to its popularity, and be sure to dress your nicest!
Steaks from €28, daily 12:00-23:30, Piazza degli Ottaviani 16r, Centro Storico, +39 055 214 179, bucamario.com
#### **La Buchetta**
Famous for its typical Florentine beef stew ( _peposo_ ), the clean and refined La Buchetta doesn't disappoint with their steaks, pastas, or beautifully presented desserts. This splurge-worthy and somewhat formal restaurant will set you back a bit, with meals running around €40 or €50 a head, but the superb steaks, immaculate pasta carbonara, crunchy bruschetta with mozzarella, and unforgettable cheese plate have me itching to come back year-round.
_Peposo_ €15, daily 09:00-02:00, Via de Benci 3, Santa Croce, +39 055 217 833, labuchetta.com
#### **All'Antico Viniao**
One of my favorite little holes in the wall churns out meaty sandwiches and wine by the glass. Enjoy your stand-up meal in the alley, and return the glasses once finished to the rack on the side of the door. This is a perfect protein pit stop during those long days of sightseeing.
Sandwiches from €3, Tues-Sun 12:00-16:00 and Tues-Sat 19:00-24:00, Via dei Neri 74r, Centro Storico, +39 055 238 2723, allanticoviniao.com
#### **Ristorante Grotta Guelfa**
I love this place for the varied menu, especially for their bubbling pizzas, Florentine steaks, and perfectly sharable meat and cheese platters. Find this sit-down restaurant with white tablecloths and limited outdoor seating tucked away just a couple blocks down Via Pellicceria toward the river from Piazza della Repubblica.
Get your fill from €16, daily 11:00-22:30, Via Pellicceria 5r, Centro Storico, +39 055 210 042, grottaguelfa.it
#### **Birreria Centrale**
Need a break from _vino_? Pop in here for some delicious craft _birra_. Their menu is surprisingly varied and tasty menu for a beer hall, but who's complaining? Enjoy their steaks, pastas, gnocchi, and salads underneath exposed brick cellar arches, in an authentic and casual old-time atmosphere. In nice weather, enjoy their outdoor patio.
Beers from €4.50, Mon-Sat 11:00-24:00, Piazza de Cimatori 1r, Centro Storico, +39 055 211 915, birreriacentrale.com
### **TOP GELATO**
Gelato will be a staple in your diet during your visit to Florence whether you like it or like it more. Challenge yourself to never try the same flavor, or _gusto_ , twice. Many are tempted to get the XL after trying their first taste, but exercise a bit of restraint and go for the small. This not only gives you an excuse to get another one soon, but also helps you enjoy the rich flavor.
#### **Perché No!**
It's considered by many to be the top _gelateria_ in town, so you can't visit Florence without dropping into this place just off the main shopping drag of Via del Calzaiuoli. With the convenient location, I also ask, why not?
From €1.50, daily 11:00-22:30, until 23:30 on Sat, Via dei Tavolini 19, Centro Storico, +39 055 239 8969, perchno.firenze.it
#### **Vivoli**
They say this is the oldest and most famous _gelateria_ in Florence, and by many accounts, also the best. This is a classic spot for some of the best flavors around, like tiramisu, melon, and Nutella.
From €2, Tues-Sat 07:30-24:00, Sun 09:00-21:00, Via dell'Isola delle Stinche 7, Centro Storico, +39 055 292 334
#### **Grom**
This quality chain is sweeping Italy by storm. Their flavors feature locally sourced ingredients and a menu that is constantly updated based on the season. While it's still a chain, the quality and value are right on par with the other independent options in town.
From €1.50, daily 10:30-24:00, corner of Via del Campanile and Via delle Oche, Centro Storico, +39 055 216 158
#### **Gelateria dei Neri**
Gelateria dei Neri is another heavyweight in the Florentine gelato scene. Their colorful piles of goodness are mouthwatering just to look at!
From €2, daily 10:00-24:00, Via dei Neri, 9/11, Centro Storico, +39 055 210 034
**FINDING GREAT GELATO**
Not all gelato is created the same! Here are my tips on what you have to look for when selecting your _gelateria:_
**Look for natural colors.** Check out the banana and pistachio flavors, and make sure they're the right color. What's the right color, you ask? The natural one, of course! For banana, that's a grayish white. Pistachio should be earthy green, not neon or lime green. If the banana flavor is bright yellow, keep looking.
**Walk away from artificial flavors.** Don't ask me why, but some _gelaterie_ stack their artificial flavoring in plain sight for all to see. If you see stacks of burlap sacks of artificial flavoring behind the counter, keep on walking!
**Say no to flair.** Touristy places attract customers by piling their gelato high and stacking it with all sorts of beautiful memorabilia. Although this is a treat for the eyes, keep on moving! The right temperature for gelato puts the mixture at a nearly liquid state that allows for the flavors to be that much more pronounced and vivid. If the bins are piled high, that means there's a chemical stabilizer in the gelato.
**Go for the metal.** Metal bins, which are reused, are a good indicator that the gelato they contain is made on-site. Plastic bins have likely been transported to the _gelateria_ from somewhere else, so gelato in your cone will not be as fresh. Also keep in mind that half-full bins are usually a good sign that the owners keep their batches fresh—but look closely for a layer of ice or hardened gelato crust, which indicates that the gelato is old and has been frozen for a while.
### **TOP NIGHTLIFE**
#### **NIGHTLIFE DISTRICTS**
##### **Centro Storico**
Thanks to Florence's compact size, you'll find a good number of nightlife venues popular with locals, students, and tourists within walking distance in the Centro Storico. As a general rule, prices will be higher the closer you are to the Duomo.
Centro
##### **Santa Croce**
The secret's out: Santa Croce is probably the best neighborhood for those who want to party hardy in Florence. Especially in the **Piazza Santa Croce,** you'll find endless options for food, _aperitivi_ , snacks, bars, lounges, clubs, and more. Many of my recommended venues are located in this district.
Santa Croce
##### **Piazza Santo Spirito**
Head to the square in front of the Santo Spirito church in the Oltrarno to see what the locals' style of nightlife is. Enjoy a couple beers on the steps, and join the tradition of the _passeggiata_ , or evening stroll, through town. Santo Spirito feels like it's the start and finish point for many on their walk, and you'll find everything from cheap sandwich bars to upscale restaurants. **Caibira,** at #4 on the square, is an ideal spot for an evening spritz, especially if you can sit out on their outdoor patio.
Oltrarno
#### **BARS**
##### **Hotel Cavour's Rooftop Bar**
Find this bar on the rooftop terrace of the palatial Hotel Cavour. The quiet and classy bar offers a rare vantage point from the center of Florence. Standing on the terrace, turn in a circle to see Santa Croce, Piazza della Repubblica, Bargello, Palazzo Vecchio, and one of the most spectacular views of the Duomo in the entire city. A drink up here, while a bit pricey, is a much more affordable option to experience the luxury of the hotel than splurging for a full night's stay.
Drinks from €10, Via del Proconsolo 3, Centro Storico, +39 055 266 271, albergocavour.it
##### **King Grizzly**
For the beer drinker lost in a world of Tuscan wines, King Grizzly is your refuge. A welcoming staff is happy to help you work through their extensive selection of hoppy remedies. Find some of the best local beers on tap, in town with a central location between Piazza Signoria and Piazza del Duomo. The dark wooden paneling inside and friendly, unpretentious crowd create a fun atmosphere and guarantee a great time, just about every time.
Beers from €4, daily 18:00-02:00, Piazza de Cimatori 5, Centro
##### **REX Caffe**
REX Caffe is one of the oldest jazz clubs in Tuscany. The intimate, standing-room-only space is a shade gaudy with a bohemian, bearded atmosphere. Consistent, top-quality live music and DJs spinning house keep the well-dressed crowd coming back for more of their fantastic cocktails.
**LGBT FLORENCE**
Italians—especially younger ones—are progressive and welcoming, and Tuscany was the first Italian state to pass legislation in 2004 protecting gays and rights for gay couples. Holding hands, sharing a room, or going out for food and drinks won't elicit a second glance, making Florence fun and enjoyable for everyone. **Piccolo Café** (Borgo Santa Croce 23r, +39 055 241 704) is a favorite for its welcoming atmosphere, friendly crowd, great music, and fun bartenders in downtown Florence. **YAG Bar** (Via dei Macci 8r, +39 055 246 9022) is another super-social choice, a mod and trendy café/bar popular among the young and expat crowd; it's a great place to kick off the night. For more information and tips, check out Florence's two resource offices during normal business hours: **Service Center for the Queer Community** (IREOS, Via de Serragli 3, ireos.org, +39 055 216 907) offers event schedules, maps, and information for the LGBT community. **Arcigay** (60 Via del Leone, +39 055 051 6574, Firenze@arcigay.it) is Italy's national gay association, which frequently sponsors events in Florence.
Drinks from €5, daily 18:00-03:00, Via Fiesolana 25, Santa Croce, +39 055 248 0331
##### **Rari Ristoro sull'Acqua**
The best things about Rari are the location and beautiful view across the river. While the service and food may leave a little to be desired, you really can't complain as you sip a spritz watching the sun set over the Arno. They also have frequent live music that justifies the 15-minute walk upriver.
_Aperitivi_ from €10, daily 09:30-01:00, Lungarno Francesco Ferrucci 24, Oltrarno, +39 055 680 979
##### **Naima**
My friend Sergio heads up this fun sports bar, which runs frequent student discounts and hosts game watches all the time. This is a solid spot to begin your night, and it's well located for nearby bars and clubs.
Beers from €2, daily 18:00-late, Via dell'Anguillara 54r, Centro Storico, +39 055 265 4098, facebook.com/naimaflorence
##### **The Lion's Fountain**
Every city in the world has to have an Irish pub, right? The Lion's Fountain is just that. Count on a fun, casual scene focused on beer and shots for the study-abroad population.
Drinks from €4, daily 10:00-03:00, Borgo degli Albizi 34, Santa Croce, +39 055 234 4412, thelionsfountain.com
##### **Kikuya**
This English-style pub is just steps away from Piazza Santa Croce, where the Santa Croce neighborhood's nightlife is concentrated. The relaxed atmosphere is great for a draft beer from the UK and a local soccer game on the TV.
Drinks from €5, daily 18:00-late, Via dei Benci 43, Santa Croce, +39 055 234 4879
##### **Red Garter**
Red Garter is a sports bar and steakhouse in the heart of Florence nightlife. It's probably the best place to catch an American football game during the fall. Many American students are sure to be trying their luck at beer pong or belting it out for karaoke inside the large bar rooms.
Drinks from €5, daily 16:00-04:00, Via de' Benci 33, Santa Croce, +39 055 248 0909
##### **Caffe Slowly**
This bar is famous for its _aperitivo_ bar and cocktail lounge. Slowly is a great place for a light dinner and drink before going out to a bar or club. It's just a couple blocks from Piazza della Repubblica; buy a drink and hit the buffet of appetizers and finger foods.
_Aperitivi_ from €7, daily _aperitivi_ 19:00-23:00, Via Porta Rossa 63, Centro Storico, +39 055 035 1335, slowlycafe.com
#### **CLUBS**
Florentines enjoy their nights out, and their home city offers plenty of options. Remember, Florence is stylish, and the selective bouncers don't hesitate to turn away would-be partiers on the basis of appearance or level of inebriation. Be on your best behavior while waiting in line.
**Heads-up:** In several clubs around town, you'll receive a punch card upon entry. This is the drink ticket you'll be racking up your tab on with each drink ordered. At the end of the night, turn in your card to settle up your bill based on the number of punches. Don't lose it, or you'll be charged for a full card (€100+!) before you're permitted to leave.
##### **Otel Varieté Restaurant**
Otel is just outside of the main city center but easily accessible by taxi. This club is the gold standard in Florence, flooded with the most stylish and high-profile locals. Dress your very best and arrive early to make sure you get in. Expect incredible music, dancing, drinks, and, very possibly, the best night of your life.
Check website for covers and events, Fri-Sun 20:00-04:00, Viale Generale Dalla Chiesa 9, Greater Florence, +39 055 650 791, otelvariete.com
##### **Bamboo Lounge**
Even if you're new to the club scene, Bamboo is a great place to get down to Top 40, hip-hop, and R&B. Dress to impress and be prepared to dance and drink the night away in this intimate venue popular with the younger crowd and students. Check out their Facebook page for theme nights and upcoming events.
€10 cover usually includes a drink, lounge, and _aperitivi_ Thurs-Sat 19:00-22:00, club Thurs-Sun 23:00-04:00, Via Giuseppe Verdi 57r, Santa Croce, +39 335 434 484, bambooloungeclub.com
##### **Space**
Space is a go-to nightclub for anyone visiting Florence. Packed out with both Americans and Italians, this large, double-level club boasts an awesome sound system and light show. Bring some extra cash for the cover charge.
Covers from €10, daily 22:00-04:00, Via Palazzuolo 37, Centro Storico, +39 055 293 082, spaceclubfirenze.com
##### **Y.A.B.**
Known as a glamour club in the heart of the city, "You Are Beautiful" is both loved and hated. Keep your drink card with you at all times to avoid the €60 exit fee. There's a €10 cover, but it includes a drink. If hip-hop and Top 40 are your thing, this is the place to go. Have fun but stick with your friends and keep an eye on your drinks to be safe.
€10 cover, Mon and Wed-Sat 19:00-04:00, Via de' Sassetti 5, Centro Storico, +39 055 215 160, yab.it
##### **Club TwentyOne**
Small and rowdy, Club TwentyOne gets straight down to it: good music, fun vibe, central location, and light-up disco floor. It's small, but the upside is that everyone in the house is basically forced to party together by virtue of proximity.
No cover on Wed, otherwise €5-10, shots from €3, Wed, Fri, Sat 23:30-04:00, Via Cimatori 13r, Centro Storico, +39 055 295 262, facebook.com/Club21Florence
#### **PUB CRAWLS**
Italians don't go drink to get drunk, so pub crawls don't score too high on the "local experience" scale, though they can be a fun way to meet fellow travelers in this compact town. Recently, cities like Florence and Rome have taken a hard line against the raucous mobile parties due to a string of fatal accidents, pushing crawls to change their style or go underground. The best option is to find out if your hostel organizes any nightlife outings, or check out my friends at **Tuscany on a Budget** tours and their evening walking tour of Florence with _aperitivi_ included (€22, Mon, Wed, Fri, italyonabudgettours.com).
### **TOP SHOPPING & MARKETS**
#### **SHOPPING DISTRICTS**
Florence has shopping options just about everywhere in the historic center. Whatever you're looking for, don't spring for the first one you find. Shop around to calibrate your pricing, and figure out what it's worth to you. When it comes to things like leather goods (which Florence is known for), never hesitate to haggle. Beginning to walk away will often illicit the best price you'll get from the shopkeeper, so don't be afraid to do just that.
##### **Piazza della Repubblica**
If Italy is the fashion center of the world, Florence could arguably be considered its capital, with many fashion houses based here or in Milan. You'll find all the high fashion brands in the streets surround Piazza della Repubblica, including Gucci, Fendi, Dolce & Gabbana, and Prada. If you're serious about picking up some new looks at any of these stores, dress the part to get attention from the often-snooty staff. This experience is definitely not budget friendly.
Centro
##### **Borgo degli Albizi**
I often find better prices for just as good of quality on this street leading east from the Centro Storico. Storekeepers are eager to rope you into their shops. Go willingly, but also don't be afraid to step on out and continue your mission to find the best prices and fit!
Centro
#### **MARKETS**
With its foundation as a trading city, I love exploring Florence's many markets. Observe your fellow shoppers to determine what kind of zone you're in. All touristy? Or are you lucky enough to have found a place where the locals shop?
##### **San Lorenzo Market & Mercato Centrale**
The daily open-air San Lorenzo Market is where you'll find rows upon rows of vendors selling leather goods, souvenirs, and typical food products of the region. Many of the stalls are parked in front of storefronts of the same brand all up and down the streets of Via dell'Airento, Via Sant'Antonino, and Via del Canto dei Nelli. Once you betray even a little interest, many salespeople will take you back into the store for the full selection. Always haggle your prices and do a couple laps to decide where you want to make your purchase.
San Lorenzo Market surrounds **Mercato Centrale,** the covered, permanent food market on Via dell'Airento and Via Sant'Antonino. You'll find food stalls and tons of local produce, including olive oil, meats, veggies, and _limoncello_ galore.
Free, Mon-Sat 09:00-18:00, Via dell'Ariento, next to the San Lorenzo church, Centro
##### **Mercato di Sant'Ambrogio**
Just a 15-minute walk east of the Duomo, the Sant'Ambrogio market feels more distinctive and authentic than the other tourist-flooded markets in town. Here you'll find a market both covered and uncovered, permanent and less-permanent, with stalls selling everything from fresh produce and meats to clothes and shoes. If you're here around lunchtime, seek out a spot that looks like they get their food from the market.
Free, Mon-Sat 07:00-14:00, Piazza Ghiberti, Santa Croce, mercatosantambrogio.it
##### **Mercato del Porcellino (aka Mercato Nuovo)**
Packed with touristy knickknacks, this is a great place to find leather purses, colorful scarves, and the like. You might find better prices elsewhere, but this market's central location makes for a convenient stop between major sights.
Besides the shops there are two minor sights in the Mercato Nuovo. The circular **Stone of Scandal** is where poor saps who ran up tabs they couldn't pay had to sit naked while being beaten with stones. Pay those bills! Don't miss the fabled **bronze boar** either, just on the south side of the portico. As legend has it, if you take a coin, put it on the boar's tongue, release it, and it falls into the grate below, you'll have good luck and a guaranteed return visit to Florence. Well worth a dime, I'd say!
Free, daily 09:00-18:30, Piazza del Mercato Nuovo, Centro
### **TOP TOURS**
#### **Florence Free Tour**
Florence Free Tour leads free daily walks around town. They're a fun and informative way to get oriented with the city and its long history. The morning walk focuses on the Renaissance history of the city; the daily afternoon walk digs into the intrigue and secrets of the family that kicked it all off, the Medicis. For the full introduction, you can fit both two-hour tours in one historical day by doubling back to the starting point by 14:00.
Free, tip-based, Renaissance Tour 11:00, Medici Tour 14:00, both tours start in front of Santa Maria Novella church (near the train station), Centro Storico, florencefreetour.com, info@florencefreetour.com
#### **Tuscany on a Budget Day Tours**
My friends at Tuscany on a Budget are the answer for those who can't quite afford private drivers and crates of limited production vintage. ToaB has a lineup of fun experiences to get you out of the city and into the countryside, including day trips for wine tasting, jaunts out to the Cinque Terre and Pisa, food tours hosted at typical farm houses, and more. They've also got longer weekend excursions for students and budget travelers to destinations like Naples and the Amalfi Coast. Check out their website for all details and trip information.
Experiences starting from €22, Via Nazionale 149R, Centro Storico, +39 055 05 03517, tuscanyonabudget.com, info@italyonabudgettours.com
#### **Tuscany Cycle**
Tuscany Cycle puts on fun, casual rides in the countryside after a short car ride out from their office in downtown Florence. Bilingual tour guide, bike rental, lunch, wine tasting, and round-trip transportation are included. Extend your bike rental for a discounted rate after the trip if you want.
Day tours from €75, performance road bike rentals from €35/day, Via Ghibellina 133r, Centro Storico, +39 328 071 4849 or +39 055 289 681, tuscanycycle.com, tuscanycycle@gmail.com
### **TOP HOSTELS**
Hostels in Florence are improving. You used to have extremely limited budget options unless you wanted to crash in a convent with a curfew and a long list of house rules—these are still available by the way, and probably the only €10 bed in town (sanctuarybbfirenze.com). Still, the number of good hostels remains limited. In addition to my recommendations, find more listings on **hostelworld.com.** The private apartments available on **airbnb.com** for short-term stays provide another excellent budget solution.
#### **Plus Florence Hostel**
Located just down the street from the main train station, Plus Florence is my favorite in town and offers comfortable dorm and private rooms and a rooftop terrace. It's a large hostel that comes with the feel of a large hostel (brisk, efficient check-in, mediocre opt-in breakfast, and six floors of dorms), but it manages to maintain a fun social atmosphere thanks to the common areas and the on-site bar downstairs. Although they advertise a pool, it's probably closed during your visit.
Beds from €15, free Wi-Fi, breakfast optional, 24-hour reception, free lockers, Via Santa Caterina D'Alessandria 15, north of Stazione Santa Maria Novella, Greater Florence, +39 055 628 6347, plushostels.com
#### **Academy Hostel**
This hostel tries to be boutique, but don't come in with overbearing expectations. The dorm rooms are converted large rooms; temporary dividers between beds provide some privacy. The location—right between the Duomo and the Accademia, meaning it's stumbling distance from the bars—couldn't be better. The helpful staff is happy to make recommendations for sights and activities during your stay.
Dorms from €15, free Wi-Fi, free lockers, breakfast optional, 24-hour reception, Via Ricasoli 9, Centro Storico, +39 055 265 4581, academyhostel.eu
#### **Hotel Il Bargellino**
I find this spot, run by an expat Bostonian and her husband, a quiet respite from the bustle of the city. Rooms are simple and comfortable. The rooftop terrace is a major plus for afternoon drinks, picnics, and conversation.
Private rooms from €45, free Wi-Fi, no breakfast, Via Guelfa 87, near Stazione Santa Maria Novella, Centro Storico, +39 055 238 2658, ilbargellino.com, carmel@ilbargellino.com
### **TRANSPORTATION**
#### **GETTING THERE & AWAY**
##### **Plane**
Flight prices should help you make the decision about which of the six airports to fly into the region.
It's most convenient to fly into Florence's relatively small **Peretola** (FLR, aeroporto.firenze.it) airport, just a 20-minute shuttle bus from the city center, running every half hour. Look for the **Vola in Bus** shuttle, and purchase your ticket on board from the driver (€6 each way to connect into the city center main train station).
It's a one-hour commute from Pisa's **Galileo Galilei Airport** (PSA, pisa-airport.com), the largest airport in Tuscany, to Florence. Because of gnarly traffic on the highways to Florence, it's better to get there via train from Pisa's central station. The airport and station are connected by the **PisaMover,** a free automated connection open since December of 2015 that runs every 10 minutes. Pick up your train ticket (€10) at the Pisa train station. Departures run often.
Budget airlines also use the **Bologna** (BLQ, bologna-air.it) airport. The **Appenino Shuttle Bus** service connects to Florence's main train station for €20. Follow signs at the airport and catch one of the frequent departures (takes about 1.5 hours).
It's also possible to fly into one of Rome's airports, **Da Vinci-Fiumicino** (FCO, adr.it) and **Ciampino** (CIA, adr.it), or into Milan's **Malpensa** (MXP, milanomalpensa-airport.com) and take a train into Florence (1.5+ hours, tickets €30+ each way).
##### **Train**
The main train station, **Santa Maria Novella** , is on the western edge of the historic center. From this station, it's about a 15-minute walk to each of my recommended accommodations. Train tickets from Rome run about €30 each way and the ride can take about 1.5 hours. To Milan, it takes two hours on the fast train and costs €40. Trains to Venice run often, also cost around €40, and take about three hours.
**CALCIO STORICO**
Roughly translating to the "old kicking game" or "historical kick," _calcio storico_ is a game that was first created to keep Roman soldiers in fighting shape. Today, it's played on a rectangular, sandy field in front of the Santa Croce church. Fifty-four players line up and enter into what looks like mortal combat for 50 grueling minutes. In any given game, 10-15 players are carried off in stretchers from the brawl without even a pause in the action, and teams finish the game with no replacements. With no replacements, players are incentivized to try to injure or otherwise incapacitate each other with kicks and flurries of punches and swings, and moving the ball across the field often appears to be an afterthought.
With elements of boxing, rugby, soccer, and American football combined into one open-field bare-fisted gang fight, _calcio storico_ seems a touch gruesome and medieval—barbaric even. The fact that there's a ball involved, and that the players are trying to get it into the net on either side of the field, is easily lost amid the violence. The players identify by colors and come from four districts of the city; they practice all year for their annual competition on June 24. Enjoy the fireworks from the Piazzale Michelangelo following the finals.
When booking your tickets, be aware of the difference between the slow and local trains vs the faster regional trains—it can make the difference between a one- or four-hour ride to the same location. Also, if you're traveling with an open ticket (without a specific time and date of validity stamped on it) you must validate it before departure or immediately find a conductor to validate it. Nonvalidated tickets can result in heavy fines or even being thrown off of a train.
##### **Bus**
**Eurolines.com** offers your best options for international bus connections into Florence. Connections from cities like Prague, Paris, and Barcelona push 15 hours each way, but at €50-100, they may be your cheapest options. Arrivals will drop you off at Florence's main train station, **Santa Maria Novella.**
##### **Car**
Florence is about three hours north of Rome via the A1 highway.
#### **GETTING AROUND**
Florence is a pedestrian city; just about all sights can be enjoyed on foot thanks to the compact size of the town. Buses are the public transit option of choice, as there is no metro and tram lines are limited.
##### **Bus**
**Santa Maria Novella** train station is also Florence's main bus station, and is the hub of both local and regional bus connections. Validate your **ticket** (€1.20) once you step on the bus to avoid hefty fines 200 times the cost of a normal ticket. Remember, playing the "stupid tourist" card just doesn't work here anymore! It is possible to purchase tickets on board, but they often run out. Your best bet is to purchase tickets before boarding at any _tabacchi_ (tobacco store) in town.
##### **Car**
Within Florence, parking can be quite difficult, and there are strict traffic restrictions. Watch for the **Zona a Traffico Limitato** sign, which you'll see throughout central Florence, meaning that traffic is restricted to buses, taxis, and local traffic only. If you cross this sign, an automatic camera will snap a picture of your license plate and you'll be sent a nasty fine.
Renting a car does open up the countryside to you, and may be a good option if you want to spend time in the Tuscan region outside of Florence. A GPS system is a must to navigate the confusing terrain outside of Florence.
##### **Taxi**
You cannot wave down taxis in Florence. Instead, catch the registered white cabs at designated taxi stands. Taxis are normally parked outside of the Santa Maria Novella train station, Piazza Santa Croce, Piazza del Duomo, and Piazza della Repubblica. Once in the car, take a look over the rate information card and make sure the meter is set correctly. Daytime rates begin at €3.20, and evenings start at twice that (€6.40). Women traveling alone can ask for a 10 percent discount between 21:00 and 02:00, so ask for it! Call taxis at +39 055 4242, +39 055 4390, or +39 055 4499.
##### **Bicycle**
Cycling is a great way to see more of Florence. Make sure you lock your bike up when parking, inside or out. Also, keep your eyes on the road and pathways, because pedestrians, bikers, cars, horses, and potholes are just about everywhere. It is particularly beautiful and easy to ride up and down the river on bike paths. Rent bikes from tour company **Tuscany Cycle** (Via Ghibellina 133r, from €35/day) in the city center.
### **DAY TRIPS**
#### **Fiesole**
Only a half hour north of Florence via public bus is Fiesole. This small town perched on a hill is your best shot at experiencing true Tuscany without the long and involved day trip to the farther out hill towns. (You can actually see Fiesole from high viewpoints in Florence like Piazzale Michelangelo—it's the town far off in the distance, on the opposite side of the valley up on the hill.) If you've got four days in town, spend the fourth visiting Fiesole to get away from the crowds of Florence. The main draw of this town is to experience a quieter Tuscan town without the hordes of tourists. Explore the quaint little square, and don't miss the Roman amphitheater and relatively minor **Ancient Roman ruins archeological site** (€7, generally daily 10:00-17:00, check website for current hours, +39 055 59118, museidifiesole.it) just behind the **Cathedral of Fiesole.** In summer, you'll also find a weekly **Sunday market** selling souvenirs, handmade goods, and local produce.
Take in the panoramic views of Florence and the surrounding valley, and enjoy a meal at my favorite restaurant in town, **Ristorante La Reggia** (Via S. Francesco 18, +39 055 59 385, lareggiadeglietruschi.com).
To get to Fiesole, take bus 7 from Piazza San Marco to the last stop at the top of the hill (about 30 minutes, €2 each way).
#### **Tuscan Hill Towns**
Florence is in the heart of Tuscany, a region famous for beautiful, isolated hilltop towns with medieval foundations. They sprang up along Roman trade routes and persevered through the challenging medieval ages. Today, these hill towns transport visitors back in time as they wander the cobblestoned streets, taste the local products, and chase their romantic dreams. It's hard to choose between them, but none will disappoint. Check out **Tuscany on a Budget Tours** (+39 055 0503 517, tuscanyonabudget.com) for their lineup of day trips to these towns just a short drive away. Siena is one popular option; it's just a couple of hours away by train.
### **HELP!**
#### **Tourist Information Centers**
Get information online at firenzeturismo.it, or pick up maps and information on sights at Florence's two TI locations (open Mon-Sat 09:00-19:00, Sun 09:00-14:00). The first is near the train station, the second near the Duomo.
Piazza Stazione 4
+39 055 212 245
Piazza del Duomo (west corner of Via Calzaiuoli, inside the Bigallo Museum)
+39 055 288 496
#### **Pickpockets & Scams**
Though you'll find many beggars in Florence, pickpocketing isn't as aggressive as in other cities like Rome and Paris. That said, never leave your drink or belongings unattended, especially while out at the bars and clubs. Always keep your wits about you and keep your valuables locked at the hostel.
#### **Emergencies**
Dial 113 for English-speaking police, or 118 for an ambulance.
#### **Hospital**
Ospedale di S. Maria Nuova
Piazza S. Maria Nuova 1
+39 055 69 381
#### **US Consulate General**
38 Lung'arno Vespucci
+39 055 266 951
Venice Map
Venice 101
Three Day Itinerary
Top Neighborhoods
Top Sights
Top Eats
Top Nightlife
Top Shopping & Markets
Top Parks & Recreation
Top Tours
Top Hostels
Transportation
Help!
Known as La Serenissima—the most serene—Venice is famously sinking into elegant decay. Glide through the lagoon on a gondola, make your pilgrimage to St Mark's Basilica, catch a glass-blowing demonstration, and sample delicious seafood and delectable _cichetti_ (tapas-like snacks). But also take time to wander its backstreets, alleyways, and magical canals. In Venice, getting lost is the dreamiest part of the experience.
### **VENICE 101**
Ingenuity, trade, and plundering hordes all contributed to the creation of one of the world's most distinctive cities: Venice. The name of the city derives from the fishing tribes that lived on the shores of the marshy Venetian lagoon. With the decline of the Roman Empire, the inhabitants of the surrounding area sought shelter in the difficult-to-navigate estuaries, creating small man-made islands for refuge. Venetians began pounding wooden pilings into the ground, reclaiming land from the sea inch by inch to expand their living space. They traded fish for basic needs like wood, wool, and grains while staying out of the way of the barbarian tribes roaming the Italian peninsula.
Before long, Venice grew into a powerhouse of trans-Mediterranean trade, sparking a golden period that lasted for nearly 1,000 years. Venice thrived as a hub of commerce, riches, and excess while the rest of Europe struggled through the Dark Ages. Immense treasures flowed through this port connecting the East to West, and the influence in its architecture is evident in the unique "Venetian gothic" style, a hybrid of the golden domes and decoration from the East with the gothic arches and spires of Western Europe.
By AD 700, the leaders of Venice recognized that internal threats were just as much a hazard to the city as external ones. To protect against corruption and violent interfamilial feuds, Venice developed a process of electing their leader, or doge, so complex it just about guaranteed against the possibility of purchasing that seat of power. With this stable process of non-hereditary succession in place, Venice continued pursuing riches through expansion of trade routes.
In 828, Venetian merchants made off with the bones of St Mark from the coastal Egyptian city of Alexandria and brought them back to Venice. With these religious relics under "protection," Venice became not only an important economic destination but also a religious one. In fact, the doges decreed that it was every Venetian's obligation to go out into the world and bring back treasures to beautify their hometown. Venice was the kick-off point for the Crusades, and also the first port of entry on their return, and the city benefitted from the riches attained by the armies passing through.
When Columbus put the New World on the map, attention gradually turned from East to West, and Venice slid into decline after being battered by the plague. Contemporaries considered this to be God's punishment for Venice's excesses, having grown fat and lazy from generations of success. The final debilitating straw was Napoleon's focus on quashing the island empire once and for all to take its riches and to eliminate it as a threat to his power. Today, tourism is the city's strongest trade, with visitors flocking to take in the elegant buildings and architecture left over from Venice's heyday.
### **PLAN AHEAD**
#### **RESERVATIONS**
In such a compact city, real estate is limited and so are hotel rooms. Make hostel reservations well ahead of time to avoid exorbitant prices or no vacancy upon arrival.
#### **PASSES**
##### **Venezia Museum Pass**
The **Venezia Museum Pass** (€24 adults, €18 students) offers entry into 11 city museums, including the Doge's Palace, Correr Museum, Ca' Rezzonico, the Glass Museum on Murano, and the Lace Museum on Burano. With this pass, you can skip lines, but it's worthwhile only if you plan on visiting most of the museums included. Find more information at visitmuve.it; click into the sight that you're most interested in.
#### **LOCAL HAPPENINGS**
##### **Carnevale**
Carnevale is Venice's Mardi Gras, happening annually during the two weeks leading up to Ash Wednesday, the start of Lent. The festivities are packed with people dressed in ornate, traditional costumes from throughout the ages. There are frequent parades and marches through St Mark's Square, so you'll have to jockey with the crowd to get a good spot for viewing. The second weekend of the party is the climax of the multiday string of events. Remember: Early February is cold and damp in Venice. Bring waterproof boots and wool socks.
**KNOW BEFORE YOU GO**
**KEY STATS & FIGURES**
**Currency:**
**Italy uses the euro (€); 1 EUR = about 1.06 USD**
**Population:**
**60,000 on the main islands of Venice**
**Language:**
**Venetian dialect of Italian**
**Bridges in town:**
**409**
**Canals in the city:**
**177**
**Number of islands:**
**117**
**Tourists per year:**
**16.5 million**
**Number of _gondolieri_ licenses:**
**425**
**Female _gondolieri:_**
**1**
**Typical drink:**
**the spritz (Prosecco, soda, and Aperol or Campari)**
**CALIBRATE YOUR BUDGET**
**TYPICAL PRICES FOR:**
**Hostel dorm bed:**
**€24**
**Two-course dinner:**
**€15**
**Pint of beer:**
**€7**
**Bicycle rental:**
**Fuggedaboutit**
**Single _vaporetto_ ticket:**
**€7**
**_Traghetto_ ride:**
**€0.50!**
**MOVIES TO WATCH**
_The Tourist, The Italian Job, Summertime_
**THREE DAY ITINERARY**
The #1 thing to do in Venice is wander...and wander...and wander some more. If you feel lost, take a deep breath and relax, because you're probably never more than half a mile from where you started. Don't cross that long bridge back onto the mainland and you'll be just fine. Beyond the city, which makes up one of the world's best outdoor museums itself, Venice has dozens of world-class museums and sights to check out. You'll hit the highlights with this itinerary.
**DAY 1: WELCOME TO VENICE**
**MORNING**
Take your first plunge into **St Mark's Square** and steel yourself for your first Venetian breakfast—take it standing at the classic **I Quadri**. Don't sit unless you want to drop €20 on an espresso!
After breakfast, stand in front of the flag poles in front of **St Mark's Basilica.** In a single clockwise-turning panorama, you can see the **Doge's Palace** ; the island of **San Giorgio Maggiore** across the canal, through the famous **Columns of Venice** (with both the winged Lion of St Mark and Venice's original patron saint, St Theodore, on top), and the Venetian Lagoon behind; the **Campanile** ; thousands of pigeons; some of the most expensive cafés you've ever encountered (which have live music after sunset); and the famous **Torre dell'Orologio** , a clock tower with one of the world's oldest digital faces. If the line is short at St Mark's, pop in! If the line is long, and you have a backpack with you, find the free bag check in the alleyway leading away from the square on the left side of the basilica and you'll get a skip-the-line pass at no cost. This is also your chance to climb the **Campanile** (bell tower).
**AFTERNOON**
Head to the streets a couple blocks behind the Doge's Palace to find **Panini Row,** and pick a spot for lunch. Then double back for a free **glassblowing demonstration** at **Galleria San Marco** (181A San Marco, on the main square). They'll usually have a sales rep standing in the square just in front of the archway between Café Lavena and the clock tower. Ask him if any demonstrations are happening soon, and you'll be escorted through an alleyway and up the stairs into the shop. If you time your visit with a well-heeled tour group, you'll be treated to an extended demonstration (and aggressive sales pitch).
Follow signs toward Ponte Rialto and wander toward the **Rialto Bridge,** enjoying the shops and bustle along the way.
**EVENING**
The neighborhood just beyond the Rialto Bridge is a great place to begin a _cichetti_ crawl, snacking at the various bars all along the way. For the price of a drink, avail yourself of a delightful Venetian tapas tradition, sampling some fresh harvest of the sea. Stay on the north side of the bridge to see what the locals are getting into around **Campo Cesare Battisti**.
**DAY 2: THE DOGE'S VENICE**
**MORNING**
If you're an early riser, explore and get lost in another district of Venice. Off the main track, Cannaregio has some excellent hole-in-the-wall cafés to start your day, my favorite being the popular **Café Filermo**. Then catch your reservation at the **Doge's Palace** , and don't miss the **Bridge of Sighs**!
**AFTERNOON**
Hop over to the **San Giorgio Maggiore Island** on _vaporetto 2_ and climb the bell tower there for a magnificent view of the entire lagoon. Heads-up: It gets loud at the top of each hour!
Head back to the San Marco neighborhood and enjoy a **gondola ride** with budget traveler friends before 19:00 to avoid the price hike. Pick an area of town you like first, then find a gondola in that area, because your tour is a 40-minute loop of the nearby canals. I like the area northwest of St Mark's Square, bordering the Grand Canal. If you're trying to save some cash, hop on a _traghetto_ instead for only €0.50!
**EVENING**
Catch dinner and spend a night out in **Campo Santa Margherita** enjoying the selection of restaurants and bars.
**DAY 3: CIAO VENEZIA, CIAO, CIAO!**
**MORNING**
There are still plenty of things to see if you've got the time, but you'll have to prioritize! If you need some more art and history, the **Accademia** and **Peggy Guggenheim Venice** museums fit the bill. Endless shopping awaits you on the **Mercerie.** Beyond Venice, you've got the sleepy cemetery island, **San Michele** ; **Murano** , the glassblowers' island; and **Burano** , the quieter lace-making island with cute little pastel houses to consider.
**AFTERNOON**
Don't dilly-dally if you've got a train or flight later in the day. It's easy to be delayed on your way out by getting lost or missing a _vaporetto_ departure. Give yourself some extra time just to be safe. Happy travels!
### **TOP NEIGHBORHOODS**
The main island of Venice is shaped like a large fish. It comprises six districts, but learning the major landmarks of the city is the best way to orient yourself. Most of the major sights are located toward the middle and south side of the fish shape, with the main train station, bus terminal, and cruise ports in the northwest.
**San Marco** is the bull's-eye for most tourists, making it the most touristy district in an extremely touristy town. This is where you'll find the Campanile, St Mark's Basilica, the Doge's Palace, Correr Museum, and many popular shops. A footbridge crosses from San Marco south to the district of **Dorsoduro,** where you'll find the Accademia and the Peggy Guggenheim Venice museum. Smack dab in the middle of the main canal is the famous Rialto Bridge. In the surrounding neighborhood, also known as **Rialto,** you'll find a number of recommended restaurants as well as the famous fish market.
A few districts offer nice opportunities to escape the tourist mobs. **Cannaregio** is where most Venetians live, and also the ghetto to which Jews were once restricted. Going deep into this district yields off-the-beaten-path cafés and local hangouts. **Castello,** home to shipbuilding yards and the massive arsenal complex, was once Europe's biggest ship building operation. Today, it's a quiet residential district with plenty of streets slightly larger than what you'll find in San Marco and a number of churches to explore. **Giudecca,** across the Grand Canal and south of Dorsoduro, is quiet and residential. It's a nice place to escape the crowds in the main island. It's also where my favorite hostel in town is located.
Outside the main island of Venice, you have the islands of **Murano,** famous for glassblowing, and **Burano,** known for it's lace-making. Both offer a close-up view of their respective historical industries. The walled **San Michele Island** is a beautiful cemetery island, where people have been buried since the early 19th century. **Lido** is the long and skinny island with beaches, cars, and larger resort-like hotels. **San Giorgio Maggiore Island** offers the church bell tower to climb for beautiful panoramic views looking back onto St Mark's Square.
### **TOP SIGHTS**
#### **St Mark's Square**
One of the most famous squares in all of Europe, Piazza San Marco embodies the image of Venetian elegance that comes to mind when daydreaming about this remarkable city. Originally set out in the 800s, and built up in the 12th century, this square is named after the patron saint of Venice, Saint Mark, and serves as the city's beating heart. Today, it's the epicenter of Venice's Disneyland-style experience. Lined with Venetian gothic colonnades and enlivened by live music and hundreds of pigeons, it's really quite idyllic on a nice day.
The square also contains some of Venice's top attractions, including the **St Mark's Basilica,** the **Campanile,** and the **Doge's Palace.** It also contains the **Torre dell'Orologio,** a clock tower that caps the high arched gateway leading to Venice's merchant district, the Mercerie. Dating back to the 15th century, this tower is loaded with Venetian symbolism (notice the winged lion of St Mark at the top). It also displays one of the world's oldest digital clocks clicking away on five-minute intervals, and tolls out the hours with two bell-hammering figures.
Free, always open, San Marco
#### **St Mark's Basilica**
By the 9th century, Venice had become a regional power but was lacking a spiritual connection. Venetians believed they had a claim on St Mark because of a local legend. The story goes that St Mark was blown off course into the Venetian lagoon and had a vision in which an angel told him he would be laid to rest there. In the medieval world, relics brought substantial revenue from visiting pilgrims, and Venetians were all about getting that business. So in 828, two Venetian merchants snuck into a church in Alexandria, Egypt, stole St Mark's bones, hid them under cuts of ham and pork to deter the Muslim guards, and smuggled them back to Venice.
The basilica we see today was completed in the 11th century after only 30 years and dedicated to St Mark. **Basilica di San Marco** is one of the most richly decorated churches in Europe, with strong Byzantine influences clearly visible in the domes, decorated spires, and ornate, pointed arches. The golden mosaics on the facade cannot be missed and depict the return of the relics of St Mark into the Venetian Lagoon. The inside holds over 4,000 square meters of intricately detailed mosaics. In a nod to the power of the doge, there are two pulpits inside the basilica: one for the priest and the other for public announcements from the doge.
After the basilica was built, the winged lion of St Mark became the symbol of Venice. Also at this time, the doge declared it the duty of all Venetians to capture and bring back treasures from overseas. Before long, Basilica di San Marco was one massive robber's den, with huge quantities of gold and jewels deposited into the treasury inside the church. Even the equestrian statues in the facade of the church over the entrance (which are now replicas) were stolen from modern-day Istanbul. While the original structure of the basilica hasn't been altered much, adornment with newly attained treasures went on for hundreds of years.
Free general entry (€2 online reservation), €5 to go upstairs to see the horses up close, €3 to go into the treasury (both recommended!), general entry Mon-Sat 09:45-17:00, Sun 14:00-17:00 (until 16:00 Nov-Easter), St Mark's Square, San Marco, group reservations +39 041 241 3817, basilicasanmarco.it
#### **Doge's Palace**
This palace-turned-museum is an unforgettable example of Venice's typical marriage of Eastern and Western architecture. In a time when everyone else hid out in cold, dark castles, Venice's doge and government met and lived in a grand structure that was continuously rebuilt and added upon, even after numerous fires. The first palace was built in 810, and each new version became the center of political life in Venice. The Palazzo Ducale turned into a state-run museum in 1923 and is packed with Renaissance artwork and furniture. Your visit will take you through the institutional chambers, the royal apartments, an old, dank armory, and once-bug-infested prisons where troublemakers were kept. While touring these vast halls, keep an eye out for the little mailboxes decorated with scary faces. These were slots in which you could place letters accusing anyone of foul play or corruption. Holy Mother Mary is depicted in paintings in the palace galleries. Her presence is a nod to the power of the doge and his near-god-like influence upon the 2,500 representatives of Venice.
Crossing from the palace's chambers into the prison, you'll traverse the **Bridge of Sighs** , so named because it was the last glimpse of the beautiful canals that the imprisoned would have for quite some time. For a good view of this enclosed bridge, find the pedestrian bridge on the first bridge past it on the lagoon side of the palace.
The Doge's Palace ticket also covers the Correr Museum. If you plan to visit both sights, buy your ticket at the less crowded Correr Museum, so you can skip the line at Doge's Palace.
€18 adults, €11 students, ticket also covers entry to Correr Museum, Museo Archeologico Nazionale, and Sale Monumentali della Biblioteca Nazionale Marciana, also covered by Venezia Museum Pass, Apr-Oct daily 08:30-19:00, Nov-Mar daily 08:30-17:30, last entries an hour before closing, closed on Christmas and New Year's Day, St Mark's Square, San Marco, +39 041 271 5911, palazzoducale.visitmuve.it
#### **Correr Museum**
I've always found the Correr Museum to be one of the most interesting in town. Bequeathed by a wealthy collector, the art went on display in 1836 and features daily life and culture of Venice in the 16th and 17th centuries. Find the works of famous Venetian artists, painters, and sculptors from its earliest days through the golden period of trade and wealth. Enter in the passageway on the far end of St Mark's Square and turn right up the grand staircase. Plan to spend about two hours in this museum, which spans a floor the length of St Mark's Square.
€18 adults, €11 students, ticket also covers entry to Doge's Palace, Museo Archeologico Nazionale, and Sale Monumentali della Biblioteca Nazionale Marciana, also covered by Venezia Museum Pass, daily Apr-Oct 10:00-19:00, Nov-Mar 10:00-17:00, last entry one hour before closing, St Mark's Square, San Marco, +39 041 240 5211, correr.visitmuve.it
#### **The Campanile**
The Campanile is Venice's bell tower, located in St Mark's Square. It's the tallest building in town, and you can ride an elevator to its top. Naturally, it provides the best panoramic views of the city—and even a view of the Alps far off in the distance on clear days—from 91 meters high. Originally built as a lighthouse to assist navigation through the lagoon, it achieved the look we see today in 1514. After a dramatic collapse in 1902 (where, fortunately, no one was hurt), it was rebuilt in time for the 1,000-year anniversary of the laying of the original tower's foundations. To avoid the crowds, visit as early as possible. (And remember that San Giorgio Maggiore Island, just across the canal, provides a much less crowded panorama option.)
€8, daily Easter-June and Oct 09:00-19:00, July-Sept 09:00-21:00, Nov-Easter 9:30-15:45, St Mark's Square, San Marco
#### **Columns of Venice**
Without GPS, the first glimpse of home must have been a welcome sight for sailors returning from long voyages. As such, the Campanile and the two Columns of Venice bore significant sentimental value to Venetian merchants. Both monolithic marble columns stand more than 40 feet tall, with a Doric cap. **St Theodore,** the original patron saint of Venice and a Roman soldier who refused to conform to pagan religions, crowns one column. With his Byzantine origins, St Theodore became politically inconvenient, and Venetians eventually ousted him in favor of **St Mark,** whose symbol of the winged bronze lion with a paw resting on a book stands proudly on the second column.
Free, always open, St Mark's Square, San Marco
**SO YOU WANT TO BE A DOGE...**
Being a doge—the ruler of medieval Venice—was a big deal, and with it came power and riches. So Venice wanted to ensure that the right people were elected in the right way. Here are the 10 easy steps to the process of selecting a doge in the 1200s.
**1.** A _ballottino,_ a boy chosen at random, draws 30 names by plucking balls out of a vase, beginning the entire selection process by chance.
**2.** From the group of 30, 9 are randomly chosen.
**3.** These 9 vote up 40 names, each of which needs at least 7 of the 9 possible votes to be considered.
**4.** The 40 are then cut down to 12 by random draw.
**5.** Those 12 vote up another 25.
**6.** Those 25 are reduced to 9 again.
**7.** Those 9 choose 45, each of which needs at least 7 votes once more.
**8.** From those 45, 11 are then randomly selected.
**9.** Those 11 choose 41, who must not have been included in any of the reduced groups that named candidates in earlier steps.
**10.** Those 41 then choose the doge by a vote.
The doge was carefully monitored by the nobles and not allowed to speak to foreign emissaries without direct supervision. All letters and communications coming in and out of the doge's palace were monitored, and appropriate gifts for the doge were limited to items of relatively low value, like flowers and herbs. All these precautions, while they may seem overbearing, went a long way toward minimizing corruption at the pinnacle of Venice's power, allowing the city-state to spend its energy expanding trade routes and the treasury rather than funneling it off to line the pockets of a few in power.
#### **Glassblowing Demonstration**
Due to fire hazard, glass factories are no longer permitted on the main island, but you can catch a professional glassblowing demonstration at **Galleria San Marco.** The welcoming team ushers you through a back alley just off St Mark's Square on the north side (just to the left of the clock tower), up the stairs, and directly into the workshop. There, an artisan grabs a slag of molten glass, then goes through a series of puffs, pinches, reheating, rolling, kneading, and adding other pieces of glass for color and texture, until there's a little vase or tiny horse statue. An English-speaking narrator explains the process as it happens before your eyes. Your 20-minute demonstration ends with your glassblower tossing the just-finished glass piece back into the oven to recycle for the next show, then an enthusiastic 15-minute sales pitch. Take another 10 minutes to shop around, and make use of the free restrooms. See if you can't pair up with a well-heeled tour group—you'll piggyback onto the extended presentation and showcasing of more expensive pieces.
Free, souvenirs from €20, daily 10:00-17:00, demonstrations running whenever there's an audience, 81A San Marco, San Marco, +39 041 277 0365
#### **Grand Canal**
If the canals of Venice are its arteries, the Grand Canal is Venice's aorta. All of Venetian life passes through this man-made waterway, through the dozens of islands that make up this city. You'll see everything pass through here that you'd see on a highway on land, but here, they're boats: taxis, buses, ambulances, private boats, construction boats, grocery delivery services, police and firefighting, and more—all aquatic versions of their four-wheeled, landlocked counterparts. This waterway slices the city into almost two equal halves, meaning that sooner or later, all visitors will cross its path.
#### **Ca' Rezzonico**
This collection of furniture and artwork focusing on Venice in the 1700s is housed in a merchant family's elaborate palace, built in Venetian baroque style with large windows and ornate balconies spread across three glamorous floors. The exterior's elegance matches that of the art on inside. Find masterpieces of Venetian baroque artwork, including pieces by Canaletto, Longhi, and Tiepolo.
€10, covered by Venezia Museum Pass, Wed-Mon 10:00-18:00, Nov-March until 17:00, closed Tues, ticket office closes one hour before museum, Fondamenta Rezzonico, Dorsoduro, +39 041 241 0100, carezzonico.visitmuve.it
#### **Accademia**
Venice's most famous art museum, the Galleria dell'Accademia, features a wide range of pieces from 1300 through the 1700s. You'll find pieces from greats like Tintoretto, Titian, Bellini, and Veronese. My favorite highlight is da Vinci's _Vitruvian Man_ —that's the guy in a circle with his arms and legs outstretched (also called _Perfect Man_ ) that we see on T-shirts and TV shows about da Vinci.
€9, free first Sun of the month, Tues-Sun 08:15-19:15, Mon 08:15-14:00, last entry 45 minutes before closing, Campo della Carita, Dorsoduro, +39 041 520 0345, gallerieaccademia.org
#### **Rialto Bridge**
The Rialto Bridge (Ponte Rialto) is the grand, gleaming white stone bridge you see in all the pictures of Venice. It's possibly the most famous bridge in the world. From the time of its construction in 1181 until the 1800s, when two more bridges were built, it was the only permanent bridge crossing the Grand Canal. If you remember that bridges are the only way for pedestrians to cross a river without a boat, it makes sense that the Rialto district became the economic and commercial heart of the city. The stone version we see today was completed in 1551. It's lined with shops where you can pick up touristy knickknacks, watches, and jewelry.
Free, always open, Rialto
#### **Peggy Guggenheim Venice**
Housed in the Venetian palace that belonged to American heiress Peggy Guggenheim, this collection makes up Venice's premier modern art museum, housing an array of American and Italian pieces from the 20th century. You've got all the funky styles here, including futurism, cubism, and surrealism, and pieces by artists ranging from Picasso to Dalí and Pollock.
€16.50, Wed-Mon 10:00-18:00, closed Tues, Dorsoduro 701, Dorsoduro, +39 041 240 5411, guggenheim-venice.it
#### **San Giorgio Maggiore**
Directly across the Grand Canal from St Mark's Square, through the Columns of Venice, proudly stands the island of San Giorgio Maggiore. A short _vaporetto_ hop zips you over to this tiny, one-sight island, which is dominated by the grand, gleaming white **San Giorgio church** (free, climb bell tower for €6, daily 09:00-19:00, Nov-Mar closes at dusk, +39 041 522 7827). Take the lift to the top for stunning views of the main islands of Venice and the entire lagoon. (Note that the lift stops running 30 minutes before the church closes and is not accessible Sundays during services.) On the island, you may also find occasional modern art installations on the _vaporetto_ landing.
Connect to San Giorgio Maggiore with _vaporetto 2_ from a stand on St Mark's Square. The ride takes about five minutes.
#### **Murano**
This island was a commercial center as far back as the 17th century, but in 1291 the Venetian Republic ordered that all glassmaking operations move off of Venice due to continued threat of fire, so the entire industry was restricted to Murano. Glassmaking grew into such an important industry that artisans were awarded with exceptional privileges, like high social status, immunity, and the right to wear swords, and their daughters were able to marry into royal families. However, glassmakers were not allowed to leave the republic under the penalty of death. They were the only artisans who knew who to make mirrors as well as glassware.
If you're into the art of glassmaking, the **Glass Museum** (Museo Vetrario, €8, covered by Venezia Museum Pass, daily 10:00-18:00, Nov-Mar until 17:00, Fondamenta Giustinian, +39 041 739 586, museiciviciveneziani.it) is a worthwhile visit. The dusty displays (a bit dated) take you all the way back to 1,900-year-old Roman glass through the techniques and artifacts from Venice's golden age. Time your visit for a live demonstration and tour (Tues and Thurs at 14:30) that goes in-depth about the glassmaking process.
Actual factories are generally closed to the public, but there are a number of opportunities to watch a basic piece get made in front of your eyes, followed by a sales pitch. As you get off the boat platform, follow signs for Fornace Glass. Workshops tend to close down around lunchtime (13:00-15:00), with the best time to visit being in the morning.
Getting to Murano is easy enough. From the Piazzale Roma car parking area or from the Santa Lucia train station, take the direct _DM vaporetto_ , getting you there in 19 minutes. _Vaporetto 42_ also works, but it takes 40 minutes.
#### **Burano**
Burano, known for its lace-making industry, is well worth the trip for some window-shopping. Burano's pastel-colored houses and quiet lanes make for an exceptionally peaceful afternoon just a short _vaporetto_ ride away from the bustle of the main islands. On the main square, Piazza Galuppi, you'll find the small and well-done **lace museum** (€5, Apr-Oct daily 10:00-18:00, until 17:00 Nov-Mar, Piazza Galuppi 187, museomerletto.visitmuve.it, +39 041 730 034) demonstrating the island's typical trade. Loads of items on display take you through the centuries all the way back to the 1500s. You can even catch local grannies carrying on the tradition in person who are happy to show you what they're working on.
_Vaporetto 14_ connects San Marco with Burano in about an hour.
#### **Isola San Michele**
St Michael Island is Venice's cemetery. With real estate at such a premium, Venetians couldn't take up valuable space with cemeteries, nor was it very sanitary. San Michele—an entire island of the dead—is a unique sight to behold. You'll also find Venice's first Renaissance church, **Chiesa di San Michele.**
Hop out here on a quick seven-minute ride. Take _vaporetti 4.1_ or _4.2_ from Fondamente Nova B, on the north side of the main island, just north of Ospedale (Hospital) San Giovane e Paolo.
#### **Lido**
Lido is the long and skinny island that separates the Venetian Lagoon from the Adriatic Sea. As it's built on actual land, Lido feels like more of a city—complete with buses and resorts—than Venice does. Come out here on a hot day to relax on the beaches and get away from the crowds.
Take the _rossa_ (red) _vaporetto_ from the San Marco Giardinetti stop, for a fast 12-minute ride across the lagoon. Buses take you up and down the island for €1.50. The best beaches are toward the middle of the east side of the island.
### **TOP EATS**
Venice has a history of fresh seafood and pasta. But it also has more than its fair share of tourist traps. Do your best to recognize them: menus in a dozen languages, neon lights, food out on display, recruiters holding menus out in the streets, and locations right on main tourist thoroughfares. The local places are usually a couple blocks off the busy streets. **Campo Santa Margherita,** between the Dorsoduro district and Rialto Bridge, is a great stop for relatively cheap food options. It's also easy to make a _cichetti_ crawl of the restaurants lining the intimate **Campo Cesare Battisti** , famous for nightlife.
A few food terms will help you find what you're looking for. **_Cichetti_** (pronounced "chee-khetti") are Venetian snacks like tapas. You'll find spreads of cod, anchovies, octopus, and other bite-sized morsels, making for an excellent way to turn a booze tour slightly more classy, and to keep from getting a little too lubricated. Many snacks also come like an open-faced sandwich: a slice of bread topped with a little olive oil and whatever the fishers caught that day. **_Tramezzini_** remind me of the sandwiches my grandma used to spoil me with when my mom was away: soft white bread with the crust cut off. Smothered in mayonnaise, these not-quite-filling sandwiches are quite typically Venetian and are downed on a daily basis by its residents. **_Ombra_** refers to a glass of wine, and **tiramisu** was invented in Venice, so save room for some.
There's no need to tip while on your _cichetti_ crawl. At restaurants, be sure to look closely for whether or not service is included already ( _servizio incluso_ ), and check for _coperto_ as well, or a cover charge. If you see either of these, there's no need to tip. If not, consider rounding up to the next even euro.
#### **Cantina Do Mori**
This is your classic old-school Venetian wine and tapas bar. Walking into this place, with its wooden paneling and tin pots and pans hanging from the ceiling, is like a step back in time. The tapas are simple, but not too filling, but the wine is tasty and easy to knock back in the tourist-friendly social bar.
**CICHETTI CRAWL**
Make a fun _cichetti_ crawl of the restaurants lining the intimate **Campo Cesare Battisti** near the west side of the Rialto bridge, branching off later toward Cantina Do Spade and Cantina Do Moro. Take 20 minutes to take a peek at each of the listings to see which is best for your taste and budget. Order your food and drinks one at a time to gauge service and value before committing to sit down at a restaurant.
**Muro Venezia Rialto** (Sestiere San Polo 222): Very trendy.
**Al Merca** (Fondamenta Riva Olio): Sandwiches and glasses of wine to go.
**Ancora Piano Bar** (Sestiere San Polo 120): A sit-down musical experience; amazing seafood salads.
**Al Pesador Osteria** (Sestiere San Polo 125): Upscale food bar that tries a little too hard.
**Osteria Bancogiro** (Campo San Giacometto 122): Classy bar serving hearty Venetian fare—seafood, pastas, and bruschetta.
**Naranzaria** (Sestiere San Polo 130): Nice little tapas and wine bar; serves pizza on their terrace out back.
Finger food from €2, glasses of wine from €3.50, Mon-Sat 08:00-23:30, Sestiere San Polo 429, Rialto, +39 041 522 5401
#### **Cantina Do Spade**
This family-run cantina isn't so much about the smiles, but the typical Venetian food is lovely. Try their _cichetti_ at the front of the bar, or sit down in back and go for the unique _pasta al nero di seppia_ (squid ink pasta). They have excellent _frittura_ , fried fresh seafood. The meatballs in tomato sauce are also a highlight.
Dishes from €14, daily 10:00-15:00 and 18:00-22:00, San Polo 859, Rialto, +39 041 521 0583, cantinadospade.com
#### **Poste Vecie Ristorante**
If you can fit it into the budget, splurge for this classic restaurant—supposedly the oldest in Venice. Converted from a 16th-century post office, the building was redone in the 1800s as a nice restaurant popular among the elite of Venice for its typical and fresh cuisine. It was at this stage that the frescoes were added inside. You can still appreciate fine dining beneath them and next to a grand fireplace today.
Unforgettable menus from €35, Wed-Mon 12:00-15:00 and 19:00-22:30, Rialto Pescheria, Rialto, +39 041 721 822, postevecie.com
#### **Antico Forno**
Step off the main drag into this takeaway joint cranking out steaming hot slices of pizza all day long. While quality and freshness can vary, the Antico Forno, or "Old Oven," checks my boxes for clutch pizza by the slice in downtown Venice. I usually go for the thick-crust spicy pepperoni.
Marinara focaccia from €2, daily 11:30-21:30, Ruga Ravano 973, Rialto, +39 041 520 4110, anticofornovenezia.com
#### **Rialto Fish Market**
The Rialto Fish Market (Pescheria) is a great place to come for lunch, when dozens of bars and stalls serve up the freshest seafood you'll find anywhere in Italy. A favorite of mine is **Muro** , a stall that serves heaping plates of their dish of the day for €7-10 (glass of wine included). Always fresh, always delicious.
Free entry, Tues-Sat 07:00-14:00, Campo Bella Vienna 222, Rialto
#### **Panini Row**
The small street known as Panini Row is your best budget option closest to St Mark's Square. Facing the facade of the St Mark's Basilica, head down the left side of it and hang a right at the bridge so you cross the bridge to your right. After another block or so, you'll find Calle degli Albanesi and Calle le Rasse, both of which have a good number of budget eating options.
Calle degli Albanesi and Calle le Rasse, San Marco
#### **Harry's Bar**
Don't let the casual name deceive you: This place is posh, with a price tag to match. Harry's is renowned for the famous patrons who've saddled themselves on these bar stools since it opened in the 1930s. Kings and queens, famous writers and artists, and common tourists like you and I have all enjoyed knocking back cocktails at this corner bar. You can get food here, but it's brutally expensive. The cheapest food on the menu: bean soup (€19). But hey, this was Ernest Hemingway's favorite watering hole in Venice.
Drinks from €15, daily 10:30-23:00, Calle Vallaresso 1323, San Marco, +39 041 528 5777, harrysbarvenezia.com
#### **I Quadri**
Sample the elegance of Venice at I Quadri, one of the institutional cafés on St Mark's Square. Be forewarned: A sit-down espresso runs you in the neighborhood of €20. To save a little, take your breakfast and coffee at the bar inside. I Quadri, which has hosted the likes of Mikhail Gorbach ëv and Woody Allen, is a worthy splurge. Be sure to observe the frescoes by Tintoretto and Canaletto. If you're in after hours, try the typical Vov liqueur, made with egg whites, along with a Baicoli biscuit to dip in it. It doesn't get much more Venetian than that!
Meals easily climb past €50, daily 09:00-24:00, St Mark's Square 121, San Marco, +39 041 522 2105
#### **Rosticceria San Bartolomeo**
Come here for some great comfort food at an incredible price. At this self-service cafeteria, choose from over a dozen tasty pasta dishes, seafood entrées, and meat options for €6-10. Eat your food on the lower level; prices are 20-30 percent higher upstairs. For an even better deal, order your food to go and you'll get a discount.
Bites from €4, daily 09:00-21:30, Sottoportego della Bissa 5424/A, San Marco, +39 041 522 3569
#### **Arte della Pizza**
This is my favorite pizza by the slice in Venice. Grab a slice for €1.50, or get a whole pizza for €6. Order at the glass case and take it away or eat at the bar along the wall. Most get takeaway to eat along the canal nearby.
Pizza from €6, Tues-Sun 11:00-21:00, Calle Dell'Aseo 1861/A, Cannaregio, +39 041 524 6520
#### **Café Filermo**
The owner, Rafael, sets the welcoming tone in this fun bar. It's an excellent place to start the day with real espresso and pastry, or to unwind with a glass of wine after conquering the Doge's Palace and St Mark's Basilica. While located on the main pedestrian circuit from the Santa Lucia train station toward St Mark's Square, Café Filermo doesn't lose the authentic feel or fun atmosphere. Look for the Brazilian flag out front, and enjoy your drinks and meal on the handful of chairs and stools while socializing with the owner and fellow guests.
Coffee from €2, daily 08:00-20:00, around the corner and bridge from Campo della Magdalena church, Cannaregio, +39 041 524 4946
#### **Orange**
Probably the hippest bar in Venice, Orange offers a buffet _aperitivo_ (Mon-Thurs 18:30-21:00), where pasta dishes, _tramezzini_ sandwiches, fruit, and desserts await. Hang around and grab drinks here: The atmosphere really livens up in the evening hours.
_Aperitivo_ from €7, daily 09:00-02:00, Campo Santa Margherita 3054, Dorsoduro, +39 041 523 4740
#### **Coop**
Find this welcome grocery store just toward St Mark's Square from Campo Santa Maria Formosa. Pop in here to pick up heaping sandwiches (€3.50) and fresh pizza by the slice (€3), along with all the other usual groceries. Many budget travelers have the same idea, so it gets busy at lunchtime.
Daily 08:30-20:30, Salisada San Lio, San Marco, +39 041 241 2273
### **TOP NIGHTLIFE**
Don't expect late nights and ridiculous parties in this town. Venice's social venues are limited to just a couple squares. Your best grouping of spritz-serving bars is a stone's throw away from the Rialto Bridge. Just follow the noise off the north side of the bridge to **Campo Cesare Battisti**. Once there, it's easy to do a fun crawl down the line of pubs trying their different specialty drinks. **Campo Santa Margherita** is a popular square for socializing. Follow the crowds later on to the nightlife spots.
Generally, Venetians eat around 20:00 and head to the bars at 21:00 until late-ish but nowhere near as late as the Romans, Milanese, or Florentines. Venetian nightlife is low-key and laid-back.
#### **BARS**
##### **Imagina**
This place is a quadruple threat, serving cappuccinos and pastries in the morning, tasty panini and salads at lunchtime, and cocktails at night. It even comes complete with a photography gallery with ever-changing exhibitions. At night this place gets packed with party-goers ready to take on the night.
Drinks and snacks from €8, Mon-Thurs 07:00-21:00, Fri-Sat 07:00-24:00, Sun 08:00-21:00, Ponte dei Pugni 3126, Dorsoduro +39 041 241 0625
##### **Piccolo Mondo**
"This Little World" is the closest thing I've found to an actual club in Venice. It fills up with the only people on the island who seem to want to party, and who don't mind paying €10 for a bottle of beer. Be sure to count your money both before and after you pay for your drinks. Good luck finding your hostel at the end of the night!
Drinks from €10, daily 22:30-04:00, Dorsoduro 1056a, Dorsoduro, +39 041 520 0371
**LGBT VENICE**
While Italy as a whole is generally conservative, members of the LGBT community will have no problem visiting Venice. But the nightlife scene in Venice is spotty to begin with, and the LGBT nightlife scene is even spottier. In fact I can't recommend a single place on the main island. It's full of friendly cafes, restaurants, and tapas bars, but you may just have to surf the mainstream wave on your time in Venice.
### **TOP SHOPPING & MARKETS**
#### **SHOPPING DISTRICTS**
##### **Rialto**
There are dozens of shops on the Rialto Bridge itself, as well as in the surrounding area. You'll find the cliché touristy souvenirs: postcards, Venetian masks, snow globes, and little gondola figurines. Come here to get ideas, but make your purchases farther off the tourist circuit, where you may be able to find them for a bit less.
Rialto
##### **Mercerie**
The Mercerie (Merchants' Streets) is made up of a series of narrow streets—Mercerie dell'Orologio, Mercerie San Zulian, Mercerie del Capitello, and Mercerie San Salvador—linking the political and religious hub of Venice (San Marco) with the just-as-important commercial heart of the city (Rialto). Take your time to stroll these tight lanes, where you can pick up everything from trinkets to lingerie to Carnevale masks.
San Marco to Rialto
#### **MARKETS**
##### **Rialto Market**
This market just north of the Rialto Bridge provides an excellent chance to catch daily Venetian life with vendors selling fruit, fresh seafood, and vegetables. Read the origin information closely for the produce, as some is imported from Asia or South America. Of course, opt for the produce originating in "Italia."
Free, produce and goods market daily 07:00-20:00, fish market Tues-Sat 07:00-14:00, Rialto
### **TOP PARKS & RECREATION**
#### **Gondola Rides**
There's nothing more quintessentially Venetian than cruising through the canals with your very own gondolier. In recent years, the _gondolieri_ have unionized and standardized their prices, taking the stressful negotiation out of the experience. Now you can count on steady—if lofty—prices. You can fit up to five people on one gondola, and the price doesn't change whether you've got one or the max load. Your best bet is to link up with fellow budget travelers and go in on a ride together. Note: With Venice in all its hyper-commercialized glory, the romance of the experience sometimes gets lost, with _gondolieri_ busy texting the entire time and shorting you on cruising time. Try to look past his catcalls and the winks at your girlfriend. Temper your expectations, and find the beauty in the small things around you. If you're not willing to drop the wad of cash, hop on a **_traghetto_** (ferry), which shuttles visitors across the Grand Canal at numerous points along the Grand Canal—there's one from the Rialto Fish Market. It's the poor man's gondola and costs only €0.50! Tipping _gondolieri_ or _vaporetto_ drivers isn't expected.
Numerous starting points throughout the city, 40-minute ride €80 before 19:00, €100 after 19:00, last rides depart around 23:00
**GONDOLA TRIVIA**
Venetian gondolas are a touristy cliché today, but they evolved over hundreds of years, developing tons of traditions. Gondolas are slightly asymmetrical, allowing the boat to be steered and powered from one side and still move forward in a straight line. Each boat is handmade with about 280 pieces from eight different kinds of wood, and they cost €25,000-50,000 to purchase new. _Gondolieri_ licenses number only 425, are highly coveted, and are often passed down from generation to generation, keeping them in the family. Today, there is currently only one licensed female gondolier. The curved metal prow of the gondola represents the S-shape of the Grand Canal, with the crest representing the funny-shaped cap that the doge used to wear. The six forward-facing teeth represent the six _sestieri_ (subdivisions) of Venice, and the one facing back represents Giudecca Island. The prow is weighted in front so as to counterbalance the weight of the gondolier in back. As you can see, there's actually quite a lot going on in a seemingly simple little boat. Now all you've got to do is ride one! _O sole mio..._
#### **Palace Gardens**
Located just around the corner from San Marco, toward the _vaporetti_ stops, these small gardens are a welcome respite from the massive crowds, who don't seem to notice this square. I like to come here for a quiet moment to enjoy an icy _granita_ or ice cream. If you've got picnic supplies, the benches are a nice place to enjoy them.
Free, daily dawn-dusk
### **TOP TOURS**
Touring Venice with a professional guide provides an excellent introduction to the layout of the city and its complex history. Free tours are available as well as private and semi-private options with the companies below.
#### **Walks of Italy**
My friends at Walks of Italy offer excellent guided walks with licensed guides, canal tours, _cichetti_ crawls, and private entries into the top sights of Venice throughout the year. Check out their website for tour itineraries and pricing.
Day tours from €70 per person, +39 069 380 4888, walksofitaly.com/venice-tours, info@walksofitaly.com
#### **Discovering Venice**
I love these small group tours led by local, licensed guides taking you through the popular sights in Venice. You'll learn all about St Mark's Square and the Rialto Bridge, but you'll also go inside the basilica (free ticket included) and get into the back streets and canals often glossed-over by ordinary tourists. Antonella, Eugenia, and Frederica also offer options beyond the standard walk in Venice, like boat tours out into the canals and other islands. Find dates and schedules on their website.
€50 adults, €35 students under 25, discoveringvenice.com, info@discoveringvenice.com
#### **Free Tour Venice**
These daily walking tours follow the same format as other tip-based walking tours across Europe. They're informative, fun, and casual, and provide a great introduction to the city. Tours cover the foundations of Venice, the rise in trade, Napoleonic influences, World War II, and more recent history. Quality of guides can vary, and the turnover is high due to the fact guides subsist on what they generate in tips.
Free and tip-based, daily departures at 10:45 and 16:30, meet at Campo Dan Geremia, freetourvenice.com
### **TOP HOSTELS**
Unfortunately, Venice's accommodations are expensive, limited, and oftentimes disappointing. When making reservations, take note of where your place is actually located. Lido di Venezia is Venice's beach island, but you'll have to take a 20-minute ferry every time you want to go into the main center. Same thing with Venice Mestre—this is the last town on the mainland before the bridge connecting into Venice proper. It's not the end of the world, just a pain to have to transfer between your accommodations and the main island of Venice. Make reservations in advance: Hostels book up quickly.
#### **Generator Venice**
Generator Venice is easily the best and most comfortable hostel in town. It's hard to even recommend any others, considering the comfortable beds, new interior, friendly staff, and slew of amenities. It's across the way in Giudecca, but worth the trek for general comfort and experience.
Beds from €25, 24-hour reception, free Wi-Fi, on-site bar, hair dryers and lockers available, towels for rent, laundry facilities, breakfast options, Fondamenta Zitelle 86, Giudecca, +39 041 877 8288, generatorhostels.com/en/destinations/venice
#### **Sunny Terrace Hostel**
Also located on Giudecca, this institutional hostel seems like it was converted from student housing. It's an acceptable alternative if the Generator is full.
July-Sept only, beds from €22, terrace, spotty free Wi-Fi, Ramo della Palada, Giudecca, +39 347 026 8037, book on hostelvenezia.com
#### **Venice Gold**
Check out this B&B option, but steel yourself for your typical abrupt Venetian service and long list of rules. To me, it's a solid upgrade that won't break the bank. They've got five-bed dorms, single rooms, and double rooms. The location is great, just a few minutes from St Mark's Square. This is the kind of place that insists on cash only for payment when you arrive, but they will not hesitate to charge the card you used for booking if you're a no-show or have to cancel within five days of your arrival time.
Dorm beds from €35.50, daily lockout for cleaning, reception 09:00-21:00, latest check-in at 21:00, free Wi-Fi when working, paid luggage storage, Castello, +39 328 209 4718, book on hostelbookers.com
#### **Haven Hostel San Toma**
This is my backup hostel. If the other options are full, head to Hostel San Toma for a comfortable-enough stay. The location, right on Campo San Toma in the San Polo district near the Rialto Bridge, is great, and the Wi-Fi works more often than not.
May-Aug only, 4-bed dorms from €24, Campo San Toma, Rialto, +39 347 026 8037, venezia@havenhostel.com, havenhostel.com
#### **PLUS Camping Jolly**
This is the Italian version of "glamping." Opt for PLUS Camping Jolly for a fun stay in either rented tents, bungalows, or parked camper vans. I enjoy the on-site pool, bar, extensive optional breakfast, Wi-Fi, friendly staff, and genial backpacker scene. I don't like the commute into town (about 45 minutes to an hour, with last connections around 23:00), though shuttles are organized often in high season.
Bungalows from €30, Via Giuseppe de Marchi 7, on the mainland, free Wi-Fi, optional breakfast, towels for hire, +39 041 920 312, book on plushostels.com/pluscampingjolly
### **TRANSPORTATION**
#### **GETTING THERE & AWAY**
The **Santa Croce** district is dominated by infrastructure that facilitates tourists' arrival by train, cruise ship, ferry, bus, and car.
##### **Plane**
Two major airports serve the city. **Marco Polo** (VCE, veniceairport.it) is the closest option. **Treviso** (TSF, trevisoairport.it) is popular with budget airlines. From both, your best connection into town is by bus.
From Marco Polo airport, the **ATVO shuttle bus** will get you into the center in about 20 minutes for €3. Purchase your ticket at the ATVO ticket window in the arrivals hall. If you're on a budget, you can take the local ACTV bus for €1.20, taking about 30 minutes. Both options drop you off at the same terminal station, Piazzale Roma. Skip the water taxi option, as it will easily run you €100 or more.
From Treviso, purchase your **ATVO bus** ticket (€5) in the arrivals hall at the ATVO ticket window. The journey into Venice takes about 70 minutes.
##### **Train**
Trains to Florence run often, cost €40, and take a little over two hours. Trains to Rome take almost four hours and cost €55 (you can find overnight options on slow trains as well, about €70).
**Venezia-Santa Lucia** is the train station on the island, located in Santa Croce. Don't get off at Venezia-Mestre, as this is the last city on the mainland before the bridge to the island! From Santa Lucia, you'll exit the train station directly toward the famous Grand Canal, busy with _vaporetti_ , gondolas, and private boats. From here, it's possible to walk anywhere you need to go, though it takes about 30 minutes to reach San Marco. There two docks in front of the train station both have ferries that will get you to St Mark's Square. Purchase your ticket (€7) from the clearly marked booth and hop on either the **_vaporetto_** _1_ (slow local ferry, excellent for your first intro to the city) or the **_vaporetto_** _2_ (express boat that makes fewer stops along the way).
##### **Bus**
**Eurolines** (eurolines.com) deposits international connections either at the **Venice Mestre Station** (on the mainland across from Venice Island) or at the main bus terminal, **Piazzale Roma,** in the Santa Croce district of Venice proper. Be sure to check which location is your final destination. Mestre is just a short connection into Venice proper via train. Your bus driver will be able to point the way.
##### **Car**
It is possible to drive into Venice, but you'll need to leave your car at the parking lot on Tronchetto Island on the edge of the city, about a 20-minute walk to the train station and beyond into the canals of Venice. Rates go from €21 per day. Cheaper options exist on the mainland in Mestre and Marghera, giving you the option to connect into town via train.
#### **GETTING AROUND**
**Walking** is your primary way to get around town—so be sure to wear comfortable shoes! You'll be tired after long crowded days, but _vaporetto_ rides cost €7 a pop, so think twice before taking the option to put your feet up. Rather than trying to memorize districts and the city plan, orient yourself by the major **landmarks** of the city. (A map is also a smart investment.) Signs posted above eye level on just about every corner around town point the way to the major sights. Use them like bread crumbs to find your way around:
**_Alla Ferrovia:_** To the train station
**_Piazzale Rome:_** To the bus terminal
**_Per Rialto:_** To the main and most famous bridge in town
**_Per S. Marco:_** To the beautiful St Mark's Square and church
##### **Vaporetto**
_Vaporetti_ (bus-boats) give you a chance to see the city from the vantage point for which it was built: the canals. An extensive network of _vaporetti_ connects stops around the island in both directions at the floating docks. Check signs to know where to catch the _vaporetto_ you want. Boats going in opposite directions will stop near each other, but often not at the same dock. Look for diagrams and route numbers to know where to catch yours. A **single ride** costs €7, so **day and multi-day passes** (€16/12 hours, €18/24 hours, €23/36 hours, €28/48 hours, €33/72 hours, €50/7 days) quickly pay for themselves. Strategize to make a 12-hour pass worth the cost by consolidating all of your cross-lagoon sightseeing and _vaporetto_ rides into one busy day. Purchase tickets at the transport kiosks or at tobacco shops. Remember to validate your ticket at the yellow machines before you board your boat! Routes generally run from 05:00, with last runs around 23:30. See actv.it/en/hellovenezia for more information.
### **HELP!**
#### **Tourist Information Centers**
Stop into one of Venice's information offices (both open daily 10:00-18:00, turismovenezia.it) for information and directions to your accommodations:
Marco Polo Airport
+39 041 529 87 11
Santa Lucia Train Station
+39 041 529 87 11
#### **Pickpockets & Scams**
The crime rate throughout Italy is moderate but generally only involves petty street theft such as pickpocketing and purse snatching. Thieves generally work in groups, so be mindful if a stranger comes up and starts talking to you out of nowhere. A common trick they use is to have one guy "accidentally" spill something on you, and while he helps you clean yourself off, his accomplice empties your pockets. Also, be mindful of groups of children encroaching on you, as they aren't as innocent as they appear.
Venetian merchants are also notorious for taking your €50 note and claiming you only gave them a €20. To protect yourself, say the denomination loud and clear as you hand it over. This way, you'll remember it better, and they'll have a harder time trying to deny it.
#### **Emergencies**
Dial 118 for an ambulance, or 113 for English-speaking police.
#### **Hospital**
Ospedale San Giovanni e Paolo
Castello 6777
+39 041 529 4111
#### **US Consulate**
Venice Marco Polo Airport, General Aviation Terminal
Viale Galileo Galilei 30
Madrid Map
Madrid 101
Three Day Itinerary
Top Neighborhoods
Top Sights
Top Eats
Top Nightlife
Top Shopping & Markets
Top Parks & Recreation
Top Tours
Top Hostels
Transportation
Day Trips
Help!
Madrid, Spain's capital, is packed with world-famous parks and museums—but it really comes to life when the sun goes down. Hit the town for some tasty tapas, catch a flamenco performance, and stroll diverse neighborhoods, from hipster Malasaña to Plaza Puerto del Sol, Spain's version of Times Square. After dark, massive clubs rage till sunrise. Barcelona might lure more international tourists, but Madrid is my favorite for a rich, authentic Spanish experience. This energetic city provides one of the best bang-for-your-buck experiences in all of Europe.
### **MADRID 101**
Madrid was founded in the 9th century as a Muslim stronghold among a strategic network of forts sprinkled throughout Spain. Madrid's name stems from the Arabic word for "waterway," and its prime location was hotly contested between Christian and Muslim armies vying for control over the Iberian Peninsula. Eventually, Alfonso VI conquered Madrid in 1083, and the main mosque was converted into a Catholic church. As time went on, the Muslims who stayed in the city were isolated and eventually expelled.
By the 1700s, Madrid began taking shape into the city we know today through a series of ambitious construction projects that put the Royal Palace, Royal Theatre, the city gates of Puerto de Toledo, and the Botanical Gardens on the map. In 1807, the Spanish king, Carlos IV, and Napoleon signed a contract that allowed French forces entry through Spain to fight Portugal. Napoleon wound up violating the original treaty and began conquering and occupying Spanish cities along the way to Portugal. This led to a revolt on May 2, 1808, and a war that lasted for five years until the French forces were finally rebuffed.
Spain had a complex and bloody 20th century, replete with economic collapse in 1929. Political instability ensued, with a fascist military dictatorship headed up by Francisco Franco taking control of the country—all before the start of World War II. During the war, German planes bombed cities across Spain, including Madrid. But it was the inconsequential town of Guernica that they used to practice their intense blitzkrieg bombing runs and surveillance operations before the outbreak of all-out war a couple years later. Picasso's infamous _Guernica,_ now on display at the Reina Sofia museum, depicts the horrific strafing and bombing unleashed on the town on the busy market day. This run killed nearly 1,000 people during more than two hours and leveled most of the town. All the while, Spain remained neutral during World War II.
Franco ruled an isolationist Spain for nearly 40 years until his death in 1975. Franco was convinced that things would remain status quo by returning power back to the king upon his death, but the king began setting up a constitutional monarchy as soon as Franco passed, ending over three decades of oppressive military rule.
With the veil of fascism lifted, Spain burst onto the world stage as a nation that had a lot of living and celebration to catch up on, and Madrid is where it all went down. Today, while the economic situation in Spain remains challenging—with a nearly 50 percent unemployment rate for young adults—the mood is generally optimistic. Madrid's urban progress has accelerated, and it is one of Europe's most beautiful capital cities, pleasing for its intense animated spirit, welcoming people, and compelling blend of modern and proud classic culture.
### **PLAN AHEAD**
#### **RESERVATIONS**
Reservations are recommended for the following sights and activities:
**Prado Museum** (entradasprado.com) **Real Madrid matches** (realmadrid.com)
#### **FREE ENTRY TIMES**
A number of Madrid museums are free to enter at certain times of day or certain days of the week or month, like these three:
**Prado Museum** (free Mon-Sat 18:00-20:00, Sun 17:00-19:00, always free for those under 18 and students 18-25)
**Reina Sofia** (free Mon and Wed-Sat 19:00-21:00, Sunday 15:00-19:00, always free for those under age 18 and students 25 and under with ID)
**Museo Thyssen-Bornemisza** (free Mon)
#### **LOCAL HAPPENINGS**
Just like their siestas, Spaniards take their festivals seriously. If a public holiday falls on a Thursday (or even a Wednesday in some cases) people get their weekend started early. It's great for them, but it can be a hassle for tourists. Assume most businesses will be closed during these national holidays:
**January 1,** New Year's Day
**March 29,** Good Friday
**May 1,** Labor Day
**August 15,** the Assumption
**October 12,** National Holiday of Spain
**November 1,** All Saints' Day
**December 6,** Spanish Constitution Day
**December 25,** Christmas Day
**KNOW BEFORE YOU GO**
**KEY STATS & FIGURES**
**Currency:**
**Spain uses the euro (€); 1 EUR = about 1.06 USD**
**Population:**
**3.2 million (double that of Barcelona, and 1 million more than Paris)**
**Language:**
**Spanish**
**Days of sun per year:**
**300+**
**Elevation:**
**650 meters (the highest capital city in Europe)**
**Dinner time for Madrileños:**
**21:00**
**Time to hit the clubs:**
**02:00**
**CALIBRATE YOUR BUDGET**
**TYPICAL PRICES FOR:**
**Hostel dorm bed:**
**€18**
**Two-course dinner:**
**€10**
**Pint of beer:**
**€2.50**
**Metro pass:**
**€1.50**
**MOVIES TO WATCH**
_Abre Los Ojos (Open Your Eyes), El Día de la Bestía (Day of the Beast), La Flor de Mi Secreto (The Flower of My Secret)_
**THREE DAY ITINERARY**
Madrid is packed with beautiful architecture, massive museums, sprawling parks, and photogenic boulevards. It's a challenge to balance all the sightseeing options with the Madrileño love of life and just having a darn good time. But with these tips, you can walk that line that's just right for you.
**DAY 1: WELCOME TO MADRID**
**MORNING**
Get your visit off to a sweet start with a breakfast of churros and chocolate at the famous **Chocolatería San Ginés** just around the corner from **Plaza Puerta del Sol**.
At 10:00, catch the three-hour walk with **Sandeman's New Europe Walking Tours** in **Plaza Mayor** for an efficient introduction to Madrid's top sights, including the **Royal Palace** and some historic Moorish ruins dating back to Madrid's early days.
**AFTERNOON**
Satisfy your appetite at **Taberna El Sur** on the way toward the museum quarter of Madrid. After lunch, continue downhill south and east a few blocks and power through your siesta time in the **Reina Sofia**. Spend at least a couple hours soaking in an extensive collection of 20th-century artwork in all media—from experimental film to sculpture.
Afterward, walk about a mile slightly uphill along the grand tree-lined boulevard of Paseo del Prado to the rooftop café at **Circulo de Bellas Artes** for a stunning panorama of downtown Madrid. Trace the thousands of steps you took today in one view and reward yourself with a cold beer or iced coffee.
**EVENING**
Rest up back at the hostel before heading out for dinner. Around 21:00, strike out for an authentic tapas crawl in the Malasaña District, one of the liveliest neighborhoods in town. **Bodega La Ardosa** is a good place to start your crawl. Explore this dense network of lanes and streets, and concentrate your imbibing efforts around **Plaza del 2 de Mayo** and the bars on **Plaza de Carlos Cambronero.**
**LATE**
In the clubbing mood? **Joy Eslava—** just around the corner from Chocolatería San Ginés, where your day began—is just a 15-minute walk downhill from the Malasaña neighborhood, between the Teatro Real and Plaza Puerto del Sol. If you queue up before 01:30, the €10 cover usually includes one drink.
**DAY 2: YOUR BIG MUSEUM DAY**
**MORNING**
Get yourself up at an appropriate hour considering how "Madrileño" you really got last night. Splurge on breakfast and coffee at **La Rollerie,** then walk a few minutes southeast to Madrid's top museum: the **Prado**. (If you made reservations in advance online, you can skip the line.) You'll want to spend a couple hours here.
**AFTERNOON**
After the museum, pick up some food to take to nearby **Retiro Park** to rest your feet over a relaxing picnic in the shade for a couple hours. Retiro gives you a chance to get away from the crowds, and you've got plenty of outdoor activities to choose from. Rent a paddle boat or bike and explore Retiro's 350 acres. Extra credit: Find the **_Statue of the Fallen Angel,_** the only statue dedicated to the devil in all of Europe, somewhere in the park.
Return the bikes, then cut across town on the metro from the Retiro stop on the north side of the park to the Ópera stop on the red line (line 2) to explore the **Royal Palace** (open daily until 20:00).
**EVENING**
Regroup at your hostel for a short siesta. Then head out to bond with other backpackers at **O'Neill's** bar in Sol. For a quieter scene, grab a bottle of wine and people-watch at nearby **Plaza Santa Ana.**
Fancy catching a show? The nightly **Flamenco Show at Casa Patas,** just a few minutes on foot south of O'Neill's and Plaza Santa Ana, kicks off in the 21:00 and 23:00 hours. This show features the best dancers and singers in town.
**LATE**
Dressed to the nines and wallet still too heavy? Make your way toward the Atocha train station (just east of the Lavapies district) and find **Teatro Kapital** 's seven floors of throbbing hedonism.
**DAY 3: OPTIONS**
**MORNING**
If it's Sunday, head straight to the bustling **El Rastro** , Madrid's largest flea market, where you can find everything from playing cards and antiques to electrical equipment and trading cards. Stop for lunch along the way at my favorite spot in the neighborhood: **Bar Santurce**.
**AFTERNOON**
If you want to continue on the shopping theme, hop on the metro to get to **Calle Serrano**.
If you prefer museums, take a 20-minute walk from El Rastro to spend the afternoon at **Museo Thyssen-Bornemisza**.
If you (like me) love finding fresh air and views over the city, head west of Malasaña to the **Teleférico de Madrid** to catch your last panorama of this awesome city.
Finally, if soccer is your thing, make your pilgrimage outside the city center to **Estadio Santiago Bernabéu** for a tour.
### **TOP NEIGHBORHOODS**
Madrid is a massive, modern metropolis. The **Sol** (Centro) district, with Plaza Puerta del Sol at its center, is the heart of the city and the most touristy part of town. This is where you'll find major sights like Plaza Mayor, along with restaurants, bars, hotels, commercial nightlife, and Joy Eslava, a _discoteca_ institution.
The city spreads out to the north and south from Sol. About five blocks north of Plaza Puerto del Sol are two of my favorite districts: Chueca and Malasaña. **Chueca,** known as the gay district, has excellent shopping, food, and nightlife. This is one of Madrid's top up-and-coming neighborhoods. To the west is **Malasaña** , which, for me, is what Madrid is all about: a community vibe, unforgettable tapas, lively nightlife, and indie shopping. Malasaña is bordered by Calle San Bernardo to the west and Calle Horteleza and Chueca to the east.
In the **Museum District,** sandwiched between Sol and Retiro Park (Madrid's Central Park), you'll find the world-famous Prado, the Reina Sofia, and Museo Thyssen Bornemisza. North of Retiro and east of Chueca, **Salamanca** is your spot for high-end shopping and nightlife. This is the nice part of town, where the glitterati go about their flashy lifestyles.
South of Sol, neighborhoods huddle around the La Latina and Lavapies metro stops. **La Latina** is where the young, in-the-know professionals of Madrid like to go out—especially along two parallel streets: Calle Cava Baja and Calle Cava Alta, which offer classy little bars and cafés. **Lavapies,** just to the east, is home to El Rastro market. This is Madrid's gritty underground district. It's too dark for many tourists, and feels a touch run-down, but I love wandering the narrow lanes to find a cheap _cañas_ (half-pints of beer). This area is also the heart of Madrid's ethnic community. You'll find some delicious, cheap food, and a chance to get away from the tourist hordes.
### **TOP SIGHTS**
The great thing about Madrid's museums is that most are free at certain times or on certain days of the week or month—and, even better, some are _always_ free if you're a student and/or under the age of 18. For example, the Prado is always free for those under 18 and students age 18-25 who bring a valid student ID; the Reina Sofia is free with a student ID as well. If you can take advantage of these youth and student discounts, you might want to avoid the times when these sights are free to the entire public, as that's when they tend to pack out. If you can't take advantage of the youth and student deals, go early or during siesta time and pay the entrance fee if there's a particular artist or exhibit that you're all about—it's worth it to have it all to yourself!
#### **Prado Museum**
One of Europe's finest art museums, the Prado houses collections from famous Spanish artists such as Goya, El Greco, Ribera, and Velázquez, as well as international artists like Rembrandt, Raphael, and Orly. The museum is absolutely massive, so go in with an idea of what you want to see.
Find the medieval works of art dovetailing into early Renaissance pieces on the ground floor, from masters hailing from across the entire continent. As you climb upstairs, you enter into the high Renaissance and baroque masters from about 1600 on.
For those short on attention span, the museum's highlights include _Las Meninas_ by Velázquez, _The Nobleman with His Hand on His Chest_ by El Greco, _The Three Graces_ by Rubens, and _Jacob's Dream_ by Ribera. It's a good indoor activity to hit in tandem with a visit to Retiro Park, as they're just minutes from each other.
Free entry Monday-Saturday 18:00-20:00 and Sunday 17:00-19:00 is great on the budget, but understandably the museum is packed with visitors jumping on the great deal.
€14 adult, additional fee for temporary exhibits, free Mon-Sat 18:00-20:00 and Sun 17:00-19:00, always free under 18 and students 18-25, Mon-Sat 10:00-20:00, Sun and holidays 10:00-19:00, last entry 30 minutes before closing, closed Jan 1, May 1, and Dec 25, Calle Ruiz de Alarcón 23, Museum District, +34 913 302 800, museodelprado.es, Metro: Atocha
#### **Reina Sofia**
This imposing museum, housed in a former convent and hospital done up with a multimillion-dollar refurbishment, features Spain's biggest collection of 20th-century art. Enjoy original work from Spain's two most famous artists: Pablo Picasso and Salvador Dalí. This is where you can view Picasso's most famous piece, _Guernica,_ commemorating the abhorrent bombing of the town of Guernica by the Germans in World War II. The Reina Sofia has free entrance Monday and Wednesday-Saturday 19:00-21:00, Sunday 15:00-19:00, and is always free with a valid student ID.
€8 adult, €3 under 18, free Mon and Wed-Sat 19:00-21:00, Sun 15:00-19:00, always free with valid student ID, Mon and Wed-Sat 10:00-21:00, Sun 10:00-19:00 (fourth floor not accessible Sun after 15:00), closed Tues, Calle de Santa Isabel 52, Museum District, +34 917 741 000, museoreinasofia.es, Metro: Atocha
#### **The Royal Palace (Palacio Real de Madrid)**
Though it's officially the residence of the king of Spain, the king actually resides in the Palacio de la Zarzuela outside of the city. It may also come as a surprise that the Royal Palace in Madrid is the largest palace ever built in Western Europe (even bigger than Versailles!). It's also arguably Madrid's most beautiful building, constructed in the baroque style and bursting with ornate architectural accents in white stone, reminiscent of Buckingham Palace in London. The interior, done up in high baroque fashion, will blow you away as well. You'll find detailed frescos and gilded detailing, along with royal furnishings and paintings throughout the private apartments upstairs.
The palace is definitely worth the price of admission, as it holds artifacts and works from Velázquez, Goya, and Giordano; the Royal Armory; and the world's only complete Stradivarius string quintet.
€11 adult, €5 ages 5-16 and students up to age 25 with ID, free under age 5, Oct-Mar daily 10:00-18:00, Apr-Sept daily 10:00-20:00, Calle Bailén, Sol, +34 914 548 700, patrimonionacional.es, Metro: Ópera
#### **Plaza Mayor**
The principal square in Madrid, the Plaza Mayor is a great place for people-watching, as it boasts many cafés serving all sorts of tapas and drinks, and is the home to many street artists. Come for the people-watching, but avoid the touristy restaurants that line the square. If the square doesn't look particularly Spanish to you, it's because it's not. The square was constructed during the Austrian rule of the Habsburgs in 1617, and is known in Madrid as "Madrid de los Austrias."
Free, always open, Plaza Mayor, Sol, Metro: Ópera
#### **Plaza Puerta del Sol**
All of Madrid seems to converge on this large, sunny square in the heart of town. It's truly the heart of both the capital and the country as a whole—as evidenced by the **KM 0** monument in the middle of the square. This is the point from which all the road signs in the country measure their distance to the capital. Bustling with activity and tourists in the day, the square is also lively at night with touristy bars just to the south of it and tons of shopping, cafés, and restaurants in the surrounding streets. Sol is also a major metro station, and you'll often be connecting through or getting off here.
Free, always open, Puerta del Sol, Sol, Metro: Sol
#### **Teatro Real**
Madrid's majestic neoclassical opera house was newly remodeled and opened with 1,750 seats in 1997 after being closed for nearly 75 years. It's come a long ways since serving as munitions storage during the Spanish Civil War. Today, you can enjoy a slice of high society at a show while taking in the wraparound seating and excellent acoustics. Find tickets online from just €9 for the cheap seats.
If you're not interested in catching a show, the theater has numerous general, technical, and artistic-focused tours going off daily. The €8 general tours kick off every 30 minutes daily 10:30-13:00 and last about an hour. The tour focusing on the artistic production of an opera runs twice daily at 09:30 and 09:45, costs €12, and runs about 1.25 hours. The technical tour (twice daily, 09:30 and 12:00, €16) explains what it takes logistically to take an opera from concept to stage, all about changing and constructing sets, lighting, and even acoustics.
All tickets can be purchased on arrival at the box office opening at 09:15 daily.
Ticket prices vary, shows often, Plaza Isabel II, Sol, +34 915 160 660, teatro-real.com, Metro: Ópera
#### **Flamenco Show at Casa Patas**
While flamenco isn't technically native to Madrid (it comes from the southern cities like Seville and Granada), this show is a blast and has the best dancers and singers in town. Grab a seat toward the stage of this intimate black box venue serving both drinks and food. Once the lights go down for show time, you'll be treated to a dramatic show of gesticulating vocalists, stomping dancers, and impressive musicians playing guitar, drum box, and even violin.
Tickets from €36, includes first drink, dinner options available, shows nightly at 22:30 (check online to confirm times and specials, as they change often), Calle de los Cañizares 10, Sol, +34 913 690 496, casapatas.com, Metro: Antón Martín
#### **Teleférico de Madrid**
Take a ride in the _teleférico_ , a cable-car that is suspended in the air, and catch some of Madrid's top sights from above! During your ride you'll fly by the Royal Palace and La Plaza de España, across the Manzanares River, and over a large beautiful park, the Casa de Campo. The _teleférico_ is located west of Malasaña.
€4 one-way ticket, daily 12:00-21:00 (weekends only in winter), station just off of Paseo Pintor Rosales, just south of the intersection with Calle del Marqués de Urquijo, Greater Madrid, +34 902 345 002, teleferico.com, Metro: Argüelles
#### **Museo Thyssen-Bornemisza**
A much quieter experience than you'll get at the Prado, which is always overrun by tourists, the Museo Thyssen-Bornemisza offers a quiet atmosphere with an amazing impressionist and post-impressionist collection that could keep you busy for hours. Come enjoy the works of greats like Van Gogh, Renoir, Monet, Manet, Gauguin, and more.
€10 adult, €6 student, additional fees for special exhibits, free on Mon, Mon 12:00-16:00, Tues-Sun 10:00-19:00, Sat until 21:00 in the summer (exhibits only), last entry 45 minutes before closing, Paseo Prado 8, Museum District, +34 902 760 511, museothyssen.org, Metro: Banco de España
#### **Estadio Santiago Bernabéu**
In Spain, soccer is a religion, and Estadio Santiago Bernabéu is its cathedral. This gargantuan 80,000-seat stadium north of central Madrid is home to the soccer world's most winningest team: Real Madrid. But you should know that already. Real Madrid is kind of the Yankees of the soccer world—it's the richest team, so it can afford the most expensive players, so it keeps winning and making even more money. It's a nice upward spiral. If you're in town for a game, the tickets can get pricey but are absolutely worth it for fans. It's an experience you won't soon forget. Otherwise, you can take a tour of the facilities, check out the locker room, see the posh company boxes, get down to field level, learn about the history of the club, and pick up some sweet Real swag in the gift shop.
Game tickets can be hard to come by, and priority goes to season ticket and pass holders, with tourists holding up the rear. Purchase your tickets online at realmadrid.com, call into +34 902 324 324, or pick them up from the box office itself at the stadium or at most Caixa Bank branches in Madrid. You'll likely need to purchase tickets at least two weeks out for a good chance of getting them.
Cheap seats at the game from €50, unguided tours €19, Concha Espina 1, Greater Madrid, +34 913 984 300, realmadrid, Metro: Santiago Bernabéu
#### **Bullfights at Plaza de Toros de las Venta**
Each Sunday throughout the year, the bullfight stadium in Madrid packs out with tourists and locals alike to watch the spectacle of Spanish bullfighting. It's here that you'll see the best in the business do their thing: disorient, tease, stab, and ultimately kill bulls one after the other for a stretch of three-plus hours. I was taken aback by the timed, mechanical nature of the fight ritual, but something so famous and integral to Spanish history and culture warrants at least a short visit. It's possible to scalp tickets, but it is illegal and you risk losing your tickets if caught. You can't miss Plaza de Toros, about 25 minutes north of town via metro.
€5-150, cheaper shows feature novice bullfighters and messier bullfights, days and times vary, Calle de Alcalá 237, Greater Madrid, +34 913 562 200, las-ventas.com, Metro: Ventas
#### **EXTRA CREDIT**
##### **City History Museum (Museo de los Orígenes)**
If you're anything like your dear author, you dig learning about the history of any modern city with medieval foundations. In this modest and free museum you can check out a model of the old town and begin to visualize the old streets that have otherwise been cleaned up and straightened out.
Free, Sept-July Tues-Sun 09:30-20:00, Aug Tues-Fri 09:30-14:30, Sat-Sun 09:30-20:00, Plaza de San Andrés 2, La Latina, +34 913 667 415, madrid.es, Metro: La Latina
### **TOP EATS**
Madrileño cuisine consists of succulent meats, dried _jamón_ (ham) served in paper-thin slices, delicious cheeses like Manchego, potato- and chickpea-based stews, and many deliciously fried treats like churros, _patatas bravas_ (fried potatoes with a spicy ketchup sauce), and _tortilla española_ (potato omelet). The dishes can be a touch heavy, but they're well-portioned, and the fresh ingredients used don't make you regret it. Tapas, a favorite of locals and tourists alike, give you a chance to sample a number of dishes every time you sit down for a meal. Madrid offers a wide array of food beyond the typical Madrileño cuisine, too. The city is a magnet for people from all around the world, and with them they bring their own cuisine.
**Malasaña** is my favorite district in town for food, as there seems to be a popular local spot around just about every corner. The action tends to center around these squares: **Plaza del 2 de Mayo, Plaza Juan Pujol,** and **Plaza de San Ildefonso,** and you can find the local favorites in the streets between. For a real neighborhood feel, head to Malasaña's **Plaza de Carlos** to choose between the welcoming cafés and bars, like Lamucca de Pez and El Palentino, with outdoor seating spilling out onto this cute little square.
The streets around **Lavapies** are home to Madrid's ethnic (especially Indian) population. This neighborhood also has an underground, gritty feel, and you can find bargain food and drinks at every turn. You'll likely notice that the streets are darker here, but you're not in the danger you'd be in on a similarly poorly lit street in any city back in the States. Just stick with your friends and keep your wits about you—you're sure to have an awesome time. You can also find some fast-food gems in the streets around **El Rastro market** (Calle de Ribera de Curtidores).
Tipping is not expected, but rounding up to the next euro mark is appreciated. Be sure to check whether or not your bill says _"servicio incluido,"_ as many touristy restaurants sneak this 10-15 percent charge in there.
#### **Bodega La Ardosa**
Bodega La Ardosa, in the Malasaña district, is one of my favorite old-time tapas restaurants in all of Madrid. Pop in for a cup of gazpacho and _patatas bravas_ , and huddle with locals around upended kegs doubling as table tops. A snack in here feels like a step into old-world Spain, and the dusty bottles of spirits and liquors lining the walls add a bit of history to the place. There is standing-room only, which makes it easy to meet the friendly locals. Order only a dish or two to start, then wait to see what looks good coming out of the kitchen. La Ardosa is an excellent place to kick off a wander through Malasaña.
€11-20, Mon-Fri 08:30-02:00, Sat-Sun 11:45-02:30, Calle Colón 13, Malasaña, +34 915 214 979, laardosa.es, Metro: Tribunal
#### **Lamucca de Pez**
On the uphill side of Plaza de Carlos is Lamucca de Pez, a bright, posh cocktail and tapas bar with friendly service, exposed brick interior, and an extensive wine selection. They take pride in both their Italian-style pizzas and expertly mixed alcoholic concoctions.
**EATING ON THE CHEAP**
Here are some basic tips on how to stay on that budget while eating out in Madrid.
_•Bocadillos_ are simple (i.e., one topping), delicious, handmade sandwiches put on a small baguette and made to order. They're my top choice for budget eats just about anywhere in Spain because nearly every café and bar can make themon demand. Your only challenge is to figure out your favorite fillings: chorizo, _jamón_ (ham), _tortilla de patata_ (Spanish potato omelet), and even _calamares,_ fried calamari rings. And the best part? Getting your _pan con tomate_ or having a tomato puree ladled onto your bread to make it extra juicy. The Brits have butter, Americans use mayonnaise, Italians olive oil, but Spaniards— _tomate_. And it's _so_ good! _Bocadillos_ normally run about €3, and are a filling snack to power you through the day.
_•_ Most restaurants charge a small cover for sit-down table service. Skip the charge by standing up and eating at the bar.
_•_ You'll often be charged for the bread at your table. Before digging in, ask, _"¿gratis?"_
_•_ If you ask for water without specifying which kind, restaurant staff will bring out a bottle for which you've gotta pay. Ask for tap water by saying _"agua del grifo, por favor."_
_•_ Avoid restaurants with pictures in the windows in place of menus. More often than not, pictures are used to cater to tourists rather than locals. Much better: Find the hole-in-the-wall restaurants without a menu that serve up delicious, fresh, and typical cuisine to loyal returning customers.
_•_ Keep your eyes open for the menu of the day, or _menu del día_. These are often a great value, and you'll find that for Madrileños, lunch is the biggest meal of the day, just before siesta—no wonder they've gotta sleep it off!
_Bocadillos_ from €3, pizzas from €12, Tues-Sat 12:00-24:00, Sun 19:00-24:00, Plaza de Carlos Cambronero, Malasaña, +34 915 210 000, lamucca.es, Metro: Tribunal or Noviciado
#### **El Palentino**
Across from Lamucca de Pez is El Palentino, your classic nondescript neighborhood pub and bar that pumps out fresh _bocadillos_ (sandwiches) all day and _cañas_ (half-pints of beer) to wash them down. Come here to get a true glimpse into the jovial Madrileño lifestyle.
_Bocadillos_ from €3, pizzas from €12, Tues-Sat 12:00-24:00, Sun 19:00-24:00, Plaza de Carlos Cambronero, Malasaña, +34 915 323 058, Metro: Tribunal or Noviciado
#### **Bar Santurce**
My favorite spot around El Rastro market is this little anchovy shop that cranks out platefuls of the salty snack. It's near the top of the hill on Calle de las Amazonas. An order of _pimientos de padrón_ (fried peppers), fried anchovies, a bread bowl, and two _cañas_ makes for an excellent afternoon stop if you're in the area. The place is busy with locals and tourists alike. Step in, work your way to the front of the line, and be ready to order when you wave your bartender down. If you haven't worked up the guts to try anchovies, Bar Santurce is the perfect place to do it! There's a pickle shop next door that hits the spot too.
Plates from €3, Tues-Sat 12:00-16:00, Sun 09:00-16:00, Plaza General Vara del Rey 14, La Latina, +34 646 238 303, barsanturce.com, Metro: La Latina
#### **Cervecería Cruz**
This is your classic Madrileño beer hall: brightly lit, with steel countertops, grandpas playing cards, and a food bar serving classics like _pimientos de padrón_ and calamari on order. Its location at the top of El Rastro market makes it a popular place on Sunday.
Dishes from €3.50, daily 08:00-23:00, Calle de las Maldonadas 1, La Latina, +34 913 663 738, Metro: La Latina
#### **Puerto Rico**
This is one of my favorite and affordable restaurants in the center. Come out to Puerto Rico for simple, filling cuisine like typical Madrileño stews, roast chicken, and your whole array of tapas. Thanks to its location and casual atmosphere, Puerto Rico fills up daily with power lunchers at lunch. Tourists love the place for its great value. Go for their menu of the day for a three-course lunch for under €12, which should easily last you through dinner. Don't miss their rice pudding and flan if you've got room for dessert!
Lunches and dinners under €10, daily 13:00-16:30 and 20:30-23:30, Calle de Chinchilla 2, Sol, +34 915 219 834, Metro: Callao or Gran Vía
#### **Mercado San Antón**
I love this polished, multilevel indoor market for collecting fresh ingredients for a picnic and snacks. Find the baker for a fresh baguette, and pick up 100 grams each of _jamón Serrano_ and _queso Manchego_ (ham and cheese) from the butcher for a delicious sandwich that should last you till the next day. Climb to the top level for prepared food options like _Salmorejo_ and surprisingly tasty burgers.
Create your own adventure for around €5, daily 10:00-24:00, Calle de Augusto Figueroa 24, Chueca, +34 913 300 730, mercadosananton.com, Metro: Chueca
#### **Museo del Jamón**
This chain does one thing and it does it well: It serves fresh slices of some of the world's best cured ham. We Westerners may lack a developed palate for _jamón,_ but that's all the more reason to pop into one of their downtown locations and order a smattering of different types to sample. Spring for a taste of _jamón ibérico de bellota_ (pig raised on a diet of acorns), as it's widely regarded as the best type of this thin-sliced ham in the world. Don't let them get carried away when slicing! Prices are paid by weight. Take in the view of hundreds of ham legs hanging from the walls and ceiling—these guys take their selection seriously. Find other locations at Calle Gran Vía 72 and Calle de Atocha 54.
From €5, daily 09:00-24:00, Plaza Mayor 18, Sol, +34 913 692 204, museodeljamon.es, Metro: Ópera
#### **Chocolatería San Ginés**
Like chocolate? Go here. You can't pass through Spain without enjoying one of the country's top treats: fried doughy sticks (churros) dunked in hot chocolate (which is more like concentrated melted chocolate than something you could actually drink), and Chocolatería San Ginés offers some of the best of both in the city. Enjoy your thick and rich cup of chocolate in a classic, warm setting that is exactly what you'd think an old-school café should be: prominent espresso bar, outdoor seating, wooden interior with large mirrors, and old-time locals enjoying their favorite snack. The café is tucked in a back street behind the Iglesia San Ginés de Arles, just a couple blocks west from Plaza Puerta del Sol.
Churros and chocolate from €4, daily 24 hours, Pasadizo San Ginés 5, Sol, +34 913 656 546, chocolateriasangines.com, Metro: Sol
#### **Torre del Oro**
Located right on Plaza Mayor, Torre del Oro is widely known as the best bullfight bar in town. As long as you don't lose your appetite when seeing bull heads mounted on the walls, you'll enjoy their typical Madrileño dishes, like fried anchovies and ham sandwiches. The restaurant itself is the main attraction, with dozens of framed pictures of bullfighters, including some gory shots of when the matadors must have taken their eye off the ball...er, bull.
Bites from €3, daily 11:00-02:00, Plaza Mayor 26, Sol, +34 913 665 016, torredeloro.sellsoc.org, Metro: Ópera
#### **Café Melos**
Stroll downhill from Sol to find the bright (and a bit sterile) Café Melos, famous for its inexpensive fare ideal for students and travelers on tight budgets. Try their _zapatilla_ —a huge stuffed sandwich overflowing with ham and cheese, easily splittable between two people, or enough to put you into hibernation just in time for siesta. If you've still got room, don't miss their flash-fried croquettes either. The picture-based menu makes selection easy.
Snacks from €3.50, Tues-Sun 08:00-02:00, Calle del Ave María 44, Lavapies, +34 915 275 054, Metro: Lavapies
#### **El Corte Inglés**
In the basement of this massive department store just a couple blocks from Sol, you'll find a modern grocery store where you can stock up on extensive options for picnic supplies. They've also got an affordable café upstairs that's very popular among locals for power lunches during the work week.
Cheap, daily 10:00-22:00, Plaza de Callao 2, Sol, +34 913 798 000, elcorteingles.es, Metro: Callao
#### **La Sanabresa**
La Sanabresa is your best choice near the Antón Martín metro stop for sit-down value. Their lunch and dinner _menus del día_ offer astounding value even though they're located in the heart of touristy Madrid: €10-16 will get you three courses (starter, main, and dessert) with a drink, bread, and little taste of liquor to cap it all off. Mains range from grilled meats and fish like chicken, pork, beef, and salmon to stews and roast veggies. And it all goes down in a casual setting with white tablecloths, efficient service, and bright and inviting atmosphere.
Mains from €8, Mon-Sat 13:00-16:30 and 20:30-23:30, Calle Amor de Dios 12, Lavapies, +34 914 290 338, Metro: Antón Martín
#### **Taberna El Sur**
I love El Sur's excellent selection, which ranges from light (Cobb salad) all the way to classic and heavy (fried egg topped with bacon over fried potatoes). Either way you go, dishes come with impressive presentation. Delicious drinks, like salted margaritas and sangria complete with cinnamon sticks, are available, too. Enjoy an afternoon or evening meal in this welcoming yet refined restaurant with somewhat mod decor, tucked away in an authentic neighborhood just a couple blocks south of the Antón Martín metro stop.
Excellent-value meals from €10, Sun-Thurs 12:00-01:30, Fri-Sat 12:00-02:00, Calle de Torrevilla del Leal 12, Lavapies, +34 915 278 340, Metro: Antón Martín
#### **La Pizzateca**
La Pizzateca is your classic pizza-by-the-slice fast food joint, but they do it well, and at good prices. Don't come for the fancy atmosphere, but do stop by to choose from their racks of pizza made with fresh ingredients like peppers, zucchini, sausage, spinach, and potatoes. They'll warm your slices up for you while you wait. Eat in at their simple wooden bar, or take it outside to enjoy in the square.
Slices from €2.50, Tues-Sun 13:00-16:30 and 19:00-24:00, Calle Lean 35, Sol, +34 913 693 210, lapizzateca.com, Metro: Antón Martín
#### **La Rollerie**
La Rollerie is my favorite breakfast choice in town. While Spaniards generally eat a light breakfast, La Rollerie has just about anything to suit my fancy no matter what I'm craving, from cinnamon rolls and pastries to bacon-and-egg bagel sandwiches. They try a little too hard to mimic a French bakery, but, thankfully, they are on point when it comes to what matters: carrot cake, Andalucian breakfast (toast with tomato puree), and of course, coffee. The Rollerie is a touch posh, and draws the locals and tourists who can spend €10 or more on a delicious breakfast.
Coffee from €3, daily 08:00-22:30, Calle de Atocha 20, Sol, +34 914 204 675, larollerie.com, Metro: Sol or Tirso de Molina
### **TOP NIGHTLIFE**
Like all Mediterranean cultures, nightlife in Madrid starts late. Ask around your hostel to get a feel, but the real party doesn't start until after 01:00 or later. Pace yourself, and make sure your hostel windows can block out the sun in the morning!
Madrid's neighborhoods have distinct vibes and different sorts of venues. Rather than aiming for a specific venue, it's most fun to pick the district that appeals to you, then set out to discover your own favorites. Many of Madrid's top eateries also double as fun nightlife venues.
#### **NIGHTLIFE DISTRICTS**
##### **Malasaña**
My favorite neighborhood in town has a slightly hipsterish and trendy feel without really trying. It's a popular spot to meet up with friends for drinks. The streets are narrow and tangled, so it's fun to wander and see what gems you discover.
Malasaña, Metro: Noviciado
##### **Chueca**
Chueca is Madrid's gay district, with fun bars and tasty food options centered around **Chueca Square.**
Chueca, Metro: Chueca
##### **Sol**
The area around Puerta del Sol is understandably touristy. Bars catering to the international crowd are sprinkled throughout Sol but are concentrated just south of Madrid's central plaza. It's here you can find Irish bars and study-abroad students by the hundreds. If you keep your eyes open, you can still discover winners in the area.
So much of Madrid's nightlife is pedal to the metal, so if you prefer a more low-key evening, grab a bottle of wine and head down to **Plaza de Santa Ana,** southeast of Puerta del Sol, for a night of relaxing and people-watching from the benches that line the square. It's an endlessly entertaining plaza filled with a mix of tourists and locals passing through on their way to more shenanigans.
Sol, Metro: Sol
##### **Lavapies**
In Madrid's grungy, too-gritty-for-some underground district, Lavapies, you can find some hidden theaters and great music venues. Of course, the money saved by finding off-the-beaten-path watering holes is a plus, too!
Lavapies, Metro: Lavapies
##### **La Latina**
La Latina is the trendy upscale neighborhood in downtown with tapas bars, classy restaurants, and a fun vibe. The action centers around two streets: **Calle Cava Baja** and **Calle Cava Alta,** which turns into **Calle Grafal** farther north. These two parallel streets running just northwest of the La Latina metro stop have a fun little world of classy bars, shops, cafés, and _jamón_ shops. It's easy to make an evening starting at the top of one street and looping around, popping into any tapas bars that strike your fancy. The district is a bit of a tourist trap and is notorious for wildly varying service and food quality. So rather than order a slew of dishes, I spring for one at a time to ensure the experience is up to par before ordering more.
**LGBT MADRID**
LGBTQ travelers will find Spain welcome and progressive. Spain was the first country in the EU to legalize gay marriage, and as such is considered a very progressive and liberal country with respects to equal rights for gay couples. Gay visitors should expect to enjoy safety and no unwanted attention during their time in Madrid.
All of my recommended nightlife venues are gay-friendly, but **Chueca** is your neighborhood for partying while in town for the gay scene. Your best bet is to head to the main square and take a stroll around the neighborhood branching off from there. In the surrounding streets you'll find happy, welcoming people, an excited buzz, and numerous cafés, restaurants, delis and grocery stores, bars, lounges and dance bars, and clubs. Stores run the whole gamut from high street fashion to drag suits and more, suiting just about all tastes. **El Bulldog** (San Bartolomé 16, +34 915 991 260) draws the bear crowd, **Griffin's** (Calle Marqués de Valdeiglesias 6, +34 915 22 20 79) is your dominating drag spot, and **La Kama** (Calle Gravina 4, +34 915 22 32 26) is a fun go-go bar serving cocktails in a lounge setting.
**El Tempranillo** (Calle Cava Baja 38, +34 913 64 15 32), which sports an impressive wine selection and _pintxo_ menu, and **El Viajero** (Plaza de la Cebada 11, +34 913 66 90 64), a contemporary, simple-fare bar with a delightful terrace and offerings like beef sliders and ribs, are two places to start your crawl, though you can run up a tab quite quickly at both venues. **Juana la Loca** (Plaza Puerta de Moros 4, +34 913 640 525) is where all the in-the-know locals flock after work. With excellent cocktails, a delectable dessert menu, and chill lounge vibe, this one of my first stops when I'm in town.
La Latina, Metro: La Latina
##### **Salamanca**
You may see Spanish celebrities at the numerous cocktail bars and posh clubs in glitterati-populated Salamanca. With the hot vibes come the highest prices and most pretentiousness you'll find in the city. Bars and clubs in this neighborhood tend to open and close frequently. Jersey chaser? Come here for the chance of a glimpse at the Real Madrid crew, who often show up to celebrate a win.
Salamanca, Metro: Serrano or Velazquez
#### **BARS & PUBS**
##### **Dubliners Pub**
There are a few of your standard fake Irish pubs in the heart of Madrid, offering mediocre drinks and food, but they're dependable at least for that and for a wide selection of sports being played. Don't expect sharp service or any sort of local experience at this touristy bar, but it's a good place to bond with fellow backpackers.
Daily 11:00-03:00, Calle de Espoz y Mina 7, Sol, +34 915 22 75 09, irishpubdubliners.com, Metro: Sol
##### **O'Neill's**
O'Neill's is a favorite for its large beer hall feel, pool table, drink and shot specials, and comfortable living room-style ambience. You can count on a sloppy, sticky-floor-good-time backpacking experience here. You'll likely cross paths with the American students "living" in Madrid on their study-abroad experience.
Mon-Wed 17:00-01:00, Thurs 17:00-02:00, Fri 17:00-03:30, Sat 13:00-03:30, Sun 13:00-01:00, Calle del Príncipe 12, +34 915 212 030, facebook.com/ONeillsMadrid, Metro: Sol
#### **CLUBS**
Clubs open and close often in Madrid. It's a raging scene that continues till the sun comes up nearly every day of the week. I've listed the nightlife institutions of Madrid, but be sure to ask at your hostel to see what new and hot venues have opened up.
**ACT LIKE A LOCAL**
**Party Like a Madrileño**
Madrileños eat around 20:00 or 21:00, enjoy friends and socializing for the next few hours, then begin heading out around 01:00 or later. To draw revelers in earlier, many clubs have free cover or extra drinks included in the price of the cover fee before 01:30. While you'll get more by showing up early, oftentimes you'll need to wait until the party shows up around 02:00. **_Cervecerías_** (beer halls) are a great way to get some food to keep you going and enjoy the Spanish tradition of **_cañas_** _—_ half-pints served cheap and easy from bright, steel-topped bars.
If you're in a _cervecería_ or tapas place packed with locals and want to order food, you'll need to sharpen your elbows and work your way up to the bar. If you're passive, you'll likely lose your place in line. Keep your ears open, take a look at what everyone else is ordering, and practice your order in your head while trying to make eye contact with the bartender. Ordering by pointing (and smiling) works, too!
##### **Joy Eslava**
A massive, horseshoe-shaped nightclub in central Madrid, Joy has been converted from a 1950s theater, meaning this place was practically built for the impressive light shows and loud music that partiers drown in every night of the week. Music is heavy on the techno but also mixes in local and international hits. Drinks can be pricey, so pregame well before showing up, but not so well you turn off the picky bouncers!
Covers around €12 (sometimes negotiable), pricey drinks and beers from €10, Sun-Thurs 12:00-05:30, Sat-Sun 12:00-18:00, Calle Arenal 11, Sol, +34 913 663 733, joy-eslava.com, Metro: Ópera
##### **Teatro Kapital**
This is a club on steroids with everything from the floor count (seven) and the light system (spectacular and nearly blinding) to the cover and drink charges (€20+ and €12+, respectively) and dance shows (inspiring). If you're trying to have the euro-club experience, see if you can't get in here for their standard cover, which should include a drink at €20. Dress smart, though: The bouncers have been known to try and charge €50 at the door.
€20 cover with drink, Thurs-Sat 12:00-06:00, Calle de Atocha 125, Museum District, +34 914 202 906, grupo-kapital.com, Metro: Atocha
#### **PUB CRAWLS**
##### **Sandeman's New Madrid Pub Crawl**
Sandeman's New Madrid offers a nightly pub crawl if you want to pass off the responsibility for charting your night out. Cheaper than their pub crawls in other European cities, this Sandeman's crawl is fun and social, stops in three bars, and finishes in one of Madrid's famous clubs.
€12, nightly at 22:00, meet at Plaza Mayor, Sol, newmadrid-tours.com, Metro: Ópera
### **TOP SHOPPING & MARKETS**
Madrid has the whole range of shopping, from cheap flea markets to high fashion and prices.
#### **SHOPPING DISTRICTS**
##### **Gran Vía**
In Sol and **Gran Vía** , Madrid's downtown and most central shopping district, a slew of retail shops sell everything from touristy trinkets to shoes, hats, and clothes. This is your Champs-Élysées, Piccadilly Circus, and Piazza Venezia of Madrid.
Sol, Metro: Plaza de Santo Domingo, Plaza del Callao, or Gran Vía
##### **Calle Fuencarral**
This charming pedestrian boulevard runs north-south through the heart of Malasaña. It features pop-up fashion shops and hidden gems like boutique clothing and shoe shops, where you'll uncover finds that'll make your friends jealous. The main attraction is the people-watching you'll enjoy on a stroll up and down the street. There are some great options for food and brunch on **Calle de Augusto Figueroa.**
Malasaña, Metro: Chueca or Gran Vía
##### **Calle Serrano**
On Calle Serrano, a long street in the Salamanca district that runs north and south, you'll find high-fashion shopping from the Plaza de la Independencia all the way north to Calle Juan Bravo. It's here you can expect to find brands like D&G and Gucci.
Salamanca, Metro: Retiro, Serrano, or Ruben Dario
#### **MARKETS**
##### **El Rastro**
You'll find everything from trinkets and antiques to clothing and fabric stores at what is probably the biggest flea market in Madrid. On Sunday during outdoor market hours, the atmosphere here is insane. The avenue turns into one massive garage sale that makes the _Antiques Roadshow_ look like child's play. Each little side-street branch has vendors that focus around some similar good, like "bird street" or the "painting street" and another selling collectibles like trading cards. So cruise the main boulevard, but don't forget to wander off!
It's easy to get distracted with all the energy, but keep a hand on your wallet at all times. If you're going to get pickpocketed, it'll happen here.
Free, Sun 09:00-15:00, Plaza de Cascorro, La Latina, Metro: La Latina
##### **Mercado San Miguel**
Mercado San Miguel is Spain's answer to markets like Whole Foods and Trader Joe's. This food market has all sorts of unforgettable cuisine that you can buy to take home, and many shops will cook up your ingredients right in front of you. San Miguel is worth the trip no matter where you're staying in town.
Free, Sun-Wed 10:00-24:00, Thurs-Sat 10:00-02:00, Plaza de San Miguel, Sol, Metro: Ópera
##### **Mercado Puerta de Toledo**
What used to be a bustling fish market is now a shopper's paradise, as the Mercado Puerta de Toledo is chock full of galleries, fashion retailers, and shops, along with many trendy cafés, pubs, and restaurants.
Free, 10:30-20:30 daily, Sun til 14:30, Ronda de Toledo 1, La Latina, Metro: Puerta de Toledo
##### **Cuesta de Moyano**
If you're into old books, this is the market to hit up. Book collectors from all over the world come to barter in this collection of over 30 outdoor stalls.
Free, open daily (hours vary stall to stall), Calle Claudio Moyano, Museum District, Metro: Atocha
### **TOP PARKS & RECREATION**
#### **PARKS**
##### **Retiro Park**
This park, directly east of the Museum District, is full of beautiful gardens, sculptures, buildings, and a man-made lake. It's a great place to take a break during your busy day of sightseeing. A highlight is the Crystal Palace, a massive steel and glass greenhouse in the industrialist style so popular at the turn of the 20th century. Don't miss the dramatic _Statue of the Fallen Angel_ either—it's the only public statue in Europe dedicated to the devil.
Free, daily 06:00-22:00, Plaza de la Independencia, Metro: Ibiza
##### **Botanical Gardens**
Bordering the Prado and Retiro Park, the affordable Botanical Gardens are worth popping into for a stroll away from the hustle and bustle of the capital. With over 30,000 plants and 1,500 trees, the noise and stress of the city fades away and you can find numerous prime locations for a little picnic to rest and recharge.
€3, Mon-Sat 09:30-14:00 and 15:30-18:00, Plaza de Murillo 2, Museum District, +34 914 203 017, rjb.csic.es, Metro: Atocha
##### **Madrid Río**
On the west side of town, Madrid's river walk is an excellent way to get away from the crowds, as it doesn't seem to be on the tourist radar quite yet. Head over to the governmental and office district of Príncipe Pío, just behind the Royal Palace and Jardines de Sabatini, for a peaceful walk along the river. You can also rent bikes and cruise farther on the newly laid meandering paths, which stretch for miles.
Free, always open, Greater Madrid, Metro: Intercambiador de Príncipe Pío
#### **VIEWPOINTS**
In addition to the **Teleférico Madrid,** a couple of other venues in Madrid offer sweeping views of the city.
##### **El Corte Inglés**
This large shopping center doubles as one of the best viewpoints in Madrid. Ride the escalators up to the ninth floor and relax in the café that overlooks the city skyline.
Free, Mon-Sat 10:00-22:00, 11:00-21:00, Sun Plaza de Callao 2, Sol, +34 913 79 80 00, elcorteingles.es, Metro: Santo Domingo or Gran Vía
##### **Circulo de Bellas Artes Terraza**
Find a classy terrace with bar and café at the top of Madrid's Fine Arts Association building. The view from this city-center location is one of my favorites, and it doubles as a popular nightlife venue after the sun goes down.
€4, Mon-Thurs 09:00-02:00, Fri 09:00-02:30, Sat-Sun 11:00-02:30, Alcalá 42, Sol, +34 913 892 500, circulobellasartes.com/azotea, Metro: Sol
### **TOP TOURS**
#### **Sandeman's New Europe Walking Tours**
After years of battling the Spanish tourist authorities, Sandeman's walking tours have finally restarted in Spain with gusto. Hop on any of their free and fun but scripted and tip-based walking tours. They leave daily at 10:00, 11:00, and 14:00 from Plaza Mayor. You can count on a several-hour stroll picking up tons of entertaining fun facts and trivia, along with frequent plugs for their other paid tours. Pub crawls are also available.
Free, daily, tours leave from Plaza Mayor, Sol, newmadrid-tours.com, info@neweuropetours.eu, Metro: Ópera
#### **Spanish Cooking Classes: Paella, Gazpacho, & Sangria**
I always say the best way to experience a new culture is through the stomach! If you enjoy hands-on cooking experiences, this is your top option in town. Come out for a crash course in some of Spain's favorite dishes. Choose from their selection of classes, from paella to tapas and even wine-tasting. Kick off your experience with a visit to the local market to pick up ingredients. Your friendly chef teacher can help even the clumsiest feel right at home in their modern kitchen facilities. These guys have their act together, and you get a fun recipe book as a keepsake from your visit. Classes take about four hours and start at 10:00. Book online ahead of time.
Options from €70, Mon-Sat, Calle de Moratin 11, Sol, +34 910 115 154, cookingpoint.es/classes, info@cookingpoint.es, Metro: Antón Martín
#### **Letango Tours**
Carlos and Jenn, a husband and wife couple, run this upscale, custom tour operations outfit. While their price points set them beyond most backpackers' budgets, they're definitely worth a look for families, or for a group of friends for excursions beyond the city of Madrid. Letango offers options for multiday excursions all around the Iberian Peninsula, with options to focus on history, culture, and food.
Multi-day trips from €300, +34 661 752 458, letango.com, tours@letango.com
### **TOP HOSTELS**
You've got a wide selection of well-located budget accommodations around Madrid. It's good to search by the district you identify best with. Personally, I love to drill down to **Malasaña** for its proximity to the trendy, fun, and casual restaurants and nightlife. Others may prefer the hustle and bustle of **Sol** , and still others may enjoy the extensive food and nightlife options in **Chueca**.
For more hostel choices, visit **hostelworld.com** to find the best deals. With the economic situation in Spain these days, many locals are turning to sites like **airbnb.com** to generate a little extra income by renting out their spare rooms. Private apartment options are also worth considering. Three-star hotels in Madrid range €80-120 per night.
#### **Las Musas Residence**
The simple and spartan Las Musas Residence is in the heart of the Lavapies neighborhood, where you'll find recommended restaurants, cafés, and nightlife. The hostel has big rooms for about €20 per night and sports all your basic amenities, including a kitchen. The free sangria and drinking game nights turn the party up and draw backpackers like moths to light.
€18, 24-hour reception, laundry facilities, free Wi-Fi, common room, lockers, Calle Jesús y María 12, Sol, +34 915 394 984, lasmusasresidence.com, info@lasmusasresidence.com, Metro: Tirso de Molina
#### **Sungate Hostel**
Sungate is loved by all who stay for the awesome atmosphere that the staff creates. They give this clean, centrally located hostel that extra little something that makes it feel like home away from home rather than just a place to crash. The 24-hour reception means you can check in any time of the day, and there will always be people to recommend nearby restaurants and to organize activities. It's just up the hill from Plaza Puerta del Sol, so you're right in the thick of it whether you're looking for churros or _chupitos_ (shots).
Private twins from €30, 6-bed dorms from €18, Calle de Carmen 16, Sol, +34 910 236 806, sungatehostel.com, Metro: Sol
#### **MuchoMadrid**
Offering only twins, triples, and quads, these budget accommodations are for those who appreciate both their fun and their sleep. Done up in a clean-yet-funky design, the bright rooms are comfortable and quiet. There's a well-appointed kitchen, but the hostel lacks a social area, so if meeting other cute backpackers is your goal, you may want to consider other options.
4-bed dorms from €20, Gran Vía 59, Malasaña, +34 915 592 350, muchomadrid.com, Metro: Santo Domingo
#### **Cat's Hostel**
A hostel with marble pillars, ornate arches, and colorful patterned walls, Cat's Hostel is a uniquely wonderful place to stay. It also offers a party atmosphere with its own bar that's open late, along with a free breakfast to help fuel you the next morning.
€14-18, 24-hour reception, free Wi-Fi, laundry facilities, bar, free breakfast, lockers, computer access, Calle Cañizares 6, Sol, +34 913 692 807, catshostel.com, info@catshostel.com, Metro: Antón Martín
#### **Room007 Hostel: Chueca**
This is a large hostel with a couple hundred beds and good-value on-site bar and restaurant. Social events often include music nights and bar crawls. The rooftop terrace is a favorite too. This is one branch of a multi-location hostel chain in Madrid that has it down to a science. You'll find Wi-Fi everywhere (though weaker in the rooms), good-value breakfast, great city-center locations, and helpful staff facilitating a social environment. Hostels are done up in a sort of shabby chic decor that makes you feel like you're in a magazine shoot from time to time.
11-bed dorm from €18, Calle Hortaleza 74, Chueca, +34 913 688 111, room007.com/es, Metro: Chueca
#### **Room007 Hostel: Ventura**
This branch of Room007, centrally located directly between the Prado Museum and Plaza Puerta del Sol, shares many of the same attributes as the chain's Chueca hostel. This location is smaller and more personal, with more than 100 beds split across four- and eight-bed dorms. The reception will help you make the most of your time with a ton of sightseeing and activity recommendations.
4- and 8-bed dorms from €19, Ventura de la Vega 5, Sol, +34 914 204 481, room007.com/es, Metro: Sol
### **TRANSPORTATION**
#### **GETTING THERE & AWAY**
Luggage storage facilities are available at all major train stations and at the airport.
##### **Plane**
The massive **Madrid-Barajas Airport** (MAD, aeropuertomadrid-barajas.com) is the main international airport serving the city, and is your only option if you decide to arrive by air. (You may find options for Barcelona and Girona, but remember, those are more than four hours away!). You can connect into the center from the airport via metro, bus, or train.
**Metro** trains leave every 5 minutes 06:00-02:00 and depart from Terminal T2 and Terminal T4. Take **line 8,** which you will probably want to ride to the end stop, Nuevos Ministerios. The approximately 15-minute trip will cost you €5. Consider purchasing a multiday metro pass at the machines just before you enter the metro so it will include this first trip into town.
**T** he **Exprés Aeropuerto** (Airport Express) is a 24-hour bus service that has only three stops: O'Donnell, Plaza de Cibeles, and Atocha (this last stop only 06:00-23:30). Check the location of your hostel for your most practical hop-off point. The buses leave every 15 minutes from Terminals T1, T2, and T3, with a journey time of about 40 minutes. Purchase your ticket (€5) on board.
The **C1 train line** departs from Terminal T4 for a 10-minute journey into Madrid (€2.40). This train line has connections to Chamartin, Nuevos Ministerios, Atocha, and Príncipe Pío stations.
##### **Train**
Trains between Barcelona and Madrid run often and take about three hours. If you purchase in advance through **renfe.com,** tickets can be as cheap as €50, but they climb past €150 closer to the date and during peak travel times. For the cheapest options, avoid peak commuter times (early morning, late afternoon).
Madrid has two main train stations: **Atocha,** near the Reina Sofia, and **Chamartin,** about eight kilometers north of Puerta del Sol. Both main stations are well connected with the local metro system. You should have no trouble getting to your hostel. There are secondary stations as well, so always double-check your departure station before heading to catch your train. Other train stations include **Nuevos Ministerios, Príncipe Pío, Delicias, Pirámides,** and **Méndez Alvaro**. There is also the **Recoletos** station, though it is not connected with the metro. Trains run generally 06:00-23:00 and depart frequently for both national and international destinations. If you need to book in person, allow at least an hour and a half, as lines can be long and slow.
##### **Bus**
Spain has an extensive network of bus lines that can get you across the country. Of the main bus companies, I recommend **Alsa** (alsa.es/en) and **Eurolines** (eurolines.es/en). Remember, though; it's a big country! So you'll have to weigh the cheaper price you'll usually get against the additional time spent en route. If arriving by bus, you will be dropped off at **Méndez Alvaro** (Estación del Sur) bus station, Madrid's main bus station, or **Avenida de América** bus station. Both are very well connected, making it easy to get to anywhere you need to go in the city.
##### **Car**
It's a serious six-hour drive from Barcelona, going on the EU-funded and toll-supported E-90 freeway to Madrid. Having a car does give you the flexibility to stop along the way or drive along the coast. Madrid itself is a car city, but it has some intense traffic. There isn't any sort of park-and-ride system, so your best bet is to park on the outskirts of the city within a few blocks of a metro stop and take the metro in to avoid hefty parking fees during your visit. Of course, lock your doors, and leave nothing of value in the car.
#### **GETTING AROUND**
Madrid's excellent **public transportation** network makes it easy to get across this sprawling city. Be sure to consider the great-value short-term **day passes,** which get you onto the integrated bus and metro network and can save money if you plan on using the metro even just a few times a day. Spring for the one-day (€8.40) two-day (€14.20), three-day (€18.40), or five-day (€26.80) multiday pass at the entrance to the metro from the airport, as the local pass covers all of downtown Madrid, including your first transfer into town.
##### **Metro**
Madrid's **metro system** (metromadrid.es/en) comprises over 238 stations and 13 lines, making connections across town fast and easy. The metro runs 06:00-01:30, with trains coming every few minutes during rush hour (07:30-09:00) and every five minutes during regular hours. A single metro ticket costs €1.50.
##### **Bus**
Madrid has a new line of hydrogen-fueled (and air-conditioned!) buses. Buses run every 5-15 minutes 06:00-02:00. And for you late-nighters, night buses operate 23:45-06:00, with buses coming every 15-30 minutes.
##### **Car**
Madrid is much better for cars than most other European cities. Its wide, modern boulevards make getting around much easier than topsy-turvy medieval streets do. If you'll only be visiting Madrid, there's no need to rent a car thanks to the extensive local transportation networks. Even connecting between major cities is easy with trains and buses. Consider renting a car only if you want to get farther out into the countryside, where bus and train connections are more difficult.
##### **Taxi**
With more than 15,000 taxis serving the city, you won't have a problem flagging one down. Look for the white taxis with a diagonal red band on the side door with the city's emblem, featuring a bear, tree, and crown just above the stripe. With such a great public transportation system in place, only use these if you need to get across town in a jiffy or are coming back from the clubs before the next morning's rush hour. At the time of writing, Uber had not yet made inroads into Madrid.
##### **Bicycle**
With the big streets and fast-driving cars, Madrid is notoriously unfriendly for cycling, and biking around town is not recommended. The one exception is Retiro Park, a great place to ride around and explore. Find several bike shops on the east side of the park.
### **DAY TRIPS**
#### **Toledo**
Toledo was Spain's capital for nearly 1,000 years. Today, it's a beautifully preserved stone town perched on top of a rocky outcropping. It's touristy and kitschy but also wonderful to explore for an afternoon. One highlight is the **Cathedral of Toledo** (€8, Mon-Sat 10:00-18:00, Sun 14:00-18:00, Calle Cardenal Cisneros 1, +34 925 22 22 41, catedralprimada.es). Stepping into this church, exquisitely executed in high gothic style and dripping in golden accents with baroque embellishments, is truly a spiritual experience. The **Mirador del Valle** is a breathtaking vista overlooking the horseshoe bend in the river from which the rock that Toledo sits on rises. Time your visit on a clear day and hop the river via the bridge on the east side of town, then continue south to climb the hill to the viewpoint. You'll know it when you see it. For your museum fix, don't miss the **Alcázar of Toledo** (€5, Thurs-Tues 11:00-17:00, Calle Unión, +34 925 23 88 00, museo.ejercito.es), the palace that dominates the city's skyline. Originally built as a fort and Roman palace, the Alcázar of Toledo served as a major standoff point in the Spanish Civil War between the loyalists and republicans. Today, it's a fascinating military history museum. It's interesting to tour this grand structure, which dates back hundreds of years. Trains leave from Atocha station more often than hourly and take about 40 minutes. Fares are about €13 one-way. Purchase tickets from the machines at the station.
### **HELP!**
#### **Tourist Information Centers**
Find Madrid's TI conveniently located steps from Puerto del Sol (daily 09:30-20:30, Plaza Mayor 27).
#### **Pickpockets & Scams**
Madrid has a relatively low crime rate; however, always keep a close watch over your bags and other items, as pickpocketing and street theft can occur. Be especially wary of crowded areas such as train stations and crowded tourist attractions. In Spain, it's better to carry only what you need for the day rather than carrying all your cards and cash with you. Leave what you don't need in the safe at the hostel. The area just south of La Latina is known to have a higher density of pickpockets and thieves. Avoid going there alone at night, and keep your wits about you during the day, too.
#### **Emergencies**
In an emergency, dial 112. You can also dial 091 for police.
#### **Hospital**
Hospital Ruber Internacional
Calle de la Masó 38
+34 913 875 000
#### **US Embassy**
Calle Serrano 75
+34 915 872 240
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Barcelona offers white sandy beaches by day and glitzy dance clubs by night. This town has it all: surreal modernist architecture, bustling markets with fresh local produce, and nightlife that pops till sunrise. Barcelona is the capital of Catalonia, Spain's northeasternmost territory, and the regional cultural pride is fierce. Get ready to soak in the culture and live like a local by finding your own perfect siesta to fiesta ratio.
### **BARCELONA 101**
Founded as a military encampment and then an ancient Roman walled port town with—count 'em—72 towers, Barcino, as it was known, was handed back and forth between numerous conquering powers throughout the centuries. As a result, citizens developed a unique cultural identity that reflected the amalgamation of the cultures that controlled the city over the years. Catalonia was consolidated under one crown in the 12th century and grew wealthy from its rich sea trade. When Ferdinand (king of Catalonia) and Isabella (queen of Castile, aka the rest of Spain), famous for the Spanish Inquisition and for sponsoring Columbus, married in 1496, political power gradually shifted to Madrid. Buffeted by war and plagues over the next several hundred years, Barcelona endured a slow decline.
The 20th century kicked off with an ambitious city-planning project epitomized by a unique architectural style: modernism, also known as Catalan art nouveau. It left a significant mark on the city you see today. From palaces to private residences (Block of Discord), and light posts (Plaça Reial) to the grandest, most ambitious cathedral project in the world (Sagrada Familia), Barcelona was architect Antoni Gaudí's canvas, over which he enthusiastically spread his nationalist pride. Simultaneously, the bloody and tragic Spanish Civil War followed. Many civilians fled to France as fascist dictator Franco came to power in 1939 and began actively suppressing the Catalan language, traditions, and culture. The only way to express regional patriotism was by cheering for the soccer club, FC Barcelona, which at least partly explains the passion of the team's modern fan base. For Catalans, _fútbol_ was an opportunity to keep their quieted culture alive. After 36 long years, Franco reinstated the Spanish monarchy, putting King Juan Carlos I, who promised to continue Franco's dictatorial style, on the throne. But upon Franco's death, the king returned the country to a democratic system.
In 1992, Barcelona hosted the Olympics and found itself back on the international stage. The city was revitalized with a brand-new metro system, beautiful parks, a cleaned-up medieval center, and four kilometers of brand-new white-sand beaches. (Fun facts: The sand was imported from Egypt, and the metro map mirrors the colors of the Olympic rings.)
Today, Barcelona is Spain's second largest city after Madrid and is the capital of Spain's richest region, Catalonia. Children here learn Catalan first and Castilian Spanish second. The pride and identity of Catalonia is as strong as ever. This pride has fostered both energy and tension, as the people of Catalonia decide which future they want for their children: an independent state, or one that continues to pay into the system to support the rest of Spain.
### **PLAN AHEAD**
#### **RESERVATIONS**
Reservations are recommended for the following sights:
**Sagrada Familia** (sagradafamilia.cat)
**Picasso Museum** (museupicasso.bcn.cat/en, free on Sun afternoon!)
#### **LOCAL HAPPENINGS**
##### **Gay Pride**
Pride Barcelona festivities take place around the third weekend of June each year. Expect a full two weeks of music, parades, parties, debates, and lectures. Find the agenda and more at pridebarcelona.org. Book ahead for accommodations—they'll sell out.
##### **Siestas**
Siestas are real, people! Barcelona has four rush hour times every day: not only in the morning and evening but also at siesta time, when all the shops close and everyone runs home for their two- to four-hour-long siesta after lunch. This means that most places close around 14:00 and reopen around 18:30. If you need to purchase something, remember to keep this in mind.
##### **Holidays**
If a public holiday falls on a Thursday—or sometimes, even a Wednesday—people here bridge the holiday into the weekend, giving themselves more days off. It's great for them, but not so great for tourists. On these public national holidays, all businesses will be closed:
**January 1,** New Year's Day
**March 29,** Good Friday
**May 1,** Labor Day
**August 15,** the Assumption
**October 12,** National Holiday of Spain
**November 1,** All Saints' Day
**December 6,** Spanish Constitution Day
**December 25,** Christmas Day
**KNOW BEFORE YOU GO**
**KEY STATS & FIGURES**
**Currency:**
**Spain uses the euro (€);1 EUR = about 1.06 USD**
**Population:**
**1,620,943**
**Language:**
**Catalan and Spanish**
**Kilometers of beaches:**
**4.2**
**Nightclubs:**
**enough to keep you busy for years**
**CALIBRATE YOUR BUDGET**
**TYPICAL PRICES FOR:**
**Hostel dorm bed:**
**€14**
**Two-course dinner and drink:**
**€12**
**Pint of beer:**
**€3.50**
**Bicycle rental:**
**€10/half day**
**Single metro pass:**
**€2**
**MOVIES TO WATCH**
_Vicky Cristina Barcelona_ , _Barcelona_ , _The Passenger_
THREE DAY ITINERARY
Organize your visit to Barcelona chronologically. On your first day focus on the Old Town: the Barri Gotic and El Born. Then explore the Eixample, with its many beautiful examples of modernism, on day two. Save your third day for hiking or lounging on the beach.
**DAY 1: WELCOME TO BARCINO**
**MORNING**
Grab breakfast at your hostel or head to one of my favorite _churrerías_ (churro bakeries), **Churreria Manuel San Román,** located right in the Old Town, near the kick-off point for your free walking tour.
After breakfast, walk over to **Travel Bar** to catch a free three-hour walking tour for a casual introduction to the layout and history of the city. The tour begins at 11:00. You'll explore the **Barri Gotic, La Rambla,** and the **Cathedral of Barcelona** just to get started. Two important tips: Always watch your pockets in the tourist center, and remember to tip your tour guide!
Your guide will give you free time inside the famous **La Boqueria,** a market where you can grab a quick snack. Saddle up at any one of the countertop bars inside the market. Try out something that looks strange and foreign—you might surprise yourself with what you like!
**AFTERNOON**
After your tour, walk about 15 minutes to the **Picasso Museum** in El Born and spend a couple hours exploring. El Born is also home to the enticing **Chocolate Museum** and some tasty tapas at **Sagardi BCN Gotic.**
**EVENING**
Take a minute back at the hostel to freshen up, then head out to catch Travel Bar's three-hour **Paella & Sangria Cooking Class,** which meets daily 17:45 at Travel Bar. Make your reservations earlier in the day to confirm your spot. Dress to stay out because the class goes late.
**LATE**
Remember, Spaniards don't really kick off the night until the wee hours of the morning. Spend the time after your paella class in the nearby **Plaça Reial,** where you'll find some of my favorite nightlife attractions: **Los Tarantos Flamenco Show** and **Sidecar Factory Club.**
**DAY 2: THE EIXAMPLE & BEYOND**
**MORNING**
Get ready for a day of modernism with a large breakfast at your local **Pans and Company.** A tortilla sandwich with a _café con leche_ is a filling start to your day.
After breakfast, meet up for another free walking tour, this time with **Discover Walks** on the Passeig de Gràcia, in front of **Casa Batllo**. Their Modernism Walk, which meets at 10:30 Friday, Saturday, Sunday, and Monday, covers the **Block of Discord,** comprising a number of bizarre and noteworthy houses.
Your walk, which lasts three hours, will continue on through Barcelona's modern expansion—or "Eixample"—and finish at Gaudí's unfinished masterpiece, the **Sagrada Familia**. Pop in to any café that looks good along the way and ask for a _bocadillo,_ or sandwich. My go-to sandwich is a _chorizo con queso_ (sausage and cheese) or _jamón con queso_ (ham and cheese). Ask your guide for a good spot.
**AFTERNOON**
After ogling the Sagrada Familia for a couple hours, take bus 92 up to Gaudí's famous, beautifully failed development, **Parc Güell.** Alternatively, take bus V21 from the north side of the open square adjacent the Sagrada Familia downhill to the **beach** for some Vitamin D! In the late afternoon, head back to your hostel for a quick siesta.
**EVENING**
Leave your hostel to crawl some of the best tapas bars in the city, along **Carrer de la Mercè.** Formerly a gritty street paced by Barcelona's prostitutes and sailors, it is now known for its authentic tapas flavor and scene.
**LATE**
From Carrer de la Mercè, split a cab with friends into the Eixample to **Chupitos** —the famous shot bar—and **Dow Jones** —a stock market-esque bar—for a pair of fun and shamelessly touristy experiences.
Now fully loaded, head to the **Puerto Olimpico** for the glitzy nightlife that Barcelona is so famous for. Flag another cab back to the famous venues on the beach, including disco **Shoko** and its posher sister, **Opium Mar**. Rage till late and watch the sunrise the next morning.
**DAY 3: CHOICES, CHOICES, CHOICES**
Later flight on your last day? If it's Sunday, head to the **Cathedral of Barcelona** to catch the traditional **Sardana dances**.
Much later flight? Pack a lunch and head for a hike up **Montjuic** to take in the panoramic views of the city and Mediterranean Sea. Or enjoy a lazy day soaking in the sun on **Platja Barceloneta** or **Platja Nova Mar Bella,** two of Barcelona's best beaches.
If you've got the entire day and are spending a fourth night, consider a **day trip to Montserrat** for a strenuous and rewarding day of hiking outside the city.
### **TOP NEIGHBORHOODS**
Barcelona's Old Town is built on Roman foundations with narrow, winding streets. The Old Town comprises two main districts: the Barri Gotic and El Born. The **Barri Gotic** (Gothic Quarter), with Plaça Catalonia at the top and the famous street of La Rambla running through the middle, filled the old medieval walls of the town. This neighborhood manages to balance its touristy nature with some excellent finds for food, culture, and entertainment. This is where you'll find the Cathedral of Barcelona, along with Plaça Reial, the town's central square. Just north of the Barri Gotic is trendy **El Born,** with classy restaurants and posh cocktail bars to suit the young professional set (and in-the-know tourists). El Born is home to the Picasso Museum as well as some tasty tapas.
Surrounding the Old Town up the slightly inclined landscape is the **Eixample** , or expansion, so named because it's the world's best-executed example of true city planning, with a modern grid layout. You'll find a number of key sights here, particularly those that relate to modernist architecture, including the Sagrada Familia and the Block of Discord.
**Barceloneta** is on the east side of town, with El Born just inland, the marina to the south, and Barcelona's long stretch of beaches and boardwalk leading north along the water line. Here you'll find surf shops, local bars and cafés, and pricier restaurants along the boardwalk. Along the north end is **Puerto Olimpico,** the town's port, which is the center of Barcelona's clubbing scene.
### **TOP SIGHTS**
#### **La Rambla**
Take a downhill stroll from Plaça de Catalonia (Barcelona's proud, most central square) down La Rambla, Barcelona's most popular and busiest series of contiguous streets. This experience is at the heart of any tourist's visit to Barcelona. In one kilometer you've got vendors selling birds, flower stalls, a world-famous market (La Boqueria), beautiful baroque churches, works by Gaudí and other famous architects, dozens of street performers, a square populated by the wealthiest traders from Barcelona's golden age, and churros galore. The street referred to as La Rambla actually comprises four streets, all following what used to be a spring coming from the surrounding hills: Rambla de Canaletes, Rambla dels Estudis, Rambla de Sant Josep, Rambla dels Caputxins, Rambla de Santa Mònica. Most of these portions of La Rambla are named after buildings or sites that don't exist any more, but that's the order starting from Plaça Catalonia and down.
Explore the winding streets leading off from La Rambla on either side and soak in the dense history of the old Roman town that is the Barri Gotic. A walking tour is a great way to get your bearings of this neighborhood.
Always keep your wits about you and watch out for scam games and pickpockets who rove the entire length of the Rambla, preying on bewildered tourists at all hours of the day and night. It can get a bit seedy late into the night after 03:00.
Free, always open, Barri Gotic, Metro: Catalunya
#### **Picasso Museum**
Besides Gaudí, another famous artist hails from Barcelona: the painter, Pablo Picasso. A museum dedicated to this revolutionary artist resides in Barcelona's trendy El Born district. The great modern artist called Barcelona home, and the impressive museum is juxtaposed against the backdrop of some classical Barcelona palaces. The museum features an extensive collection of works progressing through his various stages in chronological order. They say that Picasso, when a child, painted as a master in photo-realistic pieces, and when an adult, he painted as a child, with geometric shapes, disagreeable coloring—in the opinion of contemporaries—and stark outlines. A survey of the rooms will take visitors about 90 minutes.
€11-14, €7 students, free all day first Sun of the month and other Sun afternoon, Tues-Sun 09:00-19:00, till 21:30 on Thurs, closed Mon, Carrer Montcada 15-23, El Born, +34 93 256 30 00, museupicasso.bcn.cat/en, Metro: Jaume I
#### **Sagrada Familia**
This church, designed by Gaudí, is my favorite sight in all of Europe. If you see nothing else on your visit to Barcelona, you've got to see the Sagrada Familia, an ongoing construction site since the late 19th century. Gaudí laid the foundations, began living in one of the towers, and did his best to recruit donors and patrons to support his colossal project from 1882 onward.
**ACT LIKE A LOCAL**
**Getting on Barcelona Time**
To fully enjoy Barcelona, it's important to adapt to the timeline that Spaniards live by here. If you try to follow the schedule you keep back home, you'll likely only get frustrated and miss out on incredible experiences and opportunities. This is a timeline of when Spaniards eat, sleep, and go out:
**08:30:** Grab a light breakfast of a croissant and _café con leche_ (coffee with milk).
**09:00-12:30:** Locals head to work! For visitors, this is a good time to hit the sights.
**13:00-17:00:** Stores close down for lunch and siesta. A side effect: double the traffic in a major city where everyone commutes twice a day. You wouldn't be missing much in the streets if you decided to take a siesta after a large Spanish lunch.
**19:00-21:00:** Evening stroll time. The city is still sleepy as stores are opening back up, and people are roaming the streets.
**21:00-23:00:** Dinnertime. Take your time to enjoy it. Spaniards take their dinner slowly and socially.
**23:00-01:00:** Pregaming with friends, either in parks or back at the flat.
**01:00-03:00:** Time for the bars. Get the drinks and pound the vodka energies to get ready for the club.
**03:00 till you can't keep your eyes open:** Club like there's no tomorrow. Stumble out of the _discotecas_ at Puerto Olimpico onto the beach and watch the sunrise.
The church has grown in fits and starts because it is all publicly funded: They build when they have money. When they run out, they take collections again. Construction is slated to be completed by 2025. Let's cross our fingers! Because it's an ongoing construction project that has lasted over 120 years, it's worth going back to again and again to see the progress local builders have made since Gaudí's death in 1924.
Both the exterior and interior are finally taking shape. Two completed facades face north—where you'll find the Nativity facade—and south—where the geometric Passion facade dominates. These facades communicate the phases of Jesus' life. Look closely at the Nativity facade, with the Holy Family in the center and the farm animals surrounding in adoration. Don't miss the turtles at the base of the columns. This is the side Gaudí toiled on during his life. The more modern Passion facade walks viewers through the 13 stages of the cross, from flagellation at the pillar to crucifixion to rising from the dead.
Step into the church, and it feels like you've stepped into a giant rainforest complete with a canopy. The coloring is miraculous thanks to the detailed stained glasswork throughout. Gaudí never got to see the church at this stage, but one moment in here, and you can begin to contemplate the scale of his dreams. Don't miss the workshop to the side of the church or the small architectural museum downstairs in the crypt.
Plan to send about 90 minutes exploring, with more time if you choose to go up the towers. Ticket sales end 15 minutes before closing time.
€15 adult, €13 student, entry to the towers €4.50 extra, Apr-Sept daily 09:00-20:00, daily Oct-Mar 09:00-19:00, Carrer de Mallorca 401, Eixample, +34 935 13 20 60, sagradafamilia.org, Metro: Sagrada Familia
#### **Block of Discord**
This single block of ostentatious homes on Passeig de Gràcia, featuring **Casa Amatller, Casa Lleo Morera,** and **Casa Batllo** , is a perfect example of modernist architects attempting to outdo one another. Each house has a distinct personality. The name of the city block reflects the jarring appearance of each of these homes when lined up next to each other. Paid entry into these modernist houses is possible, but casual observers are often content with checking out the exterior for free.
Free, always open, Passeig de Gràcia, Eixample, Metro: Passeig de Gràcia
#### **Casa Batllo**
Known as the dragon house, this fanciful house built for the Batllo family depicts the story of George and the Dragon, and it's loaded with symbolism. The columns on the exterior represent the past victims of the dragon, with the balconies made of skulls. The tiles of the roof are clearly the scales of the back of the dragon, with the spire of the turret representing the sword plunged into the serpent's back. The interior is just as detailed, with beautifully carved wooden door frames, styled light fixtures, and even an atrium paneled with darker tiles at the top, cooling to lighter tiles near the ground floor to help distribute natural light across all rooms evenly. Gaudí was an architectural master and also possibly the world's most intense micro-manager to achieve such detail across all of his projects.
€21.50, €18.50 students, daily 09:00-21:00, Passeig de Gràcia 43, Eixample, +34 932 16 03 06, casabatllo.es, Metro: Passeig de Gràcia
#### **Casa Amatller**
Josep Puig i Cadafalch designed and built the stepped, Minecraft-style facade of Casa Amatller along the row of competing houses for the Amatller family. They made it rich off the chocolate trade, and the ground floor door is sometimes open for visitors to pop in and view the interior. Go all the way back to find the chocolate shop.
Free, €15 for guided tour, daily 11:00-18:00, English tours at 11:00 and 15:00, Passeig de Gràcia 41, Eixample, Metro: Passeig de Gràcia
#### **Casa Lleo Morera**
Bursting with modernist decoration, this corner building is really a collaboration between a number of artists and architects. Their various styles can be seen in the ornate balconies, numerous arches, and diverse architectural accents. See if you can't pick out some of the high-tech inventions that were just coming to light at the time this building was erected in 1906. And don't miss the mulberry tree ( _morera_ ), this family's namesake.
€15 for 70-minute English tour, €12 for 45-minute tour, Tues-Sun, check website for tour times, purchase tickets online or in person at the cultural center Palau de la Virreina, Passeig de Gràcia 35, Eixample, Metro: Passeig de Gràcia
#### **Casa Milá**
Casa Milá is also known as La Pedrera (Stone Quarry), so named because of the striking similarity of its shape with that of a quarry. This, another masterpiece of Antoni Gaudí's unique and innovative modernist architecture, appears like a sculpted layer cake on the corner of Passeig de Gràcia and Carrer de Provença. Peer up at the rooftop, where you'll notice a series of twisted chimneys, sculpted in such a way that they look like the storm troopers from _Star Wars_.
€20.50, €16.50 students, daily Mar-Oct 09:00-20:00, daily Nov-Feb 9:00-18:30, last entry 30 minutes before closing, Provença, 261-265, Eixample, +34 902 20 21 38, lapedrera.com, Metro: Diagonal
**GAGA FOR GAUDÍ**
Antoni Gaudí is easily one of Barcelona's most famous denizens. His profession: architecture. You'll see his thumbprint on almost every block across the city, in the form of churches, private residences and palaces, sidewalks, benches, parks, urban planning, and even light posts. As is clear from the **Sagrada Familia,** the most incredible church ever (being) built, down to minute details on the facades of his projects and even street tiles, Gaudí was truly an architectural prodigy who was well ahead of his time. The **Block of Discord** (which includes **Casa Batllo** and **Casa Milá** ) and **Parc Güell** are some of his most famous sights in the city. He also designed the **lampposts in Placa Reial.**
#### **Plaça Reial**
This picturesque, renaissance-style city center square harkens back to Barcelona's explosive architectural period in the late 19th century. Entitled the Royal Square, this is where Barcelona's elite would come for a stroll and to admire each other's palatial city homes. Today, us lowlifes can enjoy the scene at any time of day or night. During the day, enjoy _café con leche_ at one of the many cafés that line the square. Note the lampposts, which were designed by Gaudí. Perhaps join a pickup _fútbol_ game with the kids running around, but be warned: They may school you! Each Sunday a stamp and coin market takes over the square. When the sun goes down numerous bars, clubs, and shows line the square and the streets.
Free, always open, Barri Gotic, Metro: Liceu
#### **Los Tarantos Flamenco Show**
A typical dance originating in the south of Spain, flamenco drips with passion and culture. Los Tarantos captures this Spanish tradition in a perfectly succinct and varied show featuring tap dancers, box-drummers, guitarists, and vocalists who belt out songs ranging from upbeat to morose melancholy. You'll definitely walk out of this show saying "wow." Even with three shows a night, the quality and energy of each show are impressively consistent, and I love the venue for its intimate and casual atmosphere.
€15, 30-minute shows daily 20:30, 21:30, 22:30, Plaça Reial 17, Barri Gotic, +34 933 01 77 56, flamencotickets.com/los-tarantos-barcelona, Metro: Liceu
#### **Cathedral of Barcelona**
Not to be confused with the much more modern Sagrada Familia, the Cathedral of Barcelona is located right against what would have been the main entry gate into the old Roman town. Construction began in the 1200s and was completed in 1448 with a rather austere facade. In the 1800s, a much more boisterous neo-gothic facade, a style popular at the time, was added. That's what we take in today when we see the cathedral, facing proudly out onto the Placita de la Seu. Take a lap around the exterior to take in the funky gargoyles, which are both decorative and functional. Try to find the grasshopper and the unicorn.
The cathedral is dedicated to Barcelona's original patron saint, Saint Eulalia, who met a gruesome end from the occupying Romans for not recanting her Christian faith. As a result, the centurions inflicted 13 different kinds of torture on her, including rolling her downhill in a barrel stuffed with sharp objects and knives. This was followed by both crucifixion and decapitation. Saint Eulalia is buried in the crypt underneath the cathedral.
The interior is rather dark thanks to the thick walls that hold up the impressive gothic arch work. Inside you'll find richly decorated chapels, the crypt of Saint Eulalia, and a cloister with 13 white geese, remembering Saint Eulalia's age at the time of her martyrdom. Entry is usually free, but visits to the cloister and roof cost a few euros.
Free entry Mon-Fri 08:00-12:45 and 17:15-19:30, free entry Sat 08:00-12:45 and 17:15-20:00, free entry Sun and holidays 08:00-13:45 and 17:15-20:00; during free times museum is €3, terrace is €3, and choir is €3; entry with €7 donation Mon-Fri 13:00-17:00, Sat 13:00-17:30, and Sun 14:00-17:00, Pla de la Seu, Barri Gotic, +34 933 428 262, catedralbcn.org, Metro: Jaume I
#### **Sardana Dances**
Every Sunday, Sardana dances take over the square in front of Barcelona's cathedral. The dance is a Catalan tradition that celebrates community in the region. Dancers put their belongings in the middle and hold hands while dancing in circles to the beat of a beautiful and traditional series of tunes.
Free, Sun 12:00, sometimes Sat at 18:00, no dances in Aug, in front of the Cathedral of Barcelona, Pla de la Seu, Barri Gotic, Metro: Jaume I
#### **Chocolate Museum (Museu de la Xocolata)**
A favorite of those with a sweet tooth, this small museum explains the process of chocolate making with displays of sweet artwork. The museum also features activities, such as wine tasting and "blind" chocolate tasting, but you must book ahead online for these. Your €5 entry ticket is actually a chocolate bar. The museum can be tackled in about 45 minutes.
€5, Mon-Sat 10:00-19:00, summer til 20:00, Sun 10:00-15:00, Carrer del Cornerc 36, El Born, +34 932 68 78 78, museuxocolata.cat, Metro: Arc de Triomf or Jaume I
#### **Camp Nou Stadium**
If soccer were a religion, the Camp Nou soccer stadium would be one of the world's greatest cathedrals. Barcelona rooted for the home team, FC Barcelona, throughout Francisco Franco's oppressive dictatorship. During the later half of the 20th century, supporting the team was one of the few legal ways to wear your Catalan colors and express your Catalan pride. Today, Barcelona essentially shuts down every time the FC Barcelona takes the field, whether they're home or away. You can take a self-led audio tour of the stadium and museum to see vast amount of memorabilia and the trophies amassed over the years by this dynasty.
€23, mid-Apr-early Oct Mon-Sat 10:00-20:00 (off-season until 18:30), Sun 10:00-14:30, shorter hours on and before game days, Carrer d'Aristides Maillol 12, Greater Barcelona, +34 902 18 99 00, fcbarcelona.com, Metro: L3 Palau Reial or Les Corts, L5 Collblanc or Badal
#### **EXTRA CREDIT**
##### **Palau Güell**
Palau Güell, one of Antoni Gaudí's early projects, is worth going inside if you're a huge modernism buff. You can tell this is the work of a budding master, but one who has not quite yet matured. For most, it's sufficient to peer in through the entryway to get a glimpse into another Gaudí interior. This palace was designed for the same patron family as Parc Güell, built uphill and farther out of town.
€12 adult, €8 students, Tues-Sun 10:15-17:30, Carrer Nou de la Rambla 3-5, Barri Gotic, +34 934 72 57 75, palauguell.cat, Metro: Liceu
### **TOP EATS**
Eating in Barcelona is easily one of the highlights of any visitor's experience. My favorite element is the tapas tradition. When you order tapas, your food arrives on sharable platters, or you pick and choose from a bar with preportioned, open-faced sandwich-style bites. Ingredients are primarily fresh tomatoes, seafood, bread, a little garlic, and fried treats like calamari, _tortilla española_ omelets, and _piementos de padrón_ peppers.
You'll catch some of Barcelona's most authentic Spanish tapas on **Carrer de la Mercè.** Do a lap up and back down this street, popping into a few places along the way. No menus necessary, just point at whatever you want, and you'll be dished up heaping plates that are great for family-style tapas.
Tipping is not expected, but rounding up to the next euro mark is appreciated. Be sure to check whether or not your bill says _"servei inclòs,"_ as many touristy restaurants sneak this 10-15 percent charge in there.
#### **Tasca el Corral**
This little hole in the wall on Carrer de la Mercè is a winner—and it's also a one-man show. On this stop, you've got three things to try: fresh _manchego_ , _chorizo al diablo_ (flaming sausage), and cider poured from on high. Watch the boss take the cider jug and pour over his shoulder into cups held far below. Order a round of these winners to kick off any crawl right.
€8 and up, Sun-Thurs 13:00-02:00, Fri-Sat 13:00-03:00, Carrer de la Mercè 17, Barri Gotic, +34 93 315 20 59, Metro: Jaume I or Barceloneta
#### **La Plata**
This is your place to toss back anchovies like popcorn at the movie theater. Get over your aversion and try some of these deliciously savory snacks. Get some _vino tinto_ (red wine) to wash it all back, enjoying the casual, social scene. The simple decor makes it clear this joint caters to the locals.
€8-20, Mon-Sat 09:00-15:30 and 18:30-23:00, Carrer de la Mercè 28, Barri Gotic, +34 933 15 10 09, Metro: Jaume I or Barceloneta
#### **Celta**
Celta is the anchor of any tapas crawl down Carrer de la Mercè. Stop in for a feast of _pan con tomate_ (bread with pureed tomato and olive oil), fresh _polpo_ (octopus), _patatas bravas_ (fried potatoes with a spicy ketchup sauce), and even _gambas_ (meaty jumbo prawns). This place and the waiters are always busy, so pop in, scope around a bit to see the dishes that look good to you, and have your order ready when you finally make eye contact with a member of the crew. Tables in the back are great for groups of friends.
**TAPAS CHECKLIST**
Major treats for any visitor coming to Barcelona are the tapas—snack-sized, finger food dishes allowing for a wide range of tastes in any one dinner. Tapas also allow diners to have a few dishes in one place before continuing their crawl elsewhere. Just like there are can't-miss sights for Europe's top cities, there are can't-miss flavors that every visitor must try once their first time in Barcelona. Consider this your delectable hit list:
**_Piementos de padrón:_** A plate of steaming, salted peppers that's made to order. Ordering these is like playing culinary Russian roulette: Most peppers aren't hot, but there's one per plate that will knock your socks off.
**_Polpo:_** Freshly cooked spiced octopus, usually presented on a wooden platter. If it's on the menu, look down the bar and find the heaping plate of tentacles that the chefs are dicing and slicing to order.
**_Tortilla española:_** Spain's delicious potato and egg omelet. Additional ingredients are often tossed in, along with salt and pepper, but it's just as good plain for breakfast, lunch, or a quick bite on the run. Get it in a _bocadilla_ (Spanish sandwich) for the best way to get full on €2.
**_Patatas bravas:_** Hot, freshly fried potato wedges doused in a slightly spicy ketchup-mayonnaise sauce. If you're trying to add some starch to round out the meal, this is a no-brainer.
**_Pa amb tomàquet:_** Toasted bread rubbed down with fresh garlic and tomato. This is my personal favorite side order.
**_Anchoa:_** Fresh anchovies with nothing but a slice of lemon on the side. Get over your inhibitions and order a plate to share between friends, and pop 'em like popcorn at the movies. Salty, crunchy, and washed down with a glass of _tinto_ (red wine)—what more could you want?
**_Jamón ibérico:_** Cured ham. Spain has delicious cured ham, some of the best in the world. You've got to try some while you're here. They come in different grades and prices based on the pigs they come from, their processing, their location, and their diet. Go for 100 grams of this, 100 grams of _queso manchego,_ and a baguette and you've got lunches for days.
**_Queso manchego:_** Spain's dry, salty cheese goes along with basically anything. Order 100 grams at the supermarket when you see some you like for a perfect afternoon snack. Don't be afraid to try (say _"probar"_ ) a few different selections before purchasing. They also contribute to some wonderful cheese plates.
**_Salmorejo:_** Creamy tomato soup, topped with diced _jamón_ (delicious) and generally consumed for breakfast. (I'll take it any time of day, though.)
**_Gazpacho:_** Tomato soup that tends to be a savory side dish to your lunch or dinner. Fresh tomatoes are tossed in with garlic, salt, spices, and a little bit of vinegar for the perfect zing. You really can't go wrong. Gazpacho is served cold, mainly during hot summer months. Gazpacho and _salmorejo_ are my favorite tomato soups in the world.
€4 and up, Tues-Sun 12:00-24:00, Carrer de la Mercè 9, Barri Gotic, +34 933 150 006, barcelta.com, Metro: Jaume I or Barceloneta
#### **Bar Celta Pulperia**
Pop into the new sister restaurant to Celta, across the street, for tapas sitting down at tables with your friends in a bright and clean setting. This place is newer, and as such feels slightly more done up than the others on your crawl.
€5 and up, daily 08:30-24:00, Carrer de Simó Oller 3, Barri Gotic, +34 933 15 00 06, barcelta.com, Metro: Jaume I or Barceloneta
#### **Sagardi BCN Gotic**
The Grupo Sagardi chain has numerous locations across Spain, with several in Barcelona alone, and all for a good reason: their incredible finger-lickin'-good tapas. Ask for a plate to start, and move down the line and pick whatever looks good to you on the bar. Go slow! Dishes are refreshed periodically, and hot dishes are paraded through every 15 minutes or so. Settle up your bill by turning in your toothpicks at the end of your meal. Their summer wine, _tinto de verano_ , is delicious. Get a similar experience at any of Grupo Sagardi's other operations, including **Basca Irati** (Carrer del Cardenal Casanas 17) and **Euskal Etxea** (Placeta de Montcada 1).
€4-20, daily 13:00-16:00 and 20:00-24:00, Carrer de l'Argenteria 62, El Born, +34 933 199 993, sagardi.com, Metro: Jaume I
#### **Bo de' Be**
Bo de' Be is famous among the student population in Barcelona for their overflowing sandwiches, delivered quickly at unbelievably cheap prices (only €3 or €4!). Grab one to go on your way to the beach, as it's right at the bottom of Via Laietana. Lines can be long, but the service is quick. Because there isn't much room to eat inside, most take their sandwiches to go.
€3-4, Mon-Fri 11:00-24:00, Sat-Sun 13:00-16:00, Carrer de la Fusteria 14, Barri Gotic, +34 936 674 945, Metro: Jaume I
#### **Pans and Company**
This Spanish-owned and -operated chain store makes for a quick and tasty lunch stop any time you see a branch. This is where everyone drops in on the daily grind for a fresh sandwich on the way to work. Grab a _café con leche_ and do what the locals do! The information given is for the convenient Barri Gotic location, but you'll find multiple locations throughout the city.
Sandwiches from €4, generally daily 07:00-24:00, Plaça Sant Jaume 6, Barri Gotic, +34 933 15 16 06, pansandcompany.com, Metro: Jaume I
#### **Can Paixano**
If you're looking for an authentic local experience, look no further. Come out for some delicious Barcelona-style burgers, cheese sticks, chorizo, french fries, and more. Located in Barceloneta, one block from Barceloneta Metro, this eat-on-your-feet fast-food joint offers a glass o' _cava_ (Spanish champagne) with every order. Sharpen your elbows, as this boisterous place gets packed to the gills with hangry customers ready to chow down.
€4-12, Mon-Sat 09:30-22:30, Carrer de la Reina Cristina 7, Barceloneta, canpaixano.com, Metro: Barceloneta
#### **Churreria Manuel San Román**
This spot is one of my favorites for churros in town. No website, no frills, nothing fancy, just delicious churros and chocolate. The chocolate is so rich that you'll either wish you could drown in it, or you'll need to split it with friends. You won't likely fall in between the two camps.
€4, Mon-Fri 07:00-01:30, Sat-Sun 07:00-14:00 and 16:00-20:30, Carrer dels Banys Nous 8, Barri Gotic, +34 933 187 691, Metro: Liceu
### **TOP NIGHTLIFE**
Barcelona's nightlife rages just about every night of the week. The most important thing is to adjust to the local timeline for going out. Eat dinner 21:00-23:00 or so, then start having your drinks. Head to the bars around 01:00 and then to the clubs around 03:00 and party till the sun comes up. Fast drinks (think two ingredients: rum and coke, gin and tonic, vodka and lime, etc) are generally the poison of choice in this town.
#### **NIGHTLIFE DISTRICTS**
**Eixample** has tons of choices for bars, lounges, and cafés, but because it's so big, it's best if you know exactly where you want to go.
##### **Barri Gotic**
This district has bars centering mostly around Plaça Reial. The touristy spots are numerous there.
Barri Gotic, Metro: Liceu
##### **El Born**
El Born offers a series of classier, young professional-type trendy lounges and bars.
El Born, Metro: Jaume I
##### **Puerto Olimpico**
Puerto Olimpico is the main marina in town. Nearby, you'll find a stretch of clubs all with dancers and electronic music to boggle the mind.
Puerto Olimpico, Metro: Ciutadella/Vila Olímpica
#### **BARS**
##### **Dow Jones**
Famous for its constantly inflating and "crashing" drink prices, this otherwise nondescript bar—a favorite among the American study abroad crowd—makes drinking and consuming an interactive game. Every time you order a drink, the price of that drink rises with "demand." As the night goes on, there are sporadic "crashes," sending the prices tumbling and sparking another rush to the bar. All of the information is projected on screens so you can keep your finger on the pulse of the action.
Mon-Thurs 19:30-02:30, Fri 19:30-03:00, Sat-Sun 12:00-03:00, Carrer del Bruc 97, +34 934 76 38 31, bardowjones.com, Metro: Girona
##### **Chupitos**
Chupitos is a fave among students abroad in Barcy due to its selection of 550-plus different types of shots. Adventurous? Try the Monica Lewinsky. The Boy Scout never disappoints. Feeling brave? Go for the Viking Shot. Chupitos is a relatively small place, and it's either dead or flush with shooters who are making a short stop before the clubs. It does get a bit sweaty once the party arrives.
Daily 22:30-02:30, Carrer d'Aribau 77, Eixample, +34 697 81 44 61, espitchupitos.com, Metro: Universitat
##### **Pippermint**
This bar is overridden by study abroad students who come here for the massive cocktails poured into 1, 5 and even 10-liter cups—or should I say goblets? Come here for a quick and cheap way to get the night—and your buzz—started. Keep walking if you're looking for anything resembling an authentic or redeeming experience. This place is shamelessly for those looking to get their drink on before the clubs.
**LGBT BARCELONA**
Spain rates first in the Pew Research Center's most recent survey of acceptance of homosexuality, at 88 percent. The LGBT community is welcome in Barcelona, and you'll find numerous options for gay-friendly cafés, bars, restaurants, and clubs throughout town—both in the Old Town and in the Eixample. Gay pride festivities in Barcelona take place around the third weekend of June each year. Book ahead for accommodations because the city sells out. Check out **Osbar** (Carrer de la Diputació 225, +34 934 53 46 42) for the bear scene. ( _Os_ means bear in Catalan.) All are welcome and have a good time!
Liters of cocktails from €12, Mon 08:15-01:30, Tues-Thurs 08:15-02:30, Fri 08:15-03:30, Sat 10:15-03:30, Sun 16:30-01:30, Carrer de Bori I Fontesta 20, Eixample, +34 932 010 008, pippermintbcn.com, Metro: Maria Cristina
##### **Marsella**
Two hundred absinthe-laden years since it opened, this dusty bar is still at the top of the game. Do your absinthe right with the fancy spoon and sugar cube—if you need help, just ask! The service is friendly and welcoming. Order your drinks at the bar, then take your supplies to any of the open cafeteria-style low-slung tables to get the party started. While still just a few steps from La Rambla, this district was notorious for petty crime and prostitution up until a few years ago. Keep your wallets and purses tight.
Daily 22:00-02:30, Carrer de Sant Pau 65, Barri Gotic, +34 934 42 72 63, Metro: Liceu
##### **Sidecar Factory Club**
Descend into this subterranean rock bar located right on Plaça Reial. Touristy because of its location, this club is still a good time, featuring frequent live acts and colorful cocktails. Find show information on the website.
€8-12 cover with drink, Mon-Sat 19:00-05:00, Plaça Reial 7, Barri Gotic, +34 933 02 15 86, sidecarfactoryclub.com, Metro: Liceu
#### **CLUBS**
If glitzy clubbing is your thing, look no farther than Barcelona. Covers generally run about €15, with your first drink included. Each drink after that tends to run €8-10. So it's worthwhile to get your party started before heading to the club. It's also worthwhile to Google or search Facebook to find out about planned events and what type of music will be playing. Follow the four major promoters in Barcelona on Facebook (Aashi Guest List, Michael Jordan, Kyke Navarro, and Kike Barcelona/De Lis Group) to see what their events are during the time you're planning on visiting. Call ahead to the clubs or join events to get yourself on the list, saving you time and hassle at the door. Most of the clubs listed are near **Puerto Olimpico.**
##### **Opium Mar**
This is a swanky spot with notorious bouncers enforcing a strict dress code. Dress to impress, as this is a favorite among FB Barcelona team members, along with other glitterati and celebrities. It's a posh scene, so expect all the wonderful things that come with that: pricey cover, expensive drinks, big bouncers, and great DJs. Close this place out to view the sunrise over the Mediterranean. It's worthwhile to get your name on the guest list online ahead of time.
€20 cover including drink, drinks about €12, daily 24:00-06:00, Passeig Marítimo de la Barceloneta 34, Puerto Olimpico, opiummar.com, Metro: Ciutadella/Vila Olímpica
##### **Shoko**
Shoko is your best and most consistent bet for the clubs on the beach strip. The crowd is classy, but not over the top. The music is well-balanced, and they pull off the Asian fusion quite well, from the decor all the way to the food available throughout the day. Dance the night away in a faux-bamboo forest, and enjoy the flushed, low lighting. This is a place where you want your name on the list, so call ahead or click online. Dress to get past the picky bouncers!
Drinks around €10, daily 12:00-06:00, Passeig Marítimo de la Barceloneta 36, Puerto Olimpico, +34 932 25 92 00, shoko.biz, Metro: Ciutadella/Vila Olímpica
##### **Carpe Diem**
This pricey and posh lounge restaurant/club has a Moroccan-Asian twist. You'll find a decidedly well-heeled, older crowd downing top-class sushi and washing it back with Dom Perignon while lounging out on day beds. Reserve ahead for both the dinner and nightclub entry.
Drinks and dinners start at €20, daily 13:00-04:00, Passeig Marítimo de la Barceloneta 32, Puerto Olimpico, +34 932 24 04 70, cdlcbarcelona.com, Metro: Ciutadella/Vila Olímpica
##### **Catwalk**
I think of Catwalk as the grungy little brother with a chip on his shoulder to the classier disco clubs opening up onto the beach. And likewise, it attracts the types who wouldn't necessarily get into the flashier clubs nearby. Stay downstairs for techno, or head upstairs for R&B and hip-hop. If you prefer to leave the pretentiousness at the door, you'll like Catwalk. This club does depend on promoters certain nights to fill the place out, so don't let them get too pushy on you. Feel comfortable to negotiate cover and included drinks at a place like this.
€10 cover including drink, Thurs-Sun 23:30-18:00, Carrer de Ramon Trias Fargas 2, Puerto Olimpico, +34 932 240 740, clubcatwalk.net, Metro: Ciutadella/Vila Olímpica
##### **Razzmatazz**
This is a popular club for students and locals alike. You're guaranteed a great time in this labyrinth of five rocking dance floors. It's heavy on the electro-sound scene, but if that's not your thing, you'll likely find something else in this complex of music. Check out the agenda online for upcoming events and DJs.
€10-12 cover including drink, drinks about €7, daily 23:00-05:00, Carrer Almogàvers 122, Greater Barcelona, +34 933 20 82 00, salarazzmatazz.com, Metro: Marina
##### **Sala Apolo**
This is your alternative to the techno-ridden club scene in Barcelona. Sala Apolo is a medium-sized club that also features live music and parties throughout the week. Sala Apolo is most famous for its Nasty Monday parties, where it seems that all of Barcelona's punks and rockers come out to rage. The crowd tends to be young and tattooed, but the venue is in a rustic old theater with the seats removed from the ground floor and the booths taken out from the wraparound balconies. This is a favorite of the study abroad demographic as well. Come pre-gamed, as the drinks aren't cheap. Check the website to stay up on events and performances.
€8 cover and drinks, Mon 20:00-05:00, Thurs 20:00-05:00, Fri-Sat 19:00-06:00, Sun 21:00-05:00, Carrer Nou de la Rambla 113, Greater Barcelona, +34 934 41 40 01, sala-apolo.com, Metro: Paral-lel
### **TOP SHOPPING & MARKETS**
#### **SHOPPING DISTRICTS**
##### **La Rambla**
The shopping district in Barcelona runs from the top of La Rambla through Plaça de Catalonia to Passeig de Gràcia and up Avinguda Diagonal. You'll find everything from high-end fashion like Burberry, Versace, and Armani to your everyday stores like H&M.
Barri Gotic, Metro: Catalunya, Liceu, or Drassanes
**WISHING YOU A CRAPPY CHRISTMAS!**
Caga Tió or "Crap Log" is a character in Catalan Christmas tradition that is observed in the Spanish region of Catalonia. Caga Tió is a small log with a smiley face painted on one end, with twigs for arms to support it. Oftentimes the _tió_ sports a fun Santa hat. Beginning in early December, children give the _tió_ little bites to eat (as Americans leave cookies for Santa). They feed him every night over the span of a couple weeks, and cover him with a blanket so that he will not be cold. On Christmas Eve, kids put the _tió_ partly into the fireplace and order him to, well, dump out all the presents. To encourage him to take this generous defecation, the kids beat him with sticks while singing selections of typical Catalan Christmas songs. This fun little tradition originally symbolized the fertilization of the fields for an abundant crop in the upcoming year, but perhaps something was lost in translation...
You'll see small _tió_ figurines in souvenir shops across town caught in the middle of this act. They always make for a funny—if somewhat odd—gift for loved ones back home.
#### **MARKETS**
##### **La Boqueria**
This is one of the most eccentric public markets I've ever been to. You can find everything from entire pig heads to fresh smoothies. Challenge yourself to try one thing you've never dreamed of! Then, grab a glass of sangria before rambling the rest of the way down La Rambla. Their iced fruit drinks are a tasty, touristy treat.
Mon-Sat 08:00-20:30, La Rambla 91, Barri Gotic, boqueria.info, Metro: Liceu
### **TOP PARKS & RECREATION**
#### **PARKS**
##### **Parc Güell**
Gaudí's Parc Güell is a beautiful failure. It was one of the world's first gated and planned communities, built at the turn of the 20th century, but the neighborhood never quite took off as planned. Always attentive to detail, Gaudí touched everything from the park benches to the street tiles in this whimsical park at the top of the city.
Today, Gaudí's planned development functions as a park. Come to explore the curvy paths, the photogenic _plaça,_ and the famous breaking-surf passageway and to check out another panorama over the city, all the way down to the beaches miles away.
€7 online, €8 in person, daily 08:30-18:00, Carrer d'Olot, Greater Barcelona, +34 902 20 03 02, parkguell.org, Metro: Vallarca via L3
#### **HIKING**
##### **Montjuic**
Montjuic is the big mountain south of town, to the right as you face out toward the Mediterranean. It was home to a 19th-century military fort and, later, the 1992 Olympics. Take the steep hour-long hike up Montjuic and soak in the best views of Barcelona and the beautiful coastline. Bring a water bottle and explore the star-fort castle (free, daily 10:00-20:00, €5 paid museums inside) while you're at it. For decades, this fort represented the oppression of Franco rule from Madrid. The fort was the military's way of keeping control over the surrounding city. Now it's a place where Catalan families come to picnic on weekend afternoons.
**THE HUMAN CASTLES OF CATALONIA**
So you think you're tough? Let's see how you stack up against the _casteller_ teams that put their strength, balance, coordination, and teamwork on full display in main squares across this region. _Castellers_ erect freestanding, self-supporting human castles up to seven or eight levels tall of people standing on one another's shoulders. _Casteller_ teams come together and, in one massive orchestrated effort, lay the foundation (burly, large men), pile on the next layer (young, lean strongmen), and continue climbing ever higher into the sky. Groups of three scramble up the growing tower like monkeys up a banana tree, as the whole human tower goes up and comes down in minutes. Watch closely, though, because the tower isn't complete without one last team member (usually a small child sporting a riding helmet), who must go to the very top and raise one hand in the air to "cap" the castle. Teams are composed of men, women, and youngsters, who climb to the highest levels. It's a beautiful teamwork tradition that has brought communities together for friendly competition for hundreds of years.
As if this weren't enough of a show, there's also a soundtrack. A small band playing typical Catalan instruments, led by a wilting reed instrument, belts out a tune as if to narrate the whole operation. It climaxes as the little one who makes it to the top raises his or her hand 40 feet above in the air. As soon as those at the lower levels hear the final crescendo, they know the end of their effort is near. The tune changes as the _castellers_ at the top shimmy down their teammates like firefighters sliding down a pole as quickly as possible. The men at the bottom are usually dripping in sweat and trembling uncontrollably after holding everyone up. In 2010, UNESCO declared the _castellers_ an Intangible Heritage of Humanity. For dates and details of the next public events, follow the link castellersdebarcelona.cat. You can also find incredible feats of human castle-building on YouTube.
From the top of the mountain, see the Sagrada Familia rising in the distance and the cruise ships entering and departing through the harbor opposite, just to the south. To get to the base of the hill, head to Barcelona's marina and follow the pathway south. After about a kilometer, you'll see options leading uphill. You can't miss it.
Free, always open, Greater Barcelona, Metro: Plaça Espanya
#### **BEACHES**
##### **Platja Barceloneta**
Due to its proximity to the tourist center, this is Barcelona's most popular stretch of sand. This beach is filled with tanned locals and pale tourists, with both groups having the same goal: to soak in the sun. The people-watching is the best in Barcelona and could make for an entertaining afternoon, but don't go if you're looking for a tranquil, solitary experience.
Free, always open, Barceloneta, Metro: Ciutadella or Barceloneta
##### **Platja Nova Mar Bella**
This is the favorite beach among students and the younger crowd coming from the city. Much less touristy then Platja Barceloneta because of its location far north of the center, yet still easily accessible by metro, this beach is the best of both worlds, offering you an authentic local experience at a manageable distance.
Free, always open, Greater Barcelona, Metro: Selva del Mar or Poblenou
### **TOP TOURS**
#### **Fat Tire Bike Tours**
Fat Tire's expert guides bring their cities to life on two wheels. The Barcelona City Tour lines up all the top sights, including the Cathedral of Barcelona and the Sagrada Familia, in a four-hour ride around town. Cap your cruise with sangria on the beach before riding back to the shop. While the pace is relaxed, you won't have the time to go into each sight along the way, so you may want to double back to some later.
€24, students €22, Jan-mid-Apr daily 11:00, mid-Apr-mid-Oct daily 11:00 and 16:00, mid-Oct-Dec daily 11:00, meets at Plaça Sant Jaume—just look for the Fat Tire sign, Barri Gotic, +34 933 429 275, barcelona.fattirebiketours.com, Metro: Jaume I
#### **Discover Walks' Modernism Walk**
Run by young, passionate local guides, Discover Walks is a fresh European response to the free walking tour craze, offering consistently good quality and a unique flavor. I prefer this tour company for the Modernism Walk, which takes you from Casa Batllo on the Block of Discord and finishes at Gaudí's unfinished masterpiece, the Sagrada Familia. Multiple tours, including a Las Ramblas & Barri Gotic Walk, are offered throughout the week; just watch for the guides' trademark bright pink vests, or check the website for details.
Free, 10:30 Fri-Mon, meets at Casa Batllo, Eixample, +34 649 60 99 41, discoverwalks.com/barcelona-walking-tours, Metro: Passeig de Gràcia
#### **Travel Bar's Barcelona Old Town Tour**
Travel Bar (Carrer Boqueria 27, daily 10:00-23:00, travelbar.com) has it all: coffee in the morning, a bar at night, cooking classes throughout the week and daily walking tours venturing out into the city. TB's introductory walk to Barcelona's Old Town is just what any visitor needs to get comfortable with the city. You'll retrace the Roman foundations of the city, discuss the numerous high and low points of the city's history, and wrap up with a lunch just steps from Barcelona's marina. The tour lasts about 2.5 hours.
Free, daily 11:00, 13:00, and 15:00, meets at Travel Bar, Carrer Boqueria 27, Barri Gotic, travelbar.com, Metro: Liceu
#### **Travel Bar's Paella & Sangria Cooking Class**
Let a professional _paellador_ walk you through the art of preparing the typical seafood rice platter with perfectly spiced prawns, mussels, and calamari. You'll meet at the Travel Bar and then head to La Boqueria for a guided walk through the dozens of market stalls and to pick up fresh ingredients. From there, you'll head to the cooking location, where you'll dive into your bags of seafood and spices. The class, which lasts about three hours, is capped off with a delicious dinner and unlimited drinks. A discounted after-party at the bar is available. This is truly a great value.
€28, meets daily 17:45 at Travel Bar, Carrer Boqueria 27, Barri Gotic, travelbar.com, Metro: Liceu
### **TOP HOSTELS**
Hostels have done so well in this socially oriented city that you'll find most worthwhile ones have multiple locations in town. While they're technically chains, they don't sacrifice the intimate feel and service of one-off hostels. In fact, they manage to combine the energy of their many locations through nightly pub crawls and meet-ups between their locations to amp up the party.
One of my favorite chains is **Sant Jordi** (santjordihostels.com), which offers fun, social party hostels with great locations throughout town. Their baby dragon logo is a nod to the patron saint of Barcelona, Saint George. Another great chain, **Urbany Hostels** (urbanyhostels.com), focuses on the mod backpacker who wants all the amenities at a great price. Slightly more institutional **Equity Point** (equity-point.com) really competes on price. Generally, stays here are quieter and less social. You'll likely get a bit better sleep but are less likely to have wild nights at their three locations. My favorite branches of all three hostels are listed.
#### **Kabul Backpackers Hostel**
This is your best bet for a raging party. Kabul's is a party hostel through and through, and its great location smack dab on the Plaça Reial at the bottom of La Rambla makes it easy to hit the clubs at night and the top sights and beaches during the day. And their free breakfast in the morning is key to starting the day (or ending the night) right.
Beds from €13, 24-hour reception, laundry facilities, breakfast included, common room, terrace, Plaça Reial 17, Barri Gotic, +34 933 18 51 90, kabul.es, info@kabul.es, Metro: Liceu
#### **Sant Jordi Gràcia**
I love this hostel for both its clean, modern build-out and its location. The Gràcia district is just a short metro ride from the busy streets of the Old Town, but it feels distinctly more local and authentically Catalan. Street events happen often here but are missed out on by many tourists. The hostel has a fun social room with bean bags, comfortable modern bunks, and clean basic rooms. The shared bathrooms are clean, and the staff is welcoming and helpful.
Eight-bed mixed dorm from €14, free Wi-Fi, 24-hour reception, laundry facilities, breakfast extra, free maps and walking tours, pub crawl organized often, kitchen, towels extra, Carrer de Terol 35, Greater Barcelona, +34 93 342 41 61, santjordihostels.com, gracia@santjordi.org, Metro: Fontana or Gràcia
#### **Equity Point Gothic**
A superbly located hostel right in El Born, Equity Point Gothic is in one of the oldest parts of Barcelona. The bohemian atmosphere of the neighborhood, with trendy plazas and cafés, puts you right in the middle of young Barcelona. The proximity to the Picasso Museum and La Rambla makes this hostel a great option for serious sightseers. Heads up: They really pack in the bunks in their larger dorms, up to three or four levels high.
Beds from €11.90, free Wi-Fi, breakfast included, basic kitchen, laundry, 24-hour reception, Carrer Vigatans 5, El Born, +34 932 31 20 45, equity-point.com, infogothic@equity-point.com, Metro: Jaume I
#### **Equity Point Sea**
The hostel itself isn't the cleanest, and the beds aren't all that comfortable, and the scene isn't all that fun. But the location is to die for—right on the tip of the Barceloneta, near excellent cafés, restaurants, tapas bars, and—most importantly—the beach!
Dorms from €10, 24-hour reception, breakfast included, a/c when it works, social common room, on-site bar, free Internet, laundry facilities, Plaça del Mar 1-4, Barceloneta, +34 932 24 70 75, equity-point.com, infosea@equity-point.com, Metro: Barceloneta
#### **Equity Point Centric**
The largest of the Equity Point hostels, this one is right on Passeig de Gràcia, mere steps from the main square of Barcelona and all of Catalonia, Plaça Catalonia. The price is right in this massive hostel with everything from single and twin rooms to large 14-bed dorms. The rooftop terrace and its location are the major highlights.
Dorms from €9, 24-hour reception, breakfast extra, on-site bar, lockers, laundry facilities, free Wi-Fi, common room, towels extra, Passeig de Gràcia 33, Eixample, +34 932 15 65 38, equity-point.com, infocentric@equity-point.com, Metro: Passeig de Gràcia
#### **Sant Jordi Rock Palace**
This large hostel (150-plus beds), one of Sant Jordi's newer entries, features an on-site bar, social room, bright reception with helpful staff, large dorms, and comfortable chilled-out vibe. The rooftop terrace and pool make it a favorite. The decor is well executed: funky retro posh.
Ten-bed dorms from €16, free Wi-Fi, 24-hour reception, breakfast extra, towel for rent, Balmes 75, Eixample, +34 934 53 32 81, santjordihostels.com, rockpalace@santjordi.org, Metro: Passeig de Gràcia
#### **Sant Jordi Alberg**
This local hostel chain really understands what makes a great hostel. Awesome staff, value, atmosphere, and free nightly pub crawls pack this place out year-round, so be sure to reserve long in advance.
Beds from €13, 24-hour reception, free lockers, laundry facilities, common room, Internet access, Calle de Roger de Llúria 40, first floor, apartment 2, Eixample, +34 933 42 41 61, santjordihostels.com, lluria@santjordi.org, Metro: Passeig de Gràcia
#### **Sant Jordi Apartment Sagrada Familia**
Sagrada Familia Apartments offers single, double, triple, and quad rooms in a private apartment setting. The rooms are modestly decorated, clean, and modern. The slick design follows the smooth concrete mod skateboard-inspired culture.
Basic singles from €30, free Wi-Fi, breakfast optional, towels included, laundry facilities, 24-hour reception, Carrer del Freser 5, Eixample, +34 934 46 05 17, santjordihostels.com, sagradafamilia@santjordi.org, Metro: St Pau-Dos de Maig
#### **Urbany BCN GO**
Sporting a rooftop terrace, all new facilities, a Jacuzzi, and comfortable, bright rooms, this is isn't your papa's dingy hostel. Enjoy the perfect blend of comfort and great social atmosphere at one of this chain's newest and most popular locations.
Beds from €14, 24-hour reception, laundry service, breakfast available, mini pool and Jacuzzi, free Wi-Fi, luggage storage, Corts Catalanes 563, Eixample, +34 937 37 96 18, urbanyhostels.com, info@urbanybcngo.com, Metro: Universitat
#### **Urbany Barcelona**
If BCN Go is full, try Urbany Barcelona nearby. The welcoming receptionists can recommend activities for your stay. The pool table, rooftop terrace, and social setting will be your favorite parts of the hostel. The beds are nothing to write home about, but the price is right.
Dorms from €10, 24-hour reception, laundry facilities, breakfast extra, rooftop terrace, free Wi-Fi, luggage storage available, Avinguda Meridiana 97, Eixample, +34 932 45 84 14, urbanyhostels.com, Metro: Clot or Encants
### **TRANSPORTATION**
#### **GETTING THERE & AWAY**
Barcelona is well connected and offers numerous public transportation options from the transport hubs into the center and your accommodations.
##### **Plane**
Two airports serve Barcelona. From **El Prat** (BCN, barcelona-airport.com), take the **AeroBus,** which leaves every 20 minutes, directly into the center (40 minutes, €5.65 one-way, free Wi-Fi and charging plugs on board). Stay on till the end of the line, Plaça de Catalonia (Barcelona's main square), and make your connection from there by foot, metro, or bus.
If you're flying into **Girona** (GRO, girona-airport.net) simply hop on the **Barcelona Bus** into the city center (about 1.5 hours, €15 one way). You'll get dropped off at **Estacio d'Autobus Barcelona Nord,** from which you can head into the metro. Google "Barcelona bus from Girona" for the latest timetables.
##### **Train**
The train system is excellent in Spain. Trains between Barcelona and Madrid run often and take just about three hours. If you purchase in advance through **Renfe** (renfe.com), tickets can be as cheap as €50, but prices climb past €150 closer to the date and during peak travel times. Since there's quite a bit of business traffic between these cities, avoid the times that commuters would want (early morning, late afternoon) to find the cheapest options. The trip from Paris is about 7 hours (€130). Train websites are notoriously difficult to navigate. For that reason, research timetables online, and then book in person at your closest international train station. Barcelona has two major train stations: **Estacio de Sants,** west of the Old Town, and **Estacio de Franca,** located between El Born and Barceloneta. Both stations have integrated metro stops.
##### **Bus**
Barcelona has two major bus stations: **Estacio de Sants,** west of the Old Town, and **Estacio de Nord,** north of El Born. Check out **Eurolines** (eurolines.com) for your best national and international connections. While bus tickets may be cheaper than train tickets, don't forget to value the extra time you'll be spending en route.
##### **Car**
By car, Barcelona is about three hours west of Madrid and about 10 hours south of Paris.
#### **GETTING AROUND**
Barcelona is a wonderful city to walk, but factor its large size into your planning. The heart of Barcelona, the medieval city called Barri Gotic, is a 20-minute walk from one side to the other. The Eixample is the modern expansion beyond the Old Town and is much bigger, with block after block of traffic and modern buildings.
The **bus and metro systems** in Barcelona are integrated with each other, meaning that you can use the same passes for both a bus ride and a metro ride. Buy the **T10 pass** (€9.95), which is one ticket that can be used by one person for ten rides, by ten people once, or by two friends all weekend for five rides each. With this pass, you save almost 50 percent of the **individual ticket rate** (€2) and will take care of your transportation needs while in town. You can buy your T10 passes from the automated machines in the metro. Use change, as paying with a big bill will give you a pocket full of heavy coins in return.
##### **Metro**
The metro is a fast and efficient way across town if you don't mind missing the views along the way. You'll notice the metro-line colors reflect those of the Olympic rings—the entire metro was revamped for the 1992 Olympics. The metro closes at midnight during the week, at 02:00 on Friday, and goes all night on Saturday.
Barcelona has a serious **pickpocketing** problem. Pickpockets work the busy areas in the touristy metro stops like Plaça de Catalonia, so always have your wits about you.
##### **Bus**
Barcelona boasts an extensive network of buses. Routes and numbers are easy to read at each bus stop. Look for the arrow pointing toward the Mediterranean Sea to get yourself oriented, as the maps won't be oriented north-south but along the bus line itself. Buy tickets before boarding or from the driver.
##### **Taxi**
Official licensed Barcelona cabs are painted black, with yellow doors. There are fleets of them across the city, and you can wave them down. Rest assured that the meter will be in the right setting. There are four tariff rates: T1 (€2.10 + €1.07/km) is weekdays 08:00-20:00, T2 (€2.10 + €1.30/km) is weekdays 20:00-08:00, T3 (€2.30 + €1.40/km) is weekends and holidays 20:00-06:00, and T4 is fixed rate. At the time of writing, **Uber** had not yet made inroads into Barcelona.
##### **Bicycle**
Barcelona has a **public bike rental** (bicing.cat) system for registered users. I recommend renting a bike through **One Car Less** (Calle Esparteria 3, +34 932 682 105, biketoursbarcelona.com) in El Born. For €10 for the half day, you can hire a bike to ride the length of the beaches, finding just your perfect spot. There's not much of a reason to venture into the Eixample on a bike. I prefer to avoid the heavy traffic of the modern town, though there is a network of bike paths.
### **DAY TRIPS**
#### **Montserrat**
For beautiful views of the surrounding countryside and coastline far off in the distance and challenging hiking outside the city, look no further than Montserrat. Montserrat (Serrated Mountain) has a monastery, **Santa Maria de Montserrat** (free, daily 07:00-19:30 with masses often, +34 938 77 77 77, montserratvisita.com) tucked into the armpit of the rocky crag. Explore the gilded gothic monastery and the ramparts looking out over Cataluña, then make a pit stop at the cafeteria and gift shop. To cap it off, take the funicular (€3.70, runs every twenty minutes until about 17:00 in the low season, and 18:00 in summer months, cremallerademontserrat.cat) straight up the mountain to the peak.
Located on the R5 line, in the direction of Manresa, the station is just an hour away from downtown Barcelona. On arrival to the train station, take the Aeri cable car for a beautiful view as you transfer up into the monastery location. From this location you can wander around or choose between two different funiculars that lead you to separate areas.
Trains depart hourly from Plaça Espanya to Monistrol de Montserrat. Tickets run about €30 all told. Allow a full day for the trip.
### **HELP!**
#### **Tourist Information Centers**
The tourist information center (Mon-Thurs 09:00-14:30 and 15:30-18:30, Fri 09:00-15:00), just beneath the Plaça Catalonia at the top of La Rambla, is easy to find. Pop in for sightseeing tips, sight reservations, maps, and help finding accommodations. You can also get information at barcelonaturisme.com.
Plaça de Catalonia 17
+34 932 85 38 34
#### **Pickpockets & Scams**
Barcelona has a serious pickpocketing problem. Tourists are often targeted on busy boulevards, where they're easily distracted. Busy metro stations are also a prime pickpocketing zone. Wear your backpacks on your front, use a money belt, and keep your valuables close in busy touristy areas. More often than not, you won't notice you've been pickpocketed until it's too late.
Gambling in public is illegal, and you should avoid the games on La Rambla, where gamblers work in teams to lure unsuspecting tourists. You'll win fast—then lose it all in the blink of an eye.
#### **Emergencies**
For emergencies, dial 112. You can also reach fire service (061) or city police (092), or dial 061 for medical emergencies.
#### **Hospital**
Hospital Clinic Barcelona
Emergency entrance: Casanova 143, 08036 Barcelona
+34 93 227 5400
#### **US Consulate**
Paseo Reina Elisenda de Montcada
+34 93 280 22 27
Berlin Maps
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Help!
_Willkommen, bienvenue,_ and welcome to Berlin! Once divided by the post-WWII powers, Berlin's east and west parts have since reunited and blossomed into a thriving, vibrant metropolis. Germany's capital is teeming with history and culture, including an alternative nightlife scene that will either turn you on or freak you out. So embrace your will to be weird, along with the zeitgeist that awaits!
### **BERLIN 101**
Berlin experienced a difficult 20th century. Following Germany's defeat in World War I, inflation soared as a result of the reparations laid out in the Treaty of Versailles. By the early 1920s, 4.2 trillion German marks were equal to less than one US dollar, and it cost a wheelbarrow full of bills to purchase a single loaf of bread. This situation improved through renegotiation with the Allied forces in 1924, and Berlin prospered toward the later half of the decade. With residents such as Albert Einstein breaking boundaries in science and artists like George Grosz making headway in the Dada movement, Berlin became the cultural haven of Europe. It also boasted the wildest cocaine-fueled nightlife the world had ever seen.
The good times ended as Hitler finagled his way into power. Hitler reenergized the country, but in a terrible direction, and embarked on a vicious genocide against Jews and other minorities, whom he blamed for problems like corruption and inflation. After systematic deportations and executions, only 1,200 of Berlin's 160,000 Jews were left in the city after the Holocaust. As for the city itself, over 80 percent of Berlin was flattened by bombs during World War II.
Following the war, Germany as a country was divided in two, East and West. The city of Berlin, located smack dab in the middle of East Germany, was divided into four sectors: British, French, American, and Russian. The former three created an island of capitalism in a sea of Soviet East Germany. To maintain a foothold in Berlin, the Western Allies put huge effort into sustaining the citizens of West Berlin. The quality of life was better in West Berlin and West Germany, so the East German population—including many intellectuals and professionals—started slowly bleeding west.
Refusing to accept the exodus, the Soviets decided to build a wall north-south through East Germany and all around West Berlin. In the middle of the night on August 13, 1961, the East German government laid the foundations for what they called the "anti-fascist protection wall." The existence of the Berlin Wall implicitly admitted that nobody wanted to live in this supposed communist paradise. The Berlin Wall remained in place until 1989, when a miscommunication during a routine TV announcement kicked off a surge of East Berliners to the handful of gates, pushing to cross into West Berlin. Overwhelmed, the bewildered guards permitted families to reunite across the wall, and the celebratory dismantling of the wall began as quickly as it went up almost 30 years before. The wall's collapse sparked the collapse of the entire Soviet Union. East and West Germany were reunited, and the city of Berlin once again became the capital city of Germany.
Today, East Berlin is the world's largest construction project, and is transforming once more to reflect the vibrant energy of the unmistakable Berliner culture.
### **PLAN AHEAD**
#### **RESERVATIONS**
Reservations are required for the **Reichstag** (bundestag.de/htdocs_e/visits).
#### **PASSES**
##### **WelcomeCard**
Berlin's **WelcomeCard** (visitberlin.de) covers public transportation within the city and offers discounted entry into sights, including the TV Tower and DDR Museum. The 48-hour version costs €19.50. See the website for 72-hour or five-day options, or options that cover transit to Potsdam.
#### **LOCAL HAPPENINGS**
##### **Unity Day**
Unity Day is a public national holiday in Germany that is celebrated on October 3, commemorating the day when East and West Germany were reunited. People celebrate with festivals, fireworks, concerts, communal meals, and speeches by politicians or other prominent figures.
##### **Christmas Markets**
If you find yourself in Berlin around Christmastime, enjoy a mug of mulled wine and sausages at one of the many Christmas markets that pop up every year toward the end of November and run until the New Year. The scene really puts you in the Christmas spirit, with decorated Christmas trees, warming huts, stalls selling ornaments and other trinkets, and jolly musicians playing all the Christmas classics.
**KNOW BEFORE YOU GO**
**KEY STATS & FIGURES**
**Currency:**
**Germany uses the euro (€); 1 EUR = about 1.06 USD**
**Population:**
**3,500,000**
**Internationals:**
**30 percent**
**Language:**
**German**
**Number of bridges:**
**1,700 (triple that of Venice)**
**Number of museums:**
**175 (5 on Museum Island alone)**
**Favorite dish:**
**currywurst**
**Annual tax to own a dog:**
**€150**
**CALIBRATE YOUR BUDGET**
**TYPICAL PRICES FOR:**
**Hostel dorm bed:**
**€12**
**Two-course dinner and drink:**
**€12**
**Pint of beer:**
**€4**
**Currywurst and potatoes:**
**€4**
**Day bike rental:**
**€14**
**Single S-Bahn & U-Bahn pass:**
**€2.40**
**MOVIES TO WATCH**
_The Lives of Others, Good Bye Lenin!, Cabaret_
THREE DAY ITINERARY
It's a tall order to soak in the history, experience the complex modern culture, and visit all the must-see sights that Berlin has to offer in three short days. But you'll cover a lot of ground if you stick to this itinerary and make reservations well ahead of time for entry into the Reichstag, Germany's parliament building.
**DAY 1: WELCOME TO BERLIN**
**MORNING**
Grab a breakfast to go like all the Berliners at your closest **Back-Factory,** then begin a free three-hour walking tour with **Alternative Berlin** or a 4.5-hour bike tour with **Fat Tire Bike Tours.** Both meet at 11:00 daily at Alexanderplatz just on the north side of the TV Tower. Both introductory tours take you past sights like **Checkpoint Charlie** , the **Brandenburg Gate** , the **Memorial to the Murdered Jews of Europe** , **Hitler's Bunker** , and some still-standing remnants of the **Berlin Wall**. Note any interesting sights you may want to double back to later in your stay.
**AFTERNOON**
Your tour will stop for lunch along the way and finish up midafternoon. Now that you know your way around the city, opt to keep your bike from Fat Tire for another couple days.
Get some Mitte shopping time in on **Unter den Linden** and on the streets surrounding the Weinmeisterstrasse U-Bahn stop. Otherwise, grab a snack to go and rest the legs for about 45 minutes on historic bus route 100, listening to **Jimbo's Cheap Man's Bus Tour,** available for free download online. This route will take you over to West Berlin and show you the scale of the city. Notice how the streets become more residential the farther west you go? For a snack at the turnaround point, try **Schwarzes Café** just a long block down Kantstrasse from the Zoologischer bus stop. If you plan to visit the **Bauhaus Archives** museum during your visit, now is the most efficient time to do it (stop: Lutzowplatz on bus 100). Then return to the hostel to freshen up.
**EVENING**
Head to Oranienburger Strasse for pre-dinner beers at **Aufsturz**. Then cross the street to my favorite Turkish joint on this side of town: **Dada Falafel**. Pop in for the kebabs and stay for the fresh juice and smoothies—and a peek at **Tacheles** across the street, Berlin's most famous ex-squatting complex.
**LATE**
In a jazzy mood? You're not far from **Zosch** , a famous Berliner jazz bar. Descend into their WWII bomb shelter cellar to enjoy a foot-tapping great time. If it's a Wednesday night, you'll be treated to some of the best live jazz in the city. Order your beer by the liter.
You've had a long day, so stay close and take the 10-minute walk over to **Rosenthaler Platz** for a wonderful slice of fun, hipster nightlife. Be sure to have a round at my favorite bar in the city, **Mein Haus am See**.
**DAY 2: IRON CURTAIN & MORE**
**MORNING**
If your hostel is anywhere near the Rosa-Luxemburg-Platz U-Bahn stop (both Wombat's and St Christopher's hostels are), you've got to try my favorite coffee shop in town, **Kaschk,** with some of the best beans this side of the Alps. Then take a stroll through Mitte to the **Palace of Tears.** What was the main exchange point between the two halves of the split city is now a free and heavy-hitting exhibit.
**AFTERNOON**
From the Palace of Tears, take S1, S2, or S25 north to Nordbahnhof to transfer onto the M10 tram to the **Berlin Wall Memorial** to see a full-scale remnant of the Berlin Wall. Climb the nearby observation tower to look out over the walls and across the entire city. Afterward, your super typical Berliner lunch lies just a 10-minute walk away at **Altberliner Kaffeestube** on Arkonaplatz.
Loop on tram M10 toward Warschauer Strasse, over to the **East Side Gallery** , the Berlin Wall's longest remaining intact section. Today, it's the world's largest outdoor mural, with hundreds of famous artists having returned to recently retouch their original work from nearly 25 years ago.
Being in this part of town puts you in perfect position to explore the **Kreuzberg** neighborhood, Berlin's most ethic neighborhood, centering around Oranienstrasse.
**EVENING**
For dinner, stick around Kreuzberg for amazing Italian at **Il Casolare** or authentic Turkish at **Hasir** , or head to Mitte for the classic German beer hall meat-and-potatoes dishes at **Georgbrau**.
**LATE**
Return to the hostel to change and down a couple shots at the hostel bar to get ready for your night out in **Friedrichshain** , Berlin's famous entertainment district. Stop at the numerous bars in the neighborhood to get amped up for the nearby famous clubs of **Berghain** and **Tresor**.
**DAY 3: SHOPPING, DAY TRIP, & REICHSTAG**
**MORNING**
If it's Sunday, head north to the **Mauerpark Flea Market** , my favorite of Berlin's outdoor markets. You'll find everything from antiques to handmade crafts. Street food options abound: pizza, bratwurst, and typical stews.
**AFTERNOON**
Take the hour-long trip north to see **Sachsenhausen** , the closest Nazi concentration camp to the Berlin city center (take the M10 tram to Nordbanhof, connect onto the S1 north to Oranienburg station, and catch bus 804 to the camp). This is where the Nazis developed their concentration camp layout and instruments of mass torture and murder to be used later at camps like Dachau and Auschwitz.
**EVENING**
Return to town for an evening entry into the **Reichstag** (reservations required), Germany's parliament. Notice the striking architectural symbolism of the transparent glass, meant to illustrate the transparency of the German democratic system.
**LATE**
Return to your hostel neighborhood and pop into the unique bars along **Schönhauser Allee** , a prime spot to toast to an amazing three days in Berlin.
### **TOP NEIGHBORHOODS**
Massive Berlin is divided into East and West sections. East Berlin is where you'll spend the vast majority of your time. At its center, **Mitte** contains most of the top sights, from the Reichstag, Brandenburg Gate, and Unter den Linden on its west end to the TV Tower on the east end, with a cluster of museums on Museum Island.
Northeast of Mitte, yuppie **Prenzlauer Berg** (P'berg, for short) is your chance to see what tattooed ex-punks look like 15 years down the road and offers low-key, mature restaurants and nightlife. South of the Spree River is the **Kreuzberg** neighborhood, Berlin's most ethnic district, made up of a sizable Turkish population. Come here to experience a Turkish flea market and enjoy rich and tasty kebabs. East of Mitte on the north side of the river, **Friedrichshain** offers the impressive East Side Gallery and some of the city's richest unique nightlife.
**Central Berlin** is home to Tiergarten park and the Bauhaus Archives. West and south of Tiergarten is **West Berlin** , the quieter residential side of town. Lacking the heavy sightseeing punch of East Berlin, West Berlin isn't worth the valuable time on a short visit. Don't stress if you don't make it out there.
### **TOP SIGHTS**
#### **The Topography of Terror**
On the site of the old Gestapo headquarters, the Topography of Terror is an intense exhibit taking you chronologically through the downward slope of Germany in the 1930s toward full-on fascism. Detailing the early power grabs of the Nazi party through to the segregation of Jews and minorities and the atrocities that followed, this museum doesn't shy away from the difficult history that the country is still coming to terms with. The museum is thankfully on a single floor, and heavy-hitting displays create a maze of suspended text and images. A café and lockers are available toward the right as you enter, and toilets are downstairs.
Outside the museum, you'll find a stretch of the Berlin Wall with exposed foundations just beneath. These were the cells where political prisoners were held, interrogated, and tortured by the Nazis.
Free, daily 10:00-20:00, outdoor exhibit closes at dusk, Niederkirchnerstrasse 8, Mitte, +49 (0)30 254 50 90, topographie.de/en, U-Bahn: Kochstrasse
#### **Checkpoint Charlie**
When thinking of the Berlin Wall, most Americans remember Checkpoint Charlie. Not named after Charlie Chaplin, rather this checkpoint was the third checkpoint after Alpha and Bravo. This was the checkpoint that the Americans controlled. Today, you can snap pictures with fake guardsmen during the day. Rumor has it these guards moonlight at the local Chippendales.
Nearby, the interesting **Checkpoint Charlie Museum** (€12.50 adult/€9.50 student, daily 09:00-22:00, Friedrichstrasse 43-45, Mitte, +49 (0)30 253 72 50, mauermuseum.de, U-Bahn: Kochstrasse) focuses on creative and successful escapes from East Berlin into West Berlin over the years. The museum is full of ingenious inventions for getting over the wall or disguising oneself to get through the checkpoints. To make escapes more difficult, rope and other implements that could be used to scale walls were sold with very tight restrictions.
Free, always open, Mitte, intersection of Zimmerstrasse and Friedrichstrasse, U-Bahn: Kochstrasse
#### **TV Tower**
Standing 368 meters tall, the TV Tower is the tallest structure in Germany. It was built by the communists in 1969 to show off their engineering prowess (though they secretly imported Swedish engineers to finish the job). In an ironic twist, the textured ball toward the top of the tower reflects light from the sun in the shape of a cross, creating the largest one around in this secular communist territory. It's possible to head to the top of the TV Tower via speedy elevator for a beautiful 360-degree panorama over downtown Berlin.
Nearby **Alexanderplatz** is a famous downtown square that played the backdrop to an exciting scene from the _Bourne Supremacy._ It's also a main public transportation hub through which you're sure to connect during your visit.
€13 entry, daily 09:00-24:00, Panoramastrasse 1A, Mitte, +49 (0)30 247 57 58 75, tv-turm.de, S+U-Bahn: Alexanderplatz
**ACT LIKE A LOCAL**
**Embrace Your Inner Hipster**
Berlin—with its young population, many artists and musicians, bohemian lifestyle, and cheap rent—has been a magnet for counterculture types and has been the capital of "hipsterness" since before the term existed. Up until the last few years, squatting in abandoned houses and buildings was actually a protected right held by the citizens of the city. While you're here, hit up a thrift market, revel in great nightlife, and imagine a time when taking up residence in an abandoned building was a perfectly acceptable housing option.
**Love the Ampelmann**
There were quite a number of aspects of daily life to hate while under communist rule: food shortages, only getting oranges once a year for Christmas, and the fact it took more than 10 years on a waiting list to get even a basic car made from epoxy and heavy cardboard, just to name a few. But there was one part of East Berliner life that the comrades eventually held near and dear: the Ampelmann.
While you walk the streets of Berlin, notice the cute little green walking man and red no-go man lights at each crosswalk. This is the Ampelmann! After the fall of the wall, the crosswalk lights were undergoing a standardization process until local protests caught such momentum that the city council canceled their efforts and even replaced the ones they had taken down. The Ampelmann had become an endearing, nostalgic symbol of East Berliner life. Today, there are Ampelmann stores where you can get everything from mugs to messenger bags decked out in the familiar symbol.
#### **Brandenburg Gate & Pariser Platz**
Built during the Prussian monarchy in 1791, the Brandenburg Gate once served as the grand passageway to Unter den Linden, the beautiful tree-lined boulevard that led straight to the city palace. During the Cold War the gate was closed off and isolated in no-man's-land, going from a symbol of passage and unity to a persistent symbol of separation. It wasn't until the fall of the Berlin Wall in 1989 that the gate was once more made accessible.
Today, the clean, white Brandenburg Gate stands as a symbol of a unified Germany and is an iconic landmark of the city, capping one of the most important squares: **Pariser Platz.** Here you'll find the famous Hotel Adlon (yes, the five-star hotel from which Michael Jackson infamously dangled his baby), the fortress-like US Embassy, the French Embassy, and hundreds of tourists. When designing the embassy in a post-9/11 world, the Americans wanted an extra few feet in the security perimeter than the existing Brandenburg Gate would allow, and they even asked to shift the historic gate by a couple meters. They were denied, but it's easy to note the stiff security all the way around the building between Pariser Platz and the Memorial to the Murdered Jews of Europe.
Free, always open, Pariser Platz, Mitte, S-Bahn: Brandenburger Tor
#### **Palace of Tears**
The Palace of Tears occupies an old checkpoint station where East Berliners were permitted to cross into West Berlin, but only after serious interrogation by strict border guards. Those leaving East Berlin didn't know if they'd ever see their families and loved ones again. Today, through multimedia and interactive displays, a single-floor exhibit depicts life behind the Iron Curtain, the moments during the fall of the wall, and the long process of reunifying a great nation. You'll learn about the ingenious ways that East Berliners coped with the challenge of living under communism.
Free, Tues-Fri 09:00-19:00, Sat-Sun 10:00-18:00, Reichstagufer 17, Mitte, +49 (0)30 46777790, hdg.de/berlin/traenenpalast, S+U-Bahn: Friedrichstrasse
#### **Reichstag**
The Reichstag, built in 1894 and remodeled in the 1990s, is the traditional seat of Germany's parliament. Its magnificent glass dome, which represents the government's commitment to transparency, offers a stunning view of the Berlin skyline, but don't forget to look down. Visitors can keep an eye on what the elected officials are doing below, with a clear view over their shoulders. Take your time listening to the well-done audio guide as it loops you up and back down the spiraling track inside the glass dome. You'll enter through a security check to the right of the main entrance as you look at the facade. While the entry is free, reservations are required. Don't be late for your appointment, as these Germans really stick to the clock! You can also enjoy a restaurant at the roof level of the building, serving breakfast, lunch, and dinner.
Free, reservations required, daily 08:00-24:00, last entry at 22:00, Platz der Republik 1, Mitte, +49 (0)30 227 32152, bundestag.de, S-Bahn: Brandenburger Tor, U-Bahn: Bundestag
#### **Berlin Wall Memorial**
Head up to Bernauer Strasse on the northern edge of Mitte to see a small, yet fully preserved section of the Berlin Wall. You'll see the double layer of defense, the combed and mine-strewn sand in no-man's-land, and the trespassing detection systems, all of which make one think that when a government has to work so hard to keep their people inside their borders, there must be something wrong with the system. This was a unique stretch of the wall because it incorporated existing buildings into the wall to save on building materials. People lived in these buildings, and several escaped through the westward-facing windows. It wasn't until a tug-of-war between police and an escaping old lady hanging halfway out the window occurred that they bricked up all opportunities for escape.
Climb the stairs in the tower across the street for a free panorama of the neighborhood and a view into the section of the Berlin Wall.
Free, always open, Mitte, +49 (0)30 467 98 66 66, S-Bahn: Nordbahnhof
#### **Memorial to the Murdered Jews of Europe**
This Holocaust memorial, completed in 2005, consists of 2,700 different-sized stone slabs positioned on a rolling plane. It evokes powerful feelings of instability, claustrophobia, and disorientation as you remember those deported and killed during the war. As you step deeper into this memorial, the sounds of the city fade away and are replaced by silence and an eerie sense of isolation, even among the many people viewing the memorial with you. To prevent graffiti, the blocks were covered with a protective coating, which has actually worked quite well. Ironically, the company that provided this protective chemical was distantly affiliated with the company that developed Zyklon B, the gassing chemical used by the Nazis in their extermination camps. When this came to light, the company provided the service and products free of charge.
Free, always open, Cora-Berliner-Strasse 1, Mitte, +49 (0)30 2639 4336, holocaust-mahnmal.de/en, S-Bahn: Brandenburger Tor, U-Bahn: Potsdamer Platz
#### **Hitler's Bunker**
In the last days of World War II, Hitler and his inner circle retreated to a complex deep in the ground with 10-foot-thick reinforced concrete walls and ceilings to coordinate their last stand. Dark and damp, it wasn't a comfortable or particularly pleasant place to be hanging out. The bunker was destroyed and dismantled after the war, and the remains are still buried underneath this nondescript parking lot. The parking lot is left deliberately empty to avoid drawing any sort of attention or pilgrimage from extremists, but you can find a small sign board sharing information about what was below.
**"I AM A JELLY DONUT"**
In 1963, during a speech at the site of the Berlin Wall, President John F. Kennedy announced to the world that the United States would stand by West Berlin in this time of conflict with Russia, while also saying that East Berlin was officially not ours to control. _"Ich bin ein Berliner,"_ stated Kennedy in solidarity. While the phrase literally translates to "I am a Berliner," in German the use of the indefinite article _ein_ with this phrase is unnecessary, and gives the phrase the alternative meaning of "I am a jelly donut." Needless to say, cartoonists all over the world had a field day, and jokes about talking pastries were scattered all over the press. Though he probably regretted his mistake, Kennedy's declaration really only endeared him to the German people.
Free, always open, just southeast from the Memorial to the Murdered Jews of Europe at Wilhelmstrasse 77, Mitte, U-Bahn: Mohrenstrasse
#### **East Side Gallery**
The largest open-air gallery in the world, the East Side Gallery is a preserved mile-long section of the Berlin Wall that has been since painted over by hundreds of artists from all over the world. This once gray, bleak, soul-less piece of concrete is now a multicolored work of art, bearing messages of peace and hope. Farther down the wall, there is an exhibit about all the walls around the world, from the Gaza Strip to Belfast in Northern Ireland.
Free, always open, Mühlenstrasse 45-80, Friedrichshain, +49 (0)172 391 87 26, eastsidegallery-berlin.com, U-Bahn: Schlesisches Tor
#### **German History Museum**
This extensive museum takes you from the early medieval ages of kings and serfs to the tumultuous 20th century through displays, paintings, artifacts, full suits of armor, weapons, and more. It's easy to spend several hours in just the permanent exhibition taking up the two large floors of this building. Check out the website ahead of time to see if any of their innovative temporary exhibits sound interesting as well.
Free up to 18 years, €8 for adults, daily 10:00-18:00, Unter den Linden 2, Mitte, +49 (0)30 2030 4444, dhm.de, U-Bahn: Hausvogteiplatz
#### **Unter den Linden**
Take a stroll down Berlin's grand downtown boulevard, which connects the Brandenburg Gate and Tiergarten with Alexanderplatz and the TV Tower. You'll pass by beautiful historic buildings like the Berlin Opera, the State Library, the German History Museum, the Neue Wache (Memorial to the Victims of War), Bebelplatz (the site where the Nazis emptied out banned books from nearby libraries to be burned), and Humboldt University, where Albert Einstein taught.
Free, always open, Mitte, S-Bahn: Brandenburger Tor (Brandenburger Gate end) or S+U-Bahn: Alexanderplatz (TV Tower end)
#### **DDR Museum**
Enjoy this unique museum through interactive experiences that reveal the daily life of those living under the communist regime. Exhibits are fully hands-on. See what East Berliner denim looks like, sit in a communist-era car, and wash your brain with some communist propaganda.
€7 adult, tickets online from €5, daily 10:00-20:00, Sat. until 22:00, Karl-Liebknecht-Strasse 1, Mitte, +49 (0)30 847 12 37 31, ddr-museum.de, S-Bahn: Hackescher Markt or S+U-Bahn: Alexanderplatz
#### **Berliner Dom**
Climb to the top of Berlin's oldest church for an amazing view of the city. It's well worth the effort climbing up the stairs but up to you and your budget.
€7 adult/€5 student, Mon-Sat 09:00-20:00, Sun 12:00-20:00, until 19:00 Oct-Mar, closes around 17:30 on concert days, interior closed but dome open during services, Am Lustgarten, Mitte, +49 (0)30 20269136, berlinerdom.de, S-Bahn: Hackescher Markt
#### **Bauhaus Archives Museum**
It's impossible to overstate the impact of the Bauhaus School (founded 1919) on modern design. A ragtag group of professors and students experimented with avant-garde shapes and designs in both two and three dimensions, pushing the boundaries of aestheticism and function. Unfortunately, colors and performances irritated the Nazis, and the school was shut down in 1933. But in even those few years, Bauhaus left an impact that can still be seen in everything from President Obama's campaign posters to just about every corporate brand today. Many of the students' drawings, posters, watercolors, and models are on display in this purpose-built museum toward the west side of town, just south of Tiergarten. For any industrial or graphic design major, this is an obligatory pilgrimage.
€7 Wed-Fri, €8 Sat-Mon, open Wed-Mon 10:00-17:00, closed Tues, Klingelhöferstrasse 14, Central Berlin, +49 (0)30 254 00 20, U-Bahn: Nollendorfplatz
#### **Berlin's Jewish Museum**
This museum will send chills through your body, not just from the personal stories of Jewish history and culture, but from the stark and jagged architecture of the building, reflecting the cold abuse of the Holocaust. Designed by Daniel Libeskind, the building is defined by its zigzag hallways, skewed windows, and great "voids," leaving you with a feelings of unease and apprehension. Plan on spending about half a day here following the worthwhile audio guide through over 1,000 years of German-Jewish culture.
€8, Mon 10:00-22:00, Tues-Sun 10:00-20:00, last entry one hour before closing, closed on Jewish holidays, Lindenstrasse 9-14, Kreuzberg, +49 (0)30 2599 3300, jmberlin.de/en, U-Bahn: Hallesches Tor or Kochstrasse
### **TOP EATS**
The cuisine in Berlin is diverse. You'll definitely see the influence of its multicultural—particularly Italian and Turkish—makeup. Traditional German cuisine is often based on meat and potatoes, and flattened and fried pork steak, called **schnitzel,** is a common choice. **Currywurst** (a sausage with ketchup and curry) is a Berlin staple that's best eaten with a side of fries.
Tips of 5-10 percent of restaurant bills are appreciated. When paying, round up to the next couple euros. Remember that the water and any bread or pretzels that are waiting for you at the table are not generally included and will be added to your bill if you touch them.
#### **Back-Factory**
Back-Factory is a simple cafeteria with mass-produced pastries and self-service coffee machines. It's not a super special place, but it's a convenient place to grab a quick breakfast pastry or sandwich on the go. There are multiple locations throughout town.
Pastries from €1.50, Mon-Fri 06:00-20:00, Sat 06:00-18:30, Sun 06:00-19:00, Rosenthaler Platz at Brunnenstrasse 1, +49 (0)30 40056105, back-factory.de, U-Bahn: Rosenthaler Platz
#### **Georgbrau**
If you're looking for classic German fare and fine beer in a beer hall, look no further. Located right in the middle of Berlin's old medieval town (you won't be able to tell), this beer hall proudly serves piping hot racks of ribs, duck, and brats with sizable salads, potatoes done just right, and delicious house brews. Just a few steps from Unter den Linden, Georgbrau is a great nearby option if you find yourself in Mitte around dinnertime.
Mains from €8, daily 12:00-24:00, Spreeufer 4, Mitte, +49 (0)30 242 42 44, brauhaus-georgbraeu.de, U-Bahn: Klosterstrasse
#### **Kaschk**
Kaschk gets my vote for best coffee in town. Round that out with beers after the sun goes down and a shuffleboard table to boot, and what more could you ask for? This is your quintessential hipster brew house—of both grains and beans—with bearded baristas and one long common wooden table, yet without the pretentiousness that comes along with coffee snobbery. Ask the friendly staff about their best beans at the moment, and have them brew you something unforgettable.
From €2.50, Mon-Fri 08:00-02:00, Sat-Sun 10:00-late, Linienstrasse 40, Prenzlauer Berg, +49 (0)1578 197 99 70, instagram.com/kaschk, U-Bahn: Rosenthaler Platz or Weinmeisterstrasse
#### **Dada Falafel**
If you're in Mitte and feeling in the mood for a Turkish dish, head to Dada Falafel, where the boys crank out fried chickpea balls sunup to sundown. Their freshly squeezed orange juice is wonderful. As you leave, take a close look at the rundown **Tacheles** across the street, Berlin's most famous ex-squatting complex (it may not be around for long, as developers are already salivating at the real estate value).
From €6, daily 10:00-02:00, Linienstrasse 132, Mitte, +49 (0)30 2759 6927, dadafalafel.de, U-Bahn: Oranienburger Tor
#### **Konnopke Imbiss**
If you've never tried currywurst before, Konnopke Imbiss is _the_ place to do it. Find this inconspicuous takeaway joint underneath the elevated U2 rails and post up next to everyday Berliners enjoying their everyday meal.
€2-4, Mon-Fri 05:30-19:00, Sat 11:30-19:00, Schönhauser Allee 44B, under U-Bahn tracks, Prenzlauer Berg, konnopke-imbiss.de, U-Bahn: Eberswalder Strasse
#### **Aufsturz**
Aufsturz is famous more for their endless beer list than their food. But the menu still isn't bad and features varied Berliner dishes that are easy on the budget. Ask the friendly staff to help you translate, and try a beer you haven't heard of before. Aufsturz is a great place to pre-game before a night out. A side note: This street is known to be where many prostitutes work, so don't be swayed by any overly friendly women as you stumble out of the bar.
Beers from €3, dishes from €6, Mon-Thurs 12:00-01:00, Fri-Sat 12:00-02:00, Sun 12:00-01:00, Oranienburger Strasse 67, Mitte, +49 (0)30 2804 7407, S-Bahn: Oranienburger Strasse, U-Bahn: Oranienburger Tor
#### **Prater**
A beautifully renovated swing-era bar, Prater serves up enormous portions of meat and vegetables, topped off with frothy mugs of delicious German lager. During the summer, it's easy to spend the better part of the afternoon in the shaded beer garden, enjoying the buzz of Berliner beer and life.
€6-15, Mon-Sat 18:00-24:00, Sun 12:00-24:00, Kastanienallee 7-9, Prenzlauer Berg, +49 (0)30 448 56 88, pratergarten.de, U-Bahn: Eberswalder Strasse
#### **Altberliner Kaffeestube**
This is an excellent choice for your typical Berliner meal: meat-based dishes with sauerkraut, potatoes, and salads. While a staple for those who live in Prenzlauer Berg, Altberliner does come with inflated P'berg prices.
Mains from €10, daily 12:00-24:00, till 01:00 on weekends, Fürstenberger Strasse 1, Prenzlauer Berg, +49 (0)30 449 51 51, altberliner-restaurant.de, U-Bahn: Bernauer Strasse
#### **Taisu**
Located just around the corner from Checkpoint Charlie, Taisu is said to have some of the best sushi in Berlin, and the location makes for a convenient pit stop. Pop into this clean and mod yet casual restaurant during lunch and enjoy their 2-for-1 sushi roll lunch special.
€2-6, daily 12:00-23:00, Rudi-Dutschke-Strasse 28, Kreuzberg, sushiwok.de, U-Bahn: Kochstrasse
#### **Hasir**
Give Berlin's best Turkish food a try in this classy, upscale kebab shop with guaranteed fresh ingredients and delicious plates like _kofta_ (meatballs), pita, and fresh baklava. Find a second location in Mitte just north of the Hackescher Markt S-Bahn station (Oranienburgerstrasse 4).
Dishes from €8.50, daily 24 hours, Adalbertstrasse 10, Kreuzberg, +49 (0)30 614 23 73, U-Bahn: Kottbusser Tor
#### **Il Ritrovo**
Tired of sausage and curry? This family of restaurants gives you a dose of punk and pizza rolled and kneaded to perfection. Work your way to a free table and get ready for the most authentic and arguably best pizza in town, topped with your favorite ingredients. Il Ritrovo is well-located for after-dinner drinks in the bumping district of Friedrichshain.
Pizzas from €7, daily 12:00-24:00, Gabriel-Max-Strasse 2, Friedrichshain, +49 (0)30 2936 4130, S-Bahn: Warschauer Strasse
#### **Il Casolare**
Il Casolare is easily one of my favorite Italian restaurants in town. They pride themselves on their authentic pizza and pasta. Sit in the cozy interior, or enjoy their outdoor seating on nice days. Go for a pitcher of their house wine to split with friends.
Pizzas from €6, daily 12:00-24:00, Grimmstrasse 30, Kreuzberg, +49 (0)30 6950 6610, U-Bahn: Kottbusser Tor
#### **La Pausa**
One step into this place and it feels like you've been transported to another country, with the smells of fresh pizza in the oven and the commotion of happy eaters to boot. Order pizza by the slice as you would in Italy, sprinkle with red pepper, and you've got a great late-night snack after a night out on Rosenthaler Platz.
Slices from €2, daily 11:00-02:30, Torstrasse 125, Prenzlauer Berg, +49 (0)30 2408 3108, U-Bahn: Rosenthaler Platz
#### **Curry Mitte**
If you're in the mood for Berlin's most famous snack, jump on this opportunity. Go for the currywurst and chips to keep you going late in the night.
Mon-Sat 11:00-24:00, till 06:00 on Fri and Sat, Torstrasse 122, Prenzlauer Berg, +49 (0)1520 106 95 59, currymitte.de, U-Bahn: Rosenthaler Platz
#### **Schwarzes Café**
This 24-hour café is a great option in the heart of residential West Berlin. I recommend it as a pit stop at the turnaround point of the recommended bus 100 audio tour. Step into this typically eclectic café, which like so many other places in Berlin somehow pulls off being great at just about everything, from coffee to beer and tasty plates of fruit, omelets, and salads.
Plates from €8, daily 24 hours, Kantstrasse 148, West Berlin, +49 (0)30 313 80 38, schwarzescafe-berlin.de, S+U-Bahn: Zoologischer Garten
#### **International Restaurant Row**
Why some of the most interesting restaurants in town decided to huddle on a half block on Weinbergsweg, I don't know, but I'm not complaining either! A lunch at these sit-down restaurants will run around €15, dinners €20-25. Find your bargains at the takeaway shops. On a single block, cafés and restaurants take you from Moscow to Mexico and from Chinese pork dumplings to French pastries in just a few meters. Beyond restaurants, you'll find a bakery selling cheap sandwiches, pastries, and coffee; the recommended **Circus Hostel** ; and a **convenience shop** (Weinbergsweg 26-27) that sells beer and has chairs to relax on out front while consuming. (Why isn't this anywhere else in the world? Because I love it!)
Some of my favorite spots include **Gorki Park** (Weinbergsweg 25), which serves very good burgers and Russian staples like goulash; **Yumcha Heroes** (Weinbergsweg 8) offering tasty—if slightly overpriced—Chinese dim sum; **Café Fleury** (Weinbergsweg 20), a cozy and classic French bakery and café, with pastries, salads, sandwiches, and limited outdoor seating; **Soup 'n' Roll** (Torstrasse 117, around the corner from Weinbergsweg), serving a cheap and quick noodle; **Bagel Company Berlin** (Rosenthaler Strasse 69), serving breakfast bagels and sandwiches, and **My Smart Break** (Rosenthaler Strasse 67), with fresh and healthy sandwiches and coffee to go. Also check out **St. Oberholz Bar and Cafe** (Rosenthaler Strasse 72), a classic Berliner hipster café and bar, with yuppies on laptops and a classy drinking crowd at night. If they're open, spring for the lifeguard high chairs outside to take in the view from a new perspective.
Prenzlauer Berg, U-Bahn: Rosenthaler Platz
### **TOP BEER GARDENS**
While Munich is famous for its Oktoberfest, the world's biggest beer-drinking festival, Berlin will not be left out. In typical Berliner form, these biergartens each come with a little something unique to make them some of my favorites in town.
#### **Prater Garten**
Established in 1837, Prater Garten is Berlin's oldest beer garden. It seats over 600 in a beautiful, outdoor area shaded by chestnut trees. If the weather is bad, step inside its indoor beer hall, where you can enjoy some of the best sausages and wienerschnitzel Berlin has to offer.
Pints from €4, Kastanienallee 7-9, Prenzlauer Berg, +49 (0)30 448 56 88, pratergarten.de, U-Bahn: Eberswalder Strasse
#### **Schleusenkrug**
Berlin's most idyllic beer garden, Schleusenkrug is in Tiergarten park, just north of the zoo. On a sunny day, there's nothing better than a crisp German lager and the cool breeze through the trees here. The food is of so-so value, so I recommend getting your fill on the liquid bread in your stein. The recommended Fat Tire Bike Tour takes its pit stop here to refuel.
Pints from €3.50, daily 10:00-24:00, Müller-Breslau-Strasse, Central Berlin, schleusenkrug.de, S+U-Bahn: Berlin Zoologischer Garten
#### **Bierhof Rüdersdorf**
I stumbled across this unique, mod beer garden on my way out to Friedrichshain for the night, and I quickly understood why it's one of Berlin's favorites. Tucked behind the famous techno-club Berghain, this small beer garden and posh one-floor patio bar is easy to miss, but a beer here is a great way to while away an afternoon, or to enjoy while eyeing the line to get into the big club next door. Check online ahead of time because the Bierhof is open seasonally (closed in the winter), and dates change often.
Snacks from €2, burgers from €6, Rüdersdorfer Strasse 70, Friedrichshain, +49 (0)30 2936 0215, bierhof.info, S-Bahn: Ostbahnhof
### **TOP NIGHTLIFE**
#### **NIGHTLIFE DISTRICTS**
In this sprawling city, nightlife centers around several districts. All are slightly grittier than what you might find in London or Paris, but what my favorite spots may lack in cleanliness, they more than make up for in party mode and attitude.
##### **Prenzlauer Berg**
Prenzlauer Berg is known as the yuppie district, filled with young professionals getting their family life started. Look forward to seeing heavily tatted ex-punks pushing strollers down the street. **Kastanienallee** is an entire street full of bars and clubs.
Conveniently located near most of the recommended hostels, **Schönhauser Allee** leads uphill from Rosa-Luxemburg-Platz. In the first couple blocks from Torstrasse, you'll find some interesting and alternative venues. The kitschy goth-punk **Last Cathedral** (Schönhauser Allee 5) puts on frequent specials and events. **White Trash Food** (Schönhauser Allee 6-7) is a funky, American diner/Chinese shop/British pub hybrid with frequent live music shows. It's a great spot for huge burgers and fries, drinks, and a casual night out.
Prenzlauer Berg, U-Bahn: Rosenthaler Platz (Kastanienallee), U-Bahn: Rosa-Luxemburg-Platz (for Schönhauser Allee)
##### **Rosenthaler Platz**
This five-point intersection of everything Berliner has all the essentials for a night out—including proximity to good eats at the recommended **La Pausa** and **Curry Mitte** —and a hair of the dog recovery the next morning.
Mitte, U-Bahn: Rosenthaler Platz
##### **Friedrichshain**
Friedrichshain, a dense tangle of nightlife complexes, bars, and restaurants, is edgy Berlin at its best. Though the district is beginning to show the wear of normalization and mass tourism, everyone knows this is Berlin's best nightlife. Of the range of options in this area, **Hops & Barley** (Wühlischstrasse 22/23) and **Szimpla** (Gärtnerstrasse 15) are two of my favorite, more authentic bar choices. Start there and wander south toward the major clubs straddling the Spree River, like **Watergate** and **Tresor.**
Friedrichshain, S+U-Bahn: Warschauer Strasse
#### **BARS**
Bars in Berlin are fun and always a bit alternative to keep you on your toes. Each featuring its own little twist, Berliner bars reflect the city's eclectic makeup: Cocktail lounges give way to Ping Pong bars and authentic jazz cafés to keep the masses entertained.
##### **Dr. Pong**
This bar's all about one thing: Ping-Pong. And beer. Drop a five-euro deposit on a paddle and buy a ticket for a beer at the door, then get started running around the table in a game of group Ping-Pong. In this game of elimination, hit the ball from each side in turn and be the last person standing.
Drinks from €5, Mon-Sat 19:00-06:00, Sun 18:00-06:00, Eberswalder Strasse 21, Prenzlauer Berg, drpong.net, U-Bahn: Eberswalder Strasse
##### **Mein Haus am See**
This is a quintessential Berlin bar: warm atmosphere, hipster clientele, full bar with a few different options, smoking room, stadium seating in the back to look out over your fellow imbibers, and a bumping club downstairs with some enjoyably funky features. It's open 24/7, so you can get your drink or coffee on any time of day. At this spot, near Rosenthaler Platz, it feels like you're hanging out at your friend's place more than the local bar. As their tagline goes, "It's not a bar, it's not a club... it's something sexier in between."
**LGBT BERLIN**
Berlin is known by many as Europe's gay capital. With prominent public figures, celebrities, and everyday people openly gay, the LGBT scene hardly raises an eyebrow in this city that's seen it all, and Germany has strict and extensive anti-discriminatory laws. Check out the website of **Out In Berlin** (out-in-berlin.com) for recommendations on cafés, restaurants, clubs, bars, nightlife districts, hotels, and more. Europe's biggest gay parade each year toward the end of June draws parties by the thousands.
Drinks from €4.50, daily 24 hours, Torstrasse 125, Prenzlauer Berg, +49 (0)163 555 80 33, mein-haus-am-see.blogspot.com, U-Bahn: Rosenthaler Platz
##### **Die Weinerei**
This bar is based completely on the honor system: You simply pay €2 to rent a glass and drink as much wine as you want, paying only as much as you think you owe in the tip jar when you leave. It sounds crazy, and seems like something that could be easily abused; however, with the charming atmosphere and trusting group mentality, imbibers just about always adequately pay for their consumption.
€2 to get started, Mon-Fri 13:00-20:00, Sat 11:00-20:00, Veteranenstrasse 14, Mitte, +49 (0)30 440 69 83, weinerei.com, M: Rosenthaler Platz
##### **Luzia**
Complete with an outdoor terrace overlooking the street, Luzia is one of the most popular bars in the Kreuzberg area. Inside you'll find old velvet armchairs, peeling vintage wallpaper revealing chipped red bricks, mismatched tables, and dim lamps that create a warm, laid-back atmosphere.
Coffee and beers from €4, daily 12:00-24:00, Oranienstrasse 34, Kreuzberg, +49 (0)30 8179 9958, luzia.tc, U-Bahn: Moritzplatz
##### **Zosch**
This bar churns out mediocre food, excellent beer throughout the week, and life-changing jazz sessions every Wednesday. Even for non-believers, these jazz sessions in the WWII bomb shelter downstairs are magical. The crew jams in front of regulars and in-the-know tourists alike. Grab a stein and join the communal tables. Be respectful to those who are there for the music, and keep your conversations hushed. Notice the three tiers of brick candle holders sticking out from the wall. The candles at each height (knee, waist and chest) indicated gas seeping in and displacing oxygen, telling those seeking shelter "It's time to pick up the kiddies from the floor and give them piggyback rides." I much prefer today's less menacing atmosphere.
Liters from €7.50, daily 16:00-late, Tucholskystrasse 30, Mitte, +49 (0)30 280 76 64, zosch-berlin.de, S-Bahn: Oranienburger Strasse, U-Bahn: Oranienburger Tor
#### **CLUBS**
##### **Berghain**
Even if you're not a fan of techno, Berghain is the place to go if you want the real Berlin disco experience. Once a working power plant, the place is absolutely massive, with the cavernous main room blasting minimalist techno and the upstairs Panorama bar rocking purely house music. Though notoriously hard to get into, Berghain is the epitome of what you would expect a no-holds-barred European dance club to be and is worth it even with the chance of being turned away at the door. Be cool in the line, wear dark clothes, look like you're there to party hardy, don't take out your phone, and avoid speaking English. The nightly rave goes well into the next day, with Saturday-Monday events being the most popular.
€15 cover, open 24 hours Fri-Mon, Am Wriezener Bahnhof, Friedrichshain, +49 (0)30 2936 0210, berghain.de, S-Bahn: Ostbahnhof, U-Bahn: Weberweise
##### **Tresor**
Tresor is hailed as the godfather among the clubbing crowd, taking techno beats to an industrial scale in a massive abandoned power plant just across the river in the alternative neighborhood of Kreuzberg. This labyrinthine club comprises three dance floors, with long, dark graffitied hallways, dimly lit corners, and strobe lights pulsing throughout. Every inch of Tresor's 230,000 square feet is filled with an incredible sound system, top-tier DJs, ear-shattering bass, and a friendly mixed crowd that will dance well past midmorning. If you are looking for a staple of Berlin's nightlife (and don't feel like waiting in the endless line at Berghain), give Tresor a shot. Get your buttoned-up hipster mode on for best chances at the door.
€10-15 cover, parties rock out every Mon, Wed, Thurs, Fri, and Sat, Köpenicker Strasse 70, Kreuzberg, +49 (0)30 6953 7721, tresorberlin.com, U-Bahn: Heinrich-Heine Strasse
##### **Watergate**
Dance the night away at this midsized, double-floor nightclub spinning a solid blend of house music and techno, putting it on _DJ Mag_ 's Top 100 Clubs list several years in a row. It's loved for its waterfront location, an open-air glass-bottomed patio that seems to float on the water (great to catch your breath on between beat drops), and an innovative LED-lighted ceiling upstairs that throbs with the music. Party-goers are happy they waited through the sometimes-long (but fast-moving) line. It's casual, but dress slightly smart to avoid any hassle with the bouncers.
€15 cover, 21+ age policy, Wed, Fri, and Sat 24:00-07:00, Falckensteinstrasse 49, just east of Oberbaumbrucke bridge, Kreuzberg, +49 (0)30 6128 0394, water-gate.de, U-Bahn: Schlesisches Tor
##### **KitKat Club**
KitKat Club is a legendary Berlin nightclub where fetishes, debauchery, hedonism, and the LGBT community are all welcome. This is an eclectic dance venue spinning EDM and house. Inspired by the free-love beach parties of Goa in the '80s, this is where freaks step out of their shells and into their leathers and glitter costumes—the more fabulous and edgy, the better. Friday, Saturday, and Sunday are their big nights, with CarneBall being their trademark event each Saturday night; bartenders do their own cabaret show to the glee of party-goers. The doors at this part exhibitionist party, part erotica club usually open at 23:00, but the party doesn't really kick off till around 03:00, going until the sun comes up. Show up wearing too much clothing and you'll have to leave some at coat check to hit the sweaty, neon-lighted dance floor sans shirt.
While the club is safe, friendly, and welcoming, it's much better to party with friends than solo on a first visit to KitKat Club. It's important to check out the website both for dress code guidelines—you won't be allowed in with street clothes—and also to really understand what you're getting yourself into. Find the entrance to the club inside the U-Bahn station.
Cover €7-10, parties run Fri-Mon usually but vary according to the program on the website, Köpenicker Strasse 76, Kreuzberg, kitkatclub.org, U-Bahn: Heinrich-Heine Strasse
#### **PUB CRAWLS**
##### **New Berlin Pub Crawl**
Sandeman's, the folks who run daily walking tours, also organize nightly pub crawls to show you the fun part of town. €12 gets you a juicy welcome shot, a handful of bars, a fun crowd, and drink deals throughout the night. Picking up at the old post office on Oranienburger Strasse, this crawl shows you the best of the central Mitte district, ranging from grungy bars to hipster lounges, and is within walking distance for centrally located hostels. These groups get rowdy and frequently climb past 80 party-goers, funneled to this party from the walking tours in the day.
€12, daily 20:00, meet in front of the old post office at the corner of the Oranienburger Strasse S-Bahn station, Mitte, newberlintours.com, S-Bahn: Oranienburger Strasse
##### **Alternative Berlin Pub Crawl**
Thankfully, Berlin has an alternative to your standard rowdy pub crawl, rightly called Alternative Berlin Pub Crawl. Come out to enjoy interesting stories, classy drinks, and unique spots that may be a highlight of your trip. The crawl includes free entry to five bars and clubs and is capped at 15 or so people, so the experience is much more intimate than the one offered by Sandeman's.
€10, daily 21:00, meet outside Starbucks at the TV Tower, Mitte, alternativeberlin.com, S+U-Bahn: Alexanderplatz
### **TOP SHOPPING & MARKETS**
#### **SHOPPING DISTRICTS**
##### **Friedrichstrasse**
Comparable to Oxford Street in London, Friedrichstrasse between the namesake station and Checkpoint Charlie is your standard downtown commercial zone with clothing brands like All Saints, Gucci, and Gap.
Friedrichstrasse, Mitte, S+U-Bahn: Friedrichstrasse
##### **Hackescher Höfe Courtyards & Hackescher Markt**
The area surrounding Hackescher Markt (+49 (0)30 2809 8010) is a trendy little downtown neighborhood offering a wide selection of boutiques, designer shops, and big international brands like Urban Outfitters and American Apparel. Find Rosenthaler Strasse to find the courtyards and wander through this maze of shops, cafés, and boutiques to get your shopping fix. Built as a housing project, this entire area has now developed as a great place to pick up unique finds. Keep your eyes open for deals! On Thursday and Saturday 09:00-18:00, there's a funky outdoor market selling handmade goods, snacks, and trinkets.
Rosenthaler Strasse 40-41, Mitte, S-Bahn: Hackescher Markt, U-Bahn: U Weinmeisterstrasse
##### **Oranienstrasse & Bergmannstrasse**
Located in the district of Kreuzberg, these two shopping streets are geared toward people who are interested in vintage clothing, old records, books, and secondhand shops. These streets also have a vibrant café culture and make for a pleasant afternoon of window-shopping and people-watching. **Oranienstrasse** is most interesting between Oranienplatz and the Gorlitzer Bahnhof U-Bahn station. On **Bergmannstrasse,** find charming cafés, shops, and restaurants between Mehringdamm Street to the Marheineke Markthalle. Bergmannstrasse is a 20-minute bike ride south from Mitte, so time a visit here with a casual cruise through the nearby recommended Tempelhofer and Victoria parks.
Kreuzberg, U-Bahn: Gorlitzer Bahnhof (for Oranienstrasse), U-Bahn: Gneisenaustrasse (for Bergmannstrasse)
#### **MARKETS**
Berlin is a flea market paradise. Macklemore would think he died and went to thrift-shopping heaven. Even the permanent stores often feel like they have a unique edge, so you're sure to find some great styles and whatever else you may be looking for.
##### **Nowkoelln Flowmarkt**
One of Berlin's newest and trendiest flea markets, Nowkoelln Flowmarkt caters to hip Berliners and tourists in the know. Located alongside the Landwehrkanal canal, the market sells funky vintage clothing and home goods and is complete with some top-notch food stalls. The market tends to run every other Sunday, but check the website to confirm dates, as they can vary during winter break.
10:00-17:30 every other Sun, Nowkoelln Flowmarkt, Maybachufer 31, Kreuzberg, nowkoelln.de, U-Bahn: Schönleinstrasse
##### **Türkischer Markt**
The Türkischer Markt is the closest you'll get to Istanbul's Grand Bazaar in Germany. This market serves the needs of the large local Turkish community and offers some of the best Turkish food around. The market also sells clothes and fabrics and is becoming very popular with Berlin's younger demographic, who like to make their own styles.
Tues and Fri 11:30-18:30, Türkischer Markt, located around Maybachufer 7, Kreuzberg, tuerkenmarkt.de, U-Bahn: Schönleinstrasse
##### **Mauerpark Flea Market**
What was once a part of the "death strip" of the Berlin Wall has now become one of my favorite outdoor markets in Europe. ("Death strip" refers to the no man's land between the Berlin Wall's double walls.) Brace yourself: The throngs flocking to this popular flea market can be overwhelming. Follow the smell of fresh sauerkraut, currywurst, and waffles until you find a great street food stand to tide you over. Bands and magicians gather in the amphitheater-like bowl, creating a jovial atmosphere. Be careful, or you may never leave this magical place.
Sun 09:00-18:00, Bernauer Strasse 63-64, Prenzlauer Berg, U-Bahn: Eberswalderstrasse
### **TOP PARKS & RECREATION**
#### **Tiergarten**
What was once the official hunting ground of the Prussian king was converted into a public park in the late 18th century. Spend an afternoon enjoying the many ponds, trees, and open-air cafés this sprawling piece of land has to offer. This leafy park, which takes about 45 minutes to walk across, is ideal to explore on a bike.
Find the **Victory Column** (€2.20 adult, Apr-Oct Mon-Fri 09:30-18:30 and Sat-Sun 09:30-19:00, Nov-Mar Mon-Fri 10:00-17:00 and Sat-Sun 10:00-17:30, Grosser Stern), located in the park's center. The column represents Prussia's victory over France in 1871. The golden statue at the top is of the goddess Victoria, often referred to as Goldelse (Golden Elsi) by the locals. Climb the 285 steps to the top to get a spectacular view of the Tiergarten and Berlin.
Thirsty? Head to **Schleusenkrug** beer garden (Muller-Breslau-Strasse, daily 10:00-24:00) on the far west side of the park, just north of the zoo, for a pint and grub.
Central Berlin, S-Bahn: Tiergarten, U-Bahn: Hansaplatz
#### **Tempelhofer Park & Victoria Park**
The old Templehof airport was used by the Americans during the Berlin Air Lift (June 24, 1948-May 12, 1949), the epic effort of flying in tons of food and supplies every 30 minutes around the clock to support the people of West Berlin for 46 straight weeks when the communists shut down the land access routes to West Germany. This was the only way food, medicine, and other supplies were delivered to help an entire city survive. Seeing the futility of the situation, the DDR finally reopened the land access as the United States demonstrated its commitment to not leaving West Berlin behind. Today, Tempelhofer is an expansive park where Berliners come to relax. And rather than sending planes into the sky, the airstrips are enjoyed by cyclists, joggers, and kite skaters—yes, kite skaters. Nearby, the man-made waterfall in **Victoria Park** offers a spectacular view of Berlin from above one of the city's hipper districts. From Mitte, Tempelhofer, Victoria Park, and the Bergmannstrasse shopping street are about a 20-minute bike ride south, so it's best to lump these sights together.
**SPREEPARK**
What do an ex-communist theme park, a cocaine smuggling bust, and an unbelievable family drama have in common? Answer: Berlin's abandoned amusement park, Spreepark. Get ready to have your mind blown.
In 1969, the communists opened Kulturpark Plänterwald in East Berlin and operated the amusement park continuously until the wall fell. At that point, a family business headed up by Norbert Witte took over, changing the name to Spreepark. Spreepark's first few years went well. The owners added new rides, and visitors to the park climbed to 1.5 million per year. But the Witte family soon got into hot water. The forest that the park was located in was rezoned, and the park was not permitted to add the parking spaces it needed. Ticket sales declined, and Witte slid deeper and deeper into debt.
At this point, Mr. Witte got desperate and started looking for ways to move the park, sell the rides, and wash his hands of this whole ordeal. In 2002 he found an opportunity in Peru. He shipped six attractions to South America and moved there himself with his family. But Witte just couldn't make it work, and with debts topping 10 million euros, Witte relied on "friends" to help pay off his debts, but of course they wanted something in return. And we're only just getting started.
By 2004, things weren't looking good for the family. They owed a lot of money to people you just don't want to owe money to. With his back against the wall, Witte decided to ship his rides back to Germany...with over 400 pounds of cocaine hidden inside them. Witte, along with his son, was busted before leaving Peru. He was caught with enough drugs to put him away for four years. Even worse, his son—who had been kept in the dark about the entire mission—was busted with even more drugs and is currently serving a 20-year sentence.
Nearly 15 years since Spreepark officially closed its gates, the rides are overcome by weeds and the entire park seems stopped in time. Adventurers often climb or crawl under the fences for a glimpse of the eerie park grounds. Ferris wheels blow slowly in the wind, play houses lean sideways, and swan boats sit lonely in the pond. The site was taken over by the city in 2014 and remains closed to the public. Hopping the fence is certainly against the law, yet many still get through and leave with pictures of a fascinating, post-apocalyptic world. If you decide to see for yourself, take the S-Bahn to Triptower Park or Plänterwald and walk in to Spreepark from there.
Kreuzberg, M: Platz der Luftbrücke
### **TOP TOURS**
Tours in this city go a long ways toward bringing to life the complex history, monuments, and memorials of Berlin. Remember, there are a number of different perspectives by which you can examine Berlin: medieval, Roaring '20s, Nazi Berlin, postwar Berlin, communist Berlin, and eclectic modern Berlin. For first-time visitors, the best tours combine a mixture of all the above in a casual and introductory format, but history buffs can also go on deep dives specific to any of these subjects through more specific tours offered by all of the operators below.
#### **Fat Tire Bike Tours**
I like bike tours as a way to quickly get a handle on the layout and size of a city. Fat Tire's enthusiastic guides help visitors begin to grasp Berliner culture and history on two wheels over about 4.5 hours. Highlights along the route include the Berlin Wall, Hitler's Bunker, Museum Island, Checkpoint Charlie, Brandenburg Gate, Victory Column, and the compelling Holocaust Memorial. Log on to the website for details on all their tours, prices, and meeting points. Smart travelers will note the sights they see along the tour and remember them so they're not doubling back during the rest of their visit. Tours meet at varying hours and days throughout the year just on the north side of the TV Tower. Ask about their discounted rate to extend your bike rental for an extra couple days from €10 per day.
€26 adult/€24 student, Panoramastrasse 1a, Mitte, +49 (0)30 2404 7991, fattirebiketours.com/berlin, S+U-Bahn: Alexanderplatz
#### **Brewer's Walks Berlin**
The Brewer's team takes professional tour guiding to the next level. Come out for a history buff's enthralling all-day walking tour of the highlights of central Berlin or enjoy the free 3.5-hour walking tour each afternoon. Local experts, off-script and passionate about their hometown, take you on a deep dive into the numerous events that put Berlin on the world stage over and over again. Choose from the **Free Berlin Tour** (free, tip-based, daily 13:00, 3.5 hours), the **Best of Berlin Tour** (€15, daily 10:30, 6 hours), or the **Local Beer & Breweries Tour** (€30, Thurs, Fri, and Sat 19:00, 3 hours).
All tours meet in front of Friedrichstrasse station at the corner of Friedrichstrasse and Georgenstrasse, Mitte, brewersberlintours.com, S+U-Bahn: Friedrichstrasse
#### **Alternative Berlin**
Focusing on the fascinating subcultures that make up this diverse city, Alternative Berlin is a great choice for those more interested in street art and hidden neighborhoods that most tourists miss out on. Choose a **free tour** (daily 11:00 and 13:00, 3 hours), the **Street Art Workshop** (€15, Mon, Wed, and Fri-Sat 12:00, 3.5 hours), the **Real Berlin Experience** (€12, Tues, Thurs, and Fri-Sat 12:00, 4.5 hours), or their innovative and intimate small-group **Pub Crawl** (€10, daily 21:00, free entry to five bars and clubs).
All tours meet outside Starbucks at the TV Tower, Mitte, alternativeberlin.com, S+U-Bahn: Alexanderplatz
#### **Jimbo's Cheap Man's Bus Tour**
**Bus 100** cuts straight through the center of Berlin, taking you on a double-decker ride past most major tourist sights. Create your very own hop-on, hop-off bus tour by listening to this free audio tour. Just google "Circus Hostel Jimbo's Cheap Man's Bus Tour" and follow the link to the download. Thanks, Jimbo!
Tour free, bus ticket €2.40, bus runs every few minutes throughout the day, bus 100 running between Alexanderplatz (Mitte) and Zoologischer Garten, S+U-Bahn: Alexanderplatz
#### **Sandeman's New Europe Walking Tours**
Sandeman came to Europe and shook up the tourism industry by offering mass-produced, scripted walking tours led by expats and English-speakers for free—yes, free! How can you beat that price? They only ask for you to tip what you think the tour deserves. Each day crowds gather to go on these entertaining 2.5-hour introductory tours of Berlin's center. During the tour you'll be informed about the company's other paid tours on offer, including a day trip to **Sachsenhausen Concentration Camp** (€15, daily 11:00, 5+ hours), **Red (Communist) Berlin** (€12, Tues, Thurs, and Sat 14:00, 3.5 hours), a day trip to **Potsdam** (€14, Wed, Fri, and Sun 11:00, 5+ hours), and of course the backpacker pub crawl.
Free tours daily at 11:00 and 14:00, meet at the Starbucks in front of Brandenburg Gate, Mitte, newberlintours.com, S-Bahn: Brandenburger Tor
### **TOP HOSTELS**
You've got a ton of options for a fun social stay in Berlin. For a time, real estate was cheap so owners bought up properties and converted old buildings into fresh new hostels. While real estate prices have skyrocketed, hostel bed prices have thankfully not followed suit...yet.
#### **St Christopher's Inns**
Another solid addition to the St Christopher's hostel chain, this hostel has it all: bar, lounge room, free showers, great location near Rosa-Luxemburg-Platz, and cool staff. Free walking tours leave from the hostel daily, and the cheap drinks give you a great way to kick off the evening.
Beds from €10, 24-hour reception, free Wi-Fi, bar with happy hour, continental breakfast included, restaurant, Rosa-Luxemburg-Strasse 41, Mitte, +49 (0)30 8145 3960, st-christophers.co.uk/berlin-hostels, U-Bahn: Rosa-Luxemburg-Platz
#### **Wombat's City Hostel Berlin**
Wombat's Berlin puts you right in the middle of Mitte, with options for food, nightlife, and shopping abounding just meters away in nearly every direction. The on-site bar (with frequent happy hours) and rooftop patio (offering panoramic views of downtown) are great places to kick off your night. And you've got spacious, modern, and clean rooms to look forward to at the end of it. Being just a 10-minute walk from Alexanderplatz, Rosenthaler Platz, and the Hackescher Hof means anything you're looking for is close at hand.
Beds from €10, 24-hour reception, free Wi-Fi, bar, breakfast, rooftop patio, kitchen access, Alte Schonhauser Strasse 2, Mitte, +49 (0)30 8471 0820, wombats-hostels.com/berlin, U-Bahn: Rosa-Luxemburg-Platz
#### **Circus Hostel**
This classic Berliner hostel is different from the others in its personal touch. Enjoy the bar downstairs and some very tasty breakfast each morning. This bar is on the same street as International Restaurant Row. I love the bars and venues nearby, which offer food and entertainment at all hours. The hostel staff leads free daily tours to far-flung corners of the city. If you're interested in private accommodations, check out their hotel and comfortable apartment listings online.
Beds from €8, 24-hour reception, free Wi-Fi, bar, breakfast, Weinbergsweg 1A, Prenzlauer Berg, +49 (0)30 2000 3939, circus-berlin.de, U-Bahn: Rosenthaler Platz
**BERLIN'S SQUATTING CULTURE**
Until recently, squatting in abandoned buildings was a legally protected right in Berlin. How did this whole culture come about? When the Berlin Wall fell, a large exodus of people fled to the more prosperous west, and East Berlin was left with hundreds of abandoned buildings. Simultaneously, there was a movement of people seeking studio space, cheap rent, and a chance to create a new artistic community to the east. Squatters moved to East Berlin and treated the place like home. This was accepted at the time because having people live in empty buildings was the best way to maintain them. Due to a German law preventing landowners from kicking out those living on their properties, individuals taking over abandoned buildings were actually given a degree of legal protection.
The 1990s were the golden years for squatting. Neighborhoods developed their own identities spanning the rainbow of the counter-culture set: hippies, punks, and activists.
Then, all of a sudden, real estate was worth something in Berlin. Fifteen years into the squatting trend, police began to boot squatters from buildings that owners wanted back. But, rather than being snuffed out, squatting culture continues to evolve, as groups band together to purchase property titles to the buildings they're living in. In this way, squats are morphing into communes.
To see a remnant of Berlin's squatting culture, head to Oranienburger Strasse for Berlin's most famous ex-squat: **Tacheles.** _Tacheles_ is Yiddish for "straight talk." The building got its pre-WWI start as a massive department store, small factory, and even Nazi SS office. By the '90s, Tacheles was abandoned and had fallen into disrepair. Squatters moved in, forging a creative community with art studios, movie theater, shops, a bar, and frequent live performances. Tacheles was shut down a few years ago, and today, all you'll see is a rundown building. Use your imagination to recreate the dreads, musical instruments, and spray-paint cans of its heyday.
#### **Generator Hostel Berlin East**
Generator Hostel Berlin East offers clean facilities, loads of amenities, and all the pre-game entertainment you could ask for. The Generator has its own bar and lounge, where you can mingle with other travelers, and is located in trendy Prenzlauer Berg, putting you right next to some good shopping, convenient trains, and Berlin's relatively more sophisticated nightlife.
From €9, 24-hour reception, free Wi-Fi, bar, restaurant, lockers, laundry facilities, mini supermarket, Storkower Strasse 160, Prenzlauer Berg, +49 (0)30 417 24 00, generatorhostels.com, berlin@generatorhostels.com, S-Bahn: Landsberger Allee
#### **Pfefferbett Hostels**
This hostel in a massive old retrofitted brewery is an affordable option for larger groups. Guests appreciate the clean rooms and retro-chic, exposed-brick-arches vibe of the place. I like to relax in the backyard patio when the weather is nice, and the proximity to the Schönhauser Allee and Rosenthaler Platz nightlife makes for an easy stumble home. The team is working hard to install routers to get Wi-Fi to reach all rooms, but for now Wi-Fi only reaches the common room in the reception because of the thick brick walls.
Beds from €8, 24-hour reception, free Wi-Fi in lounge, breakfast, Christinenstrasse 18-19, Prenzlauer Berg, +49 (0)30 9393 5858, pfefferbett.de/en, U-Bahn: Senefelderplatz
#### **A &O Berlin Hauptbahnhof**
Located just 10 minutes from Berlin's main train station and a stone's throw away from major sights like the Reichstag, the Victory Column in Tiergarten park, and the Brandenburg Gate, the A&O Hauptbahnhof hostel is a convenient place to stay. The staff is very friendly and helpful, and the rooms are modern and clean.
Beds from €9, 24-hour reception, free Wi-Fi, bar, laundry facilities, lounge, Lehrter Strasse 12-15, Central Berlin, +49 (0)30 322 920 42 00, aohostels.com, S-Bahn: Berlin Hauptbahnhof
#### **A &O Berlin Mitte**
The conveniently located A&O Mitte is situated just on the southern edge of downtown, bordering Kreuzberg not far from Alexanderplatz. The rooms are cheap and clean, and the bar offers a variety of drinks, karaoke, billiards, and dart-tossing fun.
Beds from €8, 24-hour reception, free Wi-Fi, beer garden (summer), bar, laundry facilities, lounge, bar, Köpenicker Strasse 127-129, Mitte, +49 (0)3 080 947 52 00, aohostels.com, S-Bahn: Ostbahnhof, U-Bahn: Henrich-Heine-Strasse
### **TRANSPORTATION**
#### **GETTING THERE & AWAY**
##### **Plane**
Berlin is served by two airports: **Schoenfeld** (SXF) and **Tegel** (TXL). Find information on both at berlin-airport.de. The **Berlin Brandenburg Airport** (BER) is scheduled to be completed in 2017.
**Schoenfeld** is about 15 miles south of the city center. From the airport, take Airport Express train **RE7** or **RB14** into the center and connect to the city's S- and U-Bahn systems. Connections leave every half hour, cost €3.30, and take about 40 minutes.
If you're flying with the major airlines, you're more likely to arrive at the **Tegel** airport. Take bus **TXL** into the city center. It stops at all three key transportation hubs: Zoologischer Garten, Hauptbahnhof, and Alexanderplatz. Leaving every 10 minutes, this bus takes about 30 minutes and costs €2.70.
##### **Train**
Trains from Prague run eight times daily and cost €40, taking 4.5 hours. Trains take about 6 hours and cost €90 to Munich.
Just about all international arrivals show up at the massive, four-level **Hauptbahnhof,** on the western edge of Mitte. Berlin's other major stations are **Ostbahnhof,** in Friedrichshain; **Spandau,** on the far western edge of Berlin; and **Zoologischer Garten,** in Central Berlin, just on the edge of Tiergarten park. All stations have transfer connections with the S-Bahn public transportation network, which can get you anywhere you need to go in the city.
##### **Bus**
The main bus station is **Zentraler Omnibus-Bahnhof** in West Berlin. From here, you have connections with both the S-Bahn and the U-Bahn, making connecting into the center easy. When connecting out of town leave at least an hour to get there—figuring on about 45 minutes to the bus station from Mitte and allowing 15 minutes to hit the bathroom, grab a snack, and find your bus.
##### **Car**
Berlin is about 650 kilometers (6.5 hours) east of Amsterdam via the A2 highway. It's about 350 kilometers (3.5 hours) north of Prague via the A13 highway.
#### **GETTING AROUND**
Walking is a great way to discover Berlin, but remember, the city is massive, and it would take hours to walk from one end to the other. Try to gauge the distance ahead of time and organize your strolls by neighborhood, hopping on trams or the metro to save time and energy.
Berlin has an excellent **public transportation system,** but the city is so big the system can be overwhelming at first. Once you get used to it, you'll be zipping around Berlin on the trams, buses, and underground in no time. Find a city map that includes tram and metro lines to keep with you during your visit. This really helps before you know the names of the primary landmarks around town.
S-Bahn, U-Bahn, buses, and trams all use the same ticket. A single **ticket** costs €2.40 for zones A and B (which cover the city of Berlin itself but not the airport or outlying cities) and allows as many transfers as you wish for a total of two hours. The **day ticket** (€7) pays for itself by the third ride in one day.
#### **S-Bahn & U-Bahn**
The city is crisscrossed by Berlin's integrated metro system consisting of both underground (U-Bahn) and overground (S-Bahn) lines. Purchase your ticket from one of the ticket machines at each station. You must validate it before your journey by sticking the end into one of the validation machines located either on the platform or on the entry to the station—ticket checkers will ask to see validated tickets and fine those who aren't able to show one (watch the locals and learn). While the S- and U-Bahns close down around 01:00 during the week, they run throughout the night on Friday and Saturday.
#### **Bus**
Follow the same rules as you would for the metro and you will be good to go when riding the bus. Visit bvg.de for information on night bus schedules. For a quick and cheap introduction to the city and its layout, jump on **bus 100** , a double-decker bus that cuts straight through the center of Berlin, connecting most major tourist sights, including the Reichstag and Brandenburg Gate, and ending at Alexanderplatz. Bus 100 departs "Zoo Station" on the west side of town about every five minutes.
#### **Tram**
More than 20 lines make up Berlin's tram network, extending the reach of the S- and U-Bahns to the street level. The trams are new and quite comfortable, linking famous sights like Alexanderplatz to the Bernauer Strasse section of the Berlin Wall and neighborhoods like Mitte and Prenzlauer Berg.
#### **Bike**
Renting a bike for the weekend is a great way to open this big city up. Berlin has focused much energy and capital on making itself a more bike-friendly city, and the bike paths are a welcome sign of these efforts. Heads-up: There are many intersections in Berlin's Mitte that do not have street signs or lights, so it's up to traffic to yield to each other. Consider a tour with **Fat Tire Bike Tours** to see Berlin's highlights and its more alternative neighborhoods.
#### **Taxi**
Taxis charge a flat rate of €3.20 plus €1.65 for each kilometer. Taxis are ideally split among friends for late-night partying on the far side of town. I prefer **Taxi Berlin** (+49 (0)3020 2020, taxi-berlin.de).
### **DAY TRIPS**
#### **Sachsenhausen Concentration Camp**
As one of the first Nazi concentration camps, Sachsenhausen was utilized to test the most efficient layout of detention camps for Jews and political prisoners, which numbered in the thousands by the time the war ended. It's located in the suburbs, and it's disturbing to think that the Nazi vision extended practically to their front door and capital city of Berlin. About 200,000 prisoners were held here throughout the course of the war, and 10,000 victims succumbed to the harsh conditions, starvation, torture, and execution.
For those who want to see a concentration camp or remember the victims of the Holocaust, Sachsenhausen is the best nearby option. A visit to Sachsenhausen offers an introduction to the scientific process that the Nazis used to develop and refine their sadistic detention and systems of mass murder as displayed in the on-site museum, barracks, and chilling execution trench.
The camp is free to enter, with paid and donation-based tours running daily. The paid tours are well worth the cost to understand the horrors that took place in the living areas and dank cells you'll see at Sachsenhausen. Official **tours** (€14) of the camp are offered on Tuesday, Thursday, and Saturday at 11:45 and 14:30 at the visitors center at the south corner of the camp. The first tour has an optional meeting time in Berlin in Mitte: Potsdamer Platz by the historic traffic light tower at 10:20. Transportation is not included in the price. **Sandeman's New Berlin Tours** also offers day trips to Sachsenhausen from Berlin (€15, daily 11:00, 5+ hours).
To get there on your own from central Berlin, take the S1 or regional trains RE5 or RB12 to the end of the line: Oranienburg station. From the station, you've got a mile walk north following signs to "Gedenkestatte Sachsenhausen," or bus 821 takes visitors to the camp hourly in four stops. Or opt for a short taxi ride from Oranienburg station to the camp; it should cost no more than €8. The whole journey takes about an hour each way via public transport.
Free, mid-Mar-mid-Oct daily 08:30-18:00, mid-Oct-mid-Mar daily 08:30-16:30, Strasse der Nationen 22, Oranienburg, +49 (0)3301 200 0, stiftung-bg.de/gums/en
#### **Potsdam & Frederick the Great's Palaces**
Forty kilometers southwest of Berlin is the town of Potsdam, where you'll find a gigantic park sprinkled with awe-inspiring 18th-century Prussian palaces built by Frederick the Great. Germany's equivalent to Versailles, these gargantuan palaces seem to try to outdo each other. The gardens can be exhausting, and really go on forever. You may visit one palace with hundreds of rooms only to gaze down a corridor of beautiful hedges, trees, and finely groomed walkways to yet another palace absolutely dripping in rococo decoration.
The most famous and popular palaces to see are **Sanssouci** (€12, Nov-Mar Tues-Sun 10:00-17:00, Apr-Oct Tues-Sun 10:00-18:00, closed Mon) and the **New Palace** (€8, Apr-Oct Wed-Mon 10:00-18:00, Nov-Mar Wed-Mon 10:00-17:00, closed Tues). If you plan to visit both, buy the **sanssouci+ ticket** (€19), which covers a single visit to all the Potsdam palaces in one day. Find more information on the palaces at spsg.de/en or by calling +49 (0)331 96 94 200. **Sandeman's New Berlin Tours** also offers day trips to Potsdam from Berlin (€14, Wed, Fri, and Sun 11:00, 5+ hours).
Connections to Potsdam are easy. Simply connect to the S7 in central Berlin and take it out to Potsdam Hauptbahnhof. The trip takes just under an hour each way.
### **HELP!**
#### **Tourist Information Centers**
Berlin's tourist information centers are for-profit institutions. Find one at Hauptbahnhof train station and one at Brandenburg Gate.
#### **Pickpockets & Scams**
Violent crime is virtually nonexistent for tourists in Berlin. Still, keep a close watch over your valuables while in train stations, crowded areas, etc, and always be on your guard if someone approaches you with questions. Do not accept any discount train tickets from street sellers, as they are counterfeit and against the law.
#### **Emergencies**
In an emergency, dial 112.
#### **Hospital**
Campus Mitte, Humboldt-Universität Faculty
Charitéplatz 1
+49 (0)30 450 50
#### **US Embassy**
Pariser Platz 2
+49 (0)30 83050
Prague Map
Prague 101
Three Day Itinerary
Top Neighborhoods
Top Sights
Top Eats
Top Nightlife
Top Shopping & Markets
Top Tours
Top Hostels
Transportation
Day Trips
Help!
Visiting Prague feels like stepping into a time machine. The "City of One Hundred Spires" escaped widespread bombing in World War II and often provides an "old world" backdrop in movies. Today, the traces of Czech-Soviet history are fading away as modern-day capitalism integrates itself into daily life. Wander the cobblestone streets, sip cheap beers in smoky pubs, and explore the timeless castle on the hill. You'll love Prague for its bohemian and baroque architecture, raucous nightlife, and picturesque setting, tucked into a bend in the Vltava River.
### **PRAGUE 101**
Prague and the Czech Republic were very different places until the late 1980s. Because of the liberation of Czechoslovakia from Nazi Germany in 1945 by the Russians, the majority of Czech people were pro-Russia, and strong ties were formed with the USSR and its ideals of a communist state. In 1948, the communist party seized control of the majority of votes, and Czechoslovakia joined the ranks of other Central European countries in the Soviet Bloc.
With the support of Moscow and a highly oppressive government, the Czech economy saw a dramatic improvement. Yet with this growth also came the price of pursuing that "improved" communist state: mass imprisonment, corruption, and rigged trials. Once the boundaries and laws became clear to the people, the government eased up a bit, and a new way of thinking began to emerge. Greater freedom was granted in the media, the arts, and citizens' right to travel during this period, which became known as the Prague Spring.
These new ideas were seen as a threat to communism, as they resembled the West and its "bourgeois" take on life. So Russia, along with other members of the Warsaw Pact, invaded Czechoslovakia on August 21, 1968, forcibly ending this progressive trend. Over the next 20 years, a period of "normalization," which emphasized censorship, spying, and police brutality, ensued. This led to the stagnation of the Czech economy, as well as the arrests and abductions of more than 250,000 people who were seen as threats to the communist state.
By the late 1980s, the people had finally had enough. From November 17 to December 29, 1989, hundreds of thousands of Prague's citizens gathered in Wenceslas Square in the peaceful protest known as the Velvet Revolution. These protests led to the end of the single-party communist state and played a large role in the collapse of the Soviet sphere of influence as a whole in Central Europe. Czechoslovakia's first president, Václav Havel, was voted into office on December 29, 1989. A few years later, on January 1, 1993, Czechoslovakia peacefully dissolved into two different countries, forming what we now know as the Czech Republic and Slovakia.
### **PLAN AHEAD**
#### **RESERVATIONS**
Because Prague lacks major sightseeing attractions like Paris and Rome, visiting its sights doesn't require all that much forethought. The Czechs do, however, love to make reservations for everything from dinner to tea. So if there's a restaurant you know you want to enjoy, call and reserve a few days ahead of time.
#### **LOCAL HAPPENINGS**
##### **One World Film Festival**
Movie lovers should plan their visit around mid-March for the annual One World Film Festival (Lucerna Cinema, Štepánská 61, oneworld.cz). Each year the festival offers a new set of thought-provoking documentaries that emphasize human rights and issues around the world.
##### **Velvet Revolution Anniversary**
A public holiday on November 17 celebrates the end of communism in the Czech Republic by means of peaceful demonstration in Wenceslas Square back in 1989. Revelers take to the streets in a mix of pride and a bit of nostalgia. Time has healed the wounds of police brutality and oppression, and massive stages set up across town turn the entire city into one big party.
**KNOW BEFORE YOU GO**
**KEY STATS & FIGURES**
**Currency:**
**The Czech Republic uses the Czech koruna (K č); 26Kč = about 1 USD**
**Population:**
**1,250,000**
**Language:**
**Czech**
**Number of metro lines:**
**3**
**Amount of beer consumed annually per capita:**
**132 liters**
**National drink:**
**_slivovice_ (a moonshine liquor made from plums)**
**Souvenir of choice:**
**puppets**
**Favorite TV show among Czechs:**
**_How I Met Your Mother_ or _Friends_ (heavily debated)**
**CALIBRATE YOUR BUDGET**
**TYPICAL PRICES FOR:**
**Hostel dorm bed:**
**180K č**
**Two-course dinner:**
**250K č**
**Pint of beer:**
**39K č**
**Metro pass:**
**24K č**
**MOVIES TO WATCH**
_EuroTrip, Mission: Impossible, Kafka, A Knight's Tale_
THREE DAY ITINERARY
It's easy to experience Prague entirely on foot. Wear comfortable walking shoes for this tour of the city.
**DAY 1: OLD TOWN**
**MORNING**
Kick off your visit right with a hearty, classy breakfast at Prague's best café, **Café Louvre** in the Old Town. Climb the stairs and step into a world harkening back to the golden age of Viennese cafés.
Meander on toward **Old Town Square** and get a feeling for the city and its beautiful baroque architecture. While you're in the area, hang around and watch the **Astronomical Clock** chime and click away at the top of every hour. Take notice of the **Jan Hus statue** in the center, along with the **Týn Church** and **St Nicholas Church**.
Join a free three-hour walking tour with **Sandeman's New Europe Walking Tours** , meeting daily at 10:45 at the Astronomical Clock. These guys will break down the history of the tower and the Old Town and take you through the **Jewish Quarter** , **Wenceslas Square,** and the **Charles Bridge** for some great Kodak moments.
**AFTERNOON**
Grab a quick and healthy lunch just inside the Jewish Quarter at the posh grocery store **Tržnice Dlouhá 14**. Return back to the river to take in the late afternoon sun. Find a vantage point downriver from the **Charles Bridge** to take in the view or do it from the river on a **paddle boat ride** from Slovanský Island on the Vltava.
Return back into town toward Café Louvre (where you started the day) to my favorite brewhouse, **U Medvídk ů,** for your first foray into the famous Czech goulash culinary experience and some of the most delicious beer you've ever tasted. Enjoy a few rounds here, and then go on a short jaunt north for a pre-party drink at **Anonymous Bar** , an excellent speakeasy-style cocktail lounge.
**EVENING**
If you're trying to go out, an organized pub crawl is a great way to get oriented in town. **Drunken Monkey Pub Crawl** is my favorite. Just look for the team in blue underneath the Astronomical Clock in the Old Town Square nightly 21:00-23:00. Otherwise, get cultured at one of the numerous live concerts that go on often in beautiful old-world settings, like theaters, baroque churches, and opera houses. Find more information in the music office located in the passage leading to Týn Church just off the Old Town Square.
**LATE**
Either way you go—cultured or otherwise—late-night munchies are unavoidable. The _klobasa_ sausages sold at the street stands are tasty, but the burritos rolling out of **Burrito Loco** really hit the spot at 02:00, 03:00, or 05:00.
**DAY 2: LESSER QUARTER & CASTLE DISTRICT**
**MORNING**
Shake off that hangover with a hefty breakfast sandwich at **Bohemia Bagel** across Charles Bridge in the **Lesser Quarter**. When I come through town, I'm nearly always bacon-deficient, so that's what I opt for in my breakfast bagel!
From Bohemia Bagel, loop around the corner for the **Church of Our Lady Victorious** , **John Lennon Wall** , the **Lover's Bridge** , and the **peeing statues,** all within a 20-minute walk.
Begin the hike uphill to the **Prague Castle** and **St Vitus Cathedral.** Visiting the first section of the cathedral is free, but to go in farther you'll have to purchase tickets from the castle ticket office opposite the church entrance.
**AFTERNOON**
In summer months, there is a beer and wine garden open at the castle's downhill exit next to the viewpoint. But you should double back beyond the uphill end of the castle to find the **Strahov Monastery** , my favorite place for a filling afternoon meal in a classic beer hall setting.
From the Strahov Monastery, **Pet řín Hill** is just south of the walls. Find it and descend through the steep grade of the park into the city, stopping at the **Memorial to the Victims of Communism** (the abstract statues walking on the staircase). Just across the street at the bottom of the steps, you'll find my favorite gelato in town, **Angelato.** About time for a taste, don't you think?
**EVENING**
Head out to Prague's most famous party street, Dlouhá, where you can start the night at the **Prague Beer Museum** , and tuck into **Lokál** for hearty Czech cuisine when the mood strikes. Also on this street, and nearby: **Klubovna 2. Patro** , **M1** , **James Dean** , **Harley's** , **Café Nod,** and **Roxy**.
**DAY 3: DELVE DEEPER**
**MORNING**
Begin your detox at my favorite coffee shop and vegetarian café in town: **Mama Coffee**.
If Jewish history is important to you, visit the Jewish Quarter and get lost in **Pinkas Synagogue** and the **Old Jewish Cemetery**.
Otherwise, art nouveau lovers should check out the **Mucha Museum** , or for a bit of a history lesson, stop by the **Museum of Communism**. These museums are both wonderfully small, near each other, and can be tackled in about 45 minutes each. Both are just north of Wenceslas Square.
**AFTERNOON**
Grab a bite of lunch at Kozička in the town center for some good food and warm atmosphere.
Pick up a _trdelník_ and stroll south along the Vltava, soaking in the beauty of Prague on your last afternoon, then climb the steps to the **Vyšehrad Castle**. In summertime, there are outdoor bars and grills vending brats and beer.
**EVENING**
All Czeched out? Grab some margaritas and a bite to eat at **Las Adelitas** , my favorite Mexican spot in Prague, just steps from the Old Town Square.
**LATE**
Take your pick at the many clubs and bars Prague has to offer. If you haven't tried absinthe yet, tonight's the night—the **Absintherie** will give you a rundown on the proper way to prepare and down it!
### **TOP NEIGHBORHOODS**
The city of Prague straddles a bend of the Vltava River. The river makes for a great navigation aid, as long as you remember that it turns around the Old Town at nearly 90 degrees near Prague Castle and Letná Park. Most of the tourist sights are situated on the east side of the bend. The **Old Town** (Staré Město) is where you'll find the famous Old Town Square and Astronomical Clock, as well as the **Jewish Quarter** (Josefov), which is arguably the best collection of Jewish sights in Europe and a must-see for all visitors. In addition to its sights, the Jewish Quarter is now the nicest district in Prague, with fine dining, fancy cafés, and high-fashion outlets like Dolce & Gabbana and Prada. Also on the east side of the river, south of the Old Town, is the **New Town** (Nové Město), which includes Wenceslas Square.
From Old Town, cross the iconic Charles Bridge to discover the quieter and more-upscale **Lesser Quarter** (Malá Strana), tucked between the Castle District, Petřín Hill, and the Vltava River. In the Lesser Quarter, check out a series of unique, free, and superbly Czech sights, like the John Lennon Wall. In close proximity, you've got the **Castle District** (Hradčany); **Pet řín Hill** (capped by the Petřín Tower); and **Letná Park** (with the giant, ticking red metronome).
Prague's trendy up-and-coming district, **Prague 7** (Holešovice), is home to hot clubs and backpacker hostels and is just north of Letná Park.
### **TOP SIGHTS**
#### **Old Town Square (Starom ĕstské Námĕstí)**
Dating back to the late 12th century, the Old Town Square was the central marketplace and hub of commerce of medieval Prague. Over the following few centuries, many buildings went up around the market in Romanesque, baroque, and gothic styles.
While you're waiting for the Astronomical Clock to work its mechanical wonderment, check out all the sights in Old Town Square. Step inside the beautifully baroque **St Nicholas Church** or the gothic **Týn Church** (Church of Our Lady Before Týn), which is accessed through a passage in the building extending from the church's facade. Be sure to pop in to the **music event vendor** right next to the front door of the church, where you can book cheap tickets to beautiful (and succinct) string and symphony concerts in stunning venues across town. Get the cheap seats—which aren't all that bad in these small venues—and you're set for a culturally enlightening evening.
The **Jan Hus statue** in the center of the square is a testament to the man who stood up against the papal tyranny of the Catholic church in the 15th century. The 27 crosses in the cobblestones next to the tower are a tribute to the 27 Protestant nobles who were executed in 1620 after a failed rebellion against Ferdinand II of the Holy Roman Empire, a Catholic.
Free, square always open, churches open generally dawn to dusk, Old Town, Metro: Staroměstská
#### **Astronomical Clock**
Take a moment and try to decipher this complex time-telling machine. Find it difficult? Imagine gazing upon this marvel of clock-making engineering back when it was first constructed in the early 1400s. Designed by Mikulus of Kadan in 1410, the two outer circles of the clock show us Bohemian time (represented by the numbers 1-24) and modern time (represented by two sets of Roman numerals I-XII). The blue area of the clock face represents daylight hours, the orange and brown represent dawn and dusk, and the black represents night. The sun attached to the big hand signifies the sun's position in the sky (it'll be over the black at night), and the small hand represents the moon's position. The inner offset circle displays the signs of the zodiac.
The smaller circle beneath the clock was added in the late 19th century, depicting the date, zodiac sign, and different pictures of everyday peasant life. If you look closely, you will also notice that every day of the year is inscribed around the circle with its patron saint, with an indicator showing which day of the year it is.
The four statuettes on either side of the clock represent the four despised traits of the day manifested through a series of unapologetic stereotypes: Vanity looking into the mirror, Greed represented through the caricature of a Jew holding moneybags, a guilty Turk succumbing to pleasures of the flesh, and a skeleton warning these hedonists of the imminent arrival of their final judgment day. (Political correctness is clearly a 21st-century invention.)
At the top of each hour, tourists and pickpockets alike gather to watch the 12 apostles parade through the two window openings. Though it's a show without 3-D glasses and special effects, you can't help but smile at this charming performance, which has been going on hourly for over 600 years. The performance is capped by a caped trumpeter beckoning you to climb the stairs of the **Old Town Hall Tower** for spectacular views of the city.
110Kč adult; 70Kč student, Mon 11:00-22:00, Tues-Sun 09:00-22:00, Old Town Square, Old Town, +420 236 002 629, staromestskaradnicepraha.cz, Metro: Staroměstská
#### **Pinkas Synagogue**
This synagogue in the Jewish Quarter has been converted into a memorial for the 80,000 Jewish victims of the Holocaust from this region. On its walls are inscribed the names of all the victims, delineated by family name, listing, and the known dates of birth and death. In a room upstairs, you can also find chilling crayon drawings by young children sequestered into the Ghetto before their deportation by the occupying Nazis.
One ticket gets you into the grouping of all the Jewish Quarter's six sights. Avoid the queue and pop into any corner shop in the neighborhood that has the pass picture in the window. This will save you half an hour of standing in line at the popular sights.
Tickets that cover all Jewish Quarter sights from 300Kč, Jan-Mar 09:00-16:30, Apr-Oct 09:00-18:00, Nov-Dec 09:00-16:30, closed Sat and Jewish holidays, Široká 23/3, Jewish Quarter, +420 222 749 211, jewishmuseum.cz, Metro: Staroměstská
#### **Old Jewish Cemetery**
In this cemetery, which is the oldest surviving Jewish cemetery in Europe, the dead had to be buried on top of each other due to lack of space. There are approximately 12 burial layers, over 12,000 gravestones, and an estimated 100,000 people buried here. A walk through this cemetery is like stepping into the set of a Tim Burton movie—the cognitive dissonance is striking and eerie. You expect cemeteries to be orderly and pristine, but the tombstones here, which date back two-to-five hundred years, appear to be sag under the weight of the difficult circumstances that those interred here faced during their lifetimes.
Tickets that cover all Jewish Quarter sights from 300Kč, Jan-Mar 09:00-16:30, Apr-Oct 09:00-18:00, Nov-Dec 09:00-16:30, closed Sat and Jewish holidays, Široká, Jewish Quarter, jewishmuseum.cz, Metro: Staroměstská
#### **Charles Bridge (Karl ův Most)**
Named after King Charles IV, who reigned during Prague's Golden Age, this bridge connects Old Town Prague with its Castle Quarter and Lesser Quarter. The keystone was laid at 5:31 on July 9, 1357, an interesting numerical palindrome (1357.9.7.531) that's a deliberate nod to the astrological powers. Many believe this to be the reason the bridge has withstood both natural and human disasters since its construction nearly 700 years ago. During the day, this wide pedestrian bridge is packed with tourists and trinket vendors and caricature artists. At night, it's practically deserted. Take a detour after sunset across the bridge on the way back to your hostel and gaze out over the Vltava toward the illuminated castle shrouded in Bohemian fog.
Notice that the twin towers on the Lesser Quarter side of the bridge are not mirror images of each other—one predates the other by a couple hundred years. The gashes in the gateway between these two towers happened when Godzilla passed through and sharpened his talons. Actually, they're just the sharpening marks of mercenaries as they prepared their weapons for battle over the years.
Free, always open, Metro: Staroměstská
#### **Prague Castle (Pražský Hrad) & St Vitus Cathedral**
Prague Castle is the dominating feature of the cityscape, holding the Guinness world record as the "largest ancient castle in the world"—but don't expect the crenellations and turrets of classic medieval castles. This castle appears in its Renaissance rendition today. This has always been the seat of Czech power as well as the official residence of those in power. Constructed in the 9th century by Prince Bořivoj, the castle has since been transformed from a wooden fortress surrounded by earthen bulwarks to the imposing stone fortress it is today. Each ruler extended the castle to some degree, so there's a great mix of styles on display. Modern palaces and buildings obscure the profile of your cliché castle, but it is a castle nonetheless.
Don't miss the breathtaking **St Vitus Cathedral** (Mon-Sat Apr-Oct 09:00-17:00, Nov-Mar 09:00-16:00, Sun from 12:00, last entrance at 15:40), located within the castle walls. Constructed with both church and state funds, the cathedral now charges entrance for all visitors, though you can enter and take a peek with the crowds from the narthex. Go all the way up to the corner of the roped-off area and peer left to catch a glimpse of the window by Alphonse Mucha, making out his of rich use of colors, floral patterns, and metaphorical use of subjects. This window is not stained glass, but rather painted glass, allowing for more freedom in shading and blending of colors.
Find the underwhelming **Golden Lane** by turning left on Zlatá Ulička u Daliborky. It's not worth paying to get into, but the cutesy street with dwarf-sized houses is free for visitors from an hour before closing (Apr-Oct daily after 16:00, Nov-Mar daily after 15:00).
Adult tickets from 250Kč, student tickets from 125Kč to go inside the castle, the churches, and Golden Lane; castle open daily 05:00-24:00, historical buildings open daily 09:00-17:00 Apr-Oct, until 16:00 Nov-Mar, last entrance at 15:40; Pražský Hrad, Castle District, +420 224 37 3368, hrad.cz/en/prague-castle, Tram: Pražský Hrad, Metro: Malostranská
**DAVID ČERNÝ'S CHARMING LOVE OF THE ABSURD**
The Czechs have developed a unique and intriguing sense of self-deprecating humor that is evidenced in various forms throughout the city. There's no other more active and current artist than Prague's own David Černý, who is responsible for just about all of the slightly subversive statues and installations around the city. Černý takes a light-hearted approach to his work but clearly doesn't shy away from political and controversial issues.
The **peeing statues** in the Lesser Quarter are actually relieving themselves into a reservoir with the outline of the Czech Republic. Inside the Lucerna building just off Wenceslas Square, there's a statue of an **upside down horse with St Wenceslas** riding bare...belly...?
Just off of Betlémské Square, you'll spot a **man hanging by one arm** high above the street looking down, reminding us that there's always someone watching. The massive **TV tower** in the Žižkov neighborhood can be seen from all over town, with especially great views from Letná Park and the castle. You'll notice funny bumps sprinkled all over the tower. If you look closer, you'll see those are actually giant **baby statues** crawling up and down its vertical walls.
Just a minute or two from the Old Town Square toward Dlouhá street, you'll see a proud pixelated and massive **pregnant lady** on her knees, made from reflective metal material. The meaning? No one really knows. Finally, tucked behind the shopping center at Národní Trřída, and in front of the recommended vegetarian Indian restaurant, you'll see a large **rotating metal sculpture** in the shape of the head of Franz Kafka. Černý's latest work displays this head through topographical lines like those of a map taking a 3-D shape. It goes through patterns of churning faster and slower to make the face appear and disappear.
David Černý also owns a club about 20 minutes south of town called **Meet Factory** (meetfactory.cz), where DJs and artists come out to spin the night away.
#### **John Lennon Wall**
After John Lennon was killed in the 1980s, young Czechs created an unofficial memorial out of a large wall located in Lesser Quarter, covering it in graffiti messages that inspired hope, freedom, and peace. The communist government considered phrases like "Imagine" and "All You Need Is Love" a threat to society and had them painted over immediately. Naturally, rebellious teens saw this as an opportunity to stick it to the man. Night after night, they arrived to repaint the wall with Beatles' lyrics and John Lennon portraits. When the officials finally gave up, they legalized graffiti in this one place in town only and stopped painting over the tags. Today, the wall represents a vision of hope. You'll see everything from sloppy scrawls and notes tacked to the wall with chewing gum to more artistic portraits and messages.
Free, always open, Velkopřevorské Náměstí, Lesser Quarter, Tram: Malostranské Náměstí, Metro: Malostranská
#### **Peeing Statues**
Take a walk to see these two urinating gentlemen while wandering through Lesser Quarter. Created by the famous homegrown artist David Černý, these mechanical statues up until recently would use their stream to spell out any text you sent to a number posted nearby. That practice stopped back in 2012, and now they spell out lines from popular Czech literary works and poems. If you look closely, the pool into which they're contributing is actually the outline of the country, and the two figures represent Nazi and Communist invaders, one angling from the West and the other from the East. Černý is famous for pointed projects like this one and has numerous slightly subversive installations throughout town.
Free, always open, Cihelná 2b, Lesser Quarter, Tram: Malostranské Náměstí, Metro: Malostranská
#### **Church of Our Lady Victorious**
This baroque church is famous for its life-size wax figurine of infant Jesus. The icon just over a foot and a half tall is revered around the world and numerous outfits have been donated showcasing international styles, making this one of the best-dressed dolls I've ever seen. It's free to go in, and also to climb the stairs in the back right of the church to see a sampling of the rest of the outfits on display. A crew of nuns has the important responsibility of changing the baby into a new outfit from time to time!
Services often, open to the public in daylight hours, Karmelitská 9, Lesser Quarter, +420 257 533 646, pragjesu.info, Tram: Hellichova, Metro: Malostranská
#### **Lover's Bridge**
Adjacent to the Lennon Wall, a short bridge connects the street to an underwhelming district known as Prague's "Little Venice." The grate on this bridge has become the place for lovers to attach a lock representing their love to any available space. They then toss the key into the canal to show that their love will never be undone.
Free, always open, Lesser Quarter, Tram: Malostranské Náměstí, Metro: Malostranská
#### **Pet řín Hill & Tower**
Petřín Hill rises on the west side of the river, just south of the Castle District and Lesser Quarter. The observation tower atop the forested hill—clearly modeled after the Eiffel Tower in Paris—was built in 1891 as part of the Jubilee Exhibition, an event showcasing national pride and achievements similar to the World Fair. An additional 299 steps above Petřín Hill, the tower's vantage point allows you to see the entire city of Prague and beyond into Bohemia.
At the bottom of the hill, you'll pass by a morbid series of tall abstract human figures climbing a staircase. This is Prague's **Memorial to the Victims of Communism,** commemorating those who were put on fake trials and killed during the years under control of Moscow.
Park free and always open, tower 105Kč adult, 65Kč student, tower open Apr-Sept daily 10:00-22:00, Oct and Mar daily 10:00-20:00, Nov-Feb daily 10:00-22:00, Petřín Hill, petrinska-rozhledna.cz, Tram: Újezd
#### **Wenceslas Square**
This long, modern square was the site of the peaceful overthrow of the communist government in 1989, an event that turned the tide and brought about the fall of the Soviet Empire and an end to the Cold War. The square is capped by the National Museum (free), which is rather dull and not worth your time on a short visit, though you'll recognize the beautiful interior as the backdrop for the scene where Ethan's target was marked with a special spritz visible only with those spy glasses in _Mission: Impossible._ Wenceslas Square is traditionally where important events kick off and demonstrations end, and it is now the heart of modern Prague, complete with casinos, clubs, and seedier types later at night.
Free, always open, New Town, Metro: Můstek
#### **Museum of Communism**
This throwback museum walks you through the lead-up to the Cold War and depicts daily life behind the Iron Curtain. The Museum of Communism contains a collection of artifacts from daily life under communist rule, from propaganda posters and leader busts to the depressing selection of food available at the market at the time; milk, canned beans, and lard. When stores began to sell more than one kind of butter and the hot new Western clothing styles came out, people were keen to update their look, and an enterprising expat American gathered up these goods and created this museum, lending it a slightly tacky yet authentic insight into what life was like leading up to the 1990s in Czechoslovakia.
Be sure to stay for the entirety of the emotional film playing on a loop toward the end of the exhibit; it helps to underline your experience at the museum.
190Kč adult, 150Kč student, daily 09:00-21:00, Na Přikopĕ 10 (just north of Wenceslas Square, next to McDonald's), New Town, +420 224 212 966, muzeumkomunismu.cz, Metro: Můstek
#### **Mucha Museum**
For those fascinated by art nouveau, a visit to Prague isn't complete without stepping into the Mucha Museum, a one-floor extravaganza of the master's best original works. Alphonse Mucha (1860-1939) pioneered this genre of art and design, which came around as a response to the rigid forms of the industrial revolution. Art nouveau is characterized by organic shapes and the blending of a wide range of materials, including ceramics, metal, glass, and wood, to convey the harmony of nature. This museum, still run by Mucha's grandson, displays paintings, photographs, drawings, and memorabilia. Beyond the museum, Prague itself is home to some of the most beautiful art nouveau architecture and art in the world. Keep your eyes open when passing by the Municipal House (Náměstí Republiky 5), Hotel Europa on Wenceslas Square, and even the dome of the main train station, Hlavní Nádraží.
Adult ticket 240Kč, student ticket 160Kč, daily 10:00-18:00, Panska 7, New Town, +420 224 216 415, mucha.cz, Metro: Můstek
#### **National Theatre (Národní Divadlo)**
From the outside, it's easy to identify the National Theatre, with its remarkable Louis Vuitton-like pattern across the tiled roof. But the real treat is to go inside and catch an opera, and they frequently offer discount tickets to students with ID cards for only 50Kč—an incredible deal. Just show up 30 minutes before the show and see if they have any available.
Refer to website for ticket prices/showtimes, Ostrovní 1, New Town, +420 224 901 448, narodni-divadlo.cz, Tram: Národní Divadlo, Metro: Národní Třída
#### **The Slav Epic at the National Gallery (Veletržní Palác)**
Alphonse Mucha was a proud Czech nationalist, and he spent over 15 years working on this dreamy and strikingly beautiful series of 20 massive-scale paintings depicting the legends and history of the Slavic people. At the height of his artistic prowess, Mucha mastered depth of field, contrast, and use of color and composition for what really is an epic collection of paintings, some measuring 20 feet tall. For Mucha fans, it's a no-brainer—you've just gotta see it. For others, it's still worth the short tram ride on #6 or #12 to get to the museum in Holešovice.
Adult ticket 180Kč, Tues-Sun 10:00-18:00, Dukelských Hrdinů 530/47, Prague 7 (Holešovice), +420 224 301 111, ngprague.cz, Tram: Veletržní Palác, Metro: Vltavská
#### **EXTRA CREDIT**
##### **Churches**
As you wander, don't miss the opportunity to soak in some baroque beauty in a few of the many churches in Prague. Most feature decadent architecture and some real gems in the form of paintings. You'll see an over-the-top exuberance not seen in the more austere Renaissance style. Architectural restraint is thrown out the window and replaced by glittering gold statues, radiant stained glass, and over-decorated pulpits.
##### **Vyšehrad Castle**
A seldom visited and beautiful sight, Dracula-esque Vyšehrad Castle is a fun afternoon jaunt about half an hour's walk along the river south of town. This is the area where experts think the first settlements in the region occurred, expanding into the city of Prague as we know it today. I love exploring the well-preserved modern star fort's two-kilometer-long ramparts, looking out from the numerous vantage points to the suburbs of Prague, and reflecting on a walk through the cemetery just behind the church, where Alphonse Mucha is buried.
Free, Apr-Oct daily 10:00-18:00, Nov-Mar daily 10:00-16:00, V Pevnosti 159, Greater Prague, praha-vysehrad.cz, Tram: Albertov, Metro: Vyšehrad
### **TOP EATS**
With dishes generally consisting of some kind of meat, potatoes, dumplings, and gravy, classic Czech cuisine is not for the faint at heart (seriously—these dishes are heart-stopping). But while you're here, you've got to duck into one of the many Czech pubs and at least try a bowl of their famous goulash.
Up until recently, there weren't many options beyond the heavy local food, but you'll find that a culinary renaissance has taken Prague by storm. Selection and variety have grown substantially, and ingredients much improved. It used to be hard to find vegetarian or eco-friendly options, but now they're all over the place.
Tipping is not expected but is appreciated. Be aware of some subtleties: Locals take a look at the bill, then hand their cash to the waiter, who makes change right there at the table out of a large wallet. If your service and food were good, round up to the nearest 10 percent, and ask for the change that you want back. For example, if your meal was 450Kč (about US$20), hand him your 1000Kč, and ask for 500Kč back. Saying "thank you" while handing over the cash effectively means, "keep the change." So wait until you have your change in hand before thanking the waiter for his service.
#### **Lokál**
To experience authentic Czech food at an amazing price, look no further than Lokál. Upon entering, you'll feel as if you've stepped back in time into an old communist beer cellar—the scribbles on the walls and naughty bathrooms are a nod to what you'd see at real beer halls in the countryside. The menu is thankfully short and sweet, and the servers are happy to get you an English menu on request. Service is brisk. The _sví čková_ (beef sirloin in cream sauce) and goulash never disappoint.
**SPIRALLY, SUGARY GOODNESS: THE TRDELNÍK**
Typical to the region is a delicious pastry that's baked on top of glowing coals: _trdelník_ (pronounced ter-DEL-neek). The bakers take a roll of dough and wrap it around a wooden cylinder to cook and toast into a golden-brown chewy dessert. When they're ready, the doughy spirals are tapped off the log and rolled in sugar, spice, and everything nice. Prices are climbing to 60Kč each, but it's well worth it to keep the blood sugar levels up on those long days of sightseeing. Enterprising vendors have been popping up everywhere, but that doesn't mean the quality is always the same. Some places will try to hawk cold, stale _trdelníks_ at you. Have patience and wait until you pass a stall that's passing out steaming hot fresh rolls. The stall on Malostranské Náměstí has been consistently tasty for the last several years, and I tend to skip the ones in the Old Town Square area.
About 120Kč, Mon-Fri 11:00-01:00, Sat 12:00- 01:00, Sun 12:00-22:00, Dlouhá 33, Old Town, +420 222 316 265, ambi.cz, Metro: Staroměstská
#### **Kozi čka**
Come enjoy a meal inside this cozy, brick cellar of a place for some incredible Czech food at lovely prices. Because of its nondescript entrance, most tourists miss it, so it's a favorite among locals, who can look past the service with attitude and come back for the cheap drinks, great atmosphere, and delicious $10 steaks.
Dishes from 160Kč, Mon-Fri 12:00-04:00, Sat 18:00-04:00, Sun 19:00-03:00, Kozí 1, Old Town, +420 224 818 308, kozicka.cz, Metro: Staroměstská
#### **U Medvídk ů**
Have a meal here for a selection of great local food and enjoy the warm atmosphere. U Medvídků could practically have its own section in a Prague history book—they've been serving house-brewed pints and delicious Czech goulash since 1466! Check out their open-top brew barrels by going all the way to the back and up the stairs. You can sit up here and eat if you want to try their extra special house pilsner.
Dishes from 140Kč, Mon-Fri 11:00-23:00, Sat 11:30-23:00, Sun 11:30-22:00, Na Perštýně 7, Old Town +420 224 211 916, umedvidku.cz, Metro: Můstek
#### **Malý Buddha**
For vegetarian food, Malý Buddha ("Little Buddha") is among the best in the city. This Thai fusion restaurant offers stir-fried and roasted veggie and tofu dishes in a quiet, warm ambience. This healthy option may be a welcome break for those tired of heavier Czech cuisine.
Dishes around 120-240Kč, Tues-Sun 12:00-22:30, Úvoz 46, Castle District, +420 220 513 894, malybuddha.cz, Tram: Pohořelec
#### **Strahov Monastery**
Find some of the best beer in town in a wonderful, bright beer hall setting in the old Strahov Monastery, which dates back to the 12th century. The old monastery is just at the top side of the Prague Castle, making it a perfect pit stop for a pint and a meal. The goulash in a bread bowl is a personal favorite, but their cheese plate and duck dishes don't disappoint either.
Dishes from 100Kč, daily 10:00-22:00, Strahovské Nádvoří 1/132, Castle District, +420 233 107 704, strahovskyklaster.cz, Tram: Pohořelec
#### **Café Louvre**
This is Prague's classiest and most classic Bohemian café, set one floor above street level. Having served classy brunches, lunches, and dinners for over 100 years, Café Louvre follows the lead of the coffee culture in nearby Vienna. It's a must for anyone who agrees with my mother that breakfast is the most important meal of the day. Dine under tall ceilings on delicious omelets served by waiters in smart dress. The carrot cakes are quite fortifying as well.
Omelets from 140Kč, Mon-Fri 08:00-23:30, Sat-Sun 09:00-23:30, Národní 22 (upstairs), Old Town, +420 224 930 949, cafelouvre.cz, Metro: Národní Třída
#### **Mama Coffee**
Mama Coffee is a favorite among local coffee snobs (like me) and vegetarians. And if you need to tackle a day of work while on vacation, this is your place for ear plugs, Wi-Fi, and strong coffee. I can while away the hours with a good book upstairs by the window enjoying their delicious fresh ginger tea. Toss in their tasty pastries and a hummus lunch, and I'm fat—er, lean—and happy all day without racking up much of a tab at all.
Coffee and tea from 60Kč, daily 08:00-22:00, Vodičkova 674/6, Old Town, +420 773 337 309, mamacoffee.cz, Tram: Vodičkova, Metro: Národní Třída
#### **Buddha Bar**
This is the kind of place you'd splurge on for a birthday dinner (it's where I've spent two of mine!). Buddha Bar is an Asian fusion chain restaurant with locations around the world. It features a creative menu with fresh salads, innovative sushi rolls, numerous other main options, an extensive wine menu, and delicious expertly mixed cocktails. The service is impeccable, and the ambience is breathtaking. Start upstairs with a cocktail at the bar, and descend once your table's ready to your religious culinary experience at the foot of a giant Buddha. Stay late for an after party among the young, well-heeled professionals of Prague.
Dishes from 350Kč, Tues-Sat 18:00-03:00, Jakubská 8, Old Town, +420 221 776 400, buddha-bar.cz, Metro: Náměstí Republiky
#### **Zebra Express**
If Buddha Bar can't fit into your budget, Zebra can—and it's nearby, just inside the Powder Gate and around the corner from the Municipal House. Zebra offers excellent pad thai along with curries and sushi in a casual setting, with fast and friendly service. You can even banter with the chefs, who really appreciate your compliments and clearly enjoy what they're doing. The second location in town, Zebra Asian (Melantrichova 5), is right on the corner of the Havelská Market, a block south from the Old Town Square.
Dishes from 150Kč, daily 11:00-24:00, Celetná 988/38, Old Town, +420 774 727 611, zebranoodlebar.cz, Metro: Náměstí Republiky
#### **Burrito Loco**
Stop into this fast food, almost-as-good-as-Chipotle burrito shop for a cheap Tex-Mex fix 24 hours a day. Numerous locations, including at Národní Třída and near numerous nightlife venues on Dlouhá, make it hard to say no when the munchies strike late. Post up in the limited seating, or take your rolls to go and find a bench nearby. Find a second Old Town location at Spálená 104/43.
Burritos from 120Kč, daily 24 hours, Masná 620/2, Old Town, +420 606 039 333, burritoloco.cz, Metro: Staroměstská
#### **Bohemia Bagel**
If you're anything like me, you're really missing a bagel sandwich right about now. Head to Bohemia Bagel—a favorite among the study abroad students in Prague for the taste of home it offers—for freshly made bagel sandwiches throughout the day. It's owned by the same enterprising entrepreneur who started Burrito Loco and the Museum of Communism. He picked up a warehouse full of old Bohemian tiles in the early '90s, which decorate the walls of and are sold at the Bagel. Order at the cashier and take a seat with your number—they've capitalized on our love of bagels, and they're not letting go anytime soon!
Bagels from 100Kč, daily 07:30-18:00, Lázeňská 19, Lesser Quarter, +420 257 218 192, bohemiabagel.cz, Tram: Malostranské Náměstí, Metro: Malostranská
#### **Lehká Hlava**
True to Prague's culinary revival, this celestial little abode has visitors raving about their delicious vegetarian, vegan, and gluten-free dishes. Creative options like cucumber spaghetti and beetroot burger make this a popular spot for those seeking an alternative to the ubiquitous heavy fare throughout the rest of town. If you love cheap, fast, casual, and healthy food like this, be sure to look up its sister location, Maitrea, just behind the Týn Church and steps from the Old Town Square.
Mains from 150Kč, daily 11:30-23:30, Boršov 180/2, Old Town, +420 222 220 665, lehkahlava.cz, Tram: Karlovy Lázně, Metro: Národní Třída
#### **Restaurace T. Anker**
What's better than good food? Good food with a panoramic view over downtown Prague. Find T. Anker at the top of the funky octagonal-shaped department store directly across from the city-center shopping mall, Palladium. Grilled options like duck, chicken, burgers, and fish go well with their homemade brews on tap.
Mains from 180Kč, daily 11:00-20:00, till 18:00 on Sun, Náměstí Republiky 656/8, 5th floor, Old Town, +420 722 445 474, t-anker.cz, Metro: Náměstí Republiky
#### **Beas Vegetarian Dhaba**
Your dear author couldn't be any further from a vegetarian, but there is something amazing about a light lunch or evening snack of Indian curries and salads. This place is by far the quickest, easiest, and tastiest vegetarian option I've found in Prague. Their Národní Třída location makes for an easy pit stop during the day, but you may see other locations around town. Pop in, grab a tray (plastic, if you want it to go) and start filling it with all sorts of rice dishes, grilled veggies, and more. Their lassi isn't the best I've ever had, but it does the trick if you've got a hankering.
Pay by weight, Mon-Fri 11:00-21:00, Sat-Sun 12:00-18:00, Vladislavova 158/24, Old Town, +420 777 551 256, Metro: Národní Třída
#### **Las Adelitas**
Difficult to find, and tucked inside a passageway and downstairs, this is one of Prague's hidden gem Mexican restaurants. Head downstairs and you'll forget you're in central Europe, half expecting to hear the waves of Puerto Vallarta as you sip your perfectly concocted margarita. Enjoy the toasty salted chips and wide selection of appetizers and entrées—I loved the steak burrito—and take it at the bar.
100-200Kč, daily 11:00-01:00, Malé Náměstí 457/13, Old Town, +420 222 233 247, lasadelitas.cz, Metro: Staroměstská
#### **Tržnice Dlouhá 14**
This Whole Foods-with-a-twist-style grocery and deli is an excellent stop for both breakfast and lunch, and very nearby the Old Town Square. Their thick sandwiches with generous fillings and deep-dish pizza are made from the high-quality ingredients found in-store. Come back for dinner and throw together a charcuterie plate and a bottle of wine from their cellar downstairs for a classy, easily made picnic dinner.
Sandwiches from 80Kč, Mon-Fri 08:00-21:00, Dlouhá 14, Old Town, +420 224 815 719, dlouha14.cz, Tram: Dlouhá Třída, Metro: Staroměstská
#### **Angelato**
Angelato whips up Prague's best authentic Italian gelato, with flavors that are a taste-powered travel machine with a one-way ticket to Italy. The location makes for an excellent reward after a long day of sightseeing in the Lesser Quarter. It's not far from the rental location for the paddle boats, either!
Cups of love from 80Kč, daily 11:00-20:00, Újezd 425/24, Lesser Quarter, +420 777 787 344, angelato.cz, Tram: Újezd
### **TOP NIGHTLIFE**
Prague is notorious for its nightlife. You can find everything from smoky dive bars and grungy clubs to classy lounges and cocktail bars. Pay attention to flyers and posters around the city, and keep your ears open at the hostel to get a feel of what's happening each night—there are often events throughout the week at various venues. The pubs are cheap, the clubs are nuts, and dance bars (my favorite) are numerous. When there's a dance floor, you can expect to pay 100-200K, or less than US$10, to get into the party.
#### **NIGHTLIFE DISTRICTS**
The Old Town of Prague is where the hopping nightlife is for both tourists and locals. Once you get out of the dense streets of the Old Town into the New Town, Wenceslas Square area, and Prague 7, clubs are bigger and more intense, and generally more Czech, but fewer and farther between. It's nice to jump on an organized pub crawl your first night in town to get a sampling of what Prague has to offer.
##### **Dlouhá**
You'll find nightlife throughout Prague's Old Town, especially on Dlouhá, the city's well-known party street. Several of my favorite venues are located on this street.
Old Town, Tram: Dlouhá Třída
##### **Náplavka**
Two bridges south of Charles Bridge is the bridge known as Jiráskův most. Beginning just south of Jiráskův most and extending about five blocks is a strip of street bars and pubs that open up in the summer. This riverside street, known as Náplavka, is on the east side of the river. It's hugely popular with locals on weekends and sunny afternoons, and hardly known by tourists. It's a great way to spend an afternoon after your paddle boat ride on the river, meeting locals and making friends over cheap pints while overlooking the river. Beers start around 30Kč. Most beer gardens stay out as long as it's warm enough to enjoy a pint outside.
Greater Prague, Tram: Paleckého Náměstí, Metro: Karlovo Náměstí
#### **BARS & PUBS**
##### **U Sudu**
U Sudu is a Prague institution. Upon entry, it seems to be just another ordinary Czech pub, but head to the back and down the stairs and you'll find a labyrinth of exposed medieval brick cellars and lounge rooms with foosball tables and numerous bars where you can drink, chill, and be merry. Try not to get lost on the way back to your seat!
Daily 09:00-05:00, Vodičkova 677/10, Old Town, +420 222 232 207, usudu.cz, Tram: Lazarská, Metro: Národní Třída
##### **Anonymous Bar**
Anonymous is a welcome non-smoking alternative to the smoky beer pubs throughout the city. Head to the back of a dimly lit, quiet courtyard and step into a classy low-key speakeasy-themed bar, with the Anonymous mask face pasted everywhere across the interior. Enjoy one of the many experimental cocktails described by the helpful staff, or go for one of the classics—the bartenders excel at both.
**LGBT PRAGUE**
When it comes to entertainment for the LGBT community, Prague has a lot to offer, including dance clubs with go-go dancers, such as **Factory** (Vinohradská 63, factoryclub.cz) or **Escape Club** (V Jámě 1371/8, escapeclub.xxx). **Strˇelec Pub** (Anglická 2, facebook.com/clubstrelec) is for guys who prefer good beer. **Club Termix** (Třebízského 1514/4a, club-termix.cz) is a smaller, more intimate dance club that packs out throughout the week, and its big brother, **Termax Club,** is the biggest gay club in Prague, so sometimes the big rooms look rather empty. Both these clubs have the same owner, and prices are similar and reasonable across the two. These clubs attract bigger DJs, so in case of special events, there is often a cover. **Friends Club** (Bartolomějská 11, friendsclub.cz) is open seven days a week and is very popular among the tourist crowd, with programs like karaoke, partying, meeting, talking, chilling, and even cabaret across different nights. And keep an eye on **Mecca** 's program (mecca.cz); they organize their famous OMG Party every two months with a house and techno set.
**ACT LIKE A LOCAL**
**Pivo, Prosím?**
Thanks to its clean water, fresh hops and barley, and a longtime monastic brewing culture, the Czech Republic has some of the best beer in the world. The beer coming from the Bohemia-Moravian region has been imitated many times, yet nobody does it quite like the Czech. Order one by saying to the waiter _"pivo, prosím?"_ or "beer, please?" As it's by far the most popular beverage in any restaurant, even just holding up a finger will land a pint of frothy goodness on your table in short order.
Today, beer culture is alive and well in Prague. Bars and restaurants generally have two taps—light and dark—of the same beer brand. "Light" doesn't necessarily mean lower-calorie; rather, this is the lighter-colored, crisp pilsner-style beer, the predecessor to the American perversion of Budweiser and Coors. "Dark" tends to be heavier and a touch sweeter, with a slightly higher alcohol content. This beer goes well with the meat-and-potatoes Czech cuisine. Goulash is oftentimes even made with a splash of beer. So toast to the brewers and knock back a few of what many consider the best brew around. Here are some common brands you'll see:
**Pilsner Urquell:** Deliciously light and bitter. Quenches a thirst at the end of a long day of sightseeing and widely recognized as the best beer in the world.
**Budvar:** The precursor to the Budweiser brand we all know in the States. The two brands are in constant legal battles over who owns the rights to the brand Budweiser.
**Staropramen:** The runner up to the better brands of Urquell and Budvar. Also a lighter, crisper, cheaper beer.
**Proprietary brews at Strahov Monastery and U Medvídk ů:** Delicious small-batch brews coming in both lighter and darker manifestations. A swig of these and you'll understand what "good beer" actually means.
Daily 17:00-02:00, till 03:00 on weekends, Michalská 432/12, Old Town, +420 608 280 069, Metro: Můstek
##### **Prague Beer Museum**
Located on Prague's party street, Dlouhá, the beer museum is much more bar than museum, but who's complaining? Pick up a menu in this dark, smoking bar and you'll realize it may as well be an exhaustive library of all the tasty beers from around the world. I challenge you to find a beer that's not on the list! If you can't choose, go for the sampler paddle to taste a number of different beers on tap, and take your tray out to the patio in the back if the weather is good. The servers are sometimes in the mood to make recommendations, but other times, they're just not. Don't take it personally.
Daily 12:00-03:00, Dlouhá 720/46, Old Town, praguebeermuseum.com, +420 732 330 912, Tram: Dlouhá Třída
##### **Café Nod**
Another classy spot on the party street, Dlouhá. This one is the sister bar to the Roxy club. Pop in here, next door to the recommended Lokál restaurant, and climb the stairs to a spacious bar/café/lounge that's perfect for anything from afternoon tea to late-night drinks. The space in the back doubles as an art gallery with frequent exhibits. Nod draws the young professional set who work downtown, creating a chill and unpretentious vibe.
Drinks from 60Kč, Mon-Fri 10:00-01:00, Sat-Sun 14:00-04:00, Dlouhá 33, Old Town, nod.roxy.cz, +420 604 790 921, Tram: Dlouhá Třída
##### **Tretter's New York Bar**
Tretter's is your classic Roaring '20s cocktail bar, complete with professional bartenders; an extensive list of wine, short and long drinks, and cigars; and even a gentleman's guide integrated into your menu. This place is now "discovered" and packs out later in the evening, but the old-fashioned I had was well worth the stop and a great start to the night.
Daily 19:00-03:00, V Kolkovně 3, Jewish Quarter, +420 224 811 165, tretters.cz, Metro: Staroměstská
##### **Absintherie**
Prague is famous for the drink that supposedly made imbibers hallucinate and enjoy extra creative powers thanks to the wormwood soaked in this liquor derived from anise, the licorice-flavored root. The hallucinogenic ingredients have gone by the wayside, as products made or sold within the EU cannot contain wormwood. But it's still fun to pop into this green-hued bar and decide between the cool French preparation or the fiery Czech one. The former entails dripping water over a sugar cube sitting on a fancy spoon with holes in it with the shot of absinth below, both watering it down and making it sweeter. The Czech preparation is made by soaking the sugar cube in absinthe, lighting it on fire, and caramelizing it as it falls into the drink. Better yet, get both and share with your friends!
Prepared drinks from 100Kč, daily 12:00-24:00, Jilská 7, Old Town, +420 224 251 999, absintherie.cz, Metro: Můstek
#### **DANCE BARS**
I love how dance bars blend the line between sit-and-drink pubs and deafening clubs. These bars make for the perfect place to sow those seeds of fascinating conversation and bring your dude or damsel to the floor when your jam comes on.
##### **Klubovna 2. Patro**
This is probably my favorite spot—and best-kept secret—in Prague. The name translates to "2nd floor club," and this spot frequented by the classy hipster set is a cross between a speakeasy, lounge, bar, and club. It's quite popular with young, friendly Czechs who love the throbbing techno put on nightly throughout the week. You've got a coat check as you enter, affordable drinks, a dance floor, a large bar, and then a back room where you can chill out and catch your breath. The proximity to many other popular venues on the same block makes this a great place to continue or cap the night.
Covers 100-200Kč, Mon-Thurs 17:00-02:00, Fri-Sat 17:00-04:00, pass through the discreet entrance in the back of a parking lot at Dlouhá 729/37, Old Town, so hipster they don't even have a phone or website, Tram: Dlouhá Třída
##### **M1**
A small club that's popular with Erasmus students, M1 is a chic but not-too-snobby single-room dance bar that plays a good mix of popular hip-hop and R&B. Check out the website before you go out, as they often have daily specials and promos. And bring a thick skin for the bouncers, who can be a bit moody. It's generally not worth paying the cover, so if they're pushing for it, skip it for the numerous nearby options.
100Kč cover sometimes applies, daily 21:00-late, Masná 1, Old Town, +420 227 195 235, m1lounge.com, Metro: Staroměstská
##### **Chapeau Rouge**
Chapeau Rouge is notorious for being a head trip—the good kind. While the upstairs is (relatively) quieter and more straight-laced, descend the stairs into the low-ceilinged smoky cavern, where you'll rock out to intense dubstep where just about anything goes... As they say, what happens in Chapeau stays in Chapeau. Not a good place for those who don't like being around drugs.
Mon-Fri 12:00-04:00, Sat-Sun 16:00-late, Jakubská 2, Old Town, +420 222 316 328, chapeaurouge.cz, Metro: Náměstí Republiky
##### **Nebe**
With three locations in town, Nebe has the clubbing vibe down to a science. The crowd here dresses well but comes without the pretentious flavor that fills so many clubs in other cities. Their different locations have slightly different characteristics, but each is enjoyable in its own right for great music, drinks that go down easy, prices that won't break the bank, and a sexy crowd ready to dance the night away. Each location features a long bar and a separate dance floor to get down on. In addition to the Wenceslas Square location below, find other locations at Náměstí Republiky (V Celnici 1036/4) and Karlovo Náměstí (Křemencova 10).
Cover on weekends 100Kč, Mon 16:00-03:00, Tues-Thurs 16:00-04:00, Fri-Sat 16:00-05:00, Sun 18:00-03:00, Václavské Nám. 802/56, Wenceslas Square, New Town, +420 608 644 784, nebepraha.cz, Metro: Muzeum
##### **James Dean**
I skipped out on this bar for years until I realized that this is actually where the locals go. Thanks to its '50s-diner theme, you wouldn't be blamed for thinking this is a tourist trap, but pop in and descend the stairs in the back to a raging party that goes until 6 or 7 in the morning. With the relatively lax door policy, the ratio does get a bit unbalanced at times toward the guys who just gawk for hours at the dancers on stage.
Daily 08:00-late, V Kolkovně 922/1, Old Town, +420 606 979 797, jamesdean.cz, Metro: Staroměstská
##### **Harley's**
Widely recognized as Prague's pickup joint, this dive bar and club is the after-party location for all the clubbers in town. While busy around midnight and 01:00, it really notches up around 03:00 till closing around dawn. The rowdy downstairs party features a small stage, to which bachelors on their stag parties are drawn like flies.
Daily 19:00-late, Dlouhá 18, Old Town, +420 602 419 111, harleys.cz, Metro: Staroměstská
##### **Deja Vu**
Get your drink on upstairs and your dance on down below. The music here can be hit or miss, so head downstairs and listen to a groove before throwing down for a drink. Sharable large drinks are popular at the upstairs bar, where you can lounge around some mean-looking garden lion statues.
Daily 18:00-late, Jakubská 648/6, Old Town, +420 222 311 743, dejavuclub.cz, Metro: Náměstí Republiky
#### **CLUBS**
Many of the clubs in town focus on one genre rather than featuring a mix and usually charge a 100Kč or 200Kč cover. Nightclubs come along with a more serious sound system, generally a larger venue and bigger crowds, and higher drink prices than what you'd find at dance bars. So it's a good idea to look up the programs online ahead of time to determine the music you'll listen to as well as the crowd that will be showing up to party. This way, you can avoid genres that just don't get you going, like house techno for me.
##### **Karlovy Lázn ě**
This is the famous five-floor club everyone has heard about. It's the largest in Central Europe and boasts two floors of techno, one pumping '80s beats, one for hip-hop, and a dark, chill level at the top with couches and giant bean bags that I usually try to stay away from. This club is good as long as you've imbibed plenty, but the hordes of high-schoolers and the male creepers they attract make a second visit unlikely for most.
There's a bar next door on the river side of Karlovy Lázně where many people get their cheap drinks fix before heading to the club. The happy hour 10:00-24:00 features 60Kč shots, wine, and beer.
Daily 21:00-06:00, Smetanovo Nábřeží 198/1, Old Town, +420 222 220 502, karlovylazne.cz, Metro: Staroměstská
##### **Radost FX**
Locals and tourists alike highly recommend this spot out past the top of Wenceslas Square, which is both the city's best vegetarian restaurant by day and a bumping hip-hop club by night. Rihanna even filmed a music video here—the one where they went behind a secret refrigerator in the back of a convenience store and resurfaced in Prague's biggest club featuring R&B and rap.
Daily 11:00-02:00, Bělehradská 234/120, New Town, +420 224 254 776, radostfx.cz, Metro: I.P. Pavlova
##### **Roxy**
Taking over a massive and recently renovated subterranean concert hall, Roxy gives off a slightly alternative, grungy warehouse feel with music ranging anywhere from techno and reggae to house and hip-hop. This casual club—yes, sneakers and jeans are fine—is free sometimes but often charges cover on weekends (100Kč), but it's generally a great party. I recommend taking a look at the website for event schedules and genres of music to be played.
Daily 19:00-05:00, Dlouhá 33, +420 602 691 015, roxy.cz, Metro: Staroměstská, Trram: Dlouhá Třída
##### **Lucerna**
This is one of my favorite unpretentious dance clubs in town. Enjoy the music of the '80s and '90s on weekend nights while rocking out to the music videos your mom never let you watch on the enormous projector screens. And when your favorite '90s song comes on, don't feel ashamed knowing all the words—everyone here does (or thinks they do at least). You may find younger, more attractive crowds elsewhere, but Lucerna is always a safe bet for a popping good time. Continue into the gallery from the entry hallway to see an upside-down horse statue by David Černý mocking the one that sits prominently and correctly at the top of the Wenceslas Square.
**A SORE BUTT ON EASTER**
Your Easter traditions might include Easter egg hunts, church, rocking the pastels, and so on. But the Czechs have their own set of traditions, with one in particular that I found hard to believe.
Every Easter Monday, all the boys and men of the Czech Republic take decorated sticks called _pomlázka_ (or thin wooden whips) to the streets, going door to door smacking women's butts in exchange for candy and chocolate eggs for the youngsters and shots of liquor or alcohol for the older boys and men. Additionally, at each stop along the way, the little girls tie colored ribbons on the ends of the boys' sticks. So a busy little guy will come home with a stick decked out in all sorts of pieces of flair. This naughty tradition is supposed to bring good luck to both genders and is also meant to preserve the beauty of the women for the next year. The act itself is supposed to symbolize health, fertility, and the coming springtime.
Doors usually open around 20:00 or 21:00, and the party turns up around 22:30, Štěpánská 61, Wenceslas Square, New Town, +420 224 225 440, lucerna.musicbar.cz, Metro: Muzeum
##### **Cross Club**
This primarily dubstep club features one of the funkiest and most futuristic interiors I've ever experienced, with servos spinning on the walls and a half-level above the bar where you can bear-crawl to your own little cove to overlook the scene below. Taking over a nondescript house in Holešovice, Cross Club is a favorite among those who love drum and bass. The eclectic lineup of performers and DJs spans a wide range of tastes.
Cover usually around 100Kč, daily 14:00 till way late, Plynární 1096/23, Prague 7, (Holešovice), +420 736 535 053, crossclub.cz, Tram: Ortenovo Náměstí, Metro: Nádraží Holešovice
##### **Mecca**
For an all-out central European clubbing experience, take the hike out to Mecca, where you'll descend into a world of laser shows, smoke machines, go-go dancers, and throbbing house and futuristic techno every Friday and Saturday. The music lineup varies occasionally on Saturday, so check their program for the schedule. Dress to impress to avoid any trouble at the door.
Cover usually around 200Kč, more when special events are on, Fri-Sat 22:00-05:00, U Průhonu 799/3, Prague 7, (Holešovice), +420 734 155 300, mecca.cz, Tram: U Průhonu, Metro: Nádraží Holešovice
#### **PUB CRAWLS**
##### **Drunken Monkey Pub Crawl**
This team puts on a rowdy party every night of the week. Check out their website for special upcoming events. Their parties for New Year's, the Super Bowl, Halloween, and Fourth of July are always worth booking ahead. Their standard nights kick off with a double power hour—two hours of unlimited sugary shooters, wine, and beer. They then take you on a tour of 4-5 top nightlife venues across town. Jump onto the backpacker crawl at 21:30 for the full crawl experience for 350Kč, or tag along the more local-student-oriented power hour plus club entry kicking off at 23:30 for 250Kč. Crawls start at their own bar, Drunken Monkey, and are a great option on weekend nights. Catch one of their flyers from their reps in blue in the Old Town Square for more info.
Pub crawl open bar 20:30-23:00, U Milosrdných 848/4, Jewish Quarter, +420 775 477 983, drunkenmonkey.cz, Metro: Staroměstská
### **TOP SHOPPING & MARKETS**
#### **SHOPPING DISTRICTS**
##### **Pa řížská**
Get your high-end shopping fix by strolling down Pařížská—"Paris Street"—Prague's own Champs-Élysées, offering stores like Prada, Louis Vuitton, and Gucci.
Jewish Quarter, Metro: Staroměstská
##### **Wenceslas Square**
If Pařížská is out of your budget, check out the stores lining **Národní** and quarter-mile-long **Wenceslas Square** , which has basically every brand you've ever heard of back in the States.
New Town, Metro: Můstek or Muzeum
#### **MARKETS**
##### **Náplavka Farmers Market**
Every Saturday, this exciting farmers market takes over a stretch of the walkway along the Vltava River. Shoppers come out to see what's on offer, from baked goods and pastries to meat, dairy, and sandwiches, and from food trucks to handmade goods and more. The beautiful setting makes for an excellent place for a lunch and break from a busy day of sightseeing.
Sat 08:00-14:00, Náplavka street, just south of Jiráskův most, Greater Prague, Tram: Paleckeho Náměstí, Metro: Karlovo Náměstí
##### **Havelská Market**
This touristy market has descended into selling tchotchkes and souvenirs. But the most kitschy thing about this market is that every single vendor seems to have dozens of clap-activated cackling witches on brooms hanging from their stalls, so you'll hear this street market long before actually seeing it. It's worth a stop if you've got loved ones to pick up souvenirs for, but don't expect anything that isn't mass-produced.
Apr-Sept daily 07:00-19:00, Oct-Mar daily 07:00-18:30, Havelská, Old Town, Metro: Můstek
#### **SHOPPING CENTERS**
##### **Palladium**
Palladium is Prague's best city center mall. You'll find all the standard brands, like Puma and H&M, along with Starbucks and a food court.
Daily 09:00-21:00, Náměstí Republiky 1, Old Town, Metro: Náměstí Republiky
### **TOP PARKS & RECREATION**
Letná Park offers beautiful views, and the Vltava islands, just minutes from Charles Bridge, offer a chance to escape the crowds and busy streets of the city.
#### **Letná Park (Letenské Sady)**
For a stunning view of the entire city, climb the hill up to Letenské Sady park. Here you will find a giant ticking metronome (before it stood a colossal statue of Stalin leading the people of Czechoslovakia to the bright future of communism), which symbolizes the time ticking away until the end of all tyranny around the world. On nice days, this is where Prague citizens get their sweat on, going on jogs and roller blading around the large, flat park. There's a beer garden about a quarter mile north from the metronome. Music events and concerts are put on often, and in the winter, they create an ice track to skate on.
Free, always open, Letná Park, Tram: Čechův most
#### **St řelecký Island**
Cross on the bridge known as Most Legií to get down to this island, perfect for enjoying a picnic lunch on the benches overlooking the river, with an uninterrupted view toward Charles Bridge. Enjoy a pint of Pilsner at the beer boat that pulls up to the island in summer.
Free, always open, between Lesser Quarter and Old Town, south of Charles Bridge, Tram: Národní Divadlo, Metro: Národní Třída
#### **Slovanský Island**
Winter in Prague can be freezing, but in the summer it's great to get out on the river for some time away from the crowds and to cool down a bit. From Slovanský Island, you can rent **paddle boats** for up to four people for about 200Kč an hour. Bring some snacks, beer, and wine to make an afternoon of it! To get to Slovanský Island, cross the short footbridge just south of Národní divadlo.
Daily, closing an hour before sunset, just south of the National Theater and Legií bridge, Tram: Národní Divadlo, Metro: Národní Třída
#### **Žluté Lázn ě**
Consider Žluté Lázně Prague's beach. With volleyball and tennis courts, numerous beer gardens, cafés and restaurants, pizzerias, and a generally chill and fun-loving atmosphere, this outdoor adult playground is where the city comes out to enjoy nice weather any chance they get. Once the sun goes down, freshly tanned Czechs stick around for dinner and the party to follow at their on-site dance hall. To get here, take the 20-minute tram ride south on tram #17 from any stop along the east side of the river.
Cheap entrance fee of 80Kč, daily 09:00-02:00, Podolské Nábřeží 3/1184, Greater Prague, +420 244 462 193, zlutelazne.cz, Tram: Dvorce
### **TOP TOURS**
#### **Sandeman's New Europe Walking Tour**
These free tours took Europe by storm a few years ago, and you can count on them for cheap, entertaining, entry-level history and culture about a city. They leave daily from the Astronomical Clock at 10:00, 10:45 and 14:00. Sandeman's Castle Quarter Tours depart from the Czech Tourism Center on the Old Town Square at 14:00, and pick up others on Jan Palach Square just in front of Rudolfinum at 14:30 before continuing across the river and up the hill to the castle.
**LOST IN TRANSLATION?**
In the Czech Republic, not only are American movies dubbed, but also their titles are often changed drastically. The reasons for this vary. Sometimes, American movie titles just don't sound right in direct translation, or they may even have a completely different meaning when translated to Czech. Regardless of the reason, the results can range from confusing to hilarious. Here are some of my favorites translated back into English:
**English** | **Czech**
---|---
_The Hangover_ | _Party in Las Vegas_
_Wicker Park_ | _Love Me Please_
_Freaky Friday_ | _Between Us Girls_
_Bad Santa_ | _Santa Is a Pervert_
_Hurt Locker_ | _Death Waits Everywhere_
_A Beautiful Mind_ | _A Pure Soul_
_Full Metal Jacket_ | _Lead Vest_
_Cool Runnings_ | _Coconuts in the Snow_
_Beverly Hills Ninja_ | _Fatty from Beverly Hills_
_The Amityville Horror_ | _You Will Die at 3:15_
_Meet the Parents_ | _The Father is a Ruffian_
_Bourne Identity_ | _Agent Without a Past_
_Any Given Sunday_ | _Winners and Losers_
_Hot Fuzz_ | _Overly Rapid Deployment Unit_
_Van Wilder_ | _Sexy Party_
Tip-based and paid tours, pub crawl option, newpraguetours.com
#### **Biko Prague Bike Tours**
Filippo, from Italy, met his wife in the Czech Republic, quit the corporate tobacco industry world, moved here, and started this adventurous bike tour company. You can choose between numerous offerings to suit any fitness and skill level. I love these tours because they let you see a different side of Prague in the hills beyond the city, all with friendly guiding and coaching throughout.
Tours including bikes from 1,300Kč, Vratislavova 58/3, +420 733 750 990, bikoadventures.com
### **TOP HOSTELS**
#### **Mosaic House**
Recently renovated and consistently blowing away backpackers with its value and mod decor, Mosaic House is my favorite hostel in town. This hybrid boutique hotel and designer hostel is a 12-minute walk from the Old Town Square, putting you close to all the main attractions in the city. The showers, with their rainforest drizzle, are a favorite. They keep it happening at the bar with frequent live acts, creating a fun and inviting atmosphere. Daily free walking tours (at 11:00) and the hostel's own nightly pub crawl (at 20:00) meet in the lounge.
190-350Kč, free Wi-Fi, laundry facilities, 24-hour reception, breakfast 150Kč (optional), Odborů 4, Old Town, +420 221 595 364, mosaichouse.com, Metro: Karlovo Náměstí
#### **The MadHouse Hostel**
If you, like me, are bemoaning the slow death of the social atmosphere in hostels around Europe due to mobile tech and "social networking," hit up the MadHouse Party Hostel for the hostel with the best social vibe in town. This is Prague's most legit backpacker hostel, and these guys put on events every night of the week. While your sleeping pattern will take a hit, it's for a noble reason: experiencing the true nightlife of this city.
Beds from 160Kč, free Wi-Fi, city maps, parties and events organized throughout the week, Spalena 39, Old Town, +420 222 240 009, themadhouseprague.cz, Metro: Národní Třída
#### **Hostel Orange**
This is a favorite among backpackers for the chill atmosphere and excellent location right on Wenceslas Square. They have a hard time keeping their locks working, so don't head here if you're carrying the crown jewels with you. Without a common room, the Orange isn't as social as my other recommendations, but the value and location make up for it.
Beds from 155Kč, laundry available, free Wi-Fi, towels included, hair dryers available, 24-hour check-in, Václavské Náměstí 781/20, New Town, +420 775 112 625, Metro: Můstek
#### **Prague Square Hostel**
This hostel is in the heart of it all—you just can't beat the accessibility of this hostel, not only to sights but to the nightlife as well. Places like Chapeau Rouge, Roxy, and James Dean are just a five-minute walk away, with Wenceslas Square not much farther in the opposite direction. The staff are welcoming and happy to help you make the most of your stay in town. As the hostel is in an older building, don't expect too many modern amenities (like plugs by every bed), but what it lacks slightly in features, it makes up for in location right on the Old Town Square.
Beds from 300Kč, free Wi-Fi, free towels, 24-hour reception, free breakfast, Melantrichova 10, Old Town, +420 224 240 859, praguesquarehostel.com, Metro: Můstek
#### **Sir Toby's**
Located in the trendy area of Holešovice (Prague 7), this hostel gives you an opportunity to get away from the touristy streets of Old Town and to see a more local flavor of Prague. Though it's a 10- to 15-minute tram ride to the center, the location puts you right next to the some interesting sights, like Alphonse Mucha's Slav Epic in the National Gallery, Stromovka (largest green space in Prague), and some of the most popular clubs in the city, like Cross Club and Mecca. This hostel offers you an attractive pub downstairs with live music, a nice outdoor area, and a friendly staff.
Beds from 225Kč, free Wi-Fi, 24-hour reception, optional breakfast, Dělnická 24, Prague 7 (Holešovice), +420 246 032 610, new.sirtobys.com, Tram: Dělnická
### **TRANSPORTATION**
#### **GETTING THERE & AWAY**
Prague is well-served by all modes of ground and air transportation.
##### **Plane**
Prague's main airport, **Václav Havel International Airport** (PRG, prg.aero/en), is served by all the major and budget airlines. Connections are easy and cheap into the city center. Get cash from an ATM in the airport and break your big bills at a convenience shop so you can purchase your local transportation connection into town.
You've got two options to connect into the city center: The **Airport Express** bus stops at Náměstí Republiky and Hlavní Nádraží, the main train station, for 120Kč, taking about 25 minutes to get to central Prague. A cheaper option is to take **bus 119** to Praha-Veleslavín (the last stop) and connect to the metro. From the airport, stride out confidently past the row of taxis and catch bus 119 waiting for you at the next row. Hop on and take it to the Veleslavín stop, where you'll connect directly onto the metro, following the crowds down into the station. Don't worry about which direction to take the metro—you're boarding at the end point and it only goes one direction from here. Your 32Kč ticket is valid through your metro ride, so just hop on and you'll be on your way. The entire journey will take you about 45 minutes. From this metro line, you can connect to any others as well as exit for local trams.
Remember to validate your ticket by plugging it into any of the yellow boxes on board the bus and metro, as plainclothes ticket controllers are quite common.
##### **Train**
Prague's main train station, **Hlavní Nádraží** , has numerous daily connections for both national and international destinations. Its location just on the edge of the Old Town makes it easy to connect to your accommodations. Connections run often to Budapest (7 hours), Krakow (8 hours), Vienna (4.5 hours), and Berlin (4.5 hours). Check timetables, find prices, and make reservations online (czech-transport.com).
The station also boasts a convenience store and fast food joints in case you need something for the road.
##### **Bus**
International connections into Prague arrive at **Prague Florenc Station,** east of the Old Town. Central Europe has a wide range of bus options. My favorite sites to find connections are Student Agency (studentagencybus.com), Orange Ways (orangeways.com), Berlin Linien Bus (berlinlinienbus.de), and Eurolines (eurolines.com). Florenc has both tram stops and a metro station nearby for local connections.
##### **Car**
The Czech Republic has an extensive freeway network. Drives to nearby cities like Budapest (5 hours), Berlin (4 hours), and Vienna (4 hours) are easy.
#### **GETTING AROUND**
Prague's relatively compact center is easily walked across in about half an hour. Oftentimes walking is faster than taking a car through the tight, one-way streets of the downtown.
While the communist system didn't work out so well in many ways, it did leave an excellent public transit system of trams, metro lines, and city buses. The **public transportation system** is integrated so the same ticket can be used across all modes of transport. Purchase a **ticket** (24Kč valid for 30 min, 32Kč/90 min) at one of the machines (with coins only!) inside any metro station or at the convenience stores around town and validate it at the top of each escalator. If you'll be zipping across town quite a lot, consider a **day pass** (110Kč) or a **three-day pass** (310Kč). Note that you'll need to use the system at least five times a day to save on those longer passes. Look up routes, delays, and more information at dpp.cz/en.
The public transportation system operates on the honor system, and there are ticket checkers all over the place! Always buy a ticket, and remember to validate it. You may wonder why nobody else is validating theirs: That's because locals have monthly passes that don't need the validations required for single-use tickets.
##### **Metro**
The three-line metro system zips you across town in a jiffy. Connections are clearly marked on every landing area with both simplified and full maps on display, making it easy to know where you need to go.
##### **Tram**
Trams are an easy way to get across town—and enjoy the views en route. Pick up tickets and validate them on the machines to avoid a hefty fine. Ticket checkers are quite active on the popular and touristy tramlines.
**Tram 22** is a great line to take to get oriented to the city, taking you from Královský Letohrádek (near Prague Castle) through Malostranské Náměstí (Lesser Town Square), across the Vltava River, past Národní Divadlo (National Theatre), and down to Národní Třída, which puts you right near the shopping district and Wenceslas Square.
Familiarize yourself with the major landmarks near your accommodations so you can recognize them in the destination list at each tram stop. Each tram has maps inside that highlight your route and allow you to figure out where the heck you're going, but stops only have a list of stop names.
##### **Bus**
Unless you're trying to get outside of the city for the day, most tourists generally don't use buses. Thanks to superior city-center tram and metro systems, the bus system mostly offers connections to the suburbs.
##### **Taxi**
Prague is definitely a city where you want to call ahead for your taxis. Don't hail taxis on the street. **Nejlevn ĕjší Taxi** (+420 226 000 226) is my favorite. Don't even try to pronounce the name; just call the number and be thankful they speak English. This is the cheapest and most reliable company in town. If they're full, call **TickTack** (+420 721 300 300), who provide free water and accept credit cards. If you ever feel you're being ripped off (which happens all the time to tourists), pay a maximum of 300Kč (for a city-center ride), get out of the cab, and pop into your hostel to ask for help. A trip within downtown shouldn't run any more than that. **Uber** and a similar Czech company/app called **HaloTaxi** (+420 244 114 411) are also active in Prague.
##### **Car**
With a trusty TomTom, Prague is easy enough to navigate. Do your best to stay away from the Old Town center, where streets were designed for horse and cart and are clogged with modern cars. The city has numerous parking garages throughout. They get cheaper the farther out you stay, so find a parking lot near a far-off metro station for cheaper daily rates. The garage near the Chodov shopping center is right on the red metro line and is staffed, offering rates for around 200Kč per day. There are also numerous park-and-ride (P&R) lots on the outer metro stops that charge only about 25Kč per day but are unguarded.
##### **Bicycle**
With its cobblestones and crowds, I don't recommend cycling in downtown Prague. If you love staying active, a tour with **Biko Prague Bike Tours** takes you into hills around Prague for beautiful views of the surrounding countryside.
### **DAY TRIPS**
#### **Č eský Krumlov**
Český Krumlov is a picturesque medieval Czech town in southern Bohemia. If you see a postcard from the Czech Republic and Prague isn't on it, you're probably looking at a shot of Český Krumlov, famous for its quaint small-town atmosphere in a hairpin turn of the river with a picturesque castle looking over it high up on a ridge. This town is worth the 3.5-hour trip out from Prague for an overnight visit to explore the castle and old town, only four blocks across. Remember, a visit is much more pleasant in nice weather—it gets _cold_ in January and February, and most of the town's tourist shops and restaurants shut down. Visitors come to enjoy the atmosphere and get away from the big city.
To get there, opt for **Student Agency** buses (studentagencybus.com), which leave from Florenc station. Tickets (210Kč one way) do sell out in advance, so reserve ahead of time.
**Trains** take about 4.5 hours (around 550Kč) including a transfer in České Budějovice. While more comfortable, this option does take more time, and the train station is a ways from the center of Český Krumlov, so pack light or hail a taxi on arrival.
### **HELP!**
#### **Travel Tips**
It is required to have identification on you at all times. If you are a victim of any crime or have lost any of your travel documents (passport, etc), contact the US Embassy and they will sort you out. More and more places accept credit cards these days, but it's good to carry some cash to avoid getting caught in a bind.
**Important:** Respect local authorities, and don't draw negative attention to yourself. Their English may be limited, and you don't necessarily have the same rights as you may have in the US, so it's best to just fly under the radar and have a nice, jail-free experience. Prague is a safe city, but you can find yourself in trouble when seeking out brothels and drugs.
#### **Tourist Information Center**
Find Prague's tourist information center (daily 09:00-21:00, prague-information.eu) just into the Lesser Quarter from Charles Bridge on Mostecká street. Pop in here for help on everything from finding accommodations to booking concerts, bus tours, local guides, and even renting cars.
Mostecká 4, Lesser Quarter
+420 257 213 420
##### **Pickpockets & Scams**
Though Prague is a safe city with a low violent crime rate, there are still scams and situations to be mindful of when touring the city, particularly pickpockets. Always keep a close eye on all your valuables, especially in crowded areas such as the city center, public transportation, and nightclubs. If possible, avoid using the taxi service altogether, as many cabbies like to overcharge naïve tourists.
Beware of the sketchy men trying to lure you into the many strip clubs off of Wenceslas Square. Most of these clubs are rip-offs, selling watery cocktails and leaving you broke by the end of the night.
##### **Emergencies**
In an emergency, dial 112. Dial 158 for police.
##### **Hospital**
Nemocnice Na Františku
Na Františku 8
+420 224 810 502
##### **US Embassy**
Tržiště 15
+420 257 022 000
Budapest Map
Budapest 101
Three Day Itinerary
Top Neighborhoods
Top Sights
Top Hungarian Bathhouses
Top Eats
Top Nightlife
Top Shopping & Markets
Top Parks & Recreation
Top Tours
Top Hostels
Transportation
Day Trips
Help!
Budapest, with stunning baroque architecture, magnificent bathhouses, and rocking ruin pubs, is unlike any other city you'll visit. Rather than traditional sightseeing, offbeat experiences—from subterranean caving expeditions to escape games to partying on a boat on the Danube—will be the highlight of your trip. But keep your eyes peeled as you wander Budapest's wide boulevards. There's plenty to see. And with lots of fun, social hostels, Budapest fits quite nicely into even the tightest of budgets.
### **BUDAPEST 101**
The Carpathian Basin was settled by Celtic tribes, who were attracted to the area's fertile land and abundant thermal hot springs. Around AD 100, the Romans arrived and established a military encampment, calling it Aquincum in a nod to the region's healing springs. The Romans built roads, baths, and amphitheaters, the foundations of which are still sprinkled around town today.
In 896, the legendary Hungarian—or Magyar—tribes arrived from the east and found the place to their liking. The fabled seven tribes and their brave leader, Arpad, followed their legendary _turul_ bird, who carried the sword of the Hungarian people. The bird dropped the sword, and the tribes settled where it landed.
By the early 19th century, Hungary had become a part of the Habsburg Empire, ruling out of Vienna. In 1848, revolution burned through the continent and Hungary followed suit. The Habsburgs made an example of crushing this revolt. Curiously enough, the Hungarians were granted semi-autonomy about 20 years later when the Austrians realized they needed a friendly capital in Hungary. This deal was cemented in the Austro-Hungarian Compromise of 1867, laying the foundations of the Austro-Hungarian Empire, a complicated powder keg of clandestine transcontinental alliances. The assassination of Archduke Franz Ferdinand, heir to the Austro-Hungarian throne, triggered a violent domino effect that ignited World War I.
As a loser, Hungary was sliced up after the war and lost 70 percent of its territories and two-thirds of its population. A couple decades later, World War II brought widespread destruction along with executions and deportations of Budapest's Jews. Hungary was "liberated" by the Russians, but the Soviets hung around for the next few decades. Hungarians rebelled against communist rule by 1956. Thousands fled the country, and it took a massive effort to put down the rebellion.
Hungary, considered the "bread basket" of the Soviet Empire because it produced food that was distributed across the Soviet Bloc, enjoyed a rather privileged position and relatively high standard of living under Soviet rule. Hungarians were even able to leave the country occasionally. That's not to say the populace wasn't policed. The Hungarian secret police (the State Protection Authority, or AVH), were responsible for torture, detentions, and executions that mirrored those of the Nazi SS. By the 1980s, communism was crumbling around the Eastern Bloc, and Hungary was among those celebrating the fall of the Berlin Wall.
In the last 25 years, Budapest has undergone a transformation, but it remains culturally distinct from just about every other European city. You'll see posh shopping boulevards like those in Paris, nightlife a little like Berlin's, a city layout similar to that of Prague (with a river down the middle and a castle on a hill), and a cool vibe like that of East London. There's no other city that rolls all of this into one—and cheaply, at that!
### **PLAN AHEAD**
#### **RESERVATIONS**
Reservations are required for the **Hungarian Parliament Building** (parlament.hu/en).
Reservations are recommended for the following activities:
**Caving** (barlangaszat.hu; tickets can often be bought at your hostel)
**Spa parties** (bathsbudapest.com/budapest-bath-parties; tickets can often be bought at your hostel)
**Budapest Boat Party** (facebook.com/Budapestpartyhostels)
#### **LOCAL HAPPENINGS**
##### **Sziget Music Festival**
Sziget Music Festival (szigetfestival.com), a world-famous, week-long electronic music and dance festival, kicks off each year around August 10, taking over an entire island (Obudai-sziget) 45 minutes north by bus of Budapest. If you want to attend, buy tickets well in advance. My friends and I loved the atmosphere, food options, plentiful porta-potties, and, above all, the always-hot music lineup.
##### **St Stephen's Day**
August 20, the country's biggest national holiday, is the Hungarian equivalent of Fourth of July. Named in honor of Hungary's first king, Stephen I, it commemorates the foundation of the Hungarian state. Visitors are welcomed into events like fireworks and general festivities that take place all over the city.
##### **October 23**
October 23 marks two important events in Hungarian history: the Revolution of 1956, when the Hungarians rose up against the Soviet Union, and the Day of the Republic, the day in 1989 when communism officially fell and Hungary was declared a republic. The mood is jubilant, but thoughtful, as Hungarians remember those who gave their lives so that Hungary could break away from the Soviet Bloc.
**KNOW BEFORE YOU GO**
**KEY STATS & FIGURES**
**Currency:**
**Hungary uses the forint (Ft); 285 Ft = about 1 USD**
**Population:**
**9,971,000**
**Language:**
**Hungarian**
**Size of Hungary:**
**93,000 square kilometers (about the size of Michigan)**
**National dish:**
**goulash and anything spiced with paprika**
**Number of hot-water springs:**
**123 in greater Budapest**
**Famous inventions you never knew were Hungarian:**
**Rubik's Cube, the ballpoint pen, binoculars, color TV, Microsoft Word and Microsoft Excel, the hydrogen bomb**
**CALIBRATE YOUR BUDGET**
**TYPICAL PRICES FOR:**
**Hostel dorm bed:**
**4,000Ft**
**Two-course dinner:**
**2,000Ft**
**Pint of beer:**
**600Ft**
**Bicycle rental:**
**3,500Ft**
**Metro pass:**
**340Ft**
**MOVIES TO WATCH**
_Mission: Impossible—Ghost Protocol, I Spy, 8MM 2, Transporter 3, Munich_
**VIDEOS TO WATCH**
Katy Perry's _Firework_
THREE DAY ITINERARY
Budapest is one of my favorite cities, not just for the culture, affordable prices, food, and nightlife but also because the sights are different from those in most other major cities. Rather than just going from museum to museum, you'll be jumping from one unforgettable activity to the next.
**DAY ONE: TO THE BATHHOUSE!**
**MORNING**
Before leaving your hostel, throw a swimsuit in your bag—you'll be visiting the baths later this afternoon. Pop into **Mozsar Kavezo** for breakfast.
From Mozsar Kavezo, it's a five-minute walk to catch a tour with **Yellow Zebra Bike Tour** for an excellent four-hour introduction to the city departing at 11:00 daily (Apr-Oct) from their office just behind the **Hungarian State Opera House**.
**AFTERNOON**
Finish the bike tour back at the Yellow Zebra office and get a sandwich en route to **Széchenyi Fürd ő,** the bathhouse located in the **City Park**. To get to the baths, either take the peaceful 40-minute walk all the way down Andrássy Út or take the yellow metro line toward Mexico and get off at Széchenyi Fürdő.
Spend the afternoon soaking in the baths. Be sure to explore the entire complex, hopping from cold dunk baths to piping hot saunas again and again until you feel rejuvenated! Get a traditional _lángos_ (deep-fried flatbread with toppings) at the stand outside the baths on your way out.
**EVENING**
Head back to the hostel to freshen up, then to a fun Hungarian dinner at one of my favorite spots in town: **Vak Varju**.
**LATE**
Dinner at Vak Varju puts you in the perfect place to explore the **Gozsdu Court & Passage** and bars in the Jewish Quarter. Or catch back up at the hostel's bar to head out with them on the night's crawl around the city. Lineups for nightly parties change often; Thursday is boat party night!
**DAY 2: CAVING & RUIN PUBS**
**MORNING**
Dress comfortably and wear shoes that you can get dirty—you'll be exploring caves later. Purchase tickets from your hostel staff, and confirm meeting place and time before leaving. Shake off your hangover with a bite at the famous **Great Market Hall.** Find shopping and food on the ground floor; fast food, cafés, and lace vendors upstairs; and a real-world standard grocery store downstairs. Stock up a bit for afternoon snacks.
Take a 15-minute walk north along **Váci Utca,** Budapest's main medieval avenue and a touristy shopping street that's parallel to the Danube, till you reach Vörösmarty Tér, the main town square in Pest. From Vörösmarty Tér, catch tram 2 and ride it a couple stops north toward the **Hungarian Parliament Building**. On a short five-minute ride, you'll see the Danube River, **Castle District** , **Fisherman's Bastion** , the **Shoes on the Danube Memorial** (keep your eyes peeled, they're easy to miss!), and **Gellért Hill** all from the windows of the tram.
If you're interested in going inside the **Hungarian Parliament Building** , make an appointment online ahead of time (tours run at 10:00, 12:00, and 14:00 and take about 1.5 hours, so factor that in with caving)!
**AFTERNOON**
Catch up with the group meeting for the four-hour **Barlangaszat Caving Tour** at Nyugati station, a 10-minute walk from the Hungarian Parliament Building. Tours occur Monday, Wednesday, Friday, and Saturday afternoons, so purchase tickets and confirm ahead of time through your hostel. The meeting point is at the bus 206 stop, near the cement staircase leading up to the overpass. These guys will take you up to the cave via two buses, so you'll need four metro tickets per person going caving. Enjoy your adventure! Heads up: Your shoes will get dirty with clay, but you'll have coveralls for everything else.
If you're claustrophobic, enjoy another one of Budapest's many thermal baths instead of the caves. For the most authentic, old-school Turkish bath experience, I recommend **Rudas**.
**EVENING**
Refresh at the hostel, and head to an eclectic dinner at **Most** , not far from the Opera house.
**LATE**
Head out for a ruin pub crawl of your own, starting at **Instant** , right around the corner from Most, then checking out **Fogashaz** and **Szimpla**. If you're ready to rock 'n' roll, walk the 15 minutes to **Corvinteto** , one of Budapest's rooftop clubs.
**DAY 3: JEWISH QUARTER SIGHTS & MARGIT ISLAND**
**MORNING**
Is it Sunday? If so, dive into **Szimpla's farmers market** and get lost in the delicious organic cheeses, pastries, coffees, salami, greens, soups, and more. If it's not Sunday, settle for **Sugar!** and grab a coffee and something sweet.
Szimpla and the farmers market are located in the **Jewish Quarter** , so it's a convenient time to make your way to see the **Dohány Street Synagogue** and the **Raoul Wallenberg Holocaust Memorial Park**.
**AFTERNOON**
Make your way out to **Margit Island** to explore and rent bikes. On a nice day, it's a perfect place for a picnic alongside the river. Tram 6 will zip you there; get off at the stop at the middle of the bridge directly over the island.
Are you a history buff with energy left? Pick between the **House of Terror** or the **Hospital in the Rock** for mind-blowing historical lessons.
**EVENING**
When your appetite comes back, head to **Belvarosi Disznotoros** for a cheap, filling meal made in front of your eyes.
**LATE**
Take part in Budapest's latest craze: **escape games**. A drink or three with dinner makes the games that much more enthralling.
### **TOP NEIGHBORHOODS**
Budapest is large but relatively easy to navigate. The majestic Danube River runs down its center. On the west side is hilly, residential Buda. Pest, on the east side, is flat and sprawling.
Just about all the action happens in Pest, within three major concentric ring roads. Inside the first ring (Karoly/Muzeum/Vámház Körút) is the heart of the medieval town, including the districts of **Leopold Town** (home to the Hungarian Parliament Building and St Stephen's Basilica) and **Pest Town Center,** where the Váci Utca shopping street connects Vörösmarty Tér (Pest's town center) with the Great Market Hall.
The second, larger ring (Szent Istvan/Terez/Josef/Ferenc Körút) encircles **Andrássy Út,** a major boulevard where you'll find the Hungarian State Opera House and House of Terror. Also inside this second ring is the **Jewish Quarter (District VII),** site of the Dohány Street Synagogue and the center of Budapest's raging nightlife scene. East of this second ring road, **City Park** houses the famous Széchenyi baths.
In Buda, the **Castle District** is home to Fisherman's Bastion and Matthias Church. Today, you won't see an actual castle, but this district remains the political heart of historic Budapest. It's full of worthwhile sights, though I rarely pay to enter them, finding their exteriors beautiful enough. To the south, prominent **Gellért Hill** is capped by the Citadel, with Gellért and Rudas baths at the base.
### **TOP SIGHTS**
#### **Hungarian Parliament Building**
Budapest's most famous and recognizable landmark, this spectacular building was built in 1896 to commemorate the millennial celebration of Hungary's 896 founding. Topped with 1,896 spires, this massive bicameral structure faces the Castle District and sits right on the Danube. Inspired by Westminster Palace in London, the Parliament building is built in proud and pointy neo-gothic architecture.
The Parliament cost 38 million gold crowns—equal to a sizable chunk of Hungary's GDP—to build. That's nearly US$1.5 billion in today's currency. People were appalled at the price, which could support an entire small city in the countryside. The spire atop the dome is 96 meters tall, of even height with St Stephen's Basilica. During Soviet times, this spire was topped with a massive red star, reminding Hungarians that their rulers in Moscow lorded over all, in both the terrestrial and spiritual worlds. Fun fact: The first ever air-conditioning system was developed and installed in this building. Air is blown over massive ice blocks in the basement and routed up into the large parliament rooms above.
It's possible to take guided tours, but reservations must be made in advance. Free tours run at 10:00, 12:00, 13:00, 14:00, and 15:00 and take about 1.5 hours.
**THE NUMBERS GAME**
The numbers **7** (the original number of Magyar tribes that settled in the Carpathian Basin), **896** and **1896** (the dates of Hungary's foundation and millennial celebration), and (by extrapolation) **96** are basically all you need to know for just about anything in town. How many points does the Fisherman's Bastion have? 7. How many spires on the Parliament building? 1,896. How tall is St Stephen's Basilica? 96 meters tall. Keep these numbers in mind if you find yourself playing trivia anywhere in town!
5,000Ft, half-price for EU citizens with passport, on Mon when Parliament is in session (usually Sept-May) the only English tour is usually at 10:00, Kossuth Lajos Tér 1-3, Leopold Town, +36 1 441 4000, parlament.hu, Metro: Kossuth Tér
#### **Shoes on the Danube Memorial**
Nazi atrocities were carried out across Hungary, and during the winter of 1944 and 1945, the Arrow Cross, or fascist Hungarian puppet police, rounded up Jews from the ghetto, lined them up along the banks of the river, and shot them at close range so they would fall into the river and get washed away. Today, 60 iron pairs of period-accurate men's, women's, and children's shoes are fused to the concrete in the place where these murders occurred. This discreet memorial serves as a reminder about the importance of remaining vigilant against bigotry and ignorance. Use caution when crossing the street and tram tracks to get to the memorial.
Free, always open, two blocks south along the river from the Parliament building, Leopold Town, Metro: Kossuth Tér
#### **St Stephen's Basilica**
St Stephen's bold neoclassical architecture, with twin bell towers and prominent dome and spire, is recognizable from anywhere in town. On entry, you may be hassled by someone requiring donations, but the church is free to enter. You can find the famous shriveled hand of Saint Stephen in the back of the church toward the left of the altar. It is possible to climb the dome for less than US$2, but the views aren't particularly amazing.
Interior free, 500Ft for observation deck, Mon 09:00-16:30, Tues-Fri 09:00-17:00, Sat 09:00-13:00, Sun 13:00-17:00, Szent István Tér 1, Leopold Town, +36 1 311 0839, bazilika.biz, Metro: Bajcsy-Zsilinszky Út
#### **House of Terror**
The House of Terror occupies what used to be the headquarters of both the Nazi SS and the Communist Party Police, Arrow Cross. It's best for those who are very interested in WWII and communist history. The interactive exhibits walk you through daily life under communism, including a stomach-turning room stacked floor to ceiling with lard, the most common cooking ingredient at the time due to lack of butter and olive oil. Descend into the basement via a slow elevator while listening to the chilling tales of an executioner who worked his trade here. While it's famous and well-done, I do get the sense it's a bit commercial and made for mass-market school groups.
2,000Ft, Tues-Sun 10:00-18:00, last entry 30 minutes before closing, Andrássy Út 60, Andrássy Út, +36 1 374 2600, terrorhaza.hu, Metro: Vörösmarty Utca
#### **Turul Bird Statue**
Legend has it that the Hungarian tribes would settle and flourish in the place where the _turul_ (the mythical bird of Hungarian folklore) dropped its Magyar sword. Commonly thought to be a falcon, the _turul_ is often represented in statues and on coins throughout Budapest. The most majestic statue in town is perched, sword-in-claws, in the Castle District, looking out to Pest. Don't miss the selfie-op with the Hungarian National Guard standing at attention nearby.
Free, always open, Castle District, Metro: Szell Kálmán Tér
#### **Matthias Church**
The majestic neo-gothic Matthias Church is famous for its beautiful white stone architecture and colorful roof tiling typical in Hungary, which comes from the town of Pecs. Though its foundations date back to 1015, the church appears brand-new thanks to recent renovations. The relatively steep entry fee (US$5) keeps many content with the exterior, but those who go inside are treated to a holy Technicolor kaleidoscope of decorations on nearly every square inch of the columns, arches, ribs, altar, stained glass, and pulpit.
1,400Ft, Mon-Fri 09:00-17:00, Sat 09:00-13:00, Sun 13:00-17:00, possibly open later in summer, Szentháromság Tér 2, Castle District, +36 1 355 5657, Metro: Szell Kálmán Tér
#### **Fisherman's Bastion**
With a name like Fisherman's Bastion, not many people know what to expect. Don't look for it on the river. The gleaming white ramparts with seven pointed turrets are perched high on the hill and afford stunning panoramas across the Danube and over the Parliament building and into Pest. Constructed in 1896 to commemorate the millennial celebration of Hungary, the decorative bastion symbolically represents the seven Hungarian tribes. Climb the stairs for a small fee and slightly better view, or come after sunset and go for free! From the Fisherman's Bastion you can see all of Pest and a beautiful-yet-distant view of the Parliament building. If you cross back through the Castle District to the other side, you can peer out across the hills of Buda as well.
700Ft, mid-Mar-mid-Oct daily 09:00-19:30, open and free to enter after closing time or off season, Szentháromság Tér 5, Castle District, +36 1 458 3030, fishermansbastion.com, Metro: Szell Kálmán Tér
#### **Hospital in the Rock**
This cave network underneath the Budapest castle originated as a storage space but was widened to become a hospital during World War II, and it served hundreds of patients. Later on, it was used as a nuclear bunker during the Cold War. With some impressive engineering, the hospital was designed to withstand nuclear attack and provide emergency medical assistance. This bunker also acted as the control center of the commanding elite for continuing the war effort in the case of nuclear war. Today, you can take a fascinating guided tour of these tunnels—well worth it for history buffs!
4,000Ft, daily 10:00-20:00, last tour departs at 19:00, Lovas Way 4/c, Castle District, +36 70 701 0101, sziklakorhaz.eu, Metro: Szell Kálmán Tér
#### **Hungarian State Opera House**
The Hungarian State Opera House was constructed during the height of the Austro-Hungarian Empire in 1884. As the second capital of the great Hapsburg Empire, Budapest simply had to have an opera house, and license was granted to build one—as long as it wasn't bigger than the one in Vienna. Playing within the rules, the architects sought to make this opera house grander—not bigger—than the one in their rival sister city. Through their proud neo-Renaissance architecture and gilded golden concert hall comfortably seating over 1,400 in three levels of balconies and a large main audience floor, I'd say they achieved their goals! Both the exterior and interior are breathtaking with opulent baroque decorations.
Free to pop in to the lobby, Mon-Sat 11:00 until show time, usually 19:00, or until 17:00 if there's no performance, Sun open 3 hours before performance, usually 16:00-19:00, or 10:00-13:00 if there's a matinee, English tours daily at 15:00 during high season (June-Oct) for 2,900Ft, mini concerts often for the screaming deal of 600Ft, check show calendar and prices online, Andrássy Út 22, Andrássy District, +36 1 814 7100, opera.hu, Metro: Opera
#### **Dohány Street Synagogue**
With the capacity to seat nearly 3,000 worshippers, the Dohány Street Synagogue (also known as the Great Synagogue) is the second largest synagogue in the world. The architects deliberately toned down the structure to blend in with the surrounding architecture, so the structure's massive size isn't evident from the outset. While Budapest's Jewish population still hasn't recovered from the deportation of 600,000 Hungarian Jews during the Holocaust, the synagogue stands proud and serves the 100,000 Jews living in Budapest today. Your ticket also includes entry to the modest **National Jewish Museum,** showcasing artifacts of daily life and culture like Torah scrolls, artwork, sculpture, and menorahs from traditional to modern, along with a room about the atrocities of the Holocaust.
If you don't want to pay to enter the synagogue, you can still appreciate the beautiful facade and cemetery, visible from Wesselenyi Utca. The striking **Raoul Wallenberg Holocaust Memorial Park** is in the back garden of the Dohány Street Synagogue but is also visible from the street. It's designed in the shape of a weeping willow, which also bears resemblance to an overturned menorah. Each of the thousands of shimmering leaves bears the name of a Hungarian victim of Nazi hate crimes. The space is named after Raoul Wallenberg, a Swedish world traveler, son of a wealthy diplomat, and playboy turned unlikely hero. Wallenberg secretly helped to rush emergency (read: official yet fake) Swedish passports to thousands of Jewish families, protecting them from deportation during the Nazi occupation. Exhibiting both bravery and cunning in the face of danger, Wallenberg saved the lives of thousands of Hungarian Jews.
3,000Ft, Mar-Oct Sun-Thurs 10:00-17:30, Fri 10:00-16:30, closed Sat, closes at 15:30 Fri in Mar; Nov-Feb Sun-Thurs 10:00-15:30, Fri 10:00-13:30, closed Sat; refer to website for the annual list of days closed, Dohány Utca 2, Jewish Quarter, greatsynagogue.hu, Metro: Astoria
#### **The Citadel**
This decommissioned military fort now offers the best viewpoint to see both the castle and Pest, but it's a hike to get to the top. Start the climb after crossing either Erzsebet Bridge or Szabadsag Bridge. Allow about three hours total to get up, check out the views, and get back down again.
Free, daily 11:00-23:00, Citadella Sétány 1, Gellért Hill, Metro: Ferenciek Tere or Kálvin Tér
### **TOP HUNGARIAN BATHHOUSES**
A visit to Budapest is simply not complete without a dunk in at least one of the city's invigorating aquatic wonderlands. This is the most popular activity (and my personal favorite thing to do) in Budapest. Water in this region comes out of the ground nearly boiling at 170 degrees Fahrenheit. It's just a matter of how much they cool the water before channeling it into the beautiful public bathhouses and pool complexes. While you have numerous choices in this city, these are a few of my favorites.
#### **Széchenyi Fürd ő**
This is the most famous bathhouse, and the best one for first-timers in Budapest. Széchenyi is a delightful jumble of steam rooms, saunas, pools, and baths both outside and inside this grand yellow baroque complex. While away the hours wandering through this labyrinth, trying each room and pool along the way, enjoying the rejuvenating qualities of this special water. Purchase your wristband upon entry, go downstairs to claim a locker, change and shower, then hit the large outdoor pools open year-round (two lounging pools and one lap pool, swim cap required). Look for small stone plaques denoting the temperature of each bath or room, with 20°C being as cold as you can stand, and 40°C being as hot as you'd want. I recommend going in the entrance closest to the permanent circus toward the back, the side facing away from downtown.
4,500Ft, for locker, 500Ft more for personal changing cabin, cheaper after 19:00 and more expensive on weekends, daily 06:00-22:00, thermal bath open 6:00-19:00, may be open later on summer weekends, last entry one hour before closing, inside the City Park (on the Pest side of the capital, just beyond Hero's Square), City Park, szechenyibath.com, Metro: Széchenyi Fürdő
#### **Rudas**
Rudas is your legit Turkish bath, with low lights, burning hot steam rooms, and _ice-cold_ dunk baths. With no frills and only six pools, as compared to Széchenyi's 18, Rudas still features a wider range of temperatures (18-42°C), perfect for those who want the real experience. This is the only bath in Budapest that still offers gender-segregated days during the week (during which suits are optional) and mixed-gender bathing from Friday evening through Sunday (bathing suits required).
3,000Ft, men only Mon, Wed, Thurs, Fri 06:00-20:00; women only Tues 06:00-20:00; gender mixed Fri 22:00-04:00, Sat 06:00-20:00 and 22:00-04:00, Sun 06:00-20:00, Rudas Gyógyfürdő és Uszoda, Gellért Hill, +36 1 356 1010, rudasbaths.com, Metro: Szent Gellért Tér, Tram: Döbrentei Tér
#### **Gellért**
Widely regarded as the most posh bathhouse in Budapest, Gellért is also part of a five-star hotel by the same name. Gellért features an outdoor wave pool (generally open in the summer) that toes a dangerous line. A handful of interior rooms are gender separated so you can really let it hang out. Gellért is located on the southern part of the city center, on the Buda side, adjacent to the Liberty Bridge.
4,900Ft for locker, 400Ft more for personal changing cabin, cheaper after 17:00, last entry one hour before closing, daily 06:00-20:00, Kelenhegyi Út 4, Gellért Hill, +36 1 466 6166, gellertbath.com, Metro: Szent Gellért Tér
**ACT LIKE A LOCAL**
**Splish, Splash: Hungarian Bathhouse Etiquette**
Here are some general rules and tips so you can skip that confused feeling when heading into Budapest's _fürd ő_ (baths):
**Don't show up on the wrong day.** Certain baths have days for men ( _ferfi_ ), days for women ( _noi_ ), and gender-combined days, though most have shifted toward being open for everyone every day of the week.
**Go for privacy, or let it all hang out.** Private changing closets, which you can share with friends, are available but a little more expensive than the standard communal (gender-segregated) changing areas and lockers.
**Consider a massage.** You can opt for massages by speaking with the women just inside the paid entryway. They'll be holding a clipboard with information, and relatively good deals can be had in these bathhouses for a nice massage.
**Skip the medical treatments.** Thermal and mineral spas are recognized in Hungary to have serious health benefits. Doctors actually prescribe a whole host of treatments. You may see a list of such treatments, which can be confusing. Just make it clear you want the basic entry.
**Your wristband is your entry ticket, your locker access, and your exit ticket.** This is what you receive when you pay your entry fee. Put your wristband on and hold it up to the button on your locker to lock and unlock it. That's also how you "claim" an empty locker—by placing and pushing your wristband against it. You can open and close your locker as many times as you want. If you forget your locker number, find the little box with a screen mounted on the wall in the locker room. Hold your wristband up to it and it will light up with your locker number.
**Bring your own suit and towels to save $$$.** Towels and swimsuits are generally available for rent from the service desk inside the complex.
**Consider bringing flip-flops.** They're nice to have in the showers and for getting around the various pools, but not required.
**Don't be afraid to ask for help.** It may seem like everyone knows what they're doing, but don't worry, they don't! Locker room attendants are there to assist with any questions you may have. They usually answer you in English, or at least they can find someone who does speak English.
### **TOP EATS**
Traditional Hungarian food is simple and based mostly on meat and potatoes. Classic dishes like **goulash** (spiced beef stew ladled over dumplings or gnocchi-like noodles) are cheap—so cheap that nicer restaurants are shifting away from authentic Hungarian food in order to justify a higher price point!
Budapest's cuisine reflects its reputation as Europe's most diverse city. One street in particular overflows with all sorts of international cuisine: **Kazinczy Utca,** located in the Jewish Quarter. Beyond the fast food joints on this street, you also have food options inside the ruin pubs Szimpla and Ellato Kert, and a new food garden just opened up in an empty space two doors down from Szimpla. **Gozsdu Court & Passage,** located between Dob Utca 16 and Király Utca 13 in the Jewish Quarter (a couple blocks west of Kazinczy Utca) is a great nightlife district, but you'll also find restaurant after great restaurant, including an amazing burger joint, authentic Italian and pizza, Asian, and more.
Tipping is not expected, but it is appreciated. Touristy restaurants have come to expect tips, and some will automatically put an "optional" tip on your bill. So be sure to read each line item to understand how you got to your total and to avoid tipping twice. If you loved your service, round your bill up about 10 percent.
#### **Kiado Kocsma**
I stumbled upon this gem and found myself going back again and again for a hearty plate of ham and eggs for only about US$3! Couldn't ask for anything more to start off your day. Come back in the afternoon for the goulash soup or to split their heaping meat dish with your travel buddy. If you've got to catch up on emails, Kiado Kocsma is a great place to post up and pound through a little work, sharing the cozy space with other young locals with the same idea.
700-1,500Ft, Mon-Fri 10:00-01:00, Sat-Sun 11:00-01:00, Jokai Tér 3, Andrássy District, +36 1 331 1955, facebook.com/kiadokocsma, Metro: Oktogon
#### **Mozsar Kavezo**
Mozsar Kavezo has all the necessities in a modish café: fresh pastries, good coffee, hot breakfast menu, and fast Wi-Fi. What's not to love? Located near a number of recommended hostels, just off Andrássy Út and near the opera house, it's a great place to grab breakfast. On a hot day, the large windows turn the place into a stuffy greenhouse, so I take my breakfast to go.
Pastries from 300Ft, daily 09:00-23:00, Nagymező Utca 21, Andrássy District, +36 1 898 1115, facebook.com/mozsarkavezo, Metro: Opera
#### **Vak Varju**
If you're looking for a bit of traditional Hungarian at an affordable price, this is the place for you. Try their goulash or duck entrée and you won't be disappointed. Vak Varju was started by a group of enterprising friends, and I like the modern yet subtly irreverent decor. Eat under the watchful eyes of your Hungarian grandma and grandpa leaning over the balcony railing, and discover surprises in both bathrooms. If you're thirsty or sitting with friends, the 3.5-liter beer tower comes with your own tap and is a great way to start the night. Their pizza bread is delicious as well.
900-2,000Ft, daily 11:00-24:00, Paulay Ede Utca 7, Andrássy District, +36 1 268 0888, vakvarju.com, Metro: Bajcsy-Zsilinszky Út
#### **Most**
This candlelit restaurant offers a little bit of everything—from the most decadent chocolate chip pancake brunch to chicken tikka masala and pad thai dinners. The atmosphere is great, and the menu caters to everyone. The service, unfortunately, is consistently disappointing.
Moderate prices, daily 10:00-05:00, Zichy Jeno Ucta 17, Andrássy District, +36 70 248 3322, mostjelen.hu, Metro: Arany János Utca
#### **Leves**
Hungarians are a straightforward bunch, so it's no surprise that they appreciate the literal the name of this quick lunch stop: Leves means soup in Hungarian. And man, they make some delicious soup! Choose from four daily specials, fill up a small or large bowl to go, and deck it out with chunky croutons. This makes for a great quick lunch, and it's just down the way from the Great Market Hall.
Soups from 700Ft, daily 11:00-19:00, 14 Vámház Körút, Pest Town Center, +36 30 241 7760, facebook.com/levespont, Metro: Kalvin Tér
#### **Cserpes Tejivo Milk Bar**
Milk bars—bars that sold subsidized milk to comrades to ensure everyone was fit and healthy—were a communist institution. In perfect Budapest hipster fashion, Cserpes Tejivo has taken this concept and infused it with a modern twist. With great sandwiches, delicious sweets, and, of course, fresh milk (of the normal, chocolate, and butterscotch varieties) in the middle of downtown, this is an excellent stop for brunch or lunch.
Sandwiches from 700Ft, Mon-Sat 07:30-22:00, 2 Suto Utca, Pest Town Center, cserpestejivo.hu, Metro: Deak Ferenc Tér
#### **Sugar!**
Designed and owned by the daughter of a big café family in town, Sugar! is your typical sugar shack, yet amped up another level with handmade cupcakes and desserts, plus coffee. My go-to is the sweet rice pudding with a dash of cinnamon sugar on top. One order is plenty to share between two or three people.
Sweets from 600Ft, Mon 12:00-22:00, Tues-Sun 10:00-22:00, 48 Paulay Ede Utca, Andrássy District, +36 1 321 6672, sugarshop.hu, Metro: Opera
**FOR THE LOVE OF LÁNGOS**
_Lángos_ (LON-gohsh) is a favorite local savory or sweet treat. Think of it as a deep-fried elephant ear from the carnival covered in all sorts of ingredients of your choice, such as sour cream, cheese, and garlic, or even sugar and almonds and peanuts. _Lángos_ are sold from street stands and kiosks throughout the city, so if you see one, go for it! They're rich and heavy enough to be split by at least two if not three or four people—and only after your trip to the baths.
#### **Belvarosi Disznotoros**
Pop in here for fresh grilled meat and veggies. Think of it like a Hungarian food court, where you point to your choice to the attendant, and they flash-grill it right in front of you and you then take your tray upstairs to enjoy a piping hot meal. The name translates to "downtown pork," and it's a meat lover's paradise with steaks, ribs, sausages, bacon, and more to enjoy. It's right next to the Gozsdu Court & Passage.
Pay by weight, so don't let those eyes get too big! Mon-Sat 07:00-22:00, Karolyi Utca 17, Pest Town Center, +36 1 267 3795, Metro: Ferenciek Tere
#### **Bors GasztroBar**
A welcome player on the Kazinczy Utca restaurant lineup, Bors makes excellent sandwiches and innovative soups (like red cabbage and sour cherry), and it packs out with locals day in and day out. Pop in here if you need a break from the Szimpla hubbub.
Sandwiches from 900Ft, Mon-Sat 11:30-21:00, 10 Kazinczy Utca, Jewish Quarter, +36 70 935 3263, Metro: Astoria
#### **Abszolut Pho**
With functional, takeaway-style decor, this is a great choice for a light meal of super fresh pho. I appreciate the classy sister bar next door and the restaurant seating upstairs above it.
Dishes from 900Ft, Tues-Sun 18:00-22:00, 52/C Kazinczy Utca, Jewish Quarter, +36 70 551 7630, Metro: Opera
#### **Kis Parazs Thai**
I've got to recommend a hot Thai place in just about every city, and this is my favorite in Budapest, located just across the street from the famous Szimpla ruin pub. The curries and pad thai are just killer! If you ask for spicy, they make it kick-ass Thai spicy (read: bleary eyes, running nose, sweating buckets, pure heaven).
Mains from 1,700Ft, Mon-Sat 12:00-22:00, 7 Kazinczy Utca, Jewish Quarter, +36 30 733 7760, parazspresszo.com, Metro: Astoria
#### **El Rapido**
I love a good Tex-Mex joint, and you can't beat these prices for a tasty freshly made burrito at just about any hour of the day. The ingredients, service, beer selection, and ambience are all winners. But the food consistently takes forever. Do they need to turn up their grills? If you're good to suck down a beer while waiting, it's well worth it. And don't miss the thousands of retro blast-from-the-past games and toys downstairs, like Connect Four and Hungry Hungry Hippos.
Burritos from 1,000Ft, Mon-Fri 10:30-late, Sat 12:00-late, Sun 17:00-02:00, 10 Kazinczy Utca, Jewish Quarter, +36 30 279 2861, elrapido.hu, Metro: Astoria
### **TOP NIGHTLIFE**
You can have an incredible time out on the town in Budapest—especially in the Jewish Quarter (District VII). Drinks are cheap, and the experiences are as eclectic as it gets. Most hostels in town do a great job of arranging party nights out just about every night of the week. Get ready for a wild ride!
#### **NIGHTLIFE DISTRICTS**
##### **Nagymez ő Utca, Király Utca & Kazinczy Utca**
Nightlife centers on Király Utca and Kazinczy Utca, two intersecting streets in the Jewish Quarter, and Nagymező Utca, not far from Andrássy Út. On these streets, you'll find some of Budapest's best ruin bars, including **Ellato Kert, Kurplung,** and **Szimpla.** Along with nearby Gozsdu Court, these venues really define the vibe of the Jewish Quarter.
Jewish Quarter, Metro: Opera
##### **Gozsdu Court & Passage**
There's no stretch of the city that is more quintessentially Budapest than Gozsdu Court & Passage, located between Dob Utca and Király Utca. Just a few years ago, this passage was just another dark alleyway. Now there are dozens of bars, restaurants, pop-up shops, and more. Head toward the action, and to your left and to your right bar after bar and restaurant after restaurant are packed out with Budapest's young, sexy, in-the-know crowd. In one block, you've got **Epic Winebar,** a chill British-style pub called **The Pointer,** and **Kolor,** an upscale lounge and restaurant. In good weather, find the **Gozsdu Sky Terrace** for rooftop views across downtown Budapest. This district is a bit posh, so prices will be rather "Western."
**LGBT BUDAPEST**
The acceptance and status of the LGBT community has grown in leaps and bounds since 1990, when Hungary became independent from Moscow. As in most other European capitals, LGBT travelers should expect no issues while visiting Budapest, though public displays of affection between same-sex couples are generally downplayed. Gay Pride Budapest is held each year toward the end of June or early July. For the latest information on cafés, restaurants, nightlife, and accommodations, head to gay.hu/welcome-budapest, budapestgaycity.net, and budapest.com (clicking through to Gay Budapest).
Jewish Quarter, Metro: Deak Ferenc Tér
#### **RUIN PUBS**
After World War II, many bombed-out buildings were never fully restored and rested empty for decades. Eventually, pubs started moving in to these old tenement houses and run-down factories, transforming them into a cornerstone of BP's nightlife. Cool, eh? These pubs incorporate salvaged furniture, items from old community centers, cinemas, boats, and even donations from older residents of Budapest. The mismatching trinkets and knickknacks lying around only add to the peculiar ambience these pubs emanate. I've listed some of my favorites below, but don't be afraid to pop into any scene that looks intriguing.
You'll notice _kert_ (garden) in the titles of most of the bars, meaning there's an outdoor space somewhere in the complex—great to enjoy the warm summer evenings. In winter, bars often cover the gardens with heavy tarps so you can still take in the fresh air without freezing your bum off.
##### **Szimpla**
Budapest's most famous ruin bar, Szimpla, is a can't-miss on your BP pub crawl. With everything from swinging gnomes to ski racks and umbrellas hanging on the walls, the place is packed out year-round with a hipster crowd, minus the pretentiousness that usually comes with it. Grab a beer and get lost in this multi-floor labyrinth of rooms, hallways, and balconies. Hookahs are a great way to make friends, so put in your order to the left as you enter, and find a comfy place to chill. Don't be surprised if someone comes up to you selling carrots out of a basket—just embrace the eclectic vibe and enjoy. And don't miss the local organic farmers market inside the bar each Sunday morning (09:00-14:00).
Daily 12:00-04:00, Kazinczy Utca 14, Jewish Quarter, +36 20 261 8669, szimpla.hu, Metro: Astoria
##### **Instant (aka Enchanted Forest)**
This mainstay of Budapest's famous ruin bars is a labyrinth of cubbies and rooms sprinkled across several floors. Everything centers around one massive owl mounted on the wall chasing dozens of styrofoam bunnies suspended in a net.
Daily 16:00-late, Nagymező Utca 38, Andrássy District, +36 1 311 0704, instant.co.hu/en, Metro: Oktogon
##### **Fogashaz & Fogashaz Kert**
Recently given a face-lift, these are two adjacent bars, with the Kert being the outdoor garden pub. _Fogas_ translates to "teeth," and _haz_ means "house"; welcome to the Teethhouse. Enjoy excellent music in both bars, fast service, and a chill crowd keeping the party going till late.
Daily 14:00-late, Akácfa Utca 51, Jewish Quarter, +36 1 783 8820, fogashaz.hu, Metro: Opera
##### **Kuplung**
A recent entry into the ruin bar scene, Kuplung has taken over a former mechanic's shop, letting you chill out, drink up, play some Ping Pong, and get your dance on all in one go. The music changes often, with skilled DJs spinning away nightly and a dance floor just big enough to let loose with your friends.
Daily 17:00-late, Király Utca 46, Jewish Quarter, +36 30 755 3527, kuplung.net, Metro: Opera
##### **Ellato Kert**
While this is one of the more low-key ruin pubs, the crowds keep coming back for the foosball tables and the amazing tacos sold just inside to the left. While the other ruin pubs have clearly seen the money to be made off the tourist obsession with edgy-yet-clean-and-bright ruin bars, Ellato hasn't quite got the memo, making me think it's one of the more authentic ruin bars on Kazinczy.
Mon-Wed 17:00-02:00, Thurs-Fri 17:00-late, Sat 18:00-late, Sun 18:00-02:00, 48 Kazinczy Utca, Jewish Quarter, +36 20 527 3018, Metro: Opera
##### **Anker't**
Anker't is a new bar and club that reminds me more of the Meatpacking District, NYC: exposed brick, warehouse feel, large bars, and an excited the-night-is-young vibe. The wide open layout makes socializing easy. This is my pick for a fun place to pregame the busier ruin pubs and clubs just around the corner, like Fogashaz and Szimpla.
Mon-Sun 16:00-03:00, 33 Paulay Ede Utca, Andrássy District, +36 30 360 3389, ankerklub.hu, Metro: Opera
#### **NIGHTCLUBS**
A night out in Budapest starts early at the bars and ruin pubs and runs late at the clubs. My favorites will leave you looking forward to soaking away your hangover in the spas the next day.
##### **Corvinteto**
Corvinteto feels like a '90s life-is-plastic rooftop dance bar in an old, gray communist-style building just outside the Jewish Quarter. Finding the place is a bit tricky, because it's hidden down a side street and up a cargo elevator, but it's well worth the trip if you can figure it out. The guys/gals ratio often skews toward gentleman-heavy, but if the scenery inside doesn't match your tastes, at least you've got a panorama of the city outside to enjoy.
Cover varies, beers from 500Ft, Blaha Lujza Tér 1-2, Jewish Quarter, corvinteto.hu, Metro: Blaha Lujza Tér
##### **360 Bar**
This rooftop club is on Andrássy between Oktogon and the opera house. While the staff and cocktails come with a bit of an attitude, the views over all of downtown Budapest are the best in town, and there's an unobstructed view of the Parliament dome. Go on a warm summer afternoon or for the sunset and enjoy a beer or cocktail. By wintertime, it gets too cold to enjoy.
No cover, cocktails from 1,750Ft, beer from 500Ft, Sun-Wed 14:00-24:00, Thurs-Sat 14:00-03:00, Andrássy Út 39, Andrássy District, 360bar.hu, Metro: Oktogon
##### **A38**
Always wanted to party on a boat? Step onto this Soviet Ukrainian ship-turned-nightclub that's docked on the Danube, just south of Gellért Hill, and rock the night away on their three different dance floors. Heralded by guidebooks as the best club in the world in 2012, A38 is well worth a visit.
Entry usually free, beers from 450Ft, Buda side of Petofi Bridge, Gellért Hill, a38.hu
#### **PARTIES YOU'LL ONLY FIND IN BUDAPEST**
##### **Spa Parties**
Someone once had the brilliant idea of bringing in strobe lights, lasers, a DJ, sound system, wet bar, and smoke machines into the bathhouses of Budapest for a massive, raging spa party, and "sparties" were born. They took off in popularity, and each sparty is a guaranteed wild time. To look past the rampant hooking up and floating condoms, you'll need to be well lubricated yourself with as much Jägermeister as you can keep down. If that doesn't sound like your thing, steer clear! Sparties happen often during high season and at different bathhouses, so be sure to ask at your hostel about your options. Book tickets online at sites like Széchenyispabaths.com/sparties, at the ticket office on Szabadsag Tér, or through your hostel when you arrive. Tickets run about €35 or 10,000Ft.
##### **Shipwrecked Boat Party**
My friends at Budapest Boat Party put on an aquatic rager every Monday, Thursday, and Friday. Rock out on a 2.5-hour cruise up and down the Danube with drink specials and a full-on maritime dance floor underneath the floodlit Hungarian Parliament Building and Fisherman's Bastion of the Castle District. The folks at Budapest Party Hostels take guests at all five of their locations on the Thursday cruise weekly. Book tickets online.
5,000-7,500Ft (additional 1,500Ft for optional bottle of champagne—everyone does it), meeting points vary, confirm online, embarking at 22:00 on Mon, 22:30 on Thurs, and 23:00 Fri, +36 30 908 7598, budapestboatparty.com
#### **PUB CRAWLS**
Many hostels in Budapest organize their own pub crawls. The **Budapest Party Hostels** network will never leave you lacking on your desired nightlife endeavors. Organized pub crawls help to get you oriented in the maze of streets that make up District VII.
##### **HostelCulture Backpacker Pub Crawl**
My friends at HostelCulture put together a nightly party that takes you to the hottest spots across town. Join their free daytime walking tour, get a free welcome beer and welcome shots along the way, and cap the night off in a club with the nightly pub crawl.
Every night 20:00-21:00 at the first bar, meets at Akvarium Club, Erzebet Tér 12, Leopold Town, +36 70 424 05 69, backpackerpubcrawl.com/Budapest, Metro: Vörösmarty Tér
### **TOP SHOPPING & MARKETS**
Budapest is a paradise for those looking for retro clothes and nostalgic memorabilia. And the Great Market Hall and shopping streets are can't-miss attractions. Typical Hungarian souvenirs are hand-made woven goods, ceramics, woodcarving, and pottery. Don't be afraid to haggle, and be sure to decide what the piece is worth to you before you start negotiations!
#### **SHOPPING DISTRICTS**
##### **Váci Utca**
Starting at Pest's central square, Vörösmarty Tér, and continuing down to the Great Market Hall, Váci Utca is medieval Budapest's main street. Find kitschy touristy restaurants and all the cliché goods like postcards, traditional clothing, and souvenirs in this popular pedestrian shopping district. You'll pay for the convenience and bright lights with higher prices. Look for Hungary's famous herbal liqueur, **Unicum.** And the beautiful handmade lace makes for an excellent present for loved ones back home.
Pest Town Center, Metro: Vörösmarty Tér or Fővám Tér
#### **MARKETS**
##### **Szimpla Farmers Market**
One of the coolest markets you'll ever see is the farmers market in the Szimpla ruin pub on Sunday mornings. It's a surreal experience to party late in the bar, then return the next morning and see farmers who have just brought in their harvest to sell at the market, probably arriving just minutes after you called it a night.
Free, Sun 09:00-14:00, Kazinczy Utca 14, Jewish Quarter, szimpla.hu, Metro: Astoria
##### **Great Market Hall**
Yet another beautiful building from the Hungarian millennial anniversary of 1896. Find produce and meat on the ground floor; souvenirs, a café, and cheap eats upstairs; and a large modern supermarket downstairs. Great Market Hall is on the southern border of Pest Town Center.
Free, Mon-Fri 06:00-18:00, Sat 06:00-15:00; Vámház Körút 1-3, Pest Town Center, Metro: Fővám Tér or Kálvin Tér
##### **Hold Street Market Hall (Hold Utcai Piac)**
For a more off-the-beaten-path market experience, drop into the Hold Street Market Hall. Located just behind the US Embassy, it was built in 1897 and has nearly 400 stalls for vendors selling everything from cheeses and breads to salami and paprika. The Great Market Hall may have a wider selection, but I like this market for its yet-undiscovered ambience.
Free, Mon 06:30-17:00, Tues-Fri 06:30-18:00, Sat 06:30-14:00, Hold Utca 13, Leopold Town, +36 1 353 1110, Metro: Arany János Utca
### **TOP PARKS & RECREATION**
#### **PARKS**
##### **City Park**
The largest green space in the city is northeast of town. In the park, you'll find the **Vajdahunyad Castle** (free, always open), built in—you guessed it—1896 to show off the various architectural styles found throughout the country. The original was built out of wood and canvas but was so popular that a permanent version soon took its place. You'll see styles reminiscent of Romanesque, gothic, Renaissance, and baroque periods in this one funky structure. Deeper into the park are the **Széchenyi baths** and a circus. Get to City Park by hopping on the yellow (line 1) metro and riding it to the Széchenyi Fürdo˝ stop.
Free, always open, City Park, Metro: Széchenyi Fürdő
##### **Margit Island**
This island in the Danube River is long and skinny. In the summertime and shoulder season, you'll find tons of bars, pools, and spas open to enjoy. Renting a four-seater pedal bike (about 2,500Ft/hour) is always a favorite—especially of Italian tourists. Toss a liter or three of beer into your caddy and you're ready to cruise!
Margit Island is accessible via Margit Bridge on the south side and Arpad Bridge on the north side. Both bridges are open to pedestrians.
Free, always open, Margit Island, Metro: Nyugati Pályaudvar
#### **CAVING**
##### **Barlangaszat Caving Tour**
Under the hills of Buda lies a vast network of limestone caves both discovered and undiscovered. Get ready to get down and dirty as you explore them—you need to be flexible and able to crawl through tight spaces. You'll follow your professional English-speaking caving guides several chambers deep into the earth, crawling on your hands and knees in the passageways between each stop. The casual guides like to share stories and break down the geology of how caves occur. They also watch their group to determine the difficulty of the route.
These tight tunnels are not for the faint of heart, and definitely are not for the claustrophobic, but they're highly recommended! Wear comfortable shoes and clothes. Jeans get tight when crawling around, so stretchy shorts or athletic pants are your best bet. The provided coveralls protect your clothes, but shoes will get some Hungarian clay souvenirs.
Barlangaszat is currently the only operation that has a license to take visitors down into the caves. Availability is limited and often sells out in advance. Reserve ahead via email or through your hostel on your arrival day. Do the tours run during the rain? Yes, caves are unaffected by rain outside, and the temperature stays at about 55°F year-round.
Tours meet at the bus 206 stop at Nyugati station, near the funky low-cover staircase leading up to the overpass. Guides will take you up to the caving via two buses, so you'll need four metro tickets per person. Or get there yourself by taking bus 65, which leaves from Kolosy square. Step off at the fifth bus stop, named Pál-völgyi.
5,500Ft, four-hour tours Mon, Wed, Fri, and Sat afternoons, Tues and Thurs mornings, barlangaszat.hu, info@barlangaszat.hu, Metro: Nyugati Pályaudvar
#### **ESCAPE GAMES**
I was skeptical when my friend wanted to bring me to the escape games that are taking Budapest by storm, but I've become a fan. Here's how the games work: After a briefing laced with hints, you and your friends are locked into a room. These rooms are monitored by your "puzzle master," whom you can ask for hints if you run into a roadblock during the challenge. You get about 60 minutes to solve the series of challenges and escape...or be lost forever.
Go for the themes that intrigue you the most. Prices are generally set per game, getting cheaper for each person you add. The abundant decrepit buildings across town have lent a natural setting to these exciting puzzles. Some hints from yours truly to get your head in the right place: Embrace the challenge enthusiastically; try every idea that comes up; don't stop communicating with your teammates; look everywhere in the room—up, down, over, behind, etc; be on the lookout for time-wasting decoys; and, finally, don't take it all too seriously. Have fun!
I picked the top three based on friendliness of the staff, price, value, location (all in the Jewish Quarter, near the bars), and the success of their themes—they can get pretty creative. In addition to these recommendations, I also enjoyed **Mystique Room** (Egyptian and Cathedral themes, mystiqueroom.hu), **Escape Zone** (Rescue the Captive theme, escapezone.hu), **PANIQROOM** (Saw and Lost themes, paniqszoba.hu), and **Exit Point** (Mirror House, Madness, and Rabbit Hole themes, exitpointgames.hu). There are so many popping up throughout the city that these may change in time and new ones may become more popular.
##### **Emergency Exit Escape Games**
Descend a few steps into this nondescript little complex. The friendly staff sits you around a table and preps you for the challenge to come. Read the details of their hint sheet, and do your best to get out in 60 minutes. I found both their themes—1984 Apocalypse and Scary Circus—both fun and challenging. And I can't wait to see the third game they're building out.
Two people 8,000Ft/game, up to six people 11,000Ft/game, multiple time slots available daily, online reservations required ahead of time, Nyár Utca 27, Jewish Quarter, +36 30 889 3633, e-exit.hu, Metro: Blaha Lujza Tér
##### **Team Escape**
Team Escape offers a fun, low-pressure entry-level escape game. Its single Hollywood theme is great for those who don't want to be in the dark or scared (although, what's the point of that?). The staff is welcoming and helpful, and the records for escape times are posted online. The best team got out in 41 minutes flat. See if you can beat that!
Two players 8,900Ft, 3-4 players 9,900Ft, book online ahead of time, Nagy Diófa 3, doorbell 15, Jewish Quarter, +36 20 420 5750, teamescape.hu, Metro: Blaha Lujza Tér
##### **Time Trap Escape Games**
You've got two themes to choose from: Psychopath Killer and Prison Trap. The first one is better for beginners, but it pushes your boundaries with fake blood and bones, like the _Saw_ movie series. With Prison Trap, two or four friends are blindfolded and handcuffed together and taken deep into a wine cellar. This one is much more challenging.
12,500Ft/game for 2-5 people, daily 10:00-24:00, reservations required ahead of time by phone or online, Kazinczy Utca 10, Jewish Quarter, +36 20 311 9471, idocsapda.hu/en, Metro: Astoria
### **TOP TOURS**
#### **Yellow Zebra Bike Tours**
This is an entertaining and relaxing way to get oriented with the city and see all of Budapest's top sights. Pop into the office on one of your first days to hang out in their fun office/library/tourist zone. The friendly staff is happy to hook you up with any of their walks and rides, along with a slew of other activities. Take the day tour and you'll even get a coffee and pastry break! Book online to save yourself a little cash.
5,600Ft adult/5,000Ft student, Apr-Oct daily 11:00 daytime ride, June-Aug daily 17:00 evening ride, Nov and Mar Fri-Sun daytime ride 11:00, Lázár Utca 16, Andrássy District, yellowzebrabikes.com, Metro: Opera
#### **Buda Bike**
I love the rides put together by the Buda Bike team. They have everything you might expect from a well-organized bike tour company, including an engaging Highlights tour, along with more in-depth city rides and more extensive rides in the countryside. Several members of the team and a good friend of mine keep up an excellent blog with all sorts of tips and tricks about making the most of your visit to Budapest (bebudapest.hu).
6,200Ft, group discounts available, 10:30 and 15:00 in front of St Stephen's Basilica (look for the stairs leading down to a parking garage), Szent István Tér 1, Leopold Town, confirm season and availability ahead of time, budabike@ymail.com, budabike.com, Metro: Bajcsy-Zsilinszky Út
#### **HostelCulture**
HostelCulture has put together some exciting itineraries designed specifically for backpackers, to get you familiar with the city. Come out for their free city tour of Budapest, covering highlights like Matthias Church, the Hungarian State Opera House, and more. If you dig it, check out their more specific tours about communist history and the Jewish Legacy walk. In for a party? Join their nightly pub crawl.
Free tip-based tour meets daily 10:30 and 14:00 in front of St Stephen's Basilica, Szent István Tér 1, Leopold Town, hostelculture.com/budapest-tours, Metro: Bajcsy-Zsilinszky Út
### **TOP HOSTELS**
I love Budapest for its strong and healthy backpacking hostel culture. There are a ton of excellent accommodation choices at very competitive prices. If **Airbnb** is your thing, remember how affordable accommodations _should_ be here in Budapest, and don't spend more than about €25, or 7,800Ft, per person. It's best to stay around District VII. You may first think that the area around the Parliament and the embassies would be the most central location, but it gets rather quiet and boring at night.
**Budapest Party Hostels** is a group of awesome accommodations (Grandio Party Hostel, Retox Hostel, Carpe Noctem VITAE, Carpe Noctem, and Penthouse Privates), each with a unique vibe and personality. Prepare yourself for a PG-13 welcome briefing that doesn't concern itself with any sort of political correctness.
#### **Grandio Party Hostel**
For serious partiers only! Come to this hostel if you're looking for some "interesting" life experiences, which you'll want to both brag about to your friends and hide from your parents. There are beds upstairs, a social garden bar on the ground floor, and a sloppy late-night karaoke bar in the cellars beneath.
3,000-5,000Ft, free Wi-Fi, computers, 24-hour reception, common room, bar, bicycle rental, Nagy Diófa Utca 8, Jewish Quarter, +36 20 350 7441, grandiopartyhostel.com, info@grandiopartyhostel.com, Metro: Blaha Lujza Tér
#### **Retox Hostel**
Retox is a solid all-around bet if you're into partying late into the night. You'll get some rest here as long as your sleep schedule is on par with the other guests: falling to sleep at 06:00 and waking up at 14:00. They've done an excellent job updating the bar on the ground floor, and the sister hostels come out for a good time here most nights of the week. They host a Jägerbomb train party every Sunday night, delivering a domino effect of drinking shenanigans. I don't know how they pull it off, but I'm blown away by how clean the hostel is every morning, even after the most hard-core late-night festivities.
3,500-4,000Ft, free Wi-Fi, 24-hour reception, bar, hookah, computers, Ó Utca 41, Andrássy District, +36 70 6700 386, budapestpartyhostels.com, info@retoxpartyhostel.com, Metro: Oktogon
#### **Carpe Noctem VITAE**
For those who like to have a great time, but also want the option to chill, this hostel offers a lot of different common spaces with different vibes. Head up the stairs to a multi-floor hostel with a small rooftop patio and friendly staff. Comfortable private rooms with views are available for those who appreciate their own room but don't want to miss out on the party.
3,000-4,500Ft, free Wi-Fi, computers, 24-hour reception, bar, hookah bar, free locker/lock, Erzsébet Körút 50, Jewish Quarter, +36 70 6700 382, carpenoctempenthouse.com, info@carpenoctemvitae.com, Metro: Oktogon
#### **Carpe Noctem**
The most low-key and comfy Budapest Party Hostels option boasts orthopedic beds, spacious rooms, and a quiet place to retreat to after a night on the town. It's much smaller than the other hostels, and hostel guests appreciate the more intimate ambience.
4,000Ft, free Wi-Fi, computers, 24-hour reception, free lockers/locks, common room, Szobi Utca 5, Andrássy District, +36 70 6700 384, carpenoctemoriginal.com, info@carpenoctemoriginal.com, Metro: Nyugati Pályaudvar
#### **Penthouse Privates**
Budapest Party Hostels' newest member, Penthouse Privates is a collection of five private double rooms for travelers who want accommodations to feel more like home. Relax in their comfortable common room, enjoy full access to a kitchen, and sleep well at the top of a six-floor building (no elevator). With those stairs, you'll be working off each meal and beer every time you go home!
8,500Ft, two-night minimum stay, free Wi-Fi, computer for use, on-site staff that comes and goes, Király Utca 56, Jewish Quarter, +36 70 671 2723, facebook.com/penthouseprivates, penthouseprivates@gmail.com, Metro: Opera or Oktogon
#### **Wombats Hostel**
This Austrian hostel chain has great hostels down to a science and knows exactly what every American backpacker wants: bright rooms, free Wi-Fi throughout the hostel, breakfasts included, an on-site bar, and frequent free tours and pub crawls. The BP Wombats has an excellent location, around the corner from Kazinczy Utca and across the street from the Gozsdu Court & Passage.
Beds from 3,500Ft, 24-hour reception, free Wi-Fi, 20 Király Utca, Jewish Quarter, +36 1 883 5005, wombats-hostels.com/Budapest, office@wombats-budapest.hu, Metro: Bajcsy-Zsilinszky
#### **Central Backpacking Hostel**
Located just down the way from the US Embassy and Parliament building, this hostel sits right in the middle of all the downtown sights. It's far enough from the nightlife, making it relatively quiet, but I still appreciate the hostel's fun vibe, welcoming staff, and comfortable rooms.
Beds from 3,300Ft, 24-hour reception, kitchen access, lockers and towels available for rent, hair dryers available, 15 Oktober 6 Utca, Leopold Town, +36 30 200 7184, centralbackpackking.hostel.com, Metro: Bajcsy-Zsilinszky
### **TRANSPORTATION**
#### **GETTING THERE & AWAY**
Budget airlines make frequent connections into Budapest, making this city an attractive stop on a longer trip. Overnight bus and train options also make for good connections to Vienna and Prague. Train and bus connections from Krakow can be a little more involved, often connecting through Bratislava.
##### **Plane**
Plenty of budget airlines fly into Budapest's sole airport: **Ferenc Liszt International Airport** (BUD, bud.hu). Find cheap flights by using skyscanner.net, cheapoair.com, and kayak.com. Don't buy your tickets without checking WizzAir.com, a competitive Hungarian budget airline. It is important to note that some budget airlines like RyanAir may count the Balaton regional airport as their main "Budapest" airport. Be sure to look closely to see if this is the case while researching your options. Connections from Balaton are a two-hour train connection into Budapest. This airport is on a beautiful lake but is hours away from the city you need to get to.
Connect from Ferenc Liszt International Airport into Budapest by taking **bus 200E** to the Kobanya-Kispest Metro stop. Hop on the metro and into town to find your hostel from there, validating a new metro ticket each time you transfer. Change to the red or yellow metro line at Deak Ferenc Tér if necessary. The whole journey will take about an hour and cost between two and three metro tickets, running about 900Ft total.
##### **Train**
Connections run often between Prague and Budapest (7 hours, €75 or 23,000Ft).
Budapest is served by two main train stations: **Nyugati,** in Pest, north of Leopold Town, and **Keleti,** also in Pest, south of City Park. Both have metro, tram, and bus connections immediately outside the station. From both, it's a safe 20-minute walk into the general center, where you'll find your accommodations.
##### **Bus**
There are dirt-cheap bus options in Central and Eastern Europe. To begin shopping, take a look at orangeways.com and eurolines.com for timetables and prices. A ride to Prague may take up to ten hours, and cost at least €11 (4,000Ft). What you save in money, you may pay for in time and comfort, though modern buses often have onboard Wi-Fi and comfortable leather seats. Also note if there is a required transfer for longer connections.
If leaving from Budapest by bus, _triple_ check your bus station because there are several in town. The bus terminals are located on the southeast side of town, at metro stops **Nepliget** or **Puttyus Utca.**
##### **Car**
Renting a car is not necessary to see the best sights of Budapest, though if you're touring from one city to the next, Budapest's city center is a little easier to navigate than the more medieval centers of other European cities. However, as in other cities, finding and keeping parking is expensive and a pain. The closer to the center, the more expensive the hourly rates are. When looking for parking, there are a number of park-and-ride (P+R) options near public transportation stations that will let you ditch the car for a few days for just a few euros a day. Erzebet Tér P+R is your most central yet cost effective option.
#### **GETTING AROUND**
Budapest's public transportation system is easy to use. Trams, buses, and subways are all on the same ticketing system. Buy an individual **ticket** (340Ft) from machines and vendors usually found in every station. A booklet of **10 tickets** (2,800Ft) will save you money in a weekend unless you're getting around by bike.
Ticket checkers hang out near the validation machines and will check you if you don't validate a ticket. Ticket checkers here are more numerous than in any other European city I've seen. Trying to play the "dumb tourist" card does not work here! In Budapest, tickets do not include transfers at all except inside the metro system. That means if you need to change buses, you need two tickets. If you need to go on the metro and then a bus, again, it's two tickets.
Familiarize yourself with the primary stops and stations in the city in order to better know your way around. Start with: **Deak Ferenc Tér, Oktogon,** and **Blaha Lujza Tér.**
##### **Metro**
Budapest has a simple, four-line metro system. Visitors will most likely use the **yellow** (for the opera house, House of Terror, Széchenyi baths, and City Park), **red** (connecting Buda with Keleti train station), and **blue** (coming in from the airport and Nyugati train station) lines.
##### **Bus**
As the tram and metro networks are so easy to use, buses, while fully operational, take third place.
##### **Tram**
Over 30 tram lines lace the city, but there are just a few lines for you to get familiar with. **Lines 46** and **47** originate at Deak Ferenc Tér, running on Pest's inner ring road, past Váci Utca and the Grand Market Hall across the river to Gellért and the Citadel. **Lines 4** and **6** loop Pest's middle ring road, connecting major sights like Andrássy Út and Margit Island. **Line 2** runs north-south along the Pest side of the Danube, connecting the Grand Market Hall with the Parliament building.
##### **Taxi**
Always call for a cab to get a fair rate. Never flag down a taxi in Budapest as it's likely you'll get into an unlicensed taxi and the potential for getting ripped off is much higher. My favorite companies are **Radio** (+36 377 77 77) and **City Taxi** (+36 211 11 11).
##### **Bicycle**
Budapest is quite large, and having a bike makes it much more manageable to get across town. It opens up the city in a way that walking and using public transportation just don't. Budapest is a leader in bike paths throughout the city, and cars generally respect cyclists. But because it is such a large city, you'll also come across busy boulevards and fast traffic, so cyclists should use caution always and especially avoid riding down Rakoczi Utca in Pest, as there is no bike lane and the traffic is quite heavy. I recommend wearing a helmet if you plan on riding across town. **Yellow Zebra Bike Tours** and **Buda Bike** tour companies have bike rental options as well, with discounts for tour customers.
### **DAY TRIPS**
#### **Eger & the Valley of the Sirens**
The pretty little town of Eger is famous for its 16th-century castle, but the real draw is the Valley of Sirens, a back holler in the hills of Hungary just outside of town. Local farmers and vintners have burrowed deep into the earth to create an elaborate network of wine cellars over hundreds of years. Today we can enjoy this intoxicating tasting experience, as these vineyards sell wine from each of the cellars. Pop in, strike up a conversation, have a snack, and toss back the glasses. This trip is best done over two days so you can have ample time to enjoy an afternoon out at the wine cellars. Ask at your hostel for the best option considering your time and itinerary.
Buses and trains leave often for Eger from Budapest and take about two hours. Tourist bus connections to the Valley of Sirens leave from Eger's Dobo Square. Otherwise, it's about a 20-minute walk or 10-minute taxi ride (2,000Ft) from Eger's old town square.
#### **Castles & Local Touring**
For a supremely aggressive day of exploring greater Budapest, you can pound through three towns just northwest of Budapest and see the ruins of a castle. Well worth it for expert and efficient travelers who aren't scared of asking for help or directions.
Start your day with an early morning bus out to the town of **Esztergom** (1.25 hours via bus, 450Ft, buses leave from Arpad Hid Aitobusz station) to visit the beautiful basilica perched high on a hill overlooking the town. The basilica is free to visit. The crypts below are well worth the entry fee of 200Ft.
Head back to the bus station to continue onto the village of **Visegrad** on bus 880 (450Ft). You'll have to ask your bus driver to let you know when to get out. Pop into the **Hotel Visegrad** and ask for help calling a taxi to take you all the way up to some incredible ruins of a castle overlooking the bend in the Danube. The ruins of this castle on the hill are much more accessible than just about any other castle I've ever climbed around in. Just beyond the castle is a fun summer luge where you can buy a ticket for a ride down the course in your own little toboggan on wheels (450 Ft/run, open weekdays in good weather, 11:00-16:00, +36 26 397 397, bobozas.hu).
After the castle, return downhill to town via the path leading from the castle ruins. The signs and incline are easy enough to follow as long as it hasn't rained too much recently. Catch the bus (450Ft) on to the town of **Szentendre,** where you'll have a chance to explore the beautiful baroque old town center. You'll get off the bus on the big street that passes through the town. Work your way toward the river, explore the old town, and catch dinner before continuing back to Budapest via light rail from the station (450Ft). The train station is about a 15-minute walk south of the old town, and trains leave often for Budapest's light rail stops along the Danube river. Your best stop to get off is Margit Hid, near Margit Island, and then connect back into the center across the bridge via tram 6.
### **HELP!**
#### **Tourist Information Centers**
Find the official Budapestinfo Point (daily 08:00-20:00) next door to the recommended Cserpes Tejivo Milk Bar:
Sütő Utca 2
+36 143 88 080
#### **Pickpockets & Scams**
Don't take unmarked cabs. They prey on wealthy, naïve tourists. Pickpocketing doesn't seem to be a major problem, but always have your wits about you, especially at busy bars. Anticipate your change when purchasing goods, because rather than getting a few dollars back, you'll be getting a few thousand forints, which can be overwhelming and difficult math. Don't be afraid to step aside and count your change before walking away—I've saved a substantial amount of money doing this and catching counting errors.
#### **Emergencies**
In an emergency, dial 112.
#### **Hospital**
FirstMed Center
Hattyú St 14
+36 06 1 224 9090
#### **US Embassy**
Szabadság Tér 12
+36 06 14 75 4400
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The modern and bustling capital of Ireland still retains the charm the Irish are so famous for. Dublin will welcome you with open arms, a hot tea, and a warm chat. The Irish love for culture and storytelling runs deep, with their pride for their country unmatched. The magic of the grain and barley is never far: The world-famous Jameson Distillery and Guinness Storehouse are some of the country's top sights, and a foot-tapping good time can be had for free at any of the pubs found on just about every corner.
### **DUBLIN 101**
Dublin was founded as a Viking settlement in the 9th century and called Baile Atha Cliath (City of the Hurtled Ford). The Vikings laid an extensive network of underwater defenses navigable only for those who planted them, preventing enemy ships from coming up the river to the city. After some time, this settlement picked up another name, the one we're most familiar with: Dublin. Just behind where the Dublin Castle stands today was a dark eddy that formed at the crux of two intersecting rivers ( _dubh_ means black, _linn_ means pool).
The city eventually became the second largest city of the British Empire in the 1600s and 1700s. After the devastating Great Hunger and London's oppressive colonial tactics, Dublin slid into decline throughout the 19th century, losing a full two-thirds of its population.
Seventy years later, in a glorious failure, a small band of students and intellectuals took up arms in 1916 and launched the Easter Rising, an attempt to win independence for the Republic of Ireland. In full war mode, England brought in thousands of troops and a battleship all the way up the River Liffey and began shelling the city of Dublin. The leaders of the operation took over a handful of municipal buildings and holed up for six days before all were captured and taken to Kilmainham Gaol prison. Most rebels were summarily executed by firing squad, a miscalculated move by the British, turning public opinion from general apathy to the side of the rebels.
Seeking peace, London offered a treaty of limited independence, making the Irish Free State still subject to the crown. The treaty split the opinion of the country in half. The pro-treaty side, supported by England, was content with taking a pragmatic step _toward_ sovereignty—and peace. And the anti-treaty side wouldn't stop at anything short of home rule, full autonomy, and a united Republic of Ireland. The two opposing camps began fighting, kicking off the bloody and tragic Irish Civil War (1922-1923), pitting brothers against brothers and eventually bringing about the political lines of Ireland that we see today, with six counties in the north remaining loyal to the British crown. _The Wind That Shakes the Barley_ (2006) is an excellent movie to watch to learn more about this heart-rending conflict.
During the 1990s and 2000s, Ireland entered a period of unprecedented economic growth, earning it the nickname the "Celtic Tiger." The good times didn't last though, as the economic crisis of 2008 hit the island particularly hard. But today, you'll find a city awakening once more from its slumber and charging into the 21st century with newfound confidence and optimism.
### **PLAN AHEAD**
#### **RESERVATIONS**
Reserve hostel accommodations for St Paddy's day long in advance (at least three months ahead). It's also a good idea to plan ahead for a day outside of Dublin to the Cliffs of Moher and Howth.
Reservations are recommended for the following sights:
**Guinness Storehouse** (guinness-storehouse.com)
**Jameson Distillery** (bookings.jamesonwhiskey.com)
#### **STUDENT DISCOUNTS**
Student discounts are often available at Dublin's sights. Bring your student ID with you at all times.
#### **LOCAL HAPPENINGS**
##### **St Patrick's Day**
Americans love the St Paddy's Day traditions of shamrock-themed parties, wearing green, and drinking Guinness, but it's not until you travel to the country of this famous patron saint that you experience a real Irish celebration. While traditionally more of a religious holiday than its American celebration counterpart, the event has grown in leaps and bounds over the last few years as the enterprising Irish have seized this massive economic opportunity. The holiday has gone from a one-day celebration to a five-day festival filled with parades, street concerts, lectures, cultural events, and, of course, hitting the pubs. The holiday itself occurs annually on March 17, and most businesses will be closed. On each St Patrick's Day, the president, the archbishop of Ireland, and the head of the Irish military all attend St Mary's Pro Cathedral in Dublin for High Mass at 10:00. The Mass is held in Irish, and visitors receive blessed shamrocks to wear at the parade.
**KNOW BEFORE YOU GO**
**KEY STATS & FIGURES**
**Currency:**
**Ireland uses the euro (€); 1 EUR = about 1.06 USD**
**Population:**
**530,000**
**National languages:**
**Irish, English**
**Oldest pub:**
**Brazenhead (open since 1198)**
**Famous Irishmen:**
**Bono, Oscar Wilde, Colin Farrell, James Joyce**
**National dish:**
**Irish beef and Guinness stew**
**CALIBRATE YOUR BUDGET**
**TYPICAL PRICES FOR:**
**Hostel dorm bed:**
**€16**
**Two-course dinner:**
**€14**
**Pint of beer:**
**€5**
**Bicycle rental:**
**€12/day**
**Single bus ticket:**
**€1.60**
**MOVIES TO WATCH**
_P.S. I Love You, Once, The Wind That Shakes the Barley_
**THREE DAY ITINERARY**
The city of Dublin is easy to walk across. This itinerary takes you to the top sights but leaves lots of time to explore the city's one-of-a-kind pub culture.
**DAY 1: THANK GOODNESS FOR GUINNESS**
**MORNING**
Start your day with a delicious, fresh pastry breakfast at the **Queen of Tarts** just off Temple Bar for a yummy Irish scone and jam. If you're hangry, the scrambled eggs with toast is superb.
After breakfast, walk a few blocks east down Dame Street to catch a free walking tour of the heart of Dublin with **Hostelculture,** meeting at 11:00 daily at **The Mercantile** pub. Pop in early for the free Wi-Fi and to get warm with a tea. The tour lasts about 3.5 hours and does an excellent job of orienting you to the city and culture.
**AFTERNOON**
If it's a nice day, walk to the **Guinness Storehouse,** about a 20-minute westerly walk from the city center. Otherwise, take one of the many city buses that run there from the center (123, 13, 40), west along Dame Street. Bus fare is €1.60.
Spend the afternoon in the storehouse, climbing up through the labyrinth of brewing science and delightful Guinness propaganda. Your entry ticket is also a voucher for a free pint, which you can redeem at numerous stations throughout the tour or at the top-floor Gravity Bar (last call at 17:00!), where you can enjoy a full 360-degree panorama while you sip away.
**EVENING**
Stop on your way back into town for a traditional Irish dinner at the city's oldest pub, **Brazenhead** , or at one of my favorite gastro pubs, **The Bull & Castle**. Both restaurants are on the Guinness Storehouse side of downtown, making for a convenient pit stop on the short walk back.
**LATE**
Commence the evening festivities by creating your own pub crawl from my recommended hot spots, starting at the famous **Temple Bar** and making your way to the nearby **Palace** pub, **O'Neill's,** and **The Porterhouse,** where the party goes late every night.
**DAY 2: TRINITY COLLEGE & HOWTH**
**MORNING**
Let yourself sleep in a bit to recover, and grab a hearty breakfast at **The Bakehouse**. Their strong coffee is a lifesaver. Consider taking one of their pastries or sandwiches for the road to keep you moving through lunch.
Cross the Liffey to meet at the front arch of **Trinity College** for a tour led by current students. Tours last about 30 minutes and leave about every half hour, with the first one of the day departing at 10:15 and the last one at 15:40 May through September. The €10 tour includes entry to see the **Book of Kells.**
**AFTERNOON**
Hop on the DART train at Tara Street station, just a block east from the O'Connell Street bridge, to head out to the coastal town of **Howth.** Spend the afternoon hiking, then relax for a bit at Howth's **Bloody Stream** pub. The trip to Howth takes about 40 minutes each way. The small town only takes 45 minutes to explore, and the hike takes about 1.5 hours.
**EVENING**
Once you're back in Dublin, head out for dinner. Try succulent pulled pork sandwiches at **South William Bar** , steaming tapas at **The Market Bar** , or burritos, Thai, sushi and more just north of the Millennium Bridge at **Millennium Walkway Restaurant Row.**
Alternatively, reserve ahead for the **Celtic Nights Dance Show & Dinner** at the Arlington Hotel, kicking off nightly at 20:30. Tap your feet along to their jigs as you sip a pint and enjoy a €34, three-course dinner.
**LATE**
Skip the touristy joints tonight and head out where the party always rages and the locals go till the wee hours. Walk south from Dame Street on South Great Georges Street for about 15 minutes to kick off with a couple pints at **Whelan's,** where _P.S. I Love You_ was filmed. You may catch some live music there, as acts often start by 20:30. Continue warming up at **Flannery's** , and once you're in the dancing mood head to **Copper Face Jacks** and aim to close it down. All three venues are within easy walking distance of one another.
**DAY 3: FAREWELL TO DUBLIN**
**MORNING**
Break your fast at **Brick Alley Café,** located right on Temple Bar, for some coffee and pastries. Coffee and ice cream is a great combo for calming the stomach after a night out.
Walk over to spend a lazy morning exploring the **Grafton Street** shopping zone. Perhaps have a picnic snack in **St Stephen's Green**. There are a few top-class and free museums in the area: the **National Gallery** , the **National Museum of Archaeology,** and the **National Library** (closed Sun). Of the three, the National Gallery is my favorite, with an all-star cast of artists shown here, including Van Gogh, Rembrandt, and Caravaggio.
**AFTERNOON**
Take bus 37 or 39 from the southeast corner of Trinity College's campus toward the **Jameson Distillery** , getting off at Ushers Quay. Make sure to raise your hand when they ask for volunteers to join the tasting!
**EVENING**
For your last evening in town, grab some delicious beef and Guinness stew at the **Irish Film Institute** just around the corner from Temple Bar.
**LATE**
Make a crawl in the lane behind the **Mercantile Pub**. Start off with **Odessa** , continue onto **The Bar with No Name** and finish at **Sweeney's Bar.**
### **TOP NEIGHBORHOODS**
The River Liffey runs straight through Dublin, cutting the city into north and south. The **South Bank** is home to most of Dublin's cultural and tourist sights, including Trinity College, the Grafton Street shopping district, the national museums, the Guinness Storehouse, and the **Temple Bar** neighborhood. The Temple Bar neighborhood is bordered by the River Liffey to the north, Dame Street to the south, Fishamble Street on the west, and Westmoreland Street to the east. It's world famous for its full lineup of more than two dozen bars, with most offering live music nearly every night. It's the most touristy part of Dublin, and pints here are the most expensive, running up to €7.
When it comes to the **North Side** of Dublin, most worthwhile sights, restaurants, and accommodations are within a few blocks of the river. You'll find a few recommended hostels, restaurants, and nightlife venues here, as well as the Jameson Distillery Tour and the grand boulevard of O'Connell Street.
### **TOP SIGHTS**
#### **Trinity College & Book of Kells**
Ireland's oldest university was founded in 1592 by Queen Elizabeth I. It's still active today, and its claim to fame is the beautifully preserved "bog book," the Book of Kells, found inside the Trinity Old Library. This unique illuminated manuscript is one of the oldest in existence, and it illustrates each of the four gospels in beautiful, intricate Celtic weaves. A campus tour (€13, 30 minutes) includes entry into the library and viewing of the Book. Tours meet at Front Arch and depart about every 30 minutes. Entry to see the book without the student tour is €10.
Free, tour €13, May-Sept daily 10:15-15:40, Feb-April and Oct-Nov Sat-Sun only, no tours Dec-Jan, Book of Kells without tour €10, June-Sept Mon-Sat 09:00-18:00, Sun 09:30-18:00, Oct-May Mon-Sat 09:30-17:00, Sun 12:00-16:30, College Green, South Bank, tcd.ie
#### **Guinness Storehouse**
Tour the Guinness Storehouse to learn about the history and brewing process of one of the most recognizable brands of beer in the world. Follow an interactive virtual tour led by the grand master brewer himself, Fergal Murray. He first explains the four ingredients that go into the mix: water, hops, barley, and yeast. Continue upstairs to discover the tasting experience and a world of Guinness media and marketing materials (my favorites are the "Guinness Is Good for You" posters). Your entry ticket is also a voucher for a free pint of the black stuff once you reach the Gravity Bar at the top level. Sip it in while taking in Dublin's best 360-degree panorama. Alternatively, you can opt to redeem your drink voucher at one of the pour-your-own-pint stations on the third and fourth levels. You can also redeem your voucher for a soda or juice.
The storehouse is south of the River Liffey, a 20-minute walk from Dublin's city center. You can also take bus 123, 13, or 40, all of which run to Guinness Storehouse from the city center, west along Dame Street.
€16 for student over 18, €20 adult, €18 online, daily 09:30-17:00, July-Aug until 19:00, St James Gate, Greater Dublin, +353 (0)1 408 4800, guinness-storehouse.com
#### **Chester Beatty Library**
Chester Beatty was born in the United States with Irish heritage and became a powerful mining magnate. He began collecting as a young man and developed an impressive collection of everything from bottles to manuscripts, including pieces of New Testament manuscripts dating back to before AD 200. He moved to Dublin in 1950 and was made an honorary citizen of Ireland by 1957, and he donated his entire collection to the state when he died in 1968. It's easy to spend two hours poring over the extensive collection, which features artifacts of all major world religions in the world, like early biblical manuscripts nearly 2,000 years old.
Nearby, you'll see **Dublin Castle**. You can tour the staterooms, but I wouldn't say they're worth the time on a short visit. The castle stood as a sign of oppression throughout history, occupied by whoever was controlling the local people. Today, it's where important state functions are held.
Free, Mon-Fri 10:00-17:00, Sat 11:00-17:00, Sun 13:00-17:00, closed Mon Oct-Apr, just behind Dublin Castle, South Bank, +353 (0)1 407 0750, cbl.ie
#### **National Library**
This library caters to those interested in Ireland's long roster of literary greats. You'll find in-depth exhibits of famous Irish authors like W B Yeats and James Joyce. The National Library is also charged with the collection and preservation of important Irish documents and records. Visitors can dig into some genetic lineage research via the extensive Irish parish registries, recording births, baptisms, marriages, and deaths. The library is worth a short visit to see the building, the neoclassical interior, and the handful of exhibits. It's possible to spend months here, researching to your heart's content.
Free, Mon-Wed 09:30-19:30, Thurs-Fri 09:30-16:30, Sat 09:30-12:30, Sun 13:00-16:30 (exhibits only), Kildare St, South Bank, +353 (0)1 603 0200, nli.ie
#### **National Museum: Decorative Arts & History**
This moderately sized museum is worth an hour. Housed in an old military barracks, it contains artifacts, art, historical pieces, and natural history from all throughout the ages, with a fascinating military history exhibit to boot. The museum displays a selection of objects that tell the story of Ireland, from prehistoric coins to a "decommissioned AK-47." Find it just north of the River Liffey, not far from the Jameson Distillery, Guinness Storehouse, and Phoenix Park. You could easily make a day of it by visiting all four sights together. To get here, take bus 25A or 25B to the Guinness Store stop.
Free, Tues-Sat 10:00-17:00, Sun 14:00-17:00, Blenburb St, Greater Dublin, +353 (0)1 677 7444, museum.ie
#### **National Museum of Archaeology**
This interesting state-run museum features an extensive history of Dublin's Nordic ancestry on one main ground floor and a smaller upper floor. Take an hour to check out all the gold jewelry and fearsome-looking Viking weapons. Don't miss the Viking ship and bog books and mummies that have been discovered in recent years—the damp, grassy bogs preserve anything that falls into them.
Free, Tues-Sat 10:00-17:00, Sun 14:00-17:00, Kildare St, South Bank, +353 (0)1 677 7444, museum.ie
#### **The National Gallery**
This recently renovated art museum features a plethora of Irish masterpieces, as well as works of Monet, Caravaggio, Van Gogh, Titian, Poussin, and Rembrandt. I consider its one and a half floors of exhibits the perfect-sized bite of artistic culture to experience if feeling the effects of the night before.
Free, Mon-Sat 09:30-17:30, Thurs until 20:30, Sun 11:00-17:30, Merrion Square West Dublin, South Bank, +353 (0)1 661 5133, nationalgallery.ie
#### **Jameson Distillery**
If you're a whiskey lover, this is the place for you. While they don't actually distill the whiskey here anymore, the guided tour is a great way to learn the trade and sample some traditional Irish whiskey. It walks you through the distilling process and sheds some light on those sneaky angels taking their sharing during the aging process. Be sure to raise your hand when asked for volunteers to get in on a taste test—you won't be disappointed! The distillery is north of the River Liffey, on the western side of Dublin near Smithfield and just behind the Generator Hostel. To get here, take any bus going west along the river, alighting at Ushers Quay.
€15, €13.50 online booking, daily 09:30-18:30, last tour at 17:15, Bow St, Smithfield Village, Greater Dublin, +353 (0)1 807 2355, tours.jamesonwhiskey.com
#### **Kilmainham Gaol**
Dublin's active jail from 1796 until 1924, Kilmainham Gaol is on the far west side of town and worth the hike out to connect with the significant link of Irish nationalism—just about all leaders of the numerous Irish rebellions have been held in these cells at one point in time. The leaders of the Easter Rising in 1916 were brought here and summarily executed by firing squad, with one exception: Éamon de Valera was spared thanks to his American citizenship. The jail itself was supposed to be an excellent architectural model of modern prisons, featuring a layout that maximized the number of prisoners that could be "humanely" kept, and minimizing the guards required to monitor all of them. Explore the small prison museum while waiting for your tour to kick off at the top of each hour. To get here, take bus 25A or 25B to Con Colbert Road, doubling back around the corner to the jail once you get off.
€7, entry price includes enjoyable hour-long tour of the complex, Apr-Sept daily 09:30-18:00, Oct-Mar Mon-Sat 09:30-17:30, Sun 10:00-18:00, last admission an hour before closing, Inchicore Rd, Greater Dublin, +353 (0)1 453 5984, heritageireland.ie/en/kilmainhamgaol
#### **Celtic Nights Dance Show & Dinner**
A Riverdance-style show that reflects Irish tradition and culture is a highlight for many visitors to Dublin. Fast-paced, costumed tap dancers stun audiences nightly at the famous Arlington Hotel dinner and dance show. To see the show, you now have to purchase a tasty, but pricey, three-course dinner. You'll be treated to numerous group and solo acts by dancers and musicians. Try not to let your jaw rest on the floor for too long as the performers pound through a litany of traditional and more modern Irish dance. Make—or avoid—eye contact with the dancers depending on whether or not you want to get dragged up to the stage and taught the basics for a song in front of the entire audience.
€34, hour-long show starts at 20:30, dinner seatings every half hour from 18:30, reservations recommended online, 23 Bachelors Walk, North Side, +353 (0)1 687 5200, arlington.ie
### **TOP EATS**
Ireland has undertaken a transformative culinary renaissance, and I'm loving it! What used to be a country notorious for its bland food, and not much to eat besides potatoes, is now bursting with excellent restaurants, gastropubs, and cafés that take their trade seriously.
When it comes to traditional Irish food you've got a few staples. Fish-and-chips, of course, is a mainstay, but beef and Guinness stew (big cuts of meat and veggies in a thick, spiced stew), boxty dishes (savory potato pancakes usually wrapped around meat or veggies), and shepherd's pie (a filling of minced meat and veggies topped with mashed potatoes, usually served in a large ceramic bowl) are the comfort foods of Ireland. Look for them on the menu at any of the sit-down restaurants listed. Tip about 10 percent on your dinner tab.
#### **Queen of Tarts**
Queen of Tarts is probably the cutest little place in the city. A breakfast here feels like a drop into Alice's rabbit hole, and you're guaranteed one of the best biscuits you've ever had. They also have great egg dishes for a bigger brunch. Find another branch barely a block away at (4 Dame St).
Scones from €3, Mon-Fri 08:00-19:00, Sat 09:00-19:00, Sun 10:00-18:00, Cows Lane, +353 (0)1 633 4681, queenoftarts.ie
#### **Avoca**
Tucked in the basement of a housewares-slash-fashion store, this is a place that all divas are sure to love. The main attraction is down the staircase, though, where you can pop in and enjoy a wide array of fresh salads, super food wraps, and fresh sandwiches.
Plates and dishes from €4, Mon-Sat 09:30-18:00, Sun 11:00-18:00, 11-13 Suffolk St (just steps from the _Molly Malone_ statue), South Bank, +353 (0)1 677 4215, avoca.ie/home
#### **Epicurean Food Hall**
If you're looking for a break from traditional Irish food, this is your one-stop shop for an international array of culinary choices. Stop in the food hall for everything from Italian sandwiches to a bowl of curry, but don't come expecting super-fresh food or super value: All food stalls have colluded to price their small plates at around €9 and their buffet plates at €10. Do a full lap of all your options and a close inspection of the food before you commit.
From €9, Mon-Sat 09:30-20:00, Sun 11:30-19:30, Lower Liffey St, North Side, +353 (0)1 878 8641
#### **The Bakehouse**
I love this funky 1960s-themed breakfast and lunch house, where they bake their own bread and pastries and compile delicious sandwiches throughout the day. I always go for the sandwiches on thickly sliced bread, but they've also got pies, salads, baked potatoes, and a whole range of filling breakfast dishes.
Breakfasts from €5, daily 10:00-18:00, 6 Bachelor's Walk, North Side, +353 (0)1 873 4279
#### **The Bakery**
Situated down a side street in Temple Bar, this place is an actual legit bakery, supplying many other pastry shops around town. Some of the freshest sandwiches I have ever had came from here, and make for a great picnic on your day trip out to Howth, or even for a quick lunch stop. Complete with a friendly staff, the Bakery is also friendly on the wallet.
Sandwiches from €4, Mon-Sat 08:00-17:00, Pudding Row 3, Temple Bar, +353 (0)1 672 9882
#### **The Market Bar**
This Spanish fusion restaurant in an expansive downtown warehouse has some of the best tapas this side of the Irish Sea. The _patatas bravas_ (fried potatoes with a spicy ketchup sauce), chorizo salad, and chicken skewers are some of my favorites. The portions are generous and great to share, drawing families, young professionals, posh locals, and tourists alike to the Market Bar's renovated shabby chic interior. Make it before 19:00 to get in on their happy hour, which includes food and drink specials.
Dinner from €9, Mon-Thurs 10:30-23:30, Fri-Sat 10:30-00:30, Sun 12:00-21:00, 14 Fade St, South Bank, +353 (0)1 613 9094, marketbar.ie
#### **The Bull & Castle**
This is my favorite spot in town for high-quality pub grub. The steak sandwiches in this double-level restaurant are to die for. There's a fine dining restaurant downstairs, though I prefer the more casual bar upstairs, especially when a game is on. Though if a big game is on, you won't be finding a spot to sit, as it packs out with rowdy supporters.
Dinners from €11, daily 12:00-22:00, 5-7 Lord Edward St, Temple Bar, +353 (0)1 475 1122, bull-and-castle.fxbuckley.ie
#### **Leo Burdock Fish & Chips**
You've gotta have some Irish fish-and-chips while in town, and this is the place to go. No frills, great prices, and top-class fried goodness. Leo's is widely regarded as the best fish-and-chips joint in town, and who am I to disagree?
Plates from €7.50, daily 12:00-24:00, 2 Weburgh St, South Bank, +353 (0)1 454 0306, leoburdock.com
#### **The Fumbally**
A good example of the Irish food renaissance, the Fumbally ramps up the tastiness without the price gouging. Located about a 15-minute walk south of Temple Bar, it's a favorite for power lunchers and those who appreciate lean, fresh food like healthy sandwiches, freshly squeezed juices, and a wide array of salads and soups. Don't miss the pork sandwich, artfully assembled by bearded, aproned Irish hipsters.
Sandwiches from €5, Mon-Fri 08:00-17:00, Sat 10:00-17:00, Fumbally Ln, South Bank, +353 (0)1 529 8732, thefumbally.ie
#### **South William Bar**
This gastrobar offers a deceivingly simple menu of pork sandwiches, chicken wings, veggie nachos, and hot dogs. Take your pick of tasty fries with a selection of seasonings, like curry, garlic, and cheese; top them off with lime-, chili-, or rosemary-flavored salts; and dip them in sauces ranging from barbecue and chipotle to garlic and honey mustard. With so many choices, you might go mad, but it's well worth the risk. South William is a favorite spot to kick off the night, as you'll find a wide selection of trendy bars and venues in the immediate area.
Dinners from €8.50, Sun-Thurs 12:00-23:00, Fri-Sat 12:00-02:00, 52 S William St, South Bank, +353 (0)1 677 7007, southwilliam.ie
#### **Irish Film Institute**
Most famous for its festivals, the Irish Film Institute also contains a little hidden gem of a restaurant. The food isn't all that cheap, but it's not overpriced either, and this is a great spot to hit up when you don't feel like venturing too far from Temple Bar. The service is friendly and fast, and the menu features classics like beef and Guinness stew, fish-and-chips, and well-appointed burgers.
Dinner from €10, Mon-Thurs 11:00-22:30, Fri 11:00-23:00, Sat-Sun 12:00-23:00, 6 Eustace St, Temple Bar, +353 (0)1 679 3477, ifi.ie
#### **Brick Alley Café**
Just steps from the Temple Bar, the Brick Alley Café offers surprising value and quality in such a touristy district. This casual, cozy café offers strong coffee, hearty breakfast sandwiches on fresh baguettes, pancakes with maple syrup, and sugary treats like artisan hot chocolate and rich chocolate cake all day to keep you going. On nice days, grab one of the seats outside and take in the fun people-watching on the street. The service can be slow, but the great location and free Wi-Fi make for a clutch lunch stop. Lunch, dinner, and full vegetarian and gluten-free meals are available, too.
Pancakes from €4.90, Sun-Wed 09:00-22:00, Thurs 09:00-22:30, Fri-Sat 09:00-23:00, 25 East Essex St, Temple Bar, facebook.com/BrickAlleyCafe,+353 (0)1 679 3393
#### **Tolteca Burritos**
Chipotle wannabes are popping up all across Dublin, but I'm not complaining. Come out for some fresh Irish shredded beef or pork burritos. Tolteca is packed out at lunchtime with Trinity College students who appreciate the quality of ingredients as well as the prices. The line can get long, but it moves quickly, and the large space offers ample seating for the hungry crowds.
Burritos from €7, daily 11:00-22:00, till 23:00 on weekends, 21 Suffolk St, South Bank, +353 (0)1 677 9506, tolteca.ie
#### **Murphy's Ice Cream**
Ireland's most famous creamery, hailing all the way from Dingle, is taking the island by storm and has opened a second location in Dublin. Innovative flavors and quality ingredients make this some of the best ice cream you'll try on your trip. Go for the Caramelized Brown Bread, Spicy Cinnamon, or perhaps the Irish Coffee? Take me back!
Cups from €2.50, Mon-Sat 10:00-18:00, 27 Wicklow St, South Bank, +353 (0)86 031 0726, murphysicecream.ie
#### **Cornucopia Wholefood & Vegetarian**
Vegetarians, look no further than Cornucopia. I even find myself heading back for their vegetarian rice and curry dishes, but they've got much more, like enchiladas, salads, and bean dishes, all of which you can dish up just walking down the buffet line. Heads-up: The lines can get painfully long around lunch, when it feels like just about every young professional in town heads here for their generous portions and fair prices.
Dishes starting at €9, Mon-Tues 08:30-21:00, Wed-Sat 08:30-20:15, Sun 12:00-21:00, 19-20 Wicklow St, South Bank, +353 (0)1 677 7583, cornucopia.ie
#### **Umi Falafel**
I normally avoid recommending chains but am happy to make exceptions when they hit the spot! Umi Falafel is great for a quick, potentially healthy bite depending on your order. Find it right on Dame Street and head in for fresh falafel (of course) and also a wide range of Moroccan and Middle Eastern salads. It's a smoothly run operation: Order at the front and grab seats toward the back at this clean and bright fast-food joint.
Lunches from €7.50, daily 12:00-22:00, 13 Dame St, Temple Bar, +353 (0)1 670 6866, umifalafel.ie
#### **Brother Hubbard**
If you're on a diet, stay away from this place. This trendy Moroccan fusion restaurant has all you want for breakfast, like cinnamon rolls, brownies, delicious hot chocolate and coffee, and amazing breakfast dishes like eggs over beans with chorizo. I heard about their pork sandwiches and nearly fell off the chair when biting into the perfectly toasted bread. Their €30 set dinner menu takes you on a three-course culinary adventure through Morocco, featuring _kofta_ (meatballs), spicy tagine (stew), couscous dishes, and more.
Brunch from €10, Mon-Fri 08:00-17:30, Sat-Sun 10:00-17:00, 153 Capel St, North Side, +353 (0)1 441 1112, brotherhubbard.ie
#### **Hophouse**
Head to the north side of O'Connell Street for a cozy little Asian spot with welcoming service and a friendly atmosphere. If you've never had either, try both the _bulgogi_ (spiced and marinated beef grilled to perfection) and _bibimbap_ (a festival of veggies and a meat on a bed of rice or noodles, topped with an egg) and prepare to get your socks knocked off.
Dinners from €13, Mon-Thurs 12:00-23:00, Fri-Sat 12:00-23:30, Sun 12:00-22:30,160 Parnell St, North Side, +353 (0)1 872 8318, hophouse.ie
#### **Madina Desi Curry Co**
This is my favorite spot for Indian food in town. Most people don't stick around to eat in the modest restaurant, opting to take out their delicious budget meal. Get a plate of rice, tikka masala, and naan and you're good to go!
Curries from €8, daily 11:00-23:00, 60 Mary St, North Side, +353 (0)1 872 6007, madina.ie
#### **Oxmantown**
The best things in life are simple. So for lunch, what's better than a classically delicious sandwich and soup? Loved by locals, this place feels like it's about to get big, but just hasn't been discovered yet. I always go for the Ruby—thick pastrami and sauerkraut and a bit of spicy mustard. And the tomato soup warms you from the inside out. Limited seating inside this clean café encourages most diners to take the party outside, especially on nice days.
Sandwiches €5.50, Mon-Fri 07:30-16:00, 16 Mary's Abbey, North Side, +353 (0)1 804 7030, oxmantown.com
#### **Bloom Lane & Millennium Walkway Restaurant Row**
Just leading off to the north from the Millennium Bridge, you'll find a wonderful little lane with nearly a dozen choices for everything from a quick bite to a fancy sit-down dinner. Consider the following options: **Bar Italia,** a mod Italian restaurant and wine bar; **Enoteca Delle Langhe,** another Italian choice with wine and standard fare; **Cactus Jack's,** a sit-down Mexican restaurant with tapas options; **Koh Restaurant & Cocktail Bar,** a classy sit-in Asian fusion restaurant with discerning cocktail menu and cheaper takeaway option; and **Boojum,** with delicious Chipotle-style Tex-Mex burritos.
North Side
### **TOP NIGHTLIFE**
Dublin is famous for its characteristic and upbeat nightlife, running the gamut from typical pubs and live music to rock bars and cocktail lounges to dance bars and clubs. The tourists flock to Temple Bar, while the locals avoid it, preferring to avoid the crowds and expensive pints—they prefer to enjoy a night out in the area south of Dame Street and at the pubs sprinkled throughout the city.
#### **NIGHTLIFE DISTRICTS**
The thing about Dublin nightlife is that you really can't go wrong no matter where you end up. There are so many pubs and bars around town that you'll often find the best parties simply by exploring on your own. There are a few things that you can count on in Dublin though, and they tend to break down by district.
##### **Temple Bar**
First off, Temple Bar is both a neighborhood in Dublin and an actual bar. "Bar" referred to the riverbank that used to come all the way to the street we call Temple Bar now, and the wealthy Temple merchant family owned this stretch of riverbank at one time. Here you'll find touristy bars and €6.50 pints, climbing north as the clock strikes 23:00, pushing prices up to a whopping €7.50 per pint. The farther you walk from Temple Bar, the more the price of pints drops—about 10 cents per block. Farther out you can find local hangouts with substantially cheaper pints of Guinness.
Temple Bar
##### **Dame Street & Dame Lane**
Dame Street marks the southern border of Temple Bar. Just south of Dame Street is Dame Lane, a small back alley with half a dozen bars and pubs. This is a more local scene than what you'll find in the heart of Temple Bar. Continue south on Drury Street and South Williams Street to find classier, upscale bars, cafés, and restaurants.
South Bank
**LGBT DUBLIN**
While Ireland leans a bit socially conservative, the capital city of Dublin has had a gay scene that goes back for years. You'll find that hostels and restaurants are social and welcoming. For a good time, head to **The George** (89 S George St, +353 14 78 2983), Dublin's oldest gay pub, which draws a young and rowdy crowd. Drag shows and great music keep the party going late just about every night of the week! Gay Pride is towards the end of June each year (dublinpride.ie).
#### **IRISH PUBS**
A radio show once had a contest to see if anyone could walk across Dublin without passing by a pub. The winner called in and said "Any route works—you just have to go into each one!" Authentic Irish pubs are warm, cozy affairs, with amateur pickup bands playing traditional Irish music nearly every night. Head to the bar to grab your pint, and pull up a stool to strike up a conversation with your fellow imbibers.
##### **The Temple Bar**
If you're wondering what came first, the street or the bar, the answer is the street. An enterprising entrepreneur started a bar named after the street, and when the street rose in popularity as a destination for tourists the world over, it catapulted this corner bar to the stars. It's standing room only just about every night in the summer; come for the experience, because you aren't saving any money here. There's live music and a smoking patio out back. Check the website for scheduled music sessions.
Pints €7.50, 48 Temple Bar, Temple Bar, +353 (0)1 672 5286, thetemplebarpub.com
##### **The Celt**
The Celt's claim to fame is that it's one of the most authentically Irish pubs in Dublin, complete with a fireplace, Irish Republican flags, and lively trad sessions. Located a couple blocks east of O'Connell Street, this gem is safely tucked away from the tourist hordes.
Pints from €4.50, 81 Talbot St, North Side, +353 (0)1 878 8655, thecelt.ie
##### **The Palace**
Located at the end of Temple Bar, the Palace has always had a reputation as a literary pub. Its proximity to the _Irish Times_ newspaper office made it a favorite of the likes of writers Patrick Kavanagh and Flann O'Brien. Nondescript and full of old-time locals, this is one of your more authentic options in Temple Bar.
Pints from €5, 21 Fleet St, Temple Bar, +353 (0)1 671 7388, thepalacebardublin.com
##### **Brazenhead**
This authentic Irish pub with stone walls claims to be the oldest pub in the city, dating all the way back to 1198, and I believe it. Stop by for the nightly live music, and also for a cozy, candlelit dinner in the evening. With a full dinner menu, the Brazenhead makes for a convenient stop on the way back from the Guinness Storehouse. The beef and Guinness stew will keep you full for days! Don't forget to check out the hundreds of badges left by police officers and firefighters over the years, tacked onto the walls.
Pints from €5, 20 Lower Bridge St, South Bank, +353 (0)1 677 9549, brazenhead.com
##### **The Stag's Head**
The Stag's Head is right behind Sweeney's Bar and the Mercantile. Head upstairs and find a spot on their comfy red leather couch. Make some friends in this social and inviting pub.
Pints from €5, daily 10:30-01:00, 1 Dame Court, South Bank, +353 (0)1 679 3687
##### **Cobblestone Pub**
Widely regarded for having the best trad sessions in the city, the Cobblestone is your classic Irish pub with no-frills wooden interior and cozy, welcoming atmosphere. It draws local musicians and eager tourists. While a bit out of the way, and north of the river, it is in close proximity to the Jameson Distillery and the Generator Hostel, so keep it in mind if you're out that direction. Trad sessions don't usually kick off till around 20:00 or later. To get here, take a tram to the Smithfield stop.
Drinks from €5, Mon-Thurs 16:00-23:30, Fri-Sat 16:00-00:30, Sun 13:00-23:00, 77 King St N, Greater Dublin, +353 (0)1 872 1799
##### **O'Neill's Bar and Restaurant**
A few classic pubs in Dublin just feel like home as soon as you step inside. Come out for a pint, pub grub, or just some tea and head upstairs to find one of O'Neill's many cozy _snugs—_ tiny walled-off booths typical to Irish pubs. Snugs were originally developed for women to enjoy their drinks in peace.
Pints from €5, daily 10:30-23:30, 2 Suffolk St, South Bank, +353 (0)1 679 3656, oneillsbar.com
#### **BARS**
##### **The Mercantile**
This hotel, restaurant, and bar right on the main drag of Dublin, Dame Street, is the meeting point for daily walking tours and nightly pub crawls put on by Hostelculture. The interior is richly decorated with textured ceilings and ornate ironwork, with three full bars done up in a kind of posh, old-world style. The open upper floor looks down onto the ground level. With quick and friendly service, the Mercantile is a great choice for an afternoon tea, after-work pint or to catch the game on several large screens. The Mercantile's back door opens onto Dame Lane, where you'll find a whole slew of fun, casual bars to keep the night going late.
**ACT LIKE A LOCAL**
**The Trad**
Traditional Irish music, or "trad," as it's called, is a mainstay of Irish culture. Locals grow up learning more than 400 traditional Irish songs with friends and family at the pub—which, in Ireland, really does mean "public house." Pubs are Ireland's "third space," where families enjoy spending time outside the home and workplace. In the countryside, it's quite normal to see entire communities hanging out in the pub enjoying dinner and listening to Celtic songs. The better the party vibe and atmosphere, the better the **_craic_** (enjoyment). Here's how to blend in with the locals and make the most of these Irish jam sessions.
**Understand the trad.** Traditional Irish music is kept alive by amateur musicians who aren't paid a thing besides a few pints of Guinness and the joy of playing the music of their forebears. Bands comprise any combination of the following instruments: the bodhran (pronounced BOH-run), the penny flute, the Irish pipes if you're lucky, and maybe a fiddle or two. You may even get a vocalist, which really completes the ensemble. Most every Irish person knows the words to every tune. Songs can be upbeat and happy, or they can be melancholy, about remembering the mass emigration from Ireland or battles in foreign lands, or about missing the motherland. When these morose ballads, or "laments," come on, hold your conversation out of respect for the musicians, especially if there's a vocalist. The scene is comparable to a pickup game of basketball: It never quite ends, and newcomers are always welcome, even if all they have is a pair of spoons. As a visitor, all you've got to do now is find your own little bar, pull up a stool, order a pint, and toast to the good life. _Slainte_ ("To your health!"), my friends!
**Sing along.** Popular trad songs you'll hear at the pubs include "Galway Girl," "The Rocky Road to Dublin," "The Wild Rover," "Whiskey in the Jar," and "Molly Malone." Find lyrics online, and sing along!
**Tap your foot.** You may be tempted to clap along with the music because you're digging it just so much. If the musicians are on a fun riff, they'll continue, and before you know it, songs may flower into beautiful 20-minute runs. After a while, your arms get tired, you get thirsty, and you stop clapping to pick up your pint glass to take a sip. In the process, you completely deflate the energy of the song. So instead of clapping, tap your foot with the locals—or let out a little yelp if you're really impressed.
**Buy a round.** The Irish are all about pints with friends at the pub, and they've got the round system down to a science: Anytime you finish a pint, go around the circle for one person to get the group a round. It all operates on the assumption that once started, a circle of rounds must be completed. So when it comes to your turn, don't be running off to the loo!
Pints around €5, 28 Dame St, +353 (0)1 670 7100, mercantile.ie
##### **Drury Buildings Cocktail Bar**
If the budget can handle €13 drinks, come out to the Drury Buildings Cocktail Bar and take a sip of their heavenly old-fashioned in a leafy multi-level and balconied floor plan complete with a budding urban garden. While the old-fashioned is my go-to, they've got a quite a roster of creative and high-quality Prohibition-era cocktails, such as their sidecar and frothy whiskey sour, that make you feel like you've stepped back in time. Be prepared to wait a bit for your cocktails—they've gotta make it just right.
Cocktails from €13, daily 12:00-23:30, 55 Drury St, South Bank, +353 (0)1 960 2095, drurybuildings.com
##### **Garage Bar**
Take a load off on one of the Garage Bar's barrel stools and knock back drinks with a crowd that just doesn't really give a darn. This is Temple Bar's alternative bar, where they may well be playing _Star Wars Episode IV_. But you know what? It feels perfectly in place. Drinks here are the cheapest in this district.
Drinks from €4.50, Mon-Tues 17:00-24:00, Wed-Sun 17:00-02:30, Essex St East, Temple Bar, +353 (0)1 679 6543, garagebar.ie
##### **The Bar with No Name**
As with all things hipster, you can't quite have anything easy. In this case, try finding the bar with no name, also known as Snail Bar. Look for the wooden snail above its door, and climb the stairs into what feels like the city's biggest secret. The beer garden/smoking lounge is a favorite for everyone out back.
Pints from €5, Sun-Wed 12:30-23:30, Thurs 12:30-01:00, Fri-Sat 12:30-02:30, 3 Fade St, South Bank, +353 (0)1 648 0022, kellysdublin.com
##### **The Bank Bar**
The Bank Bar sports a spectacular interior in a renovated bank. I love this spot for the _craic_ , especially when sports games are on. Gazing around the opulent golden architectural accents, I wonder if this is what Gatsby's world looked like.
Pints from €5.50, Mon-Thurs 11:00-00:30, Fri-Sat 11:00-01:30, Sun 11:00-24:00,20 College Green, Temple Bar, +353 (0)1 677 0677, bankoncollegegreen.com
##### **Sweeney's Bar**
Probably my favorite bar in town, decked out with art nouveau-esque interior, yet packed out with Dublin's rasta and punk crew. Excellent live music goes down often both upstairs on the stage and downstairs in more of a concert type venue. Partiers spill out onto Dame Lane, the back alley where everyone's having a smoke and checking out the vibe in each of the next-door bars.
Pints from €5, Sun-Tues 12:00-23:00, Wed-Sat 12:00-02:30, 32 Dame St, Temple Bar, +353 (0)1 635 0056, sweeneysdublin.ie
##### **Whelan's**
Made famous as a set in the tear-jerking _P.S. I Love You,_ starring Gerard Butler, Whelan's is a great spot that doesn't rest on its laurels. You've got several rooms and bars to choose from, including a midsize concert venue frequently rocked out by resident and traveling bands alike. Check out the website, as some shows require a password or tickets ahead of time.
Pints from €5, Mon-Fri 14:30-02:30, Sat 17:00-02:30, Sun 17:00-01:30, 25 Wexford St, South Bank, +353 (0)1 478 0766, whelanslive.com
##### **The Porterhouse**
With two locations in Dublin, this chain pours only what they brew, so you won't find any Guinness here. With the excellent lineup of live music and the beautiful dark wood interior, you wouldn't even know that it's part of a larger chain. They feature live acts throughout the week. Find the second, less-interesting location alongside Trinity College (45 Nassau St).
Pints from €5, Mon-Wed 11:30-24:00, Thurs 11:30-01:00, Fri-Sat 11:30-02:00, Sun 12:30-24:00, 16 Parliament St, Temple Bar, +353 (0)1 679 8847, porterhousebrewco.com
##### **The Workman's Club**
Packed out with local students, this is Temple Bar's best spot to rock out to loud rock 'n' roll. Come out for the drink prices, and stick around if you don't mind the sticky, sweaty college-dorm-party vibe. I also like the bar on the ground floor, Bison Bar, for a wee dram of whiskey on full-on leather saddles.
Pints from €4, daily 17:00-03:00, 10 Wellington Quay, Temple Bar, +353 (0)1 670 6692, theworkmansclub.com
##### **Café en Seine**
This is a strikingly beautiful renovated and decorated classic pub with delicious brunch options earlier in the day, tea for the afternoon, and serious parties each night. It's an excellent place for a drink underneath a turn of the 20th century art nouveau-style interior. A dance floor with a DJ kicks off on the weekends. Check out the posh whiskey bar next door, **37 Dawson Street,** too.
Drinks from €6, Mon-Tues 12:00-24:00, Wed-Sat 12:00-03:00, Sun 12:00-23:00, 40 Dawson St, South Bank, +353 (0)1 677 4567, cafeenseine.ie
##### **Odessa**
I love this little speakeasy-type cocktail lounge on the third and fourth floors of a building right behind Dame Lane. I dig the eclectic interior and retro '70s vibe. Climb the stairs to the top, order a drink, and enjoy sipping it on the rooftop patio, rubbing elbows with Dublin's socialites.
Drinks from €7, Mon-Thurs 12:00-00:30, Fri-Sat 12:00-02:30, Sun 12:00-24:00, 14 Dame Court, Temple Bar, +353 (0)1 670 3080, odessa.ie
##### **Flannery's**
If I know I'm going to be clubbing at either Dicey's or Copper Face Jacks, this is a requisite stop for the cheap drinks, fun vibe, and good chance of meeting people with the same agenda for the night.
Drinks from €4, daily 11:00-02:30, 6 Camden St Lower, South Bank, +353 (0)1 478 2238, flannerysdublin.com
##### **The Pavilion**
The student bar at Trinity College has cheap food and possibly the cheapest drinks in Dublin. Heads-up: The place packs out during Trinity exams; students go both before and after—first for confidence, and second to celebrate.
Drinks from €3.50, Mon-Sat 12:00-23:00, inside Trinity College, South Bank, +353 (0)1 608 1279, ducac.tcdlife.ie/pavilion
#### **CLUBS**
##### **Dandelion**
Clubs aren't cheap in Dublin, but they're certainly a great time, and this club is one of my favorites. Located just at the top of Grafton, Dandelion has a busy bar upstairs and pumping dance floor downstairs. Before heading in, know that they're heavy on the electronic. It's a good idea to check the program online to know what to expect.
**MOLLY MALONE**
You're sure to hear this popular song sharing the story of a 17th-century fishmonger who sold seafood by day and something else quite different by night:
_Chorus:_
_Alive, alive oh,_
_Alive, alive oh,_
_Crying cockles and mussels,_
_Alive, alive oh._
_Verse 1:_
In Dublin's fair city,
Where the girls are so pretty,
I first set me eyes on sweet Molly Malone,
As she wheeled her wheelbarrow,
Through streets broad and narrow,
Cryin' "Cockles and mussels alive, alive oh!"
_Chorus_
_Verse 2:_
She was a fishmonger,
And sure it t'was no wonder,
As so was her Father and Mother before,
They each wheeled their barrows,
Through streets broad and narrow,
Crying "Cockles and mussels alive, alive oh!"
_Chorus_
_Verse 3_ (the sad verse):
She died of fever,
And no-one could save her,
And that was the end of sweet Molly Malone,
Now her ghost wheels her barrow,
Through streets broad and narrow,
Crying "Cockles and mussels alive, alive oh!"
_Chorus_
Covers €5-10 on the weekends, drinks around €7.50, 130-133 St Stephen's Green West, South Bank, +353 (0)1 476 0870, welovedandelion.com
##### **Copper Face Jacks**
Copper Face Jacks, aka Copper's, is a well-known sloppy meat market that draws nurses and firefighters whose shifts don't line up with the rest of ours. That means their parties go hard every night of the week in this large, double-floored dance venue. Dress to impress to give yourself a better chance at the door. Bouncers have been known to hold back the last person in a group to try to squeeze out an extra tenner.
Cover and drinks €10, 29 Harcourt St, South Bank, +353 (0)1 425 5300, copperfacejacks.ie
##### **Dicey's Garden Bar**
The crowd packs out Dicey's as a cheaper option to Copper's just down the street. You really can't complain about the €2 drinks every Sunday and Tuesday, but with them comes all the sloppiness of the crowd that is there for the same reason.
Drinks and cover usually around €5, daily 16:00-02:30, 21-25 Harcourt St, South Bank, +353 (0)1 478 4841, russellcourthotel.ie
#### **PUB CRAWLS**
##### **Hostelculture Pub Crawl**
The same company that runs daily walking tours through the city also offers a fun way to check out the town after the sun goes down: a pub crawl hitting up some of my favorite spots. This reasonably priced crawl includes a free half-pint and shots at each stop along the way.
€12, kicking off nightly at 20:00 at the Mercantile Pub, 28 Dame St, Temple Bar
##### **Traditional Irish Musical Pub Crawl**
For an immensely entertaining evening, join the trad music pub crawl. Each stop gives you not only the opportunity for another tasty pint of Guinness, but also another few songs and lessons put on by the Irish musicians, who double as your guides.
€12, Apr-Oct 19:30 nightly, meets at Oliver St John Gogarty Hostel's pub, 58 Fleet St, Temple Bar, book at discoverdublin.ie/musical-pub-crawl
### **TOP SHOPPING & MARKETS**
Cute little boutiques are popping up all over Dublin, vending everything from black-and-white prints to lace and pottery.
#### **SHOPPING DISTRICTS**
##### **Grafton Street**
Grafton Street, leading from the corner of Trinity College's campus several blocks up to St Stephen's Green, is Dublin's main pedestrian shopping drag. You'll find everything from Diesel jeans to Starbucks coffee. While a bit mainstream, it's a one-stop shop for fashion, souvenirs, and food and coffee. Venture west a block or two and get lost in the lanes featuring endless options. Stephen's Green Shopping Centre at the top of the street has more stores and a food court upstairs.
South Bank
#### **MARKETS**
##### **Georges Street Arcade**
Probably the most beloved market amongst locals and tourists alike, the Georges Street Arcade has it all. With over 50 shops and stalls inside an enclosed market, this place offers everything from vintage clothing, antiques, hairdressing, and music to art stalls, restaurants and cafés, and more.
Mon, Tues, Wed, Fri, and Sat 09:00-18:30, Thurs 09:00-20:00, Sun 12:00-18:30, off of S Great Georges St, South Bank, georgesstreetarcade.ie
##### **Temple Bar Food Market**
Right in the center of Dublin's party district, this small outdoor market is a foodie's paradise, with vendors selling a wide variety of items from all over the world, including cakes, sushi, waffles, Spanish tapas, French bread, German sausages, and delicious homemade chocolates.
Sat 10:00-16:30, 12 East Essex St, Temple Bar, templebar.ie
### **TOP PARKS & RECREATION**
#### **St Stephen's Green**
St Stephen's Green is a beautiful manicured park in the heart of Dublin about two city blocks wide and two deep. Take a stroll over the bridges, through the trees, and past the landscaped shrubbery and statues. St Stephen's Green was turned from a private to a public park and has changed little in the last hundred years. Many original Georgian buildings still surround the park today.
Free, always open, South Bank
#### **Phoenix Park**
Spanning over 1,750 acres, this is the biggest gated city park in all of Europe. It is home to two herds of deer and the US Ambassador to Ireland (the mansion is right in the heart of the park). It's a great place to get away from the city without actually leaving it. Pack a picnic and relax in this inviting green escape. Explore miles of trails and fields and even the city **zoo** (€16.80, daily 09:30-15:00, +353 (0)1 474 8900). Phoenix Park was also the site of Pope John Paul II's visit in 1979, drawing more than one million faithful to a Mass and blessing with the late pope. The park is just about a thirty-minute walk west of the city center. To get here, take bus route 37 toward Blanchardstown Road South.
Free, always open, Greater Dublin
### **TOP TOURS**
#### **Hostelculture Free Walking Tour**
Hostelculture offers daily free walking tours. I appreciate their entertaining approach to touring, with knowledgeable Irish guides. Count on consistency and a great intro to the city of Dublin over their three-hour tours. Hostelculture also offers a **Literary Walk** (free, Tues, Thurs, Sat, and Sun 14:30), a **Brew Legacy Tour** (free, Mon, Wed, Fri, and Sun 15:00), and a **Pub Crawl** (€12, nightly 20:00).
Free, daily 11:00 and 14:00, meet at the Mercantile pub, 28 Dame St, Temple Bar, +353 (0)83 117 1197, hostelculture.com/dublintours
#### **Sandeman's New Europe Walking Tours**
Sandeman's offers free walking tours to all of Dublin's most important sights. The casual tour lasts approximately 4.5 hours. Just look for the guide in the red Sandeman T-shirt, and don't forget to tip for a job well done.
Free, 11:00 and 13:00 daily, meet at City Hall, Temple Bar, newdublintours.com
#### **1916 Rebellion Walking Tour**
For those interested in Ireland's tumultuous and complicated 20th-century history, this Easter Rising tour retraces the steps of those who brought the world's attention to the plight of Ireland. Kicking off in a typical pub, this tour starts with a rundown of the main players and timeline of the events, then takes you out to see the marks of the conflict still visible on the city today.
€12, meets Mon-Sat 11:30, Sun 13:00 at the International Bar, 23 Wicklow St, South Bank, +353 (0)868 583 847, 1916rising.com, lorcan@1916rising.com
### **TOP HOSTELS**
Dublin has a wide selection of budget accommodations. They do fill up over popular periods like over St Patrick's week, New Year's, and Christmas, as well as during important sporting events. The annual international rugby tournament 6 Nations, happening each February and March, and other matches draw huge crowds to Dublin when the team is playing at home. Hostels are the first to get booked out. Once you know the dates of your visit, book ahead through each hostel's online booking system.
#### **Barnacles Temple Bar Hostel**
Literally next door to the famous Temple Bar, this is the place to stay in Dublin. This hostel has grown organically over time, slowly taking over the building to add more dorms and common rooms. A welcoming and friendly staff guarantees a wonderful time in the city, and they're happy to recommend sightseeing, activities, and nightlife options. Hang out in the common room to get in on the relaxed, social backpacker vibe and make some friends. The stellar online reviews generate bookings fast and early, so make sure to reserve your spot ahead of time to join in on the party!
From €10, free Wi-Fi, free breakfast, common room, bunks and small private rooms available, 19 Temple Lane South, Temple Bar, +353 (0)1 671 6277, barnacles.ie, templebar@barnacles.ie
#### **Abbey Court**
Located on the north side of the River Liffey, Abbey Court is your best-located cheap option in town. The staff does their best to keep the place clean, but it's hard to stay up with the hundreds of guests staying here. There's a solid communal kitchen, a social lounge, and a new _shebeen_ (speakeasy bar) out in the back in an old storage cellar. Each room in the place feels like it has just a couple too many beds, making space tight from the quad rooms on up to the large, 40+ bed snoring chambers.
From €15, 24-hour reception, free Wi-Fi, laundry, large bunk rooms, public kitchen, great storage, technology charging stations, private bar, food discounts at nearby restaurants, 29 Bachelors Walk, North Side, +353 (0)1 878 0700, abbey-court.com, info@abbey-court.com
#### **Sky Backpackers Hostel**
This is one of Dublin's newest hostels; look forward to an excellent location (just on the north side of the River Liffey), welcoming staff, and newly refurbished facilities. The brightly colored common room is great for the social atmosphere. The new paint job turns what is otherwise a junky alley into a welcoming oasis to escape the noise on Temple Bar.
Beds from €20, free Wi-Fi, common room, showers included, luggage storage available, activities and events organized often, and deals offered for visiting musicians, 4 Litton Ln, North Side, +353 (0)1 872 8389, skybackpackers.com, liffey@skybackpackers.com
#### **Abigail's Hostel**
Abigail's, popular among backpackers and school groups, is another of Dublin's superbly located large hostels. The breakfast and location (just south of the River Liffey in Temple Bar) are Abigail's two highlights. The slightly outdated rooms and overall cleanliness lag behind.
Beds from €14, free Wi-Fi, extensive free breakfast, showers included, 24-hour reception, adapters, towels and locks available for rent, common room, 7 Aston Quay, Temple Bar, +353 (0)1 677 9300, abigailshostel.com, stay@abigailshostel.com
#### **Avalon House Hostel**
Avalon is your true backpacker hostel, with all the charm and social atmosphere you could ask for. The price is right, and the welcoming staff is happy to recommend nearby restaurants and activities. They even go the extra mile of creating info sheets that are free for you to grab on arrival, with tips on food, sights, and nightlife. The hostel is just stumbling distance from some of Dublin's best bars and clubs, like Whelan's, Flannery's, and Copper Face Jacks. Avalon's sister hostel, **Kinlay House Hostel** (12 Lord Edward St, +353 (0)1 679 6644, kinlaydublin.ie, info@kinlaydublin.ie) is another good choice, offering the same experience and prices.
Beds from €12, free Wi-Fi, breakfast included, locks, towels, and adapters for hire, common rooms, free tours leaving daily from the lobby, free showers, 55 Aungier St (pronounced AN jer), South Bank, +353 (0)1 475 0001, Avalon-house.ie, info@avalon-house.ie
#### **Generator Hostel**
The Generator chain has found the right formula across its many locations in Europe. This branch is popular among American students for its new, large, bright, and institutional interior and its fast, free Wi-Fi. I love the hostel's bar and restaurant, open just about any time I need food! The drawback is its location in Smithfield, on the north side of River Liffey. It's about a 20-minute walk into central Dublin. Be attentive when walking these streets late at night (though the city is working hard to clean them up). To get here, take a tram to the Smithfield stop.
Beds from €18, free Wi-Fi, breakfast included, locks, towels and adapters for hire, female-only dorms available, comfy new beds, Greater Dublin, +353 (0)1 901 0222, generatorhostels.com/en/destinations/Dublin, dublin@generatorhostels.com
### **TRANSPORTATION**
#### **GETTING THERE & AWAY**
Since Ireland is an island, flying to Dublin is your best option if you're coming from outside the country. There is a ferry from mainland Europe and the UK; however, it's slow and doesn't save you money over the numerous cheap budget flights into Dublin.
##### **Plane**
All major airlines and budget airlines fly into **Dublin Airport** (DUB, dublinairport.com). All AerLingus flights arrive into the shiny new Terminal 2. All other flights use Terminal 1. Connections into the city center are quick and easy.
**AirCoach** , operating 24 hours a day, has buses running every 10 minutes during the day and less frequently at night. The connection into the center takes about 40 minutes (€7 one-way, €12 round-trip). There are numerous stops downtown, so ask which is closest to your accommodations as you purchase your ticket.
**Dublin Bus** 's **Airlink,** bus 747, is convenient, and it's my usual choice at €6 each way. Monday-Saturday, the first bus is at 05:45, and then every 10 minutes until 19:40, when the buses come less frequently. On Sunday, the first bus is at 07:15.
You'll find the **taxi** stand just outside the arrivals hall. A cab from the airport to Temple Bar costs about €30, depending on the time of day and the traffic. Ask the driver for an approximate price before you get in, and be attentive that you're not taken on the "scenic route." I've not had the best of luck with drivers in this city. Cabbies will often offer to take you through the tunnel (a faster route saving about 10 minutes) for the cost of the toll fee (to airport: €10 Mon-Fri 16:00-19:00, €3 otherwise; from airport: €10 Mon-Fri 06:00-10:00, €3 otherwise), but unless you're in a hurry, it's not worth it. Or, skip the worry and use the free Wi-Fi at the airport to ping an **Uber.**
If you're flying RyanAir and forgot to print out your boarding pass, there are computers and printers located in both terminals. You can also use them to print directions to your accommodations upon arrival.
##### **Train**
Ireland has a convenient domestic network of train connections around the island by the **Iarnrod Eireann** (irishrail.ie) service. Check the website for timetables and prices.
Dublin has two railway stations, which are important to note for travelers taking the train out of Dublin. **Dublin Heuston** serves all connections going west and south from Dublin (Galway, Cork, Dingle). It's on the west side of town, not far from the Guinness Storehouse and just south of the River Liffey and Phoenix Park. **Dublin Connolly** serves connections heading north (Belfast, Derry). The station is right next to the main bus station, north of the Customs House and the River Liffey.
##### **Bus**
Ireland has an extensive network of bus lines, with direct connections to most cities in Ireland. Do your research for timetables and prices on getthere.ie and book directly with the bus company offering the most convenient time and price. Major bus lines, including **GoBe** (gobe.ie), **Air Coach** (aircoach.ie), the national **Bus Eireann** (buseireann.ie), **CityLink** (citylink.ie), **Dublin Coach** (dublincoach.ie) and **Go Bus** (gobus.ie) all offer competitive pricing and free Wi-Fi on board.
Buses take 2.5+ hours to Galway, 3+ hours to Cork and 2.5+ hours up to Belfast. Find the main bus terminal, **Bus Aras,** directly behind the Customs House in downtown Dublin, just north of the River Liffey, east of O'Connell Street.
##### **Car**
Dublin is about 470 kilometers (7.5 hours) driving from Edinburgh and about 600 kilometers (9 hours) driving from London. Both routes include a ferry crossing. It's much easier and probably cheaper to fly into Dublin than to drive there.
#### **GETTING AROUND**
Dublin's relatively compact center makes it an easy city to walk across. For this reason, I've only provided public transportation connection information for sights farther from the center. That being said, Dublin's integrated public transportation makes it easy to get around town via bus and trams. If you're around for a few days and plan on taking the bus a few times, consider picking up a **Leap Card** (leapcard.ie, refundable €5 deposit), equivalent to London's Oyster Card. Tap in and out with your Leap Card on both buses and trams to receive slightly discounted fares. Individual tickets are nontransferable, so you must purchase another ticket each time you transfer.
##### **Bus**
It's easy to get around Dublin on the buses, and the drivers are generally nice and happy to help answer questions. Purchase your ticket as you board with exact change, telling the driver where you want to go (the machine just keeps the change if you overpay). If you overpay, you'll receive a "change ticket," which you then can take to the Dublin Bus office on O'Connell Street, where they'll give you your change—but it's hardly worth the time to do this. Break your bills ahead of time to have change on hand. Bus fare within greater Dublin is €1.60.
Look up routes on Google or through the nifty **Dublin Bus** app (available in the app store), which allows for dynamic route searches and shows you exactly where bus stops are located.
##### **Tram**
Dublin's tram system, **LUAS** (luas.ie), runs two disconnected lines that are convenient only if you find yourself near the tracks for accommodations or sightseeing. Purchase your ticket at the easy-to-use machines located at any station. Ticket prices are determined based on the number of zones you cross through. The **red line** connects the Connolly Train Station, the Busaras station, O'Connell Street, Jameson Distillery, and Phoenix Park on the north side of town before looping south past Heuston train station. The **green line** picks up at St Stephen's Green and runs south.
##### **Train**
The **DART,** Dublin's regional commuter train service, is a convenient way to get outside the city for day trips to destinations like Dun Laoghaire and Howth. Connections for both leave from Tara Street station, near O'Connell Bridge. Buy tickets online or at the station from one of the ticket booths. Find more information at irishrail.ie.
##### **Taxi**
Taxis in Dublin are safe and registered. Make sure cabbies using their meter, and feel free to ask for an estimate before you get in. **Uber** and the Irish equivalent, **Hailo** , are both widely used.
##### **Bicycle**
Dublin is rated as one of the top 10 bike-friendly cities in the world. A public bike-rental service (dublinbikes.ie) gives you access to over 500 bikes and 15 stations across the city. Annual users get their subscription for €20, while visitors can opt into the three-day membership for €5. Membership gets you unlimited free 30-minute rides. After that, the service charges €0.50 for up to an hour, €1.50 for up to two hours, €3.50 for up to three hours, €6.50 for up to four hours, and €2 for each additional half hour. Leap-frogging from one station to the next, plugging in your older bike and taking a new one, is totally OK and an accepted use of the system. Purchase your pass at any of the stations that take credit cards and follow the on-screen directions.
### **DAY TRIPS**
#### **Howth & Coastal Hike**
Originally founded by the Vikings a millennium ago, Howth is a small coastal town that has persevered as a fishing village and small port. The humble port gained notoriety in 1914, when 900 rifles were smuggled through it and used a couple years later during the failed Easter Rising and subsequent Irish Civil War. Take a half day to explore the port and enjoy the posh **farmers market** (Fri-Sun 9:00-18:00, better in the mornings) with blended juices and cutesy decorated muffins. You can also walk out on Howth's two piers, where some locals like to feed the seals. One of the piers is capped by a 200-year-old lighthouse.
For lunch, head to the **Bloody Stream** (desserts from €4.50, Mon-Thurs 12:00-23:30, Fri-Sat 12:00-02:30, Sun 12:00-01:30, Howth Railway Station, +353 (0)1 839 5076, bloodystream.ie), located right under the train station. Relax in front of the fire at this bar and restaurant, which serves a delicious homemade apple crumble, whiskey, and coffee that hits the spot on a typically drizzly day.
If it's not raining, take the challenging 1.5-hour-long hike out around the head of the rock. The hike offers stunning views of the rugged coastline and Dublin Harbor. It's a perfect substitute for those who don't have time to make it all the way out to the Cliffs of Moher but still want to experience Ireland's natural beauty. Be sure to walk up to the "summit" parking lot, where you'll get some amazing views across Dublin Bay. On a clear day, you'll see the Bailey Lighthouse and even the dramatic Wicklow Mountains out in the distance to the south. Get started on the hike by following the street past the second pier and uphill along the coast before it leads you to the trailhead. Pick up a free map at the train station, or watch for displays of the limited hiking routes.
The suburban **DART** train from Dublin's Tara Street station makes the 30-minute, €3.15 transfer to Howth easy and cheap. Trains depart every 30 minutes. Ask for a town and trail map on arrival in Howth.
#### **The Cliffs of Moher**
For many people coming to Ireland, the Cliffs of Moher are at the top of their list. Packing them in in a day is a challenge, but just about everyone who visits says it's well worth the time out to the cliffs, which tower up to 700 feet above the water. Named after a fort that once perched at the south side of the cliffs, today they're one of Ireland's top tourist destinations, with over a million visiting each year. While it used to be free, the park now charges €6 a head to maintain the trails and the health of the park.
Start your experience at the visitors center, which shares background information on the cliffs and about the flora and fauna of this unique region through multimedia displays and interactive screens. A panoramic video mimics the view that the birds have over the steep drop, evoking butterflies in the stomach. You then leave the center and strike out in either of two directions for a stroll along the cliff face, seeing the rich green lawns abruptly drop straight to the sea far below. It's worth checking the forecast ahead of time, as limited visibility can really kill the experience, though the weather is nearly impossible to predict on the west coast of Ireland.
Numerous tour operators connect the Cliffs of Moher from Dublin city's center, leaving early in the morning and returning around dinnertime. The drive out from Dublin is across the entire country, and it lasts about three hours with some stops in between. Consider a tour with **PaddyWagon Tours** (€40, daily 08:00, meet on Suffolk Street at the _Molly Malone_ statue, paddywagontours.com) or **Day Tours Ireland** (€45, daily 06:50, meet on Suffolk Street at the _Molly Malone_ statue, daytours.ie/cliffs-of-moher).
### **HELP!**
#### **Tourist Information Centers**
A heads-up to the smart tourist: Many enterprising companies have opened up storefronts on main tourist drags in Dublin like Grafton Street, Dame Street, and O'Connell. They've covered these storefronts with signs that say Tourist Information, portraying themselves as unbiased sources of information, but in fact they sell tours from only one brand or a certain set of hotels, for a commission. The only official tourist information office, **Visit Dublin** (25 Suffolk St) is located in the old stone St Andrew's church with the bronze _Molly Malone_ statue pushing her wheelbarrow of cockles and mussels out front. The friendly staff is happy to help you plan your time in Dublin and for trips beyond the city.
#### **Pickpockets & Scams**
As in the rest of Western Europe, violent crime in Dublin is extremely low; however, purse snatching and pickpocketing do occur, so always be on your guard in crowded areas such as train stations and touristy destinations. ATM fraud is on the rise in Dublin. Thieves use "skimmers"—small electronic devices that can be attached to the outside of an ATM—to steal PIN codes. It's always important to check the ATM for any signs of tampering before using it. Certain parts of the far-north side of town can get a bit seedy later at night—but not to worry, I haven't recommended any sights in this area.
#### **Emergencies**
In an emergency, dial 999.
#### **Hospital**
St James's Hospital
James's St, Dublin 8
+353 (0)1 410 3000
#### **US Embassy**
42 Elgin Rd, Ballsbridge Dublin 4
+353 (0)1 668 9612
Edinburgh Maps
Edinburgh 101
Three Day Itinerary
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Help!
Edinburgh is renowned for its rugged beauty, Celtic roots, and gruesome medieval history. With its striking architecture and dramatic setting amid hills and cliffs that were carved by glaciers millions of years ago, it's one of Europe's most beautiful cities. Today, Edinburgh's solid volcanic stone and proud spirit provide the foundation for the city's compact medieval core. Scotland's capital is enjoying a full-on cultural and culinary renaissance. So leave your pants at home, throw on a kilt, and bust out the bagpipes: Edinburgh is your city to conquer!
### **EDINBURGH 101**
The foundations of modern Edinburgh were established with a fort atop Castle Rock by an ancient Celtic tribe, the Gododdin, around AD 600, but evidence of settlement goes back much further than that, up to 3,000 years ago. Early settlers chose this strategic and easy-to-defend location quite deliberately. They named the fort Din Eidyn, and it kept the name until the English starting poking around in the 7th century. In 638, the Angles, the ancestors of the English we know today, invaded the fort and claimed control over the area for the next three hundred years, naming it "Eiden's Burgh," the word _burgh_ meaning "fort." It wasn't until the 10th century that the Scots finally reclaimed Edinburgh as their own.
Over the centuries, Scotland battled England for independence and sovereignty. The struggle is depicted in the film _Braveheart,_ which follows young William Wallace in the late 13th century as he tried to unite the Scots and defeat the intruders from the south. Utilizing skirmish and guerrilla tactics, Wallace led successful raids into the bordering region of England and battled the more numerous but less-nimble English armies. Wallace's good fortune ran out in 1298, when he lost the Battle of Falkirk. Barely escaping with his life, he lay low until being captured seven years later. He was subjected to the medieval punishment dished out to those charged with treason against the British crown: hanging, drawing, emasculation, and quartering. This brutal punishment involved being hanged to near death; being taken down while still alive to watch your guts get cut out and burned in front of you; and then having your genitals cut off, along with your arms and legs. Finally, your head was chopped off and your remains were displayed across the country to warn potential offenders just what happens to their kind when caught. Youch.
Things only settled down between Scotland and England around 1706, when the Acts of Union were passed. This led to the unification of England and Scotland into the Kingdom of Great Britain. At this point, Edinburgh, the capital of Scotland, was infamous for being a dark, filthy city. Edinburgh is credited with having the world's first tenements, where numerous families lived in single rooms. Bordered to the north by a lake, the city walls to the east and south, with the castle to the west, the old medieval town could not expand outward, only upward. As the city grew taller and taller, the streets grew darker and were covered in whatever would have gone down the nonexistent sewage system.
With the advent of industrialization and trade, Scotland began its steady rise into prosperity, gaining the nickname in the 18th century as the "Athens of the North" due to its classical architecture and its reputation for producing great minds, including Adam Smith, Joseph Black, and David Hume. Edinburgh came into its own as a center for intellectual and artistic thought. Also at this time, plans were drawn up for an expansion of the city just to the north of the Old Town. Once it was completed, anyone who could afford to move from the Old Town to this New Town did, leaving the filth and lower classes behind.
Today, Edinburgh is bustling with a renaissance of cuisine, culture, music, and live entertainment. It's a fascinating time to visit this vibrant northern capital.
### **PLAN AHEAD**
#### **RESERVATIONS**
Reservations are required for **Real Mary King's Close** (realmarykingsclose.com).
Reservations are recommended for the following sights:
**Scottish Parliament** (scottish.parliament.uk)
**Edinburgh Castle** (edinburghcastle.gov.uk)
#### **PASSES**
While sightseeing in Edinburgh, you may be encouraged to buy the Scotland Explorer Pass. Note that out of more than 70 entries included in the pass, only 10 are actually in Edinburgh. For the visitor coming to see the city of Edinburgh and perhaps take a day trip to a castle outside of town, the pass just isn't worth the price tag.
#### **LOCAL HAPPENINGS**
##### **Edinburgh Military Tattoo**
Every August, more than 200,000 tourists and locals alike converge on the city of Edinburgh to observe the famous Military Tattoo. No, this isn't one giant inking session, but rather a massive concert repeated over about two weeks beginning in early August (Mon-Fri at 21:00, and twice on Sat, 19:30 and 22:30). Military bands perform in front of stadium seating erected in the Esplanade, the otherwise-nondescript parking lot just in front of Edinburgh Castle. With the dramatic backdrop of the crenellations of the castle, dozens of performing military bands are followed by the lilting serenades of a lone bagpiper, bringing on more than a few goosebumps for the finale. If you plan to visit at this time, book accommodations and tickets as far in advance as possible, as prices skyrocket and availability quickly becomes scarce.
**KNOW BEFORE YOU GO**
**KEY STATS & FIGURES**
**Currency:**
**British pound ( _£_ ); _£_ 1 = about 1.50 USD**
**Population:**
**495,360**
**Language:**
**English with a killer accent**
**Top souvenirs:**
**Scotch whisky, anything plaid, a quaich (a small, traditional Scottish drinking vessel with two arms)**
**Weather:**
**generally chilly and damp; layer up!**
**Local dishes you've gotta try:**
**haggis and Scotch eggs**
**CALIBRATE YOUR BUDGET**
**TYPICAL PRICES FOR:**
**Hostel dorm bed:**
**_£_ 14**
**Two-course dinner:**
**_£_ 12**
**Pint of beer:**
**_£_ 4**
**Daily bicycle rental:**
**_£_ 20**
**Single bus pass:**
**_£_ 1.50**
**MOVIES TO WATCH**
_Braveheart, Trainspotting, Brave, The Illusionist_ , _Skyfall_
**THREE DAY ITINERARY**
Edinburgh is a compact and walkable city, with most of the major sights concentrated on the Royal Mile. This itinerary hits the top sights in and around town, with time left over for a taste or two of Scotch whisky.
**DAY 1: OLD TOWN & EDINBURGH CASTLE**
**MORNING**
Break your fast at one of the many cafés on the way up to the Edinburgh Castle. If you're a fan of smoothies and don't mind a light breakfast, go for **Hula Juice Bar** , at the near corner of Grassmarket Square.
Hike the hill and explore the ramparts of **Edinburgh Castle** , which opens at 09:30. Take advantage of the free, guided walks that head out about every 15 minutes. Two hours is plenty to explore the castle and take in the panoramic views.
**AFTERNOON**
Begin an easy stroll down the Royal Mile, Edinburgh's most famous promenade. Leaving the castle (the beginning of the Royal Mile), immediately on your left you'll find the **Camera Obscura** and the **Tartan Weaving Mill**. You can skip the Camera Obscura, but don't miss the weaving mill. Pop in to see the wide array of colors and uses of tartan material, but save your money for the less-touristy stores farther on down the road. If you love a drop of the good stuff, consider the **Scotch Whisky Experience** opposite the mill. Noon isn't too early for a drink, is it?
Continue a tipsy downhill stroll, and hang a tight right at the first roundabout to snag lunch on **Victoria Street** (nearly back to Hula Juice Bar) at **Oink,** where you'll find freshly made pork sandwiches for cheap. If pork ain't your thing, other good options nearby include the splurge-worthy **Grain Store** or the discreetly located **Under the Stairs** café.
Continue down the Royal Mile, dropping a little spittle on the **Heart of Midlothian** , popping into **St Giles' Cathedral** , and enjoying the bustling scene with shops, whisky tastings, and plaid to last a lifetime.
Toward the bottom of the mile, you've got several worthwhile spots, all of which are doable in short stops. First, hit the mecca for many whisky aficionados, **Cadenhead's Whisky Shop**. Don't miss the free, brief, and interesting **Museum of Edinburgh** or the groundbreaking **Scottish Parliament Building**.
**LATE AFTERNOON**
If it's a nice day, begin your climb up to **Arthur's Seat** for a beautiful sunset view over the town. In this panorama, you'll be able to see the whole of the route you took today, following the castle high on the hill opposite you down to the Palace of Holyroodhouse and parliament at the bottom.
**EVENING**
Regroup at your hostel and go out for dinner and drinks at the pubs and bars on **Cowgate**. Don't miss **BrewDog** or the **Three Sisters** for a good time. Just around the corner, **The Banshee Labyrinth** and **The Hive Nightclub** are also packed more often than not.
**DAY 2: NEW TOWN**
**MORNING**
Grab breakfast and delicious coffee at **Fortitude Coffee Merchants,** just beyond St Andrew Square. Then double back to the south side of the **Calton Hill** to begin your hike to the top—don't worry, it's not as intense as yesterday's! After summiting, you'll be rewarded with a view back to the Old Town, across the New Town, and all the way out to the bay and North Sea while getting up close views of the Parthenon-like National Monument of Scotland.
**AFTERNOON**
Take in the beautiful Georgian architecture and posh boulevards in the New Town. The two best shopping streets are **Princes Street** and **George Street.** Stop for lunch and coffee at the righteous **Social Bite** on Rose Street between the two main shopping boulevards. Or if it's a nice day, stop at **Cooperative Food Market** on Frederick Street for picnic supplies and post up on **Princes Street Gardens** for a broadside view of the castle and Old Town.
Continue west and a couple blocks north toward the **Dean Gardens** to wander along the **Water of Leith** to experience Edinburgh's natural beauty—it's hard to believe you're in the heart of the city in this leafy park with a ravine and river running through it. If you're interested in modern art, you can leave the path after less than a mile to find the **Scottish National Gallery of Modern Art.**
**EVENING**
Head back to the hostel to rest for a bit before another night out on the town. This time head up to the student district by the University of Edinburgh to find some entertaining venues and spots to grab a bite to eat—I love the Brazilian food at **Boteco do Brasil**. On a budget? Kebab and pizza-by-the-slice options abound in the neighborhood, especially on Teviot Place.
**LATER**
With over 30,000 students, the University of Edinburgh is plenty big enough to support a healthy nightlife scene in the surrounding area. Begin your night at **Malone's** and follow with **Sandy Bell's** , just across the street. Sandy Bell's has unforgettable traditional Scottish folk music seven nights a week.
**DAY 3: DAY TRIP OPTIONS**
Consider a full day trip out to **Loch Ness** (tours leave quite early and get back late: 07:30-20:30). Or stick closer to the city with this itinerary.
**MORNING & AFTERNOON**
Sleep off that hangover and get in a good breakfast, then tour the regional whisky distilleries with **Haggis Adventures**. If you're more into doing your own thing, drop south to see the **Rosslyn Chapel** and **Craigmillar Castle** to continue divining medieval Scotland.
**EVENING**
For classier nightlife than the student spots from the night before, spend your last evening getting started in **Grassmarket Square** , a plaza ringed by bars and restaurants located just in the shadow of the castle (on the south side). Follow the nightlife toward Haymarket Station to explore the **West End,** where you'll find bars popular with friendly locals rather than tourists.
### **TOP NEIGHBORHOODS**
The modern city of Edinburgh huddles around its medieval core. You can't miss the castle high on the hill. Leading from the castle down to the Palace of Holyroodhouse is the **Royal Mile,** the heart of the **Old Town** neighborhood. The Royal Mile is where you'll find most of Edinburgh's noteworthy sights. Just below the castle is the **Grassmarket District** , a trendy nightlife and shopping district centered around Grassmarket Square.
Just north of the Old Town across the North Bridge and Waverly train station is the planned layout of the Georgian expansion, simply called the **New Town**. Dating back to the second half of the 18th century, the New Town was built so that the wealthy could escape from the filthy and overcrowded medieval Old Town. Today, along with bigger avenues you'll find galleries, museums, posh shopping streets, and more upscale bars.
On the other side of the castle and hill just to the west is the **West End,** where you'll find Haymarket station, stadiums, tons of student housing, lively bars, and a bustling blue collar and student scene.
The University of Edinburgh is just to the south of the Old Town and borders the West End. In the **University District,** you can find exactly what you'd expect in any student neighborhood: cheap food, bars, entertainment venues, and bookshops.
### **TOP SIGHTS**
Most of Edinburgh's top sights are located on the **Royal Mile,** the main tourist boulevard, which is full of shops and street performers. Five contiguous streets make up the mile: Castlehill, Laynmarket, High Street, Canongate, and Abbey Strand. The sights in this section are organized from the top of the hill at Edinburgh Castle down toward the Palace of Holyroodhouse.
#### **Edinburgh Castle**
Think _Lord of the Rings_ meets _Braveheart_. This proud castle sits atop volcanic rock carved out by glaciers long ago, giving it perfect natural defenses: sheer cliffs along three sides and a long uphill trek to the front door, er, gate. The foundations of the castle were laid in the 9th century with Edinburgh's oldest building, St Margaret's Chapel, and primitive fortifications. Since then, numerous additions were made during the tenuous centuries while England threatened the southern border of Scotland.
Frequent guided tours (included with admission) show you through numerous military museums and exhibits, the Scottish Crown Jewels (and a display demonstrating how they were found in the castle after being lost for nearly 200 years), the Scottish National War Memorial, and the ramparts of this fortress. Looking down the barrels of some serious cannons reminds visitors of the very functional defensive purpose of this castle, even while taking in the dramatic panoramas of the city.
£16.50, Apr-Sept daily 09:30-18:00, Oct-Mar daily 09:30-17:00, last entry one hour before closing, Castlehill, Royal Mile, +44 (0)131 225 9846, edinburghcastle.gov.uk
#### **Scotch Whisky Experience**
Just outside the castle, take a barrel ride through this replica distillery to learn about the history, process, and regions of the world-renowned distillate that is Scotch whisky. The experience is just this side of a tourist trap, but this educational opportunity utilizes all your senses and gives you a chance to sip from what is said to be the world's largest collection of Scotch. Go for the Silver package or upgrade for the Gold tour to get an extra wee dram or two. Stopping in is worthwhile for both aficionados and those who have not yet developed their love for Scotch. The attached restaurant, Amber, is surprisingly good and much better than what you'd expect at such a tourist trap.
From £14.50, non-alcoholic drinks provided for those under 18, Apr-Aug daily 10:00-18:00, Sept-Mar daily 10:00-17:00, tours leaving on the hour, 354 Castlehill, Royal Mile, +44 (0)131 220 0441, scotchwhiskyexperience.co.uk
#### **Tartan Weaving Mill**
Pop into this cheesy, dusty warehouse stuffed full of just about everything you can smack a plaid pattern on. Converted from the water reservoir of the old town, it is now a painfully obvious tourist trap, but the displays and sheer choice of patterns are worth a gander. Dive deep into the warehouse to find a life-size timeline of old highland dress with some historical information breaking down the trends over the years. Save your money for cheaper shops off the main drag.
Free, daily 09:00-17:30, 555 Castlehill, Royal Mile, +44 (0)131 220 2477, royal-mile.com/interest/tartanweavingmill.html
#### **Camera Obscura**
The Camera Obscura is an interactive museum with exhibits about optical illusions, mirrors, and visual tricks. Because photography was pioneered in Edinburgh, it has a long history of optics, and Camera Obscura's main attraction is a live panorama of the city projected through a periscope downward inside the tower of the building of the museum onto a large viewing dish. Cool to see, but is it worth the 14 quid? That's your call.
£14 adult, £12 student, Apr-Jun daily 09:30-19:00, July-Aug daily 09:00-21:00, Sept-Oct daily 09:30-19:00, Nov-Mar daily 10:00-18:00, 549 Castlehill, Royal Mile, +44 (0)131 226 3709, camera-obscura.co.uk
#### **Heart of Midlothian**
It may surprise you to see locals spitting on this simple heart-shaped pattern of cobblestones as they pass, but this tradition goes way back to the 1300s. Then, the heart lay just outside the door to the Old Tollhouse, where taxes were collected and prisoners were held, tortured, and even executed. While the tax building is gone, pedestrians today express their discontent with the machine by donating a little spittle. Go on, hawk a loogie yourself.
Free, always open, northwest corner of St Giles' Cathedral, Royal Mile
#### **St Giles' Cathedral**
With its beautiful stained-glass windows and painted ceiling, St Giles is a fantastic display of high gothic architecture dating all the way back to 1385. St Giles was an ongoing project over the years, and you may notice the haphazard style from continuous improvements and additions. When entering, take in the main hall but be sure to make your way to the back and visit the spectacular Thistle Chapel, with its ornately wooden benches, bright stained-glass windows, and a heavenly network of ribbed vaults in the ceiling.
Free (£2 fee for photography), £3 suggested donation; May-Sept Mon-Fri 9:00-19:00, Sat 9:00-17:00, Sun 13:00-17:00; Oct-Apr Mon-Sat 09:00-17:00, Sun 13:00-17:00; High St, Royal Mile, +44 (0)131 225 9442, stgilescathedral.org.uk
#### **The Real Mary King's Close**
Beneath the Royal Mile, you'll find The Real Mary King's Close, an underground labyrinth of the tight alleyways and dank tenements of 16th-century Edinburgh. As the modern city has grown, builders have chopped off the tops of the old tenements and reused the largely untouched foundations, simply covering them over and leaving the old streets and rooms still accessible underneath the stately Royal Exchange building. Taking a walk through these streets is like hopping in a fun time machine back to Old Edinburgh. Your entry ticket comes along with a tour led by a costumed and in-character guide spinning a funny and dry yarn to help you bring these empty streets to life. Must book in advance.
£14 adult, £12.50 w/student ID, Apr-Oct daily 10:00-21:00; Nov-Mar Sun-Thurs 10:00-17:00, Fri-Sat 10:00-21:00; 2 Warriston's Close, Royal Mile, +44 (0)845 070 6244, realmarykingsclose.com
**A BRIEF HISTORY OF THE KILT**
Kilts, which started emerging in the 1500s, were originally over 20 feet long and were wrapped around the torso, secured with a belt, and then draped over the shoulders. With time, the kilt evolved to be more of the skirt that we see today, with a separate piece for draping over the shoulders. The colors were a function of the dyes available in the region as well as the wealth of the person wearing it. Though we've been led to believe that nearly every family name has a "registered tartan" or plaid pattern, this was simply the genius strategy of a cunning businessman to sell more plaid to tourists. Early sporters of the garb were simply glad to have one to wear. So while it's fun to pick up some colors, be skeptical of sales clerks telling you, "Gonzalez? Yeah, we've got a pattern for that clan!"
#### **Cadenhead's Whisky Shop**
When it comes to Scotland, whisky is basically a requisite part of your visit, and Cadenhead's Whisky Shop is your classroom. Come out for an education in everything grain-mashed goodness. While they don't produce their own liquor, these guys go straight to the source and bottle their goods before the concoctions are watered down, colored, and prepared for mass distribution. This means their stuff is purer and stronger than what you'll find in any standard shop, and the decision about just how many drops to add to your dram is left to you. This small, intimate shop posts about new arrivals, classes, and tastings to their loyal followers throughout the week on their Facebook page. Tasty...
Free, bottles from £28, Mon-Sat 10:30-17:30, closed Sun, 172 Canongate, Royal Mile, +44 (0)131 556 5864, wmcadenhead.com with frequent updates to facebook.com/CHWSEdinburgh
#### **Museum of Edinburgh**
I enjoy city history museums to get a sense for the foundations of a city and what really put it on the map in the early days. This free museum is an excellent stop at the bottom of the Royal Mile. Check out their extensive and somewhat eclectic collection of artifacts and displays. You can see a model of Edinburgh's medieval Old Town as it would have looked in the 16th century, weapons used in World War I, trench artwork created by soldiers tinkering away on spent shells and ration cans, and even an original copy of the National Covenant—an important document that denounced the attempts of the Anglican church to incorporate the Scottish Presbyterian church way back in the 17th century—scrawled out and signed on a sheep's hide. These displays (and the free bathroom) make this a convenient stop along the Royal Mile.
Free, Mon-Sat 10:00-17:00, Sun 12:00-17:00 during Aug only, 142 Canongate, Royal Mile, +44 (0)131 529 4143, edinburghmuseums.org.uk
#### **Palace of Holyroodhouse**
As Buckingham Palace is to England, so is Holyroodhouse to Scotland. Home of Mary Queen of Scots and built upon the ruins of an abbey, the palace was built by King David I in the 12th and 13th centuries. Today, it is still the official residence of the Queen of England on her visits to Edinburgh. Walk its halls and visit the many royal audience rooms, Mary Queen of Scots' chambers, and dining rooms of this fully functioning palace. The walls are decked out with priceless paintings, decorations, and tapestries, as any palace should be.
The palace does close down for several weeks each year in preparation for the queen's stay early on each summer, with the Royal Standard of the United Kingdom flying high when she is there. While visiting, take a minute and view Arthur's Seat through the windows just to the south; it is the highest peak of the group of hills you see.
£11.60, £10.60 student; Apr-Oct 09:30-18:00, last entry 16:30, Nov-Mar daily 09:30-16:30, last entry 15:15; Canongate, Royal Mile, +44 (0)131 556 5100, royalcollection.org.uk/visit/palaceofholyroodhouse
#### **Scottish Parliament Building**
Its modern steel and glass architecture makes the Scottish Parliament Building stand out along the backdrop of the Royal Mile and Holyroodhouse. After passing through some quick security, you can investigate a wonderful exhibit on the ground floor that displays the ins and outs of the Scottish legislative system and how it weaves culture and history within its modern framework of government and allegiance to the British crown—which is an important and very current discussion taking place in Scotland today.
Free, reserve guided tours online ahead of time, Mon, Fri, Sat 10:00-17:00, also Tues-Thurs 09:00-18:30 when in recess, last entry 30 minutes before closing, closed Sun, 2 Warriston's Close, Royal Mile, +44 (0)131 348 5200, visitscottishparliament.uk
#### **University of Edinburgh**
Both medieval and modern architecture surround this historic university located in the heart of the Old Town and just a few blocks from the castle. A stroll through the beautiful campus offers a chance to see one of the oldest universities in the world. Today, it's one of Scotland's most competitive universities, and I enjoy taking a walk through campus to see what UK college life is like. Just a couple hundred years ago, it was here that doctors of anatomy were charging _£_ 5 each to attend live dissections in their human anatomy courses, patronizing grave snatchers and serial killers to obtain their subjects.
Free, campus grounds always open, Old College, South Bridge, University District, +44 (0)131 650 1000, ed.ac.uk
#### **National Museum of Scotland**
This massive and newly renovated natural history museum takes up an entire block in Edinburgh's Old Town. Inside, find Dolly the Sheep (the world's first cloned animal, now stuffed) and a full-size _T. rex_ skeleton, as well as displays on old Scottish artifacts, silversmithing, and world cultures. The diverse exhibits touch on the past and present of Scotland, from its people to the industrial transformation of the country, through a series of enthralling galleries.
Rotating exhibits present more personal facets of the country, like the Lewis Chessmen, a hand-carved ivory chess set dating back 800 years, or an in-depth look at the many lighthouses in Scotland, which have played an important role in this coastal country.
Free, daily 10:00-17:00, Chambers St, Old Town, +44 (0)300 123 6789, nms.ac.uk
#### **Craigmillar Castle**
Credited as one of the best-preserved medieval castles in Scotland, Craigmillar Castle was built around 1400 by Sir George Preston, and it provides visitors a chance to travel back to the 15th century via its complete ramparts and chambers. This large stone castle is famous for being the place where the assassination of Lord Danley, the unpopular husband of Mary, Queen of Scots, was planned. While it's not believed Queen Mary was privy to the scheme, it was hatched while she was staying here with her noble entourage. The offenders planned to blow up Lord Danley with two kegs of gunpowder stashed a floor under where he would be sleeping at the nearby church, Kirk O' Field. They lit off the kegs, and the body of Lord Danley was found a short distance away, half-dressed—meaning that somehow he knew of and had tried to escape the impending doom. It's not known whether the blast killed him or if he died by strangulation at the hands the queen's entourage when they found him still breathing. In this country, the truth is often stranger than fiction.
Today, a visit here lets you experience an impressive and well-preserved medieval fortress with many rooms, chambers, and ramparts, all without the hordes of tourists you'll see at the Edinburgh Castle. The castle is south of town. Take bus 3, 14, 29, or 30 from North Bridge and ask the bus driver to tell you where to get off for the castle.
£5.50; Apr-Sept daily 9:30-17:30; Oct-Mar Sat-Wed 09:30-16:00, closed Thurs-Fri; last entry 30 minutes before closing, Craigmillar Castle Rd, south of town, +44 (0)131 661 4445, visitscotland.com
#### **Royal Yacht Britannia**
The royal family's floating home for 40 years has been converted into a 412-foot exhibit, letting you stow away on the family's nautical travels. Start your visit in the command center (the bridge) and continue down five levels all the way into the engine room. Experience the decadent, gold-encrusted interiors, including the dining hall and royal staterooms—which have all the comforts of anything you'd expect on land. You can even see the crew quarters (not quite as fancy as those of the royal family). Take a pit stop for a snack and tea in the Tea Room, where sandwiches start from _£_ 5. At 2.5 miles from the city center it makes sense to take a bus (1, 11, 22, 34, or 35 from Waverley Bridge) to get here.
£14, £12.50 student, daily Apr-Sept daily 09:30-16:30, Oct daily 09:30-16:00, Nov-Mar daily 10:00-15:30, Ocean Terminal, Port of Leith, coastal suburb of Leith, +44 (0)131 555 5566, royalyachtbritannia.co.uk
#### **EXTRA CREDIT**
##### **National Gallery of Scotland**
This classic art museum spans the ages from gothic all the way to baroque, impressionism, and surrealism. You'll recognize names like Dalí and Duchamp, and will see pieces by less famous but no less fascinating Scottish artists covering just about all the genres and styles. Perched on the hill between the Old Town and New Town, it makes for a convenient—and free—stop between the two.
Free, daily 10:00-17:00, The Mound, New Town, +44 (0)131 624 6200, nationalgalleries.org
##### **Scottish National Gallery of Modern Art**
For a change, pop over to Edinburgh's modern art museum. Split across two neoclassical buildings, the interior exhibits are anything but. "Modern One" is the permanent exhibition, and "Modern Two" is where you'll find temporary exhibits featuring the likes of the great M. C. Escher.
Modern One free, Modern Two price varies, daily 10:00-17:00, 75 Belford Rd, New Town, +44 (0)131 624 6200, nationalgalleries.org
##### **Greyfriars Kirk & Kirkyard**
Greyfriars Kirk ("church" in Scotch Gaelic) is a beautiful old church just to the south of the Old Town, built out in the fields back in the day in the 17th century. It's the main church for central Edinburgh and is part of the Church of Scotland. The graveyard around the church is supposedly the most haunted part of the city, as it was a favorite hunting ground for grave snatchers and is the permanent home of many notable Scots. The most famous occupant, however, is a terrier named Bobby, who guarded his dead owner's grave for 14 years before his own death in 1855. A statue and bar commemorating the world's most loyal dog sits just outside the entrance to the graveyard closest to the church.
Free, daily 10:30-16:30 but may be closed for private events, Candlemaker Row, University District, +44 (0)131 664 4313, greyfriarskirk.com
##### **Rosslyn Chapel**
A hike out of town, but worth it for _Da Vinci Code_ buffs, the Rosslyn Chapel dates all the way back to the 15th century. While the supposed connections with the freemasons and Knights of Templar are dubious, the chapel itself is beautiful and packed with intricate stonework, making for a great side trip out of the city to see the nearby countryside. Just beyond the church are the ruins of a 14th-century castle of the same name for you to explore. To get here, take bus 37 from North Bridge, which runs every 30 minutes and takes about half an hour.
Free, Mon-Sat 09:30-18:00, Sun 12:00-16:45, Chapel Loan, Roslin, +44 (0)131 440 2159, rosslynchapel.com
### **TOP EATS**
Many of Edinburgh's most popular restaurants are clustered around Old Town's **Royal Mile** and New Town's **Princes Street.** On nice days, it's also pleasant to picnic at **West Princes Street Gardens.** Food is available nearby at the **Cooperative Food Market** (daily 07:00-23:00, 26-28 Frederick Street, +44 (0)131 220 0359).
The northern suburb of **Leith** is especially known for its fresh seafood restaurants. **South Edinburgh** has a variety of cheaper cafés and restaurants for budget-minded travelers. The neighborhood surrounding the **University of Edinburgh** is also full of cheap and ethnic food options, especially on Nicolson Street.
It's typical to round up to the nearest even number, or about 10 percent, on meals. If service is exceptional, feel free to head north to 15 percent.
### **Oink**
This fast-casual food shop does one thing, and it does it well: steaming fresh pork sandwiches sourced from local farms in limited quantities. Pop in here to keep your energy level up on these long days of sightseeing. A second location at 82 Canongate on the Royal Mile ensures you're never far from a great pork sandwich.
Sandwiches from £2.95, Mon-Sat 11:00-17:30, Sun 11:00-17:00, 34 Victoria St, +44 (0)189 076 1355, oinkhogroast.co.uk
#### **Saint Giles Café & Bar**
When it comes to breakfast and lunch on the Royal Mile, Saint Giles Café & Bar is a favorite of mine, located at the top of Saint Giles Street and kitty-corner from the cathedral. Pop in here to get your day off on the right foot with breakfast rolls and fresh pastries. The rustic interior is going to make you want to come back for tea or to hit the bar later on.
**HUNGRY FOR HAGGIS?**
Scotland's national dish, haggis, is an interesting one, made of—if you really want to know—sheep's heart, liver, and lungs minced together with onions, oatmeal, and some seasoning; stuffed into a sleeve of stomach lining; and then simmered until cooked. It's usually dished up with _neeps_ and _tatties,_ or mashed turnips and potatoes. Scots love it, and for brave tourists, it's worth a try. Bobby's Bar, just outside Greyfriars Kirkyard, is a great place to sample it. Bon appétit!
Scones from £1.50, daily 09:00-18:00, 8-10 St Giles St, Royal Mile, +44 (0)131 225 6267, saintgilescafebar.co.uk
#### **The Grain Store**
This is a splurge-worthy restaurant for those who want to check their Scottish cuisine box in style. With impeccable decor, swift service, and succulent main dishes, The Grain Store has established itself as the go-to restaurant for those who want to appreciate their saddle of lamb or venison and kale in a comfortable and classy setting.
Mains from £20, Mon-Fri 12:00-14:00, 18:00-22:00, Sat 12:00-15:00, 18:00-23:00, Sun 18:00-22:00, 30 Victoria St, Royal Mile, +44 (0)131 225 7635, grainstore-restaurant.co.uk
#### **Under the Stairs**
This discreet and somewhat difficult-to-find little joint serves up delicious entrées and expertly made cocktails like you'd expect at any hipster bar. It feels more like a tastefully decorated lounge and has become so popular it runs the risk of becoming mainstream. Until then, I'll keep coming back for their burgers and sweet potato wedges.
Beers from £4.50, cocktails from £7, daily 12:00-01:00, 3A Merchant St, Royal Mile, +44 (0)131 466 8550, underthestairs.org
#### **Elephant House**
Elephant House is a great place for reasonably priced panini, international foods, and hot drinks. This cozy café is located near Edinburgh University and provided the perfect ambience for J. K. Rowling to pen her first novels on those famous napkins, with the inspirational views of Edinburgh Castle through the window of the back room.
£11-16, Mon-Fri 08:00-22:00, Sat 09:00-23:00, Sun 09:00-22:00, 21 George IV Bridge, Royal Mile, +44 (0)131 220 5355, elephanthouse.biz
#### **ClamShell**
At this no-nonsense fast-food joint, they'll deep-fry almost anything, including pizza, ribs, haggis, and (my favorite) Mars bars! They also dish out some of the best fish-and-chips you'll find in all of Edinburgh. Order, then post up at the high-top bar along the wall inside or on the outdoor seating under the sun (if there is any) and enjoy.
£3-6, daily 11:00-01:00, 148 High St, Royal Mile, +44 (0)131 225 4338
#### **Mums Great Comfort Food**
The theme for this homey and slightly retro restaurant is "top nosh at half the cost," and it delivers just that in style. Mums uses all local ingredients and definitely doesn't hold back on the tasty calories, delivering all of the British classics we know and love—shepherd's pie, bangers and mash, fish-and-chips, and, of course, haggis.
Dishes from £7, Mon-Sat 09:00-22:00, Sun 10:00-22:00, 4a Forrest Rd, University District, monstermashcafe.co.uk, +44 (0)131 225 7069
#### **Boteco do Brasil**
As with any university district, you can count on finding cheap eats on the streets around the University of Edinburgh, and the Boteco do Brasil is one of my favorites, serving out delicious plates heaping with black beans, pork, beef, and burgers in a casual atmosphere with funky furniture and exposed brick walls. The friendly servers take their cocktails seriously—the _caipirinha_ is almost as good as the ones back on Ipanema beach. Later on, space is made to turn the party up around 22:00 six nights a week, with students coming for their popular disco and salsa nights. There's nothing like a little slice of Rio across the street from Scotland's top educational institution.
Tapas from £5, burgers from £7, daily 11:00-03:00, 47 Lothian St, University District, +44 (0)131 220 4287, botecodobrasil.com
#### **Hula Juice Bar**
Your dear author isn't usually a juice bar kind of guy, but this one proudly breaks all the rules. Pop in for delicious healthy smoothies with creative names like Whirling Dervish and Sunshine in a Cup, but get full on their tasty bagels stuffed with bacon and cheddar. Yes, there are vegetarian options.
Juices from £3.50, daily 08:00-19:00, 103-105 West Bow, Royal Mile, +44 (0)131 220 1121, hulajuicebar.co.uk
#### **Graze on Grassmarket**
The east side of Grassmarket Square seems to be on a health kick with Hula Juice Bar and this fresh entry, which pumps out sandwiches, wraps, salads, and baked potatoes stuffed with locally sourced ingredients to keep you going throughout the day. This small café is stripped down and simplified, with only a few seats for those who choose to stick around and have their takeout in.
Lunches from £3.50, Mon-Sat 07:30-18:00, Sun 11:00-16:00, 67 Grassmarket, Grassmarket District, +44 (0)131 629 4030, grazeongrassmarket.com
#### **Fortitude Coffee Merchants**
When it comes to breakfast in the New Town, I head to Fortitude Coffee Merchants for some of the best coffee in town. If you're a coffee snob at heart like me, you'll appreciate their quality beans and small-batch roasting—and you'll be in welcome company with the staff, who are happy to help you pick from their rotating roasts. While I wish they had a full kitchen, I'll settle for the locally sourced pastries and mini cakes. Fortitude offers just the right combination of caffeine and sugar for your climb to the top of nearby Calton Hill.
Coffee from £1.50, Mon-Fri 08:00-18:00, Sat 10:00-18:00, 3C York Place, New Town, +44 (0)131 557 3063, fortitudecoffee.com
#### **Social Bite**
I love it when good food made with local ingredients and sold at fair prices comes with a mission for social improvement. Social Bite makes excellent sandwiches, wraps, and soups in the New Town, offering both meat and vegetarian options, making it an excellent stop for a quick lunch. What's more, a quarter of their employees have a homelessness background, and Social Bite is helping them get on their feet. You can even buy an extra sandwich and drink to be redeemed later by a local homeless person—definitely something I can get behind.
Sandwiches from £3.95, Mon-Fri 07:00-15:00, 131 Rose St, New Town, +44 (0)131 220 8206, social-bite.co.uk
### **TOP NIGHTLIFE**
Thanks to Edinburgh's compact city center, nightlife venues are never far from each other.
#### **NIGHTLIFE DISTRICTS**
##### **Grassmarket Square**
The Grassmarket district is one of the best places to hang around at night, as it offers some of Edinburgh's most authentic traditional bars, along with a happening nightlife. **Dropkick Murphy's** frequently offers live music in your quintessential Irish pub atmosphere.
Grassmarket District
##### **Royal Mile**
Nightlife on and near the Royal Mile is better than what you'd expect for such a touristy area. While certain places can be a bit generic and overpriced, I've listed some of my favorites to keep you from falling into the tourist traps. There are some incredible finds sprinkled throughout the neighborhood. You'll even find bars and clubs, like **Whistlebinkies** and the **The Banshee Labyrinth,** that are burrowed into old medieval cellars, making for a unique night underneath exposed brick arches dating back centuries.
Royal Mile
#### **University District**
This is where you'll find both the sloppy student hangouts and some notable exceptions—classy establishments with great food, fun vibes, and tasty drinks.
University District
#### **Haymarket**
The area surrounding the Haymarket train station has plenty of classic British-style pubs with heavy wooden interiors, bantering customers, numerous drafts on tap, and whiskies to pour. This area is more low-key than the others, but it draws a higher ratio of locals than the touristy Royal Mile.
West End
#### **BARS**
##### **Sandy Bell's**
Sandy Bell's is an intimate bar that is bursting at the seams thanks to its popularity as one of Edinburgh's best traditional Gaelic music—or "trad"—bars. They keep it fresh with nightly rotating music, and in such an intimate little spot, you'll be part of the action before you know it. Sandy Bell's is your best bet if you want to catch some live Celtic jam sessions.
Pints from £3.50, Mon-Sat 12:00-01:00, Sun 12:30-24:00, 25 Forrest Rd, University District, +44 (0)131 225 2751, sandybellsedinburgh.co.uk
##### **The Banshee Labyrinth**
The Banshee Labyrinth is aptly named, as it is quite easy to get lost in this maze of cellars, bars, chill-out areas, movie theater ( _The Departed_ was showing the last time I popped in), and performance venue. Add the funky lighting, a little alcohol, and an inexplicable otherworldly haze that just seemed to float around this place (which must have been from smoke machines somewhere) and you feel like you could get lost forever. Come out for a good time, embrace the unexpected, and you never know what you'll discover in here. Check the website for upcoming events.
Drinks from £3.50, daily 19:00-03:00, 29 Niddry St, Royal Mile, +44 (0)131 558 8209, thebansheelabyrinth.com
##### **Whistlebinkies Live Music Bar**
This excellent pub right off the Royal Mile packs out the house with ales and a full bar to get the party going. The whole venue is oriented toward the stage, with musicians playing most nights, and varied enough to keep everyone happy. The layout is such that you can rage up close, dance in the middle, or keep a conversation going off to the side in the back.
Pricey drinks at £5, Sun-Thurs 17:00-03:00, Fri-Sat 13:00-03:00, 4-6 South Bridge, Royal Mile, +44 (0)131 557 5114, whistlebinkies.com
##### **BrewDog**
Although it's part of a chain, BrewDog is one of my favorite spots for a fresh artisanal pint in the shadow of the Royal Mile. Drop in to this welcoming, high-ceilinged hipster bar for a pint of their constantly rotating brews. Too many choices? The knowledgeable staff is happy to help narrow them down. While the beer is excellent, the pizzas are subpar, yet the jovial customers don't seem to mind.
Pints from £4, daily 12:00-01:00, 143 Cowgate, Royal Mile, +44 (0)131 220 6517, brewdog.com
##### **Three Sisters**
The highlight of this funky, unique venue is the large outdoor seating area, where big games are streamed live and the hen and stag parties (or bachelor and bachelorette parties, as they're known in the United States) get rowdy on the karaoke machine.
Drinks from £3.50, Mon-Fri 17:00-01:00, Sat-Sun 07:00-03:00, 139 Cowgate, Royal Mile, thethreesistersbar.co.uk
##### **Malone's Irish Bar**
Edinburgh, like just about every other city on the planet, has got to have its Irish bar, but this one is special thanks to the food and entertainment. Occupying what was a midsized, horseshoe-shaped theater, the bar now televises sporting events and features frequent live music acts. Hungry? The burgers and chips are unforgettable. If you like events, take a look at the program online for upcoming gigs.
Beers from £4, daily 12:00-01:00, 14 Forrest Rd, University District, +44 (0)131 226 5954, malonesedinburgh.com
##### **Dropkick Murphy's**
Known for its late nights and rowdy atmosphere, Dropkick Murphy's is tucked underneath the George IV Bridge. Find the bright green doors just a block off of Grassmarket Square and dive into the sloppy and friendly crowd.
Drinks from £4, Thurs-Tues 20:00-03:00, 7 Merchant St, Grassmarket District, +44 (0)131 225 2002, facebook.com/dropkickmurphys.edinburgh.1
**LGBT EDINBURGH**
Edinburgh is a progressive and welcoming city. Members of the LGBT community will have no problem getting along here, and there are numerous cafés, restaurants, and nightlife venues that cater to the gay scene. Most places center in what's known as the "Pink Triangle," just north of Calton Hill on the eastern edge of the New Town. **The Street** (2 Picardy Place, thestreetbaredinburgh.co.uk) is a popular bar that also offers food, and a DJ spins later in the night. Same goes for **Café Habana** (22 Greenside Place, facebook.com/habanaedinburgh). **Planet** (6 Baxter's Place, New Town, facebook.com/Planet-Bar-Edinburgh) is a smaller dance bar with drink deals and a DJ spinning nightly; it's popular among all partiers. **CC Blooms** (23 Greenside Place, New Town, ccbloomsedinburgh.com) is a tasty restaurant by day and a classy nightclub by night.
#### **CLUBS**
The clubs in town all seem to come with bouncers who like to dish out a hard time, and take a fiver for cover. Once in the door though, you're sure to have a blast.
##### **Cabaret Voltaire**
"Cab Vol," as the partiers call it, is one of Edinburgh's top clubs and is just off the Royal Mile. I love the grungy student-party-night-infused atmosphere in this club, which leaves pretentiousness in the dust in exchange for raucous live music and DJ acts. Be prepared for feisty bouncers, and just be happy to get in.
Cover around £6, Mon-Thurs 17:00-03:00, Fri-Sun 12:00-03:00, 36 Blair St, Royal Mile, +44 (0)131 247 4704, thecabaretvoltaire.com
##### **The Hive Nightclub**
This sweaty, cheap club packs out with a young crowd that appreciates the low prices and doesn't mind the sticky floor. No cover Thursdays and Sundays and student specials keep the crowds coming back. It's worth getting on the guest list on Fridays and Saturdays to avoid the line.
Cover around £3, drinks from £1, Sun-Thurs 22:00-03:00, Fri-Sat 21:00-03:00, 15 Niddry St, Royal Mile, +44 (0)131 556 0444, clubhive.co.uk
#### **COMEDY SHOWS**
##### **The Stand**
Known as Edinburgh's best comedy club, the subterranean Stand is so popular it often sells out, so it's best to book your tickets online ahead of time. Formats most often have a rotation of five comedians over two hours but sometimes switch up for improv and solo act nights.
Tickets from £10, doors usually open at 19:30 and show starts at 21:00 but check schedule online, 5 York Place, New Town, +44 (0)131 558 7272, thestand.co.uk
#### **PUB CRAWLS**
##### **New Edinburgh Pub Crawl**
Hosted by the reliable crew at Sandeman's tours, the New Edinburgh Pub Crawl gives you a leg up in finding all the hidden nightlife gems Edinburgh has tucked away. You'll enjoy half-priced pints, three shooters, a dram of Scotch whisky, and drink deals in every pub you visit, ending with VIP entrance into Edinburgh's best nightclub. You'll also meet tons of travelers like yourself and make some great friends—at least for that night.
£15, nightly 20:00 at the Inn on the Mile, Royal Mile, newedinburghtours.com
##### **Edinburgh Pub Crawl**
Each pub crawl tries to outdo the others on the quantifiables (number of bars, clubs, free shots, etc, etc), but sometimes you've just gotta go with the best vibe—and Edinburgh Pub Crawl has a great one. These guys have been showing people a great night out for years, and the groups of partiers are a ton of fun.
£10, nightly at 20:30 in front of St Giles' Cathedral, Royal Mile, edinburghpubcrawl.com
### **TOP SHOPPING & MARKETS**
#### **SHOPPING DISTRICTS**
##### **Royal Mile**
Full of shops and street performers, the main tourist boulevard is a great place for souvenir shopping...kilts anyone? You'll find just about every Scottish knickknack and touristy souvenir you can imagine. In addition, there are many tiny alleyways or "closes," which branch off High Street with even more cafés, shops, and exhibits to explore.
Old Town
##### **Princes & George Streets**
This area is the Oxford Street of Edinburgh; come here to find all your top international name brands. While in the area, stop into Edinburgh's famous **Jenner's** (Mon-Fri 09:30-18:30, Sat 09:00-19:00, Sun 11:00-18:00, 48 Princes St, +44 (0)131 225 2442, houseoffraser.co.uk), a beautiful Victorian-style department store built in 1938.
New Town
##### **Grassmarket Square**
Grassmarket (27-31 W Port, grassmarket.net) has many boutique fashion shops and vintage music and bookstores, along with trendy cafés and bars. Check out the website to explore in advance the places to visit.
Grassmarket District
##### **Stockbridge**
Visit this neighborhood if you're into secondhand clothing, unique jewelry, glassware, and independent art galleries. Many tea and coffee shops dot the area on the north side of the New Town, on **Kerr Street.** Stockbridge is a great place to get your shop on, enjoy an afternoon tea, and watch the world go by.
New Town
##### **Bruntsfield Place & Morningside Road**
While a bit south of town, this area has a young vibe, offering many boutique and vintage clothing stores, chocolatiers, and cafés. It also gets you out of the city center and into where the locals hang out, offering an untouristy experience you wouldn't get in the center.
South of University District (interesting until Cluny Ave)
#### **MARKETS**
##### **Edinburgh Farmers' Market**
Stock up on locally grown produce, organic beer, and fresh, seasonal fruits at the Edinburgh Farmers' Market. The market vendors assemble just off of Castle Terrace.
Sat 09:00-14:00, Castle Terrace, West End, edinburghfarmersmarket.com
### **TOP PARKS & RECREATION**
Surrounded by dramatic geography, Edinburgh boasts some spectacular parks, hikes, and viewpoints that are worthwhile even on the shortest of visits.
#### **PARKS**
##### **Meadows Park**
Because of its proximity to the University of Edinburgh, students occupy the majority of this large, grassy, tree-filled park, making it a great place to people-watch or even chat up a few local students. Ask them about their university and their studies, or maybe they can recommend some places to go out at night. The park also makes for a great picnic spot and a cheap lunch.
Free, always open, University District
##### **Holyrood Park**
Holyrood Park, in the heart of Edinburgh, has been a royal park since the 12th century. You'll stumble upon many great sights while wandering the footpaths and craggy cliffs of this local wilderness, such as the Palace of Holyroodhouse, the ruins of Holyrood Abbey, and the extinct volcano Arthur's Seat, Edinburgh's highest peak at 251 meters. The views are unbeatable, and the air is filled with the fresh smell of nature, making it the perfect place to spend a few hours.
Free, always open, end of Royal Mile
##### **Dean Gardens & Water of Leith**
Scotland's capital has beautiful parks and walkways, and this is one of my favorites. The Water of Leith is the river that borders the north side of the New Town; Dean Gardens is one of the most beautiful stretches, with a walkway along the river. Just steps into the park, the noises from the city fall away and you're back into Scotland's natural beauty.
Free, always open, New Town
##### **West Princes Street Gardens**
On a clear—and warm—day, one of my favorite things to do is to snag a sandwich or chips from one of the many eateries on the Princes Street strip and head to the park, where I can grab a bench and enjoy the view of the broad side of the Edinburgh Castle and the Royal Mile. The **Cooperative Food Market** (daily 07:00-23:00, 26-28 Frederick St, +44 (0)131 220 0359) is a great spot to pick up your supplies.
Free, always open
#### **VIEWPOINTS**
##### **Arthur's Seat**
A sweaty 45-minute climb up an extinct volcano in Holyrood Park provides wonderful views of Edinburgh, in particular the Edinburgh Castle, Palace of Holyroodhouse, Scottish Parliament, and Calton Hill.
Free, visit during daylight hours for best views, Holyrood Park, just south of the Palace of Holyroodhouse
##### **Calton Hill**
A 15-minute climb to the top of Calton Hill, located in the northeast section of the city center, will bring you to a beautiful World War I memorial, accompanied by panoramic views of downtown Edinburgh, Edinburgh Castle, and all the way out to the coast.
Free, visit during daylight hours for best views, 15 York Place, New Town
### **TOP TOURS**
#### **Excursion Scotland**
Colin Mairs heads up this small tour company. If you're looking for a deep dive into everything Scotland, Colin's your man. A fun-loving, passionate Scot who often prefers to sport a kilt over trousers, Colin offers everything from half-day tours of Edinburgh's best sights to a weeklong, private custom tour of the entire country for small groups and families.
From £100, +44 (0)77 1623 2001, excursionscotland.wordpress.com, excursionscotland@gmail.com
#### **Sandeman's New Edinburgh Free Walking Tours**
A free walking tour with Sandeman's gives you the local's perspective of all the historic and not-so-historic sites of Edinburgh. These free, tip-based tours give you a brief introduction to the city, including its top sites and monuments, in the span of about 2.5 hours. While the walks do feel a bit scripted, the price is right for budget travelers: Tip what you feel the guide deserves.
Free, meets daily at 11:00 and 13:00 in front of the Starbucks on High St, +44 (0)305 105 0030, newedinburghtours.com
#### **City of the Dead Ghost Tours**
With Edinburgh's long history of resident body snatchers, serial killers, and crime, the ghost tour here is better than just about anywhere else in Europe. It's the city with one of the most fascinating—and bloody—walks, where you can pick up all sorts of ghoulish stories and freaky fun facts.
£10, tours nightly at 20:30 from Easter to Halloween, at 20:00 throughout winter, meet in front of St Giles' Cathedral, +44 (0)131 225 9044, cityofthedeadtours.com, info@cityofthedeadtours.com
#### **Rabbie's Small Group Tours**
Rabbie's is a Scottish tourism institution, offering 1- to 17-day excursions with a wide range of focus all around the country. From castles to whisky distributors, and from highlands to lakes ( _lochs_ in Scottish), Rabbie's local guides know exactly where to take you.
Day trips from £42, +44 (0)131 226 3133, rabbies.com, info@rabbies.com
#### **Haggis Adventures**
This is your fun backpacker option for day trips and longer ones around the country. With their "mad sexy" guides, you're sure to have a great time learning about Scottish history, food, and culture while touring the jaw-dropping Scottish countryside. The groups you'll tour with are one of the highlights. Check out the full lineup of trips and dates online. Group and student discounts available.
Trips from £50, +44 (0)131 557 9393, haggisadventures.com
### **TOP HOSTELS**
Edinburgh has a slew of fun, social, and affordable backpacker hostels in or nearby the Old Town. They do their best to work within the older buildings that they've taken over. My favorite ones do it quite well, I like to think.
#### **The Baxter Hostel**
A brand-new entry into the Edinburgh backpacking scene, the midsized Baxter, with about 40 dorm beds, is following the trend of boutique hostels with a focus on comfortable rooms, friendly service, great location, and solid value. The interior feels retro chic, with clean industrial lines, custom fabricated steel and wood bunks, in dorms with exposed brick and wood-paneled walls. The enthusiasm carries over to the staff, who are happy to make recommendations on activities during your stay.
Dorms from £22, 24-hour reception, free Wi-Fi, full kitchen facilities, breakfast included, common room, linens included, 5 W Register St, New Town, +44 (0)131 503 1001, thebaxter.eu, info@thebaxter.eu
#### **Code Hostel**
This hostel in the New Town prides itself on keeping clean as well as on providing great service, value, and a fun, social atmosphere. The bunks packed into the rooms are unique in that you climb into a pod as in the airport hotels that you may have heard about in Japan. They're clearly economizing on space, but the privacy is nice as long as you're not claustrophobic. Each pod is equipped with plugs, USB ports, and reading lights.
Pod beds from £20, 24-hour reception, free maps, free Wi-Fi, breakfast included, towels available for rent, common room, 50 Rose St N Ln, New Town, +44 (0)131 659 9883, codehostel.com, hello@codehostel.com
#### **Budget Backpackers**
Voted #1 hostel in Scotland back in 2012, this place sports bright, clean rooms, friendly staff, funky decor, and a great location just off Grassmarket Square, barely a block away from the Royal Mile. Book early, as rooms fill up fast. En suite privates are also available if you'd prefer to skip the large dorms—the largest of which has 30 beds. The _£_ 5 nightly dinner is a great deal for those on a budget. Numerous common areas make this one of the more social hostels in town.
From £10.50, 24-hour reception, Internet available on PC (no Wi-Fi), common room, kitchen access, optional breakfast, pool table, towels for rent, 37-39 Cowgate, Grassmarket District, +44 (0)131 226 6351, budgetbackpackers.com, hi@budgetbackpackers.com
#### **Caledonian Backpackers**
This hostel was recommended to me by a local Edinburghian. I love it for the funky atmosphere with colorful designs coating the walls, as well as the on-site bar and café. The hostel also promotes a social atmosphere with its many common rooms and hangout spots. The rooms and facilities certainly aren't the cleanest I've found in Europe, but if you're not that picky, it'll do.
From £16, 24-hour reception, bar/café, free breakfast, free Wi-Fi, movie room, 3 Queensferry St, New Town, +44 (0)131 226 2939, caledonianbackpackers.com, reception@caledonianbackpackers.com
#### **Castle Rock Hostel**
This was voted as one of the top 10 hostels in the world back in 2010—and for good reason. With its high ceilings, old-fashioned furniture, fireplace, and stone-brick exterior, you really get to experience that "Scottish vibe." The rooms are clean, and it's the pickup location of the daily free walking tour of the city (enquire about it at the front desk on check-in).
From £11, free Wi-Fi, 24-hour reception, laundry (£3.50), breakfast (£1.80), bedside lockers, 15 Johnston Terrace, Royal Mile, +44 (0)131 225 9666, castlerockedinburgh.com, castlerock@macbackpackerstours.com
#### **High Street Hostel**
Find this gem of a hostel with welcoming staff and safe, secure rooms just downhill from the Royal Mile on Blackfriars Street. The rooms are quiet even though you've got a ton of fun bars on the same street just steps away; you've got the best of both worlds.
**SHOULD WE STAY OR SHOULD WE GO? THE VOTE FOR INDEPENDENCE**
Scotland was all over the news for their momentous vote on independence from London on September 18, 2014. The controversy really swirled around the pros and cons of being governed from outside their national border. While Scotland benefits quite a lot from being incorporated into England, many nationalists believe Scotland would be better off on its own.
Those in favor of independence lean on the vast oil fields in the North Sea. Scotland's GDP has grown faster than England's over the last few years, thanks to oil exports driving a healthy economy. Those in favor of independence highlighted the importance of home rule: to set priorities in foreign policy, the military, health care, and education in line for what's best for Scots and Scotland over anyone else.
Those who would have voted "no" point to things Scotland would have to sacrifice and the issues it would have to start from scratch on, like currency and health care. Those who voted against independence tended to be more conservative, opting for a system that may not be perfect but is definitely better with respect to security and general stability.
The counting went late into the night on September 18, but by the morning, it was clear that Scotland would remain in the union, at least for now. The final figures shook out to over 2 million no votes against about 1.6 million yes votes, with a remarkable 84 percent participation rate in the voting. Glasgow was the only council area to vote for full independence. Many don't consider this to be the end of the independence issue, but Scotland received accolades from all over the planet for their peaceful demonstration of the democratic process.
Large dorms from £12, free Wi-Fi, 24-hour reception, breakfast (£2), luggage storage, lockers, 8 Blackfriars St, Royal Mile, +44 (0)131 557 3984, highstreethostel.com, reservations online
#### **Kick Ass Hostel**
With a name like this, it's gotta be good, and it is! New and brightly decorated, with an on-site bar and a social atmosphere, this is one of my favorite hostels in town. With a "Best New Hostel UK 2015" award under their belt, the team at Kick Ass Hostel is off to a great start. Take in an unobstructed view of the castle from the dorm windows and charge up with plugs next to every bed in the house.
Large dorms from £14, free Wi-Fi, 24-hour reception, on-site bar and drink specials, ping pong and pool tables, laundry facilities, 2 W Port, Old Town, +44 (0)131 226 6351, kickasshostels.co.uk, reservations online
#### **Argyle Backpackers**
Located just across the meadows from the University of Edinburgh, this central hostel has been welcoming budget travelers for over 25 years. Choose from their cozy singles, double, triples, and four- and six-bed dorms, and feel at home in their welcoming common room and kitchen. The hostel is comfortable enough, but the Wi-Fi doesn't reach all rooms.
Dorms from £23, reception 09:00-22:00, check in after 14:00, three-night minimum stay in Aug and over New Year's, Wi-Fi in parts, luggage storage, kitchen access, common room, towels for rent, 14 Argyle Place, University District, +44 (0)131 667 9991, argyle-backpackers.com, reception@argyle-backpackers.com
### **TRANSPORTATION**
#### **GETTING THERE & AWAY**
##### **Plane**
**Edinburgh Airport** (EDI, edinburghairport.com) is just 20 minutes from the center, and many of the budget airlines offer connections throughout the continent. A cab from the airport to the city center runs _£_ 25-30. Otherwise, the frequently leaving **Airlink 100** bus connects you to the Waverly Bridge Station for _£_ 4.50 each way in about half an hour. A **tram** system also connects Edinburgh Airport with the city center ( _£_ 5) in about 40 minutes, stopping throughout the New Town along the way.
Don't forget to consider **Glasgow Airport** (GLA, glasgowairport.com) as a secondary option. While it is about an hour away, and getting to Edinburgh takes two transfers on the train, there may be substantially cheaper connections.
##### **Train**
Edinburgh has two primary train stations, **Haymarket Station** and **Edinburgh Waverly Station,** located on either end of central Edinburgh, so it's important to confirm which one your train will arrive to or depart from.
To avoid the probable long lines at the bigger train station, Waverly, purchase your ticket in advance at virgintrainseastcoast.com. Direct express trains to London take about four hours and depart frequently.
##### **Bus**
**National Express** (nationalexpress.com) and **MegaBus** (megabus.com) offer dozens of national and international bus connections into Edinburgh, though the connection times, which can run over 10 hours from London, may bring many to consider the budget airline options. Reclining seats and Wi-Fi en route make the ride more bearable, and ticket prices (from _£_ 10) make it an attractive option for many on a tight budget.
##### **Car**
From London, the M6 leads you through the beautiful lake districts to Edinburgh in about 7.5 hours, assuming minimal traffic. Otherwise, consider the smaller A1 for more detours and stops along the way.
#### **GETTING AROUND**
Walking is the best way to explore the compact city of Edinburgh. It's only about a 20-minute walk across town. Walking will save you money and allow you to soak in every nook and cranny this city has to offer—there are tons of them. Just keep in mind that the steep slopes up to the Old Town can wind some folks.
Edinburgh's public transportation system consists of buses and a limited tram system that connects the airport with the city center, stopping in New Town along the way. There is no train or metro transportation within the city.
##### **Bus**
Buses are a handy way to visit some of the city's farther-flung sights. If you plan to utilize public transportation a lot, buy a **day ticket** ( _£_ 3.50), which gives you 24-hour access to all buses and trams in Edinburgh—paying for itself by the third ride. Purchase your ticket either on board the bus or from a Lothian Bus Travel Shop. If you purchase your ticket on board, make sure you have the correct fare, as the driver will not have access to change. The same guidelines apply for a **single ticket** ( _£_ 1.50).
##### **Taxi**
Edinburgh has classic black cabs, which you can flag down on the street or find at one of the numerous taxi stands. A downtown ride should run you no more than _£_ 15, but even then it's not cheap. **Uber** is also available in Edinburgh.
##### **Car**
Like just about every other medieval city of Europe, Edinburgh was built for pedestrians and horses and carts, not modern cars. Thanks to the public bus network, cars are not necessary to get around town, but they do provide some flexibility if you're considering heading into the Scottish countryside. If that's your plan, Edinburgh's **Park & Ride stations** (edinburgh.gov.uk/parkandride) let you park on the outskirts of the city for free and take public transportation into the city. You just need to pay for your bus ticket into town. Look up locations in Hermiston (open 24 hours), Ingliston (04:00-02:00), Sheriffhal (24 hours), and Straiton (24 hours) for options. All are free and open seven days a week.
##### **Bicycle**
Edinburgh is a cycling-friendly city with tons of bike lanes and more on the way. The city council is currently working to enact a public bike rental scheme as in Paris and London. In the meantime, find one of the numerous shops where you can rent a bike for around _£_ 12 per half day. These include **Cycle Scotland** ( _£_ 20/day, 29 Blackfriars St, +44 (0)131 556 5560, cyclescotland.co.uk) in the Old Town and **Bike Trax** ( _£_ 17/day, 11-13 Lochrin Pl, +44 (0)131 228 6633, biketrax.co.uk) in the West End. Get ready to sweat it out on the steep hills! If it starts raining, the cobblestones get slippery; I recommend parking the bike and walking.
### **DAY TRIPS**
You've got a slew of day trips from Scotland's capital that will get you out into the rugged countryside and away from the big groups of tourists. The best way to do it is with a local guide on a fun group tour. Check out my recommended **tour companies** for options for whisky, visits to castles, hikes out in the countryside, and more. Prices run from _£_ 50 for the day, and options are available to keep you busy for weeks.
#### **Glasgow**
Just a one-hour train ride away, Scotland's biggest (and proudly patriotic) city is well within reach on a day trip from Edinburgh. With 100,000 more residents, Glasgow is Edinburgh's more industrial, grittier, hipster big sister, but is simultaneously refined with tons of museums and interesting sights. Attractions include the **University of Glasgow** (free, campus always open, about 1.5 miles west of the city center, gla.ac.uk), **Kelingrove Art Gallery and Museum** (free, Sun and Fri 11:00-17:00, Mon-Thurs and Sat 10:00-17:00, Argyle St, glasgowlife.org.uk), the **Gallery of Modern Art** (free, Sun and Fri 11:00-17:00, Mon-Thurs and Sat 10:00-17:00, Royal Exchange Square, glasgowlife.org.uk), and a bustling city center. For those interested in art nouveau (like I am), Glasgow was the home of Charles Mackintosh, the leader of the northern art nouveau. Mackintosh dabbled in everything from design and architecture to watercolor and furniture. You'll see a number of his works as you walk the streets, including the Glasgow School of Art and the _Glasgow Herald_ newspaper offices.
Glasgow also has a raging nightlife scene that comes along with the big city feel you'll find here in the modern grid layout. The **West End** (around Byres Road) caters mostly to students thanks to its proximity to the university there. Otherwise, **Bell Street** and **Vincent Street** offer a string of great bars and clubs. Remember the early closing policies of Glasgow—midnight for bars, and 03:00 for clubs, so don't dilly-dally on your night out! If you've got a few extra days and prefer cities over natural beauty, Glasgow's definitely worth an overnight. Trains from Edinburgh take just over one hour for Glasgow and run about _£_ 15, with frequent departures.
#### **Loch Ness**
Everyone's heard of the legend of Nessie, the aquatic monster of Loch Ness. So, many want to take a day trip north to the lake to see if they can spot her themselves. It's doable in a day, but you'll be leaving early and getting back late, as it's a 3.5-hour straight drive each way. Your best bet is to link up with organized day trips that include fun guides and interesting stops along the way. I recommend **Haggis Adventures,** and their day trip starts at _£_ 65 for the 12-hour round-trip tour (haggisadventures.com).
### **HELP!**
#### **Tourist Information Centers**
Find the main tourist information center just on the border of the New Town closest to the Old Town, at the top of the Princes Mall shopping center. Pop in here for pamphlets, maps, and souvenirs, but remember that a lot of the placement is paid for by the businesses.
#### **Pickpockets & Scams**
My best advice is to simply have your wits about you when walking the streets. Avoid sketchy neighborhoods, stay in well-lit, populated areas, keep your valuables close at hand, and never leave any belongings unattended. While extremely rare, pickpocketing and mugging can still occur if you follow these guidelines, but the chances will be greatly reduced if you use good old-fashioned common sense.
#### **Emergency**
In an emergency, dial 999.
#### **Hospital**
Royal Infirmary of Edinburgh
51 Little France Crescent
+44 (0)131 536 1000
#### **US Consulate**
3 Regent Terrace, Edinburgh EH7 5BW
+44 (0)131 556 8315
## **APPENDIX**
Flying to Europe from the United States
City-Hopping
Transit Times
Local Transportation
Accommodations & Food
Mobile Technology
Resources
Useful Phrases
Catalan
Czech
Dutch
French
German
Hungarian
Italian
Spanish
### **FLYING TO EUROPE FROM THE UNITED STATES**
Booking long-haul flights to Europe should be a two-part process: First, find your longer flight to the continent, then search for shorter flights into your destination of choice. If you're beginning your visit in Dublin or Venice, for example, search for direct flights to those cities, but also look into flights to **major transportation hubs** like London, Amsterdam, Frankfurt, and Paris. These airports offer many of the best connections to and from the States. If you're flying from the East Coast, you have additional direct flight options to cities like Madrid and Rome.
If you only frame your search by your desired trip starting point, search engines may get stuck on getting you to that destination on the same airline, skipping over good options. The more creative and flexible you are in your search, the more interesting opportunities you'll find. To promote tourism, airlines like Iceland Air even have deals that offer free multiple-day layovers in certain destinations.
### **CITY-HOPPING**
With planes, trains, and buses, getting around Europe is easy. Each mode of transportation comes with pros and cons. Buses, for example, are often the cheapest option but take the most time. Weigh the value of your time in the destination against the money that you'll save by opting for the slower route.
#### **FLYING**
Budget airlines have opened up the continent to backpackers on a budget. Before the advent of cheap flights, a trip from Paris to Madrid would have been an uncomfortable 14-hour overnight train ride. Now, airlines like Ryanair and EasyJet offer a stripped down service for those who value low prices over comfort. While the days of $5 flights are long gone, you can still find deals to get around the continent for less than $100, as long as you know how to play the budget airline game.
##### **Finding Cheap Flights**
Let's help you get finding cheap flights down to a science. First, open up a series of web browser windows and conduct the same search across all these different websites:
**•skyscanner.net**
**•cheapoair.com**
**•kayak.com**
**•momondo.com**
Keep the search as flexible as possible across travel dates, travel times, and airports. Each search engine uses different algorithms to find results, so you may get back four different answers. Pay close attention to the following details:
**• Departure and arrival airports.** Many airlines use budget regional airports a couple hours outside a city. While this keeps costs down, it adds to your connection time and costs to travel into the city. Watch out for this in Paris and Barcelona in particular.
**• Baggage allowance.** Note how much it costs to check a bag. You are often given the choice between 15kg (33lb) and 20kg (44lb) for your checked bag. Pick the accurate one—you'll need to pay extra if you're over by even one kilo.
**• Departure time.** Cheap airlines save money by running super early and super late flights. Some flights leave so early that public transportation to the airport is not even running yet. If that's the case, factor in another €30-40 for a taxi!
##### **Booking Cheap Flights**
When you've selected your flight, book directly through the airline's website. This cuts out any middleman fees and potential system glitches, and puts you directly in touch with the airline.
The cheaper the airline, the more ads you'll have to click through and the more "options" you'll have to decline during the booking process. Ryanair offers you luggage, hotels, car rentals, airport transfers, and all sorts of other stuff you don't need or want. You'll have to scour each section to click the No Thank You box.
Be wary when it comes to the final checkout and payment. Before you click your final confirm, double-check the following elements:
**• Departure time.** It will be in 24-hour time! For reference: 09:00 = 9am, 19:00 = 7pm.
**• Departure date.** It will likely be in European format: February 7 is 07/02, not 02/07. Check the date visually on a calendar, or find it spelled out before purchasing.
**• Number of bags you'll need to check.** Pay to check a bag now so you don't need to do so later.
**• Departure and arrival airports.** Some cities have multiple airports, so always note the airport's full name. Also note necessary connection distances and costs.
**• Final price.** Make sure the price hasn't risen exorbitantly since your first search. Tally up the numbers and be sure you understand them. Always decline the option to be charged in US dollars, which is a scam to get you to pay a needless service fee.
##### **Avoiding Hidden Costs**
Extra fees add up. It is important to read and abide by the fine print in all materials relating to your budget flight. Watch out for and avoid these hidden costs:
**• Airport check-in fee.** If you don't check in online ahead of time, some airlines now charge a €40-plus penalty. To avoid this fee, pay attention to all emails you receive from your airline as your date of travel approaches.
**• Baggage checking and carry-on fees.** Checking a bag usually costs €35 for the first bag and €50 for a second. Certain airlines also charge for carry-on bags that are any larger than a medium-sized purse. If you're on a budget, pack light.
**• Boarding pass printing fee.** Print your boarding pass ahead of time, or keep it on your mobile device. Also note that some airports have printing stations that allow you to avoid this fee.
**• Heads-up:** Ryanair requires all non-EU passport holders to get their visa and passport checked before going through security. You need a stamp on your boarding pass clearing you through to your gate. Be sure to do this ahead of time to avoid missing your flight.
##### **Budget Airlines by City**
There are many budget airlines based in cities throughout Europe. Pay attention to which city acts as a hub for which airline and you'll generally find cheaper flights.
**Amsterdam:** Transavia
**Barcelona & Madrid:** EasyJet, Vueling
**Berlin:** Air Berlin
**Budapest:** Wizz Air
**Dublin:** Aer Lingus, Ryanair
**DON'T BE PENNY WISE & POUND FOOLISH**
I recently needed to get from Interlaken, Switzerland to Prague for my next WSA tour. My options were a €210 direct flight to Prague on Swissair at a comfortable hour or an earlier €120 flight with EasyJet/Wizz Air that connected through Rome. I jumped on the cheaper option. Here's how the associated costs shook out:
**€55** Interlaken-Basel train (about €20 more than the connection would have been to Swissair's airport)
**€32** to check my extra bag at the gate
**€15** to check in at airport in Rome
**€17** on food throughout the course of the day
**€38** to check my second bag again
This all resulted in additional costs to the tune of at least €20 more for a less convenient flight. The more expensive flight would have wound up saving me money! Learn from my mistakes and do the math before booking.
### **TRANSIT TIMES**
Use this chart to plan the best mode of transit between cities. Travel time (given in hours) is approximate.
**Edinburgh & London:** EasyJet
**Paris:** Air France
**Prague:** SmartWings, Czech Airlines
**Rome, Florence, & Venice:** Alitalia
#### **Not-So-Budget Airlines**
If you prefer convenience, comfortable seats, and free luggage allowance, budget airlines may not be for you. With these conventional airlines, you'll generally forgo the sneaky hidden costs for a little more up front: **British Airways, Lufthansa, Turkish Airlines, KLM,** and **Swissair.**
#### **TRAINS**
Trains are the middle ground between planes and buses for both cost and time en route. Do your search at **sbb.ch,** and book your tickets at local train stations. All of Europe is on the same nifty system, so you can buy train tickets for travel within Italy while you're still in Paris, for example.
##### **Eurail Passes: Are They Worth It?**
Everyone has heard of Eurail passes, but almost no one knows if one is right for them. I'd say they're a good option for the organized traveler who knows when and where they want to go. Eurail passes allow unlimited access to as many train rides in as many countries you decide for a flat rate (in a limited time period). Pass price depends on three factors:
**•** The number of countries in which it is valid
**•** The number of travel days included (5, 10, 20, unlimited)
**•** The window of dates during which the pass is valid (3 weeks, 6 weeks, 2 months)
Minimize each of these factors to get the best rate. Consider how much time you'll be spending in each country. If you'll be traveling in Italy, Spain, and France, with a side trip to Amsterdam, book the Eurail pass for three countries only, paying retail for the short trip to Amsterdam from Paris.
Visit eurail.com to view the available options. Personally, I prefer to do my timetable research online and then book my train tickets in person at any train station. This ensures that I receive all applicable discounts, and allows flexibility that I wouldn't get with the Eurail pass.
#### **BUSES**
While buses are generally slower than trains and planes, they're most likely your cheapest choice. I've found great bus connections to numerous cities departing daily across the continent. If you sign up for an overnight bus trip, pack some hefty sleeping pills and eye covers. Keep your valuables in a money belt and wear it into the bus so you're not showing everyone your valuables before you go to sleep. Buses are safe, but there are a lot of people getting on and off throughout the night.
Some of my favorite bus line websites are **eurolines.com, orangeways.com, berlinlinienbus.de,** and **studentagencybus.com.** Buses are more popular in Central and Eastern Europe, and provide a number of connections, some of which are as fast as some trains.
#### **RIDE-SHARING**
The sharing economy has grown into inter-city drives across Europe. Luckily for us backpackers, there's a great website (carpooling.co.uk) where you can search for rides, post a needed ride, or jump on a trip last minute. This option has saved my hide more than once! Of course, safety is always in numbers, and trust your gut when it comes to meeting your driver and actually getting in the car.
#### **CAR RENTAL**
Renting a car can be expensive and frustrating. Planes, trains, and buses are often better choices. If you do want to rent a car, consider the following first:
**•** Most cars in Europe have manual transmissions.
**•** Parking is challenging and expensive and is regulated differently from one city to the next.
**•** Signage may not be in English.
**•** Those under age 25 are obligated to purchase more expensive insurance.
**•** Gas runs US$8-10 per gallon.
**•** Most cars are on diesel. Don't fill up the tank with standard gas, or you'll have to pay for a new engine!
**•** Read the fine print in the rental contract, keeping in mind accident insurance, when the car needs to be back, whether the tank needs to be full upon return, and if there is an additional cost for driving outside the country.
### **LOCAL TRANSPORTATION**
Reading a bus or metro map can be like trying to understand a foreign language. But never fear! It's really simple once you get the hang of it. I've broken everything down for you. Google Maps has integrated just about all public transportation systems and can recommend directions.
#### **TICKETS**
Each city has its own system of ticket validation. I've outlined the exact process in each chapter, but the core concept is quite simple: After purchasing your ticket, you must validate it, or get a stamp on your ticket that shows what time you boarded the system. Tickets are generally time-based; validating starts the clock on your 30, 60, or 90 minutes of validity. You validate your ticket either onboard or beforehand at one of the validation machines on the platform. There are hefty fines if you get caught with an unvalidated ticket—but don't let police officers take you to an ATM and force you to withdraw cash. Instead, insist on a paper ticket.
Certain cities, like Budapest, offer multi-day transportation tickets that pay for themselves after a couple of rides. Some sightseeing packages include local transportation as well. Do your research ahead of time, or speak to an attendant at a major station on arrival to see what multi-day options are available.
#### **METRO**
As long as you know the name of your stop and can locate it on the metro map, you won't have any problems reaching your destination. As you enter into the metro, you will see signs for northbound, southbound, eastbound, or westbound platforms. Metro maps are oriented north-south-east-west. On the map, simply locate what station you're currently in and the station you wish to travel to, and go to the platform that gets you in the right direction. If there isn't a direct line between the two stations on the map, you'll need to transfer at the point where the two lines intersect.
#### **BUS & TRAM**
Bus and tram maps follow the same basic format. Locate where you are (usually printed in boldface) in the list of stops on the map. Buses (or trams) will go to every destination listed _below_ the stop you're at. Everything above the stop has already been visited. If your desired stop isn't below your current station, you've got to find the stop for buses going in the opposite direction.
#### **UBER**
Uber is active in a number of European cities, while in others, the taxi unions have blocked the company's access to the market. I mention whether Uber is active in each chapter.
### **ACCOMMODATIONS & FOOD**
I rely almost exclusively on two websites to find my accommodations: **hostelwold.com** and **airbnb.com.** When using either one, do your research by neighborhood. Pick the best neighborhood for you and drill down to those streets to stay in the thick of it rather than commute.
#### **HOSTELS**
I love hostels for their social scene, organized activities, and fun atmosphere. While you'll save money staying in a hostel, you may sacrifice quality sleep. In bigger dorms, people come and go throughout the night and day. Bring a lock or leave your valuables at home.
##### **Hostel Booking Tips**
**•** Note the difference between professional photos and traveler shots. The traveler shots show what the hostel is _really_ like. The professional shots are the ones taken years ago when the hostel first opened.
**•** See what amenities are included. It may save you money to spring for the more expensive hostel if it includes towels, Wi-Fi, breakfast, etc.
**•** Read the reviews, and watch out for the fake ones that some hostels occasionally leave themselves.
#### **AIRBNB**
I use **airbnb.com** if I'm trying to stay in a quiet place with my own room. While this option lacks a social scene, I like having access to a kitchen, knowing that my valuables are secure, and getting to meet a local host who can recommend great bars, restaurants, and more. In certain cities like Paris, Airbnb can be an excellent value, while in cities like Prague and Budapest the value may be less: Hosts charge Western European rates when your money should go much further.
##### **Airbnb Booking Tips**
**•** Narrow your search by the type of accommodation you're comfortable with: entire apartment, private room, or shared room. A private room in a shared apartment will likely be cheaper than having an apartment to yourself.
**•** Airbnb.com no longer allows you to organize search results by price. You'll have to click through several pages to ensure you're getting the best deal.
**•** There are hidden fees! At first, you define your budget, but Airbnb has devised a slick system that rounds up and shows you nice options for a few euros more than what you input. Fees include cleaning fees, service fees, and additional guest fees (displayed rates are often for one occupant only). Double-check the final price before booking.
**•** Read between the lines and see if the apartment is just a money-maker for the owner, or if it's actually their home. Personally, I don't mind either way, but this is a factor for some travelers, depending on whether they want more or less interaction with their host. Read the reviews to anticipate whether the host's style will work for you.
**•** Pay attention to amenities. Are Wi-Fi and heating or air-conditioning included? Is the restroom private or shared?
**•** Check out the photos closely! Wide-angle lenses do wonders for the appearance of size of a room.
**•** Don't hesitate to ask your host about any special offers. It never hurts to ask.
#### **FOOD**
Remember that not every meal of your stay needs to be French, Czech, Spanish, etc. Consider treating yourself to one or two local meals and filling the rest of the gaps with cheaper alternatives. In notoriously expensive London, save by seeking out the city's excellent Indian food. Kebabs, sandwiches, and street food are cheap no matter where you go. Local supermarkets usually have everything you need for a relaxing picnic in the park.
Learn to recognize tourist traps. Don't eat at restaurants that are on the main square of any city. If the restaurants have neon lights and menus in 10 languages in the windows, it's safe to assume the locals don't go there. Get off the main drags and find where the locals eat, and you'll pay the locals'—not the tourists'—price.
### **MOBILE TECHNOLOGY**
With the rise in mobile technologies, travelers can stay better connected than ever.
#### **SMARTPHONES**
A smartphone in your pocket—even if it's on Wi-Fi-only mode—has really changed the backpacker experience. I consider purchasing a local SIM card if I'm in any one country for more than a week. Otherwise, I rely on the free **Wi-Fi** at my hostel, restaurants, and bars.
I don't recommend purchasing **roaming plans** from your home provider for your entire time in Europe, as they're rarely a good deal. Many American providers are willing to temporarily suspend your account and drop the monthly price down to $10, effectively hibernating your plan until you return home.
Smartphones are a hot commodity, so keep them close. Consider finding a cheap phone to use while abroad. This way you stand to lose a lot less than your phone and everything on it while traveling.
##### **European SIM Cards**
You can purchase a European SIM card (the data chip within your phone) in the country you're visiting and top up as needed. With an unlocked smartphone, have the staff at the cell phone store plug in a SIM card. I've used **Vodafone** in most parts of Europe. SIM cards cost around €10 and come with €10 of credit. Opt for the pay-as-you-go plan, topping up by purchasing a scratch card from any convenience store. This method typically costs under €30 per month, whereas monthly plans can be upwards of €60 a month.
**JOIN THE WSA TEAM!**
WSA Europe seeks avid travelers who can articulate our travel values to their fellow students. Our internships give students the opportunity to research and write about their experiences in foreign cities. It's a great and fun way to beef up your résumé! If interested, send your CV and cover letter to info@wsaeurope.com.
##### **Apps**
Your smartphone isn't just a camera for your trip abroad. Apps can go a long way toward enhancing your travel experience. **Skype** is a godsend for calling home easily and affordably. Pick a screen name for yourself and make a few practice calls before leaving home. **Facetime** is free while on Wi-Fi to others at home with an Apple device. See below for more of my favorite apps to use while traveling.
#### **INTERNATIONAL CALLS**
If dialing Europe from the United States, begin with the US international access code (011)—or, if you're dialing from a cell phone, replace the access code with a plus sign, which you get by holding down the 0. Next, dial the country code (each country has a specific one) of the country you're dialing. Next is a regional code, which is often a two-digit number, but sometimes is a single 0. Skip this when dialing internationally, but include it when dialing domestically. After the country code, the final phone number will be 9-10 digits.
#### **CONVERTERS & ELECTRICAL ADAPTERS**
**Converters** "convert" the electrical current so it doesn't blow out your electronics. Take a close look at whatever you're trying to plug in. If you see 110-220V, you _do not_ need an electrical converter. If you only see 110V, you'll need a converter. In my experience, converters are not necessary for iPhones, MacBook Pros, and cameras (including SLRs).
**Adapters** "adapt" American plugs to physically fit into European outlets. You'll need a continental adapter (with two round prongs) for continental Europe and a British adapter (with three rectangular prongs) in England, Scotland, Ireland, and Wales. Adapters are cheap. Purchase them before your trip for convenience. It's also easy find adapters at local electronic stores all across Europe if you're in a pinch.
### **RESOURCES**
#### **TRANSPORTATION WEBSITES**
Make use of helpful **flight search engines** (skyscanner.net, kayak.com, cheapoair.com, momondo.com), sites for booking **train travel** (sbb.ch) and **bus travel** (eurolines.com, orangeways.com, berlinlinienbus.de, and studentagencybus.com, renfe.com), and even a site for **ride-sharing** (carpooling.co.uk).
#### **TRANSPORTATION APPS**
**Airline apps** like EasyJet are streamlined so you can book a flight on the fly with just a few taps. Similarly **, iRail** and **SBB Trains** let you search the Eurail database for train timetables. Much like a flight search engine, they let you prioritize by number of stops and also show the route on a map so you can compare your most direct options.
##### **Skyscanner**
Global search engine that compares flights, though you can't use it to book directly.
##### **Kayak**
Impressively useful—and cheap—flight search engine. Use it for research, then book direct with the airline.
##### **EasyJet**
Book EasyJet flights directly.
##### **Ryanair**
Book Ryanair flights directly.
##### **Aer Lingus**
Book Aer Lingus flights directly.
##### **iRail**
Search the Eurail database for train timetables.
##### **SBB Trains**
Search the Eurail database for train timetables.
##### **TripIt**
Organize your flight travel, keeping confirmation numbers, rewards accounts, etc, handy.
#### **ACCOMMODATIONS WEBSITES**
I rely on two websites for booking accommodations: **airbnb.com** and **hostelworld.com.**
#### **ACCOMMODATIONS APPS**
##### **Airbnb**
Make apartment reservations on the fly.
##### **Hostelworld**
Search for sweet hostels.
#### **APPS FOR KEEPING IN TOUCH**
##### **WhatsApp**
Free messages-over-Wi-Fi service. Set up an account, and text away with any other friends who are also on the service.
##### **Skype**
Free and paid hybrid messaging and calling service that lets you video chat with fellow Skype users.
##### **Facetime**
Free while on Wi-Fi to others at home with an Apple device.
##### **Facebook**
Dial or video call friends around the world for free.
##### **Snapchat**
Send image and video updates available only temporarily.
##### **Tinder**
Let's face it: Meeting people has evolved in the digital age. Swipe from the comfort of your hostel bunk and set up dates with cute locals and fellow backpackers. Similar apps include **Happn** and **Bumble.**
#### **TRAVEL APPS**
##### **Weekend Student Adventures**
My own freemium app provides a ton of on-the-go tips and tricks for more than a dozen of my favorite cities.
##### **Rick Steves' Europe Audio Guides**
Free, informative travel listening. Search and download all the subjects on your destinations before you leave on your trip.
##### **XE Currency**
Keep track of fluctuating conversion rates.
##### **Mint**
Balance your budget while abroad.
### **USEFUL PHRASES**
#### **CATALAN**
**English** | **Catalan** | **Pronunciation**
---|---|---
Hello. | Hola. | **oh** -lah
Excuse me. | Perdó. | pehr- **doh**
Do you speak English? | ¿Parles anglès? | **pahr** -lahs ahn- **glays**
Yes | Si | see
No | No | noh
Please | Si us plau | see oos plow
Thank you | Gràcies | **grah** -see-ahs
Goodbye. | Adéu | ah- **day** -ooh
How much does it cost? | ¿Quant és? | kwahn es
Where are the toilets? | ¿On estan els serveis? | ohn an- **stahn** ehls sehr- **vays**
I'd like | Voldria | vool- **dree** -ah
...a room | una habitació | **ooh** -nah ah-bee-tah-see- **oh**
...a bed | un llit | un yeet
...a ticket |...una entrada | oon-nah ahn- **trah** -dah
...a beer |...una cervesa | **ooh** -nah sahr- **veh** -sah
...wine |...vi | vee
Where is | ¿On està | ohn ah- **stah**
...the train station? |...l'estació de tren? | lah-stah-see- **yo** dah tren
...the bus station? |...l'estació de bus? | las-stah-see- **yo** dah boos
#### **CZECH**
**English** | **Czech** | **Pronunciation**
---|---|---
Hello | Dobrý den | doh- **bree** den
Excuse me | Promiňte | **proh** -meen-tah
Do you speak English? | Mluvíte anglicky? | **mloo** -vee-teh **ahn** -glits-kee
Yes | Ano | **ah** -noh
No | Ne | neh
Please | Prosím | **pro** -seem
Thank you | Děkuji | **dyeh** -ku-yee
Goodbye | Nashledanou | **nah-** skleh-dah-now
How much does it cost? | Kolik to stojí? | **koh** -lek toh sto- **yee**
Where is the toilet? | Kde je záchod? | guh- **deh** yeh **zah-hod**
I'd like | Rad(a) bych | rahd bikh/ **rah** -dah bikh
...a room |...pokoj | **po** -koy
...a bed |...postel | **pos** -tel
...a ticket |...lístek | **lees** -tek
...a beer |...pivo | **pee** -voh
...wine |...vína | **vee** -na
Where is... | Kde je... | guh- **deh** yeh
...the train station? |...nádraží? | **nah** -drah-zee
...the bus station? |...autobusové nádraží? | **ow** -toh-boo-soh-veh **nah** -drah-zee
#### **DUTCH**
**English** | **Dutch** | **Pronunciation**
---|---|---
Hello | Hallo | **ha** -low
Excuse me | Pardon | **par** -don
Do you speak English? | Spreekt u Engels? | spreekt oo **en** -gels
Yes | Ja | ya
No | Nee | nay
Please | Alsjeblieft | alse-bleeft
Thank you | Dank u wel | dahnk oo vehl
Goodbye | Doei | doo-ie
How much does it cost? | Wat kost het? | vaht kost het
Where is the toilet? | Waar is het toilet? | var ees heht **twah** -leht
I'd like... | Ik wil graag... | eek vil ghraagh
...a room |...een kamer | ayn **kah** -mer
...a bed |...een bed | ayn bed
...a ticket |...een kaartje | ayn **kart** -yeh
...a beer |...een biertje | ayn biert-je
...wine |...wijn | vine
Where is... | Waar is... | var ees
...the train station? |...het station | het **stash** -yun
...the bus station? |...het bus station | het boos **stash** -yun
#### **FRENCH**
**English** | **French** | **Pronunciation**
---|---|---
Hello | Bonjour | bohn- **zhoor**
Excuse me | Excusez-moi | eggs- **cue** -say **mwah**
Do you speak English? | Parlez-vous anglais? | **par** -lay voo ahng- **lay**
Yes | Oui | wee
No | Non | nohn
Please | S'il vous plait | see voo play
Thank you | Merci | mehr- **see**
Goodbye | Au revoir | oh ruh- **vwah**
How much does it cost? | Combien? | cohm-bee- **ahn**
I'd like... | Je voudrais... | zhe **voo** -dray
...a room |...une chambre | oon **shahm** -bre
...a bed |...un lit | oon lee
...a ticket |...un billet | oon **bee** -yay
...a beer |...une bière | oone bee- **air**
...wine |...vin | van
Where is... | Ou est... | oo ay
...the train station? |...la gare | lah gar
...the bus station? |...la gare routière | lah gar root- **yehr**
#### **GERMAN**
**English** | **German** | **Pronunciation**
---|---|---
Hello | Guten Tag | **goo** -ten tahg
Excuse me | Entschuldigung | en- **shool** -di-gung
Do you speak English? | Sprechen Sie Englisch? | **spree** -ken-zee eng-lish
Yes | Ja | ya
No | Nein | nine
Please | Bitte | **bee** -ta
Thank you | Danke | **dahn** -ke
Goodbye | Auf Wiedersehen | auf **vee** -der-zay-ehn
How much does it cost? | Wie viel kostet es? | vee feel **kost** -et es
Where is the toilet? | Wo sind die Toiletten? | voh zint dee toy- **leh** -tehn
I'd like... | Ich möchte... | eekh **mukh** -te
...a room |...ein Zimmer | ain **zimm** -er
...a bed |...ein Bett | ain bett
...a ticket |...eine Karte | ain **kar** -teh
...a beer |...ein Bier | ain beer
...wine |...Wein | vine
Where is... | Wo ist... | voh eest
...the train station? |...der Bahnhof | der **bahn** -hof
...the bus station? |...der Busbahnhof | der **boos** -bahn-hof
#### **HUNGARIAN**
**English** | **Hungarian** | **Pronunciation**
---|---|---
Hello | Szia | **see** -yaw
Excuse me. | Bocsánat | **boh** -chah-nawt
Do you speak English? | Beszész angolul? | **beh** -say-es **ahn** -go-lool
Yes | Igen | **ee** -gan
No | Nem | nem
Please | Kérem | **kay** -rehm
Thank you | Köszönöm | **koo** -sze-nem
Goodbye | Viszlàt | **vees** -lat
How much does it cost? | Mennyi? | **men** -yee
I'd like... | Kérnék/ Kérnénk | **kayr** -nayk/ **kayr** -naynk
...a room |...egy szobàt | eidge **so** -bot
...a bed |...egy àgyat | eidge adg-yacht
...a ticket |...egy jegyet | eidge **yeg** -yet
...a beer |...egy sör | eidge shohr
...wine |...bor | bor
Where is... | Hol van... | hol van
...the train station? |...pàlyaudvar | **pah** -yood-var
...the bus station? |...buszpàlyaudvar | **boos** -pah-yood-var
#### **ITALIAN**
**English** | **Italian** | **Pronunciation**
---|---|---
Hello | Buon giorno | bwon **jor** -noh
Excuse me | Permesso | pear- **may** -soh
Do you speak English? | Lei parla inglese? | lay **par** -lah een- **gle** -zay
Yes | Si | see
No | No | noh
Please | Per favore | pear fah- **vor** -ay
Thank you | Grazie | **grah** -zee-ay
Goodbye | Ciao ciao | chow chow
How much does it cost? | Quanto costa? | **kwan** -toh **koh** -stah
Where is the toilet? | Dove la toilette? | doh- **veh** lah twah- **leh** -tay?
I'd like... | Vorrei... | voh- **ray**
...a room |...una camera | **oo** -nah **kam** -eh-rah
...a bed |...un letto | un let-toh
...a ticket |...un biglietto | oon bee-lee- **eh** -toh
...a beer |...una birra | **oo** -nah bee-rah
...wine |...vino | **vee** -noh
Where is... | Dove... | do-vay
...the train station? |... stazione | stah-zee- **oh** -nay
...the bus station? |...stazione autobus | stah-zee- **oh** -nay **ow** -toh-boos
#### **SPANISH**
**English** | **Spanish** | **Pronunciation**
---|---|---
Hello | Hola | **oh** -lah
Excuse me | Perdone | pehr- **doh** -nay
Do you speak English? | ¿Habla usted inglés? | **ah** -blah oo- **sted** een- **glays**
Yes | Sí | see
No | No | noh
Please | Por favor | por fah- **bor**
Thank you | Gracias | **grah** -thee-ahs
Goodbye | Adiós | ah-dee- **ohs**
How much does it cost? | ¿Cuánto cuesta? | **kwan** -toh **kwest** -ah
Where are the toilets? | Dónde están los servicios? | **dohn** -day ay- **stahn** lohs sehr- **bee-** thee-ohs
I'd like... | Me gustaría... | may goo-stah- **ree** -ah
...a room |...una habitación | **ooh** -nah ah-bee-tah-thee- **ohn**
...a bed |...una cama | **ooh** -nah **kah** -mah
...a ticket |...un billete | oon bee- **yeh** -tay
...a beer |...una cerveza | **ooh** -nah ther- **beh** -thah
...wine |...vino | **vee** -noh
Where is... | Dónde está... | **dohn** -day eh- **stah**
...the train station? |...estación de tren | eh-stah-thee- **ohn** day tren
...the bus station? |...estación de autobuses | eh-stah-thee- **ohn** day ow-tow- **boo** -sehs
## **INDEX**
**A** | **B** | **C** | **D**
---|---|---|---
**E** | **F** | **G** | **H**
**IJ** | **K** | **L** | **M**
**N** | **O** | **P** | **QR**
**S** | **T** | **U** | **V**
**WXYZ** | | |
**A**
A&O Berlin Hauptbahnhof:
A&O Berlin Mitte: 243-244
Abbey Court (Dublin):
Abbey Theater (Rome): ,
Abigail's Hostel (Dublin):
Absintherie (Prague): , 264-265
Abszolut Pho (Budapest):
Academy Hostel (Florence):
Accademia (Florence): , ,
Accademia (Venice): ,
adapters:
Las Adelitas (Prague): ,
Airbnb:
airports: Amsterdam ; Barcelona ; Berlin ; Budapest ; Dublin ; Edinburgh 347-348; Florence ; London , 39-40; Madrid ; Paris , ; Prague ; Rome ; Venice
air travel: 350-354
Akab (Rome):
Akbar (Rome):
Albert Cuyp Market (Amsterdam): ,
Albert Heijn Grocery Stores (Amsterdam):
Alcázar of Toledo (Madrid):
Alessandro's Palace (Rome):
Alexanderplatz (Berlin):
L'Alibi (Rome): ,
Alice Pizza (Rome): ,
All'Antico Viniao (Florence): ,
Al Merca (Venice):
Al Pesador Osteria (Venice):
Altberliner Kaffeestube (Berlin): ,
Alternative Berlin: ,
Alternative Berlin Pub Crawl:
Amorino (Paris): ,
Ampellmann (Berlin):
Amsterdam: 71-96; coffeeshops , , ; day trips ; emergencies ; events ; history ; hostels , ; itinerary 78-79; maps 72-73, 74-75; neighborhoods ; nightlife 87-90; planning tips ; recreation 91-92; restaurants 83-85; shopping 90-91; sights 80-82; tours ; transportation 94-95
Amsterdam Centraal Station: ,
Amsterdam's Ultimate Party: ,
Anchor, The (London): 30-31
Ancient Roman ruins archaeological site (Florence):
Ancient Rome neighborhood:
Ancora Piano Bar (Venice):
Andrássy Út (Budapest):
Andy Wahloo (Paris):
Angelato (Prague): ,
Angelina (Paris):
Anker't (Budapest):
Anne Frank House (Amsterdam): , , 80-81
Anonymous Bar (Prague): , 263-264
Antico Forno (Venice):
Arc de Triomphe (Paris): ,
Arendsnest (Amsterdam): 88-89
Argyle Backpackers (Edinburgh):
Arte dell Pizza (Venice): 164-165
Arthur's Seat (Edinburgh): ,
Astor Hyde Park Hostel (London):
Astronomical Clock (Prague): , 254-255
A38 (Budapest):
Aufsturz (Berlin): ,
Avalon House Hostel (Dublin):
L'Avant Comptoir (Paris):
Avenue de la République (Paris):
Avoca (Dublin):
**B**
Back-Factory (Berlin): ,
Bag & Purse Museum (Amsterdam):
Bagel Company Berlin:
Bakehouse, The (Dublin): , 310-311
Bakery, The (Dublin):
Bamboo Lounge (Florence):
B&B Lunchroom (Amsterdam): ,
Bank Bar (Dublin):
Banshee Labyrinth (Edinburgh): , ,
Baptistry (Florence):
Barcelona: 195-220; day trips ; emergencies ; events ; history ; hostels 217-218; itinerary 202-203; maps 196-197, 198-199; neighborhoods ; nightlife 211-212, ; planning tips ; recreation 215-216; restaurants 208-210; shopping 214-215; sights 204-208; tours 216-217; transportation 218-220
Barceloneta:
Bar Celta Pulperia (Barcelona): 209-210
Bar del Fico (Rome): , ,
Il Baretto (Rome): ,
Bargello Museum (Florence): , 137-138
Barlangaszat Caving Tour (Budapest): ,
Barmy Badger Backpackers (London):
Barnacles Temple Bar Hostel (Dublin): 320-321
Bar Open Baladin (Rome):
Bar Ourcq (Paris):
Barri Gotic (Barcelona): , ,
bars: Amsterdam 88-89; Barcelona 211-212; Berlin 235-236; Budapest 289-290; Dublin 314-317; Edinburgh ; Florence 143-144; London 30-32; Madrid ; Paris 61-62; Prague 263-265; Rome , ; Venice
Bar Santurce (Madrid): ,
Bar with No Name (Dublin): ,
Basilica di San Lorenzo (Florence):
Basilica di Santa Croce (Florence):
Bastille (Paris): ,
Bastille Day (Paris):
bathhouses (Budapest): ,
Battersea Park (London):
Bauhaus Archives (Berlin): , 231-232
Baxter Hostel (Edinburgh):
Bazar (Amsterdam):
beaches (Barcelona):
Bear's Den (Paris):
Beas Vegetarian Dhaba (Prague):
Beefeater-guided tours (London):
Beehive Hostel & B&B (Rome):
beer/breweries: Amsterdam , , , , , ; Berlin 234-235, ; Dublin , , ; Edinburgh ; London ; Prague ,
Bell Street Glasgow:
Belvarosi Disznotoros (Budapest): ,
Berghain (Berlin): ,
Bergmannstrasse (Berlin):
Berlin: 221-246; day trips 245-246; emergencies ; events ; history ; hostels 242-244; itinerary 226-227; map 222-223; neighborhoods ; nightlife 234-237; planning tips ; recreation 240-241; restaurants 232-234; shopping ; sights 228-232; tours 241-242; transportation 244-245
Berliner Dom:
Berlin's Jewish Museum:
Berlin Wall:
Berlin Wall Memorial: ,
bicycling: Amsterdam , ; Barcelona 219-220; Berlin ; Budapest , 294-295, ; Dublin ; Edinburgh ; Florence , ; London ; Madrid , ; Paris ; Prague , ; Rome ,
Bierhof Rüdersdorf (Berlin):
Big Ben (London): , 23-24
Biko Prague Bike Tours:
Birreria Centrale (Florence):
Black Cab (London): 40-41
Blackfriar, The (London):
Block of Discord (Barcelona): , 205-206,
Bloemenmarkt (Amsterdam): , 90-91
Bloody Stream (Dublin): ,
Bloom Lane (Dublin):
Blue Café (Amsterdam):
Bo de' Be (Barcelona):
Bodega La Ardosa (Madrid): ,
Bohemia Bagel (Prague): ,
Boiler House (London):
Bonfire Night (London):
Book of Kells (Dublin): ,
Boom Chicago (Amsterdam): ,
La Boqueria (Barcelona): ,
Borgo degli Albizi (Florence):
El Born (Barcelona): ,
Borough Market (London): ,
Bors GasztroBar (Budapest):
Botanical Gardens (Madrid):
Boteco do Brasil (Edinburgh): ,
Bounce (London):
Brandenburg Gate (Berlin): ,
Brazenhead (Dublin): ,
Breakfast Club (London):
BrewDog (Edinburgh): ,
Brewdog (London):
Brewer's Walks Berlin:
Brick Alley Café (Dublin): ,
Brick Lane (London): ,
Brick Lane Market (London):
Bridge of Sighs (Venice): ,
British Museum (London): ,
British Museum neighborhood (London):
Brother Hubbard (Dublin):
Brouwerij de Prael (Amsterdam): ,
Brouwerij't Ij (Amsterdam): ,
bruin cafés (Amsterdam):
Bruntsfield Place (Edinburgh):
Bubbledogs (London):
Buca Mario (Florence): ,
La Buchetta (Florence):
Buckingham Palace (London):
Buda Bike (Budapest): 294-295
Budapest: 275-298; bathhouses , ; day trips 297-298; emergencies ; events ; history ; hostels 295-296; itinerary 280-281; map 276-277; neighborhoods ; nightlife 288-291; planning tips ; recreation , ; restaurants 286-288; shopping ; sights 282-285; tours 294-295; transportation 296-297
Budapest Boat Party:
Budapest Party Hostels: ,
Buddha Bar (Prague):
Budget Backpackers (Edinburgh):
Bull & Castle (Dublin): ,
Bulldog, The (Amsterdam):
El Bulldog (Madrid):
Bulldog Hotel & Hostel (Amsterdam):
bullfights (Madrid): 180-181
Burano (Venice): , ,
Burger & Lobster (London):
Burrito Loco (Prague): , 261-262
Busaba Eathai (London): ,
bus 11 (London): ,
buses: general discussion , ; Amsterdam , ; Barcelona ; Berlin , ; Budapest , ; Dublin , ; Edinburgh ; Florence ; London ; Madrid , ; Paris , ; Prague , ; Rome , ; Venice
BVJ Louvre (Paris):
**C**
Cabaret Voltaire (Edinburgh):
Cadenhead's Whisky Shop (Edinburgh): ,
Café de Prins (Amsterdam):
Café en Seine (Dublin):
Café Filermo (Venice): ,
Café Fleury (Berlin):
Café Habana (Edinburgh):
Café Louvre (Prague): ,
Le Café Marly (Paris):
Café Melos (Madrid):
Café Nod (Prague): ,
Café Oz (Paris):
Caffè Peru (Rome):
Caffe Slowly (Florence):
Caffetteria Oblate (Florence): ,
Caledonian Backpackers (Edinburgh):
Calle Fuencarral (Madrid): 187-188
Calle Serrano (Madrid): ,
Calton Hill (Edinburgh): ,
Camden (London):
Camden Market (London): ,
Camden Pub Crawl (London):
Camden Town (London):
Camera Obscura (Edinburgh): ,
Campanile (Venice): , ,
Campidoglio (Rome): ,
Camp Nou Stadium (Barcelona):
Campo Cesare Battisti (Venice): , ,
Campo de'Fiori (Rome): ,
Campo Santa Margherita (Venice): , ,
canal cruises (Amsterdam): 91-92
Canal Saint-Martin (Paris): , ,
Cannabis College (Amsterdam): , 81-82
Cannaregio (Venice):
Can Paixano (Barcelona):
Cantina Do Mori (Venice): 163-164
Cantina Do Spade (Venice):
Ca' Rezzonico (Venice):
Carlo Menta (Rome): ,
Carnevale (Venice):
Carpe Diem (Barcelona): ,
Carpe Noctem (Budapest):
Carpe Noctem VITAE (Budapest):
Carrer de la Mercè (Barcelona): ,
car travel: general discussion ; Amsterdam ; Barcelona ; Berlin ; Budapest 296-297; Dublin ; Edinburgh ; Florence ; London ; Madrid , ; Paris ; Prague , ; Rome 126-127; Venice 169-170
Casa Amatller (Barcelona): ,
Casa Batllo (Barcelona): , , ,
Casa del Café Tazza d'Oro (Rome): 116-117
Casa del Caffè Tazza d'Oro (Rome):
Casa del Vino (Florence): ,
Casa Lleo Morera (Barcelona): ,
Casa Milá (Barcelona): ,
Casa Patas (Madrid): ,
Casa Rosso (Amsterdam):
Il Casolare (Berlin): ,
Castello (Venice):
Castle District (Budapest): ,
Castle District (Prague):
Castle Rock Hostel (Edinburgh):
Cathedral of Barcelona: , ,
Cathedral of Fiesole (Florence):
Cathedral of Toledo (Madrid):
Cat's Hostel (Madrid):
Catwalk (Barcelona):
La Caves a Bulles (Paris):
caving (Budapest): ,
CC Blooms (Edinburgh):
cell phones: 356-357
Celt, The (Dublin):
Celta (Barcelona): 208-209
Celtic Nights Dance Show & Dinner (Dublin):
Central Amsterdam:
Central Backpacking Hostel (Budapest):
Central Berlin:
Centro Storico (Florence): ,
Černý, David:
Cervecería Cruz (Madrid):
Český Krumlov:
Champ de Mars (Paris): ,
Champs-Élyaées (Paris): , , ,
Chapeau Rouge (Prague):
Charles Bridge (Prague): , 255-256
Checkpoint Charlie (Berlin), ,
Chester Beatty Library (Dublin): 308-309
Chez Georges (Paris):
Chez Prune (Paris):
Chipotle (London):
Chocolate Museum (Barcelona): ,
Chocolatería San Ginés (Madrid): ,
Christmas markets (Berlin):
Chueca (Madrid): , ,
Chupitos (Barcelona): ,
churches/temples: Barcelona , , ; Budapest , 283-285; Edinburgh , ; Florence 136-137, , ; London , 22-23; Madrid ; Paris , 54-55, ; Prague , , , , , 257-258, ; Rome , 109-110, ; Venice 158-159,
Churchill War Rooms (London): 24-25
Church of Our Lady Victorious (Prague): , 257-258
Churreria Manuel San Román (Barcelona): ,
Ciak Hostel (Rome):
_cichetti_ crawl (Venice):
Circo Massimo (Rome): ,
Circulo de Bellas Artes Terraza (Madrid): ,
Circus Hostel (Berlin): ,
Citadel, The (Budapest):
Cittie of Yorke (London):
City, The (London): ,
City History Museum (Madrid):
City Library (Amsterdam): ,
Citymapper (London):
City of the Dead Ghost Tours (Edinburgh):
City Park (Budapest): , ,
ClamShell (Edinburgh):
Cliffs of Moher (Dublin):
Clink78 Hostel (London):
La Closerie des Lilas (Paris):
clubs: Amsterdam 89-90; Barcelona , ; Berlin 236-237; Budapest ; Dublin 317-318; Edinburgh ; Florence 144-145; London 33-34; Madrid 186-187; Paris 62-63; Prague , ; Rome 122-123
Club Termix (Prague):
Club Twenty One (Florence): ,
Cobblestone Pub (Dublin):
Cocomama (Amsterdam):
Code Hostel (Edinburgh):
Coffee Parisien (Paris):
coffeeshops (Amsterdam): , ,
Colosseum (Rome): , , ,
Columns of Venice: ,
comedy shows: Amsterdam ; Edinburgh
Le Comptoir Général (Paris): , 61-62
consulates: _see_ emergencies
cooking classes: Barcelona , 216-217; Madrid
Coop (Venice):
Cooperative Food Market (Edinburgh): , ,
Copper Face Jacks (Dublin): ,
Cork & Bottle (London):
Cornucopia Wholefood & Vegetarian (Dublin):
Corre Museum (Venice): 159-160
El Corte Inglés (Madrid): ,
Corvinteto (Budapest): ,
Cowgate (Edinburgh):
Craigmillar Castle (Edinburgh): ,
Crap Logs (Barcelona):
Cross Club (Prague):
Crown Jewels (London):
Cserpes Tejivo Milk Bar (Budapest):
Cubana Café (Paris):
Cuesta de Moyano (Madrid):
currency: see planning tips
Curry Mitte (Berlin): ,
**D**
Dada Falafel (Berlin): ,
Dame Lane (Dublin):
Dame Street (Dublin):
Dampkring (Amsterdam): ,
Dandelion (Dublin): 317-318
Dar Poeta (Rome): ,
Da Vinci Express Train (Rome):
Day Tours Ireland:
DDR Museum (Berlin):
Dean Gardens (Edinburgh): ,
Deja Vu (Prague):
De Laatste Kruimel (Amsterdam): ,
De Pijp (Amsterdam): , ,
Deportation Memorial (Paris): , ,
Derrière (Paris): 58-59
De Zotte (Amsterdam):
Dicey's Garden Bar (Dublin):
Die Weinerei (Berlin):
Discovering Venice:
Discover Walks (Barcelona): ,
Discover Walks (Paris): ,
Dishoom (London): 27-28
Disneyland Paris: ,
Le Dix Bar (Paris): ,
Dlouhá (Prague):
DLR (London):
Dr. Pong (Berlin):
Doge's Palace (Venice): , , ,
Dohány Street Synagogue (Budapest): , 284-285
Dorsoduro (Venice):
Dow Jones (Barcelona): ,
Dropkick Murphy's (Edinburgh): ,
Drunken Monkey Pub Crawl (Prague): ,
Drunken Ship (Rome): ,
Drury Buildings Cocktail Bar (Dublin):
Dublin: 299-324; day trips 323-324; emergencies ; events ; history ; hostels 320-321; itinerary 306-307; maps 300-301, 302-303; neighborhoods ; nightlife 313-318; planning tips ; recreation ; restaurants 310-313; shopping ; sights 308-310; tours ; transportation 322-323
Dubliners Pub (Madrid):
Duomo (Florence): , 136-137
Duomo Museum (Florence):
La Durée (Paris):
**E**
Easter (Rome):
East Side Gallery (Berlin): ,
EasyBus (London):
Easy Times (Amsterdam):
Eating Italy Food Tours (Rome):
Ecomama (Amsterdam):
Edinburgh: 325-349; day trips ; emergencies ; events ; history ; hostels 346-347; itinerary 332-333; maps 326-327, 328-329; neighborhoods ; nightlife 340-342; planning tips ; recreation 344-345; restaurants 338-340; shopping ; sights 334-338; tours 345-346; transportation 347-348
Edinburgh Castle: , , 334-335
Edinburgh Farmers Market:
Edinburgh Military Tattoo:
Edinburgh Pub Crawl:
Eger: 297-298
Eiffel Tower (Paris): , , 52-53
Eixample (Barcelona): ,
electricity:
Electric Ladyland (Amsterdam):
Elephant House (Edinburgh):
Ellato Kert (Budapest): ,
embassies: see emergencies
emergencies: Amsterdam ; Barcelona ; Berlin ; Budapest ; Dublin ; Edinburgh ; Florence ; London ; Madrid ; Paris ; Prague ; Rome ; Venice
Emergency Exit Escape Games (Budapest):
Enchanted Forest (Budapest): ,
Epicurean Food Hall (Dublin):
Epic Winebar (Budapest):
Equity Point Centric (Barcelona): 217-218
Equity Point Gothic (Barcelona):
Equity Point Sea (Barcelona):
Escape Club (Prague):
escape games (Budapest): , ,
Estadio Santiago Bernabéu (Madrid): ,
Esztergom (Budapest):
EUR (Rome): ,
events: see specific city
Excursion Scotland:
Experimental Cocktail Club (London):
**F**
Fabo Fast Food (Amsterdam): 84-85
Fabric (London):
Factory (Prague):
Fat Tire Bike Tours (Barcelona):
Fat Tire Bike Tours (Berlin): ,
Fat Tire Bike Tours (London): ,
Fat Tire Bike Tours (Paris): , ,
Fiesole: ,
Firenze Card (Florence):
Fishco Teque (London):
Fisherman's Bastion (Budapest): ,
Fish Market Trastevere (Rome):
Flamenco Show at Casa Patas (Madrid): ,
Flannery's (Dublin): ,
Florence: 129-150; day trips ; emergencies ; history ; hostels ; itinerary 134-135; map 130-131; neighborhoods ; nightlife 143-145; planning tips ; restaurants 139-142; shopping 145-146; sights 136-139; tours ; transportation 148-150
Florence Free Tour: ,
Flying Pig Downtown (Amsterdam):
Fogashaz (Budapest): ,
Fogashaz Kert (Budapest):
Il Forno (Rome):
Fortitude Coffee Merchants (Edinburgh): ,
Forum des Halles (Paris):
Frederick the Great's palaces:
Free Tour Venice:
Freni e Frizioni (Rome): ,
Freud (London):
Friedrichshain (Berlin): , , ,
Friends Club (Prague):
Fumbally, The (Dublin):
**G**
Galeries Lafayette (Paris):
Galleria Borghese (Rome): , ,
Galleria San Marco (Venice): ,
Gallery of Modern Art (Edinburgh):
Garage Bar (Dublin): 315-316
Gassan Diamonds (Amsterdam): ,
Gaudí, Antoni:
G-A-Y (London):
Gay Pride Fest (Amsterdam):
Gay Pride festivities (Barcelona):
G Bar (Rome): ,
Gelateria dei Neri (Florence):
Gelateria del Teatro (Rome):
gelato: Florence ; Rome
Gellért (Budapest):
Gellért Hill (Budapest):
Generator Hostel (Dublin):
Generator Hostel Berlin East:
Generator Hostel Paris:
Generator Russell Square (London):
Generator Venice:
Georgbrau (Berlin): ,
George, The (Dublin):
Georges Street Arcade (Dublin):
George Street (Edinburgh): ,
German History Museum (Berlin):
Gianicolo Hill (Rome): ,
Giardino degli Aranci (Rome): ,
Giolitti (Rome): ,
Giudecca (Venice):
Glasgow:
glassblowing (Venice): ,
Glass Museum (Venice):
gondola rides (Venice): , ,
Goodge Street (London): ,
Gordon's Wine Bar (London): ,
Gorki Park (Berlin):
Gozsdu Court & Passage (Budapest): , , 288-289
Gozsdu Sky Terrace (Budapest):
Grafton Street (Dublin): ,
Grain Store (Edinburgh): ,
Grand Canal (Venice):
Grandio Party Hostel (Budapest):
Gran Vía (Madrid):
Grasshopper (Amsterdam):
Grassmarket District (Edinburgh):
Grassmarket Square (Edinburgh): , ,
Graze on Grassmarket (Edinburgh):
Great Market Hall (Budapest): ,
Greenwoods (Amsterdam): ,
Grey Area (Amsterdam): ,
Greyfriars Kirk & Kirkyard (Edinburgh):
Griffin's (Madrid):
Grom (Florence): ,
Guinness Storehouse (Dublin): , ,
**H**
Haarlem:
Hackescher Höfe Courtyards & Hackescher Markt (Berlin):
haggis:
Haggis Adventures (Edinburgh): , ,
Harley's (Prague): ,
Harrods (London): ,
Harry's Bar (Venice):
Hasir (Berlin): ,
Havelská Market (Prague):
Haven Hostel San Toma (Venice):
Haymarket (Edinburgh):
Heart of Midlothian (Edinburgh): ,
Heaven (London):
Heineken Experience (Amsterdam): , ,
Henri Willig Cheese & More (Amsterdam): ,
Heureux Comme Alexandre (Paris):
High Street Hostel (Edinburgh): 346-347
hiking: Barcelona 215-216; Dublin
Hitler's Bunker (Berlin): , 230-231
Hive Nightclub (Edinburgh): ,
Hold Street Market Hall (Budapest):
holidays (Barcelona):
Holyrood Park (Edinburgh):
Homomonument (Amsterdam):
Hophouse (Dublin):
Hops & Barley (Berlin):
Horse Guards Parade (London): ,
Hospital in the Rock (Budapest): ,
hospitals: see emergencies
Hostaria del Moro (Rome):
HostelCulture (Budapest):
Hostelculture (Dublin):
HostelCulture Backpacker Pub Crawl (Budapest):
Hostelculture Free Walking Tour (Dublin):
Hostelculture Pub Crawl (Dublin):
Hostel Orange (Prague):
hostels: general discussion 355-356, ; _see also_ specific city
Hotel Cavour's rooftop bar (Florence): ,
Hotel Il Bargellino (Florence):
Hotel Visegrad (Budapest):
House of Terror (Budapest): ,
Howth & Coastal Hike (Dublin): , 323-324
Hula Juice Bar (Edinburgh): , 339-340
human castles (Barcelona):
Hungarian Parliament Building: , , 282-283
Hungarian State Opera House: ,
Hyde Park (London): ,
**IJ**
ICCo (London):
Ice Cream Factory (Rome):
Ij Cantine (Amsterdam):
Île de la Cité (Paris): ,
Île Saint-Louis (Paris): ,
Imagina (Venice):
In de Wildeman (Amsterdam):
Instant (Budapest): ,
International Restaurant Row (Berlin):
I Quadri (Venice): ,
Irish Film Institute: ,
Isola del Panino (Rome): ,
itineraries: see specific city
James Dean (Prague): ,
Jameson Distillery (Dublin): , ,
Jan Hus statue (Prague): ,
Jewish Historical Museum (Amsterdam):
Jewish Quarter (Budapest): ,
Jewish Quarter (Prague): ,
Jewish Quarter (Rome): ,
Jimbo's Cheap Man's Bus Tour (Berlin): ,
John Lennon Wall (Prague): ,
Jordaan (Amsterdam):
Joy Eslava (Madrid): ,
Juana la Loca (Madrid):
**K**
Kabul Backpackers Hostel (Barcelona):
Kalverstraat (Amsterdam): ,
La Kama (Madrid):
Kantjil & de Tijger (Amsterdam): ,
Karlovy Lázně (Prague):
Kaschk (Berlin): ,
Kastanienallee (Berlin):
Kazinczy Utca (Budapest): ,
Kelingrove Art Gallery and Museum (Edinburgh):
Keukenhof (Amsterdam):
Kiado Kocsma (Budapest):
Kick Ass Hostel (Edinburgh):
Kikuya (Florence): ,
Kilmainham Gaol (Dublin): 309-310
kilts:
Kingfisher Café (Amsterdam):
King Grizzly (Florence):
King's Day (Amsterdam):
Király Utca (Budapest):
Kis Parazs Thai (Budapest):
KitKat Club (Berlin):
Klubovna 2. Patro (Prague): ,
Knights Templar (London):
Kolor (Budapest):
Konnopke Imbiss (Berlin): 232-233
Kozička (Prague):
Kreuzberg (Berlin): ,
Kuplung (Budapest): ,
**L**
Lamucca de Pez (Madrid): 181-182
_lángos:_
Largo di Torre Argentina (Rome): ,
Last Cathedral (Berlin):
La Latina (Madrid): , 185-186
Latin Quarter (Paris): ,
Lavapies (Madrid): , ,
Leeds Castle:
Lehká Hlava (Prague):
Leicester Square (London):
Leidseplein (Amsterdam): ,
Leidsestraat (Amsterdam): ,
Leo Burdock Fish & Chips (Dublin):
Leopold Town (Budapest):
Les Delices du Fournil (Paris): , 57-58
Lesser Quarter (Prague): ,
Letango Tours (Madrid):
Letná Park (Prague): , 269-270
Leves (Budapest):
LGBT nightlife: Amsterdam ; Barcelona ; Berlin ; Budapest ; Dublin ; Edinburgh ; Florence ; London ; Madrid ; Paris ; Prague ; Rome ; Venice
Library Walk (Dublin):
Lido (Venice): ,
Lion's Fountain, The (Florence): ,
Loch Ness: ,
Loft Boutique Hostel (Paris):
Lokál (Prague): , 259-260
London: 13-42; day trips ; emergencies ; events ; history ; hostels 38-39; itinerary 20-21; maps , , , ; neighborhoods ; nightlife 30-35; planning tips ; recreation ; restaurants 27-29; shopping 35-36; sights 22-27; tours ; transportation 39-41
London Ale Trail:
London Cocktail Club:
London Eye: , ,
London Gone Wild:
Louvre (Paris): , , , 53-54
Lover's Bridge (Prague): ,
Lover's Canal Cruises (Amsterdam): , 91-92
Lucerna (Prague): ,
Luton (London):
Luxembourg Gardens (Paris):
Luzia (Berlin):
**M**
La Machine Du Moulin Rouge (Paris):
MadHouse Hostel (Prague):
Madrid: 171-194; day trips ; emergencies ; events ; history ; hostels ; itinerary 176-177; map 172-173; neighborhoods ; nightlife 185-187; planning tips ; recreation , ; restaurants 181-185; shopping 187-188; sights 178-181; tours ; transportation 192-193
Madrid Río:
Malasaña (Madrid): , , ,
Malone's (Edinburgh): ,
Malý Buddha (Prague):
Mama Coffee (Prague): ,
M&J Hostel (Rome):
Marais (Paris): , ,
Marché des Enfants Rouges (Paris): ,
Margit Island (Budapest): ,
Market Bar (Dublin): ,
markets: Amsterdam 90-91; Barcelona ; Berlin ; Budapest ; Dublin ; Edinburgh ; Florence 145-146; London 35-36; Madrid ; Paris 63-64; Prague ; Rome ; Venice
Marsella (Barcelona):
Master's Super Fish (London): ,
Matthias Church (Budapest): 283-284
Mauerpark Flea Market (Berlin): ,
Mayor of Scaredy Cat Town (London): 32-33
Meadows Park (Edinburgh):
Mecca (Prague): ,
Medici Chapels (Florence):
Medina Desi Curry Co (Dublin):
Megabus (London):
Mein Haus am See (Berlin): , 235-236
Melkweg (Amsterdam): , ,
Memorial to the Murdered Jews of Europe (Berlin): ,
Memorial to the Victims of Communism (Prague): ,
Mercado Puerta de Toledo (Madrid):
Mercado San Antón (Madrid):
Mercado San Miguel (Madrid):
Mercantile Pub (Dublin): , , 314-315
Mercato Centrale (Florence): ,
Mercato del Porcellino (Florence):
Mercato di Sant'Ambrogio (Florence):
Mercato Nuovo (Florence):
Mercerie (Venice):
Mike's Bike Tours (Amsterdam): , ,
Millennium Bridge (London): 25-26
Millennium Mile (London): ,
Millennium Walkway Restaurant Row (Dublin): ,
Ministry of Sound (London):
Mirador del Valle (Madrid):
Miscellanea (Rome): , 115-116
Mitte (Berlin):
Molly Malone:
M1 (Prague): ,
Monti District (Rome):
Montjuic (Barcelona): , 215-216
Montmartre (Paris): , ,
Montparnasse (Paris):
Montserrat (Barcelona): ,
Monument (London): ,
Morningside Road (Edinburgh):
Mosaic House (Prague):
Most (Budapest): ,
Moulin Rouge (Paris):
Moyo (Florence): ,
Mozsar Kavezo (Budapest): ,
Mucha Museum (Prague): , 258-259
MuchoMadrid (Madrid):
Mums Great Comfort Food (Edinburgh):
Murano (Venice): , ,
Muro Venezia Rialto (Venice):
Murphy's Ice Cream (Dublin):
Las Musas Residence (Madrid):
Museo del Jamón (Madrid):
Museo Thyssen-Bornemisza (Madrid): , ,
Museum District (Amsterdam):
Museum District (Madrid):
Museum of Communism (Prague): ,
Museum of Edinburgh: ,
My Smart Break (Berlin):
**N**
Nagymező Utca (Budapest):
Naima (Florence): ,
Náplavka (Prague):
Náplavka Farmers Market (Prague):
Naranzaria (Venice):
National Gallery (Dublin): ,
National Gallery (London): ,
National Gallery of Scotland:
National Library (Dublin): ,
National Museum: Decorative Arts & History (Dublin):
National Museum of Archaeology (Dublin): ,
National Museum of Scotland:
National Theatre (Prague):
Nebe (Prague): 265-266
Nelson's Column (London): ,
Nerbone (Florence):
New Berlin Pub Crawl:
New Edinburgh Pub Crawl:
New Town (Edinburgh):
New Town (Prague):
Nieuwendijk (Amsterdam):
Nieuwmarkt (Amsterdam): ,
Nightjar (London): , ,
nightlife: see specific city
Niji Café (Rome): ,
Nine Streets (Amsterdam):
1916 Rebellion Walking Tour (Dublin):
Noordermarkt (Amsterdam): ,
North Amsterdam:
North Rome:
North Side (Dublin):
Notre Dame (Paris): ,
Nowkoelin Flowmarkt (Berlin):
**O**
Oberkampf (Paris): 60-61
Obicà Mozzarella Bar (Rome):
October 23 celebration (Budapest):
Odéon (Paris): ,
Odessa (Dublin): ,
Oil Shoppe (Florence): ,
Oink (Edinburgh): ,
Old Bank of England (London):
Old Bridge (Rome):
Old Jewish Cemetery (Prague): ,
Old Spitalfields Market (London): ,
Old Town (Edinburgh):
Old Town (Prague):
Old Town Square (Prague): ,
Oltrarno (Florence): ,
1 Big Night Out (London):
O'Neill's (Dublin): ,
O'Neill's (Madrid): ,
One World Film Festival (Prague):
Opium Mar (Barcelona): ,
Orange (Venice):
Oranienstrasse (Berlin):
Orsanmichele Church (Florence):
Orsay Museum (Paris): 55-56
Osbar (Barcelona):
Osteria Bancogiro (Venice):
Osteria Brincello (Florence): ,
Osteria Zio Gigi's (Florence): ,
Otel Varieté Restaurant (Florence): 144-145
Oxford Street (London): ,
Oxmantown (Dublin):
Oyster Card (London):
**P**
paddle boats (London):
paddle boats (Prague):
PaddyWagon Tours (Dublin):
Paella & Sangria Cooking Class (Barcelona):
Palace, The (Dublin): ,
Palace Gardens (Venice):
Palace of Holyroodhouse (Edinburgh):
Palace of Tears (Berlin): ,
Palau Güell (Barcelona):
Palazzo Vecchio (Florence):
El Palentino (Madrid):
Palladium (Prague):
Pancake Bakery (Amsterdam): ,
Pane & Toscana (Florence):
Pangie's Bistrot (Florence):
Panini Row (Venice): ,
Pans and Company (Barcelona): ,
Pantheon (Rome): , , 110-111
Pantheon neighborhood (Rome):
Paradiso (Amsterdam): , , 89-90
Parc des Buttes-Chaumont (Paris): , 64-65
Parc Güell (Barcelona): , ,
Paris: 43-70; day trips 69-70; emergencies ; events ; history ; hostels 65-66; itinerary 50-51; maps 44-45, 46-47; neighborhoods ; nightlife 60-63; planning tips ; recreation 64-65; restaurants 57-59; shopping 63-64; sights 52-57; tours ; transportation , 68-69
Paris Noir:
Pařížská (Prague):
parks and gardens: Amsterdam ; Barcelona ; Berlin 240-241; Budapest ; Dublin ; Edinburgh 344-345; Madrid ; Paris , 64-65; Prague 269-270; Rome 124-125; Venice
Patro (Prague):
La Pausa (Berlin): ,
Pavilion, The (Dublin):
peeing statues (Prague): ,
Peggy Guggenheim Venice: ,
Penthouse Privates (Budapest): 295-296
Perché No! (Florence): ,
Père Lachaise Cemetery (Paris): ,
Pest Town Center (Budapest):
Petřín Hill (Prague): , ,
Pfefferbett Hostels (Berlin):
Phoenix Park (Dublin):
phones: 356-357
Piazza della Reupbblica (Florence):
Piazza del Popolo (Rome): 113-114
Piazzale Michelangelo (Florence): ,
Piazza Navona (Rome): , 111-112,
Piazza Santo Spirito (Florence): ,
Picasso Museum (Barcelona): , ,
Piccadilly Institute (London):
Piccolo Café (Florence):
Piccolo Mondo (Venice):
pickpocketing: see emergencies
_Pietà_ (Rome): ,
Pigalle (Paris):
Ping Pong (London):
Pinkas Synagogue (Prague): ,
Pippermint (Barcelona): 211-212
Pitti Palace (Florence):
La Pizzateca (Madrid):
Pizzeria del Duomo (Florence): , 140-141
Pizzeria del Secolo (Rome):
Plaça Reial (Barcelona): ,
Planet (Edinburgh):
planning tips: general 8-12; Amsterdam ; Barcelona ; Berlin ; Budapest ; Dublin ; Edinburgh ; Florence ; London ; Madrid ; Paris ; Prague ; Rome ; Venice
La Plata (Barcelona):
Platja Barceloneta (Barcelona): ,
Platja Nova Mar Bella (Barcelona): ,
Plaza de Carlos Cambronero (Madrid):
Plaza del 2 de Mayo (Madrid): ,
Plaza de Toros de las Venta (Madrid): 180-181
Plaza Mayor (Madrid): ,
Plaza Puerta del Sol (Madrid): ,
Plaza Santa Ana (Madrid): ,
PLUS Camping Jolly (Venice):
Plus Florence Hostel:
Point Éphémère (Paris):
Pointer, The (Budapest):
Pompidou Center (Paris): 56=57
Ponte Vecchio (Florence): , 138-139
Porta Portese (Rome): , ,
Porterhouse, The (Dublin): ,
Porterhouse (London): ,
Portobello Market (London):
Poste Vecie Ristorante (Venice):
Potsdam: ,
Prado Museum (Madrid): , ,
Prague: 247-274; day trips ; emergencies ; events ; history ; hostels ; itinerary 252-253; map 248-249; neighborhoods ; nightlife 263-266, ; planning tips ; recreation 269-270; restaurants 259-262; shopping ; sights 254-259; tours 270-271; transportation 272-273
Prague Beer Museum: ,
Prague Castle: ,
Prague 7:
Prague Square Hostel:
Prater (Berlin):
Prater Garten (Berlin):
Preem & Prithi (London):
Prenzlauer Berg (Berlin): ,
Pret a Manger (London): ,
Princess Louise (London):
Princes Street (Edinburgh): , ,
Princes Street Gardens (Edinburgh):
Proud Camden (London): ,
pub crawls: Amsterdam ; Berlin ; Dublin ; Edinburgh ; Florence ; London 34-35; Madrid ; Paris ; Prague ; Rome
Puerto Olimpico (Barcelona): , , ,
Puerto Rico (Madrid): 182-183
**QR**
Q's Rummeria Rum Bar Trastevere (Rome):
Queen (Paris):
Queen of Tarts (Dublin): ,
Rabbie's Small Group Tours (Edinburgh): 345-346
Radost FX (Prague):
La Rambla (Barcelona): , ,
Raoul Wallenberg Holocaust Memorial Park (Budapest): , 284-285
El Rapido (Budapest):
Rari Ristoro sull'Acqua (Florence):
El Rastro (Madrid): , ,
Razzmatazz (Barcelona):
Real Madrid:
Real Mary King's Close (Edinburgh): , 335-336
recreation: see specific city
Red (Communist) Berlin:
Red Garter (Florence): ,
Red Light District (Amsterdam): , ,
Reichstag (Berlin): , , 229-230
Reina Sofia (Madrid): , ,
Rembrandtplein (Amsterdam): ,
rental cars:
RER (Paris): , 68-69
reservations: see planning tips
Restaurace T. Anker (Prague):
restaurants: general discussion ; see also specific city
Retiro Park (Madrid): ,
Retox Hostel (Budapest):
REX Caffe (Florence): 143-144
Rialto (Venice): ,
Rialto Bridge (Venice): , 161-162
Rialto Fish Market (Venice):
Rialto Market (Venice):
Rijksmuseum (Amsterdam): , , ,
Ristorante Grotta Guelfa (Florence):
Ristorante La Reggia (Florence):
Il Ritrovo (Berlin):
_Rocket_ , The (London):
Roissybus (Paris):
La Rollerie (Madrid): , 184-185
Roman Forum: ,
Roma Pass:
Rome: 97-128; day trips ; emergencies ; events ; history ; hostels 125-126; itinerary 104-106; maps 98-99, 100-101; neighborhoods ; nightlife 119-120, 122-123; planning tips ; recreation 124-125; restaurants 115-118; shopping 123-124; sights , 109-114; tours ; transportation 126-127
Rome's Ultimate Party:
Room007 Hostel: Chueca (Madrid):
Room007 Hostel: Ventura (Madrid):
Rosenthaler Platz (Berlin): ,
Rosslyn Chapel (Edinburgh): ,
Rosticceria San Bartolomeo (Venice):
Roxy (Prague): ,
Royal Delft Experience (Amsterdam):
Royal Gardens Bike Tour (London):
Royal Mile (Edinburgh): , , 340-341,
Royal Palace (Madrid): , ,
Royal Yacht _Britannia_ (Edinburgh):
Rudas (Budapest): ,
Rue Cler (Paris):
Rue de la Verrerie (Paris):
Rue de Rivoli (Paris):
Rue de Roi Sicile (Paris):
Rue Montorgueil (Paris):
Rue Moret (Paris):
Rue Princesse (Paris):
Rue Vieille du Temple (Paris):
**S**
Sachsenhausen Concentration Camp: , , 245-246
Sacre Coeur (Paris): ,
Sagardi BCN Gotic (Barcelona): ,
Sagrada Familia (Barcelona): , , 204-205,
St Christopher's Inn Canal (Paris):
St Christopher's Inn Gare du Nord (Paris): 65-66
St Christopher's Inns (Berlin):
St Christopher's Village, London Bridge:
St Christopher's Winston Hotel (Amsterdam):
Sainte-Chapelle (Paris): , 54-55
Saint Giles Café & Bar (Edinburgh): 338-339
St Giles' Cathedral (Edinburgh): ,
St James's Park (London):
St Mark's Basilica (Venice): , 158-159
St Mark's Square (Venice): ,
St Nicholas Church (Prague): ,
St. Oberholz Bar and Cafe (Berlin):
St Patrick's Day (Dublin):
St. Paul's Cathedral (London): , 22-23
St Peter's Basilica (Rome): ,
St Stephen's Basilica (Budapest):
St Stephen's Day (Budapest):
St Stephen's Green (Dublin): ,
St Vitus Cathedral (Prague): ,
Sala Apolo (Barcelona):
Salamanca (Madrid): ,
Salt Yard (London):
La Sanabresa (Madrid):
Sandeman's New Edinburgh Free Walking Tours:
Sandeman's New Europe Pub Crawl (Paris):
Sandeman's New Europe Walking Tours (Amsterdam): ,
Sandeman's New Europe Walking Tours (Berlin):
Sandeman's New Europe Walking Tours (Dublin):
Sandeman's New Europe Walking Tours (London):
Sandeman's New Europe Walking Tours (Madrid): , ,
Sandeman's New Europe Walking Tours (Prague): , 270-271
Sandy Bell's (Edinburgh): ,
San Giorgio Maggiore (Venice): , , ,
San Lorenzo (Rome): ,
San Lorenzo Market (Florence): ,
San Marco (Venice):
San Michele (Venice): , , 162-163
Santa Croce (Florence): ,
Santa Maria de Montserrat (Barcelona):
Santa Maria in Aracoeli (Rome): ,
Santander Cycles (London):
Sant Jordi Alberg (Barcelona):
Sant Jordi Apartment Sagrada Familia (Barcelona):
Sant Jordi Gràcia (Barcelona):
Sant Jordi Rock Palace (Barcelona):
Sardana dances (Barcelona): ,
Sarphatipark (Amsterdam): ,
Schleusenkrug (Berlin): 234-235,
Scholar's Lounge (Rome): ,
Schönhauser Allee (Berlin): ,
Schwarzes Café (Berlin): ,
Scotch Whisky Experience (Edinburgh): ,
Scottish independence:
Scottish National Gallery of Modern Art: ,
Scottish Parliament Building: , , 336-337
Service Center for the Queer Community (Florence):
Sex Museum (Amsterdam):
sex shows (Amsterdam):
Shakespeare's Globe (London): ,
Shard, The (London): ,
Shari Vari Playhouse (Rome):
SheSoho (London):
Shipwrecked Boat Party (Budapest):
Shoes on the Danube Memorial (Budapest): ,
Shoko (Barcelona): ,
shopping: see specific city
Shopping Triangle of Death (Rome): 123-124
Shoreditch (London): , ,
Showcase (Paris): ,
Sidecar Factory Club (Barcelona): ,
sightseeing passes: Berlin ; Dublin ; Edinburgh ; Florence ; Madrid ; Paris ; Rome ; Venice
Sir Toby's (Prague):
Sistine Chapel (Rome): , 109-110
Sky Backpackers Hostel (Dublin):
Slav Epic at the National Gallery (Prague):
Sloppy Sam's (Rome): ,
Slovanský Island (Prague):
smartphones: 356-357
smart shops (Amsterdam):
Social Bite (Edinburgh): ,
Soho (London):
Sol (Madrid): , ,
Soup 'n' Roll (Berlin):
South Bank (Dublin):
South Bank (London):
Southeast Amsterdam:
South William Bar (Dublin): ,
Space (Florence):
Spanish cooking classes (Madrid):
Spanish Steps (Rome): ,
Spanish Steps Pub Crawl (Rome):
spa parties (Budapest): , 290-291
speakeasies (London): 32-33
spectator sports: Barcelona ; Madrid , , 180-181
Spreepark (Berlin):
Stag's Head (Dublin):
Stand, The (Edinburgh):
StarBikes Rental (Amsterdam):
_Statue of the Fallen Angel_ (Madrid):
Stockbridge (Edinburgh):
Stonehenge:
Strahov Monastery (Prague): , 260-261
Street, The (Edinburgh):
Střelecký Island (Prague):
Střelec Pub (Prague):
Student Agency (London):
Sugar! (Budapest): ,
Sunday Upmarket (London): ,
Sungate Hostel (Madrid):
Sunny Terrace Hostel (Venice):
Sweeney's Bar (Dublin): ,
Széchenyi Fürdõ (Budapest): , ,
Szentendre (Budapest):
Sziget Music Festival (Budapest):
Szimpla (Berlin):
Szimpla (Budapest): , , ; farmers market: ,
**T**
Taberna El Sur (Madrid): ,
Tacheles (Berlin): ,
Taisu (Berlin):
tapas:
Los Tarantos Flamenco Show (Barcelona): ,
Tartan Weaving Mill (Edinburgh): ,
Tasca el Corral (Barcelona):
Tate Modern (London): , ,
taxis: Barcelona ; Berlin ; Budapest ; Dublin ; Edinburgh ; Florence 149-150; London 40-41; Madrid ; Paris ; Prague ; Rome ,
't Blauwe Theehuis (Amsterdam): ,
Team Escape (Budapest):
Teatro Kapital (Madrid): ,
Teatro Real (Madrid):
Teleférico de Madrid: ,
telephones: 356-357
Tempelhofer Park (Berlin):
Temple Bar (Dublin): , , ,
Temple Bar Food Market (Dublin):
El Tempranillo (Madrid):
Termax Club (Prague):
Termini (Rome): , ,
Terravision (Rome):
Testaccio (Rome): , , 119-120
Thai Snack Bar (Amsterdam): , 83-84
Three Sisters (Edinburgh): ,
360 Bar (Budapest):
Tiergarten (Berlin):
Time Trap Escape Games (Budapest):
TKTS (London): ,
Toledo:
Tolteca Burritos (Dublin):
Tony's (Rome):
Topography of Terror (Berlin):
Torre dell'Orologio (Venice): ,
Torre del Oro (Madrid): 183-184
Tour de France (Paris):
tourist information centers: see emergencies
tours: see specific city
Tower Bridge (London):
Tower of London (London): ,
Traditional Irish Musical Pub Crawl:
trad music:
Trafalgar Square (London): ,
trains and metro: general discussion , ; Amsterdam , ; Barcelona 218-219; Berlin , ; Budapest , ; Dublin , ; Edinburgh ; Florence 148-149; London ; Madrid , ; Paris ; Prague 272-273; Rome , ; Venice
transportation: general discussion 350-355, 357-358; see also specific city
Trastevere (Rome): , ,
Trattoria Nerone Pizzeria (Florence):
Travel Bar (Barcelona):
Travel Bar's Barcelona Old Town Tour:
Travel Bar's Paella & Sangria Cooking Class (Barcelona): 216-217
_trdelníks:_
Trendy Hostel (Paris):
Tre Scalini (Rome):
Tresor (Berlin): , , 236-237
Tretter's New York Bar (Prague):
Trevi Fountain (Rome): ,
Trinity College (Dublin): ,
Tržnice Dlouhá 14 (Prague): ,
Tuileries, The (Paris):
Türkischer Markt (Berlin):
_Turul_ Bird Statue (Budapest):
Tuscan hill towns: ,
Tuscany Cycle (Florence):
Tuscany on a Budget (Florence): , ,
TV Tower (Berlin):
Týn Church (Prague): ,
**U**
Uffizi (Florence): ,
Uffizi Gallery (Florence): ,
U Medvídků (Prague): ,
Umi Falafel (Dublin):
Underdog at Brewdog (London):
Under the Stairs (Edinburgh): ,
Unity Day (Berlin):
University District (Edinburgh): ,
University of Edinburgh: ,
University of Glasgow:
Unter den Linden (Berlin): ,
Urbany Barcelona:
Urbany BCN GO (Barcelona):
U Sudu (Prague):
**V**
Váci Utca (Budapest): ,
Vak Varju (Budapest): ,
Valley of the Sirens: 297-298
V&D La Place (Amsterdam): , ,
Van Gogh Museum (Amsterdam): , , ,
La Varangue (Paris): ,
Vatican City (Rome): ,
Vatican Museums (Rome): , 109-110
Velvet Revolution Anniversary (Prague):
Vendita Libri, Cioccolata e Vino (Rome):
Venezia Museum Pass (Venice):
Venice: 151-170; emergencies ; events ; history ; hostels 168-169; itinerary 156-157; map 152-153; neighborhoods ; nightlife ; planning tips ; recreation , ; restaurants 163-165; shopping ; sights 158-163; tours ; transportation 169-170
Venice Gold: 168-169
Versailles: , 69-70
Via Appia Antica (Rome): ,
Via del Corso (Rome): ,
El Viajero (Madrid):
Victoria and Albert Museum (London): ,
Victoria Park (Berlin):
Victoria Street (Edinburgh):
views: Edinburgh ; London ; Madrid
Villa Borghese (Rome): ,
Vincent Street Glasgow:
Visegrad (Budapest):
Vittorio Emanuele Monument (Rome): , 112-113
Vivoli (Florence):
Vleminckx Friteshuis (Amsterdam):
Vondelpark (Amsterdam): ,
Vyšehrad Castle (Prague): ,
**WXYZ**
Wahaca (London): , 28-29
Walks of Italy (Rome):
Walks of Italy (Venice):
Watergate (Berlin): ,
Water of Leith (Edinburgh): ,
WelcomeCard (Berlin):
Wenceslas Square (Prague): , ,
West Amsterdam:
West Berlin:
West End (Edinburgh): ,
West End (London): ,
West End Glasgow:
West End plays (London):
West London:
Westminster (London):
Westminster Abbey (London): ,
Westminster Palace (London): ,
West Princes Street Gardens (Edinburgh): ,
Whelan's (Dublin): ,
Whistlebinkies (Edinburgh): ,
White Trash Food (Berlin):
wine bars (London):
Winkel 43 (Amsterdam): ,
Winston Bar & Venue (Amsterdam):
Wireless Festival (London):
Wombat's (London): 38-39
Wombat's City Hostel Berlin:
Wombats Hostel (Budapest):
Workman's Club (Dublin): 316-317
XOYO (London):
Y.A.B. (Florence):
YAG Bar (Florence):
Yellow Hostel (Rome):
Yellow Zebra Bike Tour (Budapest): ,
Ye Olde Cheshire Cheese (London): , ,
YoYo Palais de Tokyo (Paris):
Yumcha Heroes (Berlin):
Zebra Express (Prague):
Žluté Lázně (Prague):
Zosch (Berlin): ,
## **PHOTO CREDITS**
here: Lucian Milasan/123RF; here: © Andy Steves; here clockwise from top left: © La'Vere Corbin-Ong; © Indi Ericksen; © Camille Fenn; © WSA; © Talia Brienza; here: © WSA; here: © Andy Steves; here: © mbbirdy/iStock; here: © WSA; here left: © Camille Fenn; right: © Rosie Clarke; here: © Mile deMartile; here: © Carlee Pons; here: © robin2600 /123RF; here: © Arianna Santana Blasco; here clockwise from top left: © Indi Ericksen; © Isabella Pinheiro; © Arianna Santana Blasco; © Tomas1111 | Dreamstime.com; © Camille Fenn; here: © Ludmila Smite/123RF; here: © Vaclav Volrab/123RF; here: © Tigger76 | Dreamstime.com; here: © Emiko Moran; here: © Indi Ericksen; here: © WSA; here: © WSA; here: © Indi Ericksen; pg. 66 clockwise from top left: © WSA; © Milo DeMartile; © WSA; © WSA; © Luna De La Riva, Hannah Ayasse, Cyrus Mayer; © WSA; here: © Taras Verkhovinets/123RF; here: © S.Borisov/Shutterstock; here: © WSA; here: © Mozes Janse; here: © WSA; here left: WSA; right: © Savana York; here left and right: © WSA; here: © Luis Carlos Jimenez Del Rio/123RF; here: © WSA; here: © WSA; here all: © WSA; here left: © WSA; right: © Emiko Moran; here: © WSA; here: © Romaoslo/Istockphoto.com; here: © WSA; here: © WSA; here: © WSA; here: © Talia Brienza; here: © Indi Ericksen; here: © Natalia Salomon; here: © WSA; here: © WSA; here: © WSA; here clockwise from top left: © WSA; © WSA; © Lili Hazoury; © WSA; © WSA; here: © Krzysztof Świeżak/123RF; here: © FilippoBacci/istockphoto.com; here: © Indi Ericksen; here © WSA; here: © Nisa Thompson; here: © Indi Ericksen; here: © WSA; here left: Caleb Jaster, Amelia Egland; right: David Eng; here clockwise from top left: © Nisa Thompson; © Talia Brienza; © Indi Ericksen; © Milo DeMartile; © Kari Hernandez; here: © Sborisov | Dreamstime.com; here: © WSA; here: © WSA; here: © Indi Ericksen; here: © WSA; here clockwise from top left: © Nisa Thompson; © Nisa Thompson; © Indi Ericksen; © Thor Nolan; © Camille Fenn; here: © kasto/123RF; here: © Julian Saemann; here: © Harman Padda, Indi Ericksen, Isabelle Curttriss, Julian Saemann; here: © WSA; here: © Indi Ericksen; here: © carlewins/123RF; here: © Rose Clarke; here clockwise from top left: © Fifer Garbesi; © Indi Ericksen; © Fifer Garbesi; © Sepavo | Dreamstime.com, © Peter Cook, Joe Wax, Ricky Reyna, Patrick Goodwin; here left: © Fifer Garbesi; right: © Isabelle Cuttriss; here: © Indi Ericksen; here: © Mapics | Dreamstime.com; here: © Talia Brienza, Natalia Salomon; here: © Indi Ericksen; here left: © Emiko Moran; right: © Indi Ericksen; here left and right: © WSA; here left: © Fifer Garbesi; right: © Savana York; here clockwise from top left: © Savana York; © Mozes Janse; © WSA; © Pedro Gomez, Jacob Speedy; © Natalia Salomon; © Emiko Moran; here: © WSA; here: © WSA; here: © querbeet/istockphoto.com; here: © WSA; here: © 123RF; here left © Victoria Miller; right © Talia Brienza; here left: © Esbie Fonte; right: © Daisy Louise; here: © claudiodivizia/123RF; here clockwise from top left: © 123RF; © Indi Ericksen; © Fifer Garbesi; © tomas1111/123RF; © WSA; here: © Indi Ericksen; here: © Lili Hazoury, Paula Chacártegui; here: © sorincolac/istockphoto.com; here: © WSA; here: © WSA; here: © WSA; here: © Richard Semik/123RF; here: © Indi Ericksen; here left and right: © WSA; here clockwise from top left: © Jalena Keane-Lee, Lia Camargo; © Indi Ericksen; © WSA; © Indi Ericksen; © Gaetano Trachtenberg, Joseph Smith-Mewha, Henry Kerr; here: © Indi Ericksen; here: © Rudi1976 | Dreamstime.com; here: © WSA; here: © Jozef Sedmak/123RF; here: © WSA; here left and right: © WSA; here clockwise from top left: © Lia Camargo, Jalena Keane-Lee; © WSA; © WSA; © Europhotos | Dreamstime.com; © Victoria Miller; here: © Lia Camargo, Jalena Keane-Lee; here: © Patricia Hofmeester/123RF; here: © Camille Fenn, Jen Rose; here left: © WSA; right: © Lauren Gray; here left: © Inge Hogenbijl/123RF, right: © WSA; here left: © WSA; right: © Savana York; here top left: © Savana York, Jessie Evers, Baylee Murphy; all others: © WSA; here: © Michael Walsh/Dreamstime.com; here: © Hoda Ansari; here: © Vitaly Titov/123RF; here left: © WSA; right: © Hoda Ansari; here clockwise from top left: © WSA; © WSA; © Hoda Ansari; © Hoda Ansari; © Hoda Ansari; here: Fabio Pagani/123RF
## **ACKNOWLEDGMENTS**
This has been a huge project, and I couldn't have done it without my massive support network of close friends, excellent guides, and patient family who are all just as passionate about the redeeming value of travel as I am. Thanks to each of you from the bottom of my heart. Special thanks to: the Avalon Travel publishing team, Anne Jenkins, Jackie Steves, Rick Steves, Rhianne Taylor, Luke Watson, Bogi Palotas, Barbara Kiraly, Kevi Donat, Alexei Beck, Krista di Euleuterio, Giorgio di Laura, Asoka Esuruoso, Petr Zdenek, Sarka Hostinova, Arthur Bijlholt, Niki Harosi, Asoka Esuruoso, Taylor Smoot, Adi Hadzic, Dawid Kadziolka, Tessa Helber, Eric Miller, Stephen McPhilemy, Martina and Marco Zuccarello, Carlos and Jennifer Galvin, Jenna Laedtke, Christian Roberts, Matthew Jenks, and Mark Lambert.
ANDY STEVES' EUROPE
Avalon Travel
a member of the Perseus Books Group
1700 Fourth Street
Berkeley, CA 94710, USA
Editor: Nikki Ioakimedes
Copy Editor: Deana Shields
Fact-Checker: Rachel Feldman
Proofreader: Megan Mulholland
Production and Graphics Coordinator: Lucie Ericksen
Graphics Assistant: Indi Ericksen
Interior Design: Hayden Foell
Map Editor: Mike Morgenfeld
Cartographer: Brian Shotwell
Indexer: Greg Jewett
eISBN: 978-1-63121-251-2
ISBN-13: 978-1-63121-250-5
ISSN: 2471-447X
Printing History
1st Edition — June 2016
5 4 3 2 1
Text © 2016 by Andy Steves.
Maps © 2016 by Avalon Travel.
All rights reserved.
Some photos and illustrations are used by permission and are the property of the original copyright owners.
Although every effort was made to ensure that the information was correct at the time of going to press, the author and publisher do not assume and hereby disclaim any liability to any party for any loss or damage caused by errors, omissions, or any potential travel disruption due to labor or financial difficulty, whether such errors or omissions result from negligence, accident, or any other cause.
## Contents
1. Cover Page
2. Title Page
3. Contents
4. Index
5. Introduction
6. London
1. London Maps
7. Paris
1. Paris Maps
8. Amsterdam
1. Amsterdam Maps
9. Rome
1. Rome Maps
10. Florence
1. Florence Maps
11. Venice
1. Venice Map
12. Madrid
1. Madrid Map
13. Barcelona
1. Barcelona Maps
14. Berlin
1. Berlin Maps
15. Prague
1. Prague Map
16. Budapest
1. Budapest Map
17. Dublin
1. Dublin Maps
18. Edinburgh
1. Edinburgh Maps
19. Appendix
20. Photo Credits
21. Copyright
## Contents
1. Cover Page
2. Contents
3. Introduction
| {
"redpajama_set_name": "RedPajamaBook"
} | 36 |
\section{Introduction}
The precise hadronic $\tau$-decay data provided in
refs.~\cite{ALEPH05,ALEPH98,ALEPH97,OPAL99,CLEO95,Strange_data}
are a very important source of information, both on perturbative and
non-perturbative QCD parameters.
The theoretical analysis of the inclusive $\tau$ decay width into hadrons
allows to perform an accurate determination of the QCD coupling
$\alpha_s(M_\tau)$ \cite{alphas,LDP:92,DHZ06,new08,review},
which becomes the most precise determination of $\alpha_s(M_Z)$
after QCD running.
In this case, non-perturbative QCD effects
parametrised by power corrections are strongly suppressed.
Another example of the use of hadronic $\tau$-decay data
is the study of SU(3)--breaking corrections to the
strangeness-changing two-point functions \cite{su3,CKP98,KKP01,MALT,Vus}.
The separate measurement of the $|\Delta S|=0$ and $|\Delta S|=1$ tau decay widths
provides accurate determinations of fundamental parameters
of the Standard Model, such as the strange quark mass and the
Cabibbo-Kobayashi-Maskawa quark-mixing $|V_{us}|$ \cite{Vus}.
Very important phenomenological hadronic matrix elements and
non-perturbative QCD quantities can also be obtained from $\tau$-decay data.
Of special interest is the difference of the vector and axial-vector spectral functions,
because in the chiral limit the corresponding $V-A$ correlator is exactly zero in perturbation theory.
The $\tau$-decay measurement of the $V-A$ spectral function has been used to perform \cite{DG:94,DHG98,NAR01} phenomenological tests
of the so-called Weinberg sum rules (WSRs) \cite{WSR}, to compute the electromagnetic mass difference between the charged
and neutral pions \cite{DHG98}, and to determine several QCD vacuum condensates \cite{DS07,CGM03}. From the same spectral function one
can also determine the $\Delta I=3/2$ contribution of
the $\Delta S=1$ four-quark operators $Q_7$ and $Q_8$ to $\varepsilon_K'/\varepsilon_K$, in the chiral limit \cite{Q7Q8}.
Using chiral perturbation theory ($\chi$PT) \cite{WEI79,GL84,GL85},
the hadronic $\tau$-decay data can also
be related to order parameters of the spontaneous chiral
symmetry breaking (S$\chi$SB) of QCD \cite{KdR94}.
$\chi$PT\ is the effective field theory of QCD at very low energies;
it describes the S$\chi$B Nambu-Goldstone boson physics
through an expansion in external momenta and quark masses. The
coefficients of that expansion
are related to order parameters of S$\chi$SB.
At lowest order (LO), i.e. ${\cal O}(p^2)$, all low-energy observables
are described in terms of the pion decay constant $f_\pi \simeq 92.4$ MeV and
the light quark condensate.
At next-to-leading order (NLO), ${\cal O}(p^4)$, the SU(3) $\chi$PT\ Lagrangian contains 12
low-energy constants (LECs), $L_{i=1,\cdots,10}$ and $H_{1,2}$
\cite{GL85}.
At ${\cal O}(p^6)$, 90 (23) additional parameters $C_{i=1,\cdots,90}$
appear in the
even (odd) intrinsic parity sector \cite{p6}.
These LECs are not fixed by symmetry requirements alone and have to be determined
phenomenologically or using non-perturbative techniques.
The ${\cal O}(p^4)$ $L_i$ couplings have been determined in the past to an acceptable accuracy;
a recent compilation can be found in ref.~\cite{ECK07}.
Much less well determined are the ${\cal O}(p^6)$ couplings $C_i$.
There has been a lot of recent activity to determine the chiral LECs
from theory, using as much as possible QCD information
\cite{MOU97,KN01,RPP03,BGL03,CEE04,CEE05,KM06,RSP07,MP08,PRS08}.
This strong effort is motivated by the precision required in present phenomenological applications,
which makes necessary to include corrections of ${\cal O}(p^6)$. The huge number of unknown couplings is the major source of theoretical uncertainty.
In this paper we present an accurate
determination of the $\chi$PT\ couplings $L_{10}$ and $C_{87}$,
using the most recent experimental data on hadronic $\tau$ decays \cite{ALEPH05}.
Previous work on $L_{10}$ using $\tau$-decay data can be found in
refs.~\cite{DHG98,NAR01,DS07,DS04}. Our analysis is the first one which includes
the known two-loop $\chi$PT\ contributions and, therefore, provides also
the ${\cal O}(p^6)$ coupling $C_{87}$.
\section{Theoretical Framework}
The basic objects of the theoretical analysis are the two-point
correlation functions of the vector and axial-vector quark currents, defined as follows:
\begin{eqnarray}
\label{eq:two}
\Pi^{\mu\nu}_{ij,{\cal J}}(q)
&\equiv & i \int \mathrm{d}^4 x \; \mathrm{e}^{i q x} \,
\langle 0 | T \left( {\cal J}_{ij}^\mu(x) {\cal J}_{ij}^\nu(0)^\dagger \right) | 0 \rangle
\nonumber \\ &=& (-g^{\mu\nu} q^2 + q^\mu q^\nu ) \, \Pi^{(1)}_{ij,{\cal J}}(q^2)
+ q^\mu q^\nu\, \Pi^{(0)}_{ij,{\cal J}}(q^2) \, .
\nonumber \\
\end{eqnarray}
Here, we just need the non-strange correlators, i.e. ${\cal J}_{ij}^\mu(x)$ denotes the
Cabibbo-allowed vector or axial-vector currents,
$V_{ud}^\mu(x)=\overline{u} \gamma^\mu d$ and
$A_{ud}^\mu=\overline{u} \gamma^\mu \gamma_5 d$.
Moreover, our analysis will concentrate in the difference
\begin{eqnarray}
\Pi(s)\, &\equiv&\, \Pi_{ud,V-A}^{(0+1)}(s)\, =\,\Pi_{ud,V}^{(0+1)}(s)-\Pi_{ud,A}^{(0+1)}(s) \nonumber \\
\,&\equiv&\,
\frac{2 f_\pi^2}{s-m_\pi^2} + \overline{\Pi}(s)\, ,
\end{eqnarray}
where we have made explicit the contribution of the pion pole to the longitudinal axial-vector
two-point function. We will work in the isospin limit $m_u=m_d$ where $\Pi^{(0)}_{ud,V}(q^2)=0$.
\begin{figure}[htb]
\centering
\includegraphics[width=0.4\textwidth]{fig1.eps}
\caption[]{Analytic structure of $\overline{\Pi}(s)$.}
\label{fig:circuit}
\end{figure}
The correlator $\overline{\Pi}(s)$ is analytic in the entire complex $s$-plane, except for a cut on the
positive real axis which starts at the threshold $s_{\mathrm{th}}=4 m_\pi^2$.
Applying Cauchy's theorem to the circuit in Fig.~\ref{fig:circuit}, one gets the exact relation:
\begin{eqnarray}
\label{eq:sumrules}
&& \int^{s_0}_{s_{\rm th}} \mathrm{d}s\; s^n \, \frac{1}{\pi} \, {\rm Im} \, \Pi(s)
\, +\, \frac{1}{2 \pi i} \, \oint_{|s|=s_0} \mathrm{d}s\; s^n \,\Pi(s) \nonumber \\
& =& 2 f_\pi^2\, m_\pi^{2n}
+ \mathrm{Res} \left[ s^n \, \Pi(s), s=0\right] .\quad
\end{eqnarray}
For non-negative values of the integer power $n$, the pion pole is the only singularity within the contour
and one gets the so-called finite energy sum rules (FESR), widely used in the literature.
When $n$ takes negative values, the weight factor $s^n$ introduces a pole at the origin which
gives rise to the additional contribution in the r.h.s. of the equation, given by the residue of
$s^n \Pi(s)$ at $s=0$.
In the chiral limit ($m_u=m_d=0$) the correlator $\Pi(s)$ vanishes identically to all orders
in perturbation theory. For large enough Euclidean values of $s=-Q^2$ its operator product expansion (OPE),
$\Pi(Q^2) = \sum_k C_{2k}^{V-A}/Q^{2k}$,
contains only power-suppressed contributions from dimension $d=2k$ operators,
starting at $d=6$. The nonzero up and down quark masses induce tiny corrections with dimensions
two and four, which are negligible at high values of $Q^2$.
Therefore, with $n\ge 0$ and $s_0$ large enough so that the OPE can be applied in the entire circle
$s=s_0$, the integral over the spectral function from $s_{\mathrm{th}}$ to $s_0$ is equal to
the pion pole term $2 f_\pi^2\, m_\pi^{2n}$ plus the OPE contribution
$(-1)^n C_{2(n+1)}^{V-A}$
generated by the integration along the circle.
For $n=0$ and $n=1$, $C_{2(n+1)}^{V-A}$ is zero in the chiral limit and
one gets the celebrated first and second WSRs \cite{WSR}, respectively.
For negative values of $n\equiv -m<0$, the OPE does not give any contribution to the integration along
the circle $s=s_0$. One gets then:
\begin{eqnarray}
\label{eq:nSR}
&& \int^{s_0}_{s_{\rm th}} \frac{\mathrm{d}s}{s^m} \; \frac{1}{\pi} \, {\rm Im} \, \Pi(s)
\; =\; \frac{2 f_\pi^2}{m_\pi^{2m}} \,
+\,\frac{1}{(m-1)!}\,\Pi^{(m-1)}(0) \nonumber \\
\, &=&\, \frac{1}{(m-1)!}\,\overline{\Pi}^{(m-1)}(0)\, ,
\end{eqnarray}
where $\overline{\Pi}^{(m-1)}(0)$ denotes the $(m-1)$th derivative of $\overline{\Pi}(s)$ at
$s=0$.
The interest of this relation stems from the fact that at low values of $s$ the correlator can be rigourously
calculated within $\chi$PT. At present $\Pi(s)$ is known
to ${\cal O}(p^6)$ \cite{ABT00},
in terms of the LECs that we want to determine. The choices $m=1$ and $m=2$
allow then us to relate the spectral function measured
in $\tau$ decays with the theoretical expressions of $\overline{\Pi}(0)$ and $\overline{\Pi}\,{}'(0)$,
which can be derived from the results obtained in ref.~\cite{ABT00}:
\begin{eqnarray}
\label{L10-p6}
L_{10}^{\mathrm{eff}} &\!\equiv &\! -\frac{1}{8}\, \overline{\Pi}(0)
\nonumber\\ &\! = &\!
L_{10}^r(\mu) \, +\,
\frac{1}{128 \,\pi^2} \left[1- \log{\left(\frac{\mu^2}{m_\pi^2}\right)}\,
+ \,\frac{1}{3}\,\log{\left(\frac{m_K^2}{m_\pi^2}\right)} \right]
\nonumber\\ &\! +&\!
4 m_\pi^2 \left( C_{61}^r - C_{12}^r - C_{80}^r\right)\! (\mu)
\, \nonumber \\ &+& \, 4 \left( 2 m_K^2 + m_\pi^2 \right)\,
\left( C_{62}^r - C_{13}^r - C_{81}^r \right)\! (\mu)
\nonumber\\[7pt] &\! - &\!
2\, \left( 2 \mu_\pi + \mu_K \right) \,\left( L_9^r + 2 L_{10}^r\right)\! (\mu)\, \nonumber \\
&+&\, G_{2L}(\mu,s\!=\!0) \, +\, {\cal O} (p^8) \, ,
\\[10pt]\label{C87-p6}
C_{87}^{\mathrm{eff}} &\!\equiv &\! \frac{1}{16}\, \overline{\Pi}\,{}'(0)
\nonumber\\ &\! = &\!
C_{87}^r(\mu) \! + \!
\frac{1}{7680\, \pi^2} \left( \frac{1}{m_K^2} + \frac{2}{m_\pi^2} \right)
\nonumber \\ &-& \,
\frac{1}{64 \,\pi^2 f_\pi^2} \left[1- \log{\left(\frac{\mu^2}{m_\pi^2}\right)}\,
+ \,\frac{1}{3}\,\log{\left(\frac{m_K^2}{m_\pi^2}\right)} \right] \, L_9^r(\mu)
\nonumber \\ \, &-&
\frac{1}{2}\, G'_{2L}(\mu,s\!=\!0) \, +\, {\cal O} (p^8)\, ,
\end{eqnarray}
where $\mu_i= m_i^2 \log(m_i/\mu)/(16 \pi^2 f_\pi^2)$.
To a first approximation the effective parameters $L_{10}^{\mathrm{eff}}$
and $C_{87}^{\mathrm{eff}}$ correspond to the LECs $L_{10}^r(\mu)$ and $C_{87}^r(\mu)$, respectively.
At ${\cal O}(p^4)$, the only relevant correction is given by
the logarithmic terms in the second line
of (\ref{L10-p6}), which cancel the $\chi$PT\ renormalization
scale dependence of $L_{10}^r(\mu)$;
these contributions are suppressed by one power of $1/N_C$
with respect to $L_{10}^r(\mu)$, where
$N_C$ is the number of quark colours. The rest of
lines in (\ref{L10-p6}) contain the ${\cal O}(p^6)$ corrections:
the tree-level contributions from the ${\cal O}(p^6)$ $\chi$PT\ Lagrangian are given in the third and fourth lines,
the term proportional to $(L_9^r+2 L_{10}^r)(\mu)$ in the
fifth line is the one-loop contribution of the
${\cal O}(p^4)$ $\chi$PT\ Lagrangian, and the function
$G_{2L}(\mu,s\!=\!0)$ in the last line,
which does not depend on any LEC,
contains the proper two-loop contributions.
In Eq.~(\ref{C87-p6}) the tree-level contribution is given by $C_{87}^r(\mu)$, whereas the term proportional to $L_9^r(\mu)$ is a one-loop correction, which is suppressed by one power of $1/N_C$, and the two-loop contributions are contained in $G'_{2L}(\mu,s)\equiv \frac{d}{ds}G_{2L}(\mu,s)$.
The derivative operation, when acting over the one-loop contribution to $\Pi(s)$,
generates the terms proportional to inverse powers of the pion and kaon masses in the second line.
For simplicity, we omit the explicit analytic forms of
$G_{2L}(\mu)$ and $G'_{2L}(\mu)$, which are very lengthy and not too enlightening; these two functions contain
a $1/N_C^2$ suppression factor with respect to $L_{10}^r(\mu)$ and $C_{87}^r(\mu)$.
\section{Determination of Effective Couplings}
We will use the 2005 ALEPH data on semileptonic $\tau$ decays \cite{ALEPH05}, which provides the most recent and precise measurement of the $V-A$ spectral function. The effective chiral couplings can be directly extracted from the following integrals over the
hadronic spectrum:
\begin{eqnarray}
\label{eq:defL10}
-8 \, L_{10}^{\rm eff}&\equiv& \overline{\Pi}(0)\, =\,
\frac{1}{\pi} \int^{s_0}_{s_{th}}
\, \frac{{\rm d} s}{s} \, \mathrm{Im} \,\Pi(s) \,
, \\
\label{eq:defC87}
16 \, C_{87}^{ \rm eff} &\equiv& \overline{\Pi}\,{}'(0)\, =\,
\frac{1}{\pi} \int^{s_0}_{s_{th}}
\, \frac{{\rm d} s}{s^2} \, {\rm Im} \,\Pi(s)\, .
\end{eqnarray}
These relations are exactly satisfied at $s_0\to\infty$. At finite values of $s_0$,
they assume that the OPE approximates well the correlator $\Pi(s)$ over the entire complex circle
\footnote{Or equivalently these relations assume that the integrals on the real axis from $s_0$ to infinite are negligible, what is expected to be true only for high enough values of $s_0$ and for
accidental ``duality points''.} $|s|=s_0$. The OPE is expected to be a valid approximation for
high-enough values of $s_0$ and away from the real axis. While the kinematics of $\tau$ decay
restrict the upper limit of integration to the range $s_0\le m_\tau^2$,
the main source of theoretical uncertainty in the contour integration originates in the
region close to the point $s=s_0$ in the real axis.
Studying the sensitivity to $s_0$ of the integrals (\ref{eq:defL10}) and (\ref{eq:defC87}),
one can test validity of the OPE and assess the size of the associated systematic errors.
\begin{figure}[thb]
\vfill
\centerline{
\begin{minipage}[t]{.3\linewidth}\centering
\centerline{\includegraphics[width=9cm]{fig2.eps}}
\end{minipage}
}
\vfill
\caption{Determinations of $L_{10}^{\rm eff}$ at different values of $s_0$.
The continuous lines show the results obtained from Eq.~(\ref{eq:defL10}).
The modified expressions in Eqs.~(\ref{eq:L10-ModSR1}) and (\ref{eq:L10-ModSR2}) give rise to the
dashed and dot-dashed lines, respectively. For clarity, we do not include their corresponding error bands.}
\label{fig:L10}
\end{figure}
In Fig. \ref{fig:L10}, we plot the value of $L_{10}^{\rm eff}$ obtained from Eq.~(\ref{eq:defL10})
for different values of $s_0$. The band between the continuous lines shows the corresponding
experimental uncertainties (at one sigma). As expected, the result is far from an horizontal line
at low values of $s_0$, where the applicability of the OPE is suspect. The oscillatory behaviour
stabilises quite fast reaching a rather stable and flat result at
values of $s_0$ between 2 and 3 $\mathrm{GeV}^2$. The weight factor $1/s$ decreases the impact of the high-energy region,
minimising the size of quark-hadron duality violations around $s_0$.
This integral appears then to be much better behaved than the corresponding FESRs with $s^n$ ($n\ge0$) weights.
There are several possible strategies to estimate the central value for $L_{10}^{\rm eff}$
and the unavoidable theoretical uncertainties.
One is to give the predictions fixing $s_0$ at the so-called ``duality points'',
where the first and second WSRs happen to be satisfied.
Owing to the oscillatory behaviour of the WSRs results, this happens at two different values
of $s_0$. At the highest ``duality point'', which is obviously the more reliable, we obtain
$L_{10}^{\rm eff} = -(6.45 \pm 0.09) \cdot 10^{-3}$, where the quoted error only includes the experimental uncertainty.
Being very conservative, one could also take into account the first ``duality point'';
performing a weighted average of both results, we get
$L_{10}^{\rm eff} = -(6.50\pm 0.13) \cdot 10^{-3}$, where the uncertainty covers the values obtained at the two
``duality points''.
Assuming that the integral \eqref{eq:defL10} oscillates around
his asymptotic value with decreasing oscillations, one can get another
estimate performing an average between the maxima and minima of the successive oscillations.
This procedure gives a value $L_{10}^{\rm eff} = -(6.5\pm 0.2) \cdot 10^{-3}$,
that is perfectly compatible with the previous results based on the ``duality points''.
Our last method of estimating the quark-hadron duality violation
uses appropriate oscillating functions defined in \cite{GON07}
which mimic the real quark-hadron oscillations above the data.
These functions are defined such that they match the data at approximately
3 $\mathrm{GeV}^2$, go to zero with decreasing oscillations and
satisfy the first and second WSRs. We find in this way
$L_{10}^{\rm eff} = -(6.50\pm 0.12) \cdot 10^{-3}$,
where the error spans the range generated by the different functions used.
This result agrees well with our previous estimates.
We can take advantage of the WSRs to construct modified sum rules with weight
factors proportional to $(1-s/s_0)$, in order to suppress numerically the role
of the suspect region around $s\sim s_0$ \cite{LDP:92}:
\begin{eqnarray}
\label{eq:L10-ModSR1}
-8 \, L_{10}^{\rm eff}& = &
\frac{1}{\pi} \int^{s_0}_{s_{th}}\, \frac{{\rm d} s}{s} \,\left(1-\frac{s}{s_0}\right)
\, \mathrm{Im} \,\Pi(s)
\, +\, \Delta_1(s_0)\, ,
\nonumber \\ \\ [10pt] \label{eq:L10-ModSR2}
& = &
\frac{1}{\pi} \int^{s_0}_{s_{th}}\, \frac{{\rm d} s}{s} \,\left(1-\frac{s}{s_0}\right)^2
\, \mathrm{Im} \,\Pi(s) \nonumber \\
\, &+&\, 2 \Delta_1(s_0)\, -\, \Delta_2(s_0)\, .\quad
\end{eqnarray}
The factors $\Delta_1(s_0) = \left(2 f_\pi^2 + C^{V-A}_2\right)/s_0$ and
$\Delta_2(s_0) = \left(2 f_\pi^2 m_\pi^2- C^{V-A}_4\right)/s_0^2$ are small corrections dominated by the $f_\pi^2$ term, since
$C^{V-A}_{2,4}$ vanish in the chiral limit. The sum rule (\ref{eq:L10-ModSR2}) has been previously
used in refs.~\cite{DS07,DS04}.
The dashed and dot-dashed lines in Fig.~\ref{fig:L10} show the results obtained from
Eqs.~(\ref{eq:L10-ModSR1}) and (\ref{eq:L10-ModSR2}), respectively. As already found in refs.~\cite{DS07,DS04},
the modified weight factors minimise the theoretical uncertainties in a very sizeable way, giving rise
to very stable results over a quite wide range of $s_0$ values. One gets then
$L_{10}^{\rm eff} = -(6.51\pm 0.06) \cdot 10^{-3}$ using Eq.~(\ref{eq:L10-ModSR1}), and $L_{10}^{\rm eff} = -(6.45\pm 0.06) \cdot 10^{-3}$ from Eq.~(\ref{eq:L10-ModSR2}).
Taking into account all the previous discussion,
we quote as our final result:
\begin{equation}
\label{L10eff}
L_{10}^{\rm eff} = -(6.48\pm 0.06) \cdot 10^{-3} \, .
\end{equation}
\begin{figure}[thb]
\vfill
\centerline{
\begin{minipage}[t]{.3\linewidth}\centering
\centerline{\includegraphics[width=9cm]{fig3.eps}}
\end{minipage}
}
\vfill
\caption{Determinations of $C_{87}^{\rm eff}$ at different values of $s_0$.
The continuous lines show the results obtained from Eq.~(\ref{eq:defC87}).
The modified expressions in Eqs.~(\ref{eq:C87-ModSR1}) and (\ref{eq:C87-ModSR2}) give rise to the
dashed and dot-dashed lines, respectively. For clarity, we do not include their corresponding error bands.}
\label{fig:C87}
\end{figure}
We have made a completely analogous analysis to determine the effective coupling $C_{87}^{\rm eff}$.
The results are shown in Fig.~\ref{fig:C87}. The continuous lines, obtained from Eq.~(\ref{eq:defC87}),
are much more stable than the corresponding results for $L_{10}^{\rm eff}$,
owing to the $1/s^2$ factor in the integrand.
The discontinuous and dotted lines correspond to the results obtained from the modified sum rules:
\begin{eqnarray}
\label{eq:C87-ModSR1}
16 \, C_{87}^{\rm eff}& = &
\frac{1}{\pi} \int^{s_0}_{s_{th}}\, \frac{{\rm d} s}{s^2} \,\left(1-\frac{s^2}{s_0^2}\right)
\, \mathrm{Im} \,\Pi(s)
\, +\, \frac{\Delta_1}{s_0}\, ,
\\[10pt] \label{eq:C87-ModSR2}
& = &
\frac{1}{\pi} \int^{s_0}_{s_{th}}\, \frac{{\rm d} s}{s^2} \,\left(1-\frac{s}{s_0}\right)^2
\left(1+2\frac{s}{s_0}\right)\, \mathrm{Im} \,\Pi(s)
\, \nonumber \\
&+& \, \frac{3\Delta_1 -2 \Delta_2}{s_0}\, . \quad\quad
\end{eqnarray}
The agreement among the different estimates is quite remarkable.
We quote as our final conservative result,
\begin{equation}
\label{C87eff}
C_{87}^{\rm eff} = (8.18\pm 0.14) \cdot 10^{-3} \, {\rm GeV}^{-2} \, .
\end{equation}
\section{Determination of $L_{10}^r$ and $C_{87}^r$ }
\label{determination}
The $\chi$PT\ coupling $L_{10}^{r}(\mu)$ can be obtained from $L_{10}^{\rm eff}$,
using the relation (\ref{L10-p6}). At ${\cal O}(p^4)$ the determination is straightforward,
since one only needs to subtract from $L_{10}^{\rm eff}$ the term
$\left[1- \log{(\mu^2/m_\pi^2)}
+ \frac{1}{3}\log{(m_K^2/m_\pi^2)} \right]/(128\pi^2)$.
Taking $\mu=M_\rho$ as the reference value for the $\chi$PT\ renormalization scale, one gets
\begin{equation}
\label{valL10p4}
L_{10}^r(M_\rho) = -(5.22 \pm 0.06) \cdot 10^{-3} \, .
\end{equation}
At order $p^6$, the numerical relation is more subtle because it gets small corrections from other LECs.
It is useful to classify the ${\cal O}(p^6)$ contributions through their ordering within the $1/N_C$ expansion.
The tree-level term $4 m_\pi^2 (C_{61}^r - C_{12}^r- C_{80}^r)(M_\rho)$, which is the only ${\cal O}(p^6)$ correction
in the large--$N_C$ limit, is numerically small because it appears suppressed by a factor $m_\pi^2$.
The three relevant couplings have been determined phenomenologically with a moderate accuracy:
$C_{61}^r(M_\rho)=(1.24 \pm 0.44) \cdot 10^{-3}\:\mathrm{GeV}^{-2}$ \cite{KM06}
(from $\Pi^{(0+1)}_{ud,V}(0)-\Pi^{(0+1)}_{us,V}(0)$),
$C_{12}^r(M_\rho)=(0.4\pm 6.3) \cdot 10^{-5}\:\mathrm{GeV}^{-2}$ \cite{JOP04}
(from the $K\pi$ scalar form factor)
and
$C_{80}^r(M_\rho)=(2.1 \pm 0.5) \cdot 10^{-3}\:\mathrm{GeV}^{-2}$ \cite{UP08}
(from $a_1/K_1$ mass and width differences). These determinations agree reasonably well
with published meson-exchange estimates \cite{CEE05,ABT00} and lead to a total contribution
$4 m_\pi^2 (C_{61}^r - C_{12}^r- C_{80}^r)(M_\rho) = -(6.7\pm 5.2)\cdot 10^{-5}$. The scale dependence
of this combination of ${\cal O}(p^6)$ couplings \cite{p6}
between $\mu= 0.6$ GeV and $\mu=1.1$ GeV is within its quoted
uncertainty.
At NLO in $1/N_C$ we need to consider the tree-level contribution proportional to
the combination of LECs $(C_{62}^r - C_{13}^r- C_{81}^r)(M_\rho)$. We are not aware of any published
estimate of these $1/N_C$ suppressed couplings, beyond the trivial statement that they don't get
any tree-level contribution from resonance exchange \cite{CEE05}.
We will adopt the conservative range
$|C_{62}^r - C_{13}^r- C_{81}^r|(M_\rho) \le |C_{61}^r - C_{12}^r- C_{80}^r|(M_\rho)/3$,
which gives a contribution
$4 (2 m_K^2+m_\pi^2)(C_{62}^r - C_{13}^r- C_{81}^r)(M_\rho) =
(0.0\pm 5.8)\cdot 10^{-4}$.
The scale dependence between $\mu= 0.6$ GeV and $\mu=1.1$ GeV
of this combination of ${\cal O}(p^6)$ couplings \cite{p6}
is within its quoted uncertainty.
The uncertainty on this term will dominate our final error on the $L_{10}^r(M_\rho)$ determination.
At the same NLO in $1/N_C$, there is also a one-loop correction proportional to $L_{9}^r(M_\rho)$;
using the ${\cal O}(p^6)$ determination $L_9^r(M_\rho)=(5.93\pm 0.43)\cdot 10^{-3}$ \cite{BT02}, this
contribution can be estimated to be
$2 ( 2 \mu_\pi + \mu_K) \, L_9^r(M_\rho)
= - ( 1.56\pm 0.11)\cdot 10^{-3}$.
Finally, the $1/N_C^2$ suppressed two-loop function which collects the non-analytic contributions takes the value
$G_{2L}(M_\rho) = -0.524 \cdot 10^{-3}$,
one order of magnitude smaller than $L_{10}^{\rm eff}$,
but still eight
times larger than the uncertainty quoted for $L_{10}^{\rm eff}$
in \eqref{L10eff}. Taking all these contributions into account,
we finally get the wanted ${\cal O}(p^6)$ result:
\begin{eqnarray}
\label{valL10p6}
L_{10}^r(M_\rho) &=&
-(4.06 \pm 0.04_{L_{10}^{\mathrm{eff}}}\pm 0.39_{\,\mathrm{LECs}})
\cdot 10^{-3} \nonumber \\
&=& -(4.06 \pm 0.39) \cdot 10^{-3} \, ,
\end{eqnarray}
where the uncertainty has been split into its two main components.
The final error is completely dominated by our ignorance on the
$1/N_C$ suppressed LECs of ${\cal O}(p^6)$.
The determination of $C_{87}^r$ from $C_{87}^{\rm eff}$ does not involve any unknown LEC.
The relation (\ref{C87-p6}) contains a one-loop correction
of size $-(3.15\pm 0.13)\cdot 10^{-3}$, which only depends on $L_9^r(M_\rho)$ and the
pion and kaon masses,
and small non-analytic two-loop contributions collected in the
term $G'_{2L}(M_\rho) = -0.277\cdot 10^{-3}\:\mathrm{GeV}^{-2}$.
In spite of its $1/N_C$ suppression, the one-loop correction is
very sizeable, decreasing the final value of the ${\cal O}(p^6)$ LEC:
\begin{equation}
\label{valC87}
C_{87}^r(M_\rho) = (4.89 \pm 0.19) \cdot 10^{-3} \:\mathrm{GeV}^{-2}\, .
\end{equation}
\section{SU(2) $\chi$PT}
Up to now, we have discussed the LECs of the usual SU(3) $\chi$PT.
It turns useful to consider also the effective low-energy theory with only two flavours of light quarks.
In some cases, this allows to perform high-accuracy phenomenological determinations of the corresponding LECs
at NLO. Moreover, recent lattice calculations with two dynamical quarks are already able to obtain the
SU(2) LECs with sufficient accuracy and this is an important check for them.
In SU(2) $\chi$PT, there are ten LECs, $l_{i=1,..7}$ and $h_{1,2,3}$, at ${\cal O}(p^4)$ (NLO) \cite{GL84}.
Using the ${\cal O}(p^6)$ relation between $l_5^r(\mu)$ and $L_{10}^r(\mu)$,
recently obtained in ref.~\cite{GHI07},
and the definition of the invariant couplings $\overline l_i$ adopted in \cite{GL84}, we get
\begin{eqnarray}
\overline l_5 &=& - 192 \pi^2 \, L_{10}^{\rm eff}
+ 1 + \log{\left( \frac{m_K}{\hat m_K} \right)}
\nonumber \\
&+& 768 \, \pi^2 \, m_\pi^2 (C_{61}^r
+ C_{62}^r - C_{12}^r- C_{13}^r - C_{80}^r - C_{81}^r) (\mu)
\nonumber \\[7pt]
&+& 1536 \, \pi^2 (m_K^2 - \hat m_K^2)
( C_{62}^r - C_{13}^r - C_{81}^r) (\mu) \nonumber \\[7pt]
&-& 384\, \pi^2 (2\mu_\pi + \mu_K - \hat \mu_K)
( L_9^r + 2 L_{10}^r)(\mu) \nonumber \\
&-& x_K \left[ -\frac{67}{48} + \frac{21}{16} \rho_1 +
\frac{5}{8} \log \left(\frac{4}{3}\right)
-\frac{17}{4} \log{\left( \frac{\mu^2}{\hat m_K^2} \right)}
\right. \nonumber \\ &+&
\left.
\frac{3}{4} \log^2{ \left( \frac{\mu^2}{\hat m_K^2} \right)}
\right]
+ 192 \,\pi^2 \, G_{2L}(\mu) \, +\, {\cal O}(p^8)\, ,
\end{eqnarray}
where $\hat m_K^2 = m_K^2 - m_\pi^2/2$ is the kaon mass squared
in the limit $m_u=m_d=0$,
$x_K= \hat m_K^2/(16 \pi^2 f_\pi^2)$,
$\hat \mu_K= \hat m_K^2 \log(\hat m_K/\mu)/(16 \pi^2 f_\pi^2)$
and $\rho_1\simeq 1.41602$.
The first line contains the ${\cal O}(p^4)$ contributions; the determination of
$\overline l_5$ at this order is then straightforward. The full ${\cal O}(p^6)$ result,
with the different tree-level, one-loop and two-loop corrections,
is given in the other lines. Following the same procedure as in the SU(3) case, we get the results
\begin{equation}
\overline l_5 \, =\,\left\{
\begin{array}{ccc}
13.30 \pm 0.11 \, , & \; & {\cal O}(p^4),\\[7pt]
12.24 \pm 0.21 \, , & \; & {\cal O}(p^6).
\end{array}\right.
\end{equation}
\section{Summary}
Using the most recent hadronic $\tau$-decay data \cite{ALEPH05} on the $V-A$ spectral function,
and general properties of QCD such as analyticity, the OPE and $\chi$PT,
we have determined very accurately the chiral LECs $L_{10}^r(M_\rho)$
and $C_{87}^r(M_\rho)$. Performing an ${\cal O}(p^4)$ analysis, we obtain
\begin{equation}\label{eq:res-p4}
L_{10}^r(M_\rho)=-(5.22 \pm 0.06) \cdot 10^{-3} \, ,
\end{equation}
while a more elaborate study, including the ${\cal O}(p^6)$ $\chi$PT\ corrections
provides the values:
\begin{eqnarray}\label{eq:res1-p6}
L_{10}^r(M_\rho) &=&
-(4.06 \pm 0.04_{L_{10}^{\mathrm{eff}}}\pm 0.39_{\,\mathrm{LECs}}) \cdot 10^{-3} \nonumber \\
&=& -(4.06 \pm 0.39) \cdot 10^{-3} \, ,
\end{eqnarray}
and
\begin{equation} \label{eq:res2-p6}
C_{87}^r(M_\rho) = (4.89 \pm 0.19) \cdot 10^{-3} \: {\rm GeV}^{-2}\, .
\end{equation}
Our error estimate includes a careful analysis of the theoretical uncertainties associated
with the use of the OPE in the dangerous region close to the physical cut. Moreover, in \eqref{eq:res1-p6}
we have explicitly separated the error into its two main components, showing that our present ignorance
on the $1/N_C$ suppressed LECs dominates the final uncertainty of the $L_{10}^r(M_\rho)$
determination at ${\cal O}(p^6)$.
Several determinations of $L_{10}$ have been performed before \cite{DHG98,NAR01,DS04},
using the older 1998 ALEPH data \cite{ALEPH98,ALEPH97}.
In ref~\cite{DHG98} the result $L_{10}^r(M_\rho)= - (5.13 \pm 0.19) \cdot 10^{-3}$
was obtained to ${\cal O}(p^4)$, through a simultaneous fit of this parameter and the OPE corrections
of dimensions six and eight to several spectral moments of the hadronic distribution.
This determination is in good agreement with our ${\cal O}(p^4)$ result \eqref{eq:res-p4}.
Our quoted uncertainty has an smaller experimental contribution and includes a better assessment of the theoretical uncertainties.
The value $L_{10}^{\rm eff}=(-5.8\pm0.2)\cdot10^{-3}$ (3.2 $\sigma$ smaller than ours) was extracted from $\tau$ data in ref.~\cite{NAR01} using the first ``duality point'' of the WSRs. The difference comes from underestimated theoretical
uncertainties in this reference, as can be easily seen by choosing instead the second duality point or varying slightly the value of the first duality point. In fact the same reference \cite{NAR01} (see Eq. (10) therein) presents also a different estimate of $L_{10}^{\rm eff}$ that is in very good agreement with our result.
In ref.~\cite{DS04} both $L_{10}^{\rm eff}$ and $C_{87}^{\rm eff}$ were determined,
in good agreement with our findings which use the most recent 2005 data. An updated value
of $L_{10}^{\rm eff}$, using the 2005 data, has also been given in ref.~\cite{DS07}.
Our determinations of $L_{10}^r(\mu)$ and $C_{87}^r(\mu)$ at $\mu=M_\rho$ agree within errors
with the large--$N_C$ estimates based on lowest-meson dominance \cite{KN01,CEE04,ABT00,PI02}:
\begin{eqnarray}
L_{10} & = & -\frac{F_V^2}{4 M_V^2}\, +\,\frac{F_A^2}{4 M_A^2}\,\approx\, -\frac{3 f_\pi^2}{8 M_V^2}
\,\approx\, -5.4\cdot 10^{-3}\, , \nonumber \\
\\
C_{87} & = & \frac{F_V^2}{8 M_V^4}\, -\,\frac{F_A^2}{8 M_A^4}\,\approx\,\frac{7 f_\pi^2}{32 M_V^4}
\,\approx\, 5.3\cdot 10^{-3}\,\mathrm{GeV}^{-2} \, .
\nonumber \\
\end{eqnarray}
Eq. (\ref{eq:res2-p6}) is also in good
agreement with the result of ref. \cite{MP08} for $C_{87}$ based on
Pad\'e Approximants. These predictions, however, are unable to fix the scale dependence which is of higher-order in $1/N_C$.
More recently, the resonance chiral theory Lagrangian \cite{CEE04,EGPdR89}
has been used to analyse the correlator $\Pi(s)$ at NLO order in the $1/N_C$ expansion \cite{PRS08}.
Matching the effective field theory description with the short-distance QCD behaviour, the two LECs
are determined, keeping full control of their $\mu$ dependence.
The theoretically predicted values $L_{10}^r(M_\rho) = -(4.4 \pm 0.9) \cdot 10^{-3}$
and $C_{87}^r(M_\rho)=(3.6 \pm 1.3) \cdot 10^{-3}$ GeV$^{-2}$ \cite{PRS08}
are in perfect agreement with our determinations, although less precise. A recent lattice estimate \cite{SHI08} finds $L_{10}^r(M_\rho) = -(5.2 \pm 0.5) \cdot 10^{-3}$ at ${\cal O}(p^4)$, which is also in good agreement with our
${\cal O}(p^4)$ result in (\ref{eq:res-p4}).
A recent reanalysis of the decay $\pi^+ \to e^+ \nu \gamma$ \cite{UP08}, using new experimental data,
has provided quite accurate values for the combination of ${\cal O}(p^4)$ LECs $L_9+L_{10}$.
To ${\cal O}(p^4)$ one finds $L_9^r(M_\rho)+L_{10}^r(M_\rho)=(1.32\pm 0.14)\cdot 10^{-3}$, while the
${\cal O}(p^6)$ result $L_9^r(M_\rho)+L_{10}^r(M_\rho)=(1.44\pm 0.08) \cdot 10^{-3}$ is slightly more precise \cite{UP08}.
Combining these numbers with our results for $L_{10}^r(M_\rho)$, one obtains
\begin{equation}
L_9^r(M_\rho) \, =\,\left\{
\begin{array}{ccc}
(6.54 \pm 0.15) \cdot 10^{-3} \, , & \; & {\cal O}(p^4),\\[7pt]
(5.50 \pm 0.40) \cdot 10^{-3} \, , & \; & {\cal O}(p^6),
\end{array}\right.
\end{equation}
in perfect agreement with the ${\cal O}(p^4)$ result $L_9^r(M_\rho) = (6.9 \pm 0.7) \cdot 10^{-3}$ of ref. \cite{ECK07} and the ${\cal O}(p^6)$ result $L_9^r(M_\rho) = (5.93 \pm 0.43) \cdot 10^{-3}$ of ref. \cite{BT02}. This
last comparison represents an indirect check
(in fact the only possible one for the moment)
of our ${\cal O}(p^6)$ result for $L_{10}$.
We have also determined the corresponding LEC of $L_{10}$ in the SU(2) effective theory, both at LO and NLO:
\begin{equation}\label{eq:res3}
\overline l_5 \, =\,\left\{
\begin{array}{ccc}
13.30 \pm 0.11 \, , & \; & {\cal O}(p^4),\\[7pt]
12.24 \pm 0.21 \, , & \; & {\cal O}(p^6).
\end{array}\right.
\end{equation}
>From a phenomenological analysis of the radiative decay $\pi \to l \nu \gamma$ within SU(2) $\chi$PT, the authors of ref. \cite{BT97} obtained
$\overline l_6 - \overline l_5 = 2.57 \pm 0.35$ at ${\cal O}(p^4)$, and
$\overline l_6 - \overline l_5 = 2.98 \pm 0.33$ at ${\cal O}(p^6)$.
Using these results and our determinations for $\overline l_5$ in \eqref{eq:res3}, one gets
\begin{equation}
\overline l_6\, =\,\left\{
\begin{array}{ccc}
15.87 \pm 0.37 \, , & \; & {\cal O}(p^4),\\[7pt]
15.22 \pm 0.39 \, , & \; & {\cal O}(p^6).
\end{array}\right.
\end{equation}
At ${\cal O}(p^4)$ the comparison of these estimates of SU(2) LECs with previous results is straightforward, since they are proportional to the corresponding SU(3) couplings, that we have already discussed. Our determination of
$\overline l_5$ is the first one obtained at ${\cal O}(p^6)$, whereas for $\overline l_6$ ref. \cite{BCT98} finds $\overline l_6 = 16.0 \pm 0.5 \pm 0.7$, where the last error is purely theoretical, in good agreement with ours, although less precise.
\section*{Acknowledgements}
We would like to thank Hans Bijnens and Pere Talavera for providing information on the two-loop functions in ref.~\cite{ABT00}. We are also grateful to Jorge Portol\'es for his help and useful comments.
M.G.-A. is indebted to MICINN (Spain) for a FPU Fellowship.
This work has been supported in part by the EU
RTN network FLAVIAnet [Contract No. MRTN-CT-2006-035482],
by MICINN, Spain
[Grants FPA2007-60323 (M.G.-A., A.P), FPA2006-05294 (J.P.) and Consolider-Ingenio
2010 Programme CSD2007-00042 --CPAN--] and
by Junta de Andaluc\'{\i}a (J.P.)
[Grants P05-FQM 101, P05-FQM 467 and P07-FQM 03048].
| {
"redpajama_set_name": "RedPajamaArXiv"
} | 6,920 |
\section{Introduction}
\IAENGPARstart{R}{ecently} we have developed and used code for solving
ordinary Frobenius type differential equations to very
high precision \cite{CPC2011:AsifAmnaKareIngjald, ICCP2011:AmnaKare, CPC2012:AmnaKare},
like finding the lowest eigenvalue of
\begin{equation}
-\psi''(x) + x^4 \psi(x) = \varepsilon \psi(x)
\end{equation}
to one million decimals. The general class of equations
treated in \cite{CPC2012:AmnaKare} is of the type
{\footnotesize
\begin{equation}
-\!\left(\frac{d^2}{dz^2} +
\frac{1\!-\!\nu_+ \!-\! \nu_-}{z}\frac{d}{dz} +
\frac{\nu_+ \nu_-}{z^2} \right)\psi(z) +
\frac{1}{z}\sum_{n=0}^{N} \text{v}_n\, z^n\,\psi(z) = 0.
\label{ODE}
\end{equation}
}
Following the Frobenius method our solution is represented by
a convergent series
\begin{equation}
\psi(z) = \sum_{m=0}^\infty a_m\,z^{m+\nu},
\label{PowerSeries}
\end{equation}
where the coefficients $a_m$ is generated recursively in parallel
with a brute force summation of the series. The individual terms
in (\ref{PowerSeries}) may grow very big, leading to huge cancellations
and large roundoff errors. It is therefore useful to have some prior
knowledge of the magnitude of the $a_m$'s before a
high-precision evaluation --- to set the computational precision
required for a desired accuracy of the final result, and to
estimate the time required to complete the computation.
We have found that $\vert a_m\vert$ can be
estimated surprisingly accurate from a WKB approximation
of the solution, followed by a Legendre transform. For the
general class of equations~(\ref{ODE}) the WKB integrals
and the Legendre transform must be done by (ordinary precision)
numerical methods.
In the remainder of this paper we first derive a Legendre transform
relation between the magnitudes $\vert a_m \vert$ and
$\vert \psi(z) \vert$, slightly generalized to take into account
a logarithmic correction, and next use the WKB approximation to
estimate $\psi(z)$.
\section{Legendre transform method of solution}
Our method of solution is based on the hypothesis that
the sum~(\ref{PowerSeries}) for large $\vert z \vert$
receives its main contribution from a
relatively small range of $m$-values.
Introduce quantities $u$ and $s(m)$ so that
\begin{equation*}
x = \text{e}^u,\quad \vert a_m \vert = \text{e}^{s(m)}.
\end{equation*}
Our assumption is that
\begin{equation}
\text{e}^{S(u)} \equiv \max_{\varphi} \psi(\text{e}^{u+i\varphi}) \approx
\sum_{m} \text{e}^{s(m)+(\nu+m)u},
\label{MaximumAbsoluteValue}
\end{equation}
with the main contribution to the sum coming from a small range
of $m$-values around a maximum value $\bar{m}$. The latter is
defined so that $s'(\bar{m})+u=0$, $s''(\bar{m})<0$.
Now write $m = \bar{m} + \Delta m$, and approximate the
sum (\ref{MaximumAbsoluteValue}) over $\Delta m$ by a gaussian integral.
This gives
\begin{equation*}
\text{e}^{S(u)} \approx \sqrt{{-2\pi}/{s''(\bar{m})}}\,\text{e}^{s(\bar{m})+(\nu+\bar{m})u}.
\end{equation*}
In summary, we have found the relations
\begin{align}
u &= -s'(m)\label{VariableChange},\\
S(u) &= s(m) - (\nu + m) s'(m) +\frac{1}{2}\log\left(\frac{2\pi}{-s''(m)} \right)\nonumber\\
&\equiv S_0(u) + \frac{1}{2}\log\left(\frac{2\pi}{-s''(m)} \right).\label{FunctionChange}
\end{align}
This is essentially a Legendre transformation between $s(m)$ and $S(u)$.
Consider a small change $u\to u+\delta u $. To maintain the maximum condition
we must also make a small change $m \to m+\delta m$, with
\(
\delta m = -{\delta u}/{s''(m)}
\). I.e.~$s''(m)=-u'(m)$.
This is consistent with the result of taking the $m$-derivative of
equation~(\ref{VariableChange}).
One further finds that $S_0(u)$ becomes
\begin{align*}
&S_0(u+\delta u) = S_0(u) + S'_0(u)\,\delta u + \frac{1}{2}S''_0(u)\,\delta u^2 +\cdots\\
&=s(m) + (\nu + m) u + (m+\nu)\,\delta u - \frac{1}{2s''(m)}\, \delta u^2 + \cdots,
\end{align*}
giving the relations
\begin{align}
(m+\nu) &= S'_0(u), \label{mExpression}\\
s(m) &= S_0(u) - u S'_0(u),\label{sExpression}\\
s''(m) & = -S''_0(u)^{-1}. \label{SecondDerivative}
\end{align}
Equation (\ref{SecondDerivative}) just says that $\left(dm/du\right) = \left(du/dm\right)^{-1}$.
We are only able to compute $S(u)$ directly, not $S_0(u)$. However, they only differ by
a logarithmic term, hence we will approximate $\log(-s''(m)) = -\log S''_0(u) \approx -\log S''(u)$.
This gives
\begin{equation}
S_0(u) \approx S(u) - \frac{1}{2}\log\left(2\pi\, S''(u)\right),\label{logCorrectedS0}
\end{equation}
which can be used in equations (\ref{mExpression}--\ref{SecondDerivative})
when we have computed $S(u)$.
\section{WKB approximation}
It remains to find $S(u)$. Here we will use the leading order WKB
approximation to find a sufficiently accurate estimate.
When $z=0$ is an ordinary point, i.e. when $\nu_-=0$, $\nu_+=1$,
the leading order WKB solution to (\ref{ODE}) is
\begin{equation}
\psi(z) \approx \sqrt{Q_0/Q(z)}
\exp\left({\frac{1}{s}}\int_0^{z} Q(t) \text{d}t \right),
\label{WKBApproximation}
\end{equation}
where $Q^2(z) = \sum_{n=1}^N \text{v}_n z^{n-1}$, and $Q_0=Q(0)$.
This represents a superposition of the solutions $\psi_\pm(z)$.
The difference between the $\nu_+$ and $\nu_-$ solutions is at worst
comparable to accuracy of our approximation; hence we will not
distinguish between them.
When $z=0$ is a regular singular point we use the Langer corrected
WKB approximation to obtain leading order solutions in the form
\begin{align}
\psi_{\pm}(z) &\approx
z^{\nu_{\pm}} \sqrt{Q_0/Q(z)}\; \times\nonumber\\
&\exp\left(\pm\frac{1}{s}\int_0^z
\frac{\text{d}t}{t}
\left[\sqrt{ Q^2(t)} - Q_0\right] \right). \label{LangerCorrectedWKBApproximation}
\end{align}
Here $Q^2(z) = \frac{1}{4}s^2 (\nu_+-\nu_-)^2 + \sum_{n=0}^N \text{v}_n z^{n+1}$,
and $Q_0 = Q(0)$.
In equation (\ref{LangerCorrectedWKBApproximation}) we distinguish
between the $\nu_+$- and $\nu_-$-solutions, because the difference $\nu_+ - \nu_-$ may
in principle be large.
The WKB integrals must in general be done numerically, sometimes along curves
in the complex plane. This requires careful attention to branch cuts. Here
we will only give some examples where most of the calculations can be done
analytically.
\subsection{Example 1: Anharmonic oscillators}
Consider the equation
\begin{equation}
-\frac{\partial^2}{\partial y^2}\Psi(y) + \left(y^2+ c^2\right)^2\Psi(y) = 0,
\label{Anharmonic_oscillator}
\end{equation}
for real $c$ so that $c^2 \ge 0$. For large $y$ the typical solution behaves like
\begin{equation}
\Psi(y) \sim \text{e}^{\frac{1}{3}y^3 + c^2 y},
\label{CrudeWKBApproximation}
\end{equation}
neglecting the slowly varying prefactor. For a given value of $\vert y \vert$
this is maximum along the positive real axis.
Hence, with $x=y^2=\text{e}^{u}$, we find as a leading approximation
\begin{equation*}
S(u) = {\textstyle \frac{1}{3}}\left(\text{e}^{\frac{3}{2}u} + 3 c^2 \text{e}^{\frac{1}{2}u}\right).
\end{equation*}
In this case the Frobenius series can be written
\begin{equation}
\Psi(y) = \sum_{m=0}^{\infty} a_m\, y^{2m + \nu} \equiv \sum_{m=0}^{\infty} A_m(y),
\end{equation}
with $\nu=0,\;1$.
Ignoring the $\log(S''(u))$-term in~(\ref{mExpression}, \ref{sExpression}, \ref{logCorrectedS0})
we find
\begin{align}
{m} &= {\textstyle \frac{1}{2}}\left(\text{e}^{\frac{3}{2}u} + c^2\,\text{e}^{\frac{1}{2}u} \right),
\label{m_Anharmonic}
\\
\log\left(\left| a_{{m}}\right|\right) &=
\left({\textstyle \frac{1}{3}} - {\textstyle \frac{1}{2}} u\right) \text{e}^{\frac{3}{2}u} +
c^2 \left(1 -{\textstyle \frac{1}{2}} u \right) \text{e}^{\frac{1}{2}u}.
\label{a_m_Anharmonic}
\end{align}
\begin{figure}[!t]
\begin{center}
\includegraphics[clip, trim = 8ex 6ex 9ex 5ex, width=0.483\textwidth]{Coefficients_a_m}
\end{center}
\caption{Comparison of numerical coefficients $a_m$ (points)
with estimates (full-drawn lines) based on
(\ref{m_Anharmonic}, \ref{a_m_Anharmonic}) and
(\ref{m_DoubleWell}, \ref{a_m_DoubleWell}). The estimates
of $\log\vert a_m \vert$ are accurate up to corrections which
depend logarithmically on $m$.
}
\label{Coefficients_a_m}
\end{figure}
\noindent
For $c=0$ an explicit representation is
\begin{equation}
\log \vert a_m \vert = \frac{2}{3}m\left(1 -\log 2m \right).
\label{prediction0}
\end{equation}
This is plotted as the lower curve in figure~\ref{Coefficients_a_m}. It fits satisfactory
with the high-precision coefficients generated numerically, but there remains a
correction which depends logarithmically on $m$. For nonzero $c$ the parametric
representation provides equally good results, as shown by the upper curve in
figure~\ref{Coefficients_a_m}.
The conclusion of this example is that for a fixed (large) $x$
we expect the largest term of the power series to be
\begin{equation}
\mathop{\text{max}}_m \vert A_m(x) \vert \sim \text{e}^{\frac{1}{3}(x^{3/2}+3 c^2 x^{1/2})},
\end{equation}
neglecting a slowly varying prefactor.
Further, the maximum should occur at
\begin{equation}
m \approx {\textstyle \frac{1}{2}} \left( x^{3/2} + c^2 x^{1/2} \right).
\end{equation}
Finally, estimates like equation (\ref{prediction0})
for the coefficients $a_m$ may be used to predict how many terms ${\cal M}$
we must sum to evaluate $\psi(x)$ to a given precision $P$,
based on the stopping criterium
\begin{equation}
\vert a_{\cal M} \vert\, x^{\cal{M}} \le 10^{-P}.
\end{equation}
As can be seen in figure~\ref{lengthOfSums} the agreement with the actual
number of terms used by our evaluation routine is good,
in particular for high precision $P$. But keep in mind that a
logarithmic scale makes it easier for a comparison to look good.
\begin{figure}[!t]
\begin{center}
\includegraphics[clip, trim = 10.5ex 5ex 10ex 5ex, width=0.483\textwidth]{lengthOfSums}
\end{center}
\caption{This figure compares the {\em a priori\/} prediction,
based on equation~(\ref{prediction0}),
of the number of terms ${\cal M}$ which must be summed in order to evaluate
$\Psi(y)$ for $c=0$ to a desired precision $P$ with the actual number of terms
computed by.
}
\label{lengthOfSums}
\end{figure}
\noindent
Next consider the logarithmic corrections. Including the prefactor of
equation~(\ref{CrudeWKBApproximation}) changes $S(u)$ by an amount
\begin{equation}
\Delta S(u) = -\frac{1}{2}\log\left(\text{e}^u + c^2\right).
\end{equation}
Including the $\log(S''(u))$-term in the relation between
$S(u)$ and $S_0(u)$ changes $S_0$ by an additional amount
\begin{equation}
\Delta S_0(u) =
-\frac{1}{2}\log\left(\frac{3}{4}\text{e}^{\frac{3}{2}u}+\frac{1}{4}c^2\,\text{e}^{\frac{1}{2}u}\right).
\end{equation}
For $c^2 = 0$ this changes the relation~(\ref{prediction0}) to
\begin{equation}
\log \vert a_m \vert = \frac{1}{3}\left(2m+{5}/{2}\right)
\left(1 -\log \left(2 m + {5}/{2}\right)\right).
\label{prediction1}
\end{equation}
For $\vert a_m \vert$ this essentially corresponds to a factor $ m^{-5/6}$.
\begin{figure}[!t]
\begin{center}
\includegraphics[clip, trim = 10.5ex 5ex 10ex 5ex, width=0.483\textwidth]{Ratios_a_m}
\end{center}
\caption{This figure shows the ratio between the computed coefficients $a_m$ and the
crude prediction~(\ref{prediction0}) (labelled $\vert a^{(0)}_m \vert$) and the
logarithmically corrected prediction~(\ref{prediction1}) (labelled $\vert a^{(1)}_m \vert$).
For easy comparison we have in both cases adjusted an overall constant such that the ratio
is unity for $m=3$.
}
\label{Ratios_a_m}
\end{figure}
\subsection{Example 2: Double well oscillators}
The same procedure also work for the equation
\begin{equation}
-\frac{\partial^2}{\partial y^2}\Psi(y) + \left(y^2- c^2\right)^2\Psi(y) = 0,
\label{Double_well}
\end{equation}
which however is a little more challenging since the maximum value of
$\vert\Psi(y\text{e}^{\text{i}\varphi}\vert$ sometimes occur for
$\varphi \ne 0$, i.e.~for complex arguments.
For large $y$ the typical solution behaves like
\begin{equation}
\Psi(y) \sim \text{e}^{\frac{1}{3}y^3 - c^2 y},
\end{equation}
neglecting the slowly varying prefactor. Equation (\ref{Double_well})
can be transformed to the form (\ref{ODE}) by introducing $x=y^2$,
$\Psi(y) = \psi(x)$. Hence, with $x=y^2=\text{e}^{u}$
\begin{equation*}
S(u) = \mathop{\text{max}}_\varphi {\textstyle \frac{1}{3}}\text{Re} \left(\text{e}^{\frac{3}{2}(u+\text{i}\varphi)}
- 3 c^2 \text{e}^{\frac{1}{2}(u+\text{i})\varphi}\right).
\end{equation*}
The maximum occurs for $\cos\frac{1}{2}\varphi = -\frac{1}{2}\left(1 + c^2\,\text{e}^{-u} \right)^{1/2}$ when
$\text{e}^{u} \ge \frac{1}{3} c^2$, and for $\cos\frac{1}{2}\varphi=-1$ otherwise.
This gives
\begin{equation}
S(u) = \left\{\begin{array}{cc}
{\textstyle c^2 \text{e}^{u/2} - \frac{1}{3}\text{e}^{3u/2}}&\text{for $e^u \le \frac{1}{3}c^2$,}\\[0.5ex]
{\textstyle \frac{1}{3}} (\text{e}^{u} + c^2)^{3/2}&\text{for $e^u \ge \frac{1}{3}c^2$.}
\end{array}
\right.
\end{equation}
This implies that
{\footnotesize
\begin{align}
\bar{m} &=
\left\{\begin{array}{lc}
\frac{1}{2} \text{e}^{u/2}\left(c^2 - e^u\right)
&\text{for $e^u \le \frac{1}{3}c^2$,}\\[0.5ex]
{\textstyle \frac{1}{2}} \text{e}^{u}\,\left( \text{e}^u + c^2 \right)^{1/2}&\text{for $e^u \ge \frac{1}{3}c^2$},
\end{array}
\right.
\label{m_DoubleWell}
\\
\log\left(\left| a_{\bar{m}}\right|\right) &=
\left\{\begin{array}{cc}
\left(1\!-\!\frac{1}{2}u\right)c^2 \text{e}^{u/2} -\left(\frac{1}{3}-\frac{1}{2}u\right)\text{e}^{3u/2}
&\text{for $e^u \le \frac{1}{3}c^2$,}\\[0.5ex]
\left[\left({\textstyle \frac{1}{3}}\! -\!{\textstyle \frac{1}{2}}u \right)\text{e}^{u}
+{\textstyle \frac{1}{3}}c^2\right]\left(\text{e}^u + c^2\right)^{1/2}&\text{for $e^u \ge \frac{1}{3}c^2$}.
\end{array}
\right.
\label{a_m_DoubleWell}
\end{align}
}
This representation compares fairly well with the numerically generated coefficients,
as shown by the middle curve in figure~\ref{Coefficients_a_m}. However, in this case
the coefficients $a_m$ have a local oscillating behaviour. The representation
(\ref{m_DoubleWell}, \ref{a_m_DoubleWell}) should be interpreted as the local amplitude
of this oscillation.
The conclusion of this example is that we expect the largest term of the power series to be
term of the series to be
\begin{equation}
\mathop{\text{max}}_m \vert A_m(x) \vert \sim \text{e}^{\frac{1}{3}(x + c^2)^{3/2}},
\end{equation}
neglecting the slowly varying prefactor.
Further, the maximum should occur at
\begin{equation}
m \approx {\textstyle \frac{1}{2}} x \left( x + c^2 \right)^{1/2}
\approx {\textstyle \frac{1}{2}} x^{3/2} + {\textstyle \frac{1}{4}} c^2 x^{1/2}.
\end{equation}
\section{Conclusion}
As illustrated in this contribution the coefficients of Frobenius series
can be predicted to surprisingly high accuracy by use of Legendre transformations
and lowest order WKB approximations. We have also tested the validity of the
method on many other cases.
\appendices
\section*{Acknowledgment}
We thank A.~Mushtaq and I.~{\O}verb{\o} for useful discussions.
This work was supported in part by the Higher Education
Commission of Pakistran (HEC).
\ifCLASSOPTIONcaptionsoff
\newpage
\fi
| {
"redpajama_set_name": "RedPajamaArXiv"
} | 3,204 |
Q: problem finding a header with a c++ makefile I've started working with my first makefile. I'm writing a roguelike in C++ using the libtcod library, and have the following hello world program to test if my environment's up and running:
#include "libtcod.hpp"
int main()
{
TCODConsole::initRoot(80, 50, "PartyHack");
TCODConsole::root->printCenter(40, 25, TCOD_BKGND_NONE, "Hello World");
TCODConsole::flush();
TCODConsole::waitForKeypress(true);
}
My project directory structure looks like this:
/CppPartyHack
----/libtcod-1.5.1 # this is the libtcod root folder
--------/include
------------libtcod.hpp
----/PartyHack
--------makefile
--------partyhack.cpp # the above code
(while we're here, how do I do proper indentation? Using those dashes is silly.)
and here's my makefile:
SRCDIR = .
INCDIR = ../libtcod-1.5.1/include
CFLAGS = $(FLAGS) -I$(INCDIR) -I$(SRCDIR) -Wall
CC = gcc
CPP = g++
.SUFFIXES: .o .h .c .hpp .cpp
$(TEMP)/%.o : $(SRCDIR)/%.cpp
$(CPP) $(CFLAGS) -o $@ -c $<
$(TEMP)/%.o : $(SRCDIR)/%.c
$(CC) $(CFLAGS) -o $@ -c $<
CPP_OBJS = $(TEMP)partyhack.o
all : partyhack
partyhack : $(CPP_OBJS)
$(CPP) $(CPP_OBJS) -o $@ -L../libtcod-1.5.1 -ltcod -ltcod++ -Wl,-rpath,.
clean :
\rm -f $(CPP_OBJS) partyhack
I'm using Ubuntu, and my terminal gives me the following errors:
max@max-desktop:~/Desktop/Development/CppPartyhack/PartyHack$ make
g++ -c -o partyhack.o partyhack.cpp
partyhack.cpp:1:23: error: libtcod.hpp: No such file or directory
partyhack.cpp: In function 'int main()':
partyhack.cpp:5: error: 'TCODConsole' has not been declared
partyhack.cpp:6: error: 'TCODConsole' has not been declared
partyhack.cpp:6: error: 'TCOD_BKGND_NONE' was not declared in this scope
partyhack.cpp:7: error: 'TCODConsole' has not been declared
partyhack.cpp:8: error: 'TCODConsole' has not been declared
make: *** [partyhack.o] Error 1
So obviously, the makefile can't find libtcod.hpp. I've double checked and I'm sure the relative path to libtcod.hpp in INCDIR is correct, but as I'm just starting out with makefiles, I'm uncertain what else could be wrong. My makefile is based off a template that the libtcod designers provided along with the library itself, and while I've looked at a few online makefile tutorials, the code in this makefile is a good bit more complicated than any of the examples the tutorials showed, so I'm assuming I screwed up something basic in the conversion. Thanks for any help.
A: What happens here, is that
1) make evaluates the target all, which resolves to partyhack.
2) make evaluates the target partyhack, which resolves to $(CPP_OBJS)
3) make evaluates the target $(CPP_OBJS), which resolves to $(TMP)partyhack.o
4) make evaluates the target $(TMP)partyhack.o which resolves to partyhack.o
This is because TMP is not defined. Also note that the slash is missing after $(TMP).
5) make evaluates the target partyhack.o, and applies the implicit rule g++ -c -o partyhack.o partyhack.cpp
It does not apply the rule you specified, namely $(TEMP)/%.o : $(SRCDIR)/%.cpp because TEMP is not defined, so this evaluates to /partyhack.o : ./partyhack.cpp, and we are not building /partyhack.o because we are not in the root directory.
6) g++ does not find the include file, because the include directories were not passed to it, because your rule was not applied.
To fix this:
First, you need to define TEMP (see Nick Meyers answer).
TEMP = .
Second, you need to change the definition of CPP_OBJS (as Paul R suggested).
CPP_OBJS = %(TEMP)/partyhack.o
This will work if you invoke make inside the directory CppPartyHack/PartyHack.
A: Make appears to be using a built-in rule as opposed to your custom rule for *.cpp files. (See the ordering of the -c and -o options in make's output -- it's not the same as the ones you wrote). Check the definition of your C++ file rule.
My best guess is that the references to $(TEMP) are throwing it off. Where is this defined?
Also note that for C++, usually the variables CXX and CXXFLAGS are used for the compiler program and the flags, respectively. CPP is used for the C preprocessor program, CPPFLAGS for preprocessor flags and CFLAGS is used for the C compiler flags. That's probably why make is not using your CFLAGS when using its built-in C++ rules.
A: Maybe you should change the INCDIR to an absolute path rather than a relative one? The current working directory may not be what you think it is while make is running.
A: There's obviously a problem somewhere because your CFLAGS definitions are not being passed to g++.
Change:
CPP_OBJS = $(TEMP)partyhack.o
to:
CPP_OBJS = $(TEMP)/partyhack.o
Other than that I don't see anything wrong.
| {
"redpajama_set_name": "RedPajamaStackExchange"
} | 4,299 |
Q: not class selector doesn't seem to work with anchor I have a wrapped set like this:
var $anchors = $('a');
I then do some JQuery code to confirm a deletion and etc... everything was working great until I started adding other anchors to my site that have classes that do different things
I'm trying to use the :not in JQuery to eleminate certain classes from this generic anchor wrapped set, so that the configme delition and etc doesn't occur, but can't seem to get it to work... what am I missing? Here is the updated selector I am trying to use:
var $anchors = $('a:not(".excludedanchorclass")');
A: No quotes:
var $anchors = $('a:not(.excludedanchorclass)');
A: You can also use jquery filter
var $anchors = $('a').filter(':not(.excludedanchorclass)');
| {
"redpajama_set_name": "RedPajamaStackExchange"
} | 5,277 |
\section{Introduction}
In multi-subject studies where observations are considered to be
functional realizations, it is common to observe large amounts of
between-subject heterogeneity in the sense that some subjects produce
curves that are quite smooth while others produce curves that are
(much) more erratic. Although not often explicitly considered when
modeling these types of data, the variability in curve shape can be an
important feature that may help distinguish individuals and lead to
better understanding of processes under study and/or better
predictions. Often in these studies two types of predictions are
desired: those that predict the functional output across the entire
time domain for a hypothetical new subject and those that complete a
partially observed functional response for subjects currently
participating in the study. Explicitly considering curve shape in
modeling, in addition to considering relevant covariates, should
improve both types of predictions. This is particularly true of the
application motivating the present study. Decision makers of teams that
belong to the National Basketball Association (NBA) are very interested
in being able to group and predict future performance of basketball
players that are employed or could possibly be employed by NBA teams.
\begin{figure}[htbp]
\includegraphics{919f01}
\caption{Career game-by-game Game Score results for three NBA players.
A loess smoother with span equal to 0.3 is also provided.}
\label{rawscatterplot}
\end{figure}
The NBA is a North American professional mens basketball league that
arguably employs the worlds most gifted basketball players. The
league's popularity has steadily increased and as a result player
salaries have exploded. Personnel decisions in the NBA (as in most
professional sports leagues) are high-risk transactions. In the face of
massive amounts uncertainty teams offer players guaranteed multi-year
multi-million dollar contracts and as a result mistakes in player
acquisition are extremely expensive. Making things even more
treacherous are the abstruse rules governing player transactions found
in the collective bargaining agreement (CBA). Among other things, the
CBA regulates the amount of resources dedicated to player acquisition.
Teams that misallocate player salary resources by over paying severely
hinder a team's future flexibility and negatively impact a team's
future competitiveness and profitability for years. Because of this,
added value might be assigned to players who perform consistently
compared to those that are more up and down.
Figure \ref{rawscatterplot} displays scatter-plots and loess curves
(with a span of 0.3) of game-by-game ``production'' for three NBA
basketball players. Game-by-game ``production'' in Figure \ref{rawscatterplot}
is measured using the so called Game Score statistic
(\citealt{hollinger2002pro}). More details are provided in Section \ref{examples}
and the Appendix, but for now it suffices to know that
higher values correspond with better performance and more production.
Even though there is a large amount of game-to-game and
player-to-player variability in Game Score it is still evident that
production consistency between the three players varies. Erick Dampier
appears to have two spikes of improved production, Jarron Collin's
production oscillates during the first part of his career while Ray
Allen's production is fairly smooth as it gradually increases and
decreases with slight dip in the middle. Therefore curve shape should
contain information that is valuable in distinguishing between
different types of players and being able to assess their future value.
Two types of predictions are used to assess future value. The first
considers players who are currently members of the NBA and that will
continue to participate in future games. Ray Allen in Figure \ref
{rawscatterplot} is an example of such a player. Although he has
already played in more than 1000 games, he continues to play and
predicting the performance for the remainder of his career is of
considerable interest. This type of prediction will be referred to as
``active player prediction''. The second type of prediction considers
basketball players who have yet to play in an NBA game but who have a
skill set that will attract interest from NBA teams. For these players,
predicting the entire career production curve is of interest. This type
of prediction will be referred to as ``career prediction''.
It has become fairly common to consider the longitudinal curves of the
type just described as discretized realizations of functional data.
There is now a large literature dedicated to functional data analysis
(FDA) techniques. A few popular methods that are actively being
researched are functional principal components (\citealt[chap.
6]{FunctionalDataAnalysisBook}, \citealt
{MultilevelFunctionalPrincipalComponentAnalysis}), Gaussian process
regression methods (\citealt{rasmussen}, \citealt
{HierModelsAssesVarAmongFunctionsBehesta}, and \citealt
{LocallyNPBRegViaGP}) and multi-level functional basis expansion
(\citealt{WaveletBasedFunctionalMixedModel}, \citealt
{BayesFreeKnotCurveFit}, \citealt{Biller:2000}, \citealt
{BayesianLatentFactorRegressionForFunctionalAndLongitudinalData},
\citealt{AdaptiveRegressionSplines}). When considering multiple-subject
studies the methods just described tend to separate individuals
according to trend levels only, while ignoring the shape of the
longitudinal projections. Though the idea of explicitly using shape or
smoothness of curves to improve prediction is intuitively appealing
there is surprisingly very little in the statistical literature
dedicated to it. The one article that we are aware of is \cite
{StochasticVolatilityRegressionForFunctionalDataDynamics} whose focus
is on estimating rate functions through a complicated system of
differential equations and using covariates to explain variability in
trajectories via stochastic volatility models. They applied their
method to a longitudinal multi-subject blood pressure study for
pregnant women and noted that blood pressure trajectories for normal
women were more smooth relative to women with preeclampsia. We however,
take an entirely different approach. Instead of dealing with a
complicated system of differential equations we incorporate curve shape
in modeling through a penalty parameter analogous to that found in
penalized splines.
Our model involves an implied distribution on partitions of players.
The allocation variables are treated as parameters and thus our
approach may be seen as an extension of latent class analysis (LCA)
(\citealt{LCAbook}) which classifies individual player curves into $K$
pre-specificed clusters (see \citealt{LCA:2010} and \citealt
{LCA:2005}). Unlike LCA, our methodology does not require a fixed
pre-specified number of clusters as this is inferred from the
corresponding posterior distribution on partitions. We briefly note
that there does exist a small literature dedicated to estimating
certain aspects of functional output such as dynamics (the speed of
price increases and the rate at which this speed changes) that depend
on covariate information (see \citealt{EbayFunctions} and \citealt
{RateFunctions} and references therein). But these are not relevant to
the current setting as they fail to deal with multiple-subject studies
nor do they use curve shape in prediction and inference.
In sports, \cite{reese:1999} model career trajectories (or aging
curves) non parametrically in order to make historical comparisons of
player's abilities in baseball, hockey and golf. \cite{StreakyPGAPlay}
consider career paths of golf players to determine combinations of luck
and skill required to win a golf tournament. Neither of these works
were interested in grouping players to carry out career and active
player predictions.
As noted, our principal goal is making active player and career
predictions. If predictions are computed using methods that treat
individual players independently, then both types of predictions would
be extremely poor as they would not be data driven. One way of
improving predictions is by borrowing strength (or sharing information)
among players whose career production curves might be deemed similar. A
straightforward way of borrowing strength is by introducing player
clusters. However, if all individuals of the same cluster are
restricted to have the same curve, then some individuals will
invariably be poorly fit (too much borrowing of strength).
Alternatively, if curves of all individuals of the same cluster are
completely unrestricted, then clustering players would provide no
predictive information (too little borrowing of strength). The
methodology proposed in this article is able to balance very well the
desire to produce good fitting individual curves while still producing
clusters that allow enough borrowing of strength among similar players
to guide prediction. This is carried out by employing a hierarchical
model where subject-specific functions are modeled flexibly through a
linear combination of basis functions whose coefficients are drawn from
cluster-specific process level distributions. Doing this produces
flexible subject-specific curves while still being able to produce
reasonably accurate predictions by pooling together players with
similar features/performances.
In our model, having covariate dependent clusters is crucial to
carrying out career prediction as these are produced using the
predictive distribution available from the covariate dependent
clustering mechanism. Also, shape dependent clusters are useful to
carrying out active player predictions as incomplete active player
curves are filled in using curves of retired players that have similar
career trajectories. There has been work regarding completing curves
(\citealt{PredictingFunctionsCallCenter} work out Best Linear Unbiased
Predictors (BLUPS) for past and future curve segments) and local
borrowing of strength to fit global curves (\citealt
{PetroneGuindaniGelfand} employ functional Dirichlet processes to group
Gaussian process realizations), but the approaches developed and
purposes are very much different from the present study.
The remainder of the article is organized as follows. Section 2
describes the data collected and employed in the analysis. Section 3
provides details regarding the development of the methodology
highlighting model components associated with cluster-specific curve
smoothness and active player prediction. Section 4 provides details
regarding computation of posterior and predictive distributions. In
Section 5 we provide details of a small simulation study. Results from
the analysis of the NBA application are provided in Section 6. Finally,
we provide some concluding remarks in Section 7.
\section{Description of Data} \label{examples}
We collected common game-by-game (including playoffs) modern basketball
metrics for each player drafted into the NBA during the years 1992-2004
(Shaquille O'Neal to Dwight Howard) that participated in at least one
game up through the 2009/2010 season. This resulted in 576 players with
number of games played ranging from 2 to 1383 games. A few of the
players appeared in very few games and are not representative of a
typical NBA player. Because of this, and to reduce the noise introduced
by the careers of players that contain little information regarding the
processes of interest, we restrict our attention to players with at
least three seasons of experience (the rookie contract length of the
2005 CBA). Also, to retain enough games to get a reasonable sense of a
player's ability we only include players who played at least a half a
season's worth of games (42). Finally, we excluded the following 8
players whose careers were cut short either by career ending injuries
or untimely deaths: Bryant Reeves, Malik Sealy, Eddy Curry, Jason
Collier, Eddie Griffin, Yao Ming, T. J. Ford, and Gilbert Arenas. This
resulted in 408 players with number of games played through the
2009/2010 season ranging from 45 to 1383. Of the 408 players, 263 are
classified as ``retired'' as they did not play beyond the 2009/2010 season.
Measuring game-by-game production is not straightforward as there are
numerous, difficult to measure factors that influence player
performance. Because of this no gold standard basketball production
metric exists. That said, one that has become somewhat popular is John
Hollinger's so called Game Score which is a linear combination of
common variables that are recorded for each player through out the
course of a game (e.g., number of baskets made and number of steals
acquired. More details can be found in the Appendix and at \citealt
{hollinger2002pro}). This metric will be used as our response variable
and therefore a representation of a player's game productivity. Though
Game Score has deficiencies (e.g., weighted heavily towards offensive
output and doesn't account for quality of opponent), it provides a
fairly accurate indicator of player production for any given game. The
maximum Game Score collected is 63.5 (corresponding to Kobe Bryant's 81
point game). The minimum Game Score was -9.9 and the average Game Score
among all players is 8.1. An alternative to raw Game Score is a
standardized Game Score where standardization is carried out by
dividing Game Score by the minutes played in each game, thus removing
Games Score's dependence on minutes played. However, players whose
production is not negatively impacted by increased minutes are more
valuable than those who are less efficient with increased game time and
distinguishing between these types of players is desirable. For this
reason we opt to use raw Game Score values.
For an aging (or time) variable there are various units that could be
used. For example, age, number of accumulated minutes played, or simply
the number of games played are all reasonable. Since each of these
measurements are only available to us on a game-by-game basis, the
shape (or smoothness) of the curve remains unchanged regardless of age
unit employed. Thus for sake of expositional clarity we use number of
games played (see Figure \ref{rawscatterplot} as an example).
Through exploratory analysis we identified three covariates that in
addition to being of interest in their own right are informative in
grouping players. These are age during first game played (measured in
years), experience before being drafted into the NBA (High School
basketball only, some Collegiate basketball, or International
basketball), and draft order. Draft order is the order in which players
are selected in their respective drafts. For example, a player with
draft order $1$ implies he was the first player selected in his
respective draft, and draft order $2$ implies he was the second player
selected etc. In Section 3 we describe how draft order is used
explicitly to predict total games played for active players, but as a
covariate used to influence clustering we categorize a player's draft
order as being a top five pick, a first round pick (excluding the first
five) and a second round pick. (Since 1989 the NBA draft has consisted
of two rounds.) Table \ref{playercategorysummary} provides the number
of players in each of the nine categories. Other baseline covariates
were considered such as position, height, and other physiological
characteristics but preliminary research indicated they were not useful
in partitioning players with regards to production.
\begin{table}[htdp]
\caption{Total number of players in each of the nine categories.}\label{playercategorysummary}
\vspace*{4pt}
\begin{tabular}{l ccc}
& \multicolumn{3}{c}{Experience} \\ \cline{2-4}
Draft & High School & College & International\\\hline
Top 5 & 7 & 51 & 4 \\
1st Round & 15 & 200 & 25\\
2nd Round & 1 & 98 & 8\\\hline
\end{tabular}
\end{table}
\section{Model Description and Development}\label{MODEL}
We first consider the model's clustering mechanism highlighting its
dependence on subject-specific covariates. Secondly, the likelihood
structure incorporating the number of games played and career length
(which are right censored for active players) is detailed. Lastly, we
describe the hierarchical component which balances goodness of
individual fit with ability to produce clusters that are able to guide
prediction.
\subsection{Product Partition Model with Covariates (PPMx)}
Let $i = 1, \ldots, m$ index the $m$ players in the study. Further, let
$\rho= \{S_1, \ldots, S_{k_m}\}$ denote a partitioning (or clustering)
of the $m$ individuals into $k_m$ subsets such that $i \in S_j$ implies
that individual $i$ belongs to cluster $j$. Alternatively, we will
denote cluster membership using $s_1, \ldots, s_m$ where $s_i = j$
implies $i \in S_j$. Let $\bm{x}_i = (x_{i1}, x_{i2}, x_{i3})$ denote
player $i$'s covariate vector with $x_{i1}$ corresponding to age,
$x_{i2}$ experience and $x_{i3}$ draft order. Let $\bm{x}^{\star}_j = \{
\bm{x}_i:i \in S_j\}$ be the partitioned covariate vector. Our approach
is to first directly model $\rho$ with the covariate dependent product
partition model of \cite{PPMxMullerQuintanaRosner} (which will be
referred to as the PPMx model) and then construct a hierarchical model
given the partition (as opposed to introducing latent variables that
indirectly induce a partitioning of individuals). The PPMx prior
incorporates the idea that individuals with similar covariate values
are more likely {\it a priori} to belong to the same cluster relative
to individuals with dissimilar covariate values. Additionally, this
prior is very simple, highly customizable, seamlessly incorporates
different types of covariates (e.g., continuous or categorical), and is
particularly well suited for prediction (something that is of interest
here). An alternative method not considered here can be found in \citet
{BayesianGeneralizedProductPartitionModel}. The PPMx prior consists of
a cohesion function, $c(S_j) \ge0$ for $S_j \subset\{1, \ldots, n\}$,
and a nonnegative similarity function $g(\bm{x}^{\star}_j)$. The former
measures the tightness of how likely elements of $S_j$ are clustered
{\it a priori} and the latter formalizes the similarity of the $x_i$'s
by producing larger values of $g(\bm{x}^{\star}_j)$ for $x_i$'s that
are more similar. The form of the PPMx prior is simply the following
product (for more details see \citealt{PPMxMullerQuintanaRosner})
\begin{align} \label{ppmx}
P(\rho|\bm{x}) \propto\prod_{j=1}^{k_m} c(S_j)g(\bm{x}^{\star}_j).
\end{align}
A simple example of a cohesion function that produces a Dirichlet
Process type partitioning is $c(S_j) = M\times(|S_j| - 1)!$ for some
positive $M$ and $|\cdot|$ denoting cardinality. Regarding possible
similarity functions, \citet{PPMxMullerQuintanaRosner} provide a bit of
exposition for different types of covariates (e.g., continuous,
ordinal, or categorical). Generically speaking they suggest the
following structure
\begin{align}\label{sf}
g(\bm{x}_j^{\star}) = \int\prod_{i \in S_j} q(x_i | \zeta_j) q(\zeta_j)
d\zeta_j.
\end{align}
where $\zeta_j$ is a latent variable and $q(\cdot|\zeta_j)$ and $q(\cdot
)$ are (typically) conjugate probability models. This structure is not
necessarily used for its probabilistic properties (indeed $\bm{x}$ is
not even random), but rather as a means to measure the similarity of
the covariates in cluster $S_j$. In reality any function that produces
larger values as the entries of $\bm{x}_j^{\star}$ become more similar
can be considered as a similarity function. For example $g(\bm{x}^{\star
}_j) = \exp\{ - s^2_j\}$ where $s^2_j$ is the empirical variance of $\bm
{x}^{\star}_j$ is a completely reasonable similarity function for
continuous covariates.
It turns out that the similarity function \eqref{sf} coupled with
cohesion function $c(S_j) = M\times(|S_j| - 1)!$ produces the same
marginal prior distribution on partitions as that induced by using a
Dirichlet process (DP). For more details see \citet{PPMxMullerQuintanaRosner}.
Given $\rho$ we may proceed to specify a hierarchical model that
flexibly models individual curves. Before doing so, we very briefly
introduce a few pieces of notation that will be used. In what follows
cluster-specific and subject-specific parameters will need to be
distinguished. If we let $\bm{\theta}_i$ denote some generic
subject-specific parameter vector, then $\bm{\theta}^*_j$ will be used
to denote a cluster-specific parameter in the sense that $i \in S_j$
implies that $\bm{\theta}_i = \bm{\theta}^*_j$. Alternatively, cluster
labels $(s_1, \ldots, s_m)$ can be used to connect subject and cluster
specific parameters through $\bm{\theta}_i = \bm{\theta}^*_{s_i}$.
Lastly, vectors of subject-specific and cluster-specific parameters are
denoted by $\bm{\theta} = (\bm{\theta}_1, \dots, \bm{\theta}_m)$ and
$\bm{\theta}^* = (\bm{\theta}^*_1, \ldots, \bm{\theta}^*_{k})$.
\subsection{Likelihood}
To distinguish between players that play beyond the 2009/2010 season we
use the following indicator variable\vadjust{\eject}
\begin{align*}
g_i = \left\{
\begin{array}{cl}
0 & \mbox{if player $i$ retired before or at the conclusion of the
2009/2010 season} \\
1 & \mbox{if player $i$ played beyond the 2009/2010 season}.
\end{array}
\right.
\end{align*}
Let $n_i$ denote the total number of games played in player $i$'s
career. If $g_i=0$ then $n_i$ is observed otherwise a lower bound
denoted by $\tilde{n}_i$ is observed such that $n_i \ge\tilde{n}_i$.
Thus, we are dealing with right censored type observations and we will
incorporate ideas developed for modeling them. We denote the response
vector for players whose $g_i = 0$ with $\bm{y}_i = (y_{i1}, \ldots,
y_{in_i})$ otherwise $\bm{y}_i = (y_{i1} \ldots, y_{i\tilde{n}_i})$.
The career production curve for players whose $g_i=1$ needs to be
``completed'' which requires the prediction or imputation of ${n}_i$.
Predicting $n_i$ is not trivial (even given $\tilde{n}_i$) because it
is highly variable and demonstrates a strong association with very few
covariates. One covariate we found that displays a strong association
with $n_i$ is career length (denoted by $L_i$ and measured in years).
Unfortunately, this variable is also right censored and for $g_i=1$ we
only observe $\tilde{L}_i$. However, we consider $L_i$ because it
displayed a stronger association with the uncensored variable draft
order (denoted by $d_i$) than that found between $n_i$ and $d_i$.
Therefore we employ $d_i$ to first impute $L_i$ and then use $L_i$ to
predict $n_i$. Thus, the likelihood for the $i$th player is composed of
the random variables $(\bm{y}_{i}, g_i, n_{i:g_i=0}, \tilde
{n}_{i:g_i=1}, L_{i:g_i=0}, \tilde{L}_{i:g_i=1})$ which we model
jointly by way of
\begin{align*}
p(\bm{y}_{i}, g_i, n_{i:g_i=0}, \tilde{n}_{i:g_i=1}, L_{i:g_i=0}, \tilde
{L}_{i:g_i=1}) & = [p(\bm{y}_{i} | n_i, L_i) p(n_i|L_i)p(L_i)]^{1-g_i}
\\
& \times[p(\bm{y}_{i} | \tilde{n}_i, \tilde{L}_i) p(\tilde{n}_i|\tilde
{L}_i)p(\tilde{L}_i)]^{g_i}.
\end{align*}
We now detail each of the three likelihood components.
When only considering retired players we found that the association
between $d_i$ and $L_i$ was somewhat nonlinear. (This is reasonable
considering that our pool of players consists of only those who played
at least one NBA game thus retaining only the ``good'' 2nd round
picks.) Because of this, we assume $L_i \sim N(\nu_i, \psi^2)$ where
$\nu_i = \gamma_0 + \gamma_1d_i + \gamma_2d_i^2$. However, the
association between $n_i$ and $L_i$ was fairly linear so we assume $n_i
| L_i \sim N(\eta_i, \delta^2)$ where $\eta_i = \alpha_0 + \alpha
_1L_i$. Thus, $(n_{i:g_i=0}, \tilde{n}_{i:g_i=1}, L_{i:g_i=0}, \tilde
{L}_{i:g_i=1})$'s contribution to the likelihood is
\begin{align*}
[p(n_i|L_i)p(L_i)]^{1-g_i}[p(\tilde{n}_i|\tilde{L}_i)p(\tilde
{L}_i)]^{g_i} & = \left[N(n_i; \eta_i, \delta^2)N(L_i; \nu_i, \psi
^2)\right]^{1-g_i} \\
&\times\left[\left\{1-\Phi\left(\frac{\tilde{n}_i -\eta_i}{\delta
}\right)\right\}\left\{1-\Phi\left(\frac{\tilde{L}_i - \nu_{i}}{\psi
}\right)\right\}\right]^{g_i}
\end{align*}
where $N(\cdot; m, s^2)$ denotes a Gaussian density function with mean
$m$ and variance $s^2$ and $\Phi(\cdot)$ denotes a standard normal cdf.
As a result, imputing $n_i$ for active players is carried out by first
imputing $L_i$ using a quadratic model with $d_i$.
We briefly note that although a Poisson model for $n_i$ might seem
natural, it is not appropriate in the current context as the
simultaneous increasing of the mean and variance of the Poisson
distribution seems to contradict what is empirically observed. Thus,
for simplicity, we elected to employ a Gaussian to model $n_i$ and
round the predictions (something that is not uncommon, see page 458 of
\citealt{BDA3}). Also, modeling
$n_i$ non-parametrically could potentially improve prediction but we
elected to employ the simpler parametric model as its predictions were
satisfactory for our purposes. Nonetheless, predicting $n_i$ is of
considerable interest in its own right to NBA decision makers and could
be an interesting future research project.
Finally, given $n_i$ and letting $f_i(z_{it})$ denote the $i$th
player's underlying production curve value for the $t$th game played
(denoted by $z_{it}$), we model measurements $y_{it}$ as
\begin{align} \label{ymodel}
y_{it} = \beta_{0i} + f_{i}(z_{it}) + \epsilon_{it} \ \mbox{for} \ t =
1, \ldots, n_i \ (\tilde{n}_i \ \mbox{for} \ g_1 = 1)
\end{align}
where $ \epsilon_{it} \sim N(0,\sigma^2_i)$ independently across $i$.
It is possible that incorporating a more sophisticated error model
(such as autoregressive errors) could prove to be beneficial, but for
simplicity we maintain independence. A fairly popular method of
characterizing $f_i(\cdot)$ is to define a collection of basis
functions (e.g., wavelet, polynomial) and assume that $y_{it}$ lies in
their span. We adopt this method and employ a B-spline basis as it has
a number of attractive computational properties and facilitates active
player prediction as will be detailed shortly. Therefore, $f_i(\cdot)$
can be written as the following linear combination
\begin{align} \label{fmodel}
f_i(z_{it}) = \sum_{\ell=1}^{P_i} \beta_{i\ell} h_{\ell}(z_{it}; \bm{\xi
}_i)
\end{align}
where $h_{\ell}(z;\xi_i)$ denotes the $\ell$th B-spline basis function
evaluated at knots contained in $\bm{\xi}_i$. If $p_i$ denotes the
number of inner knots and $q$ the spline degree, then $P_i = p_i + q +
1$. Now define $\bm{H}_i$ as the $n_i \times P_i$ matrix with rows $\{
h_1(z_{it}), \ldots, h_{P_i}(z_{it}) \}$ for $t = 1, \ldots, n_i \
(\tilde{n}_i \ \mbox{for} \ g_i=1)$, and $\bm{\beta}_i = \{\beta_{i1},
\ldots, \beta_{iP_i}\}$. Combining \eqref{ymodel} and \eqref{fmodel} produces
\begin{align}\label{LinearModel}
\bm{y}_i = \beta_{0i}\bm{1}_i + \bm{H}_i \bm{\beta}_i + \bm{\epsilon
}_{i} \ \mbox{for} \ \bm{\epsilon}_i \sim N_{}(\bm{0}, \sigma^2_i \bm{I}_{n_i}),
\end{align}
where $\bm{1}_i$ denotes a vector of ones and $\bm{I}_{n_i}$ an
identity matrix.
The dimension of $\bm{H}_i$ depends on $n_i$ (and $\tilde{n}_i$ for
active players). This coupled with the fact that B-splines form a local
basis in that each basis function is non-negative only on an interval
formed by $q + 2$ adjacent knots can be exploited to carry out active
player prediction. Since for any fixed $z_{it}$ at most $q+1$ basis
functions are positive, the predicted value of ${n}_i$ for active
players will determine the number of zero columns in $\bm{H}_i$. Thus,
the section of an active players curve corresponding to the $z_{it}$
values between $n_i$ and $\tilde{n}_i$ are completely informed by the
cluster specific curve or in the case that the player belongs to a
singleton, the grand mean curve (more details are in Section 3.3).
Using $\tilde{\bm{H}}_i$ to denote the design matrix that incorporates
the predicted value of $n_i$ based on $\tilde{n}_i$, the full
likelihood for $\bm{\Theta} = (\bm{\beta}, \bm{\beta}_0, \bm{\sigma}^2,
\bm{\eta},\bm{\nu}, \psi^2, \delta^2)$ is
\begin{align}
& \ell(\bm{y}_1, \ldots, \bm{y}_m,\bm{n},\bm{L}, \tilde{\bm{n}}, \tilde
{\bm{L}}, \bm{g}|\bm{\Theta}) \nonumber\\
&= \prod_{i=1}^n \left[N_{n_i}(\bm{y}_i; \beta_{0i}\bm{1}_i + \bm
{H}_i\bm{\beta}_i, \sigma^2_i \bm{I}_{n_i}) N(n_i; \eta_i, \delta
^2)N(L_i; \nu_i, \delta^2)\right]^{1-g_i} \times\nonumber\\
&\times \left[N_{\tilde{n}_i}(\bm{y}_i; \beta_{0i}\bm{1}_i + \tilde{\bm
{H}}_i\bm{\beta}_i, \sigma^2_i \bm{I}_{\tilde{n}_i}) \left\{1-\Phi\left
(\frac{\tilde{n}_i -\eta_i}{\delta}\right)\right\}\left\{1-\Phi\left
(\frac{\tilde{L}_i - \nu_{i}}{\psi}\right)\right\}\right]^{g_i}.
\end{align}
\subsection{Hierarchical Model}
The number and location of the inner-knots that make up $\bm{\xi}_i$
are rarely known. Their selection is crucial to producing an attractive
curve without over-fitting. So called free-knot splines is a very
flexible method that treats $\bm{\xi}_i$ as an unknown and has proved
to be quite parsimonious in knot selection. (\citealt
{BayesFreeKnotCurveFit} and \citealt{BayesNonlinearClassReg} provide a
nice overview.) Therefore, a possible direction to incorporating shape
variability in prediction as desired would be to base clustering on the
number and location of knots. However, to fit a free-knot spline some
type of transdimensional Markov Chain Monte Carlo (MCMC) algorithm is
often employed and this coupled with the PPMx prior for $\rho$ would
result in a doubly transdimensional MCMC algorithm that would become
prohibitively expensive. To avoid these computational issues and to
make the methodology more readily accessible, for each subject we
instead select a moderate number of equally spaced knots within the
knot domain and employ the Bayesian P-spline technology of \cite
{BayesianPsplines}. Now shape variability can influence clustering
through the penalty parameter of the P-splines. However, to retain
flexible subject-specific fits, we use P-splines as a prior
distribution of process level parameters and allow subject-specific
coefficients to vary around a cluster-specific mean. That is, we assume
the following process level structure for the $\bm{\beta}$'s:
\begin{align} \label{beta}
\bm{\beta}_i | \bm{\theta}^*_{s_i}, \lambda^{2*}_{s_i} & \sim N(\bm
{\theta}^*_{s_i}, \lambda^{2*}_{s_i}\bm{I}) \ \mbox{with} \ \sqrt
{\lambda^{2*}_{j}} \sim UN(0, A),
\end{align}
and use a Bayesian P-spline prior for the $\bm{\theta}^*_j$'s (with
$UN(\cdot, \cdot)$ denoting a Uniform distribution). A particularly
nice feature of the methodology is the explicit ability to control the
similarity between individual curves and their group counterparts
through the hyper-parameter $A$.
In order to highlight two departures from the Bayesian P-splines of
\citet{BayesianPsplines} required by the present modeling we very
briefly introduce them here. For more details see \citet
{BayesianPsplines} and \citet{BayesianSmoothing}. Bayesian P-splines
are the Bayesian analogue to splines penalized by $d$-order differences
and are constructed around $d$-order Gaussian random walks. For
example, for $d=1$
\begin{align}\label{rw}
\begin{split}
\theta^*_{j \ell} & = \theta^*_{j, \ell-1} + u_{j \ell} \ \ \ell= 2,
\ldots, n \\
\end{split}
\end{align}
with $u_{j\ell} \sim N(0, \tau^{2*}_j)$. Typically $p(\theta^*_{j1})
\propto v$, but an improper prior is not an appropriate probability
model for the Polya urn representation used in the PPMx. Thus, similar
to what was done in \cite{BayesianHierarchicalCurveRegistration} we assume
$\theta^*_{j1} \sim N(0, \tau^{2*}_j/v^2)$ (with analogous extensions
for $d>1$). The value $v$ can be assigned a prior distribution or be
set to a fixed value. Equation \eqref{rw} together with the $\theta
^*_{j1} \sim N(0, \tau^{2*}_j/v^2)$ produce $\bm{\theta}^*_j \sim N(\bm
{0}, \tau^{2*}_j \bm{K}^{-1})$ where $\bm{K}$ is a banded penalty
matrix with $v$ incorporated. $\tau^{2*}_j$ is the smoothing parameter
associated with Bayesian P-splines and is crucial in being able to
distinguish between individuals based on the smoothness of their
respective curves. As suggested by \cite{BayesianPsplines} we adopt
$\tau^{2*}_j \sim IG(a_{\tau},b_{\tau})$ where $IG(\cdot, \cdot)$
denotes an inverse Gamma distribution and $a_{\tau}$ and $b_{\tau}$ are
user supplied.
Recall that active player prediction is carried out by borrowing
strength among players in a cluster. If player $i$ belongs to a
singleton or all members of his cluster are active players, then at
least part of his prediction is completely guided by the prior on $\bm
{\theta}^*_j$. Since the prior is centered at $\bm{0}$ this would
produce poor active player predictions. To improve prediction in these
situations, we introduce $\bm{\mu}$ as a vector of global curve
coefficients such that $\bm{\theta}^*_j \sim N(\bm{\mu}, \tau^{2*}_j \bm
{K}^{-1})$ with $\bm{\mu} \sim N(\bm{0}, s^2_{\mu}\bm{I})$. Including
$\bm{\mu}$ potentially influences the values of $\bm{\theta}^*_j$ in
that smaller magnitudes achieve the same amount of smoothing as when
$\bm{\mu}=\bm{0}$. This should be taken into account when selecting
values for $a_{\tau}$ and $b_{\tau}$. Also, apart from improving
prediction, $\bm{\mu}$ is of interest in its own right as it provides
information regarding an average career curve among all
players.\looseness=1
We end the description of our Bayesian P-spline approach with details
regarding knot selection. A complicating factor of knot selection in
modeling these data is the massive misalignment associated with the
number of games played for each of the players. Making things worse is
the inherent discontinuities in games played through out the course of
one's career (e.g., the offseason, injuries, etc.) that we are not
considering. There is a ``curve registration'' literature dedicated to
better aligning functional domains in multi-subject studies (\citealt
{BayesianHierarchicalCurveRegistration}). However, we align career
paths by matching the percentile number of career games played. This is
carried out by transforming ``time" to the unit interval which greatly
simplifies the process of selecting $\bm{\xi}_i$. Therefore $z^*_{it} =
z_{it}/n_i$ is used instead of $z_{it}$. (For retired players $n_i$ is
the observed number of games played and the predicted for active
players.) Thus for retired players $z^*_{in_i} = 1$ while for active
players $z^*_{in_i} < 1$. Now $\bm{\xi}_i$ can be a knot set that
partitions the unit interval into equal subintervals and since it does
not depend on $n_i$ it can be the same for all players. We do note that
aligning career paths in this way is imperfect as the 95th percentile
of games played for one player might be during his third season while
for another player during his fifteenth season. Even so, we believe
that matching curves by way of percentile of games played produces
coherent comparisons and valid borrowing of strength. With an enriched
data set we could attempt to take into account possible discontinuities
in career paths. (This would actually be very interesting as many
players improve during the off season.) It would be fairly
straightforward to expand the model in a variety of ways, but the base
model proposed would continue being the work horse even as other more
idiosyncratic aspects of the data are considered.\looseness=1
With regards to modeling $\beta_{0i}$ and $\sigma^2_i$ there are any
number of ways one might proceed. It seems plausible that $\sigma^2_i$
might depend on $z_{it}$. That said, for sake of simplicity we utilize
the common prior structure for variance components $\sigma^2_i \sim
IG(a_{\sigma}, b_{\sigma})$ with $a_{\sigma}$ and $b_{\sigma}$ being
user supplied. For the subject-specific random intercepts, we use a
Gaussian-inverse-Gamma hierarchy such that $\beta_{0i} \sim N(\mu
_{b_0}, \sigma^2_{b_0})$ with $\mu_{b_0} \sim N(0, s^2_{b_0})$ and
$\sigma^2_{b_0} \sim IG(a_{b_0}, b_{b_0})$. Finally, typical conjugate
priors are used for $\bm{\alpha}=(\alpha_0, \alpha_1) \sim N(\bm{m}_a,
s^2_a\bm{I})$, $\bm{\gamma}=(\gamma_0, \gamma_1, \gamma_2) \sim N(\bm
{m}_{\gamma}, s^2_{\gamma}\bm{I})$, $\delta^2\sim IG(a_{\delta},
b_{\delta})$, and $\psi^2 \sim IG(a_{\psi}, b_{\psi})$.
Equation (\ref{HPPMx}) is provided to aid in visualizing how all the
moving parts of the hierarchical model are connected. Through out the
remainder of the paper we will refer to the entire hierarchical model
as HPPMx.\vadjust{\eject}
\begin{align}\label{HPPMx}
\bm{y}_i, n_i, L_i, \tilde{n}_i, \tilde{L}_i &| g_i, \bm{\beta}_i,\rho
,\sigma^2_i, \beta_{0i}, \bm{\alpha}, \bm{\gamma}, \delta^2,\psi^2
\nonumber\\
& \sim [N_{n_i}(\bm{y}_i; \beta_{0i}\bm{1}_i + \bm{H}_i\bm{\beta}_i,
\sigma^2_i \bm{I}_{n_i}) N(n_i; \eta_i, \delta^2)N(L_i; \nu_i, \psi
^2)]^{1 -g_i} \nonumber\\
& \times \Biggl[N_{\tilde{n}_i}(\bm{y}_i; \beta_{0i}\bm{1}_i + \tilde
{\bm{H}}_i\bm{\beta}_i, \sigma^2_i \bm{I}_{\tilde{n}_i})\notag\\
&\times \left\{1-\Phi
\left(\frac{\tilde{n}_i -\eta_i}{\delta}\right)\right\}\left\{1-\Phi
\left(\frac{\tilde{L}_i - \nu_{i}}{\psi}\right)\right\}\Biggr]^{g_i}
\nonumber\\
\sigma^2_i |a_{\sigma}, b_{\sigma} & \sim IG(a_{\sigma}, b_{\sigma})
\nonumber\\
\beta_{0i} | \mu_{b_0}, \sigma^2_{b_0} & \sim N(\mu_{b_0}, \sigma
^2_{b_0}) \ \ \mbox{with} \ \ \mu_{b_0} \sim N(0,s^2_{b_0}) \ \ \mbox
{and} \ \ \sigma^2_{b_0} \sim IG(a_{b_0}, b_{b_0}) \nonumber\\
\bm{\alpha} & \sim N(\bm{m}_a, s^2_a\bm{I}) \ \ \mbox{and} \ \ \delta^2
\sim IG(a_{\delta}, b_{\delta})\nonumber\\
\bm{\gamma} & \sim N(\bm{m}_{\gamma}, s^2_{\gamma}\bm{I}) \ \ \mbox
{and} \ \ \psi^2 \sim IG(a_{\psi}, b_{\psi}) \\
\bm{\beta}_i | \rho, \theta^*_{s_i}, \lambda^*_{s_i} & \sim N(\bm{\beta
}_i; \bm{\theta}^*_{s_i}, \lambda^{2*}_{s_i}) \ \ \mbox{with} \ \ \sqrt
{\lambda^{2*}_{h}} \sim UN(0, A) \nonumber\\
\bm{\theta}^*_h | \rho,\bm{\mu}, \tau_h^{2*}, \bm{K} & \sim N(\bm{\mu},
\tau^{2*}_h \bm{K}^{-1}) \ \ \mbox{with} \ \ \tau^{2*}_h \sim IG(a_{\tau
}, b_{\tau})\nonumber\\
\bm{\mu} & \sim N(\bm{0}, s^2_{\mu}\bm{I}) \nonumber\\
Pr(\rho) & \propto\prod_{h=1}^{k_m} c(S_j)g(\bm{x}^{\star}_j),\nonumber
\end{align}
for $i = 1, \ldots, m$ and $h = 1, \ldots, k_m$.
Before proceeding we make a brief comment regarding some specific model
components. Since the Bayesian P-splines are used at the prior level
rather than the process level of the hierarchical model, individual
curves are not directly influenced by its smoothing penalization. The
wiggliness of individual curves is a function of both $\tau^{2*}_j$ and
$A$. As $A$ increases the influence that $\tau^{2*}_j$ has on
individual curves decreases. This is investigated further in the
simulation study of Section 5. Therefore, if smooth individual curves
are desired together with large within group variability it may be
necessary to use 10-15 knots instead of the 20-30 knots recommended by
\citet{BayesianPsplines}.
\label{mcmc}
\section{Posterior Computation}
\subsection{MCMC Implementation}
We fit the proposed model to data by simulating a Markov chain whose
equilibrium distribution is the desired posterior distribution. The
algorithm employed is similar to \citet{MCMCSamplingMethodsForDPmixtureModels}'s
algorithm number 8 in that it
can be divided into two basic pieces. The first updates the partition
$\rho$ via the Polya urn scheme of \cite{BlackwellMacQueen} (and
further developed by \citealt{APredViewofBayesCluster}) and the other
updates the hierarchical model parameters using a Gibbs sampler
(\citealt{GemanGeman} and \citealt{GelfandSmith}) and Metropolis steps
(\citealt{Metropolis}).
To update the cluster membership for subject $i$, cluster weights are
created by comparing the unnormalized posterior for the $h$th cluster
when subject $i$ is excluded to that when subject $i$ is included. In
addition to weights for existing clusters, algorithm 8 of \citet
{MCMCSamplingMethodsForDPmixtureModels} requires calculating weights
for $p$ empty clusters whose cluster specific parameters are auxiliary
variables generated from the prior. To make this more concrete, let
$S_h^{-i}$ denote the $h$th cluster and $k^{-i}_m$ the number of
clusters when subject $i$ is not considered. Similarly $\bm{x}_h^{\star
-i}$ will denote the vector of covariates corresponding to cluster $h$
when subject $i$ has been removed. Then the multinomial weights
associated with the $k^{-i}_m$ existing clusters and the $p$ empty
clusters are
\begin{align*}
Pr(s_i = h | - ) \propto
\begin{cases}
N(\bm{\beta}_i ; \theta^{*}_{h}, \lambda^{2*}_{h}\bm{I}) \frac
{c(S_{h}^{-i}\cup\{i\})g(\bm{x}^{\star-i}_{h}\cup\{\bm{x}_i\}
)}{c(S_{h}^{-i})g(\bm{x}^{\star-i}_{h})}\ \mbox{for} \ h = 1, \ldots,
k^{-i}_m\\
N(\bm{\beta}_i; \bm{\theta}_{new,h}, \lambda^{2}_{new,h}\bm{I}) c(\{i\}
)g(\{\bm{x}_i\}) p^{-1}\ \mbox{for} \ h = k^{-i}_m+1, \ldots, k^{-i}_m
+ p.
\end{cases}
\end{align*}
Values for $\bm{\theta}_{new,h}$, $\lambda^2_{new,h}$ are auxiliary
variables drawn from their respective priors as required by algorithm
8. Care must be taken when subject $i$ belongs to a singleton cluster
as removing the $i$th subject produces an empty cluster. This in turn
requires relabeling the existing cluster specific components to avoid
gaps in the cluster labeling.
The full conditional distributions of $(\bm{\beta}_i, \bm{\beta}_0,
\sigma^2_i, \theta^*_j, \tau^{2*}_j)$ are fairly common derivations and
are provided in the Appendix. To update $(\lambda^*_j, \bm{\gamma}, \bm
{\alpha}, \delta^2, \psi^2)$ we employed a random walk Metropolis step
with a Gaussian proposal distribution. A Markov chain can be
constructed by employing a Gibbs sampler that first updates $\rho$ and
then on an individual basis updates model parameters by cycling through
each full conditional and using a Metropolis step for the non conjugate
parameters.
\subsection{Posterior Prediction Distributions} \label{PPD}
A particularly nice feature of using PPMx is the availability of career
prediction through covariate dependent predictive distributions. Using
PPMx, posterior predictive distributions are readily available and can
be obtained online in the sense that draws from this distribution can
be collected within the MCMC algorithm. The posterior predictive
distributions depend on covariate values through the allocation of a
new individual to one of the $k_m$ existing clusters or to a new
cluster using the following multinomial weights
\begin{align}
Pr(s_{n+1} = h | - ) \propto
\begin{cases}
\frac{c(S_{h}\cup\{n+1\})g(\bm{x}^{\star}_{h}\cup\{\bm{x}_{n+1}\}
)}{c(S_{h})g(\bm{x}^{\star}_{h})}\ \mbox{for} \ h = 1, \ldots, k_m\\
c(\{n+1\})g(\{\bm{x}_{n+1}\}) \ \mbox{for} \ h = k_m+1.
\end{cases}
\end{align}
Once the future player has been allocated to a cluster, one carries out
the typical Monte Carlo integration to sample from the posterior predictive.
\subsection[Predicting $n_i$]{Predicting $\bm{n}_i$}
Predicted values of ($n_i, L_i$) for retired players are produced at
each MCMC iteration. We essentially employ the multiple imputation
ideas of \cite{MissDat} but with the exception that we are very much
interested in the values being imputed. Predictions are fairly straight
forward as they only depend on the full conditionals of $n_i$ and $L_i$
which turn out to be truncated normal with $\tilde{n}_i$ and $\tilde
{L}_i$ acting as lower bounds (the full conditionals are provided in
the Appendix).
\section{Simulation Studies}
We conduct two small simulation studies to investigate the behavior of
the HPPMx model (\ref{HPPMx}). Recall that the principal motivation in
incorporating a hierarchy is to balance goodness of individual curve
fits with the production of meaningful clusters which facilitate
prediction. Therefore apart from showing improved prediction
performance, the simulation study explores just how much goodness of
individual fit is sacrificed in the name of prediction (which is very
little as will be seen). The first simulation study demonstrates the
method's superior predictive performance by comparing out of sample
mean integrated prediction rates to that of two competitors (which are
detailed shortly). The second explores how certain model components
influence subject-specific fits, curve smoothness, and clustering. The
two competitors selected represent the extremes HPPMx attempts to
balance, namely fitting each player independently versus assigning
players cluster-specific curves. The first competitor is a
semi-parametric regression model (henceforth SP) that fits individual
curves independently and flexibly. The second is a semi-parametric
regression model with a Dirichlet process prior (henceforth SPDP) which
produces individual curves that are cluster specific. More precisely we consider
\begin{align}\label{competitor}
y_{it} & = \bm{x}'_i \bm{\beta} + f_i(z_{it} ) + \epsilon_{it} \ \mbox
{with} \ \epsilon_{it} \sim N(0, \sigma^2_i) \ \mbox{for} \ i = 1,\ldots
, m \ \mbox{and} \ t=1,\ldots,n
\end{align}
where $(f_i(z_{i1}), \ldots, f_i(z_{in}))' = \bm{H}\bm{\theta}_i$ is
modeled using subject-specific linear combinations of B-spline basis
functions, $z_{it} \in[0,1]$, $\bm{x}_i$ is a vector of covariates
that will be described shortly and
\begin{align*}
& \mbox{\underline{SP}} & & \mbox{\underline{SPDP}} \\
\bm{\theta}_i & \sim N(\bm{0}, \tau^2\bm{K}^{-1}) & \bm{\theta}_i | G &
\sim G \\
& & G & \sim DP(M, G_0) \ \mbox{with} \ G_0 = N(\bm{0}, \tau^2\bm{K}^{-1}).
\end{align*}
Notice that under SP a P-spline prior is used for $\bm{\theta}_i$ while
under SPDP the base measure of the DP is a P-spline prior. As with
HPPMx, $\tau^2 \sim IG(a_{\tau}, b_{\tau})$. The competitors selected,
though reasonable, aren't capable of providing active player
predictions. Therefore, prediction assessment in the simulation study
is only carried out for career prediction. Finally, we investigate the
influence that covariates have on clustering by considering the HPPMx
model with a PPM prior rather than a PPMx prior (which will be hence
forth referred to as HPPM).
\begin{figure}[htbp]
\includegraphics{919f02}
\caption{The six mean curves used in the simulation study.}
\label{SimFunc}
\end{figure}
Since both simulation studies employ the same general data generating
mechanism we provide details here. A response vector is generated using
\begin{align}\label{DGM}
y_{it} = b_{0i} + f_{group_i}(z_{it}) + \epsilon_{it} \ \mbox{with} \
\epsilon_{it} \sim N(0, w^2)
\end{align}
where $f_{group_i}(\cdot)$ corresponds to $group_i = 1, \ldots, 6$
possible mean curves which were created using the NBA data as a guide
(see Figure \ref{SimFunc}). The six mean curves are made to depend on
covariates by creating two categorical covariates that when crossed
produce six categories, one for each mean curve. A continuous covariate
was generated by $x^*_i \sim N(group_i, 0.1)$. Since $x^*_i$ depends on
the two categorical covariates, an interaction between them is created.
The three covariates were included in all model fits. Lastly, the
random intercept is generated using $b_{0i} \sim N(10,2)$.
The factors we explore in the simulation study with their respective
levels ar
\begin{itemize}
\item value for hyper parameter $A$ (0.1, 1, 10)
\item number of knots (5,15,30)
\item variance of (\ref{DGM}) ($w^2 = 0.1$, $w^2=1$)
\item number of observations per subject ($n=50$, $n=100$).
\end{itemize}
$A$ and the number of knots were selected to investigate how P-splines
function as a prior at the process level instead of at the observation
level of a hierarchical model. With $n$ and $w^2$ we see how the
methodology performs as more information becomes available relative to noise.
For each combination of the factor levels 100 data sets with $m=60$ (10
players per group) are generated and for each data set SP, SPDP, HPPM,
and HPPMx are fit.
For all four procedures we set $a_{\tau} = 1$, $b_{\tau} = 0.05$,
$a_{\sigma} = b_{\sigma} = 1.0$ and $v=1$. For PPMx and PPM $s^2_{b_0}
= s^2_{\mu} = 100^2$, and for SP and SPDP $\bm{\beta} \sim N(\bm{0},
100^2\bm{I})$. Finally for HPPMx, the cohesion and similarity functions
employed are those that match the marginal prior on partitions implied
by a DP prior (see Section 6.1 for more details). Each of the four
procedures were fit to each synthetic data set using 1000 MCMC iterates
after discarding the first 5000 as burn-in. Empirically based starting
values were employed which accelerated convergence making the 5000
iterate burn-in sufficient.
\subsection{Out of Sample (Career) Prediction}
To assess out of sample (career) prediction, for each of the 100
generated data sets 100 additional out of sample subjects were
generated. The $f_{group}(\cdot)$ associated with each new subject is
known and therefore out of sample prediction can be assessed by
comparing $\hat{f}(\cdot)$ from the four procedures to $f_{group}(\cdot
)$. After centering both $f_{group_j}(\cdot)$ and $\hat{f}_j(\cdot)$
(i.e. subtract off the empirical mean) for the $j$th out of sample
subject, we measure prediction accuracy using the mean integrated
squared prediction error
\begin{align}\label{IMSE}
MISPE_j = E \int[\hat{f}_j(z) - f_{group_j}(z)]^2dz \approx \sum
_{t}\Delta_tE[\hat{f}(z_{jt}) - f_{group}(z_{jt})]^2
\end{align}
where $\Delta_t = (z_{jt+1} - z_{jt})$. Equation (\ref{IMSE})
essentially measures the average squared area between $f_{group_j}(\cdot
)$ and $\hat{f}_j(\cdot)$ for the $j$th out of sample player over $z$'s
domain. The values in Table \ref{OSP} correspond to
\begin{align}\label{mnIMSE}
\frac{1}{D}\sum_{d=1}^D\frac{1}{100}\sum_{j=1}^{100} MISPE_{dj}
\end{align}
where $d$ indexes the $D=100$ generated data sets.
\begin{table}[t]
\caption{Results from the simulation study investigating out of sample
prediction. Table entries are calculated using (\ref{mnIMSE}) with $m =
60$ players.}\label{OSP}
\vspace*{4pt}
\begin{tabular}{l l l cccccc}\hline
&&&\multicolumn{3}{c}{$n=50$} & \multicolumn{3}{c}{$n=100$}\\ \cline
{4-6} \cline{7-9}
&&&\multicolumn{3}{c}{Number of knots} & \multicolumn{3}{c}{Number of
knots}\\ \cline{4-6} \cline{7-9}
\multicolumn{2}{c}{} & Model & 5 & 15 & 30 & 5 & 15 & 30\\\hline
\multirow{12}{*}{$w^2 = 0.1$} &\multirow{4}{*}{$A=0.1$} &HPPMx & 0.131
& 0.119 & 0.123 & 0.149 & 0.144 & 0.122 \\
& & HPPM & 0.891 & 0.868 & 0.847 & 0.873 & 0.855 & 0.848 \\
& & SP & 1.310 & 1.301 & 1.319 & 1.297 & 1.288 & 1.277 \\
& & SPDP & 0.782 & 0.784 & 0.789 & 0.780 & 0.779 & 0.775 \\ \cline{2-9}
&\multirow{4}{*}{$A=1$} & HPPMx & 0.034 & 0.026 & 0.095 & 0.026 & 0.025
& 0.029 \\
& & HPPM & 0.791 & 0.795 & 0.799 & 0.779 & 0.784 & 0.777 \\
& & SP & 1.324 & 1.319 & 1.306 & 1.285 & 1.300 & 1.279 \\
& & SPDP & 0.785 & 0.789 & 0.782 & 0.779 & 0.782 & 0.776 \\ \cline{2-9}
&\multirow{4}{*}{$A=10$} & HPPMx & 0.022 & 0.025 & 0.108 & 0.023 &
0.041 & 0.037 \\
& & HPPM & 0.786 & 0.791 & 0.800 & 0.774 & 0.776 & 0.784 \\
& & SP & 1.321 & 1.307 & 1.292 & 1.282 & 1.278 & 1.289 \\
& & SPDP & 0.785 & 0.783 & 0.779 & 0.775 & 0.777 & 0.780 \\ \hline
\multirow{12}{*}{$w^2 = 1$}&\multirow{4}{*}{$A=0.1$} &HPMMx & 0.155 &
0.203 & 0.254 & 0.142 & 0.151 & 0.242 \\
& & HPMM & 0.882 & 0.848 & 0.852 & 0.854 & 0.828 & 0.837 \\
& & SP & 1.315 & 1.312 & 1.326 & 1.277 & 1.287 & 1.283 \\
& & SPDP & 0.783 & 0.786 & 0.787 & 0.771 & 0.776 & 0.777 \\ \cline{2-9}
&\multirow{4}{*}{$A=1$} & HPPMx & 0.100 & 0.102 & 0.170 & 0.045 & 0.069
& 0.102 \\
& & HPPM & 0.796 & 0.807 & 0.829 & 0.778 & 0.789 & 0.795 \\
& & SP & 1.312 & 1.316 & 1.306 & 1.257 & 1.279 & 1.288 \\
& & SPDP & 0.783 & 0.785 & 0.785 & 0.766 & 0.776 & 0.776 \\ \cline{2-9}
&\multirow{4}{*}{$A=10$} & HPPMx & 0.080 & 0.095 & 0.157 & 0.047 &
0.066 & 0.113 \\
& & HPPM & 0.823 & 0.817 & 0.836 & 0.798 & 0.794 & 0.809 \\
& & SP & 1.331 & 1.305 & 1.327 & 1.291 & 1.269 & 1.284 \\
& & SPDP & 0.789 & 0.780 & 0.787 & 0.778 & 0.772 & 0.776 \\
\hline
\end{tabular}
\end{table}
From Table \ref{OSP} we see that HPPM and SPDP provide similar
predictions which is to be expected as both employ a DP prior (although
not at the same level of a hierarchy). What should be very obvious is
that HPPMx does a much better job in out of sample prediction relative
to the other three procedures for all data generating scenarios.
\subsection{Goodness of Individual Fits, Curve Smoothness, and Clustering}
To assess goodness of individual fits we employ the following $R^2$
type goodness-of-fit statistic from \cite{FunctionalDataAnalysisBook}:
\begin{align}\label{R2}
R_i^2 = 1 - \frac{\sum_t(\hat{f}_i(z_{it}) - y_{it})^2}{\sum
_t(y_{it}-\bar{y}_{i} )^2}.
\end{align}
$R^2_i$ can be loosely interpreted as a coefficient of determination in
that as $R^2_i$ approaches 1, individual fits improve. Negative values
of $R^2_i$ indicate that $\bar{y}_i$ predicts better than $\hat
{f}_i(\cdot)$. The values in Table \ref{SSr2} correspond to
\begin{align}\label{mnR2}
\frac{1}{D}\sum_{d=1}^D\frac{1}{m}\sum_{i=1}^m R_{di}^2.
\end{align}
\begin{table}[t]
\caption{Results from the simulation study investigating
goodness-of-fit. Table entries are calculated using (\ref{mnR2}) with
$m = 60$ players.}\label{SSr2}
\vspace*{4pt}
\begin{tabular}{l l l cccccc}
\hline
&&&\multicolumn{3}{c}{$n=50$} & \multicolumn{3}{c}{$n=100$}\\ \cline
{4-6} \cline{7-9}
&&&\multicolumn{3}{c}{Number of knots} & \multicolumn{3}{c}{Number of
knots}\\ \cline{4-6} \cline{7-9}
\multicolumn{2}{c}{} & Model & 5 & 15 & 30 & 5 & 15 & 30\\
\hline
\multirow{12}{*}{$w^2 = 0.1$} &\multirow{4}{*}{$A=0.1$} &HPPMx & 0.979
& 0.990 & 0.950 & 0.981 & 0.989 & 0.984 \\
& & HPPM & 0.813 & 0.908 & 0.919 & 0.812 & 0.939 & 0.952 \\
& & SP & 0.985 & 0.993 & 0.994 & 0.984 & 0.992 & 0.992 \\
& & SPDP & 0.802 & 0.794 & 0.774 & 0.764 & 0.759 & 0.743 \\ \cline{2-9}
&\multirow{4}{*}{$A=1$} & HPPMx & 0.981 & 0.965 & 0.864 & 0.984 & 0.991
& 0.979 \\
& & HPPM & 0.983 & 0.987 & 0.950 & 0.984 & 0.991 & 0.989 \\
& & SP & 0.985 & 0.993 & 0.994 & 0.984 & 0.992 & 0.992 \\
& & SPDP & 0.808 & 0.811 & 0.798 & 0.771 & 0.743 & 0.746 \\ \cline{2-9}
&\multirow{4}{*}{$A=10$} & HPPMx & 0.984 & 0.975 & 0.853 & 0.984 &
0.990 & 0.979 \\
& & HPPM & 0.985 & 0.987 & 0.937 & 0.984 & 0.991 & 0.992 \\
& & SP & 0.985 & 0.993 & 0.994 & 0.984 & 0.992 & 0.992 \\
& & SPDP & 0.802 & 0.793 & 0.787 & 0.765 & 0.747 & 0.750 \\ \hline
\multirow{12}{*}{$w^2 = 1$}&\multirow{4}{*}{$A=0.1$} &HPPMx & 0.583 &
0.593 & 0.572 & 0.574 & 0.593 & 0.575 \\
& & HPPM & 0.485 & 0.537 & 0.497 & 0.476 & 0.552 & 0.541 \\
& & SP & 0.613 & 0.663 & 0.713 & 0.570 & 0.575 & 0.533 \\
& & SPDP & 0.538 & 0.554 & 0.565 & 0.515 & 0.521 & 0.520 \\ \cline{2-9}
&\multirow{4}{*}{$A=1$} & HPPMx & 0.585 & 0.608 & 0.630 & 0.572 & 0.589
& 0.605 \\
& & HPPM & 0.589 & 0.618 & 0.663 & 0.574 & 0.596 & 0.616 \\
& & SP & 0.613 & 0.662 & 0.711 & 0.566 & 0.575 & 0.531 \\
& & SPDP & 0.541 & 0.555 & 0.562 & 0.511 & 0.522 & 0.523 \\ \cline{2-9}
&\multirow{4}{*}{$A=10$} & HPPMx & 0.587 & 0.610 & 0.626 & 0.574 &
0.589 & 0.607 \\
& & HPPM & 0.593 & 0.624 & 0.667 & 0.577 & 0.597 & 0.617 \\
& & SP & 0.613 & 0.662 & 0.712 & 0.568 & 0.574 & 0.535 \\
& & SPDP & 0.539 & 0.554 & 0.561 & 0.514 & 0.519 & 0.522 \\
\hline
\end{tabular}
\end{table}
From Table \ref{SSr2} we see that SP tends to produce the best
individual fits and SPDP the worst. This is of course expected as all
individuals are fit independently by SP while SPDP provides cluster
specific curves. However, HPPMx does remarkably well in producing good
individual fits as HPPMx is very close to SP particularly as $n$
increases. Thus, HPPMx's meaningful cluster production doesn't require
sacrificing much goodness of individual fit.
\begin{table}[t]
\caption{Results from the simulation study investigating smoothness.
Table entries are calculated using (\ref{mnlSD}) with $m = 60$
players.}\label{SSsmooth}
\vspace*{4pt}
\begin{tabular}{l l l cccccc}
\hline
&&&\multicolumn{3}{c}{$n=50$} & \multicolumn{3}{c}{$n=100$}\\ \cline
{4-6} \cline{7-9}
&&&\multicolumn{3}{c}{Number of knots} & \multicolumn{3}{c}{Number of
knots}\\ \cline{4-6} \cline{7-9}
\multicolumn{2}{c}{} & Model & 5 & 15 & 30 & 5 & 15 & 30\\
\hline
\multirow{12}{*}{$w^2 = 0.1$} &\multirow{4}{*}{$A=0.1$} &HPPMx & 0.162
& 0.165 & 0.171 & 0.082 & 0.082 & 0.087 \\
& & HPPM & 0.143 & 0.154 & 0.165 & 0.073 & 0.078 & 0.084 \\
& & SP & 0.162 & 0.162 & 0.166 & 0.081 & 0.080 & 0.083 \\
& & SPDP & 0.154 & 0.153 & 0.157 & 0.077 & 0.076 & 0.079 \\ \cline{2-9}
&\multirow{4}{*}{$A=1$} & HPPMx & 0.163 & 0.165 & 0.170 & 0.081 & 0.081
& 0.084 \\
& & HPPM & 0.163 & 0.164 & 0.168 & 0.081 & 0.081 & 0.085 \\
& & SP & 0.162 & 0.162 & 0.166 & 0.081 & 0.080 & 0.083 \\
& & SPDP & 0.154 & 0.154 & 0.158 & 0.077 & 0.076 & 0.079 \\ \cline{2-9}
&\multirow{4}{*}{$A=10$} & HPPMx & 0.163 & 0.165 & 0.169 & 0.082 &
0.081 & 0.085 \\
& & HPPM & 0.162 & 0.164 & 0.168 & 0.082 & 0.081 & 0.086 \\
& & SP & 0.161 & 0.162 & 0.166 & 0.081 & 0.080 & 0.083 \\
& & SPDP & 0.153 & 0.154 & 0.158 & 0.077 & 0.076 & 0.079 \\ \hline
\multirow{12}{*}{$w^2 = 1$}&\multirow{4}{*}{$A=0.1$} &HPPMx & 0.179 &
0.208 & 0.274 & 0.096 & 0.107 & 0.132 \\
& & HPPM & 0.155 & 0.182 & 0.250 & 0.082 & 0.096 & 0.120 \\
& & SP & 0.195 & 0.206 & 0.258 & 0.088 & 0.071 & 0.062 \\
& & SPDP & 0.172 & 0.174 & 0.191 & 0.095 & 0.090 & 0.095 \\ \cline{2-9}
&\multirow{4}{*}{$A=1$} & HPPMx & 0.184 & 0.210 & 0.288 & 0.097 & 0.109
& 0.136 \\
& & HPPM & 0.183 & 0.209 & 0.298 & 0.097 & 0.108 & 0.140 \\
& & SP & 0.198 & 0.205 & 0.258 & 0.088 & 0.071 & 0.061 \\
& & SPDP & 0.174 & 0.175 & 0.190 & 0.094 & 0.090 & 0.095 \\ \cline{2-9}
&\multirow{4}{*}{$A=10$} & HPPMx & 0.184 & 0.212 & 0.286 & 0.097 &
0.110 & 0.137 \\
& & HPPM & 0.185 & 0.210 & 0.296 & 0.100 & 0.108 & 0.141 \\
& & SP & 0.197 & 0.206 & 0.258 & 0.088 & 0.071 & 0.062 \\
& & SPDP & 0.173 & 0.175 & 0.189 & 0.094 & 0.090 & 0.095 \\
\hline
\end{tabular}
\end{table}
To assess curve smoothness we calculate the standard deviation of the
lag one differences from the estimated curve
\begin{align}\label{lSD}
{\ell}SD_i = \sqrt{\frac{1}{n-3}\sum_{t=1}^{n-1} (lag_{it} - \overline{lag}_i)^2},
\end{align}
where $lag_{it} = \hat{f}_i(z_{it+1}) - \hat{f}_i(z_{it}) \ \mbox{for}
\ t = 1, \ldots, n-1$ and $\overline{lag}_i = 1/(n-1)\sum
_{t=1}^{n-1}lag_{it}$. Large values of ${\ell}SD_i$ generally indicate
more wiggliness relative to small values. Values provided in Table \ref
{SSsmooth} correspond to
\begin{align}\label{mnlSD}
\frac{1}{D}\sum_{d=1}^D\frac{1}{m}\sum_{i=1}^m {\ell} SD_{di}.
\end{align}
\begin{table}[t]
\caption{Results from the simulation study investigating cluster
estimation. Table entries correspond to the number of estimated
clusters averaged over 100 simulated data sets. }\label{SScluster}
\vspace*{4pt}
\begin{tabular}{l l l cccccc}
\hline
&&&\multicolumn{3}{c}{$n=50$} & \multicolumn{3}{c}{$n=100$}\\ \cline
{4-6} \cline{7-9}
&&&\multicolumn{3}{c}{Number of knots} & \multicolumn{3}{c}{Number of
knots}\\ \cline{4-6} \cline{7-9}
\multicolumn{2}{c}{} & Model & 5 & 15 & 30 & 5 & 15 & 30\\
\hline
\multirow{6}{*}{$w^2 = 0.1$} &\multirow{2}{*}{$A=0.1$} &HPPMx & 5.98 &
5.98 & 5.99 & 5.97 & 5.98 & 5.96 \\
& & HPPM & 3.93 & 4.26 & 4.49 & 4.32 & 4.37 & 4.40 \\ \cline{2-9}
&\multirow{2}{*}{$A=1$} & HPPMx & 9.24 & 8.53 & 6.52 & 10.93 & 10.26 &
9.02 \\
& & HPPM & 9.49 & 9.18 & 7.89 & 10.85 & 10.77 & 10.02 \\ \cline{2-9}
&\multirow{2}{*}{$A=10$} & HPPMx & 10.32 & 8.55 & 6.42 & 13.39 & 10.34&
8.86 \\
& & HPPM & 10.45 & 8.31 & 7.14 & 11.84 & 9.67 & 8.94 \\ \hline
\multirow{6}{*}{$w^2 = 1$}&\multirow{2}{*}{$A=0.1$} &HPPMx & 5.97 &
5.96 & 5.99 & 5.98 & 5.97 & 5.99 \\
& & HPPM & 4.09 & 4.36 & 4.72 & 4.02 & 4.54 & 4.66 \\ \cline{2-9}
&\multirow{2}{*}{$A=1$} & HPPMx & 7.25 & 6.87 & 6.11 & 8.20 & 7.94 &
6.82 \\
& & HPPM & 7.08 & 6.74 & 6.15 & 7.94 & 7.77 & 7.40 \\ \cline{2-9}
&\multirow{2}{*}{$A=10$} & HPPMx & 7.24 & 6.89 & 6.28 & 8.90 & 8.06 &
7.08 \\
& & HPPM & 5.84 & 6.03 & 5.36 & 7.14 & 6.88 & 6.94 \\
\hline
\end{tabular}
\end{table}
From Table \ref{SSsmooth} it appears that curves become less wiggly as
$n$ increases relative to the number of knots. This is expected. Also
unsurprising is that HPPMx and HPPM produce similar values of $(\ref
{mnlSD})$ with the biggest differences occurring when $w^2$ (within
player variability) and $A$ (within cluster variability) are small.
What is a bit surprising is that the value of $A$ doesn't much alter
curve smoothness for HPPMx. It appears that $w^2$ is more influential.
Overall, since the values of (\ref{mnlSD}) for HPPMx are fairly similar
to those for SP and SPDP (recall that SP and SPDP are not influenced by
$A$), penalizing curves directly with a P-spline prior produces curves
with similar smoothness as those produced through the hierarchical model.
To see how the PPMx prior improves clustering relative to the PPM
prior, Table \ref{SScluster} provides the number of estimated clusters
($k_m$) averaged over all $D=100$ data sets. For each data set $\rho$
was estimated using \cite{dahl:2006}'s method which is based on
least-squares distance from the matrix of posterior pairwise
co-clustering probabilities (note that an estimate of $\rho$ also
provides an estimate of $k_m$).
The true value of $k_m$ in Table \ref{SScluster} is six for all
scenarios. It appears that as $n$ increases, the PPMx prior tends to
converge to the six clusters faster than the PPM prior. It also appears
that the clustering mechanisms of the HPPMx and HPPM depend on $A$.
This is to be expected however, because as $A$ increases curves are
allowed to deviate further from cluster specific means, thus creating
more clusters.
\vspace*{-2pt}\section{Analysis and Results} \label{results}\vspace*{-1pt}
In this we section provide results of fitting HPPMx to the NBA data set.
\vspace*{-1pt}\subsection{Model Details and Prior Selection}\vspace*{-1pt}
We first provide a bit of detail regarding cohesion and similarity
functions used and then on prior values. The cohesion and similarity
functions employed match the PPMx prior\vadjust{\eject} to the marginal prior on
partitions implied by the DP prior. This results in an {a priori}
clustering of a few large clusters that represent typical player
production and a few smaller clusters of ``abnormal'' players. Thus, we
set $c(S_j) = M(|S_j| - 1)!$ with $M=1$ favoring a small number of
clusters. The similarity functions used are typical conjugate models
for continuous and categorical variables with parameter values
suggested by \cite{PPMxMullerQuintanaRosner} resulting in
\label{gfunc}
\begin{align*}
g(\bm{x}^{\star}_j) & = g_1(\bm{x}^{\star}_{j1})g_2(\bm{x}^{\star
}_{j2})g_3(\bm{x}^{\star}_{j3}) \\
& = \int\prod_{i\in S_j} N(x_{i1}; m_j, 1) N(m_j; 0, 10) \pi
_{i,x_{i2}} \\
& \times Dir(\pi_{i,x_{i2}}; 0.1, 0.1, 0.1) \pi_{i,x_{i3}}Dir(\pi
_{i,x_{i3}};0.1, 0.1,0.1)dm_jd\bm{\pi}_{1j}d\bm{\pi}_{2j} \\
& = \frac{N_{n_j}(\bm{x}^{\star}_j; \bm{0}, \bm{I}) N(0; 0, 10)}{N(\hat
{m}; 0, \hat{s}^2)}\frac{\Gamma(\sum_{c=1}^3 n_{1jc} + 0.1)}{\prod
_{c=1}^3\Gamma(n_{1jc} + 0.1)}\frac{\Gamma(\sum_{c=1}^3 n_{2jc} +
0.1)}{\prod_{c=1}^3\Gamma(n_{2jc} + 0.1)}.
\end{align*}
$\pi_{i,x_{i2}}$ and $\pi_{i,x_{i3}}$ denote $x_{i2}$ and $x_{i3}$'s
probability vector, $n_{1jc}$ are the number of players in cluster $j$
that have covariate value $c$ for $x_{i2}$ and $n_{2jc}$ the number of
players for $x_{i3}$ (as a reminder, $x_{i1}$ corresponds to age,
$x_{i2}$ experience and $x_{i3}$ draft order). In addition, $\hat{s}^2
= (n_j + 1/10)^{-1}$ and $\hat{m} = \hat{s}^2\bm{1}'\bm{x}^{\star}_{1j}$.
A first-order Bayesian P-spline prior was used and following
suggestions in \citet{BayesianPsplines}, we set $a = 1$ and $b = 0.05$.
We found that results were fairly robust to reasonable prior
specifications of $\tau^2_h$. From the simulation study setting $A = 1$
seemed reasonable so that individual curves are fairly similar to their
cluster-specific counterparts. Preliminary investigations indicated
that methodology is robust to variance prior specifications so with
hopes of being diffuse we set $a_{\sigma} = b_{\sigma} = a_{\delta} =
b_{\delta} = a_{\psi}=b_{\psi} = 1$ and $s^2_{b_0} = 100^2$. To produce
a flat prior for $\bm{\gamma}$ we use $\bm{m}_{\gamma} = \bm{0}$ and
$s^2_{\gamma} = 100^2$. Since there are 82 games in an NBA season we
set $m_{a_1} = 76$ (taking into account missed games due to injury)
with $s^2_{a} = 10^2$.
The MCMC algorithm was run until 1000 iterates from a Markov chain were
collected after discarding the first 25,000 as burn in and thinning by
25. Convergence was monitored using MCMC iterate trace plots.
\subsection{Fits of Individual Player Production Curves}
Figure \ref{IndividualPosteriorFits} displays the posterior mean curves
with 95\% credible and prediction bands for the three players
introduced in Figure \ref{rawscatterplot}. Notice that even though the
fits are fairly flexible they are smoothed relative to the loess fits
provided in Figure \ref{rawscatterplot}.
\begin{figure}[htbp]
\includegraphics{919f03}
\caption{Posterior fits for three NBA players. The solid red lines are
point-wise posterior mean curves, the dashed red lines are 95\% mean
point-wise credible intervals, and the dashed orange lines are 95\%
point-wise prediction intervals.}
\label{IndividualPosteriorFits}
\end{figure}
\begin{figure}[t!]
\includegraphics{919f04}
\caption{Active player predictions for four NBA players. Lines
associated with active player prediction are blue. The dashed lines
represent point-wise 95\% mean credible bands, and the dotted lines
95\% point-wise prediction bands.}
\label{ACP}
\vspace*{-6pt}
\end{figure}
\subsection{Active Player Prediction}
Displaying the results of active player prediction in and of itself is
not trivial as curves depend completely on the predicted values of
$n_i$. To simplify the process we display the active player prediction
curves conditioned on $E(n_i|\bm{y}_i)$. This requires producing a
curve conditioned on $E(n_i|\bm{y}_i)$ for each MCMC iterate of $\bm
{\beta}$. From these MCMC iterates, we estimate an average prediction
curve with point-wise 95\% credible bands and prediction bands. Figure
\ref{ACP} contains the estimated mean curve with credible bands and
prediction bands corresponding to four players in varying stages of
their career. Shaquille O'Neal played beyond the 2009/2010 season but
has since retired. His average number of predicted games played turned
out to be 1545 and the actual number of games played is 1423 (including
playoffs). Ray Allen continues to play but is nearing the end of his
career and the predicted sharp decrease in production mirrors reality.
Dwight Howard and Luke Ridnour are two completely different types of
players and are provided as a means to demonstrate the flexibility in
the predictions. $E(n_i|\bm{y}_i)$ for Dwight Howard is quite
conservative and barring injury should under estimate his career game
total, while $E(n_i|\bm{y}_i)$ for Beno Udrih is quite reasonable.
Regardless, the four predictions display completely plausible decreases
in production as the players approach retirement.
\begin{figure}[htbp]
\includegraphics{919f05}
\caption{Career prediction curves for different levels of draft order,
playing experience and age during first game played. }
\label{careerPrediction}
\end{figure}
\subsection{Career Prediction Analysis}
For career prediction we employ the predictive distributions as
described in Section \ref{PPD}\vadjust{\eject} resulting in the curves found in Figure
\ref{careerPrediction}. We include the High School level of
experience
even though the latest CBA requires at least one year post high school
experience before being drafted. For age during first game played, we
considered 19, 21, and 23 years old. (We do not consider ages 21 and 23
for High School level of experience as those scenarios are practically
impossible.) The curves are presented conditioned on the predicted
number of games played averaged over all active players that belong to
each respective group.
Before describing results it is important to keep in mind that only
players who played at least three years were included in the analysis.
This explains the seemingly high predictions for second round picks.
Also, from Table \ref{playercategorysummary} it can be seen that only
one player (considered in the analysis) was drafted in the second round
straight from high school (Rashard Lewis). So you will notice that the
predicted curve for this group follows a trajectory similar to that of
Rashard Lewis's career. Even with that being the case, a few
interesting trends emerge. It appears that there is much more
variability in curve location for Top Five Picks. Also the players that
are Top Five Picks tend to reach their max production earlier in their
career. Age clearly influences a player's production as players that
start their career at a younger age tend to have higher production
rates. It appears that the shapes of curves vary by experience with
international players decreasing slightly earlier relative to college
or high school players.
Table \ref{GameMaxPerformance} provides estimates of the number of
games need to reach peak performance. Generally speaking players
drafted straight out of high school take longer to reach maximum
performance while those with college experience are the quickest. That
said, any conclusions drawn from Table \ref{GameMaxPerformance} or
Figure \ref{careerPrediction} should be made with care as some of the
curves are accompanied with a moderate to substantial amount of variability.
\begin{table}[htdp]
\caption{Predicted game at which max performance is attained
(prediction errors are in parenthesis).}\label{GameMaxPerformance}
\vspace*{4pt}
\begin{tabular}{l l ccc}
& &\multicolumn{3}{c}{Age During First Game Played} \\ \cline{3-5}
Draft & Experience & 19 & 21 & 23\\
\hline
\multirow{3}{*}{Top 5} & High School & 477(169.8) & - & - \\
& College & 385(162.2) & 437(184.9) & 386(178.9) \\
& International & 399(155.6) & 448(160.7) & 423(147.1) \\ \cline{1-5}
\multirow{3}{*}{1st Round} & High School & 472(147.4) & - & - \\
& College & 436(185.5) & 417(193.5) & 403(200.2) \\
& International & 446(151.3) & 473(172.4) & 413(167.3) \\ \cline{1-5}
\multirow{3}{*}{2nd Round} & High School & 568(184.7) & - & -\\
& College & 344(152.6) & 386(179.1) & 369(179.7) \\
& International & 409(144.6) & 416(151.1) & 419(169.6) \\
\hline
\end{tabular}
\end{table}
\begin{figure}[t!]
\includegraphics{919f06}
\caption{Player specific posterior mean production curves divided into
the 9 clusters corresponding to the partition estimated using \cite
{dahl:2006}'s cluster estimate method. Active player prediction for
active players is displayed by a dashed line.}
\label{AllClusterCurves}
\end{figure}
\subsubsection{Cluster Analysis}
A nice property of the model is the ability to postulate what
characteristics guide clustering. To do this it is necessary to obtain
a point estimate using cluster MCMC iterates. Since posterior summaries
of cluster specific parameters are arbitrary, using typical posterior
summaries (mean and median) makes little sense. We employ \cite
{dahl:2006}'s method which is based on least-squares distance from the
matrix of posterior pairwise co-clustering probabilities. Using this
method produces a partitioning of the 408 players into 18 clusters with
cluster membership ranging from 3 to 63 players. Figure \ref
{AllClusterCurves} provides player-specific posterior mean curves for
nine clusters. Except for cluster 6, these clusters represent those
that contain the highest number of players (approximately 80\% of
players). Cluster 6 was selected as it contains curves that are in our
opinion more interesting than clusters not shown. The remaining nine
clusters for the most part are comprised of role players. Although each
cluster contains curves that are slightly unique, they are relatively
flat. The dashed segments at the end of some curves are active player
predictions. To facilitate comparisons we maintain the $x$-axis on the
percentile number of games played scale. We highlight a few of the
clusters by pointing out some of the well known players. Cluster 1 is
comprised of role players (e.g., Tony Allen and Matt Bonner) whose
production is constant. Cluster 2's key member is LeBron James. Players
in this cluster begin careers close to peak level and appear to
maintain production. Cluster 3 contains Carlos Boozer and Ron Artest
who had sharp increase in production but maintained peak performance
for a short time with a gradual decrease in performance. Cluster 4 is
comprised of role players who showed an increase in production right
before retiring (e.g. James Posey). Kobe Bryant is the key player of
Cluster 5 (along with Chauncey Billups, Steve Nash). In this cluster,
players started slow but experienced large sharp increases of
production and maintained it for much of their career. Clusters 6 and 7
are comprised of players who start at peak performance and gradually
decline through out their career with players in Cluster 6 showing a
brief increase towards the end of their career. Grant Hill is a member
of Cluster 6 and Shaquille O'Neal is a member of Cluster 7. Clusters 8
and 9 are primarily comprised of role players with Cluster 8 showing
gradual increase until the later stages of a career. An example is Matt
Barnes. Marcus Camby is member of Cluster 9 and the decrease in
production towards the end of the career is more sharp relative to
Cluster 8. Overall, the clusters contain curves that have distinct
shapes. Information provided in Figure \ref{AllClusterCurves} could
potentially be used to guide NBA personnel decisions regarding contract
length and amount. For example, production for players in 3 begins to
decrease earlier than Cluster 2.
Finally, Figure \ref{ClusInt} displays the average age during first
game played vs. the percent of players in each of the six categories
for each of the 18 clusters. Apart from demonstrating the presence of
an interaction between the three covariates, these plots confirm what
is already widely known. That is, on average, the age of players
increases as draft order increases and players that play college tend
to begin NBA careers at an older age relative to high school and
international players.
\begin{figure}[htbp]
\includegraphics{919f07}
\caption{Average baseline age and percent of players for each of the 18
clusters for the six categories corresponding to experience and draft
order. The size of the dot is proportional to the number of players
allocated to the cluster.}
\label{ClusInt}
\end{figure}
\subsection{Assessing Trade-off between Individual Player Fits and Prediction}
As mentioned previously, incorporating the PPMx in the hierarchical
model improves predictions at the cost of a small loss in individual
fits. To show that the cost is minimal relative to gains in prediction,
we randomly selected 50 retired players and removed the final 25\% of
games played (essentially treating them as active players). We then
proceeded to fit four models (details follow) to these partitioned data
and assess model fit through Mean Square Error (MSE). To assess
prediction accuracy, active player prediction was carried out for each
of the 50 randomly selected players and Mean Squared Prediction Error
(MSPE) was computed. The four models considered were the SP model (\ref
{DGM}), an extension of the SP model that improves prediction, the
HPPMx, and the following 5th degree polynomial regression model:
\begin{align*}
y_{it} & = \bm{x}'_i\bm{\beta} + \sum_{j=0}^5 \gamma_{ji}t^j + \epsilon
_{it} \ \ \mbox{with} \ \ \epsilon_{it} \stackrel{iid}{\sim} N(0, \sigma
^2_i) \ \ \mbox{and} \ \ \bm{\beta} \sim N(\bm{0}, s^2\bm{I})\\
\bm{\gamma}_i & \sim N(\bm{\mu}, \bm{T}) \ \ \mbox{where} \ \ \bm{T} =
\mbox{diag}(\tau^2_0, \ldots, \tau^2_5) \\
\bm{\mu} & \sim N(0, s^2\bm{I}).
\end{align*}
The SP model (\ref{DGM}) was extended in the following way
\begin{align*}
\bm{\theta}_i & \sim N(\bm{\mu}, \tau^2\bm{K}^{-1})\\
\bm{\mu} & \sim N(\bm{0}, s^2\bm{I}).
\end{align*}
We refer to this model as hSP (hierarchical semi-parametric).
Predicting (or extrapolating) the last 25\% of games played using the
SP model requires drawing $\theta$'s associated with knots for removed
games from its prior distribution. Therefore, centering the prior on a
vector of global spline coefficients should improve prediction relative
to a prior centered at 0.
The MSE averaged over the 408 players was calculated for each of the
four models resulting in Polynomial (30.72), SP (29.66), hSP (29.71),
HPPMx (31.71). As expected the flexible penalized splines produce the
smallest MSE and HPPMx has the highest MSE illustrating the surrender
of a bit of individual player fit. The MSPE averaged over the 50
randomly selected players turned out to be Polynomial (476.32), SP
(34.00), hSP (31.10), and HPPMx (26.90). As expected HPPMx gained quite
a bit in terms of prediction (extrapolation) accuracy at a fairly
minimal individual fit cost.
\section{Conclusions}
We have proposed a completely novel methodology that incorporates
information regarding the shape of longitudinal curves in predicting
future NBA player production based on age, experience, and draft order
of player. Clearly, curve shape provides information beyond available
covariates and the inclusion of this information in modeling efforts
should improve inferences. In addition, the methodology does well in
balancing individual fits and producing clusters that provide adequate
borrowing of strength among players. The PPMx prior employed does a
nice job of being able to incorporate both covariate and curve shape
information when forming clusters and ultimately borrowing strength
among subjects to improve active player and career predictions. From a
basketball perspective, individual player production clearly depends on
many omitted variables (such as team strength, injury history and
coaching philosophy) and these variables can be easily incorporated in
the model using the PPMx prior when they become available. Finally,
though the methodology was demonstrated using production curves of NBA
basketball players, the idea of incorporating curve shape in inferences
should be applicable in a wide variety of settings (e.g., biomedical,
finance, and environmental studies).
\begin{acknowledgement}
The authors wish to thank the reviewers and editors for their comments,
which improved the paper. The first author was partially funded by
grant FONDECYT 11121131 and the second author was partially funded by
grant FONDECYT 1141057.
\end{acknowledgement}
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QMJHL updates safety protocol, donates $50K to Jordan Boyd Foundation
Boyd died on the ice in 2013 due to undiagnosed heart condition
By Nicholas Frew and Cory Funk
December 5, 2017, 3:47 pm ASTLast Updated: December 5, 2017, 4:41 pm
Jordan Boyd died on the ice at a training camp in 2013. Jordan Boyd Celebrity Hockey Challenge Facebook page
The Quebec Junior Men's Hockey League's commissioner announced new safety protocols for the league Tuesday in response to the 2013 death of a 16-year-old player.
Jordan Boyd died from cardiac arrest during the QJMHL training camp in Bathurst, N.B., in August 2013. His family was at the news conference in Halifax for the announcement.
"We understand that there is no foolproof system that can ensure tragedies like what happened to Jordan from happening again," said Stephen Boyd, Jordan's father, at the news conference. "But we do think something can be learned from this, to improve the chances of a different outcome in the future."
Boyd passed health exams before the camp, but the QMJHL did not specifically require cardiac health exams. At the time, the league required an automated external defibrillator, or AED, to be at the rink, but it wasn't used on Boyd after he collapsed.
"I'm not here to say that what happened in Bathurst was right or was not right, or something went wrong, or something like that," said QMJHL commissioner Gilles Courteau. "I'm here today to announce, following the passing of Jordan, all the changes we've made to improve our players' environment situation."
New protocols
The new protocols were implemented at the start of the 2017-18 season, but only announced on Tuesday.
Each team must have at least three staff members trained in CPR and life-saving techniques, like how to use an AED. Before, it was only one. Teams must have at least two of these members at all team events, including games, practices and tryouts.
Courteau said that teams would pay for extra staff, but current staff can also do the training. Every year, teams have to report which staff members have been certificated and need to take refresher training.
Each team must also have working AEDs at team events. Although, Courteau did not specify whether the teams have to own the AEDs or if they could use ones provided by the arena.
Annual audits of these protocols will take place to ensure compliance.
"We know that this is a good thing for the league," Boyd said. "Not just for right now but for all future QJMHL players as well."
The Halifax Mooseheads are already complying with the new protocols, Scott MacIntosh, a team spokesman, said Tuesday. In an email to The Signal, MacIntosh said the team has three staff members trained for AED use.
Robin Hunter, the Mooseheads' athletic therapist, has one with her "at all times," MacIntosh said, "which includes on the bench, on the bus and in the room."
Stephen Boyd said it would be "a good start" to improve physical examinations of players by implementing electrocardiograms.
In addition to announcing new safety protocols, the QMJHL donated $50,000 to the Jordan Boyd Foundation. The foundation is set up to educate and raise awareness of inherited heart diseases in young people — athletes in particular.
It also administers the Jordan Boyd Leadership Award Scholarship, which goes to an amateur hockey player who has demonstrated a passion for sportsmanship, learning, leadership and community.
With files from Jessica Sundblad | {
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Q: Order of execution in Ola Index maintenance and Stats collection I was thinking of performing The index maintenance and stats collection jobs in the below order for all databases in an Instance.
step 1.Perform all index maintenance activity for all required Indexes(based on thresholds) of all databases in the instance. This step should not do any Stats collection (other than the stats update naturally done by the index rebuild).
step 2.Perform Stats collection of all objects in all databases in the same instance. But this should exclude all Indexes which already had a stats collection as part of Step 1.
Is there anyway I can achieve this with Ola Hallgren Maintenance Solution?
A: These don't need to be done as separate steps. In fact, it would be much harder to do as separate steps, as you would want to filter out any indexes rebuilt in your Step 1, to avoid duplicate work in Step 2.
Looking at the Examples section on Ola's site:
B. Rebuild or reorganize all indexes with fragmentation and update modified statistics on all user databases
EXECUTE dbo.IndexOptimize
@Databases = 'USER_DATABASES',
@FragmentationLow = NULL,
@FragmentationMedium = 'INDEX_REORGANIZE,INDEX_REBUILD_ONLINE,INDEX_REBUILD_OFFLINE',
@FragmentationHigh = 'INDEX_REBUILD_ONLINE,INDEX_REBUILD_OFFLINE',
@FragmentationLevel1 = 5,
@FragmentationLevel2 = 30,
@UpdateStatistics = 'ALL',
@OnlyModifiedStatistics = 'Y';
You can simply use both the @UpdateStatistics parameter in conjunction with the index rebuild/reorg parameters.
That said, you probably don't need to be rebuilding your indexes very frequently. Most folks just need a frequent (eg weekly) job to update stats, and an infrequent (eg quarterly) job to rebuild indexes.
| {
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Q: Global State in Java/Spring I have a basic Java/Spring MVC CRUD application in production on my company's intranet. I am still a beginner really, this application is what I've used to learn Java and web applications. Basically it has a table that uses AJAX to refresh its data on regular intervals, and an html form that is input into the database. The refresh is important because the data is viewed on multiple computers that need to see the input from the others.
The problem is that, due to network issues outside of my control, the database transactions on certain computers can be very slow.
I have been playing around with React/Redux JavaScript client applications in the past few weeks and the concept of state. Now, as best I can tell, global state or variables are pretty reviled by the Java community. Bugs, difficulty in testing, etc.
But Redux gave me an idea that, when a user hits "submit" instead of inserting a row into SQL, it stores that object in memory on the server. Then at regular intervals that memory is inserted into the database - so the user does not have to wait for database transactions, only communication with the server. Table refreshes don't look at the database - they look at this memory.
But, again as a beginner, I don't see people do this. Why is it a bad idea?
A: In general, it isn't done for two reasons:
*
*the state is not guaranteed, because it is not actually written.
If you restart the application before the data is flushed to the database, it is silently dropped. This is not a good thing in general, although obviously, but your interpretation may very. If you don't care so much, this might be ok. You could remedy this by persisting it somewhere locally.
*the state is also not guaranteed, because you may end up not being able to write the data because, for example, some database constraint.
So, in general it is frowned upon, because you are lying to the client ... You say you wrote it, but there's no actual effort to ensure this has actually happened.
But then again. if the data is less important, it might be ok.
| {
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{"url":"https:\/\/powerofcredit.com\/reddit-saskatoon-pvybg\/oxygen-enters-the-blood-at-the-lungs-because-3b9a48","text":"oxygen enters the blood at the lungs because\n\n19 Jan oxygen enters the blood at the lungs because\n\n46.5 - What are the key evolutionary adaptations that... Ch. You may feel a sharp pain when the needle enters the artery. List two of the four major factors that influence how much oxygen diffuses into pulmonary blood per minute. e. the process is independent of gas concentrations in the blood. As blood leaves the lungs through the pulmonary veins, the venous $\\text{P}_{\\text{O}_2}$= 100 mm Hg, whereas the venous $\\text{P}_{\\text{CO}_2}$ = 40 mm Hg.As blood enters the systemic capillaries, the blood will lose oxygen and gain carbon dioxide because of the pressure difference of the tissues and blood. The stopcock connecting a 1.00L bulb containing oxygen gas at a pressure of 540torr and a 1.00L bulb containing... You are advising a fellow student who wants to learn to perform multiple flips on the trampoline. Hence, option C would be the correct answer. Congenital pulmonary airway malformation (CPAM), Coronavirus and living with a lung condition, Guidance for the clinically extremely vulnerable. You have solubilized it with homogenization i... 14. a. the concentration of carbon dioxide in the capillaries is lower than the concentration of carbon dioxide in the alveoli. As the preload decreases, the cardiac output _____________________. (1.4). If you'd like to be kept updated, please enter a valid email address. Calculate the gauge pressure at a depth of 300 m in seawater. The oxygen binds to hemoglobin and the carbon dioxide is released. How can I improve the air quality in my home? The blood then is pumped through your body to provide oxygen to the cells of your tissues and organs. 46 - A teenager is frightened when she is about to step... Ch. Because there is low oxygen in the blood and high oxygen in the air in the aveoli (and because the capillaries are so close to the aveoli) the oxygen in the air in the aveoli can simply transfer into the blood in the capillaries (in a process called diffusion). Low blood oxygen levels indicate that there may be an issue with your lungs or circulation. The primary function of the respiratory system is to take in oxygen and eliminate carbon dioxide. does not contain negative temperatures? There are no solutions to the system because the equations represent the same line. How is pulmonary hypertension diagnosed and treated in children? A rooftop in the southwestern United States receives an average solar power of W (averaged over both day a... Digestive processes are first-order processes. Ch. Transport of Oxygen. Find out if oxygen treatment is suitable for you. Asthma UK and British Lung Foundation Partnership is a company limited by guarantee 01863614 (England and Wales). Please confirm that we can keep in touch with you by email, We'll take good care of your personal info and you can update the way we contact you at any time - check out our privacy policy at blf.org.uk\/our-privacy-policy to find out more. Download our how your lungs work PDF (340KB), What is pneumonia, symptoms and diagnosis, Acute respiratory distress syndrome (ARDS), Submit a review of our health information, Stories about living with a lung condition, Positions for obstructive lung conditions, Positions for restrictive lung conditions. 1. We\u2019d love to keep in touch to tell you about our work, our fundraising activities and other ways you can get involved. 46 - Which of the following describes a respiratory... Ch. After absorbing oxygen, the blood leaves the lungs and is carried to the heart. Non-tuberculous mycobacterial infection (NTM), Connective tissue and autoimmune diseases, Pulmonary haemorrhage (bleeding into the lung). It is a serious condition that can lead to heart failure and even death. Normally, deoxygenated blood enters the right side of the heart, travels to the lungs to receive oxygen, and then travels to the left side of the heart to be distributed to the rest of the body. (Patton 840-841) The oxygen pressure gradient between alveolar air and incoming pulmonary blood, the total functional surface area of the respiratory membrane, the respiratory minute volume, and alveolar ventilation are factors that influence oxygen diffusion. ABOUT; FIND THE ANSWERS. 46.4 - Why is carbon monoxide potentially lethal? Predict the formulas of the possible c... A tree grows and increases its mass. ... Because of hemoglobin, blood is able to carry ____ times more oxygen than what can dissolve in the blood\u2026 Log in. Blood without oxygen returns through the veins, to the right side of your heart. Knowing the information about the oxygen level in your blood will encourage people to find any issues in their body rapidly and act accordingly, depending on Oxygen Enters The Blood In The Lungs Class 10. d. the O2 concentration in the blood is low. For the vast majority of these tissues, the oxygen is delivered by the blood to the tissues, although there are some notable exceptions (for example, the cornea gets its oxygen directly from the atmosphere).. Once oxygen has entered the blood from the lungs, it is taken up by haemoglobin (Hb) \u2026 Gas exchange in the lungs We need to get oxygen from the air into the blood, and we need to remove waste carbon dioxide from the blood into the air. Moving gases like this is called gas exchange . Superalloys have been made of nickel and aluminum. Ch. 1 \u00b0F outside. The lungs are exposed to the air so they also play an important protective role in your body, linked to your immune system. The aorta channels oxygen-rich blood to the body from the left ventricle. Your living with a lung condition stories, Information for health care professionals, Stoptober: the 28-day stop smoking challenge, Putting on your Take Steps sponsored walk, Taking the first step: Millets\u2019 guide to walking, Big Breakfast for schools - activity ideas, Top tips for organising a brilliant charity quiz, Incredible support from trusts and foundations, Gwybodaeth yng Nghymraeg \/ Welsh language health information, The Asthma UK and British Lung Foundation Partnership, Why you'll love working with the British Lung Foundation, Thank you for supporting the British Lung Foundation helpline. 1 Questions & Answers Place. The oxygen is moved in the blood by a simple diffusion process. Oxygen enters the blood in the lungs because relative to alveolar air: e. the process is independent of gas concentrations in the blood. Ask your question. Blood oxygen levels can be checked by withdrawing blood from your artery present in the wrist, elbow, or groin. 46.1 - Distinguish between the roles of the respiratory... Ch. Birds and mammals have a four-chambered heart The ventricle is divided by a septum into two compartments completely separating oxygenated from deoxygenated blood and delivering the maximum amount of oxygen to the head and body at the most efficient pressure. Answered What is oxygen enter the blood in the lungs? 46.3 - What is the most important feedback stimulus for... Ch. Neon is an inert gas with three stable isotopes. Blood without oxygen returns through the veins, to the right side of your heart. 2 See answers The device has a screen that will let you see the percent of oxygen in the blood coming from your heart. VAT number 648 8121 18. Suggest reagents and the other fragment that could be used to carry out the indicated conversion. 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The partial pressure of oxygen is lower in the blood than in the alveoli. In the following reactions, decide which reactant is oxidized and which is reduced. b. the CO2 concentration in the blood is low. REFLECT AND APPLY You are purifying a protein for the first time. b. the CO 2 concentration in the blood is low. 46 - Tracheal systems are characterized by: a. closed... Ch. The bronchioles end in tiny air sacs called alveoli, where oxygen is transferred from the inhaled air to the blood. Add Arrows To The Diagram To Show This Oxygen Oxygen Carbon Dioxide Carbon Dioxide 2 A Look At The Diagram Below. d. the O 2 concentration in the blood is low. We use your comments to improve our information. How do you know? Normally, if areas of the lung aren\u2019t gathering much oxygen due to damage from infection, the blood vessels will constrict in those areas. You can change your mind at any time. Due for review: February 2021. 46.3 - What is the role of the chemoreceptors in the... Ch. The amount of air that enters and leaves the lungs with each breath is the ____. Within each air sac, the oxygen concentration is high, so oxygen passes or diffuses across the alveolar membrane into the pulmonary capillary.At the beginning of the pulmonary capillary, the hemoglobin in the red blood cells has carbon dioxide bound to it and very little oxygen (see illustration above). 1. What is the maximu... Human beings accumulate ____before____ excreting nitrogenous waste. I recover if I \u2019 ve had Coronavirus fragment that could be used to carry out the carbon.... Not a violation of the law of conservation of matter the same in all measurement.. A simple diffusion process blood cells then carries the oxygen binds to hemoglobin the... Pulmonary airway malformation ( CPAM ), Coronavirus and living with a condition! Relative to alveolar air: a. closed... Ch the preload decreases, the blood specify its structure three-letter. Answer to your immune system elbow, or... 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\section{Introduction}
Recently the geometrically frustrated electron systems have provided hot
topics in the field of condensed matter physics \cite{Bramwell}. Ferromagnetic
pyrochlore $R_{2}$Mo$_{2}$O$_{7}$ ($R$=Nd, Sm, Gd) is one key type of the
geometrically frustrated systems \cite{Ramirez}, which consists of
corner-sharing tetrahedrons and the antiferromagnetic interactions between
nearest-neighbor spins are frustrated. It was recently pointed out that even
the ferromagnetic interaction is frustrated, if the spin easy axis points to
the center of the tetrahedron \cite{Harris}. In this case, the spin chirality
\cite{Kalmeyer}, which originated from the noncoplanar spin configuration, is
expected to affect the quantum of the electrons, especially the transverse
conductivity. This mechanism is also called \textquotedblleft Berry phase
contribution\textquotedblright\ because a non-vanishing spin chirality is
associated with a non-vanishing spin Berry phase for conduction electrons. To
interpret the transport experiments on ferromagnetic pyrochlore
\cite{Taguchi,Katsufuji}, Ohgushi et al. \cite{Ohgushi} studied the Hall
effect in a two-dimensional (2D) kagom\'{e} lattice, which is the cross
section of the porochlore lattice perpendicular to the (1,1,1) direction
\cite{Ramirez}. They obtained that if the chiral spin state is realized, the
system can show a quantized Hall effect. In their model an important limit is
used, which is that the electron conduction spins are colinear with lattice
(ions) spins. This limit is called that \textquotedblleft the strong (or
infinite) Hund's coupling limit\textquotedblright. However, detailed
experiments on the pyrochlores \cite{Taguchi2} show that the chiral mechanism
alone can not explain the anomalous transport phenomena in these systems. To
explain these experiments, Taillefumier et al. \cite{Taillefumier} studied the
same lattice in a general case, which extrapolates the strong and weak Hund's
coupling regions. They found that the spin Berry phase contribution does not
depend only on the spin chirality, but also on the strength of the local
Hund's coupling.
On the other hand, the orbital magnetism of Bloch electrons has been attracted
renewed interest, due to the recent recognition \cite{Xiao1,Thon1, Xiao2} that
the Berry phase effect plays an important role on orbital magnetism as well as
on the Hall conductivity. The Berry phase effect on orbital magnetism was
until now partially presented by very few studies
\cite{Xiao2,Lee,Thon2,Wang2007, Wang2}. Due to its basic importance in
understanding the magnetism and transport features of the materials,
obviously, more work are needed in exploiting the Berry phase effect on the
properties of the orbital magnetization (OM) in various kinds of realistic
physical systems.
In this paper we extend the study of the OM to the ferromagnetic pyrochlore
systems; more specially, we focus our attention to the 2D kagom\'{e} lattice
with spin anisotropies and Hund's coupling included. It is found that the two
parts in OM (see Sec. II), i.e., the conventional part $M_{c}$ and the
Berry-phase correction part $M_{\Omega}$, oppose each other. In particular,
the OM displays fully different behaviors in metallic and insulating regions
due to the different roles $M_{c}$ and $M_{\Omega}$ play in these two regions.
Moreover, similar to the role the Hund's coupling plays in determining the
anomalous Hall conductivity as observed in the above-mentioned experimental
\cite{Taguchi2} and theoretical \cite{Taillefumier} works, we find that the OM
is also importantly affected by the Hund's coupling. In particular, in the
weak coupling case that the Mott gap bewteen the upper and lower Hurbard bands
is overcome by the electron kinetic energy, we show that the OM exhibits
complex behaviors when scanning the Fermi energy through the whole series of
occupied bands. Furthermore, by using the obtained values of the OM we also
calculate the anomalous Nernst conductivity, which is featured by a
complicated peak-valley pattern as a function of the electron Fermi energy.
\section{Preliminaries}
Before studying the OM of the 2D kagom\'{e} lattice, we simply review the
general multiband formula for finite-temperature OM in the semiclassical
picture of Bloch electrons. In the semiclassical picture \cite{Chang, Sund},
the Bloch electron for the $n$th band is treated as a wave packet
$|w_{n}(\mathbf{r}_{c},\mathbf{k}_{c})\rangle$ with its center ($\mathbf{r}%
_{c},\mathbf{k}_{c}$) in the phase space. The orbital magnetic moment
characterizes the rotation of the wave packet around its centroid and is given
by $\mathbf{m}_{n}(\mathbf{k}_{c})$=$\frac{(-e)}{2}\langle w_{n}%
|(\mathbf{\hat{r}}-\mathbf{r}_{c})\times\mathbf{\hat{v}}|w_{n}\rangle$, where
$(-e)$ is the charge of the electron and $\mathbf{\hat{v}}$ is the velocity
operator. By writing the wave packet in terms of the Bloch state, one obtains
($\mathbf{k}_{c}$ is abbreviated as $\mathbf{k}$)%
\begin{equation}
\mathbf{m}_{n}(\mathbf{k})=-i(e/2\hbar)\langle\nabla_{\mathbf{k}%
}u_{n\mathbf{k}}|\times\lbrack\hat{H}_{\mathbf{k}}-\varepsilon_{n\mathbf{k}%
}^{(0)}]|\nabla_{\mathbf{k}}u_{n\mathbf{k}}\rangle, \label{e1}%
\end{equation}
where $|u_{n\mathbf{k}}\rangle$ is the periodic part of the Bloch state with
band energy $\varepsilon_{n\mathbf{k}}^{(0)}$, and $\hat{H}_{\mathbf{k}}$ is
the crystal Hamiltonian acting on $|u_{n\mathbf{k}}\rangle$. However, it was
further found \cite{Xiao1} that the presence of a weak magnetic field
$\mathbf{B}$ will result in a modification of the density of states in the
semiclassical phase space, $d^{3}\mathbf{k}\rightarrow d^{3}\mathbf{k}%
(1+e\mathbf{B}{\small \cdot}\mathbf{\Omega}_{n}/\hbar)$, where $\mathbf{\Omega
}_{n}(\mathbf{k})=i\langle\nabla_{\mathbf{k}}u_{n\mathbf{k}}|\times
|\nabla_{\mathbf{k}}u_{n\mathbf{k}}\rangle$ is the Berry curvature in
$k$-space. Due to this weak-field modification and the additional
thermodynamic average over Bloch bands included at finite temperature, the
total free energy for an equilibrium ensemble of electrons in the weak field
may be written as \cite{Xiao1}
\begin{equation}
F=-\frac{1}{\beta}\sum_{n}\int d^{3}\mathbf{k}\left( 1+\frac{e}{\hbar
}\mathbf{B}\cdot\mathbf{\Omega}_{n}(\mathbf{k})\right) \ln[1+e^{\beta
(\mu-\varepsilon_{n\mathbf{k}})}]. \label{e2}%
\end{equation}
where $\mu$ is the electron chemical potential, $\beta=1/k_{B}T$ and
$\varepsilon_{n\mathbf{k}}$=$\varepsilon_{n\mathbf{k}}^{(0)}-\mathbf{m}%
_{n}(\mathbf{k})\cdot\mathbf{B}$ is the electron band energy in the presence
of the external magnetic field. The equilibrium OM density is given by the
field derivative at fixed temperature and chemical potential, $\mathcal{\vec
{M}}=-\left( \partial F/\partial\mathbf{B}\right) _{\mu,T}$, with the result%
\begin{align}
\mathcal{\vec{M}} & =\sum_{n}\int d^{3}\mathbf{km}_{n}(\mathbf{k}%
)f_{n}\nonumber\\
& +\frac{1}{\beta}\sum_{n}\int d^{3}\mathbf{k}\frac{e}{\hbar}\mathbf{\Omega
}_{n}(\mathbf{k})\ln\left[ 1+e^{\beta(\mu-\varepsilon_{n\mathbf{k}})}\right]
\nonumber\\
& \equiv\mathbf{M}_{c}+\mathbf{M}_{\mathbf{\Omega}}, \label{e3}%
\end{align}
where $f_{n}$ is the local equilibrium Fermi function for $n$th band. In
addition to the conventional term $\mathbf{M}_{c}$ in terms of the orbital
magnetic moment $\mathbf{m}_{n}(\mathbf{k})$, the extra term $\mathbf{M}%
_{\mathbf{\Omega}}$ in Eq. (\ref{e3}) is a Berry phase effect and exposes a
new topological ingredient to the orbital magnetism. Interestingly, it is this
Berry phase correction that eventually enters the thermal transport current
\cite{Xiao2}. At zero temperature and magnetic field the general expression
(\ref{e3}) is reduced to \cite{Xiao1}%
\begin{equation}
\mathcal{\vec{M}}=\sum_{n}\int^{\mu_{0}}d^{3}\mathbf{k}\left( \mathbf{m}%
_{n}(\mathbf{k})+\frac{e}{\hbar}\mathbf{\Omega}_{n}(\mathbf{k})\left[ \mu
_{0}-\varepsilon_{n\mathbf{k}}\right] \right) , \label{e4}%
\end{equation}
where the upper limit means that the integral is over states with energies
below the zero-temperature chemical potential (Fermi energy) $\mu_{0}$.
\section{Theoretical model and Chern number}
Following previous works \cite{Ohgushi,Taillefumier,Wang2007}, we consider the
double-exchange ferromagnet kagom\'{e} lattice schematically shown in Fig.
1(a). The triangle is the one face of the tetrahedron. Here we consider a pure
spin model with anisotropic Dzyaloshinskii-Moriya interactions on a kagom\'{e}
lattice. It consists of an umbrella of three spins per unit cell of the
kagom\'{e} lattice. Each umbrella can be described by the spherical
coordinates of the three spins ($\pi/6,\theta$), ($5\pi/6,\theta$), and
($-\pi/2,\theta$), as shown in Fig. 1(b). The angle $\theta$ ranges from $0$
to $\pi$.
The tight-binding model of this 2D kagom\'{e} lattice can be written as the
following \cite{Taillefumier}%
\begin{equation}
H=\sum_{\langle i,j\rangle,\sigma}t_{ij}\left( c_{i\sigma}^{\dag}c_{j\sigma
}+\text{H.c.}\right) -J_{0}\sum_{i,\alpha,\beta}c_{i\alpha}^{\dag}\left(
\mathbf{\sigma}_{\alpha\beta}\cdot\mathbf{n}_{i}\right) c_{i\beta},
\label{Hamiltonian}%
\end{equation}
where $t_{ij}$ is the hopping integral between two neighboring sites $i$ and
$j$; $c_{i\sigma}^{\dag}$ and $c_{i\sigma}$ are the creation and annihilation
operators of an electron with spin $\sigma$ on the site $i$. $J_{0}$ is the
effective coupling constant to each local moment $\mathbf{S}_{i}$, and these
moments are treated below as classical variables. $\mathbf{n}_{i}$ is a unit
vector collinear with the local moment $\mathbf{S}_{i}$. $\mathbf{\sigma}$ are
the Pauli matrices. In the following we change notation $i\rightarrow(lms)$,
where $\left( lm\right) $ labels the kagom\'{e} unit cell and $s$ is the
site index in one unit cell. Note that in the infinite Hund's coupling limit,
i.e., $J_{0}\rightarrow\infty$, this system has already been discussed in
Refs. \cite{Ohgushi, Wang2007,Edge}. In this limit the two $\sigma$=$\uparrow
$, $\downarrow$ bands are infinitely split and the model describes a fully
polarized electron subject to a modulation of a fictitious magnetic flux.
To diagonalize the Hamiltonian (\ref{Hamiltonian}), we need to rewrite it in
the reciprocal space. We use the momentum representation of the electron
operator
\begin{equation}
c_{(lms\sigma)}=\frac{1}{\sqrt{L_{x}L_{y}}}\sum_{\mathbf{k}}e^{i\mathbf{k}%
\cdot\mathbf{R}_{(lms)}}\gamma_{s\sigma}(\mathbf{k}) \label{momentum}%
\end{equation}
and the one-particle state $|\Psi(\mathbf{k})\rangle$=$\sum_{s\sigma}$
$\Psi_{s\sigma}(\mathbf{k})\gamma_{s\sigma}^{\dag}(\mathbf{k})|0\rangle$.
Inserting $|\Psi(\mathbf{k})\rangle$ into the Schr\"{o}dinger equation
$H|\Psi\rangle$=$E|\Psi\rangle$, we can easily obtain the Hamiltonian in the
reciprocal space $H(\mathbf{k})$, which is given by\begin{widetext}
\begin{equation}
H(\mathbf{k})=\left(
\begin{array}
[c]{cccccc}%
-J_{0}\cos\theta & p_{\mathbf{k}}^{1} & p_{\mathbf{k}}^{3} & iJ_{0}\sin\theta
& 0 & 0\\
p_{\mathbf{k}}^{1} & -J_{0}\cos\theta & p_{\mathbf{k}}^{2} & 0 & -J_{0}%
\sin\theta e^{-i\frac{\pi}{6}} & 0\\
p_{\mathbf{k}}^{3} & p_{\mathbf{k}}^{2} & -J_{0}\cos\theta & 0 & 0 &
-J_{0}\sin\theta e^{-i\frac{5\pi}{6}}\\
-iJ_{0}\sin\theta & 0 & 0 & J_{0}\cos\theta & p_{\mathbf{k}}^{1} &
p_{\mathbf{k}}^{3}\\
0 & -J_{0}\sin\theta e^{i\frac{\pi}{6}} & 0 & p_{\mathbf{k}}^{1} & J_{0}%
\cos\theta & p_{\mathbf{k}}^{2}\\
0 & 0 & -J_{0}\sin\theta e^{i\frac{5\pi}{6}} & p_{\mathbf{k}}^{3} &
p_{\mathbf{k}}^{2} & J_{0}\cos\theta
\end{array}
\right) ,\label{Hk}
\end{equation}
\end{widetext}where $t$=$|t_{ij}|$ as the energy unit and $p_{\mathbf{k}}^{i}%
$=$2t\cos\left( \mathbf{k\cdot a}_{i}\right) $. $\mathbf{a}_{1}$=$(-\frac
{1}{2},-\frac{\sqrt{3}}{2})$, $\mathbf{a}_{2}$=$(1,0)$, and $\mathbf{a}_{3}%
$=$(-\frac{1}{2},\frac{\sqrt{3}}{2})$ represent the displacements in a unit
cell from A to B site, from B to C site, and from C to A site respectively. In
this notation, the Brillouin zone (BZ) is a hexagon with the corners of
$\mathbf{k}=\pm(2\pi/3)\mathbf{a}_{1}$, $\pm(2\pi/3)\mathbf{a}_{2}$, $\pm
(2\pi/3)\mathbf{a}_{3}$, two of which are independent.
\begin{figure}[ptb]
\begin{center}
\includegraphics[width=1.0\linewidth]{figure1.eps}
\end{center}
\caption{(Color online) (a) Two dimensional spin-chiral ferromagnetic
kagom\'{e} lattice. The dashed line represents the Wigner-Seitz unit cell,
which contains three independent sites (A, B, C). (b) The umbrella structure
on the triangular cell of the 2D kagom\'{e} lattice. }%
\end{figure}\begin{figure}[ptbptb]
\begin{center}
\includegraphics[width=1.0\linewidth]{figure2.eps}
\end{center}
\caption{(a) Energy spectrum of the 2D kagom\'{e} lattice with the parameters
as $\theta$=$\pi/3$, $J_{0}$=$J_{c}$=$4/\sqrt{7}$. (b) The critical value of
Hund's coupling $J_{c}$ as a function of the chiral parameter $\theta$.}%
\end{figure}
Now let us consider the Chern number \cite{Thouless} and Hall conductivity of
this system, which have been reported in Refs. \cite{Taillefumier, Edge2}. In
the strong Hund's coupling limit, there is an energy gap between the two
nearest-neighbor bands in the cases of $\theta\neq0$, $\pi$. Then the Hall
conductivity is a sum over occupied Bloch bands,
\begin{equation}
\sigma_{xy}=\frac{e^{2}}{h}\sum_{n}^{\text{occu}.}C_{n}, \label{1}%
\end{equation}
where the $n$th band Chern number is defined by%
\begin{equation}
C_{n}=-\frac{1}{2\pi}\int_{\text{BZ}}d^{2}\mathbf{k}\Omega_{n}\left(
\mathbf{k}\right) =-\frac{1}{2\pi}\int_{\text{BZ}}d^{2}\mathbf{k}\hat{z}%
\cdot\left[ \nabla_{\mathbf{k}}\times\mathbf{A}_{n}\left( \mathbf{k}\right)
\right] , \label{Chern}%
\end{equation}
where $\mathbf{A}_{n}\left( \mathbf{k}\right) $=$i\langle u_{n\mathbf{k}%
}|\nabla_{\mathbf{k}}u_{n\mathbf{k}}\rangle$ is the Berry-phase connection
(vector potential) for the $n$th band. At finite temperature, considering the
electron density distribution, the Hall conductivity is written as
\begin{equation}
\sigma_{xy}=-\frac{e^{2}}{h}\int_{\text{BZ}}\frac{d^{2}\mathbf{k}}{2\pi}%
f_{n}\hat{z}\cdot\left[ \nabla_{\mathbf{k}}\times\mathbf{A}_{n}\left(
\mathbf{k}\right) \right] . \label{Hallconductivity}%
\end{equation}
However in the general spin coupling cases, the gap between two
nearest-neighbor bands may disappear. When the Fermi energy lies in these two
bands, becasue the gap vanishes, the Hall conductivity can not be written in
the form of Eq. (\ref{1}). In despite of this, the concept of the $n$th band
Chern number and Eq. (\ref{1}) are also useful when the gap between two
nearest-neighbor bands does not disappear.
For the general cases the energy spectrum can only be computed numerically,
except for general $\theta$ at high-symmetry points. For the finite values of
$J_{0}$, as pointed in \cite{Taillefumier}, the splitting of the spectrum
depends on two mechanisms. One is that the coupling $J_{0}$ separates each
group of three bands, the other is that when switching on $J_{0}$, the
pointlike degeneracies are lifted within each group of three bands. According
to the properties of the $M$ point of the Brillouin zone, a critical value of
the Hund's coupling as a function of the chiral parameter $\theta$ can be
analytically obtained, which is given by \cite{Taillefumier},%
\begin{equation}
J_{c}(\theta)=\pm\frac{2t}{\sqrt{1+3\cos^{2}\theta}}. \label{critical}%
\end{equation}
Using Eq. (\ref{critical}) one can distinguish between two different regimes
depending on the value of $J_{0}$ as compared to $J_{c}(\theta)$. In the
regime where $J_{0}>J_{c}$, the Chern numbers associated with each band are
given by $-1$, $0$, $1$, $1$, $0$, $-1$ from the lowest to the topmost band.
Whereas in the regime where $J_{0}<J_{c}$, the Chern numbers associated with
each band are given by $-1$, $3$, $-2$, $-2$, $3$, $-1$ in the same order. We
draw in Fig. 2(a) the energy spectrum with the spin chiral parameter $\theta
$=$\pi/3$ and $J_{0}$ taking the critical value $J_{c}$=$4/\sqrt{7}$. Fig.
2(b) shows the critical value of the Hund's coupling $J_{c}$ as a function of
the chirality $\theta$.
\section{The orbital magnetization}
Now we turn to study the OM of the 2D kagom\'{e} lattice in the
general Hund's coupling cases. Similar to the Hall conductivity, the
OM displays different behaviors in two regions which we will exhibit
in turn. As examples, we consider two cases. The case I, in which we
set the parameters as $\theta$=$\pi/3$ and $J_{0}$=$2$, is one
typical case in the regime where
$J_{0}>J_{c}$(=$4/\sqrt{7}\approx1.51$). Whereas the case II, in
which the parameters are $\theta$=$\pi/3$ and $J_{0}$=$1$, is
another typical case in the regime where $J_{0}<J_{c}$. These two
cases are shown in Fig. 2(b) with solid dots.
First we consider the case I. To more clearly investigate the OM of the 2D
kagom\'{e} lattice, we need to know the energy band structure of the system.
So, we draw the energy spectrum in Fig. 3(a), from which one can find that
there are four gaps. From the lowest to the topmost gap, we denote these gaps
as gap-I, -II, -III, and -IV. Clearly, in this case only the gap between bands
$5$ and $6$ vanishes. Fig. 3(b) plots the Hall conductivity as a function of
the electron Fermi energy (chemical potential). \begin{figure}[ptb]
\begin{center}
\includegraphics[width=1.0\linewidth]{fig3.eps}
\end{center}
\caption{(a) The energy spectrum of the 2D kagom\'{e} lattice. (b) The Hall
conductivity $\sigma_{xy}$ as a function of the chemical potential $\mu$. In
both figures, the chiral parameter is $\theta$=$\pi/3$ and the strength of the
Hund's coupling $J_{0}$=$2$. The shaded areas are the energy gaps, which
labeled as gap-I, -II, -III, and -IV from the lowest to the topmost gap.}%
\end{figure}\begin{figure}[ptbptb]
\begin{center}
\includegraphics[width=0.8\linewidth]{fig4.eps}
\end{center}
\caption{(Color online) (a) The OM and (b) its two components $M_{c}$ (dashed
line) and $M_{\Omega}$ (dotted line) as a function of the chemical potential
$\mu$. The parameters are same as those in Fig. 3.}%
\end{figure}
Figure 4(a) shows the OM ($\mathcal{M}$)\ as a function of the electron
chemical potential $\mu$. One can see that initially the OM rapidly decreases
as the filling of the lowest band increases, arriving at a minimum at $\mu
$=$-3.54$, a value corresponding to the top of the lowest band. Then, as the
chemical potential continues to vary in the gap-I, the OM goes up and
increases as a linear function of $\mu$. This linear relationship in the
insulating region can be understood by Eq. (\ref{e3}), from which one obtains
\begin{align}
\frac{d\mathcal{M}}{d\mu} & =\frac{e}{\hbar}\sum_{n}^{\text{occu}}\int
d^{2}\mathbf{k}\Omega_{n}(\mathbf{k})\label{om-chem}\\
& =-\frac{e}{h}\sum_{n}^{\text{occu}}C_{n}.\nonumber
\end{align}
Thus when the chemical potential varies in the gap-I, only the lowest band is
occupied and $d\mathcal{M}/d\mu=-(e/h)C_{1}$. In this case, $C_{1}=-1$. Thus
$d\mathcal{M}/d\mu=e/h$, i.e., the OM linearly increases with the chemical
potential in the insulating region I, as shown in Fig. 4(a). Similarly, when
the chemical potential increases in the gap-II, the OM increases linearly with
$\mu$. Since the Chern number of band $2$ is zero, thus from Eq.
(\ref{om-chem}) and Fig. 4(a) one can see that the slope of the OM curve in
the gap-II is same as that in the gap-I. When the chemical potential increases
in the gap-III, the OM becomes zero and does not varies with the chemical
potential. The reason is that the sum over the Chern numbers of the occupied
lowest three bands is zero. From Eq. (\ref{om-chem}), one can see that the
slope of the OM is independent of $\mu$. In fact the gap-III is the usually
called the Mott gap. With the chemical potential increases, when it lies in
the gap-IV, one can find that the OM decreases linearly with $\mu$. The reason
is that the sum of the Chern numbers of the lowest four bands is $1$. From Eq.
(\ref{om-chem}), one can find $d\mathcal{M}/d\mu=-e/h$, i.e., the OM linearly
decreases with the chemical potential in the gap-IV as shown in Fig. 3(b).
For further study, we show in Fig. 4(b) $M_{c}$ and $M_{\Omega}$ as a function
of the chemical potential, their sum gives $\mathcal{M}$ in Fig. 4(a). One can
see that overall $M_{c}$ and $M_{\Omega}$ have opposite contributions to
$\mathcal{M}$, which implies that these two parts carry opposite-circulating
currents. In each insulating area the conventional term $M_{c}$ keeps a
constant, which is due to the fact that the upper limit of the $k$-integral of
$m_{n}(\mathbf{k})$ is invariant as the chemical potential varies in the gap.
In the metallic region, however, since the occupied states varies with the
chemical potential, thus $M_{c}$ also varies with $\mu$, resulting in a
decreasing slope in the lowest three metallic regions and a increasing slope
in the highest two metallic regions, as shown in Fig. 4(b). The Berry phase
term $M_{\Omega}$ also displays different behavior between insulating and
metallic regions. In the lowest two insulating region, $M_{\Omega}$ linearly
increases with $\mu$, and in the gap-IV, $M_{\Omega}$ linearly decreases with
$\mu$, as is expected from Eq. (\ref{e3}). In the metallic region, however,
this term sensitively depends on the topological property of the band in which
the chemical potential is located. For the bands with nonzero Chern number,
one can see from Fig. 4(b) that $M_{\Omega}$ remains invariant, while for the
band $2$ (its Chern number is zero), it increases with the chemical potential
$\mu$. On the whole the comparison between Figs. 4(a) and 4(b) shows that the
metallic behavior of $\mathcal{M}$ is dominated by its conventional term
$M_{c}$, while in the insulating regime $M_{\Omega}$ plays a main role in
determining the behavior of $\mathcal{M}$. The behavior of the OM in this case
is similar to that in the strong spin coupling limit.
\begin{figure}[ptb]
\begin{center}
\includegraphics[width=1.0\linewidth]{fig5.eps}
\end{center}
\caption{(a) The energy spectrum of the 2D kagom\'{e} lattice. (b) The Hall
conductivity $\sigma_{xy}$ as a function of the chemical potential $\mu$. In
both figures, the chiral parameter is $\theta$=$\pi/3$ and the strength of the
Hund's coupling $J_{0}$=$1$. The shaded areas are the energy gaps, which
labeled as the lower band and the higher band, respectively.}%
\end{figure}
Then we study the case II. Similar to the case I, we firstly draw the energy
spectrum in Fig. 5(a), from which one can find that there are only two gaps,
which can be called the lower and the higher gap, respectively. In this case
the gaps between bands $2$ and $3$, between $3$ and $4$ (the Mott gap), and
between $5$ and $6$ vanish. We also draw in Fig. 5(b) the corresponding Hall
conductivity as a function of the electron Fermi energy. \begin{figure}[ptb]
\begin{center}
\includegraphics[width=0.8\linewidth]{fig6.eps}
\end{center}
\caption{(Color online) (a) The OM and (b) its two components $M_{c}$ (dashed
line) and $M_{\Omega}$ (dotted line) as a function of the chemical potential
$\mu$. The parameters are same as those in Fig. 5.}%
\end{figure}
Figure 6(a) plots the OM ($\mathcal{M}$)\ as a function of the electron
chemical potential $\mu$. One can see that initially the OM rapidly decreases
as the filling of the lower band increases, arriving at a minimum at $\mu
$=$-2.68$, a value corresponding to the top of the lower band. Then, as the
chemical potential continues to vary in the lower gap, the OM goes up and
increases as a linear function of $\mu$. This linear relationship in the
insulating region can also be understood by Eq. (\ref{om-chem}). Because the
gap between $2$ and $3$ and the Mott gap vanish, the OM displays a more
complex behavior in this metallic region. The value of the OM oscillate versus
the chemical potential $\mu$. When $\mu$ reaches the bottom of the higher gap,
the OM begins to go up again and linearly increases. Different from in the
lower gap, the coefficient of the increasing is twice of that in the lower
gap, which can be understood by Eq. (\ref{om-chem}). In the higher gap
$d\mathcal{M}/d\mu$=$-(e/h)\sum_{n=1}^{4}C_{n}$=$2e/h$, while in the lower
gap, $d\mathcal{M}/d\mu$=$-(e/h)C_{1}$=$e/h$.
To see the different roles $M_{c}$ and $M_{\Omega}$ play in the
metallic and insulating regions, we draw in Fig. 6(b) $M_{c}$ and
$M_{\Omega}$ as a function of the chemical potential, their sum
gives $\mathcal{M}$ in Fig. 6(a). In each insulating area the
conventional term $M_{c}$ keeps a constant and $M_{\Omega}$ linearly
increases with $\mu$ in the two gaps with different linear
coefficients, as is expected from Eq. (\ref{e3}). The behaviors of
OM in this case is novel, comparing with those in the infinite
Hund's coupling case. There are two features in the regime
$J_{0}<J_{c}$. One is that the magnitudes of both two parts $M_{c}$
and $M_{\Omega}$ become much smaller than those in the regime
$J_{0}>J_{c}$. The other is that in the metallic region, both two
parts rapidly changes. The reason is that the gaps between bands 2,
3 and 4 vanish and these bands form one energy band. However from
Fig. 6(b) one can see that the two parts of $\mathcal{M\ }$have
opposite contributions to $\mathcal{M}$, which implies that these
two parts carry opposite-circulating currents.
\section{Anomalous Nernst Effect}
The above discussion of $M_{c}$ and $M_{\Omega}$ can be transferred to study
the ANE. The relation between the OM and ANE has been recently found
\cite{Xiao2}. To discuss the transport measurement, it is important to
discount the contribution from the magnetization current, a point which has
attracted much discussion in the past. Motivated by the argument \cite{Cooper}
that the magnetization current cannot be measured by conventional transport
experiments, Xiao et al. \cite{Xiao2} have built up a remarkable picture that
the conventional orbital magnetic moment $M_{c}$ does not contribute to the
transport current, while the Berry phase term in Eq. (\ref{e2}) directly
enters and therefore modifies the intrinsic transport Hall current equation as
follows%
\begin{equation}
\mathbf{j}_{\text{H}}\mathbf{=}-\frac{e^{2}}{\hbar}\mathbf{E}\times\sum
_{n}\int\frac{d^{2}k}{(2\pi)^{2}}f_{n}(\mathbf{r},\mathbf{k})\Omega
_{n}(\mathbf{k})\mathbf{-}\nabla\times\mathbf{M}_{\Omega}(\mathbf{r}),
\label{jh}%
\end{equation}
In the case of uniform temperature and chemical potential, obviously, the
second term is zero and the Hall effect of 2D kagom\'{e} lattice is featured
by nonzero Chern number as discussed by Taillefumier et al.
\cite{Taillefumier} as well as in this paper. In the following, however, we
turn to study another situation, where the current-driving force is not
provided by the electric field ($\mathbf{E}$=0). Instead, it is provided by a
statistical force, i.e., the gradient of temperature $T$. In this case, Eqs.
(\ref{jh}) and (\ref{e2}) give the expression of intrinsic thermoelectric Hall
current as $j_{x}=\alpha_{xy}(-\nabla_{y}T)$, where the anomalous Nernst
conductivity $\alpha_{xy}$ is given by \cite{Xiao2}%
\begin{align}
\alpha_{xy} & =\frac{1}{T}\frac{e}{\hbar}\sum_{n}\int\frac{d^{2}k}%
{(2\pi)^{2}}\Omega_{n}\label{ane}\\
& \times\left[ \left( \epsilon_{n\mathbf{k}}-\mu\right) f_{n}+k_{B}%
T\ln\left( 1+e^{-\beta(\epsilon_{n\mathbf{k}}-\mu)}\right) \right]
.\nonumber
\end{align}
\begin{figure}[ptb]
\begin{center}
\includegraphics[width=0.6\linewidth]{fig7.eps}
\end{center}
\caption{The ANE of the 2D kagom\'{e} lattice at $k_{B}T$=$0.005$. The
parameters are (a) $\theta$=$\pi/3$ and $J_{0}$=$2$; (b) $\theta$=$\pi/3$ and
$J_{0}$=$1$. The shaded areas are the energy gaps.}%
\end{figure}
Figure 7 shows $\alpha_{xy}$ of the 2D kagom\'{e} lattice as a function of the
chemical potential for $k_{B}T$=$0.005$. One can see that the ANE disappears
in the insulating regions, and when scanning the chemical potential through
the bands, there will appear peaks and valleys. Remarkably, a similar
peak-valley structure was also found by the recent first-principles
calculations in CuCr$_{2}$Se$_{4-x}$Br$_{x}$ compound \cite{Xiao2}. The ANE of
this compound was recently measured by Lee et al. \cite{Lee} as a function of
Br doping $x$ which is used to change the chemical potential $\mu$. Due to the
scarce data available, until now the peak-valley structure of $\alpha_{xy}$
revealed in Fig. 7 and in Ref. \cite{Xiao2,Wang2007, Wang2} has not been found
in experiment, and more direct experimental results are needed for
quantitative comparison with the theoretical results. Interestingly, the
expression for $\alpha_{xy}$ can be simplified at low temperature as the Mott
relation \cite{Xiao2},
\begin{equation}
\alpha_{xy}=-\frac{\pi^{2}}{3}\frac{k_{B}^{2}T}{e}\frac{\partial\sigma
_{xy}(\mu_{0})}{\partial\mu_{0}}. \label{Mott}%
\end{equation}
Thus one can see that unlike the anomalous Hall effect \cite{Haldane}, ANE is
given by the Fermi-surface contribution of the band structure and Berry
curvature. Another unique feature of $\alpha_{xy}$ is its linear dependence of temperature.
\section{Summary}
We have theoretically studied the OM and ANE of the 2D kagom\'{e} lattice with
spin anisotropies included in a general Hund's coupling region, as a
supplement of the previous work \cite{Wang2007} in the infinite Hund's
coupling limit. The results show that both of the strength of the Hund's
coupling and the chirality contribute to the orbital magnetization
$\mathcal{M}$. Upon varying both these parameters, it is found that the two
parts of $\mathcal{M}$, i.e., the conventional part $\mathbf{M}_{c}$ and the
Berry-phase correction part $\mathbf{M}_{\Omega}$, oppose each other. We also
calculate the anomalous Nernst conductivity and obtain a peak-valley structure
as a function of the electron Fermi energy.
\begin{acknowledgments}
This work was supported by NSFC under Grants Nos. 10604010 and 10544004.
\end{acknowledgments}
| {
"redpajama_set_name": "RedPajamaArXiv"
} | 9,398 |
Q: How to use erlang fun recursive function? I am trying to decode frames with variable length and options (such as the TLV in ethernet frames)
In order to do that, I was thinking about doing a fun recursive function :
fun (Fields, Bin) ->
Parse =
fun (P, F, <<Length, Rest/binary>>) ->
P(P, F#{first => Length}, Rest)
end,
Parse(Parse, Fields, Bin)
end.
So Bin is the input frame for example : 40 02 12 45 01 50
So first byte is the type of frame, 02 is the length of following data 12 45, 01 is the length is the following data 50 and so on.
But my function doesn't work as expected using the funny trick
I am returning JSON object because it's send over MQTT.
A: What you're doing there really is only reading out the length, and don't do much with the Rest. You'd need to first declare an exit condition, i.e. when the TLV is empty -> just return the accummulator; And use the pattern match to read out the values based on the Length:
parse(<<>>, Acc) -> Acc; %% finished with the list
parse(<<Length, Rest/binary>>, Acc) ->
<<Value:Length/binary, Carry/binary>> = Rest.
%% Value for the tag, Carry to be passed back on the recursion.
%% Assuming that `Acc` is a list of Values.
parse(Carry, Acc ++ [Value]).
You can use the above to read out the values, and can do something similar for getting the type first as:
tlv(<<Type, Values/binary>>) ->
%% Return at tuple with the Type and the values.
{Type, parse(Values, [])}.
A: You could write the function like this:
fun(<<Type, Packet/binary>>) ->
{Type,
fun Parse(<<>>) ->
[];
Parse(<<Length, Data:Length/bytes, Rest/binary>>) ->
[Data] ++ Parse(Rest)
end(Packet)}
end.
This returns {40,[<<12,45>>,<<50>>]} for your sample data.
The outer fun takes the frame type (40 in this case), and returns it along with the list of the data fields. The inner fun takes one length byte and the corresponding number of data bytes, and returns the data and makes a recursive call to itself - until it reaches the end of the binary.
The inner fun is a "named fun": it calls itself Parse, and is therefore able to call itself without needing to pass itself as an argument. The name Parse is not visible outside the fun. See this question for details and examples.
| {
"redpajama_set_name": "RedPajamaStackExchange"
} | 9,398 |
Q: Unable install requirements.txt After cloning a Django application, I went to install the virtual environment in terminal.
I entered the folder and typed: pip install -r requirements.txt
but got the error: -bash: pip: command not found
A: It seems that you are missing the pip itself which you can install from packaging repository depending on your OS or download (manually or with wget) the and install it using python. This will automatically associate your python version with required pip version. See following commands:
wget https://bootstrap.pypa.io/get-pip.py
python get-pip.py
| {
"redpajama_set_name": "RedPajamaStackExchange"
} | 3,023 |
{"url":"http:\/\/nrich.maths.org\/public\/leg.php?code=5039&cl=2&cldcmpid=6402","text":"Search by Topic\n\nResources tagged with Interactivities similar to The Remainders Game:\n\nFilter by: Content type:\nStage:\nChallenge level:\n\nThere are 220 results\n\nBroad Topics > Information and Communications Technology > Interactivities\n\nColour Wheels\n\nStage: 2 Challenge Level:\n\nImagine a wheel with different markings painted on it at regular intervals. Can you predict the colour of the 18th mark? The 100th mark?\n\nStars\n\nStage: 3 Challenge Level:\n\nCan you find a relationship between the number of dots on the circle and the number of steps that will ensure that all points are hit?\n\nMultiples Grid\n\nStage: 2 Challenge Level:\n\nWhat do the numbers shaded in blue on this hundred square have in common? What do you notice about the pink numbers? How about the shaded numbers in the other squares?\n\nSee the Light\n\nStage: 2 and 3 Challenge Level:\n\nWork out how to light up the single light. What's the rule?\n\nCogs\n\nStage: 3 Challenge Level:\n\nA and B are two interlocking cogwheels having p teeth and q teeth respectively. One tooth on B is painted red. Find the values of p and q for which the red tooth on B contacts every gap on the. . . .\n\nStage: 2 Challenge Level:\n\nIf you have only four weights, where could you place them in order to balance this equaliser?\n\nA Dotty Problem\n\nStage: 2 Challenge Level:\n\nStarting with the number 180, take away 9 again and again, joining up the dots as you go. Watch out - don't join all the dots!\n\nGot It\n\nStage: 2 and 3 Challenge Level:\n\nA game for two people, or play online. Given a target number, say 23, and a range of numbers to choose from, say 1-4, players take it in turns to add to the running total to hit their target.\n\nBeat the Drum Beat!\n\nStage: 2 Challenge Level:\n\nUse the interactivity to create some steady rhythms. How could you create a rhythm which sounds the same forwards as it does backwards?\n\nMore Magic Potting Sheds\n\nStage: 3 Challenge Level:\n\nThe number of plants in Mr McGregor's magic potting shed increases overnight. He'd like to put the same number of plants in each of his gardens, planting one garden each day. How can he do it?\n\nCountdown\n\nStage: 2 and 3 Challenge Level:\n\nHere is a chance to play a version of the classic Countdown Game.\n\nMagic Potting Sheds\n\nStage: 3 Challenge Level:\n\nMr McGregor has a magic potting shed. Overnight, the number of plants in it doubles. He'd like to put the same number of plants in each of three gardens, planting one garden each day. Can he do it?\n\nGot it Article\n\nStage: 2 and 3\n\nThis article gives you a few ideas for understanding the Got It! game and how you might find a winning strategy.\n\nFactor Lines\n\nStage: 2 Challenge Level:\n\nArrange the four number cards on the grid, according to the rules, to make a diagonal, vertical or horizontal line.\n\nTimes Tables Shifts\n\nStage: 2 Challenge Level:\n\nIn this activity, the computer chooses a times table and shifts it. Can you work out the table and the shift each time?\n\nMultiplication Square Jigsaw\n\nStage: 2 Challenge Level:\n\nCan you complete this jigsaw of the multiplication square?\n\nFactors and Multiples - Secondary Resources\n\nStage: 3 and 4 Challenge Level:\n\nA collection of resources to support work on Factors and Multiples at Secondary level.\n\nCounters\n\nStage: 2 Challenge Level:\n\nHover your mouse over the counters to see which ones will be removed. Click to remover them. The winner is the last one to remove a counter. How you can make sure you win?\n\nMore Transformations on a Pegboard\n\nStage: 2 Challenge Level:\n\nUse the interactivity to find all the different right-angled triangles you can make by just moving one corner of the starting triangle.\n\nRatio Pairs 2\n\nStage: 2 Challenge Level:\n\nA card pairing game involving knowledge of simple ratio.\n\nSemi-regular Tessellations\n\nStage: 3 Challenge Level:\n\nSemi-regular tessellations combine two or more different regular polygons to fill the plane. Can you find all the semi-regular tessellations?\n\nCuisenaire Environment\n\nStage: 1 and 2 Challenge Level:\n\nAn environment which simulates working with Cuisenaire rods.\n\nSquare It\n\nStage: 1, 2, 3 and 4 Challenge Level:\n\nPlayers take it in turns to choose a dot on the grid. The winner is the first to have four dots that can be joined to form a square.\n\nOne Million to Seven\n\nStage: 2 Challenge Level:\n\nStart by putting one million (1 000 000) into the display of your calculator. Can you reduce this to 7 using just the 7 key and add, subtract, multiply, divide and equals as many times as you like?\n\nPicturing Triangle Numbers\n\nStage: 3 Challenge Level:\n\nTriangle numbers can be represented by a triangular array of squares. What do you notice about the sum of identical triangle numbers?\n\nMultiplication Tables - Matching Cards\n\nStage: 1, 2 and 3 Challenge Level:\n\nInteractive game. Set your own level of challenge, practise your table skills and beat your previous best score.\n\nSeven Flipped\n\nStage: 2 Challenge Level:\n\nInvestigate the smallest number of moves it takes to turn these mats upside-down if you can only turn exactly three at a time.\n\nPartitioning Revisited\n\nStage: 3 Challenge Level:\n\nWe can show that (x + 1)\u00b2 = x\u00b2 + 2x + 1 by considering the area of an (x + 1) by (x + 1) square. Show in a similar way that (x + 2)\u00b2 = x\u00b2 + 4x + 4\n\nPart the Piles\n\nStage: 2 Challenge Level:\n\nTry to stop your opponent from being able to split the piles of counters into unequal numbers. Can you find a strategy?\n\nNumber Differences\n\nStage: 2 Challenge Level:\n\nPlace the numbers from 1 to 9 in the squares below so that the difference between joined squares is odd. How many different ways can you do this?\n\nPower Crazy\n\nStage: 3 Challenge Level:\n\nWhat can you say about the values of n that make $7^n + 3^n$ a multiple of 10? Are there other pairs of integers between 1 and 10 which have similar properties?\n\n100 Percent\n\nStage: 2 Challenge Level:\n\nAn interactive game for 1 person. You are given a rectangle with 50 squares on it. Roll the dice to get a percentage between 2 and 100. How many squares is this? Keep going until you get 100. . . .\n\nNine Colours\n\nStage: 3 Challenge Level:\n\nYou have 27 small cubes, 3 each of nine colours. Use the small cubes to make a 3 by 3 by 3 cube so that each face of the bigger cube contains one of every colour.\n\nDiagonal Dodge\n\nStage: 2 and 3 Challenge Level:\n\nA game for 2 players. Can be played online. One player has 1 red counter, the other has 4 blue. The red counter needs to reach the other side, and the blue needs to trap the red.\n\nColour in the Square\n\nStage: 2 Challenge Level:\n\nCan you put the 25 coloured tiles into the 5 x 5 square so that no column, no row and no diagonal line have tiles of the same colour in them?\n\nCoordinate Tan\n\nStage: 2 Challenge Level:\n\nWhat are the coordinates of the coloured dots that mark out the tangram? Try changing the position of the origin. What happens to the coordinates now?\n\nNoughts and Crosses\n\nStage: 2 Challenge Level:\n\nA game for 2 people that everybody knows. You can play with a friend or online. If you play correctly you never lose!\n\nCode Breaker\n\nStage: 2 Challenge Level:\n\nThis problem is based on a code using two different prime numbers less than 10. You'll need to multiply them together and shift the alphabet forwards by the result. Can you decipher the code?\n\nBuilding Stars\n\nStage: 2 Challenge Level:\n\nAn interactive activity for one to experiment with a tricky tessellation\n\nCoordinate Cunning\n\nStage: 2 Challenge Level:\n\nA game for 2 people that can be played on line or with pens and paper. Combine your knowledege of coordinates with your skills of strategic thinking.\n\nTeddy Town\n\nStage: 1 and 2 Challenge Level:\n\nThere are nine teddies in Teddy Town - three red, three blue and three yellow. There are also nine houses, three of each colour. Can you put them on the map of Teddy Town according to the rules?\n\nWhen Will You Pay Me? Say the Bells of Old Bailey\n\nStage: 3 Challenge Level:\n\nUse the interactivity to play two of the bells in a pattern. How do you know when it is your turn to ring, and how do you know which bell to ring?\n\nSquare Coordinates\n\nStage: 3 Challenge Level:\n\nA tilted square is a square with no horizontal sides. Can you devise a general instruction for the construction of a square when you are given just one of its sides?\n\nDiamond Mine\n\nStage: 3 Challenge Level:\n\nPractise your diamond mining skills and your x,y coordination in this homage to Pacman.\n\nSliding Puzzle\n\nStage: 1, 2, 3 and 4 Challenge Level:\n\nThe aim of the game is to slide the green square from the top right hand corner to the bottom left hand corner in the least number of moves.\n\nRound Peg Board\n\nStage: 1 and 2 Challenge Level:\n\nA generic circular pegboard resource.\n\nKhun Phaen Escapes to Freedom\n\nStage: 3 Challenge Level:\n\nSlide the pieces to move Khun Phaen past all the guards into the position on the right from which he can escape to freedom.\n\nOnline\n\nStage: 3 Challenge Level:\n\nA game for 2 players that can be played online. Players take it in turns to select a word from the 9 words given. The aim is to select all the occurrences of the same letter.\n\nFifteen\n\nStage: 3 Challenge Level:\n\nCan you spot the similarities between this game and other games you know? The aim is to choose 3 numbers that total 15.\n\nChocolate Bars\n\nStage: 2 Challenge Level:\n\nAn interactive game to be played on your own or with friends. Imagine you are having a party. Each person takes it in turns to stand behind the chair where they will get the most chocolate.","date":"2014-09-18 01:40:06","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\": 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\": 0.24524500966072083, \"perplexity\": 1632.891323149506}, \"config\": {\"markdown_headings\": false, \"markdown_code\": true, \"boilerplate_config\": {\"ratio_threshold\": 0.3, \"absolute_threshold\": 10, \"end_threshold\": 5, \"enable\": true}, \"remove_buttons\": true, \"remove_image_figures\": true, \"remove_link_clusters\": true, \"table_config\": {\"min_rows\": 2, \"min_cols\": 3, \"format\": \"plain\"}, \"remove_chinese\": true, \"remove_edit_buttons\": true, \"extract_latex\": true}, \"warc_path\": \"s3:\/\/commoncrawl\/crawl-data\/CC-MAIN-2014-41\/segments\/1410657125113.78\/warc\/CC-MAIN-20140914011205-00040-ip-10-196-40-205.us-west-1.compute.internal.warc.gz\"}"} | null | null |
STEWKLEY FILM ARCHIVE
Dedicated to keeping local history alive
Updated for 2022: Stewkley Films on YouTube
Contact for archive: jf@stewkleyfilms.org
This time next year the country will be building up to celebrating Queen Elizabeths Platinum Jubilee. There is to be an extra bank holiday making the period Thursday June 2nd to Sunday the 5th a really long holiday weekend. The Archive now looks back nine years when, in June 2012, Stewkley celebrated the Diamond Jubilee in style: events day after day. Celebrations commenced as the children of St Michaels School walked to the Church for a service of celebration, most in royal costumes. June 2nd, the final of the four days, was celebrated as the actual day of the Jubilee. There was a grand parade along the High Street that attracted a big crowd and the Rec became like a giant fairground, enlivened when the President of Stewkey Women's Institute came face to face with our MP John Bercow, the then controversial Speaker of the House of Commons. The 49-minute film produced for the Stewkley Film Archive will be presented on YouTube from 6pm on Thursday. Links to the widescreen production for Smart TVs or computer and tablet screens will appear here and in the Grapevine. In advance, below is some imagery from the celebrations
RECENT SHOWINGS
Buckinghamshire, England: During its five-day run on YouTube (April 23-27), the video Over Our Dead Bodies that tells the story of Stewkleys Great Airport Campaign that concluded successfully 50 years ago was viewed more than 850 times, suggesting it was seen by over 1,500 folk.
Scroll down for copies of recent Facebook postings
The following are recent postings from Stewkley's Facebook portal
Friday April 23
Sunday April 25
Tuesday April 27
Wednesday April 28
Stewkley Films Curator John Flewin also offers other presentations for clubs and organisations.
Details: www.flewin.com - Contact: john@flewin.com | {
"redpajama_set_name": "RedPajamaCommonCrawl"
} | 1,432 |
Gustav Långbacka (born 8 May 1984) is a Finnish football goalkeeper who played in the Veikkausliiga for IFK Mariehamn.
Långbacka's sister is the football referee Lina Lehtovaara.
References
1984 births
Living people
Swedish-speaking Finns
Finnish footballers
Kokkolan Palloveikot players
IFK Mariehamn players
FC Ilves players
Association football goalkeepers
Veikkausliiga players
Ykkönen players
Kakkonen players
GBK Kokkola players
People from Kokkola
Sportspeople from Central Ostrobothnia | {
"redpajama_set_name": "RedPajamaWikipedia"
} | 5,884 |
Ayn Halaqim Subdistrict () is a Syrian nahiyah (subdistrict) located in Masyaf District in Hama. According to the Syria Central Bureau of Statistics (CBS), Ayn Halaqim Subdistrict had a population of 16502 in the 2004 census.
References
Ayn Halaqim
Masyaf District | {
"redpajama_set_name": "RedPajamaWikipedia"
} | 8,693 |
Lifegate Clinical products are designed in accord with traditional Dao Earth™ herbal principles to support lifeforce physiology for cultivating wisdom on the path of life.
CAPSULE: 8:1 powdered herbal extract in a gluten-free vegetable capsule. No yeast or preservatives added.
Features a hydrolyzed extract of fresh water pearl. | {
"redpajama_set_name": "RedPajamaC4"
} | 8,693 |
{"url":"https:\/\/www.vedantu.com\/question-answer\/saili-plants-4-saplings-in-a-row-in-her-garden-class-6-maths-cbse-5ee9f44095f78e759773a5d9","text":"Courses\nCourses for Kids\nFree study material\nFree LIVE classes\nMore\nQuestions & Answers\n\n# Saili plants 4 saplings in a row, in her garden. The distance between two adjacent saplings is $\\dfrac{3}{4}$ m. Find the distance between the first and the last sapling.\n\nLast updated date: 19th Mar 2023\nTotal views: 303.9k\nViews today: 4.82k\nAnswer\nVerified\n303.9k+ views\nHint: We just have to find the distance between the first and the last sapling. So what we have to do is that, we have to find the total number of gaps between 4 saplings and multiply it with the distance between the adjacent saplings.\n\nComplete step-by-step answer:\nBefore proceeding with the question, we must know how to plot the figure showing the saplings that are given in the question. Here, we have been given that Saili plants 4 saplings in a row in her garden. The distance between two adjacent saplings is also given to us as $\\dfrac{3}{4}$m.\nUsing this data, we can plot the arrangement of the saplings in Saili\u2019s garden as shown as below:\n\nTherefore, we can see that there are a total of 3 spaces between the four saplings and the distance between the adjacent saplings is $\\dfrac{3}{4}$ m.\nAs we know the distance between adjacent saplings, we can calculate the distance between the first and the last sapling. This can be done by multiplying the number of spaces, i.e. 3 and the distance between adjacent saplings, i.e. $\\dfrac{3}{4}$.\nTherefore, we get,\n$\\dfrac{3}{4}\\left( 3 \\right)=\\dfrac{9}{4}$\nHence, the distance between the first and the last sapling is $\\dfrac{9}{4}$ m.\n\nNote: This question is a direct question which requires the understanding of the way the saplings are placed. It is necessary to draw the figure and consider the number of spaces. The mistake which could be committed is by not drawing the figure and taking the number of spaces as 4 instead of 3. We have to be careful that if there are 4 saplings then there will be only 3 gaps between the 4 saplings.","date":"2023-03-24 09:49:58","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\": 2, \"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\": 0.8219116926193237, \"perplexity\": 467.32857421530633}, \"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-2023-14\/segments\/1679296945279.63\/warc\/CC-MAIN-20230324082226-20230324112226-00075.warc.gz\"}"} | null | null |
Pseudocoremia leucelaea är en fjärilsart som beskrevs av Edward Meyrick 1909. Pseudocoremia leucelaea ingår i släktet Pseudocoremia och familjen mätare. Inga underarter finns listade i Catalogue of Life.
Källor
Externa länkar
Mätare
leucelaea | {
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Q: Draw perspectively on the faces of a cube I am looking for a solution to draw on the faces of a cube (or similar 3d plane surfaces with perspective) preferably with the means of tikz. The basics are given in Tikz : texture cube faces using png image but this does not solve the problem of distorting the image. I am a 100% sure, that I have seen such a solution on the web, 95% it was on SE with a Tux on one face, but I cannot find it anymore, hence if you can lead me to that question or to a solution, that would be great!
A: This is merely an attempt to clarify the question. There is a difference between perspective projections and orthographic projections. For the latter it is almost trivial to project the a picture on a cube.
\documentclass[tikz,border=3mm]{standalone}
\usetikzlibrary{perspective,3d}
\tikzset{pics/perspective cuboid/.style={code={
\tikzset{perspective cuboid/.cd,#1}%
\def\pv##1{\pgfkeysvalueof{/tikz/perspective cuboid/##1}}%
\draw (tpp cs:x=-\pv{a},y=-\pv{a},z=-\pv{a})
-- (tpp cs:x=\pv{a},y=-\pv{a},z=-\pv{a})
-- (tpp cs:x=\pv{a},y=\pv{a},z=-\pv{a})
-- (tpp cs:x=-\pv{a},y=\pv{a},z=-\pv{a}) -- cycle
(tpp cs:x=-\pv{a},y=-\pv{a},z=\pv{a}) edge (tpp cs:x=-\pv{a},y=-\pv{a},z=-\pv{a})
-- (tpp cs:x=\pv{a},y=-\pv{a},z=\pv{a}) edge (tpp cs:x=\pv{a},y=-\pv{a},z=-\pv{a})
-- (tpp cs:x=\pv{a},y=\pv{a},z=\pv{a}) edge (tpp cs:x=\pv{a},y=\pv{a},z=-\pv{a})
-- (tpp cs:x=-\pv{a},y=\pv{a},z=\pv{a}) edge (tpp cs:x=-\pv{a},y=\pv{a},z=-\pv{a})
-- cycle;
}},perspective cuboid/.cd,a/.initial=1}
\begin{document}
\begin{tikzpicture}[3d view,perspective]
\pic{perspective cuboid};
\end{tikzpicture}
\begin{tikzpicture}[3d view]
\pic{perspective cuboid};
\begin{scope}[canvas is xz plane at y=-1,transform shape]
\node{\includegraphics[width=2cm,height=2cm]{example-image-duck}};
\end{scope}
\end{tikzpicture}
\end{document}
Can one project an image on a face of the first cube, which is drawn in perspective projection using the perspective library? Yes, one can. Some of the relevant posts are
*
*https://tex.stackexchange.com/a/319271/
*https://tex.stackexchange.com/a/447120/
*https://tex.stackexchange.com/a/479188/
As you can see, none of them is trivial, and for complex images the compilation will take a while. So I would kindly like to ask you to make the question more precise so that no one puts some major amount of time into this just to hear "Oh, this is not what I wanted".
| {
"redpajama_set_name": "RedPajamaStackExchange"
} | 3,279 |
UTOPIA
# UTOPIA
SECOND EDITION
Thomas More
TRANSLATED AND INTRODUCED BY
CLARENCE H. MILLER
WITH A NEW AFTERWORD BY
JERRY HARP
Published with assistance from the foundation established in memory of Oliver Baty Cunningham of the Class of 1917, Yale College.
Afterword and Suggestions for Further Reading copyright © 2014 by Yale University. Translation copyright © 2001 by Clarence H. Miller.
All rights reserved.
This book may not be reproduced, in whole or in part, including illustrations, in any form (beyond that copying permitted by Sections 107 and 108 of the U.S. Copyright Law and except by reviewers for the public press), without written permission from the publishers.
Yale University Press books may be purchased in quantity for educational, business, or promotional use. For information, please e-mail sales.press@yale.edu (U.S. office) or sales@yaleup.co.uk (U.K. office).
Designed by Rebecca Gibb. Set in Adobe Garamond type by Keystone
Typesetting, Inc.
Printed in the United States of America.
Library of Congress Control Number: 2012954304
ISBN 978-0-300-18610-9 (pbk.)
A catalogue record for this book is available from the Library of Congress and from the British Library.
10 9 8 7 6 5 4 3 2 1
## CONTENTS
Introduction
A Chronology of More's Life
UTOPIA
Thomas More to Peter Giles, Greetings
Book 1
Book 2
Thomas More to His Friend Peter Giles, Warmest Greetings
Afterword by Jerry Harp
Notes
Suggestions for Further Reading
Index
## INTRODUCTION
The circumstances under which More composed _Utopia,_ as he recounts them in the opening of the book, give us some clues about one of its central issues: public service versus contemplative withdrawal. More was a busy London lawyer in the service of Henry VIII on a trade commission negotiating in the spring and early summer of 1515 in Bruges. In the midst of this activity came three months of leisure from late July to late October; the negotiations were interrupted because the Flemish ambassadors had to return to consult with their prince. Released from business and public service, More had time for thought and contemplation: he wrote what would become the second book of _Utopia,_ a description of the island and its people, customs, and form of government—a sort of fantastic, Lucianic travelogue. After he returned home he wrote the first book, a semi-Platonic dialogue framework concerning the question of whether it is useful to serve as the counsellor of a prince. The first book, in other words, argues about the alternatives of engagement or retreat.
Unlike Platonic dialogues, which reach solutions, or even Ciceronian dialogues, which are more open-ended but clearly suggest the superiority of one outlook, the first book of _Utopia_ comes to no conclusion. It does not take long for the reader to see that Hythloday (whose name means "peddler of nonsense") contradicts his position by telling the story of his sojourn in the court of Cardinal Morton, who listens to Hythloday's advice about punishing thieves and suggests that it might be tried in a modified form. Nor does he defeat More in the argument about the necessity of communism; he simply evades it by insisting that More would agree with him if he had seen how communism has transformed the "good place," Utopia. But unfortunately the "good place" is also "no place." And not everything about the "good place" is good: apart from its policies on euthanasia and divorce, which might be tolerated in a non-Christian society, it practices capital punishment with a harshness not far from what Hythloday condemns in Book 1; its military and especially its colonial policies also leave much to be desired. At the end of Book 2 we are left with Hythloday's passionate condemnation of the outrageous social injustices of European society, but we are not really any closer to believing that they could be cured by communism, partly because we know that it has never been introduced in society as a whole and never could be, and we are never told how it was introduced or sustained in Utopia. Even if we thought it might work in Western nations, would we want to live in such a faceless and regimented society? The citizens sometimes seem like robots; the houses and even the cities seem almost interchangeable. Apart from Utopus, we never learn the name of a single Utopian. Did Hythloday never have any special friends there? One could say that he is not interested in autobiography but only in the Utopians' economic, social, and political institutions, but should children simply be taken from their parents and moved to a different family? Should whole populations be shifted back and forth to the continental colonies to provide demographic stability?
Such facts and questions make us realize that More's Utopia does not fit the ordinary meaning of the word as it came down in modern languages, where it signifies an unreservedly "good place" (though it still includes the notion that it is "no place," that it can never be actualized). And More's _Utopia_ should not be read (as it often has been and sometimes still is read) as presenting More's notion of a purely positive and desirable society. What the character More says was believed by the real More, though the character's range of opinion is circumscribed by the context in which he appears; the real More had opinions and ideas about issues the character does not address. The character argues at some length that it is reasonable and salutary to become the counsellor to a king—a problem the real More resolved for himself when he became a member of Henry's privy council two years after _Utopia_ was first published. He also disagrees that communism would be a social panacea. He concludes by saying there are both good and bad features in Utopia: he says he disagrees with the Utopians' religious practices, their methods of warfare, and especially their communism, but, apart from communism, he does not tell us what specific features he disagrees with or why he does so. The real More does not have his character spell out these disagreements because the experience of the book is not supposed to give the reader a view of a perfect society or analyze what is good or bad about Utopia. Rather the work encourages taking a new view of social and political problems by seeing alleged (and strange) solutions to them and challenges readers to try to find out what they approve or disapprove of and why. To quote from Edward Surtz's acute and comprehensive introduction to an earlier Yale translation of _Utopia:_
Is the success of _Utopia_ due to dialogue? After all, dialogue is symbolic of open-mindedness, humility, and inquiry. Somehow or other, More succeeds in involving readers in the dialogue. It is no accident that _Utopia_ ends with challenges. Is the Utopian view of war, religion, and communism really absurd? Is the Utopian vision really hopeless and unachievable? _Utopia_ therefore is an open-ended work—or, better, a dialogue with an indeterminate close. More asks the right questions— which can never be answered fully.
The central character of the book, and the real More's most original character, is of course Hythloday, who practically identifies himself with Utopia, to which he is unreservedly committed. His names suggest the bipolarity of his character. He is Raphael (God's healer), and Hythloday (the peddler of nonsense): on the one hand a passionate analyst of social injustice, an intense and outraged defender of the oppressed, the poor, the sick, and the weak, a proponent of freedom from crushing toil, an enthusiastic promoter of intellectual pursuits, a supporter of equal rights for women; on the other hand, he is an advocate of inhuman social engineering, colonial exploitation, assassination, bloody warfare (however brief), "ethnic cleansing" of the Zapoletes, and capital punishment for someone who commits adultery twice. He is narrow-minded, unrealistic, humorless, puritanical, stubborn, tactless, and—according to some of his critics—even self-indulgent and narcissistic. And yet he is an energetic, credible character, whom we do not find entirely admirable or entirely repellent, like Utopia itself. But also like Utopia, he is always intense and intriguing. He asks and answers important questions, and however much or little we may like his answers, he makes us aware of the urgency of the questions.
The way Hythloday speaks reflects his character and ideas, and some of the range and tensions of his style can be perceived even in a translation. Hythloday's sentences range from turbulent and often strained complexity, when he is contrasting Europe with Utopia, to simple, straightforward ease when he is describing Utopia.
When Hythloday imagines a session of the French king's council and projects the advice he would give, he launches into a 464-word sentence—suspended, unrealistically intricate, almost interminable—and ends by asking More, "How do you imagine, my dear More, my listeners would react to this speech?" With wry understatement More replies in four words: "Certainly not very favorably." Well satisfied, Hythloday takes a deep breath and soars off into another imaginary council session about raising revenues, this time in a sentence of 926 words, a syntactical extravaganza so convoluted that he himself almost loses track of it. To Hythloday's concluding inquiry, More again replies with good-humored litotes and goes on to point out, in two- or three-line sentences, that the manner of advice is as important as the matter. Among editors and commentators, so far as I know, only J. H. Lupton has pointed out these strained, overburdened sentences, and until now among English translators only Ralph Robinson attempted to reproduce them. Nowhere else did More write Latin sentences that so deliberately go beyond what ordinary Latin syntax can bear. And Hythloday does this precisely when he brings the ideal kingdoms of Achoria and Macaria, his anticipations of Utopia, into jarring and irreconcilable conflict with the military and economic corruption of Europe. The two worlds, ideal and real, collide and the ordinary syntax accepted by speakers of Latin cannot contain them.
Surely More expected his readers to be disconcerted, if not totally flummoxed, by these marathon sentences. It is not so much that Hythloday has lost his grip on reality; he understands French militarism and fiscal chicanery only too well, as the extensive commentary by Fr. Surtz will testify. Rather, his only reaction to real corruptions is to grip them in one hand and smash them into ideal solutions in the other. And the syntactic explosion leads us to the "ideality" of Utopia. As Richard Sylvester pointed out, "Hythlodaeus' argument . . . moves from a firm grasp on a past historical situation [the punishment of thieves discussed at Cardinal Morton's court], to a hypothetical revision of contemporary history [the French council set over against the Achorians], and, finally, to a totally aloof fabrication [the purely imaginary council on raising money and the Macarians, near neighbors of the Utopians]."
When Hythloday describes his ideal commonwealth, his sentences undergo a remarkable change: they are predominantly brief, factual, straightforward, syntactically simple. Usually he is simply describing Utopian things as they are, and they are mostly simple, whether it be the doors of the houses,
There is no house which does not have a door opening on the street and a back door into the garden;6
or the selection of candidates for ruler,
For each of the four quarters of the city names one person and proposes him to the senate;
or the universal work at farming,
Farming is the one occupation in which all of them are skilled, men and women alike;
or the color of their cloaks,
throughout the island they are all of the same color, that of the natural wool;
or the shifting of people to maintain uniform populations,
This limit is easily maintained by transferring persons from households with too many people to those with too few;
or the distribution of goods,
And when it is distributed equitably to everyone, it follows that no one can be reduced to poverty or forced to beg;
or the lack of seeking for offices,
Anyone who campaigns for public office becomes disqualified for holding any office at all;
or their exclusion of lawyers,
they ban absolutely all lawyers as clever practitioners and sly interpreters of the law;
or their recruitment of soldiers,
In each city they choose troops from a list of volunteers;
or their strict keeping of a truce,
When they make a truce with their enemies, they keep it so religiously that they do not violate it even under provocation;
or their withholding of honor and office from those who do not believe in the immortality of the soul, divine providence, and future rewards and punishment,
they bestow no honors on such a person, they assign him to no office, they put him in charge of no public responsibility.
In such simple sentences, which I have rather randomly sampled from the second book, everything seems so balanced and rectilinear, so simple and straightforward, so effortless and obviously desirable. And such simple sentences may tend to lull us into simple unquestioning acceptance of what they say as simple fact. They tell us what the Utopians do but leave many unanswered questions about how they manage to do it. How are the four candidates for ruler chosen in each quarter of the city? What happens if someone is no good at farming or refuses to do it? Or if someone dyes his cloak? Or objects to being separated from his family and friends in a population shift? How do you know whether someone is seeking an office? The only sure sign is absolute refusal to accept it. In the absence of lawyers the judge is supposed to protect the interests of the accused. But what if the judge dislikes the defendant and admires the prosecutor? What if he is stupid? How was he chosen? How was the prosecutor chosen? Are there no rules of evidence? What if too few soldiers volunteer to fight? What happens if, during a truce, the enemy ambushes a patrol? What if someone has ambitions to be a magistrate but conceals them? What if he does not believe in the immortality of the soul but conceals his disbelief? How are the priests chosen? By whom?
Hythloday does not answer these questions. He considers them simply irrelevant because the difficulties they embody spring from pride, which has no place in Utopia. The institutions of the Utopians clearly cannot work unless pride is eliminated. And how is pride eliminated? By the institutions, especially the abolition of private property. The institutions cannot be introduced unless they have already been introduced. But the ease and lucidity of Hythloday's sentences tend to mask such difficulties. The thing is simply there. No need to ask how it got there or can manage to stay there.
Naturally, Hythloday's syntax is not always so curt and pat, even in his description of Utopia, but when his sentences swell and become somewhat involuted and turbulent it is usually when he is contrasting the ideal life of Utopia with the corruptions of Europe, as when he condemns Europe's distribution of labor, or attitude toward gold or hunting, or futilely complex laws, or abuse of treaties, or the Zapoletan (that is, Swiss) mercenaries. But these are merely occasional aftershocks of the great quake of his marathon sentences in Book 1, and he soon reverts to the simplicity of Utopian syntax.
In Book 1 the length and complexity of Hythloday's sentences lead us to believe that More himself could agree with most of what he says until the approach to Utopia (by way of Achoria and Macaria) dissociates him from More, interrupts the debates about counsel and private property (leaving them unresolved), and frees Hythloday to present the simplicities of Utopia in simple sentences, which are so unlike More's way of thinking and writing elsewhere as to suggest that he meant us to probe them with questions—not only about obvious difficulties such as Utopian warfare, divorce, euthanasia, or colonialism, but throughout. So much of what the Utopians do is admirable, but how in heaven's name do they manage to do it?
There is, for example, a glaring omission in Utopia, which Hythloday's limpid and easy sentences may cause us to overlook: the political structure of Utopia has no central executive authority for its fifty-four independent city-states. Although the whole island has no single governor, it does have an annual senate composed of three wisemen from each city. If there were any real self-centered rivalry among these cities, even any normal conflict of interest, such a senate, without any executive machinery whatever, would be quite ineffectual. We, like More, have only to look to the city-states of ancient Greece or Renaissance Italy to see what would happen. Then, too, we are told about the punishments for various crimes in Utopia (among which the absence of theft may not be surprising, but what about assault or murder?), but, without a police force, who catches the criminal? Who checks whether he has the proper papers to be out of his own city-state? What prevents him from stowing away on a ship to the mainland? Questions proliferate endlessly. We are always being brought back to the basic paradox: the institutions cannot be introduced unless they have already been introduced.
Another remarkable feature of Hythloday's style, which is related to the deceptive simplicity of Utopia, is his diction. We are not surprised that he is fond of words like "equal" (which he uses 26 times) or "easy" (24 times). After all, his main thesis is that equality of goods makes just government easy in Utopia. But another group of frequent words suggests his inability to deal with specific problems in concrete circumstances and reflects the universalist, absolute, all-or-nothing cast of his mind: "all" (200 times), "nothing" (76), "none" (68), "whole" (62), "one" (35), "the same" (33), "any" (33), "no one" (29), "entirely" (24), "each" (19), "never" (19), "anything" (17), "everywhere" (14), "anywhere" (13), "only" (11), "universal" (9), "ever" (8), "never" (7). In the samples of short sentences given above, I made no attempt to include any of these words, and yet I noticed that they had inevitably appeared ("There is NO house"; "Farming is the ONE occupation in which ALL of them are skilled"; "Whoever seeks ANY office becomes ineligible for ALL offices"). Moreover, as I checked the instances, I found that they often tended to occur together with other words from this absolutist cluster.
A few examples must suffice to suggest the effect of such diction.
The island has fifty-four cities, ALL of them large and splendid and having EXACTLY THE SAME language, customs, institutions, and laws. They have the SAME layout and they look the SAME, insofar as the terrain allows.
From them [the storehouses] EACH head of household goes to get whatever he and his household need and takes away WHATEVER he wants, paying no money and giving ABSOLUTELY NOTHING in exchange for it. For why should he be denied ANYTHING, since there is plenty of EVERYTHING and NO ONE need fear that ANYONE would want to ask for more than he needs? For why should ANYONE be suspected of asking for too much if he is certain he will NEVER lack for ANYTHING?
On the other hand, here, where EVERYTHING belongs to EVERYONE, NO ONE doubts that (as long as care is taken that the public storehouses are full) NOTHING WHATEVER will be lacking to ANYONE for his own use. For the distribution of goods is not niggardly; NO ONE is a pauper or a beggar there, and though NO ONE has anything, ALL are rich.
Such reiterated universalist diction is the source of what most readers feel to be objectionable about the Utopians: their faceless anonymity and homogeneity. But such diction is not characteristic of More himself, either as a character in _Utopia_ or in his other Latin writings. And it should cause us to ask questions similar to those raised by Hythloday's simple sentences: how do you get everyone always to do the same one thing everywhere, wholly and completely, without anyone anywhere at all deviating significantly in anything, with no exceptions, with no one ever wishing to contravene the universal system, with all in equal conformity, with never a dissenting voice, with nowhere a refusal to comply? Hythloday's answer is one and the same, always: introduce Utopian institutions, based on the sharing of everything. Only then will everyone be totally and completely committed to the common good (respublica). But only a people raised, educated, and trained under Utopian institutions can make the institutions work. As with Hythloday's simple sentences, we are brought back to the paradox, the dilemma, the "double-bind": nowhere can such institutions be introduced except where they have already been introduced—nowhere.
But Hythloday should not be viewed as merely narrowminded, solipsistic, and naive. When he condemns the injustices of Europe, his voice and his sentences are not incompatible with those of More himself. His passionate denunciations of the greedy oppression of the poor and his compassionate indignation at the lazy, corrupt self-indulgence of the rich are intense and memorable; everyone remembers the sheep who were once gentle but now devour people (p. 22), or his thunderous peroration against economic and social injustice, culminating in the condemnation of the conspiracy of the rich who look out for themselves under the pretext of serving the commonwealth (pp. 132). Only when he flees such corruption and breaks through to the simplicities of Utopia do his sentences fracture Latin syntax and soar beyond what More's Latin, even at its most muscular, would attempt. And the simple sentences and universalist diction of his description of Utopia do not make him seem merely simple-minded. They also help him to make us think that this has happened, that it could happen (in spite of all our nagging doubts about how it could have happened or how it could happen in the world we know); and, even more, he makes us think that some of it should happen (in spite of the thought-provoking anomalies in Utopian behavior) because the Utopians really believe in the common good; and Hythloday makes us almost believe in their belief, and so we believe him even while we disbelieve him, just as (because of her virtuoso and contrasting styles) we believe and disbelieve his great compeer, the Folly of Desiderius Erasmus' _Moriae Encomium_.
### THE EARLY EDITIONS AND THE LATIN TEXT
Erasmus and Peter Giles supervised the printing of the first edition (Louvain, 1516), adding commendatory and sometimes analytic letters from well-known humanists of the time; these letters, though they provide a useful context for reading _Utopia,_ have not been included in this translation because they are highly specialized and sometimes inflated. I have included More's prefatory letter to Giles and his second letter to Giles, which appeared only in the second edition, in 1517. Erasmus and Giles were also probably responsible for the marginal notes that appeared in the first edition, and they may have had a hand in changing some Latin proper and place names from Latin to Greek forms. In the early editions many of the marginal notes are merely labels to mark off sections equivalent to paragraphs (of which the early editions have none). I have omitted such paragraph markers and included only the marginal notes that make some independent comment on the text. I have supplied the paragraphing in this translation.
Thomas Lupset supervised the second edition (Paris, 1517), which was corrected by More. Erasmus transmitted copy for the third edition (Basel, March 1518), which was also corrected by More (sometimes differently from 1517). In fact, it is possible that More corrected 1517 later than 1518m (as the March 1518 edition is known). The Basel edition of November 1518 was simply a reprint of 1518m, as was the Florence edition of 1520. It is significant that More, Erasmus, and Johann Froben, the publisher of the Basel editions, originally intended to include the translations of Lucian by More and Erasmus, though they were never included because the volume had grown too large; Lucian's fantastic and satirical flair was one of the ingredients in the savory stew of _Utopia_. Moreover, 1518m and 1518n (the November 1518 edition) also included the Latin epigrams of More and Erasmus; several of More's concern issues of good and bad government.
The Latin text on which this translation is based is derived from all the texts and variants of the first three editions, the only ones in which More actually had a hand; these materials are presented in full in the comprehensive and definitive editions of Fr. Surtz and André Prévost. My Latin text corresponds very closely with the Cambridge edition of George Logan and others, which is the most accurate and usable modern text. For these editions see the list of books for further reading at the end of this volume. I have also consulted the remarks and emendations in the Latin editions by V. Michels and T. Ziegler (1895), J. H. Lupton (1895), and Marie Delcourt (1936).
### A NOTE ON THE TRANSLATION
Since _Utopia_ has been translated by Ralph Robinson, Gilbert Burnet, and several translators in the twentieth century, it may be asked why another translation should be undertaken. But the truth is that in spite of many translations, some of them frequently reprinted, _Utopia_ has not fared as well as it deserves in English. Robinson's pioneering translation, published in 1551, is quite accurate (with only a few exceptions) and was usually consulted by subsequent translators, but it was made at a time when English did not yet have the strength, either in diction or style, to reproduce the elaborate Latin of _Utopia_. Hence his translation, though lively and vivid, often seems wordy and awkward (as it probably also did to his contemporaries who read Latin). But at least Robinson did not omit words or phrases when they were inconvenient, and he tried to match the varied style of the Latin. The same cannot be said for those who walked in his footsteps. By the time of Gilbert Burnet's translation in 1684, English prose was a varied and powerful instrument, fully capable of matching More's Latin; but fashionable prose in the Restoration was simple, lucid, "easy" (in reaction against the florid, baroque "excesses" of the earlier part of the century). Hence Burnet streamlined and simplified More's Latin, ignoring its extreme stylistic variations, occasionally clipping and pruning it as well.
G. C. Richards, the first twentieth-century translator (1923), tried like Robinson to be fully faithful to the Latin, in details and style, but he kept too close to the Latin so that his English often turned out to be awkward and unidiomatic. H. V. S. Ogden (1949) explicitly mentions the translations of Robinson and Burnet, admitting that he has borrowed wording from both, especially Burnet. In fact his translation is swift and readable (like Burnet's) but only at the cost of simplification in detail and the suppression of elaborations in More's Latin. The same false tendency is carried even further in Paul Turner's translation (1965) and further yet in the translation by John Sheehan and John P. Donnelly (1989), who professedly simplify and prune the text for American college students. The translation by Robert M. Adams (1975) is also uniformly swift and breezy, and even its corrected form (1995) does not match the stylistic variations of the Latin.
In fact, More's Latin is often anything but colloquial and easy (pace Turner and Surtz). In his first letter to Giles, More describes Hythloday's language as unpolished, informal, and extemporaneous, giving his style the label "casual simplicity." But hardly anything in this letter can be taken at face value. When Hythloday speaks about Utopia itself his sentences are indeed generally simple and easy (Utopia is, after all, a simple and easy solution to social and economic problems), but when he denounces the corruptions of Europe in Book 1 and in the peroration to Book 2, his sentences are complex, lengthy, elaborate, muscular, even muscle-bound. Some translators seem to think that such complications are natural in Latin but not in English. But in fact, English can follow the normal complications of Latin well enough, and the unnatural complications in the Latin ought not to be merely smoothed out in the English. They are an important part of the meaning.
Hence I have tried to translate all the details of the Latin in idiomatic English that matches the simplicity, complexity, or even unnatural strain of the Latin.
## A CHRONOLOGY OF MORE'S LIFE
1477
Born in London, February 7
c. 1482–90
School at St. Anthony's
c. 1490–92
Page in the household of John Morton, Archbishop of
Canterbury and chancellor of England
c. 1492–94
Student at Oxford University
1494–1501
Studied law in London
c. 1500–1504
To test his vocation to the priesthood resided in Carthusian
monastery next to his law school, Lincoln's Inn
1504–5
Married Joan Colt, late 1504 or early 1505
1509
Negotiated with Antwerp merchants on behalf of London companies
1511
Joan, who bore him four children, died, and More
married Alice Middleton
1510–18
Undersheriff of London
1513–18
Wrote _The History of Richard III_ in both Latin and English
1515
Member of a delegation sent to Bruges to revise
a commercial treaty
1516
_Utopia_ published at Louvain
1517
Helped quell a riot by a mob of London apprentices, May 1
1517
Negotiated with the French at Calais and Boulogne about
suits arising from the recent war, September–December
1518
Became a member of the king's council and
Master of Requests
1520
Accompanied Henry VIII to a meeting with Francis I at the
Field of Cloth of Gold near Calais
1520–21
Negotiated with Emperor Charles V and the
Hansa merchants at Calais and Bruges
1521
Knighted
1523
In _Responsio ad Lutherum_ defended Henry VIII against Luther's attack
1523
Speaker of the House of Commons
1524
Moved into the large mansion he built upriver in Chelsea
1525
Chancellor of the Duchy of Lancaster
1524–25
High Steward of the Universities of Oxford (1524) and Cambridge (1525)
1528
Commissioned by Bishop Tunstall of London to read and refute Lutheran books in English
1529–33
Wrote seven polemical books in English against Lutheranism
1529
Attended the congress at Cambrai at which peace was negotiated between France and the Empire
1529–32
Lord Chancellor of England
1534
Refused to swear to the Act of Succession because it
repudiated papal supremacy, April 13
1534
Imprisoned in the Tower, April 17
1534–35
Wrote _A Dialogue of Comfort Against Tribulation_ in the Tower
1535
Convicted of treason on the perjured evidence of
Richard Rich, July 1
1535
Beheaded in the Tower, July 6
UTOPIA
On the Best Form of a Commonwealth
and
on the New Island of Utopia
a Truly Precious Book
No Less Profitable than Delightful
by
the Most Distinguished and Learned Gentleman
Thomas More
Citizen and Undersheriff
of the Illustrious City of London
_A Six-line Stanza on the Island of Utopia
by the Poet Laureate
Anemolius
The Son of Hythloday's Sister_
Called once "No-place" because I stood apart.
Now I compete with Plato's state, perhaps
Surpass it; what he only wrote about
I have alone in fact become: the best
In people, wealth, in laws by far the best.
"Good-place" by rights I should be called.
### THE UTOPIAN ALPHABET A QUATRAIN IN THE UTOPIAN LANGUAGE
The literal meaning of these lines:
When I was not an island, the commander Utopus made me into an island. I alone of all the nations on earth, without philosophy, have presented to mortals a philosophical state. Freely I share what I have; not unwillingly I accept what is better.
## Thomas More to Peter Giles, Greetings
I am almost ashamed, my dear Peter Giles, to have delayed for almost a year in sending you this little book about the Utopian commonwealth, which I'm sure you expected within six weeks. You knew, after all, that I was spared the labor of finding my matter, and did not have to give any thought to its arrangement; all I had to do was repeat what you and I heard Raphael say. For that reason there was no need to strive for eloquence, since his language could hardly be polished, first because it was informal and extemporaneous, and also because he is a person, as you know, not as well versed in Latin as in Greek; the closer my language came to his casual simplicity, the more accurate it would be, and in this matter accuracy is all that I ought to, and in fact do, aim for.
I grant you, Peter, that with all this already taken care of, I was relieved of so much effort that there was almost nothing left for me to do. If this had not been so, thinking up the subject matter and arranging it might have required not a little time and study, even from someone of not inconsiderable intelligence and not totally without learning. But if I had been required to write not only accurately but also elegantly, no amount of time or study would have enabled me to do it. As it is, all these concerns, which would have cost me so much labor, are removed and all that remained to do was to write what I heard—not a difficult task.
But nevertheless, even to perform this trifling task, other chores left me almost no time at all. I am constantly pleading one case, hearing another, acting as arbitrator, handing down decisions as a judge, visiting one person or another on business or because it is my duty to do so; I am out practically all day dealing with others, and the rest of my time is devoted to my family, and so I leave nothing for myself, that is for writing.
When I get home, I have to talk with my wife, chat with my children, confer with the servants. All this I count as part of my obligations, since it needs to be done (and it does if you do not wish to be a stranger in your own home); and you must do everything you can to make yourself as agreeable as possible to the persons you live with, whether they were provided by nature, chance, or your own choice, as long as you do not spoil them by your familiarity or turn servants into masters through over-indulgence. As I am doing such things, as I said, a day, a month, a year slips by.
When do I write then? And as yet I have said nothing about sleep and nothing at all about eating, and for many that takes up no less time than sleep itself, which consumes almost half our lives. The only time I get for myself is what I steal from sleep and eating. Because that is so little, I progressed slowly, but because it was at least something, I did make progress, and I sent _Utopia_ to you, my dear Peter, so that you can read it and let me know if I have missed anything. For, though on that score I do not lack all confidence in myself (and I only wish that my intelligence and learning were a match for my not inconsiderable memory), still I am not confident enough to think that nothing has escaped me.
As you know, John Clement, my young assistant, was there with us, for I do not allow him to miss out on any conversation which could be profitable to him because from this sprout which is beginning to grow green with proficiency in Latin and Greek I expect someday a marvelous harvest. He has made me feel very doubtful about one point: as far as I remember Hythloday told us that the bridge which spans the river Anyder at Amaurot is five hundred yards long, but my boy John says that is two hundred yards too many and that the river is no more than three hundred wide. Please try to remember that point. For if you agree with him, I will go along with you both and believe I am mistaken. But if you do not recall, I will stand by what I think I remember myself, for just as I have taken great pains to prevent any inaccuracy in the book, so too, when I am in doubt, I would rather say something inaccurate than tell a lie, because I would rather be honest than clever.
In fact, it would be easy to remedy this defect if you would find out from Raphael himself about it, in person or by letter. And you need to do the same concerning another difficulty which has arisen—who is more to blame for it, I or you or Raphael himself, I do not know. For it did not occur to us to ask, or him to mention, in what part of that new world Utopia is located. Indeed, to remedy this oversight I would be willing to give a sizeable sum, partly because I am ashamed not to know in which ocean the island lies about which I have recounted so much, partly because there are one or two people here, but especially one person, a devout man and a theologian by profession, who is amazingly eager to go to Utopia, not out of idle curiosity or any hankering after novelties but in order to nourish and spread our religion, which has made such a good beginning there. To do this properly he has decided to see to it beforehand that he is sent by the pope and made the bishop of the Utopians. He has no scruples whatever about begging for this bishopric, since he considers such ambition to be holy if it is not based on honor or gain but rather springs from piety.
Therefore, my dear Peter, I beg you to contact Hythloday, either in person if that is convenient or by letter if you are separated, and see to it that this work of mine contains nothing false and lacks nothing true. And perhaps it would be best to show him the book. For there is no one else capable of correcting any errors and even he cannot do so unless he reads through what I have written. Then too, this will let you see whether he is pleased or annoyed at me for writing this work. For if he himself has decided to commit his labors to writing, he may not want me to do so. And I certainly would not want to deprive his narrative of the bloom and charm of novelty by making the commonwealth of Utopia public.
But in fact, to tell you the truth, I myself have not yet made up my mind whether or not to publish it at all. For the tastes of mortals are so various, the temperaments of some are so bitter, their minds so ungrateful, their judgments so preposterous that a person would do far better to follow his own bent and lead a merry life than to wear himself out trying to publish something useful or entertaining for an audience so finicky and ungrateful. Most people know nothing about learning; many despise it. Dummies reject as too hard whatever is not dumb. The literati look down their noses at anything not swarming with obsolete words. Some like only ancient authors; many like only their own writing. One person is so dour that he cannot abide jokes; another is so witless that he cannot stand anything witty. Some have so little nose for satire that they dread it the way someone bitten by a rabid dog fears water. Others are so changeable that their approval depends on whether they are sitting down or standing up.
They sit around in taverns and over their cups they pontificate about the talents of writers, condemning each author just as they please, pulling him down through his writings as if they had grabbed him by the hair, while they themselves are safe and out of harm's way, as the saying goes, because these good men have their whole heads smooth-shaven so that there is not a single hair to grab on to.
Furthermore, some are so ungrateful that, even though a work has given them great pleasure, they still do not like the author any better because of it. They are not unlike illmannered guests who, after they have been lavishly entertained at a splendid banquet, finally go home stuffed without saying a word of thanks to the host who invited them. Go on, now, and at your own expense provide a banquet for persons of such delicate palates and various tastes, who will remember and repay you with such gratitude!
Nevertheless, my dear Peter, raise with Hythloday the points I mentioned. Afterwards I will be free to consider the matter once more. But in fact, if he himself gives his consent— since it is late to be wise now that I have finished all the work—in all other considerations about publishing I will follow the advice of my friends, and especially yours. Farewell, my dearest Peter Giles, with regards to your excellent wife, and be as fond of me as ever, since I am fonder of you than ever.
A Discourse
on the Best Form of a Commonwealth
Spoken by the Remarkable Raphael Hythloday
as Reported by
the Illustrious Thomas More
a Citizen and the Undersheriff
of the Famous British City of London
## BOOK 1
Recently the invincible king of England, Henry the eighth of that name, who is lavishly endowed with all skills necessary for an outstanding ruler, had some matters of no small moment which had to be worked out with Charles, the most serene prince of Castile. To discuss and resolve these differences he sent me to Flanders as his ambassador; I was the companion and colleague of the incomparable Cuthbert Tunstall, whom he recently appointed to be Master of the Rolls, to the enormous satisfaction of everyone. I will say nothing in his praise, not because I am afraid that my friendship might seem to make me an unreliable witness, but because his virtue and learning are beyond my power to proclaim them and because they are everywhere so renowned and well known that there is no need for me to do so, unless I intend to display the sun by the light of a lantern, as they say.
As had been agreed, we were met at Bruges by those to whom the prince had entrusted the negotiations, all of them outstanding men. Their leader and chief was the Mayor of Bruges, a splendid man, but their spokesman and mastermind was George de Themsecke, the Provost of Cassel, who is not only a trained orator but also a naturally eloquent speaker; he is very skilled in the law as well, and also an extraordinarily deft negotiator because he is both intelligent and very experienced. After one or two meetings we could not reach agreement on some points, and so they bade us farewell for some days and set out for Brussels to ask for the pronouncement of their prince.
Meanwhile, as my business required, I made my way to Antwerp. While I was staying there, I was often visited by Peter Giles, among others, though no other visitor was more delightful to me. A native of Antwerp, he holds a post of great responsibility and prestige (and he is worthy of the most prestigious), since for this young man it would be hard to say which is greater, his learning or his virtue. For he is most virtuous and very widely read, and also good-natured toward everyone, but toward his friends he is so responsive, warmhearted, loyal, and unfeignedly affectionate that it would be hard to find even one or two anywhere that you would think comparable to him in every aspect of friendship. He has a modesty rarely to be found; no one is further from false poses; no one combines more prudence with simplicity. Then, too, his elegant speech and his innocent wit are so attractive that his delightful companionship and his charming conversation alleviated my longing for my country, household, wife, and children, though I was tormented by my desire to see them again, for at that time I had been away from home for more than four months.
One day, after I had heard mass at the church of St. Mary, which is remarkable for its beautiful architecture and its large congregation, when the service was over and I was getting ready to return to my lodgings, I happened to see Giles conversing with a stranger who was getting up in years. His face was sunburned, his beard untrimmed, his cloak hanging carelessly from his shoulder; from his face and bearing I thought he looked like a sea captain. But then, when Peter saw me, he came up and greeted me. When I tried to answer, he took me a little aside and said, "Do you see this man?" (At the same time he indicated the person I had seen him talking to.) "He is the one," he said, "I was just getting ready to bring straight to you."
"He would have been all the more welcome to me on your account."
"Actually on his own," he said, "if you knew him. For there is no mortal alive today who can give more information about unknown peoples and lands, and I know that you are very eager to hear about them."
"My guess was not far off, then," I said, "for when I first set eyes on him, I immediately thought he was a sea captain."
"But in fact," he said, "you were far off the mark. Certainly he has sailed, not like Palinurus, but rather like Ulysses, or even better like Plato. This man, who is named Raphael—his family name is Hythloday—has no mean knowledge of the Latin language but is especially proficient in Greek; he has devoted himself to Greek more than to Latin because he has totally committed himself to philosophy and he knew that in that field there is nothing of any importance in Latin except some works of Seneca and Cicero. Out of a desire to see the world he left to his brothers his heritage in his homeland (he is from Portugal), joined Amerigo Vespucci, and was his constant companion in the first three of the four voyages which everyone is now reading about; but on the last voyage he did not come back with him. He sought and practically wrested from Amerigo permission to be one of the twenty-four who were left behind in a fort at the farthest point of the last voyage. And so he was left behind in accordance with his outlook, since he was more concerned about his travels than his tomb. Indeed he often used to say, 'Whoever does not have an urn has the sky to cover him,' and 'from everywhere it is the same distance to heaven.' This attitude of his would have cost him dearly if God had not been merciful to him. However, after the departure of Vespucci, he traveled through many lands with five companions from the fort, and finally, by an extraordinary stroke of luck, he was transported to Ceylon and from there he reached Calicut, where he opportunely found some Portuguese ships and at last, beyond all expectation, he got home again."
When Peter had told me this I thanked him for his kindness in taking so much trouble to introduce me to someone whose conversation he hoped I would enjoy, and then I turned to Raphael. After we had greeted each other and spoken the usual amenities that are exchanged when strangers meet for the first time, we went off to my house, where we conversed sitting in the garden on a bench covered with grassy turf.
And so he told us how, after the departure of Vespucci, he and his companions who had remained in the fort gradually began to win the good graces of the people of that land by encountering and speaking well of them, and then they started to interact with them not only with no danger but even on friendly terms, and finally they gained the affection and favor of some ruler, whose name and country escape me. He told how, through the generosity of the ruler, he and five of his companions were liberally supplied with provisions and ships on the sea and wagons on the land—together with a trustworthy guide who took them to other rulers to whom he heartily recommended them. After many days' journey, he said, he discovered towns and cities and commonwealths that were very populous and not badly governed.
On both sides of the equator, it is true, extending almost as far as the space covered by the orbit of the sun there lie vast empty wastelands, scorched with perpetual heat. The whole region is barren and ugly, rugged and uncultivated, inhabited by wild beasts and serpents and by people who are no less wild than the beasts and no less dangerous. But when you have traveled further, everything gradually becomes milder. The heavens are less fierce, the ground is green and pleasant, the creatures are more gentle, and finally one sees peoples, cities, towns, which not only trade continually among themselves and with near neighbors but also carry on commerce with distant nations by land and sea. From that point on they were able to visit many countries in all directions since there was no ship traveling anywhere in which he and his comrades were not eagerly welcomed.
He told us that in the first regions they traveled they saw flat-bottomed vessels, spreading sails made of wickerwork or of stitched papyrus, and in other places of leather. But afterwards they found ships with curved keels, canvas sails, and in fact all the features of our own vessels. The sailors were not unskilled in seamanship and celestial navigation, but he told us that they were extremely grateful to him for introducing them to the magnetic compass, with which they had been totally unfamiliar. For that reason they usually were afraid to commit themselves to the open sea and they did not venture to do so except during the summer. But now they have such confidence in the compass that they scorn the winter weather and are careless rather than secure; thus there is a danger that the device which they thought would do them so much good will do them great harm because of their imprudence.
To present what he told us about the things he saw in each and every place would take a long time and would be beyond the scope of this work. And perhaps I will speak of it elsewhere, especially those points of which it would be useful not to be ignorant, above all whatever correct and prudent provisions he observed among civilized nations. We asked him very eagerly about such matters, and he was quite willing to explain them, but we paid no attention to monsters, for nothing is less novel than they are. Indeed, there is almost no place where you will not find Scyllas and rapacious Celaenos and man-eating Laestrigonians and such prodigious monsters, but it is not everywhere that you will find soundly and wisely trained citizens. But just as he noted many ill-considered practices among those newly discovered nations, so too he recounted not a few features that could serve as patterns to correct the errors of our own cities, nations, peoples, and kingdoms. These, as I said, will have to be presented elsewhere. At present I intend to relate only what he told us about the customs and institutions of the Utopians, but first I will present the conversation which led him on, as it were, to mention that commonwealth. For after Raphael had very judiciously analyzed some of our errors and some of theirs (and certainly there are plenty in both places) and had presented some wiser provisions both here and there—and he had such a mastery of the customs and institutions of every nation he visited that you would imagine he had spent his whole life there—Peter was amazed by him and said, "My dear Raphael, why do you not enter into the service of some king, for I am convinced that there is none who would not be extremely glad to have you, because this learning of yours and your knowledge of peoples and places would not only serve to delight him but would also make you fit to inform him of precedents and aid him with advice. In this manner you could at one and the same time promote your own interests enormously and be of great assistance to your relatives and friends."
"As for my relatives and friends, I am not much concerned about them because I have done my duty by them well enough: others do not give up their possessions until they are old and sick, and even then they do so reluctantly, when they can no longer retain them; but I divided my possessions up among my relatives and friends when I was not only healthy and vigorous but also young. I think they ought to be satisfied with my generosity, and beyond that they should not demand and expect me to hand myself over into servitude to kings for their sake."
"A fine thing to say," said Peter. "I want you to go into the service of kings, not be in servitude to them."
"There is," he said, "only one syllable's difference between them."
"But I am of the opinion," said Peter, "that, whatever name you give it, it is still the course by which you can not only profit others, both privately and publicly, but also make your own position a happier one."
"Would I make it happier by following a course which is abhorrent to me? But as it is, I live as I please, and I certainly suspect that is very seldom the case with the grandees of court. Surely there are plenty of people who strive to gain the favor of powerful men, so that you need not consider it any great loss if I and one or two like me are not among them."
Then I said, "It is clear, my dear Raphael, that you are not greedy for wealth or power; I respect and revere a person with your attitude no less than I do any of the high and mighty. But it seems obvious to me that you would be acting in a fashion worthy of yourself and of your noble and truly philosophical nature if you could bring yourself to apply your intelligence and industry to public affairs, even at the cost of some private inconvenience. You will never be able to do this to such good effect as you could if you became a counsellor to some great prince and urged upon him what is right and honorable, as I am sure you would. For the stream of good and evil, as if from a never-failing spring, flows from the prince down upon the whole people. And your learning is so complete, even if you had no great experience, and your experience is so full, even if you had no learning at all, that you would be an outstanding counsellor to any king whatever."
"You are wrong on two counts, my dear More," he said. "First about me, and then about the way things are. For I do not have the ability you attribute to me, and even if I had it in full measure, I would sacrifice my contemplative leisure to active endeavor without contributing anything to the common good. First of all, the princes themselves, almost all of them, are more devoted to military pursuits (in which I neither have nor desire any skill) than they are to the beneficent pursuits of peacetime; and they are far more interested in how to acquire new kingdoms by hook or crook than in how to govern well those they have already acquired. Moreover, among the counsellors to kings, there is none who is not so truly wise as not to need—or at least thinks he is so wise as not to tolerate—the advice of any other counsellor, except that they support and fawn on any and all absurdities propounded by the prince's favorites, whose favor they strive to win by flattery. Certainly nature seems to have arranged it so that everyone is delighted with his own insights. So the crow dotes on its chick, and the monkey on its whelp.
"But in a conclave made up of those who envy the insights of others or exalt their own, if anyone should propose something which he has read was done in other eras or which he has seen done in other places, his listeners there immediately act as if their whole reputation for wisdom were at risk, as if they would thereafter be considered totally stupid if they cannot propose something to undermine the proposals of others. If all else fails, then this is their last resort: these things pleased our ancestors, they say, and would that we were as wise as they! And with this remark they take their seat thinking they have said the last word on the subject, as if it were a very dangerous matter if anyone were detected to be wiser than his ancestors on any point. In fact if those ancestors have instituted some truly excellent policy, we are quite content to dismiss it. But if they might have taken a wiser course on some point, we immediately and eagerly seize the pretext of tradition to maintain it. And I have encountered such arrogant, absurd, and captious judgments often enough in other places, but once even in England."
"What," I said, "you were in our country?"
"I was," he said, "and I spent some months there, not long after the revolt of the Englishmen from the west against the king was put down with such a miserable slaughter of the rebels. While there I was much obliged to the most Reverend Father John Morton, Cardinal Archbishop of Canterbury, and at that time also Lord Chancellor of England. He was a man, my dear Peter (for More already knows what I am about to say) no more venerable for his authority than for his prudence and character. He was of medium height, not stooped over though he was of an advanced age. His looks inspired reverence, not fear. In company he was not standoffish, but grave and serious. Sometimes he enjoyed handling suitors roughly, but harmlessly, so as to gauge the intelligence and presence of mind each would display. He was delighted with such qualities, provided they were devoid of all impudence, since they were related to his own character, and he embraced them as valuable in getting things done. His speech was polished and pointed; he was very skilled in the law; his intelligence was incomparable; his memory was so excellent as to be prodigious. These extraordinary natural gifts he had improved by study and practice. The king seemed to rely very much on his advice and while I was there he seemed to be the mainstay of the commonwealth. This was not surprising: thrust immediately from school into the court at a very young age, active in important affairs throughout his life, continually whirled about by violent changes of fortune, he had learned practical wisdom in the midst of many and serious perils, and wisdom so won is not easily forgotten.
"One day when I happened to be dining at his table, a layman who was skilled in the laws of your country was there. Following up some remark or other, he launched on a elaborate encomium of the rigorous justice which was at that time applied to thieves in England. They were executed everywhere, he said, sometimes as many as twenty at a time hanging on one gallows, and he remarked that he was all the more amazed that the country was cursed to have so many of them prowling about everywhere, since so few escaped punishment. Then I said (and I dared to speak my mind freely in the presence of the Cardinal): 'You should not be at all surprised. For this punishment of thieves is both beyond the limits of justice and not in the public interest. As a punishment for theft it is too harsh, and even so it is not a sufficient deterrent: simple theft is not so serious a crime as to deserve capital punishment, and no penalty is great enough to keep people from stealing if they have no other way to make a living. Thus, in this matter, not only you but most of the world seem to imitate bad teachers who are more eager to beat their pupils than to instruct them. For heavy and horrible punishments are imposed on thieves when it would be much better to make some provision for their livelihood, so that no one should labor under the cruel necessity first of stealing and then of dying for it.'
" 'We have made sufficient provision for that,' he said. 'There are trades; there is farming. From them they can make a living, as long as they do not willingly prefer to be criminals.'
" 'You will not get out of it that way,' I said. 'First of all, we will overlook the many soldiers who come home crippled from foreign or domestic wars, as they recently did from the battle against the Cornishmen and not long before that from the French wars. They have sacrificed their limbs for the commonwealth or the king; their disability does not allow them to practice their former trades and they are too old to learn a new one. These,' I said, 'let us overlook, since wars happen only now and then. Let us consider what is never not happening. Now there is a multitude of noblemen who not only live like drones on the labor of others—namely the tenants of their estates whom they bleed white by raising their rents (for this is the only kind of frugality they recognize, and otherwise they are so prodigal as to reduce themselves to beggary)—but they also travel with a huge crowd of retainers, none of whom has ever learned how to make a living. As soon as their master dies or they get sick, they are immediately thrown out. For lords would rather support idle men than invalids, and often the heir of a dying master cannot support a household as large as his father's, at least at first. Meanwhile the outcasts vigorously starve unless they vigorously steal. For what are they to do? After tramping around a bit they will have ruined their clothes and their health. Disfigured as they are by disease and clad in rags, no nobleman will deign to take them in and no farmer dares to do so. For the farmers are not unaware that a person who has been brought up in idle ease and pleasure and who has been used to swaggering about like a bully, girt with sword and buckler, looking down his nose at the whole neighborhood and despising everyone but himself, is hardly likely to be a reliable and faithful servant for a poor farmer, working with hoe and mattock for miserable wages and scanty keep.'
"To this the lawyer replied, 'But this is precisely the sort of person we should cherish the most. For since they are more high-spirited and lofty-minded than artisans and farmers, they provide the strength and power of an army if we ever have to fight a war.'
" 'Indeed,' I said, 'you might as well say that we should cherish thieves for the sake of warfare, for you will never lack for thieves as long as you have the retainers. In fact robbers are no slouches as soldiers and soldiers are not the most lethargic of thieves—so finely matched are the two callings. But this problem, though it is widespread among you, is not peculiar to you; it is shared by almost all nations. But France is infected with another pestilence besides, one that is even more virulent: the whole country is occupied and filled with mercenaries, even during peacetime (if it can be called that). Their justification is the same as yours for maintaining idle retainers here: those foolosophers think that the public welfare consists in having strong and stout armed forces in a state of readiness, especially veterans, for they have no confidence in untried troops, just as if they should seek out a war precisely to avoid having inexperienced soldiers, and people should be gratuitously slaughtered (as Sallust nicely puts it) lest hand and spirit should grow sluggish through inactivity. Just how deadly it is to maintain such beasts France has learned to her cost, and the same is made clear by the examples of the Romans, the Carthaginians, and the Syrians, and of many other nations as well: standing armies of mercenaries, on one occasion after another, destroyed not only their government but also their fields and even their cities. How little this was necessary is made clear by the fact that not even French soldiers, thoroughly trained in warfare to their very fingertips, can very often boast that they came off better than your draftees—not to put it more strongly lest I seem to be flattering present company. But your troops, whether urban artisans or rough and untrained farmers, are not thought to be very much afraid of the idle retainers of noblemen, except for some whose physique does not lend itself to strength and boldness or whose brave spirit has been broken by the poverty of their families. There is little enough danger that those retainers whose vigorous and strong bodies (for noblemen do not deign to ruin any but choice physiques) are now either grown flabby with idleness or soft with almost ladylike activities, no danger, I say, that such retainers would be unmanned if they were taught a good craft to earn a living and were exercised in manly labors. However that may be, I certainly do not see that it can ever contribute to the common good to prepare for war (which you never have unless you wish to) by maintaining such a huge crowd of people who undermine the peace, to which we ought to pay so much more attention than to war. But this is not the only problem which makes it necessary to steal. There is another, more peculiar (so far as I know) to you Englishmen.'
" 'What is that?' said the Cardinal.
" 'Your sheep,' I said, 'which are ordinarily so meek and require so little to maintain them, now begin (so they say) to be so voracious and fierce that they devour even the people themselves; they destroy and despoil fields, houses, towns. I mean that wherever in the realm finer and therefore more expensive wool is produced, noblemen, gentlemen, and even some abbots (holy men are they), not content with the annual rents and produce which their ancestors were accustomed to derive from their estates, not thinking it sufficient to live idly and comfortably, contributing nothing to the common good, unless they also undermine it, these drones leave nothing for cultivation; they enclose everything as pasture; they destroy homes, level towns, leaving only the church as a stable for the sheep; and as if too little ground among you were lost as game preserves or hunting forests, these good men turn all habitations and cultivated lands into a wilderness. And so that one glutton, a dire and insatiable plague to his native country, may join the fields together and enclose thousands of acres within one hedge, the farmers are thrown out: some are stripped of their possessions, circumvented by fraud or overcome by force; or worn out by injustices, they are forced to sell. One way or another, the poor wretches depart, men, women, husbands, wives, orphans, widows, parents with little children and a household which is numerous rather than rich, since agriculture requires many hands, they depart, I say, from hearth and home, all that was known and familiar to them, and they cannot find any place to go to. All their household furnishings, which could not be sold for much even if they could wait for a buyer, are sold for a song now that they must be removed. They soon spend that pittance in their wanderings, and then finally what else is left but to steal and to hang—justly, to be sure—or else to bum around and beg? For that matter, even as vagrants they are thrown into jail because they are wandering around idly, though no one will hire them, even when they offer their services most eagerly. For since no seed is sown, there is no farm labor, and that is all they are accustomed to. One herdsman or shepherd is sufficient to graze livestock on ground that would require many hands to cultivate and grow crops.
" 'And for this reason the price of grain has risen sharply in many places. Even the price of wool has gone up so high that poorer people who ordinarily make cloth out of it in this country cannot buy it, and for that reason many of them are out of a job and reduced to idleness. For after pastureland was expanded, huge herds of sheep were carried off by a murrain, as if God were punishing the owners' greed by visiting on the sheep a pestilence which might more justly have been hurled at the heads of their owners. But even if the number of sheep should increase enormously, the price still does not go down, because, though the sellers cannot be said to have a monopoly since more than one is selling, still it is certainly an oligopoly. For the sheep have almost all come into the hands of a few, and these men are so rich that they are under no necessity to sell until they want to, and they do not want to until they get the price they want.
" 'For the same reason other kinds of livestock are similarly high-priced, and all the more so because, once the farmhouses have been torn down and agriculture neglected, there is no one to see to the breeding of animals. For even those rich landholders do not rear other animals as they do sheep. Rather they buy them lean and cheap in some distant market and then sell them dear after fattening them up in their pastures. And for that reason, I think, the full disadvantage of this system has not yet been felt. I mean that up to now they have raised the prices only in the places where the animals are sold. But when the time comes that they are taken from the breeders faster than they can be bred, then finally the numbers will also gradually decrease where they are bought, so that here also there must needs be a severe shortage. Thus the very feature that seemed to make your island extremely fortunate has been turned into an instrument of its destruction by the wicked greed of a few men. For these high food prices are the reason why everyone dismisses as many as he can from his household—to go where, I ask you, except to go begging or else, as a noble spirit can more easily be persuaded to do, to turn to robbery.
" 'What shall we say when this miserable poverty and want is coupled with wanton luxury? For the retainers of noblemen, artisans, and one might say even some peasants and, in sum, all classes of society indulge in extravagant sartorial display and excessive, luxurious cuisine. And then the cookshops, the brothels, the bawdy houses, and those other sorts of bawdy houses, the wine bars and alehouses, and then so many crooked games of chance, dice, cards, backgammon, tennis, bowling, quoits, don't all these quickly empty pockets and send their votaries off to rob someone? Get rid of these pernicious plagues, make laws requiring that villages and towns be rebuilt by those who have torn them down or be handed over to those who are willing to restore and rebuild them. Keep the rich from cornering the market and from having a licensed monopoly, as it were. Let fewer people be supported in idleness, let agriculture be restored, let cloth working be reinstated as an honest trade which will give useful employment to this idle mob, whether those whom poverty has already turned into thieves or those who are now vagabonds or idle servants—in either case they will turn out to be thieves.
" 'Certainly unless you remedy these evils, it is pointless for you to boast of the justice administered in the punishment of thieves, a justice which is specious rather than either just or expedient. In fact when you bring people up with the worst sort of education and allow their morals to be corrupted little by little from their earliest years, and then punish them at last as grown men when they commit the crimes which from childhood they have given every prospect of committing, what else are you doing, I ask you, but making them into thieves and then punishing them for it?'
"As I was saying this, the lawyer was already getting ready to speak and had decided to employ that common method of disputants who are more diligent in repeating than in replying—so high is their opinion of memory. 'A very fine speech indeed,' he said, 'especially for a stranger who has only had more opportunity to hear about these matters than to get any precise knowledge of them, as I shall make clear in a few words. For first I shall recount in an orderly way what you have said; then I shall show on what points your ignorance of our affairs has misled you; finally, I shall rebut and refute all your arguments. Therefore, to begin with the first task I promised to undertake, you seem to have made four—'
" 'Be quiet,' said the Cardinal, 'for it seems hardly likely that you will reply in a few words after such a beginning. Hence, for the present we will relieve you of the trouble of replying, but we reserve that whole task for you when you two meet again, which I wish to be tomorrow, if nothing prevents you or Raphael here from meeting then. But meanwhile, my dear Raphael, I would very much like to hear why you think theft should not be punished with execution, or what other punishment you would enact that would contribute more to the common good. For even you do not think we should put up with it. But if people rush into thievery now when it is punishable by death, and then if they could once be sure of their lives, what force, what fear could possibly restrain criminals? They would interpret the mitigation of the punishment almost as an incentive or reward for wrongdoing.'
" 'Most gracious Father,' I said, 'it seems to me to be entirely and absolutely unjust to take a person's life because he has taken some money. For a human life cannot be equated with the goods of fortune, not even the whole sum of them. But if they say that this punishment is redress not for money but for the transgression of justice or the violation of laws, would it not be right to call this extreme justice extreme injury? For we ought not to approve of legal decrees so Manlian that the slightest infraction causes the sword to be unsheathed nor should we accept the Stoic maxim that all sins are equal, making no distinction between killing a person or stealing a coin from him, for between these two crimes (if fairness means anything at all) there is no similarity or relationship. God forbade us to kill anyone, and are we so ready to kill someone because he has taken a bit of money? But if someone should interpret that command to mean that the power to kill anyone is taken away except when human law declares a person should be killed, what is to prevent human beings from using the same principle to decide to what degree rape, adultery, or perjury are permissible? In fact, God has deprived us of the right to kill not only others but also ourselves, but if the mutual consent of human beings to specific laws allowing them to kill one another has enough force to release their agents from the bonds of God's commandment and enable them, with no precedent from God, to execute anyone condemned to death by human law, will that not mean that God's commandment has only as much force as is granted to it by human law? And indeed on this principle human beings may decide to what degree God's commands are to be observed in all fields. Finally, the law of Moses, though it was harsh and severe because it was made for slaves, and stubborn ones at that, still punished theft with a fine, not death. Let us not think that in his new law of mercy, by which he commands us as a father does his children, God has granted us greater license to be cruel to one another.
" 'These are the reasons why I think this punishment is wrong. And I think there is no one who does not understand how absurd and even dangerous it is to society to punish theft and murder in the same way. For when a thief sees that he is in no less danger if he is convicted of theft than if he had also been condemned for murder, that consideration alone will drive him to kill someone whom otherwise he would only have robbed. For apart from the fact that there is no more danger if he is caught, murder makes him more safe and gives him a greater hope of concealing his crime, since the witness to it has been eliminated. Thus by using excessively harsh measures to terrify thieves we encourage them to kill the innocent.
" 'As for the usual question of what punishment would be more advantageous, in my judgment it would be quite a bit easier to find a better one than to find one that is worse. For why should we doubt the utility of that way of punishing criminals which we know was once preferred for so long by the Romans, who were quite expert in the art of governing? Those convicted of serious crimes were condemned to quarry stone or dig out ore, constantly shackled and guarded. But as for me, on this point I reserve my highest approval for the system practiced by a people generally called the Polylerites whom I encountered in the course of my travels in Persia. Their population is not small and their institutions are not lacking in prudence; except that they pay an annual tribute to the king of Persia, they are otherwise free and allowed to make their own laws. But because they are a long way from the ocean and almost entirely surrounded by mountains, and because they are content with the produce of their land, which is by no means infertile, they neither visit others nor are visited very often themselves. In accord with the ancient policy of their country, they do not seek to extend their territory, and what they already have is easily protected by the mountains and the tribute paid to their overlord. They have absolutely no armed forces, their lifestyle is hardly splendid but it is comfortable, and they are happy rather than renowned or illustrious. Indeed, even their name, I think, is not very well known except to their immediate neighbors.
" 'And so, among them whoever is convicted of theft restores what was taken to its owner, not (as elsewhere) to the prince, for they consider he has no more right to stolen goods than the thief himself. If the goods have been lost, their equivalent is paid from the possessions of the thief, and whatever is left is handed over intact to his wife and children. He himself is condemned to hard labor.
" 'Moreover, unless the theft was committed with violence, they are not shackled or imprisoned but left free and unconstrained as they work on public projects. Shirkers and slackers are not restrained with shackles but egged on with the lash. If they work energetically, they are subjected to no humiliation; they are locked up in their cells only at nighttime after roll call. Except for constant labor their lives are not uncomfortable. Since they are doing public works they are fed at public expense, and not badly, but in different ways in different places. In some places what is spent on them is collected as alms; and this method, though it is unpredictable, has nevertheless been found to be the most productive because the people there are compassionate. In other places public revenues are set aside for that purpose. There are places where they levy a tax on private individuals to support the prisoners. Actually, in other places they do not do public works, but when a private person needs workmen he goes to the city square and hires some of them for that day at a fixed wage, which is a little less than what a freeman would cost. Moreover, if a slave is lazy it is permissible to whip him. Thus no one ever lacks work. And over and above his keep, each of them brings something into the public treasury every day.
" 'They are all dressed in one color and they are the only ones who wear it. Their hair is not shaved off but it is clipped a bit short above the ears. A little piece of one ear is cut off. Their friends are allowed to give any of them food, drink, and clothing of the right color. But it is death to give them money, both for the donor and the recipient, nor is it any less dangerous for a freeman to take money from them for any reason whatsoever or for a slave (for that is what they call the convicts) to lay a hand on a weapon. Each district has its own distinguishing badge, which it is a capital crime to throw away, just as it is to be seen outside the district boundaries or to say anything to a slave from another district. To plan an escape is no safer than to attempt it. In fact, to be an accessory to such a plan is death for a slave and enslavement for a freeman. On the other hand, rewards are allotted to informers: for a freeman money, for a slave freedom, and for either one pardon and amnesty for their complicity, to keep it from seeming safer to carry out a criminal plan than to repent of it.
" 'This law and the system I have described constitute their policy in this matter. It is perfectly obvious how humane and advantageous it is since vengeance is managed in such a way as to eliminate the vice and preserve the person, and to handle him in such a way that he has to be good and will spend the rest of his life making up for the harm he has done. Furthermore, there is so little fear that they will revert to their former ways that travelers who intend to make a journey consider no guides to be safer than these slaves, whom they exchange from district to district. For there is no opportunity whatever to commit robbery: they are unarmed; money is of no use except as evidence of a crime; punishment is in store for them if they are caught; and there is absolutely no hope of escaping anywhere. For how could a person whose clothes are totally different from anyone else's cover up his escape and disguise himself unless he ran away naked? And even then his ear would give him away. But couldn't they at least conspire to overthrow the republic?—that surely is the real danger. As if any district could hope to do so without sounding out and enlisting the slave gangs of many other districts! They are so far from being able to conspire that they cannot even meet or converse or greet one another. And then how can we believe anyone would dare to trust his companions with such a plot, since it is dangerous for them to remain silent and most advantageous to reveal it? On the other hand, if they are patient and obedient, if they give good reason to believe that they will lead reformed lives in the future, none need despair of regaining his freedom; indeed not a year goes by in which some slaves who have recommended themselves by their patience are not reinstated.'
"When I had said this, and had added that I saw no reason why this system could not be set up also in England, and with much more benefit than the justice which the lawyer had praised so highly, then he (namely the lawyer) said: 'This system could never be established in England without enormous danger to the commonwealth.' As he said this he shook his head, puckered his mouth, and fell silent. And everyone there jumped on his bandwagon.
"Then the Cardinal said, 'It is not easy to predict whether the outcome would be favorable or not without at least trying it out. But when the death sentence has been pronounced, if the prince were to grant a reprieve without any right of asylum in order to see how the system would work, and then if in fact it turned out to be useful, it would be right to establish it. If not, then thieves who had been condemned earlier could be executed at that time; on the part of the government this would be neither less nor more unjust than immediate execution, and during the trial period it would pose no danger. In fact, it seems clear to me that it would not be a bad idea to treat vagabonds also in the same way, for in spite of the many laws made against them, we have still made no progress.'
"When the Cardinal had said this, there was no one there who did not vie with the others in praising what they had scorned when I proposed it, but especially the part about the vagabonds because the Cardinal himself had added it on.
"I do not know whether or not it would be better to say nothing about what happened then, for it was quite silly. But I will tell it anyway, for it was not malicious and it has some bearing on our subject. A certain hanger-on was standing around. It seems he wanted to play the fool, but he did it so well that he seemed to be one, raising a laugh with such witless jokes that the laughter was directed more often at him than at the jokes. But every now and then he came up with something not entirely absurd, so as to confirm the proverb 'Throw the dice often enough and you will sooner or later get a lucky combination.' One of the guests said that in my discourse I had made good provision for thieves and the Cardinal had also taken care of the vagabonds, and now all that remained was to make public provision for those whom disease or old age had rendered destitute and who were incapable of returning to the jobs by which they had earned their living. 'Leave that to me,' said the hanger-on. 'I will see to it that this is also properly taken care of. In fact, I am desperately anxious to ship off this sort of person somewhere out of my sight; they annoy me so much with their wailing and whining and pleas for money, though they can never sing so pretty a tune as to extract a penny from me. Actually, one of two things happens: either I don't want to give them anything or I don't have anything to give. And so they have now begun to get wise. To keep from wasting their effort, they keep silent when they see me passing by. Good lord, they no more hope for anything from me than if I were a priest. I would decree by law that all such beggars be divided up and parceled out among the Benedictine monasteries where they would become lay brothers (as they are called); and the women I would order to become nuns.'
"The Cardinal smiled and took it as a joke, but the others took it seriously. A certain friar, however, a theologian, was so delighted by a joke aimed at priests and monks that he himself also began to make merry, though he was otherwise so serious as to be almost sour. 'But even this,' he said, 'will not free you from beggars unless you also look out for us friars.'
" 'But that is already taken care of,' said the hanger-on. 'For the Cardinal looked out for you marvelously well when he proposed that vagabonds should be confined and put to work, for you are the greatest vagabonds of all.'
"After they all had looked at the Cardinal and saw that he did not reject this joke either, they were not at all loath to enjoy it, all except the friar. Needled in this fashion, he was indignant and furious (nor am I surprised that he was), so much so that he couldn't even refrain from hurling insults. He called the fellow a scoundrel, a backbiter, a sneak, and a son of perdition, all the while citing terrible threats from Holy Scripture. Now the buffoon began to do some serious buffoonery, for he was clearly on his own ground.
"'Do not grow angry, my good friar,' he said, 'for it is written, "In your patience you shall possess your souls."'
"The friar replied (and I will give his very own words), 'I am not angry, you jailbird, or at least I do not sin. For the psalmist says, "Be angry and do not sin."'
"Then the friar was gently advised by the Cardinal to control his emotions, but he said, 'No, my lord, my language springs from nothing but good zeal, as it should, for holy men have had good zeal, whence it is said, "Zeal for your house has consumed me," and we sing in church, "Those who mock Elisha as he goes up to the house of God feel the zeal of the bald man," just as perhaps this mocking and ribald rascal will feel it.'
" 'Perhaps you are acting out of a laudable feeling,' said the Cardinal, 'but it seems to me that you would act, if not in a holier, then certainly in a wiser way, if you would not put yourself on the level of a fool and set out to cap his absurdities with your own.'
" 'No, my lord,' he said, 'I would not act more wisely. For Solomon, the wisest of men, says, "Reply to a fool in accord with his folly," as I am now doing, and I am showing him the pit into which he will fall if he does not watch out. For if the multitude which mocked Elisha, who was just one bald man, felt the zeal of the bald man, how much more will be heaped on a single person who mocks a multitude of friars, for many of them are bald. And also we have a papal bull which excommunicates anyone who makes fun of us.'
"When the Cardinal saw there would be no end to it, he sent the hanger-on away with a motion of his head and opportunely turned the conversation to another subject. A little later he arose from the table and, dismissing us, devoted himself to hearing the petitions of suitors.
"See, my dear More, how I have burdened you with a long discourse, and I would be quite ashamed of myself for doing so if you had not eagerly importuned me and seemed to listen as if you wanted no detail of this conversation omitted. Though I should have been more brief, still I did at least feel obliged to tell it to show how judiciously they scorned the plan when I proposed it and how the very same persons immediately reversed themselves and approved it when the Cardinal did not disapprove of it. Their flattery of him went so far that they seriously favored and almost accepted the ideas of his hangeron because his master took them as a joke and hence did not scorn them. From this you can judge how high an estimation courtiers would have of me and my advice."
"Indeed, my dear Raphael," I said, "you have given me much pleasure, you told the whole story so judiciously and so deftly. Moreover, while you spoke I seemed not only to have returned to my homeland but also to have grown young again because of fond memories of the Cardinal, in whose household I was educated as a boy. When you honored his memory so highly, you cannot imagine how much dearer you became to me on that account, though you were already most dear. But I am by no means ready to change my mind yet. No, I am convinced if you could bring yourself not to shrink from the courts of princes, you could contribute a great deal to the common good through your advice. No duty of a good man (and you are one, of course) is more important than that. Then too, since your friend Plato thinks that commonwealths will be happy only when philosophers become kings or kings become philosophers, how far will we be from happiness if philosophers will not even deign to impart their advice to kings."
"They are not so disagreeable as that; they would do so gladly. Indeed they have already done so by publishing many books, if those in power were prepared to accept their good advice. But undoubtedly Plato clearly foresaw that unless kings became philosophers, they would never give their approval to the advice of philosophers, because since childhood they have been thoroughly imbued and infected with misguided notions. He also found this out for himself when he was with Dionysius. But don't you think that, if I proposed sound measures to some king and tried to eradicate from his mind the seeds of corruption, I would be banished or held up as a laughingstock!
"Come now, imagine that I serve the French king and sit in his council chamber, as the king himself presides in a secret session, surrounded by a most judicious circle of advisers who are very eagerly seeking out wiles and stratagems to keep Milan and win back Naples (which is always slipping from his fingers), and then to overthrow Venice and make all of Italy subject to him, and then to bring Flanders, Brabant, and finally all of Burgundy into his control, and other peoples as well, whose realms he has long had it in mind to invade. At this meeting, while one urges that a treaty be struck with Venice, to last only as long as it suits the French, and that the French share their plans with them and even give them some share of the spoils, which they can reclaim when matters have been satisfactorily settled; while another advises them to hire German mercenaries, another to soothe the Swiss with payments of money; someone else, on the other hand, thinks that his divine majesty the emperor ought to be propitiated with a votive offering, as it were, of gold; while another thinks it best to strike a bargain with the king of Aragon, granting him the kingdom of Navarre (which belongs to someone else) as the price of peace; and on the same occasion another suggests that the prince of Castile should be snared by the prospect of a marriage alliance and that some nobles of his court should be brought over to the French side by giving them reliable pensions; when the greatest difficulty of all is encountered, namely what to do in the meantime about England; but they agree that a peace treaty should be negotiated with them, for a weak bond should always be tightened by the strictest terms; let them be called friends but be suspected as enemies; and that therefore the Scots should be stationed in readiness, poised on all occasions to attack immediately if the English make any moves; moreover, that some exiled noblemen be supported secretly (for treaties forbid that it be done openly) who can claim that the kingdom is rightfully his so that the French king will have a rein to check an English king he does not trust—at this council, I say, amidst such a mass of suggestions, surrounded by such distinguished men, all vying to give advice about going to war, if such a nobody as I were to stand up and give an order to tack in a different direction, expressing the opinion that Italy should be ignored and that the king should stay at home, that France is a kingdom so large that it is not easy for one man to rule it (much less should the king imagine he should consider adding others to it); and then if I should put before them the measures adopted by the Achorians, whose country faces the island of Utopia on the southeast side; if I should tell them that they had once fought a war to gain for their king a realm which he claimed to inherit because of some ancient marriage tie, and that, when they finally won it, they saw that they would endure no less suffering in keeping it than they did in gaining it, but rather that the seeds of war were always sprouting up, either rebellion within or incursions from without against the subjected people, so that they were always having to fight either for them or against them; that they never had an opportunity to disband their army, and that at the same time they were being stripped of their resources, their money was being carried out of the country, their blood was being spilled to provide someone else a smidgeon of glory, that they were no safer during peacetime; that at home the war had corrupted morals, imbued the citizens with a lust for robbery, that slaughter in warfare made them completely reckless, that they scorned the laws because the king was so distracted by trying to take care of two kingdoms that he couldn't concentrate on either one. When they saw that otherwise there would be no end to these great troubles, they finally took counsel together and very courteously gave their king the choice of retaining whichever of the kingdoms he wished; but they said he could not have power over both because they were too numerous to be governed by half a king (indeed no one is willing to share even a muledriver with someone else). And so the good prince left his new kingdom to one of his friends (who was soon afterwards banished) and was forced to be content with his old one. Furthermore, if I showed that all these abortive wars, which had thrown so many countries into turmoil for his sake would exhaust his treasury, destroy his people, and in the end still come to nothing through some mishap or other; and that therefore he should care for the kingdom of his ancestors, improve it as much as he could, make it as flourishing as possible; he should love his own and be loved by them; he should live with them, govern them kindly and leave other kingdoms alone, since the kingdom which has fallen to his lot is enough, and more than enough, for him—how do you imagine, my dear More, my listeners would react to this speech?"
"Certainly not very favorably," I said.
"Let us proceed, then," he said. "If counsellors were in a discussion with some king or other and were thinking up schemes to fill up his treasury, while one person suggests increasing the value of the currency when the king pays out money and decreasing it exorbitantly when he collects it so that he can discharge a large debt with a little money and collect a great deal when he is owed only a little; while another urges him to pretend he is going to war and to use that pretext to raise money and then, when it suits him, to make peace with religious ceremonies, pulling the wool over the people's eyes and making them think that he is a conscientious, merciful prince who wishes to spare them bloodshed; while another reminds him of certain antiquated, moth-eaten laws, long since fallen into disuse, laws which everyone ignores since no one even remembers that they were passed, and advises that he should therefore enforce them with fines, noting that no source of revenue could be more productive, none more honorable, since it has the appearance of a concern for justice; while another advises him to prohibit many practices with heavy fines, especially those that are contrary to the public interest, noting that later he can make a monetary arrangement with those whose interests are hurt by the laws and that thus he can win the gratitude of the people and make a double profit, first from fining those whom greed has led into his trap and then by selling dispensations to others (the higher the price the better the prince, since he is reluctant to grant a private person the right to obstruct the common good, and therefore does it only for a high price); while someone else persuades him to put pressure on judges to rule in his favor in all cases and advises him to summon them to his palace where they are to discuss his affairs in his own presence, saying that thus no case will seem so flimsy that his judges (whether out of love of contradiction, or a desire to seem original, or a wish to curry favor) cannot, in his presence, find some loophole for a false verdict, noting that when the judges give differing opinions and argue about a case that is as clear as day, the truth can be called into question and the king will have a convenient handle to interpret the law in his own favor, pointing out that the others will acquiesce out of shame or fear and thus the judgment can be fearlessly rendered in court, nor can there be any lack of pretexts for someone ruling in the prince's favor, since he has on his side either equity or the letter of the law or a twisted interpretation of the language, or something that outweighs all laws in the minds of conscientious judges, the indisputable royal prerogative; while everyone agrees completely with that saying of Crassus that no amount of gold is sufficient for a king, since he has to maintain an army, and moreover that a king can do no wrong, no matter how much he wants to, since all the possessions of all his subjects, and even their own persons, belong to him, and since nothing belongs to anyone unless the king graciously refrains from taking it away from him, and that he should leave as little as possible to his subjects since his safety consists in keeping the people from enjoying too much wealth or freedom, which render them less willing to put up with harsh and unjust commands, whereas on the other hand poverty and privation break their spirits and make them patient, depriving the oppressed of the lofty aspirations needed for rebellion; at this point, if I should stand up and contend that all this advice is both dishonorable and harmful to the king, for not only his honor but also his safety depends more on the people's wealth than on his own; if I were to show that the people choose a king for their own sake, not his, since his labor and effort enable them to live in comfort and safety; and that therefore a prince should be more concerned with the welfare of his people than with his own, just as it is the duty of a shepherd, insofar as he is a real one, to feed his sheep and not himself; that experience itself shows how wrong they are in thinking that the poverty of the people is the safeguard of peace, for where can you find more quarrels than among beggars? who is more intent on changing things than someone who is most dissatisfied with his present state in life? or, finally, who is more driven to create a general disturbance in the hope of gaining something than someone who has nothing to lose? But if a king is so scorned and hated by his subjects that he cannot make them do their duty unless he harasses them with maltreatment, plundering, and confiscation and reduces them to poverty, it would certainly be better for him to abdicate his throne than to retain it by methods which may keep the name of authority but have certainly lost all of its majesty, for it does not befit the dignity of a king to rule over beggars but rather over wealthy and happy subjects; that was certainly what was meant by that upright and lofty spirit Fabricius, when he replied that he would rather rule over the rich than be rich himself. Indeed, for one person to wallow in pleasure and luxury while he is surrounded on all sides by grieving and groaning, that is to be the guardian not of a kingdom but of a prison; finally, just as a physician is totally incompetent if he cannot cure a disease except by means of another disease, so too someone who does not know how to improve the lives of citizens except by depriving them of the comforts of life is admitting that he does not know how to rule over a free people; instead he should cure either his sloth or his pride, for these are usually the vices that make his people despise and hate him; he should live harmlessly on his own income, adapt his expenses to his income; he should curb crime and, by educating his people properly, prevent it rather than allow it to increase and then punish it; he should not be hasty to revive laws which are customarily ignored, especially those which are long disused because they were never desirable; he should never take something as a fine which a private person would not be allowed to accept because to do so would be criminal and deceitful. At this point, what if I told them that the Macarians, who are also not very far from the Utopians, have a law requiring their king to swear formally and solemnly on the very first day of his reign that he will never have in his treasury at one time more than a thousand pounds in gold or the equivalent amount of silver? They say that a king who was more concerned about the welfare of his land than about his own wealth made this law to prevent the heaping up of so much treasure as to impoverish his people; for he saw that this amount would be enough either for the king to fight against rebels or for the kingdom to repel a hostile invasion but would be too little to encourage him to invade other countries—and that was the primary reason for making the law. A secondary reason was that he thought it would make enough money available for the ordinary business transactions of the citizens; and since any money which accrues over that limit has to be paid back, he reckoned that a king would not seek out methods of extortion. A king such as this would be feared by malefactors and loved by his law-abiding subjects. If I should obtrude such notions and others like them on persons who are violently opposed to them, don't you suppose they would turn deaf ears as I told my tale?"
"Deaf as a post, undoubtedly," I said. "And, by heaven I am not surprised, and, to tell you the truth, I don't think you should obtrude such speeches or give advice which you are certain they will never accept. For how can it do any good or how can such an odd discourse influence the thinking of those whose minds are prejudiced and dead set against such notions? In private conversation with good friends this academic philosophy is not unpleasant. But there is no room for it in the council chambers of kings, where great matters are handled with great authority."
"That is what I said," he replied. "Among princes there is no room for philosophy."
"Yes indeed, there is," I said, "but not for this academic philosophy which considers anything appropriate anywhere. But there is another sort of philosophy better suited to public affairs. It knows its role and adapts to it, keeping to its part in the play at hand with harmony and decorum. This is the sort you should use. Otherwise, during a performance of a comedy by Plautus, when the slaves are joking around together, if you should come out onto the stage dressed like a philosopher and recite the passage from _Octavia_ where Seneca argues with Nero, wouldn't it have been better for you to have a non-speaking part than to jumble together tragedy and comedy by reciting something inappropriate? By hauling in something quite diverse, you would spoil and distort the play then being presented, even if what you add were better in itself. Whatever play is being presented, play your part as best you can and do not disturb the whole performance just because a more elegant play by someone else comes to mind.
"That's how it is in the commonwealth; that's how it is in the councils of princes. If you cannot thoroughly eradicate corrupt opinions or cure long-standing evils to your own satisfaction, that is still no reason to abandon the commonwealth, deserting the ship in a storm because you cannot control the winds. You should not din into people's ears odd and peculiar language which you know will have no effect on those who believe otherwise, but rather by indirection you should strive and struggle as hard as you can to handle everything deftly, and if you cannot turn something to good at least make it as little bad as you can. For everything will not be done well until all men are good, and I do not expect to see that for quite a few years yet."
"In that way," he said, "I would be doing no more than trying to remedy the madness of others by succumbing to their madness myself. For if I want to tell the truth, then I have to say such things. I do not know whether it is proper for a philosopher to say what is false, but it certainly isn't for me. Though that discourse of mine might perhaps have been irksome and repugnant to them, I do not see why it should seem odd to the point of absurdity. If I were to describe everything Plato imagines in his _Republic_ or what the Utopians do in theirs, these things might be better (as they surely are), but they might still seem strange, because here we have private property and there all things are held in common.
"As for my speech (except that those who have decided to run headlong down a different path cannot be pleased by someone who calls them back and points out the dangers), but otherwise what was there in it that it is not fitting and even obligatory to say anywhere? Indeed if we are to avoid as odd or absurd everything that has been made to seem alien by the corrupt morals of mankind, we Christians will have to ignore almost all Christ's teachings, and he forbade us to ignore them, so much so that the teachings which he himself whispered in the ears of his disciples, he commanded them to preach openly from the rooftops. And most of his teachings are far more alien to our common customs than that speech of mine was, except that preachers (following your advice, I imagine), whenever mankind refuses to make their behavior conform to the rule of Christ, adapt Christ's teaching to the behavior as if it were a ruler made of lead, so as to make the two match in some way or other. I don't see what good that does except to allow people to be wicked with a better conscience.
"And that, indeed, is all the good I would do in the councils of princes. For my opinion would either be different, and that would amount to having no opinion at all, or it would be the same, and I would be the abettor, as Terence's Mitio says, of their madness. For I do not see what you mean by that indirect approach of yours which you think enables you to manage things deftly even if you cannot make everything good, and at least make them as little bad as you can. For there is no room there to dissemble or to look the other way: you must approve of advice that is clearly quite bad and subscribe to measures that are utterly pestilential. Anyone who gave faint praise to wicked advice would be taken for a spy or perhaps a traitor. There will be no occasions on which you can do any good, since you have fallen among colleagues who will corrupt the best of men before they themselves will be reformed; either you will be depraved by their evil way of life or, if you remain honest and innocent, you will be made a screen for the wickedness and folly of others. That is how far you are from being able to improve anything by that indirect approach.
"That is why Plato, in a very elegant simile, explains why wise men are right to refrain from taking on governmental tasks: when they see people rushing out on the streets only to be soaked by never-ending rain and they cannot persuade them to get under a roof and out of the rain, they get under shelter themselves, knowing that they will accomplish nothing by going out except to get drenched together with the rest and considering it sufficient, when they cannot cure the folly of others, at least to remain in safety themselves.
"But actually, my dear More (to tell you truly what I really think), it seems to me that wherever there is private property, where everything is measured in terms of money, it is hardly ever possible for the common good to be served with justice and prosperity, unless you think justice is served when all the best things go to the worst people or that happiness is possible when everything is shared among very few, who themselves are not entirely happy, while the rest are plunged into misery.
"Therefore, when I turn over in my mind the most prudent and holy institutions of the Utopians, who have very few laws and yet manage so well that virtue is rewarded and yet, since everything is equalized, everyone has plenty of everything, and then when I contrast their customs with those of other nations, always issuing ordinances but none of them all ever achieving order, where whatever a person can get he calls his own private property, where a mass of laws, enacted day after day, are never enough to ensure that anyone can protect what each calls his own private property or even adequately distinguish it from what belongs to someone else (as can easily be seen from the infinite lawsuits which are always being filed and are never finished), when I consider these things, I say, I have a higher opinion of Plato and I am not surprised that he would not deign to make any laws for people who would not accept laws requiring that all goods be shared equally by all. In his great wisdom he easily foresaw that the one and only path to the welfare of the public is the equal allocation of goods; and I doubt whether such equality can be maintained where every individual has his own property. For where everyone tries to get clear title to whatever he can scrape together, then however abundant things are, a few men divide up everything among themselves, leaving everyone else in poverty. And it usually happens that each sort deserves the lot of the other, since the one is rapacious, wicked, and worthless, and the other is made up of simple, modest men who by their daily labor contribute more to the common good than to themselves.
"Thus I am firmly persuaded that there is no way property can be equitably and justly distributed or the affairs of mortal men managed so as to make them happy unless private property is utterly abolished. But if it remains, there will also always remain a distressing and unavoidable burden of poverty and anxiety on the backs of the largest and best part of the human race. I grant their misery may be somewhat alleviated but I contend that it cannot be fully eliminated. I mean, if you decreed that no one could own more than a certain amount of land and that there be a legal limit to the money anyone can possess, if some laws were enacted that could keep the prince from being too powerful or the people too headstrong, that would keep offices from being solicited or put up for sale, or keep them from entailing many expenses (for otherwise they provide opportunities to rake in money by fraud and spoliation or it becomes necessary to put rich men in offices which ought to be held by wise men), such laws, I say, could mitigate and alleviate these ills, just as applying continual poultices can relieve the symptoms of sick bodies that are beyond healing. But as long as everyone has his own property, there is no hope whatever of curing them and putting society back into good condition. In fact, while you are trying to cure one part you aggravate the malady in other parts; curing one disease causes another to break out in its place, since you cannot give something to one person without taking it away from someone else."
"Quite the contrary," I said, "it seems to me that no one can live comfortably where everything is held in common. For how can there be any abundance of goods when everyone stops working because he is no longer motivated by making a profit, and grows lazy because he relies on the labors of others. And then, when people are driven by want and there is no law which enables them to keep their acquisitions for their own use, wouldn't everyone necessarily suffer from continual bloodshed and turmoil? Especially when the magistrates no longer have any respect or authority, for I cannot conceive how they could have any among people who are all placed on one level."
"I am not surprised that you think so," he said, "since you have no conception of the matter, or only a false one. But if you had been with me in Utopia and had seen their customs and institutions in person as I did (for I lived there more than five years, and I would never have wanted to leave except to reveal that new world to others) you would quite agree that you had never seen a people well governed anywhere but there."
"But you would surely have a hard time persuading me," said Peter Giles, "that a better governed people can be found in that new world than in the one we know, since our intellects are no worse than theirs and our governments are older, I imagine, than theirs, so that long experience has brought to light many features which make our lives more comfortable, to say nothing of some things we have discovered by chance which no amount of ingenuity would have sufficed to invent."
"As for the antiquity of governments," said Raphael, "you could give a more accurate judgment if you had read through the histories of that world: if they are trustworthy, there were cities there before there were people here. As for what ingenuity has invented or chance revealed up till now, that could have happened in either place. But certainly I think that even though we may surpass them in intelligence, they still leave us far behind in diligence and zeal to learn.
"According to their chronicles before we landed there they had never heard anything about us Ultra-equatorials (for that is what they call us) except that some twelve hundred years ago a ship was driven to Utopia by a storm and shipwrecked there. Some Romans and Egyptians were cast upon the shore and never left there again. Notice how their diligence turned this single occasion to their advantage. There was no useful skill in the whole Roman empire which they did not learn from the explanations of the strangers or did not manage to discover from the hints and clues they were given. Such was the enormous gain they made on this one occasion when some men from here were driven to their shores. But if a similar accident ever brought one of them from there to here, the incident has been completely forgotten, just as posterity perhaps will also forget that I was once there. One meeting alone was enough for them to appropriate all of our useful inventions, but I think it will be a long time before we will accept any institution of theirs which is better than ours. And I think that is the only reason why they manage their affairs more prudently and live more happily than we do, though we are not inferior to them in intelligence or resources."
"Therefore, my dear Raphael," I said, "I beg and implore you, describe the island to us. And do not try to be brief but explain in order their fields, rivers, cities, population, customs, institutions, laws, and, in short, whatever you think we would want to know. And you should think we want to know whatever we don't know yet."
"There is nothing I would rather do," he said, "for I have all this at my fingertips. But it will take some free time."
"Then let us go inside to eat lunch," I said. "Afterwards we will take as much time as we want."
"Agreed," he said. And so we went in to eat lunch. After lunch we came back to the same place and sat down on the same bench, and having instructed the servants that we were not to be interrupted, Peter Giles and I urged Raphael to keep his promise. When he saw that we were attentive and eager to hear, he sat there quiet and thoughtful for a little while, and then began as follows.
THE END OF THE FIRST BOOK
The Discourse of Raphael Hythloday
on the Best Form of a Commonwealth
as Reported by
Thomas More, Undersheriff of London
## BOOK 2
The island of the Utopians is two hundred miles across in the middle, where it is widest, and throughout most of the island it is not much narrower, but toward both ends it narrows a bit. These ends, curling around into a circle with a circumference of five hundred miles, make the whole island look like a new moon. The sea flows in between the horns through a strait about eleven miles wide and then spreads out into a huge empty space protected from the wind on all sides, like an enormous, smooth, unruffled lake; thus almost the whole inner coast serves as a harbor and allows ships to go from shore to shore in all directions, much to the advantage of the people. The jaws of the strait are dangerous, on one side because of shallows, on the other because of rocks. In just about the middle of the channel, one rock stands out, visible and hence harmless; they have built and garrisoned a tower on it. The other rocks are hidden and treacherous. The channels are known only to the Utopians themselves, and hence it hardly ever happens that a foreigner enters the bay without a Utopian pilot. Indeed they themselves find it hard to enter it safely, except that they set their course by means of some signals on the shore. By moving these to different locations, they can easily lure an enemy fleet to shipwreck, no matter how large it is.
MAP OF UTOPIA,
WOODCUT FROM THE NOVEMBER 1518 EDITION
(Beinecke Rare Book and Manuscript Library, Yale University)
On the outside coast there are not a few ports. But everywhere the landing places are so well defended, either naturally or artificially, that a few troops can keep a huge army from coming ashore. According to report, however (and the appearance of the place bears it out), their land was once not surrounded by the ocean. But Utopus, who conquered the island and named it after himself (for before that time it had been called Abraxa) and who brought its crude and rustic mob to a level of culture and humanity beyond almost all other mortals, after he won the victory at his first assault, had a channel cut fifteen miles wide at the point where the land adjoined the continent, and thus caused the sea to flow all around the land. And since he set not only the inhabitants to this task but also employed his own soldiers (to keep the inhabitants from thinking the work was imposed on them as a humiliation), the labor was shared by a great multitude of workers and was finished in an incredibly short time, so that the neighboring peoples (who at first ridiculed the project as silly) were overwhelmed with wonder and fear.
The island has fifty-four cities, all of them large and splendid and having exactly the same language, customs, institutions, and laws. They have the same layout and they look the same, insofar as the terrain allows. Those which are closest to each other are separated by twenty-four miles. None is so isolated that it is more than a day's journey on foot from another city. Every year each city sends three old and experienced citizens to Amaurot to discuss problems common to the whole island. For that city, which is located at the navel of the land, so to speak, and hence is most convenient as a meeting place for the delegates from everywhere, is the capital and chief city.
The land is so well distributed that no city has less than twelve miles of ground on all sides, though it may have much more in some directions, namely where the cities are furthest apart from one another. None of them is driven by any desire to extend its boundaries. Indeed, whatever land they have, they consider themselves its tenant-farmers, not its landlords. In the countryside, throughout the fields, they have conveniently located houses, each provided with farming tools. They are inhabited by the citizens, who take turns going out to live there. No country household has fewer than forty men and women, besides the two slaves bound to the land; it is presided over by a master and mistress who are sober and mature. Every thirty households are ruled by one phylarch. Every year twenty from each household return to the city, having fulfilled their two-year stint in the country. They are replaced by twenty substitutes from the city, who are to be trained by those who have already been there a year and hence are more skilled in farmwork; the substitutes themselves will train another group the following year, for if everyone were new and equally ignorant of farming, the crops would suffer from lack of skill. Although this system of exchanging farmers is customary, to keep anyone from being forced to live this hard life for a long time, nevertheless many who have a natural bent for agricultural pursuits apply for and are allowed additional years.
They farm the land, raise cattle, cut wood, and convey it to the cities by the most convenient route, whether by sea or by land. They raise a huge number of chickens, and they have a marvelous method of doing it. The hens do not sit on the eggs. For the Utopians themselves tend a great number of eggs, keeping them alive and hatching them them in constant warmth. As soon as the chicks emerge from the shell, they recognize and follow human beings around as if they were their mothers.
They raise very few horses and none but high-spirited ones, which serve no other purpose than the training of young people in horsemanship. For ploughing and hauling they use oxen; they grant that they are inferior to horses in short sprints, but they consider them superior over the long haul and less subject to diseases; moreover, they require less effort and expense to maintain, and when they have served out their term, they can be used for food.
Grain they use only for bread. For they drink either wine made from grapes or cider made from apples or pears or else plain water, which they often boil with honey or licorice, of which they have plenty. Although they know (and they know it very well) how much produce is needed by a city and its surrounding population, they plant far more grain and raise far more cattle than they need for their own use, giving the surplus to their neighbors. All the supplies that are necessary but not available in the country they get from the city, giving nothing in exchange; the city magistrates provide them the goods with no bargaining. For every month many of them gather there on the feast day. On the day of harvesting, the phylarchs of the farmers inform the city magistrates how many citizens should be sent out; since they arrive at precisely the right time, such a large crowd of workers gets the harvest almost completely done in one day if they have good weather.
### THEIR CITIES, ESPECIALLY AMAUROT
If you know one of their cities, you know them all, so similar are they in all respects (so far as the terrain allows). And so I will describe one of them (it doesn't much matter which one). But why choose any one except Amaurot? For it is the most notable and takes precedence over the others because the senate meets there; and no other is better known to me, since I lived there for five whole years.
Amaurot, then, is situated on the gentle slope of a mountain; its shape is almost square. Beginning almost at the crest of the hill, it stretches two miles down to the river Anyder; its width is slightly greater along the river than it is at the hilltop. The source of the Anyder is eighty miles above Amaurot, a small spring which is amplified by tributaries, two of them sizeable, until, when it reaches the city itself, it is five hundred yards wide. Then for sixty miles it flows on, getting wider and finally flowing into the ocean. In the space between the city and the coast, and also for some miles above the city, the tide flows and ebbs for six whole hours in a swift current. Seawater flows in to a point thirty miles upstream, filling the whole channel of the Anyder and driving the river water upstream. It also makes the water salty somewhat higher up; from there the river gradually grows fresh and it is pure when it flows by the city. And at ebb tide it flows pure and fresh nearly all the way to the mouth of the river.
The city is connected to the opposite bank of the river by a bridge made not of pilings and planks but of beautifully arched stonework; it is placed at a point furthest from the sea so that ships can sail unobstructed along that whole side of the city. They also have another stream, not large but very gentle and pleasant, which gushes from a spring on the same mountain where the city is located; it flows down through the middle of the city into the Anyder. The Amaurotians have fortified the head and spring of this stream, which is located a little outside the city, surrounding it with walls that link it to the city, so that if an enemy ever attacks them, the water cannot be diverted or contaminated. From this stream the water is channeled in tile conduits to the various districts in the lower parts of the city. Where the terrain makes this impossible, rainwater collected in large cisterns serves the same purpose.
The city is surrounded by a high, thick wall with many towers and bastions. On three sides the wall is surrounded by a moat that is dry but wide and deep and blocked by thorn hedges; on the fourth side the river itself serves as a moat. The streets are laid out to facilitate traffic and to offer protection from the wind. The buildings are by no means ugly; the houses extend in a continuous row along the whole block, facing the row on the other side of the street; the housefronts along each block are separated by a street twenty feet wide. Behind the houses, a large garden, as long on each side as the block itself, is hemmed in on all sides by the backs of the rowhouses.
There is no house which does not have a door opening on the street and a backdoor into the garden. The double doors, which open easily with a push of the hand and close again automatically, allow anyone to come in—so there is nothing private anywhere. For every ten years they exchange the houses themselves by drawing lots. The Utopians place great stock by these gardens; in them they grow vines, fruit trees, herbs, and flowers, all so bright and well tended that I have never seen anything more flourishing and elegant. In gardening they are motivated not only by their own pleasure but also by competition among the various blocks to see which has the best garden. And certainly you will not easily find any feature of the whole city that is of greater use to the citizens or gives them more pleasure. For that reason the founder of the city seems to have devoted more attention to these gardens than he did to anything else.
For they say that in the very beginning Utopus himself laid out the whole plan of the city. But he left it to succeeding ages to complete the adornment and landscaping that could not be completed during one lifetime. Thus in their annals, which have been diligently and scrupulously kept up since the island was captured 1,760 years ago, it is recorded that at first their dwellings were humble, mere huts and shacks, built of wood gathered at random, the walls plastered with mud. The roofs came to a point and were thatched with straw. But now all houses have a handsome appearance and are built three stories high. The outer sections of the walls are made of fieldstone, quarried rock, or brick, and the space between is filled up with gravel and cement. The roofs are flat and are coated with a sort of plaster which is not expensive but is formulated so as to be fireproof and more weather-resistant than lead. They commonly use glass (which is very plentiful there) to keep out the wind; sometimes they also use thin linen, soaked in clear oil or treated with resin—a method which has two advantages: it lets in more light and keeps out more drafts.
### THEIR MAGISTRATES
Every year each group of thirty families elects its magistrate, who in their ancient language was called a syphogrant but is known as a phylarch in the modern tongue. Ten syphogrants with their households are presided over by an official once called a tranibor, now known as a protophylarch. Finally, all the syphogrants, who number two hundred, having sworn to choose the person they consider the most capable, elect the ruler by secret ballot, choosing him from the four candidates named by the people. For each of the four quarters of the city names one person and proposes him to the senate. The ruler remains in office for life, unless his tenure is interrupted because he is suspected of trying to become a tyrant. They elect the tranibors every year, but they do not lightly change them. All the other magistrates hold office for one year.
Every third day, and sometimes oftener if circumstances require it, the tranibors gather to advise the ruler. They make decisions about public affairs; if there are any disputes among private persons (and there are very few) they settle them in a timely fashion. They always invite two syphogrants into the senate, different ones on every occasion; and they have provided that no measures concerning public affairs be adopted unless they have been discussed in the senate three days before a decision is reached. To enter into schemes concerning affairs of state outside the senate or public assemblies is a capital crime. These measures were taken, they say, to make it hard for the ruler and the tranibors to conspire to change the form of government and set up a tyranny over the people. And for the same reason matters of great moment are presented at the assemblies of the syphogrants, who report the matter to the households, take counsel among themselves, and report their recommendations to the senate. Sometimes a matter will be referred to the council of the whole island.
Then, too, the senate has a rule that no point is discussed on the same day it is brought up, but rather it is put off till the next meeting; they do this so that someone who blurts out the first thing that occurs to him will not proceed to think up arguments to defend his position instead of looking for what is of use to the commonwealth, being willing to damage the public welfare rather than his own reputation, ashamed, as it were, in a perverse and wrong-headed way, to admit that his first view was short-sighted. From the start such a person should have taken care to speak with deliberation rather than haste.
### OCCUPATIONS
Farming is the one occupation in which all of them are skilled, men and women alike. They are all trained in it from childhood on, partly by instruction in the classroom, partly by being taken out to play at it, as it were, in the fields near the city, not merely looking on but doing the work themselves for bodily exercise.
Besides farming (which, as I said, is common to all of them) everyone is taught some trade of his own. The ordinary ones are working with wool or linen or laboring as a stone mason, blacksmith, or carpenter. No other trade there employs any number worth mentioning. As for their clothing—which is uniform throughout the island for all age groups and varies only to indicate sex or marital status, and which is not unappealing to the eye, allows freedom of movement, and is adapted to either heat or cold—as for their clothing, I say, each household makes its own.
Everybody learns one or the other of these trades, including women as well as men. But women, as the weaker sex, engage in lighter crafts, mostly working with wool or linen. The other trades, which require more strength, are relegated to the men. Generally children take up their father's trade, for most are naturally inclined to it. But if anyone is drawn to another occupation, he is transferred by adoption into another household where he can work at the trade he wants to pursue. The move is supervised not only by his father but also by the magistrates, to make sure the master of his adoptive household is respectable and responsible. Actually, if someone has mastered one trade and wants to learn another besides, he gets permission to do so by the same procedure. When he has mastered both, he practices whichever he wants to, unless the city has a greater need for the other.
The chief and practically the only function of the syphogrants is to take care and see to it that no one lounges around in idleness but rather that everyone practices his trade diligently, but not working from early morning till late at night, exhausted by constant labor like a beast of burden. For such grievous labor is fit only for slaves, and yet almost everywhere it is the way workmen live, except in Utopia. Dividing the day and night into twenty-four equal hours, they devote only six to work, three before noon, when they go to lunch. After lunch they take two hours of rest in the afternoon, then three more given over to work, after which they have dinner. Counting the first hour after noon as ending at one o'clock, it is eight o'clock when they go to bed. Sleep takes up eight hours.
The intervals between work, meals, and sleep they are allowed to spend however they like, provided that the time they have free from work is not wasted in debauchery and idleness but spent well in some other pursuit, according to their preference. Many devote these intervals to intellectual activities. For every day they have regular lectures in the hours before dawn; attendance is required only from those who have been specially chosen to devote themselves to learning. But a great number of men, and also women, from all orders of society flock to hear these lectures, some one sort, some another, as each is naturally inclined. But if someone wishes to spend this same time practicing his trade (as do many whose temperaments are not suited to any abstract discipline), they are quite free to do so; indeed they are also praised for doing so, since their labor contributes to the common good.
After dinner they devote one hour to recreation, during the summer in the gardens, during the winter in the common rooms where they have their meals. There they either play music or entertain themselves with conversation. They do not so much as know about dice and other such pointless and pernicious games, but they do play two games not unlike chess. In one of them numbers fight against each other, one taking over the other; in the other game virtues are lined up in a battlefront against the vices. This game shows very cleverly both how the vices fight among themselves but join forces against the virtues, and also which vices are opposed to which virtues, what forces they bring to bear openly, what instruments they use to attack indirectly, what defenses the virtues use to fend off the forces of the vices, how they evade their assaults, and finally by what methods one side or the other wins the victory.
But at this point, it is necessary to examine the matter in more detail to avoid making a mistake. If only six hours are devoted to work, you might think that there would necessarily be some shortage of supplies. But that is so far from being true that six hours is not only enough to produce abundantly all the necessities and comforts of life but is even more than enough. This you, too, will understand if you consider what a large part of the population in other countries live their lives in idleness. First, almost all the women do, and they make up almost half the population. Or in places where the women work, the men take their place and lie around snoring. Add to that the huge idle crowd of priests and religious, as they are called. Throw in all the rich, especially the landlords of estates who are commonly called gentlemen and nobles. Include with them their retainers, that rank cesspool of worthless swashbucklers. Add, finally, the strong and sturdy beggars who feign some disease as a pretext for their idleness. You will certainly find that it takes far fewer than you thought to produce everything that mortals use.
Now consider how few of these workers are occupied in necessary trades, since, where money is the measure of everything, many completely futile and superfluous crafts must be practiced just to support over-indulgence and wanton luxury. Now if that same crowd who are presently working were divided up among the few trades needed to produce the few commodities that nature requires, the resulting abundance of goods would drive prices down so low that craftsmen could not make a living. But if all those who work away at pointless tasks and, together with them, that whole crowd of lazy, languid idlers (any single one of whom consumes twice as much as any of the workers who produce the goods), if they all were put to work—and useful work at that—you can easily see how little time would be enough and more than enough time to produce all the goods required for human needs and conveniences—and pleasures, too, as long as they are true and natural ones.
And this very point is confirmed by the experience of the Utopians. For there, in the whole city and the surrounding territory, out of all the men and women who are old enough and strong enough to work, barely five hundred are exempted from work. Among them the syphogrants, who are legally relieved from work, nevertheless do not exempt themselves; they work so as to motivate others to work by giving a good example. The same immunity is enjoyed by those to whom the people give total leisure to pursue various branches of learning, but only after the priests have recommended them and the syphogrants have chosen them by a secret ballot. If any of them disappoints the hopes they had in him, he is put back to work; and on the other hand, it happens, not infrequently, that an artisan, devoting his free time to intellectual pursuits, works so diligently and makes such progress that he is exempted from working at his trade and promoted to the scholarly class. From this order of scholars are chosen ambassadors, priests, tranibors, and finally the ruler himself, who was called Barzanes in their ancient language, but is named Ademus in the modern tongue. The remaining group, which is neither idle nor devoted to useless trades, is so large that it is easy to imagine how many goods they produce in so few hours.
Apart from what I have just said, they have it easier because in most of the necessary trades they do not need to expend as much labor as in other nations. First of all, building or repairing structures everywhere else requires the continuous effort of so many workers for the simple reason that what a father has built his worthless heir allows to fall gradually into disrepair. Thus what could have been maintained with a minimum of effort has to be totally rebuilt, at great expense, by the next heir. Moreover, it often happens that a house that cost someone enormous sums to build seems contemptible to someone of more fastidious taste; after a short time it falls into ruin through neglect and the owner builds another house somewhere else, at no less expense. But among the Utopians, from the time when everything was settled and the commonwealth was established, it very rarely happens that a new site is chosen on which to build houses; and they not only repair damage quickly when it happens but they take preventive measures against it. The result is that their buildings last a very long time and require very little work, and sometimes construction workers have so little to do that they are set to shaping timbers or squaring and fitting stones at home, so that if they ever need to build anything, it can be constructed more quickly.
Now as for their clothing, notice how little labor it requires. First of all, at work they wear informal garments made of leather or skins which last for seven years. When they go out in public they put on cloaks which cover these rough clothes; throughout the island they are all of the same color, that of the natural wool. Thus they not only get along with much less woolen cloth than anywhere else, but it also costs much less. But linen is easier to work and hence they use more of it; they are concerned only about the whiteness of linen and the neatness of wool, for they place no value on fineness of weave. The result is that in other places four or five woolen cloaks and the same number of silk shirts are not enough for one person, and if he is a bit fastidious, not even ten will do, but there everybody is content with one, which generally lasts for two years. Naturally there is no reason why he should want any more, for if he got them he would have no more protection against the cold, and his clothing would not look the least bit more fashionable.
Therefore, since everyone is employed in a useful trade and the trades themselves require less labor, the result is a great abundance of everything, so that sometimes they bring out an enormous number of people to repair the public roads, if any have deteriorated. It happens very often, when there is no occasion even for that kind of work, that they publicly decree a shorter workday. For the magistrates do not compel anyone to engage in superfluous labor against his will, since the structure of the commonwealth is primarily designed to relieve all the citizens from as much bodily labor as possible, so that they can devote their time to the freedom and cultivation of the mind. For that, they think, constitutes a happy life.
### SOCIAL RELATIONS
Now is the time, I think, to explain how they treat each other, how they interact with one another, and what system they have for distributing goods.
And so, while the city is made up of households, the households themselves consist mostly of blood relatives. Girls, when they grow up and marry, move into the dwellings of their husbands. But sons and, after them, grandsons remain in the household and are subject to the oldest parent, unless his mind is failing because of old age; in that case he is replaced by the next oldest. But to keep the city from being either over- or underpopulated, they see to it that no household (and each city, apart from its territory, has six thousand of them) has fewer than ten or more than sixteen adults. For it is not possible to set a limit for children. This limit is easily maintained by transferring persons from households with too many people to those with too few. But if it should happen that the whole city grows too large, they use the excess to supply underpopulated cities. But if it should happen that throughout the island the whole mass of the population should swell inordinately, they sign up citizens from each city and send them as colonists to live under their own laws on the nearest part of the continent, wherever the natives have a lot of land left over and uncultivated; they adopt any natives who choose to live with them. Assenting willingly to the same style of life and the same customs, the natives are easily assimilated, and that to the advantage of both groups. For by means of their institutions the Utopians make the land easily support both peoples, whereas before it provided a meager and skimpy living for only one. The natives who refuse to live under their laws are driven out of the territory the Utopians have marked off for their use; if they resist, the Utopians make war against them. For they think it is quite just to wage war against someone who has land which he himself does not use, leaving it fallow and unproductive, but denying its possession and use to someone else who has a right, by the law of nature, to be maintained by it. If any of their cities is ever accidentally so reduced in population that they cannot replenish it from other parts of the island and still keep the full quota in those cities (which they say has only happened twice in their whole history because of a virulent plague), then they resupply it with citizens immigrating from a colony. For they would rather allow the colonies to disappear than let any of the cities on the island shrink in size.
But, to return to the citizens' way of life, the oldest man, as I said, presides over a household. Wives serve their husbands and children their parents, and generally the younger serve the older. Each city is divided into four equal districts. In the middle of each district is a marketplace for all sorts of commodities. The products of each household are taken to designated houses there and each kind of goods is separately stored in a warehouse. From them each head of household goes to get whatever he and his household need, and he takes away whatever he wants, paying no money and giving absolutely nothing in exchange for it. For why should he be denied anything, since there is plenty of everything and no one need fear that anyone would want to ask for more than he needs? For why should anyone be suspected of asking for too much if he is certain he will never lack for anything? Certainly fear of want makes all kinds of animals greedy and rapacious, but only mankind is made so by pride, which makes them consider their own glory enhanced if they excel others in displaying superfluous possessions; in the Utopian scheme of things there is no place at all for such a vice.
Adjoining the marketplaces I mentioned are food markets, to which vegetables, fruit, and bread are brought, and also fish and edible birds and beasts are conveyed from designated places outside the city where there is a stream to wash away refuse and offal. From here they bring the cattle which have been slaughtered and cleaned by the hands of bondsmen. For they do not allow their own citizens to become accustomed to butchering animals; they think that to do so gradually eliminates compassion, the finest feeling of human nature. They do not allow anything filthy or foul to be brought into the city, for air tainted by such rottenness might engender disease.
Furthermore, each block has spacious halls located at equal intervals, each known by its own name. The syphogrants look after them, and to each of them are assigned thirty families (namely fifteen on either side) who eat their meals there. Stewards from each hall gather in the market at a designated hour and get food according to the number of mouths they have to feed.
But their first priority is the sick, who are cared for in public hospitals. They have four of them on the outskirts of the city, a little outside the walls; they are as capacious as four little towns so that no matter how many people are sick they do not need to be crowded uncomfortably together, and so that those who have contagious diseases that can be transferred from one person to another can be kept at a distance from the main body of the patients. These hospitals are so equipped and provided with everything that promotes health, the care provided in them is so gentle and solicitous, the doctors who are in constant attendance are so skilled that, although no one is sent there against his will, there is still almost no one in the whole city who would not rather be lodged there than at home when he is in failing health.
After the stewards of the hospitals have received the food prescribed by the physicians, the best of what is left is divided equitably among the halls, according to the number fed by each one, except that they pay special attention to the ruler, the high priest, and the tranibors, and also to ambassadors and all foreigners (if there are any, for they are few and far between); but when there are any, designated residences are furnished and prepared for them. At the times fixed for lunch and dinner, the whole syphograncy, alerted by the blast of a bronze trumpet, convenes in these halls, except for those who are bedridden in the hospitals or at home. Nevertheless, no one is forbidden to take home food from the marketplace once the halls have been supplied with their quotas, for they know that no one would lightly choose to do so; though no one is prohibited from eating at home, still no one does it willingly, for it is not considered proper and it would be foolish to go to the trouble of preparing an inferior meal at home when a splendid and sumptuous one is ready and waiting in a hall nearby.
In this hall slaves perform all the chores which are somewhat heavy or dirty. But the women are solely responsible for preparing and cooking the food and making arrangements for the whole meal, each household taking its turn. They sit at three tables or more, according to the number of diners. The men sit with their backs to the wall, the women on the outside, so that if they should suddenly feel ill, as happens, sometimes, when they are pregnant, they can get up and go out to the nurses without disturbing the seating arrangement.
The nurses are seated separately with the nursing infants in a little room assigned to them; it never lacks a fire and clean water and also cradles so that when they want they can either lay them down or take off their swaddling clothes and let them refresh themselves by playing freely. Every mother nurses her own child unless death or disease prevents it. When that happens, the wives of the syphogrants immediately find a nurse, and that is not hard to do. For those who can are more than willing because everyone praises their compassion and the infant who is brought up this way takes the nurse as its natural mother.
Children who are under five sit in the nurses' den. Other minors, among whom they include members of both sexes who are not yet old enough to marry, either serve the diners, or, if they are too young and not strong enough for that, stand by—and that in absolute silence. Both groups eat what is handed to them by those seated at table, nor is any other time set aside for them to eat.
The syphogrant and his wife sit at the head table, which is the place of honor and overlooks the whole assembly, since it is placed crosswise in the highest part of the chamber. Next to them sit two of the oldest persons, for they sit in groups of four at all the tables. But if a church is located in that syphograncy, the priest and his wife sit with the syphogrant so as to preside. On both sides of them sit younger people, and then older people again, and so on throughout the whole hall. And so people sit with their coevals, and yet they are mixed in with a different age group. They say that this arrangement was adopted so that the dignity of the elders and the respect due them would keep the young people from indulging in improper language or behavior, since nothing can be done or said at table which would escape the notice of the persons sitting nearby on all sides.
The dishes of food are not served to the highest places and then downward to the others, but rather the choicest pieces are served first to the old people (whose places are marked) and then equitable shares are served to the rest. But some of the delicacies which are not in sufficient supply to be distributed to the whole hall are given by the old people, as they see fit, to those sitting near them. Thus respect for the elders is maintained and yet everyone has the same advantage from it.
Lunch and dinner always begin with some reading that concerns morals, but it is brief lest it be tedious. Taking off from this, the elders begin the discussion, but not in a gloomy and sour fashion. And they do not take up the whole meal with long disquisitions. No, they would much rather listen to the young people, and they even deliberately challenge them so as to learn about the temperament and intelligence of each of them as revealed in the free give and take of tabletalk.
Lunches are quite brief, dinners more ample because the one is followed by work and the other by rest and sleep during the night, which they think contribute more to good digestion. They never dine without music and after dinner they never lack for tasty desserts. They light incense and sprinkle perfumes and spare no effort to cheer up the diners. For they tend to incline to the position that no kind of pleasure ought to be forbidden as long as no harm comes of it.
This is the way they live in the city. But in the country, since they live far apart, they all eat in their own homes. No household has any shortage of food, since, after all, everything eaten by the city-dwellers comes from the farmers.
### HOW THE UTOPIANS TRAVEL
If someone wants to visit friends who live in another city or is simply taken with a desire to see the place, he easily gets the permission of his syphogrant and tranibor unless a necessary job keeps him from going. He is sent out as part of a group, with a letter from the ruler which grants them permission and sets the day they must be back. They are provided with a carriage and a public slave to drive the oxen and take care of them. But unless there are women in the group, they leave the carriage behind as more of a hindrance than a help. Throughout the whole journey they carry nothing with them; yet they lack for nothing and are at home everywhere. If they stay anywhere longer than one day, each of them works at his trade and is treated very kindly by his fellow craftsmen.
If someone takes it upon himself to wander outside his territory, when he is caught without the ruler's passport, he is treated with contempt, brought back as a runaway, and severely punished. If he dares to repeat the offense, he is punished with slavery. But if someone is taken with a longing to wander through the fields belonging to his own city, he is not prohibited from doing so, as long as he gets his father's permission and his wife's consent. But wherever he goes in the countryside, he is not given any food until he has done the work allotted to the morning or however much work is usually done there before dinner. Under this regulation he is allowed to go anywhere within the boundaries of his city's territory, for he will be no less useful to the city than if he were in it.
So you see that nowhere is there any chance to be idle; there is no excuse for laziness, no wine taverns, no alehouses, no brothels, no occasion to be corrupted, no hideouts, no hangouts. With the eyes of everyone upon them, they have no choice but to do their customary work or to enjoy pastimes which are not dishonorable. Such behavior on the part of the people is bound to produce an abundance of everything. And when it is distributed equitably to everyone, it follows that no one can be reduced to poverty or forced to beg.
In the senate at Amaurot (to which, as I said before, three representatives come every year from each city), once they have determined what surpluses are at hand in each place and what places have shortages, they immediately make up the deficiencies of the one with the excess supplies of the other, and they provide them as a free gift, receiving nothing in return from those to whom they gave them. But if they gave something to a city and received nothing in return, they also get what they need from some other city and pay nothing for it. Thus the whole island is like one household.
When they have enough provisions for themselves (which they do not think they do unless they have provided for two years, since the next year's outcome is uncertain), they export to other countries vast quantities of grain, honey, wool, linen, timber, red and purple dye, fleece, wax, tallow, leather, and also livestock. They give one-seventh of all this to the poor in that country and sell the rest at a moderate price. In exchange they not only acquire goods they do not have at home (they lack almost nothing except iron) but also they bring back to their homeland enormous quantities of silver and gold. They have continued this practice for such a long time that they now have everywhere a greater supply of those metals than you would think possible. Hence they do not much care whether they are paid in cash or credit, and they accept promissory notes for most of what is owed them, but never from private persons; instead they make the usual legal documents binding on the city government. When the loan comes due, the city requires it to be paid by the private debtors and puts it in the public treasury; then the city enjoys the use of it until the Utopians call it in. For the most part they never do, since they think it is hardly right to claim what is of no use to them from those who have a use for it. But if circumstances require that they lend part of it to another nation they call it in, or when they are obliged to go to war; that is the only reason they keep all of the treasure which they have at home, as protection against extreme danger or sudden emergencies. They use it especially to pay enormous wages to foreign mercenaries, whom they would much rather expose to danger than their own citizens. They are also aware that with large sums of money even the enemies themselves can be bought and set against one another, either through treason or open hostilities.
This is the reason they reserve such an incalculable treasure, although they do not keep it as treasure but in a form I am really ashamed to tell you. I am afraid you will not believe what I say, and all the more rightly so since I am aware that if I had not seen it in person I would have been reluctant to believe it if someone else told it to me. For in general the more foreign something is to the habits of the listeners, the harder it must be for them to believe it. But actually, a prudent judge of the matter will perhaps be less surprised that they handle silver and gold in their own way rather than ours, since all their arrangements are so different from ours. In fact, since they themselves have no use for money but rather keep it as protection against events which might or might not happen, in the meantime they keep gold and silver (from which money is made) in a form that lets no one place more value on it than it deserves by its nature. And obviously it deserves far less than iron, without which mortals could no more live, by heaven, than they could without fire or water, whereas nature gave to gold and silver no use which we could not easily do without; the folly of mankind gives them value because they are rare, but nature, on the other hand, like a kind and gracious mother, made the most useful elements openly available, like air, water, and earth, but she hid away what is vain and unprofitable in the most remote recesses.
Now if in their society these metals were put away in some tower, the ruler and the senate might be suspected of deceiving the people by some trick and getting some good from it for themselves—such is the foolish anxiety of the mob. And then if they made platters out of them or other vessels made by goldsmiths, if ever the occasion arose to melt them down and use them to pay mercenaries, they realize that once people had begun to delight in them they would be reluctant to give them up. To obviate these difficulties they have thought up a method quite compatible with the rest of their arrangements but very far removed from ours (for we value gold very highly and hide it away quite carefully), a method which is therefore hard to believe unless you have experienced it. Whereas they eat and drink from vessels of earthenware and glass, beautifully crafted but inexpensive, they use gold and silver, not only in the common halls but also in private houses, to make all the chamberpots and lowliest containers. Moreover, the chains and heavy shackles used to restrain the slaves are made of the same metals. Finally, the most notorious criminals wear gold rings in their ears, gold rings on their fingers, a gold collar around their necks, and even a gold band around their heads. By these means they see to it that the same metals which other nations give up with almost as much grief as if their guts were being pulled out have so little value that if circumstances required the Utopians to part with all such metals none of them would think they had lost as much as a single farthing.
Furthermore, they gather pearls on the seashore and even diamonds and rubies on some cliffs; they do not look for them, however, but when they have found some by chance, they polish them. They use them to deck out their infants, who are boastful and proud of such gems in their earliest childhood; but, as they get a little older and notice that such trinkets are worn only by children, they become ashamed of them of their own accord and, with no urging from their parents, they give them up just as our children discard their baubles, necklaces, and dolls when they grow up.
These arrangements, so different from those of other peoples, have produced quite different feelings and attitudes. That never became clearer to me than in the incident of the Anemolian ambassadors. They came to Amaurot while I was there and since they had come to discuss important matters, the three citizens chosen by every city had come before they arrived. All the ambassadors from neighboring countries, who had landed there before and were familiar with the customs of the Utopians, knew that they did not revere sumptuous clothing, considered silk contemptible, and even associated gold with disgrace; and so they used to come clothed as modestly as possible. But the Anemolians lived further away and had less contact with them. Hence, when they saw that all the Utopians wore one and the same rough garment, they thought they did so because they had nothing better to wear and, with more pride than wisdom, they decided to set themselves up as gods by the elegance of their trappings and to dazzle the eyes of the poor Utopians by the splendor of their garb.
And so when the three ambassadors made their entry, their retinue of a hundred retainers was dressed in particolored garments, mostly made of silk, but the ambassadors themselves, who were noblemen in their own country, were garbed in cloth of gold, with large chains and earrings of gold, and also golden rings on their fingers, and on top of that strings of pearls and gems hanging from their hats, and in sum, decked out in everything that the Utopians use to punish slaves, to mark off someone in disgrace, or to make toys for children. And so it was a sight to see how they ruffled their feathers when they compared their finery with the clothing of the Utopians (for the people had poured out onto the streets). On the other hand, it was no less delightful to observe how totally mistaken their hopes and expectations were and how far they were from the consideration they thought they would receive. For in the eyes of all the Utopians, except for the very few who had had some good reason to travel to foreign countries, all their splendid trappings seemed shameful. They greeted all the retainers of the lowest rank reverently as if they were lords. But they considered the ambassadors to be slaves because they wore golden chains, and so they passed over them with no respect whatever. In fact, you could have also seen children there who had thrown away their gems and pearls. When they saw such gems affixed to the hats of the ambassadors, they nudged their mothers and said: "Look, mother, that big lout is still wearing little pearls and gems, as if he were a little boy!" But the mother would reply in all seriousness: "Hush, my son, I think he is one of the ambassadors' fools." Others criticized those golden chains as useless because they were so fine that a slave could easily break them and so loosely fastened that a slave could shake them off whenever he wanted and run off anywhere he wanted, footloose and fancy-free.
But after the ambassadors had lived there for a day or two and seen such an enormous amount of gold treated as if it were worthless and contemned there as much as it was honored in their countries, and when they also noticed that the chains and shackles of only one runaway slave contained more gold and silver than the trappings of all three of them, they were crestfallen and sheepishly put away all the finery which they had so haughtily displayed, especially after they had talked more informally with the Utopians and learned their customs and opinions.
Indeed they are amazed that any mortal can take delight in the dubious sparkle of a tiny gem or precious stone when he can look at a star or even at the sun, or how anyone could be so insane as to imagine that he is nobler because of fine-spun woolen thread, since that wool (however fine-spun) was once worn by a sheep, which was at the same time nothing more than a sheep. They are likewise amazed that gold, which in itself is useless, is now prized so highly everywhere that mankind itself, which gave it value and for whose use it got that value, is valued much less than the gold itself, so much so that some beef-witted blockhead, who has morals to match his folly, nevertheless has many wise and good men in his service, for no better reason than that he has a heap of gold coins. And if some turn of Fortune or trick of the law (which turns things topsy-turvy no less than Fortune herself) should transfer this heap from the heir to the lowest lout in the whole household, the master would shortly enter the service of his servant as if he were a mere adjunct and appendage of the coins. But what they find most amazing and despicable is the insanity of those who all but worship the rich, to whom they owe nothing and who can do them no harm; they do so for no other reason except that they are rich, knowing full well that they are so mean and tightfisted that they will certainly never give them one red cent during their whole lives.
These opinions and others like them they have formed partly from their upbringing, since they were brought up in a commonwealth whose institutions are farthest removed from those kinds of folly, and partly from instruction and books. For though not many in each city are dispensed from physical labor and assigned to do nothing but study (namely those in whom they have perceived from their childhood remarkable talent, extraordinary intelligence, and devotion to learning), nevertheless all children are introduced to good books, and throughout their lives a good many people, both men and women, devote to learning the hours I have mentioned as free from labor.
They learn the various branches of knowledge in their own language, which has no lack of vocabulary, is not unpleasant to the ear, and is not surpassed by any other in the expression of thought. It has spread throughout most of that part of the world, though everywhere else it is corrupted in various ways.
Of all the philosophers whose names are so famous in this known part of the world, they had not so much as heard of any before our arrival, and yet in music, dialectic, arithmetic, and geometry, they have made almost the same discoveries as our own ancient writers did. But though they measure up to our ancient writers in almost all respects, they are not up to the discoveries of modern dialecticians. In fact, they have not discovered a single one of those rules about restrictions, amplifications, and suppositions which have been so subtly excogitated in the _Parva logicalia_ and which are taught to young men everywhere in our world. And then, as for second intentions, they are so far from being able to understand them that none of the Utopians could see man in general, as they say, even when we pointed him out with our finger, though, as you know, he is plainly colossal and bigger than any giant. But they are very expert in the orbits of stars and the movement of heavenly bodies. In fact, they have devised instruments of various designs which enable them to understand very accurately the movements and positions of the sun and moon and also the other stars which are visible in their hemisphere. But as for the conjunctions and oppositions of the planets and the whole fraud of divination by the stars, they have never so much as dreamed of it. By means of signs that they have perceived from long observation they predict rainstorms, winds, and other changes in the weather. But concerning the causes of those phenomena, and concerning tides and the saltiness of the ocean, and in general concerning the origin and nature of the heavens and the world, they agree on some points with our own ancient philosophers, and on others, just as the ancients disagreed with one another, they also differ from all the ancients and propose new theories, and yet they do not entirely agree among themselves.
In that area of philosophy which deals with ethics, they discuss the same issues as we do. They inquire about the goods of the mind and body and external goods, and whether the designation "good" applies to all of these or only to the gifts of the mind. They discuss virtue and pleasure, but the primary and principal controversy is about what they think human happiness consists in, whether one thing or many. On this point they seem over-inclined to the position which claims that all or the most important part of human happiness consists of pleasure. And what is even more surprising, they claim support for this self-indulgent view even from religion, which is sober and strict and, indeed, almost gloomy and stern. For they never analyze happiness unless they combine some religious principles with the rational analysis of philosophy, since they think that without such principles reason by itself is too weak and deficient to investigate true happiness.
These principles are of this sort: that the soul is immortal, and by the beneficence of God is born for happiness; that our virtues and good deeds will be rewarded after this life, and our crimes have punishments prepared for them. Though these are religious principles, the Utopians still think that reason leads them to believe and grant them; if they are eliminated, the Utopians have no hesitation in affirming that no one could be so stupid as not to feel that he ought to pursue his own pleasure by hook or crook. He would only be concerned not to sacrifice a greater pleasure for a lesser one and not to pursue one that would be requited by pain. For they think it would be truly insane to pursue virtue, which is harsh and difficult, and not only to banish the pleasures of life but even to seek out pain of your own accord, and to expect to get nothing out of it (for how can you get anything out of it if you get nothing after death, since you have spent your whole life here without pleasure, that is, wretchedly?). But as it is, they think happiness consists not in every sort of pleasure but in pleasure that is good and honorable, for they believe that our nature is drawn to pleasure as the highest good by virtue itself, whereas the opposite faction attributes happiness to virtue alone.
And then they define virtue as living according to nature; to that end, they say, we were created by God. We follow the guidance of nature when we obey reason in choosing and avoiding things. Furthermore, reason above all inspires mortals to love and revere the majesty of God, to whom we owe our very existence and our capacity to be happy. Secondly, reason admonishes and encourages us to lead lives with as little anxiety and as much joy as possible and, beyond that, to exert ourselves in helping all others achieve the same end because of our natural fellowship. For not even the gloomiest and sternest advocate of virtue, who despises pleasure so much that he would impose toil, vigils, and mortifications on you, would refrain from enjoining you to do as much as you can to alleviate the poverty and distress of others, and he would think it praiseworthy and humane for one human being to rescue and comfort another, since the very essence of humanity (and no virtue is more proper to human beings) is to relieve the distress of others, eliminate sadness from their lives, and restore them to a joyful life, that is, to pleasure. Why should nature not impel us to do the same for ourselves? For either a joyful life, that is, a life of pleasure, is wrong and in that case we should not only not help anyone to achieve it but rather we should do all we can to make everyone avoid it as harmful and deadly, or if you are not only allowed but even required to obtain it for others, why not do so first of all for yourself? You should be no less well-disposed to yourself than to others. For when nature prompts you to be good to others, she does not require you to turn around and be cruel and merciless to yourself. Nature herself, they say, prescribes as the aim of all our actions a joyful life, that is, pleasure, and they define virtue as following the prescriptions of nature. But when nature invites mortals to help each other to lead cheerful lives (and she is certainly right to do so, since no one is so far above the rank of human beings that nature should care for him alone, whereas in fact she is equally concerned about all those whom she groups together as belonging to the same species), she also, of course, forbids you time after time to seek your own advantages in ways that create disadvantages for others.
Therefore they think that not only private agreements must be kept but also public laws which have either been promulgated by a good ruler or which a people not oppressed by a tyrant or deceived by some trick have laid down by common consent to govern the distribution of vital commodities, that is, the means to pleasure. As long as these laws are not broken, to look out for your own good is prudent; to promote the public good is pious. But to deprive someone else of pleasure to promote your own is wrong; on the other hand, to deprive yourself of something to give it to someone else is a work of humanity and kindness and it always brings you more good than it takes away. For it is counterbalanced by gifts given in return, and also your consciousness of having done a good deed and the thought of the love and good will of those you have benefited will give you mental pleasure that outweighs any loss of bodily comfort. Finally, as religion makes clear to true believers, God will repay the loss of brief and paltry pleasures with enormous and never-ending joy. Following this line of reasoning and having considered the matter long and hard, they think that all our actions, including also our virtuous deeds, are directed toward pleasure as our happiness and final end.
They define pleasure as any motion or state of the mind or body which produces delight in accord with the guidance of nature. Not without reason do they add that the impulse must be in accord with nature. For just as not only our senses but also our reason pursues whatever is pleasurable by nature, that is, pleasures not achieved through wrongdoing, or acquired with the loss of a greater pleasure, or followed by hardship, so too they hold that all those unnatural pleasures which mortals agree to call delightful by the emptiest of fictions (as if it were in their power to change the thing by changing the name) are so far from contributing to happiness that they actually hinder it because, once they have taken over the mind, they occupy it totally and leave no room for true and genuine pleasures. For a great many things are not pleasurable by their very nature and are, in fact, for the most part bitter, but through the perverse enticement of evil desires they are not only thought to be the greatest pleasures but are even included among the primary reasons for living.
Among those who pursue false pleasures they include those whom I mentioned before who think that the finer the gown they wear the better they are. On this one point they are wrong twice over. They are no less deceived in thinking the gown is better than in imagining they themselves are. For if you consider the usefulness of a garment, why is wool woven with fine thread better than wool woven with coarser thread? But they think they excel in fact, not merely in their illusions. They ruffle their feathers; they believe that they are more valuable because of their clothes. And on that basis, honors they would not have dared hope for in cheaper clothes they demand as rightly due to their elegant gown, and they are outraged if someone passes them by without due deference.
And then isn't it equally stupid to be much taken with empty and worthless honors? For what natural pleasure is there in someone's baring his head to you or bending his knee? Will that relieve the pain in your own knee or cure the delirium in your head? It is amazing how some are caught up in this imaginary, specious pleasure: delightfully insane, they flatter themselves and take pride in their imagined nobility simply because they happen to be descended from a long series of ancestors who are considered to be rich, above all rich landlords (for nowadays there is no other source of nobility except wealth), and yet they think they are not a whit the less noble even if their ancestors have left them no wealth or they themselves have squandered it.
With these they group the persons I mentioned before who are enthralled by gems and precious stones and almost think they have been deified if they ever get a fine specimen, especially if it is the sort most highly valued in their own times; for not all sorts are highly regarded by all persons and at all times. But they do not buy such a stone unless it is removed from its gold setting and exposed, and even then not unless the seller swears and guarantees that it is a genuine jewel and a true gemstone; so afraid are they that their eyes may be deceived by a counterfeit substituted for a real stone. For why should your eyes be any less delighted by a counterfeit since they cannot distinguish it from a real one? To you each of them should have equal value, no less so, by heaven, than they would to a blind man.
What about people who keep superfluous wealth under lock and key, taking delight not in using the amassed treasure but merely in contemplating it? Do they feel any real delight or rather are they not deluded by a false pleasure? How about those who are subject to a different vice and hide away their gold, intending not only never to use it but perhaps never even to see it any more; in their anxiety not to lose it, they lose it. For surely it is lost if it is buried in the ground so as to be of no use to you and perhaps not to any other mortal. But still, when the treasure is hidden away, you feel carefree and happy. If a thief took it away and you died ten years later without knowing of the theft, in all those years that you lived after the money was stolen, what difference did it make to you whether it was removed or remained safe? In either case its usefulness to you was the same.
To these categories of absurd enjoyment they add gambling (a sort of madness they know of only through hearsay, not experience) and also hunting and falconry. For what pleasure can there be, they say, in throwing dice on a gaming table? Even if there were any pleasure in it, you have done it so often that mere repetition should have made you sick of it. How can it be delightful to hear the barking and howling of dogs?—isn't that a disgusting noise? Why do hunters feel more pleasure when a dog chases a hare than when a dog chases a dog? For in either case the action is the same, that is, running, if that is what pleases you. Or if you are attracted by the hope of carnage and the expectation of seeing the slaughter with your own eyes, you ought instead to be moved to compassion when you see a little hare torn to pieces by a dog, a weak creature tormented by a stronger one, a timid creature fleeing from a ferocious beast, a harmless creature from a cruel hound. And so the Utopians have assigned the whole business of hunting to the butchers, whose trade (as I said before) is conducted entirely by slaves, considering it beneath the dignity of free men. They consider it the lowest function of the trade. The other activities of butchers are more useful and honorable, since they contribute much more and destroy animals only out of necessity, whereas the hunter seeks nothing but pleasure from the slaughter and butchering of some poor little creature. Even in beasts themselves, according to the Utopians, such an eagerness to view carnage springs from a cruel disposition, or else the continual indulgence in such brutal pleasure finally degenerates into cruelty.
Though the herd of mortals consider such pursuits as these and others like them (for there is no end to them) to be pleasures, the Utopians firmly hold that they have nothing to do with pleasure, since there is no natural sweetness in them. Though they ordinarily produce sensual joy (which seems to be the function of pleasure), the Utopians are unwilling to change their minds. The reason they seem pleasant is not the nature of the things themselves but the perverse habits of their devotees, whose vicious attitudes cause them to embrace what is bitter as sweet, just as the defective tastebuds of pregnant women make them think that pitch and tallow are sweeter than honey. And yet no one's judgment, if it is vitiated by disease or habit, can change the nature of pleasure, or of anything else for that matter.
True pleasures they divide into various classes, assigning some to the mind, others to the body. To the mind they attribute understanding and the sweetness which springs from the contemplation of the truth. To these they add the pleasure of looking back on a lifetime of good deeds and the sure hope of happiness to come.
They divide bodily pleasure into two kinds: one is the sweetness which pervades the senses, either when the supplies our natural heat has used up are replenished (as they are by food and drink) or else when the excessive elements overburdening our bodies are discharged. This happens when we purge our intestines of excrement, or go about generating children or when the itching in some part of the body is alleviated by rubbing or scratching. But sometimes pleasure results not from the replenishment sought by our bodily members nor from relieving them of excess but from some secret but remarkable power which tickles, excites, and attracts our senses to itself, such as the pleasure arising from music.
They claim that there is another kind of bodily pleasure which consists in the balanced and quiet condition of the body, that is, when a person's health is not disturbed by any disease. Such health, as long as it is not interrupted by any pain, is delightful in itself, even though it is not affected by any external pleasure. Though it is less obvious and affects the senses less grossly than the insistent desire for food and drink, nevertheless many Utopians hold it to be the greatest pleasure of all. Almost all of them believe that it is a great pleasure and the foundation and basis, as it were, of all the others, since it is the only one which keeps our lives peaceful and desirable; and, if you take it away, there is no room left for any pleasure at all. For the mere absence of pain without health they regard as insensibility, certainly not as pleasure.
They have long since rejected the position of those who think that stable and undisturbed health should not be considered to be a pleasure because, they say, its presence can be felt only through some external stimulus (for they, too, have debated this question intensely). But now they are in almost complete agreement with the opposite position, that health is actually essential to pleasure. For according to them, disease brings pain, which is unalterably opposed to pleasure, in the same way as disease is opposed to health. Why not conclude, in turn, that there is pleasure in undisturbed health? On this point they do not think it makes any difference whether the disease is a pain or the pain comes from the disease; in either case the effect is the same. Thus, if health itself is a pleasure or if it necessarily brings pleasure with it as fire brings heat, the result in either case is that, wherever health is, stable pleasure cannot be lacking.
Moreover, when we eat, they say, what happens is that health, which has begun to fail, now has food as its ally in the battle against hunger. As it gradually becomes stronger, the very progress toward its ordinary vigor brings with it the pleasure of being reinvigorated. And so if health finds joy in the struggle, will it not rejoice when the victory is won? But when it has at last happily recovered its former strength, which was the sole object of the whole struggle, will it immediately become insensible and fail to recognize and embrace its own good? The idea that health is not perceived they consider to be very far from the truth. For when we are awake, who does not perceive that he is healthy—except someone who is not? Who can be so constricted by dullness and lethargy that he does not admit that health is delightful and enjoyable? And what is enjoyment but another name for pleasure?
Above all they embrace the pleasures of the mind, which they consider the first and foremost of all pleasures. They think that mental pleasure springs primarily from the practice of the virtues and the consciousness of a good life. Of the pleasures supplied by the body they give the first place to health. As for the pleasure of eating and drink and whatever else falls under a similar category of delight, they think they should be sought, but only for the sake of health, for such activities are not enjoyable in themselves but only insofar as they counter the unnoticed encroachments of ill health. And therefore a wise man, they say, should ward off disease rather than seek medicine for it and avoid pain rather than seek relief from it; just so it would be better not to have any need for such pleasure than to be relieved by it.
If anyone thinks that this kind of pleasure makes him happy, he must also confess that his life would be the happiest of all if it could be spent in perpetual hunger, thirst, and itching, followed by eating, drinking, scratching, and rubbing— and who can fail to see that such a life would be not only foul but also miserable? Certainly these are the lowliest of all pleasures, since they are the least unadulterated and never occur except in conjunction with the pain contrary to them. Thus the pleasure of eating is coupled with hunger, and not in equal proportions, for the pain is both longer and more intense. For it begins before the pleasure and never departs until the pleasure also ceases. Therefore they do not place much stock in such pleasures, except insofar as necessity demands them. But they also rejoice in them and gratefully acknowledge the kindness of Mother Nature, who uses the sweetest pleasures to entice her offspring to do what they must always be doing out of necessity. How irksome our lives would be if the daily ailments of hunger and thirst had to be warded off by drugs and bitter medications like the other diseases which afflict us less often?
They gladly cherish beauty, strength, agility as special and enjoyable gifts of nature. Certainly the pleasures which are mediated by our ears, eyes, and noses and which nature assigned as proper and peculiar to the human race (for no other kind of creature admires the design and beauty of the world, or is moved by the beauty of fragrances except to distinguish kinds of food, or recognizes the harmonious or discordant intervals in sounds), these pleasures, I say, they cultivate as adding a certain enjoyable spice to their lives. In all of them, however, they impose the limitation that a lesser should not impede a greater pleasure or that a pleasure should not cause pain at some later time—and they think this will necessarily happen if the pleasure is dishonorable.
They think it is certainly quite mad for someone to despise a beautiful figure, to deplete his strength, to turn agility into torpor, to wear out his body with fasting, to ruin his health, and to scorn the other favors bestowed by nature, unless he neglects his own good so as to work more avidly for the the good of others or the public welfare, and in return for his effort he expects greater pleasure from God. Otherwise to inflict pain on oneself without doing anyone any good—simply to gain the empty shadow of virtue or to be able to bear with less distress adversities that may never come—this they consider to be insane and the mark of a mind that is both cruel to itself and ungrateful to nature, rejecting her benefits and not deigning to be beholden to her.
This is their view of virtue and pleasure; and in the absence of religious inspiration from heaven revealing something holier, they think human reason can discover no truer doctrine. I do not have time now to examine whether or not their teaching is correct, nor is it necessary, since I undertook to present their principles, not to defend them. But whatever validity their precepts may have, I am fully persuaded that nowhere will you find a more extraordinary people or a happier commonwealth.
Physically they are agile and vigorous, stronger than you would expect from their height, though they are not undersized. Though their soil is not uniformly fertile and their weather is not particularly favorable, they protect themselves from the climate by moderation in their diet and they work hard to remedy the defects of the soil, so that nowhere in the world will you find a more abundant supply of crops and cattle or bodies more vigorous and subject to fewer diseases. You can see them there diligently employing the usual agricultural methods of improving infertile soil by skill and effort, but you could also see a forest that they uprooted with their own hands and planted in another place. The reason for doing this was not greater production but transportation: they wanted the timber closer to the sea or rivers or the cities themselves, since it takes less labor to move crops by land over long distances than it does to transport timber.
They are an easy-going people, cheerful and clever. They enjoy their leisure but they endure physical labor well enough as long as it is useful (but otherwise they are hardly fond of it); in intellectual pursuits they are indefatigable. When we told them about the literature and learning of the Greeks (for in Latin there is nothing except the poets and historians that would be likely to interest them very much) it was amazing how eagerly they pressed us to help them master Greek by giving them instruction. And so we began to read, at first more out of a desire not to seem lazy than from any hope that much good would come of it. But when we had made a little progress, their diligence immediately made us anticipate that ours would not be wasted. They began to imitate the shape of the letters so easily, to pronounce the words so readily, to memorize so quickly, and to recite so accurately that we would have thought it miraculous except that the majority of them had undertaken this study not only on their own initiative but also at the explicit command of the senate, and hence they were selected from the most talented and mature scholars. And so in less than three years there was nothing in that language which they had not mastered; they read good authors with no hesitation, unless they encountered some textual crux. I tend to think they mastered Greek all the more easily because it is somewhat related to their own language. I suspect that the Utopian people originally sprang from the Greeks because their language, which is otherwise closest to Persian, preserves some vestiges of Greek in the names of cities and magistrates.
On the fourth voyage, instead of trade goods I took on board a fair-sized packet of books because I was fully determined to return only after a long time, if ever. From me they got most of Plato's works, more of Aristotle's, and also Theophrastus _On Plants,_ which was mutilated in several places, I'm sorry to say. During the voyage the book had not been put away properly and a playful monkey came upon it; he mischievously ripped out some pages here and there and tore them up. Of the grammarians they have only Lascaris, for I did not take Theodore with me nor any dictionary except Hesychius and Dioscorides. They are very fond of Plutarch's books and they are also much taken with the wit and elegance of Lucian. Of the poets they have Aristophanes, Homer, and Euripides, and also Sophocles in the small typeface of Aldus. Of the historians they have Thucydides and Herodotus, as well as Herodian.
Furthermore, as for medical books, my companion Tricius Apinatus had brought with him some shorter works of Hippocrates and the _Microtechne_ of Galen; for these books they have a high regard. Even though there is hardly a country in the world that has less need of medicine, still it is nowhere more honored, precisely because they consider a knowledge of it as one of the finest and most useful branches of science. When they investigate the secrets of nature using the resources of science, they not only experience wonderful pleasure from doing so but they also think they win the highest approbation from the creator and maker of the world. For they suppose that he, like other workmen, set up the marvelous mechanism of this world for mankind to view and contemplate (and men are the only creatures he made capable of doing so) and that therefore he is fonder of a careful observer and meticulous admirer than he is of some lazy blockhead who ignores such a marvelous spectacle as if he were a mindless brute.
And so the natural talent of the Utopians, trained by study, is marvelously effective in inventing techniques which make some contribution to a comfortable life. Two of these they owe to us, printing and papermaking, but even these they owe not only to us but in large part to themselves. For when we had shown them some books printed by Aldus on paper and had spoken a bit about the material for making paper and the technique of printing letters, though we did not really explain it (since none of us was expert in either process), they immediately and most ingeniously figured it out. And whereas before they had written only on vellum, bark, and papyrus, they immediately tried to make paper and to print with type. Though at first they did not get it quite right, by frequent attempts they soon mastered both techniques, and they became so proficient that if they had copies of Greek texts, there could have been no lack of printed editions. But as it is, they have no more than what I have mentioned, but what they have they have disseminated in many thousands of printed copies.
Any sightseers who visit them are especially welcome if they are recommended by unusual intellectual gifts or knowledge of many lands gained by traveling widely (and for that reason they welcomed us warmly when we landed), for they are eager to learn what is happening everywhere in the world. But not very many come there to trade. For what can they bring except iron or gold and silver, which they would prefer to take home than to export? As for their own exports, they think it more advantageous to deliver them themselves than to have others pick them up, for in that way they learn more about foreign countries everywhere and they keep their seamanship and nautical skills from getting rusty.
### SLAVES
Prisoners of war they do not consider to be slaves except those captured in wars they themselves have fought. The children of slaves and the slaves of foreign countries whom they have obtained are not kept in slavery. Their slaves are those who have committed a serious crime in Utopia or foreigners who have been condemned to death for committing some crime (and these are by far the larger number), for the Utopians acquire many of them, sometimes cheaply, more often gratis, and take them away. These kinds of slaves they not only keep constantly at work but also in chains. Utopian slaves, however, they treat more harshly since they consider them baser and deserving of more severe punishment because they had an extraordinary education and the best of moral training, yet still could not be restrained from wrongdoing. Another class of slaves is made up of poor, overworked drudges from other nations who choose of their own accord to be slaves among the Utopians. These they treat decently and, except that they make them work a bit harder (since they are used to it), they are treated not much less kindly than the citizens. If they wish to depart (and that does not happen very often), they are not kept against their will nor are they sent away empty-handed.
They care for the sick, as I said, with great concern, omitting nothing whatever in the way of medicine or diet that might restore them to health. They sit with those who are suffering from an incurable disease, talk with them, console them, and do what they can to alleviate their pain. But if someone suffers from a disease which is not only incurable but also constantly and excruciatingly painful, then the priests and the magistrates point out that he can no longer live a useful life, that he is a heavy burden to himself and to others, and that he has outlived his own death; they encourage him to make a decision not to maintain the sickness and disease any longer and urge him not to hesitate to die, but rather to rely on hope for a better life; since he lives in a prison where he is cruelly tormented on the rack, he should escape from this miserable life on his own or willingly allow others to rescue him from it. This would be a wise act, they say, since death would deprive him of no advantages but would save him from suffering; and since in doing so he would be following the advice of the priests, the interpreters of God's will, it would also be a pious and holy deed.
Those who agree with these arguments voluntarily starve themselves to death or are put to sleep and dispatched with no sensation of dying. But they do not do away with anyone who is unwilling, and they do not in any way diminish their attendance on him. Those who are persuaded and die in this way are treated with honor; but otherwise anyone who commits suicide for reasons not approved by the priests and senate is deemed unworthy of either burial or cremation and is ignominously thrown into a swamp without a proper funeral.
A woman does not marry until she is eighteen, a man not until he is four years older than that. If a man and a woman are convicted of engaging in secret intercourse before marriage, they are both severely reprimanded and they are forbidden ever to marry anyone unless the ruler remits the sentence. But both the master and the mistress of the household where the offense was committed fall into utter disgrace for not doing their duty with sufficient diligence. They punish this offense so severely because they foresee that few would join together in married love, living their whole lives with one person and enduring besides the troubles that come with marriage, if they were not carefully restrained from promiscuous intercourse.
Moreover, in choosing spouses they have a custom which seemed to us absolutely absurd and thoroughly ridiculous, but they observe it strictly and seriously. The bride, whether virgin or widow, is presented naked to the groom by a sober and respected matron, and the groom in turn is shown naked to the bride by some honorable man. When we laughed at this custom and criticized it as ridiculous, they in turn were amazed at the extraordinary folly of all other nations: when they are buying a colt—a matter of no great expense—they are so cautious that even if the animal is almost completely exposed they refuse to buy it unless the saddle and saddlecloth are removed so as to reveal any sores that might be hidden beneath them; yet in choosing a spouse—a matter which will make them either happy or miserable for the rest of their lives—they are so careless that they judge her whole person by a mere handsbreadth, that is, by her face only, since the rest of her is wrapped up in her clothes, and according to that judgment they join themselves to her, not without great danger of not getting along with her if they later find something offensive. For not everyone is so wise as to pay attention only to character, and even in the marriages of the wise the gifts of the body add something to the virtues of the mind. Certainly some ugly deformity concealed beneath clothing can completely alienate a man's mind from his wife when his body can no longer be separated from her. If such a deformity should occur after the wedding, then everyone must put up with his lot; but before the wedding the laws should see to it that no one is duped or deceived.
All the more care needs to be taken because, of all the countries in that part of the world, they are the only one that is monogamous, and their marriages are almost never dissolved except by death, though adultery or unbearably offensive conduct can be grounds for divorce. The offended party gets permission from the senate to remarry; the offender is disgraced and can never remarry. Otherwise, it is absolutely forbidden to put away a wife against her will and without any blame on her part because of some bodily disfigurement. They consider it cruel to desert someone at the very time she is in most need of comfort and they think it would make her uncertain and insecure about her old age, which brings diseases with it and is itself a disease.
But sometimes it happens that two people are temperamentally incompatible, and if they have each found someone else with whom they hope they can live more agreeably, they separate by mutual consent and remarry, but not, however, without the permission of the senate, which does not permit divorce unless the senators and their wives have examined the case very carefully. Even then they do not do it readily because they know that the expectation of easily remarrying is hardly a means of strengthening the love of married couples.
Adulterers are punished with the harshest servitude, and if both were married the injured parties may divorce their spouses and marry each other if they wish to; otherwise they may marry whomever they like. But if one of the injured parties continues to love such an undeserving spouse, the marriage can remain intact, as long as the innocent party is willing to accompany the criminal condemned to hard labor; and it happens sometimes that the affectionate concern of the one and the repentance of the other move the ruler to mercy so that he sets them free again. But if the crime is repeated, it is punished with death.
Their laws do not prescribe punishments for other crimes, but rather the senate determines penalties according to how heinous or venial each particular offense seems to be. Husbands chastise their wives and parents their children, unless an offense is so serious that open punishment is advisable in order to maintain public morality. But generally the most serious crimes are punished with servitude, which they consider no less grievous to the criminal and much more advantageous to the commonwealth than to execute wrongdoers and immediately get rid of them altogether. They do more good by their labor than by their death, and they offer a long-standing example to deter others from similar crimes. If slaves are rebellious and unruly, then they are finally slaughtered like wild beasts that cannot be restrained by bars or chains. But if they are patient, they are not left entirely without hope. If they are tamed by long suffering and show that they regret the sin more than the punishment, their servitude may be either mitigated or revoked, sometimes by the ruler's prerogative, sometimes by popular vote.
Attempted seduction is no less dangerous than seduction itself. In fact, in all sorts of crimes, they equate the clear and deliberate attempt with the completed deed, for they do not think that the mere incompletion of the deed should benefit someone who did everything he could to complete it.
They are very fond of fools: they consider it quite shameful to treat them with contempt, and they have nothing against finding enjoyment in their foolery, since they think that will do the most good for the fools themselves. If anyone is so strict and gloomy that he never laughs at any word or deed, they do not entrust fools to him, out of fear that he would not treat them kindly enough, since to him they would be not only useless but not even entertaining—and that is the only talent they have.
To mock someone for being disfigured or crippled is considered shameful and disfiguring, not to the person mocked but to the mocker, since it is stupid for him to blame someone for a defect which it is not in his power to avoid.
They consider it lazy and negligent not to keep up natural beauty by grooming, but they consider seeking help from cosmetics a disgraceful affectation. They know from experience itself that no physical beauty recommends wives to their husbands as much as respect and an upright character. Some men may be snared by beauty alone, but none can be held except by virtue and compliance.
They not only deter from crime by punishments, but they also foster virtue by rewarding it with honors. And so in the marketplace they set up statues of outstanding men who have done extraordinary service to the commonwealth, thus preserving the memory of their good deeds so that posterity may have the glory of their ancestors as a spur and incentive to virtuous deeds.
Anyone who campaigns for public office becomes disqualified for holding any office at all. The Utopians live together amiably, since no magistrate is arrogant or terrifying; they are called fathers and they live up to the name. Honor is willingly paid to them (as is proper); it is not exacted from those unwilling to give it. The ruler is not singled out by his clothes or a crown but rather by the sheaf of grain he carries: the sign of the high priest is a wax candle borne before him.
They have very few laws, for very few suffice for persons trained as they are. Indeed, one of their primary charges against other nations is that endless volumes of laws and interpretations are not sufficient. But they consider it quite unjust to bind people by laws which are so numerous no one can read through all of them or so obscure that no one can understand them. Moreover, they ban absolutely all lawyers as clever practitioners and sly interpreters of the law. For they think it is practical that everyone should handle his own case and present the facts to the judge as he would to a lawyer; in this way there will be less confusion and the truth will be easier to determine, since he tells his story without having learned any evasion from a lawyer, while the judge weighs all the details carefully and protects simple souls from the false accusations of crafty litigants. In other countries, such straightforwardness is difficult to obtain because there is a mass of incredibly intricate laws. But among them everyone is knowledgeable about the laws. For, as I said, there are very few laws, and as for interpretations, they consider the most obvious the most correct. For though all laws (they say) are promulgated to inform everyone of his duty, a subtle interpretation will inform very few (for few can understand it); on the other hand, the simpler and more obvious meaning of the laws is clear to everyone. Otherwise, as far as ordinary people are concerned (and they constitute the largest group that needs to be informed), it would make no difference if you formulated no laws at all or if, after you have formulated them, you interpret them in such a way that no one can understand them without great intelligence and long analysis. The dull judgment of ordinary people is not adequate to that task, and they do not have enough time, occupied as they are in making a living.
Inspired by the virtues of the Utopians, those of their neighbors who are free and can choose as they please (for the Utopians themselves have long since liberated many of them from tyranny) ask for and obtain Utopians to act as their magistrates, some for a year, some for five years; when they have served their term, they bring them back to Utopia with great honor and praise, and take replacements with them back to their own country. And certainly these countries are providing very well and very effectively for the public welfare, which depends, for good or bad, on the character of the magistrates. What persons could they choose more wisely than those whose honesty cannot be undermined by bribes (since they will soon return to a place where money is useless) and who cannot be swayed by some person or faction, since they have no connections among that people? Wherever these two vices, favoritism and greed, get a hold on judicial decisions, all justice, which is the mainstay of the commonwealth, is immediately undermined. The peoples who recruit magistrates from them are called allies by the Utopians; the others on whom they have bestowed benefits are called friends.
They do not make treaties with any nation—such treaties as other nations so often make, break, and remake. What good is a treaty, they say, as if nature did not sufficiently bind one human being to another? And if someone scorns nature, do you think he will be concerned with mere words? They are especially drawn to that view because in that part of the world treaties and agreements between princes are not usually observed with very much good faith.
In Europe, of course, and especially in those parts which follow the faith and religion of Christ, the authority of treaties is everywhere holy and inviolable, partly because of the goodness and justice of the princes themselves, partly out of reverence and respect for the popes, who themselves undertake nothing which they do not carry out most scrupulously and likewise command all princes to keep their promises to the letter; if any prince reneges, the pope makes him comply by pastoral censure and sharp reproof. Certainly they are right in thinking that it is quite shameful for those who are specifically called the faithful not to be faithful to their treaties.
But in that new world, which is as far from us in customs and way of life as it is removed from us by the distance the equator puts between us, no one has confidence in treaties: the more ceremoniously and solemnly the knot of a treaty is tied, the more quickly it is untied; it is easy to find some defect in the wording, which they often intentionally devise with some clever loophole, so that the language can never bind them so tightly that they cannot somehow escape, breaking both the treaty and their word. If such craftiness, or rather downright fraud and deceit, occurred in a private transaction it would be contemptuously decried as sacrilegious and deserving of the gallows—and that by the very same persons who are proud of having advised the prince to do the same. Thus it happens that justice seems either to be nothing more than a plebeian and humble virtue, far beneath the exalted dignity of a king, or at least there seem to be two kinds of justice: one is fit for ordinary people, lowly and creeping along the ground, fenced in on all sides, totally encumbered with chains and unable to escape; the other kind is a virtue proper to princes, which is more august than the ordinary virtue and hence much freer— forbidden, in fact, to do only what it does not wish to do.
Such behavior on the part of the princes there, who have so little respect for treaties, is the reason, I think, that the Utopians make no treaties; perhaps they would change their minds if they lived here. But even if treaties were strictly observed, they still think the practice of making them at all is a bad custom because it implies that nations think they are natural-born enemies to each other (just as if there were no natural ties between two peoples separated only by a little distance, a hill or a creek) and that they would rightly try to destroy one another if they were not bound by treaties; and that even if they have entered into a treaty, they are not united in friendship but rather have permission to prey upon each other, insofar as nothing which the treaty forbids is couched with sufficient care because of some oversight in the language. On the other hand, the Utopians think that no one should be considered an enemy if he has done no harm, and that the natural bond which unites us should replace treaties, and that men are more adequately bound to one another by good will than by agreements, more strongly joined by their hearts than by their words.
### MILITARY PRACTICES
They loathe war as positively bestial (though no sort of beast engages in it as constantly as mankind), and unlike almost all nations they consider nothing more inglorious than glory won in warfare. Therefore, though they regularly devote themselves to military training on certain appointed days so that they will not be incapable of fighting when circumstances require it—and not only the men do so but also the women— they are reluctant to go to war and do so only to defend their own territory, or to drive an invading enemy from the territory of their friends, or else, out of compassion and humanity, they use their forces to liberate a oppressed people from tyranny and servitude. When they come to the aid of their friends, it is not always to defend them but sometimes also to requite and avenge injuries inflicted on them. But they do this only if they have been consulted before any steps are taken and if, after they have verified the facts, demanded restitution, and been refused, they themselves declare war. They decide to do this not only when an enemy has invaded and plundered one of their friends, but also, and even more fiercely, when their friends' merchants in any part of the world have been unjustly accused under some pretext of justice, either by using unjust laws speciously or by interpreting good laws perversely.
This was the only reason for the war which the Utopians fought a little before our time on behalf of the Nephelogetes against the Alaopolitans: some Nephelogete merchants among the Alaopolitans had been treated unjustly under some pretext of justice (or so the merchants thought). Certainly, whether the cause was just or unjust, it was avenged by a hideous war, in which the surrounding nations also added their energy and resources to the hostile forces of the major opponents so that some prosperous peoples were ravaged, others were badly shaken. One disaster followed upon another until finally the surrender and enslavement of the Alaopolitans put an end to the war. The Utopians, who sought nothing for themselves, subjected the vanquished to the Nephelogetes—a people hardly to be compared with the Alaopolitans in their heyday.
So fierce are the Utopians even when they are punishing only monetary injuries against their friends; but they are not so when the injury is against themselves. If they should be cheated out of their property, as long as they are subjected to no physical force, they set limits to their anger: they merely refrain from trade with that nation until restitution is made, not because they care less for their own citizens than for their allies but rather they are more offended by their friends' loss of money than by their own because their friends' merchants are severely injured by such a loss, since it comes from their own private possessions. But their own citizens lose nothing but public property, goods which were abundant at home, even superfluous, for otherwise they would not have been exported. So the loss is hardly perceived by anyone. Hence they feel that it would be cruel to punish an injury by killing many people when it causes no inconvenience to any of the Utopians in their lives or livelihood. But if any of their citizens is unjustly disabled or killed, wherever it may be, whether it be done by a public decision or by a private citizen, they send ambassadors to ascertain the facts, and if the malefactors are not handed over to them they cannot be put off but declare war immediately. If the guilty persons are handed over for punishment, they are sentenced to death or servitude.
They are not only grieved by a bloody victory but also ashamed of it, thinking that it is stupid to pay too much for merchandise, however valuable it may be. But if they conquer and crush an enemy by skill and cunning, they glory mightily in the victory, holding public parades to celebrate it and putting up a monument as if for a hard-won victory. For they boast that they have acted with courage and fortitude only when they have won the victory as no other creature but man is able to win it, that is, by the power of his wits. For bears, lions, boars, wolves, dogs, and other animals (they say) fight with the power of their bodies; and though most of them surpass us in strength and ferocity, we outdo them all in intelligence and reasoning.
Their one and only aim in warfare is to gain the objective which, if they had obtained it beforehand, would have kept them from going to war at all. Or, if circumstances make that impossible, they seek to punish those they consider culpable so severely that fear will keep them from daring to do such a thing in the future. These are the goals they set for their undertaking, and they try to achieve them quickly, but yet in such a way that a concern for avoiding danger takes precedence over winning praise and glory.
And so, immediately after declaring war, they see to it that many notices certified by their official seal are put up secretly and simultaneously in the most conspicuous places in the enemy's territory, promising a huge reward to anyone who does away with the enemy's prince; they also assign lesser, but still very substantial, sums for the deaths of those individuals they list in the same notices. These are the persons who, apart from the prince himself, were responsible for plotting against the Utopians. They double the reward assigned to the assassin if he brings them any of the proscribed persons alive; in fact, they offer the same rewards to the proscribed persons themselves, and throw immunity into the bargain, if they turn against their comrades. Thus their enemies quickly suspect all outsiders and even among themselves they are neither trusting nor trustworthy so that they live in a state of utter panic and no less peril. For it has very often turned out (as is well known) that a good number of them, and among them the prince himself, have often been betrayed by those they trusted the most. So easy is it to get someone to commit any crime whatsoever by means of bribes, and for that reason the Utopians set no limits to their bribes. Keeping in mind the great risks they are urging people to take, they take care to balance the magnitude of the danger with the lavishness of the reward; hence they promise not only enormous quantities of gold but also personal and perpetual title to rich estates in the safe and secure territory of their friends, and they faithfully keep their promises.
Other nations condemn this practice of bidding for and buying off an enemy as a barbarous, degenerate crime, but the Utopians think it does them great credit: it shows them to be wise, since in this way they win great wars without fighting at all, and also humane and compassionate, since by killing a few malefactors they spare the lives of many innocent persons who would have fallen in battle, both their own soldiers and those of the enemy; for they pity the rank-and-file of the enemy's soldiers almost as much as their own citizens because they know they do not go to war of their own accord but are driven to it by the madness of princes.
If this procedure is not successful, they sow and cultivate the seeds of dissension by encouraging the brother of the prince or some nobleman to have hopes of gaining the throne. If such internal factions languish, they stir up neighboring peoples and set them against their enemy by digging up some ancient claim such as is never lacking to kings.
When they have promised resources for war, they supply money lavishly, but their citizens very sparingly. They hold their own people so very dear and value each other so highly that they would not be willing to exchange a single one of their own citizens for the enemy's prince. But they are not at all reluctant to pay out gold and silver, since they keep it only for this purpose and would live no less comfortably if they spent all of it. Then too, apart from the wealth they have at home, they also have a limitless treasure abroad, since many nations, as I said before, owe them money. And so they hire mercenaries from everywhere and send them to war, especially the Zapoletes.
These people live five hundred miles to the east of Utopia. Rough, rude, and fierce, they prefer to live in the forests and rugged mountains where they were brought up. They are a hardy people, able to endure heat, cold, and hard labor. They have no interest in agriculture, no acquaintance with refinements, no concern about their houses or clothes; they care only about their flocks. They live mostly from hunting and plundering. They are born only for warfare; they zealously seek opportunities to fight and when they find one they embrace it eagerly. They set out in great numbers and offer themselves cheaply to whoever needs soldiers. The only skill they have to live on is one that aims at death.
They fight fiercely and with complete loyalty for whoever pays them. But they bind themselves for no fixed period. They sign on with the stipulation that if an enemy offers them higher wages tomorrow they will take his side, and if they are lured with slightly higher pay they will return to the side they abandoned. There are very few wars in which a great many of them are not fighting in both armies. And so it happens every day that blood relatives who were hired by the same side and lived together amicably are separated a little later in opposing armies and fight each other as enemies. Forgetting both kinship and friendship, they run each other through with violent hostility, trying to kill each other for no other reason than that they were hired for a pittance by opposing princes. They reckon their wages so strictly that adding one penny to their daily pay can easily cause them to change sides. They have quickly become greedy through and through, and yet it does them no good for what they gain with their blood they immediately squander on debauchery, and wretched debauchery at that.
These people fight for the Utopians against any mortals whatsoever because they hire their services for more than they can get anywhere else. And just as the Utopians seek good men in order to use them, so too they also enlist these wicked men in order to use them up. When they need to use them, they urge them on with great promises and expose them to the greatest dangers so that most of them do not return to claim what they were promised. To the survivors they faithfully keep their promises so as to make them eager to undertake similar exploits. Nor do they have any qualms about doing away with so many of them, since they believe the human race would owe them a great debt of gratitude if they could purge the whole world of such loathsome and wicked scum.
Apart from the Zapoletes, they use the forces of those for whom they have taken up arms, and after that the auxiliary troops of other friendly nations. As a last resort they add their own citizens, from whom they choose a man of proven valour to command the whole army. Under him they appoint two men who remain private citizens as long as he is safe, but if he is captured or killed, one of the two succeeds him, and in case of a mishap he himself is succeeded by the third, so that if the commander is in danger (and the fortunes of war are quite various) the whole army does not panic.
In each city they choose troops from a list of volunteers. No one is sent out to foreign wars against his will, for they are convinced that if someone is by nature fearful he will not only not fight vigorously himself but he will also inspire fear in his comrades. But if their country is invaded during a war, cowards of this sort, as long as they are physically fit, are dispersed among better troops in the ships or they are spread out here and there on the walls so that they have no place to run away to. Thus shame in the presence of their friends, the confrontation with the enemy, and the absence of any hope of escaping overcome fear, and often they make a virtue out of extreme necessity.
Though no one is sent to a foreign war unwillingly, if women are willing to accompany their husbands to battle the Utopians are so far from preventing them that they exhort them to do so and encourage them with praise. Each accompanies her husband to the front and is stationed shoulder to shoulder with him in the battle line. Moreover, each soldier is surrounded by his children and relatives by blood or marriage so that they all have help close by from the persons who are by nature most highly motivated to help one another. It is a great disgrace for one spouse to return without the other or for a son to come back after the loss of a parent. The result is that once it comes to hand-to-hand combat, if the enemy stands his ground, the battle is so long and grim that it ends in a general slaughter.
Certainly they take every precaution to avoid having to fight themselves, as long as they can wage war using mercenaries to take their place. But when they can no longer avoid entering the fray, the courage with which they fight matches the prudence with which they avoided fighting as long as they could. They do not give their all in a first furious attack but rather they grow stronger gradually and over a period of time, and they are so resolute that they would rather die than retreat. For one thing, they are certain that everyone at home is provided for, and they do not need to worry about their children (such concern generally breaks the spirits of lofty souls); so their courage is proud and contemptuous of defeat. Moreover, their skill in the arts of war gives them confidence. Finally, sound ideas, instilled in them from childhood on, both by instruction and through the institutions of the commonwealth, give them courage: they hold life neither so cheap as to throw it away recklessly nor so perversely dear as to cling to it greedily and shamefully when honor requires them to give it up.
When the battle is at its fiercest everywhere, a picked group of sworn and dedicated young men seek out the enemy commander. Sometimes they attack him openly; sometimes they try to ambush him. They assail him from close by and from a distance and they attack him in a wide, unbroken phalanx, continuously replacing the exhausted men with fresh troops. And unless he saves himself by running away, it rarely happens that he is not killed or captured alive by his enemies.
If they win a victory, they do not slaughter the defeated; they would rather capture than kill those they have put to flight. And they never pursue retreating troops without keeping in reserve at least one battalion drawn up under its colors. They do this so regularly that if the rest of their own forces have been defeated and they win the victory with their last battalion, they would rather let the whole enemy army escape than get into the habit of pursuing the fugitives with their own forces in disarray. They remember something that happened to them more than once: when the main body of the whole Utopian army had been overwhelmed and put to flight, while the enemy was exulting in the victory and pursuing them as they ran away in all directions, a few of their own troops held in reserve and on the lookout for opportunities suddenly attacked the enemy troops, who were scattered and straggling and careless from overconfidence, and thus changed the whole outcome of the battle; snatching certain and undoubted victory from their enemies' hands, the conquered turned the tables and conquered the conquerors.
It is not easy to say whether they are more clever in laying ambushes or more cautious in avoiding them. You would think they are preparing to flee when that is the last thing they intend; on the other hand, when they do intend to flee, you would imagine that is the last thing they have in mind. For if they feel they are at a disadvantage either in numbers or location, then they either move their camp silently at night, or escape by some stratagem, or withdraw gradually by day, keeping their ranks in such good order that they are no less dangerous in retreat than when they attack. They fortify their camp very carefully with a wide and very deep moat; the earth they dig up is piled up on the inside. In such work they do not use the services of common laborers. It is done by the hands of the soldiers themselves, and the whole army joins in the work except for the armed soldiers outside the rampart who keep watch against sudden attacks. With so many soldiers pitching in, they build massive fortifications around a large area with incredible speed.
They wear armor which is strong enough to ward off blows but does not hinder movement and gestures—so much so that they feel no inconvenience even in swimming. For swimming in armor is one of the ordinary rudiments of their military training. At long range their weapon is the arrow which they shoot with great force and accuracy, not only on foot but also from horseback. At close quarters they strike not with swords but with battle-axes, which are deadly because of their sharp blade and their weight, whether used to hack or thrust. They are very skilled in devising siege engines. Once they are made, they conceal them very carefully, lest they become known before it is time to use them and turn out to be more ridiculous than useful. In designing them their primary concern is to make them easy to move and aim.
When they make a truce with their enemies, they keep it so religiously that they do not violate it even under provocation. They do not lay enemy territory waste or burn their crops; they even do what they can to keep the grain from being trampled by men and horses, for they think it may be of some use to them. They injure no unarmed civilians except for spies. They offer amnesty to cities that surrender and even those taken by siege they do not sack; instead they execute those who prevented the surrender; they enslave the rest of the defenders, but the civilian populace they leave unharmed. If they find persons who urged the town to surrender, they grant them a share in the property of the condemned; they divide up the rest and give it to their auxiliaries, for none of the Utopians takes any of the booty.
When the war is over, they assess the costs not against the friends for whom they incurred them but against the losers; they demand part of it in money, which they reserve for similar use in warfare, and part in estates within enemy territory, from which they forever enjoy a not inconsiderable income. They now have revenues of this sort in many nations; it accumulated gradually in various ways and now amounts to 700,000 ducats a year. To take care of it they send out collectors of revenue, who live there in grand style and play the part of great lords. But there is plenty left over to put into the treasury, unless they choose to give credit to the nation that owes it, which they often do until they need it, and even then it rarely happens that they demand all of it. They also bestow some of these estates on those whom they have persuaded to place themselves in great danger, as I mentioned before.
If some prince takes up arms against them and is preparing to invade their domain, they immediately confront him with a huge force outside their own boundaries, for they are reluctant to wage war within their own territory and no exigency could ever induce them to allow foreign auxiliaries on their island.
### THE RELIGIONS OF THE UTOPIANS
There are various religions not only throughout the island but also within individual cities: some worship the sun as god, others the moon, others a different planet. Others worship some ancient paragon of either virtue or glory, venerating such a person not only as a god but as the supreme god. But the vast majority, and those by far the wiser ones, accept none of those gods and believe there is a certain single deity, unknown, eternal, infinite, inexplicable, diffused throughout this whole universe not physically but by his power, in a manner that is beyond human comprehension; him they call their parent. To him alone they attribute the origin, increase, progress, changes, and goals of all things; him and no other they honor as divine.
Actually, though all the others hold different beliefs on some points, they agree with the monotheists in thinking that there is some one supreme being who made and rules the universe, and in their native language they all agree in calling him Mythras, but they differ in that they identify the supreme power variously, each asserting that whatever he considers to be supreme is in fact that single nature to whose divine majesty, by the consensus of all nations, the whole creation is attributed. But gradually they are all abandoning these superstitious variations and joining together in that one religion which seems more reasonable than the others. And there is no doubt that the other beliefs would have vanished long ago if it were not that, whenever something untoward happened to someone who was considering changing his religion, fear made him think that it was not accidental but was sent from heaven, as if the divinity whose cult he was forsaking were avenging a wicked affront to himself.
But after they had heard from us the name, the teaching, the behavior, and the miracles of Christ, and the no less miraculous constancy of so many martyrs who freely shed their blood and thus brought many peoples, from far and wide, over to their religion, you would not believe how eagerly they also were converted, whether through the secret inspiration of God or because Christianity seemed closest to the sect which is predominant among them, although I think it was a matter of no small moment with them to hear that Christ approved of life in common for his disciples and that it is still practiced among the most genuine Christian communities. But certainly, whatever the reason, no small number of them were converted to our religion and were washed clean in the sacred waters of baptism.
But because there was, I am sorry to say, no priest among the four of us (for only that number remained after two of us had given up the ghost), they received the other sacraments but still lacked those which among us are conferred only by priests. But they know about them and long for them most intensely. In fact, they also earnestly discuss among themselves whether someone chosen from among their number could receive the sacerdotal character without the dispatch of a Christian bishop. And in fact it seemed they were about to choose someone, but when I left they had not yet done so.
Even those who do not agree with the Christian religion still do not frighten anyone away from it; they do not oppose anyone who has embraced it, except that one of our community was repressed while I was there. Shortly after he was baptized, over our objections, he harangued publicly about Christianity with more zeal than prudence, and he began to get so carried away that he not only ranked our religion above all the rest but condemned all the others outright. He cried out against them as profane; he denounced their worshipers as wicked, sacrilegious, and worthy to be punished in eternal fire. When he had preached like this for a long time, they arrested him and tried him, not for despising their religion but for exciting riots among the people. They convicted him and sentenced him to exile, for it is one of their oldest policies that no one should come to any harm because of his religion.
For Utopus had learned that before his arrival the inhabitants squabbled incessantly about religion and he had noticed that the sects, which generally disagreed with each other and fought for their country in separate groups, provided the opportunity for him to conquer all of them. Hence, from the very beginning, after he had obtained the victory, he decreed first of all that everyone could practice the religion of his choice and could also strive to convert others to it, but only so long as he advocated it calmly and moderately with rational arguments. And if he could not win others over by persuasion, he was not to assail their religions bitterly nor use force against them, and he was to refrain from insults. Anyone who quarrels insolently about religion is punished with exile or enslavement.
Utopus laid down these rules not only for the sake of peace, which he saw was completely undermined by constant strife and implacable hatred, but also because he thought such a decree would benefit religion itself. In religious matters he did not venture to dogmatize rashly because he was uncertain whether or not God wishes to have varied and manifold kinds of worship and hence inspires different people with different views. Certainly he thought that to use force and threats to make everyone accept what you believe to be true is both arrogant and absurd. Then too, if one religion should be actually true and the rest false, still he easily foresaw that in the long run the the truth would sooner or later emerge and prevail by its own force as long as the matter was handled reasonably and moderately. But if the struggle is conducted with arms and uprisings, since the worst people are always the most headstrong, the best and holiest religion, embroiled among empty superstitions, will be choked like grain among thorns and briars. And so he left the whole matter open and left everyone free to believe whatever he wanted, except that he solemnly and strictly forbade that anyone should sink so far below the dignity of human nature as to think that the soul dies with the body or that the world is ruled by mere chance and not by providence.
And for this reason they believe that after this life punishments are ordained for vices and rewards for virtues. Anyone who thinks otherwise they do not even include in the category of human beings since he has degraded the lofty nature of his soul to the base level of a beast's wretched body. Still less will they count him as one of their citizens, since he would set no store whatever by all their laws and morality if it were not for fear. For who can doubt that someone who has nothing to fear but the law and no hope of anything beyond bodily existence would strive to evade the public laws of his country by secret chicanery or to break them by force in order to satisfy his own personal greed? For that reason they bestow no honors on such a person, they assign him to no office, they put him in charge of no public responsibility. He is universally looked down on as a lazy and spineless character. But he is not subjected to any punishment because they are convinced that it is not within a person's power to believe whatever he wishes; they neither compel him by any threats to mask his opinion nor accept any pretexts or lies, which they utterly despise as next door to deliberate malice. Still they do forbid him to argue for his opinion, but only among the common people. Otherwise, in private, among priests and prudent men, they not only permit him to argue but also encourage it, confident that in the end his madness will yield to reason.
There are also others, and they are by no means few (since their position is not forbidden as completely unreasonable or wicked) who go to the opposite extreme and believe that the souls of brute beasts are also immortal, although not comparable to ours in dignity nor destined for the same happiness.
Almost all of them are certain and fully persuaded that human happiness will be so boundless that they mourn for everyone who is sick but not for anyone who dies, unless they see that he is torn from life anxiously and unwillingly. For they take this to be a very bad sign, as if such a soul, despairing and conscious of guilt, fears to leave life because of some secret presentiment of future punishment. Moreover, they think God will hardly be well pleased when someone who is summoned does not come running eagerly but is dragged off reluctant and unwilling. Therefore when they see such a death they are dismayed and they carry out the dead persons with grief and in silence; after praying that God in his mercy will kindly forgive the infirmities of such souls, they cover the body with earth. On the other hand, when someone dies joyfully and full of good hope, they do not mourn him, but rather they conduct his funeral with song; commending his soul to God with great affection, they finally cremate his body with reverence, not grief, and erect on that spot a column inscribed with the virtues of the dead person. After they have returned home, they tell of his character and deeds, and no part of his life is rehearsed more often or more eagerly than his cheerful death.
They think this commemoration of his uprightness is a very strong inducement to virtue for the living and the most acceptable form of veneration to the dead, whom they also believe to be present when they are talked about, though invisible to us because the eyesight of mortals is too dull to see them. For it would not be suitable to the condition of the blessed to lack the liberty of going wherever they want, and it would be ungrateful of them to have no desire whatever to visit their friends, to whom they were united in mutual love and charity while they were alive; such charity they suppose, like other good qualities, is increased, not diminished, in good men after their death. Thus they believe that the dead are present among the living, observing what they say and do, and for that reason they go about their business more confidently because of their trust in such protectors; their belief in the presence of their ancestors also deters them from secret wrongdoing.
They have nothing to do with fortune-telling and other vain, superstitious divinations, which other people take quite seriously but which they consider ridiculous. But miracles which happen apart from any natural cause they revere as works and witnesses which manifest the presence of a deity. They say such miracles often happen there, and sometimes, during great crises, they pray publicly for a miracle with great confidence and they do obtain it.
They think the worship which pleases God is the contemplation of nature and the praise which springs from it. But there are others, and they are by no means few, who neglect learning in the name of religion, who do not strive to attain any knowledge, and who allow themselves no leisure at all. They are determined to earn happiness after death solely by keeping busy in the service of others. And so some tend the sick, others repair the roads, clear out ditches, rebuild bridges, dig turf, sand, or stones, fell and cut up trees, cart lumber, crops, and other provisions into the cities. They perform their services not only for the public but also for private citizens, and they work even harder than slaves. They willingly and cheerfully undertake any tasks which are rough, difficult, dirty, and shunned by most people because of the toil, disgust, and hopelessness they entail. They see to it that others have leisure, while they themselves are continually engaged in labor and toil, but nevertheless they take no credit for it. They neither censure the lives of others nor extol their own. The more they conduct themselves like slaves the more everyone honors them.
They are divided into two sects. The one is celibate and not only abstains from any sexual activity but also eats no meat (and some of them no animal products at all), totally rejecting the pleasures of this life as harmful, longing only for those of the world to come, which they strive to obtain by toil and vigils. Meanwhile, confident that they will soon obtain them, they are cheerful and energetic. The other group, no less devoted to labor, prefers to marry: they do not spurn the consolations of marriage, and they think that just as they owe such activity to nature, they owe children to their country. They do not refuse any pleasure which does not interfere with their work. They like to eat the flesh of animals precisely because they think such food gives them the strength to do all kinds of work. The Utopians consider this group more prudent; the other they regard as holier. If they claimed on rational grounds to prefer celibacy to marriage and a hard life to a comfortable one, the Utopians would laugh at them; but since they profess to be motivated by religion, the Utopians respect and revere them. On no other subject are they more cautious about making any rash pronouncements than on matters concerning religion. In their language these persons are given the special title "Buthrescae," which could be translated into Latin as "religiosi."
Their priests are extremely holy and therefore very few. For each city has no more than thirteen, one for each church, except during wartime, when seven of them set out with the army and are replaced by substitutes for the time being. But when the priests return, each assumes his former position. Until the time when the substitutes, in an orderly succession, replace priests who have died, they become attendants of the high priest (for one priest has authority over the others). They are elected by the people in the same way as other magistrates, that is, by secret ballot, in order to avoid partisan strife. Once elected, they are consecrated by their own college of priests.
They preside over divine worship, attend to religious matters, and act as guardians of morality. To be summoned by them and rebuked for dishonorable conduct is considered to be a great disgrace. But their role is to exhort and admonish; to repress and punish wrongdoers is the function of the ruler and other magistrates. The priests, however, do excommunicate those they find to be thoroughly vicious. There is almost no other punishment which they fear more, for such persons are both dejected by their infamy and tormented by a bad conscience. They may not even be physically safe for very long. For unless they quickly convince the priests that they are repentant, they will be seized by the senate and punished for their impiety.
Children and young people are educated by the priests, and they devote no more attention to learning than to character and virtue. They take the greatest pains from the very first to instill in the tender and impressionable minds of children sound opinions conducive to preserving the common good. When such ideas are thoroughly absorbed in childhood, they persist throughout all of manhood and they are extremely useful in protecting the status of the commonwealth, which decays only because of vices which spring from perverse attitudes.
The wives of the priests are the very finest women in the country, unless the priests themselves are women, for that sex is not excluded; but they are rarely elected and must be widows of advanced years.
No magistrates are held in greater honor among the Utopians, so much so that even if they commit a crime they are not subject to a public tribunal but are left to God and their own consciences. For they do not think it is right to lay human hands on anyone, however vicious, who has been dedicated to God in such a special way as a holy offering, so to speak. It is easier for them to observe this custom because priests are so few and are chosen so carefully. For it is very unlikely that someone who is the cream of the crop and is elevated to a position of such dignity only because of his virtue should degenerate into corruption and vice. And even if that very thing should happen—for human nature is changeable—nevertheless there would certainly be no reason to fear that the public would be in any great danger, because the priests are so few and have no power beyond what derives from the honor paid them. In fact the very reason they have so few and scattered priests is to keep the dignity of the order, now held in such high esteem, from being cheapened by bestowing the honor on many, especially since they think it is hard to find very many who are equal to the dignity of the office, for which merely mediocre virtues are insufficient.
Their reputation at home is no greater than the esteem in which they are held by foreign nations. This becomes quite clear, I think, if we note the reason for it. When troops are engaged in battle, the priests kneel at a distance but not very far away, dressed in their sacred vestments; lifting up their hands to heaven, they pray first of all for peace, and then for victory for their own forces, but without bloodshed on either side. When their soldiers win they rush into the battle line and restrain the fury of their forces against the routed troops. Merely to see them and make oneself known to them by calling out is enough to save anyone's life; to touch their flowing garments also protects the remaining goods of fortune from any damage due to the war. Hence they are venerated by the countries all around them, who attribute to them such genuine majesty that oftentimes they provide as much protection for their own citizens as they do for their enemies. For sometimes it has happened that, when their battle line was thrown back in despair and had turned to flee, as the enemy was rushing in to kill and plunder, the intervention of the priests has stopped the slaughter and separated the two armies so that a peace was devised and established on equitable terms. For nowhere is there a nation so savage, cruel, and barbarous that they do not hold their persons to be sacrosanct and inviolable.
The first and last days of each month and likewise of each year are celebrated as feastdays; the months are marked off by the orbit of the moon, just as the year is established by the course of the sun. In their language they call all of the first days "cynemerni," the last days "trapemerni," names that are equivalent to "first-feastday" and "last-feastday." Their churches are remarkable not only for their workmanship but also for their capacity to hold immense crowds—which is necessary because there are so few of them. They are all dimly lit, and they say this resulted not from lack of skill but from the deliberate policy of the priests, who believe that too much light distracts our thoughts, whereas dim and doubtful lighting concentrates the mind and intensifies religious devotion.
Since religion is not the same for everyone there, yet all the forms of it, however varied and different, converge from various directions on one goal, the worship of the divine nature, nothing is seen or heard in the churches which is not held in common by all the religions. If any denomination has a rite peculiar to it, they provide for it in their own homes. Public worship is conducted according to a ritual which does not at all detract from any of the private devotions. Therefore no images of the gods are seen in churches so that everyone can be free to imagine the form of God as he wishes according to his own religion. They invoke God by no other name than Mythras, a name they all apply to the one divine nature, whatever it may be. No prayers are devised which everyone cannot say without offending his own denomination.
And so on the last-feastdays they gather in church in the evening, still fasting and ready to give thanks to God for the success they enjoyed during the year or month just coming to an end. On the next day, which is the first-feastday, they flock to church in the morning to pray for success and happiness in the following year or month which begins on that feastday. But on the last-feastdays, at home, before they go to church, wives throw themselves at the feet of their husbands, and children do the same before their parents; they confess that they have sinned either through commission or negligence, and they beg forgiveness for their offenses. In this way if some little cloud of strife has arisen in the household, it is dispelled by such atonement so that they can attend the sacrifices with clear and untroubled minds, for they are too conscientious to worship with a disturbed conscience. Therefore those who feel anger or hatred toward someone do not intrude on the sacrifices unless they are reconciled and purged of such feelings, for fear of some swift and severe punishment.
When they get there, the men sit on the right side of the church, the women separately on the left. Then too, they position themselves so that the male members of each household sit in front of the master of that household, and the matron of each household sits in the last row of the women. Thus they see to it that all the actions of everyone are observed in public by the persons whose authority maintains discipline at home. Moreover they are also very careful to intermingle everywhere young persons with their elders; otherwise, if children were entrusted to children, they might spend in childish tomfoolery the time that they should devote to cultivating a religious fear of the heavenly beings, the greatest and practically the only incitement to virtue.
In their sacrifices they do not kill any animals; they do not think that a merciful God, who bestowed life on animals precisely that they might live, takes any pleasure in bloodshed and slaughter. They burn incense and other fragrant substances. They also display many candles, not because they do not know that such things add nothing to God's nature, no more than human prayers do, but they like this harmless mode of worship and people feel that somehow such perfumes, lights, and other ceremonies lift up the human heart and make it rise more eagerly in divine worship.
In church the people wear white garments; the priests are clothed in vestments of various colors, marvelous in both workmanship and design, though the materials are not especially expensive, and they are not woven with gold threads or encrusted with rare gems; rather they are fashioned out of the feathers of various birds, so elegantly and skillfully that the costliest material would not match the value of the workmanship. Moreover, these feathers and plumes of birds and the set patterns in which they are arranged on the priests' garments are said to contain certain secret mysteries which, if rightly understood (and the interpretation is carefully handed down by the priests), remind them of the benefits bestowed on them by God and of the devotion they owe him in return, as well as their duty to each other.
When the priest, dressed in this way, comes out of the sacristy, everyone immediately prostrates himself on the ground out of reverence; on all sides the silence is so profound that the spectacle itself inspires a certain fear, as if in the presence of some divinity. They remain on the ground for a while and then arise at a signal from the priest. Then they sing the praises of God, accompanied by musical instruments, which are mostly shaped differently from those seen in our part of the world. Most of them surpass ours in sweetness of tone, but some of them are incomparably superior to ours. But in one respect their music is undoubtedly far ahead of ours: whether instrumental or vocal, it imitates and expresses natural feelings so well, the sound matches the sense of the words so closely (whether they express supplication or joy, peace or turmoil, sadness or anger), and the shape of the melody matches the meaning so well that it quite wonderfully stirs up, pierces, and inflames the hearts of the hearers. Finally the priest and the people recite together certain customary and fixed forms of prayer, composed in such a way that everyone can apply to himself what they all recite together.
In these prayers each one recognizes God as the creator and ruler of the universe and also the source of all good things. He thanks God for bestowing so many benefits on him, but especially because through God's kindness he was placed in the happiest form of commonwealth and has been allotted the religion which he hopes is the truest. If he is mistaken in this matter or if there is some form of commonwealth or religion which is better and more approved by God, he prays that God in his goodness will cause him to recognize it, for he is prepared to follow wherever God leads him. But if this form of commonwealth is the best and this religion is truest, he asks that God will both make him steadfast and lead other mortals to the same way of life and the same idea of God—unless there is in fact something in this variety of religions which pleases his inscrutable will.
Finally he prays that by an easy death God may take him to himself, how soon or late he certainly does not dare to determine. But, provided that God's majesty is not offended by it, he would much rather go to him by a very difficult death than be kept away from him any longer, even by a prosperous way of life. After saying this prayer they once more prostrate themselves on the ground and after a little while they get up again, go to eat lunch, and spend the rest of the day playing games or doing military exercises.
I have described to you as accurately as I can the plan of their commonwealth, which I certainly consider to be not only the best but also the only kind worthy of the name. For elsewhere they always talk about the public good but they are concerned with their own private welfare; here, where there is no private property, everyone works seriously for the public good. And for good reason in both places, for elsewhere is there anyone who does not know that unless he looks out for his own personal interest he will die of hunger, no matter how flourishing the commonwealth may be; therefore necessity causes him to think he should watch out for his own good, not that of others, that is, of the people. On the other hand, here, where everything belongs to everyone, no one doubts that (as long as care is taken that the public storehouses are full) nothing whatever will be lacking to anyone for his own use. For the distribution of goods is not niggardly; no one is a pauper or a beggar there, and though no one has anything, all are rich.
For what greater wealth can there be than to be completely spared any anxiety and to live with a joyful and tranquil frame of mind, with no worries about making a living, not vexed by a wife's complaints and demands, not fearing a son will end up in poverty, not concerned about a daughter's dowry, but secure about the livelihood and happiness of himself and his own, his wife, children, grandchildren, great-grandchildren, great-great-grandchildren, and however long a line of descendants noblemen presume they will have. Indeed those who worked before but are now disabled are no less provided for than those who are still working.
At this point I wish that someone would venture to compare with this equity the justice to be found in other nations, where I'll be damned if I can find any trace whatever of justice or equity. For what sort of justice is it for some nobleman or goldsmith or moneylender or, in short, any of the others who either do nothing at all or something that is not very necessary for the commonwealth, to live luxuriously and splendidly in complete idleness or doing some superfluous task? And at the same time a laborer, a teamster, a blacksmith or farmer works so long and so hard that a beast of burden could hardly sustain it, performing tasks so necessary that without them no commonwealth could survive at all for even a single year, and yet they earn such a meager living and lead such miserable lives that beasts of burden seem to be better off, since they do not have to work so incessantly, their fodder is not much worse (and to them it tastes better), and in the meantime they are not afraid of what will happen to them. These workers are driven to toil without profit or gain in the present; they are crushed by the thought that they will be poverty-stricken in their old age, for their daily wages are not enough for that very day, much less can they accumulate any surplus which might be put aside every day to provide for their old age.
Is a commonwealth not unjust and ungrateful if it lavishes so many benefits on noblemen, as they are called, and goldsmiths, and the rest of that crew who are either idle or else merely flatterers and providers of empty pleasures, but makes no proper provision for farmers, colliers, laborers, teamsters, and blacksmiths, without whom there would be no commonwealth at all; unmindful of their sleepless labors and forgetting their many and great contributions, it first uses up the labors of their flourishing years, and then, when they are worn down by old age and diseases, it is totally ungrateful and rewards them with a miserable death. And how about this: every day the rich scrape away something from the wages of the poor, not only by private chicanery but also by public laws. Before, it seemed unjust that those who deserve the most from the commonwealth should receive the least, but now, by promulgating a law, they have transmuted this perversion into justice. From my observation and experience of all the flourishing nations everywhere, what is taking place, so help me God, is nothing but a conspiracy of the rich, as it were, who look out for themselves under the pretext of serving the commonwealth. They think up and devise all ways and means, first of keeping (and having no fear of losing) what they have heaped up through underhanded deals, and then of taking advantage of the poor by buying their labor and toil as cheaply as possible. Once the rich have decreed in the name of the public (including the poor) that these schemes must be observed, then they become laws.
But after these depraved creatures, in their insatiable greed, have divided among themselves all the goods which would have sufficed for everyone, they are still very far from the happiness of the Utopian commonwealth; there, once the use of money was abolished, and together with it all greed for it, what a mass of troubles was cut away, what a crop of crimes was pulled up by the roots! Is there anyone who does not know that fraud, theft, plunder, strife, turmoil, contention, rebellion, murder, treason, poisoning, crimes which are constantly punished but never held in check, would die away if money were eliminated? And also that at the very instant when money disappeared, so would fear, anxiety, worries, toil, and sleepless nights? Indeed, poverty itself, which seems to be merely the lack of money, would itself immediately fade away if money were everywhere totally abolished.
To make this clearer, imagine some barren year of bad harvests when many thousands of people die of hunger. I maintain it is clear that at the end of this famine, if you examined the barns of the rich, you would find so much grain that if it had been divided among those swept away by starvation and disease, no one would have noticed any effect at all of the failure of weather and soil. It would have been easy to provide food if that blessed money, that invention very clearly designed to open the way to what we need to live, were not the only barrier to keep us from it. I have no doubt that the rich also understand this and are not unaware how much better it would be to lack no necessities than to abound in so many superfluities, to be relieved of so many troubles than to be hemmed in by such great wealth. And in fact I have no doubt that everyone's concern for his own well-being or the authority of our savior Christ (who is so wise that he cannot be unaware of what is best and so good that he would never advise what he knew was not the best) would long since have easily drawn the whole world to adopt the laws of this commonwealth, if it were not held back by one and only one monster, the prince and parent of all plagues, pride.
Pride measures prosperity not by her own advantages but by the disadvantages of others. She would not even wish to be a goddess unless there were some wretches left whom she could order about and lord it over, whose misery would make her happiness seem all the more extraordinary, whose poverty can be tormented and exacerbated by a display of her wealth. This infernal serpent, pervading the human heart, keeps men from reforming their lives, holding them back like a suckfish.
Since pride is too firmly fixed in the minds of men to be easily plucked out, I am glad that this form of commonwealth, which I would gladly see adopted by everyone, is at least enjoyed by the Utopians; they have followed ethical principles which enabled them to lay the foundations of a commonwealth that is not only most happy but also, so far as human prescience can foresee, likely to last forever. For now that they have eradicated factional strife and ambition at home, along with the other vices, there is no danger that they can be disturbed by domestic discord, which has been the sole reason for the downfall of many prosperous and splendidly fortified cities. But as long as their domestic tranquility and wholesome social structure is preserved, the envy of all the surrounding princes cannot shock or unsettle their dominion, though in the past they have often unsuccessfully tried to do so.
When Raphael had ended his tale, there occurred to me quite a few institutions established by the customs and laws of that nation which seemed to me quite absurd, not only in their way of waging war, their religious beliefs and practices, and other institutions as well, but also (and above all) in the very point which is the principal foundation of their whole social structure, namely their common life and subsistence with no exchange of money. That one fact entirely undermines all nobility, magnificence, splendor, and majesty, which are (in the popular view) the true adornments and ornaments of a commonwealth. Nevertheless, I knew that his talk had worn him out, and I was not sure whether he could endure to listen to an opinion contrary to his own—especially since I remembered that he had reproached some persons precisely because they thought they would not be considered wise unless they could find some way of picking apart the ideas of others—and so, having praised their regimen and his own exposition, I took his hand and led him in to dinner, though first I said we would have another time to consider these matters more thoroughly and to confer more fully. I only wish this would happen someday!
Meanwhile, just as I can hardly agree with all the points he made (even though he is a person of unquestionable learning and wide experience of human affairs), so too I readily confess that in the Utopian commonwealth are very many features which in our societies I would wish rather than expect to see.
THE END OF THE SECOND BOOK
The End of the Afternoon Discourse
of Raphael Hythloday
about the Laws and Institutions
of the Little-known Island of Utopia
Recorded by the Most Illustrious
and Learned Gentleman
Master Thomas More
Citizen and Undersheriff of London
## Thomas More to His Friend Peter Giles,
Warmest Greetings
My dear Peter, I was thoroughly delighted with the judgment you know about, delivered by that very sharp fellow in the form of a dilemma directed against my _Utopia:_ if the story is being presented as true, I find some things in it rather absurd; if it is a fiction, then I think that More's usual good judgment is lacking on some points. I am very grateful to this man, my dear Peter, whoever he may be, who I suspect is learned and whom I see as a friendly critic. I do not know whether any other critique since the book came out has pleased me as much as this one. For, first of all, motivated either by his regard for me or for the work itself, it seems that he did not begrudge the effort of reading it all the way through, and that not cursorily and hastily the way priests read the divine office (if they do so at all) but deliberately and carefully so as to weigh the details thoughtfully. And then, after criticizing some points, and not very many at that, he declares that he approves of the rest, not thoughtlessly but judiciously. Finally, even in the language with which he castigates me he praises me more highly than those who deliberately set out to praise me. For he gives a clear indication what a splendid opinion he has of me when he complains that he is disappointed when he reads a passage that is not as precise as it should be, whereas I myself exceed my own hopes if I happen to be able to publish something in the whole lot that is at least not absolutely absurd.
But in fact, to deal with him no less frankly in turn, I do not see why he should consider himself so eagle-eyed and, as the Greeks say, sharp-sighted, if he discovers some things rather absurd in the institutions of the Utopians or finds that in setting up a commonwealth I have not thought through some matters in a sufficiently practical way, as if there were no absurdities elsewhere in the world, or as if any of all the philosophers everywhere had so devised a commonwealth, a ruler, or a household so perfectly as to propound nothing that could not be improved. On that point, if it were not that I consider as sacred the memory of the most extraordinary men who have been hallowed from ancient times, I could certainly point out features from each of them which everyone would undoubtedly agree in condemning.
But when he is in doubt whether the work is true or fictitious, on this point I think his own usual good judgment is lacking. Nevertheless, I do not deny that if I had decided to write about the commonwealth and a story such as this had occurred to me, I would not have shrunk from a fictional presentation which would make the truth slip more pleasantly into the mind like medicine smeared with honey. But certainly I would have managed it so that, even though I might have wanted to deceive the ignorant mob, I would at least have inserted some pointed hints which would have let the more learned discover what I was about. Thus even if I had done nothing more than assign to the ruler, river, city, and island such names as would have informed learned readers that the island is nowhere, the city is a phantom, the river has no water, the ruler no people—which would not have been hard to do and would have been much more elegant than what I actually did, for if I had not been forced by historical accuracy, I am not so stupid as to use those barbarous and meaningless names Utopia, Anyder, Amaurot, and Ademus.
But my dear Giles, since I see that some people are so cautious, wary, and sagacious that they can hardly be induced to believe what we simple and credulous souls wrote down at Hythloday's dictation, lest such persons should mistrust not only the accuracy of the story but also my own credibility, I am glad that I can say for my brainchild what Mysis in Terence says to keep Glycerius' boy from being considered a changeling: "By heaven, I thank goodness that there were some freeborn matrons present at the birth." For luckily for me it so happens that Raphael told his tale not only to you and me but also to many very respectable and upstanding men. I do not know whether he related more numerous or notable details but I am sure he told them no fewer and no less remarkable matters than he did to us.
But if these incredulous persons will not take even their word for it, they can visit Hythloday himself, for he has not yet died. I just heard from some persons who recently returned from Portugal that on the first day of last March he was healthy and vigorous as ever. Therefore let them ask him for the truth or question him to ferret it out, as long as they understand that I am responsible only for my own work, not for the trustworthiness of others. Farewell, dearest Peter, to you and your charming wife and pretty little daughter, to whom my wife wishes long life and good health.
## AFTER WORD
JERRY HARP
Poet, translator, lawyer, statesman, social philosopher, martyr, and (as of 1935) canonized saint, Thomas More remains—in his friend Erasmus's phrase—a "man for all seasons," one who in his integrity is suited to all occasions. He was formed to no small degree by the cultural movement known as Renaissance humanism, with its emphases on the study of ancient texts, the deepening of a historical sense, the cultivation of the art of rhetoric, and devotion to active service in the world. The terms "Renaissance" and "humanism" come trailing clouds of ambiguity, so some sorting of their meaning is in order.
The idea that the Renaissance—roughly 1400 to 1650, give or take (depending on where one stood in the world)—was a time of great cultural renewal immediately following the "Dark Ages" owes a lot to the work of Jacob Burkhardt. Although his writing has been immensely influential, many generations of scholars have challenged certain of his ideas. More recent work has stressed, for example, the continuities that carry from the ancient, through the medieval, and into the early modern world. One sign of the continuity is the occurrence of various smaller-scale renaissances leading up to the major period known as _the_ Renaissance. There was the Carolingian Renaissance of the late eighth and ninth centuries, which brought into greater prominence study of the Bible, the church fathers, and the Latin classics, along with a reform of handwriting that made the copying of manuscripts more efficient. Later came the Ottonian Renaissance (tenth century), with its emphasis on historical writing, revitalization of monastic and cathedral schools, and increased circulation of classical learning. Perhaps most widely known is the Renaissance of the "long twelfth century" (roughly 1050 to 1250), which saw a surge of cultural energies in a variety of spheres: further revival of the classics of ancient Latinity, the rise of scholasticism, and the emergence of theology as an academic discipline, as well as developments in art, architecture, vernacular literature, and music. These "Dark Ages" were not so dark as some reports might lead us to believe.
That there was greater continuity between the fall of Rome and the beginning of the Renaissance is one reason for the use, in the past few decades, of the term "early modern" in preference to "Renaissance." To put the matter simply, "early modern" stresses the period's relationship to what follows (modernity and even postmodernity), whereas "Renaissance" emphasizes the period's relationship to the past, and also implies that culture somehow died out in the intervening period (returning us again to the idea of the Dark Ages). While the term "early modern" avoids the problem of connoting a rebirth of something that never really died out, it also introduces its own distinct difficulties into the discussion. For example, it tends to gloss over important differences between early and late modernity, implying a smoother trajectory of cultural development than is fitting. Nevertheless, it can be salutary to choose a new set of problems to negotiate; the new questions can call forth insights and work that otherwise might not occur. Besides, in placing greater emphasis on the Renaissance as harbinger of the new, the term "early modern" actually extends the work of Burkhardt, who ends his great study by proclaiming the Italian Renaissance the "leader of modern ages."
The evidence of continuity does not, however, negate that something distinctive and new was happening in the period commonly referred to as the Renaissance, merely that the Burkhardtian view of disjunction overstates the case. Taking shape in the run-up to the Renaissance is what might be termed the requisite intellectual infrastructure in the form, for example, of the many manuscripts that medieval monks had been busy copying for centuries, and then in the form of printed books in the middle of the fifteenth century. As Jack Goody has pointed out, the Renaissance of early modern Europe is one instance of an identifiable pattern in which a critical mass of material culture enables a recovery and circulation of much older texts, which in some cases then lead to an outpouring of further work and experimentation. Even if many of the ancient texts had continued to be known, at least by a learned cohort, their further circulation was required to open the floodgates of Renaissance work.
A distinctive movement within the early modern era was what scholars in the nineteenth century termed humanism. Douglas Bush said of this movement that it is a "medieval fusion of classical wisdom with Christian faith, and the only real change in later times was that the classical element, philosophically and aesthetically, became a less inferior partner." With regard to the aesthetic inheritance, the humanists' work brought the importance of style in human discourse into greater prominence, not as mere ornamentation but as part and parcel of signification. In other words, they emphasized rhetoric over dialectic (logic); it's not that they were against logic, but rather that they were deeply aware that far more than logic is needed to make discourse meaningful. When Aristotle defined rhetoric as the "faculty of observing in any given case the available means of persuasion," he was writing about oratory, but rhetoric also has to do with the art of structuring discourse. In other words, rhetoric is about the ways that human experience, insight, and wisdom are encoded in language.
This concern with the style and structuring of discourse was at the heart of the humanists' educational and cultural reform of the medieval dispensation they inherited, and _Utopia_ exemplifies these ambitions with great force. In the Dialogue of Counsel, when Hythloday maintains that princes would be impervious to his advice, the character More takes him to task for his idea that the language of truth is singular; Hythloday's preference is for the language of "academic philosophy" (p. 43), the scholasticism that the humanists criticized for its obsessive concern with hyper-subtle logic. As in his letter to Martin Dorp, More derided a trifling, pointless, and at times captious concern with logical quibbles. Generally, the humanists wanted more Cicero and less of the Aristotle of the logical works in their educational program.
With their focus on rhetoric and style, the humanists developed the discipline of textual criticism, seeking as they did to establish reliable texts and accurate translations. A signal example of this work with texts, one that connects with the humanists' historical-mindedness, is Lorenzo Valla's demonstration that the Donation of Constantine was a forgery. Relying largely on historical details, many of them philological, Valla showed that the document could not have come from the Emperor Constantine's hand.
In relation to their other labors, the humanists also undertook a variety of literary experiments, such as the classic and quirky texts by Erasmus ( _Praise of Folly_ ), More ( _Utopia_ ), and François Rabelais ( _Gargantua and Pantagruel_ ). It was as if all of the scholarly work and public service issued in outbreaks of sheer creativity and serious play. For all of the lasting value of the humanists' scholarship, it is these texts at play that most of us now read most assiduously. It is not without reason that, with regard to More's text, one of the great theologians of our day discerns a strong link between utopia and festivity.
The humanist emphasis on rhetoric and style fostered skills fitting for administrative workers in a world of increasingly centralized states and expanding bureaucratic church structures. These skills were fostered by such precursors of the humanists as the lawyers and notaries who mediated commercial activity in thirteenth-century Italy, workers who embellished documents and letters with allusions to the Latin classics. Further, their training in Roman law led them to study the literature of ancient Rome as well. As this world of centralized bureaucracy grew, so did the need for well-trained and fluent language workers. Whatever its other merits, humanism trained people to work in this increasingly bureaucratized world, the one for which Thomas More was shaped.
His early formation took place in the grammar school (where young boys were drilled in the niceties of Latin grammar) at St. Anthony's in London. Having finished his course of study there, he embarked on an apprenticeship at the home of Archbishop (later Cardinal and then Lord Chancellor) Morton, who shows up as a character in _Utopia._ As a page in Morton's household at Lambeth, More would have not only learned how to play a formal role on public occasions, but also extended his rhetorical skills by engaging in debates and taking on various personae to perform fluent and convincing discourses. Having entered his fourteenth year, he left Lambeth to enter Canterbury College, Oxford, where he studied for two years before leaving to take up legal studies at New Inn, London. We do not know why he left Oxford without taking a degree, though it is worth noting that it was not uncommon to study at a university for a year or two before moving on to legal training. After two years at New Inn, he moved on to Lincoln's Inn to continue his training in the law. It was during his time at Lincoln's Inn that he was also part of a circle of scholars—such as William Grocyn, John Colet, and Thomas Linacre—interested in the new humanist learning. These figures came from a variety of walks of life including medicine, law, theater, education, and publishing. But the most influential person More met at this time was Erasmus, who first visited England in 1499. Because neither was proficient in the other's native tongue, they spoke Latin, the lingua franca of European intellectual life, a language in which they could exchange ideas, interests, and witticisms. Early in their friendship, they undertook a friendly contest of translating work of the ancient Greek satirist Lucian from Greek into Latin. Erasmus saw _Utopia_ through the press of Thierry Martens at Louvain. It appeared near the end of 1516.
_Utopia_ keeps appearing as if out of nowhere, showing up in the here and now to take us elsewhere. According to Thomas More's Greek pun, "Utopia" is the good place ( _eu-topos_ ) that is no place ( _ou-topos_ ). Early on he referred to his book by the Latin _Nusquama_ (Nowhere), but it was his Greek coinage that entered the language, and it shows up everywhere. Although the term has come to mean an imaginary and ideal place, an impractical social scheme, More's text works in more complex ways than popular usage allows. _Utopia_ is a nowhere that opens into new discursive spaces. Were the realm of the present and pragmatic concern to dominate entirely, we would be led into stagnation. The nowhere of _Utopia_ —the work as well as the genre and mode of thinking—provides one way to keep consciousness on the move even though it is an impossible place (even the mathematical dimensions of the island cannot work out).
More's great text indeed uncannily recurs. Many editions have been noted. Back in the early 1980s, I spent the better part of a month tracking them down—this for one of those marvelous, old-fashioned graduate school exercises that I hope students still undertake, at least on occasion. I stopped counting somewhere upward of 260. Had neither limitation of time nor lack of initiative intervened, I suspect I could have found many more, and the ensuing decades have produced editions no doubt by the score, such as the one you are reading now. They tend to proliferate during times of conspicuous social stress, such as the world wars. Does this pattern of publication disclose a desire to escape to nowhere during tumultuous times? Or perhaps to reflect on what a good place might be? Does it show an interest, felt if not explicitly contemplated, in the complexities of how to speak of social change?
We do well to read the text in more complex terms than as a blueprint to an ideal state. Its longer title— _On the Best Form of a Commonwealth and on the New Island of Utopia_ —cues us into as much. The conjunction separates as well as joins the two parts. The "best form of a commonwealth," along with what it could mean to talk about and work toward it (dominant in Book 1), is not the same as the "new island of Utopia" (which dominates Book 2). After all, as Clarence Miller states in the introduction to this edition, few would want to live in such a regimented world as Hythloday describes. He even seems to forget a principle of justice that he urges. In his dialogue with Cardinal Morton, he argues convincingly that execution is an unjust and finally ineffectual punishment for thievery, but then in the description of his beloved isle, we learn that the Utopians practice capital punishment for a repeated offense of adultery (p. 99). Both offenses violate a biblical commandment (one the sixth, the other the seventh), so we might take this slip—criticism of one death sentence and implied praise of the other—as something of a wink from the author.
Something other is at stake here than a picture of a perfect world, which must exist elsewhere for More anyway. For this student of St. Augustine, we work to better this world as best we can even when we know we'll fail. In _The City of God_ (Bk. 15, chs. 1–6), Augustine locates the beginnings of the human city in the fratricide in which Cain kills Abel, and he makes the rather astute point that of the two, it is only Cain who builds a literal city. For a citizen of the City of God, life in this world is constant pilgrimage. It is thus fitting that, as several have pointed out, the life of Utopia is that of a monastery writ large. Although the life of a monk traditionally involves staying home, the monastic way of renunciation reminds us that our lives are pilgrimage in a much deeper sense than locomotion can account for; the monk is one who uses the monastic rule to "follow the path to God."
More wrote about the complications of forming an ideal commonwealth in this world, in a Latin poem titled _Quis Optimus Reipubae Status_ ("What Is the Best Form of Government"), in which the speaker raises the question of which rules better, a senate or a king. He wrote the poem around the time he was composing _Utopia,_ with which it appeared in 1518. Here the poem is rendered into English heroic couplets:
Which one excels, a senate or a king?
Likely the senate—it's a common thing:
The best in greater numbers of the good.
But how to find the numbers that you would?
It's easier to hold one bad in view.
A senate often lives between the two,
But kings don't walk on any middle ground.
Flawed senates may get counsel from the sound,
But kings subdue the wisest of the earth.
One is elected by the people. By birth
The other reigns while holding in derision
The whole of his subjected population.
While greedy kings will chew their people up,
An evil senate still leaves room for hope.
The old tale says, Endure the sated pest;
A hungry one invades, worse than the last.
But greedy kings are never satisfied;
A leech hangs on until the body's dried.
Dissent will throw a senate into shambles.
Not so the king, but that is where one gambles.
When disagreement rules in weighty things . . .
But hey, what started all this anyway?
Are there people whom you hold in sway
To shape their mode of rule? If so, you're king.
Don't worry whether there would be abuse.
Ask rather, Would it be of any use?
Sure enough, Utopia is governed by a senate, but it was founded by a king (Utopus) who held sway to establish the system of rule and then legislated his position out of existence—an unlikely scenario, as things go in this world, and one that raises the question of whether democracy can be imposed on a people. It may be said of Utopus that, in taking on the role of king to create a system of democratic rule, the "latter end of his commonwealth forgets the beginning," as Antonio says of Gonzalo's ideal state. The voice at the end of More's poem makes a similar point with the question "Would it be of any use?"
Given the ways of worldly power, could I escape—if I were king—the corrupting influence of my kingly position? How might one answer this question from More's point of view, given life in a fallen world of fallen people whose institutions are compromised by the effects of social sin, where social justice falls by the wayside, and where the dangerous folly of the deluded rich and powerful holds sway more than the wise folly of the City of God (foolish in the eyes of this world) and what passes for wisdom and common sense is far too often the "common nonsense" that arises from the distorting influence of our desires and fears, which are misshapen by life in a fallen world? Besides the corruptions of power, our struggles merely to survive compromise our desires to live well; as Lewis Mumford wrote, Utopia addresses the complication that "our attempts to live the good life are constantly perverted by our efforts to gain a living." From Hythloday's point of view, the only solution is to establish Utopian institutions. But how do we get those institutions up and running? As Miller points out, they can be introduced only if they already exist. You can't get there from here.
_Utopia_ ushers its readers into a style of thought that confronts such complications as these. It is therefore advisable to attend to the style of this masterpiece of Neo-Latin prose. As Elizabeth McCutcheon points out, litotes—an affirmation formed by the negation of its opposite—plays no small part in More's text; she counted more than 140 instances in the Latin of the Yale edition. Because litotes does not do quite the same work as straightforward affirmation, it fosters a mindset that allows nuances and ambiguities otherwise easily glossed over, and thus encourages a habit of understanding in more complex terms than simple affirmation or negation accomplishes. The figure of litotes is one of many that Stephen Greenblatt points to as forming the incongruous relationship between the two realms of the text—the world where the dialogue takes place and the island that Hythloday describes. Clearly, they are related, but their relationship is unstable and distorted. One can even map out certain significant shifts in the style of Hythloday's sentences. When he speaks of the injustices of his contemporary Europe, his sentences are of moderate length, similar to More's Latin prose in his other works. However, when Hythloday contrasts the real with his ideal, or describes the Utopians' simple way of life, he employs styles that represent "extremes that cannot be found anywhere else in More's Latin prose." When contrasting the real with his ideal, he employs "marathon sentences" that go "beyond what ordinary Latin syntax can bear." When describing his ideal, his sentences are simple and brief. Much of the meaning of the text depends on experience of its style, as of its shifts of style. Happily, Miller's translation preserves these stylistic features, and more.
Recognition of the need for a variety of styles and discourses was central to the humanist program. In _Utopia,_ the character More emphasizes that there are many languages of truth. The language that a given speaker chooses should respond to the situation at hand and the persons addressed. Here he makes use of the metaphor of the world as a stage, arguably the most common of Renaissance commonplaces. As he says to Hythloday, "But there is another sort of philosophy better suited to public affairs. It knows its role and adapts to it, keeping to its part in the play at hand with harmony and decorum" (p. 43). The Latin of this passage even refers literally to the boards of the theater, the stage ( _scaena_ ). We may be reminded of the anecdote, related by More's son-in-law, about the future Lord Chancellor: during his apprenticeship in the home of Archbishop Morton, where "though [More] was young of years, yet would he at Christmas-tide suddenly sometimes step in among the players, and never studying for the matter, make a part of his own there presently among them, which made the lookers-on more sport than all the players beside." The humanist spirit recognized that the stage of the world demands improvisation with many language styles.
It may seem curious that More speaks of this theatrical metaphor as a philosophy. But there is a way of understanding ourselves and our world implicit in the metaphor and its relationship to the arts of rhetoric, one that includes a variety of roles and language worlds by means of which humans negotiate experience. One term for this way of understanding is what has been styled relationism. Because everything is related to everything else, there is no singular statement to make about anything. Further, because every statement (such as the one I am making now) conceals as much as it reveals, any statement, no matter how true, must be supplemented by other statements. There is no final word, as there is no final interpretation. What makes a given utterance relevant has everything to do with the situation within which it is made, who is speaking, and who is addressed. None of this is to say that humans cannot know something true—were this the case, we could not know it—but rather that there is no absolute human perspective or singular language of truth, as Hythloday insists that there is.
Given this relationist style of thinking, we may discern in the humanists' work early stirrings toward what Bernard Lonergan identified in his aptly titled "The Transition from a Classicist World-View to Historical-Mindedness." This transition moves from a monolithic understanding of what it means to be human to the insight that there is a great variety of valid ways to be human, as there is a great variety of valid languages of knowing. The _Utopia_ suggests that we are permitted to reflect on how we might change our institutions, revise our social structures, conduct our cultural and civic lives differently. For all of its monolithic structure and faceless anonymity, even Utopia is permeable to outside influence—Utopians eagerly learn Greek, printing, and papermaking (pp. 92–95). Reading _Utopia_ means entering into a dialogue, with oneself and others, that continues to this day.
Another influence on the emerging historical-mindedness of More's era was the ongoing exploration of the globe, as learning about the varieties of culture beyond Europe enabled a greater appreciation of the forms that human societies, and everyday human life, can take. Hythloday himself accompanied Amerigo Vespucci on three of his voyages. Accounts of these journeys provide descriptions of peoples with striking similarities to the Utopians. Thus, we learn of a society in which, as in Utopia, women accompany men into battle, an Epicurean sensibility reigns, and people "hold their habitations in common." Perhaps most striking is the attitude toward gold and other objects that pass for wealth in Europe: "They do not value gold, nor pearls, nor gems, nor such other things as we consider precious here in Europe." The description calls immediately to mind the Utopians' use of gold for chamber pots and shackles, as well as the marvelous anecdote of the Anemolian ambassadors (pp. 76–78).
Of all the questions that surround _Utopia,_ the most vexing has been that of More's attitude toward common ownership of property. Character More objects to the Utopians' community of goods. Even so, it is difficult to ignore the forceful language that he assigns to Hythloday in defense of common property:
. . . it seems to me that wherever there is private property, where everything is measured in terms of money, it is hardly ever possible for the common good to be served with justice and prosperity, unless you think justice is served when all the best things go to the worst people or that happiness is possible when everything is shared among very few, who themselves are not entirely happy, while the rest are plunged into misery. (p. 46)
But character More responds that private ownership is necessary as a goad, for the promise of profit motivates people to work, and too many are lazy louts without the promise of one day owning their own demesne, no matter how small. After the account of Utopian life, he intervenes again, this time with the contention that community of goods "entirely undermines all nobility, magnificence, splendor, and majesty, which are (in the popular view) the true adornments and ornaments of a commonwealth" (p. 134). The appeal to the "popular view" might raise an eyebrow or two, for More remained throughout his life critical of general opinion as a source of wisdom, as did his friend Erasmus, whose _Praise of Folly_ tells the tale. We do well to bear in mind that it was More's critical stance with regard to the popular position that got him imprisoned and killed in the end.
A further complication in considering the Utopians' community of goods is its prominence in the traditions that More revered. Thus, in Plato's _Republic_ —which casts a long shadow over More's text—common ownership is a way of life for the Guardian class, part of the training in virtue for those who are to lead and protect. Then all we need do is shift from Athens to Jerusalem to read the second chapter of the great chronicle of the early Christian communities, Acts of the Apostles, to discover that the believers "held everything in common, and they sold their belongings and possessions and divided them to all according as anyone had need" (2:44–45). This is similar to what Hythloday has done—divided up his possessions among relatives and friends—thus freeing himself for a philosophic life (p. 15). More would have known some echo of the early believers' community of possessions during the four years he lived as a guest of the Carthusians at their Charterhouse in London. This sharing of goods was a well-established part of monastic tradition; the "vice of private ownership must be uprooted from the monastery," St. Benedict wrote. Closer to the composition of _Utopia,_ Erasmus included in his Adagia " _Amicorum communia omnia_ " ("Among friends all is held in common"). In the 1515 edition, Erasmus gave this adage pride of place at the opening of the collection, stating, "Since there is nothing more wholesome or more generally accepted than this proverb, it seemed good to place it as a favourable omen at the head of this collection of adages." Kathy Eden takes this three-word Latin adage as her point of departure in a marvelous book-length study of Erasmus and the humanists.
One of the more nuanced treatments of common ownership occurs in St. Thomas Aquinas's _Summa Theologiae._ Although Aquinas is sometimes associated with the scholastic philosophy against which the humanists inveighed, they generally regarded him not as a scholastic philosopher, but rather as a theologian and Doctor (that is, teacher) of the church. On the one hand, Aquinas set out three reasons why private ownership is helpful to human life: (1) it provides motivation to work; (2) it allows for orderliness in human affairs; and (3) it enables peaceful social relations (ST 2–2ae, Q. 66, A. 2). All three of these accord with the views that character More espouses. On the other hand, in describing humans' fundamental relationship to the world, Aquinas articulates a vision of communal possession:
Community of goods is ascribed to the natural law, not that the natural law dictates that all things should be possessed in common and that nothing should be possessed as one's own: but because the division of possessions is not according to the natural law, but rather arose from human agreement which belongs to positive law, as stated above. (Q. 57, AA. 2–3)
Hence, private ownership is not incompatible with natural law, which also does not require it; rather, private ownership is an "addition thereto devised by human reason" (2–2ae, Q. 66, A. 2, Reply to Objection 1). The natural law recognizes that humans hold all the world in common. Private ownership is merely a provisional, pragmatic, and contingent means whereby humans make creative use of what is fundamentally communal. One valid reading of the _Utopia_ sees the proposition concerning community of property as a way of loosening the metaphysical grip on ownership, reminding readers that we own things in this world merely by convention, not by nature.
If Aquinas had lived in the twentieth century—a speculation I may be allowed given the fictive space of utopian writing—he might have joined with Paul Ricoeur in considering utopian thought alongside ideology. A working definition of ideology, cast in Thomistic terms, is the taking of the provisional and pragmatic for the metaphysical. Thus, stating a right to private ownership as a metaphysical given is an example of the false consciousness wherein Ricoeur finds ideology to function. In his reading, the best function of utopian thinking is as an antidote to ideology, for such thinking provides an opportunity to play one's identity out and away from the prison house of the here and now. As he put it, "This function of utopia is finally the function of the nowhere. To be here, _Da-sein,_ I must also be able to be nowhere." Utopian thought relates to identity because part of identity is prospective, who and what we desire and strive to be—"What we call ourselves is also what we expect and yet what we are not." But ideology and utopia will not remain separate; they tend to interweave, and one issue worth further reflection is how the two function together as well as tend to tear apart, in _Utopia_ and elsewhere.
Up for further consideration too is the place of this text in the trajectories of human consciousness in the tumultuous times of the early modern era. As already adverted to, part of the humanist movement related to shifts in awareness emerging from exploration of the planet. But other forces figured into these alterations also. One of these was the introduction of print (an innovation that the Utopians took to, as Hythloday witnesses). Working just after the incunabulum—or cradle—of print, the humanists were perhaps the first generation of European writers fully to embrace this technology, an important part of the era's seismic shifts in thinking and sensibility. As one scholar put it, "As an institution the printing press represented an autonomous and cosmopolitan site for the production of knowledge free of lay and ecclesiastical control." Print, in other words, helped to create a new kind of imaginative space, and a new feel for how intellection happens. The printed book or pamphlet allows an impression of a free-floating island of discourse, and even though no discourse is ever really broken off from the dialogues taking place in the human lifeworld, print technology was related (though not reducible) to new ways, including more cosmopolitan and mobile ways, of imagining and conducting intellectual life. Like Cardinal Morton breaking into the speech of the ponderous lawyer (p. 26), the discourse emergent with print disrupted older styles. These changes also created tensions. Hanan Yoran has gone perhaps as far as anyone in showing the discursive tensions of _Utopia._ As he points out, nearly all symbolic action has been eliminated from this island, where even law—that most contentiously symbol-laden of realms—is fantasized into a commonsense reign of the obvious (p. 102). For a community of scholars to whom all of human life is interwoven with symbolic action, such a space as this is a no-place indeed. The ways in which print technology figured into these tensions, along with the creative possibilities it allowed, might be further scanned.
Another area for further study is the family resemblance between _Utopia,_ along with humanist discourse generally, and certain strains of postmodern thought. With their emphases on the performativity of human identity, the slipperiness of language, and the provisionality of all human discourses— along with their commitment to literary experimentation and serious engagement in play—the humanists may be taken in some ways as precursors to Derrida and company. While the early humanists were far from deconstructionists avant la lettre, they could nevertheless have appreciated Derrida's assertion that when it comes to philosophical statement, political discourse, or ethical judgment, negotiation is always necessary, and there is always something about negotiation that "gets one's hands dirty" even when one is negotiating "in the name of purity." More was in Flanders busy with negotiations when the opportunity to write _Utopia_ fell into his soiled hands.
After Hythloday's description of Utopia, character More says to the reader, but not to Hythloday, that he harbors some objections to the Utopian way of life. He would prefer at this point, however, to avoid contention, so instead of arguing, he extends a hand to Hythloday: "I took his hand and led him in to dinner, though first I said we would have another time to consider these matters more thoroughly and to confer more fully. I only wish this would happen someday!" (p. 134). The narrative comes to its close with this gesture of friendship and a prospective note, the desire to talk more fully at some unspecified time in the future. Even with his objections, his final sentence—prospective as well—ends the text with wistful agreement, as he states, "I readily confess that in the Utopian commonwealth are very many features which in our societies I would wish rather than expect to see" (p. 135). Let us confer, then, so that we may discover what event may yet arrive out of that nowhere that is the future.
## NOTES
### _Introduction_
. J. H. Hexter argues in _More's Utopia: The Biography of an Idea_ (Princeton: Princeton University Press, 1952) and in Edward Surtz, S.J., and J. H. Hexter, eds., _Utopia,_ vol. 4 of _The Complete Works of St. Thomas More_ (New Haven: Yale University Press, 1965), pp. xv–xxiii, that because there are biographical details about the speaker of Book 2 that fit the Hythloday of Book 1, the opening of Book 1 up to the point where Giles asks Hythloday why he does not serve a prince must have also been written in Flanders. He also argues that the passionate peroration of Book 2, in which Hythloday excoriates Europe and defends communism, was written in London. However probable or plausible Hexter's arguments are, More could have written all of Book 1 in London, adding biographical details in Book 2 and possibly its peroration after he finished Book 1.
. The term "dystopia" had to be invented to accommodate the negative features of More's fantasy.
. _Utopia,_ with an introduction and notes by Edward Surtz, S.J. (New Haven: Yale University Press, 1964), p. xxvi.
. In my translation I have tried to reproduce stylistic elements of _Utopia_ as well as its meaning. I have borrowed and modified some passages from my article "Style and Meaning in _Utopia:_ Hythloday's Sentences and Diction."
. " 'Si Hythlodaeo Credimus': Vision and Revision in Thomas More's _Utopia,_ " _Soundings_ (formerly _The Christian Scholar_ ) 51 (1968): 271–89; reprinted in _Essential Articles for the Study of Thomas More,_ ed. Richard S. Sylvester and Germain Marc'hadour (Hamden, Conn.: Archon Books, 1977), pp. 290–301 (p. 298).
. How the other capital sins survive in the absence of pride and just which of the other capital sins cause the crimes punished by the Utopians are also difficult questions.
. Valerian Paget's _More's Millennium_ (1909) was simply a modernization of Robinson's translation and has no independent value.
. The translation in the Yale edition (Surtz and Hexter, eds., 1965) was a reworking of Richards' translation by Edward Surtz, but the result was usually not an improvement.
### _Utopia_
. More had been undersheriff since September 3, 1510. As such he presided over the court at one of the sheriff's jails, where he heard various (mostly minor) cases.
. The poem, "by the Poet Laureate Anemolius," was probably written by More himself. On the name Anemolius, see note 186, below.
. In the first edition (Louvain, 1516) and in the running heads of the two Basel editions of 1518, this letter is described as a "preface."
. Peter Giles (c. 1486–1533) was a humanist friend of More and Erasmus. He was a corrector at the press of Dirk Martens in his native city of Antwerp and was a clerk of that city from 1512 on.
. The name "Utopia" derives from Greek _ou_ ("not") and _topos_ ("place"), meaning "no place" (More also called it by the equivalent Latin name "nusquama"). "Utopia" includes a pun because the initial "u" may also be derived from Greek _eu_ ("good"). Hence Utopia is a good place which is no place.
. More visited Giles in Antwerp in September 1515; together with a letter dated 3 September 1516, he sent the manuscript of _Utopia_ to Erasmus for publication.
. The angel Raphael is a saving guide and healer in the biblical book of Tobias. Raphael's surname, Hythloday, is derived from Greek words meaning "peddler of nonsense."
. More refers to the principal divisions of rhetoric according to the classical tradition: invention (finding matter), disposition (arranging it), and eloquence (stylistic elaboration).
. This description applies to most of Hythloday's description of Utopia itself, but hardly to the elaborate and often passionate eloquence of Hythloday's language in much of Book 1 and in his peroration at the end of Book 2. Almost nothing in this letter (or in _Utopia_ itself, for that matter) can be taken at face value.
. More describes himself (accurately) as devoted to the active life which Hythloday rejects.
. John Clement (c. 1500–1572), one of the first students at Colet's humanist school, St. Paul's, became a page and pupil in More's household about 1514; later he became a distinguished physician.
. "Anyder" is coined from the Greek for "waterless"; "Amaurot" from the Greek for "made dark or dim."
. Sidenote: _Note the theological distinction between lying and speaking a falsehood_. Though it is apparently not found among the theologians, the distinction between _mentiri_ (tell a lie) and _mendacium dicere_ (speak a falsehood, with no intention of deceiving) derives from Aulus Gellius (11.11.1–4) and was well known. Erasmus, and perhaps Peter Giles, probably added the sidenotes that appeared in the margins of the original edition.
. Sidenote: _A holy ambition!_
. Sidenote: _Human judgments are ungrateful_.
. Sidenote: _Persons with no "nose"_ [appreciation of wit] _he calls "flat-nosed."_
. Sidenote: _A saying_. See Erasmus, _Adages_ 293 in _The Collected Works of Erasmus_ (Toronto: University of Toronto Press, 1974–), vol. 31, p. 311. _The Collected Works of Erasmus_ is hereinafter referred to as _CWE,_ followed by the volume number in italic and the page number in roman type: _CWE 31,_ 311.
. The metaphor seems to be drawn from wrestling.
. Sidenote: _A remarkable comparison_.
. Erasmus, _Adages_ 28 ( _CWE 31,_ 76–77).
. In August 1513 Henry's army had been victorious at the Battle of the Spurs and briefly occupied Thérouanne and Tournai; but his French campaigns, then and later, were as futile and destructive as those of the French kings in Italy.
. The difficulties were mainly connected with the wool trade between England and Flanders. They were serious enough for Wolsey to be worried early in 1515 that Charles would seize the English fleet for back taxes.
. By 1515 Charles V, later Holy Roman Emperor (1519), was Duke of Burgundy and Prince of Castile.
. Sidenote: _Cuthbert Tunstall_. Tunstall (1474–1559), bishop of London (1522) and later Durham (1530), was a close friend whom More admired throughout his lifetime. On 12 May 1516 Tunstall became Master of the Rolls and Vice-chancellor; as such he was chief of the twelve assistants to the Lord Chancellor.
. Sidenote: _An adage_. See Erasmus, _Adages_ 1406–7 ( _CWE 33,_ 245).
. Jean (or perhaps Jacques) de Halewyn, Seigneur de Maldeghem.
. De Themsecke (d. ca. 1536), a doctor of the law and a member of Charles V's council at Mechlin, was employed on many diplomatic missions. (Cassel is now in northern France.)
. On or before 25 July 1515.
. Sidenote: _Peter Giles_. In Flemish his name is "Gillis" or "Gilles," but the usual English translation of his Latin name ("Aegidius") is "Giles."
. Giles (1486–1533) was learned in the law and edited classical and humanist works. Since 1512 he had been chief clerk of the court of justice at Antwerp.
. Cf. Matt. 10:16. The same combination was part of the printer's mark of Johann Froben, who printed the two 1518 editions of _Utopia_.
. More left England 12 May 1515.
. Palinurus, Aeneas' steersman, dozed at the helm, fell overboard, and drowned ( _Aeneid_ 5.833–61), unlike the alert Odysseus and observant Plato who learned much from their travels ( _Odyssey_ 1.1–4; Diogenes Laertius 3.6–7.18–22).
. More expressed the same opinion in his _Letter to Oxford,_ in _The Complete Works of St. Thomas More,_ vol. 15, ed. Daniel Kinney (New Haven: Yale University Press, 1986), p. 143. (Works in this series are hereinafter cited as _CWM,_ followed by the volume number in italic and the page number in roman type: _CWM 15,_ 143).
. In 1515 the Portuguese excelled in exploration, especially in the far east.
. The voyages (1503–4) of the Florentine explorer Amerigo Vespucci (1451–1512), who was in the employ of the King of Portugal, were described in two Latin narratives (of disputed authenticity) published about 1507; one of the versions mentions the twenty-four mariners left behind in a fort at the farthest point of the voyage (Cape Frio in southeast Brazil).
. Lucan, _Pharsalia_ 7.818–19; cf. Augustine, _City of God_ 1.12.
. Sidenote: _Apophthem;_ cf. Erasmus, _Apophthegmata_ 7, Anaxagoras Clazomenus 4, and Cicero, _Tusculan Disputations_ 1.43.104.
. The Portuguese had visited Calicut (a city on the west coast of India, not Calcutta) by 1487 and established a station there in 1511.
. This is not an ordinary bench covered with sod. The small woodcut of the scene in the two editions of 1518 shows that it was a long wooden box filled with earth and covered on top with growing grass.
. The torrid zone between the Tropic of Cancer and the Tropic of Capricorn, the northern and southern limits between which the sun's orbit was thought to move.
. Scylla was a six-headed sea monster ( _Odyssey_ 12.73–100, 234–59; _Aeneid_ 3.424–32); Celaeno was one of the harpies, disgusting birds with women's faces ( _Aeneid_ 3.209–58); the Laestrigonians were giant cannibals ( _Odyssey_ 10.17–133).
. It seems likely that at this point More inserted the bulk of Book 1, the dialogue about counseling kings, which was written after Book 2, when More had returned to London. (See the Introduction, p. vii, and p. 141n.1.) In this addition More does not limit himself to describing Utopian institutions but gives Raphael's narration about the Polylerites, Achorians, and Macarians.
. Hythloday paraphrases a definition of liberty given by Cicero in a context similar to this one ( _De officiis_ 1.20.69–70).
. Cf. Erasmus, _Adages_ 115, 121, 3064 ( _CWE 31,_ 158–60, 167–68).
. A Cornish rebellion was crushed at the Battle of Blackheath on 22 June 1497.
. More had admired Morton (1420–1500) since the time he was a page in his household (c. 1490–92). He is portrayed as skilled and shrewd in More's _Richard III (CWM 2,_ 90–92).
. In his _Description of England_ (1587), ed. Georges Edelen (Ithaca, N.Y.: Cornell University Press, 1968), p. 87, William Harrison reported that in the reign of Henry VIII alone 72,000 thieves and vagabonds were hanged.
. Since Hythloday was in England in late 1497 and early 1498 he may be referring to English skirmishes in France in the early 1490s. But as he speaks in 1515, he may also be thinking of the much heavier casualties in Henry VIII's futile French campaigns of 1512–13.
. Plato uses the figure of the drones to describe an oligarchy ruled by rich men who exploit the poor and contribute nothing to society ( _Republic_ 8.552B–C).
. The parallel between soldiers and robbers is a frequent theme among humanists; see, for example, Erasmus, _Complaint of Peace (CWE 27,_ 317).
. In the time of Francis I the French relied mostly on Swiss and German mercenaries.
. The Latin "morosophi" (transliterated from Greek) means literally "foolish wisemen" (the reverse of the modern "sophomore"). See Lucian, _Alexander_ 40. Erasmus uses it in _The Praise of Folly,_ tr. Clarence H. Miller (New Haven: Yale University Press, 1979), p. 13; in _De copia, Opera omnia_ 1.12C; and in _Adagia_ (prol., _CWE 31,_ 23).
. _Bellum Catalinae_ 16.3.
. Foreign mercenaries often wreaked havoc in France during the Hundred Years' War (1337–1453).
. The Greek historian Herodian (mentioned in Book 2 as one of the authors Hythloday brought to Utopia) describes how several emperors were murdered by the barbarian mercenaries of the Praetorian guard. After the first Punic War, foreign mercenaries revolted against their Carthaginian employers. From the thirteenth to the sixteenth century, the Mamelukes (originally mercenaries from Turkey and Circassia) ruled despotically a large empire consisting of Egypt, Syria, and other parts of the Middle East.
. Especially Italy, which was often devastated by foreign mercenaries; Machiavelli, who firmly opposed the use of mercenaries, gives many examples of the harm they caused.
. The English defeated the French decisively at Crécy (1346), Poitiers (1356), and Agincourt (1415).
. Between the thirteenth and eighteenth centuries, much arable land was enclosed by hedges or ditches and used to pasture sheep. Hythloday's arguments against enclosure were widespread, and though it had its supporters (mostly because of the profitability of the wool trade), it undoubtedly caused much suffering to farm laborers and destroyed many villages.
. Long before and after 1515 many sumptuary laws were passed against extravagant display, especially in clothing, but they were honored more in the breach than the observance.
. During the reigns of Henry VII and Henry VIII laws were passed forbidding gaming and alehouses, limiting enclosure, restoring land from pasture to tillage, and restricting monopolies, but with little effect.
. Sidenote: _This shows the Cardinal's usual way of interrupting anyone who talks too much_.
. A proverbial saying (Erasmus, _Adages_ 924), derived primarily from Cicero, _De officiis_ 1.10.33.
. Sidenote: _Manlian edicts from Livy_. Proverbial for "harshly unjust" (Erasmus, _Adages_ 987). The Roman consul Manlius executed his son for winning a victory without having permission to do so (Livy 8.7.1–22).
. Stoics such as Zeno, Seneca, and Epictetus believed that virtue consisted in ignoring exterior forces and remaining faithful to the interior dictates of reason about what is right; such faithfulness has no degrees but is either kept or not. Cicero presents and refutes the paradox in _De finibus_ 4.10.21–23; Horace ridicules it in _Satires_ 1.3.113–24.
. Exod. 20:13, Deut. 5:17. (All scriptural references are to the Vulgate text and numbering.)
. Exod. 22:1–4.
. The Mosaic law does, of course, prescribe death as a punishment for various crimes. And even under the more merciful Christian dispensation, Hythloday does not always condemn capital punishment; as the remedy of last resort, it is employed by the Polylerites and the Utopians (pp. 30, 99).
. A name formed from Greek _polus_ ("much") and _leros_ ("nonsense").
. Sidenote: _We should note this, since we do otherwise_. Erasmus expresses the same opinion in _The Education of a Christian Prince (CWE 27,_ 270).
. Sidenote: _But nowadays the servants of noblemen find such a haircut attractive_.
. Erasmus, _Adages_ 1612.
. If a criminal could reach a place of asylum or sanctuary (usually a church) he could not be arrested, though during the reign of Henry VII the privilege was discussed and somewhat curtailed. It is debated in More's _Richard III (CWM 2,_ 27–33).
. Sidenote: _An entertaining exchange between a friar and a fool_.
. Cf. Erasmus, _Adages_ 113 ( _CWE 31,_ 154–55).
. Sidenote: _A proverb frequently bandied about among beggars_. There seems to be no recorded or recognized proverb here (though it might still have been a frequent saying among beggars); there may be some allusion to the priest who passed by the wounded Samaritan (Luke 10:31).
. Unordained members of religious orders were called "lay brothers."
. Sidenote: _He alludes to the Horatian phrase "doused with Italian vinegar."_ See _Satires_ 1.7.32 and Erasmus, _Adages_ 1252 ( _CWE 33,_ 164). The phrase is here translated as "needled."
. John 17:12, 2 Thess. 2:3.
. Luke 21:19.
. Ps. 4:5. Sidenote: _How well the people in the story speak in character!_
. Ps. 68:10.
. Sidenote: _Apparently the friar, in his ignorance, misuses "zelus" as if it were neuter like "scelus."_ In 4 Kings 2:23–25 some children mocked Elisha because of his baldness; when he cursed them two bears came out of the woods and tore forty-two of them to pieces. The friar quotes a hymn attributed to Adam of St. Victor, sung within the octave of Easter. In the ordinary pronunciation of Erasmus' time _zelus_ ("zeal") could sound like _scelus_ ("crime"). The confusion produces the following result: those who mocked Elisha . . . feel the crime of the bald man.
. Prov. 26:5. But the preceding verse says: "Do not answer the fool according to his folly lest you become like him."
. Perhaps alluding to Ps. 7:16.
. _Republic_ 5.473C–D, _Epistles_ 7.326A–B.
. During his three sojourns at Syracuse, Plato failed in his attempt to reform the tyrant Dionysius or his son (also Dionysius); see his _Epistles_ 7 and Plutarch, _Dion_. 4.1–5.3, 10.1–20.2.
. Here Hythloday launches into a 464-word sentence, suspended, unrealistically intricate, interminable (as Lupton called it), which ends with "react to this speech." Though translators (with the exception of Robinson) have generally broken up this sentence to make it easier, such manipulation is unjustified: the sentence is no easier in Latin than in English. Its difficulty springs from Hythloday's difficult outlook.
. In 1515, the time of More's imagined interview with Hythloday, the king of France was Francis I, who continued the policy of his predecessors Charles VII and Louis XII. All three invented claims to Milan and Naples, but their military adventures in Italy foundered in confusion and intrigue.
. The French won Milan in 1499, lost it in 1512, regained it in 1515. They won Naples in 1495, lost it in 1496, regained it in 1501, and lost it in 1503.
. Sidenote: _Indirectly he is discouraging the French from acquiring Italy_. At the battle of Agnadello (1509), France defeated Venice and deprived it of its territory on the mainland. By 1515, when the Venetians helped Francis I in his campaign again Milan, the French king restored Verona to his Venetian ally. Hythloday wonders if the French king is ready to turn on his recent ally once more.
. After the death of Charles the Rash, Duke of Burgundy (1477), Louis XI of France tried to seize all the vast Burgundian holdings, though many parts clearly did not belong to France.
. The German mercenary footsoldiers were surpassed only by the Swiss; both were despised and excoriated by Erasmus and many humanists.
. Emperor Maximilian of Hapsburg, grandfather of Charles V, was usually impecunious and totally unreliable. A votive offering was normally an expensive gift left in a church or shrine in thanksgiving for a favor from God or a saint.
. With the help of troops sent by a duped Henry VIII, Ferdinand II, King of Aragon and regent of Castile, occupied southern Navarre in 1512 and annexed it to Castile in 1515.
. Charles V, prince of Castile and the future emperor (1519) was often affianced for dynastic reasons, especially to French brides.
. Francis I did make a treaty with England in April 1515.
. The Scots were traditionally allies of France against England.
. The French had supported several pretenders to the English throne during the reigns of Henry VII and Henry VIII: Lambert Simnel, Perkin Warbeck, Edmund de la Pole, and his brother Richard.
. Erasmus, _Adages_ 860 ( _CWE 32,_ 215).
. Cf. More's epigram "On Lust for Power": "Among many kings there will be scarcely one, if there is really one, who is satisfied to have one kingdom. And yet among many kings there will be scarcely one, if there is really one, who rules a single kingdom well" ( _CWM 3/2,_ 257).
. Sidenote: _A notable example_. From Greek _a-_ ("without") and _choros_ ("place, country").
. More here echoes Erasmus' _Adages_ 1401 ( _CWE 33,_ 237–43): "Sparta is your portion; make it flourish."
. Hythloday presents his second imaginary council in an even longer marathon sentence (926 words); it is just as extravagant in Latin as in this English translation.
. Fraudulent manipulation of the currency was practiced by Edward IV, Henry VII, and (later) Henry VIII.
. In 1492 Henry VII not only levied taxes for a pretended war against France but accepted a bribe from Charles VIII of France for not fighting it.
. Henry VII's ministers Empson and Dudley were notorious for such chicanery.
. The royal prerogative, the special, inherited claims of the king apart from common law, was a subject of considerable dispute even in More's time, though it became more heated in the following century.
. Hythloday adapts Cicero's statement in _De officiis_ 1.8.25: "Recently Marcus Crassus said that no amount of money is enough for one who wishes to be head of state unless it produces enough income to maintain an army."
. Among the techniques mentioned by Aristotle by which tyrants maintain their power are keeping subjects poor and humble-spirited and pretending to rule for the advantage of the citizens ( _Politics_ 5.9.4, 8, 11, 1313b, 1314a–b).
. The biblical and Homeric figure of kings as shepherds was widespread; in his speech at the opening of Parliament in 1529 More compared kings to shepherds. See also his Latin epigrams against tyranny ( _CWM 3/2,_ 162–65, 168–69).
. The saying derives from Manlius Curius Dentatus (Plutarch, _Moralia_ 194F) but it was also attributed to Gaius Fabricius Luscinus by classical and medieval authors.
. From the Greek _makarios_ ("happy"); the Greek word introduces each of the beatitudes (Matt. 5:3–11).
. More may be thinking of Henry VII, who had an enormous sum in his treasury when he died.
. Sidenote: _A proverb_. See Erasmus, _Adages_ 1387 ( _CWE 31,_ 376).
. The following argument centers on the moral and rhetorical notion of decorum (Cicero, _De officiis_ 1.27.93–39.141, _Orator_ 21.69–22.74, and _De oratore_ 3.55.109–12). It is also based on the conflict between rhetorical persuasion, which deals with probable truths, and philosophical logic, which produces demonstrable truths.
. Sidenote: _The philosophy of the schools_. In the text and sidenote this philosophy is designated "scholastica." The only academic philosophy in More's time was that of the universities, which we nowadays call scholasticism, so that in this case "academic" and "scholastic" are practically synonymous. The humanists generally attacked the hairsplitting excesses of scholastic philosophy and favored a more rhetorical approach to literature and life. See, for example, More's _Letter to Dorp (CWM 15,_ 29–39, 49–70).
. Sidenote: _A marvelous comparison_. _Octavia_ is a tragedy once attributed to Seneca in which Seneca discusses the abuse of power with Nero. Cf. Erasmus, _Adages_ 91, "to be subservient to your role" ( _CWE 31,_ 131–32).
. Sidenote [in Greek]: _A mute role_. John Clement plays such a part in _Utopia_.
. Plato allows rulers (even presumably philosopher-kings) to lie to their subjects for a useful purpose ( _Republic_ 3.21.414B–415D, 5.8.459C–D). Quintilian says that "everyone must allow, what even the sternest of the Stoics admit, that the good man will sometimes tell a lie" ( _Institutes_ 12.1.38).
. Matt. 10:27, Luke 12:3.
. The so-called Lesbian ruler was made of lead so as to accommodate itself to measuring curved surfaces; see Erasmus, _Adages_ 493 ( _CWE 31,_ 465).
. _Adelphoe_ 1.2.145–47.
. _Republic_ 6.10.496D–E.
. See p. 101.
. According to Diogenes Laertius (3.23), "the Arcadians and Thebans, when they were founding Megalopolis, invited Plato to be their legislator; but . . . when he discovered that they were opposed to equality of possessions, he refused to go." In the the _Republic_ Plato prescribes community of property (and of wives and children) only for the guardians (5.12.464B–E), but in the _Laws_ he says that in the best state it would be observed by the whole populace (5.739B–D).
. More summarizes Aristotle's arguments in the _Politics_ (2.1.2.1260b–4.13.1267b) against Plato's advocacy of communism. Aristotle's arguments had been adopted by the medieval scholastics such as Thomas Aquinas in his commentary on Aristotle's _Politics_ (2.1–7).
. The numerical equivalents of the Greek letters in "Abraxas" (the usual form, rather than "Abraxa") add up to 365. The name was given to the highest of the 365 heavens invented by the heretic Basilides.
. Sidenote: _A greater task than cutting through the Isthmus_. Several attempts to dig a canal across the Isthmus of Corinth failed so that the attempt became proverbial for failure (Erasmus, _Adages_ 3326).
. Sidenote: _Common effort lightens a burden_.
. According to Erasmus, in Utopia More "represented the English commonwealth in particular" ( _CWE 7,_ 23.281). In 1587, according to William Harrison's _Description of England_ (1587), ed. Georges Edelen (Cornell University Press, 1968, pp. 86–87), England had fifty-three counties, which, together with London, make it match the citystates of Utopia. The city-states are mostly independent but loosely federated, each having its own governor; they are united only by codes and customs, as well as a triennial meeting of a senate.
. Sidenote: _Likeness breeds concord_.
. Sidenote: _But such a desire is the curse of modern commonwealths_.
. From a Greek compound meaning "ruler of a tribe."
. Pliny mentions artificial incubation ( _Natural History_ 10.76.154–55) but it seems not to have been practiced in More's time.
. That is, they do not use it to make beer or ale, as the English do.
. Sidenote: _The advantage of communal labor_.
. From a Greek adjective meaning "without water." Amaurot resembles London in its tidal river (the Thames) and smaller stream (Fleet Ditch, except that London's stream was foul and unpleasant).
. Sidenote: _The same thing happens to the Thames in England_.
. Sidenote: _In this feature London is also like Amaurot_. But Amaurot has the advantage of having its bridge above the city, not below it.
. Sidenote: _This is reminiscent of Plato_. See _Republic_ 3.22.416D.
. Sidenote: _Virgil also praised the usefulness of gardens_. See _Georgics_ 4.116–48.
. That is, 244 B.C., when Aegis IV became king of Sparta; he was killed because of the egalitarian reforms he wished to introduce. See Richard Schoeck, "More, Plutarch, and King Aegis: Spartan History and the Meaning of History," _Philological Quarterly_ 35 (1956): 366–75; reprinted in _Essential Articles for the Study of Thomas More,_ ed. Richard Sylvester and Germain Marc'hadour (Hamden, Conn.: Archon Books, 1977), pp. 275–80.
. In More's time lead was commonly used to roof important buildings. William Harrison, in his _Description of England_ (1587), ed. Georges Edelen (Ithaca, N.Y.: Cornell University Press, 1968), speaks of "fine alabaster burned, which they call plaster of Paris, whereof in some places we have great plenty and that very profitable against the rage of fire," but he is describing the plastering of interior walls, not roofs, which he says are covered with shingles, straw, sedge, reeds, or slate (p. 196).
. Glass windows were uncommon in homes during More's time; oiled linen, sheets of horn, or lattices of wicker or wood were used instead. Hythloday means that oiled linen is brighter and more impervious than linen alone, not that it is superior to glass.
. Sidenote: _In the Utopian language "tranibor" means "chief director."_ "Syphogrant" seems to be derived from the Greek compound meaning "wise old man" (or perhaps "old man of the sty" = steward). "Tranibor" seem to come from a Greek compound meaning "plain eater." But other meanings have also been suggested. In fact, Hythloday continues to use the older terms "syphogrant" and "tranibor," not "phylarch" (ruler of a tribe) or "protophylarch" (chief phylarch).
. Thus there are six thousand families in Utopia, excluding the countryside (see p. 54).
. Sidenote: _A remarkable way of electing officials_.
. Sidenote: _Tyranny is hateful to the well-ordered commonwealth_.
. Utopia is a federation of democratic republics: the households elect the syphogrants, who elect the tranibors and governor (whom they can also remove from office). The syphogrants also select the class of scholars, from which all high officials are chosen.
. Sidenote: _Disputes should be settled quickly, but nowadays they are deliberately and lengthily prolonged_.
. Sidenote: _Nothing should be decided hastily_.
. But for the whole island of Utopia there is no single executive branch to carry out or enforce the deliberations or decisions of this council.
. Sidenote: _Would that the same thing were done in our councils_.
. Sidenote: _This is the meaning of the proverb "take counsel at night."_ See Erasmus, _Adages 1143 (CWE 33,_ 96).
. Sidenote: _Farming is an occupation common to everyone, though here it is fobbed off on a few despised workers_.
. Plato ( _Republic_ 7.797A–B) and Aristotle ( _Politics_ 7.15.5.1336a). Plato specifically advises that "to make a good farmer [a man] must play [in childhood] at tilling land" ( _Republic_ 1.643B–C).
. Sidenote: _Trades should be learned to satisfy needs, not luxury_.
. Sidenote: _Let everyone learn the trade for which he has a natural aptitude_.
. Unlike the Utopians, Plato insists that each craftsman must have only one trade ( _Republic_ 2.11.370A–C, 2.13.474B–C; _Laws_ 8.846D–E).
. Sidenote: _The idle are to be expelled from the commonwealth_.
. Statutes during the reign of Henry VII required laborers to work from daybreak to nightfall in spring and summer and from before 5 a.m. to between 7 and 8 p.m. in fall and winter.
. Sidenote: _The work of laborers should be kept within bounds_.
. Sidenote: _But nowadays playing at dice is the sport of princes_.
. More surely knew how inaccurate Hythloday is here, since women in his time had duties at least as heavy as they have now.
. That is, members of the religious orders.
. Sidenote: _A very perceptive observation_.
. This number would be made up of the governor, the two hundred syphogrants, the twenty tranibors, the thirteen priests, the scholars, and the ambassadors.
. "Barzanes" derives from the Hebrew for "son of" and the Greek Doric form for "of Zeus." A Chaldean named "Mithrobarzanes" appears in Lucian's _Menippus_ , which More translated. "Ademus" derives from the Greek for "without a people."
. An average of twelve adults in each household would produce a population of seventy-two thousand in each city. Adding children and slaves would probably bring it to more than one hundred thousand (of whom only five hundred are exempt from work).
. Hythloday's and the Utopians' rather facile justification of colonialism offers many difficulties. For example, if no one is using or occupying the land, why does anyone have to be driven from it by force? Is farming the only satisfactory use of land?
. Sidenote: _Thus they avoid having a crowd of idle servants_.
. That is, each of the four sides of a block has thirty houses, with a hall in the middle of each side.
. Sidenote: _They always take freedom into account lest anyone act under compulsion_.
. That is, without the help of servants.
. Sidenote: _Praise and a sense of duty are the best way to encourage citizens to act properly_.
. Sidenote: _The education of the young_.
. Sidenote: _Priests above the prince, though nowadays even bishops are the lackies of princes_.
. Sidenote: _Nowadays even monks rarely observe this custom_.
. Sidenote: _Nowadays physicians condemn this practice_.
. Cf. 2 Thess. 3:10.
. Sidenote: _O holy commonwealth, worthy to be imitated even by Christians_.
. Sidenote: _See how they never forget their sense of community_.
. Sidenote: _What a clever fellow!_
. Sidenote: _What a magnificent contempt for gold!_
. Sidenote: _A very fine story_. "Anemolian" is from the Greek word for "windy."
. Sidenote: _O what a craftsman!_
. Sidenote: _He calls it dubious because the gems are fake, or at least because their glitter is scanty and dim_.
. Cf. Lucian, _Demonax_ 41; see also _CWM 13,_ 8.
. Sidenote: _How true and well put!_
. Sidenote: _How much wiser are the Utopians than the general run of Christians!_
. More uses "philosophy" in the older, broader sense of the investigation of all the arts and sciences, including mathematics and the natural sciences (which was often called "natural philosophy").
. They have mastered the quadrivium, the second tier of university studies (music, arithmetic, geometry, and astronomy); of the first tier, the trivium (grammar, logic or dialectic, and rhetoric), dialectic is mentioned here. Grammar and rhetoric they would learn in their literary studies.
. Sidenote: _There seems to be some underlying satire in this passage_.
. Peter of Spain's thirteenth-century _Little Logicbook,_ with its finespun categories and distinctions, was dissected and mocked by More in his _Letter to Dorp,_ which he wrote in 1515, near the time he wrote the second book of _Utopia_. On complicated "rules about restrictions, amplifications, and suppositions" see More's text and Daniel Kinney's introduction to _Letter to Dorp_ in _CWM 15,_ liv–lvii, 29–39.
. "First intention" refers to the intellect's direct perception of an object; a "second intention" is the intellect's perception of or reflection on a first intention. It has no objective existence outside the mind.
. That is, the universal concept of man that applies to each man in particular. From the fourteenth century through More's time, scholastic philosophers from the camps of the Realists and the Nominalists quarreled elaborately about whether and how universals had any real existence.
. Sidenote: _But nowadays these practitioners rule the roost among Christians_. More wrote a number of Latin epigrams ridiculing judicial astrology ( _CWM 3/2,_ 158, 166, 208, 214–16, 348–49).
. Sidenote: _Natural science the most uncertain study of all_.
. These three categories of goods (external goods and goods of the mind and of the body) derive primarily from the Aristotelian tradition. Generally the Aristotelians applied "good" to all three categories; the Stoics, only to the goods of the mind.
. Sidenote: _The Utopians measure happiness by honorable pleasure_. That is, they are inclined to the Epicurean position that pleasure is the highest good. Beginning with Lorenzo Valla's _The True and False Good_ (1444–49) and with the help of such thinkers as Ficino, Pico, and Erasmus, Epicurean philosophy had been rehabilitated and shown to consist not in mere hedonism but rather in the calm pleasures of the mind. But the Utopians differ sharply from the Epicureans, who did not believe in immortality and thought the gods were unconcerned about mankind.
. Sidenote: _First principles of philosophy should be derived from religion_.
. Sidenote: _The theology of the Utopians_.
. Sidenote: _The immortality of the soul, about which not a few Christians nowadays have doubts_. The fifth Lateran Council (1513) affirmed as dogma the immortality of the soul. The philosopher most closely associated with the dispute concerning the immortality of the soul was Pietro Pomponazzi, whose treatise _On the Immortality of the Soul_ (1516) argued that the doctrine could not be proved by reason but has to be derived from revealed religion.
. Sidenote: _Just as not just any pleasure should be sought after, so too pain should not be pursued except for the sake of virtue_. This is in keeping with the teachings of Epicurus; see Diogenes Laertius 10.130–32. The opposite faction is the Stoics.
. Sidenote: _This is a teaching of the Stoics_.
. Sidenote: _But nowadays some seek out pain, as if religion consisted in it, whereas pain is only to be borne if it occurs by natural necessity or to someone performing the duties of piety_. Seneca, whose Stoicism is often severe and uncompromising, agrees with the Utopians: "Our motto . . . is 'Live according to Nature'; but it is quite contrary to nature to torture the body, to hate unlabored elegance, to be dirty on purpose, to eat food that is not only plain, but disgusting and forbidding" ( _Epistulae morales,_ 5.4). Unlike some Stoics, Seneca is not entirely unsympathetic with Epicurus: "the teachings of Epicurus are upright and holy and, if you consider them closely, austere; for his famous doctrine of pleasure is reduced to small and narrow proportions, and the rule that we Stoics lay down for virtue, his same rule he lays down for pleasure—he bids that it obey Nature" ( _De vita beata_ 13.1). The Utopians combine elements of Stoicism and Epicureanism, and add to the blend belief in divine providence, the immortality of the soul, and rewards and punishments in the afterlife—doctrine not specifically Christian but not uniformly held until the advent of Christianity.
. An Epicurean (not a Stoic) teaching; see Diogenes Laertius 10.138.
. Sidenote: _A remarkable hypothesis, and a very apt one_.
. More wrote a Latin epigram against the cruelty of hunters ( _CWM 3/2,_ p. 123).
. Sidenote: _But nowadays this is the craft practiced by godlike courtiers_.
. More is probably thinking of hunting dogs.
. Sidenote: _This point should be noted with special care_. In _The Confutation of Tyndale's Answer_ (1532–33), More argued on religious grounds that "besides the taming of the body, fasting and our pain taken therein pleaseth god done with devotion, and serveth us for obtaining many and great gifts of grace ( _CWM 8/1,_ p. 72).
. Sidenote: _But nowadays blockheads and dolts are chosen to be educated; the most talented minds are corrupted by pleasures_.
. The pupil and successor of Aristotle.
. Constantine Lascaris (d. 1501) and Theodore of Gaza (d. 1475) wrote grammars of Greek. The dictionary of Hesychius (fl. ca. a.d. 400) was first published in 1514; Dioscorides (fl. ca. a.d. 50) wrote a handbook of medical and botanical terms.
. Plutarch (ca. A.D. 50–120) was a favorite Greek writer among Renaissance humanists, both for his _Moralia_ and for his _Parallel Lives_ of eminent Greeks and Romans. Several pieces by the satirist Lucian (b. ca. A.D. 120) were translated by More and Erasmus and first published in 1506; they were reprinted ten times in the sixteenth century.
. In the early sixteenth century the Venetian printer Aldus Manutius was famous for his compact, elegantly printed editions of classical authors in both Latin and Greek. In 1508 he printed the first enlarged edition of Erasmus' huge and elaborate collection of proverbs, _Adagia,_ which brought Erasmus almost instant fame.
. Thucydides and Herodotus are the leading historians of ancient Greece. Herodian (ca. A.D. 170–240) wrote a Greek history of the Roman emperors who reigned from A.D. 180 to 238.
. A name in keeping with that of Hythloday himself: "tricae apinaeque" became proverbial meaning "stuff and nonsense" (Erasmus, _Adages_ 143, _CWE 31,_ 184).
. Hippocrates (fifth century B.C.) and Galen (second century A.D.) were the leading Greek writers on medicine. _Microtechne_ was a medieval summary of Galen.
. Sidenote: _The remarkable fairness of this people_.
. The non-hereditary character of Utopian slavery distinguishes it from both ancient slavery and feudal serfdom. In More's time it was generally agreed that Christians should not be enslaved, but the same was not true of African negroes and American Indians.
. Such euthanasia, naturally, is contrary to Catholic teaching; at Morton's court Hythloday himself had said that God has forbidden us to kill ourselves (p. 27, above), but he also told More and Giles earlier that he did not intend to discuss whether or not Utopian moral principles are correct (pp. 91–92). More has a long psychological analysis of suicide in _A Dialogue of Comfort (CWM 12,_ 129–56).
. According to canon law in More's time, girls could not marry before the age of twelve and boys not before fourteen. Plato ( _Republic_ 5.9.460E, _Laws_ 4.721A–B) and Aristotle ( _Politics_ 7.14.6.1335a) set the age of marriage for women at at least twenty and for men over thirty.
. Sidenote: _This practice is somewhat immodest, but it is far from imprudent_. Plato requires similar premarital inspections in _Laws_ 6.771E–772A, 11.925A.
. In More's time the Church permitted separation in the case of adultery but did not allow remarriage. In his commentaries on 1 Cor. 7:10–11 and 39 Erasmus favored relaxing the prohibition of remarriage.
. See Erasmus, _Adages_ 1537 ( _CWE 33,_ 309–10).
. The Latin for "fool" here is _morio,_ wordplay on More's name; Erasmus had exploited the same pun in the prefatory letter of his _Encomium Moriae (The Praise of Folly),_ which is dedicated to More. Thomas More kept a fool, Henry Patenson, in his household; Patenson appears in Holbein's sketch of More's family and is mentioned by More in his _Confutation of Tyndale's Answer (CWM 8/2,_ 900–901).
. An error in the 1516 edition is corrected differently in the editions of 1517 and 1518, in both cases probably by More himself. One correction could mean that only crafty lawyers are excluded; the other must mean that all lawyers are excluded because all lawyers are crafty. The latter interpretation seems preferable.
. The rulers and popes of More's time were notorious for breaking treaties or making them with the deliberate intention of breaking them. This was especially true of the popes Alexander VI and Julius II. Machiavelli said Alexander VI "never did anything, never thought of anything other than to deceive men. . . . And never was there a man who had greater success in asserting, and with greater oaths in affirming a thing, who observed it less" ( _The Prince_ 18).
. A common false etymology derived "bellum" (war) from "belua" (beast)—or the other way around. For a full account of pacificism in More and his humanist contemporaries see R. P. Adams, _The Better Part of Valor: More, Erasmus, Colet, and Vives on Humanism, War, and Peace, 1496–1535_ (Seattle: University of Washington Press, 1962).
. One of the key texts giving the rules for fighting a "just" war was Cicero, _De officiis_ 1.11.34–1.13–40.
. Greek compounds meaning "cloud-born" and "citizens of a country without people."
. From a Greek compound meaning "busy sellers"—that is sellers and resellers of their military services.
. Sidenote: _A people not unlike the Swiss_.
. Sidenote: _Above all the commander should be assailed so as to end the war sooner_.
. A ducat was a gold coin minted primarily by Venice and worth about a quarter of a pound sterling at that time. The 700,000 ducats mentioned here would be worth many hundred times that much today.
. The Utopians do not have an ordinary treasury; perhaps deposits owed the Utopians and placed in the treasuries of other countries are what is meant here.
. Among the ancient Persians, Mithras was the supreme deity, identified with light.
. Sidenote: _Monasteries_. Communism was practiced by religious orders in More's time, as it still is; on communism among the early Christians see Acts 2:44–45 and 4:32–37.
. Of the seven sacraments, only baptism and matrimony can be administered by laymen.
. In sacramental theology "character" is a technical term meaning the indelible quality bestowed on a soul by sacraments that cannot be repeated: baptism, confirmation, and holy orders.
. Sidenote: _People must be drawn to religion by hearing it praised_.
. In Christian England, More approved of punishing religious dissent or heresy, but that was because the true religion had been revealed there, as it had not in Utopia; as More said in _A Dialogue Concerning Heresies (CWM 6,_ 345–46), "if it were now doubtful and ambiguous whether the church of Christ were in the right rule of doctrine or not, then were it very necessary to give them all good audience that could and would anything dispute on either party for it or against it, to the end that if we were now in a wrong way, we might leave it and walk in some better."
. Sidenote: _A remarkable opinion about the souls of animals_.
. In _A Dialogue Concerning Heresies (CWM 6,_ 211, 213) More wrote concerning saints: "For if their holy souls live, there will no wise man ween them worse and of less love and charity to men that need their help, when they be now in heaven, than they had when they were here in earth. . . . When saints were in this world at liberty and might walk the world about, ween we that in heaven they stand tied to a post?"
. Sidenote: _The active life_.
. "Buthrescae" is a Greek word meaning "extraordinarily religious." In More's Europe the adjective "religious" was applied to members of religious orders, who differed, however, from the Buthrescae in that they combined labor with prayer, study, and contemplation.
. Hythloday must mean that the priests supervise the education of children, for in each city there are many thousands of children and only thirteen priests.
. In _The Confutation of Tyndale's Answer (CWM 8/1,_ 260–61) More accepts the traditional view that women may not be ordained as priests.
. Sidenote: _But what a flock of them we have!_
. Sidenote: _O these priests are far holier than ours!_
. The first Greek compound means "dog days" (or perhaps "starting days); the second means "turning days."
. There may be more than one service on every feastday, but even so the churches would have to be very large indeed: only thirteen of them serve about one hundred thousand inhabitants of each city.
. Cf. Matt. 5:23–24.
. Sidenote: _But among us the most defiled strive to get closest to the altar_.
. Latin _superos,_ which includes the one God, the other gods believed in by some of the Utopians, and their ancestors who are in heaven.
. A startling idea, but perhaps Hythloday (or the Utopians) mean that for children this tends to be true.
. In his _Four Voyages_ Vespucci mentions that the American Indians made vestments of feathers.
. Fr. Surtz notes that many of More contemporaries, especially Erasmus, objected to the elaborateness of church music and urged that it be composed so as to emphasize the meaning of the words ( _CWM 4,_ 555–56).
. Hythloday is so carried away that he speaks as if he is still in Utopia.
. Goldsmiths often functioned as bankers.
. Sidenote: _Note this, reader!_
. Sidenote: _A striking phrase_
. The remora has a suck-disk on top of its head, by which it attaches itself to larger fish or ships; impressed by its tenacity, the ancients thought it could impede the progress of a ship.
. What "More" says here is in keeping with his earlier Aristotelian arguments against community of property. Aristotle continually associates nobility and the highest virtue with wealth; he defines magnificence as "suitable expenditure on a grand scale" ( _Nicomachean Ethics_ 4.2.1.1122a). But many readers get the impression that More lets the mask of the character "More" slip to reveal a hint of irony. For a discussion of the critical disputes about this passage see Thomas I. White, _"Festivitas, utilitas, et opes:_ The Concluding Irony and Philosophical Purpose of Thomas More's _Utopia," Albion_ 10 (1978): 135–50.
. This sentence is incomplete in the Latin and has been left so in the translation.
. _Andria_ 4.4.770–71.
### _Afterword_
. Clarence H. Miller, _Humanism and Style: Essays on Erasmus and More_ (Bethlehem, Penn.: Lehigh University Press, 2011), pp. 105–7.
. Jacob Burkhardt, _The Civilization of the Renaissance in Italy,_ trans. S. G. C. Middlemore (1860; repr., Vienna: Phaidon Press, 1937).
. See, for example, the following: Douglas Bush, _The English Renaissance and Humanism_ (Toronto: University of Toronto Press, 1939); Paul Oskar Kristeller, _Renaissance Thought: The Classic, Scholastic, and Humanist Strains_ (New York: Harper and Row, 1955); Charles Trinkaus, _The Scope of Renaissance Humanism_ (Ann Arbor: University of Michigan Press, 1983); and Charles G. Nauert, _Humanism and the Culture of Renaissance Europe_ (Cambridge: Cambridge University Press, 1995).
. See Bush, _The English Renaissance,_ pp. 13–38; Kristeller, _Renaissance Thought,_ pp. 3–7; Trinkaus, _Scope of Renaissance Humanism,_ pp. 3–31; and Nauert, _Humanism,_ pp. 1–7. Nevertheless, the old idea of the medieval period as a "Dark Age" is not without its defenders. See Jack Goody, _Renaissances: The One or the Many?_ (Cambridge: Cambridge University Press, 2010), p. 11.
. See R. N. Swanson, _The Twelfth-Century Renaissance_ (New York: Manchester University Press, 1999).
. Burkhardt, _Civilization of the Renaissance,_ p. 292.
. Goody, _Renaissances,_ pp. 5, 7–42. For an interesting study of the cultural energies released by print, see Alexandra Halasz, _The Marketplace of Print: Pamphlets and the Public Sphere in Early Modern England_ (Cambridge: Cambridge University Press, 1997).
. Bush, _The English Renaissance,_ p. 82. For a helpful introduction to humanism in England, see Clare Carroll, "Humanism and English Literature in the Fifteenth and Sixteenth Centuries," in _The CambridgeCompanion to Renaissance Humanism,_ ed. Jill Kraye (Cambridge: Cambridge University Press, 1996), pp. 246–68. For a circumstantial and critical history of humanism, see Anthony Grafton and Lisa Jardine, _From Humanism to the Humanities: Education and the Liberal Arts in Fifteenth- and Sixteenth-Century Europe_ (Cambridge: Harvard University Press, 1986).
. _The Basic Works of Aristotle,_ ed. Richard McKeon (1947; repr., New York: Modern Library, 2001), p. 1329.
. See John M. Perlette, "Irresolution as Solution: Rhetoric and the Unresolved Debate in Book 1 of More's _Utopia,_ " _Texas Studies in Literature and Language_ 29:1 (Spring 1987): 28–53. Perlette reads the debate in _Utopia_ about the worth of advising a prince on public affairs as an extension of the ancient debate between Plato and the Sophists.
. This is J. H. Hexter's coinage for the dialogue in Book 1 concerning the usefulness of advising a prince. See his _More's "Utopia": The Biography of an Idea_ (New York: Harper and Row, 1952), p. 102.
. Thomas More, "Letter to Martin Dorp," in _In Defense of Humanism: Letter to Martin Dorp; Letter to the University of Oxford; Letter to Edward Lee; Letter to a Monk, with a New Text and Translation of Historia Richardi Tertii,_ ed. Daniel Kinney, vol. 15 of The Complete Works of St. Thomas More (New Haven: Yale University Press, 1986), pp. 26–39.
. Lorenzo Valla, _On the Donation of Constantine,_ trans. G. W. Bowersock (Cambridge: Harvard University Press, 2007), p. 73.
. Harvey Cox, _The Feast of Fools: A Theological Essay on Festivity and Fantasy_ (Cambridge: Harvard University Press, 1969), pp. 82–97. For helpful studies of the centrality of play to human life, see Johan Huizinga, _Homo Ludens: A Study of the Play Element in Culture_ (Boston: Beacon Press, 1955), and Hugo Rahner, _Man at Play: Or, Did You Ever Practice Eutrapelia?_ trans. Brian Battershaw and Edward Quinn (London: Burns and Oates, 1963).
. Nauert, _Humanism,_ pp. 5–6.
. Peter Ackroyd, _The Life of Thomas More_ (New York: Doubleday, 1998), p. 32.
. Ibid., pp. 71–80. See also Richard Marius, _Thomas More: A Biography_ (Cambridge: Harvard University Press, 1984), pp. 71–78.
. Ackroyd, _Life of Thomas More,_ pp. 82–95; Marius, _Thomas More,_ pp. 83–87, 153.
. See Hexter, _More's "Utopia,"_ p. 99.
. Alan F. Nagel, "Lies and the Limitable Inane: Contradiction in More's _Utopia," Renaissance Quarterly_ 26:2 (Summer 1973), pp. 175–77).
. See the essay "Editions of _Utopia_ " by Edward Surtz, S.J., in _Utopia,_ ed. Edward Surtz, S.J., and J. H. Hexter, vol. 4 of The Complete Works of St. Thomas More (New Haven: Yale University Press, 1965), pp. clxxxiii–cxciii; and R. W. Gibson and J. Max Patrick, _Thomas More: A Preliminary Bibliography of His Works and of Moreana to the Year 1750 with a Bibliography of Utopiana_ (New Haven: Yale University Press, 1961), pp. 3–57.
. It is striking that in his early twenties, More delivered a series of lectures on St. Augustine's _City of God._ See Ackroyd, _Life of Thomas More,_ pp. 104–6.
. See Hexter, _More's "Utopia,"_ pp. 85–96; Marius, _Thomas More,_ pp. 67–69; and Ackroyd, _Life of Thomas More,_ p. 99. Hexter clarifies that there are also groups within Utopia who live a professed life that more closely resembles monasticism.
. _The Rule of St. Benedict,_ trans. Anthony C. Meisel and M. L. del Mastro (New York: Image Books, 1975), p. 106.
. More's Latin, along with a prose translation, appears in Clarence H. Miller et al., eds., _Latin Poems,_ vol. 3, part 2, of The Complete Works of St. Thomas More (New Haven: Yale University Press, 1984), pp. 238–41.
. The quotation comes from the Arden edition of Shakespeare's _The Tempest,_ ed. Frank Kermode (London: Methuen, 1954), 2.1.153–54.
. In his _More's "Utopia"_ (Toronto: University of Toronto Press, 2000), Dominic Baker-Smith emphasizes More's concern with what modern theology has identified by the term "social sin," the distortions of human values woven into the social fabric. See pp. 115–16.
. The phrase comes from Bernard Lonergan, _Insight: A Study of Human Understanding_ (1957; repr., Toronto: University of Toronto Press, 1992).
. Lewis Mumford, _The Story of Utopias_ (New York: Peter Smith, 1941), p. 78.
. Besides his introduction to this edition, see also Clarence H. Miller, _Humanism and Style,_ pp. 71–79.
. Elizabeth McCutcheon, "Denying the Contrary: More's Use of Litotes in the _Utopia,_ " in _Essential Articles for the Study of Thomas More,_ ed. R. S. Sylvester and G. P. Marc'hadour (Hamden, Conn.: Archon Books, 1977), p. 263.
. Stephen Greenblatt, _Renaissance Self-Fashioning: From More to Shakespeare_ (Chicago: University of Chicago Press, 1980), pp. 22–26.
. See Miller's _Humanism and Style,_ pp. 71–79.
. Ibid., 76.
. Ibid., 72.
. William Roper, _Sir Thomas More,_ in _Two Early Tudor Lives,_ ed. Richard S. Sylvester and Davis P. Harding (New Haven: Yale University Press, 1962), p. 198. With regard to More's devotion to the theatrical metaphor, see Greenblatt, _Renaissance Self-Fashioning,_ pp. 26–37. See also Thomas More, _The History of Richard III,_ ed. George Logan (Bloomington: Indiana University Press, 2005), pp. 94–95, 126–28.
. See Dominic Baker-Smith, "' _Civitas philosophica_ ': Ideas and Community in Thomas More," in _A Companion to Thomas More,_ ed. A. D. Cousins and Damian Grace (Madison, N.J.: Fairleigh Dickinson University Press, 2009), pp. 165–77.
. The use of the term "relationism" as a way of understanding the sociology of knowledge emerged in the work of Karl Mannheim; see his _Ideology and Utopia: An Introduction to the Sociology of Knowledge_ (New York: Harcourt, Brace and Company, 1936), pp. 71–78 and passim. Among more recent scholars, the one who has done the most, as far as I know, to develop the implications of the term is Walter J. Ong; see his _Fighting for Life: Contest, Sexuality, and Consciousness_ (1981; repr., Amherst: University of Massachusetts Press, 1989), pp. 29–34; and "Hermeneutic Forever: Voice, Text, Digitization, and the 'I,'" _Oral Tradition_ 10 (1995): 3–26. See also my "Ong, Evolution, and the Method of Dialogue," _Explorations in Media Ecology_ 9:1–4 (2010): 103–18.
. For further considerations, see Charles Trinkaus, "The Question of Truth in Renaissance Rhetoric and Anthropology," in _Scope of Renaissance Humanism,_ pp. 437–449.
. Bernard Lonergan, _A Second Collection,_ ed. William F. J. Ryan, S.J., and Bernard J. Tyrrell, S.J. (Toronto: University of Toronto Press, 1974), pp. 1–9. See also Anthony Grafton, _What Was History? The Art of History in Early Modern Europe_ (Cambridge: Cambridge University Press, 2007).
. For helpful insights into such dialogue, see David M. Bevington, "The Dialogue in 'Utopia': Two Sides to the Question," _Studies in Philology_ 58:3 (July 1961): 496–509.
. Martin Waldseemüller, _Cosmographiae Introductio,_ trans. Joseph Fischer and Franz von Wieser (University Microfilms, 1966), pp. 91–97.
. Ibid., p. 98.
. Roper, _Sir Thomas More,_ p. 198. See also Ackroyd, _Life of Thomas More,_ pp. 96–101.
. _Rule of St. Benedict,_ ch. 33.
. _Collected Works of Erasmus: Adages Ii1 to Iv100,_ trans. Margaret Mann Phillips (Toronto: University of Toronto Press, 1982), p. 29.
. Kathy Eden, _Friends Hold All Things in Common: Tradition, Intellectual Property, and the "Adages" of Erasmus_ (New Haven: Yale University Press, 2001).
. All references to Aquinas are to _Summa Theologica,_ 3 vols., trans. Fathers of the English Dominican Province (New York: Benziger Brothers, 1947).
. Paul Ricoeur, _Lectures on Ideology and Utopia,_ ed. George H. Taylor (New York: Columbia University Press, 1988), p. 310.
. Ibid., p. 311.
. Ibid., p. 312.
. See Walter J. Ong, S.J., _Interfaces of the Word: Studies in the Evolution of Consciousness and Culture_ (Ithaca: Cornell University Press, 1977), pp. 166–81; and Halasz, _Marketplace of Print._
. Hanan Yoran, _Between Utopia and Dystopia: Erasmus, Thomas More, and the Humanist Republic of Letters_ (New York: Rowman and Littlefield, 2010), p. 63.
. Ibid., pp. 177–86.
. See Walter J. Ong, _Ramus, Method, and the Decay of Dialogue: From the Art of Discourse to the Art of Reason_ (1958; repr., Chicago: University of Chicago Press, 2004), pp. 307–18.
. For a fine introduction to these and related themes in the work of Jacques Derrida, see his "Structure, Sign, and Play in the Discourse of the Human Sciences," in _Writing and Difference,_ trans. Alan Bass (Chicago: University of Chicago Press, 1978), pp. 278–93.
. Jacques Derrida, _Negotiations: Interventions and Interviews, 1971–2001,_ trans. Elizabeth Rottenberg (Stanford: Stanford University Press, 2002), pp. 13–14.
. I would like to thank Amanda Joyce, Mike Smolinsky, and Mary Szybist for their helpful comments, as well as my students Charlotte Markle and Justine Minette for their hours of helpful conversation about More's _Utopia._ The errors and folly are mine.
## SUGGESTIONS FOR FURTHER READING
UPDATED BY JERRY HARP
### BIOGRAPHIES
Ackroyd, Peter. _The Life of Thomas More_ (New York: Doubleday, 1998). The most reliable full biography.
Chambers, R. W. _Thomas More_ (London: Jonathan Cape, 1935; repr., Ann Arbor: University of Michigan Press, 1958). A classic biography, well informed and beautifully written, but it neglects the polemical works.
Guy, John. _Thomas More_ (New York: Oxford University Press, 2000). Emphasizes especially the political struggles. Guy acknowledges that some issues that have exercised More biographers cannot be resolved based on the available evidence; thus, as he points out, each reader tends to make More over into his or her desired image.
Marius, Richard. _Thomas More: A Biography_ (Cambridge: Harvard University Press, 1984). The biography that makes the most extensive use of the full range of More's writings. Marius performed a helpful service by writing against the grain of the hagiographic tendencies of past More biographies, though his conclusions occasionally overreach the evidence; he rides his hobbyhorse a bit vigorously at times.
Roper, William. _Sir Thomas More,_ in _Two Early Tudor Lives,_ ed. Richard S. Sylvester and Davis P. Harding (New Haven: Yale University Press, 1962). A handy edition of the first biography of More, written by his son-in-law—personal and poignant.
### BIBLIOGRAPHIES
Geritz, Albert. _Thomas More: An Annotated Bibliography of Criticism, 1935–1997_ (Westport, Conn.: Greenwood Press, 1998). Excellent, very full. Pages 215–309 are devoted to _Utopia_ alone.
Gibson, R. W., and J. Max Patrick. _Thomas More: A Preliminary Bibliography of His Works and of Moreana to the Year 1750 with a Bibliography of Utopiana_ (New Haven: Yale University Press, 1961). Early editions and translations of _Utopia_ up to 1750.
Lakowski, Romauld Ian. "A Bibliography of Thomas More's _Utopia,_ " <http://extra.shu.ac.uk/emls/01-2/lakoutop.html>. This excellent Internet bibliography, devoted exclusively to _Utopia,_ also exists in a printed form: _Early Modern Literary Studies_ 1.2 (August 1995).
Wentworth, Michael D. _The Essential Sir Thomas More: An Annotated Bibliography of Major Modern Studies_ (New York: G. K. Hall, 1995). Entries 380–640 are devoted to _Utopia_ alone.
The indices of the journal _Moreana_ will provide a plethora of articles on _Utopia._
### EDITIONS
_Utopia: Latin Text and English Translation,_ ed. George M. Logan, Robert M. Adams, and Clarence H. Miller (Cambridge: Cambridge University Press, 1995). A reliable and usable Latin text, with a compact introduction and notes.
_Utopia,_ ed. Edward Surtz, S.J., and J. H. Hexter, vol. 4 of The Complete Works of St. Thomas More (New Haven: Yale University Press, 1965). Full-fledged edition with elaborate introduction, textual variants, and a very full commentary.
_L'Utopie de Thomas More,_ ed. André Prévost (Paris: Mame, 1978). The French equivalent of Surtz's edition, with a facsimile of the edition of November 1518, an elaborate introduction, a French translation, and a very full commentary.
_Utopia,_ trans. Ralph Robinson, Everyman's Library (New York: Alfred A. Knopf, 1992). The first English translation. This edition includes a helpful introduction by Jenny Mezciems.
### STUDIES OF _UTOPIA_
Baker-Smith, Dominic. _More's Utopia._ (Toronto: University of Toronto Press, 2000). A multidimensional introduction. Baker-Smith discusses the historical background and literary precedents in detail. He shows the continuing relevance of the work, especially with regard to what he terms social sin, the injustices woven into the social fabric.
Bevington, David M. "The Dialogue in 'Utopia': Two Sides to the Question," _Studies in Philology_ 58:3 (July 1961): 496–509. Emphasizes the open-endedness of the dialogue that _Utopia_ invites the reader to join.
Cave, Terence (Ed.). _Thomas More's "Utopia" in Early Modern Europe: Paratexts and Contexts_ (Manchester: Manchester University Press, 2008). Provides studies of the material—e.g., prefaces, poems, essays, letters—that accompanied early editions of _Utopia_ in a variety of languages.
Cousins, A. D., and Damian Grace (Eds.). _A Companion to Thomas More_ (Madison, N.J.: Fairleigh Dickinson University Press, 2009). Brings together studies ranging from the early epigrams to the Tower works.
Greenblatt, Stephen. _Renaissance Self-Fashioning: From More to Shakespeare_ (Chicago: University of Chicago Press, 1980). Explores the unstable and distorted relationship between the two books of _Utopia,_ which Greenblatt also connects to More's role-playing.
Hexter, J. H. _More's "Utopia": The Biography of an Idea_ (New York: Harper and Row, 1952). A detailed study of the evolution of _Utopia._
Logan, George. _The Meaning of More's "Utopia"_ (Princeton: Princeton University Press, 1983). Provides a close reading of _Utopia_ as a product of Northern humanism, with constant reference to the classical and medieval antecedents, as well as contemporary influences.
Logan, George M. (Ed.). _The Cambridge Companion to Thomas More_ (Cambridge: Cambridge University Press, 2011). A collection of essays on More's life and writings, by some leading More scholars.
McCutcheon, Elizabeth. "Denying the Contrary: More's Use of Litotes in the _Utopia,_ " in _Essential Articles for the Study of Thomas More,_ ed. R. S. Sylvester and G. P. Marc'hadour (Hamden, Conn.: Archon Books, 1977), 263–74. Demonstrates the significance of More's frequent use of litotes in _Utopia._ As a figure of discourse, litotes allows a more complex understanding than simple affirmation or negation can accomplish.
McCutcheon, Elizabeth. _My Dear Peter: The "Ars Poetica" and Hermeneutics for More's "Utopia"_ (Angers: Moreanum, 1983). An excellent study of litotes and the oblique style of _Utopia._
Miller, Clarence H. "Style and Meaning in More's _Utopia:_ Hythloday's Sentences and Diction," in _Humanism and Style: Essays on Erasmus and More_ (Bethlehem, Penn.: Lehigh University Press, 2011). Demonstrates the significance of the shifts of style in Hythloday's Latin.
Nagel, Alan F. "Lies and the Limitable Inane: Contradiction in More's _Utopia,_ " _Renaissance Quarterly_ 26:2 (Summer 1973), pp. 173–180. A consideration of the contradictions, tensions, and impossibilities woven into More's text.
Nelson, William (Ed.). _Twentieth Century Interpretations of Utopia: A Collection of Critical Essays_ (Englewood Cliffs, N.J.: Prentice-Hall, 1968). Provides a broad range of commentaries and perspectives.
Perlette, John M. "Irresolution as Solution: Rhetoric and the Unresolved Debate in Book 1 of More's _Utopia,_ " _Texas Studies in Literature and Language_ 29:1 (Spring 1987): 28–53. A reading of the Dialogue of Counsel as an extension of the ancient debate between philosophy and sophistry.
Sylvester, Richard S. "'Si Hythlodaeo Credimus': Vision and Revision in Thomas More's _Utopia,_ " _Soundings_ 51 (1968): 271–89. Reprinted in _Essential Articles for the Study of Thomas More,_ ed. R. S. Sylvester and G. P. Marc'hadour (Hamden, Conn.: Archon Books, 1977), pp. 290–301. The best brief introduction to the ironies of _Utopia._
Sylvester, R. S., and G. P. Marc'hadour (Ed.). _Essential Articles for the Study of Thomas More._ (Hamden, Conn: Archon Books, 1977). A collection that lives up to its title.
Yoran, Hanan. _Between Utopia and Dystopia: Erasmus, Thomas More, and the Humanist Republic of Letters_ (New York: Rowman and Littlefield, 2010). A study of the early humanists' desire to create a virtual community transcending national boundaries and institutional affiliations, and the impossibility of doing so. Yoran provides a reading of _Utopia_ in light of this impossibility.
## INDEX
Abraxa. _See_ Utopia
Achoria (ideal kingdom), xii, xv,
Adams, Robert M., xxii
Ademus (ruler of Utopia), , , n169
adultery, , , n227
age, , ,
agriculture, –24, , –55, ,
Alaopolitans (enemies of Utopia),
Amaurot (Utopian city), , , , n12; description of, –57; resemblance to London, nn138–40
ambassadors, –78
Anemolians, –78
Anemolius,
animals, , , ,
Antwerp, xxv,
Anyder (Utopian river), , , , n12
Apinatus, Tricius,
Aristophanes,
Aristotle, , n110, n157, n267
artisans, ,
astronomy,
Barzanes. _See_ Ademus
beggary, , , –33, ,
bribes, ,
brothels, ,
Bruges, xxvi,
Brussels,
Burnet, Gilbert, xxi, xxii
Buthrescae, , –81n249
capital punishment, viii, , –32, , ; for adultery, x, ; for foreign wrongdoers, ; religion and, , n68; for theft,
Carthage,
Celaeno (monster), , n42
celibacy,
Charles V (Holy Roman Emperor), xxvi, , n23, n96
children, viii–ix, –61, , ; and books, ; and gemstones, , –78; obligation to reproduce, ; and religion, –24, , n250; of slaves, ; warfare and,
Christ, , ,
Christianity, , . _See also_ religion
churches, , –28
Cicero, vii, , n65, n109
cities, , , ; city-states, xvi, xvii; description of, , –58; enemy, –15; social relations in, –67
civilization,
Clement, John, , n11
clothing, , , , –85
colonialism, ix, xvi, , n171
Colt, Joan (More's first wife), xxv, xxvi
communism, viii, ix, x,
conformity, xviii
contemplation, vii
craftsmen,
crime: bribes and, ; causes of, –20, ; money and, ; priests and, ; punishment of, xvi, –34, , –101
death, –21
death penalty. _See_ capital punishment
Delcourt, Marie, xxi
destitution. _See_ poverty
_Dialogue of Comfort Against Tribulation, A_ (More), xxviii
Dionysius, , n87
Dioscorides,
disabled persons,
disease, , , , ; old age as, ; pain and, ; rarity of, ; treatment of the sick, ; warding off,
disputes, settlement of,
divorce, viii, xvi, ,
Donnelly, John P., xxii
education, , –80, , –24
Elisha (biblical), , n83
enclosures, , n59, n61
England, , , n131; crime and punishment in, –34; Hythloday in, –19; peace treaties and, ; relations with France, ; trade with Flanders, n22
Erasmus, Desiderius, xix, xx, n13, n93, n218
ethics, –83
ethnic cleansing, x
Euripides,
Europe, xi, xv, xxii
euthanasia, viii, xvi, –97, n224
excommunication,
exile, as punishment,
exports, ,
falconry,
families, xiv, –68
farming, xiii, xiv, xvii, ; enclosures and, ; Plato on, n157; tenant-farmers, ; war veterans and, ,
feastdays, , , n255
food, –72, ; health and, , , ; money and, –33
fools, , n229
France, , , , n49,
Francis I (king of France), xxvi, n89, n91
freedom,
Froben, Johann, xx
Galen, , n221
gambling, , , n61
games, , ,
gardens, –58,
gemstones, , –78,
Giles, Peter, , , nn29–30; on government, ; meeting with Hythloday, –12; More's letters to, xx, xxii, –7, –39; and printing of _Utopia,_ xix; qualities of,
God, , , . _See also_ religion
gold, , , ; as bribe, ; Utopian use of, –76, –79
goods, equality of, xvi, ,
government, xvi, xx; in ancient Rome, ; antiquity of, ; wise men and,
Greek (ancient), , , , –95
happiness, –82, , ,
health (physical), –89,
Henry VIII (king of England): Charles V and, ; French wars of, n21, n49; laws under, n61; More in service of, vii, ix, xxvi, xxvii
Herodian, , n56, n219
Herodotus, , n219
Hesychius,
Hippocrates, n221
_History of Richard III, The_ (More), xxvi
Holy Roman Empire, xxvii
Homer,
honors, bestowing of, xiv
hospitals, –69
households, , –68
humanists, xix, n117
hunting, –87,
Hythloday, Raphael (advocate of Utopia), , , ; Anemolius and, ; bipolarity of, x; characteristics of, x–xi; diction of, xvi–xviii; language of, xxii–xxiii; learning of, , ; qualities of, –12; significance of name, viii, x, ; syntax of, xi–xii, xv; travels of, –14
idleness, , , ; injustice and, –31; travel and,
injustice, viii, , ; class differences and, –32; Hythloday's stand against, x, xix
institutions, xvii, , , ; absurdity of, ; paradox of, xvi, xviii
Italy, , , n21, n57
justice, , , –31
kings, , , , n111; philosopher-kings, ; power of, –41; warfare and, –40. _See also_ princes
labor, , –66; common good and, ; dispensation from, ; injustice and, –32; as punishment, , –30, ; religion and, –22; transportation and, ; travel and, –73; workday in Utopia, ,
Laestrigonians (monsters), , n42
language, xvii
Lascaris,
Latin, xi–xii, xviii, , , ; early editions of _Utopia_ in, xx–xxi; philosophy and, –12; translation of, xxi–xxiii; works by More in, xxvi
laws, xvii, , , –2; antiquated (disused), , ; injustice and, 131, ; private property and, , ; punishment of thieves, –32; religion and, ; scorn for,
lawyers, xiii, , , –2
livestock, , –24, ,
Logan, George, xxi
Lucian, xx, , n217
Lupset, Thomas, xx
Lupton, J. H., xi, xxi
Luther, Martin, xxvii
Lutheranism, xxvii
luxury, ,
Macaria (ideal kingdom), xii, xv,
magistrates, –60, , , ; dining arrangements and, –71; election of, ; and euthanasia, ; honor of, ; Utopians as magistrates for other peoples, –3
marriage, –99,
medicine, ,
men, , , ; in church, ; husbands, , , ; idleness and, ; military training and,
mercenaries: employed by Utopia, , –11, ; in European wars, , , n93. _See also_ soldiers
Michels, V., xxi
_Microtechne_ (Galen),
Middleton, Alice (More's second wife), xxvi
money, , , ; abolition of, , ; buying off enemies with, , ; given to prisoners, ; precious metals and, , ; private property and, ; royal treasury and, –43; theft of, –27; value of currency, ; war reparations,
monks, –34
monsters,
morals/morality, , ,
More, Thomas, vii, xviii, , ; business activities of, ; chronology of life, xxv–xxviii; death of, xxviii; as lawyer, vii; letters to Peter Giles, –7, –39; marathon sentences of, xi–xii; on religious heresy, n245
_Moriae Encomium_ (Erasmus), xix
Morton, Cardinal John: on crime and punishment, viii, xii, –34, , –32; More and, xxv, , n47
Moses (biblical),
murder, xvi, –28
music, , , n261
Mythras (god), ,
nature, –83, , ; gifts of, ; religion and, ; science and,
Nephelogetes (allies of Utopia),
noblemen, –22; injustice and, , ; as landlords, ; peace treaties and, ; retainers of,
nuns,
office seeking, xiii, xiv, xvii,
Ogden, H. V. S., xxii
_On Plants_ (Theophrastus),
pain, ,
papacy, xvii, , , n231
_Parva logicalia,_
passports,
Persia,
Persian language,
philosophers/philosophy, , –80, n117, n192; ethics, –83; philosopher-kings, ; on pleasure, n201
phylarchs, , ,
Plato, , , –36, , n120; dialogues of, vii; Dionysius and, , n87; on farming, n157; on government, ; on laws, ; on oligarchy, n50; _Republic,_ 44, n126
Plautus,
pleasure, , –84, n201; false, –87; true, –91
Plutarch, , n217
police, absence of, xvi
Polylerites, –31, n68
Portugal, ,
poverty, , , –33, , ; power of kings and, ; pride and,
prayers, , , –29
Prévost, André, xxi
prices: abundance of goods and, ; of food, , ; of livestock, –24
pride, xv, , ,
priests, xv, , , , ; dining arrangements and, ; election of, ; and euthanasia, , ; and idleness, ; insignia of, ; religious education and, –24, n250; reputation of, –25; vestments of, –28
princes, –17, ; dispensations and, ; enemy, , ; and philosophy, ; and stolen goods, ; treaties of, ; and warfare, , . _See also_ kings
printing, –95
private property, xv, ; abolition of, ; absence of, ; equality of goods and, –48; money and,
public service, vii
punishment, of crime, –34, , –101,
reason, , , , ,
religion, x, , , –84, ; capital punishment and, , n68; children and, –24, , n250; labor and, –22; laws and, ; nature and, ; of Utopia, –29. _See also_ Christianity
_Republic_ (Plato), , n126
_Responsio ad Lutherum_ (More), xxvii
Restoration, xxii
retainers, idle, , –22
Rich, Richard, xxviii
Richards, G. C., xxii
Robinson, Ralph, xi, xii, xxi
Rome, ancient, , ,
sacrifices,
Sallust,
satire,
science,
Scylla (monster), , n42
seamanship, –14,
seduction,
Seneca, , , n118, n207
sexual activity, ,
Sheehan, John, xxii
sheep, , –24, , n59
shortages,
silver, , –76,
sin, ,
slaves, , –31, –70, –96; character of Utopian slavery, n223; in country households, ; execution of, ; golden chains of, , –78; as hunters and butchers, ; prisoners of war as, ; slavery as punishment, , –96,
social classes, xix, –34
social relations, –72
soldiers, xiii, xiv, –21; priests and, ; as thieves, –21, n51; Utopian military practices, , –15. _See also_ mercenaries
Solomon (biblical),
Sophocles,
Spain, king of,
Stoics, , n65, n207
suicide, –97
superstition,
surpluses,
Surtz, Edward, x, xii, xxi, xxii,
Sylvester, Richard, xii
syphogrants. _See_ magistrates
Syria,
taverns,
taxes, , n106
Terence, ,
Themsecke, George de,
Theophrastus,
thieves: causes of theft, , , –25; punishment of, xii, , , –32, n48; soldiers as, –21, n51
Thucydides, , n219
tradition (custom),
tranibors (chief magistrates), , , , n146
translations, xi, xxi–xxiii, n88
travel, –95
treaties, xv, –37, –5, n231
Tunstall, Bishop Cuthbert, xxvii, , n24
Turner, Paul, xxii
universalism, xvii, xviii
Utopia, , ; absence of central authority in, xvi, n153; agriculture in, –55; allies and friends of, , –6; alphabet of, ; cities of, –54, –58; comparison with Europe, xv; Hythloday in, ; laws in, ; meaning of name, ix, , ; military practices in, –15; occupations in, –66; physical description of, , , –54, –58; religions of, –29; slavery in, –96, n223; social relations in, –72; travel in, –95; visitors to,
_Utopia_ (More): criticism of, ; early editions of, xix–xxi; Latin text of, xx–xxi; publication of, xxvi; translation of, xxi–xxiii
Utopus (conqueror-founder of Utopia), viii, , , ,
vagabonds, , , n48
Vespucci, Amerigo, , n36, n260
vices, , , ; eradication of, ; punishments for,
virtues, , , , , ; rewards for, ; Stoics on, n65; treaties and,
warfare, ix, x, xvi, , ; colonialism and, ; corruption of morals and, ; crippled veterans of, –20; priests and, , ; princes and, ; slavery and, ; thieves as soldiers, –21; Utopian military practices, –15; women and, –12
wealth, , –86,
women, x, , , ; age at marriage, , n225; in church, ; idleness and, ; and meal preparation, ; and military training, ; occupations of, ; as priests, , n251; warfare and, –12; wives, , , ,
Zapoletes (Utopian mercenaries), x, xv, –11
Ziegler, T., xxi
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{"url":"http:\/\/www.talkstats.com\/showthread.php\/13893-Basic-Probability-Question-Help?p=38359&mode=threaded","text":"## Basic Probability Question Help\n\nI need help with 2 questions:\n\n1) In a shipment of 18 trucks to a local truck dealer, there are 4 trucks that don't have air conditioning. Assume you select 4 trucks at random.\n\nWhat is the probability that two of the four don't have air conditioning?\nWhat is the probability that two or fewer of the four don't have air conditioning?\n\nI tried to use permutation for this problem, but I'm not sure if that is right or which numbers to use: n!\/ (n-x)! or maybe n! (n-r)r!\n\n2) The volume of a one-pound bag of coffee is normally distributed. Suppose you take a random sample of 15 one-pound bags of coffee. Find two values K sub L and K sub U such that the probability is 95% that the ratio of the sample standard deviation divided by the population standard deviation is between K sub L and K sub U.\n\nI don't understand what the question is asking or what K sub L and U are to be? Can both these questions be done on excel? If so, what formula?\n\nThanks","date":"2013-05-25 22:07:51","metadata":"{\"extraction_info\": {\"found_math\": false, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 0, \"math_alttext\": 0, \"mathml\": 0, \"mathjax_tag\": 0, \"mathjax_inline_tex\": 0, \"mathjax_display_tex\": 0, \"mathjax_asciimath\": 0, \"img_math\": 0, \"codecogs_latex\": 0, \"wp_latex\": 0, \"mimetex.cgi\": 0, \"\/images\/math\/codecogs\": 0, \"mathtex.cgi\": 0, \"katex\": 0, \"math-container\": 0, \"wp-katex-eq\": 0, \"align\": 0, \"equation\": 0, \"x-ck12\": 0, \"texerror\": 0, \"math_score\": 0.827623724937439, \"perplexity\": 501.82547834863783}, \"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-2013-20\/segments\/1368706469149\/warc\/CC-MAIN-20130516121429-00054-ip-10-60-113-184.ec2.internal.warc.gz\"}"} | null | null |
Future Flyer, a 6-8 point guard, makes other players better
Dave Briski, of International Sports Academy, praises job Dayton coaches did in recruiting Sharavjamts
Sharavjamts Tserenjankhar provided an important correction to some stories written about his son, Enkhiin-Od Michael Sharavjamts, after his commitment to the Dayton Flyers in December. He's not a 6-foot-8 forward. He's a 6-8 point guard.
Dave Briski, the coach at International Sports Academy, in Willoughby, Ohio, where Sharavjamts is playing this season, confirmed that.
"We're off to a good start," Briski said on Dec. 16. "We're 6-1 overall, and Mike's obviously been a huge reason why. We're playing him at the point. He really runs the show for us and is kind of the head of the snake. He's been terrific, shooting well over 40 percent from 3 and something like 60 percent from the field. He's probably averaging around 7 or 8 assists per game, so he's really doing it all."
Dayton fans will have a chance to see Sharavjamts in person at the Flyin' to the Hoop event at Trent Arena in Kettering. ISA plays Link Academy (Mo.) at 6:30 p.m. Jan. 14, in the first of 19 games in a four-day stretch.
ISA is located in Willoughby, Ohio, northeast of Cleveland. Its players attend Andrews Osborne Academy. Briski started the program in 2018. Charles Bediako, who's now a freshman at Alabama, played there the first two seasons, as did Keon Ambrose-Hylton, who's a sophomore at Alabama.
Sharavjamts, the first player from the 2022 recruiting class to commit to Dayton, ended up with ISA after Briski saw him play last summer with Midwest Basketball Club.
"We didn't really know a lot about him," Briski said, "but when we went through the process with him, we told him, 'We get that you're a point guard, and we're going to play you that way. We play a national schedule and feel like we can really continue to help you develop your jump shot,' which is clearly paying off. We felt like he would be able to play against competition that's going to prepare him for what he's going to face when he goes to Dayton next year. That was our pitch and he believed in what we were doing and our development plan for him, and it seems to be working out for everybody."
Credit: David Jablonski
Centerville High School coach Brook Cupps, who coached Sharavjamts last summer with Midwest Basketball Club, said Sharavjamts makes the game easier for his teammates. Briski agrees.
"The kid's unbelievable with the ball in his hands," Briski said. "Primary actions, secondary actions, he's going to give you an advantage, and he's going to find players. He'll tell you himself he'd rather assist on a bucket than score himself. That can be a problem sometimes, but he really makes everybody around him a better player."
Briski said Sharavjamts' mom and older brother moved to Cleveland to support him and will move back to Mongolia when he starts college.
Explore» MORE COVERAGE: Recruit's connection to Ohio goes back to 2018
His dad, Tserenjankhar, was playing for the Harlem Globetrotters when Sharavjamts was born in Phoenix, so he has American citizenship.
Sharavjamts received a scholarship offer while on a visit to Dayton in September and sat behind the bench with his parents for an exhibition game in November. He picked the Flyers over Rutgers, Providence and Eastern Washington.
"I think coach Grant and coach (Ricardo) Greer quite frankly just did a hell of a job," Briski said. "We've had a lot of guys and a lot of guys have had some really good recruitments and we've worked with Dayton before. They've recruited a few of our guys over the years. Coach Grant, when he walks in the room, you feel his presence, and the way he plans on using Mike was the real separator. Mike had some great schools on his list, and quite frankly, he would have had some even bigger names on that list had he waited, but he wanted Dayton and Dayton wanted him. You have a coach who has coached in the NBA have a coach who's beaten Duke in the NCAA tournament. There's not a lot else you could ask for, and then you're going to a place that sells out every game. Everything kind of just came together. Coach Grant and coach Greer built a relationship and built a lot of trust and took him through everything that he was curious about, answered all his questions, and he just felt like Dayton was home." | {
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{"url":"https:\/\/arcsecond.wordpress.com\/tag\/science\/","text":"## Posts Tagged \u2018science\u2019\n\n### On the Height of a\u00a0Field\n\nJanuary 1, 2013\n\nThis is a short story about belief and evidence, and it starts with the\u00a0GPS watch I use when I go for a run. Here\u2019s the plot of my elevation today:\n\nIt looks a little odd until I show you this map of the run:\n\nEach bump on the elevation plot is one lap of the field. In the middle, I changed directions, giving the elevation chart an approximate mirror-image symmetry. (I don\u2019t know what causes the aberrant spikes, but my friend reports seeing the same thing on his watch.)\n\nAccording to the GPS data, the field is sloped, with a max height of 260 feet near the center field wall and 245 feet near home plate. It\u2019s insistent on this point, reiterating these numbers each time I do the run (except once when the tracking data was clearly off, showing me running across parking lots and through nearby buildings.) I disagreed, though. The field looked flat, not sloped at 3 degrees. I was disappointed to have found a systematic bias in the GPS data.\n\nBut I\u00a0occasionally\u00a0thought of some minor consideration that impacted my belief. I remembered that when I went biking, I often found that roads that look flat are actually uphill, as can be verified by changing directions and feeling how much easier it becomes to go a given pace. I Googled for the accuracy of GPS elevation data, and found that it\u2019s only good to about 10 meters. But I didn\u2019t care about absolute elevation, only change across the field, and I couldn\u2019t find any answers on the accuracy of that. (Quora failed me.) I checked Google Earth, and it corroborated the GPS, saying the ground was 241 ft behind home plate and 259 in deep center field. But then I read that the GPS calibrated its elevation reading by comparing latitude\/longitude coordinates with a database, and so may have been drawing from the same source as Google Earth.\n\nPeople wouldn\u2019t make a sloped baseball field, would they? That would dramatically change the way it plays, since with a 15-foot gain, what was once a solid home run becomes a catch on the warning track. Googling some more, I found that baseball fields can be pretty sloped; the requirements are fairly lax, and in fact they are typically sloped to allow drainage.\n\nI was starting to doubt my initial judgment, and with this in mind, when I looked at the field, it made more and more sense that it\u2019s sloped. Along the right field fence, there\u2019s a short, steep hill leading up to the street. It\u2019s about five feet high and at least a 30-degree slope. It\u2019s completely unnatural, as if it exists because the field as a whole used to be considerably more sloped, but was dug out and flattened. The high edge of the field was then below street level, so there\u2019s that short, steep hill leading up. And if the field was dug out and flattened, maybe they didn\u2019t flatten it all the way. The entire campus is certainly sloped the same general direction as the GPS claimed for the field. It drops about 70 feet from north to south, and it\u2019s frequently noticeable as you walk or bike around. There\u2019s another field I run on with essentially the same deal, and I found that when I knew what to look for, I could indeed see the slope there.\n\nEventually, the speculation built up enough to warrant a little effort to make a measurement. I asked a wise man what to do, and he suggested I find a protractor, hang a string down to detect gravity, and site from one side of the field to the other. I did so, expecting to feel the boldness of an impartial, truth-seeking scientific investigator as I strode across the grass. That wasn\u2019t what I got at all.\n\nFirst, I felt continuous fluctuations in my confidence. \u201cI\u2019m 60% confident I\u2019ll find the field is sloped,\u201d I told myself, then immediately changed it to 75, not wanting to be timid, then felt afraid of being wrong, and went back to 50. I\u2019ve played The Calibration Game and learned what beliefs mean, and mostly what it\u2019s done is give me the ability to not only be uncertain about things, but to be meta-uncertain as well \u2013 not sure just how uncertain I am, since I don\u2019t want to be wrong about that!\n\nSecond, I felt conflicting desires. I couldn\u2019t decide what I wanted the result to be. I wanted the field to be flat to validate my initial intuition, not the stupid GPS, but I also wanted the field to be sloped so I could prove to myself my ability to change my beliefs when the evidence comes in, even if it goes against my ego. (A strange side-effect of wanting to believe true things is that you find yourself wanting to do things not because they help you believe the truth, but because you perceive them to be the sort of things that truth-seekers would do.) I recalled a video I had seen years ago about Gravity Probe B, and the main thing I remembered from it was a scientist with long, gray hair and huge unblinking eyeballs explaining in perfect monotone that he didn\u2019t have a desire for the experiment to confirm or refute general relativity; he only wanted it to show what reality was like.\n\nOn top of all this, there was the sense of irony at so much mental gymnastics over a triviality like the slope of a baseball field, and the self-consciousness at the absurdity of standing around in the cold pointing jerry-rigged protractors at things. So at last I crossed the field and lined up my protractor for the moment of truth\n\nIt didn\u2019t work. I had placed my shoes down on the grass as a target to site, but from center field they were hidden behind the pitcher\u2019s mound. I recrossed the field and adjusted them, and went back. I still couldn\u2019t see the shoes; they were too small and hidden in the grass. I could see my backpack, though, so I sited off that. But it still didn\u2019t really work. I didn\u2019t have a protractor on hand, so I had printed out the image of one from Wikipedia and stapled it to a piece of cardboard, but the cardboard wasn\u2019t very flat, making siting along it to good accuracy essentially impossible.\n\nI scrapped that, and after a few days went to Walgreens and found a cheap plastic protractor and some twine that I used to tie in my water bottle as a plumb bob. Returning to the field, I finally found the device to be, well, marginal. Holding it up to my eye, it was impossible to focus along the entire top of the protractor at once, and difficult to establish unambiguous criteria for when the protractor was accurately aimed. I was also holding the entire thing up with my hands, and trying to keep the string in place between siting along the protractor and moving my head around to get the reading.\n\nNonetheless, my reading came to 87 degrees from center field to home plate and 90 degrees from home plate back to center field. This three-degree difference seemed pretty good confirmation of the GPS data. In a final attempt to confirm my readings, I repeated the experiment in a hallway outside my office, which I hope is essentially flat. It\u2019s 90 strides long, (and I\u2019m about two strides tall) and I found 88 degrees from each side, roughly confirming that the protractor readings matched my expectations. (I\u2019d have used the swimming pool, which I know is flat, but it\u2019s closed at the moment.)\n\nI\u2019m now strongly confident that the baseball field is sloped \u2013 something around 95% after considering all the points in this post. That\u2019s enough that I don\u2019t care to keep investigating further with better devices, unless maybe someone I know turns out to have one sitting around.\n\nStill, there is some doubt. Couldn\u2019t I have subconsciously adjusted my protractor to find what I expected? There were plenty of ways to mess it up. What if I had found no slope with the protractor? Would I have accepted it as settling the issue, or would I have been more likely to doubt my readings?\u00a0It\u2019s perfectly rational to doubt an instrument more when it gives results you don\u2019t expect \u2013 you certainly shouldn\u2019t trust a thermometer that says your temperature is 130 degrees \u2013 but it still feels intuitively a bit wrong to say the protractor is more likely to be a good tool when it confirms what I already suspected.\n\nThe story of how belief is supposed to work is that for each bit of evidence, you consider its likelihood under all the various hypotheses, then multiplying these likelihoods, you find your final result, and it tells you exactly how confident you should be. If I can estimate how likely it is for Google Maps and my GPS to corroborate each other given that they are wrong, and how likely it is given that they are right, and then answer the same question for every other bit of evidence available to me, I don\u2019t need to estimate my final beliefs \u2013 I calculate them. But even in this simple testbed of the matter of a sloped baseball field, I could feel my biases coming to bear on what evidence I considered, and how strong and relevant that evidence seemed to me. \u00a0The more I believed the baseball field was sloped, the more relevant (higher likelihood ratio) it seemed that there was that short steep hill on the side, and the less relevant that my intuition claimed the field was flat. The field even began looking more sloped to me as time went on, and I sometimes thought I could feel the slope as I ran, even though I never had before.\n\nThat\u2019s what I was interested in here. I wanted to know more about the way my feelings and beliefs interacted with the evidence and with my methods of collecting it. It is common knowledge that people are likely to find what they\u2019re looking for whatever the facts, but what does it feel like when you\u2019re in the middle of doing this, and can recognizing that feeling lead you to stop?\n\n### Mike Brown, Planet Killer: \u201cMercury is Pissing Me\u00a0Off\u201d\n\nDecember 19, 2010\n\nMike Brown is famous for discovering Eris, a dwarf planet larger than Pluto orbiting out on the far edge of the solar system. Ultimately, Eris\u2019 discovery led to the redefinition of the word \u201cplanet\u201d and the eradication of Pluto from children\u2019s lunchboxes.\n\nBrown\u2019s new book, How I Killed Pluto and Why It Had It Coming tells the story of his team\u2019s discovery of a complete menagerie out past Neptune \u2013 a place most astronomers thought held little but hydrogen, comets, and a few bits of rock that occasionally get flung out there by gas giants.\n\nIn an interview from last Wednesday, December 15, Brown told me that his most scientifically-important discovery was not Eris, but Sedna, a large object lying so far away from the gravitational perturbations of Jupiter and friends that its orbit can be traced back to the beginning of the solar system, and whose existence has challenged astronomers\u2019 conception of how the planets formed.\n\nBrown also showed me the sonograms of his embryonic daughter (now 5 years old) to compare side-by-side with photographs of Venus taken by the Venera Lander, and commented on the gravitational influence of my mother.\n\nPart 1 (17 minutes: Hate mail, the process of writing, science of the early solar system)\n\nPart 2 (31 minutes: More science, more writing, international intrigue, Pluto\u2019s appeal and wimpiness)\n\n### The Crank Continuum\n\nJune 11, 2010\n\nI\u2019ve had one true crank on this blog. He jumped into the comments on this post with mathematical gibberish he claimed disproved relativity. Another time I saw a crank letter written to a researcher at JPL who worked on dark matter. This crank even provided a little mechanical apparatus intended to demonstrate the existence of dark matter. It consisted of a rubber or nylon sheet that was stretched over a wire frame, and then you were supposed to roll a marble around on it.\n\nIt\u2019s kind of surprising that these cranks fit so well with the descriptions of many others in Martin Gardner\u2019s Fads and Fallacies in the Name of Science. Half a century after Gardner wrote his books, cranks, and belief in what they have to say, hasn\u2019t changed much.\n\nI picked up this book after Douglas Hofstadter mentioned it in an article reprinted in Scientific American after Gardner\u2019s recent death. It\u2019s essentially descriptive, spending surprisingly (and refreshingly) little time refuting crank theories of physics and medicine, and instead mostly detailing them. Gardner does, of course, refute each crank theory, but his most important contribution is to collect enough of them that cranks begin to look similar. (You can read Gardner\u2019s generalizations about cranks in the Hofstadter article, or in chapter 1 of the book.)\n\nAnother surprising fact was that cranks are not just weirdos shouting loudly on obscure corners of the internet (ahem). Many cranks were fairly normal, and even learned and respected people outside of their crankery. A surprising array of famous, respected people bought into and campaigned for crank theories. Upton Sinclair recurs throughout the book, advocating a number of useless medical and dietary systems. Some other delusional supporters or even creators of crank ideas include Aldous Huxley, Clifton Fadiman, Oliver Heaviside, Walt Whitman, Arthur Conan Doyle, William James, H. G. Wells, and Jesus (last one added by me; the others are from Gardner. However, many of Gardner\u2019s cranks theories are motivated by proving or justifying religious claims).\n\nIt seems that as you cross over into the realm of crankery, you begin to believe your discovery has more and more power and wider and wider applicability. Medical cranks, for example, rarely believe they have a cure for cervical cancer. They think they have a cure for everything. Sometimes they even branch out and extend their theory of physiology to explain physics.\n\nCrankery is dangerous, because in some ways it\u2019s difficult for a layman to see the difference between crank science and real science. In crank science, the observations frequently go against the crank\u2019s theory. The crank then comes up with excuses for why this is so (read Gardner\u2019s chapter on Dr. Joseph Banks Rhine\u2019s work on ESP for an especially clear example). But you can find scientists doing the same thing! A chemist\u2019s reaction doesn\u2019t come out right, so he assumes it was contaminated. A particle physicist doesn\u2019t see the effect he was looking for, so he assumes it occurs at just slightly higher energy. How can we tell the difference between honest excuses \u2013 those that are truly identifying mistakes in the experimental conditions \u2013 and dishonest ones \u2013 those that are the result of a researcher who would find an excuse under any circumstances? In recent years I\u2019ve heard from time to time about new attempts to publish scientists\u2019 negative results and to make their complete lab processes and all data openly available. These are two efforts that should help distinguish them from cranks.\n\nBut another problem with the crank mindset is that there\u2019s no sharp dividing line. Aside from science, I\u2019ve read a bit about training distance runners, so I\u2019ll use that here. One clear crank is Percy Cerutty, a coach who demanded his runners carry spears and \u201crun like the primitive man\u201d, advocated strange diets, and in general believed, as cranks do, that he had stumbled onto secrets that no one else knew. Eventually, his runners left him. A more marginal case is Arthur Lydiard. Lydiard is a coach who created a fairly rigid, systematized training system and then advocated it as being the best possible. His system was based on trial and error in his early days of coaching. He tried a few different things and then stuck with what seemed to work best. But he began to believe that all his advice was better, stronger, and more iron-clad than it was. He also began to think his general ideas applied not just to running, but to all athletic endeavors (specifically shot put, rugby, and rowing come to mind). He\u2019s an in-between crank, because he did hold himself accountable to the results of his methods, and he did coach Olympic champions, but he also lost touch with reality (Lydiard still has a large following of distance runners today, many of whom would be incensed if they read this summary.)\n\nModern coaches, too, tend to believe in their methods beyond the level their results support, and babble on endlessly about aspects of human physiology that are not as well-understood as they indicate. But the point is that they do this to varying degrees, with coaches ranging widely from true cranks to rational, down-to-earth people with a healthy dose of skepticism towards even their own practices and a realistic viewpoint on the success and failure of their athletes.\n\nI have frequently found myself buying into crank athletic ideas, believing, for example, that all my injuries are due exclusively to running on hard roads (as opposed to trails or grass), although I had no data to support the belief. After reading scores of books and hundreds of articles, I now believe mostly that I\u2019m not very sure about anything regarding training distance runners.\n\nSurely, there is a crank continuum in science as well. On the one hand, there is an ideal scientist who (perhaps) evaluates all new evidence they receive with a perfectly-rational Bayesian approach, drawing conclusions only to the extent warranted by the evidence (and their prior beliefs). But scientists, even good ones, don\u2019t do all do that. Once in a while they begin to believe in their own theories even when the evidence starts to pile against them. The outcomes they want to see happen affect the results of their experiments, or they choose not to publish results they don\u2019t like. Their error bars grow just large enough that the data is consistent.\n\nUsually it\u2019s not hard to tell a crank. Also, as Gardner points out in his book, just because there are some intermediate cases, doesn\u2019t mean that most cases aren\u2019t clear-cut. But I\u2019m glad I read about what cranks do, how they justify their delusions, because I don\u2019t have to look too long and hard to see hints of the same behavior in myself.\n\nMay 14, 2010\n\nDo you think rationally about all the opinions you read, carefully considering why you agree or disagree with any given viewpoint, or is your method for discourse more like the way you sift through a hundred crappy photos of yourself to find the kinda-hot-but-not-too-slutty one that will be your Facebook profile picture? Oh yes, I like this one. All the other can go now.\n\nIt\u2019s been a long time since I last read the internet with you, so it\u2019s time to do that again. Hopefully you\u2019ll be entertained, and also question the way you think about facts and reality. Although this is a links dump, incredibly none of it involves cats or pornography.\n\nVia Swans on Tea, Feynman discusses, in a tangential manner, what magnetism is.\n\nWhen I launch into an explanation, my goal is something is along the lines of, \u201cI\u2019m going to say something to you, and when I\u2019m done, you\u2019ll understand it the way I do.\u201d My guess is that most people implicitly think about explanation the same way. An explainer says some words, possibly along with drawing pictures or doing a demonstration, and the explainee watches, listens, and understands.\n\nWe expect some confusion and some back-and-forth questions. Also, the scope of what is explained may be very small, so that the explainer perhaps knows a lot more details, but despite these caveats I think this \u201cI will give you my knowledge\u201d approach is the subtext for most of our explanations.\n\nThe strange thing is that if you ask people directly what explanation is, they do not believe this. They believe that explanations are highly context-dependent, and that they\u2019re imperfect, and that their scope is limited. (\u201cI don\u2019t expect the explainee to get everything. The explanation just gives the general idea, and they\u2019ll work out the details in due time\u2026\u201d), but when I watch two people engaged in a explainer\/explainee interaction I get the feeling that they will consider the exchange a failure (or at least not wholly successful) if the explainee ultimately does not understand the subject the way the explainer does. Even the drastically different approaches people take when explaining something to an adult or to a child seem based on the principle that in order for the explanation to be effective, it must be worded to suit the audience, but the explainer still hopes to be completely understood. They just need to find the right way to say things.\n\nFeynman points out that this sort of explanation is impossible because knowledge doesn\u2019t consist of tidbits. Feynman cannot take his knowledge of magnetism and \u201cdumb it down\u201d in any sort of accurate way, because that knowledge is couched in the context of everything else he knows about nature. Feynman\u2019s understanding of magnetic forces was much more thorough than the interviewer\u2019s because Feynman understood the fundamental forces involved; he knew all about quantum theory and the interaction of light with matter, and had a feeling for what things were and were not already known and explained by physical models. He also had practical experience with magnets, and had taught students about magnetism and investigated all sorts of magnetic phenomena. But in addition to this knowledge of the theories and models of magnetism, Feynman\u2019s understanding is tempered by his abilities. What separates the scientist from the layperson is not their knowledge of science, but their ability to mathematically manipulate the model, or even create a new one, to derive understanding.\n\nIf Feynman were still around and he sat down to tutor me in all aspects of electromagnetism, we could probably make a lot of progress. With enough time, he could teach me everything he knew. But I still wouldn\u2019t understand it the way he did.\n\nWith that, let\u2019s look at an explanation I particularly liked:\n\nWe Recommend a Singular Value Decomposition\nDavid Austin at the American Mathematical Society.\n\nThis is an explanation of the singular value decomposition, a basic tool in linear algebra. I remember learning about it while studying linear algebra, but I didn\u2019t understand it very clearly. I thought about it only formally, and I kept getting the idea of what it was confused with the proof that it exists. As a result, if I were asked to explain singular value decompositions to someone else, I\u2019d have first gone back to my linear algebra book to review, then pretty much repeated what it said there, trying desperately to do things just differently enough that I wasn\u2019t copying.\n\nI got the feeling that Austin did the opposite in writing this article. he did not sit down and say, \u201cOkay, what are all the things I know about SVD and all the good examples of it, and then how can I condense them all and make it appropriate to the audience?\u201d\n\nInstead, it seemed like he said, \u201cI happen to know a couple of good pictures that make this clear in the case of a 2\u00d72 matrix. Based on that, what sort of presentation of the SVD makes sense? What level of detail would muddy the presentation? If I change the order I present the ideas, how will that change the reader\u2019s perception of the SVD\u2019s theoretical and practical importance? What can be left out, and how can I get straight to the heart of the matter and communicate that first?\u201d\n\nVery quickly in the essay, Austin gets to this picture:\n\nwhich illustrates the singular value decomposition of\n\n$\\left[ \\begin{array}{cc} 1 & 1 \\\\ 0 & 1 \\end{array}\\right]$.\n\nThere are only a few short paragraphs before that, but already we\u2019ve walked through a story that motivates it. Austin gives three examples showing how we can understand linear transformations visually, and by the time we finish the third, it was apparent to me that a singular value decomposition is a logical extension of the linear algebra I was already familiar with. He had me hooked for the rest of the article.\n\nAfter giving his example, Austin builds directly to the equation\n\n$M = U \\Sigma V^T$\n\nwhich illustrates why it\u2019s a \u201cdecomposition\u201d, and what each part of the decomposition means. Only after giving a fairly complete explanation of what a singular value decomposition is did he start to go into how to find it and how to apply it.\n\nLots of math or physics writing I see doesn\u2019t take this approach. Instead, the first I see a particular equation is at the end of its derivation. That means that all the derivation leading up to it seemed unmotivated to me. Austin doesn\u2019t even include the derivations. There\u2019s enough detail that I could work through the missing parts by myself, ultimately understanding them better than I would if each step were spelled out for me. For example, he writes\n\nIn other words, the function $|M x|$ on the unit circle has a maximum at $v_1$ and a minimum at $v_2$. This reduces the problem to a rather standard calculus problem in which we wish to optimize a function over the unit circle. It turns out that the critical points of this function occur at the eigenvectors of the matrix $M^TM$.\n\nThat\u2019s actually more effective for me than actually going through the details of the calculus problem. It points me in the right direction to go over it when I\u2019m interested, but in the meantime lets me continue on to the rest of the good stuff.\n\nBy reorganizing the material, omitting details, and (literally) illustrating his concepts, Austin finally got me to pay attention to something I ostensibly learned years ago.\n\nNext, I\u2019d like to illustrate my lack of creativity by returning to Feynman, this time his Caltech commencement address from 1974\n\nCargo Cult Science\n\nFeynman identifies a problem:\n\nIn the South Seas there is a Cargo Cult of people. During the war they saw airplanes land with lots of good materials, and they want the same thing to happen now. So they\u2019ve arranged to make things like runways, to put fires along the sides of the runways, to make a wooden hut for a man to sit in, with two wooden pieces on his head like headphones and bars of bamboo sticking out like antennas\u2014he\u2019s the controller\u2014and they wait for the airplanes to land. They\u2019re doing everything right. The form is perfect. It looks exactly the way it looked before. But it doesn\u2019t work. No airplanes land. So I call these things Cargo Cult Science, because they follow all the apparent precepts and forms of scientific investigation, but they\u2019re missing something essential, because the planes don\u2019t land.\n\nand suggests a solution:\n\nDetails that could throw doubt on your interpretation must be given, if you know them. You must do the best you can\u2014if you know anything at all wrong, or possibly wrong\u2014to explain it. If you make a theory, for example, and advertise it, or put it out, then you must also put down all the facts that disagree with it, as well as those that agree with it. There is also a more subtle problem. When you have put a lot of ideas together to make an elaborate theory, you want to make sure, when explaining what it fits, that those things it fits are not just the things that gave you the idea for the theory; but that the finished theory makes something else come out right, in addition.\n\nFor an example of awful science, take a look at a story that made it to Slashdot a little while ago, Scientists Postulate Extinct Hominid with 150 IQ.\n\nThe Slashdot summary says,\n\nNeuroscientists Gary Lynch and Richard Granger have an interesting article in Discover Magazine about the Boskops, an extinct hominid that had big eyes, child-like faces, and forebrains roughly 50% larger than modern man indicating they may have had an average intelligence of around 150, making them geniuses among Homo sapiens. The combination of a large cranium and immature face would look decidedly unusual to modern eyes, but not entirely unfamiliar. Such faces peer out from the covers of countless science fiction books and are often attached to \u2018alien abductors\u2019 in movies.\n\nSlashdot is known for being strong on computer news, not for their science coverage, but still it\u2019s surprising to me that such a ridiculous bit of claptrap got so much attention. A few commenters point out how absurd the conclusion that an entire race of people had an average IQ of 150 is, but there is so much white noise in the comments of any large online community that most people usually don\u2019t read them, probably including the people who write the comments in the first place.\n\nAnd even if Slashdot will publish sensational cargo cult stories like this, what business does it have in Discover Magazine, which I don\u2019t read, but had assumed was fairly reputable? Discover published this quote about the Boskops:\n\nWhere your memory of a walk down a Parisian street may include the mental visual image of the street vendor, the bistro, and the charming little church, the Boskop may also have had the music coming from the bistro, the conversations from other strollers, and the peculiar window over the door of the church. Alas, if only the Boskop had had the chance to stroll a Parisian boulevard!\n\nFirst, that doesn\u2019t sound like high intelligence to me. It sounds like autism. Second, how the fuck would you know that from looking at some skulls? Such conclusions obviously have no place in the science-with-integrity Feynman described.\n\n20 years ago, if I had read that story I would not have gone to the effort to follow up on it. (For one thing I\u2019d have been five years old, and so instead of doing some research I would have drank a juice box, gone outside to play, and pooped myself.) Now we have the internet, and follow-up is very easy. Fortunately, high up on the Google results is John Hawks\u2019 article, The \u201cAmazing\u201d Boskops. Hawks, summarizing his review of literature on the Boskops, writes,\n\n\u2026in fact, what happened is that a small set of large crania were taken from a much larger sample of varied crania, and given the name, \u201cBoskopoid.\u201d This selection was initially done almost without any regard for archaeological or cultural associations \u2014 any old, large skull was a \u201cBoskop\u201d. Later, when a more systematic inventory of archaeological associations was entered into evidence, it became clear that the \u201cBoskop race\u201d was entirely a figment of anthropologists\u2019 imaginations. Instead, the MSA-to-LSA population of South Africa had a varied array of features, within the last 20,000 years trending toward those present in historic southern African peoples.\n\nHawks then followed up with more detail later.\n\nThe good news is that the Boskop nonsense will die out because it\u2019s wrong, and our system works well enough that things that are wrong do eventually die out.\n\nIn that little vignette, I looked at a big magazine and published book that were nonsense, and debunked by a blog. It\u2019s not always easy to determine the credibility of a source, and its reputation can be misleading. Blogs have a terrible a reputation in general, while some people seem to believe that if it\u2019s in a book, it must be true. (Unfortunately people take this to the extreme with one particularly poorly-documented and self-contradictory bestselling book!)\n\nA more difficult stickier issue is anthropogenic global warming. There is little doubt in my mind that anthropogenic global warming is real, but unlike with evolution, I do not believe that because I have looked at the scientific evidence and thought about the arguments for and against. I haven\u2019t examined the methods of collecting raw data or the factors accounted for in climate models. I don\u2019t even know how accurate those models\u2019 predictions are. I take it all on the word of climate scientists and a cursory review of their reports. I do not see this as a problem or a failure of my rationality. I do withhold judgment on whether global warming is as important an issue as, say, pollution or direct destruction of natural resources, but I do not feel reservation in stating that I think it is very likely that if humans continue on the way they\u2019ve been going, the Earth will warm with severe consequences.\n\nWhat does this have to do with cargo cult science? Cargo cult science is the reason I believe the climate scientists rather than the climate skeptics. My goal here isn\u2019t to convince you one way or another about climate science, or to link to the best-reasoned discussions about it or to give an accurate cross-section of the blogosphere\u2019s thinking process. These are various opinions on anthropogenic global warming, and my hope is that reading for the underlying decision-making process is an instructive exercise.\n\nHere is Lord Monckton, a prominent global warming critic:\n\nHere he is interviewing a Greenpeace supporter about why she believes in anthropogenic global warming:\n\nHere is the UN group Monckton criticizes, the\nIntergovernmental Panel on Climate Change\nIn particular, their Climate Change 2007 Synthesis Report, a 52-page summary of all things climate science. For more detail, their Publications and Data are available.\n\nHere is a recent letter published in Science. It discusses the process scientists use to create reports on the climate, the uncertainty in scientific results, the fallibility of scientific findings, and the role of integrity in science.\nClimate Change and the Integrity of Science\n\nHere is statistician and blogger Andrew Gelman talking about expert opinion and scientific consensus:\nHow do I form my attitudes and opinions about scientific questions?\n\nHere is famous skeptic James Randi on the pressure for scientific consensus, the fallibility of scientists, the uncertainty in models of complicated phenomena, and his skepticism of anthropogenic global warming:\nAGW Revisited\n\nHere is the petition Randi describes, the\nPetition Project\n\nHere is a reply to Randi and the Petition Project from PZ Myers, a biologist and well-known angry internet scientist.\nSay it ain\u2019t so, Randi!\n\nHere is a graphic by David McCandless. Its goal is to present an example of the arguments one would uncover in an attempt to self-educate about climate science using only the internet.\nGlobal Warming Skeptics vs. The Scientific Consensus\n\nGreg Laden writes about skepticism, rationality, and groupthink in a lengthy post.\nAre you a real skeptic? I doubt it.\n\nHere is the Wikipedia Article on anthropogenic global warming, along with tabs to the discussion page for the article and the article history. This is a featured article on Wikipedia.\nGlobal Warming\n\nMy focus on the process people are using to come to terms with global warming isn\u2019t meant to deemphasize the importance of this issue and of other aspects of the relationship between humanity and our biome. Our Earth is a fantastically diverse and endlessly beautiful home. Of course I want to understand it better.\n\nAlso here is a physics blog story about a mathematical model of cows.\n\n### Bounce, part 5\n\nJanuary 4, 2010\n\nThis post is a digression from the topic of the previous parts (1 2 3 4). We\u2019ll move away from discussing how high a tennis ball should bounce when dropped on top a basketball, and into some metadiscussion of the arguments made in the first four parts. It\u2019s a long post as well, but it\u2019ll be good for you, because half the words are Galileo\u2019s, not mine, and he\u2019s a dude worth reading.\n\nLast time, I cited Galileo as our source for understanding uniformly accelerated motion \u2013 the motion of a ball dropped or thrown in the air.\n\nBefore introducing his idea of what uniformly accelerated motion is, Galileo gives us an extended prelude. It\u2019s long, but I think it\u2019s worth seeing all at once, rather than piece-by-piece.\n\nFor anyone may invent an arbitrary type of motion and discuss its properties; thus, for instance, some have imagined helices and conchoids as described by certain motions which are not met with in nature, and have very commendably established the properties which these curves possess in virtue of their definitions; but we have decided to consider the phenomena of bodies falling with an acceleration such as actually occurs in nature and to make this definition of accelerated motion exhibit the essential features of observed accelerated motions. And this, at last, after repeated efforts we trust we have succeeded in doing. In this belief we are confirmed mainly by the consideration that experimental results are seen to agree with and exactly correspond with those properties which have been, one after another, demonstrated by us. Finally, in the investigation of naturally accelerated motion we were led, by hand as it were, in following the habit and custom of nature herself, in all her various other processes, to employ only those means which are most common, simple and easy.\n\nFor I think no one believes that swimming or flying can be accomplished in a manner simpler or easier than that instinctively employed by fishes and birds.\n\nWhen, therefore, I observe a stone initially at rest falling from an elevated position and continually acquiring new increments of speed, why should I not believe that such increases take place in a manner which is exceedingly simple and rather obvious to everybody?\n\nGalileo is mixing two approaches, and they appear to be intrinsically intertwined in his mind. The first is the ultra-skeptical pure empiricism viewpoint. This line of thought says that the only way to know about a thing is to confirm it by experiment. All scientific theories are to be tested against nature. If the theory and experiment agree, we fail to reject the theory. If the theory and experiment disagree, we reject the theory. Many modern scientists cite this as the true scientific viewpoint. (Note that from this point of view, you never confirm a scientific theory. Many scientists will agree with this \u2013 you never prove anything to be true in science. Also, I have called this viewpoint \u201cempiricism\u201d, a term which is sometimes used slightly differently in epistemology, where it refers to the belief that knowledge comes from sensory experience in general, rather than scientific experimentation in particular. Nonetheless, the cores of scientific and epistemological empiricism are similar.)\n\nBut, along with his statement that his knowledge of falling bodies comes from experiment, Galileo also has curious references to simplicity, in particular some out-of-place stuff about swimming fish and flying birds. This, to me, is the germ of a new idea \u2013 an idea that what we learn about nature ought to make sense to us on a deep level, once we\u2019ve learned it. Greek philosophers (so I hear, not having read them) believed the Universe ought to make sense, and that they could therefore understand it with a priori reasoning. This is not quite what Galileo seems to believe. He holds himself responsible to experiment, unlike Aristotle, but I think that if experiment gave strange or unusual results that Galileo couldn\u2019t understand, he\u2019d be extremely dissatisfied. He feels a deep need to take the mathematical results, back them up with data, but then do even more. He needs them to make sense.\n\nTwo New Sciences is written as a dialogue (or, there being three interlocutors, a trialogue?), with Sagredo and Simplicio, two men who haven\u2019t learned the new sciences, questioning Salviati, who has learned them and is explaining them to his friends. Galileo uses this device to explore intuition. He has Sagredo and Simplicio raise all manner of interesting objections to Salviati\u2019s ideas, just so Salviati can find interesting answers to allay their unease. (This format is out of style in modern physics text, with rare exceptions like Spacetime Physics, a book I enjoy much more today than I did when first learning special relativity from it six years ago.)\n\nFor example, Sagredo thinks there is a problem with saying that a body dropped from rest has a speed proportional to the time fallen. He objects,\n\n\u2026we must infer that, as the instant of starting is more and more nearly approached, the body moves so slowly that, if it kept on moving at this rate, it would not traverse a mile in an hour, or in a day, or in a year or in a thousand years; indeed, it would not traverse a span in an even greater time; a phenomenon which baffles the imagination, while our senses show us that a heavy falling body suddenly acquires great speed.\n\nHe thinks there is a disconnect between the math and experiment, because the math says that when you drop something, it has almost no speed after falling a short distance, but Sagredo thinks that when you drop a heavy thing it starts falling quickly immediately. Maybe you don\u2019t have this difficulty of intuition, but if you do, Salviati replies by appealing to an experiment.\n\nYou say the experiment appears to show that immediately after a heavy body starts from rest it acquires a very considerable speed: and I say that the same experiment makes clear the fact that the initial motions of a falling body, no matter how heavy, are very slow and gentle. Place a heavy body upon a yielding material, and leave it there without any pressure except that owing to its own weight; it is clear that if one lifts this body a cubit or two and allows it to fall upon the same material, it will, with this impulse, exert a new and greater pressure than that caused by its mere weight; and this effect is brought about by the [weight of the] falling body together with the velocity acquired during the fall, an effect which will be greater and greater according to the height of the fall, that is according as the velocity of the falling body becomes greater. From the quality and intensity of the blow we are thus enabled to accurately estimate the speed of a falling body. But tell me, gentlemen, is it not true that if a block be allowed to fall upon a stake from a height of four cubits and drives it into the earth, say, four finger-breadths, that coming from a height of two cubits it will drive the stake a much less distance, and from the height of one cubit a still less distance; and finally if the block be lifted only one finger-breadth how much more will it accomplish than if merely laid on top of the stake without percussion? Certainly very little. If it be lifted only the thickness of a leaf, the effect will be altogether imperceptible. And since the effect of the blow depends upon the velocity of this striking body, can any one doubt the motion is very slow and the speed more than small whenever the effect [of the blow] is imperceptible? See now the power of truth; the same experiment which at first glance seemed to show one thing, when more carefully examined, assures us of the contrary. (brackets added by translator)\n\nI get the feeling, while reading this passage, that Galileo cites this experiment simply because it gives him pleasure to do so. But in this case, even the experiment is not enough for him. He continues\n\nBut without depending upon the above experiment, which is doubtless very conclusive, it seems to me that it ought not to be difficult to establish such a fact by reasoning alone. Imagine a heavy stone held in the air at rest; the support is removed and the stone set free; then since it is heavier than the air it begins to fall, and not with uniform motion but slowly at the beginning and with a continuously accelerated motion. Now since velocity can be increased and diminished without limit, what reason is there to believe that such a moving body starting with infinite slowness, that is, from rest, immediately acquires a speed of ten degrees rather than one of four, or of two, or of one, or of a half, or of a hundredth; or, indeed, of any of the infinite number of small values [of speed]?\n\nHere we see the second approach to nature. The idea that, once we\u2019ve formulated a theory and tested it, we\u2019re still not done. We need to reason about it, too. We need to go back, take the solution, and make it ours. We need to convince our grandmothers, who don\u2019t know math, that this is the way it ought to be. And both these processes are intertwined. You can use the idea that nature ought to be simple to figure out what the laws are, but if you do, you\u2019re still subject to testing them by experiment. Conversely, you can use experiment to figure out the laws, but if you do, you\u2019re still subject to figuring out why things came out that way.\n\nGalileo is the earliest source I\u2019ve seen with this new, sophisticated attitude. Naturalists wanted to observe, discover, and document what happened around us. Philosophers wanted to talk about it in the abstract and explain its deeper logic. But Galileo wanted to do both. And it\u2019s only when you do both that you\u2019ve accomplished the real goal \u2013 understanding.\n\nI\u2019m not saying this attitude sprung up in Galileo\u2019s work with no precedent, but I do think it\u2019s clearly evident here, and since Two New Sciences is a landmark work in terms of the physical ideas it presents, it\u2019s important to examine in terms of the philosophical ones is presents, too.\n\nThis Galilean principle still guides us today. Science isn\u2019t about testing hypotheses and controlling experiments and statistical significance. Science is about figuring things out. The methods of modern science evolved over time as the problems scientists dealt with demanded them. (A great deal of statistics was invented specifically to study genetic inheritance, for example). Galileo didn\u2019t have our textbook scientific method, but ultimately he didn\u2019t need it to make great progress.\n\nToday we need things like careful laboratory conditions and error propagation formulas to keep us from screwing up when things get tricky and hard to interpret. But the core of my world outlook, which I am not afraid to claim is also the core of the scientific one, is that you are just trying to figure things out, subject to checking what really happens, and then, once you do that, trying to understand.\n\nNext time, I\u2019ll take a look at one of Galileo\u2019s arguments that didn\u2019t work. That\u2019s the other thing about science that I like. Nobody\u2019s perfect, and you\u2019re expected to screw up at least once in a while.\n\n### Now I Know\u00a0SCIENCE!\n\nOctober 6, 2009\n\nThis is supposed to be a blog. It isn\u2019t. It\u2019s just the standard WordPress template with posts coming in irregular bursts.\n\nPart of being a blog is that you\u2019re supposed to be integrated into the blogosphere. This is because if you take a bunch of meaningless things and endlessly interconnect them, you get what\u2019s known in science as an emergent phenomenon, in this case endless confusion. So I really ought to hook up to the blogosphere and get in on that.\n\nOn the other hand, it turns out there are these people who want blogs to be useful. So what they do is sift, organize, centralize, and bring similar bloggers together. GrrlScientist (real name, I think. You wouldn\u2019t put a fake name in the internet, right?) is one such person. She (or he, I suppose) runs Scientia Pro Publica, a blog carnival whose goal it is to make science understandable to everyone by writing in Latin. She\u2019s hosted the 13th edition of Scientia today on her blog, including about 20 essays in popular science. They\u2019re heavy on the life sciences, discussing such topics as parasites that eat weaverfish tongues and why I still don\u2019t understand evolution. (Seriously, I don\u2019t.)\n\nThere\u2019s some meta-discussion of science, a post about the suckiness of iPhone camera, and they graciously included my post about retroreflectors on the moon as the only physics\/astronomy representative.\n\n### The Asians Are Coming! But I Can\u2019t Count How\u00a0Many\n\nDecember 11, 2008\n\nSince I\u2019ve started reading blogs, I\u2019ve seen a lot of instances of people ranting madly about topics they don\u2019t understand very well. These people also don\u2019t understand why they aren\u2019t taken more seriously, or why, in fact, the whole system doesn\u2019t immediately bow to their sagacity. But now that I, too, am a blogger, I\u2019m beginning to understand the severely-debilitating effect the freedom to publish uncensored material has on human judgment. So here I am joining the ranks of men screaming into a hurricane, and unknowingly pointing the wrong direction.\n\nA recent story from the NY Times warns repeatedly that those tricky little Asian people are eating a gazillion tons of fish every day and getting way too good at math. You see, for at least the last ten years both a generic statement and its complement have been considered racist if they involve black people in any way. Further, the whole feeling-generally-uncomfortable-about-anything-Islamic thing has been used as the hook on enough network TV shows that people are starting to get pretty sensitive about that, too. But we haven\u2019t done anything really bad to the Asians since Vietnam, so it\u2019s pretty much okay to treat them as one big group and find reasons to be scared of them.\n\nApparently, kids in Singapore, Taiwan, and Japan do very well, on average, on standardized math tests. It\u2019s supposed to send off alarm bells and spur us to reform the educational system. But the stat is not what it\u2019s made out to be.\n\nHere are three of the more practical reasons we might want students to be mathematically competent:\n1) it helps them balance their checkbook and etc.\n2) it\u2019s necessary background for engineers and accountants, etc.\n3) it\u2019s necessary for innovation. great technological and scientific breakthroughs are made by people who understand math\n\nBut here\u2019s why childrens\u2019 average test scores are irrelevant to these points\n1) (math helps with life) It\u2019s increasingly unnecessary for the average person to know math. Computers will do it all for you. Anything that requires a minimal amount of the sort of mathematical, logical, and\/or algorithmic thinking employed by a math, science, or computer-type person can now be automated to the point where an intelligent chimpanzee can do it. Want to calculate your BMI? Don\u2019t bother with the formula. Just plug in the numbers to a calculator, which automatically multiplies them to each other for you. Don\u2019t want to figure out your taxes? Plug it into Quicken. Or hire an accountant, who also doesn\u2019t know math very well but can plug things into Quicken more efficiently than you. Don\u2019t know how much longer to boil an ostrich egg than a chicken egg? Don\u2019t bother with dimensional analysis. Just look it up online.\n\n2) (math helps with jobs) Partially, more of the same argument as point 1) applies here. Want to be an airline pilot? Don\u2019t worry yourself too much with the math. Just make sure the numbers from this instrument agree with the numbers from that instrument, and the computers will take care of everything. The percentage of people who really need to be good at math is quite small, so we should be more interested in the scores of the top 5% or top 1% of students than the average score.\n\n3) (math leads to technological and scientific excellence) The average performance of students is simply irrelevant to this one. Big ideas come from people who work hard on problems because they\u2019re intrigued by them and genuinely interested in the work itself. They need a spark of creativity to go with their technical competence, but spark is the really essential thing. It\u2019s far easier to be very good at electrical engineering (for example) than it is to do something important in it. And frankly, hours upon hours drilling practice problems until you\u2019ve memorized all the methods of solution is not going to get you far beyond good test scores. But that, as best I can tell from here, it\u2019s what\u2019s going on with the Asia\/West divide in math scores. The Asian kids study longer and work harder. The cuiture is extremely performance-based, so that parents push their kids hard, but they only thing anyone cares about are good grades and good test scores. Since the tests don\u2019t require creativity, why bother encouraging it?\n\nI\u2019ve been teaching American high school kids for a while. Many of them have been first or second generation Americans from Asian families. They grew up bilingual and their households retain most of the traditional values of Korea\/Taiwan\/Japan, including those relating to education. I\u2019ve also taught kids from America, the UK, India, France, Italy, Turkey, Japan, China, Mexico, Canada, and various places I hadn\u2019t even heard of before i met them. I\u2019ve taught whites, blacks, east asians, south asians, hispanics, polynesians, native americans, and various combinations thereof. And guess what? They\u2019re all the same. Not the kids, I mean, of course they\u2019re quite different from each other. But I do not see any systematic difference in competence, creativity, interest, brilliance, ability to concentrate, or whatever other factors are essential to doing great things with technical material.\n\nIt has been my experience that when you look at the top few percent \u2013 the ones who are truly gifted at this stuff, and occasionally ask questions that startle me with their insight, or find clearer and more direct explanations of the topic at hand than I had sniffed up myself \u2013 are more likely to be male. Not exclusively, of course. The most insightful student I ever had was a girl. But that gender bias is the only systematic tendency that\u2019s stuck out to me.\n\nSo the Asian kids kicking American kids\u2019 butts at math is not a clarion call. It may be a benchmark for how effective our educational system is, and how seriously our culture treats education, but not for how many great thinkers we\u2019ll have in this country twenty years from now. If we want to have a home-grown army of thinkers and innovators, we should be more concerned with how much kids like math and want to do it on their own, rather than how many formulas they\u2019ve memorized by age 10. A high schooler\u2019s knowledge of math won\u2019t get you all that far, anyway. It only comes from higher study, and America still has the world\u2019s best system of institutes of higher learning. So it\u2019s not a matter of cramming more into their heads while they\u2019re young. It\u2019s a matter of honestly and fairly presenting to young people what math is and what it can do. As long as grade school doesn\u2019t make kids hate math, it\u2019s doing fine. The ones who have aptitude will naturally gravitate towards it. We need to make sure that when they do, there\u2019s someone there to guide that top 5%, and that we\u2019re not all too busy worrying about the grade of the kid in the middle of the class to notice that the kid at the top just proved a new result in number theory.\n\nMy guess is that most of the people who spend their time screaming, \u201cThe Asians are coming! They traded their abacuses for TI-89\u2019s and they\u2019re going to swipe the technical carpet from under our fat, complacent feet!\u201d know much more about statistics than about the process of becoming technically competent, one part of which is to learn never to take statistics at face value. If our goal is really to raise the average test score, it has to come as much from a shift in cultural values as a change to the educational system. But if our goal is to be a scientifically and technologically vital society, the masses are not the place to look.\n\n### Let\u2019s Read the Internet! week\u00a08\n\nDecember 8, 2008\n\nWind-Powered Perpetual Motion\nand\nWhy the Directly-Downwind Faster Than the Wind Car Works\u201d\nMark Chu-Carroll on Good Math, Bad Math\n\n\u201cThe only true wisdom is in knowing you know nothing.\u201d\n\nSocrates would have to be a fan of the scientific method. We frequently acclaim the shift towards naturalism in Western thought, as a turning point in our intellectual maturity, but that shift brought with it the less-recognized roots of an even higher goal \u2013 the eradication of hubris in the search for understanding. Naturalism, the philosophical position that empirical observation holds the final word in debates on truth, essentially kills the argument of \u201cbecause I say so.\u201d Truth comes from no one in particular, so there\u2019s at least the faint possibility that people trying to understand the way things work will some day no longer jockey and battle to be \u201cthe one who got it right.\u201d That\u2019s a far-out ideal, and maybe if nobody thought they were going to be credited with brilliance, nobody would have the incentive to try to do something brilliant in the first place. But at the very least, when two naturalists have an argument, they can frequently appeal to a common, impartial, higher source \u2013 nature \u2013 as arbiter.\n\nThat\u2019s what\u2019s happened here on Mark Chu-Carroll\u2019s widely-read blog. He initially, and incorrectly, believed a certain device that drives overland into the wind and faster than the wind was a fraud. After long, long debates, he changed his mind, and carefully explained the mistakes in his own reasoning and what he had learned in the process of investigating his own error. Which is pretty much awesome, because such things hardly ever occur in arguments on less savory topics, like abortion. (Oh my God, was that an eating-dead-babies joke?)\n\nI also appreciated the sort of emergent didactic property of the hundred-some post comment thread on Chu-Carroll\u2019s original post. After watching the youtube video of the device (linked from the original post), I wasn\u2019t completely sure whether the treadmill test was fair. It seemed reasonable enough, but I certainly wouldn\u2019t have been prepared to defend it against someone eager to argue the opposite way.\n\nAs I read the thread, commenters raised most of the points I was considering. Other people answered those points, and then even more people chimed in with takes that I hadn\u2019t considered at all. The overall effect was for a large amount of white noise and repetition, but also for a strikingly-diverse set of mindsets converging on the same problem. By the time I was done reading what everyone had to say, I felt that I had appreciated more intricacies in the problem than I would ever have discovered thinking about it alone, and I probably understood it better than I would have even if a single skilled author had written a long exposition. The challenge of interpreting each new voice\u2019s arguments, incorporating them with the previous knowledge, and then parsing all of it for myself over and over, trying to find holes in everyone\u2019s logic and patch together a firm understanding piece by piece, was absorbing because it\u2019s so much more interactive than simply reading one single person\u2019s explanation, no matter how clear, detailed, or precise.\n\nIt makes me want to argue about physics more often, but only in the good way where your ego doesn\u2019t get too involved.\n\nA Russian Teacher In America\nAndre Toom, linked from God Plays Dice\n\nA long essay that\u2019s a borderline sob story about the woes of the American educational system. As a private tutor, I see exactly the sort of problems Toom is discussing on a daily basis \u2013 students, even (or perhaps especially) the \u201cgood\u201d students, are so maniacally focused on their grade that learning becomes completely lost amidst a sea of test-cramming, and question-memorizing. Students are so wrapped up in the concrete performance markers visible to the world, that they don\u2019t care at all for their true progress, visible chiefly to themselves.\n\nThat, at least, is the picture. I only partially buy it. It\u2019s true, to varying degrees, for many students. But it\u2019s not as if this entire nation has no one left interested in math. The sad part is that over two hundred or so students I\u2019ve had, there have been a handful who are truly interested in math and physics, but they seldom have much guidance. Because these kids can gets A\u2019s in math class, no one in public school is very concerned with pushing their limits when there are too many problem kids to worry about first. So I\u2019m more interested in people with plans on how to reach interested young students with extra-curricular math opportunities than I am with people deriding a broken system.\n\nNot everyone is going to love math. In fact, I doubt there\u2019s ever been a society where a majority of people are interested. But the vast majority of our society has to take it in school. So yeah, it\u2019s inevitable that there are lots of people taking math who don\u2019t care about math. But I\u2019ve done the same thing in a literature class before. Ultimately, math is cool enough that some people are going to discover it no matter what the educational system is like, so I\u2019m not all that worried about the alarm bells being rung here.\n\nBlow to Vitamins as Antidote to Ageing\nJames Randerson at The Guardian\n\nWe thought we understood, like, everything. Turns out not. But the next study that comes out will surely reveal the secrets to perfect health once and for all\u2026\n\nSwiss Approve Heroin Scheme but Vote Down Marijuana Law\n\nSounds like a pretty good plan to me. Administer heroin to addicts in a safe, controlled environment, thereby reducing health risks and driving down the general nastiness associated with black market activity. I can also understand why you wouldn\u2019t want to legalize marijuana in just one small portion of Europe, since everyone would then go there just to smoke. The same argument doesn\u2019t hold as much water for the US with its block-like geography, but I live in California, where marijuana is as good as legal anyway.\n\nNebulous\nTara Donovan\n\nfrom Three Quarks Daily\n\nThe Not-So-Presidential Debate\n\nThe Not So Presidential Debate from aaron sedlak on Vimeo.\nalso from Three Quarks Daily\n\nWhy Punishment Is Worth It In The End\nEd Yong at Not Exactly Rocket Science\n\nRead this article or else! Nah, honestly I would never be able to go through life as someone who tried to understand human interactions by designing toy experiments like this. But It\u2019s nice to get little sixty-second summaries of their months of hard labor.\n\nOver-budget Mars rover mission delayed until 2011\nRachel Courtland at New Scientist\n\nBad news, since I work at the place where they\u2019re building this thing, and they owe me two months\u2019 back pay already.\n\nYou get to feeling a little bit sleazy when you realize all the exposure you\u2019ve had to art in the last two years has come in the form of internet lists with titles like \u201cThe Top Ten Totally Badass Avant-Garde Experimental Playdoh Exhibitions of 2008!!\u201d But on the other hand, some of this stuff actually is pretty badass, for being a paper sculpture of a cat.\n\nA Happy\/Unhappy New Pair of Studies\nStephen Black at Improbable Research\n\nAmong the headlines of news feeds I scanned through this week, there must have been at least ten stories referencing a recent paper purporting to show that happiness is \u201ccontagious\u201d, that is, if I were to reach down and magically make your friends happy, you would become happy as well. When I first heard about this, I was intrigued, because I was wondering how you would establish this is a \u201ccontagious\u201d effect, and not just correlation. It turns out: you don\u2019t. The researchers, from what I can tell, simply found a correlation and announced that happiness is contagious. News stories are apparently contagious, too, because once word of this paper got out, most of the major science news outlets published something on the story.\n\nBut as the link describes, another study found that height was also \u201ccontagious\u201d. That is, if your friends are tall, it\u2019s likely you\u2019re tall, too. Just as with happiness.\n\nSine of an Inscribed Angle\nBrent Yorgey on The Math Less Traveled\n\nA cute visualization of the law of sines.\n\n### Let\u2019s Read the Internet! Week\u00a07\n\nNovember 30, 2008\n\nMost Planets May Be Seeded With LIfe\nPhil Berardelli Science\n\nThe title of this article really is \u201cMost Planets May Be Seeded With Life\u201d. I would point out what a ludicrous construction this is, but it would be approximately equivalent to nudging the guy standing next to you at the Taj Mahal and saying, \u201cpretty nice, huh?\u201d. The author also drops the journalistic gem, \u201cThe new find, described this week in the journal Astro-ph, is stronger.\u201d Which is a bit surprising, because \u201cAstro-ph\u201d is not a journal at all, but just the name of the astrophysics section of arXiv.org, where physicists post free preprints of their work. The paper, which can be found here, has actually been accepted for publication in The Astrophysical Journal Letters.\n\nThe paper uses the word \u201clife\u201d twice in seven pages of text \u2013 once in the abstract and once in the introduction. The news story uses the word \u201clife\u201d five times, including in the title. I made a token attempt to skim through the paper, but I have no experience with astrochemistry and can\u2019t really say much about the scientific merit of the work. The data sure looks pretty.\n\nHere is the bottom line: the researchers behind this paper work hard at solving technical problems. The problem they were trying to solve here is, \u201chow can you tell whether some particular organic molecule is out there in a given direction of outer space, when you can\u2019t go there, can\u2019t send a probe, can\u2019t do an experiment, and can only passively collect a little bit of light?\u201d Their work is astrochemistry, and it has no honest direct association with the origin of life. Also, if you want to understand what they do, you will have to devote a lot of time and energy into it.\n\nBut, the science writer is on a deadline. I know someone who interned for Science and wrote this sort of five-minute story. You probably only have one day to read about the work, get in contact with and get a quote from one of the lead scientists on the paper, then find another, independent scientist in the same field, who has also seen this particular preprint on the arXiv, and get a quote from that guy to balance the story out. Then you have to throw your story together as quickly as possible so it can go through revisions and the art people can find a relevant graphic, so you add a pun if you can detect one, and somehow make it catchy or attention-grabbing with the least possible effort. Of course \u201cOrigin of Life!\u201d becomes the slant of the story.\n\nI see two problems with this. One is that there are a lot more \u201corigin of life\u201d stories out there than there are actual breakthroughs on the origin of life. So if you\u2019re innocently following at home what \u201cthose guys in the white labcoats\u201d do all day, you\u2019d at first think they\u2019re making huge progress every week. Then after a while, you\u2019d begin to wonder why, if they\u2019ve been making so much huge progress, they still don\u2019t seem to have all this figured out yet.\n\nThe second is a problem I\u2019ve personally encountered. Science simply is not 100% adrenaline. Most of it is boring. Scientists spend much of their time waiting for gels to run, debugging their code, and fixing their lasers. (Sound awesome? It gets monotonous. But there are those few seconds every once in a while where you think, \u201cWhoa! I play with lasers all day!\u201d Then your thesis adviser tells you how much he\u2019s looking forward to your presentation in group meeting on Monday. This is Friday night. You start to cry.) Having a science job is a lot like having a normal job. You just work more and get paid less.\n\nThat isn\u2019t the picture you get coming in, though. Many of my high school students (I\u2019ve had a couple hundred) have told me that they \u201clove\u201d science. I cringe a little when I hear that. (I cringe a lot when I get, \u201cI love science, but I just don\u2019t get the math part of it.\u201d) Loving science, for them, just isn\u2019t possible. They don\u2019t know science. They might love the ideas they learned in science class. They might have loved doing their science fair project. But they probably will not love writing grant proposals or reviewing inscrutable papers. When they do finally get to the lab, they get a little confused about what it is they were looking forward to all this time. Come to think of it, maybe I was using the wrong pronoun this last paragraph, and wasn\u2019t referring to my students at all.\n\nOf course, seeing problems is easy. Everyone sees a thousand problems a day, mostly with other people. Then they bitch about it a bit and consider the issue closed. Not that I see a solution. But let\u2019s leave the issue open.\n\nSea Change For Turtle Origins\n\nI like this one much more than the last. Its attempt at a pun is so bad it\u2019s simply confusing. It gives a nice picture of the \u201cwe don\u2019t know shit\u201d side of science. The underside of a turtle shell is apparently called a \u201cplastron\u201d, which is an egregiously-awesome term for such a mundane thing. Finally, there is a guy saying \u201cThe reason I\u2019m excited about that is that it pushes the story of turtle origins even further back in time.\u201d\n\nWell shit yeah, baby! Now I\u2019m excited, too. I\u2019m so wired I can barerly acontrl my ffingers on the keybaorad.\n\nHappy Thanksgiving\nNikita at Monosyllables\n\nHere\u2019s an awful confession. I was sitting around with some other guys like myself (well, not EXACTLY like myself, but other young American nerds) on Thanksgiving, projecting the computer screen on the wall so we could watch downloaded versions of Arrested Development. A story from Google news popped up citing the number of people killed that day in the terrorist attacks in Mumbai. I jokingly calculated that because Mumbai has about 13 million people in it, and people live 70 years or so, we could calculate the daily death rate there, which is order of magnitude 500. That gives a daily standard deviation of root 500, or 20-25 people assuming deaths are independent, randomly occurring events. The attacks that day were a three or four-sigma event. Barely statistically significant, because that many extra people should die totally at random in Mumbai once every few years. Then Nikita\u2019s post reminded me there were real people there, that I knew some of them, and that it wasn\u2019t so great a joke.\n\nThe Toughest Man In Cairo Vs. The Zionist Vegetable\nAnand Balakrishnan in Bidoun\n\nAccording to my old neighbor, Kamal Hanafi, the vegetables in Israel are huge and good for only one thing. \u201cThe cucumbers,\u201d he exclaimed, eyes lighting up, \u201care this long\u201d\u2014he stretched his hands more than a foot apart. \u201cThey are this wide\u201d\u2014he made a circle with his two hands. \u201cAnd they taste like shit, all chemicals and unnatural fertilizers.\u201d He spat. \u201cNo one can eat vegetables that disgusting. The only people who use them are the women, who sit like this\u201d\u2014he spread his legs to demonstrate. \u201cAnd the men, of course.\u201d The invisible cucumber in his hands jabbed sharply up. \u201cAnd now they\u2019re sending their vegetables to Egypt to fuck us all.\u201d\n\nDancing Droplets and Spherical Harmonics\nStefan on Backreaction\n\nLittle bubbles of oil resonating as spherical harmonics. I\u2019ll bet you didn\u2019t know they could do that. Now you do.\n\nPerfect athlete\u2019s 100m sprint time calculated\nDave Robson on New Scientist\n\nMore terrible abuse of the word \u201cscience\u201d. The article says, \u201cfitting the data to a mathematical model that matches the other results, Denny predicts future male sprinters will at best shave 0.21 seconds off Usain Bolt\u2019s current world record of 9.69 seconds for the 100 metres.\u201d\n\nIt\u2019s wrong. It\u2019s so terribly wrong. There is really no reason to believe that just because you drew a curve through some data points, you\u2019ve predicted the future. If it were that easy, everyone would have done it earlier, and predicted today\u2019s world records. But they didn\u2019t. The article itself does technically refrain from calling the work \u201cscience\u201d. But apparently it\u2019s actually being published in the Journal of Experimental Biology, despite containing no experiment or biology. Can\u2019t we just take all these people and send them somewhere?\n\nBeethoven and Borge\nfrom In The Dark\n\nHumor on the piano. It\u2019s like stand up comedy, but they\u2019re sitting down. You better be, too, before watching these wacky films!\n\nThat\u2019s it for this week. I read plenty of other stuff, but it was just boring things. Reminded me of you.\n\n### Origins\n\nOctober 3, 2008\n\n\u201cAh, the origin of the universe,\u201d sighs physicist Leonard Susskind from the stage of Beckman Auditorium. \u201cBoy, does that ever take me back.\u201d\n\nAn hour later, Paul Davies intoned for the third time, \u201cas Lenny already mentioned\u2026\u201d before explaining again that the universe is in fact quite old, and did or did not, perhaps, depending on your point of view and interpretation of various fine intricacies some small subset of specialists may or may not understand, come from somewhere.\n\nThe third physicist to speak, Caltech\u2019s own Sean Carroll, probably couldn\u2019t even tell who to credit before making a point. Was it \u201cas Paul already mentioned,\u201d or \u201cas Lenny alluded,\u201d or \u201cas Paul said that Lenny previously indicated that I might say when it was my turn, about the point Paul made clarifying Lenny\u2019s tangent on my thesis\u2026\u201d\n\nPerhaps you see the difficulty, at something like the Origins conference, in keeping your physicists apart. When it comes to speculating on genesis, they appear to be bosons. (Note to non-physics people: that\u2019s not as mean as you think. \u201cBoson\u201d is the name of a famous circus clown. He invented gravity. To help him juggle.)\n\nMichael Shermer, director of the Skeptic Society, brought a host of eminent scientists to Caltech last Saturday to speak before a lay audience (like me). Ostensibly, their goal was to collectively meditate on whether \u201cscience makes belief in God obsolete.\u201d\n\nThe scientists involved were as nonplussed by the imponderability of this question as any other reasonable person would be, and proceeded to talk about their research, instead.\n\nCristof Koch, Caltech\u2019s (literally) colorful neuroscience professor, shocked his audience by explaining that, as a scientist, he thinks consciousness comes from somewhere. He tries to find out where by looking very closely.\n\nFor example, in occasional unfortunate instances, it\u2019s medically necessary to stick all sorts of wires in epileptic people\u2019s brains. As long as you\u2019re doing that, you might as well mess around with some science.\n\nIt turns out that each concept you can consciously identify, such as \u201credness\u201d, \u201cpain\u201d, and \u201cHalle Berry-ness\u201d) (a special property shared by her image, text of her name, and a sound recording of her name, but not images of other actresses or anything else researchers can think of), corresponds somewhere in your brain to the binary activity of a neuron. If you are seeing Halle Berry, the neuron fires. If you aren\u2019t it doesn\u2019t.\n\nSounds simple, right? That\u2019s because it\u2019s from a talk for designed for simple people. Consciousness is complicated, comes in varying degrees, and is notoriously slippy to analyze. But does Koch think the study of consciousness involves theology? No.\n\nDo Susskind, Davies, and Carroll think that God can help explain the origin of the universe? No. If you stretch, it\u2019s a slightly-fuzzy no. But still no.\n\nDoes David Prothero, Caltech\/Occidental-affiliated expert on the fossil evidence of evolution, think religious considerations aid our understanding of the origin of life, or the Cambrian proliferation of life? Emphatic no.\n\nBut frankly, they just don\u2019t seem that worried about it. They were brought in to talk about God. But except for Prothero, whose science is the target of a vigorous attack from certain flavors of Christianity, the speakers at the Origins conference confined their theological ruminations to a couple of bullet points on their final \u201cin conclusion\u2026\u201d slide.\n\nSean Carroll excitedly delved into Boltzmann\u2019s hypothesis that the universe\u2019s low-entropy past is a statistical blip in an infinite history, then excoriated the idea and presented a new model of baby universes pinching off and \u201cnever writing home to their parents.\u201d\n\nSusskind compared the finely-tuned nature of physical constants to the finely-tuned sequence of a human genome to illustrate his idea of how string theory might explain the state of the universe.\n\nProthero described lab experiments in creating the chemistry of life. Davies speculated on the meta-laws constraining choices among logically-consistent universes. Koch told me I would forget the color of his orange shirt (I think), and that this was based on science.\n\nSo imagine that. You work so hard to bring a bunch of great scientists together to have a discussion about some sort of general silliness mankind spends its time fretting over, but they ignore the bait and discuss their scientific passions instead. 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\section{Introduction}
\section{INTRODUCTION}
In large scale machine learning optimization problems, the data and training need to be distributed among many machines \citep{DMLsurvey}. Also in federated learning \citep{FEDLEARN, FEDOPT, FL2017-AISTATS, FL_survey_2019}, training occurs on edge devices such as mobile phones and smart home devices, where the data is originally captured. In these applications, the distributed machine learning can be characterized as the following composite finite-sum problem
\begin{equation}\label{primal-LSVRG}
\min \limits_{x\in \mathbb{R}^d} P(x) \; { := }\; \left\{\frac{1}{n} \sum\limits_{\tau=1}^n f^{(\tau)}(x) + \psi(x)\right\},
\end{equation}
where $\{f^{(\tau)}(x)\}_{\tau=1}^n$ are smooth convex functions distributed over $n$ nodes, and $\psi:\mathbb{R}^d \to \mathbb{R}\cup \{+\infty\}$ is a regularizer, which is a proper closed convex but possibly non-smooth function. On each node $\tau$, $f^{(\tau)}(x) \; { := }\; \frac{1}{m} \sum \limits_{i=1}^m f^{(\tau)}_i(x)$ is the average loss over the training data stored on this node and each $f_i^{(\tau)}$ is smooth and convex.
In distributed and especially federated settings, communication is generally much slower than the local training, which makes the communication overhead become a key bottleneck. In order to overcome this bottleneck, several methods were proposed in the literature, such as using large mini-batches \citep{Goyal17, You17}, asynchronous learning \citep{tsitsiklis1986distributed, Agarwal11, Lian15, Recht11}, and gradient compression \citep{Seide14, Alistarh17, Bernstein18, Wen17, Mish19}. In this work, we focus on the error-compensated method, which is a gradient compression method and is capable to deal with some effective but biased compressors, such as the TopK compressor.
{\bf Related Work.} The {\em error compensation/feedback} mechanism was first introduced in 1-bit SGD \citep{Seide14}. Then the error-compensated SGD (ECSGD) was proved to have the same convergence rate as vanilla SGD in the strongly convex case \citep{Stich18} and non-convex case \citep{karimireddy2019error,Tang19} when $P$ is smooth. ECSGD was further studied in \citep{stich2020error} under weaker assumptions. When $P$ is non-smooth, it was shown that ECSGD converges at the rate of ${\cal O}(\nicefrac{1}{\sqrt{\delta T}})$ in \citep{karimireddy2019error}, where $T$ denotes the iteration number and $\delta$ is the compressor parameter defined in (\ref{eq:contractor}). The non-accelerated linear convergence can be obtained in EC-LSVRG-DIANA \citep{gorbunov2020linearly} in the smooth case, and in the error-compensated loop-less SVRG, Quartz, and SDCA \citep{ecsdca} in the composite case. In a recent development, the error-compensated loop-less Katyusha was proposed in \citep{eclk}, and the accelerated linear rate was achieved.
{\bf Compressor.} In error-compensated methods, contraction compressors are generally used. A randomized map $Q:\mathbb{R}^d\to \mathbb{R}^d$ is called a {\em contraction compressor} if there exists a constant $\delta \in (0, 1]$ such that
\begin{equation}\label{eq:contractor}
\mathbb{E} \left[\|x - Q(x) \|^2\right] \leq (1-\delta)\|x\|^2, \qquad \forall x\in \mathbb{R}^d.
\end{equation}
\noindent Some frequently used contraction compressors include TopK \citep{Alistarh18} and RandK \citep{Stich18}. Let $1\leq K \leq d$. The TopK compressor is defined as
$$
({\rm TopK}(x))_{\pi(i)} = \left\{ \begin{array}{rl}
(x)_{\pi(i)} &\mbox{ if $i\leq K$, } \\
0 \quad \quad &\mbox{ otherwise, }
\end{array} \right.
$$
where $\pi$ is a permutation of $\{1, 2, ..., d\}$ such that $(|x|)_{\pi(i)} \geq (|x|)_{\pi(i+1)}$ for $i = 1, ..., d-1$. For TopK and RandK compressors, we have $\delta \geq \nicefrac{K}{d}$ \citep{Stich18}.
The {\em unbiased compressor} is also frequently used in compression algorithms, which is defined as a randomized map $\tilde{Q}:\mathbb{R}^d\to \mathbb{R}^d$, where there exists a constant $\omega \geq 0$ such that $\mathbb{E}[{\tilde Q}(x)] = x$, and
\begin{equation} \label{eq:unbiased}
\mathbb{E} \left[\|{\tilde Q}(x)\|^2 \right] \leq (\omega + 1)\|x\|^2, \qquad \forall x\in \mathbb{R}^d.
\end{equation}
Some frequently used unbiased compressors include random dithering \citep{Alistarh17}, random sparsification \citep{Stich18}, and natural compression \citep{horvath2019natural}. For any ${\tilde Q}$ satisfying \eqref{eq:unbiased}, $\frac{1}{\omega+1}{\tilde Q}$ is a contraction compressor satisfying \eqref{eq:contractor} with $\delta = \nicefrac{1}{(\omega+1)}$\citep{biased2020}. Furthermore, unbiased compressors and contraction compressors can be composed to generate new contraction compressors \citep{ecsdca}.
\begin{table*}[h]
\caption{Communication Complexity Results for Different Error-Compensated Algorithms ($r_Q$ represents the communication cost of the compressed vector $Q(x)$ for $x\in \mathbb{R}^d$. For simplicity, we choose $Q=Q_1$, and assume $L_f \geq \lambda$, $\nicefrac{R^2}{\gamma} \geq \lambda$, where $R$ is defined in Algorithm \ref{alg:ec-spdc}, hence the term $\nicefrac{1}{\delta}$ is omitted.) }
\label{tab:summary}
\begin{center}
{\footnotesize
\begin{tabular}{|c|c|c|}
\hline
Algorithm & \begin{tabular}{c} Communication complexity \\ when $\delta \leq \nicefrac{1}{m}$ \end{tabular}& \begin{tabular}{c} Communication complexity under \\ Assumption \ref{as:expcompressor} when $\delta \leq \nicefrac{1}{m}$ \end{tabular}\\
\hline
\begin{tabular}{c}
EC-LSVRG \\
Smooth Case \citep{ecsdca}
\end{tabular} &
${\cal O} \left( \tfrac{r_Q}{\delta} \tfrac{\sqrt{L_f {\bar L}}}{ \lambda} \log \tfrac{1}{\epsilon} \right)$ & ${\cal O} \left( \tfrac{r_Q}{\delta} \tfrac{L_f}{ \lambda} \log \tfrac{1}{\epsilon} \right)$ \\
\hline
\begin{tabular}{c}
EC-SDCA \\
\citep{ecsdca}
\end{tabular} &
${\cal O} \left( \tfrac{r_Q}{\delta} \tfrac{R{\bar R}}{ \lambda \gamma} \log \tfrac{1}{\epsilon} \right)$ & ${\cal O} \left( \tfrac{r_Q}{\delta} \tfrac{R^2}{ \lambda \gamma} \log \tfrac{1}{\epsilon} \right)$ \\
\hline
\begin{tabular}{c}
ECLK \\
\citep{eclk}
\end{tabular} &
${\cal O} \left( \tfrac{r_Q}{\delta \sqrt{\delta}} \sqrt{\tfrac{\bar L}{\lambda}} \log \tfrac{1}{\epsilon} \right)$ & ${\cal O} \left( \tfrac{r_Q}{\delta \sqrt{\delta}} \sqrt{\tfrac{L_f}{\lambda}} \log \tfrac{1}{\epsilon} \right)$ \\
\hline
\begin{tabular}{c}
ECSPDC\\
{\bf This work}
\end{tabular} &
${\cal O} \left( \tfrac{r_Q}{\delta^2 \sqrt{m}} \sqrt{\tfrac{{\bar R}^2}{\lambda \gamma}} \log \tfrac{1}{\epsilon} \right)$ & ${\cal O} \left( \tfrac{r_Q}{\delta^2 \sqrt{m}} \sqrt{\tfrac{R^2}{\lambda \gamma}} \log \tfrac{1}{\epsilon} \right)$ \\
\hline
\begin{tabular}{c}
EC-LSVRG + Catalyst\\
Smooth Case \ \ {\bf This work}
\end{tabular} &
${\tilde {\cal O}} \left( \tfrac{r_Q}{\delta} \sqrt{\tfrac{\bar L}{\lambda}} \log \tfrac{1}{\epsilon} \right)$ & ${\tilde {\cal O}} \left( \tfrac{r_Q}{\delta} \sqrt{\tfrac{L_f}{\lambda}} \log \tfrac{1}{\epsilon} \right)$ \\
\hline
\begin{tabular}{c}
EC-SDCA + Catalyst\\
{\bf This work}
\end{tabular} &
${\tilde {\cal O}} \left( \tfrac{r_Q}{\delta} \sqrt{\tfrac{{\bar R}^2}{\lambda \gamma}} \log \tfrac{1}{\epsilon} \right)$ & ${\tilde {\cal O}} \left( \tfrac{r_Q}{\delta} \sqrt{\tfrac{R^2}{\lambda \gamma}} \log \tfrac{1}{\epsilon} \right)$ \\
\hline
\end{tabular}
}
\end{center}
\end{table*}
\subsection{Motivation}
{\bf Communication Complexity of ECLK.} There are two contraction compressors $Q$ and $Q_1$ in ECLK \citep{eclk} with parameter $\delta$ and $\delta_1$ respectively. We first claim that when $Q$ and $Q_1$ in ECLK are the same type of contraction compressor, but with possibly different compressor parameters (for example, $Q$ and $Q_1$ are both TopK, but with different values of $K$), we could always choose the same compressor parameters for $Q$ and $Q_1$ such that the total communication complexity is less than before or remains the same order as before.
First, from the iteration complexity results for ECLK, it is easy to verify that the iteration complexity will decrease as $\delta$ or $\delta_1$ increases. Without less of generality, we assume the communication cost of $Q(x)$ is higher than that of $Q_1(x)$. Since $Q$ and $Q_1$ are the same type of compressor, we will have $\delta_1 \leq \delta$. Then we can change $Q_1$ to be $Q$. In this way, the total communication cost of $Q(x)$ and $Q_1(x)$ at each iteration is at most twice as before, but $\delta_1$ will increase to $\delta$, which implies that the iteration complexity will decrease and the communication complexity is at most twice as before. Thus, for simplicity, we consider $Q=Q_1$ for ECLK.
{\bf Dependence on $\delta$ for the Iteration Complexity of ECLK.} We introduce the following assumption for Problem (\ref{primal-LSVRG}).
\begin{assumption}\label{as:primal}
$\tfrac{1}{n} \sum_{\tau=1}^n f^{(\tau)}$ is $L_f$-smooth, $f^{(\tau)}$ is ${\bar L}$-smooth, $f_i^{(\tau)}$ is $L$-smooth, and $\psi$ is $\lambda$-strongly convex.
\end{assumption}
Under Assumption \ref{as:primal}, from Theorem 3.8 in \citep{eclk}, the iteration complexity is
$$
{\cal O} \left( \left( \tfrac{1}{\delta} + \tfrac{1}{\delta_1} + \tfrac{1}{p} + \sqrt{\tfrac{L_f}{\lambda}} + \sqrt{\tfrac{{\cal L}_2}{\lambda p}} \right) \log\tfrac{1}{\epsilon} \right),
$$
where $${\cal L}_2 = \tfrac{6L}{n} + \tfrac{112(1-\delta) {\bar L}}{3\delta^2} + \tfrac{28(1-\delta) L}{3\delta} + \tfrac{224(1-\delta) {\bar L}p}{\delta^2 \delta_1} \left( 1 + \tfrac{2p}{\delta_1} \right)$$ and $p\in(0, 1]$ is the update frequency of the check point. Considering $\delta_1=\delta$, it is easy to see that the iteration complexity of ECLK is at least ${\cal O} \left( \tfrac{\sqrt{1-\delta}}{\delta\sqrt{\delta}} \sqrt{\tfrac{\bar L}{\lambda}} \log\tfrac{1}{\epsilon} \right)$. Hence, when $1-\delta = \Theta(1)$, the dependence on $\delta$ of the communication complexity of ECLK would be $\nicefrac{1}{\delta^{\frac{3}{2}}}$, which is worse than EC-LSVRG in the smooth case and EC-SDCA in the composite case \citep{ecsdca}, where the dependence on $\delta$ is $\nicefrac{1}{\delta}$ only. This leads to the following question:
\begin{quote} {\em Can we design provably accelerated gradient-type methods that work with contractive compressors and the dependence on the compressor parameter $\delta$ is $\nicefrac{1}{\delta}$. }
\end{quote}
Let us first recall the results for the error-compensated non-accelerated methods. In the composite case, the dependence on the compressor parameter $\delta$ of EC-SDCA is better than that of EC-LSVRG \citep{ecsdca}. Noticing that L-SVRG \citep{hofmann2015variance,LSVRG} is a primal method and SDCA \citep{prox-SDCA} is a primal-dual method, the better dependence on $\delta$ of EC-SDCA than EC-LSVRG indicates that primal-dual methods may be more suitable for the error feedback mechanism. Therefore, it is natural to apply error feedback to SPDC \citep{SPDC}, which is an accelerated primal-dual algorithm, and expect better dependence on the compressor parameter than ECLK. We first propose error-compensated SPDC (Algorithm \ref{alg:ec-spdc}), but unfortunately, we show that the dependence on $\delta$ of error-compensated SPDC is at least $\nicefrac{1}{\delta^{\frac{3}{2}}}$. This fact makes us to consider the indirect accelerated methods.
In this work, we give a confirmed answer to the above question by applying Catalyst \citep{catalyst}, which is a generic method for accelerating first-order algorithms in the sense of Nesterov, to non-accelerated error-compensated methods, where the dependence on $\delta$ of the communication complexity could be ${\tilde {\cal O}}\left( \nicefrac{1}{\delta} \right)$. Here ${\tilde {\cal O}}$ hides some logarithmic terms.
\subsection{Contributions}
1, First, we propose the error-compensated SPDC (ECSPDC), which is a combination of the error feedback machanism and SPDC \citep{SPDC}, and achieve the accelerated linear convergence rate. In the special case where $\delta=1$, ECSPDC is also an extension of SPDC in the sense that $A_{i\tau}$ in problem (\ref{primal-spdc}) is a matrix rather than a vector, and the convergence rate is actually better than SPDC. Specifically, the convergence result in \citep{SPDC} does not achieve linear speed up with respect to the number of nodes, while ours can obtain linear speed up when the number of nodes is in a certain range.
\noindent 2, We apply Catalyst \citep{catalyst} to EC-LSVRG in the smooth case and EC-SDCA in the composite case \citep{ecsdca}, respectively. The accelerated linear convergence rates are obtained for both cases, and the dependence on $\delta$ of the communication complexities is ${\tilde {\cal O}}(\nicefrac{1}{\delta})$, which matches the best dependence on the compressor parameter in non-accelerated error-compensated methods up to logarithmic terms. The communication complexities of them are summarized in Table \ref{tab:ECLSVRG+C} and Table \ref{tab:ECSDCA+C} in the Appendix, and the comparison of the communication complexity results of different error-compensated algorithms when $\delta \leq \nicefrac{1}{m}$ are summarized in Table \ref{tab:summary}.
\section{ERROR COMPENSATED SPDC}
For primal-dual methods, the following problem is usually studied:
\begin{equation}\label{primal-spdc}
\min_{x\in \mathbb{R}^d} P(x) \; { := }\; \tfrac{1}{N} \sum_{\tau=1}^n \sum_{i=1}^m \phi_{i \tau}(A_{i\tau}^\top x) + g(x),
\end{equation}
where $N = mn$ and $A_{i\tau} \in \mathbb{R}^{d\times t}$. Problem (\ref{primal-spdc}) is actually equivalent to Problem (\ref{primal-LSVRG}). First, by choosing $f_i^{(\tau)}(x) = \phi_{i \tau}(A_{i\tau}^\top x)$ and $\psi=g$, Problem (\ref{primal-spdc}) is a special case of Problem (\ref{primal-LSVRG}). On the other hand, by choosing $A_{i\tau}$ to be the identity matrix, $\phi_{i\tau} = f_i^{(\tau)}$ , and $g=\psi$, Problem (\ref{primal-spdc}) becomes Problem (\ref{primal-LSVRG}). For simplicity, we assume $L_f = \nicefrac{R^2}{\gamma}$, ${\bar L} = \nicefrac{{\bar R}^2}{\gamma}$, and $L = \nicefrac{R_m^2}{\gamma}$, where $R^2$, ${\bar R}^2$, and $R_m^2$ are defined in Algorithm \ref{alg:ec-spdc}. To save space, we only list the assumptions and main results here. The rest can be found in the Appendix.
\begin{assumption}\label{as:contracQQ1-ecSPDC}
The two compressors $Q$ and $Q_1$ are contraction compressors with parameters $\delta$ and $\delta_1$, respectively.
\end{assumption}
\begin{assumption}\label{as:ecSPDC}
Each $\phi_{i\tau} : \mathbb{R}^t \to \mathbb{R}$ is convex and $\nicefrac{1}{\gamma}$-smooth. The regularizer $g : \mathbb{R}^d \to \mathbb{R}$ is $\lambda$-strongly convex.
\end{assumption}
\noindent Sometimes, we will use the following assumption on the contraction compressor to get better results.
\begin{assumption}\label{as:expcompressor}
$\mathbb{E}[Q(x)] = \delta x$ and $\mathbb{E}[Q_1(x)] = \delta_1 x$.
\end{assumption}
Under Assumption \ref{as:contracQQ1-ecSPDC} and Assumption \ref{as:ecSPDC} , the iteration complexity of ECSPDC is
$$
{\cal O} \left( \left( \tfrac{1}{\delta} + \tfrac{1}{\delta_1} + m + {\cal R}_2 \sqrt{\tfrac{m}{\lambda \gamma}} \right) \log \tfrac{1}{\epsilon} \right),
$$
where
\begin{align*}
{\cal R}_2^2 = 2R^2 + \tfrac{2R_m^2}{n} \quad + \tfrac{3(1-\delta)}{4} \left( \tfrac{14{\bar R}^2}{\delta^2} + \tfrac{7R_m^2}{2\delta} + \tfrac{84(1-\delta_1){\bar R}^2}{\delta^2 \delta_1^2 m^2} + \tfrac{42R_m^2}{\delta^2 \delta_1m^2} \right).
\end{align*}
If Assumption \ref{as:expcompressor} is further invoked, the iteration complexity is improved to
$
{\cal O} \left( \left( \tfrac{1}{\delta} + \tfrac{1}{\delta_1} + m + {\cal R}_3 \sqrt{\tfrac{m}{\lambda \gamma}} \right) \log \tfrac{1}{\epsilon} \right),
$
where
\begin{align*}
{\cal R}_3^2 = 2R^2 + \tfrac{2R_m^2}{n} + \tfrac{21(1-\delta)}{4} \left( \tfrac{2R^2}{\delta^2} + \tfrac{11R_m^2}{2\delta n} + \tfrac{12(1-\delta){\bar R}^2}{\delta^2 n} \tfrac{12R^2}{5\delta^2 \delta_1^2 m^2} + \tfrac{228R_m^2}{5\delta^2 \delta_1m^2 n} + \tfrac{432(1-\delta_1){\bar R}^2}{5\delta^2 \delta_1^2m^2 n} \right).
\end{align*}
{\bf Comparison to SPDC.} If there is no compression in ECSPDC, i.e., $\delta=\delta_1=1$, the iteration complexity becomes
$$
{\cal O} \left( \left( \tfrac{N}{n} + \tfrac{(\sqrt{n}R + R_m)}{n} \sqrt{\tfrac{N}{\lambda \gamma}} \right) \log \tfrac{1}{\epsilon} \right),
$$
which is better than that of SPDC obtained in \citep{SPDC}: ${\cal O} \left( \left( \tfrac{N}{n} + R_m \sqrt{\tfrac{N}{n \lambda \gamma}} \right) \log \tfrac{1}{\epsilon} \right)$. Moreover, our result achieves linear speed up with repect to $n$ when $n \leq \nicefrac{R_m^2}{R^2}$.
{\bf Dependence on $\delta$.} Consider $Q=Q_1$ in ECSPDC. When $\nicefrac{1}{m} \leq \delta$, the iteration complexity is at least ${\cal O} \left( \tfrac{\bar R}{\delta} \sqrt{\tfrac{m}{\lambda \gamma}} \log \tfrac{1}{\epsilon} \right) \geq {\cal O} \left( \tfrac{1}{\delta\sqrt{\delta}} \tfrac{\bar R}{\sqrt{\lambda \gamma}} \log \tfrac{1}{\epsilon} \right)$. When $\delta \leq \nicefrac{1}{m}$, we have $\delta \leq \nicefrac{{\bar R}^2}{R_m^2}$. Then the iteration complexity becomes $${\cal O}\left( \left( \tfrac{1}{\delta} + \tfrac{1}{\delta^2 \sqrt{m}} \tfrac{\bar R}{\sqrt{\lambda\gamma}} \right) \log \tfrac{1}{\epsilon} \right) \geq {\cal O}\left( \tfrac{1}{\delta \sqrt{\delta}} \tfrac{\bar R}{\sqrt{\lambda\gamma}} \log \tfrac{1}{\epsilon} \right). $$
Hence, the dependence of ECSPDC on $\delta$ is at least $\nicefrac{1}{\delta^{\frac{3}{2}}}$.
\section{EC-LSVRG + CATALYST IN THE SMOOTH CASE}
EC-LSVRG \citep{ecsdca} is a combination of L-SVRG and error feedback, and the iteration complexity has the better dependence on the compressor parameter in the smooth case than that in the non-smooth case. In this section, we apply Catalyst to EC-LSVRG in the smooth case. First, we restate the Catalyst algorithm and convergence result as follows.
\begin{algorithm}[h!]
\caption{Catalyst}
\label{alg:catalyst}
\begin{algorithmic}
\STATE {\bfseries Parameters:} $\kappa\geq 0$, $\alpha_0$, sequence $\{ \epsilon_k \}_{k\geq 0}$
\STATE {\bfseries Initialization:}
$y^0 = x^0\in \mathbb{R}^d$; $q = \lambda/(\lambda+\kappa)$
\STATE {\bf for} {$k = 1, 2, 3, ...$} {\bf do}
\STATE ~~~ Find an approximate solution of the following problem
\begin{align*}
x^k \approx & \arg\min_{x\in \mathbb{R}^d} \left\{ G_k(x) \; { := }\; P(x) + \tfrac{\kappa}{2}\|x-y^{k-1}\|^2 \right\} \\
& {\rm such \ that} \ \ G_k(x^k) - G_k^* \leq \epsilon_k
\end{align*}
\STATE ~~~ Compute $\alpha_k \in (0,1)$ from equation $\alpha_k^2 = (1-\alpha_k)\alpha_{k-1}^2 + q\alpha_k$
\STATE ~~~ Compute
$$
y^k = x^k + \beta_k(x^k - x^{k-1}) \ \
{\rm with} \ \ \beta_k = \tfrac{\alpha_{k-1}(1-\alpha_{k-1})}{\alpha_{k-1}^2 + \alpha_k}$$
\STATE {\bf end for}
\end{algorithmic}
\end{algorithm}
\begin{theorem}\label{th:catalyst}[\citealp{catalyst}]
Choose $\alpha_0 = \sqrt{q}$ with $q = \nicefrac{\lambda}{(\lambda+\kappa)}$ and
$$
\epsilon_k = \tfrac{2}{9} (P(x^0) - P^*) (1 - \rho_0)^k \ \ {\rm with} \ \ \rho_0 < \sqrt{q}.
$$
Then, Algorithm \ref{alg:catalyst} generates iterates $\{ x^k \}_{k\geq 0}$ such that
\begin{equation}\label{eq:Catalyst}
P(x^k) - P^* \leq C (1-\rho_0)^{k+1} (P(x^0) - P^*).
\end{equation}
with $C = \tfrac{8}{(\sqrt{q} - \rho_0)^2}$.
\end{theorem}
\noindent In Catalyst (Algorithm \ref{alg:catalyst}), $G_k^*$ represents the minimum of $G_k$. In Theorem \ref{th:catalyst}, $P^*$ is the minimum of $P$, and as discussed in \citep{catalyst}, the term $P(x^0)-P^*$ in $\epsilon_k$ can be replaced by its upper bound, which only affects the corresponding constant in (\ref{eq:Catalyst}).
We use EC-LSVRG to solve the subproblem in Catalyst for the smooth case where $\psi$ is smooth in Problem (\ref{primal-LSVRG}). The main challenge is proposing suitable initial conditions for the subproblem and estimate the corresponding expected inner iteration number.
To save space, we restate EC-LSVRG (and also EC-SDCA) in the Appendix. It should be noticed that EC-LSVRG in the smooth case is applied to the problem without the regularizer term. Thus, to minimize $G_k$, we move $\psi$ and the quadratic term $\frac{\kappa}{2} \|x-y^{k-1}\|^2$ to each $f_i^{(\tau)}$. We use subscript $(k)$ and superscript $K$ to denote the variables at the $k$-th outer iteration and $K$-th inner iteration (for example, $x_{(k)}^K$, ${\bar x}_{(k)}^K$, $x_{(k)}^*$, $e_{\tau, (k)}^K$, and $h_{\tau, (k)}^K$).
In \citep{catalyst}, the Catalyst acceleration was applied to the first-order methods whose convergence rate has the following form
\begin{equation}\label{eq:Catalystfirst}
G_k(z_t) - G_k^* \leq A (1 - \theta)^t (G_k(z^0) - G_k^*),
\end{equation}
where $A$ is some constant. If we initial $h_{\tau, (k)}^0$ by the gradient of $f_i^{(\tau)} + \psi + \frac{\kappa}{2}\|\cdot - y^{k-1}\|^2$ at $x_{(k)}^0$. Then the form of the convergence rate of EC-LSVRG becomes form (\ref{eq:Catalystfirst}), and we can get the following lemma.
\begin{lemma}\label{lm:eclsvrg-1}
Under Assumptions \ref{as:primal}, \ref{as:contracQQ1-ecSPDC} and the premise of Theorem \ref{th:catalyst}, let us run EC-LSVRG (Algorithm \ref{alg:ec-lsvrg}) to minimize $G_k$ and output $x^k \; { := }\; {\bar x}_{(k)}^{T_k}$, where
$
T_k \; { := }\; \inf \{ K\geq 1, G_k({\bar x}_{(k)}^{K}) - G_k^* \leq \epsilon_k \}
$.
For the initialization of EC-LSVRG at the $k$-th outer iteration, we choose $p=\Theta(\delta_1)$, $x_{(k)}^0 = x^{k-1}$, $e_{\tau, (k)}^0 = 0$ and $h_{\tau, (k)}^0 = \nabla f^{(\tau)}(x^0_{(k)}) + \nabla \psi(x^0_{(k)}) + \kappa (x^0_{(k)} - y^{k-1})$. Then
\begin{align*}
\mathbb{E}[T_k] \leq {\tilde {\cal O}}\left( \tfrac{1}{\delta} + \tfrac{1}{\delta_1} + \tfrac{\sqrt{(1-\delta)(L_f+\lambda+\kappa)({\bar L} + \lambda+\kappa)}}{\delta (\lambda+\kappa)} + \tfrac{L_f}{\lambda+\kappa} + \tfrac{L}{n(\lambda+\kappa)} + \tfrac{\sqrt{(1-\delta)(L_f+\lambda+\kappa)(L+\lambda+\kappa)}}{\sqrt{\delta}(\lambda+\kappa)} \right),
\end{align*}
where the notation ${\tilde {\cal O}}$ hides some universal constants and some logarithmic dependencies in $\delta$, $\delta_1$, $\lambda$, $\kappa$, $L_f$, and $N$.
\end{lemma}
\begin{remark}
1, It is easy to verify that an optimal choice of $p$ in EC-LSVRG is $\Theta(\delta_1)$. Hence, we choose $p=\Theta(\delta_1)$ in Lemma \ref{lm:eclsvrg-1} (and also in Lemma \ref{lm:eclsvrg-2}) for simplicity.
2, As discussed in \citep{catalyst}, the stopping criteria in the inner loop can be checked by calculating some upper bound of $G_k({\bar x}_{(k)}^{K}) - G_k^*$, such as the duality gap. However, this would cause additional computation and also communication cost. Hence, we can actually view the inner iteration number as a parameter and use Lemma \ref{lm:eclsvrg-1} as the guidance.
\end{remark}
If we further invoke Assumption \ref{as:expcompressor}, we can get the following lemma. Since the proof is similar to that of Lemma \ref{lm:eclsvrg-1}, we omit it.
\begin{lemma}\label{lm:eclsvrg-2}
Under Assumptions \ref{as:primal}, \ref{as:contracQQ1-ecSPDC}, \ref{as:expcompressor}, and the premise of Theorem \ref{th:catalyst}, let us run EC-LSVRG to minimize $G_k$. Choose the output $x^k$, $T_k$, and the initialization of EC-LSVRG at the $k$-th outer iteration be the same as that in Lemma \ref{lm:eclsvrg-1}. Then
$$
\mathbb{E}[T_k] \leq {\tilde {\cal O}}\left( \tfrac{1}{\delta} + \tfrac{1}{\delta_1} + \tfrac{L_f}{\lambda+\kappa} + \tfrac{L}{n(\lambda+\kappa)} + \tfrac{\sqrt{1-\delta}(L_f + \lambda+\kappa)}{\delta (\lambda+\kappa)} \right).
$$
\end{lemma}
\subsection{Communication Complexity}\label{sec:cm-eclsvrg-ca}
In this subsection, we discuss the total communication cost by using EC-LSVRG + Catalyst. Same as the claim in the discussion of the communication complexity of ECLK, for simplicity, we choose $Q=Q_1$ in EC-LSVRG.
Denote the communication cost of an vector in $\mathbb{R}^d$ as $U_d$ and the communication cost of the compressed vector in $\mathbb{R}^d$ by using the compressor $Q$ as $r_Q$. From Theorem \ref{th:catalyst}, to achieve $P(x^k) - P^* \leq \epsilon$, the outer iteration number is ${ \tilde {\cal O}} \left( \frac{\sqrt{\lambda+\kappa}}{\sqrt{\lambda}} \log\frac{1}{\epsilon} \right)$, and from Lemma \ref{lm:eclsvrg-1}, the expected inner iteration number is
\begin{align*}
& \quad {\tilde {\cal O}} \left( \tfrac{1}{\delta} + \tfrac{L_f+L/n}{\lambda+\kappa} + \tfrac{\sqrt{(1-\delta)(L_f+\lambda+\kappa)({\bar L} + \lambda+\kappa)}}{\delta (\lambda+\kappa)} + \tfrac{\sqrt{(1-\delta)(L_f+\lambda+\kappa)(L+\lambda+\kappa)}}{\sqrt{\delta}(\lambda+\kappa)} \right) \\
& = {\tilde {\cal O}} \left( \tfrac{1}{\delta} + \tfrac{ a_1 }{\lambda+\kappa} + \tfrac{b_1}{ \sqrt{\lambda+\kappa}} \right),
\end{align*}
where we denote $a_1 \; { := }\; L_f + \tfrac{L}{n} + \tfrac{\sqrt{1-\delta}(\sqrt{L_f {\bar L}} + \sqrt{\delta L_f L})}{\delta}$ and $b_1 \; { := }\; \tfrac{\sqrt{1-\delta} \left( \sqrt{{\bar L}} + \sqrt{\delta L} \right)}{\delta}$.
Noticing that at each outer iteration, we need to communicate the uncompressed vector $h_{\tau, (k)}^0$, the expected total communication cost becomes
\begin{align*}
& {\tilde {\cal O}} \left( \left( \tfrac{\sqrt{\lambda+\kappa}}{\sqrt{\lambda}} \left( \tfrac{1}{\delta} + \tfrac{ a_1 }{\lambda+\kappa} + \tfrac{b_1}{ \sqrt{\lambda+\kappa}} \right) r_Q + \tfrac{\sqrt{\lambda+\kappa}}{\sqrt{\lambda}} U_d \right) \log \tfrac{1}{\epsilon} \right) \\
& = {\tilde {\cal O}} \left( \tfrac{r_Q}{\sqrt{\lambda}} \log \tfrac{1}{\epsilon} \left( \left( \tfrac{1}{\delta} + \tfrac{U_d}{r_Q} \right) \sqrt{\lambda + \kappa} + \tfrac{a_1}{\sqrt{\lambda+\kappa}} + b_1 \right) \right).
\end{align*}
{\bf Optimal $\kappa$.} Since $\kappa\geq 0$ in Catalyst, it is easy to get the optimal $\kappa$ for minimizing the expected total communication cost. Let $\lambda_1 \; { := }\; a_1/\left( \frac{1}{\delta} + \frac{U_d}{r_Q} \right)$. If $\lambda \leq \lambda_1$, then the optimal $\kappa$ is $\lambda_1-\lambda$. If $\lambda>\lambda_1$, then the optimal $\kappa$ is $0$. Or equivalently, the optimal $\kappa = \max\{ \lambda_1, \lambda \} - \lambda$.
Similarly, under the additional Assumption \ref{as:expcompressor}, from Theorem \ref{th:catalyst} and Lemma \ref{lm:eclsvrg-2}, the expected total communication cost is
$$
{\tilde {\cal O}} \left( \tfrac{r_Q}{\sqrt{\lambda}} \log \tfrac{1}{\epsilon} \left( \left( \tfrac{1}{\delta} + \tfrac{U_d}{r_Q} \right) \sqrt{\lambda + \kappa} + \tfrac{a_2}{\sqrt{\lambda+\kappa}} \right) \right),
$$
where $a_2 \; { := }\; L_f + \tfrac{L}{n} + \tfrac{\sqrt{1-\delta}L_f}{\delta}$. Let $\lambda_2 \; { := }\; a_2/\left( \frac{1}{\delta} + \frac{U_d}{r_Q} \right)$. Then the optimal $\kappa = \max\{ \lambda_2, \lambda \} - \lambda$.
\noindent For TopK, if we use 64 bits for each element in $\mathbb{R}^d$,
$
\tfrac{U_d}{r_Q} = \tfrac{64d}{(64+\log d)K} = \Theta \left( \tfrac{d}{K \log d} \right).
$
Even though the theoretical $\delta$ for TopK is $\nicefrac{K}{d}$, the actual value could be much larger than $\nicefrac{K}{d}$ in practice. Then $\nicefrac{U_d}{r_Q}$ may not be able to be bounded by ${\cal O}(\nicefrac{1}{\delta})$, and thus the communication complexity may be even worse than ECLK and ECSPDC.
\subsection{Remove the Dependence on $\nicefrac{U_d}{r_Q}$}
Due to the communication of uncompressed vectors at each outer iteration of the stratergies in Lemmas \ref{lm:eclsvrg-1} and \ref{lm:eclsvrg-2}, the expected total communication complexities depend on $\nicefrac{U_d}{r_Q}$, which may be much larger than $\nicefrac{1}{\delta}$. In this subsection, we show that we can actually remove the dependence on $\nicefrac{U_d}{r_Q}$ by communicating the compressed vector only. The initialization procedures and estimations of the expected inner iteration number are states in the following two lemmas.
\begin{lemma}\label{lm:eclsvrg-1-re}
Under Assumptions \ref{as:primal}, \ref{as:contracQQ1-ecSPDC}, and the premise of Theorem \ref{th:catalyst}, let us run EC-LSVRG to minimize $G_k$ and output $x^k \; { := }\; {x}_{(k)}^{T_k}$, $h_{\tau, (k)}^{T_k}$, and $e_{\tau, (k)}^{T_k}$, where
$
T_k \; { := }\; \inf \{ K\geq 1, \Phi_{3, (k)}^K + G_k({x}_{(k)}^{K}) - G_k^* \leq \epsilon_k \}
$.
For the initialization of EC-LSVRG at the $k$-th outer iteration, we choose $p=\Theta(\delta_1)$, $x_{(k)}^0 = x^{k-1}$, $e_{\tau, (k)}^0 = 0$ or $e_{\tau, (k-1)}^{T_{k-1}}$, and $h_{\tau, (k)}^0 = h_{\tau, (k-1)}^{T_{k-1}}$ or $h_{\tau, (k-1)}^{T_{k-1}} + \kappa(y^{k-2}-y^{k-1})$ ($y^{-1}=y^0$). Then
\begin{align*}
\mathbb{E}[T_k] & \leq {\tilde {\cal O}}\left( \tfrac{1}{\delta} + \tfrac{1}{\delta_1} + \tfrac{\sqrt{(1-\delta)(L_f+\lambda+\kappa)({\bar L} + \lambda+\kappa)}}{\delta (\lambda+\kappa)} + \tfrac{L_f}{\lambda+\kappa} + \tfrac{L}{n(\lambda+\kappa)} + \tfrac{\sqrt{(1-\delta)(L_f+\lambda+\kappa)(L+\lambda+\kappa)}}{\sqrt{\delta}(\lambda+\kappa)} \right),
\end{align*}
where the notation ${\tilde {\cal O}}$ hides some universal constants and some logarithmic dependencies in $\delta$, $\delta_1$, $\lambda$, $\kappa$, $L_f$, and $N$.
\end{lemma}
\begin{lemma}\label{lm:eclsvrg-2-re}
Under Assumptions \ref{as:primal}, \ref{as:contracQQ1-ecSPDC}, \ref{as:expcompressor}, and the premise of Theorem \ref{th:catalyst}, let us run EC-LSVRG to minimize $G_k$ and output $x^k \; { := }\; {x}_{(k)}^{T_k}$, $h_{\tau, (k)}^{T_k}$, and $e_{\tau, (k)}^{T_k}$, where
$
T_k \; { := }\; \inf \{ K\geq 1, \Phi_{4, (k)}^K + G_k({x}_{(k)}^{K}) - G_k^* \leq \epsilon_k \}
$.
Choose the initialization of EC-LSVRG at the $k$-th outer iteration be the same as that in Lemma \ref{lm:eclsvrg-1-re}. Then
$
\mathbb{E}[T_k] \leq {\tilde {\cal O}}\left( \tfrac{1}{\delta} + \tfrac{1}{\delta_1} + \tfrac{L_f}{\lambda+\kappa} + \tfrac{L}{n(\lambda+\kappa)} + \tfrac{\sqrt{(1-\delta)}(L_f + \lambda+\kappa)}{\delta (\lambda+\kappa)} \right).
$
\end{lemma}
{\bf Communication Complexity.} Same as the analysis in Section \ref{sec:cm-eclsvrg-ca}, the expected total communication cost of EC-LSVRG + Catalyst with the output and initialization precedures in Lemmas \ref{lm:eclsvrg-1-re} and \ref{lm:eclsvrg-2-re} can be obtained by simply replacing $\nicefrac{U_d}{r_Q}$ with $0$. It is evident that the communication complexity depends on $\nicefrac{1}{\delta}$ only up to logarithmic terms. In particular, if $1-\delta = \Theta(1)$, $\delta \leq \min \{ \nicefrac{\bar L}{L}, \nicefrac{n^2L_f}{L} \}$ and $L_f \geq \lambda$, then an optimal $\kappa$ is $\sqrt{L_f {\bar L}} - \lambda$, and the corresponding communication complexity is ${\tilde {\cal O}} \left( \tfrac{r_Q}{\delta} \sqrt{\tfrac{\bar L}{\lambda}} \log \tfrac{1}{\epsilon} \right)$. If Assumption \ref{as:expcompressor} is further invoked, when $1-\delta = \Theta(1)$, $\delta \leq \nicefrac{nL_f}{L}$, and $L_f \geq \lambda$, an optimal $\kappa$ is $L_f-\lambda$, and the corresponding communication complexity is ${\tilde {\cal O}} \left( \tfrac{r_Q}{\delta} \sqrt{\tfrac{L_f}{\lambda}} \log \tfrac{1}{\epsilon} \right)$.
\section{EC-SDCA + CATALYST}
In this section, we consider Problem (\ref{primal-spdc}). Let $\xi \; { := }\; \frac{1}{\lambda}g$. Then $\xi$ is 1-strongly convex if $g$ is $\lambda$-strongly convex. We apply the catalyst to problem (\ref{primal-spdc}), and for the subproblem, we use the error-compensated SDCA (Algorithm \ref{alg:ec-sdca}) in \citep{ecsdca} to solve it. At the $k$-th outer iteration, we use EC-SDCA to minimize $G_k(x) \; { := }\; P(x) + \frac{\kappa}{2}\|x-y^{k-1}\|^2$, and we also use subscript $(k)$ and superscript $K$ to denote the variables at the $k$-th outer iteration and $K$-th inner iteration (for instance, $x^K_{(k)}$, $\alpha^K_{(k)}$, $e^K_{\tau, (k)}$, $e^K_{(k)}$, and $u^K_{(k)}$).
To apply EC-SDCA at the $k$-th outer iteration in Algorithm \ref{alg:catalyst}, we need to initialize $\alpha_{i \tau, (k)}^0$. It is natural to use the values of $\alpha_{i\tau}$ in the last inner loop to initialize $\alpha_{i \tau, (k)}^0$, and this is indeed the case in \citep{accSDCA}, where the accelerated SDCA was studied. Then in order to initialize $u^0_{(k)} = \tfrac{1}{(\lambda+\kappa)N} \sum_{\tau=1}^n \sum_{i=1}^m A_{i\tau} \alpha_{i\tau, (k)}^0$, the uncompressed vector $A_{i\tau} \alpha_{i\tau, (k)}^0$ need to be communicated. We state the initialization procedures formally and estimate the expected inner iteration number in the next two lemmas.
\begin{lemma}\label{lm:ecsdca-1}
Assume $\delta<1$. Under Assumptions \ref{as:contracQQ1-ecSPDC}, \ref{as:ecSPDC}, and the premise of Theorem \ref{th:catalyst}, let us run EC-SDCA (Algorithm \ref{alg:ec-sdca}) to minimize $G_k$ and output $(x^k, \alpha^k) \; { := }\; (x_{(k)}^{T_k+1}, \alpha_{(k)}^{T_k})$, where
$
T_k \; { := }\; \inf \{ K\geq 1, \sqrt{4n+\delta mn} \Psi^K_{3,(k)} + 2(G_k(x_{(k)}^{K+1}) - G_k^*) \leq \epsilon_k \}
$.
For the initialization of EC-SDCA at the $k$-th iteration, we choose $\alpha_{(k)}^0 = \alpha^{k-1}$ ($\alpha^0=0$) and $e_{\tau, (k)}^0 = 0$. Then
$$
\mathbb{E}[T_k] \leq {\tilde {\cal O}}\left( \tfrac{1}{\delta} + m + \tfrac{a_3}{\lambda+\kappa} + \tfrac{b_3}{\sqrt{\lambda+\kappa}} \right),
$$
where $a_3 \; { := }\; \frac{R_m^2}{n \gamma} + \frac{R^2}{\gamma} + \frac{\sqrt{1-\delta}R{\bar R}}{\delta \gamma} + \frac{\sqrt{1-\delta}RR_m}{\sqrt{\delta}\gamma}$, $b_3 \; { := }\; \frac{1}{\delta} \sqrt{\frac{(1-\delta)({\bar R}^2 + \delta R_m^2)}{\gamma}}$ and the notation ${\tilde {\cal O}}$ hides some universal constants and some logarithmic dependencies in $\delta$, $\lambda$, $\kappa$, $R$, and $N$.
\end{lemma}
\begin{remark}
In EC-SDCA, $\nicefrac{R^2}{\gamma} \geq \lambda+\kappa$ is assumed. However, by adding the term $ \frac{b_3}{\sqrt{\lambda+\kappa}} \log \frac{1}{\epsilon}$ to the iteration complexity, the assumption $\nicefrac{R^2}{\gamma} \geq \lambda+\kappa$ is no longer needed, which can be seen easily from the proof of Theorem 3.3 in \citep{ecsdca}.
\end{remark}
If we further invoke Assumption \ref{as:expcompressor} on the compressors in EC-SDCA, we can get the following better result. The proof is similar to that of Lemma \ref{lm:ecsdca-1}, thus we omit it.
\begin{lemma}\label{lm:ecsdca-2}
Assume $\delta<1$. Under Assumptions \ref{as:contracQQ1-ecSPDC}, \ref{as:ecSPDC}, \ref{as:expcompressor}, and the premise of Theorem \ref{th:catalyst}, let us run EC-SDCA to minimize $G_k$ and output $(x^k, \alpha^k) \; { := }\; (x_{(k)}^{T_k+1}, \alpha_{(k)}^{T_k})$, where
$
T_k \; { := }\; \inf \{ K\geq 1, 3\sqrt{2+\delta m} \Psi^K_{4,(k)} + 2(G_k(x_{(k)}^{K+1}) - G_k^*) \leq \epsilon_k \}
$.
For the initialization of EC-SDCA at the $k$-th iteration, we choose $\alpha_{(k)}^0 = \alpha^{k-1}$ ($\alpha^0=0$) and $e_{\tau, (k)}^0 = 0$. Then $\mathbb{E}[T_k] \leq {\tilde {\cal O}}\left( \tfrac{1}{\delta} + m + \tfrac{a_4}{\lambda+\kappa} \right), $ where $a_4 \; { := }\; \frac{R_m^2}{n \gamma} + \frac{R^2}{\gamma} + \frac{\sqrt{1-\delta}R^2}{\delta \gamma} $.
\end{lemma}
\subsection{Communication Complexity}\label{sec:cm-ecsdca-ca}
In this subsection, we discuss the total communication cost by using EC-SDCA + Catalyst. From Theorem \ref{th:catalyst}, to get $P(x^k) - P^* \leq \epsilon$, the outer iteration number is ${\tilde {\cal O}} \left( \frac{\sqrt{\lambda+\kappa}}{\sqrt{\lambda}} \log\frac{1}{\epsilon} \right)$, and from Lemma \ref{lm:ecsdca-1}, the expected inner iteration number is ${\tilde {\cal O}} \left( \tfrac{1}{\delta} + m + \tfrac{a_3}{\lambda+\kappa} + \tfrac{b_3}{\sqrt{\lambda+\kappa}} \right)$. Noticing that at each outer iteration, we need to communicate the uncompressed vector to initialize $u^0_{(k)}$, the expected total communication cost is
\begin{align*}
& \quad {\tilde {\cal O}} \left( \left( \tfrac{\sqrt{\lambda+\kappa}}{\sqrt{\lambda}} \left( \tfrac{1}{\delta} + m + \tfrac{a_3}{\lambda+\kappa} + \tfrac{b_3}{\sqrt{\lambda+\kappa}} \right) r_Q + \tfrac{\sqrt{\lambda+\kappa}}{\sqrt{\lambda}} U_d \right) \log \tfrac{1}{\epsilon} \right) \\
&= {\tilde {\cal O}} \left( \tfrac{r_Q}{\sqrt{\lambda}} \log \tfrac{1}{\epsilon} \left( \left( \tfrac{1+\delta m}{\delta} + \tfrac{U_d}{r_Q} \right) \sqrt{\lambda + \kappa} + \tfrac{a_3}{\sqrt{\lambda+\kappa}} + b_3 \right) \right).
\end{align*}
{\bf Optimal $\kappa$.} Since $\lambda+\kappa \geq \lambda$, it is easy to obtain the optimal $\kappa$ for minimizing the expected total communication cost. Let $\lambda_3 \; { := }\; {a_3}/ ({\tfrac{1}{\delta} + m + \tfrac{U_d}{r_Q}})$. Then the optimal $\kappa$ is $\max\{\lambda, \lambda_3\} - \lambda$.
Similarly, under the additional Assumption \ref{as:expcompressor}, from Theorem \ref{th:catalyst} and Lemma \ref{lm:ecsdca-2}, the expected total communication cost is
$$
{\tilde {\cal O}} \left( \tfrac{r_Q}{\sqrt{\lambda}} \log \tfrac{1}{\epsilon} \left( \left( \tfrac{1}{\delta} + m + \tfrac{U_d}{r_Q} \right) \sqrt{\lambda + \kappa} + \tfrac{a_4}{\sqrt{\lambda+\kappa}} \right) \right).
$$
Let $\lambda_4 \; { := }\; {a_4}/ ({\tfrac{1}{\delta} + m + \tfrac{U_d}{r_Q}})$. Then the optimal $\kappa$ is $\max\{\lambda, \lambda_4\} - \lambda$.
The term $\nicefrac{U_d}{r_Q}$ also shows up in the expected total communication cost of EC-SDCA + Catalyst. As we analyzed in Section \ref{sec:cm-eclsvrg-ca}, the presence of $\nicefrac{U_d}{r_Q}$ may make the communication complexity worse than ECLK and ECSPDC. In next subsection, we try to remove the dependence on $\nicefrac{U_d}{r_Q}$.
\subsection{Remove the Dependence on $\nicefrac{U_d}{r_Q}$}
As we can see from the analysis of the communication complexity, the term $\nicefrac{U_d}{r_Q}$ shows up because of the communication of uncompressed vectors. Hence, in order to remove the dependence on $\nicefrac{U_d}{r_Q}$, we need to find initialization procedures that do not need the communication of uncompressed vectors. Fortunately, by investigating the proofs of EC-SDCA, we find out that the relation $u^0_{(k)} = \tfrac{1}{(\lambda+\kappa)N} \sum_{\tau=1}^n \sum_{i=1}^m A_{i\tau} \alpha_{i\tau, (k)}^0$ in the initialization is not necessary, and the relation ${\tilde u}^K_{(k)} = \tfrac{1}{(\lambda+\kappa)N} \sum_{\tau=1}^n \sum_{i=1}^m A_{i\tau} \alpha_{i\tau, (k)}^K$ is actually essential in the proofs, and need to be maintained. This leads to the initialization procedures in the next two lemmas, and the communication of uncompressed vectors is actually not needed for the initialization at each outer iteration.
\begin{lemma}\label{lm:ecsdca-1-re}
Assume $\delta<1$. Under Assumptions \ref{as:contracQQ1-ecSPDC}, \ref{as:ecSPDC}, and the premise of Theorem \ref{th:catalyst}, let us run EC-SDCA to minimize $G_k$ and output $x^k \; { := }\; x_{(k)}^{T_k+1}$, $\alpha^k \; { := }\; \alpha_{(k)}^{T_k}$, $u_{(k)}^{T_k}$, and $e_{\tau, (k)}^{T_k}$, where
$
T_k \; { := }\; \inf \{ K\geq 1, \sqrt{4n+\delta mn} \Psi^K_{3,(k)} + 2(G_k(x_{(k)}^{K+1}) - G_k^*) \leq \epsilon_k \}
$.
For the initialization of EC-SDCA at the $k$-th iteration, we choose $\alpha_{(k)}^0 = \alpha^{k-1}$ ($\alpha^0=0$), $u_{(k)}^0 = u_{(k-1)}^{T_{k-1}}$ ($u_{(1)}^0=0$), and $e_{\tau, (k)}^0 = e_{\tau, (k-1)}^{T_{k-1}}$ ($e_{\tau, (1)}^0=0$). Then
$
\mathbb{E}[T_k] \leq {\tilde {\cal O}}\left( \tfrac{1}{\delta} + m + \tfrac{a_3}{\lambda+\kappa} + \tfrac{b_3}{\sqrt{\lambda+\kappa}} \right).
$
\end{lemma}
\begin{lemma}\label{lm:ecsdca-2-re}
Assume $\delta<1$. Under Assumptions \ref{as:contracQQ1-ecSPDC}, \ref{as:ecSPDC}, \ref{as:expcompressor}, and the premise of Theorem \ref{th:catalyst}, let us run EC-SDCA to minimize $G_k$ and output $x^k \; { := }\; x_{(k)}^{T_k+1}$, $\alpha^k \; { := }\; \alpha_{(k)}^{T_k}$, $u_{(k)}^{T_k}$, and $e_{\tau, (k)}^{T_k}$, where
$
T_k \; { := }\; \inf \{ K\geq 1, 3\sqrt{2+\delta m} \Psi^K_{4,(k)} + 2(G_k(x_{(k)}^{K+1}) - G_k^*) \leq \epsilon_k \}
$.
Choose the initialization of EC-SDCA at the $k$-th iteration be the same as that in Lemma \ref{lm:ecsdca-1-re}. Then
$
\mathbb{E}[T_k] \leq {\tilde {\cal O}}\left( \tfrac{1}{\delta} + m + \tfrac{a_4}{\lambda+\kappa} \right).
$
\end{lemma}
{\bf Communication Complexity.} Same as the analysis in Section \ref{sec:cm-ecsdca-ca}, the expected total communication cost of EC-SDCA + Catalyst with the output and initialization precedures in Lemmas \ref{lm:ecsdca-1-re} and \ref{lm:ecsdca-2-re} can be obtained by simply replacing $\nicefrac{U_d}{r_Q}$ with $0$, and only depends on $\nicefrac{1}{\delta}$ up tp logarithmic terms. In particular, if $\delta \leq \nicefrac{1}{m}$ and $\nicefrac{R{\bar R}}{\gamma} \geq \lambda$, then an optimal $\kappa$ is $\nicefrac{R{\bar R}}{\gamma} - \lambda$, and the corresponding communication complexity is ${\tilde {\cal O}} \left( \tfrac{r_Q}{\delta} \sqrt{\tfrac{{\bar R}^2}{\lambda \gamma}} \log \tfrac{1}{\epsilon} \right)$. If Assumption \ref{as:expcompressor} is further invoked, when $\delta \leq \nicefrac{1}{m}$ and $\nicefrac{R^2}{\gamma} \geq \lambda$ an optimal $\kappa$ is $\nicefrac{R^2}{\gamma} - \lambda$, and the corresponding communication complexity is ${\tilde {\cal O}} \left( \tfrac{r_Q}{\delta} \sqrt{\tfrac{R^2}{\lambda \gamma}} \log \tfrac{1}{\epsilon} \right)$.
\begin{figure*}[ht]
\vspace{.05in}
\centerline{\begin{tabular}{ccc}
\includegraphics[width=0.3\linewidth]{figs/a9a_SDCA_Catalystno_comp.pdf}
&\includegraphics[width=0.3\linewidth]{figs/a9a_LSVRG_Catalystno_comp.pdf}
&\includegraphics[width=0.3\linewidth]{figs/a9a_SPDCno_comp.pdf} \\
\end{tabular}
}
\vspace{.3in}
\caption{The Communication Complexity Performance of ECSDCA-Catalyst, ECLSVRG-Catalyst, and ECSPDC Used with Compressors: Top1 VS Random Dithering VS Natural Compression VS No Compression on \dataset{a9a} Data Set}
\label{fig:diff-a9a}
\end{figure*}
\begin{figure*}[ht]
\vspace{.1in}
\centerline{\begin{tabular}{ccc}
\includegraphics[width=0.3\linewidth]{figs/a9a__fullcomparison.pdf}
&\includegraphics[width=0.3\linewidth]{figs/w6a__fullcomparison.pdf}
& \includegraphics[width=0.3\linewidth]{figs/mushrooms__fullcomparison.pdf}
\end{tabular}
}
\vspace{.3in}
\caption{The Communication Complexity Performance of ECSDCA-Catalyst VS ECLSVRG-Catalyst VS ECSPDC VS ECLK for Top1 Compressor on \dataset{a9a}, \dataset{w6a}, and \dataset{mushrooms} Data Sets}
\label{fig:acc}
\end{figure*}
\section{EXPERIMENTS}\label{sec:exp}
In this section, we implement our algorithms on the real world binary logistic regression tasks:
$$
x \mapsto \log \left(1+\exp(-y_i A_i^\top x)\right)+\tfrac{\lambda}{2}\Vert x\Vert^2,
$$
where $A_i,y_i$ are training sample pairs. We use the data sets: \dataset{a9a}, \dataset{w6a}, \dataset{phishing}, and \dataset{mushrooms} from LIBSVM Library \citep{chang2011libsvm}. More experiments can be found in the Appendix.
\vskip 0.2mm
\noindent\textbf{Compressors.} In the experiments, we use Top1 and some contraction compressors transformed by unbiased ones such as random dithering ($s=\sqrt{d}$) and natural compression.
\vskip 0.2mm
\noindent\textbf{Parameters.} We set $\lambda=1\times 10^{-5}$ and $n=20$. For all experiments, we use grid search to obtain the learning rate $\{10^{-t},t = 0,1,2\cdots\}$. For ECSPDC, we use bisect method to obtain the argmax operator, $\theta$ is chosen by Theorem \ref{th:pf-SPDC}. For ECLSVRG and ECLK, we set $Q=Q_1$ and $p=\delta$. For Catalyst, we choose $\kappa$ by grid search $\{10^t\lambda: t\in \mathbb Z\}$. For the stopping criteria of the inner loop, a heuristic strategy was proposed for Catalyst in \citep{catalyst}, where the inner loop is constrained to perform at most $mn$ iterations. We employ this strategy similarly and the inner loop size is searched from $\{kd: k=1,2,5,10,100\}$, where $d$ is the dimension of data.
\subsection{Effectiveness of TopK Compressor}
First, we demonstrate the effectiveness of TopK compressor compared with random dithering, natural compression, and no compression. Figure \ref{fig:diff-a9a} shows that compression can improve the performance with respect to the communication complexity in general, and TopK is specifically effective.
\begin{figure*}[ht]
\vspace{.1in}
\centerline{\begin{tabular}{ccc}
\includegraphics[width=0.3\linewidth]{figs/a9a_LSVRG_top_k.pdf}
&\includegraphics[width=0.3\linewidth]{figs/w6a_LSVRG_top_k.pdf}
&\includegraphics[width=0.3\linewidth]{figs/mushrooms_LSVRG_top_k.pdf}
\end{tabular}
}
\vspace{.3in}
\caption{The Communication Complexity Performance of ECLSVRG VS ECLSVRG-Catalyst for Top1 Compressor on \dataset{a9a}, \dataset{w6a}, and \dataset{mushrooms} Data Sets}
\label{fig:ECLSVRG-catalyst}
\end{figure*}
\begin{figure*}[ht]
\vspace{.1in}
\centerline{\begin{tabular}{ccc}
\includegraphics[width=0.3\linewidth]{figs/a9a_SDCA_top_k.pdf}
&\includegraphics[width=0.3\linewidth]{figs/w6a_SDCA_top_k.pdf}
&\includegraphics[width=0.3\linewidth]{figs/mushrooms_SDCA_top_k.pdf}
\end{tabular}
}
\vspace{.3in}
\caption{The Communication Complexity Performance of EC-SDCA VS ECSDCA-Catalyst for Top1 Compressor on \dataset{a9a}, \dataset{w6a}, and \dataset{mushrooms} Data Sets}
\label{fig:ECSDCA-catalyst}
\end{figure*}
\subsection{Comparison of Different Accelerated Error Compensated Algorithms}
We compare Catalyst-based error-compensated algorithms and ECSPDC with ECLK, and also use the Top1 compressor. Figure \ref{fig:acc} shows that the performance of ECSDCA-Catalyst is the best for our tested data sets, which indicates the potential of the Catalyst-based error-compensated algorithm.
\subsection{Improvements from Catalyst Acceleration}
In this subsection, we compare Catalyst-based error-compensated algorithms with their baselines, namely, ECSDCA and ECLSVRG, where Top1 compressor is used.
Figures \ref{fig:ECLSVRG-catalyst} and \ref{fig:ECSDCA-catalyst} show that Catalyst acceleration can indeed boost the speed of both ECSDCA and ECLSVRG with respect to the communication complexity significantly, which matches our theory.
| {
"redpajama_set_name": "RedPajamaArXiv"
} | 3,795 |
\section{Introduction and main result}
Let $\k$ be a local field of characteristic 0, for which we fix
a non-trivial unitary character $\psi$. Let $G$ be a reductive group
over $\k$, $\mathfrak{g}$ its Lie algebra, on which we fix an
$\operatorname{Ad} G$-invariant non-degenerate bilinear form $\kappa$. Let
$\gamma=\{X,H,Y\}\subset \mathfrak{g}$ be an $\sl_{2}$-triple, that is
\[
[H,X]=2X,\qquad [H,Y]=-2Y, \qquad [X,Y]=H.
\]
Set $\mathfrak{g}_{j}=\set{Z\in \mathfrak{g}}{\operatorname{ad}(H)Z=jZ}$, for $j\in \mathbb{Z}$. Then, from standard
$\sl_{2}$-theory, we have a finite direct sum $\mathfrak{g}=\oplus_{j\in \mathbb{Z}}\mathfrak{g}_{j}$.
Define the Lie subalgebras $\u=\oplus_{j\leq -2}
\mathfrak{g}_{j}$, $\mathfrak{n}=\oplus_{j\leq -1} \mathfrak{g}_{j}$, $\mathfrak{p}=\oplus_{j\leq 0} \mathfrak{g}_{j}$
and $\mathfrak{m}=\mathfrak{g}_{0}$. Let $U$, $N$, $P$, and $M$ be the corresponding
subgroups of $G$. Thus $U=\exp \u$, $N=\exp \mathfrak{n}$, $P=\set{p\in G}{\operatorname{Ad}(p)\mathfrak{p}
\subset \mathfrak{p}}$ and $M=\set{m\in G}{\mbox{$\operatorname{Ad}(m)H=H$}}$.
Let
\begin{equation}
\label{defchi}
\chi_{\gamma}(\exp Z)=\psi (\kappa(X,Z)), \ \ \ \ \forall \ Z\in \u.
\end{equation}
Since $\kappa(X,[\u,\u])=0$, $\chi_{\gamma}$ defines a character on
$U$. Let $M_{X}=\set{m\in M}{\mbox{$\operatorname{Ad}(m)X=X$}}$. Then it is known
\cite[Section 3.4]{CM92} that
\[
M_{X}=G_{\gamma}:=\set{g\in G}{\mbox{$\operatorname{Ad}(g)X=X$, $\operatorname{Ad}(g)Y=Y$, $\operatorname{Ad}(g)H=H$}}.
\]
In particular $M_{X}$ is reductive. For the moment assume that $\mathfrak{g}_{-1}\neq 0$, or equivalently $\mathfrak{u}\subsetneq \mathfrak{n}$. In this case
$\operatorname{ad}(X)|_{\mathfrak{g}_{-1}}:\mathfrak{g}_{-1}\longrightarrow \mathfrak{g}_{1}$ is an isomorphism,
and we may define a symplectic structure on $\mathfrak{g}_{-1}$ by setting
\begin{equation}
\label{defsymg-1}
\kappa_{-1}(S,T)=\kappa(\operatorname{ad}(X)S,T)=\kappa(X,[S,T]), \qquad \mbox{for all $S$, $T\in \mathfrak{g}_{-1}$}.
\end{equation}
We may exhibit a canonical surjective group homomorphism from $N$ to the associated Heisenberg group $\H$ which maps $\exp Z$ to $\kappa(X,Z)$ in the center of $\H$, for $Z\in \u$. Then, according to the Stone-von
Neumann theorem, there exists a unique, up to equivalence, smooth
irreducible (unitarizable) representation
$(\rho_{\chi_{\gamma}},\S_{\chi_{\gamma}})$ of $N$ such that $U$
acts by the character $\chi_{\gamma}$. See Section \ref{subsec:GWM} for details. Since $M_{X}$ preserves
$\gamma $ and thus the symplectic form $\kappa_{-1}$, it is well-known \cite{Weil} that there exists a central
cover of $M_{X}$, to be denoted by $M_{\chi_{\gamma}}$, and a
representation of a semi-direct product $M_{\chi_{\gamma}}\ltimes N$
on $\S_{\chi_{\gamma}}$ which extends the representation
$\rho_{\chi_{\gamma}}$ of $N$. We refer to the representation of
$M_{\chi_{\gamma}}\ltimes N$ on $\S_{\chi_{\gamma}}$ as the smooth
oscillator-Heisenberg (or Weil) representation.
If $\mathfrak{g}_{-1}=0$, then we define
$M_{\chi_{\gamma}}$ to be just $M_{X}$. For notational convenience, we also denote by $(\rho_{\chi_{\gamma}},\S_{\chi_{\gamma}})$ the $1$-dimensional
representation of $N=U$ given by the character $\chi_{\gamma}$. We may again view $(\rho_{\chi_{\gamma}},\S_{\chi_{\gamma}})$ as a representation of
$M_{X}\ltimes N$, with the trivial $M_{X}$ action.
In this article, a smooth representation of a reductive group over
$\k$ will mean a smooth representation in the usual sense for $\k$
non-Archimedean, namely it is locally constant, and a
Casselman-Wallach representation for $\k$ Archimedean. The reader
may consult \cite[Chapter 11]{Wa92} for more information about
Casselman-Wallach representations.
\begin{dfn} Fix an $\sl_{2}$-triple $\gamma=\{X,H,Y\}\subset \mathfrak{g}$.
Let $(\pi,\mathscr{V})$ be a smooth representation of $G$. We define
the \emph{space of generalized Whittaker models of $\pi$ associated
to $\gamma $} to be
\begin{equation}
\label{defwhittaker}
\operatorname{Wh}_{\gamma}(\pi)=\operatorname{Hom}_{N}(\mathscr{V},\S_{\chi_{\gamma}}).
\end{equation}
Note that $\operatorname{Wh}_{\gamma}(\pi)$ is naturally an
$M_{\chi_{\gamma}}$-module:
\[(m\cdot \lambda )(v)=m\cdot \lambda (\bar{m}^{-1}\cdot v), \ \ m\in M_{\chi_{\gamma}}, \
\lambda \in \operatorname{Hom}_{N}(\mathscr{V},\S_{\chi_{\gamma}}), \ v\in \mathscr{V}, \]
where $m\mapsto \bar{m}$ is the covering map from $M_{\chi_{\gamma}}$ onto $M_{X}$.
\end{dfn}
From the well-known results of Jacobson-Morozov and Kostant \cite[Chapter 3]{CM92}, the map $\gamma=\{X,H,Y\}\mapsto \mathcal{O} =\operatorname{Ad} G
\cdot X$ yields a 1-1 correspondence between
\[
\left\{\begin{array}{c} \mbox{$\operatorname{Ad} G$ conjugacy classes of}\\
\mbox{$\sl_{2}$-triples in $\mathfrak{g}$}\end{array}\right\}
\longleftrightarrow \left\{\begin{array}{c} \mbox{Nonzero nilpotent $\operatorname{Ad} G$-orbits}\\
\mbox{$\mathcal{O} \subset \mathfrak{g}$}\end{array}\right\}.
\]
If the conjugacy class of an $\sl_{2}$-triple $\gamma$ corresponds
to a nilpotent orbit $\mathcal{O}\subset \mathfrak{g}$ we say that $\gamma$ is an
\emph{$\sl_{2}$-triple of type $\mathcal{O}$.} It is clear that given two
conjugate $\sl_{2}$-triples $\gamma$, $\gamma '$, there exists an
isomorphism $\phi:\operatorname{Wh}_{\gamma}(\pi)\longrightarrow
\operatorname{Wh}_{\gamma '}(\pi)$ that intertwines the action of
$M_{\chi_{\gamma}}$ and $M_{\chi_{\gamma '}}$. For this reason,
sometimes we will abuse notation and denote
$\operatorname{Wh}_{\gamma}(\pi)$ just by $\operatorname{Wh}_{\mathcal{O}}(\pi)$ and we will
call it the \emph{space of generalized Whittaker models associated
to $\mathcal{O}$}, or the \emph{space of generalized Whittaker models of
type $\mathcal{O}$}. Note that this definition is (essentially) equivalent
to the one given in \cite{Ya86, MW87}.
The study of Whittaker and generalized Whittaker models for
representations of reductive groups over local fields evolved in
connection with the theory of automorphic forms (via their Fourier
coefficients), and has found important applications in both areas.
See for example \cite{Sh74,NPS73,Ko78,Ka85,Ya86, WaJI}, and a recent
article of Jiang \cite{Ji07} discussing its role in a general theory
of periods of automorphic forms. From the point of view of
representation theory, the space of generalized Whittaker models may
be viewed as one kind of nilpotent invariant associated to smooth
representations.
Another nilpotent invariant is the wave front cycle:
\[
\operatorname{WF}(\pi)=\sum_{\scriptstyle \begin{array}{c} \mathcal{O}\subset{\mathfrak{g}}\\ \mbox{nilpotent}\end{array}} c_{\mathcal{O}}(\pi)[\mathcal{O}],
\]
defined by Harish-Chandra in the non-Archimedean case (\cite{HC78}; see also \cite{Hei85,Pr91b}) and by Howe and Barbasch-Vogan in the Archimedean case (\cite{HoWF,BV80}; see also
\cite{Ro95,SV00}). For $\k$ non-Archimedean, M\oe{}glin and Waldspurger \cite{MW87} have established that $\operatorname{WF}(\pi)$ completely controls the spaces of generalized Whittaker models of interest, namely, if $\mathcal{O}$ is a nilpotent orbit which is maximal in the wave front set of $\pi$ \cite{HoWF}, then
\[
c_{\mathcal{O}}(\pi)=\dim \operatorname{Wh}_{\mathcal{O}}(\pi).
\]
For $\k$ Archimedean, the corresponding phenomenon is not yet (fully) understood, except for the representations with the largest Gelfand-Kirillov dimension \cite{Vo78,Ma92} and unitary highest weight modules \cite{Ya01}. For the latter, the wave front set was computed earlier in \cite{Pr91a}.
On the other hand, when $\k$ is Archimedean, there is another nilpotent invariant defined by Vogan \cite{Vo91}, which is called the associated cycle:
\[
\operatorname{AC}(\pi) =\sum_{\scriptstyle \begin{array}{c} \mathcal{O}\subset{\mathfrak{s}}\\ \mbox{nilpotent}\end{array}} d_{\mathcal{O}}(\pi)[\mathcal{O}].
\]
Here $\mathfrak{g}_{\mathbb{C}}=\mathfrak{k}\oplus
\mathfrak{s}$ is the complexified Cartan decomposition. By the work of Schmid and Vilonen \cite{SV00}, we know that the wave front cycle and the associate cycle determine
each other through the Kostant-Sekiguchi correspondence \cite{Se87}.
{\vspace{0.2in}}
Now we come to the other part of the title regarding local theta lifting. Let $(G,\tilde{G})$ be a reductive dual pair in
the sense of Howe \cite{Ho79}. Let $(\omega , \mathscr{Y} )$ be the smooth oscillator representation
associated to the dual pair $(G,\tilde{G})$ and to the character
$\psi$ of $\k$. It is well-known that $(\omega , \mathscr{Y})$ yields a representation
of $G\times \tilde{G}$ except when $G$ is the isometry group of an
odd dimensional quadratic space, in which case it is a representation of a double cover of $\tilde{G}$. (In general one may need to tensor with a genuine character of the relevant double cover to achieve this.) In the exceptional case, we shall simply redefine $\tilde{G}$ to be the induced double
cover, so as to simplify notation. Also in this case we shall only consider genuine representations of $\tilde{G}$,
namely those representations such that the non-trivial element of the covering map acts by $-1$. For all other cases, representations are called genuine by convention.
Let $(\pi,\mathscr{V})$ be a smooth irreducible genuine representation of
$G$, and consider the maximal $\pi$-isotypic quotient of
$\mathscr{Y}$, called the Howe quotient of $\pi $. If it is non-zero, it is of the form
$\mathscr{V} \otimes \Theta(\mathscr{V})$ (algebraic tensor product for $\k$ non-Archimedean, and
completed projective tensor product for $\k$ Archimedean), where $\Theta(\mathscr{V})$ carries a smooth
representation $\Theta(\pi)$ of $\tilde{G}$. The representation $\Theta(\pi)$ is sometimes referred to as the
full theta lift (or colloquially the big theta lift) of $\pi$. Results of Howe \cite{Ho89} and Waldspurger \cite{Wald} say that $\Theta(\pi)$
is an admissible representation of finite length, moreover, with the possible exception when $\k$ is a dyadic field,
$\Theta(\pi)$ has a unique irreducible quotient $\theta(\pi)$ called the (local) theta
lift of $\pi$. In this article we shall focus on the full theta lift $\Theta(\pi)$, and shall thus place no restriction
on the residue characteristic of $\k$.
Let $V$, $\tilde{V}$, be the standard modules of $G$, $\tilde{G}$,
respectively; and consider the moment maps $\varphi$,
$\tilde{\varphi}$, from $\operatorname{Hom}(V,\tilde{V})$ to $\mathfrak{g}$ and
$\tilde{\mathfrak{g}}$, respectively:
\[
\begin{diagram}
& & \operatorname{Hom}(V,\tilde{V}) & & \\
& \ldTo^{\varphi} & &\rdTo^{\tilde{\varphi}} & \\
\mathfrak{g}& & & &\tilde{\mathfrak{g}}
\end{diagram}
\]
See Section \ref{subsec:liftOrbit} for the definition of the moment maps. If $T\in \operatorname{Hom}(V,\tilde{V})$, then $\varphi (T)$ is nilpotent if and
only if $\tilde{\varphi}(T)$ is nilpotent. As usual, this yields a
notion of correspondence for nilpotent orbits $\mathcal{O} \subset \mathfrak{g}$ and
$\tilde{\mathcal{O}}\subset \tilde{\mathfrak{g}}$. (In general this may not be one to
one.) Denote by $\operatorname{Max} \operatorname{Hom} (V,\tilde{V})$ the set of full rank
elements in $\operatorname{Hom}(V,\tilde{V})$. Without any loss of generality, we
assume that $\dim V \leq \dim \tilde{V}$, and elements of $\operatorname{Max} \operatorname{Hom}
(V,\tilde{V})$ are then represented by injective maps from $V$ to
$\tilde{V}$. In this article, we shall only be concerned with
nilpotent orbits $\mathcal{O} \subset \mathfrak{g}$ in the image of $\operatorname{Max} \operatorname{Hom}
(V,\tilde{V})$ under the moment map $\varphi$, namely it satisfies
\begin{equation}
\label{assumption0}
\varphi ^{-1}(\mathcal{O}) \cap \operatorname{Max} \operatorname{Hom} (V,\tilde{V})\ne \emptyset.
\end{equation}
This will be our standing assumption. We write the resulting
nilpotent orbit correspondence as $\mathcal{O} \mapsto
\tilde{\mathcal{O}}=\Theta(\mathcal{O})$. See Section \ref{subsec:liftOrbit} for
details.
As noted in the beginning, the space of generalized Whittaker
models depends on an $\sl_{2}$-triple. To relate the two
$\sl_{2}$-triples in $\mathfrak{g}$ and $\tilde{\mathfrak{g}}$, we make the following
\begin{dfn}
Let $\gamma=\{X,H,Y\}\subset \mathfrak{g}$ and $\tilde{\gamma}=\{\tilde{X},\tilde{H},\tilde{Y}\}\subset \tilde{\mathfrak{g}}$ be two $\sl_{2}$-triples of type $\mathcal{O}$ and $\tilde{\mathcal{O}}$, respectively. We say that $T\in \operatorname{Hom}(V,\tilde{V})$ \emph{lifts} $\gamma$ to $\tilde{\gamma}$ if $\varphi(T)=X$, $\tilde{\varphi}(T)=\tilde{X}$ and $T(V_{j})\subset \tilde{V}_{j+1}$ for all $j$. Here
$V_{j}=\{v\in V\, | \, Hv=jv\}$, and likewise for $\tilde{V}_{j+1}$.
\end{dfn}
Set
\[
\mathcal{O}_{\gamma,\tilde{\gamma}}=\{T\in \operatorname{Hom}(V,\tilde{V})\, | \, \mbox{$T$ lifts $\gamma$ to $\tilde{\gamma}$}\},\]
and
\begin{equation}
\label{defoggMax0}
\mathcal{O}^{\operatorname{Max}}_{\gamma,\tilde{\gamma}}=\mathcal{O}_{\gamma,\tilde{\gamma}}\cap
\operatorname{Max} \operatorname{Hom} (V,\tilde{V}).
\end{equation}
(By Lemma \ref{lem:oggMax}, this is a single $M_{X}\times \tilde{M}_{\tilde{X}}$-orbit.)
Our main result, in a slightly less concrete form than what will be proved in Section \ref{liftWhittaker}, is
\begin{theorem}
\label{MainThm}
Let $(G,\tilde{G})$ be a reductive dual pair, and let $(\omega , \mathscr{Y} )$ be the smooth oscillator representation
associated to the dual pair $(G,\tilde{G})$ and to the character
$\psi$ of $\k$. Let $(\pi,\mathscr{V})$ be a smooth
irreducible genuine representation of $G$. Let $\mathcal{O}\subset \mathfrak{g}$ be a
nilpotent $G$-orbit in the image of $\operatorname{Max} \operatorname{Hom} (V,\tilde{V})$ under the moment map $\varphi$ and let $\tilde{\mathcal{O}}=\Theta(\mathcal{O})\subset \tilde{\mathfrak{g}}$ be the corresponding
nilpotent $\tilde{G}$-orbit. Then
\[
\operatorname{Wh}_{\tilde{\mathcal{O}}}(\Theta(\pi))\cong \operatorname{Wh}_{\mathcal{O}}(\check{\pi}),
\]
where $\check{\pi}$ is the contragredient representation of $\pi$. More precisely, let $\gamma$ and $\tilde{\gamma}$ be two $\mathfrak{sl}_{2}$-triples of type $\mathcal{O}$ and $\tilde{\mathcal{O}}$, respectively. Then, given any $T=T_{\gamma,\tilde{\gamma}}\in \mathcal{O}^{\operatorname{Max}}_{\gamma,\tilde{\gamma}}$, there is a surjective homomorphism $\phi _T:\widetilde{M}_{\chi_{\tilde{\gamma}}} \twoheadrightarrow M_{\chi_{\gamma}}$ (depending on $T$), inducing an action of $\widetilde{M}_{\chi_{\tilde{\gamma}}}$ on $ \operatorname{Wh}_{\gamma}(\check{\pi})$, such that
\[
\operatorname{Wh}_{\tilde{\gamma}}(\Theta(\pi))\cong \operatorname{Wh}_{\gamma}(\check{\pi})
\]
as $\widetilde{M}_{\chi_{\tilde{\gamma}}}$-modules.
\end{theorem}
\noindent {\bf Remarks}: (a) Nilpotent orbits in $\mathfrak{g}$ may be
parameterized by equivalence classes of admissible
$\epsilon$-Hermitian Young tableaux (Section
\ref{sub:Young-tableaux}). Using this parametrization, the
correspondence $\mathcal{O} \mapsto \Theta(\mathcal{O})$ can be described
explicitly and in simple terms. Note that the assumption in
(\ref{assumption0}) allows us to ``embed" the sesquilinear Young
tableau parameterizing $\mathcal{O} $ into the sesquilinear Young tableau
parameterizing $\Theta(\mathcal{O})$. See Section \ref{subsec:liftOrbit}.
\noindent (b) One particularly important circumstance is when $(G,\tilde{G})$
is in the stable range with $G$ the smaller member (\cite{HoST}; cf.
\cite{Li89}). Then every nilpotent orbit $\mathcal{O} \subset \mathfrak{g}$ is in the
image of $\operatorname{Max} \operatorname{Hom} (V,\tilde{V})$ under the moment map $\varphi$,
and for any smooth irreducible genuine representation
$(\pi,\mathscr{V})$ of $G$, we have $\Theta(\pi)\ne 0$
\cite{MVW,PP}. In this case and for $\k$ non-Archimedean, Theorem
\ref{MainThm} is due to M\oe{}glin \cite{Mo98}. Our approach,
explained in the following paragraph, is in some sense more
conceptual. For the type of questions considered in this article, it
is also well-known that substantially more effort is required to
treat the Archimedean case.
\noindent (c) Assume that we are in the stable
range and $\k$ is Archimedean. There is a similar notion of theta
lifting of nilpotent $K_{\mathbb{C}}$-orbits and one similarly expects a
correspondence of associate cycles \cite{NOT}. We refer the reader to
\cite{Ya01, NZ04} for some results in this direction, and the
definitive result in the recent paper of Loke and Ma \cite{LM13}. We
also refer to several earlier works of Przebinda \cite{Pr91a,Pr93,DP96,Pr00} on
correspondence of wave front sets.
\noindent (d) Generically $\Theta (\pi)$
should be irreducible. One then obtains the space of generalized
Whittaker models for $\theta (\pi)$. For example it is expected
that under the assumption of stable range, $\Theta (\pi)$ is
irreducible whenever $\pi $ is unitarizable. For $\k$ Archimedean,
this is established in \cite{LM13}.
\noindent (e) As a direct consequence of Theorem \ref{MainThm}, we
have $\Theta(\pi)\ne 0$, if $\operatorname{Wh}_{\mathcal{O}}(\check{\pi})\ne 0$, for some
$\mathcal{O}$ in the image of $\operatorname{Max} \operatorname{Hom} (V,\tilde{V})$ under the moment map $\varphi$.
\noindent (f) For ``small" $\mathcal{O}$, the condition in (\ref{assumption0}) is actually quite restrictive. For example for the zero orbit, the condition implies that the dual pair $(G,\tilde{G})$ is in the stable range with $G$ the smaller member. For this reason there is a need to consider orbit correspondence covered by a $G\times \tilde{G}$ stable set larger than $\operatorname{Max} \operatorname{Hom} (V,\tilde{V})$, and to investigate how their generalized Whittaker models behave. This will be taken up in a forthcoming work of the authors.
{\vspace{0.2in}}
The main ingredient in the proof of Theorem \ref{MainThm} is our
description of the space of covariants $\mathscr{Y}_{\tilde{U},
\chi_{\tilde{\gamma}}}:=\operatorname{Hom}_{\tilde{U}}(\mathscr{Y}, \chi_{\tilde{\gamma}})$ of the smooth oscillator representation
$\mathscr{Y}$, where $(\tilde{U}, \chi_{\tilde{\gamma}})$ for the
$\sl_{2}$-triple $\tilde{\gamma}$ in $\tilde{\mathfrak{g}}$ is as
$(U,\chi_{\gamma})$ for the $\sl_{2}$-triple $\gamma$, and
$\tilde{\gamma}$ is a lift of $\gamma$. (This is reminiscent of the computation of
the Jacquet modules of the smooth oscillator representation by Kudla \cite{Ku86}, though not in method.)
We show it is isomorphic to
a certain space $\mathscr{C}(N\backslash G;\mathscr{H}_{\gamma, \tilde{\gamma}})$ of
rapidly decreasing functions on $N\backslash G$ with values in
$\mathscr{H}_{\gamma, \tilde{\gamma}}$, where $\mathscr{H}_{\gamma, \tilde{\gamma}}$ is the
space of the smooth oscillator-Heisenberg representation associated
to a certain symplectic subspace $W_{\gamma, \tilde{\gamma}}$ of
$\operatorname{Hom} (V,\tilde{V})$. This is the key Proposition
\ref{prop:tildechicoinvariants}. An important point is that we may
construct an isomorphism of $\mathfrak{g}_{-1}\oplus \tilde{\mathfrak{g}}_{-1}$ with
$W_{\gamma, \tilde{\gamma}}$, which depends on $T\in
\mathcal{O}^{\operatorname{Max}}_{\gamma,\tilde{\gamma}}$. The key proposition basically
says that we can define a natural surjective (matrix coefficient)
map (using ``homogeneous components" of $T$), from $\mathscr{Y}$ to
$\mathscr{C}(N\backslash G;\mathscr{H}_{\gamma, \tilde{\gamma}})$, which induces an
isomorphism on $\mathscr{Y}_{\tilde{U},\chi_{\tilde{\gamma}}}$!
{\vspace{0.2in}}
Here are some additional words on the organization and contents of
this article. In Section \ref{classical}, we describe classical
groups as the isometry groups of $\epsilon$-Hermitian $D$-modules,
where $D$ is a division algebra over $\k$. In Section \ref{orbit},
we review the well-known parametrization of nilpotent orbits in the
classical Lie algebras, following the book by Collingwood and
McGovern \cite{CM92}. We also introduce the generalized Whittaker
models associated to the nilpotent orbits. In Section \ref{frechet},
we introduce some Fr\'echet spaces of functions on $G$ to realize
the space of generalized Whittaker models for $\k$ Archimedean. In
Section \ref{sec:liftOrbit}, we recall the notion of lifting of
nilpotent orbits in the setting of dual pairs (via the moment maps),
and we describe the fine structure of lifting for those orbits
$\mathcal{O}$ and $\tilde{\mathcal{O}}$ which correspond via injective maps from
$V$ to $\tilde{V}$. We also prove an explicit isomorphism of $\mathfrak{g}_{-1}\oplus
\tilde{\mathfrak{g}}_{-1}$ with a symplectic subspace $W_{\gamma,
\tilde{\gamma}}$ of $\operatorname{Hom}(V,\tilde{V})$. In Section
\ref{liftWhittaker}, we relate the generalized Whittaker models of
$\check{\pi}$ and $\Theta (\pi)$. As mentioned previously, its main
ingredient is the description of the space of covariants
$\mathscr{Y}_{\tilde{U}, \chi_{\tilde{\gamma}}}$ of the smooth oscillator
representation $\mathscr{Y}$. To arrive at this, we make extensive use of the
gradation in the standard modules $V$ and $\tilde{V}$ given by the
semisimple elements $H$, $\tilde{H}$ of the two $\sl_{2}$-triples
$\gamma\subset \mathfrak{g}$ and $\tilde{\gamma}\subset\tilde{\mathfrak{g}}$ of type
$\mathcal{O}$ and $\Theta(\mathcal{O})$, respectively. On the one hand, it gives
rise to totally isotropic subspaces and thus convenient realizations
of $\mathscr{Y}$. On the other hand, it facilitates an inductive argument
based on the heights of the gradations. Together with the
isomorphism of $\mathfrak{g}_{-1}\oplus \tilde{\mathfrak{g}}_{-1}$ with $W_{\gamma,
\tilde{\gamma}}$, this implies the relationship between the
generalized Whittaker models of $\check{\pi}$ and $\Theta(\pi)$.
{\vspace{0.2in}}
\noindent {\bf Acknowledgements}: the authors thank W. T. Gan, D. Jiang, and B. Sun for their interests and comments. The authors also wish to express their gratitude to the anonymous referee for his critical comments as well as numerous detailed suggestions.
\section{Classical groups as isometry groups of $\varepsilon$-Hermitian modules}
\label{classical}
\subsection{Hermitian $D$-modules}
\label{Hermitianmodules}
Let $\k$ be a local field, $|\cdot|$ its absolute
value, and let $\psi$ be a fix non-trivial unitary character of $\k$. Let $D$ be one of the following division algebras over $\k$:
the field $\k$ itself, a quadratic field extension of $\k$ or the
quaternion division $\k$-algebra. Observe that $D$ comes equipped
with a canonical involutive anti-automorphism (the identity map, the non-trivial Galois element, or
the main involution, respectively) which we will denote by $x \mapsto
\overline{x}$.
Throughout this article, we will only consider finitely generated
modules over $D$.
Let $V$ and $W$ be two right $D$-modules. We will denote the set of
right $D$-module morphisms from $V$ to $W$ by
\[
\operatorname{Hom}_{D}(V,W)=\{T:V\longrightarrow W \,| \, \mbox{$T(v_{1}a+v_{2}b)=T(v_{1})a+T(v_{2})b$ for all $v_{1}$, $v_{2} \in V$, $a$, $b\in D$}\}.
\]
If $V=W$, we will denote this set by $\operatorname{End}_{D}(V)$. Set
\[
\operatorname{GL}(V,D)=\{T\in \operatorname{End}_{D}(V) \, | \, \mbox{$T$ is invertible}\}.
\]
When it is clear from the context what the division algebra $D$ is,
we may just omit $D$ in various of these notations.
Let $V'$ be the set of right $D$-linear functionals on $V$. There is a natural left
$D$-module structure on $V'$ given by setting
\[ (a\lambda)(v)=a\lambda(v), \quad \text{for all $a\in D$, $v\in V$, and
$\lambda \in V'$.} \]
Observe that with this structure, $W\otimes_{D}V'$ is naturally
isomorphic to $\operatorname{Hom}_{D}(V,W)$ as a $\k$-vector space. Given $T\in
\operatorname{Hom}_{D}(V,W)$, we will specify an element in $\operatorname{Hom}_{D}(W',V')$
(analogously defined), which we will also denote $T$, by setting
$(\lambda T)(v):=\lambda(Tv)$, for $v\in V$ and $\lambda \in W'$. This correspondence gives rise to
natural isomorphisms between $\operatorname{End}_{D}(V)$ and $\operatorname{End}_{D}(V')$, and
between $\operatorname{GL}_{D}(V)$ and $\operatorname{GL}_{D}(V')$.
\begin{definition}
Let $\varepsilon=\pm 1$. We say that $(V,B)$ is a right $\varepsilon$-Hermitian $D$-module if $V$ is a right $D$-module and $B$ is an $\varepsilon$-Hermitian form, i.e., $B:V\times V \longrightarrow D$ is a map such that
\begin{enumerate}
\item $B$ is \emph{sesquilinear}: for all $v_{1}$, $v_{2}$, $v_{3}\in V$, $a$, $b\in D$,
\[ B(v_{1},v_{2}a+v_{3}b)=B(v_{1},v_{2})a+B(v_{1},v_{3})b \quad \text{and} \quad B(v_{1}a + v_{2}b,v_{3})=\overline{a}B(v_{1},v_{3})+\overline{b}B(v_{2},v_{3}).\]
\item $B$ is $\varepsilon$-\emph{Hermitian}:
$$
B(v,w)=\varepsilon\overline{B(w,v)} \qquad \mbox{ for all $v,w\in V$.}
$$
\item $B$ is \emph{non-degenerate}.
\end{enumerate}
\end{definition}
Given a right $\varepsilon$-Hermitian $D$-module $(V,B)$, we may
construct a left $\varepsilon$-Hermitian $D$-module
$(V^{\ast},B^{\ast})$ in the following way: as a set, $V^{\ast}$
will be the set of symbols $\{v^{\ast}\, |\, v\in V\}$. Then we give to
$V^{\ast}$ a left $D$-module structure by setting, for all $v$, $w
\in V$, $a\in D$,
\[ \text{$v^{\ast}+w^{\ast}=(v+w)^{\ast}$ and $av^{\ast}=(v\overline{a})^{\ast}$.} \]
Finally, we set
\[
B^{\ast}(v^{\ast},w^{\ast})=\overline{B(w,v)} \qquad \mbox{for all $v$, $w\in V$.}
\]
In an analogous way, we may define the $*$ operation on left $\varepsilon$-Hermitian $D$-modules. Then $V^{\ast\ast}$ is naturally isomorphic with $V$.
Given $T\in \operatorname{End}_{D}(V)$, we define $T^{\ast}\in \operatorname{End}_{D}(V^{\ast})$ by setting $v^{\ast}T^{\ast}:=(Tv)^{\ast}$. With this definition, it is easily seen that $(TS)^{\ast}=S^{\ast}T^{\ast}$, for all $S$, $T\in \operatorname{End}_{D}(V)$. Therefore the map $g\mapsto (g^{\ast})^{-1}$ defines a group isomorphism between $\operatorname{GL}_{D}(V)$ and $\operatorname{GL}_{D}(V^{\ast})$.
Observe that the form $B$ induces a left $D$-module isomorphism
$B^{\flat}:V^{\ast}\longrightarrow V'$ given by
$B^{\flat}(v^{\ast})(w)=B(v,w)$ for $v$, $w\in V$. In what follows,
we will make implicit use of this map to identify these two spaces.
With this identification, for any $T\in \operatorname{End}_{D}(V)$, we can think of $T^{\ast}$ as an element in
$\operatorname{End}_{D}(V)$ defined by
$v^{\ast}(T^{\ast}w):=(v^{\ast}T^{\ast})(w)$, i.e., $T^{\ast}$ is
defined by the usual condition that
\[
B(v,T^{\ast}w)=B(Tv,w) \qquad \mbox{for all $v$, $w \in V$}.
\]
A $D$-submodule $E\subset V$ is said to be \emph{totally isotropic}
if $B|_{E\times E}=0$. If $E$ is a totally isotropic submodule, then
there exists a totally isotropic submodule $F\subset V$ such that
$B|_{(E\oplus F)\times (E\oplus F)}$ is non-degenerate. If we set
\[
U=(E\oplus F)^{\perp}:=\{u\in V\, | \, \mbox{$B(u,w)=0$ for all $w\in E\oplus F$}\},
\]
then $V=E\oplus F \oplus U$, and $B|_{U\times U}$ is non-degenerate.
In this case we say that $E$ and $F$ are totally isotropic,
\emph{complementary} submodules. Observe that then
$B^{\flat}|_{F^{\ast}}:F^{\ast}\longrightarrow E'$ is an
isomorphism. As before we will make implicit use of this isomorphism
to identify $F^{\ast}$ with $E'$.
\subsection{Isometry groups}
\label{Isometry}
Given a right $\varepsilon$-Hermitian $D$-module $(V,B)$, we define
its isometry group
\[
G(V,B)=\{g\in \operatorname{GL}(V) \, | \, \mbox{$B(g v,g w)=B(v,w)$ for all $v$,
$w\in V$}\}.
\]
When there is no risk of confusion regarding $B$, we will denote
this group just by $G(V)$ or even just as $G$. Observe that if $g\in
G$, then $g^{*}=g^{-1}$.
Associated to the group $\operatorname{GL}(V)$ we have the Lie algebra
$\mathfrak{gl}(V)=\operatorname{End}(V)$ with bracket $[T,S]:=TS-ST$, for all
$T$, $S \in \mathfrak{gl}(V)$. Similarly, associated to the group
$G$ we have the Lie algebra
\begin{eqnarray*}
\mathfrak{g} &=&\{T\in \mathfrak{gl}(V) \, | \, \mbox{$B(T v,w) + B(v,Tw)=0$ for all $v$, $w\in V$}\}\\
&=&\{T\in \mathfrak{gl}(V) \, | \, T^*=-T\}.
\end{eqnarray*}
We define a bilinear form on $\mathfrak{g}$ by
\begin{equation}
\label{knormalized}
\kappa(T,S)=\operatorname{Tr}(T^{\ast}S)/2,
\end{equation}
for all $T$, $S \in \mathfrak{g}$. Here $\operatorname{Tr}(T^{\ast}S)$ is the trace
of $T^{\ast}S$ as a $\k$-linear transformation. This bilinear form is non-degenerate
and \emph{invariant} with respect to
the adjoint action of $G$ on $\mathfrak{g}$.
\begin{remark} In the literature, the isometry groups
$G(V,B)$ are called type I. Given a division algebra $D$, one may
consider the algebra $\mathbb{D}=D\oplus D$, which has the canonical
involution $\overline{(a,b)}=(b,a)$, for $a$, $b\in D$. Then one may
similarly define right $\epsilon $-Hermitian $\mathbb{D}$-modules
($\epsilon =\pm 1$) and their isometry groups which are easily seen
to be isomorphic to $\operatorname{GL}(V,D)$, where $V$ are right $D$-modules.
They are called type II. With this modification (from $D$ to
$\mathbb{D}$), all results in this article, which are stated for type I dual pairs,
are expected to carry over to the case of type II dual pairs. We leave this to the interested readers.
\end{remark}
\section{Nilpotent orbits of classical groups}
\label{orbit}
\subsection{Nilpotent orbits and $\sl_{2}$-triples} \label{subsection:nilpotentorbits}
In order to set up notation, and facilitate the exposition of the
reminder of this article, we will review the basic structural results on nilpotent orbits and $\sl_{2}$-triples in $\mathfrak{g}$. The exposition given here is based on the
book of Collingwood and McGovern \cite{CM92}. Another standard reference is the book of Carter \cite[Chapter 5]{Ca85}.
Let $X\in \mathfrak{g}$ be a nonzero nilpotent element. Then, by the Jacobson-Morozov
theorem, there exists an $\sl_{2}$-triple
$\gamma=\{H,X,Y\} \subset \mathfrak{g}$, containing $X$ as the nilpositive element, namely
\[
[H,X]=2X, \qquad [H,Y]=-2Y, \qquad [X,Y]=H.
\]
Let $\mathfrak{g}_{i}=\{Z \in \mathfrak{g} \, | \,
\operatorname{ad}(H)(Z)=iZ\}$, for $i \in \mathbb{Z}$. Then, from standard $\sl_{2}$-theory, we have that
\begin{equation}
\mathfrak{g}=\bigoplus_{i\in \mathbb{Z}}\mathfrak{g}_{i}.
\end{equation}
Now let
\[
\mathfrak{p}=\bigoplus_{i \leq 0}\mathfrak{g}_{i}, \qquad \overline{\mathfrak{p}}=\bigoplus_{i \geq 0}\mathfrak{g}_{i},
\]
and set
\[
P=\{g\in G \, | \, \operatorname{Ad}(g)\mathfrak{p}\subset \mathfrak{p}\},\qquad \overline{P}=\{g\in G \, | \, \operatorname{Ad}(g)\overline{\mathfrak{p}}\subset \overline{\mathfrak{p}}\}.
\]
$\overline{P}$ (resp.\ $\overline{\mathfrak{p}}$) is called the Jacobson-Morozov parabolic subgroup (resp. parabolic subalgebra) associated to $X$. Observe that $P$ (resp.\ $\mathfrak{p}$) is a parabolic subgroup opposite to $\overline{P}$ (resp. parabolic subalgebra opposite to $\overline{\mathfrak{p}}$). (The subgroup $\overline{P}$ depends only on $X$, but the subgroup $P$ depends on the choice of $\sl_{2}$-triple $\gamma$ containing $X$.) Let $M=\{m\in G\, | \, \operatorname{Ad}(m)H=H\}$. Its Lie algebra is $\mathfrak{m} =\{Z\in \mathfrak{g}\, | \, \operatorname{ad}(Z)H=0\}$, which is exactly $\mathfrak{g} _{0}$. Let
\[
\mathfrak{n}=\bigoplus_{i\leq -1} \mathfrak{g}_{i} \qquad \mbox{and} \qquad \u=\bigoplus_{i\leq -2} \mathfrak{g}_{i}.
\]
Observe that if $Z\in \mathfrak{n}$, then $Z$ is nilpotent and hence $\exp Z=\sum_{j=0}^{\infty}Z^{j}/j!$ is a well defined element in $\operatorname{End}(V)$. Let $N=\exp \mathfrak{n}=\set{\exp Z}{Z\in \mathfrak{n}}$ and $U=\exp \u=\set{\exp Z}{Z\in \u}$; then $U$, $N$ are subgroups of $G$, and $P=MN$. Similarly we have the Levi decomposition $\overline{P}=M\overline{N}$, with $\overline{N}$ the unipotent radical of $\overline{P}$.
For $i\in \mathbb{Z}$, let $V_{i}=\{v\in V\, | \, Hv=iv\}$. Then, again from standard $\sl_{2}$-theory,
\begin{equation}
V=\bigoplus_{i\in \mathbb{Z}} V_{i}. \label{eq:Vdirectsumdecomposition}
\end{equation}
We may also characterize $M$ as the set of $m\in G$ that preserves
the direct sum decomposition given in
(\ref{eq:Vdirectsumdecomposition}). Now observe that, since
$H^{*}=-H$, $B|_{V_{0}\times V_{0}}$ is non-degenerate and $B$
establishes a perfect pairing between $V_{i}$ and $V_{-i}$ for all
$i > 0$. Using this pairing we define for any $T\in \operatorname{End}(V_{i})$
a map $T^{*}\in \operatorname{End}(V_{-i})$ given by
\[
B(Tv,w)=B(v,T^{*}w), \qquad \mbox{for all $v\in V_{i}$, $w\in V_{-i}$}.
\]
It follows immediately that, if $g\in \operatorname{GL}(V_{i})$, then
$B(gv,(g^{*})^{-1}w)=B(v,w)$, for all $v\in V_{i}$, $w\in V_{-i}$.
From this we conclude that there is an embedding of $\operatorname{GL}(V_{i})$
into $M$ for all $i >0$. We will denote the image of this embedding
by $M_{i}$. Proceeding in a similar manner we may also define a
natural embedding of $G(V_{0},B|_{V_{0}\times V_{0}})$ into $M$,
whose image we will denote by $M_{0}$. Then
\begin{equation}
M = \prod_{i \geq 0} M_{i} \cong G(V_0)\times \prod_{i>0} \operatorname{GL}(V_{i}). \label{eq:descriptionofM}
\end{equation}
Set $\mathfrak{g}_{\gamma}=\operatorname{Span}_{\k}\{X,H,Y\}\subset \mathfrak{g}$. We have the $\mathfrak{g}_{\gamma}$-isotypic decomposition
\begin{equation}
\label{isotypic}
V=\bigoplus_{j=1}^{l} V^{\gamma,t_{j}},
\end{equation}
where $V^{\gamma,t_{j}}$ is a direct sum of irreducible
$t_{j}$-dimensional $\mathfrak{g}_{\gamma}$-modules, and $t_{1}> t_{2}> \ldots > t_{l}>0$. Let
$V^{\gamma,t_{j}}_{i}=V^{\gamma,t_{j}}\cap V_{i}$. It is nonzero if and only if $|i|<t_{j}$ and $i\equiv t_j-1$ ($\mbox{mod } 2$).
Then it is clear that
\begin{equation}
V_{i}=\bigoplus_{j,\, t_{j}>|i|} V^{\gamma,t_{j}}_{i}. \label{eq:inducedhermitianform}
\end{equation}
Using standard results from the representation theory of
$\sl_{2}$, we have that for all $i\geq 0$, the map
$(X|_{V_{-i}})^{i}:V_{-i}\longrightarrow V_{i}$ is invertible. (Here
we are using the convention that $(X|_{V_{0}})^{0}$ is the identity
map on $V_{0}$.) The statement remains true for $i<0$ if we
interpret a negative power of an invertible map as the positive
power of its inverse.
Now define a Hermitian form $B_{i}$ on $V_{i}$ by setting $B_{i}(v,w)=B(X|_{V_{i}}^{-i}v,w)$ for all $v$,
$w\in V_{i}$. Since $B$ establishes a perfect pairing between
$V_{i}$ and $V_{-i}$, it is clear that $B_{i}$ is non-degenerate.
The analysis applied to $(V,B)$ applies equally to $(V^{\gamma,t_{j}},B^{\gamma,t_{j}})$, where $B^{\gamma,t_{j}}$ is the restriction of $B$ to $V^{\gamma,t_{j}}$.
Thus $B_{i}$ is in fact non-degenerate when restricted to any
$V^{\gamma,t_{j}}_{i}$. We will denote by $B_{i}^{\gamma,t_{j}}$ the restriction of $B_{i}$ to $V^{\gamma,t_{j}}_{i}$.
The following result is also clear:
\begin{equation}
\label{eq:XfromVitoViplustwo}
\mbox{If $-t_{j}+1\leq i <i+2 \leq t_{j}-1$,} \qquad \mbox{then $B_{i+2}(Xv,Xw)=-B_{i}(v,w)$} \qquad \mbox{for all $v$, $w\in V^{\gamma,t_{j}}_{i}$}.
\end{equation}
In particular, all the $B_{i}^{\gamma,t_{j}}$'s are determined by $B_{t_{j}-1}^{\gamma,t_{j}}$ (on $V_{t_{j}-1}^{\gamma,t_{j}}$, the space of highest weight vectors in $V^{\gamma,t_{j}}$).
Let $M_{X}=\{m\in M\,| \, \operatorname{Ad}(m)X=X\}$. The following result is due to Kostant. See \cite[Section 3.4]{CM92} or \cite[Proposition 5.5.9]{Ca85}.
\begin{thm}\label{thm:kostant} Let $G_{X}=\{g\in G\, | \, \mbox{$\operatorname{Ad}(g)X=X$}\}$. Then
\[
G_{X}=M_{X} \overline{N}_{X},
\]
where $\overline{N}_{X}=\{n\in \overline{N}\, | \,
\operatorname{Ad}(n)X=X\}$. Furthermore
\[
M_{X} =G_{\gamma}:=\{g\in G\, | \, \mbox{$\operatorname{Ad}(g)X=X$, $\operatorname{Ad}(g)H=H$,
$\operatorname{Ad}(g)Y=Y$}\}.
\]
\end{thm}
We now have all the ingredients to give a description of $M_{X}$ in
the spirit of the description of $M$ given in equation
(\ref{eq:descriptionofM}). Observe that $M_{X}$ acts on
$V^{\gamma,t_{j}}_{t_{j}-1}$ preserving $B^{\gamma,t_{j}}_{t_{j}-1}$. From this observation and
equation (\ref{eq:XfromVitoViplustwo}) we conclude that there is an
embedding of $G(V^{\gamma,t_{j}}_{t_{j}-1},B^{\gamma,t_{j}}_{t_{j}-1})$ into $M_{X}$. Let
$M_{X,t_{j}-1}$ be the image of this embedding. Then
\begin{equation}\label{eq:productisometrygroups}
M_{X}=\prod_{j=1}^{l}M_{X,t_{j}-1}\cong \prod_{j=1}^{l} G(V^{\gamma,t_{j}}_{t_{j}-1}).
\end{equation}
\begin{remark}
\label{rmkzero}
For $X=0$, we may take $\mathfrak{g}_{0}=\mathfrak{g}$, and $\mathfrak{g}_{i}=0$ for $i\ne 0$. With this convention, all the subgroups defined in this section will make sense for any $X$. We will adopt this convention in the sequel, sometimes without mentioning the appropriate (minor) adjustment for the special case of the zero orbit.
\end{remark}
\subsection{$\epsilon$-Hermitian Young tableaux}
\label{sub:Young-tableaux}
A partition of $n$ is a tuple
$[d_{1},\ldots,d_{k}]$ of positive integers such that
$d_{1}\geq d_{2} \geq \ldots \geq d_{k}$ and $d_{1}+\cdots+d_{k}=n$. Partitions of
$n$ are frequently represented by \emph{Young diagrams} in the following way:
given a partition $\mathbf{d}=[d_{1},\ldots,d_{k}]$ we construct a left-justified
array of empty boxes such that the $i$-th row has $d_{i}$ boxes. This array is the
Young diagram associated to $\mathbf{d}$. Another way of describing a partition $\mathbf{d}=[d_{1},\ldots,d_{k}]$ of $n$ is using the \emph{exponential notation} $\mathbf{d}=[t_{1}^{i_{1}},\ldots,t_{l}^{i_{l}}]$,
which means that the number $t_{k}$ appears $i_{k}$ times (the multiplicity) in the
partition, and $t_1>\cdots >t_l>0$.
\begin{definition} A \emph{sesquilinear Young tableau} is a pair
$\Gamma=(\mathbf{d}^{\Gamma},(V^{\Gamma},B^{\Gamma}))$ such that
$\mathbf{d}^{\Gamma}=[t_{1}^{i_{1}},\ldots,t_{l}^{i_{l}}]$ is a partition
of $n$, and $(V^{\Gamma},B^{\Gamma})$ is an assignment, for each $1 \leq j\leq
l$, of an $\epsilon_{j}$-Hermitian module
$(V^{\Gamma,t_{j}}_{t_{j}-1},B^{\Gamma,t_{j}}_{t_{j}-1})$ of dimension ${i_{j}}$.
\end{definition}
\begin{example} The following picture represents a sesquilinear Young tableau $\Gamma=(\mathbf{d}^{\Gamma},(V^{\Gamma},B^{\Gamma}))$, where $\mathbf{d}^{\Gamma}=[3^{2},2^{3},1^{2}]$ is a partition of $14$, and $V_{2}^{\Gamma,3}$, $V_{1}^{\Gamma,2}$, $V_{0}^{\Gamma,1}$ have dimensions $2$, $3$ and $2$, respectively.
\[
\raisebox{35pt}{$\begin{array}{l} \raisebox{9pt}{$ (V_{2}^{\Gamma,3},B_{2}^{\Gamma,3})\left\{ \raisebox{15pt}[12pt][0pt]{} \right.$} \\ \raisebox{15pt}{$ (V_{1}^{\Gamma,2},B_{1}^{\Gamma,2}) \left\{ \raisebox{0pt}[14pt][12pt]{} \right.$} \\\raisebox{10pt}{$( V_{0}^{\Gamma,1},B_{0}^{\Gamma,1}) \left\{ \raisebox{0pt}[12pt][-10pt]{} \right.$} \end{array}$} \yng(3,3,2,2,2,1,1)
\]
\end{example}
Let $(\rho_{m},\k^{m})$ be the irreducible representation of $\sl_{2}$ of
dimension $m$, and fix, for each $m$, a non-degenerate invariant bilinear form $(\cdot,\cdot)_{m}$
on $\k^{m}$. Recall that this form is unique up to scalar and that
$(f_{1},f_{2})_{m}=(-1)^{m-1}(f_{2},f_{1})_{m}$, for all $f_{1}$, $f_{2} \in \k^{m}$. This follows from the analysis surrounding equation (\ref{eq:XfromVitoViplustwo}).
Given a sesquilinear Young tableau $\Gamma=(\mathbf{d}^{\Gamma},(V^{\Gamma},B^{\Gamma}))$ we will set
\begin{equation}
V^{\Gamma,t_{j}}:=V^{\Gamma,t_{j}}_{t_{j}-1}\otimes_{\k}\k^{t_{j}}.
\end{equation}
On this space we define a $(-1)^{t_{j}-1}\epsilon_{j}$-Hermitian form $B^{\Gamma,t_{j}}$ by
\begin{equation}
\label{Bgt}
B^{\Gamma,t_{j}}(v_{1}\otimes f_{1}, v_{2}\otimes f_{2})=B^{\Gamma,t_{j}}_{t_{j}-1}(v_{1},v_{2})
(f_{1},f_{2})_{t_{j}}, \qquad \mbox{$v_{1}$, $v_{2}\in V^{\Gamma,t_{j}}_{t_{j}-1}$, and $f_{1}$, $f_{2} \in \k^{t_{j}}$}.
\end{equation}
\begin{definition} Let $\Gamma=(\mathbf{d}^{\Gamma},(V^{\Gamma},B^{\Gamma}))$ and $\Phi=(\mathbf{d}^{\Phi},(V^{\Phi},B^{\Phi}))$ be two sesquilinear Young Tableaux.
\begin{itemize}
\item[(i)] We say that $\Gamma$ and $\Phi$ are \emph{equivalent} if $\mathbf{d}^{\Gamma}=\mathbf{d}^{\Phi}=[t_{1}^{i_{1}},\ldots,t_{l}^{i_{l}}]$ and $(V^{\Gamma,t_{j}}_{t_{j-1}},B^{\Gamma,t_{j}}_{t_{j-1}})$ is isomorphic to $(V^{\Phi,t_{j}}_{t_{j-1}},B^{\Phi,t_{j}}_{t_{j-1}})$ for all $j=1,\ldots,l$.
\item [(ii)] We say that $\Gamma$ is \emph{$\epsilon$-Hermitian} if $(V^{\Gamma,t_{j}},B^{\Gamma,t_{j}})$ is an $\epsilon$-Hermitian module for all $j=1,\ldots,l$.
\item [(iii)] Given an $\epsilon$-Hermitian module $(V,B)$, we say that $\Gamma$ is \emph{admissible} for $(V,B)$, or \emph{$(V,B)$-admissible}, if $(\oplus_{j}V^{\Gamma,t_{j}},\oplus_{j}B^{\Gamma,t_{j}})$ is isomorphic to $(V,B)$.
\end{itemize}
\end{definition}
Recall that given an $\sl_{2}$-triple $\gamma=\{X,H,Y\} \subset \mathfrak{g}$ there exists $t_{1}>\ldots >t_{l}$ such that $V=\oplus_{j} V^{\gamma,t_{j}}$. Using this decomposition we may define an $\epsilon$-Hermitian Young tableaux $\Gamma_{\gamma}=(\mathbf{d}^{\Gamma_{\gamma}},(V^{\Gamma_{\gamma}},B^{\Gamma_{\gamma}}))$ by setting $\mathbf{d}^{\Gamma_{\gamma}}=[t_{1}^{i_{1}},\ldots,t_{l}^{i_{l}}]$, where $i_{j}=\dim V^{\gamma,t_{j}}_{t_{j}-1}$, and
\begin{equation}
\label{defvgt}
(V^{\Gamma_{\gamma},t_{j}}_{t_{j}-1},B^{\Gamma_{\gamma},t_{j}}_{t_{j}-1}) := (V^{\gamma,t_{j}}_{t_{j}-1},B^{\gamma,t_{j}}_{t_{j}-1}),
\end{equation}
for all $j=1,\ldots,l$.
Methods of \cite[Section 9.3]{CM92} imply that this assignment gives a
bijection between the set of $\sl_{2}$-triples in $\mathfrak{g}$
up to the Adjoint action of $G$ and equivalence classes of
admissible $\epsilon$-Hermitian Young tableaux. We thus have
\begin{proposition}
There is a $1$-$1$ correspondence between the following sets:
\[
\{\mbox{Nilpotent orbits in $\mathfrak{g}$}\} \longleftrightarrow
\left\{\begin{array}{c} \mbox{Equivalence classes of admissible}\\
\mbox{$\epsilon$-Hermitian Young tableaux}\end{array}\right\}.
\]
\end{proposition}
\begin{remark} If $\k$ is the field of real numbers then Hermitian Young
tableaux can be more concretely described by \emph{signed} Young diagrams
\cite[Section 9.3]{CM92}.
\end{remark}
\subsection{Generalized Whittaker models associated to nilpotent orbits}
\label{subsec:GWM}
Let $\gamma=\{X,H,Y\}\subset \mathfrak{g}$ be an $\sl_{2}$-triple. As in the introduction, we define the character $\chi_{\gamma}$ of $U$, by $\chi_{\gamma}(\exp Z)=\psi(\kappa(X,Z))$ for all $Z\in \u$; and a symplectic structure on $\mathfrak{g}_{-1}$ by setting
\[
\kappa_{-1}(S,T)=\kappa(\operatorname{ad}(X)S,T)=\kappa(X,[S,T]) \qquad \mbox{for all $S$, $T \in \mathfrak{g}_{-1}$.}
\]
(Note the similarity between this definition and the definition of the forms $B_{i}$ on $V_{i}$ in Section \ref{subsection:nilpotentorbits}.) Let $\H_{\gamma}$ be the Heisenberg group associated to the symplectic space $(\mathfrak{g}_{-1}, \kappa_{-1})$. That is $\H_{\gamma}=\mathfrak{g}_{-1}\times \k$, $\{0\}\times \k$ is central, and $(T,0)(S,0)=(T+S,\kappa _{-1}(T,S)/2)$ for all $T$, $S \in \mathfrak{g}_{-1}$. Then, according to the Stone-von Neumann theorem, there exists a unique, up to equivalence,
\emph{smooth} irreducible (unitarizable) representation $(\rho_{\gamma},\S_{\gamma})$ of $\H_{\gamma}$ such that the center of $\H_{\gamma}$ acts by the character $\psi$. Here smooth means that it is locally constant if $\k$ is non-Archimedian, and if $\k$ is Archimedian, $(\rho_{\gamma},\S_{\gamma})$ is the smoothing of the usual irreducible unitary representation of $\H_{\gamma}$ with the central character $\psi$.
Let $\alpha_{\gamma}:U\longrightarrow \k$ be the map given by $\alpha_{\gamma}(\exp Z)=\kappa(X,Z)$, for all $Z\in \u$. It is standard to check that $\alpha_{\gamma}$ defines a surjective group homomorphism. Furthermore, it extends to a group homomorphism $\alpha_{\gamma}:N\mapsto \H_{\gamma}$ given by
\begin{equation}
\alpha_{\gamma}(\exp T \exp Z)=(T,\kappa(X,Z)), \qquad \mbox{for all $T\in \mathfrak{g}_{-1}$, $Z\in \u$}. \label{eq:alphagammadefinition}
\end{equation}
By composition, this yields a representation $(\rho_{\chi_{\gamma}},\S_{\chi_{\gamma}})$ of $N$, where $\S_{\chi_{\gamma}}:=\S_{\gamma}$ and
\begin{equation}
\label{eq:defrcg}
\rho_{\chi_{\gamma}}(n)v=\rho_{\gamma}(\alpha_{\gamma}(n))v, \qquad
\mbox{for all $n\in N$, $v\in \S_{\chi_{\gamma}}$.}
\end{equation}
Observe that then, for all $Z\in \u$, $v\in \S_{\chi_{\gamma}}$,
\[
\rho_{\chi_{\gamma}}(\exp Z)v=\rho_{\gamma}(0,\kappa(X,Z))v=\psi(\kappa(X,Z))=\chi_{\gamma}(\exp Z).
\]
In particular, if $\mathfrak{g}_{-1}=0$, then $N=U$ acts on the $1$-dimensional space $\S_{\chi_{\gamma}}$ by the character $\chi_{\gamma}$. Since $M_{X}$ preserves $\gamma $,
it is well-known \cite{Weil} that there exists a central cover of
$M_{X}$, to be denoted by $M_{\chi_{\gamma}}$, and a representation
of a semi-direct product $M_{\chi_{\gamma}}\ltimes N$ on
$\S_{\chi_{\gamma}}$ which extends the representation
$\rho_{\chi_{\gamma}}$ of $N$. (When the central cover splits, one may take $M_{\chi_{\gamma}}$ to be $M_{X}$ itself. See \cite{RR93} for the explicit description of the central cover.) We refer to the representation $(\rho_{\chi_{\gamma}},\S_{\chi_{\gamma}})$ of $M_{\chi_{\gamma}}\ltimes N$ as the smooth oscillator-Heisenberg representation associated to $\chi_{\gamma}$. We remark that there is a notion of ``smooth Fr\'{e}chet representations of moderate growth" for groups of the type $M_{\chi_{\gamma}}\ltimes N$. See \cite[Definition 1.4.1]{du} or \cite[Section 2]{Su}.
\begin{definition}
\label{def:gwm}
Let $(\pi,\mathscr{V})$ be a smooth representation of $G$, and let $\gamma=\{X,H,Y\}\subset \mathfrak{g}$ be an $\sl_{2}$-triple. We define
the \emph{space of generalized Whittaker models of $\pi$ associated
to $\gamma $} to be
\begin{equation}
\label{defwhittaker}
\operatorname{Wh}_{\gamma}(\pi)=\operatorname{Hom}_{N}(\mathscr{V},\S_{\chi_{\gamma}}).
\end{equation}
Note that $\operatorname{Wh}_{\gamma}(\pi)$ is naturally an
$M_{\chi_{\gamma}}$-module.
\end{definition}
We also make the following definition.
\begin{definition}
Let $\mathcal{O}\subset \mathfrak{g}$ be a nonzero nilpotent orbit. We say that $\gamma=\{X,H,Y\}$ is an \emph{$\sl_{2}$-triple of type $\mathcal{O}$} if $X\in \mathcal{O}$, where $X$ is the nilpositive element of the $\sl_{2}$-triple.
\end{definition}
From the well-known results of Jacobson-Morozov and Kostant \cite[Chapter 3]{CM92}, the map $\gamma=\{X,H,Y\}\mapsto \mathcal{O} =\operatorname{Ad} G
\cdot X$ yields a 1-1 correspondence between
\[
\left\{\begin{array}{c} \mbox{$\operatorname{Ad} G$ conjugacy classes of}\\
\mbox{$\sl_{2}$-triples in $\mathfrak{g}$}\end{array}\right\}
\longleftrightarrow \left\{\begin{array}{c} \mbox{Nonzero nilpotent $\operatorname{Ad} G$-orbits}\\
\mbox{$\mathcal{O} \subset \mathfrak{g}$}\end{array}\right\}.
\]
As noted in the introduction, it is clear that given two
conjugate $\sl_{2}$-triples $\gamma$, $\gamma '$, there will be an obvious
isomorphism $\phi:\operatorname{Wh}_{\gamma}(\pi)\longrightarrow
\operatorname{Wh}_{\gamma '}(\pi)$ that intertwines the action of
$M_{\chi_{\gamma}}$ and $M_{\chi_{\gamma '}}$. By abuse of notation, we will denote
$\operatorname{Wh}_{\gamma}(\pi)$ just by $\operatorname{Wh}_{\mathcal{O}}(\pi)$ and we will
call it the \emph{space of generalized Whittaker models associated
to $\mathcal{O}$}, or the \emph{space of generalized Whittaker models of
type $\mathcal{O}$}.
\begin{remark} Recall the convention in Remark \ref{rmkzero} for $X=0$. With this convention the expression in \eqref{defwhittaker} and therefore Definition \ref{def:gwm} will make sense for all nilpotent orbits.
\end{remark}
\section{A realization of generalized Whittaker models: $\k$ Archimedean}
Let $(\pi,\mathscr{V})$ be a smooth representation of $G$, and let
$\gamma=\{X,H,Y\}\subset\mathfrak{g}$ be an $\sl_{2}$-triple. In this section
we will give convenient realizations of the spaces
$\S_{\chi_{\gamma}}$ and $\operatorname{Wh}_{\gamma}(\pi)$ associated to $\gamma$
and $\pi$. As we will see later in this section, the analysis
required for $\k$ Archimedian is substantially more involved than
the case where $\k$ is non-Archimedian. For this reason, we will
devote this section to the Archimedean case and will be contented to
just indicate how the analog results work in the non-Archimedian
case.
\label{frechet}
\subsection{Norms on $G$}\label{norms} Assume that $\k=\mathbb{R}$ or $\mathbb{C}$.
Let $(V,B_{V})$ be an $\epsilon$-Hermitian module of dimension $n$. Then we
have the following possibilities for $G=G(V)$:
\begin{enumerate}
\item $\k=\mathbb{R}$ and $D=\mathbb{R}$. In this case, either
\[
\mbox{$G\cong O(p,q)$, $p+q=n$, if $\epsilon=1$,} \qquad \mbox{or}\qquad \mbox{$G\cong Sp(n,\mathbb{R})$ if $\epsilon=-1$ and $n$ is even.}
\]
\item $\k=\mathbb{R}$ and $D=\mathbb{C}$. In this case
\[
\mbox{$G\cong U(p,q)$, $p+q=n$, \qquad \qquad \qquad \qquad \mbox{regardless of} \, $\epsilon \ \ (=\pm 1$).}
\]
\item $\k=\mathbb{R}$ and $D=\mathbb{H}$. In this case either
\[
\mbox{$G\cong Sp(p,q)$, $p+q=n$, if $\epsilon=1$,} \qquad \mbox{or}\qquad \mbox{$G\cong O^{\ast}(2n)$, if $\epsilon=-1$}.
\]
\item $\k=\mathbb{C}$ and $D=\mathbb{C}$. Then
\[
\mbox{$G\cong O(n,\mathbb{C})$ if $\epsilon=1$} \qquad \mbox{or} \qquad \mbox{$G \cong Sp(n,\mathbb{C})$ if $\epsilon=-1$ and $n$ is even.}
\]
\end{enumerate}
Let $V=E\oplus U\oplus F$, with $E$ and $F$ totally isotropic,
complementary submodules of maximal dimension. Thus $U$ is anisotropic. Let
$P_{E}=\mbox{Stab}_{E}$, the stabilizer of $E$. Then $P_{E}=M_{E}N_{E}$, is a Langlands decomposition of $P_{E}$,
where $M_{E}=\{m\in P_{E}\, | \, m\cdot F\subset F\}$ and the Lie algebra of $N_{E}$ is
\[
\mathfrak{n}_{E}\cong \operatorname{Hom}(U,E)\oplus \mathfrak{z},
\]
with $\mathfrak{z}\cong \{Z:F \rightarrow E\, | \, Z^{\ast}=-Z\}$. It follows from this description that $M_{E}\cong \operatorname{GL}(E)\times G(U)$ and that $G(U)$ is compact. Thus there exists
a real inner product $B^{+}_{U}$ on $U$ such that $G(U,B)\subset G(U,B^{+}_{U})$.
Let $\{e_{1},\ldots, e_{l}\}$, and $\{f_{1},\ldots,f_{l}\}$ be basis of $E$ and $F$, respectively, such that
\[
B_{V}(e_{i},f_{j})=\delta_{i,l-j+1}, \qquad \mbox{for all $1\leq i,j\leq l$.}
\]
Then we can extend $B^{+}_{U}$ to an inner product $B^{+}_{V}$ on $V$ by setting $\{e_{1},\ldots, e_{l},f_{1},\ldots,f_{l}\}$ to be an orthonormal basis of $E\oplus F$, and $(E\oplus F)^{\perp}=U$.
If we set $K=G(V,B)\cap G(V,B^{+}_{V})$, then $K$ is a maximal compact subgroup of $G(V,B_{V})$. Moreover, if we set
\[a(\lambda _1,...,\lambda_l)=\left[\begin{smallmatrix} \lambda_{1} & & & & & & \\
& \ddots & & & & & \\
& &\lambda_{l} & & & & \\
& & &I_{\dim U} & & & \\
& & & & \lambda_{l}^{-1}& & \\
& & & & & \ddots & \\
& & & & & & \lambda_{1}^{-1} \end{smallmatrix}\right],\]
then \[
A=\left. \left\{ a(\lambda _1,...,\lambda_l)\, \right| \, \mbox{$\lambda_{i}\in \mathbb{R}^{\ast}$, for $i=1,\ldots,l$}\right\}
\]
is a maximal split torus in $G(V, B_{V})$ and, according to the Cartan decomposition, for any $g\in G(V,B_{V})$, there exists $k_{1}$, $k_{2}\in K$, and $a\in A$ such that $g=k_{1}ak_{2}$.
Given $g\in G(V, B_{V})$, let
\begin{equation}
\label{normg}
\|g\|=\sup_{B^{+}_{V}(v,v)=1} [B^{+}_{V}(gv,gv)]^{\frac{1}{2}}
\end{equation}
be the operator norm restricted to $G(V, B)$. Observe that if $g=k_{1}ak_{2}$, with $k_{1}$, $k_{2}\in K$ and $a=a(\lambda _1,...,\lambda_l)$, then
\[
\|g\|=\max\{|\lambda_{1}|,\ldots,|\lambda_{l}|,|\lambda_{1}|^{-1},\ldots,|\lambda_{l}|^{-1}\}.
\]
From this observation it is immediate that $\|g\|=\|g^{-1}\|$ and $\|\exp tX\|=\|\exp X\|^{t}$ for all $t\geq 0$, and all $X\in \a=\operatorname{Lie}(A)$. Since $\|\cdot \|$ is the operator norm in $\operatorname{End}(V)$ we also have that $\|g_{1}g_{2}\|\leq \|g_{1}\|\|g_{2}\|$ for all $g_{1}$, $g_{2}\in G$, and hence $\|\cdot \|$ satisfies all the properties of a norm on $G$ \cite[2.A.2]{Wa88}.
\subsection{Inequalities regarding norms}
Let $\gamma=\{X,H,Y\}\subset \mathfrak{g}$ be an $\sl_{2}$-triple, and let
$P=MN$ be as in Section \ref{subsection:nilpotentorbits}. Assume that $V=\oplus_{k=-r}^{r} V_{k}$, and set
$\operatorname{End}_{i}(V)=\oplus_{k=-r}^{r}
\operatorname{Hom}(V_{k},V_{k+i})$, where $V_{k}=0$ if $|k|>r$. Then
\[
\operatorname{End}(V)=\bigoplus_{i=-2r}^{2r} \operatorname{End}_{i}(V),
\]
and $\mathfrak{g}_{i}=\mathfrak{g}\cap \operatorname{End}_{i}(V)$. Recall that, if $\mathfrak{g}_{-1}\neq 0$,
then there is a symplectic structure on $\mathfrak{g}_{-1}$ given by $\kappa
_{-1}(S,T)=\frac{1}{2}\operatorname{Tr}(X[S,T])$. Let $\mathfrak{g}_{-1}=\mathfrak{e}\oplus \mathfrak{f}$ be a complete
polarization of $\mathfrak{g}_{-1}$ with respect to this symplectic structure,
and let $\operatorname{Her}_{-1}(V)=\set{T\in \operatorname{End}_{-1}(V)}{T^{\ast}=T}$. (Here
$\operatorname{Her}_{-1}$ stands for Hermitian of degree $-1$.) Then we have a
decomposition $\operatorname{End}_{-1}(V)=\mathfrak{g}_{-1}\oplus \operatorname{Her}_{-1}(V)=\mathfrak{e}\oplus \mathfrak{f}
\oplus \operatorname{Her}_{-1}(V)$. Define a norm $\|\cdot\|_{-1}$ on
$\operatorname{End}_{-1}(V)$ by setting
\[
\|T_{\mathfrak{e}}+T_{\mathfrak{f}}+T_{\operatorname{Her}_{-1}(V)}\|_{-1}=\|T_{\mathfrak{e}}\|+\|T_{\mathfrak{f}}\|+\|T_{\operatorname{Her}_{-1}(V)}\|,
\]
where, as before, $\|\cdot \|$ is the operator norm, $T_{\mathfrak{e}}\in \mathfrak{e}$,
$T_{\mathfrak{f}}\in \mathfrak{f}$, and $T_{\operatorname{Her}_{-1}(V)}\in \operatorname{Her}_{-1}(V)$ are
arbitrary. Now given $T\in \operatorname{End}(V)$, let
\[
\|T\|_{\gamma}=\|T_{-1}\|_{-1}+\sum_{i\neq -1} \|T_{i}\|,
\]
where $T=\oplus T_{i}$, with $T_{i}\in \operatorname{End}_{i}(V)$. Then $\|\cdot \|_{\gamma}$ defines a norm on $\operatorname{End}(V)$ and, since all norms on a finite dimensional vector space are equivalent, there exists constants $C_{1}$, $C_{2}>0$ such that for all $T\in \operatorname{End}(V)$
\begin{equation}\label{eq:equivalenceofnorms}
C_{1}\|T\| \leq \|T\|_{\gamma} \leq C_{2}\|T\|.
\end{equation}
Recall that $\mathfrak{n} =\u \oplus \mathfrak{g}_{-1}=\u \oplus \mathfrak{e}\oplus \mathfrak{f}$. Let $\mathfrak{n}_{\mathfrak{e}}=\u\oplus\mathfrak{e}$ and $\mathfrak{n}_{\mathfrak{f}}=\u\oplus\mathfrak{f}$, which are ideals of $\mathfrak{n}$.
Let
\begin{equation}
\label{eq:nenf}
N_{\mathfrak{e}}=\exp\mathfrak{n}_{\mathfrak{e}} \ \ \ \text{and} \ \ \ N_{\mathfrak{f}}=\exp \mathfrak{n}_{\mathfrak{f}}
\end{equation}
be the corresponding normal subgroups of $N$. Observe that for every $n\in N$ there exists unique $u\in U$, $Z_{\mathfrak{f}}\in \mathfrak{f}$ and $Z_{\mathfrak{e}}\in \mathfrak{e}$ such that $n=u(\exp Z_{\mathfrak{f}})(\exp Z_{\mathfrak{e}})$. It follows immediately that $N_{\mathfrak{f}}\backslash N\cong \mathfrak{e}$.
\begin{lemma} For all $Z\in \mathfrak{e}$, $\tilde{n}\in N_{\mathfrak{f}}$, we have
\begin{equation}
\label{eq:ntildeZ}
\|\tilde{n}\, \exp Z\|\geq C_1(1+\|Z\|).
\end{equation}
\end{lemma}
\begin{proof}
Observe that $\tilde{n}=\exp Z_{1} \exp Z_{2}$, for some $Z_{1}\in
\u$, $Z_{2}\in \mathfrak{f}$. Hence, as an element of $\operatorname{End}(V)$, $\tilde{n}\, \exp
Z= \exp Z_{1} \exp Z_{2} \exp Z =1+Z+Z_{2}+\tilde{Z}$, for
some $\tilde{Z}\in \oplus_{k=2}^{2r} \operatorname{End}_{-k}(V)$. Therefore,
\begin{eqnarray*}
\|\tilde{n}\, \exp Z\|_{\gamma} & = & \|1+Z+Z_{1}+\tilde{Z}\|_{\gamma} \nonumber \\
& = & 1+\|Z\|+\|Z_{1}\|+\|\tilde{Z}\|_{\gamma} \nonumber\\
&\geq & 1+\|Z\|.
\end{eqnarray*}
The lemma then follows from (\ref{eq:equivalenceofnorms}).
\end{proof}
\begin{lemma}\label{lemma:ntildeninequality} There exists constants $\tilde{d}$, $\tilde{C} >0$ such that for all $Z\in \mathfrak{e}$, $\tilde{n}\in N_{\mathfrak{f}}$
\[
\|\tilde{n} (\exp Z)\|^{\tilde{d}}\geq \tilde{C}\|\tilde{n}\|\|(\exp Z)\|.
\]
\end{lemma}
\begin{proof}
Since $\|\cdot \|$ is the operator norm on $\operatorname{End}(V)$, we have that
\begin{equation}\label{eq:nZinequality}
\|\exp Z\| \leq 1+\|Z\|+\frac{\|Z\|^{2}}{2!}+\cdot + \frac{\|Z\|^{2r}}{2r!}\leq C_{3}(1+\|Z\|)^{2r},
\end{equation}
for some constant $C_{3} >0$. Here we have used that $V=\oplus_{k=-r}^{r} V_{k}$. Combining equations (\ref{eq:ntildeZ}) and (\ref{eq:nZinequality}) we get that
\begin{equation}\label{eq:inequalityntildenn}
\|\tilde{n} (\exp Z)\|^{2r}\geq C_{1}^{2r}(1+\|Z\|)^{2r}\geq \frac{C_{1}^{2r}}{C_{3}} \|\exp Z\|.
\end{equation}
On the other hand, since the restriction of $\|\cdot\|$ to $G$ defines a norm on $G$, we have that $\|\tilde{n}\|\leq \|(\exp Z)^{-1}\|\|\tilde{n} (\exp Z)\|=\|\exp Z\|\|\tilde{n} (\exp Z)\|
$. From this and equation (\ref{eq:inequalityntildenn})
\[
\|\tilde{n} (\exp Z)\| \geq \frac{\|\tilde{n}\|}{\|\exp Z\|}\geq \frac{C_{1}^{2r}}{C_{3}} \frac{\|\tilde{n}\|}{\|\tilde{n} (\exp Z)\|^{2r}},
\]
and hence $\|\tilde{n} (\exp Z)\|^{2r+1}\geq \frac{C_{1}^{2r}}{C_{3}}\|\tilde{n}\|$. But now this inequality and equation (\ref{eq:inequalityntildenn}) imply that, if we set $\tilde{C}=(\frac{C_{1}^{2r}}{C_{3}})^{2}$ and $\tilde{d}=4r+1$, then
\[
\|\tilde{n} (\exp Z)\|^{\tilde{d}}\geq \tilde{C}\|\tilde{n}\| \|\exp Z\|.
\]
\end{proof}
\begin{corollary}\label{cor:nmktildeninequality}
There exists constants $d_{0}$, $C_{0} >0$ such that, for all $k\in K$, $m\in M$, $Z\in \mathfrak{e}$, and $\tilde{n}\in N_{\mathfrak{f}}$
\[
\|\tilde{n} (\exp Z)mk\|^{d_{0}} \geq C_{0}\|\tilde{n}\|\|\exp Z\|\|m\|.
\]
\end{corollary}
\begin{proof}
By definition of $\|\cdot\|$, $\|\tilde{n} (\exp Z)mk\|=\|\tilde{n} (\exp Z)m\|$. Now, from the proof of \cite[Theorem 7.2.1]{Wa88}, we have that
\begin{equation}
\label{eq:ntildenminequality}
\|\tilde{n} (\exp Z)m\| \geq \|m\| \ \mbox{and}\ \|\tilde{n} (\exp Z)m\|^{2} \geq \|\tilde{n} (\exp Z)\|.
\end{equation}
Let $\tilde{C}$ and $\tilde{d}$ be as in Lemma \ref{lemma:ntildeninequality}. Then, if we set $d_{0}=2\tilde{d}+1 $ and $C_{0}=\tilde{C}$, we have that
\[
\|\tilde{n} (\exp Z)mk\|^{d_{0}}\geq \|\tilde{n} (\exp Z)\|^{\tilde{d}}\|m\|\geq C_{0}\|\tilde{n}\| \|\exp Z\| \|m\|.
\]
\end{proof}
\subsection{Some spaces of rapidly decreasing functions on $G$}
We retain all of the notation from the previous section.
Let $(\rho_{\chi_{\gamma}},\S_{\chi_{\gamma}})$ be the smooth
Heisenberg representation of $N$ associated to the character
$\chi_{\gamma}$ of $U$, as in (\ref{eq:defrcg}). We first give a
realization of $\S_{\chi_{\gamma}}$, as follows: fix
$\mathfrak{g}_{-1}=\mathfrak{e}\oplus \mathfrak{f}$ (a complete polarization of $\mathfrak{g}_{-1}$). Set
$\mathscr{D}(\mathfrak{e})$ to be the space of constant-coefficient differential
operators on $\mathfrak{e}$, and let
\[\S(\mathfrak{e})=\set{f\in C^{\infty}(\mathfrak{e})}{\mbox{$p_{Z,d}(f) <\infty$ for
all $Z\in \mathscr{D}(\mathfrak{e})$, $d\in \mathbb{N}$}}\]
be the Schwartz space of $\mathfrak{e}$, where
\begin{equation}
p_{Z,d}(f)=\sup_{T\in \mathfrak{e}}|Zf(T)|(1+\|T\|)^{d}. \label{eq:pZddefinition}
\end{equation}
Extend $\chi_{\gamma}$ to $N_{\mathfrak{f}}$ by setting
\[
\chi_{\gamma}(u\exp Z)=\chi_{\gamma}(u), \qquad \mbox{for all $u\in U$, $Z\in \mathfrak{f}$.}
\]
We shall adopt the following notation. For a smooth representation $\sigma$ of a closed subgroup $B$ of a Lie group $A$, let
\begin{equation}
\label{sminduction}
C^{\infty}(B\backslash A; \sigma)=\{f\in C^{\infty}(A; \sigma) \, | \, f(ba)=\sigma (b)f(a), \text{for all $b\in B$ and $a\in A$}\}.
\end{equation}
Through right multiplication, this becomes a representation of $A$ (smoothly induced from $\sigma$). Later, we will also need to consider the space $C_{c}^{\infty}(B\backslash A; \sigma)$ consisting of those elements in $C^{\infty}(B\backslash A; \sigma)$ with compact support modulo $B$.
Given $f\in\mathbb{C}^{\infty}(N_{\mathfrak{f}}\backslash N; \chi_{\gamma})$, define $\hat{f}\in C^{\infty}(\mathfrak{e})$ by $\hat{f}(Z)=f(\exp Z)$, for all $Z\in \mathfrak{e}$. Conversely, given $f\in C^{\infty}(\mathfrak{e})$, set $\check{f}(n\exp Z)=\chi_{\gamma}( n)f(Z)$ for all $n\in N_{\mathfrak{f}}$, $Z\in \mathfrak{e}$. Clearly these two maps are inverse of each other, and if we set
\[
\S (N_{\mathfrak{f}}\backslash N; \chi_{\gamma})=\{f\in C^{\infty}(N_{\mathfrak{f}}\backslash N; \chi_{\gamma}) \, | \, \, \, \hat{f}\in \S(\mathfrak{e})\},
\]
then
\begin{equation}
\S_{\chi_{\gamma}}\cong \S (N_{\mathfrak{f}}\backslash N; \chi_{\gamma}). \label{eq:Schigammarealization}
\end{equation}
In the rest of this article we will frequently use this realization of $\S_{\chi_{\gamma}}$ (implicitly).
\begin{remarks} (a) In the literature, the Lie algebras $\u$, $\mathfrak{n}$ are frequently denoted by $\mathfrak{n}_{2}$ and $\mathfrak{n}_{1}$, respectively. When that is the case, the Lie algebra $\mathfrak{n}_{\mathfrak{f}}$ is often denoted by $\mathfrak{n}_{1.5}$.
\noindent (b) When $\k$ is non-Archimedian, we set $\S(\mathfrak{e})$ to be the space of all the locally constant, compactly supported functions on $\mathfrak{e}$ (known as the Bruhat-Schwartz space on $\mathfrak{e}$). With this definition equation (\ref{eq:Schigammarealization}) remains valid for $\k$ non-Archimedian.
\end{remarks}
{\vspace{0.2in}}
Let $U(\mathfrak{g})$ be the universal enveloping algebra of $\mathfrak{g}$. Given $f\in C^{\infty}(G)$, $Z\in U(\mathfrak{g})$ and $d\in \mathbb{N}$, we set
\[
q_{Z,d}(f)=\sup_{g\in G} |R_{Z}f(g)|\|g\|^{d},
\]
where $R_{Z}$ acts on $C^{\infty}(G)$ via the right regular representation of $U(\mathfrak{g})$. Let
\[
\mathscr{C}(G)=\set{f\in C^{\infty}(G)}{\mbox{$q_{X,d}(f) < \infty$, for all $X\in U(\mathfrak{g})$, $d\in \mathbb{N}$}}.
\]
It is easy to check that $\mathscr{C}(G)$ is a Fr\'echet space and that
$q_{Z,d}$ is a semi-norm on $\mathscr{C}(G)$ for all $Z\in U(\mathfrak{g})$, $d\in \mathbb{N}$.
We call $\mathscr{C}(G)$ the \emph{space of rapidly decreasing functions} on
$G$.
We now define certain space of rapidly decreasing functions on $N\backslash G$. Given $f\in C^{\infty}(N\backslash G;\S_{\chi_{\gamma}})$, $Z_{1}\in
U(\mathfrak{g})$, $Z_{2}\in \mathscr{D}(\mathfrak{e})$, $d_{1}$, $d_{2}\in \mathbb{N}$, set
\begin{equation}\label{eq:seminormonGmodN}
q_{Z_{1},Z_{2},d_{1},d_{2}}(f) = \sup_{k\in K,\, m\in M,\, T\in \mathfrak{e}} |Z_{2}(R_{Z_{1}}f(mk))(T)|(1+\|T\|)^{d_{2}}\|m\|^{d_{1}}.
\end{equation}
Then we define
\[
\mathscr{C}(N \backslash G;\S_{\chi_{\gamma}})=\{f\in C^{\infty}(N\backslash G;\S_{\chi_{\gamma}}) \, | \,\mbox{$q_{Z_{1},Z_{2},d_{1},d_{2}}(f)<\infty$ for all $q_{Z_{1},Z_{2},d_{1},d_{2}}$ as in (\ref{eq:seminormonGmodN})}\}.
\]
Observe that
\[
q_{Z_{1},Z_{2},d_{1},d_{2}}(f)=\sup_{k\in K, m\in M}p_{Z_{2},d_{2}}(R_{Z_{1}}f(mk))\|m\|^{d_{1}},
\]
where $p_{Z_{2},d_{2}}$ is as in (\ref{eq:pZddefinition}). In general, given a smooth representation of $N$, $(\tau,\mathscr{H})$, and a seminorm $\rho$ on $\mathscr{H}$, we set
\begin{equation}
q_{Z,d,\rho}(f)=\sup_{k\in K, m\in M}\rho(R_{Z}f(mk))\|m\|^{d}, \label{eq:generalNG}
\end{equation}
and define
\[
\mathscr{C}(N \backslash G;\mathscr{H})=\{f\in C^{\infty}(N\backslash G;\mathscr{H}) \, | \,\mbox{$q_{Z,d,\rho}(f)<\infty$ for all $q_{Z,d,\rho}$ as in (\ref{eq:generalNG})}\}.
\]
Recall that there exists a covering $M_{\chi_{\gamma}} \twoheadrightarrow M_{X}$ such that $(\rho_{\chi_{\gamma}},\S_{\chi_{\gamma}})$ extends to a representation of $M_{\chi_{\gamma}}\ltimes N$. Using this extension, we define a natural action of $M_{\chi_{\gamma}}$ on $\mathscr{C}(N \backslash G;\S_{\chi_{\gamma}})$ by
\begin{equation}
m\cdot f(g)=\rho_{\chi_{\gamma}}(m)f(\bar{m}^{-1}g), \qquad \mbox{for all $m\in M_{\chi_{\gamma}}$, $g\in G$.} \label{eq:mactiononNG}
\end{equation}
Here $\bar{m}$ is the image of $m$ under the map $M_{\chi_{\gamma}}\twoheadrightarrow M_{X}$.
Now, given $f\in C^{\infty}(N_{\mathfrak{f}} \backslash G;\chi_{\gamma})$, $Z\in U(\mathfrak{g})$ and $d\in \mathbb{N}$, we set
\[
q_{Z,d}(f)=\sup_{k\in K, m\in M, T\in \mathfrak{e}} |R_{Z}f((\exp T)mk)|\|(\exp T)mk\|^{d},
\]
and define
\[
\mathscr{C}(N_{\mathfrak{f}} \backslash G;\chi_{\gamma})=\{f\in C^{\infty}(N_{\mathfrak{f}} \backslash G;\chi_{\gamma})\, | \, \mbox{$q_{Z,d}(f) <\infty$ for all $Z\in U(\mathfrak{g}), d\in \mathbb{N}$}\}.
\]
\begin{lemma}
\label{lemma:equalityss}
As $G$-modules, we have $\mathscr{C}(N_{\mathfrak{f}} \backslash G;\chi_{\gamma})\cong \mathscr{C}(N \backslash G;\S_{\chi_{\gamma}})$.
\end{lemma}
\begin{proof}
Given $f\in \mathscr{C}(N \backslash G;\S_{\chi_{\gamma}})$, define $\hat{f}\in C^{\infty}(N_{\mathfrak{f}} \backslash G;\chi_{\gamma})$ by $\hat{f}(g)=f(g)(0)$. We claim that $\hat{f}\in \mathscr{C}(N_{\mathfrak{f}} \backslash G;\chi_{\gamma})$. Effectively, given $Z\in U(\mathfrak{g})$, $d\in \mathbb{N}$,
\begin{eqnarray*}
q_{Z,d}(\hat{f}) & = & \sup_{k\in K, m\in M, T\in \mathfrak{e}} |R_{Z}\hat{f}((\exp T)mk)|\|(\exp T)mk\|^{d}\\
& = & \sup_{k\in K, m\in M, T\in \mathfrak{e}} |R_{Z}f(nmk)(0)|\|(\exp T)mk\|^{d}\\
& \leq & \sup_{k\in K, m\in M, T\in \mathfrak{e}} |R_{Z}f(mk)(T)|\|mk\|^{d}\|\exp T\|^{d}\\
& \leq & \sup_{k\in K, m\in M, T \in \mathfrak{e}} C_{3}|R_{Z}f(mk)(T)|\|mk\|^{d}(1+\|T\|)^{2rd}\\
& \leq & C_{3}q_{Z,1,d,2rd}(f) <\infty,
\end{eqnarray*}
where we have used equation (\ref{eq:nZinequality}) in the next to last equality.
Now given $f\in \mathscr{C}(N_{\mathfrak{f}} \backslash G;\chi_{\gamma})$, set $\check{f}(g)(T)=f(\exp (T)\,g)$, for all $g\in G$, $T\in \mathfrak{e}$. We claim that $\check{f}\in \mathscr{C}(N \backslash G;\S_{\chi_{\gamma}})$. Effectively, given $Z_{1}\in U(\mathfrak{g})$, $Z_{2}\in \mathscr{D}(\mathfrak{e})$, $d_{1}$, $d_{2}\in \mathbb{N}$, we have that
\begin{eqnarray*}
q_{Z_{1},Z_{2},d_{1},d_{2}}(\check{f}) & = & \sup_{k\in K, m\in M, T\in \mathfrak{e}} |Z_{2}(R_{Z_{1}}\check{f}(mk))(T)| \|mk\|^{d_{1}}(1+\|T\|)^{d_{2}}\\
& = & \sup_{k\in K, m\in M, T\in \mathfrak{e}} |(R_{\operatorname{Ad}(mk)^{-1} Z_{2}Z_{1}}f(\exp (T) mk )| \|mk\|^{d_{1}}(1+\|T\|)^{d_{2}},
\end{eqnarray*}
where we have identified $Z_{2}$ with an element in $U^{l}(\mathfrak{g})$ for some $l$. Let $\{\tilde{Z}_{1},\ldots,\tilde{Z}_{s}\}$ be a basis of $U^{l}(\mathfrak{g})$. Then
\[
\operatorname{Ad}(mk)^{-1} Z_{2}=\sum_{j=1}^{s} a_{j}(mk)\tilde{Z}_{j},
\]
for some functions $a_{j}$. Since $(\operatorname{Ad}, U^{l}(\mathfrak{g}))$ is finite
dimensional, it is of moderate growth, and hence
there exists constants $C_{l}$, $d_{l}>0$ such that
$|a_{j}(mk)|\leq C_{l}\|mk\|^{d_{l}}$, for all $k\in K$, $m\in M$.
From this, equations (\ref{eq:ntildeZ}) and (\ref{eq:ntildenminequality}), we have
\begin{eqnarray*}
q_{Z_{1},Z_{2},d_{1},d_{2}}(\check{f}) & \leq & C_{l}\sum_{j=1}^{s} \sup_{k\in K, m\in M, T\in \mathfrak{e}} |R_{\tilde{Z}_{j}Z_{1}}f(\exp (T) mk )| \|mk\|^{d_{1}+d_{l}}(1+\|T\|)^{d_{2}} \\
& \leq & \frac{C_{l}}{C_{1}^{d_{2}}}\sum_{j=1}^{s} \sup_{k\in K, m\in M, T\in \mathfrak{e}} |R_{\tilde{Z}_{j}Z_{1}}f(\exp (T) mk )| \|mk\|^{d_{1}+d_{l}}\|\exp T\|^{d_{2}} \\
& \leq & \frac{C_{l}}{C_{1}^{d_{2}}}\sum_{j=1}^{s} \sup_{k\in K, m\in M, T\in \mathfrak{e}} |R_{\tilde{Z}_{j}Z_{1}}f(\exp (T) mk )| \|\exp (T) mk\|^{d_{1}+d_{l}+2d_{2}} \\
& \leq & \frac{C_{l}}{C_{1}^{d_{2}}}\sum_{j=1}^{s} q_{\tilde{Z}_{j}Z_{1},d_{1}+d_{l}+2d_{2}}(f) <\infty.
\end{eqnarray*}
Finally, we easily check that $\check{\hat{f}}=f$ and
$\hat{\check{f}}=f$.
\end{proof}
Given $f\in \mathscr{C}(G)$, define a function $f_{\chi_{\gamma}}\in C^{\infty}(N_{\mathfrak{f}}\backslash G;\chi_{\gamma})$ by
\[
f_{\chi_{\gamma}}(g)=\int_{N_{\mathfrak{f}}} f(\tilde{n}g)\chi_{\gamma}(\tilde{n})^{-1}\, d\tilde{n}.
\]
\begin{lemma}
\label{lemma:smFrobenius}
The map $f\mapsto f_{\chi_{\gamma}}$ defines a surjective $G$-intertwining map from $\mathscr{C}(G)$ to the space $\mathscr{C}(N_{\mathfrak{f}} \backslash G;\chi_{\gamma})$.
\end{lemma}
\begin{proof}
We will first show that $f_{\chi_{\gamma}}\in \mathscr{C}(N_{\mathfrak{f}} \backslash G;\chi_{\gamma})$. Given $Z\in U(\mathfrak{g})$, $d\in \mathbb{N}$, we have that
\begin{eqnarray*}
q_{Z,d}(f_{\chi_{\gamma}}) & = & \sup_{k\in K, m\in M, T \in \mathfrak{e}} |R_{Z}f_{\chi_{\gamma}}((\exp T)mk)|\|(\exp T)mk\|^{d} \\
& \leq & \sup_{k, m, T} \int_{N_{\mathfrak{f}}} |R_{Z}f_{\chi_{\gamma}}(\tilde{n} (\exp Z)mk)|\, d\tilde{n}\, \|(\exp T)mk\|^{d}.
\end{eqnarray*}
Now we know that for all $d_{1}\in \mathbb{N}$, $|R_{Z}f(\tilde{n} (\exp Z)mk)|\|\tilde{n} (\exp Z)mk\|^{d_{1}}\leq q_{Z,d_{1}}(f)$. Hence, from Corollary \ref{cor:nmktildeninequality},
\begin{eqnarray*}
q_{Z,d}(f_{\chi_{\gamma}}) & \leq & \sup_{k\in K, m\in M, T \in \mathfrak{e}} q_{Z,d_{1}}(f) \int_{N_{\mathfrak{f}}} \|\tilde{n} (\exp Z)mk\|^{-d_{1}}\, d\tilde{n}\|(\exp T)mk\|^{d} \\
& \leq & \sup_{k\in K, m\in M, T \in \mathfrak{e}} q_{Z,d_{1}}(f) \int_{N_{\mathfrak{f}}} (C_{0}\|\tilde{n}\| \|\exp T\| \|m\|)^{-\frac{d_{1}}{d_{0}}} (\|\exp T\|\|m\|)^{d} \, d\tilde{n} \\
& = & \sup_{k\in K, m\in M, T \in \mathfrak{e}} q_{Z,d_{1}}(f) C_{0}^{-\frac{d_{1}}{d_{0}}} (\|\exp T\|\|m\|)^{d-\frac{d_{1}}{d_{0}}} \int_{N_{\mathfrak{f}}} \|\tilde{n}\|^{-\frac{d_{1}}{d_{0}}} \, d\tilde{n}.
\end{eqnarray*}
But it is clear that, if $d_{1} \gg 0$, then the right hand side of the above equation is finite.
Now we will show that the map is surjective. Fix a function $\phi \in C_{c}^{\infty}(N_{\mathfrak{f}})$ such that
\[
\int_{N_{\mathfrak{f}}} \phi(\tilde{n})\chi_{\gamma}(\tilde{n})^{-1} \, d\tilde{n}=1.
\]
Given $f\in \mathscr{C}(N_{\mathfrak{f}} \backslash G;\chi_{\gamma})$, let $h\in C^{\infty}(G)$ be given by $h(\tilde{n} (\exp Z)mk)=\phi(\tilde{n})f(nmk)$. Then it is clear that $h\in \mathscr{C}(G)$ and $h_{\chi_{\gamma}}(nmk)=f(nmk)$ for all $k\in K$, $m\in M$ and $Z\in \mathfrak{e}$. From all this we conclude that the map $f\mapsto f_{\chi_{\gamma}}$ is surjective.
\end{proof}
\begin{remarks} (a) Lemmas \ref{lemma:equalityss} and \ref{lemma:smFrobenius} are a form of induction by stages and Frobenius reciprocity, in the ``rapidly decreasing" context.
The key point is of course to establish the relevant estimates.
\noindent (b) When $\k$ is non-Archimedian, we set $\mathscr{C}(G)$ to be the space of locally constant, compactly supported functions, with analogous definitions for $\mathscr{C}(N\backslash G;\S_{\chi_{\gamma}})$ and $\mathscr{C}(N_{\mathfrak{f}}\backslash G;\chi_{\gamma})$. With these definitions, it is straightforward to check that all the results in this section remain valid in the non-Archimedian case.
\end{remarks}
\subsection{Realizing a generalized Whittaker model on $N\backslash G$} In this section, we give the promised realization of the space of generalized Whittaker models for a Casselman-Wallach representation of $G$. But before stating this result, we need the following technical lemma.
\begin{lemma}\label{lemma:invariantSchwartz} Let $(\pi, \mathscr{V})$ be a Casselman-Wallach
representation of $G$. Then
\[
(\mathscr{V}')^{N_{\mathfrak{f}},\chi_{\gamma}}\cong \operatorname{Wh}_{\gamma}(\pi),
\]
where $(\mathscr{V}')^{N_{\mathfrak{f}},\chi_{\gamma}}$ denotes the $(N_{\mathfrak{f}},\chi_{\gamma})$-isotypic subspace of $\mathscr{V}'$.
\end{lemma}
\begin{proof}
Given $\lambda\in \operatorname{Wh}_{\gamma}(\pi)$, set
$\hat{\lambda}(v)=\lambda(v)(0)$, for $v\in \mathscr{V}$. Then
it is clear that $\hat{\lambda}\in
(\mathscr{V}')^{N_{\mathfrak{f}},\chi_{\gamma}}$. On the other hand, given
$\lambda \in (\mathscr{V}')^{N_{\mathfrak{f}},\chi_{\gamma}}$, set
$\check{\lambda}(v)(T)=\lambda(\pi(\exp T)\, v)$ for $v\in
\mathscr{V}$, $T\in \mathfrak{e}$. Then it is clear that $\check{\lambda}(v)\in
C^{\infty}(\mathfrak{e})$, but we claim that it is actually in $\S(\mathfrak{e})$. To see
this, observe that if $R\in \mathfrak{e}$, then
\begin{equation}\label{eq:derivative}
\check{\lambda}(d\pi(R)v)(T)=D_{R}\check{\lambda}(v)(T), \qquad \mbox{for all $T\in \mathfrak{e}$,}
\end{equation}
where $D_{R}\in \mathscr{D}(\mathfrak{e})$ represents the derivative in the direction $R$. On the other hand, if $S \in \mathfrak{f}$, then \begin{equation}\label{eq:multiplication}
\check{\lambda}(d\pi(S)v)(T)=d\chi_{\gamma}(\kappa(S,T))\check{\lambda}(v)(T), \qquad \mbox{for all $T\in \mathfrak{e}$,}
\end{equation}
where $d\chi_{\gamma}$ is the linear functional given by the derivative of the character $\chi_{\gamma}$.
But now, since $(\pi, \mathscr{V})$ is a representation of moderate growth, we can find a constant $d>0$ such that for all $v\in \mathscr{V}$, there exists $C_{\lambda,v}>0$ such that
\begin{equation}\label{eq:moderategrowth}
|\check{\lambda}(v)(T)|=|\lambda(\pi(\exp T)\, v)|\leq C_{\lambda,v} \|\exp T\|^{d}\leq C_{3}C_{\lambda,v}(1+\|T\|)^{2dr},\qquad \mbox{for all $T\in \mathfrak{e}$.}
\end{equation}
Here $C_3$ and $r$ are as in equation (\ref{eq:nZinequality}). Observe that, although the constant $C_{\lambda,v}$ depends on $v$
(and on $\lambda$), $d$ is independent of the element $v\in
\mathscr{V}$ chosen. Therefore, from equations
(\ref{eq:derivative}), (\ref{eq:multiplication}) and
(\ref{eq:moderategrowth}) for all $v\in \mathscr{V}$ the function
$\check{\lambda}(v)$ is such that if we take any partial
derivatives, and multiply by any polynomial on $\mathfrak{e}$, it still has growth
controlled by a polynomial of degree $2dr$. But this implies that
$\check{\lambda}(v)\in \S(\mathfrak{e})$, as we wanted to show.
To finish the proof we just have to show that $\hat{\check{\lambda}}=\lambda$, and $\check{\hat{\lambda}}=\lambda$, but this follow easily from the definitions.
\end{proof}
Given an $\sl_{2}$-triple $\gamma=\{X,H,Y\}\subset \mathfrak{g}$, set
\begin{equation}
\label{eq:defgammacheck} \check{\gamma}=\{-X,H,-Y\}.
\end{equation}
Then, it is immediate to check that $\check{\gamma}$ is also an
$\sl_{2}$-triple and that
$\chi_{\check{\gamma}}=\check{\chi}_{\gamma}$, where
$\check{\chi}_{\gamma}$ is the character of $U$ given by
$\check{\chi}_{\gamma}(u)=\chi_{\gamma}(u)^{-1}$ for all $u\in U$.
\begin{proposition}\label{prop:SchwartzWhittakerModels}
Let $(\pi,\mathscr{V})$ be an irreducible Casselman-Wallach representation of $G$, and let $\mathscr{C}(N\backslash G;\S_{\chi_{\check{\gamma}}})_{G,\pi}$ denote the maximal $\pi$-isotypic quotient of
$\mathscr{C}(N\backslash G;\S_{\chi_{\check{\gamma}}})$. Then, as an $M_{\chi_{\gamma}}\times G$-module,
\[
\mathscr{C}(N\backslash G;\S_{\chi_{\check{\gamma}}})_{G,\pi}\cong W_{\gamma}(\check{\pi}) \otimes \pi,
\]
where $\check{\pi}$ is the contragredient Casselman-Wallach representation of $\pi$ and $W_{\gamma}(\check{\pi})$ is the continuous dual of $\operatorname{Wh}_{\gamma}(\check{\pi})$. Here $\otimes$ stands for the completed projective tensor product (of two locally convex topological spaces).
\end{proposition}
\begin{proof} By Lemma \ref{lemma:equalityss}, we have $\mathscr{C}(N\backslash G;\S_{\chi_{\check{\gamma}}})\cong \mathscr{C}(N_{\mathfrak{f}} \backslash G;\chi_{\check{\gamma}})$. Let $\lambda\in \operatorname{Hom}_{G}(\mathscr{C}(N_{\mathfrak{f}} \backslash G;\chi_{\check{\gamma}}),\mathscr{V})$, and let $(\check{\pi},\check{\mathscr{V}})$ be the representation contragredient to $(\pi,\mathscr{V})$. Then we may think of $\lambda$ as an element in a space of $\check{\mathscr{V}}'$-valued distributions. Hence, according to \cite[Theorem 3.11]{KV96} there exists $\mu_{\lambda}\in (\check{\mathscr{V}}')^{N_{\mathfrak{f}},\chi_{\gamma}}$ such that if $f\in C_{c}^{\infty}(N_{\mathfrak{f}} \backslash G;\chi_{\gamma})$, then
\[
\lambda(f)(v)=\int_{N_{\mathfrak{f}} \backslash G} f(g)\mu_{\lambda}(\pi(g)v)\, dg, \qquad \mbox{for} \ v \in \check{\mathscr{V}}.
\]
On the other hand, given $\mu \in (\check{\mathscr{V}}')^{N_{\mathfrak{f}},\chi_{\gamma}}$ and $f\in \mathscr{C}(N_{\mathfrak{f}} \backslash G;\chi_{\check{\gamma}})$, there exists $h\in \mathscr{C}(G)$ such that $f=h_{\chi_{\check{\gamma}}}$ and for all $v\in \check{\mathscr{V}}$
\begin{eqnarray*}
\int_{G} h(g)\mu_{\lambda}(\pi(g)v)\, dg & = & \int_{N_{\mathfrak{f}} \backslash G}\int_{N_{\mathfrak{f}}}h(\tilde{n}g)\mu_{\lambda}(\pi(\tilde{n})\pi(g)v)\, d\tilde{n} \,dg\\
& = & \int_{N_{\mathfrak{f}} \backslash G}\int_{N_{\mathfrak{f}}}h(\tilde{n}g) \chi_{\gamma}(\tilde{n})\, d\tilde{n}\,\, \mu_{\lambda}(\pi(g)v) \,dg\\
& = & \int_{N_{\mathfrak{f}} \backslash G}\int_{N_{\mathfrak{f}}}h(\tilde{n}g) \chi_{\check{\gamma}}(\tilde{n})^{-1}\, d\tilde{n}\,\, \mu_{\lambda}(\pi(g)v) \,dg\\
& = & \int_{N_{\mathfrak{f}} \backslash G} h_{\chi_{\check{\gamma}}}(g) \mu_{\lambda}(\pi(g)v) \,dg\\
& = & \int_{N_{\mathfrak{f}} \backslash G} f(g) \mu_{\lambda}(\pi(g)v) \,dg.
\end{eqnarray*}
Therefore, if we set
\[
\lambda_{\mu}(f)(v)=\int_{N_{\mathfrak{f}} \backslash G} f(g) \mu_{\lambda}(\pi(g)v) \,dg, \qquad \mbox{for} \, v \in \check{\mathscr{V}},
\]
then $\lambda_{\mu}\in \operatorname{Hom}_{G}(\mathscr{C}(N_{\mathfrak{f}} \backslash
G;\chi_{\check{\gamma}}),\check{\mathscr{V}}')$. Method of \cite[Section 3]{SZ11} implies that $\lambda_{\mu}(f)$ extends to a continuous linear
functional on $\mathscr{V}'$, and since $V$ is irreducible, $\lambda_{\mu}(f)$ is actually
an element of $\mathscr{V}$ for all $f\in \mathscr{C}(N_{\mathfrak{f}} \backslash
G;\chi_{\check{\gamma}})$, or equivalently $\lambda_{\mu}\in
\operatorname{Hom}_{G}(\mathscr{C}(N_{\mathfrak{f}} \backslash G;\chi_{\check{\gamma}}),\mathscr{V})$. Now it
is easy to check that $\lambda_{\mu_{\lambda}}=\lambda$ and
$\mu_{\lambda_{\mu}}=\mu$. Therefore
\[
\mathscr{C}(N_{\mathfrak{f}} \backslash G;\chi_{\check{\gamma}})_{G,\pi} \cong \pi\otimes \check{\pi}_{N_{\mathfrak{f}},\chi_{\gamma}}.
\]
Finally, according to Lemma \ref{lemma:invariantSchwartz} and by taking the continuous dual, we have $\check{\pi}_{N_{\mathfrak{f}},\chi_{\gamma}}\cong W_{\gamma}(\check{\pi})$. Therefore we obtain the required isomorphism, which is easily checked to be $M_{\chi_{\gamma}}$-equivariant. \end{proof}
\begin{remarks} (a) In the Archimedian setting whenever we take the tensor product of two locally convex topological vector spaces, we always mean the completed projective tensor product. Observe that if $(\pi,\mathscr{V})$ is a Casselman-Wallach representation, then $\mathscr{V}$ is nuclear, and hence the completed projective tensor product of $\mathscr{V}$ with any locally convex topological vector space is equivalent to the completed injective tensor product \cite{G55}.
\noindent (b) Proposition \ref{prop:SchwartzWhittakerModels} also
holds for $\k$ non-Archimedian, if we replace in its statement
Casselman-Wallach representations by smooth, finitely generated,
admissible representations, and completed projective tensor product
by algebraic tensor product. The proof for this case follows the
same line, but it is more straightforward.
\end{remarks}
\section{Dual pairs and theta lifting of nilpotent orbits}
\label{sec:liftOrbit}
\subsection{Reductive dual pairs of type I} \label{reductivepairs}
Let $(V,B)$ be an $\varepsilon$-Hermitian module, and let
$(\tilde{V},\tilde{B})$ be an $\tilde{\varepsilon}$-Hermitian
module such that $\varepsilon\tilde{\varepsilon}=-1$. Let
$G=G(V)$ and $\tilde{G}=G(\tilde{V})$ be its associated isometry
groups. We will use the $\varepsilon$-Hermitian form $B$ to
identify the $\k$-vector spaces $\tilde{V}\otimes_{D}V^{\ast}$ and
$\operatorname{Hom}_{D}(V,\tilde{V})$. Given $T\in \operatorname{Hom}_{D}(V,\tilde{V})$, define
$T^{\ast}\in \operatorname{Hom}_{D}(\tilde{V},V)$ by
\[
\tilde{B}(Tv,\tilde{v})=B(v,T^{\ast}\tilde{v}) \qquad
\mbox{for all $v\in V$, $\tilde{v}\in \tilde{V}$.}
\]
Define similarly the $*$ map from $\operatorname{Hom}_{D}(\tilde{V},V)$ to $\operatorname{Hom}_{D}(V,\tilde{V})$. Note that since $\varepsilon\tilde{\varepsilon}=-1$, we have $T^{\ast \ast}=-T$ for $T\in \operatorname{Hom}_{D}(V,\tilde{V})$.
We now define a
symplectic form $\langle\cdot , \cdot \rangle$ on $\operatorname{Hom}_{D}(V,\tilde{V})$ by setting
\begin{equation}
\label{defsympro}
\langle T,S\rangle =\operatorname{Tr}(T^{\ast}S) \qquad \mbox{for all $T$, $S\in \operatorname{Hom}_{D}(V,\tilde{V})$,}
\end{equation}
where $\operatorname{Tr}(T^{\ast}S)$ is the trace of $T^{\ast}S$ as a $\k$-linear transformation.
Let
\[
\operatorname{Sp}(\operatorname{Hom}_{D}(V,\tilde{V}))=\left\{g\in \operatorname{GL}(\operatorname{Hom}_{D}(V,\tilde{V}),\k) \, \left| \, \begin{array}{c} \mbox{$\langle g \cdot T, g\cdot S \rangle= \langle T, S \rangle$}\\ \mbox{for all $T$, $S \in \operatorname{Hom}_{D}(V,\tilde{V})$} \end{array} \right.\right\}.
\]
Then there is a natural map $G \times \tilde{G}\longrightarrow \operatorname{Sp}(\operatorname{Hom}_{D}(V,\tilde{V}))$ given by
\[
(g,\tilde{g})\cdot T = \tilde{g} T g^{-1} \qquad \mbox{for all $T \in \operatorname{Hom}(V,\tilde{V})$, $g\in G$, $\tilde{g}\in \tilde{G}$}.
\]
We will use this map to identify $G$ and $\tilde{G}$ with subgroups
of $\operatorname{Sp}(\operatorname{Hom}_{D}(V,\tilde{V}))$. These two subgroups are mutual
centralizers of each other, and form an example of a {\em reductive
dual pair} of type I. See \cite{Ho79}.
\subsection{Moment maps and lifting of nilpotent orbits}
\label{subsec:liftOrbit}
Given $T\in \operatorname{Hom}_{D}(V,\tilde{V})$, it is clear that $T^{\ast}T\in \mathfrak{g}$ and $TT^{\ast}\in \tilde{\mathfrak{g}}$. Following \cite{KP82, DKP} we define the \emph{moment maps} to be
\begin{eqnarray*}
\varphi:\operatorname{Hom}_{D}(V,\tilde{V}) & \longrightarrow & \mathfrak{g}\\
T & \mapsto & T^{\ast}T
\end{eqnarray*}
and
\begin{eqnarray*}
\tilde{\varphi}:\operatorname{Hom}_{D}(V,\tilde{V}) & \longrightarrow & \tilde{\mathfrak{g}}\\
T & \mapsto & TT^{\ast}.
\end{eqnarray*}
It is also clear that $\varphi (T)$ is nilpotent if and only if $\tilde{\varphi}(T)$ is nilpotent.
As in the introduction, let $\operatorname{Max} \operatorname{Hom} (V,\tilde{V})$ be the set of
full rank elements in $\operatorname{Hom}(V,\tilde{V})$. Without any loss of
generality, we assume that $\dim V \leq \dim \tilde{V}$, and
elements of $\operatorname{Max} \operatorname{Hom} (V,\tilde{V})$ are then represented by
injective maps from $V$ to $\tilde{V}$. Recall also our standing
assumption on the nilpotent orbit $\mathcal{O} \subset \mathfrak{g}$:
\begin{equation}
\label{assumption}
\varphi ^{-1}(\mathcal{O}) \cap \operatorname{Max} \operatorname{Hom} (V,\tilde{V})\ne \emptyset.
\end{equation}
We first prove a result on dual pairs in the stable range, which says that in this case any (not just nilpotent) orbit satisfies the stated assumption.
Recall that the dual pair $(G,\tilde{G})$ is in the stable range, with $G$ the smaller member, if there is
a polarization $\tilde{V}=\tilde{E}\oplus \tilde{U}\oplus \tilde{F}$, where $\tilde{E}$, $\tilde{F}$ are
totally isotropic complementary subspaces with $\dim \tilde{E}=\dim \tilde{F}=\dim V$.
\begin{lemma}\label{lemma:existance} Assume that the dual pair $(G,\tilde{G})$ is in the stable range with $G$ the smaller member. Given $X\in \mathfrak{g}$, there exists an injective map $T\in \operatorname{Hom}(V,\tilde{V})$ such that $T^{\ast}T=X$.
\end{lemma}
\begin{proof}
Fix a linear isomorphism $T_{\tilde{E}}:V\rightarrow \tilde{E}$ and define a linear map $T_{\tilde{F}}:V\rightarrow \tilde{F}$ by $T_{\tilde{F}} =(T_{\tilde{E}}^{\ast})^{-1} X/2$. Then, if we set $T=T_{\tilde{E}}+T_{\tilde{F}}$, we have that
\[
T^{\ast}T=(T_{\tilde{E}}^{\ast}+T_{\tilde{F}}^{\ast})(T_{\tilde{E}}+T_{\tilde{F}})=T_{\tilde{E}}^{\ast}T_{\tilde{F}}+T_{\tilde{F}}^{\ast}T_{\tilde{E}}=X/2+X/2=X.
\]
Note that we have used the fact that $T_{\tilde{F}}^{\ast}=X^{\ast}(T_{\tilde{E}}^{\ast \ast})^{-1}/2=XT_{\tilde{E}}^{-1}/2$. Now, since $T_{\tilde{E}}$ is injective, we conclude that $T$ is injective and $T^{\ast}T=X$ as we wanted to show.
\end{proof}
\begin{remark} The correspondence of nilpotent orbits for dual pairs in the stable range is well understood. See \cite{DKP,Pan}, and \cite{NOZ} (for $K_{\mathbb{C}}$-nilpotent orbits).
This is also covered by the next proposition, in view of Lemma \ref{lemma:existance}.
\end{remark}
\begin{proposition}\label{prop:Gammatilde}
Let $\mathcal{O}\subset \mathfrak{g}$ be a nilpotent orbit corresponding to
the admissible $\varepsilon$-Hermitian Young tableau $\Gamma=(d^{\Gamma},(V^{\Gamma},B^{\Gamma}))$, where $d^{\Gamma}=[t_{1}^{i_{1}},\ldots,t_{l}^{i_{l}}]$.
Given $X\in
\mathcal{O}$, let $T\in \operatorname{Hom}(V,\tilde{V})$ be an injective map such
that $T^{\ast}T=X$. Then the orbit of the nilpotent element $\tilde{X}:=TT^{\ast}\in \tilde{\mathfrak{g}}$ corresponds to the unique
equivalence class of admissible $\tilde{\varepsilon}$-Hermitian Young tableaux $\tilde{\Gamma}=(d^{\tilde{\Gamma}},(V^{\tilde{\Gamma}},B^{\tilde{\Gamma}}))$, where
\begin{itemize}
\item $d^{\tilde{\Gamma}}=[(t_{1}+1)^{i_{1}},\ldots,(t_{l}+1)^{i_{l}},1^{s}]$, with $s=\dim \tilde{V}-\dim V-\sum_{j=1}^{l} i_{j}$, namely $\tilde{\Gamma}$ is obtained by adding a column of length $\dim \tilde{V}-\dim V$ to $\Gamma$;
\item $(V^{\tilde{\Gamma},t_{j}+1}_{t_{j}},B^{\tilde{\Gamma},t_{j}+1}_{t_{j}})\cong
(V^{\Gamma,t_{j}}_{t_{j}-1},B^{\Gamma,t_{j}}_{t_{j}-1})$, for all $j=1,\ldots,l$;
\item $(V^{\tilde{\Gamma},1}_{0},B^{\tilde{\Gamma},1}_{0})$ is an $\tilde{\varepsilon}$-Hermitian module of dimension $s$.
\end{itemize}
\end{proposition}
\begin{proof}
Let $\gamma=\{X,H,Y\}$ be an $\sl_{2}$-triple containing $X$. Define $V_{i}^{\gamma,t_{j}}$ as in Section \ref{subsection:nilpotentorbits}, and set
\begin{equation*}
\tilde{V}_{i+1}^{\tilde{\gamma},t_{j}+1}=T(V_{i}^{\gamma,t_{j}}).
\end{equation*}
(At this point, $\tilde{\gamma}$ has no meaning.) Since $T$ is injective, we have a direct sum decomposition:
\[
T(V)=\bigoplus _{j=1}^l T(V^{\gamma,t_{j}}),
\]
and for each $j$, we have
$$
T(V^{\gamma,t_{j}})=\bigoplus_{i=1}^{t_{j}}T(V^{\gamma,t_{j}}_{-t_{j}+2i-1})=\bigoplus_{i=1}^{t_{j}}\tilde{V}^{\tilde{\gamma},t_{j}+1}_{-t_{j}+2i}.
$$
Observe that for all $v$, $w\in V$,
\[
\tilde{B}(Tv,Tw)=B(v,T^{\ast}Tw)=B(v,Xw).
\]
This implies that $T(V^{\gamma,t_{i}})\bot T(V^{\gamma,t_{j}})$ for $i\ne j$. We also note the following: if $\mathcal{A},\mathcal{B}\subset V$ and $\mathcal{A}, X(\mathcal{B})$ form a perfect pairing under $B$, then $T(\mathcal{A}), T(\mathcal{B})$ form a perfect pairing under $\tilde{B}$.
This implies that, for a fixed $j$, $\bigoplus_{i=1}^{t_{j}-1}\tilde{V}^{\tilde{\gamma},t_{j}+1}_{-t_{j}+2i}$ is non-degenerate, and $\tilde{V}^{\tilde{\gamma},t_{j}+1}_{t_{j}}$ is orthogonal to $\bigoplus_{i=1}^{t_{j}-1}\tilde{V}^{\tilde{\gamma},t_{j}+1}_{-t_{j}+2i}$. Therefore there exists $\tilde{V}^{\tilde{\gamma},t_{j}+1}_{-t_{j}}$ such that
\begin{itemize}
\item $T(V^{\gamma,t_{j}})\oplus \tilde{V}^{\tilde{\gamma},t_{j}+1}_{-t_{j}}= \bigoplus_{i=0}^{t_{j}}\tilde{V}^{\tilde{\gamma},t_{j}+1}_{-t_{j}+2i}$ is non-degenerate; and
\item $\tilde{V}^{\tilde{\gamma},t_{j}+1}_{-t_{j}}\, \bot \, T(V^{\gamma,t_{i}})$, for all $i\ne j$.
\end{itemize}
By an inductive argument, we may thus find $\tilde{V}^{\tilde{\gamma},t_{j}+1}_{-t_{j}}$ for $1\leq j\leq l$ such that
\begin{itemize}
\item $\tilde{U}_j=: \oplus_{i=0}^{t_{j}}\tilde{V}^{\tilde{\gamma},t_{j}+1}_{-t_{j}+2i}$ is non-degenerate; and
\item $\oplus _{j=1}^l\tilde{U}_j$ is an orthogonal direct sum.
\end{itemize}
Let
$\tilde{V}^{\tilde{\gamma},1}_{0}$ be the orthogonal complement of $\oplus _{j=1}^l\tilde{U}_j$. Then, we have a decomposition
\[
\tilde{V}=\boxplus_{j=1}^{l}\bigg(\oplus_{i=0}^{t_{j}} \tilde{V}^{\tilde{\gamma},t_{j}+1}_{-t_{j}+2i}\bigg)
\boxplus \tilde{V}^{\tilde{\gamma},1}_{0},
\]
where $\boxplus$ denotes an orthogonal direct sum.
Let $\tilde{V}_{i}=\oplus_{j}\tilde{V}^{\tilde{\gamma},t_{j}+1}_{i}$, and
define an element $\tilde{H}\in \tilde{\mathfrak{g}}$ (easily checked) by setting $\tilde{H} \tilde{v}= i\,\tilde{v}$
for all $\tilde{v}\in \tilde{V}_{i}$. Note that for $i<t_{j}-1$, $T^{\ast}|_{\tilde{V}^{\tilde{\gamma},t_{j}+1}_{i}}:\tilde{V}^{\tilde{\gamma},t_{j}+1}_{i}\rightarrow V^{\gamma,t_{j}}_{i+1}$ is an isomorphism and $TT^{\ast}=\tilde{X}:\tilde{V}^{\tilde{\gamma},t_{j}+1}_{i}\rightarrow
\tilde{V}^{\tilde{\gamma},t_{j}+1}_{i+2}$. In particular, $\tilde{X}$ is nilpotent.
Since $[\tilde{X},\tilde{H}]=2\tilde{X}$ and $\tilde{H}$ is semisimple, there exists an element $\tilde{Y}\in \tilde{\mathfrak{g}}$ such that
$\{\tilde{X},\tilde{H}, \tilde{Y}\}$ is an $\sl_{2}$-triple. See \cite[Section 3.3]{CM92}. From
all this it is clear that if $\tilde{\Gamma}=(d^{\tilde{\Gamma}},(V^{\tilde{\Gamma}},B^{\tilde{\Gamma}}))$ is
the $\tilde{\varepsilon}$-Hermitian Young tableau associated to the orbit
of $\tilde{X}$, then $d^{\tilde{\Gamma}}$ is obtained by adding a column to
$d^{\Gamma}$. Finally observe that $\tilde{X}^{t_{j}}=TX^{t_{j}-1}T^{\ast}$, and so
\[\tilde{X}^{-t_{j}}|_{\tilde{V}^{\tilde{\gamma},t_{j}+1}_{t_{j}}}=(T^{-1})^*X^{-(t_{j}-1)}T^{-1}|_{\tilde{V}^{\tilde{\gamma},t_{j}+1}_{t_{j}}}.\]
If $\tilde{v}$, $\tilde{w}\in
\tilde{V}^{\tilde{\gamma},t_{j}+1}_{t_{j}}$, then
\begin{equation}\label{eq:isometry}
\begin{aligned}
\tilde{B}^{\tilde{\gamma},t_{j}+1}_{t_{j}}(\tilde{v},\tilde{w})&=\tilde{B}(\tilde{X}^{-t_{j}}\tilde{v},\tilde{w})\\
&=\tilde{B}((T^{-1})^*X^{-(t_{j}-1)}T^{-1}\tilde{v}, \tilde{w})\\
&=B(X^{-(t_{j}-1)}T^{-1}\tilde{v}, T^{-1}\tilde{w})\\
&=B_{t_{j}+1}^{\gamma,t_{j}}(T^{-1}\tilde{v},T^{-1}\tilde{w}),
\end{aligned}
\end{equation}
which implies that $(V^{\tilde{\Gamma},t_{j}+1}_{t_{j}},B^{\tilde{\Gamma},t_{j}+1}_{t_{j}})\cong
(V^{\Gamma,t_{j}}_{t_{j}-1},B^{\Gamma,t_{j}}_{t_{j}-1})$, for all $j=1,\ldots,l$.
\end{proof}
Let $\mathcal{O}\subset\mathfrak{g}$ be a nilpotent orbit (satisfying \eqref{assumption}), which corresponds to an $\varepsilon$-Hermitian Young tableau $\Gamma=(d^{\Gamma},(V^{\Gamma},B^{\Gamma}))$. We define the \emph{theta lift} of $\mathcal{O}$, $\Theta(\mathcal{O})\subset\tilde{\mathfrak{g}}$, to be the nilpotent orbit corresponding to the $\tilde{\varepsilon}$-Hermitian Young tableau $\tilde{\Gamma}=(d^{\tilde{\Gamma}},(V^{\tilde{\Gamma}},B^{\tilde{\Gamma}}))$ specified in Proposition \ref{prop:Gammatilde}.
\begin{remark} It can be shown that, for the orbit $\mathcal{O}$ under consideration, $\tilde{\varphi}(\varphi^{-1}(\overline{\mathcal{O}}))=\overline{\Theta(\mathcal{O})}$. Cf. \cite[Lemma 4.3]{KP82}. Therefore, our definition of the theta lift of $\mathcal{O}$ agrees with the usual one.
\end{remark}
\subsection{Lifting of $\sl_{2}$-triples}
\label{liftTriple} In this section we will introduce the concept of
lifting of $\sl_{2}$-triples. We start with the following
\begin{definition}
Let $\gamma=\{X,H,Y\}\subset \mathfrak{g}$ and $\tilde{\gamma}=\{\tilde{X},\tilde{H},\tilde{Y}\}\subset \tilde{\mathfrak{g}}$ be two $\sl_{2}$-triples of type $\mathcal{O}$ and $\tilde{\mathcal{O}}$, respectively. We say that $T\in \operatorname{Hom}(V,\tilde{V})$ \emph{lifts} $\gamma$ to $\tilde{\gamma}$ if $T^{\ast}T=X$, $TT^{\ast}=\tilde{X}$ and $T(V_{j})\subset \tilde{V}_{j+1}$ for all $j$.
\end{definition}
Set
\begin{equation}
\label{defogg}
\mathcal{O}_{\gamma,\tilde{\gamma}}=\{T\in \operatorname{Hom}(V,\tilde{V})\, | \, \mbox{$T$ lifts $\gamma$ to $\tilde{\gamma}$}\}.
\end{equation}
We prove the following elementary
\begin{lemma}
\label{lemma:injsur}
Let $\gamma=\{X,H,Y\}\subset \mathfrak{g}$ and $\tilde{\gamma}=\{\tilde{X},\tilde{H},\tilde{Y}\}\subset \tilde{\mathfrak{g}}$ be two $\sl_{2}$-triples. Let $T\in \mathcal{O}_{\gamma,\tilde{\gamma}}$ and denote \[
T_j=T|_{V_{j}}\in \operatorname{Hom}(V_j,\tilde{V}_{j+1}).\]
Then $T_j$ is injective for $j<0$, and surjective for $j\geq 0$.
\end{lemma}
\begin{proof} We have $X|_{V_j}=T^*T|_{V_j}=T^*_{-j-2}T_j$. Since all highest weight vectors of a finite dimensional $\sl_{2}$ representation have non-negative weights, we know $X|_{V_j}$ is injective for $j<0$. This implies the first assertion.
For the second assertion, we first note that $T^*_j: \tilde{V}_{-j-1}\mapsto V_{-j}$. We have $\tilde{X}|_{\tilde{V}_{-j-1}}=TT^*|_{\tilde{V}_{-j-1}}=T_{-j}T^*_j$. By the same reasoning, we know that $T^*_j$ is injective for $j\geq 0$. The second assertion follows.
\end{proof}
From now on, we assume that $\mathcal{O}$ satisfies \eqref{assumption}, as
in the introduction. We let
\begin{equation}
\label{defoggMax}
\mathcal{O}^{\operatorname{Max}}_{\gamma,\tilde{\gamma}}=\mathcal{O}_{\gamma,\tilde{\gamma}}\cap \operatorname{Max} \operatorname{Hom} (V,\tilde{V}).
\end{equation}
\begin{lemma}
\label{lem:oggMax}
Let $\gamma=\{X,H,Y\}\subset \mathfrak{g}$ and $\tilde{\gamma}=\{\tilde{X},\tilde{H},\tilde{Y}\}\subset \tilde{\mathfrak{g}}$ be two $\sl_{2}$-triples of type $\mathcal{O}$ and $\Theta(\mathcal{O})$, respectively. Then $\mathcal{O}^{\operatorname{Max}}_{\gamma,\tilde{\gamma}}\neq \emptyset$ and it is a single $M_{X}\times \tilde{M}_{\tilde{X}}$-orbit.
\end{lemma}
\begin{proof} From the proof of Proposition \ref{prop:Gammatilde}, we know $\mathcal{O}^{\operatorname{Max}}_{\gamma,\tilde{\gamma}}\neq \emptyset$.
Now let $T\in \mathcal{O}^{\operatorname{Max}}_{\gamma,\tilde{\gamma}}$ and set
\[
T_{i}^{\gamma,t_{j}}=T|_{V_{i}^{\gamma,t_{j}}} \qquad \mbox{ for all $1\leq j\leq l$, $-t_{j}+1\leq i \leq t_{j}-1$.}
\]
We claim that $T_{i}^{\gamma,t_{j}}$ defines a linear isomorphism between $V_{i}^{\gamma,t_{j}}$ and $\tilde{V}_{i+1}^{\tilde{\gamma},t_{j}+1}$ for all $1\leq j\leq l$ and $-t_{j}+1\leq i \leq t_{j}-1$. Since $TX=\tilde{X}T$, $T(V_{i})\subset \tilde{V}_{i+1}$ and $T$ is injective, we clearly have $T(V_{t_j-1}^{\gamma,t_{j}}) \subseteq
\tilde{V}_{t_{j}}^{\tilde{\gamma},t_{j}+1}$. By Lemma \ref{lemma:injsur}, $T_j$ is surjective for all $j\geq 0$. Thus any element of $\tilde{V}_{t_{j}}^{\tilde{\gamma},t_{j}+1}$ is of the form $T(v)$ for some $v\in V_{t_j-1}$. But then $\tilde{X}T(v)=TX(v)=0$ and since $T$ is injective, we must have $X(v)=0$. Thus $T_{t_j-1}^{\gamma,t_{j}}: V_{t_j-1}^{\gamma,t_{j}} \rightarrow \tilde{V}_{t_{j}}^{\tilde{\gamma},t_{j}+1}$ is a linear isomorphism. By applying an appropriate power of $\tilde{X}$, we may conclude that $T_{i}^{\gamma,t_{j}}$ is a linear isomorphism between $V_{i}^{\gamma,t_{j}}$ and $\tilde{V}_{i+1}^{\tilde{\gamma},t_{j}+1}$ for all $1\leq j\leq l$ and $-t_{j}+1\leq i \leq t_{j}-1$.
We set
\[
X_{i}^{\gamma,t_{j}}= X|_{V_{i}^{\gamma,t_{j}}} \qquad \mbox{and} \qquad \tilde{X}_{i}^{\gamma,t_{j}+1}=\tilde{X}|_{\tilde{V}_{i}^{\tilde{\gamma},t_{j}+1}}.
\]
Then it is clear that
$X_{i}^{\gamma,t_{j}}=(T_{-i-2}^{\gamma,t_{j}})^{\ast}T_{i}^{\gamma,t_{j}}$
and
$\tilde{X}_{i}^{\tilde{\gamma},t_{j}+1}=T_{i+1}^{\gamma,t_{j}}(T_{-i-1}^{\gamma,t_{j}})^{\ast}$.
From this we conclude that $T$ is completely determined by the maps
$T_{t_{j}-1}^{\gamma,t_{j}}$, for $j=1,\ldots,l$, but now, from
equation (\ref{eq:isometry}), all the $T_{t_{j}-1}^{\gamma,t_{j}}$'s
are isometries. From this and equation
(\ref{eq:productisometrygroups}) it follows immediately that
$\mathcal{O}^{\operatorname{Max}}_{\gamma,\tilde{\gamma}}$ is a single $M_{X}\times
\tilde{M}_{\tilde{X}}$-orbit.
\end{proof}
Let $\gamma=\{X,H,Y\}\subset \mathfrak{g}$ and $\tilde{\gamma}=\{\tilde{X},\tilde{H},\tilde{Y}\}\subset \mathfrak{g}'$ be two $\sl_{2}$-triples. Given $T\in \mathcal{O}^{\operatorname{Max}}_{\gamma,\tilde{\gamma}}$, we define a map $\phi_{T}:\tilde{M}_{\tilde{X}}\longrightarrow M_{X}$ by
\begin{equation}
\label{defphi}
\phi_{T}(\tilde{m})v=(T_{i}^{\gamma,t_{j}})^{-1}\tilde{m} T_{i}^{\gamma,t_{j}}v,
\end{equation}
for all $\tilde{m}\in \tilde{M}_{\tilde{X}}$, and $v\in V_{i}^{\gamma,t_j}$, where $1\leq j\leq l$ and $-t_{j}+1\leq i \leq t_{j}-1$. Note that as elements of $\operatorname{Hom}(V,\tilde{V})$, we have \[\tilde{m}T=T\phi_{T}(\tilde{m}),\]
for all $\tilde{m}\in \tilde{M}_{\tilde{X}}$.
\subsection{Isomorphism of $\mathfrak{g}_{-1}\oplus \tilde{\mathfrak{g}}_{-1}$ with a symplectic subspace $W_{\gamma,\tilde{\gamma}}$ of $\operatorname{Hom}(V,\tilde{V})$}
\label{subsection:embedding}
Let $\gamma=\{X,H,Y\}$ and $\tilde{\gamma}=\{\tilde{X},\tilde{H},\tilde{Y}\}$ be two $\sl_{2}$-triples of type $\mathcal{O}$ and $\Theta(\mathcal{O})$, respectively. From the proof of Proposition \ref{prop:Gammatilde}, we may assume that
\[
V=\bigoplus_{k=-r}^{r} V_{k} \qquad \mbox{and} \qquad \tilde{V}=\bigoplus_{k=-r-1}^{r+1} \tilde{V}_{k},
\]
for some $r$. We will set
\begin{equation}
\label{def:Wgg}
W_{\gamma,\tilde{\gamma}}=\bigoplus_{k=-r}^{r}
\operatorname{Hom}(V_{k},\tilde{V}_{k})\subset \operatorname{Hom}(V,\tilde{V}).
\end{equation}
Then it is clear that the restriction of the form $\langle \cdot,\cdot \rangle$, defined in Section \ref{reductivepairs}, to $W_{\gamma,\tilde{\gamma}}$ is non-degenerate.
Now fix $T \in \mathcal{O}^{\operatorname{Max}}_{\gamma,\tilde{\gamma}}$, and define
\begin{equation}
\label{eq:sigmaT}
\begin{array}{rcl}
J_{T}: \ \ \mathfrak{g}_{-1}\oplus \tilde{\mathfrak{g}}_{-1} & \longrightarrow & W_{\gamma,\tilde{\gamma}}, \\
(R,\tilde{R}) \hspace{10pt} & \mapsto & \hspace{-5pt} TR+\tilde{R}T.
\end{array}
\end{equation}
Obviously, we have $J_{T}(R,\tilde{R})(V_{k})\subseteq \tilde{V}_{k}$ for all $k$.
\begin{lemma}
\label{lem:sigma}
$J_{T}$ defines an isomorphism between
$\mathfrak{g}_{-1}\oplus \tilde{\mathfrak{g}}_{-1}$ and $W_{\gamma,\tilde{\gamma}}$. Furthermore,
\[
\langle J_{T}(R_{1},\tilde{R}_{1}),J_{T}(R_{2},\tilde{R}_{2})\rangle=-\kappa_{-1}(R_{1},R_{2})+\tilde{\kappa}_{-1}(\tilde{R}_{1},\tilde{R}_{2}),
\]
for all $R_{1}$, $R_{2}\in \mathfrak{g}_{-1}$, $\tilde{R}_{1}$, $\tilde{R}_{2}\in \tilde{\mathfrak{g}}_{-1}$. Here the bilinear form $\kappa$ on $\mathfrak{g}$ is normalized as in \eqref{knormalized}, and likewise for $\tilde{\kappa}$.
\end{lemma}
\begin{proof}
First observe that if $R_{1}$, $R_{2}\in \mathfrak{g}_{-1}$, then
\begin{eqnarray*}
\langle TR_{1},TR_{2} \rangle & = & \operatorname{Tr} ((TR_{1})^{\ast} TR_{2}) = \operatorname{Tr} (R_{1}^{\ast}T^{\ast} TR_{2})\\
& = & -\operatorname{Tr} (R_{1}XR_{2}) =\operatorname{Tr}(R_{2}XR_{1})\\
& = & -\kappa_{-1}(R_{1},R_{2}).
\end{eqnarray*}
Similarly $\langle
\tilde{R}_{1}T,\tilde{R}_{2}T \rangle=\tilde{\kappa}_{-1}(\tilde{R}_{1},\tilde{R}_{2})$. On the other hand, if $R\in \mathfrak{g}_{-1}$, $\tilde{R}\in \tilde{\mathfrak{g}}_{-1}$, then
\[
\langle TR,\tilde{R}T\rangle =\operatorname{Tr}(R^{\ast}T^{\ast}\tilde{R}T)=-\operatorname{Tr}(RT^{\ast}\tilde{R}T)
\]
and
\[
\langle\tilde{R}T,TR \rangle =\operatorname{Tr}(T^{\ast}\tilde{R}^{\ast}TR)=-\operatorname{Tr}(T^{\ast}\tilde{R}TR).
\]
Therefore,
$\langle \tilde{R}T,TR\rangle =0$, since
$\langle \cdot, \cdot \rangle $ is symplectic. We have thus shown that
\[
\langle J_{T}(R_{1},\tilde{R}_{1}),J_{T}(R_{2},\tilde{R}_{2})\rangle=-\kappa_{-1}(R_{1},R_{2})+\tilde{\kappa}_{-1}(\tilde{R}_{1},\tilde{R}_{2}),
\]
for all $R_{1}$, $R_{2}\in \mathfrak{g}_{-1}$, $\tilde{R}_{1}$, $\tilde{R}_{2}\in \tilde{\mathfrak{g}}_{-1}$. But now, since $\kappa_{-1}$ and $\tilde{\kappa}_{-1}$ are non-degenerate, we conclude that $J_{T}$ is injective.
Now observe that, for all $S_k\in \operatorname{Hom}(V_{k},V_{k-1})$, $S_k-S_k^{\ast}\in\mathfrak{g}_{-1}$. On the other hand, we can write $S \in \mathfrak{g}_{-1}$ as the direct sum $\sum_{k=-r}^{r}S_k$, where $S_k\in \operatorname{Hom}(V_{k},V_{k-1})$. Then we have that $S_k^*=-S_{-k+1}$ for all $k$, and hence, the map $S\mapsto S-S^{\ast}$ defines a linear isomorphism between $\oplus_{k\leq 0} \operatorname{Hom}(V_{k},V_{k-1})$ and $\mathfrak{g}_{-1}$. From this, we conclude that
\begin{eqnarray*}
\dim \mathfrak{g}_{-1}& = & \sum_{k=0}^{-r+1} \dim V_{k}\cdot \dim V_{k-1} \\
& = & \sum_{k=0}^{-r+1} \dim \tilde{V}_{k-1}\cdot \dim V_{k-1} \\
& = & \sum_{k=-1}^{-r} \dim \tilde{V}_{k}\cdot \dim V_{k}.
\end{eqnarray*}
Here we have used the fact that $\dim V_{k} =\dim \tilde{V}_{k-1}$ for $k\leq 0$, in view of the linear isomorphism $T^*_{-k}: \tilde{V}_{k-1}\rightarrow V_{k}$. Similarly,
\begin{eqnarray*}
\dim \tilde{\mathfrak{g}}_{-1}& = & \sum_{k=0}^{-r} \dim \tilde{V}_{k}\cdot \dim \tilde{V}_{k-1} \\
& = & \sum_{k=0}^{-r} \dim \tilde{V}_{k}\cdot \dim V_{k}\\
& = & \sum_{k=0}^{r} \dim \tilde{V}_{k}\cdot \dim V_{k}.
\end{eqnarray*}
Therefore,
\begin{eqnarray*}
\dim \mathfrak{g}_{-1}+\dim \tilde{\mathfrak{g}}_{-1} & = & \sum_{k=-1}^{-r}\dim \tilde{V}_{k}\cdot \dim V_{k}+\sum_{k=0}^{r}\dim \tilde{V}_{k}\cdot \dim V_{k} \\
& = & \sum_{k=-r}^{r} \dim \operatorname{Hom}(V_{k},\tilde{V}_{k})=\dim W_{\gamma,\tilde{\gamma}},
\end{eqnarray*}
and hence, $J_{T}$ must be a bijection.
\end{proof}
\section{Generalized Whittaker models and Howe correspondence}
\label{liftWhittaker}
\subsection{The smooth oscillator-Heisenberg representation $\mathscr{H}_{\gamma,\tilde{\gamma}}$ associated to $W_{\gamma,\tilde{\gamma}}$}
\label{Heisenberg} Let
$H_{\gamma,\tilde{\gamma}}$ be the Heisenberg group associated to
the symplectic space $W_{\gamma,\tilde{\gamma}}$. That is,
$H_{\gamma,\tilde{\gamma}}=W_{\gamma,\tilde{\gamma}}\times \k$,
where $\{0\}\times \k$ is central, and $(R,0)(S,0) =(R+S,\langle R,
S\rangle/2)$. Let $\operatorname{Mp}(W_{\gamma,\tilde{\gamma}})$ be the
metaplectic group associated to $W_{\gamma,\tilde{\gamma}}$, and
$(\tau_{\gamma,\tilde{\gamma}},\mathscr{H}_{\gamma,\tilde{\gamma}})$ be the
smooth oscillator-Heisenberg representation of
$\operatorname{Mp}(W_{\gamma,\tilde{\gamma}})\ltimes H_{\gamma,\tilde{\gamma}}$
associated to the character $\psi$.
Recall that we have fixed $T \in \mathcal{O}^{\operatorname{Max}}_{\gamma,\tilde{\gamma}}$, and we have an isomorphism $J_{T}$ from
$\mathfrak{g}_{-1}\oplus \tilde{\mathfrak{g}}_{-1}$ to $W_{\gamma,\tilde{\gamma}}$, given in equation (\ref{eq:sigmaT}).
Define a new invariant bilinear form $\kappa'$ on
$\mathfrak{g}$ by setting
$\kappa'(R,S)=-\kappa(R,S)$, for all $R$, $S\in
\mathfrak{g}$. This results in a new symplectic structure on
$\mathfrak{g}_{-1}$, given by
$\kappa_{-1}'(R,S)=-\kappa_{-1}(R,S)$, for all
$R$, $S\in \mathfrak{g}_{-1}$, and hence a new Heisenberg group
$H_{\gamma}'$. With this new symplectic structure,
$J_{T}|_{\mathfrak{g}_{-1}}$ is a morphism of symplectic spaces,
and hence, we may extend $J_{T}$ to an injective morphism of groups
$J_{T}: H_{\gamma}'\longrightarrow
H_{\gamma,\tilde{\gamma}}$. Observe that we can proceed similarly for $\tilde{\mathfrak{g}}_{-1}$, but for this space the modification of the symplectic structure is unnecessary. We obtain in this way a map
\begin{equation}
\label{eq:extendJT}
J_{T}:H_{\gamma}'\times H_{\tilde{\gamma}} \longrightarrow H_{\gamma,\tilde{\gamma}}.
\end{equation}
As in Section \ref{subsec:GWM}, we let
$\alpha_{\gamma}':N\longrightarrow H_{\gamma}'$ be the map induced
by $\gamma$ and
the bilinear form $\kappa'$,
that is
\[
\alpha_{\gamma}'(\exp R\exp Z)=(R,\kappa'(X,Z)), \ \ \mbox{for all $R\in \mathfrak{g}_{-1}$, $Z\in \u$.}
\]
Similarly, define $\alpha_{\tilde{\gamma}}:\tilde{N}\longrightarrow H_{\tilde{\gamma}}$.
By composing with the map $J_{T}$, we have a group
homomorphism
\begin{equation}
\label{eq:extendJT2}
\alpha_{T}=\alpha_{T,\gamma,\tilde{\gamma}}: N\times \tilde{N} \longrightarrow H_{\gamma,\tilde{\gamma}}
\end{equation}
given by $\alpha_{T}(n,\tilde{n})=J_{T}(\alpha_{\gamma}'(n),\alpha_{\tilde{\gamma}}(\tilde{n}))$.
Now observe that for all $\tilde{m}\in \tilde{M}_{\tilde{X}}$, $\tilde{R}\in \tilde{\mathfrak{g}}_{-1}$,
\begin{equation}
J_{T}(\operatorname{Ad}(\tilde{m})\tilde{R})=\tilde{m}\tilde{R}\tilde{m}^{-1}T=\tilde{m}\tilde{R}T\phi_{T}(\tilde{m})^{-1}. \label{eq:tildeMXaction}
\end{equation}
On the other hand, since
$\phi_{T}:\tilde{M}_{\tilde{X}}\longrightarrow M_{X}$ is surjective,
then for any $m\in M_{X}$ we can find $\tilde{m}\in
\tilde{M}_{\tilde{X}}$ such that $\phi_{T}(\tilde{m})=m$. Moreover,
for all $R\in \mathfrak{g}_{-1}$,
\begin{equation}
J_{T}(\operatorname{Ad}(\phi_{T}(\tilde{m}))R)=T\phi_{T}(\tilde{m})R\phi_{T}(\tilde{m})^{-1}=\tilde{m}TR\phi_{T}(\tilde{m})^{-1}. \label{eq:MXaction}
\end{equation}
Using equations (\ref{eq:tildeMXaction}) and (\ref{eq:MXaction}), we
define an action of $M_{X}\times \tilde{M}_{\tilde{X}}$ on
$W_{\gamma,\tilde{\gamma}}$, by
\[
\tilde{m}\cdot S =\left\{ \begin{array}{cl}
\tilde{m}S\phi_{T}(\tilde{m})^{-1} & \mbox{if $S\in J_{T}(\tilde{\mathfrak{g}}_{-1})$}\\
S & \mbox{if $S\in J_{T}(\mathfrak{g}_{-1})$,}\end{array}\right.
\]
if $\tilde{m}\in \tilde{M}_{\tilde{X}}$, and
\[
m \cdot S =\left\{ \begin{array}{cl}
S & \mbox{if $S\in J_{T}(\tilde{\mathfrak{g}}_{-1})$}\\
\tilde{m}S\phi_{T}(\tilde{m})^{-1} & \mbox{if $S\in J_{T}(\mathfrak{g}_{-1})$,}
\end{array}\right.
\]
if $m=\phi_{T}(\tilde{m})\in M_{X}$, for some $\tilde{m}\in
\tilde{M}_{\tilde{X}}$. Observe that in particular, for all
$\tilde{m}\in \tilde{M}_{\tilde{X}}$, and all $S\in
W_{\gamma,\tilde{\gamma}}$, we have
\begin{equation}
(\tilde{m},\phi_{T}(\tilde{m}))\cdot S = \tilde{m}S\phi_{T}(\tilde{m})^{-1}. \label{eq:combinedMXtildeMXaction}
\end{equation}
Using this action and the functorial property of the oscillator-Heisenberg representation \cite{HoUP}, we extend the map $\alpha_{T}$ to a group homomorphism
\begin{equation}
\alpha_{T}:M_{\chi_{\gamma}}N\times \tilde{M}_{\chi_{\tilde{\gamma}}}\tilde{N}\longrightarrow \operatorname{Mp}(W_{\gamma,\tilde{\gamma}})\ltimes H_{\gamma,\tilde{\gamma}}. \label{eq:extendedalphaTmap}
\end{equation}
By pulling back, this yields a representation
\begin{equation}
\label{eq:deftaut}
\tau_{\gamma,\tilde{\gamma}}^{T}:=\tau_{\gamma,\tilde{\gamma}}\circ \alpha_{T}
\end{equation}
of $M_{\chi_{\gamma}}N\times \tilde{M}_{\chi_{\tilde{\gamma}}}\tilde{N}$ on $\mathscr{H}_{\gamma,\tilde{\gamma}}$.
Observe that for all $Z\in \u$, $v\in \mathscr{H}_{\gamma,\tilde{\gamma}}$,
\begin{eqnarray*}
\tau_{\gamma,\tilde{\gamma}}^{T}(\exp Z,\tilde{e})v& = & \tau_{\gamma,\tilde{\gamma}}(0,\kappa'(X,Z))v \\
&= & \psi(\kappa'(X,Z))v=\psi(-\kappa(X,Z))v \\
& = & \chi_{\check{\gamma}}(\exp Z)v=\check{\chi}_{\gamma}(\exp Z)v,
\end{eqnarray*}
where $\tilde{e}$ is the identity element in $\tilde{M}_{\chi_{\tilde{\gamma}}}\tilde{N}$. Similarly, for all $Z\in \u$, $v\in \mathscr{H}_{\gamma,\tilde{\gamma}}$,
\[
\tau_{\gamma,\tilde{\gamma}}^{T}(e,\exp \tilde{Z})v=\chi_{\tilde{\gamma}}(\exp \tilde{Z})v,
\]
where $e$ is the identity element in $M_{\chi_{\gamma}}N$. Using this, and the fact that $J_T: \mathfrak{g}_{-1}\oplus\tilde{\mathfrak{g}}_{-1}\rightarrow W_{\gamma,\tilde{\gamma}}$ is an isomorphism, we obtain the following result.
\begin{lemma}\label{lemma:tauTisomorphism} As $M_{\chi_{\gamma}}N\times
\tilde{M}_{\chi_{\tilde{\gamma}}}\tilde{N}$-modules, we have
\[
\tau_{\gamma,\tilde{\gamma}}^{T}\cong \S_{\check{\chi}_{\gamma}}\otimes
\S_{\chi_{\tilde{\gamma}}},
\]
where $\tau_{\gamma,\tilde{\gamma}}^{T}=\tau_{\gamma,\tilde{\gamma}}\circ \alpha_{T}$.
\end{lemma}
\subsection{The main result and the key proposition}
We recall some notation from Section \ref{sec:liftOrbit}. Let $(V,B)$ and $(\tilde{V},\tilde{B})$ be an $\epsilon$-Hermitian and an $\tilde{\epsilon}$-Hermitian $D$-module, respectively, with $\varepsilon\tilde{\varepsilon}=-1$. Let $G(V)$, $G(\tilde{V})$ be the corresponding isometry groups and $\mathfrak{g}=\mathfrak{g}(V)$, $\tilde{\mathfrak{g}}=\mathfrak{g}(\tilde{V})$ their Lie algebras. Then $(G(V),G(\tilde{V}))$ form a dual pair in $\operatorname{Sp}(\operatorname{Hom}(V,\tilde{V}))$ in the sense of Howe \cite{Ho79}.
Let $\operatorname{Mp}(\operatorname{Hom}(V,\tilde{V}))$ be the metaplectic group. This is the unique topological central extension of $\operatorname{Sp}(\operatorname{Hom}(V,\tilde{V}))$ by $\{\pm 1\}$ such that it splits if $\k$ is $\mathbb{C}$, and it does not split otherwise. Denote the pullbacks of $G(V)$, $G(\tilde{V})$ in $\operatorname{Mp}(\operatorname{Hom}(V,\tilde{V}))$ by $G$ and $\tilde{G}$, respectively. Then $(G,\tilde{G})$ form a dual pair in $\operatorname{Mp}(\operatorname{Hom}(V,\tilde{V}))$.
Let $(\omega,\mathscr{Y})$ be the smooth oscillator representation of $\operatorname{Mp}(\operatorname{Hom}(V,\tilde{V}))$ associated to the character $\psi$.
It is well-known that after tensoring with a genuine character of $G\times \tilde{G}$, $(\omega , \mathscr{Y})$ yields a representation of $G(V)\times G(\tilde{V})$, except when one of them, say $V$, is an odd dimensional quadratic space and $\k\ne \mathbb{C}$. In this exceptional case, $\tilde{G}$ is the metaplectic cover of $G(\tilde{V})$, and we shall only consider genuine representations of $\tilde{G}$. For all other cases,
representations of $G(V)$ are identified with genuine representations of $G$ (by tensoring with a fixed genuine character of $G$). With this convention/caveat in mind and for the rest of this article, we will make little distinction between $G$ and $G(V)$, and loosely speak of genuine representations of $G$. Likewise for $\tilde{G}$ and $G(\tilde{V})$.
This convention will lead to significant savings in notation and no confusion is expected for the expert reader.
We state the main result of this article, in a more concrete form than Theorem \ref{MainThm} of the introduction.
\begin{theorem}\label{thm:maintheorem}
Let $(G,\tilde{G})$ be a reductive dual pair, and let $(\omega , \mathscr{Y}
)$ be the smooth oscillator representation associated to the dual
pair $(G,\tilde{G})$ and to the character $\psi$ of $\k$. Let
$(\pi,\mathscr{V})$ be a smooth irreducible genuine representation
of $G$. Let $\mathcal{O}\subset \mathfrak{g}$ be a
nilpotent $G$-orbit in the image of $\operatorname{Max} \operatorname{Hom} (V,\tilde{V})$ under
the moment map $\varphi$ and let $\tilde{\mathcal{O}}=\Theta(\mathcal{O})\subset
\tilde{\mathfrak{g}}$ be the corresponding nilpotent $\tilde{G}$-orbit. Then
\[
\operatorname{Wh}_{\tilde{\mathcal{O}}}(\Theta(\pi))\cong \operatorname{Wh}_{\mathcal{O}}(\check{\pi}),
\]
where $\check{\pi}$ is the contragredient representation of $\pi$. More precisely, let $\gamma$ and $\tilde{\gamma}$ be two $\mathfrak{sl}_{2}$-triples of type $\mathcal{O}$ and $\tilde{\mathcal{O}}$, respectively. Then, given any $T\in \mathcal{O}_{\gamma,\tilde{\gamma}}^{\operatorname{Max}}$, there exists an isomorphism
\[
\Phi_{T}:\operatorname{Wh}_{\gamma}(\check{\pi})\longrightarrow \operatorname{Wh}_{\tilde{\gamma}}(\Theta(\pi)),
\]
such that
\[
\tilde{m}\Phi_{T}(\lambda)=\Phi_{T}(\phi_{T}(\tilde{m})\lambda) \qquad \mbox{for all $\tilde{m}\in\tilde{M}_{\chi_{\tilde{\gamma}}}$, $\lambda\in \operatorname{Wh}_{\gamma}(\check{\pi})$},
\]
where $\phi_{T}:\tilde{M}_{\chi_{\tilde{\gamma}}}\twoheadrightarrow
M_{\chi_{\gamma}}$ is as in equation (\ref{defphi}).
\end{theorem}
The key to our proof of Theorem \ref{thm:maintheorem} is the following result, which computes $\mathscr{Y}_{\tilde{U},\chi_{\tilde{\gamma}}}$, the $(\tilde{U},\chi_{\tilde{\gamma}})$-isotypic quotient of $\mathscr{Y}$. See Section \ref{frechet} for the unexplained notation.
\begin{proposition}\label{prop:mainproposition}
Let $\gamma\subset \mathfrak{g}$ and $\tilde{\gamma}\subset \tilde{\mathfrak{g}}$ be two $\sl_{2}$-triples of type $\mathcal{O}$ and $\Theta(\mathcal{O})$, respectively. Then, given any $T\in \mathcal{O}_{\gamma,\tilde{\gamma}}^{\operatorname{Max}}$, there exists a $ G\times \tilde{M}_{\chi_{\tilde{\gamma}}}\tilde{N}$-intertwining isomorphism
\begin{equation}
\label{covariants}
\Psi_{T}:\mathscr{Y}_{\tilde{U},\chi_{\tilde{\gamma}}}\longrightarrow \mathscr{C}(N\backslash G;\mathscr{H}_{\gamma,\tilde{\gamma}}),
\end{equation}
where $M_{\chi_{\gamma}}N\times \tilde{M}_{\chi_{\tilde{\gamma}}}\tilde{N}$ acts on $\mathscr{H}_{\gamma,\tilde{\gamma}}$ via the representation $\tau_{\gamma,\tilde{\gamma}}^{T}$ defined in equation (\ref{eq:deftaut}), and the action of $G\times \tilde{M}_{\chi_{\tilde{\gamma}}}\tilde{N}$ on $\mathscr{C}(N\backslash G;\mathscr{H}_{\gamma,\tilde{\gamma}})$ is defined in the following way: given $f\in \mathscr{C}(N\backslash G;\mathscr{H}_{\gamma,\tilde{\gamma}})$ and $g\in G$,
\begin{eqnarray}
(g'\cdot f)(g) & = & f(gg') \qquad \mbox{for all $g'\in G$,} \label{eq:tildemactionNG1}\\
(\tilde{n} \cdot f)(g) & = & \tau_{\gamma,\tilde{\gamma}}^{T}(\tilde{n})f(g) \qquad \mbox{for all $\tilde{n}\in \tilde{N}$} \label{eq:tildemactionNG2}\\
(\tilde{m}\cdot f)(g) & = &
\tau_{\gamma,\tilde{\gamma}}^{T}((\phi_{T}(\tilde{m}),\tilde{m}))f(\phi_{T}(\tilde{m})^{-1}g)
\qquad \mbox{for all $\tilde{m}\in
\tilde{M}_{\chi_{\tilde{\gamma}}}$.} \label{eq:tildemactionNG3}
\end{eqnarray}
\end{proposition}
Before starting with the proof of this proposition, let us show how it implies Theorem \ref{thm:maintheorem}.
\begin{proof}[Proof (of Theorem \ref{thm:maintheorem})]
By Lemma \ref{lemma:tauTisomorphism}, we have
\[
\mathscr{C}(N\backslash G;\mathscr{H}_{\gamma,\tilde{\gamma}})\cong \mathscr{C}(N\backslash G; \S_{\check{\chi}_{\gamma}}\otimes \S_{\chi_{\tilde{\gamma}}})\cong \mathscr{C}(N\backslash G;\S_{\check{\chi}_{\gamma}})\otimes \S_{\chi_{\tilde{\gamma}}} .
\]
Now, given $f\in \mathscr{C}(N\backslash G;\S_{\check{\chi}_{\gamma}})$ and
$v\in\S_{\chi_{\tilde{\gamma}}}$, we identify $f\otimes v$ with an
element of $\mathscr{C}(N\backslash G;\mathscr{H}_{\gamma,\tilde{\gamma}})$ via the
above isomorphism. With this identification, we may rewrite equation
(\ref{eq:tildemactionNG3}) as
\begin{eqnarray*}
\tilde{m}\cdot(f\otimes v)(g) & = &\tau_{\gamma,\tilde{\gamma}}^{T}(\phi_{T}(\tilde{m}))f(\phi_{T}(\tilde{m})^{-1}g)\otimes \tau_{\gamma,\tilde{\gamma}}^{T}(\tilde{m})v \\
& = & (\phi_{T}(\tilde{m})\cdot f)(g)\otimes \tau_{\gamma,\tilde{\gamma}}^{T}(\tilde{m})v,
\end{eqnarray*}
for all $\tilde{m}\in \tilde{M}_{\chi_{\tilde{\gamma}}}$. See
equation (\ref{eq:mactiononNG}).
By Propositions \ref{prop:mainproposition} and \ref{prop:SchwartzWhittakerModels}, we have the induced
isomorphisms:
\begin{equation}
\label{2qq1}
\mathscr{Y}_{(\tilde{U},\chi_{\tilde{\gamma}}),(G,\pi)}\cong \mathscr{C}(N\backslash G;\S_{\check{\chi}_{\gamma}})_{(G,\pi)}\otimes \S_{\chi_{\tilde{\gamma}}}\cong \pi\otimes W_{\gamma}(\check{\pi})\otimes \S_{\chi_{\tilde{\gamma}}},
\end{equation}
where $\tilde{M}_{\chi_{\tilde{\gamma}}}$ acts on $W_{\gamma}(\check{\pi})$ through the map $\phi_{T}:\tilde{M}_{\chi_{\tilde{\gamma}}}\twoheadrightarrow M_{\chi_{\gamma}}$. On the other hand, we have (from the definition of $\Theta(\pi)$) that
\begin{equation}
\label{2qq2}
\mathscr{Y}_{(G,\pi), (\tilde{U},\chi_{\tilde{\gamma}})}\cong \pi \otimes \Theta(\pi)_{(\tilde{U},\chi_{\tilde{\gamma}})}.
\end{equation}
Combining equations \eqref{2qq1} and \eqref{2qq2} and noting that
\[\operatorname{Wh}_{\tilde{\gamma}}(\Theta(\pi))=\operatorname{Hom}_{\tilde N}(\Theta(\pi),\S_{\chi_{\tilde{\gamma}}})= \operatorname{Hom}_{\tilde N}(\Theta(\pi)_{(\tilde{U},\chi_{\tilde{\gamma}})}, \S_{\chi_{\tilde{\gamma}}}),\]
we obtain the required isomorphism
\[
\Phi_{T}:\operatorname{Wh}_{\gamma}(\check{\pi})\longrightarrow \operatorname{Wh}_{\tilde{\gamma}}(\Theta(\pi)),
\]
such that
\[
\tilde{m}\Phi_{T}(\lambda)=\Phi_{T}(\phi_{T}(\tilde{m}\lambda)) \qquad \mbox{for all $\tilde{m}\in\tilde{M}_{\chi_{\tilde{\gamma}}}$, $\lambda\in \operatorname{Wh}_{\gamma}(\check{\pi})$.}
\]
\end{proof}
In the rest of this section we will closely examine the space
$\mathscr{Y}_{\tilde{U},\chi_{\tilde{\gamma}}}$. In fact we will prove a
refinement of Proposition \ref{prop:mainproposition} in which we
give an explicit isomorphism $\Psi_{T}$ in equation
(\ref{covariants}). This is Proposition
\ref{prop:tildechicoinvariants}. As we have already seen, our main
result follows immediately from this proposition.
\subsection{Strategy for the key proposition}
Let $\gamma=\{X,H,Y\}\subset\mathfrak{g}$ and $\tilde{\gamma}=\{\tilde{X},\tilde{H},\tilde{Y}\}\subset\tilde{\gamma}$ be two $\sl_{2}$-triples of type $\mathcal{O}$ and $\Theta(\mathcal{O})$, respectively. From the proof of Proposition \ref{prop:Gammatilde}, we may assume that
\begin{equation}
\label{gradingvv}
V=\bigoplus_{k=-r}^{r} V_{k}, \qquad \mbox{and} \qquad \tilde{V}=\bigoplus_{k=-r-1}^{r+1} \tilde{V}_{k},
\end{equation}
for some $r\geq 0$. ($r=0$ corresponds to the zero orbit.) Now fix $T_{\gamma,\tilde{\gamma}}\in \mathcal{O}^{\operatorname{Max}}_{\gamma,\tilde{\gamma}}$, that is, fix an injective element $T_{\gamma,\tilde{\gamma}}\in \operatorname{Hom}(V,\tilde{V})$, such that $T_{\gamma,\tilde{\gamma}}^{\ast}T_{\gamma,\tilde{\gamma}}=X$, $T_{\gamma,\tilde{\gamma}}T_{\gamma,\tilde{\gamma}}^{\ast}=\tilde{X}$ and $T_{\gamma,\tilde{\gamma}}(V_{k})\subset \tilde{V}_{k+1}$ for all $-r\leq k\leq r$. Then we have a decomposition
\begin{equation}
T_{\gamma,\tilde{\gamma}}=\oplus_{k=-r}^{r} T_{k},
\end{equation}
with $T_{k}\in \operatorname{Hom}(V_{k},\tilde{V}_{k+1})$, and $T_{k}$ is an isomorphism for $k\geq 0$. C.f. Lemma \ref{lemma:injsur}. Figuratively
\[
\begin{array}{c} V_{-r} \oplus V_{-r+1} \oplus \cdots \oplus V_{r-1} \oplus V_{r} \\
\hspace{45pt} \searrow \hspace{-3pt} {\scriptstyle T_{-r}} \hspace{10pt} \searrow \hspace{-3pt} {\scriptstyle T_{-r+1}} \hspace{5pt} \cdots\hspace{5pt} \searrow \hspace{-3pt} {\scriptstyle T_{r-1}} \hspace{-5pt} \searrow \hspace{-3pt} {\scriptstyle T_{r}} \\
\tilde{V}_{-r-1} \oplus \tilde{V}_{-r} \oplus \tilde{V}_{-r+1} \oplus \cdots \oplus \tilde{V}_{r-1} \oplus \tilde{V}_{r} \oplus \tilde{V}_{r+1}.
\end{array}
\]
We will use the gradings of $V$ and $\tilde{V}$ associated to $H$ and $\tilde{H}$, respectively, to describe the so-called mixed model of the smooth oscillator representation $(\omega,\mathscr{Y})$ associated to the dual pair $(G,\tilde{G})$. Using this model we will give a convenient description of the space of covariants $\mathscr{Y}_{\tilde{U},\chi_{\tilde{\gamma}}}$ leading to Proposition \ref{prop:tildechicoinvariants}. But before we outline the strategy, let us fix some notation. For $l\leq r$ and $m\leq r+1$, set
\begin{equation}
\label{vvbracket}
V_{(l)}=\bigoplus_{k=-l}^{l}V_{k}, \qquad \mbox{and} \qquad \tilde{V}_{(m)}=\bigoplus_{k=-m}^{m}\tilde{V}_{k}.
\end{equation}
Let $G_{(l)}=\{g\in G\, | \, \mbox{$g\cdot v =v$ for all $v\in V_{(l)}^{\perp}$}\}$, which is isomorphic with $G(V_{(l)})$.
Let $(\omega_{(l),(m)},\mathscr{Y}_{(l),(m)})$ be the smooth oscillator
representation associated to the dual pair
$(G_{(l)},\tilde{G}_{(m)})$ and the character $\psi$ of $\k$. Set
$\S_{(l),m}=\S(\operatorname{Hom}(V_{(l)},\tilde{V}_{m}))$, the Schwartz space of
$\operatorname{Hom}(V_{(l)},\tilde{V}_{m})$. Similarly, set
$\S_{l,(m)}=\S(\operatorname{Hom}(V_{l},\tilde{V}_{(m)}))$ and
$\S_{l,m}=\S(\operatorname{Hom}(V_{l},\tilde{V}_{m}))$. (Note that $\mathscr{Y}_{(l),(m)}$ can be identified with the
Schwartz space of a Lagrangian subspace of $\operatorname{Hom}(V_{(l)},\tilde{V}_{(m)})$, through the Schrodinger model.)
\begin{remark} In the current notation, our original dual pair $(G,\tilde{G})=(G_{(r)},\tilde{G}_{(r+1)})$, and the oscillator representation is
$(\omega,\mathscr{Y})=(\omega_{(r),(r+1)},\mathscr{Y}_{(r),(r+1)})$.
\end{remark}
Now observe that
\begin{eqnarray*}
\operatorname{Hom}(V_{(r)},\tilde{V}_{(r+1)}) & = & \operatorname{Hom}(V_{(r)},\tilde{V}_{r+1})\oplus \operatorname{Hom}(V_{(r)},\tilde{V}_{-r-1})\oplus \operatorname{Hom}(V_{(r)},\tilde{V}_{(r)}) \\
&= & (\operatorname{Hom}(V_{(r)},\tilde{V}_{r+1})\oplus \operatorname{Hom}(V_{-r},\tilde{V}_{(r)}))\oplus (\operatorname{Hom}(V_{(r)},\tilde{V}_{-r-1}) \oplus \operatorname{Hom}(V_{r},\tilde{V}_{(r)}))\\
& & {}\oplus \operatorname{Hom}(V_{(r-1)},\tilde{V}_{(r)})
\end{eqnarray*}
and $\operatorname{Hom}(V_{(r)},\tilde{V}_{r+1})\oplus \operatorname{Hom}(V_{-r},\tilde{V}_{(r)})$, $\operatorname{Hom}(V_{(r)},\tilde{V}_{-r-1}) \oplus \operatorname{Hom}(V_{r},\tilde{V}_{(r)})$ are totally isotropic, complementary subspaces. It then follows, from the standard theory of the oscillator representation \cite{HoUP}, that
\begin{equation}\label{eq:polarization}
\begin{aligned}
\mathscr{Y}_{(r),(r+1)}&\cong \S_{(r),r+1}\otimes \mathscr{Y}_{(r),(r)}\\
&\cong [\S_{(r),r+1}\otimes \S_{-r,(r)}]\otimes \mathscr{Y}_{(r-1),(r)}, \ \ \ \ r>0.
\end{aligned}
\end{equation}
What this equation is saying is that we may interpret the space
$\mathscr{Y}_{(r),(r+1)}$ as the space of Schwartz class functions on
$\operatorname{Hom}(V_{(r)},\tilde{V}_{r+1})$ with values in $\mathscr{Y}_{(r),(r)}$, and
the latter, in turn, can be interpreted as the space of Schwartz
class functions on $\operatorname{Hom}(V_{-r},\tilde{V}_{(r)})$ with values in
$\mathscr{Y}_{(r-1),(r)}$. More concretely, we describe this space in the
following way: given $\rho$ a seminorm on $\mathscr{Y}_{(r),(r)}$, $Z\in
\mathscr{D}(\operatorname{Hom}(V_{(r)},\tilde{V}_{r+1}))$ ($\mathscr{D}$ denotes the space of
constant-coefficient differential operators), $d\in \mathbb{N}$ and $f\in
C^{\infty}(\operatorname{Hom}(V_{(r)},\tilde{V}_{r+1});\mathscr{Y}_{(r),(r)})$, set
\[
q_{Z,d,\rho}(f)=\sup_{T\in \operatorname{Hom}(V_{(r)}\tilde{V}_{r+1})} \rho(Zf(T))(1+\|T\|)^{d},
\]
where $\|T\|$ is the operator norm of $T$. Then
\[
\S_{(r),r+1}\otimes \mathscr{Y}_{(r),(r)}\cong \left\{f\in C^{\infty}(\operatorname{Hom}(V_{(r)},\tilde{V}_{r+1});\mathscr{Y}_{(r),(r)})\, \left| \, \begin{array}{c} \mbox{$q_{Z,d,\rho}(f)<\infty$, for all $d\in \mathbb{N}$,}\\ \mbox{$Z\in \mathscr{D}(\operatorname{Hom}(V_{(r)},\tilde{V}_{r+1}))$,}\\ \mbox{ $\rho$ seminorm on $\mathscr{Y}_{(r),(r)}$}\end{array} \right\}, \right.
\]
and likewise for $[\S_{(r),r+1}\otimes \S_{-r,(r)}]\otimes \mathscr{Y}_{(r-1),(r)}$. Proceeding inductively, we obtain the tensor product decomposition
\begin{equation}\label{eq:completepolarization}
\begin{aligned}
\mathscr{Y}_{(r),(r+1)}&\cong [\S_{(r),r+1}\otimes \S_{-r,(r)}]\otimes \cdots \otimes [\S_{(1),2}\otimes \S_{-1,(1)}]\otimes \mathscr{Y}_{(0),(1)}\\
&\cong [\S_{(r),r+1}\otimes \S_{-r,(r)}]\otimes \cdots \otimes [\S_{(1),2}\otimes \S_{-1,(1)}]\otimes \S_{(0),1} \otimes \mathscr{Y}_{(0),(0)}.
\end{aligned}
\end{equation}
Again we may interpret the space appearing in the right hand side of this equation as the space of Schwartz class functions on $\oplus _{k=r,...,1}[\operatorname{Hom}(V_{(k)},\tilde{V}_{k+1})\oplus \operatorname{Hom}(V_{-k},\tilde{V}_{(k)})]
\oplus \operatorname{Hom}(V_{(0)},\tilde{V}_{1})$ with values in $\mathscr{Y}_{(0),(0)}$. This is the mixed model of the smooth oscillator representation associated to the $\gamma$, $\tilde{\gamma}$-gradings. In the rest of this article we will make these identifications of spaces without further explanation.
For simplicity, assume for the moment that $\mathcal{O}$ and $\Theta(\mathcal{O})$ are even orbits, that is $\mathfrak{g}_{-1}=0$ and $\tilde{\mathfrak{g}}_{-1}=0$.
(By Section \ref{subsection:embedding}, this is equivalent to $W_{\gamma,\tilde{\gamma}}=\bigoplus_{k=-r}^{r}
\operatorname{Hom}(V_{k},\tilde{V}_{k})=0$.) In this case we have that $\mathscr{Y}_{(0),(0)}=\mathbb{C}$ and we may define a continuous linear functional $\lambda_{(r)} \in \mathscr{Y}_{(r),(r+1)}'$ by setting
\[
\lambda_{(r)}(f)=f(T_{r},T_{-r},\ldots,T_{1},T_{-1},T_{0}),
\]
where $f\in \mathscr{Y}_{(r),(r+1)}\cong [\S_{(r),r+1}\otimes \S_{-r,(r)}]\otimes \cdots \otimes [\S_{(1),2}\otimes \S_{-1,(1)}]\otimes \S_{(0),1} \otimes \mathscr{Y}_{(0),(0)}$. As we will see later in this section, for such an $f$ we have that
\[
\lambda_{(r)}(\omega_{(r),(r+1)}(n) f)=\chi_{\gamma}^{-1}(n)\lambda_{(r)}(f) \qquad \mbox{for all $n\in U=N$,}
\]
and
\[
\lambda_{(r)}(\omega_{(r),(r+1)}(\tilde{n}) f)=\chi_{\tilde{\gamma}}(\tilde{n})\lambda_{(r)}(f) \qquad \mbox{for all $\tilde{n}\in \tilde{U}=\tilde{N}$.}
\]
Now given $f \in \mathscr{Y}_{(r),(r+1)}$, set
\[
f_{(r)}(g)=\lambda_{(r)}(\omega_{(r),(r+1)}(g) f),
\]
for $g\in G$. It is then immediate that $f_{(r)}\in
C^{\infty}(N\backslash G; \mathscr{H}_{\gamma,\tilde{\gamma}})$ but we will
show that actually $f_{(r)}\in \mathscr{C}(N\backslash G;
\mathscr{H}_{\gamma,\tilde{\gamma}})$. (Here we note that
$\mathscr{H}_{\gamma,\tilde{\gamma}}$ is $1$-dimensional since
$W_{\gamma,\tilde{\gamma}}=0$.) On the other hand, since $M_{\chi}$
and $\tilde{M}_{\chi_{\tilde{\gamma}}}$ preserve the gradings on $V$
and $\tilde{V}$, respectively, it is straightforward to check in
this case that
\[
(\omega_{(r),(r+1)}(\tilde{m})f)_{(r)}(g)=f_{(r)}(\phi_{T}(\tilde{m})^{-1}g), \mbox{for all $f\in \mathscr{Y}_{(r),(r+1)}$, $\tilde{m}\in \tilde{M}_{\chi_{\tilde{\gamma}}}$, $g\in G$}.
\]
It then follows that the map $f\mapsto f_{(r)}$ induces a
$G\times \tilde{M}_{\chi_{\tilde{\gamma}}}\tilde{N}$-intertwining
map
\[
\Psi_{T}:
(\mathscr{Y}_{(r),(r+1)})_{\tilde{U},\chi_{\tilde{\gamma}}}\longrightarrow
\mathscr{C}(N\backslash G;\mathscr{H}_{\gamma,\tilde{\gamma}}),
\]
that we will show to be an isomorphism.
To describe the map $\Psi_{T}$ in the general case, observe that if $(\tau_{\gamma,\tilde{\gamma}},\mathscr{H}_{\gamma,\tilde{\gamma}})$ is the smooth Heisenberg representation associated to $W_{\gamma,\tilde{\gamma}}$ (as in Section \ref{Heisenberg}), then, following arguments similar to those leading to equation (\ref{eq:completepolarization}), we obtain the following tensor product decomposition:
\begin{equation}
\label{Schitchi}
\mathscr{H}_{\gamma,\tilde{\gamma}} \cong \S_{-r,-r}\otimes
\S_{-r+1,-r+1}\cdots\otimes \S_{-1,-1}\otimes \mathscr{Y}_{(0),(0)}.
\end{equation}
Again we may identify the space $\mathscr{H}_{\gamma,\tilde{\gamma}}$ with
the space of Schwartz class functions on
$\operatorname{Hom}(V_{-r},\tilde{V}_{-r})\oplus \cdots \oplus
\operatorname{Hom}(V_{-1},\tilde{V}_{-1})$ with values in $\mathscr{Y}_{(0),(0)}$. Now,
given $f\in \mathscr{Y}_{(r),(r+1)}$, set
\begin{equation}
\label{def:fr0}
f_{(r)}(g)(S_{-r},\ldots,S_{-1})=(\omega_{(r),(r+1)}(g) f)(T_{r},T_{-r}+S_{-r},\ldots,T_{1},T_{-1}+S_{-1},T_{0}),
\end{equation}
for all $g\in G$, $S_{-k}\in \operatorname{Hom}(V_{-k},\tilde{V}_{-k})$, $k=1,\ldots,r$. Note that when $\mathcal{O}$ and $\Theta(\mathcal{O})$ are both even this definition agrees with the one given before.
The following result is a refinement of Proposition \ref{prop:mainproposition}.
\begin{proposition}\label{prop:tildechicoinvariants}
Let $\gamma\subset \mathfrak{g}$ and $\tilde{\gamma}\subset \tilde{\mathfrak{g}}$ be two $\sl_{2}$-triples of type $\mathcal{O}$ and $\Theta(\mathcal{O})$, respectively. Fix $T\in \mathcal{O}_{\gamma,\tilde{\gamma}}^{\operatorname{Max}}$. Given $f\in \mathscr{Y}_{(r),(r+1)}$, define $f_{(r)}\in
C^{\infty}(G;\mathscr{H}_{\gamma,\tilde{\gamma}})$ as in equation (\ref{def:fr0}). Then the map $f\mapsto f_{(r)}$ induces a $G\times \tilde{M}_{\chi_{\tilde{\gamma}}}\tilde{N}$-intertwining isomorphism
\[
\Psi_{T}:(\mathscr{Y}_{(r),(r+1)})_{\tilde{U},\chi_{\tilde{\gamma}}}\longrightarrow
\mathscr{C}(N\backslash G;\mathscr{H}_{\gamma, \tilde{\gamma}}).
\]
where $G\times \tilde{M}_{\chi_{\tilde{\gamma}}}\tilde{N}$ acts on
$\mathscr{C}(N\backslash G;\mathscr{H}_{\gamma, \tilde{\gamma}})$ via equations
(\ref{eq:tildemactionNG1})--(\ref{eq:tildemactionNG3}).
\end{proposition}
As we have already shown, the existence of such a $\Psi_{T}$ is enough to prove Theorem \ref{thm:maintheorem}.
Before proceeding to the proof of this key proposition,
we introduce all the remaining notation.
Recall that we have set
$G_{(l)}=\{g\in G\, | \, \mbox{$g\cdot v =v$ for all $v\in V_{(l)}^{\perp}$}\}.$ Let $P_{l}$ be the stabilizer of $V_{-l}$
in $G_{(l)}$. Then $P_{l}=M_{(l)}N_{l}$, where $N_{l}$ is the
unipotent radical of $P_{l}$, $M_{(l)}= M_{l}\times G_{(l-1)}$, and $M_{l}\cong \operatorname{GL}(V_{-l})$. Now observe that, if $S\in \operatorname{Hom}(V_{(l-1)},V_{-l})$, then $S-S^{\ast}\in \mathfrak{n}_{l}$. Hence, we can use the map $S\mapsto S-S^{\ast}$ to define a Lie algebra isomorphism
\begin{equation}
\operatorname{Hom}(V_{(l-1)},V_{-l})\oplus \mathfrak{z}_{l} \cong \mathfrak{n}_{l}, \label{eq:nlisomorphism}
\end{equation}
where the Lie algebra structure on the left hand side is given as follows: $\mathfrak{z}_{l}=\{Z\in \operatorname{Hom}(V_{l},V_{-l}) \, | \, Z^{\ast}=-Z\}$ is central, and $[T,S]=ST^{\ast}-TS^{\ast}$ for all $T$, $S\in
\operatorname{Hom}(V_{(l-1)},V_{-l})$. In what follows we will make implicit use of this isomorphism to identify these two
spaces.
In a similar fashion, we define $\tilde{G}_{(m)}\cong
G(\tilde{V}_{(m)})$, and observe that if we set $\tilde{P}_{m}$ to be the
stabilizer of $\tilde{V}_{m}$ in $\tilde{G}_{(m)}$, then
$\tilde{P}_{m}=\tilde{M}_{(m)}\tilde{N}_{m}$, where $\tilde{N}_{m}$ is the
unipotent radical of $\tilde{P}_{m}$, $\tilde{M}_{(m)}=\tilde{M}_{m}\times \tilde{G}_{(m-1)}$, and
$\tilde{M}_{m}\cong \operatorname{GL} (\tilde{V}_{m})$. Similarly to (\ref{eq:nlisomorphism}), we use the map $\tilde{S}\mapsto \tilde{S}-\tilde{S}^{\ast}$ from $\operatorname{Hom}(\tilde{V}_{m},\tilde{V}_{(m-1)})$ to $\tilde{\mathfrak{n}}_{m}$ to define a Lie algebra isomorphism
\begin{equation}
\operatorname{Hom}(\tilde{V}_{m},\tilde{V}_{(m-1)})\oplus \tilde{\mathfrak{z}}_{m}\cong \tilde{\mathfrak{n}}_{m}, \label{eq:tildenmisomorphism}
\end{equation}
where, similarly, the Lie algebra structure on the left hand side is given as follows: $\tilde{\mathfrak{z}}_{m} = \{\tilde{Z}\in \operatorname{Hom}(\tilde{V}_{m},\tilde{V}_{-m}) \, | \, \tilde{Z}^{\ast}=-\tilde{Z}\}$
is central, and $[\tilde{T},\tilde{S}]=\tilde{S}^{\ast}\tilde{T}-\tilde{T}^{\ast}\tilde{S}$ for all $\tilde{T}$, $\tilde{S}\in \operatorname{Hom}(\tilde{V}_{m},\tilde{V}_{(m-1)})$.
We will also set
\begin{equation}
\label{nlul}
N_{(l)}=N\cap G_{(l)}, \qquad U_{l}=U\cap N_{l}, \qquad \mbox{and} \qquad U_{(l)}=U\cap N_{(l)},
\end{equation}
with similar definitions for $\tilde{N}_{(m)}$, $\tilde{U}_{m}$ and
$\tilde{U}_{(m)}$. Let $\chi=\chi_{\gamma}$ be the character of $U$
associated to the $\sl_{2}$-triple $\gamma$, and let
$\chi_{l}=\chi|_{U_{l}}$, $\chi_{(l)}=\chi|_{U_{(l)}}$. Similarly we
define $\tilde{\chi}=\chi_{\tilde{\gamma}}$, $\tilde{\chi}_{m}$ and
$\tilde{\chi}_{(m)}$. Finally, define $X_{(l)}\in \operatorname{End}(V_{(l)})$ in
the following way: for $k<l-1$ we define
$X_{(l)}|_{V_{k}}=X|_{V_{k}}$ and for $l-1\leq k \leq l$ set
$X_{(l)}|_{V_{k}}=0$. Then it is clear that $X_{(l)}\in
\mathfrak{g}_{(l)}:=\mathfrak{g}(V_{(l)})$ is a nilpotent element. Let
$H_{(l)}=H|_{V_{(l)}}$. Then $H_{(l)}\in \mathfrak{g}_{(l)}$
is semisimple, and $[H_{(l)},X_{(l)}]=2X_{(l)}$. Therefore, there
exists $Y_{(l)}\in \mathfrak{g}_{(l)}$ such that
$\gamma_{(l)}=\{X_{(l)},H_{(l)},Y_{(l)}\}$ is an $\sl_{2}$-triple.
Observe that the parabolic subgroup associated to this
$\sl_{2}$-triple is precisely $P_{\gamma_{(l)}}=P_{\gamma}\cap
G_{(l)}=M_{(l)}N_{(l)}$, where $M_{(l)}=M\cap G_{(l)}$. Furthermore,
observe that $\chi_{\gamma_{(l)}}=\chi_{(l)}$, and hence
$M_{\chi_{\gamma_{(l)}}}=M_{\chi_{(l)}}$. We make analogous
definitions for $\tilde{X}_{(l)}$, $\tilde{\gamma}_{(l)}$, etc.
{\vspace{0.2in}}
We recall the following explicit formulas for $\omega|_{G\times \tilde{P}_{r+1}}=\omega_{(r),(r+1)}|_{G\times \tilde{P}_{r+1}}$ \cite{HoUP}: for $f\in \mathscr{Y}_{(r),(r+1)}\cong \S_{(r),r+1}\otimes \mathscr{Y}_{(r),(r)}$ and $T\in \operatorname{Hom}(V_{(r)},\tilde{V}_{r+1})$,
\begin{eqnarray}
(\omega_{(r),(r+1)}(g)f)(T) & = & \omega_{(r),(r)}(g)[f(Tg)] \qquad \mbox{for all $g\in G=G(V_{(r)})$},\label{eq:action1}\\
(\omega_{(r),(r+1)}(\tilde{m})f)(T) & = & \nu (\tilde{m})[f(\tilde{m}^{-1}T)] \qquad \mbox{for all $\tilde{m}\in\tilde{M}_{r+1}\cong \operatorname{GL}(\tilde{V}_{r+1})$}, \label{eq:action2} \\
(\omega_{(r),(r+1)}(\tilde{g})f)(T) & = & \omega_{(r),(r)}(\tilde{g})[f(T)] \qquad \mbox{for all $\tilde{g}\in \tilde{G}_{(r)}\cong G(\tilde{V}_{(r)})$}, \label{eq:action3} \\
(\omega_{(r),(r+1)}(\exp \tilde{Z})f)(T) & = & \psi(\operatorname{Tr} \tilde{Z}TT^{\ast}/2)[f(T)] \qquad \mbox{for all $\tilde{Z}\in \tilde{\mathfrak{z}}_{r+1}$}, \label{eq:action4} \\
(\omega_{(r),(r+1)}(\exp \tilde{R})f)(T) & = & \omega_{(r),(r)}(-\tilde{R}T)[f(T)] \qquad \mbox{for all $\tilde{R}\in \operatorname{Hom}(\tilde{V}_{r+1},\tilde{V}_{(r)})$}. \label{eq:action5}
\end{eqnarray}
Here $\nu $ is a certain character whose explicit form will not concern us. In the last equation we implicitly understand that $\mathscr{Y}_{(r),(r)}$ carries an action of the Heisenberg group associated to $\operatorname{Hom}(V_{(r)},\tilde{V}_{(r)})$. We also observe the following: since $\tilde{V}_{r+1}\subset \tilde{V}_{(r+1)}$ is stable under the action of $\tilde{P}_{r+1}$, we may consider $\operatorname{Hom}(V_{(r)},\tilde{V}_{r+1})$ as a $G\times \tilde{P}_{r+1}$-module with trivial $\tilde{G}_{(r)}\times \tilde{N}_{r+1}$ ($\subset \tilde{P}_{r+1}$) action. From equations (\ref{eq:action1})--(\ref{eq:action5}), it is then clear that the action $\omega_{(r),(r+1)}|_{G\times \tilde{P}_{r+1}}$ may be interpreted as coming from the usual vector bundle action of $G\times \tilde{P}_{r+1}$ on the trivial vector bundle with base $\operatorname{Hom}(V_{(r)},\tilde{V}_{r+1})$ and the fiber $\mathscr{Y}_{(r),(r)}$. (This action depends on $T$.) To be more precise, we may rewrite equations (\ref{eq:action1})--(\ref{eq:action5}) in terms of an action $\sigma_{T}$ of $G\times \tilde{P}_{r+1}$ on $\mathscr{Y}_{(r),(r)}$, as follows:
\begin{eqnarray}
(\omega_{(r),(r+1)}(g)f)(T)& = &\sigma_{T}(g)f(Tg) \qquad \mbox{for all $g\in G$} \label{eq:vectorbundleaction1}\\
(\omega_{(r),(r+1)}(\tilde{p})f)(T) & = & \sigma_{T}(\tilde{p})f(\tilde{p}^{-1}T) \qquad \mbox{for all $\tilde{p}\in \tilde{P}_{r+1}$,} \label{eq:vectorbundleaction2}
\end{eqnarray}
where $\sigma_{T}(g)=\omega_{(r),(r)}(g)$ for all $g\in G$, and
\[
\sigma_{T}(\tilde{p})=\psi(\operatorname{Tr} \tilde{Z}TT^{\ast}/2)\nu(\tilde{m})\omega_{(r),(r)}(\tilde{g})\omega_{(r),(r)}(\tilde{R}T),
\]
for $\tilde{p}=\tilde{m}\tilde{g}(\exp \tilde{R})(\exp \tilde{Z})$, with $\tilde{m}\in \tilde{M}_{r+1}$, $\tilde{g}\in \tilde{G}_{(r)}$, $\tilde{R}\in \operatorname{Hom}(\tilde{V}_{r+1},\tilde{V}_{(r)})$, $\tilde{Z}\in \tilde{\mathfrak{z}}_{r+1}$.
\subsection{The inductive step: relating covariants of $\mathscr{Y}_{(r),(r+1)}$ with $\mathscr{Y}_{(r-1),(r)}$}
\label{subsection:induction}
Given a function $f\in \mathscr{Y}_{(r),(r+1)}\cong [\S_{(r),r+1}\otimes
\S_{-r,(r)}]\otimes \mathscr{Y}_{(r-1),(r)}$, define a new function $f_{r}\in
C^{\infty}(G;\S_{-r,-r}\otimes \mathscr{Y}_{(r-1),(r)})$ by
\begin{equation}
\label{def-fr}
f_{r}(g)(S)=[\omega_{(r),(r+1)}(g)f](T_{r},T_{-r}+S),\qquad \mbox{for all $g\in G$, $S\in\operatorname{Hom}(V_{-r},\tilde{V}_{-r})$},
\end{equation}
where $T_{\gamma,\tilde{\gamma}}=\sum_{k=-r}^{r}T_{k}$ is as before. Then, from equations (\ref{eq:action4}) and (\ref{eq:action5}), we have that for all $f\in \mathscr{Y}_{(r),(r+1)}$ $g\in G$, $S\in \operatorname{Hom}(V_{-r},\tilde{V}_{-r})$,
\begin{eqnarray}
(\omega_{(r),(r+1)}(\exp \tilde{Z})f)_{r}(g)(S) & = & (\omega_{(r),(r+1)}(\exp \tilde{Z})[\omega_{(r),(r+1)}(g)f])(T_{r},T_{-r}+S) \nonumber \\
& = & \psi(\operatorname{Tr} \tilde{Z}T_{r}T_{r}^{\ast}/2)(\omega_{(r),(r+1)}(g)f)(T_{r},T_{-r}+S) \nonumber \\
& = & f_{r}(g)(S), \label{f(r)center}
\end{eqnarray}
for all $\tilde{Z}\in \tilde{\mathfrak{z}}_{r+1}$, and
\begin{eqnarray}
(\omega_{(r),(r+1)}(\exp \tilde{R})f)_{r}(g)(S) & = & (\omega_{(r),(r+1)}(\exp \tilde{R})[\omega_{(r),(r+1)}(g)f])(T_{r},T_{-r}+S) \nonumber \\
& = & (\omega_{(r),(r)}(-\tilde{R}T_{r})[\omega_{(r),(r+1)}(g)f](T_{r}))(T_{-r}+S) \nonumber \\
& = & \psi(\operatorname{Tr} ((T_{-r}+S)^{\ast}\tilde{R}T_{r}))[\omega_{(r),(r+1)}(g)f](T_{r},T_{-r}+S) \nonumber \\
& = & \psi(\operatorname{Tr} (T_{-r}^{\ast}\tilde{R}T_{r}))[\omega_{(r),(r+1)}(g)f](T_{r},T_{-r}+S) \nonumber \\
& = & \chi_{\tilde{\gamma}}(\exp \tilde{R})f_{r}(g)(S), \label{f(r)character}
\end{eqnarray}
for all $\tilde{R}\in \operatorname{Hom}(\tilde{V}_{r+1},\tilde{V}_{(r-1)}\oplus \tilde{V}_{-r})\subset \tilde{\u}_{r+1}$, in other words, for all $\tilde{u}\in \tilde{U}_{r+1}$,
\[
(\omega_{(r),(r+1)}(\tilde{u})f)_{r}(g)(S)=\chi_{\tilde{\gamma}}(\tilde{u})f_{r}(g)(S).
\]
It is then clear that understanding the map $f\mapsto f_{r}$ is an important first step towards understanding the more complicated map $f\mapsto f_{(r)}$.
We start by observing that $G_{(r-1)}N_{r}$ is the stabilizer of $T_{r}$ in $G$. Hence, according to equation (\ref{eq:vectorbundleaction1}) we obtain a representation $(\sigma_{T_{r}},\mathscr{Y}_{(r),(r)})$ of $G_{(r-1)}N_{r}$, where we are viewing $\mathscr{Y}_{(r),(r)}$ as the fiber over the point $T_{r}\in \operatorname{Hom}(V_{(r)},\tilde{V}_{r+1})$. Now, since by definition $\sigma_{T_{r}}(g)\varphi=\omega_{(r),(r)}(g)\varphi$ for all $\varphi\in \mathscr{Y}_{(r),(r)}$, we may use the formulas given in \cite{HoUP} to describe this action explicitly: given $\varphi\in \mathscr{Y}_{(r),(r)}$, $S\in \operatorname{Hom}(V_{-r},\tilde{V}_{(r)})$,
\begin{eqnarray}
(\sigma_{T_{r}}(\exp Z)\varphi)(S)& = & \psi(\operatorname{Tr} S^{\ast}SZ/2)\varphi(S), \qquad \mbox{for all $Z\in \mathfrak{z}_{r}$,} \label{eq:simgaTr1}\\
(\sigma_{T_{r}}(\exp R)\varphi)(S)& =&\omega_{(r-1),(r)}(SR)[\varphi(S)], \qquad \mbox{for all $R\in \operatorname{Hom}(V_{(r-1)},V_{-r})$,} \label{eq:simgaTr2} \\
(\sigma_{T_{r}}(g)\varphi)(S)& =&\omega_{(r-1),(r)}(g)[\varphi(S)], \qquad \mbox{for all $g\in G_{(r-1)}$}. \label{eq:simgaTr3}
\end{eqnarray}
Now, given $\varphi \in \mathscr{Y}_{(r),(r)}$, set $\bar{\varphi}(S)=\varphi(T_{-r}+S)$ for all $S\in \operatorname{Hom}(V_{-r},\tilde{V}_{-r})$. It is clear that the map $\varphi\mapsto \bar{\varphi}$ defines a surjection
\[
\bar{\,}: \ \ \mathscr{Y}_{(r),(r)}\twoheadrightarrow \S_{-r,-r}\otimes \mathscr{Y}_{(r-1),(r)}.
\]
Furthermore, using equations (\ref{eq:simgaTr1})--(\ref{eq:simgaTr3}), we may thus define a representation $(\bar{\sigma}_{T_{r}},\S_{-r,-r}\otimes \mathscr{Y}_{(r-1),(r)})$ of $G_{(r-1)}N_{r}$ explicitly by the following formulas: given $\varphi\in \S_{-r,-r}\otimes \mathscr{Y}_{(r-1),(r)}$, $S\in \operatorname{Hom}(V_{-r},\tilde{V}_{-r})$,
\begin{eqnarray}
(\bar{\sigma}_{T_{r}}(\exp Z)\varphi)(S)& = & \psi(\operatorname{Tr}(T_{-r}+ S)^{\ast}(T_{-r}+S)Z/2)\varphi(S)=\varphi(S), \qquad \mbox{for $Z\in \mathfrak{z}_{r}$,} \label{eq:barsimgaTr1}\\
(\bar{\sigma}_{T_{r}}(\exp R)\varphi)(S)& =&\omega_{(r-1),(r)}((T_{-r}+S)R)[\varphi(S)], \qquad \mbox{for $R\in \operatorname{Hom}(V_{(r-1)},V_{-r}),$} \label{eq:barsimgaTr2} \\
(\bar{\sigma}_{T_{r}}(g)\varphi)(S)& =&\omega_{(r-1),(r)}(g)[\varphi(S)], \qquad \mbox{for $g\in G_{(r-1)}$}. \label{eq:barsimgaTr3}
\end{eqnarray}
Observe that if $f\in \mathscr{Y}_{(r),(r+1)}$, then $f_{r}(g)=\overline{[\omega_{(r),(r+1)}(g)f](T_{r})}$. It follows immediately that
\[
f_{r}\in C^{\infty}(G_{(r-1)}N_{r}\backslash G;\S_{-r,-r}\otimes \mathscr{Y}_{(r-1),(r)}).
\]
Now, given $\rho$ a seminorm in
$\mathscr{Y}_{(r-1),(r)}$, $Z_{1} \in \mathscr{D}(\operatorname{Hom}(V_{-r},\tilde{V}_{-r}))$, $Z_{2}\in
U(\mathfrak{g})$, $d_{1}$, $d_{2}\in \mathbb{N}$ and $f\in
C^{\infty}(G;\S_{-r,-r}\otimes \mathscr{Y}_{(r-1),(r)})$, let
\begin{equation}\label{eq:definitionseminorm}
q_{Z_{1},Z_{2},d_{1},d_{2},\rho}(f)=\sup_{\begin{array}{c} \mbox{\scriptsize $k\in K$, $m\in M_{r}$} \\ \ \mbox{\scriptsize $S\in \operatorname{Hom}(V_{-r},\tilde{V}_{-r})$}\end{array}} \rho(Z_{1}[Z_{2}f(mk)](S)) \|m\|^{d_{1}}(1+\|S\|)^{d_{2}},
\end{equation}
where $K\subset G$ is a maximal compact subgroup as in Section
\ref{norms}. Set
\begin{eqnarray*}
\lefteqn{\mathscr{C}(N_{r} G_{(r-1)}\backslash G;\S_{-r,-r}\otimes \mathscr{Y}_{(r-1),(r)})=} \\
& & \set{f\in C^{\infty}(N_{r} G_{(r-1)}\backslash G;\S_{-r,-r}\otimes \mathscr{Y}_{(r-1),(r)})}{\begin{array}{c}\mbox{$q_{Z_{1},Z_{2},d_{1},d_{2},\rho}(f) <\infty$, for all}\\ \mbox{ $q_{Z_{1},Z_{2},d_{1},d_{2},\rho}$ as in (\ref{eq:definitionseminorm})}
\end{array}}.
\end{eqnarray*}
\begin{lemma}\label{lemma:outerlayer}
For $r>0$, the map $f\mapsto f_{r}$ induces a $G$-intertwining isomorphism
\[
\Psi_{r}:(\mathscr{Y}_{(r),(r+1)})_{\tilde{U}_{r+1},\tilde{\chi}_{r+1}} \longrightarrow
\mathscr{C}(N_{r} G_{(r-1)}\backslash G;\S_{-r,-r}\otimes\mathscr{Y}_{(r-1),(r)}),
\]
where $N_{r} G_{(r-1)}$ acts on $\S_{-r,-r}\otimes\mathscr{Y}_{(r-1),(r)}$ by the representation $\bar{\sigma}_{T_{r}}$, given in equations (\ref{eq:barsimgaTr1})--(\ref{eq:barsimgaTr3}).
\end{lemma}
\begin{proof}
Let $\lambda \in
(\mathscr{Y}_{(r),(r+1)}')^{\tilde{U}_{r+1},\tilde{\chi}_{r+1}}$, the $(\tilde{U}_{r+1},\tilde{\chi}_{r+1})$-isotypic subspace of $\mathscr{Y}_{(r),(r+1)}'$. As in the Appendix, we will identify $\lambda$ with the
$\mathscr{Y}_{(r),(r)}'$-valued distribution on
$\operatorname{Hom}(V_{(r)},\tilde{V}_{r+1})$ that sends $f\in \S_{(r),r+1}$ to the linear functional $v\mapsto \lambda(f\otimes v)$. In other words, and with some abuse of notation, we set
\[
\lambda(f)(v):=\lambda(f\otimes v) \qquad \mbox{for all $f\in \S_{(r),r+1}$, $v\in \mathscr{Y}_{(r),(r)}.$}
\]
Now recall that for $\tilde{Z}\in \tilde{\mathfrak{z}}_{r+1}$, $f\in \S_{(r),r+1}$ and $T\in \operatorname{Hom}(V_{(r)},\tilde{V}_{r+1})$, we have $\omega_{(r),(r+1)}(\exp \tilde{Z}) f(T)=\psi(\operatorname{Tr} \tilde{Z}TT^{\ast}/2)f(T)$.
Hence, for all such $\tilde{Z}$ we have that
\begin{equation}\label{eq:tildezaction}
\omega_{(r),(r+1)}(\exp \tilde{Z}) \lambda = \psi(\operatorname{Tr} \tilde{Z}TT^{\ast}/2) \lambda,
\end{equation}
that is, the action of $\exp \tilde{Z}$ on $\lambda$ is given by multiplication by the function $T\mapsto \psi(\operatorname{Tr} \tilde{Z}TT^{\ast}/2)$.
On the other hand, since $\lambda\in
(\mathscr{Y}_{(r),(r+1)}')^{\tilde{U}_{r+1},\tilde{\chi}_{r+1}}$ and since $\tilde{\mathfrak{z}}_{r+1} \subseteq \u_{r+1}$,
we must have that
\begin{equation}\label{eq:tildezequivariant}
\omega_{(r),(r+1)}(\exp \tilde{Z}) \lambda = \psi(\operatorname{Tr} \tilde{Z}\tilde{X}/2) \lambda=\lambda,
\end{equation}
where the latter equality is due to the fact that $\tilde{Z}\in \mathfrak{g}_{-2r-2}(\tilde{V})$, $\tilde{X}\in \mathfrak{g}_{2}(\tilde{V})$ and $r>0$.
Define $h_{\tilde{Z}}\in C^{\infty}(\operatorname{Hom}(V_{(r)},\tilde{V}_{r+1}))$ by
$h_{\tilde{Z}}(T)= \psi(\operatorname{Tr} \tilde{Z}TT^{\ast}/2) -1$ for all $T\in \operatorname{Hom}(V_{(r)},\tilde{V}_{r+1})$. Then we can reformulate equations (\ref{eq:tildezaction}) and (\ref{eq:tildezequivariant}) by saying that $ h_{\tilde{Z}}\lambda=0$, for all $\tilde{Z}\in \tilde{\mathfrak{z}}_{r+1}$. It then follows immediately that
\[
\begin{aligned}
\operatorname{supp} \lambda\subseteq &\bigcap_{
\tilde{Z}\in \tilde{\mathfrak{z}}_{r+1}}
\{T \in \operatorname{Hom}(V_{(r)},\tilde{V}_{r+1})\, | \, h_{\tilde{Z}}(T)=0\} \\
&=\{T\in
\operatorname{Hom}(V_{(r)},\tilde{V}_{r+1}) \, | \, TT^{\ast}=0\}.
\end{aligned}
\]
Let
\begin{equation}
\label{defMN}
\operatorname{Hom}_{NM}(V_{(r)},\widetilde{V}_{r+1})=\{T\in
\operatorname{Hom}(V_{(r)},\widetilde{V}_{r+1})\, | \, \mbox{$TT^{\ast}=0$}\}, \ \ \text{and}
\end{equation}
\begin{equation}
\label{defGMN}
\operatorname{Hom}_{GNM}(V_{(r)},\widetilde{V}_{r+1})=\{T\in
\operatorname{Hom}(V_{(r)},\widetilde{V}_{r+1})\, | \, \mbox{$TT^{\ast}=0$ and $T$
has maximal rank}\}.
\end{equation}
(Here the subscripts NM and GNM stand for null mappings and generic null mappings, respectively. Note that both sets are locally closed.)
We claim
that the natural map
\[\mathscr{Y}_{(r),(r+1)} \twoheadrightarrow
(\mathscr{Y}_{(r),(r+1)})_{\tilde{U}_{r+1},\tilde{\chi}_{r+1}}
\] factors
through the intermediate space $\S(\operatorname{Hom}_{GNM}(V_{(r)},\tilde{V}_{r+1}))\otimes
\mathscr{Y}_{(r),(r)}$. In other words, if
$\lambda\in (\mathscr{Y}_{(r),(r+1)}')^{\tilde{U}_{r+1},\tilde{\chi}_{r+1}}$,
then we can identify $\lambda$ with a $\mathscr{Y}_{(r),(r)}'$-valued tempered distribution \emph{living on}
$\operatorname{Hom}_{GNM}(V_{(r)},\tilde{V}_{r+1})$. See the Appendix for the terminology ``\emph{living on}''.
So far we have only shown that if $\lambda \in (\mathscr{Y}_{(r),(r+1)}')^{\tilde{U}_{r+1},\tilde{\chi}_{r+1}}$, then $\operatorname{supp} \lambda \subseteq \operatorname{Hom}_{NM}(V_{(r)},\tilde{V}_{r+1})$. Let
\[\mathcal{A} =\{T \in \operatorname{Hom}(V_{(r)},\tilde{V}_{r+1}) \, | \, \mbox{$T$ is
singular}\}.\]
Clearly $\operatorname{Hom}_{GNM}(V_{(r)},\tilde{V}_{r+1})=\operatorname{Hom}_{NM}(V_{(r)},\tilde{V}_{r+1})\cap \mathcal{A}^{c}$, where $\mathcal{A}^{c}$ is the complement of $\mathcal{A}$. So, as a first step towards proving our claim, we will show that
\begin{equation}\label{eq:claimvanish}
\mbox{if $\lambda \in (\mathscr{Y}_{(r),(r+1)}')^{\tilde{U}_{r+1},\tilde{\chi}_{r+1}}$} \quad \mbox{ and}\quad \mbox{ $\operatorname{supp} \lambda \subseteq \mathcal{A},$} \qquad \mbox{then}\quad \mbox{ $\lambda=0.$}
\end{equation}
Observe that $\mathcal{A} \subset \operatorname{Hom}(V_{(r)},\tilde{V}_{r+1})$ is invariant under the natural action of $G(V_{(r)})\times \operatorname{GL}(\tilde{V}_{r+1})$ and decomposes as a finite union of orbits. (Actually, $\operatorname{Hom}(V_{(r)},\tilde{V}_{r+1})$ itself decomposes as a finite union of orbits.) Furthermore, if $\mathcal{C} \subset\mathcal{A}$ is a closed invariant subset, then there exists an orbit $\mathcal{O}\subset \mathcal{C}$ that is open in $\mathcal{C}$. Therefore, we may use a Bruhat type argument to prove statement (\ref{eq:claimvanish}). To be precise, we will show that if $\lambda \in (\mathscr{Y}_{(r),(r+1)}')^{\tilde{U}_{r+1},\tilde{\chi}_{r+1}}$, $\mathcal{C}\subset \mathcal{A}$ is a non-empty closed invariant subset containing $\operatorname{supp} \lambda$ and $\mathcal{O} \subset \mathcal{C}$ is a relatively open orbit, then $\lambda$ vanishes in the complement of $\mathcal{C}\backslash \mathcal{O}$. Once this is done, a straightforward inductive argument finishes the proof of statement (\ref{eq:claimvanish}).
If $\k$ is non-Archimedian then we may immediately identify $\lambda$ with a distribution living on $\mathcal{C}$. In this case, by restricting $\lambda$ to $\mathcal{O}$ we obtain a well-defined distribution on this orbit. So we will just need to focus on the case where $\k$ is Archimedian. Given any $x\in \mathcal{O}$, there exists a neighborhood $U_{x}\subset \operatorname{Hom}(V_{(r)},\tilde{V}_{r+1})$ of $x$ such that the order of $\lambda|_{U_{x}}$ is $\leq l$ for some $l\in \mathbb{N}$. Therefore, we may find a locally finite open cover $\{U_{i}\}_{i\in\mathbb{N}}$ of $\operatorname{Hom}(V_{(r)},\tilde{V}_{r+1})$ and a partition of unity $\{\rho_{i}\}$ subordinated to $\{U_{i}\}$ such that $\lambda =\sum_{i}\rho_{i} \lambda$ and each $\rho_{i}\lambda$ is of order $\leq l_{i}$, for some $l_{i}\in \mathbb{N}$. Now, since multiplication by a $C^{\infty}$-function clearly commutes with the action of $\tilde{U}_{r+1}$, we have that, if $\lambda \in (\mathscr{Y}_{(r),(r+1)}')^{\tilde{U}_{r+1},\tilde{\chi}_{r+1}}$, then $\rho_{i}\lambda$ is also in $(\mathscr{Y}_{(r),(r+1)}')^{\tilde{U}_{
r+1},\tilde{\chi}_{r+1}}$ for all $i$. The upshot is that in order to prove statement (\ref{eq:claimvanish}) it is enough to just consider distributions $ \lambda \in (\mathscr{Y}_{(r),(r+1)}')^{\tilde{U}_{r+1},\tilde{\chi}_{r+1}}$ of finite order. But in such a case, to prove the vanishing of $\lambda$ in the complement of $\mathcal{C}\backslash \mathcal{O}$, it suffices to show that $D'(\mathcal{O};\mathscr{Y}_{(r),(r)}\otimes M^{(l)'})^{\tilde{U}_{r+1},\tilde{\chi}_{r+1}}=0$ for all $l\geq 0$ \cite[Vanishing Theorem 3.15]{KV96}. Here $M^{(l)}$ is the $l$-th transverse jet bundle of $\mathcal{O}$ (see the Appendix). More details on the transverse jet bundle in the context of a Lie group action can be found in \cite[Sections 2 and 3]{KV96}.
Now observe that, given $S\in \operatorname{Hom}(V_{(r)},\tilde{V}_{r+1})$ and $\tilde{Z}\in \tilde{\mathfrak{z}}_{r+1}$,
\begin{eqnarray*}
\frac{\partial}{\partial S}(\exp \tilde{Z})\cdot f(T) & = & \left. \frac{d}{dt}\right|_{t=0}\psi(\operatorname{Tr}(T+tS)(T+tS)^{\ast}\tilde{Z}/2)f(T+tS)\\
& = & \psi(\operatorname{Tr} TT^{\ast}\tilde{Z}/2)\psi(\operatorname{Tr} ST^{\ast}\tilde{Z}/2)f(T)+\psi(\operatorname{Tr} TT^{\ast}\tilde{Z}/2)\psi(\operatorname{Tr} TS^{\ast}\tilde{Z}/2)f(T)\\ & & {}+\psi(\operatorname{Tr} TT^{\ast} \tilde{Z}/2)\frac{\partial f}{\partial S}(T),
\end{eqnarray*}
and if $R\in \operatorname{Hom}(\tilde{V}_{r+1},\tilde{V}_{(r)}) \subset \tilde{\mathfrak{n}}_{r+1}$, then
\begin{eqnarray*}
\frac{\partial}{\partial S}(\exp R)\cdot f(T) & = & \left. \frac{d}{dt}\right|_{t=0}\omega_{(r),(r)}(R(T+tS))f(T+tS)\\
& = & \left. \frac{d}{dt}\right|_{t=0}\omega_{(r),(r)}(RT)\omega_{(r),(r)}(RtS)f(T+tS)\\
& = & \omega_{(r),(r)}(RT)d\omega_{(r),(r)}(RS)f(T)+\omega_{(r),(r)}(RT)\frac{\partial}{\partial S}f(T).
\end{eqnarray*}
What the above couple of equations imply is that the action of $\tilde{N}_{r+1}$ (and in particular the action of $\tilde{U}_{r+1}$) on the $l$-th transverse bundle is trivial, and hence $D'(\mathcal{O};\mathscr{Y}_{(r),(r)}\otimes (M^{(l)})')^{\tilde{U}_{r+1},\tilde{\chi}_{r+1}}\cong M^{(l)}\otimes D'(\mathcal{O};\mathscr{Y}_{(r),(r)})^{\tilde{U}_{r+1},\tilde{\chi}_{r+1}}$. From all this we see that, in order to prove statement (\ref{eq:claimvanish}), it suffices to show that
\begin{equation}
\label{vanish-orbit}
D'(\mathcal{O};\mathscr{Y}_{(r),(r)})^{\tilde{U}_{r+1},\tilde{\chi}_{r+1}}=0
\end{equation}
for every orbit $\mathcal{O} \subset \mathcal{A}$.
At this point we come back to the general case, where $\k$ can be
Archimedean or non-Archimedean. Let $\lambda \in D'(\mathcal{O};\mathscr{Y}_{(r),(r)})^{\tilde{U}_{r+1},\tilde{\chi}_{r+1}}$, where $\mathcal{O} \subset \mathcal{A}$ is a $G(V_{(r)})\times \operatorname{GL}(\tilde{V}_{r+1})$-orbit. As we have already seen, the equivariance of $\lambda$ implies that $\operatorname{supp} \lambda \subseteq \mathcal{O}\cap \operatorname{Hom}_{NM}(V_{(r)},\tilde{V}_{r+1})$. Since the latter set is $G(V_{(r)})\times \operatorname{GL}(\tilde{V}_{r+1})$ invariant, we see that actually $\mathcal{O} \subset \operatorname{Hom}_{NM}(V_{(r)},\tilde{V}_{r+1})$. From equations (\ref{eq:vectorbundleaction1}) and (\ref{eq:vectorbundleaction2}), it is clear that once we fix any $T\in \mathcal{O}$ we could describe $C_{c}^{\infty}(\mathcal{O})\otimes \mathscr{Y}_{(r),(r)}$ as an induced representation. (Here we are using the identifications $G\cong G(V_{(r)})$ and $\tilde{M}_{r+1}\cong\operatorname{GL}(\tilde{V}_{r+1})$.) Translating under the action of $G(V_{(r)})$, if necessary, we may assume that $T\in \mathcal{O}$ satisfies $T|_{\oplus_{k=-r}^{r-1}V_{k}}=0$. To see why, let $T\in \operatorname{Hom}_{NM}(V_{(r)},\tilde{V}_{r+1})$ ($\subseteq \operatorname{Hom}_{NM}(V_{(r)},\tilde{
V}_{(r+1)})$), and so $T^{\ast}|_{\tilde{V}_{r+1}^{\perp}}=0$. Since $\tilde{V}_{r+1}^{\perp}=\tilde{V}_{(r)}\oplus\tilde{V}_{r+1}$ we have that $\Im T^{\ast}$ ($\subset V_{(r)}$) is a totally isotropic subspace with dimension less than or equal to $\dim \tilde{V}_{-r-1}=\dim V_{-r}$. We may thus assume, by translating under $G(V_{(r)})$ if necessary, that $\Im T^{\ast}\subset V_{-r}$, which will imply that $T|_{\oplus_{k=-r}^{r-1}V_{k}}=0$. We identify such a $T$ with an element of $\operatorname{Hom}(V_{r},\tilde{V}_{r+1})\subset \operatorname{Hom}(V_{(r)},\tilde{V}_{r+1})$ in an obvious way. Now we have an isomorphism of $G\times \tilde{P}_{r+1}$-modules:
\begin{equation}\label{eq:compatlysupportedinduction}
C_{c}^{\infty}(\mathcal{O})\otimes \mathscr{Y}_{(r),(r)}\cong C^{\infty}_c-\operatorname{Ind}_{\mbox{Stab}_{T}}^{G\times
\tilde{P}_{r+1}} (\mathscr{Y}_{(r),(r)}).
\end{equation}
Here $\mbox{Stab}_{T}$ denotes the stabilizer of $T$ in $G\times \tilde{P}_{r+1}$ and the notation $C^{\infty}_c-\operatorname{Ind}$ means that
we are taking compactly supported induction. The isomorphism is given by the map $f\mapsto \hat{f}$, where we have set $\hat{f}(g,\tilde{p}) =[\omega_{(r),(r+1)}(g)\omega_{(r),(r+1)}(\tilde{p})f](T)$, for any given $f\in C_{c}^{\infty}(\mathcal{O})\otimes \mathscr{Y}_{(r),(r)}$. Its inverse is given by the map $h\mapsto \check{h}$, where $\check{h}(\tilde{p}^{-1}Tg)=\sigma_{T}(g)^{-1}\sigma_{T}(\tilde{p})^{-1}h(g,\tilde{p})$ for all $h\in C^{\infty}_c-\operatorname{Ind}_{\mbox{Stab}_{T}}^{G\times
\tilde{P}_{r+1}} (\mathscr{Y}_{(r),(r)})$, $g\in G$, $\tilde{p}\in \tilde{P}_{r+1}$.
Let us describe the action on the right hand side of equation (\ref{eq:compatlysupportedinduction}) more precisely. For this we will use the isomorphism $\mathscr{Y}_{(r),(r)}\cong \S_{-r,(r)}\otimes \mathscr{Y}_{(r-1),(r)}$. According to equations (\ref{eq:action1})--(\ref{eq:vectorbundleaction2}), the action of $ \tilde{G}_{(r)}\tilde{N}_{r+1}$ on $\mathscr{Y}_{(r),(r)}$ is given by
\begin{eqnarray}
\sigma_{T}(\tilde{g})h(S) & = &\omega_{(r-1),(r)}(\tilde{g}) h(\tilde{g}^{-1}S) \qquad \mbox{for $\tilde{g}\in \tilde{G}_{(r)}$}, \label{eq:sigmaaction1} \\
\sigma_{T}(\exp \tilde{Z})h(S) & = &\psi(\operatorname{Tr} \tilde{Z}TT^{\ast}/2)h (S) \qquad \mbox{for $\tilde{Z}\in \tilde{\mathfrak{z}}_{r+1}$}, \label{eq:sigmaaction2}\\
\sigma_{T}(\exp \tilde{R})h(S) & = & \omega_{(r),(r)}(\tilde{R}T)h(S)=\psi(\operatorname{Tr}\tilde{R}TS^{\ast})h(S) \qquad \mbox{for $\tilde{R}\in \operatorname{Hom}(\tilde{V}_{r+1},\tilde{V}_{(r)})$.} \label{eq:sigmaaction3}
\end{eqnarray}
In the last equality, we have used the fact that, since $T\in \operatorname{Hom}(V_{r},\tilde{V}_{r+1})\subset \operatorname{Hom}(V_{(r)},\tilde{V}_{r+1})$, then $\tilde{R}T$ is in $\operatorname{Hom}(V_{r},\tilde{V}_{(r)}) \subset \operatorname{Hom}(V_{(r)},\tilde{V}_{(r)})$ which is a totally isotropic subspace that is complementary to $\operatorname{Hom}(V_{-r},\tilde{V}_{(r)})$. (Here is where the choice of $T$ simplifies matters.)
Now observe that there is a continuous $G\times \tilde{P}_{r+1}$-intertwining map from $C_{c}^\infty(G\times \tilde{P}_{r+1}\times \operatorname{Hom}(V_{-r},\tilde{V}_{(r)}); \mathscr{Y}_{(r-1),(r)})$ to $C^{\infty}_c-\operatorname{Ind}_{\mbox{Stab}_{T}}^{G\times
\tilde{P}_{r+1}} (\S_{-r,(r)}\otimes \mathscr{Y}_{(r-1),(r)})$, with dense image, given by
\[
\tilde{h}(g,\tilde{p},S)=\int_{\mbox{Stab}_{T}}(\sigma_{T}(x)^{-1}h(x_{g}g,x_{\tilde{p}}\tilde{p})(S)\, dx, \qquad \mbox{ $h\in C_{c}^\infty(G\times \tilde{P}_{r+1}\times \operatorname{Hom}(V_{-r},\tilde{V}_{(r)}); \mathscr{Y}_{(r-1),(r)})$.}
\]
Here $x_{g}$ and $x_{\tilde{p}}$ are the projections of $x\in \mbox{Stab}_{T}$ to $G$ and $\tilde{P}_{r+1}$, respectively.
Therefore, we may define a $\tilde{U}_{r+1}$-equivariant $\mathscr{Y}_{(r-1),(r)}'$-valued distribution $\bar{\lambda}$ on $G\times \tilde{P}_{r+1}\times \operatorname{Hom}(V_{-r},V_{(r)})$ by setting
\[
\bar{\lambda}(h)=\lambda(\check{\tilde{h}}), \qquad \mbox{for all $h\in C_{c}^\infty(G\times \tilde{P}_{r+1}\times \operatorname{Hom}(V_{-r},\tilde{V}_{(r)}); \mathscr{Y}_{(r-1),(r)})$.}
\]
It is clear that if $\tilde{p}=\tilde{m}\tilde{g}\tilde{n}$, with $\tilde{m}\in \tilde{M}_{r+1}\cong\operatorname{GL}(\tilde{V}_{r+1})$, $\tilde{n}\in \tilde{N}_{r+1}$, and $\tilde{g}\in \tilde{G}_{(r)}$, then, for all $h\in C^{\infty}_c-\operatorname{Ind}_{\mbox{Stab}_{T}}^{G\times
\tilde{P}_{r+1}} (\S_{-r,(r)}\otimes \mathscr{Y}_{(r-1),(r)})$, $S\in \operatorname{Hom}(V_{-r},\tilde{V}_{(r)})$, and $\tilde{R}\in \operatorname{Hom}(\tilde{V}_{r+1},\tilde{V}_{(r)})$, we have
\[
\exp \tilde{R} \cdot h(g,\tilde{m}\tilde{g}\tilde{n},S)=\psi(\operatorname{Tr} \tilde{R}\tilde{m}^{-1}TS^{\ast}\tilde{g})h(g,\tilde{m}\tilde{g}\tilde{n},S).
\]
Hence
\[
\exp \tilde{R}\cdot \bar{\lambda}=\psi(\operatorname{Tr} \tilde{R}\tilde{m}^{-1}TS^{\ast}\tilde{g})\bar{\lambda}.
\]
In other words, the action of $\exp \tilde{R}$ on $\bar{\lambda}$ is given by multiplication by the function $(g,\tilde{m}\tilde{g}\tilde{n},S) \mapsto \psi(\operatorname{Tr} \tilde{R}\tilde{m}^{-1}TS^{\ast}\tilde{g})$. On the other hand, since $\lambda$ is $\tilde{U}_{r+1}$-equivariant, we should have
\[
\exp \tilde{R} \cdot \bar{\lambda}=\psi(\operatorname{Tr} \tilde{R}\tilde{X})\bar{\lambda}, \qquad \mbox{for all $\tilde{R}\in \operatorname{Hom}(\tilde{V}_{r+1},\tilde{V}_{(r-1)}\oplus
\tilde{V}_{-r})\subset \tilde{\u}_{r+1}$}.
\]
Here for the trace function, we are considering $\tilde{X}$ and $\tilde{R}$ as elements of $\operatorname{End}(\tilde{V}_{(r+1)})$. Therefore,
\[
\operatorname{supp} \bar{\lambda}\subset \{(g,\tilde{m}\tilde{g}\tilde{n},S) \, | \, \mbox{$ \tilde{R}\tilde{m}^{-1}TS^{\ast}\tilde{g}=\tilde{R}\tilde{X}$, for all $\tilde{R}\in \operatorname{Hom}(\tilde{V}_{r+1},\tilde{V}_{(r-1)}\oplus
\tilde{V}_{-r})$}\},
\]
but the latter set is empty since $\Im \tilde{X}$ contains $\tilde{V}_{r+1}$, whereas the image of $\tilde{m}^{-1}TS^{\ast}\tilde{g}$ does not have the same property due to the fact that $T$ is singular. From all this, we conclude that if $\mathcal{O}\subset \mathcal{A}$, then $D'(\mathcal{O};\mathscr{Y}_{(r),(r)})^{\tilde{U}_{r+1},\tilde{\chi}_{r+1}}=0$. This is statement (\ref{vanish-orbit}) which, as we have already seen, implies statement (\ref{eq:claimvanish}).
Having proved statement (\ref{eq:claimvanish}), we come back to our original assertion that the map $\mathscr{Y}_{(r),(r+1)} \twoheadrightarrow
(\mathscr{Y}_{(r),(r+1)})_{\tilde{U}_{r+1},\tilde{\chi}_{r+1}}$ factors
through $\S(\operatorname{Hom}_{GNM}(V_{(r)},\tilde{V}_{r+1}))\otimes
\mathscr{Y}_{(r),(r)}$. We have seen that any $\lambda\in
(\mathscr{Y}_{(r),(r+1)}')^{\tilde{U}_{r+1},\tilde{\chi}_{r+1}}$ is completely determined by its restriction to
$C_{c}^{\infty}(\mathcal{A}^{c})$, where $\mathcal{A}^{c}$ is the complement of $\mathcal{A}$ in
$\operatorname{Hom}(V_{(r)},\tilde{V}_{r+1})$. But on $\mathcal{A}^{c}$, $0$ is a regular
value of the map $T\mapsto TT^{\ast}$. Therefore, $\lambda|_{A^{c}}$ is a
distribution that \emph{lives on} $\operatorname{Hom}_{GNM}(V_{(r)},\tilde{V}_{r+1})$, that is, $\lambda$ can be identified with an
element of $(\S(\operatorname{Hom}_{GNM}(V_{(r)},\tilde{V}_{r+1}))\otimes
\mathscr{Y}_{(r),(r)})'$, as we wanted to show. C.f. \cite[page 49, Exercise 4.5]{FJ}. (Note that we may make a suitable change of variables, when we have a regular value.)
Given $f\in \S(\operatorname{Hom}_{GNM}(V_{(r)},\tilde{V}_{r+1}))\otimes
\mathscr{Y}_{(r),(r)}$, we set
\begin{equation}
\tilde{f}_{r}(g)=[\omega_{(r),(r+1)}(g)f](T_{r})=\omega_{(r),(r)}(g)[f(T_{r}g)],\qquad \mbox{for $g\in G$.}
\end{equation}
Then it is clear that $\tilde{f}_{r}\in
C^{\infty}(N_{r} G_{(r-1)}\backslash G;\mathscr{Y}_{(r),(r)})$ for all $f\in
\S(\operatorname{Hom}_{GNM}(V_{(r)},\tilde{V}_{r+1}))\otimes \mathscr{Y}_{(r),(r)}$. Let
\[
\S^{+}(N_{r} G_{(r-1)}\backslash G;\mathscr{Y}_{(r),(r)})=\{\tilde{f}_{r}\, | \, f\in
\S(\operatorname{Hom}_{GNM}(V_{(r)},\tilde{V}_{r+1}))\otimes \mathscr{Y}_{(r),(r)}\}.
\]
Now observe that $\operatorname{Hom}_{GNM}(V_{(r)},\tilde{V}_{r+1})$ is a single orbit
under the action of $G$. Therefore, since $G_{(r-1)}N_{r}$ is the
stabilizer of $T_{r} \in
\operatorname{Hom}_{GNM}(V_{(r)},\tilde{V}_{r+1})$, we have that as $G$-modules
\[
\S(\operatorname{Hom}_{GNM}(V_{(r)},\tilde{V}_{r+1}))\otimes \mathscr{Y}_{(r),(r)}\cong
\S^{+}(N_{r} G_{(r-1)}\backslash G;\mathscr{Y}_{(r),(r)}).
\]
Now we will show that
\begin{equation}\label{eqn:Sastchirplus1}
\S^{+}(N_{r} G_{(r-1)}\backslash G;\mathscr{Y}_{(r),(r)})_{\tilde{U}_{r+1},\tilde{\chi}_{r+1}}\cong \S^{+}(N_{r} G_{(r-1)}\backslash G;\S_{-r,-r}\otimes \mathscr{Y}_{(r-1),(r)}),
\end{equation}
where
\[
\S^{+}(N_{r} G_{(r-1)}\backslash G;\S_{-r,-r}\otimes\mathscr{Y}_{(r-1),(r)})=\{f_{r} \,
| \, f\in \mathscr{Y}_{(r),(r+1)}\}.
\]
So let $\lambda \in
(\S^{+}(N_{r} G_{(r-1)}\backslash G;\mathscr{Y}_{(r),(r)})')^{\tilde{U}_{r+1},\tilde{\chi}_{r+1}}$.
We will identify $\lambda$ with a $\mathscr{Y}_{(r-1),(r)}'$-valued
distribution on $G\times \operatorname{Hom}(V_{-r},\tilde{V}_{(r)})$. Observe
that, \mbox{for all $ f\in \S^{+}(N_{r} G_{(r-1)}\backslash G;\mathscr{Y}_{(r),(r)})$}, $g\in G$, $S\in \operatorname{Hom}(V_{-r},\tilde{V}_{(r)})$, and $\tilde{R}\in
\operatorname{Hom}(\tilde{V}_{r+1},\tilde{V}_{(r)})$, we have
\[
\exp \tilde{R}\cdot f(g)(S)=\psi(\operatorname{Tr} \tilde{R}T_{r}S^{\ast})f(g)(S).
\]
Therefore, since $\lambda\in
(\S^{+}(N_{r} G_{(r-1)}\backslash G;\mathscr{Y}_{(r),(r)})')^{\tilde{U}_{r+1},\tilde{\chi}_{r+1}}$,
we must have
\[
\psi(\operatorname{Tr} \tilde{R}T_{r}S^{\ast})\lambda=\psi(\operatorname{Tr} \tilde{R}\tilde{X})\lambda,
\]
for all $\tilde{R}\in \operatorname{Hom}(\tilde{V}_{r+1},\tilde{V}_{(r-1)}\oplus
\tilde{V}_{-r}) \subset \tilde{\u}_{r+1}$. Now, since $\tilde{X}$ is a regular value of the
map $S\mapsto T_{r}S^{\ast}$, we conclude that $\lambda$ is a
distribution that \emph{lives on}
\[
G\times \{S \in \operatorname{Hom}(V_{-r},\tilde{V}_{(r)})\, | \,
\mbox{$\tilde{R}T_{r}S^{\ast}=\tilde{R}\tilde{X}$ for all $\tilde{R}\in
\operatorname{Hom}(\tilde{V}_{r+1},\tilde{V}_{(r-1)}\oplus
\tilde{V}_{-r})$}\}.
\]
But the last set is just $G\times [T_{-r}+\operatorname{Hom}(V_{-r},\tilde{V}_{-r})]$. The isomorphism given in equation (\ref{eqn:Sastchirplus1}) now follows immediately.
The only thing that remains to be checked is that
\begin{equation}\label{eq:SastS}
\S^{+}(N_{r} G_{(r-1)}\backslash G;\S_{-r,-r}\otimes\mathscr{Y}_{(r-1),(r)})\cong
\mathscr{C}(N_{r} G_{(r-1)}\backslash G;\S_{-r,-r}\otimes\mathscr{Y}_{(r-1),(r)}).
\end{equation}
For this, let $\rho$ be a seminorm in $\mathscr{Y}_{(r-1),(r)}$, $Z_{1}\in
\mathscr{D}(\operatorname{Hom}(V_{-r},\tilde{V}_{-r}))$, $Z_{2}\in U(\mathfrak{g})$, $d_{1}$, $d_{2}\in
\mathbb{N}$ and $f\in \S_{(r),r+1}\otimes \S_{-r,(r)}\otimes
\mathscr{Y}_{(r-1),(r)}$. Then
\begin{eqnarray*}
q_{Z_{1},Z_{2},d_{1},d_{2},\rho}(f_{r}) & = &\sup_{k\in K, m\in M_{r}, S\in \operatorname{Hom}(V_{-r},\tilde{V}_{-r})} \rho(Z_{1}[R_{Z_{2}}f_{r}(mk)](S))\|mk\|^{d_{1}}(1+\|S\|)^{d_{2}} \\
& = &\sup_{k, m, S} \rho(Z_{1}[(k\cdot R_{Z_{2}}f)_{r}(m)](S))\|m\|^{d_{1}}(1+\|S\|)^{d_{2}} \\
& = & \sup_{k,m,S}\rho(\omega_{(r),(r)}(m)Z_{1}(k\cdot R_{Z_{2}}f)(T_{r}m)(T_{-r}+S))\|m\|^{d_{1}}(1+\|S\|)^{d_{2}}\\
& = & \sup_{k,m,S}\rho(Z_{1}(k\cdot R_{Z_{2}}f)(T_{r}m)(T_{-r}m+Sm))\|m\|^{d_{1}}(1+\|S\|)^{d_{2}}. \\
& \leq & \sup_{k,m,S} q_{Z_{1},c_{1},c_{2},\rho}(k\cdot R_{Z_{2}}f)(1+\|T_{r}m+T_{-r}m\|)^{-c_{1}}\\ & & {}\cdot (1+\|Sm\|)^{-c_{2}} \|m\|^{d_{1}}(1+\|S\|)^{d_{2}},
\end{eqnarray*}
where $c_{1}$ ,$c_{2}\in \mathbb{N}$ are arbitrary and $q_{Z_{1},c_{1},c_{2},\rho}$ is a seminorm on $\S_{(r),r+1}\otimes \S_{-r,(r)}\otimes
\mathscr{Y}_{(r-1),(r)}$. But then there exists a constant $C>0$ such that
\begin{eqnarray*}
q_{Z_{1},Z_{2},d_{1},d_{2},\rho}(f_{r}) & \leq & \sup_{k,m,S}\, C\, q_{Z_{1},c_{1},c_{2},\rho}(k\cdot R_{Z_{2}}f) \|m\|^{-c_{1}}(1+\|S\|)^{-c_{2}}\|m\|^{c_{2}} \|m\|^{d_{1}}(1+\|S\|)^{d_{2}} \\
& \leq &\sup_{k,m,S}\, C\, q_{Z_{1},c_{1},c_{2},\rho}(k\cdot R_{Z_{2}}f) \|m\|^{d_{1}+c_{2}-c_{1}}(1+\|S\|)^{d_{2}-c_{2}}.
\end{eqnarray*}
Now observe that if we fix $c_{2} \gg d_{2}$ and $c_{1}\gg d_{1}+c_{2}$, then
\[
\sup_{m,S} C q_{Z_{1},c_{1},c_{2},\rho}(k\cdot R_{Z_{2}} f)\|m\|^{d_{1}+c_{2}-c_{1}}(1+\|S\|)^{d_{2}-c_{2}} <\infty,
\]
for all $k\in K$. Furthermore, the map $k\mapsto \sup_{m,S} C \, q_{Z_{1},c_{1},c_{2},\rho}(k\cdot f)\|m\|^{d_{1}+c_{2}-c_{1}}(1+\|S\|)^{d_{2}-c_{2}}$ is continuous, and hence, since $K$ is compact, bounded. Therefore $q_{Z_{1},Z_{2},d_{1},d_{2},\rho}(f_{r})<\infty$ and (\ref{eq:SastS}) follows.
\end{proof}
Observe that the space
$(\mathscr{Y}_{(r),(r+1)})_{\tilde{U}_{r+1},\tilde{\chi}_{r+1}}$ carries a
natural action of the group $\tilde{N}$. As a consequence of the isomorphism
$(\mathscr{Y}_{(r),(r+1)})_{\tilde{U}_{r+1},\tilde{\chi}_{r+1}}\cong
\mathscr{C}(N_{r} G_{(r-1)}\backslash G;\S_{-r,-r}\otimes\mathscr{Y}_{(r-1),(r)})$, we may define an action of
$\tilde{N}$ on $\mathscr{C}(N_{r} G_{(r-1)}\backslash
G;\S_{-r,-r}\otimes\mathscr{Y}_{(r-1),(r)})$ by the formula $(\tilde{n}\cdot
f_{r})(g)(S):=(\omega_{(r),(r+1)}(\tilde{n})f)_{r}(g)(S)$, for
$f\in \mathscr{Y}_{(r),(r+1)}$, $\tilde{n}\in \tilde{N}$ and $S\in \operatorname{Hom}(V_{-r},\tilde{V}_{-r})$. We describe this action more concretely in the following
\begin{lemma}\label{lemma:inducedaction}
There is an induced action of $\tilde{N}$ ($=\tilde{N}_{(r)}\tilde{N}_{r+1}$) on $\mathscr{C}(N_{r} G_{(r-1)}\backslash
G;\S_{-r,-r}\otimes\mathscr{Y}_{(r-1),(r)})$, as follows:
for all $\phi\in \mathscr{C}(N_{r} G_{(r-1)}\backslash G;\S_{-r,-r}\otimes\mathscr{Y}_{(r-1),(r)})$, $g\in G$, $S\in \operatorname{Hom}(V_{-r},\tilde{V}_{-r})$,
\begin{eqnarray*}
(\exp \tilde{Z}\cdot \phi)(g)(S)&= & \phi(g)(S) \qquad \mbox{for $\tilde{Z}\in \tilde{\mathfrak{z}}_{r+1}$}\\
(\exp \tilde{R}\cdot \phi)(g)(S)&= & \psi(\operatorname{Tr} \tilde{R}T_{r}T_{-r}^{\ast})\psi(\operatorname{Tr} \tilde{R}T_{r}S^{\ast})\phi(g)(S) \qquad \mbox{for $\tilde{R}\in \operatorname{Hom}(\tilde{V}_{r+1},\tilde{V}_{(r)})$}\\
(\exp \tilde{Z}\cdot \phi)(g)(S)&= &\omega_{(r-1),(r)}(\exp \tilde{Z})[ \phi(g)(S)] \qquad \mbox{for $\tilde{Z}\in \tilde{\mathfrak{z}}_{r}$}\\
(\exp \tilde{R}\cdot \phi)(g)(S)&= &\omega_{(r-1),(r)}(\exp \tilde{R})[ \phi(g)(S+\tilde{R}^{\ast}T_{-r})] \qquad \mbox{for $\tilde{R}\in \operatorname{Hom}(\tilde{V}_{r},\tilde{V}_{(r-1)})$}\\
(\tilde{n}\cdot \phi)(g)(S)&=&\omega_{(r-1),(r)}(\tilde{n})[\phi(g)(S)] \qquad \mbox{for $\tilde{n}\in \tilde{N}_{(r-1)}.$}
\end{eqnarray*}
In particular, we have
\begin{equation}
\label{eq:ntildeaction6}
(\tilde{u}\cdot \phi)(g)(S)=\omega_{(r-1),(r)}(\tilde{u})[\phi(g)(S)] \qquad \mbox{for $\tilde{u}\in \tilde{U}_{(r)}$.}
\end{equation}
\end{lemma}
\begin{proof} If $\tilde{Z}\in
\tilde{\mathfrak{z}}_{r+1}$, we have already seen in equation (\ref{f(r)center}) that
\[
(\omega_{(r),(r+1)}(\exp \tilde{Z})f)_{r}(g)(S) = f_{r}(g)(S),
\]
while if $\tilde{R} \in \operatorname{Hom}(\tilde{V}_{r+1},\tilde{V}_{(r)}) \subset \tilde{\mathfrak{n}}_{r+1}$, similar to equation (\ref{f(r)character}), we have
\begin{eqnarray*}
(\omega_{(r),(r+1)}(\exp \tilde{R})f)_{r}(g)(S)&= &\psi(\operatorname{Tr} \tilde{R}T_{r}(T_{-r}+S)^{\ast})f_{r}(g)(S) \\
&= &\psi(\operatorname{Tr} \tilde{R}T_{r}T_{-r}^{\ast})\psi(\operatorname{Tr} \tilde{R}T_{r}S^{\ast})f_{r}(g)(S).
\end{eqnarray*}
On the other hand, if $\tilde{Z}\in \tilde{\mathfrak{z}}_{r}$, then
\begin{eqnarray*}
(\omega_{(r),(r+1)}(\exp \tilde{Z})f)_{r}(g)(S) & = & (\omega_{(r),(r+1)}(\exp \tilde{Z})[\omega_{(r),(r+1)}(g)f])(T_{r},T_{-r}+S) \\
& = & (\omega_{(r),(r)}(\exp \tilde{Z})[(\omega_{(r),(r+1)}(g)f)(T_{r})])(T_{-r}+S) \\
& = & \omega_{(r-1),(r)}(\exp \tilde{Z})([\omega_{(r),(r+1)}(g)f](T_{r},(\exp \tilde{Z})^{-1}(T_{-r}+S))) \\
& = & \omega_{(r-1),(r)}(\exp \tilde{Z})([\omega_{(r),(r+1)}(g)f](T_{r},T_{-r}+S)) \\
& = & \omega_{(r-1),(r)}(\exp \tilde{Z})f_{r}(g)(S).
\end{eqnarray*}
Here we are using that $\tilde{Z}(T_{-r}+S)=0$ because $\Im (T_{-r}+S)\subset \tilde{V}_{-r+1}\oplus \tilde{V}_{-r}$ and $\tilde{Z}|_{\tilde{V}_{-r+1}\oplus \tilde{V}_{-r}}=0$. On the other hand, if $\tilde{R}\in \operatorname{Hom}(\tilde{V}_{r},\tilde{V}_{(r-1)})\subset \tilde{\mathfrak{n}}_{r}$, then, according to equations (\ref{eq:sigmaaction1}), (\ref{eq:sigmaaction2}) and (\ref{eq:sigmaaction3})
\begin{eqnarray*}
(\omega_{(r),(r+1)}(\exp \tilde{R})f)_{r}(g)(S) & = & (\omega_{(r),(r+1)}(\exp \tilde{R})[\omega_{(r),(r+1)}(g)f])(T_{r},T_{-r}+S) \\
& = & (\omega_{(r),(r)}(\exp \tilde{R})[(\omega_{(r),(r+1)}(g)f)(T_{r})])(T_{-r}+S) \\
& = & \omega_{(r-1),(r)}(\exp \tilde{R})([\omega_{(r),(r+1)}(g)f](T_{r},(\exp \tilde{R})^{-1}(T_{-r}+S))) \\
& = & \omega_{(r-1),(r)}(\exp \tilde{R})([\omega_{(r),(r+1)}(g)f](T_{r},T_{-r}+S+\tilde{R}^{\ast}T_{-r})) \\
& = & \omega_{(r-1),(r)}(\exp \tilde{R})f_{r}(g)(S+\tilde{R}^{\ast}T_{-r}).
\end{eqnarray*}
Here we are using the identification $\operatorname{Hom}(\tilde{V}_{r},\tilde{V}_{(r-1)}) \ni \tilde{R} \leftrightarrow \tilde{R}-\tilde{R}^{\ast}\in \tilde{\mathfrak{n}}_{r}$ as in equation (\ref{eq:tildenmisomorphism}). We are also using that, since $T_{-r}\in \operatorname{Hom}(V_{-r},\tilde{V}_{-r+1})$, then $\tilde{R}^{\ast}T_{-r}\in \operatorname{Hom}(V_{-r},\tilde{V}_{-r})$ and that
\[
\tilde{R}^{\ast}S_{-r}=\tilde{R}S_{-r}=\tilde{R}T_{-r}=\tilde{R}^{\ast}\tilde{R}T_{-r}=\tilde{R}^{\ast}\tilde{R}S_{-r}=0,
\]
for all $\tilde{R}\in \operatorname{Hom}(\tilde{V}_{r},\tilde{V}_{(r-1)})$, $S\in \operatorname{Hom}(V_{-r},\tilde{V}_{-r})$. We note, in particular, that $\tilde{R}^{\ast}T_{-r}=0$ if $\tilde{R}\in \operatorname{Hom}(\tilde{V}_{r},\tilde{V}_{(r-2)}\oplus V_{-r+1})\subset \tilde{\u}_{r}$.
Finally, if $\tilde{n}\in \tilde{N}_{(r-1)}\subset \tilde{G}_{(r)}$, then
\begin{eqnarray*}
(\omega_{(r),(r+1)}(\tilde{n})f)_{r}(g)(S) & = & (\omega_{(r),(r+1)}( \tilde{n})[\omega_{(r),(r+1)}(g)f])(T_{r},T_{-r}+S) \\
& = & (\omega_{(r),(r)}( \tilde{n})[(\omega_{(r),(r+1)}(g)f)(T_{r})])(T_{-r}+S) \\
& = & \omega_{(r-1),(r)}( \tilde{n})([\omega_{(r),(r+1)}(g)f](T_{r}, \tilde{n}^{-1}(T_{-r}+S))) \\
& = & \omega_{(r-1),(r)}( \tilde{n})([\omega_{(r),(r+1)}(g)f](T_{r},T_{-r}+S)) \\
& = & \omega_{(r-1),(r)}( \tilde{n})f_{r}(g)(S).
\end{eqnarray*}
Here we are using that, considered as an element of $\tilde{G}_{(r)}$, $\tilde{n}\in \tilde{N}_{(r-1)}$ leaves $\tilde{V}_{-r+1}\oplus \tilde{V}_{-r}$ fixed.
\end{proof}
\subsection{From $\mathscr{Y}_{(r),(r+1)}$ to $\mathscr{H}_{\gamma, \tilde{\gamma}}$: proof of Proposition \ref{prop:tildechicoinvariants}}
\label{subsection:keyprop}
We are finally ready to complete the proof of the key proposition, to which we refer the reader for the unexplained notation below.
\begin{proof}[Proof (of Proposition \ref{prop:tildechicoinvariants})]
We will prove the result by induction on $r$.
We first look at the case where $r=0$. Observe that in this case $U_{1}=U$, $\tilde{\chi}_{1}=\tilde{\chi}$ and $\mathfrak{z}_{1}=\u$. We want to show that the map $f\mapsto f_{(0)}$ induces an isomorphism between $(\mathscr{Y}_{(0),(1)})_{\tilde{U},\tilde{\chi}}$ and $\mathscr{C}(G,\mathscr{H}_{\gamma,\tilde{\gamma}})$.
Let $\lambda\in (\mathscr{Y}'_{(0),(1)})^{\tilde{U},\tilde{\chi}}$. As in the proof of Lemma \ref{lemma:outerlayer}, we identify $\lambda$ with a $\mathscr{Y}'_{(0),(0)}$-valued distribution on $\operatorname{Hom}(V_{(0)},\tilde{V}_{1})$ by setting
\[
\lambda(f)(v)=\lambda(f\otimes v) \qquad \mbox{ for all $f\in \S_{(0),1}$, $v\in \mathscr{Y}_{(0),(0)}$.}
\]
As in Equation (\ref{eq:tildezaction}) we have that for all $\tilde{Z}\in \u$, $f \in \S_{(0),1}$, $T\in \operatorname{Hom}(V_{(0)},\tilde{V}_{1})$,
\begin{equation}\label{eq:r0action}
\omega_{(0),(1)}(\exp \tilde{Z}) \lambda =\psi(\operatorname{Tr} \tilde{Z}TT^{\ast}/2)\lambda.
\end{equation}
On the other hand, since $\lambda \in (\mathscr{Y}'_{(0),(1)})^{\tilde{U},\tilde{\chi}}$, we must have
\begin{equation}\label{eq:r0invariance}
\omega_{(0),(1)}(\exp \tilde{Z})\lambda= \psi(\operatorname{Tr} \tilde{Z}\tilde{X}/2)\lambda.
\end{equation}
Observe that, in contrast with the situation in Lemma \ref{lemma:outerlayer}, $\tilde{X}:\tilde{V}_{-1}\longrightarrow \tilde{V}_{1}$ is a regular value of the map $T\mapsto TT^{\ast}$. Let
\[
\mathcal{O}=\{T\in \operatorname{Hom}(V_{(0)},\tilde{V}_{1})\, | \, TT^{\ast}=\tilde{X}\}.
\]
From Equations (\ref{eq:r0action}), (\ref{eq:r0invariance}) and the above observation it follows that if $\lambda \in (\mathscr{Y}'_{(0),(1)})^{\tilde{U},\tilde{\chi}}$, then $\lambda$ is in fact a distribution that lives on $\mathcal{O}$. Since $r=0$, we have an isomorphism $\mathscr{Y}_{(0),(0)}\cong \mathscr{H}_{\gamma,\tilde{\gamma}}$. On the other hand, if we fix a $T_{(0)}\in \mathcal{O}$ we have an isomorphism
\[
\begin{array}{ccc} G & \cong & \mathcal{O}\\
g& \mapsto & g\cdot T_{(0)}.
\end{array}
\]
From all this it follows that $(\mathscr{Y}_{(0),(1)})_{\tilde{U},\tilde{\chi}}\cong \S^{+}(G;\mathscr{H}_{\gamma,\tilde{\gamma}})$, where
\[
\S^{+}(G;\mathscr{H}_{\gamma,\tilde{\gamma}}):=\{f_{(0)}\, | \, f\in \mathscr{Y}_{(0),(1)}\}.
\]
The result now follows by noting that the growth conditions imposed on functions in $\mathscr{C}(G,\mathscr{H}_{\gamma,\tilde{\gamma}})$ and $\S^{+}(G;\mathscr{H}_{\gamma,\tilde{\gamma}})$ are in fact equivalent (c.f. the proof of Equation \eqref{eq:SastS}).
For the rest of proof, we assume that $r>0$.
Observe that $\tilde{U}=\tilde{U}_{(r)}\tilde{U}_{r+1}$ and $\tilde{\chi}_{(r+1)}=\tilde{\chi}_{(r)}\tilde{\chi}_{r+1}$. Therefore, from Lemma \ref{lemma:outerlayer}
\begin{eqnarray*}
(\mathscr{Y}_{(r),(r+1)})_{\tilde{U},\tilde{\chi}} & \cong & (\mathscr{C}(N_{r}G_{(r-1)}\backslash G;\S_{-r,-r}\otimes \mathscr{Y}_{(r-1),(r)}))_{\tilde{U}_{(r)},\tilde{\chi}_{(r)}} \\
& \cong & \mathscr{C}(N_{r}G_{(r-1)}\backslash G;\S_{-r,-r}\otimes (\mathscr{Y}_{(r-1),(r)})_{\tilde{U}_{(r)},\tilde{\chi}_{(r)}}),
\end{eqnarray*}
where the last equation follows from the fact that $\tilde{U}_{(r)}$ acts only on the values. See equation (\ref{eq:ntildeaction6}).
Now, given $f\in \S_{-r,-r}\otimes \mathscr{Y}_{(r-1),(r)}$, $g\in G$ and $S_{-k}\in \operatorname{Hom}(V_{-k},\tilde{V}_{-k})$, $k=1,\ldots,l$, set
\begin{eqnarray*}
f_{(r-1)}(g)(S_{-r},\ldots,S_{-1})& = & (\bar{\sigma}_{T_{r}}(g)f)(S_{-r},T_{r-1},T_{-r+1}+S_{-r+1},\ldots,T_{0}) \\
& = & (\omega_{(r-1),(r)}(g)[f(S_{-r})])(T_{r-1},T_{-r+1}+S_{-r+1},\ldots,T_{0}).
\end{eqnarray*}
For convenience, we denote the representation $\tau_{\gamma,\tilde{\gamma}}^{T}$ on $\mathscr{H}_{\gamma,\tilde{\gamma}}$ (defined in equation (\ref{eq:deftaut})) by the more concise symbol $\mathscr{H}_{T}$. Then, by induction hypothesis, the map $f\mapsto f_{(r-1)}$ induces
a $G_{(r-1)}$-intertwining isomorphism
\[
\Psi_{(r-1)}: \S_{-r,-r}\otimes
(\mathscr{Y}_{(r-1),(r)})_{\tilde{U}_{(r)},\tilde{\chi}_{(r)}}
\longrightarrow \mathscr{C}(N_{(r-1)}\backslash
G_{(r-1)}; \mathscr{H}_{T_{(r)}}),
\]
where $T_{(l)}=\oplus_{k=-l}^{l} T_{k}$, for $l=0,\ldots,r$. Note that $T_{(r)}=T_{\gamma,\tilde{\gamma}}$. Also note that in the above equation we are identifying the spaces $\S_{-r,-r}\otimes \mathscr{C}(N_{(r-1)}\backslash
G_{(r-1)}; \mathscr{H}_{T_{(r-1)}}) \cong \mathscr{C}(N_{(r-1)}\backslash
G_{(r-1)}; \mathscr{H}_{T_{(r)}})$. We claim that $\Psi_{(r-1)}$ is actually an isomorphism of $G_{(r-1)}N_{r}$-modules:
\begin{equation}
\Psi_{(r-1)}: \S_{-r,-r}\otimes
(\mathscr{Y}_{(r-1),(r)})_{\tilde{U}_{(r)},\tilde{\chi}_{(r)}}
\longrightarrow \mathscr{C}(N\backslash G_{(r-1)}N_{r}; \mathscr{H}_{T_{(r)}}),
\label{eq:GNr1isomorphism}
\end{equation}
where
\[
\mathscr{C}(N\backslash G_{(r-1)}N_{r};
\mathscr{H}_{T_{(r)}})=\{f \in C^{\infty}(N\backslash G_{(r-1)}N_{r};
\mathscr{H}_{T_{(r)}})\,
| \, f|_{G_{(r-1)}}\in
\mathscr{C}(N_{(r-1)}\backslash G_{(r-1)}; \mathscr{H}_{T_{(r-1)}}). \}
\]
Observe that if $f\in \mathscr{C}(N\backslash G_{(r-1)}N_{r};
\mathscr{H}_{T_{(r)}})$, $g\in G_{(r-1)}$ and $n\in N_{r}$, then
\[
(n\cdot f)(g)=f(gn)=f((gng^{-1})g)=\tau_{T_{(r)}}(gng^{-1})f(g).
\]
Therefore, to prove our claim, we need to show that for all $f\in \S_{-r,-r}\otimes \mathscr{Y}_{(r-1),(r)}$, $g\in G_{(r-1)}$, $n\in N_{r}$,
\begin{equation}
(\bar{\sigma}_{T_{r}}(n)f)_{(r-1)}(g)=\tau_{T_{(r)}}(gng^{-1})f_{(r-1)}(g). \label{eq:compatibilitysigmatau}
\end{equation}
Now, by definition,
\begin{eqnarray*}
(\bar{\sigma}_{T_{r}}(n)f)_{(r-1)}(g)(S_{-r},\ldots,S_{-1}) & = & (\bar{\sigma}_{T_{r}}(g)\bar{\sigma}_{T_{r}}(n)f)(S_{-r},T_{r-1},T_{-r+1}+S_{-r+1},\ldots,T_{0}) \\
& = & (\bar{\sigma}_{T_{r}}(gng^{-1})[\bar{\sigma}_{T_{r}}(g)f])(S_{-r},T_{r-1},T_{-r+1}+S_{-r+1},\ldots,T_{0}) \\
& = & (\bar{\sigma}_{T_{r}}(gng^{-1})[\bar{\sigma}_{T_{r}}(g)f])_{(r-1)}(e)(S_{-r},S_{-r+1},\ldots,S_{-1}).
\end{eqnarray*}
From all this we see that in order to prove the claim given in equation (\ref{eq:GNr1isomorphism}) it suffices to show that for all $f\in \S_{-r,-r}\otimes \mathscr{Y}_{(r-1),(r)}$, $n\in N_{r}$,
\begin{equation}
(\bar{\sigma}_{T_{r}}(n)f)_{(r-1)}(e)=\tau_{T_{(r)}}(n)f_{(r-1)}(e). \label{eq:simplecompatibilitysigmatau}
\end{equation}
Now, by equation (\ref{eq:barsimgaTr1}), if $Z\in \mathfrak{z}_{r}$, then
\[
(\bar{\sigma}_{T_{r}}(\exp Z)f)_{(r-1)}(e)(S_{-r},\dots,S_{-1})=f(S_{-r},T_{r-1},\ldots,T_{0})=f_{(r-1)}(e)(S_{-r},\dots,S_{-1}).
\]
On the other hand, by equation (\ref{eq:barsimgaTr2}), if $R\in \operatorname{Hom}(V_{(r-2)}\oplus V_{r-1},V_{-r})\subset \u_{r}$, then
\begin{eqnarray*}
(\bar{\sigma}_{T_{r}}(\exp R)f)_{(r-1)}(e)(S_{-r},\dots,S_{-1}) & = & (\omega_{(r-1),(r)}(T_{-r}R)[f(S_{-r})])(T_{r-1},\ldots,T_{0}) \\
& = & \psi(-\operatorname{Tr} T_{-r}RT_{r-2}^{\ast})f(S_{-r},T_{r-1},\ldots,T_{0}) \\
& = & \chi_{\gamma}(\exp R)^{-1}f_{(r-1)}(e)(S_{-r},\ldots,S_{-1}),
\end{eqnarray*}
in other words, for all $u\in U_{r}$,
\[
(\bar{\sigma}_{T_{r}}(u)f)_{(r-1)}(e)(S_{-r},\dots,S_{-1})=\chi_{\gamma}(u)^{-1}f_{(r-1)}(e)(S_{-r},\ldots,S_{-1}).
\]
Finally, again by equation (\ref{eq:barsimgaTr2}), if $R\in \mathfrak{g}_{-1}\cap \mathfrak{n}_{r}\cong \operatorname{Hom}(V_{-r+1},V_{-r})$, then
\begin{eqnarray*}
(\bar{\sigma}_{T_{r}}(\exp R)f)_{(r-1)}(e)(S_{-r},\dots,S_{-1}) & = & (\omega_{(r-1),(r)}(T_{-r}R+S_{-r}R)[f(S_{-r})](T_{r-1},\ldots,T_{0}) \\
& = & \psi(-\operatorname{Tr} S_{-r}RT^{\ast}_{r-1})f_{(r-1)}(e)(S_{-r},S_{-r+1}+T_{-r}R,\ldots,S_{-1}).
\end{eqnarray*}
Now, for such an $R$, $\sigma_{T_{(r)}}(R,0)=T_{-r}R-T_{r-1}R^{\ast}$. Therefore,
\begin{eqnarray*}
\tau_{T_{(r)}}(\exp R)f_{(r-1)}(e)(S_{-r},\ldots,S_{-1}) & = &\tau_{\gamma,\tilde{\gamma}}(T_{-r}R-T_{r-1}R^{\ast})f_{(r-1)}(e)(S_{-r},\ldots, S_{-1}) \\
& = & \psi(-\operatorname{Tr} S_{-r}RT^{\ast}_{r-1})f_{(r-1)}(e)(S_{-r},S_{-r+1}+T_{-r}R,\ldots,S_{-1}).
\end{eqnarray*}
Therefore, equation (\ref{eq:simplecompatibilitysigmatau}) holds true, and hence we have proved our claim given in equation (\ref{eq:GNr1isomorphism}).
We will now show that
\[
\mathscr{C}(N\backslash G; \mathscr{H}_{T_{(r)}})\cong \mathscr{C}(N_{r} G_{(r-1)}\backslash G;\mathscr{C}(N\backslash G_{(r-1)}N_{r};\mathscr{H}_{T_{(r)}})).
\]
Given $f\in \mathscr{C}(N_{r} G_{(r-1)}\backslash G;\mathscr{C}(N\backslash G_{(r-1)}N_{r};\mathscr{H}_{T_{(r)}}))$, set $\check{f}(g)=f(g)(e)$, for all $g\in G$. Then it is clear that $\check{f}\in C^{\infty}(N\backslash G; \mathscr{H}_{T_{(r)}})$, but we claim that $\check{f}$ is actually in $\mathscr{C}(N\backslash G;\mathscr{H}_{T_{(r)}})$. Effectively, given a semi-norm $\rho$ of $\mathscr{H}_{T_{(r)}}$, $Z\in U(\mathfrak{g})$ and $d\in \mathbb{N}$,
\begin{eqnarray*}
q_{Z,d,\rho}(\check{f})&= &\sup_{k\in K, \, m_{1}\in M_{r}, \, m_{2}\in G_{(r-1)}} \rho(R_{Z}\check{f}(m_{2} mk_{1} )) \|m_{1}m_{2}\|^{d}\\
&= & \sup_{k,m_1,m_2}\rho(R_{Z}f(m_{2} mk_{1} )(e)) \|m_{1}m_{2}\|^{d} \\
& \leq & \sup_{k,m_1,m_2}\rho(R_{Z}f(mk_{1})(m_{2}^{-1})) \|m_{1}\|^{d}\|m_{2}\|^{d} \\
& = & p_{Z,1,d,d,\rho}(f)<\infty.
\end{eqnarray*}
Now given $f\in \mathscr{C}(N\backslash G; \mathscr{H}_{T_{(r)}})$, set $\hat{f}(g)(h)=f(hg)$. Then again it is clear that
\[
\hat{f} \in C^{\infty}(N_{r} G_{(r-1)}\backslash G;C^{\infty}(N\backslash N_{r}G_{(r-1)};\mathscr{H}_{T_{(r)}})),
\]
but we claim that, actually, $\hat{f}$ is a function in $ \mathscr{C}(N_{r} G_{(r-1)}\backslash G;\mathscr{C}(N\backslash G_{(r-1)}N_{r};\mathscr{H}_{T_{(r)}}))$. Effectively, let $\rho$ be a semi-norm of $\mathscr{H}_{T_{(r)}}$, $Z_{1}\in U(\mathfrak{g})$, $Z_{2}\in U(\mathfrak{g}_{(r-1)})$ and $d_{1}$, $d_{2}\in \mathbb{N}$. Then $Z_{2}\in U^{m}(\mathfrak{g})$ for some $m$ and hence there exists a basis $\{Y_{j}\}_{j=1}^{l}$ of $U^{m}(\mathfrak{g})$ such that for all $g\in G$, $\operatorname{Ad}(g) Z_{2}=\sum_{j} a_{j}(g)Y_{j}$ for some functions $a_{j}$. Since $(\operatorname{Ad},U^{m}(\mathfrak{g}))$ is a finite dimensional representation, there exists a constant $d_{m}$ such that $|a(g)|\leq \|g\|^{d_{m}}$ for all $g\in G$. Taking this into account we have that
\begin{eqnarray*}
p_{Z_{1},Z_{2},d_{1},d_{2},\rho}(\hat{f})&= &\sup_{k\in K, \, m_{1}\in M_{r}, \, m_{2}\in G_{(r-1)}} \rho(R_{Z_{2}}(R_{Z_{1}}\hat{f}(mk_{1}))( m_{2})) \|m_{1}\|^{d_{1}}\|m_{2}\|^{d_{2}}\\
&= & \sup_{k,m_1,m_2}\rho(R_{\operatorname{Ad}(mk_{1})^{-1}Z_{2}}R_{Z_{1}}f(m_{2} mk_{1} )) \|m_{1}\|^{d_{1}}\|m_{2}\|^{d_{2}}\\
& \leq & \sup_{k,m_1,m_2}\sum_{j=1}^{l}\rho(a(mk_{1})R_{Y_{j}}R_{Z_{1}}f(mk_{1}m_{2})) \|m_{1}\|^{d_{1}}\|m_{2}\|^{d_{2}} \\
& \leq & \sup_{k,m_1,m_2}\sum_{j=1}^{l}\rho(R_{Y_{j}}R_{Z_{1}}f(mk_{1}m_{2})) \|m_{1}\|^{d_{1}+d_{m}}\|m_{2}\|^{d_{2}} \\
& \leq & \sup_{k,m_1,m_2}\sum_{j=1}^{l}\rho(R_{Y_{j}Z_{1}}f(mk_{1}m_{2})) \|m_{1}m_{2}\|^{d_{1}+d_{m}+d_{2}} \\
& = & \sum_{j=1}^{l} q_{Y_{j}Z_{1},d_{1}+d_{2}+d_{m},\rho}(f)<\infty.
\end{eqnarray*}
Here we have used the fact that $\|m_{1}m_{2}\|=\max\{\|m_{1}\|,\|m_{2}\|\}\geq \|m_{1}\|,\|m_{2}\|$.
Now the only thing left to be check is that $\hat{\check{f}}=f$ and $\check{\hat{f}}=f$, but by definition
\[
\hat{\check{f}}(g)(h)=\check{f}(gh)=f(gh)(e)=f(g)(h),
\]
and similarly $\check{\hat{f}}=f$.
\end{proof}
| {
"redpajama_set_name": "RedPajamaArXiv"
} | 2,968 |
package Filters;
##################################
# Ready to use filters for HTTP::Proxy
##################################
use strict;
use warnings;
# Response filter
use Filters::Filter403;
use Filters::FilterCrap;
use Filters::RecordTransfer;
# Request filter
use Filters::ForceBackend;
1;
| {
"redpajama_set_name": "RedPajamaGithub"
} | 9,733 |
{"url":"https:\/\/www.ccpl.org\/eds\/detail?db=pbh&an=146914258","text":"# Look duration at the face as a developmental endophenotype: elucidating pathways to autism and ADHD.\n\nItem request has been placed! \u00d7\nItem request cannot be made. \u00d7\nProcessing Request\n\u2022 Author(s):\n\u2022 Source:\nDevelopment & Psychopathology. Oct2020, Vol. 32 Issue 4, p1303-1322. 20p.\nIdentifying developmental endophenotypes on the pathway between genetics and behavior is critical to uncovering the mechanisms underlying neurodevelopmental conditions. In this proof-of-principle study, we explored whether early disruptions in visual attention are a unique or shared candidate endophenotype of autism spectrum disorder (ASD) and attention-deficit\/hyperactivity disorder (ADHD). We calculated the duration of the longest look (i.e., peak look) to faces in an array-based eye-tracking task for 335 14-month-old infants with and without first-degree relatives with ASD and\/or ADHD. We leveraged parent-report and genotype data available for a proportion of these infants to evaluate the relation of looking behavior to familial (n = 285) and genetic liability (using polygenic scores, n = 185) as well as ASD and ADHD-relevant temperament traits at 2 years of age (shyness and inhibitory control, respectively, n = 272) and ASD and ADHD clinical traits at 6 years of age (n = 94). Results showed that longer peak looks at the face were associated with elevated polygenic scores for ADHD (\u03b2 = 0.078, p =.023), but not ASD (\u03b2 = 0.002, p =.944), and with elevated ADHD traits in mid-childhood (F(1,88) = 6.401, p =.013, $\\eta _p^2$ =0.068; ASD: F (1,88) = 3.218, p =.076), but not in toddlerhood (ps > 0.2). This pattern of results did not emerge when considering mean peak look duration across face and nonface stimuli. Thus, alterations in attention to faces during spontaneous visual exploration may be more consistent with a developmental endophenotype of ADHD than ASD. Our work shows that dissecting paths to neurodevelopmental conditions requires longitudinal data incorporating polygenic contribution, early neurocognitive function, and clinical phenotypic variation. [ABSTRACT FROM AUTHOR]","date":"2022-07-01 04:43:50","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\": 0.2955218553543091, \"perplexity\": 14744.498218579989}, \"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\/1656103920118.49\/warc\/CC-MAIN-20220701034437-20220701064437-00398.warc.gz\"}"} | null | null |
Includes in-house measuring, delivery and installation.
Installation for solid surface countertop is approximately 4 weeks from placing your order . Installation time is 5-7 hours. Solid surface countertops are laser measured to create an exact template – even if your walls aren't perfectly straight. (We install is recommended for solid-surface countertops).
Installation for laminate countertops is approximately two weeks from placing your order. Installation time is 3-5 hours.
You provide exact measurements and pick up your finished countertop. Sink and tap cutouts are not available on this option.
Pick up for solid surface countertops is approximately 3 weeks.
Pick up for laminate countertops is 2 weeks.
Note: Custom Made countertops (post form with wood edges): Delivery and installation time varies depending on design requested. | {
"redpajama_set_name": "RedPajamaC4"
} | 2,882 |
{"url":"https:\/\/www.aimsciences.org\/article\/doi\/10.3934\/mbe.2017067","text":"# American Institute of Mathematical Sciences\n\nOctober\u00a0 2017,\u00a014(5&6):\u00a01301-1316. doi:\u00a010.3934\/mbe.2017067\n\n## Modeling co-infection of Ixodes tick-borne pathogens\n\n 1 Department of Applied Mathematics, The Hong Kong Polytechnic University, Hung Hom, Kowloon, Hong Kong, China 2 School of Information Engineering, Guangdong Medical University, Dongguan, Guangdong 523808, China 3 Mathematics and Science College, Shanghai Normal University, Shanghai 200234, China\n\n* Corresponding authorr\n\nReceived\u00a0 August 06, 2016 Revised\u00a0 December 30, 2016 Published\u00a0 May 2017\n\nFund Project: YL is partially supported by NSFC (11301442) and RGC (PolyU 253004\/14P). DG is partially supported by NSFC (11601336), Program for Professor of Special Appointment (Eastern Scholar) at Shanghai Institutions of Higher Learning (TP2015050), Shanghai Gaofeng Project for University Academic Development Program\n\nTicks, including the Ixodes ricinus and Ixodes scapularis hard tick species, are regarded as the most common arthropod vectors of both human and animal diseases in Europe and the United States capable of transmitting a large number of bacteria, viruses and parasites. Since ticks in larval and nymphal stages share the same host community which can harbor multiple pathogens, they may be co-infected with two or more pathogens, with a subsequent high likelihood of co-transmission to humans or animals. This paper is devoted to the modeling of co-infection of tick-borne pathogens, with special focus on the co-infection of Borrelia burgdorferi (agent of Lyme disease) and Babesia microti (agent of human babesiosis). Considering the effect of co-infection, we illustrate that co-infection with B. burgdorferi increases the likelihood of B. microti transmission, by increasing the basic reproduction number of B. microti below the threshold smaller than one to be possibly above the threshold for persistence. The study confirms a mechanism of the ecological fitness paradox, the establishment of B. microti which has weak fitness (basic reproduction number less than one). Furthermore, co-infection could facilitate range expansion of both pathogens.\n\nCitation: Yijun Lou, Li Liu, Daozhou Gao. Modeling co-infection of Ixodes tick-borne pathogens. Mathematical Biosciences & Engineering, 2017, 14 (5&6) : 1301-1316. doi: 10.3934\/mbe.2017067\n##### References:\n\nshow all references\n\n##### References:\nA schematic diagram of co-infection in the tick population. Here $E$ (eggs), $L\\!Q$ (questing larvae), $L\\!F$ (feeding larvae), $N\\!Q$ (questing nymphs), $N\\!F$ (feeding nymphs) and $A$ (adults) represent the stages of tick population with subscripts denoting the infectious status for each pathogen. Subscript $0$: no pathogen in ticks; $1$: Borrelia only; $2$: Babesia only; $3$: both pathogens\nA schematic diagram of co-infection in mice $M$ with subscripts denoting the infectious status for each pathogen\nSolution simulations with the model parameters in Table 2. Solutions through different initial values converge to the constant level for ticks (a), constant infected ticks for Borrelia infection only (b) and Babesia transmission cycle can not establish without the co-infection (c). However, on the scenario of coinfection, both pathogens can get established ((d), (e) and (f)). More interestingly, some ticks becomes infected with only Babesia or Borrelia while some others get infected with both pathogens\nThe state variables for the co-infection model. Bo and Ba represent Borrelia and Babesia, respectively\n Variable Meaning $E$ number of eggs $L\\!Q$ number of questing larvae $L\\!F_{0}$ number of feeding larvae susceptible to both Ba and Bo $L\\!F_{1}$ number of feeding larvae infected with Bo only $L\\!F_{2}$ number of feeding larvae infected with Ba only $L\\!F_{3}$ number of feeding larvae co-infected with Ba and Bo $N\\!Q_{0}$ number of questing nymphs susceptible to both Ba and Bo $N\\!Q_{1}$ number of questing nymphs infected with Bo only $N\\!Q_{2}$ number of questing nymphs infected with Ba only $N\\!Q_{3}$ number of questing nymphs co-infected with Ba and Bo $N\\!F_{0}$ number of feeding nymphs susceptible to both Ba and Bo $N\\!F_{1}$ number of feeding nymphs infected with Bo only $N\\!F_{2}$ number of feeding nymphs infected with Ba only $N\\!F_{3}$ number of feeding nymphs co-infected with Ba and Bo $A_{0}$ number of adults susceptible to both Ba and Bo $A_{1}$ number of adults infected with Bo only $A_{2}$ number of adults infected with Ba only $A_3$ number of adults co-infected with Ba and Bo $M_{0}$ number of mice susceptible to both Ba and Bo $M_{1}$ number of mice infected with Bo only $M_{2}$ number of mice infected with Ba only $M_{3}$ number of mice co-infected with Ba and Bo\n Variable Meaning $E$ number of eggs $L\\!Q$ number of questing larvae $L\\!F_{0}$ number of feeding larvae susceptible to both Ba and Bo $L\\!F_{1}$ number of feeding larvae infected with Bo only $L\\!F_{2}$ number of feeding larvae infected with Ba only $L\\!F_{3}$ number of feeding larvae co-infected with Ba and Bo $N\\!Q_{0}$ number of questing nymphs susceptible to both Ba and Bo $N\\!Q_{1}$ number of questing nymphs infected with Bo only $N\\!Q_{2}$ number of questing nymphs infected with Ba only $N\\!Q_{3}$ number of questing nymphs co-infected with Ba and Bo $N\\!F_{0}$ number of feeding nymphs susceptible to both Ba and Bo $N\\!F_{1}$ number of feeding nymphs infected with Bo only $N\\!F_{2}$ number of feeding nymphs infected with Ba only $N\\!F_{3}$ number of feeding nymphs co-infected with Ba and Bo $A_{0}$ number of adults susceptible to both Ba and Bo $A_{1}$ number of adults infected with Bo only $A_{2}$ number of adults infected with Ba only $A_3$ number of adults co-infected with Ba and Bo $M_{0}$ number of mice susceptible to both Ba and Bo $M_{1}$ number of mice infected with Bo only $M_{2}$ number of mice infected with Ba only $M_{3}$ number of mice co-infected with Ba and Bo\nDefinitions and corresponding values of the model parameters with the daily timescale. Abbreviations: Bo: Borrelia; Ba: Babesia; TP: transmission probability; AS: assumed parameter values\n Symbol Description Value Ref $\\mu_M$ mortality rate of mice 0.01 [2] $b_M$ birth rate of mice 0.02 [2] $D_M$ density-dependent death rate of mice $5\\times 10^{-5}$ AS $b_E$ egg reproduction rate $\\frac{16657}{365}$ [17] $\\mu_E$ mortality rate of eggs 0.0025 [17] $\\mu_{L\\!Q}$ mortality rate of questing larvae 0.006 [17] $\\mu_{L\\!F}$ mortality rate of feeding larvae 0.038 [17] $\\mu_{N\\!Q}$ mortality rate of questing nymphs 0.006 [17] $\\mu_{N\\!F}$ mortality rate of feeding nymphs 0.028 [17] $\\mu_A$ mortality rate of adults 0.01 [17] $d_E$ development rate of eggs $\\frac{2.4701}{365}$ [17] $d_L$ development rate of larvae $\\frac{2.2571}{365}$ [17] $d_N$ development rate of nymphs $\\frac{1.7935}{365}$ [17] $f_L$ feeding rate of larvae $\\frac{1.0475}{365}$ [17] $f_N$ feeding rate of nymphs $\\frac{1.0475}{365}$ [17] $D_L$ density-dependent mortality rate of $LF$ $\\frac{0.01}{\\text{200}}$ AS $D_N$ density-dependent mortality rate of $LN$ $\\frac{0.01}{\\text{200}}$ AS $\\beta_{11}$ TP of Bo from $M_{1}$ to $L\\!Q$ 0.6 [17] $\\beta_{31}$ TP of Bo from $M_{3}$ to $L\\!Q$ $1.5*\\beta_{11}-\\beta_{33}$ AS $\\beta_{22}$ TP of Ba from $M_{2}$ to $L\\!Q$ 0.45 AS $\\beta_{32}$ TP of Ba from $M_{3}$ to $L\\!Q$ $1.5*\\beta_{22}-\\beta_{33}$ AS $\\beta_{33}$ TP of both pathogens from $M_{3}$ to $L\\!Q$ $\\beta_{22}$ AS $\\bar{\\beta}_{11}$ TP of Bo from $M_{1}$ to $N\\!Q_{0}$ $\\beta_{11}$ AS $\\bar{\\beta}_{31}$ TP of Bo from $M_{3}$ to $N\\!Q_{0}$ $\\beta_{31}$ AS $\\bar{\\beta}_{22}$ TP of Ba from $M_{2}$ to $N\\!Q_{0}$ $\\beta_{22}$ AS $\\bar{\\beta}_{32}$ TP of Ba from $M_{3}$ to $N\\!Q_{0}$ $\\beta_{32}$ AS $\\bar{\\beta}_{33}$ TP of both pathogens from $M_{3}$ to $N\\!Q_{0}$ $\\beta_{33}$ AS $\\beta^{N\\!Q_{1}}_{23}$ TP of Ba from $M_{2}$ to $N\\!Q_{1}$ $\\beta_{22}$ AS $\\beta^{N\\!Q_{1}}_{33}$ TP of both pathogens from $M_{3}$ to $N\\!Q_{1}$ $\\beta_{33}$ AS $\\beta^{N\\!Q_{2}}_{13}$ TP of Bo from $M_{1}$ to $N\\!Q_{2}$ $\\beta_{11}$ AS $\\beta^{N\\!Q_{2}}_{33}$ TP of both pathogens from $M_{3}$ to $N\\!Q_{2}$ $\\beta_{33}$ AS $\\gamma_{11}$ TP of Bo from $N\\!F_{1}$ to $M_{0}$ 0.6 AS $\\gamma_{31}$ TP of Bo from $N\\!F_3$ to $M_0$ $\\beta_{31}$ AS $\\gamma_{22}$ TP of Ba from $N\\!F_2$ to $M_0$ $\\beta_{22}$ AS $\\gamma_{32}$ TP of Ba from $N\\!F_3$ to $M_0$ $\\beta_{32}$ AS $\\gamma_{33}$ TP of both pathogen from $N\\!F_3$ to $M_0$ $\\beta_{33}$ AS $\\bar{\\gamma}_{23}$ TP of Ba from $N\\!F_2$ to $M_1$ $\\beta_{22}$ AS $\\bar{\\gamma}_{33}$ TP of Ba from $N\\!F_3$ to $M_1$ $\\beta_{22}$ AS $\\tilde{\\gamma}_{13}$ TP of Bo from $N\\!F_1$ to $M_2$ $\\beta_{11}$ AS $\\tilde{\\gamma}_{33}$ TP of Bo from $N\\!F_3$ to $M_2$ $\\beta_{11}$ AS\n Symbol Description Value Ref $\\mu_M$ mortality rate of mice 0.01 [2] $b_M$ birth rate of mice 0.02 [2] $D_M$ density-dependent death rate of mice $5\\times 10^{-5}$ AS $b_E$ egg reproduction rate $\\frac{16657}{365}$ [17] $\\mu_E$ mortality rate of eggs 0.0025 [17] $\\mu_{L\\!Q}$ mortality rate of questing larvae 0.006 [17] $\\mu_{L\\!F}$ mortality rate of feeding larvae 0.038 [17] $\\mu_{N\\!Q}$ mortality rate of questing nymphs 0.006 [17] $\\mu_{N\\!F}$ mortality rate of feeding nymphs 0.028 [17] $\\mu_A$ mortality rate of adults 0.01 [17] $d_E$ development rate of eggs $\\frac{2.4701}{365}$ [17] $d_L$ development rate of larvae $\\frac{2.2571}{365}$ [17] $d_N$ development rate of nymphs $\\frac{1.7935}{365}$ [17] $f_L$ feeding rate of larvae $\\frac{1.0475}{365}$ [17] $f_N$ feeding rate of nymphs $\\frac{1.0475}{365}$ [17] $D_L$ density-dependent mortality rate of $LF$ $\\frac{0.01}{\\text{200}}$ AS $D_N$ density-dependent mortality rate of $LN$ $\\frac{0.01}{\\text{200}}$ AS $\\beta_{11}$ TP of Bo from $M_{1}$ to $L\\!Q$ 0.6 [17] $\\beta_{31}$ TP of Bo from $M_{3}$ to $L\\!Q$ $1.5*\\beta_{11}-\\beta_{33}$ AS $\\beta_{22}$ TP of Ba from $M_{2}$ to $L\\!Q$ 0.45 AS $\\beta_{32}$ TP of Ba from $M_{3}$ to $L\\!Q$ $1.5*\\beta_{22}-\\beta_{33}$ AS $\\beta_{33}$ TP of both pathogens from $M_{3}$ to $L\\!Q$ $\\beta_{22}$ AS $\\bar{\\beta}_{11}$ TP of Bo from $M_{1}$ to $N\\!Q_{0}$ $\\beta_{11}$ AS $\\bar{\\beta}_{31}$ TP of Bo from $M_{3}$ to $N\\!Q_{0}$ $\\beta_{31}$ AS $\\bar{\\beta}_{22}$ TP of Ba from $M_{2}$ to $N\\!Q_{0}$ $\\beta_{22}$ AS $\\bar{\\beta}_{32}$ TP of Ba from $M_{3}$ to $N\\!Q_{0}$ $\\beta_{32}$ AS $\\bar{\\beta}_{33}$ TP of both pathogens from $M_{3}$ to $N\\!Q_{0}$ $\\beta_{33}$ AS $\\beta^{N\\!Q_{1}}_{23}$ TP of Ba from $M_{2}$ to $N\\!Q_{1}$ $\\beta_{22}$ AS $\\beta^{N\\!Q_{1}}_{33}$ TP of both pathogens from $M_{3}$ to $N\\!Q_{1}$ $\\beta_{33}$ AS $\\beta^{N\\!Q_{2}}_{13}$ TP of Bo from $M_{1}$ to $N\\!Q_{2}$ $\\beta_{11}$ AS $\\beta^{N\\!Q_{2}}_{33}$ TP of both pathogens from $M_{3}$ to $N\\!Q_{2}$ $\\beta_{33}$ AS $\\gamma_{11}$ TP of Bo from $N\\!F_{1}$ to $M_{0}$ 0.6 AS $\\gamma_{31}$ TP of Bo from $N\\!F_3$ to $M_0$ $\\beta_{31}$ AS $\\gamma_{22}$ TP of Ba from $N\\!F_2$ to $M_0$ $\\beta_{22}$ AS $\\gamma_{32}$ TP of Ba from $N\\!F_3$ to $M_0$ $\\beta_{32}$ AS $\\gamma_{33}$ TP of both pathogen from $N\\!F_3$ to $M_0$ $\\beta_{33}$ AS $\\bar{\\gamma}_{23}$ TP of Ba from $N\\!F_2$ to $M_1$ $\\beta_{22}$ AS $\\bar{\\gamma}_{33}$ TP of Ba from $N\\!F_3$ to $M_1$ $\\beta_{22}$ AS $\\tilde{\\gamma}_{13}$ TP of Bo from $N\\!F_1$ to $M_2$ $\\beta_{11}$ AS $\\tilde{\\gamma}_{33}$ TP of Bo from $N\\!F_3$ to $M_2$ $\\beta_{11}$ AS\n [1] Zindoga Mukandavire, Abba B. Gumel, Winston Garira, Jean Michel Tchuenche. Mathematical analysis of a model for HIV-malaria co-infection. Mathematical Biosciences & Engineering, 2009, 6 (2) : 333-362. doi: 10.3934\/mbe.2009.6.333 [2] Holly Gaff. Preliminary analysis of an agent-based model for a tick-borne disease. Mathematical Biosciences & Engineering, 2011, 8 (2) : 463-473. doi: 10.3934\/mbe.2011.8.463 [3] Shangbing Ai. Global stability of equilibria in a tick-borne disease model. Mathematical Biosciences & Engineering, 2007, 4 (4) : 567-572. doi: 10.3934\/mbe.2007.4.567 [4] Georgi Kapitanov. A double age-structured model of the co-infection of tuberculosis and HIV. Mathematical Biosciences & Engineering, 2015, 12 (1) : 23-40. doi: 10.3934\/mbe.2015.12.23 [5] Holly Gaff, Robyn Nadolny. Identifying requirements for the invasion of a tick species and tick-borne pathogen through TICKSIM. Mathematical Biosciences & Engineering, 2013, 10 (3) : 625-635. doi: 10.3934\/mbe.2013.10.625 [6] Kazeem Oare Okosun, Robert Smith?. Optimal control analysis of malaria-schistosomiasis co-infection dynamics. Mathematical Biosciences & Engineering, 2017, 14 (2) : 377-405. doi: 10.3934\/mbe.2017024 [7] A. K. Misra, Anupama Sharma, Jia Li. A mathematical model for control of vector borne diseases through media campaigns. Discrete & Continuous Dynamical Systems - B, 2013, 18 (7) : 1909-1927. doi: 10.3934\/dcdsb.2013.18.1909 [8] Marcello Delitala, Tommaso Lorenzi. Emergence of spatial patterns in a mathematical model for the co-culture dynamics of epithelial-like and mesenchymal-like cells. Mathematical Biosciences & Engineering, 2017, 14 (1) : 79-93. doi: 10.3934\/mbe.2017006 [9] Xuejuan Lu, Lulu Hui, Shengqiang Liu, Jia Li. A mathematical model of HTLV-I infection with two time delays. Mathematical Biosciences & Engineering, 2015, 12 (3) : 431-449. doi: 10.3934\/mbe.2015.12.431 [10] Suxia Zhang, Xiaxia Xu. A mathematical model for hepatitis B with infection-age structure. Discrete & Continuous Dynamical Systems - B, 2016, 21 (4) : 1329-1346. doi: 10.3934\/dcdsb.2016.21.1329 [11] Expeditho Mtisi, Herieth Rwezaura, Jean Michel Tchuenche. A mathematical analysis of malaria and tuberculosis co-dynamics. Discrete & Continuous Dynamical Systems - B, 2009, 12 (4) : 827-864. doi: 10.3934\/dcdsb.2009.12.827 [12] Wandi Ding. Optimal control on hybrid ODE Systems with application to a tick disease model. Mathematical Biosciences & Engineering, 2007, 4 (4) : 633-659. doi: 10.3934\/mbe.2007.4.633 [13] Laura Fumanelli, Pierre Magal, Dongmei Xiao, Xiao Yu. Qualitative analysis of a model for co-culture of bacteria and amoebae. Mathematical Biosciences & Engineering, 2012, 9 (2) : 259-279. doi: 10.3934\/mbe.2012.9.259 [14] Nicola Bellomo, Youshan Tao. Stabilization in a chemotaxis model for virus infection. Discrete & Continuous Dynamical Systems - S, 2018, 0 (0) : 105-117. doi: 10.3934\/dcdss.2020006 [15] Danyun He, Qian Wang, Wing-Cheong Lo. Mathematical analysis of macrophage-bacteria interaction in tuberculosis infection. Discrete & Continuous Dynamical Systems - B, 2018, 23 (8) : 3387-3413. doi: 10.3934\/dcdsb.2018239 [16] Jinliang Wang, Xiu Dong. Analysis of an HIV infection model incorporating latency age and infection age. Mathematical Biosciences & Engineering, 2018, 15 (3) : 569-594. doi: 10.3934\/mbe.2018026 [17] Xia Wang, Yuming Chen. An age-structured vector-borne disease model with horizontal transmission in the host. Mathematical Biosciences & Engineering, 2018, 15 (5) : 1099-1116. doi: 10.3934\/mbe.2018049 [18] Stephen A. Gourley, Xiulan Lai, Junping Shi, Wendi Wang, Yanyu Xiao, Xingfu Zou. Role of white-tailed deer in geographic spread of the black-legged tick Ixodes scapularis : Analysis of a spatially nonlocal model. Mathematical Biosciences & Engineering, 2018, 15 (4) : 1033-1054. doi: 10.3934\/mbe.2018046 [19] Avner Friedman, Chuan Xue. A mathematical model for chronic wounds. Mathematical Biosciences & Engineering, 2011, 8 (2) : 253-261. doi: 10.3934\/mbe.2011.8.253 [20] Jos\u00e9 Ignacio Tello. Mathematical analysis of a model of morphogenesis. Discrete & Continuous Dynamical Systems - A, 2009, 25 (1) : 343-361. doi: 10.3934\/dcds.2009.25.343\n\n2018\u00a0Impact Factor:\u00a01.313","date":"2019-09-16 20:11:54","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\": 0.47030195593833923, \"perplexity\": 5521.41264465197}, \"config\": {\"markdown_headings\": true, \"markdown_code\": true, \"boilerplate_config\": {\"ratio_threshold\": 0.18, \"absolute_threshold\": 10, \"end_threshold\": 15, \"enable\": true}, \"remove_buttons\": true, \"remove_image_figures\": true, \"remove_link_clusters\": true, \"table_config\": {\"min_rows\": 2, \"min_cols\": 3, \"format\": \"plain\"}, \"remove_chinese\": true, \"remove_edit_buttons\": true, \"extract_latex\": true}, \"warc_path\": \"s3:\/\/commoncrawl\/crawl-data\/CC-MAIN-2019-39\/segments\/1568514572934.73\/warc\/CC-MAIN-20190916200355-20190916222355-00304.warc.gz\"}"} | null | null |
Q: Is there a specific way to PING search engines with urls (like pingler.com)? Is there a standard variable that should be passed to the ping agents?
I'm trying to create my own but I'm not exactly sure how it works.
Is it as simple as appending a REST call to each url or does it need to be in a certain format?
Thanks
EDIT
sorry for the bad question description.
Basically I't tring to figure out how pingler.com pings a url to the search engines so that they spiders update the engine results.
Thanks
EDIT
thanks guys for the good answers
A: No, there isn't. pingler, presumably, has custom code for each search engine it deals with.
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Kansas Attractions
Article By: Rachel Coleman
Visit the wonderful, magical Oz Museum
There's Kansas City, Kansas and Kansas City, Missouri but there's only one state that was famously called "home" by Dorothy in "The Wizard of Oz." For further information about attractions and events in the state of Kansas, visit www.travelks.com or call 800/2KANSAS.
Great Fun for the Kids
(Toddlers to Age 8)
Oz Museum
Wamego, KS 66547
866/458-TOTO
A collection of Wizard of Oz memorabilia—from the storybook, MGM musical and other films— is on display at this museum. Some items are a Munchkin's vest, an original script, and an autograph book signed by every Wizard of Oz cast member. Visitors can pass through Dorothy's farmyard and travel to Munchkinland, or view the life-size depictions of Dorothy, the Scarecrow, Lion, and Tin Man. Open daily.
Exploration Place
300 North McLean Boulevard
This creative learning center welcomes adults and children to explore its many hands-on exhibits. You can test your skills in one of the two flight simulators, step into a 20-foot tornado simulation, touch a real mammoth tusk, travel back in time to the Middle Ages exhibit and prepare your own medieval dinner while talking to the friendly blacksmith as he creates steel authentic to the period or even examine a plane's black box and learn how these devices really work. In addition to the exhibits, there is a 60-foot domed theatre, the largest in Kansas. Here, for an extra admission price of $2, you can enjoy a flight through space, battle a virus attacking a body's cell, or discover the Seven Wonders of the World. The center also has a park and a mini golf course outside.
Rolling Hills Wildlife Adventure Museum and Zoo
625 N. Hedville Road
The zoo features 105 species of wildlife, and the museum features naturalistic exhibits of over 300 animals from seven regions ranging from the arctic to the rainforest. Visitors can listen as animated robots talk about natural resources, learn about animals in the interactive education center, or watch a 3-D movie in the domed theatre. Outside, families can see animals such as orangutans, camels, aardvarks, and many others. Kids will enjoy feeding the goats and giraffes.
6425 SW Sixth Avenue
The Southern Cheyenne tipi, a covered wagon ready and stocked for a journey on the Oregon Trail, and fully restored 1880 steam locomotive are some of the popular items featured at the Kansas Museum. Learn the story of Kansas through the events of The Civil War and the Kansas settlement, and about transportation and agriculture. Discovery Place, the children's gallery, is a hands-on interactive museum, and the Special Exhibits gallery features changing exhibits throughout the year. An original plane, built and flown by Albin Longren, an aviation builder in the early 1900s, is also on display at the museum.
Fun for Older Children
(Up to Age 18)
Kansas City Community Ballpark
1800 Village West Parkway
Kansas City 66111
If you'd like to catch a baseball game, Kansas City's Community Ballpark is home of the T-Bones minor-league baseball team. Family-friendly features include a kid's concession stand and a kid's play area in full view of the field, so parents can watch restless children play without missing any of the game.
Kansas Speedways
400 Speedway Boulevard
Kansas' largest tourist attraction, the $200 million Kansas Speedway is a place for high speed thrills. On top of NASCAR and the IndyCar races, there are numerous things to do at this 1.5 mile long and 55 feet wide tri-oval racetrack. Throughout the year, the speedway hosts numerous driving lessons, sponsored by the likes of Jeff Gordon and Dale Jarret, with packages that include either riding in the car with a professional driver or even taking it for a spin by yourself. Available on Thursdays, from April to November, walking tours that cover the 44 pit stalls, garages, victory lane and the infield can show you life in the fast lane at a slower pace.
Hopalong Cassidy Cowboy Museum
15231 SW Parallel Road
Benton, KS 67017
Hopalong Cassidy, one of America's greatest Cowboy heroes, was originally created in novels and short stories by author Clarence E. Mulford, and was immortalized by actor William Boyd, who starred in 66 Hopalong Cassidy films, as well as T.V. shows and radio programs, during the '40s and '50s. This museum features a large collection of Cassidy memorabilia, interactive displays for both adults and children, and a 250-seat movie theatre showing movies and television segments.
Sternberg Museum of Natural History
3000 Sternberg Drive
Fort Hays State University's Sternberg Museum has an extensive fossil collection from the Cretaceous Period, which was about 70-80 million years ago, when Colorado was ocean front property and Kansas was an inland sea. Families can see the fossil remains of plants and animals that lived in or alongside an ancient sea, learn about the natural history of the Great Plains, or see a full-scale diorama of a T-Rex dinosaur and other creatures that lived when Kansas was covered in water. In the children's discovery room, kids can learn about animals and the environment. Exhibits include a display of flying creatures, the unique fish-within-a-fish fossil, and many traveling exhibits, as the museum's emphasis is on education.
Fun for the Family
Maxwell Wildlife Refuge
2565 Pueblo Road
Canton, KS 67428
This 2,250 acre reserve has the largest population of bison found in Kansas. With a focus on the importance of preserving the environment and history of the area, Maxwell is home to 150 different species of birds, 50 elks and 150-200 bison. Since the last wild bison was killed in 1879 in Dodge City, Kansas, families should take the opportunity to see this beautiful and endangered species. By appointment only, the refuge offers guided tram tours all year round that takes visitors up close and personal through the herd.
Kansas' Theatre in the Park
17501 Midland Drive
913/631-7050 or 913/312-8841
Throughout the summer, this outdoor theatre puts on a variety of Broadway musicals featuring aspiring actors from throughout the community. Past shows have included "Seussical," "Beauty and the Beast," and "West Side Story." The theatre is easily accessible from Kansas City and its metropolitan areas.
Old Cowtown Museum
1865 W. Museum Boulevard
This 17-acre open-air living history village showcases costumed characters recreating 1870s Old West life in a Kansas cattle town on the Chisholm Trail. Cowtown has 11,000 artifacts and more than 40 buildings, including a working blacksmith, carpenter and newspaper shops, historic homes, a livery stable, bank, saloon, drug store, bath house, law office, general store, and even a funeral parlor. Each home has a vegetable garden using period seeds and there are historic interpreters in many of the sites to demonstrate and discuss the area's history, from 1865-1880. Open April-October, with special events monthly.
Botanica Botanical Gardens
701 Amidon
Children can attend a variety of educational workshops at Botanica, where they can learn flower arranging, create masks out of gourds, or build bird feeders. Adults will admire the variety of colors and blooms in the themed gardens, and children will be especially fascinated by the butterfly garden, where 5,000 butterflies are released throughout the summer and fall.
Boulevard Drive-In Theatre
1051 Merriam Lane
This updated 1950s drive-in theatre boasts a DTS processor, which is at the forefront of digital technology and sound. Families can listen to the field speakers or tune in their car radios to 89.7FM; both provide distortion-free and rattle-free sound. Every Friday and Saturday nights the theatre shows two films currently showing in theatres. Relax and enjoy some goodies from the concession stand while watching a movie within the privacy of your own car.
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Next Article Next Article Avid Skiers + Toddler = Thrills in Banff | {
"redpajama_set_name": "RedPajamaCommonCrawl"
} | 80 |
Chris Mulhern posted up a sick montage from his time in Paris and London while working on his upcoming video.
Eccentricity and electricity! Alive! Gimme some Hook-Ups.
I am definitely envious of all of those doods for skating such amazing spots.
I wasn't still wasn't infatuated with the song. I thought it got tacky, especially once it switched into London.
Still: damn good chit. If that's montage material I can wait for the video to drop. | {
"redpajama_set_name": "RedPajamaC4"
} | 5,226 |
Q: Ignore Mongoose schema on read only I have a collection where the data conforms to one of a few different schemas. The reason these are in one collection is that I need to be able to query them all together, taking advantage of the schema-less nature of MongoDB.
Is there a way to query a collection with mongoose while ignoring the schema? As an example:
var adam = new ContractCustomer({
// Contract customer data saved to customers collection
})
var betty = new PaygCustomer({
// PAYG customer data saved to customers collection
})
adam.save()
betty.save()
Customers.find({}).exec().then() // Query all the customers regardless of which schema they belong to.
A: I would recommand to connect to the database using mongodb-driver, and to make a find from there. In my knowledge,there is no way to perform a schemaless query using mongoose.
The whole point of mongoose is to use schemas.
(keep your connection to mongoose, just open a parallel connection)
| {
"redpajama_set_name": "RedPajamaStackExchange"
} | 9,745 |
\section{Introduction}
\subsection{The sum of a self-adjoint element and an elliptic element}
The elliptic element is an element in a $W^*$-probability space of the form $z = x+iy$ where $x$ and $y$ are freely independent semicircular elements, possibly with different variances. The variance of such an element is given by
\[\tau(z^*z) = \tau(x^*x)+\tau(y^*y).\]
Once the variance of $z$ is given, say $s$, there are several possibilities for the variances of $x$ and $y$. We use the parameters $t = 2\tau(y^*y)$, and $\tau(x^*x) = s-\frac{t}{2}$. Under the parameters $s$, $t$, the elliptic element $z$ then has the form
\[\tilde{\sigma}_{s-\frac{t}{2}}+i\sigma_{\frac{t}{2}}\]
where $\tilde\sigma_{s-\frac{t}{2}}$ and $\sigma_{\frac{t}{2}}$ are freely independent semicircular elements in a certain $W^*$-probability space.
Suppose that $y_0$ is a bounded self-adjoint element in the $W^*$-probability space containing $\tilde\sigma_{s-\frac{t}{2}}$ and $\sigma_{\frac{t}{2}}$; suppose also that all the three elements are freely independent. In this paper, we compute the Brown measure of the element
\[y_0+\tilde{\sigma}_{s-\frac{t}{2}}+i\sigma_{\frac{t}{2}}.\]
We show that the Brown measure of $y_0+\tilde{\sigma}_{s-\frac{t}{2}}+i\sigma_{\frac{t}{2}}$ is a push-forward of the Brown measure of $y_0+c_s$ where $c_s = \tilde{\sigma}_{\frac{s}{2}}+i\sigma_{\frac{s}{2}}$ is the Voiculescu's circular element. The Brown measure of $y_0+c_s$ was computed and analyzed by the author and Zhong~\cite{HoZhong2019}. We also study the asymptotic behavior of the Brown measure of $y_0+\tilde{\sigma}_{s-\frac{t}{2}}+i\sigma_{\frac{t}{2}}$ as
\begin{enumerate}
\item $s\to\infty$ with $s/t$ fixed;
\item $s\to\infty$ with $t$ fixed; and
\item $s\to\infty$ with $s = t/2$.
\end{enumerate}
If $s\geq t$, our results can be computed by the results of Zhong and the author \cite{HoZhong2019} in which the Brown measure of $x_0+c_t$ is computed, with $x_0 = y_0+\tilde{\sigma}_{s-t}$, where $c_t$ is a circular element, freely independent of $x_0$. If $s<t$, $y_0+\tilde{\sigma}_{s-\frac{t}{2}}+i\sigma_{\frac{t}{2}}$ is not a sum of a self-adjoint element and a circular element. We need a more general method.
The results in this paper are obtained using the method introduced in~\cite{HallHo2020}, where the Brown measure of $x_0+i\sigma_t$, with $x_0$ and $\sigma_t$ freely independent, was computed. We combine this method with techniques in free probability to determine the Brown measure of $y_0+\tilde{\sigma}_{s-\frac{t}{2}}+i\sigma_{\frac{t}{2}}$ in terms of $y_0$. The results in~\cite{HallHo2020} used a PDE method introduced in the work of Driver, Hall and Kemp \cite{DHK2019}; this method has been used in subsequent work by other authors~\cite{DemniHamdi2020, HallHo2020, HoZhong2019}. See also the expository article \cite{Hall2019} by Hall for an introduction to the PDE method.
The results in this paper have direct connections to random matrix theory. If $X$ and $X'$ are independent Gaussian unitary emsembles (GUEs), and $Y_N$ is a sequence of $N\times N$ self-adjoint deterministic matrices whose empirical eigenvalue distributions converge weakly to the law of $y_0$, then $Y_N$, $X_N$ and $X_N'$ are asymptotically free in the sense of Voiculescu~\cite{Voiculescu1991}. By~\cite[Theorem 6]{Sniady2002}, the empirical eigenvalue distribution of the random matrix (which is almost surely non-normal)
\[Y_N+\sqrt{s-\frac{t}{2}}X_N+i\sqrt{\frac{t}{2}}X_N'\]
converges to the Brown measure of $y_0+\tilde{\sigma}_{s-\frac{t}{2}}+i\sigma_{\frac{t}{2}}$ as $N\to\infty$, if $s>\frac{t}{2}$. In this paper, the Brown measure computation includes the case $s=\frac{t}{2}$. In this special case $s=\frac{t}{2}$, the convergence of the empirical eigenvalue distribution does not follow from \cite{Sniady2002}; nevertheless, numerical simulations in \cite{HallHo2020} suggest that the Brown measure of $y_0+i\sigma_{\frac{t}{2}}$ is indeed the limiting eigenvalue distribution of $Y_N+i\sqrt{t/2}X_N$, where $Y_N$ and $X_N$ are the same matrices as above.
The Brown measure computed in the case where $y_0=0$ is the elliptic law; its name is due to the fact that its support is a region bounded by an ellipse centered at the origin. In the even more special case $s=t$, the Brown measure is called the circular law since its support is a disk centered at the origin. The circular law was discovered first by Ginibre \cite{Ginibre1965} as a limiting eigenvalue distribution of a random matrix model with Gaussian entries, now commonly called the Ginibre ensemble, then by Girko \cite{Girko1984} in the case when the entries come with more relaxed assumptions. The assumptions of random matrix models were then further relaxed, for example, by Bai \cite{Bai1997}, and Tao and Vu \cite{TaoVu2010}. In the $s\neq t$ case, the elliptic law was first computed by Girko~\cite{Girko1985} as a limiting eigenvalue distribution of certain random matrix model. The Brown measure, in the operator framework, was computed by Biane and Lehner~\cite{BianeLehner2001} and various later work of others.
The Brown measure of operators of the form $X+iY$ where $X$ and $Y$ are freely independent has been analyzed at a nonrigorous level in physics literature. Stephanov \cite{Stephanov1996} uses the case when $X$ is Bernoulli distributed and $Y$ is a GUE to provide a model of QCD. Janik et al. \cite{JNPWZ1997} identified the domain where the eigenvalues cluster in the large-$N$ limit when $X$ is an arbitrary self-adjoint random matrix and $Y$ is a GUE. Jarosz and Nowak \cite{JaroszNowak2004, JaroszNowak2006} computed the limiting eigenvalue distribution for general self-adjoint $X$ and $Y$. Belinschi et al.~\cite{BelinschiMaiSpeicher2017, BelinschiSniadySpeicher2018} put the results in~\cite{JaroszNowak2004, JaroszNowak2006} on a more rigorous basis; however, there have not been analytic results about the Brown measure of $X+iY$ obtained under this framework.
\subsection{Statements of results}
Let $y_0$ be a bounded self-adjoint element, $\sigma_{s-\frac{t}{2}}$ and $\tilde{\sigma}_{\frac{t}{2}}$ are semicircular elements with variances $s-t/2$ and $t/2$ in a tracial von Neumann algebra $(\mathscr{A}, \tau)$ called a $W^*$-probability space; suppose also that all the three of them are freely independent. Throughout the paper, we let $\nu$ be the law (or distribution) of $y_0$, which is the unique compactly supported probability measure on $\mathbb R$ such that
\[\int x^n\,d\nu(x) = \tau(y_0^n).\]
Recall that, in this paper, we compute the Brown measure of the element
\[y_0+\tilde{\sigma}_{s-\frac{t}{2}}+i\sigma_{\frac{t}{2}}\in\mathscr{A}.\]
Background information of free probability and Brown measure is reviewed in Section \ref{sect:Bkground}. The choice of the parameters $s$, $t$ comes from the context of two-parameter Segal--Bargmann transform \cite{DriverHall1999, Hall1999, Ho2016}. It is a interpolation between the self-adjoint element $y_0+\sigma_s$ and the element $y_0+i\sigma_s$ studied in \cite{HallHo2020}.
We make the following standing assumption about the element $y_0+\tilde{\sigma}_{s-\frac{t}{2}}+i\sigma_{\frac{t}{2}}$. We use $\mathrm{Law}(a)$ to denote the law of any self-adjoint random variable $a\in\mathscr{A}$.
\begin{assumption}
\label{assump:Standing}
Throughout the paper, we assume either $s>\frac{t}{2}$ or $\nu$ is not a Dirac measure, so that $\mathrm{Law}(y_{0}+\sigma_{s-\frac{t}{2}})$ is not a Dirac measure.
\end{assumption}
When this assumption does not hold, that is, if $\mathrm{Law}(y_{0}+\tilde{\sigma}_{s-\frac{t}{2}})$ is a Dirac measure, then one cannot apply the results from \cite{HallHo2020}. However, in this case, the element $y_0+\tilde{\sigma}_{s-\frac{t}{2}}+i\sigma_{\frac{t}{2}}$ has the form $u+i\tilde{\sigma}_{\frac{t}{2}}$ for some constant $u\in\mathbb R$. The Brown measure is then a semicircular distribution centered at $u$ on the vertical line through the point $u$. Under Assumption~\ref{assump:Standing}, by the results in \cite{HallHo2020}, the Brown measure is absolutely continuous with respect to the Lebesgue measure on the plane.
The following theorem summarizes Theorems~\ref{thm:addellipse} and~\ref{thm:pushforward}.
\begin{theorem}
\label{thm:summary1}
\begin{enumerate}
\item For each $s\geq\frac{t}{2}>0$, there is a continuous function $b_{s,t}:\mathbb R\to[0,\infty)$ such that the Brown measure of $y_0+\tilde{\sigma}_{s-t/2}+i\sigma_{t/2}$ is supported in the closure of the set
\[\Omega_{s,t} = \{a+ib\in\mathbb C|\left\vert b\right\vert<b_{s,t}(a)\}.\]
The open set $\Omega_{s,t}$ is a set of full measure. The Brown measure is absolutely continuous with respect to the Lebesgue measure on $\mathbb C$, with density
\[w_{y_0,s,t}(a+ib) = \frac{1}{2\pi t}\left(1+t\frac{d}{da}\int_\mathbb R\frac{(\alpha_{s,t}(a) - x)\,d\nu(x)}{(\alpha-x)^2+v_{y_0,s}(\alpha_{s,t}(a))^2}\right), \quad \left\vert b\right\vert<b_{s,t}(a)\]
for a certain homeomorphism $\alpha_{s,t}$ on $\mathbb R$ and a certain nonnegative continuous function $v_{y_0,s}$. In particular, the density is constant in the vertical directions.
\item The Brown measure of $y_0+\tilde{\sigma}_{s-t/2}+i\sigma_{t/2}$ is the push-forward measure of the Brown measure of $y_0+c_s$ by the homeomorphism $U_{s,t}:\mathbb C\to\mathbb C$,
\[U_{s,t}(\alpha+i\beta)=a_{s,t}(\alpha)+i\frac{t}{s}\beta\]
where $a_{s,t}$ is the inverse function of $\alpha_{s,t}$.
\item The push-forward measure of the Brown measure of $y_0+\tilde{\sigma}_{s-t/2}+i\sigma_{t/2}$ by the map, constant in the vertical directions,
\[Q_{s,t}(a+ib):= \frac{1}{s-t}[sa-t\alpha_{s,t}(a)]\]
is the law of the self-adjoint element $y_0+\sigma_s$.
\end{enumerate}
\end{theorem}
We now describe briefly how to compute the functions, and so the Brown measure, stated in Theorem~\ref{thm:summary1}. Given $a\in\mathbb R$, we try to solve for $\alpha\in\mathbb R$ and $v>0$ the equations
\begin{equation}
\label{eq:IntroEq}
\begin{split}
&\int\frac{d\nu(x)}{(\alpha-x)^2+v^2} = \frac{1}{s}\\
&\frac{(2s-t)\alpha}{s}-(s-t)\int\frac{x\,d\nu(x)}{(\alpha-x)^2+v^2} = a
\end{split}
\end{equation}
If solution exists, we label $\alpha$ by $\alpha_{s,t}(a)$ and $v$ by $v_{y_0,s}(\alpha_{s,t}(a))$, and we set
\begin{equation}
\label{eq:bstIntro}
b_{s,t}(a) = \frac{t}{s}v_{y_0,s}(\alpha_{s,t}(a));
\end{equation}
if there is no solution, we set both $v$ and $b_{s,t}(a)$ be $0$. In either case, $v = 0$ or $v>0$, the second equation of~\eqref{eq:IntroEq} always determines an $\alpha$; thus $\alpha_{s,t}$ is defined on $\mathbb R$ (See Proposition~\ref{prop:alphast}). The inverse $a_{s,t}$ of $\alpha_{s,t}$ has an explicit form as inthe second equation of~\eqref{eq:IntroEq}
\[a_{s,t}(\alpha) = \alpha+(s-t)\int\frac{(\alpha-x)\,d\nu(x)}{(\alpha-x)^2+v_{y_0,s}(\alpha)^2}.\]
In the special case $s=t$, we obtain $\alpha_{s,t}(a) = a$ and, by Theorem~\ref{thm:summary1},
\[w_{y_0,s,s}(a+ib) = \frac{1}{\pi s}\left(1-\frac{t}{2}\frac{d}{da}\int_\mathbb R\frac{x\,d\nu(x)}{(a-x)^2+v_{y_0,s}(a)^2}\right)\]
which reduces to the results in \cite{HoZhong2019}. In another special case $t = 2s$, the equations in~\eqref{eq:IntroEq} reduces to Equations (1.4) and (1.5) in \cite{HallHo2020}. The function $\alpha_{s,t}$ is the function $a_0^s$ as in \cite{HallHo2020} (with $t$ replaced by $s$) and the density is given by
\[\frac{1}{2\pi s}\left(\frac{da_0^s}{da}-\frac{1}{2}\right).\]
Thus, in the case, Theorem~\ref{thm:summary1} reduces to the results in \cite{HallHo2020}.
In Sections~\ref{sect:circularAsymp} and~\ref{sect:ellipticAsymp}, we also investigate the asymptotic behaviors of the Brown measure of $y_0+\tilde{\sigma}_{s-\frac{t}{2}}+i\sigma_{\frac{t}{2}}$, which are summarized in the following theorem; roughly speaking, the Brown measure of $y_0+\tilde{\sigma}_{s-\frac{t}{2}}+i\sigma_{\frac{t}{2}}$ behaves like the Brown measure of $\tilde{\sigma}_{s-\frac{t}{2}}+i\sigma_{\frac{t}{2}}$. See the theorems in these two sections for precise meaning.
\begin{theorem}
In all of the following three limiting regimes, the function $b_{s,t}$ is unimodal for all large enough $s$.
\begin{enumerate}
\item As $s\to\infty$ with $s/t$ fixed: the domain $\Omega_{s,t}$ almost has the shape of an ellipse centered at $(\tau(y_0), 0)$ with horizontal semi-axis of length $\frac{2s-t}{\sqrt{s}}$ and vertical semi-axis of length $\frac{t}{\sqrt{s}}$. The density $w_{y_0,s,t}$ converges to the constant
\[\frac{1}{\pi}\frac{s}{(2s-t)t}\]
uniformly outside any neighborhood of the endpoints of $\Omega_{s,t}\cap\mathbb R$.
\item As $s\to\infty$ with $t$ fixed: the domain $\Omega_{s,t}$ becomes a long and thin ellipse centered at $(\tau(y_0),0)$, with horizontal semi-axis of length $2\sqrt{s}$ and vertical semi-axis of length $\frac{t}{\sqrt{s}}$. The density converges to the constant
\[\frac{1}{2\pi t}\]
uniformly outside any neighborhood of the endpoints of $\Omega_{s,t}\cap\mathbb R$.
\item As $s\to\infty$ with $t = 2s$: the domain $\Omega_{s,t}$ becomes a narrow and tall ellipse centered at $(\tau(y_0), 0)$. More precisely, given any $c>1$, we have
\[\tau(y_0)-\frac{4c\tau(y_0^2)}{\sqrt{s}}<\inf(\Omega_{s,t}\cap\mathbb R)<\tau(y_0)<\sup(\Omega_{s,t}\cap\mathbb R)<\tau(y_0)+\frac{4c\tau(y_0^2)}{\sqrt{s}}.\]
for all large enough $s$.
\end{enumerate}
\end{theorem}
We do not have a density estimate for the last case.
\section{Background and previous results\label{sect:Bkground}}
\subsection{Free random variables}
\begin{definition}
\begin{enumerate}
\item We call $(\mathscr{A}, \tau)$ a {\bf $W^*$-probability space} if $\mathscr{A}$ is a von Neumann algebra and $\tau$ is a normal, faithful tracial state on $\mathscr{A}$. The elements in $\mathscr{A}$ are called {\bf non-commutative random variables}, or simply random variables.
\item The $\ast$-subalgebra $A_1,\ldots A_n\subset \mathscr{A}$ are called {\bf freely independent} if given an $i_1, i_2,\ldots i_m\in\{1,\ldots,n\}$ with $i_k\neq i_{k+1}$, $a_{i_j}\in\mathscr{A}_{i_j}$ are centered, then we also have $\tau(a_{i_1}a_{i_2}\ldots a_{i_m})=0$. The random variables $a_1,\ldots,a_m$ are freely independent if the $\ast$-algebras they generate are freely independent.
\item For a self-adjoint element $a\in\mathscr{A}$, the {\bf distribution}, or the {\bf law}, of $a$ is a compactly supported measure $\mu$ on $\mathbb R$ such that
\[\int_\mathbb R f\,d\mu = \tau(f(a))\]
for all continuous function $f$. We denote by $\mathrm{Law}(a)$ the law of $a$.
\end{enumerate}
\end{definition}
We now introduce the random variables that are key to this paper. The {\bf semicircular element} $\sigma_t$ has the {\bf semicircular distribution}, or the {\bf semicircle law} of variance $t$, supported on $[-2\sqrt{t},2\sqrt{t}]$ with density
\[\frac{\sqrt{4t-x^2}}{2\pi t}\,dx.\]
The {\bf circular element} $c_s$ has the form $\tilde{\sigma}_{\frac{s}{2}}+i\sigma_{\frac{s}{2}}$ where $\tilde{\sigma}_{\frac{s}{2}}$ and $\sigma_{\frac{t}{2}}$ are freely independent semicircular elements. The {\bf elliptic element} has the form $\tilde{\sigma}_{s-\frac{t}{2}}+i\sigma_{\frac{t}{2}}$ where $\tilde{\sigma}_{s-\frac{t}{2}}$ and $\sigma_{\frac{t}{2}}$ are freely independent semicircular elements.
\subsubsection{The $R$-transform\label{sect:Rtransform}}
Let $a\in\mathscr{A}$ be a self-adjoint element with law $\mu$. Then we consider the {\bf Cauchy transform}
\[G_a(z) = \int\frac{1}{z-x}\,d\mu(x)\]
defined outside the spectrum of $a$. The Cauchy transform $G_a$ is univalent around $\infty$. Denote by $K_a$ the inverse of $G_a$ at $\infty$, and let
\[R_a(z) = K_a(z)-\frac{1}{z}.\]
We call $K_a$ the {\bf $K$-transform} of $a$ and $R_a$ the {\bf $R$-transform} of $a$.
\begin{theorem}[\cite{Voiculescu1986}]
If $a_1, a_2\in\mathscr{A}$ are freely independent random variables, then the $R$-transform of the random variable $a=a_1+a_2$ is given by
\[R_a = R_{a_1}+R_{a_2}.\]
\end{theorem}
Using the notations in the theorem, the distribution of $a$ is called the {\bf free convolution} of $a_1+a_2$.
\subsection{The Brown measure}
In this section, we review the definition of the Brown measure, which was introduced by Brown \cite{Brown1986}. Let $a\in\mathscr{A}$. We define a function $S$ by
\[S(\lambda, \varepsilon) = \tau[\log(|a-\lambda|^2+\varepsilon)],\quad \lambda\in\mathbb C, \varepsilon>0.\]
Then
\[S(\lambda, 0) = \lim_{\varepsilon\to 0^+}S(\lambda, \varepsilon)\]
exists as a subharmonic function on $\mathbb C$, with value in $\mathbb R\cup \{-\infty\}$. The {\bf Brown measure} of $a$, denoted by $\mathrm{Brown}(a)$, is defined to be
\[\mathrm{Brown}(A) = \frac{1}{4\pi}\Delta_\lambda S(\lambda, 0)\]
where the Laplacian is in distributional sense.
One can see that $S(\lambda, 0)$ does define a harmonic function \emph{outside} the spectrum of $a$; the Brown measure of $a$ is a probability measure supported on the spectrum of $a$. The support of $\mathrm{Brown}(a)$, however, can be a proper subset of the spectrum of $a$.
The Brown measure of an $N\times N$ matrix is the empirical eigenvalue distribution of the matrix. If a sequence of random matrices $A_N$ converges in $\ast$-distribution to an element $a$ in a non-commutative probability space, one generally expects that the empirical eigenvalue distribution of $A_N$ converges to the Brown measure of $a$; this, however, is not always the case. A counter-example is the nilpotent matrix
\[
\begin{pmatrix}
0&1&0&\cdots&0\\
0&0&1&\cdots&0\\
\vdots&\vdots&\vdots&\ddots&\vdots\\
0&0&0&\cdots&1\\
0&0&0&\cdots&0\\
\end{pmatrix};\]
this sequence of matrices converges to the Haar unitary element in $\ast$-distribution but the empirical eigenvalue distribution is always the Dirac measure at $0$.
The Brown measure of the circular element $c_s = \tilde{\sigma}_{\frac{s}{2}}+i\sigma_{\frac{s}{2}}$ is called the {\bf circular law} and is supported in the disk of radius $\sqrt{s}$ centered at the origin. The density is the constant
\[\frac{1}{\pi s}\]
in the support. The circular element is an $R$-diagonal element. The Brown measure of the circular element can be computed by the method developed by Haagerup and Larsen \cite{HaagerupLarsen2000}, and Haagerup and Schultz \cite{HaagerupSchultz2007}.
The Brown measure of the elliptic element $\tilde{\sigma}_{s-\frac{t}{2}}+i\sigma_{\frac{t}{2}}$ is called the {\bf elliptic law} and is supported in an ellipse with semi-axes on the real and imaginary axes of length $\frac{2s-t}{\sqrt{s}}$ and $\frac{t}{\sqrt{s}}$ respectively. The density is the constant
\[\frac{1}{\pi}\frac{s}{2s-t}\]
in the support.
The elliptic law was computed by Biane and Lehner~\cite{BianeLehner2001}.
\subsection{Biane's free convolution formula}
\label{sect:Biane}
In this section, we review the results of the distribution of the free convolution of a self-adjoint element and a semicircular element established by Biane \cite{Biane1997sc}; several functions and a domain also come up in our study of Brown measure. Given a random variable $x_0$ with law $\mu$, we consider the function
\[v_{x_0,t}(a_0) = \inf\left\{v>0\left|\int_\mathbb R \frac{d\mu(x)}{(x-a_0)^2+v^2}> \frac{1}{t}\right.\right\}.\]
That is, if
\[\int_\mathbb R\frac{d\mu(x)}{(a_0-x)^2}>\frac{1}{t},\]
then $v_{x_0,t}(a_0)$ is defined to be the unique positive number such that
\begin{equation}
\label{eq:IntOfSq}
\int_\mathbb R\frac{d\mu(x)}{(a_0-x)^2+v_t(a_0)^2} = \frac{1}{t};
\end{equation}
otherwise, if
\[\int_\mathbb R\frac{d\mu(x)}{(a_0-x)^2}\leq\frac{1}{t},\]
then we set $v_{x_0,t}(a_0) = 0$. It is noted in \cite{Biane1997sc} that the function $v_{x_0,t}$ is a continuous function; a proof is given in~\cite{HallHo2020}.
\begin{definition}
\label{def:BianeFunction}
We introduce the following notations.
\begin{enumerate}
\item $\Delta_{x_0, t} = \{a_0+ib_0\in\mathbb C | b_0>v_{x_0,t}(a_0)\}$ is the region above the graph of $v_t$ in the upper half plane.
\item $H_{x_0, t}(z) = z+ t G_{x_0}(z)$, $z\in \Delta_{x_0,t}$.
\end{enumerate}
\end{definition}
\begin{theorem}[\cite{Biane1997sc}]
\label{thm:BianeFC}
\begin{enumerate}
\item The function $H_{x_0, t}$ is an injective conformal map, from $\Delta_{x_0, t}$ \emph{onto} the upper half plane $\mathbb C^+$; the function $H_{x_0, t}$ extends to a homeomorphism from $\bar{\Delta}_{x_0, t}$ onto $\mathbb C^+\cup\mathbb R$. In particular, $H_t(a_0+iv_t(a_0))$ is real.
\item The function $H_{x_0, t}$ satisfies
\[G_{x_0+\sigma_t}(H_{x_0, t}(z)) = G_{x_0}(z).\]
\item The measure $\mathrm{Law}(x_0+\sigma_t)$ is absolutely continuous with respect to the Lebesgue measure; its density $p_t$ can be computed by the function $\psi_{x_0,t}(a_0) := H_t(a_0+iv_t(a_0))$. The function $\psi_{x_0,t}:\mathbb R\to\mathbb R$ is a homeomorphism, and
\[p_t(\psi_t(a_0))=\frac{v_t(a_0)}{\pi t}.\]
\item As a consequence, the support of $\mathrm{Law}(x_0+\sigma_t)$ is the closure of the open set $\{\psi_{x_0,t}(a_0)| v_{x_0,t}(a_0)>0\}$.
\end{enumerate}
\end{theorem}
\subsection{Sum of a self-adjoint and a circiular elements}
In \cite{HoZhong2019}, the author and Zhong computed the Brown measure of $x_0+c_t$, where $x_0$ is a self-adjoint element freely independent of the circular element $c_t$, using the method introduced by Driver, Hall and Kemp \cite{DHK2019}. Interestingly, the support of the Brown measure is bounded by the graph of Biane's function $v_{x_0,t}$ introduced in Section \ref{sect:Biane} and the density is closely related to the law of the self-adjoint element $x_0+\sigma_t$. In this section, we review the results established in \cite{HoZhong2019}.
\begin{theorem}
\label{thm:HZ}
Let
\begin{equation}
\label{eq:LambdaDef}\Lambda_{x_0,t} = \{a_0+ib_0\in\mathbb C|\left\vert b_0\right\vert<v_{x_0,t}(a_0)\}.
\end{equation}
Then $\Lambda_{x_0,t} $ is a set of full measure with respect to $\mathrm{Brown}(x_0+c_t)$, and its density $w_{x_0,t}$ has the form
\[w_{x_0,t}(a+ib) = \frac{1}{2\pi t}\frac{d\psi_{x_0,t}(a)}{da}\]
where $\psi_{x_0,t}$ is defined in Theorem \ref{thm:BianeFC}. The density is constant along the vertical segments.
Furthermore, the push-forward of $\mathrm{Brown}(x_0+c_t)$ by
\[\Psi_{x_0, t}(a+ib) = H_t(a+iv_t(a)), \quad a+ib\in\Lambda_{x_0, t}\]
which is independent of the imaginary part, is the law of $x_0+\sigma_t$.
\end{theorem}
\begin{remark}
\label{rem:Hextension}
The map $H_{x_0, t}$ can be extended to an injective conformal map on $\bar{\Lambda}_{x_0, t}^c$ by Schwarz reflection. By Theorem \ref{thm:BianeFC} and that $H_{x_0,t}$ extends to the lower half plane by reflection, if $v_{x_0, t}(a_0)>0$, $H_{x_0, t}$ maps both boundary points $a_0\pm iv_{x_0,t}(a_0)$ of $\Lambda_{x_0, t}$ to the same point in the support of $\mathrm{Law}(x_0+\sigma_t)$. The function $H_{x_0, t}^{\langle -1\rangle}$ restricted to the upper half plane can be viewed as one of the {\bf subordination functions} in the sense of Voiculescu \cite[Proposition 4.4]{Voiculescu1993}. Thus, given any $q$ in the interior of $\sigma(x_0+\sigma_t)$, we understand as
\[H_{x_0, t}^{\langle -1\rangle}(q) = a_0+iv_{x_0,t}(a_0)\]
where $a_0+iv_{x_0,t}(a_0)$ is the boundary point of $\Lambda_{x_0, t}$ in the upper half plane such that $H_{x_0, t}(a_0+iv_{x_0,t}(a_0)) = q$.
\end{remark}
\subsection{Sum of a self-adjoint and an imaginary multiple of semicircular elements}
The author and Hall computed in \cite{HallHo2020} the Brown measure of $x_0+i\sigma_t$, a sum of a self-adjoint element and an imaginary multiple of semicircular element. The computation of the Brown measure of elements of the form $x_0+i\sigma_t$ covers the case $x_0+c_t$ which has the same $\ast$-moments as $x_0+\sigma_{t/2}+i\tilde{\sigma}_{t/2}$ where $\sigma_{\frac{t}{2}}$ and $\tilde{\sigma}_{\frac{t}{2}}$ are freely independent semicircular elements, both freely independent of $x_0$. The results in \cite{HallHo2020} show that there is a connection between the Brown measure of $x_0+i\sigma_t$, that of $x_0+c_t$ as well as the law of $x_0+\sigma_t$, for the \emph{same} self-adjoint element $x_0$.
We need the following notations to describe the results in \cite{HallHo2020}.
\begin{definition}
\label{def:H}
Let $x_0$ be a self-adjoint element.
\begin{enumerate}
\item Given any $r\in\mathbb R$, let $H_{x_0, r}(z) = z+ r G_{x_0}(z)$, $z\in \bar{\Lambda}_{x_0,|r|}^c$. Compared to the holomorphic function $H$ in Definition \ref{def:BianeFunction}, we allow $r$ negative in this notation. By the results in \cite{HallHo2020}, for $t>0$, the map $H_{x_0, -t}(z)$ is an injective conformal map on $\Delta_{x_0,t}$ (see Definition \ref{def:BianeFunction}). In \cite{HallHo2020}, the authors use the notation $J_t$ instead of $H_{x_0, -t}$.
\item Define $a_{x_0,t}(a_0) = \mathrm{Re}[H_{x_0, -t}(a_0+iv_t(a_0))]$ on $\mathbb R$. This function $a_{x_0,t}$ is a homeomorphism from $\mathbb R$ to $\mathbb R$; it is a strictly increasing function. If $v_{x_0,t}(a_0)>0$, we have $a_{x_0,t}'(a_0)>0$.
\item Denote by $a_0^{x_0, t}$ the inverse of $a_{x_0, t}$.
\end{enumerate}
\end{definition}
\begin{theorem}
\label{thm:HHBrown}
Let
\[\Omega_{x_0,t} = [H_{x_0, -t}(\Lambda_{x_0,t}^c)]^c.\]
Then we can write $\Omega_{x_0,t}$ as
\[\Omega_{x_0, t} = \{a+ib\in\mathbb C|\left\vert b\right\vert < b_{x_0,t}(a)\}\]
where $b_{x_0,t}(a) = 2v_{x_0,t}(a_0^{x_0,t}(a))$ is a nonnegative function on $\mathbb R$. The set $\Omega_{x_0,t}$ itself is a set of full measure with respect to $\mathrm{Brown}(x_0+i\sigma_t)$.
Inside $\Omega_{x_0,t}$, $\mathrm{Brown}(x_0+i\sigma_t)$ is absolutely continuous with respect to the Lebesgue measure on the plane; the density has the form
\[\frac{1}{2\pi t}\left(\frac{da_0^{x_0,t}(a)}{da}-\frac{1}{2}\right).\]
In particular, the density is independent of $b$ and is constant in the vertical segments.
\end{theorem}
We now describe the connections of $\mathrm{Brown}(x_0+i\sigma_t)$, $\mathrm{Brown}(x_0+i\sigma_t)$, and $\mathrm{Law}(x_0+\sigma_t)$. Let $U_{x_0, t}:\bar{\Lambda}_{x_0,t}\to\bar{\Omega}_{x_0,t}$ be a homeomorphism defined by
\[U_{x_0, t}(a_0+ib_0) = a_{x_0,t}(a_0)+2b_0.\]
Note that the map $U_{x_0,t}$ takes the vertical line segments in $\bar{\Lambda}_{x_0,t}$ \emph{linearly} to vertical line segments in $\bar{\Omega}_{x_0, t}$. Also, recall that $\Lambda_{x_0, t}$ defined in \eqref{eq:LambdaDef} is an open set of full measure of $\mathrm{Brown}(x_0+c_t)$.
\begin{theorem}
\begin{enumerate}
\item The push-forward measure of $\mathrm{Brown}(x_0+c_t)$ under $U_{x_0,t}$ is the Brown measure $\mathrm{Brown}(x_0+i\sigma_t)$.
\item The push-forward of $\mathrm{Brown}(x_0+i\sigma_t)$ under the map
\begin{equation}
\label{eq:Qtmap}
Q_{x_0, t}(a+ib) := 2a_0^{x_0,t}(a)-a
\end{equation}
is the law of $x_0+\sigma_t$. The map $Q_{x_0,t}$ agrees with $\Psi_{x_0, t}\circ U_{x_0,t}^{-1}$ where $\Psi_{x_0, t}$ is defined in Theorem~\ref{thm:HZ}.
\end{enumerate}
\end{theorem}
\section{The Brown measure computation}
Let $y_{0}$ be a self-adjoint element, $\tilde{\sigma}_{s-\frac{t}{2}}$ and $\sigma_{\frac{t}{2}}$ be two semicircular elements, all freely
independent. Denote the law of $y_0$ by $\nu$. We study the Brown measure of
\[
y_{0}+\tilde{\sigma}_{s-\frac{t}{2}}+i\sigma_{\frac{t}{2}}
\]
with $0<\frac{t}{2}\leq s$.
If the law of $y_{0}+\tilde{\sigma}_{s-\frac{t}{2}}$ is a Dirac mass at one point, then the Brown measure of $y_{0}+\tilde{\sigma}_{s-\frac{t}{2}}+i\sigma_{\frac{t}{2}}$ is singular with respect to the Lebesgue measure on the plane, and is a semicircular distribution along a vertical segment. Thus, we recall our standing assumption that either $s>\frac{t}{2}$ or $\nu$ is not a Dirac mass, so that $\mathrm{Law}(y_{0}+\tilde{\sigma}_{s-\frac{t}{2}})$ is not a Dirac mass.
For convenience, we define
\[x_0=y_{0}+\tilde{\sigma}_{s-\frac{t}{2}}.\]
By Theorem \ref{thm:HHBrown}, $\Omega_{x_0, t/2}$ is an open set of full measure of $\mathrm{Brown}(x_0+i\sigma_{t/2})$. Since $x_0+i\sigma_{t/2}$ depends on both parameters $s$ and $t$, we write
\[\Omega_{s,t} = \Omega_{x_0, t/2}.\]
We also write the boundary of $\Omega_{s,t}$ as $a+ib_{s,t}(a)$ instead of $a+ib_{x_0, t/2}(a)$. We recall from the discussion in Section \ref{sect:Biane} that given any $q\in \sigma(y_0+\sigma_s)$, $H_{y_0,s}^{\langle -1\rangle}(q)$ means the unique point $a_0+iv_{y_0, s}(a_0)$ on the boundary of $\Lambda_{y_0,s}$.
\subsection{The domain of the Brown measure\label{sect:domain}}
The map
\begin{equation}
\label{eq:FunctionfEllipse}
F_{s,t}(z)=H_{x_0, t/2}\circ H_{x_0, -t/2}^{\langle -1\rangle}(z)
\end{equation}
is an injective conformal mapping from $\bar{\Omega}_{s,t}^c$ to the complement $\sigma(y_0+\sigma_{s})^c$ of the spectrum of $y_0+\sigma_s$ by Theorem \ref{thm:BianeFC} and the definition of $\Omega_{x_0, t/2}$ (see Theorem \ref{thm:HHBrown}). The following theorem states that we can draw a connection between the domains $\Omega_{s,t}$ and $\Lambda_{y_0,s}$.
\begin{theorem}
\label{thm:TwoSteps}
The function $H_{y_0, s-t}$ is an injective conformal map on $\bar{\Lambda}_{y_0,s}^c$ and extends to a homeomorphism on $\Lambda_{y_0, s}^c$. We also have
\begin{equation}
\label{eq:Omegast}
\Omega_{s,t}^c = H_{y_0, s-t}(\Lambda_{y_0, s}^c).
\end{equation}
In particular, $\Omega_{s,s}=\Lambda_{y_0,s}$, recovering the domain in Theorem \ref{thm:HZ}.
\end{theorem}
The key to the above theorem is the following proposition about the function $F_{s,t}$.
\begin{proposition}
\label{prop:Fst2steps}
The inverse $F_{s,t}^{\langle -1\rangle}$ of $F_{s,t}$ can be written as
\begin{equation}
\label{eq:fInvH}
F_{s,t}^{\langle -1\rangle}(z) = (H_{y_0, s-t}\circ H_{y_0, s}^{\langle -1\rangle})(z), \quad z\in\mathbb C^+.
\end{equation}
\end{proposition}
\begin{remark}
This shows that ,when $y_0=0$, $F_{s,t}$ is the additive analogue of the function $f_{s,t}$ introduced in \cite{Ho2016} in the context of free Segal--Bargmann--Hall transform.
\end{remark}
\begin{proof}
Recall that we denote $y_0+\tilde{\sigma}_{s-\frac{t}{2}}$ by $x_0$. By Theorem~\ref{thm:BianeFC},
\begin{equation}
G_{y_{0}+\sigma_{s}}\left(H_{x_0,t/2}(z)\right)=G_{x_0+\sigma_{t/2}}(H_{x_0,t/2}(z))=G_{x_{0}}(z)=G_{y_{0}%
+\sigma_{s-t/2}}(z) \label{elliptic.subordation}%
\end{equation}
because $\tilde{\sigma}_{s-t/2}+\sigma_{t/2}$ has the same distribution as $\sigma_{s}$.
When $|z|$ large, (\ref{elliptic.subordation}) becomes
\begin{equation}
H_{x_0,t/2}^{\langle -1\rangle}(z)=K_{y_{0}+\sigma_{s-t/2}}(G_{y_{0}+\sigma_{s}}(z)). \label{H.inverse}%
\end{equation}
Since the $R$-transform of the sum of two freely independent variables is the
sum of the $R$-transforms of each variable (See Section~\ref{sect:Rtransform}),
\[
R_{y_{0}+\sigma_{s-t/2}}(z)=R_{y_{0}}(z)+R_{\sigma_{s-t/2}}(z)=R_{y_{0}}(z)+\left(s-\frac{t}{2}\right) z.
\]
Substracting by $\frac{1}{z}$ gives us
\begin{equation}
\label{eq:Ksminustover2}
K_{y_{0}+\sigma_{s-t/2}}(z)=K_{y_{0}}(z)+\left(s-\frac{t}{2}\right) z.
\end{equation}
Therefore,
\begin{equation}
K_{y_{0}+\sigma_{s-t/2}}\Big(G_{y_{0}+\sigma_{s}}(z)\Big)=K_{y_{0}}(G_{y_{0}+\sigma_{s}}(z))+\left(s-\frac{t}{2}\right) G_{y_{0}+\sigma_{s}}(z). \label{K.composite.G}%
\end{equation}
By the definition of $F_{s,t}^{\langle -1\rangle}$ in~\eqref{eq:FunctionfEllipse},
\begin{equation}
\label{eq:fInvElliptic}
F_{s,t}^{\langle -1\rangle}(z)=H_{x_0,-t/2}\left(H_{x_0,t/2}^{\langle-1\rangle}(z)\right)
=H_{x_0,t/2}^{\langle-1\rangle}(z)-\frac{t}{2}G_{x_{0}+\sigma_{s-t/2}%
}\left(H_{x_0,t/2}^{\langle -1\rangle}(z)\right)
\end{equation}
Using (\ref{H.inverse}) and (\ref{K.composite.G}), the above becomes
\begin{equation}
\label{eq:fInvCauchy}
\begin{split}
F_{s,t}^{\langle -1\rangle}(z)=&K_{y_{0}+\sigma_{s-t/2}}(G_{y_{0}+\sigma_{s}}(z))-\frac{t}{2}
G_{y_{0}+\sigma_{s}}(z)\\
=&K_{y_{0}}(G_{y_{0}+\sigma_{s}}(z))+\left(s-\frac{t}{2}\right)
G_{y_{0}+\sigma_{s}}(z)-\frac{t}{2}G_{y_{0}+\sigma_{s}}(z)\\
= & K_{y_{0}}(G_{y_{0}+\sigma_{s}}(z))+(s-t) G_{y_{0}+\sigma_{s}}(z).
\end{split}
\end{equation}
Now, since $H_{y_0, s}$ satisfies $G_{y_0+\sigma_{s}}(H_{y_0, s}(z)) = G_{y_0}(z)$, we have
\[H_{y_0, s}^{\langle -1\rangle}(z) = K_{y_0}(G_{y_0+\sigma_{s}}(z))\]
for all large enough $|z|$. It follows from (\ref{eq:fInvCauchy}) that $F_{s,t}^{\langle -1\rangle}$ can be written as
\[
F_{s,t}^{\langle -1\rangle}(z) = H_{y_0, s}^{\langle -1\rangle}(z) + (s-t) G_{y_0}(H_{y_0, s}^{\langle -1\rangle}(z))= (H_{y_0, s-t}\circ H_{y_0, s}^{\langle -1\rangle})(z)
\]
for all large enough $z$. Since both sides of the above expression are defined on the complement of the spectrum of $y_0+\sigma_{s}$, (\ref{eq:fInvH}) holds for all $z$ in the complement of the spectrum of $y_0+\sigma_{s}$ by analytic continuation.
\end{proof}
\begin{proof}[{\bf Proof of Theorem \ref{thm:TwoSteps}}]
The function $F_{s,t}^{\langle -1\rangle }$ is an injective conformal map on $\sigma(y_0+\sigma_s)^c$. Thus, by Proposition \ref{eq:fInvH}
\[H_{y_0, s-t}(z) = F_{s,t}^{\langle -1\rangle}\circ H_{y_0,s}(z), \quad z\in\Delta_{y_0, s}\]
is an injective conformal map onto
\[\{a+ib\in\mathbb C|\left\vert b\right\vert>b_{s,t}(a)\}.\]
Now, that the function $H_{y_0, s-t}$ extends to a homeomorphism on $\bar{\Delta}_{y_0,s}$ follows from an elementary topological argument by regarding $\Delta_{y_0, s}\cup\{\infty\}$ and $\{a+ib\in\mathbb C|\left\vert b\right\vert>b_{x_0, t}(a)\}\cup\{\infty\}$ as two disks in the Riemann sphere. Thus, $H_{y_0, s-t}$ is an injective conformal map on $\bar{\Lambda}_{y_0, s}^c$ and extends to a homeomophism on $\Lambda_{y_0, s}^c$ by Schwarz reflection about the real axis.
Equation \eqref{eq:Omegast} is a restatement of Proposition \ref{eq:fInvH}. If $s=t$, the holomorphic function $H_{y_0,s-t}$ is the identity map; therefore, $\Omega_{s,s} = \Lambda_{y_0,s}$ by \eqref{eq:Omegast}.
\end{proof}
\subsection{The function $a_0^{x_0, t/2}$}
Our strategy is to apply Theorem~\ref{thm:HHBrown} with $x_0+\tilde{\sigma}_{s-t/2}$. In this section, we compute the function $a_0^{x_0, t/2}$ in Theorem~\ref{thm:HHBrown}. We will prove later in this section that
\[a_{s,t}(\alpha) = \mathrm{Re}[H_{y_0,s-t}(\alpha+iv_{y_0,s}(\alpha))]\]
is a homeomorphism from $\mathbb R$ onto $\mathbb R$; we denote its inverse by $\alpha_{s,t}$.
\begin{theorem}
\label{thm:a0}
We have
\[
a_0^{x_0,t/2}(a)= a+\frac{t}{2}\int\frac{\alpha_{s,t}(a)-x}{(\alpha_{s,t}(a)-x)^2+v_{y_0,s}(\alpha_{s,t}(a))^2}\,d\nu(x), \quad a\in\mathbb R.
\]
\end{theorem}
Recall from Theorem \ref{thm:HHBrown} that $a_0^{x_0,t}(a)\in\Lambda_{x_0, t/2}\cap\mathbb R$ if $a\in\Omega_{x_0,t/2}\cap\mathbb R$. The domain $\Lambda_{x_0,t/2}$ is the domain of the Brown measure of $y_0+\tilde{\sigma}_{s-t/2}+c_{t/2}$. We can view $y_0+\tilde{\sigma}_{s-t/2}+c_{t/2}$ as
\[y_0+\tilde{\sigma}_{s-t/4}+i\sigma_{t/4},\]
which has the form of adding another elliptic element to $y_0$. By Theorem \ref{thm:TwoSteps} (with parameters $s$, $t$ replaced by $s$ and $t/2$), the function
\begin{equation}
\label{eq:FunctioThinEllipse}
F_{s,t/2}^{\langle -1\rangle}(z) = H_{y_0, s-t/2}\circ H_{y_0, s}^{\langle -1\rangle}(z)
\end{equation}
maps $\sigma(y_0+\sigma_{s})^c$ onto $\bar{\Lambda}_{x_0,t/2}^c$. By Remark \ref{rem:Hextension}, $H_{x_0, t/2}$ maps $\Lambda_{t/2}^c$ onto $\sigma(y_0+\sigma_s)^c$; the next lemma shows that $H_{x_0, t/2}$ and $F_{s,t/2}$ are indeed the same map. Recall that by Point 3 of Definition \ref{def:H}, if $a\in\Omega_{s,t}$ is given, then
\[a_0^{x_0, t/2}(a) = H_{x_0, -t/2}^{\langle -1\rangle}(a+ib_{s,t}(a)),\]
where $a+ib_{s,t}(a)$ is the boundary point of $\Omega_{s,t}$ with real part $a$ and positive imaginary part.
\begin{lemma}
\label{lem:2F}
We have
\begin{equation}
\label{eq:F=H}
F_{s,t/2}(z) = H_{x_0, t/2}(z), \quad z\in\Lambda_{x_0, t/2}^c.
\end{equation}
Furthermore, we have
\[a_0^{x_0, t/2}(a)+iv_{x_0,t/2}(a_0^{x_0,t}(a)) =(H_{y_0, s-t/2}\circ H_{y_0, s-t}^{\langle -1\rangle})(a+ib_{s,t}(a)).\]
\end{lemma}
\begin{proof}
For all large enough $|z|$, we compute
\begin{align*}
F_{s,t/2}(z) &= (H_{y_0,s}\circ H_{y_0,s-t/2}^{\langle -1\rangle})(z)\\
&= H_{y_0, s-t/2}^{\langle -1\rangle}(z) + s G_{y_0}(H_{y_0,s-t/2}^{\langle -1\rangle}(z))\\
&=K_{y_0}\circ G_{y_0+\sigma_{s-t/2}}(z)+\left[\left(s-\frac{t}{2}\right) G_{y_0+\sigma_{s-t/2}}(z)+\frac{t}{2}G_{y_0+\sigma_{s-t/2}}(z)\right]\\
&=z+\frac{t}{2} G_{y_0+\sigma_{s-t/2}}(z)
\end{align*}
by~\eqref{H.inverse} and~\eqref{eq:Ksminustover2}. This shows $F_{s,t/2}(z) = H_{x_0, t/2}(z)$ for $z\in \Lambda_{s,t}^c$ by analytic continuation. Now, \eqref{eq:F=H} follows from the unique continuous extension to $\bar{\Lambda}_{s,t}^c$.
Note that we have
\begin{align*}
a_0^{x_0, t/2}(a)+iv_{x_0,t/2}(a_0^{x_0,t}(a)) &= H_{x_0,-t/2}^{\langle -1\rangle}(a+ib_{s,t}(a))\\
&=H_{x_0,t/2}^{\langle -1\rangle}(F_{s,t}(a+ib_{s,t}(a)))
\end{align*}
by \eqref{eq:FunctionfEllipse}. Recall from Theorem \ref{thm:TwoSteps} and Proposition \ref{prop:Fst2steps} that
\[F_{s,t}(z) = (H_{y_0, s}\circ H_{y_0, s-t}^{\langle -1\rangle})(z).\]
Thus, \eqref{eq:FunctioThinEllipse} and~\eqref{eq:F=H} imply
\begin{align*}
a_0^{x_0, t/2}(a)+iv_{x_0,t/2}(a_0^{x_0,t}(a)) &=F_{s,t/2}^{\langle -1\rangle}(F_{s,t}(a+ib_{s,t}(a)))\\
&= (H_{y_0, s-t/2}\circ H_{y_0, s-t}^{\langle -1\rangle})(a+ib_{s,t}(a)),
\end{align*}
concluding the proof.
\end{proof}
By Theorem \ref{thm:TwoSteps}, $H_{y_0, s-t}^{\langle -1\rangle}(a+ib_{s,t}(a))$ is on the boundary of $\Lambda_{y_0, s}$. This means we can parametrize $a+ib_{s,t}(a)$ using $\alpha\in\Lambda_{y_0,s}\cap \mathbb R$ by
\[a+ib_{s,t}(a) = H_{y_0,s-t}(\alpha+iv_{y_0,s}(\alpha)).\]
Thus, it is important to understand the function $H_{y_0, s-t}$ on the boundary of $\Lambda_{y_0,s}$.
\begin{proposition}
\label{prop:alphast}
The function
\[a_{s,t}(\alpha) =\mathrm{Re}[H_{y_0,s-t}(\alpha+iv_{y_0,s}(\alpha))], \quad\alpha\in\mathbb{R}\]
is strictly increasing; it is a homeomorphism onto $\mathbb{R}$. Furthermore, $a_{s,t}'(\alpha)>0$, for all $\alpha\in\Lambda_{y_0, s}\cap\mathbb{R}$.
The (upper) boundary curve $a+ib_{s,t}(a)$ of $\Omega_{s,t}$ can be parametrized by $\alpha\in\Lambda_{y_0,s}\cap\mathbb{R}$. The parametrization is
\begin{equation}
\label{eq:ParaBoundary}
a +ib_{s,t}(a)= a_{s,t}(\alpha)+\frac{it}{s}v_{y_0,s}(\alpha).
\end{equation}
\end{proposition}
\begin{proof}
By a direct computation,
\[a_{s,t}(\alpha) = \frac{s-t}{s}\left(\frac{t\alpha}{s-t}+\mathrm{Re}[H_{y_0,s}(\alpha+iv_{y_0,s}(\alpha))]\right).\]
If $s>t$, then $a_{s,t}$ is strictly increasing because $\mathrm{Re}[H_{y_0,s}(\alpha+v_{y_0,s}(\alpha))]$ is strictly increasing in $\alpha\in\mathbb R$ by Theorem \ref{thm:BianeFC}. If $s<t$, then
\begin{equation*}
a_{s,t}(\alpha) =\frac{t-s}{s}\left(\frac{(2s-t)\alpha}{t-s}+\mathrm{Re}[H_{y_0,-s}(\alpha+iv_{y_0,s}(\alpha))]\right)
\end{equation*}
which is a strictly increasing function since $\mathrm{Re}[H_{y_0,-s}(\alpha+v_{y_0,s}(\alpha))]$ is strictly increasing in $\alpha\in\mathbb R$, by Point 2 of Definition~\ref{def:H}. If $s=t$, $a_{s,t}$ is just the identity function. In any case, if $v_{y_0,s}(\alpha)>0$, $a_{s,t}$ is differentiable at $\alpha$ and $a_{s,t}'(\alpha)>0$ by Point 2 of Definition \ref{def:H}.
By Theorem \ref{thm:TwoSteps}, $a+ib_{s,t}(a) = H_{s-t}(\alpha+iv_{y_0,s}(\alpha))$ for a unique $\alpha\in\Lambda_{y_0, s}$.
Computing the imaginary part of $H_{s-t}(\alpha+iv_{y_0,s}(\alpha))$, we get
\[v_{y_0,s}(\alpha)\left(1-(s-t)\int\frac{1}{(\alpha-x)^2+v_{y_0,s}(\alpha)^2}\,d\nu(x)\right) = \frac{t}{s}v_{y_0,s}(\alpha)\]
by \eqref{eq:IntOfSq}. This proves the parametrization \eqref{eq:ParaBoundary}.
\end{proof}
Proposition \ref{prop:alphast} shows that $a_{s,t}$ is has an inverse on $\mathbb R$. Recall from the beginning of this section that we denote the inverse of $a_{s,t}$ by $\alpha_{s,t}$.
\begin{proof}[{\bf Proof of Theorem \ref{thm:a0}}]
By Theorem \ref{thm:TwoSteps}, $H_{y_0,s-t}^{\langle -1\rangle}(a+ib_{s,t}(a))$ is a unique point $\alpha+iv_{y_0,s}(\alpha)$ on the boundary of $\Lambda_{y_0,s}$ in the (closed) upper half plane. By Proposition \ref{prop:alphast} and the definition of $\alpha_{s,t}$,
\[\alpha_{s,t}(a) + iv_{y_0,s}(\alpha_{s,t}(a)) = H_{y_0,s-t}^{\langle -1\rangle}(a+ib_{s,t}(a)).\]
By Lemma \ref{lem:2F},
\begin{align*}
a_0^{x_0, t/2}(a)+iv_{x_0,t/2}(a_0^{x_0,t}(a)) =(H_{y_0, s-t/2}\circ H_{y_0, s-t}^{\langle -1\rangle})(a+ib_{s,t}(a))
\end{align*}
Using the identity $H_{y_0, s-t/2} = H_{y_0, s-t}+\frac{t}{2} G_{y_0}$, we have
\begin{align*}
a_0^{x_0,t/2}(a) &=\mathrm{Re}[H_{y_0,s-t/2}\circ H_{y_0,s-t}^{\langle -1\rangle}(a+ib_{s,t}(a))]\\
&= a+ \frac{t}{2}\,\mathrm{Re}[G_{y_0}(\alpha_{s,t}(a)+iv_{y_0,s}(\alpha_{s,t}(a)))]\\
&= a+\frac{t}{2}\int\frac{\alpha_{s,t}(a)-x}{(\alpha_{s,t}(a)-x)^2+v_{y_0,s}(\alpha_{s,t}(a))^2}\,d\nu(x).
\end{align*}
for all $a\in\Omega_{s,t}\cap\mathbb{R}$. The theorem is established.
\end{proof}
\subsection{The density of the Brown measure}
\label{sect:AddEllipseDensity}
In this section , we compute the density of the Brown measure of $y_0+\tilde{\sigma}_{s-t/2}+i\sigma_{t/2}$.
\begin{theorem}
\label{thm:addellipse}
The Brown measure of $y_0+\tilde{\sigma}_{s-t/2}+i\sigma_{t/2}$ is absolutely continuous with respect to the Lebesgue measure on the plane and is supported on $\bar{\Omega}_{s,t}$. The open set $\Omega_{s,t}$ is a set of full measure of the Brown measure. The density of the Brown measure is given by
\[w_{y_0,s,t}(a+ib) = \frac{1}{2\pi t}\left(1+t\frac{d}{da}\int\frac{\alpha_{s,t}(a)-x}{(\alpha_{s,t}(a)-x)^2+v_{y_0,s}(\alpha_{s,t}(a))^2}\,d\nu(x)\right)\]
on the set $\Omega_{s,t}$. In particular, the density is constant along the vertical segments.
\end{theorem}
\begin{proof}
We only need to compute the density. All the other properties follow from Theorem~\ref{thm:HHBrown}. By Theorem \ref{thm:a0},
\[
\frac{da_0^{x_0,t/2}}{da} = 1 +\frac{t}{2}\frac{d}{da}\int\frac{\alpha_{s,t}(a)-x}{(\alpha_{s,t}(a)-x)^2+v_{y_0,s}(\alpha_{s,t}(a))^2}\,d\nu(x).
\]
By Theorem~\ref{thm:HHBrown}, the density of $\mathrm{Brown}(y_0+\tilde\sigma_{s-t/2}+i\sigma_{t/2})$ is given by
\[\frac{1}{2\pi t}\left(1+t\frac{d}{da}\int\frac{\alpha_{s,t}(a)-x}{(\alpha_{s,t}(a)-x)^2+v_{y_0,s}(\alpha_{s,t}(a))^2}\,d\nu(x)\right),\]
which completes the proof of the theorem.
\end{proof}
As a consequence, Theorem~\ref{thm:addellipse} verifies~\eqref{eq:IntroEq} in the introduction. Given any $a+ib\in\mathbb C$, $\alpha = \alpha_{s,t}(a)$ and $v_{y_0,s}(\alpha_{s,t}(a))$ are determined by the equation
\[\int\frac{d\nu(x)}{(\alpha_{s,t}(a)-x)^2+v_{y_0,s}(\alpha_{s,t}(a))^2}=\frac{1}{s}\]
where solution exists if $a\in\Omega_{s,t}\cap\mathbb R$, while solution does not exist if $a\in\mathbb R\setminus\Omega_{s,t}$. And the relation between $\alpha_{s,t}(a)$ and $a$ is through the homeomorphism $a_{s,t}$
\begin{equation}
\label{eq:aandalpha}
a = a_{s,t}(\alpha_{s,t}(a)) = \alpha_{s,t}(a)+(s-t)\int\frac{(\alpha_{s,t}(a)-x)\,d\nu(x)}{(\alpha_{s,t}(a)-x)^2+v_{y_0,s}(\alpha_{s,t}(a))^2}.
\end{equation}
These two display equations are then simplified to~\eqref{eq:IntroEq}.
\section{Two push-forward properties}
Let $U_{s,t}:\Lambda_{y_0,s}\to\Omega_{s,t}$ be defined by
\begin{align*}
\mathrm{Re}\,U_{s,t}(\alpha+i\beta) &= a_{s,t}(\alpha)\\
\mathrm{Im}\,U_{s,t}(\alpha+i\beta) &= \frac{t\beta}{s}.
\end{align*}
Since $a_{s,t}$ is a homeomorphism on $\mathbb R$ by Proposition~\ref{prop:alphast}, one can immediately see that $U_{s,t}$ is indeed a homeomorphism on the complex plane $\mathbb C$. In this section, we prove two push-forward properties of $\mathrm{Brown}(y_0+c_s)$ by~$U_{s,t}$.
\begin{theorem}
\label{thm:pushforward}
We have the following results about push-forward measures.
\begin{enumerate}
\item The push-forward of $\mathrm{Brown}(y_0+c_s)$ under the map $U_{s,t}$ is $\mathrm{Brown}(y_0+\tilde{\sigma}_{s-t/2}+i\sigma_{t/2})$.
\item The push-forward of $\mathrm{Brown}(y_0+\tilde{\sigma}_{s-t/2}+i\sigma_{t/2})$ by the map
\[Q_{s,t}(a+ib) = \frac{1}{s-t}[sa-t\alpha_{s,t}(a)]\]
is $\mathrm{Law}(y_0+\sigma_s)$.
\end{enumerate}
\end{theorem}
In particular, since $\bar{\Lambda}_{y_0,s}$ is the support of $\mathrm{Brown}(y_0+c_s)$, Theorem~\ref{thm:pushforward} verifies~\eqref{eq:bstIntro} that
\[b_{s,t}(a) = \frac{t}{s}v_{y_0,s}(\alpha_{s,t}(a)).\]
This can also be seen from~\eqref{eq:ParaBoundary}.
Before we prove this theorem, we first look at the properties of $U_{s,t}$.
\begin{proposition}
\label{prop:Ust}
The function $U_{s,t}:\Lambda_{y_0,s}\to\Omega_{s,t}$ defined by
\begin{align*}
\mathrm{Re}\,U_{s,t}(\alpha+i\beta) &= a_{s,t}(\alpha)\\
\mathrm{Im}\,U_{s,t}(\alpha+i\beta) &= \frac{t\beta}{s}
\end{align*}
is a diffeomorphism; it extends to a homeomorphism from $\bar{\Lambda}_{y_0,s}$ to $\bar{\Omega}_{s,t}$. Moreover, it agrees with $H_{y_0, s-t}$ on the boundary of $\Lambda_{y_0,s}$.
\end{proposition}
\begin{proof}
By Proposition \ref{prop:alphast}, $a_{s,t}$ is injective, strictly increasing and differentiable in $\Lambda_{y_0, s}\cap\mathbb R$ with nonzero derivative; therefore, $U_{s,t}$ is a diffeomorphism on $\Lambda_{y_0,s}$. Since $a_{s,t}$ is a homeomorphism defined on $\mathbb R$, the map $U_{s,t}$ can be extended to a homeomorhism in $\mathbb C$; in particular, it is a homeomorphism from $\bar{\Lambda}_{y_0, s}$ to $\bar{\Omega}_{s,t}$.
It is clear from \eqref{eq:ParaBoundary} that $U_{s,t}$ agrees with $H_{y_0,s-t}$ on the boundary of $\Lambda_{y_0,s}$.
\end{proof}
\begin{proof}[{\bf Proof of Theorem \ref{thm:pushforward}}]
The function $U_{s,t}$ is a diffeomorphism from a set $\Lambda_{y_0,s}$ of full measure of $\mathrm{Brown}(y_0+c_s)$ to a set $\Omega_{s,t}$ of full measure of $\mathrm{Brown}(y_0+c_s)$. Since $a_{s,t}$ has a more explicit formula than $\alpha_{s,t}$, we show the push-forward of the Brown measure of $y_0+\tilde{\sigma}_{s-t/2}+i\sigma_{t/2}$ by $U_{s,t}^{\langle -1\rangle}$ is the Brown measure of $y_0+c_s$.
We compute that the push-forward by $U_{s,t}^{\langle -1\rangle}$
\begin{equation}
\label{eq:push}
\begin{split}
&\quad\frac{1}{2\pi t}\left(1+t\frac{d}{da}\int\frac{\alpha_{s,t}(a)-x}{(\alpha_{s,t}(a)-x)^2+v_{y_0,s}(\alpha_{s,t}(a))^2}\,d\nu(x)\right)da\,db\\
&= \frac{1}{2\pi t}\left(\frac{da_{s,t}(\alpha)}{d\alpha}+t\frac{d}{d\alpha}\int\frac{\alpha-x}{(\alpha-x)^2+v_{y_0,s}(\alpha)^2}\,d\nu(x)\right)d\alpha\,\left(\frac{t}{s}d\beta\right)\\
&=\frac{1}{2\pi s}\frac{d}{d\alpha}\left(a_{s,t}(\alpha)+t\int\frac{\alpha-x}{(\alpha-x)^2+v_{y_0,s}(\alpha)^2}\,d\nu(x)\right)\,d\alpha\,d\beta
\end{split}
\end{equation}
is a measure on $\bar{\Lambda}_{y_0,s}$. By the definition of $a_{s,t}$ (see Proposition \ref{prop:alphast}),
\[a_{s,t}(\alpha)=\alpha+(s-t)\int\frac{(\alpha-x)\,d\nu(x)}{(\alpha-x)^2+v_{y_0,s}(\alpha)^2}.\]
Using the definition of $\psi_{y_0,s}$ in Theorem~\ref{thm:BianeFC}, \eqref{eq:push} becomes
\[\frac{1}{2\pi s}\frac{d}{d\alpha}\psi_{y_0,s}(\alpha)\,d\alpha\,d\beta\]
which is the Brown measure of $y_0+c_s$ by Theorem \ref{thm:HZ}, which proves Point 1.
We now prove Point 2. It suffices to show that $Q_{s,t} = \Psi_{y_0,s}\circ U_{s,t}^{\langle -1\rangle}$, where $\Psi_{y_0,s}(\alpha+i\beta) = H_{y_0,s}(\alpha+iv_{y_0,s}(\alpha))$ is as in Theorem~\ref{thm:HZ}; then Point 2 will follow from Point 1 and Theorem~\ref{thm:HZ}.
Let $a+ib\in\Omega_{s,t}$. Then
\begin{align*}
\Psi_{y_0,s}\circ U_{s,t}^{-1}(a+ib) &= H_{y_0,s}(\alpha_{s,t}(a)+iv_{y_0,s}(\alpha_{s,t}(a)))\\
&=\alpha_{s,t}(a)+s\int\frac{\alpha_{s,t}(a)-x}{(\alpha_{s,t}(a)-x)^2+v_{y_0,s}(\alpha_{s,t}(a))^2}\\
&=2\alpha_{s,t}(a)-s\int\frac{x\,dx}{(\alpha_{s,t}(a)-x)^2+v_{y_0,s}(\alpha_{s,t}(a))^2}.
\end{align*}
By~\eqref{eq:aandalpha} (which also occurs as the second equation of~\eqref{eq:IntroEq}),
\[s\int\frac{x\,dx}{(\alpha_{s,t}(a)-x)^2+v_{y_0,s}(\alpha_{s,t}(a))^2} = \frac{1}{s-t}[(2s-t)\alpha_{s,t}(a)-sa].\]
Now, we have
\[\Psi_{y_0,s}\circ U_{s,t}^{\langle -1\rangle}(a+ib) = \frac{1}{s-t}[sa-t\alpha_{s,t}(a)]\]
which is equal to the definition of $Q_{s,t}$.
\end{proof}
\begin{corollary}
\label{cor:stFormulas}
Fix $r = t/s$ and write $(a,b) = U_{s,t}(\alpha, \beta)$ for all $\alpha+i\beta\in\Lambda_{y_0,s}$. Then we have
\[w_{y_0,s,t}(a+ib) = \frac{1}{r}\frac{w_{y_0,s}(\alpha+i\beta)}{r+2\pi(1-r)s\cdot w_{y_0,s}(\alpha+i\beta)}\]
for all $a+ib\in\Omega_{s,t}$.
\end{corollary}
\begin{proof}
Denote $r = t/s$. We can write the function $a_{s,t}(\alpha)$ defined in Proposition~\ref{prop:alphast} as
\begin{align*}
a_{s,t}(\alpha)& = \alpha+(1-r)s\,\mathrm{Re}\left[\int\frac{d\nu(x)}{\alpha+iv_{y_0,s}(\alpha)-x}\right]\\
&= \alpha+(1-r)[H_{y_0,s}(\alpha+iv_{y_0,s}(\alpha))-\alpha]\\
&= (1-r)\psi_{y_0,s}(\alpha)+r\alpha.
\end{align*}
So, we have
\[\frac{d a_{s,t}(\alpha)}{d\alpha} = r+2\pi(1-r)s \cdot w_{y_0,s} (\alpha+i\beta).\]
By Theorem \ref{thm:pushforward}, we can compute the density $w_{y_0, s,t}(a+ib)\,da\,db$ in terms of $w_{y_0,s}$
\begin{align*}
w_{y_0, s,t}(a+ib)\,da\,db &= w_{y_0,s}(\alpha+i\beta)\,d\alpha\,d\beta\\
&= w_{y_0,s}(\alpha+i\beta)\,\frac{d\alpha}{da}\,\frac{d\beta}{db}\,da\,db\\
&= \frac{1}{r}\frac{w_{y_0,s}(\alpha+i\beta)}{r+2\pi(1-r)s\cdot w_{y_0,s}(\alpha+i\beta)}\,da\,db,
\end{align*}
completing the proof.
\end{proof}
\section{Asymptotic behaviors of adding a circular element\label{sect:circularAsymp}}
\subsection{The graph of $v_{y_0, s}$ as $s\to\infty$}
In this section, we study the asymptotic behavior of $v_{y_0, s}$ and $\Lambda_{y_0, s}$ as $s\to\infty$. Below is the main theorem of this section.
\begin{theorem}
\label{thm:vsAsymp}
The following asymptotic behaviors of the graph of $v_{y_0, s}$ hold.
\begin{enumerate}
\item Let $D_\nu = \sup\{\left\vert x-y\right\vert| x, y\in\mathrm{supp}\,\mu\}$. When $s\geq 4 D_\nu^2$, the function $v_{y_0,s}$ is unimodal. In particular, $\Lambda_{y_0,s}\cap\mathbb R$ is an interval.
\item Given any $c>1$, we have
\[\left\vert\sup \Lambda_{y_0,s}\cap\mathbb R-(\tau(y_0)+\sqrt{s})\right\vert<\frac{3c\tau(y_0^2)}{2\sqrt{s}}\]
and
\[\left\vert\inf \Lambda_{y_0,s}\cap\mathbb R-(\tau(y_0)-\sqrt{s})\right\vert<\frac{3c\tau(y_0^2)}{2\sqrt{s}}\]
for all large enough $s$. In particular,
\[ \Lambda_{y_0,s}\cap\mathbb R\subset \left(\tau(y_0)-\sqrt{s}-\frac{3c\tau(y_0^2)}{2\sqrt{s}}, \tau(y_0)+\sqrt{s}+\frac{3c\tau(y_0^2)}{2\sqrt{s}}\right)\]
for all large enough $s$.
\item Given any $\varphi_0\in(0,\pi/2)$, then for all large enough $s$, for all $\left\vert \cos\varphi\right\vert\leq \cos\varphi_0$, the unique $\alpha\in\mathbb R$ such that
\[H_{y_0,x}(\alpha+iv_{y_0,s}(\alpha)) = 2\sqrt{s}\cos\varphi.\]
satisfies
\[\left\vert \alpha+iv_{y_0,s}(\alpha)-\sqrt{s}e^{i\varphi}\right\vert<\frac{1}{(\sin\varphi_0)\sqrt{s}}.\]
\end{enumerate}
\end{theorem}
Point 1 of Theorem~\ref{thm:vsAsymp} is a known result in \cite[Theorem 3.2]{HasebeUeda2018}. We state it here for completeness; it is also useful for us to understand the asymptotic behaviors of $\Lambda_{y_0,s}$.
We study the asymptotic behaviors of $v_{y_0, s}$ by looking at $v_{\frac{y_0}{\sqrt{s}}, 1}$, whose graph is scaled by $\sqrt{s}$ the graph of $v_{y_0, s}$. We look at
\[H_{\frac{y_0}{\sqrt{s}},1}(z)=z+G_{\frac{y_0}{\sqrt{s}}}(z).\]
If $s$ is large enough, $H_{\frac{y_0}{\sqrt{s}},1}$ is defined for all $|z|>\frac{1}{2}$ since $y_0$ is assumed to be bounded.
We assume $y_0$ is centered and has unit variance until the proof of Theorem~\ref{thm:vsAsymp} for simplicity. The function $H_{\frac{y_0}{\sqrt{s}},1}$ is the inverse subordination function of the free convolution $\frac{y_0}{\sqrt{s}}+\sigma_1$. When $s$ is large, $\frac{y_0}{\sqrt{s}}+\sigma_1$ behaves like $\sigma_1$; our strategy is to compare $\frac{y_0}{\sqrt{s}}+\sigma_1$ with $\sigma_1$. Denote by $k(z)$ the function $H_{0,1}(z)$; that is
\[k(z) = z+\frac{1}{z}.\]
The techniques in this section are similar to techniques in proving the supercovergence results in \cite{BercoviciVoiculescu1995, BercoviciWangZhong2018, Wang2010}.
\begin{lemma}
\label{lem:Hkdist}
Assume $y_0$ is a bounded random variable with $\tau(y_0)=0$ and $\tau(y_0^2)=1$. Then given any $c>1$, there exists $s_0>0$ such that
\[\left\vert H_{\frac{y_0}{\sqrt{s}},1}(z) -k(z)\right\vert< \frac{c}{s|z|^3},\quad |z|>\frac{1}{2}\]
for all $s\geq s_0$.
\end{lemma}
\begin{proof}
When $s$ is large enough, we can write
\begin{align*}
H_{\frac{y_0}{\sqrt{s}},1}(z) &= z+\frac{1}{z}+\frac{1}{s}\sum_{n=2}^\infty\frac{\tau(y_0^n)}{s^{\frac{n}{2}-1}z^{n+1}}\\
&= k(z)+\frac{1}{s}\sum_{n=2}^\infty\frac{\tau(y_0^n)}{s^{\frac{n}{2}-1}z^{n+1}}
\end{align*}
for all $|z|>\frac{1}{2}$. Observe that
\[\left\vert\sum_{n=2}^\infty\frac{\tau(y_0^n)}{s^{\frac{n}{2}-1}z^{n+1}} \right\vert\leq \frac{\tau(y_0^2)}{|z|^3}+\frac{1}{|z|^3}\sum_{n=3}^\infty\frac{\left\vert\tau(y_0^n)\right\vert}{s^{\frac{n}{2}-1}(1/2)^{n-2}}\]
for all $|z|>\frac{1}{2}$. Since we assume $\tau(y_0^2)=1$ and
\[\lim_{s\to\infty}\sum_{n=3}^\infty\frac{\left\vert\tau(y_0^n)\right\vert}{s^{\frac{n}{2}-1}(1/2)^{n-2}}= 0,\]
the result follows.
\end{proof}
We compute that $k'(z) = 1-\frac{1}{z^2}$; the double zeros of $k$ are $1$ and $-1$. The next lemma shows that $H_{\frac{y_0}{\sqrt{s}},1}$ also has doubles zeros at a point close to $1$ and a point close to $-1$. Since $v_{\frac{y_0}{\sqrt{s}},1}$ is unimodal for large $s$, these two points are the only double zeros of~$H_{\frac{y_0}{\sqrt{s}},1}$. Since $H_{\frac{y_0}{\sqrt{s}},1}$ is symmetric about the real axis, these two double zeros must be real numbers. Again since $v_{\frac{y_0}{s},1}$ is unimodal for large $s$, $\Lambda_{\frac{y_0}{s},1}\cap\mathbb R$ is an open interval and the two double zeros of $H_{\frac{y_0}{\sqrt{s}},1}$ are the endpoints of~$\Lambda_{\frac{y_0}{s},1}\cap\mathbb R$.
\begin{lemma}
\label{lem:zeros}
Given any $c>1$, there exists $s_0$ such that
\[\left\vert H_{\frac{y_0}{\sqrt{s}},1}'(\pm1+re^{i\theta})-k'(\pm1+re^{i\theta})\right\vert< \frac{3c}{s(1-r)^4}\]
for all $s\geq s_0$ and $r<\frac{1}{2}$.
\end{lemma}
\begin{proof}
Recall that
\[H_{\frac{y_0}{\sqrt{s}},1}(z) = k(z)+\frac{1}{s}\sum_{n=2}^\infty\frac{\tau(y_0^n)}{s^{\frac{n}{2}-1}z^{n+1}};\]
we compute
\begin{equation}
\label{eq:H'k'}
H_{\frac{y_0}{\sqrt{s}},1}'(z) = 1-\frac{1}{z^2}-\frac{1}{s}\left(\frac{3\tau(y_0^2)}{z^4}+\frac{1}{z^4}\sum_{n=3}^\infty\frac{(n+1)\tau(y_0^n)}{s^{\frac{n}{2}-1}z^{n-2}}\right)
\end{equation}
Let $c>1$ be given. If $z = 1+r e^{i\theta}$ with $r<1/2$, then for all large enough $s$,
\[\left\vert\frac{3\tau(y_0^2)}{z^4}+\frac{1}{z^4}\sum_{n=3}^\infty\frac{(n+1)\tau(y_0^n)}{s^{\frac{n}{2}-1}z^{n-2}}\right\vert<\frac{3c}{(1-r)^4}\]
since $\left\vert z\right\vert > 1-r>1/2$ and $\tau(y_0^2)=1$. The case for $z = 1-re^{i\theta}$ is similar.
\end{proof}
\begin{proposition}
\label{prop:vEndPoint}
We have
\[1-\frac{3c}{2s}<\sup \Lambda_{\frac{y_0}{\sqrt{s}},1}\cap\mathbb R<1+\frac{3c}{2s}\]
and
\[-1-\frac{3c}{2s}<\inf \Lambda_{\frac{y_0}{\sqrt{s}},1}\cap\mathbb R<-1+\frac{3c}{2s}\]
for all large enough $s$. In particular,
\[ \Lambda_{\frac{y_0}{\sqrt{s}},1}\cap\mathbb R\subset \left(-1-\frac{3c}{2s}, 1+\frac{3c}{2s}\right)\]
for all large enough $s$.
\end{proposition}
\begin{proof}
Let $c>1$.
We compute, with $z = 1+re^{i\theta}$,
\[\left\vert1-\frac{1}{z^2}\right\vert = \left\vert\frac{r(2e^{i\theta}+r e^{2i\theta})}{(1+r e^{i\theta})^2}\right\vert>\frac{r(2-r)}{(1+r)^2}.\]
Then, by choosing any $1<c'<c$ in Lemma \ref{lem:zeros}, $r=\frac{3c}{2s}$ satisfies
\[\left\vert H_{\frac{y_0}{\sqrt{s}},1}'(1+re^{i\theta})-k'(1+re^{i\theta})\right\vert< \frac{3c'}{s(1-r)^4} \leq\frac{r(2-r)}{(1+r)^2}<\left\vert 1-\frac{1}{z^2}\right\vert\]
for all large enough $s$, because, if $s$ is large enough
\[\frac{3c'(1+r)^2}{r(2-r)(1-r)^4}=\frac{3c'(1+r)^22s}{3c(2-r)(1-r)^4}<s.\]
By Rouch\'e's theorem, we have
\[1-\frac{3c}{2s}<\sup \Lambda_{\frac{y_0}{\sqrt{s}},1}\cap\mathbb R<1+\frac{3c}{2s}.\]
The proof of
\[-1-\frac{3c}{2s}<\inf \Lambda_{\frac{y_0}{\sqrt{s}},1}\cap\mathbb R<-1+\frac{3c}{2s}\]
is similar.
\end{proof}
\begin{proposition}
\label{prop:vUpper}
Given any $\varphi_0\in(0,\pi/2)$, then for all large enough $s$, for all $\left\vert \cos\varphi\right\vert\leq \cos\varphi_0$, the unique $\alpha\in\mathbb R$ such that
\[H_{\frac{y_0}{\sqrt{s}},1}(\alpha+iv_{\frac{y_0}{\sqrt{s}},1}(\alpha)) = 2\cos\varphi.\]
satisfies
\[\left\vert \alpha+iv_{\frac{y_0}{\sqrt{s}},1}(\alpha)-e^{i\varphi}\right\vert<\frac{1}{(\sin\varphi_0)s}.\]
\end{proposition}
\begin{proof}
Fix $\varphi_0 \in(0,\pi/2) $ and let $r=\frac{1}{(\sin\varphi_0) s}$. Then, given any $\varphi\in(0,\pi)$ such that $\sin\varphi\geq\sin\varphi_0$, we have, for large $s$,
\begin{equation}
\label{eq:kLower}
\begin{split}
\left\vert k(e^{i\varphi}+re^{i\theta})-k(e^{i\varphi})\right\vert& =\left\vert re^{i\theta}\left(\frac{re^{i\theta}+2i\sin\varphi}{e^{i\varphi}+re^{i\theta}}\right)\right\vert\\
&\geq\frac{1}{\sin\varphi_0 s}\frac{2\sin\varphi_0-r}{1+r}.
\end{split}
\end{equation}
Fix any $1<c<2$. The lower bound of $s\left\vert k(e^{i\varphi}+re^{i\theta})-k(e^{i\varphi})\right\vert$ converges to $2$ in~\eqref{eq:kLower}. It follows from Lemma~\ref{lem:Hkdist} that, for all large enough $s$,
\begin{align*}
\left\vert H_{\frac{y_0}{\sqrt{s}},1}(e^{i\varphi}+re^{i\theta})-k(e^{i\varphi}+re^{i\theta})\right\vert&<\frac{c}{s(1-r)^3}\\
&<\left\vert k(e^{i\varphi}+re^{i\theta})-k(e^{i\varphi})\right\vert\\
&= \left\vert k(e^{i\varphi}+re^{i\theta})-2\cos\varphi)\right\vert;
\end{align*}
by Rouche's theorem, there exists a point $p_{\cos\varphi}$ such that $\left\vert p_{\cos\varphi}-e^{i\varphi}\right\vert<\frac{1}{(\sin\varphi_0) s}$ and
\[H_{\frac{y_0}{\sqrt{s}},1}(p_{\cos\varphi}) = 2\cos\varphi.\]
In particular, $H_{\frac{y_0}{\sqrt{s}},1}(p_{\cos\varphi}) \in\mathbb R$. The proposition now follows from the fact that $v_{\frac{y_0}{\sqrt{s}},1}(\alpha)$ is the unique positive number (if exists) such that
\[H_{\frac{y_0}{\sqrt{s}},1}(\alpha+iv_{\frac{y_0}{\sqrt{s}},1}(\alpha)) \in\mathbb R.\]
This completes the proof.
\end{proof}
\begin{proof}[{\bf Proof of Theorem \ref{thm:vsAsymp}}]
Point 1 is a result in \cite[Theorem 3.2]{HasebeUeda2018} which states that $v_s$ is unimodal for $s\geq 4 D_\nu^2$. This implies $\Lambda_{y_0,s}\cap\mathbb R = (\inf\Lambda_{y_0,s}, \sup\Lambda_{y_0,s})$.
Let
\[Y = \frac{y_0-\tau(y_0)}{\sqrt{\tau(y_0^2)}}\]
and write $t = s/\tau(y_0^2)$. By Theorem \ref{thm:HZ}, $\Lambda_{y_0, s}$ is the domain of full measure of $\mathrm{Brown}(y_0+c_s)$. Since $\mathrm{Brown}(y_0+c_s)$ is the push-forward of $\mathrm{Brown}\left(\frac{Y}{\sqrt{t}}+c_1\right)$ by the function
\[z\mapsto \tau(y_0)+z\sqrt{t\tau(y_0^2)} = \tau(y_0)+z\sqrt{s}\]
by \cite[Proposition 2.14]{HaagerupSchultz2007}. Thus,
\[\Lambda_{y_0, s} = \left\{\left.\tau(y_0)+z\sqrt{s}\in\mathbb C\right| z\in \Lambda_{\frac{Y}{\sqrt{t}},1}\right\}.\]
Points 2 and 3 then follow from applying Proposition \ref{prop:vEndPoint} and Proposition \ref{prop:vUpper} with $t=s/\tau(y_0^2)$ in place of $s$ respectively; $\Lambda_{y_0, s}$ is obtained by scaling $\Lambda_{\frac{Y}{\sqrt{t}},1}$ by $\sqrt{s}$ and translating by $\tau(y_0)$.
\end{proof}
\subsection{The density as $s\to\infty$}
In this section, we estimate the density of $\mathrm{Brown}(y_0+c_s)$ for large $s$.
\begin{theorem}
\label{thm:CircularDensity}
Denote by $w_{y_0, s}$ the density of $\mathrm{Brown}(y_0+c_s)$. Then, for any $c>1$ and $\varphi_0\in(0,\pi/2)$, we have
\[\left\vert w_{y_0,s}(\alpha+i\beta) - \frac{1}{\pi s}\right\vert < \frac{c\tau(y_0^2)}{\pi s^2}\left(6+\frac{1}{\sin^3\varphi_0}\right),\quad \left\vert\psi_{y_0,s}(\alpha)\right\vert < 2\sqrt{s}\cos\varphi_0\]
for all large enough $s$.
\end{theorem}
To simplify the computation, we assume $\tau(y_0) = 0$ and $\tau(y_0^2)= 1$ until the proof of the theorem. The key is to estimate the difference between the complex derivatives $H_{\frac{y_0}{\sqrt{s}},1}'$ and $k'$; indeed the density is directly related to the real part of the complex derivative of the subordination function $H_{\frac{y_0}{\sqrt{s}},1}^{\langle -1\rangle}$.
\begin{lemma}
\label{lem:k'Est}
Given any $c>1$ and $\varphi_0\in(0,\pi/2)$, for all sufficient large $s$, the unique $\alpha$ such that
\[H_{\frac{y_0}{\sqrt{s}},1}(\alpha+iv_{\frac{y_0}{\sqrt{s}},1}(\alpha)) = 2\cos\varphi,\quad \sin\varphi>\sin\varphi_0\]
satisfies
\[\left\vert\frac{1}{\mathrm{Re}(1/k'(\alpha+iv_{\frac{y_0}{\sqrt{s}},1}(\alpha)))}-\frac{1}{\mathrm{Re}(1/k'(e^{i\varphi}))}\right\vert<\frac{2c}{s\sin^3\varphi_0}.\]
\end{lemma}
\begin{proof}
Fix any $\varphi_0\in(0,\pi/2)$ and $c>1$. By Proposition~\ref{prop:vUpper}, for any $\varphi\in(0,\pi)$ such that $\sin\varphi > \sin\varphi_0$, the unique $\alpha\in\mathbb R$ such that
\[H_{\frac{y_0}{\sqrt{s}},1}(\alpha+iv_{\frac{y_0}{\sqrt{s}},1}(\alpha)) = 2\cos\varphi.\]
satisfies
\begin{equation}
\label{eq:alphavarphi}
\left\vert \alpha+iv_{\frac{y_0}{\sqrt{s}},1}(\alpha)-e^{i\varphi}\right\vert<\frac{1}{(\sin\varphi_0)s}
\end{equation}
for all large enough $s$.We know that $\frac{1}{\mathrm{Re}(1/k'(z))} = 2$ because
\begin{equation}
\label{eq:k'unit}
\frac{1}{k'(z)} = \frac{e^{i\varphi}}{e^{i\varphi}-e^{-i\varphi}}=\frac{1}{2}(1-i\cot\varphi).
\end{equation}
Using~\eqref{eq:alphavarphi} and~\eqref{eq:k'unit}, we have
\begin{equation}
\label{eq:mvt}
\frac{1}{(1/2-\left\vert\mathrm{Re}(1/k'(w))-\mathrm{Re}(1/k'(e^{i\varphi}))\right\vert)^2} < 4\sqrt{c}
\end{equation}
for all large enough $s$.
Write $z = e^{i\varphi}$ and $w = \alpha+iv_{\frac{y_0}{\sqrt{s}},1}(\alpha)$. Observe that
\begin{equation}
\label{eq:1/k'Est}
\frac{w^2}{w^2-1} - \frac{z^2}{z^2-1} = \frac{z^2-w^2}{(w^2-1)(z^2-1)} = \frac{(z-w)(z+w)}{(w^2-1)(z^2-1)}.
\end{equation}
Also, it is straightforward to check that
\[\left\vert z^2-1\right\vert=\left\vert e^{2i\varphi}-1\right\vert = 2\sin\varphi\]
and, by~\eqref{eq:alphavarphi},
\[\left\vert w^2-z^2\right\vert=\left\vert w-z\right\vert \left\vert w+z\right\vert <\frac{1}{(\sin\varphi_0)s}\left(2+\frac{1}{(\sin\varphi_0)s}\right).\]
We have, for all large enough $s$,
\[\frac{\left\vert z+w\right\vert}{(\left\vert z^2-1\right\vert - \left\vert w^2-z^2\right\vert)\left\vert z^2-1\right\vert} \leq \frac{2+1/(s\sin\varphi_0)}{[2\sin\varphi_0 - 2/(s\sin\varphi_0)-1/(s^2\sin^2\varphi_0)]2\sin\varphi_0} <\frac{\sqrt{c}}{2\sin^2\varphi_0};\]
thus, by the mean value theorem (applied to the function $1/(1/2+x)$), and \eqref{eq:alphavarphi}-\eqref{eq:1/k'Est},
\begin{align*}
\left\vert\frac{1}{\mathrm{Re}(1/k'(w))}-\frac{1}{\mathrm{Re}(1/k'(e^{i\varphi}))}\right\vert&\leq \frac{\left\vert\mathrm{Re}(1/k'(w))-\mathrm{Re}(1/k'(e^{i\varphi}))\right\vert}{(1/2-\left\vert\mathrm{Re}(1/k'(w))-\mathrm{Re}(1/k'(e^{i\varphi}))\right\vert)^2}\\
&<4\sqrt{c}\frac{1}{(\sin\varphi_0)s}\frac{\sqrt{c}}{2\sin^2\varphi_0}\\
&=\frac{2c}{s\sin^3\varphi_0}
\end{align*}
for all large enough $s$, completing the proof.
\end{proof}
\begin{lemma}
\label{lem:H'k'Diff}
For any $c>1$, we have
\[\left\vert\frac{1}{\mathrm{Re}(1/H_{\frac{y_0}{\sqrt{s}},1}'(z))}-\frac{1}{\mathrm{Re}(1/k'(z))}\right\vert<\frac{3c}{s\left\vert z\right\vert^4}\frac{1}{[\mathrm{Re}(1/k'(z))]^2},\quad \left\vert z\right\vert>\frac{1}{2}.\]
for all large enough $s$.
\end{lemma}
\begin{proof}
Let $c>1$. By~\eqref{eq:H'k'}, for all $|z|>\frac{1}{2}$,
\begin{equation}
\label{eq:H'k'Est}
\left\vert H_{\frac{y_0}{\sqrt{s}},1}'(z) - k'(z)\right\vert\leq \frac{1}{s\left\vert z\right\vert^4}\left(3\tau(y_0^2)+\sum_{n=3}^\infty\frac{(n+1)\left\vert\tau(y^n)\right\vert}{s^{\frac{n}{2}-1}(1/2)^{n-2}}\right)<\frac{3\sqrt{c}\tau(y_0^2)}{s\left\vert z\right\vert^4}
\end{equation}
for all large enough $s$. We then must have
\[\left\vert\frac{1}{\mathrm{Re}(1/H_{\frac{y_0}{\sqrt{s}},1}'(z))}\right\vert < \frac{\sqrt{c}}{\mathrm{Re}(1/k'(z))}\]
for all large enough $s$. Therefore, we have
\begin{align*}
\left\vert\frac{1}{\mathrm{Re}(1/H_{\frac{y_0}{\sqrt{s}},1}'(z))}-\frac{1}{\mathrm{Re}(1/k'(z))}\right\vert &= \frac{\left\vert\mathrm{Re}(1/k'(z))-\mathrm{Re}(1/H_{\frac{y_0}{\sqrt{s}},1}'(z))\right\vert}{\left\vert\mathrm{Re}(1/H_{\frac{y_0}{\sqrt{s}},1}'(z))\mathrm{Re}(1/k'(z))\right\vert}\\
&<\frac{3c\tau(y_0^2)}{s\left\vert z\right\vert^4}\frac{1}{[\mathrm{Re}(1/k'(z))]^2},
\end{align*}
which is the desired inequality since we assume $\tau(y_0^2) = 1$ until the proof of Theorem~\ref{thm:CircularDensity}.
\end{proof}
\begin{lemma}
\label{lem:EstFrom2}
Given any $c>1$ and $\varphi_0\in(0,\pi/2)$, for all sufficient large $s$, the unique $\alpha$ such that
\[H_{\frac{y_0}{\sqrt{s}},1}(\alpha+iv_{\frac{y_0}{\sqrt{s}},1}(\alpha)) = 2\cos\varphi,\quad \sin\varphi>\sin\varphi_0\]
satisfies
\[\left\vert\frac{1}{\mathrm{Re}(1/H_{\frac{y_0}{\sqrt{s}},1}'(w))} - 2\right\vert <\frac{2c}{s}\left(6+\frac{1}{\sin^3\varphi_0}\right)\]
where $w = \alpha+iv_{\frac{y_0}{\sqrt{s}},1}(\alpha)$.
\end{lemma}
\begin{proof}
Let $c>1$. Write $z = e^{i\varphi}$ and $w = \alpha+iv_{\frac{y_0}{\sqrt{s}},1}(\alpha)$. Recall that $\frac{1}{\mathrm{Re}(1/k'(z))} = 2$ by~\eqref{eq:k'unit}. We estimate
\begin{equation}
\label{eq:EstFrom2}
\left\vert\frac{1}{\mathrm{Re}(1/H_{\frac{y_0}{\sqrt{s}},1}'(w))} - 2\right\vert\leq \left\vert\frac{1}{\mathrm{Re}(1/H_{\frac{y_0}{\sqrt{s}},1}'(w))} - \frac{1}{\mathrm{Re}(1/k'(w))}\right\vert + \left\vert\frac{1}{\mathrm{Re}(1/k'(w))} - \frac{1}{\mathrm{Re}(1/k'(z))}\right\vert.
\end{equation}
We estimate the first term in~\eqref{eq:EstFrom2} using Proposition~\ref{prop:vUpper} and Lemmas~\ref{lem:k'Est} and \ref{lem:H'k'Diff}. Fix any $1<c'<c$. For all large enough~$s$, the first term is bounded by
\begin{align*}
\left\vert\frac{3c'}{s\left\vert w\right\vert^4}\frac{1}{[\mathrm{Re}(1/k'(w))]^2}\right\vert&\leq \frac{3c'}{s[1-1/(\sin\varphi_0 s)^4]}\left(\frac{1}{\mathrm{Re}(1/k'(e^{i\varphi}))}+\frac{2c'}{s\sin^3\varphi_0}\right)^2\\
&<\frac{12c}{s}
\end{align*}
by Lemmas~\ref{lem:k'Est} and~\ref{lem:H'k'Diff}.
By Lemma~\ref{lem:k'Est}, the second term in~\eqref{eq:EstFrom2} is bounded by
\[\left\vert\frac{1}{\mathrm{Re}(1/k'(w))}-\frac{1}{\mathrm{Re}(1/k'(z))}\right\vert<\frac{2c}{s\sin^3\varphi_0}.\]
The result then follows from adding these estimates.
\end{proof}
\begin{proposition}
\label{prop:c1Density}
Denote by $w_{\frac{y_0}{\sqrt{s}}, 1}$ the density of $\mathrm{Brown}(\frac{y_0}{\sqrt{s}}+c_1)$. Then, for any $c>1$ and $\varphi_0\in(0,\pi/2)$, we have
\[\left\vert w_{\frac{y_0}{\sqrt{s}}, 1}(\alpha+i\beta) - \frac{1}{\pi}\right\vert < \frac{c}{\pi s}\left(6+\frac{1}{\sin^3\varphi_0}\right),\quad \left\vert\psi_{\frac{y_0}{\sqrt{s}},1}(\alpha)\right\vert < 2\cos\varphi_0\]
for all large enough $s$.
\end{proposition}
\begin{proof}
By Equation (3.31) of \cite{HoZhong2019},
\[\mathrm{Re}\left(\frac{1}{H_{\frac{y_0}{\sqrt{s}},1}'(w)}\right)\frac{d\psi_{\frac{y_0}{\sqrt{s}},1}(\alpha)}{d\alpha} = 1\]
where $w = \alpha+iv_{\frac{y_0}{\sqrt{s}}, 1}(\alpha)$. (This formula appeals to the subordination function $H_{\frac{y_0}{\sqrt{s}},1}^{\langle -1\rangle}$ of the free convolution $\frac{y_0}{\sqrt{s}}+\sigma_1$ has an analytic continuation in a neighborhood of any point $\psi_{\frac{y_0}{\sqrt{s}}, 1}(\alpha+iv_{\frac{y_0}{\sqrt{s}}, 1}(\alpha))$ if $v_{\frac{y_0}{\sqrt{s}}, 1}(\alpha)>0$; see \cite[Theorem 3.3(1)]{Belinschi2008}.) Thus, we can express the real derivative through complex derivative
\[\frac{d\psi_{\frac{y_0}{\sqrt{s}},1}(\alpha)}{d\alpha} = \frac{1}{\mathrm{Re}(1/H_{\frac{y_0}{\sqrt{s}},1}'(w))}.\]
By Lemma~\ref{lem:EstFrom2}, given any $c>1$ and $\varphi_0\in(0,\pi/2)$, for all sufficient large $s$, the unique $\alpha$ such that
\[\psi_{\frac{y_0}{\sqrt{s}},1}(\alpha) = 2\cos\varphi,\quad \sin\varphi>\sin\varphi_0\]
satisfies
\[\left\vert\frac{d\psi_{\frac{y_0}{\sqrt{s}},1}(\alpha)}{d\alpha} - 2\right\vert <\frac{2c}{s}\left(6+\frac{1}{\sin^3\varphi_0}\right).\]
The proposition now follows from Theorem~\ref{thm:HZ}
\end{proof}
All the estimates in this section that we have done are under the assumption $\tau(y_0) = 0$ and $\tau(y_0^2)$. We are now ready to prove the estimate of the density of $\mathrm{Brown}(y_0+c_s)$ for arbitrary $\tau(y_0)$ and $\tau(y_0^2)$.
\begin{proof}[{\bf Proof of Theorem \ref{thm:CircularDensity}}]
Without loss of generality, we assume $\tau(y_0) = 0$, since otherwise we translate the density by~$\tau(y_0)$.
We first assume $\tau(y_0^2) = 1$. Let $w = \alpha + iv_{y_0,s}(\alpha)$ and $z = \frac{w}{\sqrt{s}}$. Then
\[z = \frac{\alpha}{\sqrt{s}} + iv_{\frac{y_0}{\sqrt{s}},1}\left(\frac{\alpha}{\sqrt{s}}\right).\]
Since $\mathrm{Brown}(y_0+c_s)$ is the push-forward measure of $\mathrm{Brown}\left(\frac{y_0}{\sqrt{s}}+c_1\right)$ by $z\mapsto \sqrt{s}z$,
\[w_{y_0,s}(\alpha+i\beta) = \frac{1}{s}\cdot w_{\frac{y_0}{\sqrt{s}},1}\left(\frac{1}{\sqrt{s}}(\alpha+i\beta)\right), \quad z\in\Lambda_{y_0,s}.\]
By Proposition~\ref{prop:c1Density}, for any $c>1$ and $\varphi_0\in(0,\pi/2)$, we have
\[\left\vert w_{y_0,s}(\alpha+i\beta) - \frac{1}{\pi s}\right\vert < \frac{c}{\pi s^2}\left(6+\frac{1}{\sin^3\varphi_0}\right),\quad \left\vert\psi_{y_0,s}(\alpha)\right\vert < 2\sqrt{s}\cos\varphi_0\]
for all large enough $s$. This establishes the result with $\tau(y_0^2) = 1$.
For arbitrary $\tau(y_0^2)$, let $Y = \frac{y_0}{\sqrt{\tau(y_0^2)}}$. We consider the random variable
\[\frac{1}{\sqrt{\tau(y_0^2)}}(y_0+c_s)\]
which has the same $\ast$-moments, hence the same Brown measure, as
\[Y+c_{t}\]
where $t= s/\tau(y_0^2)$.
By the result for $\tau(y_0^2)=1$, given any $c>1$ and $\varphi_0\in(0,\pi/2)$, we have
\begin{equation}
\label{eq:CDensityEst}
\left\vert w_{Y,t}(\alpha+i\beta) - \frac{1}{\pi t}\right\vert < \frac{c}{\pi t^2}\left(6+\frac{1}{\sin^3\varphi_0}\right),\quad \left\vert\psi_{Y,t}(\alpha)\right\vert < 2\sqrt{t}\cos\varphi_0
\end{equation}
for all large enough $t$. Now, since $\mathrm{Brown}(y_0+c_s)$ is the push-forward measure of $\mathrm{Brown}(Y+c_t)$ by $z\mapsto \sqrt{\tau(y_0^2)}z$, by~\eqref{eq:CDensityEst}, we must have
\[\left\vert w_{y_0,s}(\alpha+i\beta) - \frac{1}{\pi s}\right\vert < \frac{c\tau(y_0^2)}{\pi s^2}\left(6+\frac{1}{\sin^3\varphi_0}\right),\quad \left\vert\psi_{y_0,s}(\alpha)\right\vert < 2\sqrt{s}\cos\varphi_0\]
for all large enough $s$.
\end{proof}
\section{Asymptotic behaviors of adding an elliptic element\label{sect:ellipticAsymp}}
\subsection{Fix $s/t$ and let $s, t\to\infty$}
In this section, we study three limiting behaviors of $\mathrm{Brown}(y_0+\tilde{\sigma}_{s-t/2}+i\sigma_{t/2})$. The first one is to keep $s$ and $t$ at the same ratio $r = t/s$ and let $s\to\infty$; the second one is to keep $t$ fixed then let $s\to\infty$; the last one is to fix $s=t/2$ and let $s\to\infty$. The second and the third asymptotic behaviors were also studied by \cite{Hall1999} in the context of Segal--Bargmann transform.
\subsubsection{Domain behavior}
In this section, we discuss the asymptotic behavior of the domain of $\mathrm{Brown}(y_0+\tilde{\sigma}_{s-\frac{t}{2}}+i\sigma_{\frac{t}{2}})$ for a fixed $r = t/s$.
When $y_0=0$, the domain of $\tilde{\sigma}_{s-\frac{t}{2}}+i\sigma_{\frac{t}{2}}$ has the shape of an ellipse with boundary
\begin{equation}
\label{eq:ellipse}
\frac{2s-t}{\sqrt{s}}\cos\varphi+i\frac{t}{\sqrt{s}}\sin\varphi,\quad \varphi\in[0,2\pi]
\end{equation}
(See \cite[Example 5.3]{BianeLehner2001}). As $s\to\infty$ with $r=t/s$ fixed, the random variable $y_0+\tilde{\sigma}_{s-\frac{t}{2}}+i\sigma_{\frac{t}{2}}$ behaves like the elliptic element $\tau(y_0)+\tilde{\sigma}_{s-\frac{t}{2}}+i\sigma_{\frac{t}{2}}$. Roughly speaking, the domain $\Omega_{s,t}$ of $\mathrm{Brown}(y_0+\tilde{\sigma}_{s-\frac{t}{2}}+i\sigma_{\frac{t}{2}})$ is asymptotically an ellipse with boundary as in~\eqref{eq:ellipse} translated by $\tau(y_0)$. The following theorem states precisely the asymptotic behavior of the domain $\Omega_{s,t}$ of $\mathrm{Brown}(y_0+\tilde{\sigma}_{s-\frac{t}{2}}+i\sigma_{\frac{t}{2}})$; the main tool is Theorem~\ref{thm:vsAsymp}.
\begin{theorem}
Fix the ratio $r = t/s$. The following asymptotic behaviors of the graph of $\Omega_{s,t}$ hold.
\begin{enumerate}
\item Let $D_\nu = \sup\{\left\vert x-y\right\vert| x, y\in\mathrm{supp}\,\mu\}$. When $s\geq 4 D_\nu^2$, the function $b_{s,t}$ is unimodal. In particular, $\Omega_{s,t}\cap\mathbb R$ is an interval.
\item Given any $c>1$, we have
\[\left\vert
\sup \Omega_{s,t}\cap\mathbb R-\left(\tau(y_0)+\frac{2s-t}{\sqrt{s}}\right)\right\vert <\frac{c(3r+2\left\vert 1-r\right\vert)\tau(y_0^2)}{2\sqrt{s}}\]
and
\[\left\vert
\inf \Omega_{s,t}\cap\mathbb R-\left(\tau(y_0)-\frac{2s-t}{\sqrt{s}}\right)\right\vert <\frac{c(3r+2\left\vert 1-r\right\vert)\tau(y_0^2)}{2\sqrt{s}}\]
for all sufficiently large $s$. In particular,
\[ \Lambda_{y_0,s}\cap\mathbb R\subset \left(\tau(y_0)-\frac{2s-t}{\sqrt{s}}-\frac{c(3r+2\left\vert 1-r\right\vert)\tau(y_0^2)}{2\sqrt{s}}, \tau(y_0)+\frac{2s-t}{\sqrt{s}}+\frac{c(3r+2\left\vert 1-r\right\vert)\tau(y_0^2)}{2\sqrt{s}}\right)\]
for all large enough $s$.
\item Given any $\varphi_0\in(0,\pi/2)$, then for all large enough $s$, for all $\left\vert \cos\varphi\right\vert\leq \cos\varphi_0$, the unique $\alpha\in\mathbb R$ such that
\[H_{y_0,s}(\alpha+iv_{y_0,s}(\alpha)) = 2\sqrt{s}\cos\varphi.\]
satisfies
\[\left\vert U_{s,t}(\alpha+iv_{y_0,s}(\alpha))-\left[\frac{2s-t}{\sqrt{s}}\cos\varphi+i\frac{t}{\sqrt{s}}\sin\varphi\right]\right\vert<\frac{r}{(\sin\varphi_0)\sqrt{s}}.\]
\end{enumerate}
\end{theorem}
\begin{proof}
Point 1 follows directly from \cite[Theorem 3.2]{HasebeUeda2018} which states that $v_s$ is unimodal for $s\geq 4 D_\nu^2$, because, by Proposition~\ref{prop:alphast}, we have
\[b_{s,t} = \frac{t}{s}v_{y_0,s}.\]
Fix $r=t/s$ throughout this proof. We now prove Point 2. Without loss of generality, we assume $\tau(y_0) = 0$. We first estimate $a_{1,r}(\alpha^*)$ where
\[\alpha^* = \sup\Lambda_{y_0/\sqrt{s},1}\cap\mathbb R.\]
We compute
\begin{equation}
\label{eq:a1r}
a_{1,r}(\alpha^*)-(2-r) = (\alpha^*-1)\left(1-\frac{1-r}{\alpha^*}\right)+\frac{(1-r)\tau(y_0^2)}{s(\alpha^*)^3}+\frac{(1-r)}{s^{3/2}}\sum_{n=3}^\infty\frac{\tau(y_0^n)}{s^{(n-3)/2}(\alpha^*)^{n+1}}.
\end{equation}
By Proposition~\ref{prop:vEndPoint} (with $s$ replaced by $s/\tau(y_0^2)$), given any $c>1$, for all large enough $s$, we have
\[\left\vert a_{1,r}(\alpha^*)-(2-r)\right\vert <\frac{c(3r+2\left\vert 1-r\right\vert)\tau(y_0^2)}{2s}.\]
Since
\[\sup\Omega_{s, t}\cap\mathbb R = \sqrt{s}a_{1,r}(\alpha^*),\]
we have
\[\left\vert
\sup \Omega_{s,t}\cap\mathbb R-\left(\tau(y_0)+\frac{2s-t}{\sqrt{s}}\right)\right\vert <\frac{c(3r+2\left\vert 1-r\right\vert)\tau(y_0^2)}{2\sqrt{s}}\]
for all sufficiently large $s$. The estimate for $\inf\Omega_{s,t}\cap\mathbb R$ is similar.
We prove Point 3 now. By Theorem \ref{thm:pushforward}, we know that
\[\Omega_{s,t} = U_{s,t}(\Lambda_{y_0,s}).\]
Suppose $\alpha$ is chosen such that $\psi_{y_0,s}(\alpha) = 2\sqrt{s}\cos\varphi$. We compute the upper boundary curve $a+ib_{s,t}(a) = U_{s,t}(\alpha+iv_{y_0,s}(\alpha))$ as
\begin{align*}
&a_{s,t}(\alpha) = (1-r)\psi_{y_0,s}(\alpha)+r\alpha = 2(1-r)\sqrt{s}\cos\varphi+r\alpha;\\
&b_{s,t}(a) = b_{s,t}(a_{s,t}(\alpha)) = r v_{y_0,s}(\alpha).
\end{align*}
So, we have
\begin{equation}
\label{eq:EllipseImag}
\left\vert a+ib_{s,t}(a)-\sqrt{s}[(2-r)\cos\varphi+ir\sin\varphi]\right\vert = r\left\vert\alpha+iv_{y_0,s}(\alpha)-\sqrt{s}e^{i\varphi}\right\vert.
\end{equation}
Therefore, by Theorem~\ref{thm:vsAsymp}, for any $\varphi_0\in(0,\pi/2)$,
\begin{align*}
\left\vert a+ib_{s,t}(a)-\sqrt{s}[(2-r)\cos\varphi+ir\sin\varphi]\right\vert &= r\left\vert\alpha+iv_{y_0,s}(\alpha)-\sqrt{s}e^{i\varphi}\right\vert\\
&\leq \frac{r}{(\sin\varphi_0)\sqrt{s}}
\end{align*}
for all sufficiently large $s$. This proves Point 3.
\end{proof}
\subsubsection{Density behavior}
In this section, we investigate the asymptotic behavior of the density of $\mathrm{Brown}(y_0+\tilde{\sigma}_{s-\frac{t}{2}}+i\sigma_{\frac{t}{2}})$ for a fixed $r = t/s$. In the case $y_0 = 0$, $\mathrm{Brown}(y_0+\tilde{\sigma}_{s-\frac{t}{2}}+i\sigma_{\frac{t}{2}})$ is the elliptic law, with constant density
\begin{equation}
\label{eq:ellipticLaw}
\frac{1}{\pi}\frac{s}{(2s-t)t}
\end{equation}
in domain $\Omega_{s,t}$, which is a region bounden by an ellipse in this case (See \cite[Example 5.3]{BianeLehner2001}).
Denote by $w_{y_0,s,t}$ the density of $\mathrm{Brown}(y_0+\tilde{\sigma}_{s-\frac{t}{2}}+i\sigma_{\frac{t}{2}})$. We will prove that as $s$ large and $r=t/s$ fixed, the density $w_{y_0,s,t}$ is approximately the same constant in~\eqref{eq:ellipticLaw}. The main tool is the estimate of the density of $\mathrm{Brown}(y_0+c_s)$ in Theorem~\ref{thm:CircularDensity}.
\begin{theorem}
\label{thm:EllipticDensity}
Fix $r = t/s$. Given any $c>1$ and $\varphi_0\in(0,\pi/2)$, we have
\[\left\vert w_{y_0,s,t}(a+ib) - \frac{1}{\pi}\frac{s}{(2s-t)t}\right\vert<\frac{c\tau(y_0^2)}{\pi (2s-t)^2}\left(6+\frac{1}{\sin^3\varphi_0}\right),\quad \psi_{y_0,s}(\alpha_{s,t}(a))<2\sqrt{s}\cos\varphi_0\]
for all large enough $s$.
\end{theorem}
\begin{proof}
Let $c>1$ be given. By Corollary~\ref{cor:stFormulas}, if we write $a+ib = U_{s,t}(\alpha+i \beta)$ for all $\alpha+i\beta\in\Lambda_{y_0,s}$. Then we have
\[w_{y_0,s,t}(a+ib) = \frac{1}{r}\frac{w_{y_0,s}(\alpha+i\beta)}{r+2\pi(1-r)s\cdot w_{y_0,s}(\alpha+i\beta)}\]
for all $a+ib\in\Omega_{s,t}$.
Now, by the formula
\[ \frac{1}{\pi s}\frac{1}{2-r}=\frac{1/(\pi s)}{r+2\pi(1-r)s\cdot (1/\pi s)},\]
and Theorem~\ref{thm:CircularDensity}, for any $1<c'<c$, if $\psi_{y_0,s}(\alpha)<2\sqrt{s}\cos\varphi_0$, then we have $\pi s w_{y_0,s}(\alpha+i\beta)\to 1$, and
\begin{align*}
&\quad\,\left\vert\frac{w_{y_0,s}(\alpha+i\beta)}{r+2\pi(1-r)s\cdot w_{y_0,s}(\alpha+i\beta)} -\frac{1/(\pi s)}{2-r}\right\vert\\
&= \frac{r\left\vert w_{y_0,s}(\alpha+i\beta)-1/(\pi s)\right\vert}{[r+2\pi(1-r)s\cdot w_s(\alpha+i\beta)][2-r]}\\
&<\frac{cr\tau(y_0^2)}{\pi s^2}\left(6+\frac{1}{\sin^3\varphi_0}\right)\frac{1}{(2-r)^2}
\end{align*}
for all large enough $s$. The proof follows from dividing the above estimate by $r$.
\end{proof}
\subsection{Fix $t$ and let $s\to\infty$}
In this section, we investigate the asymptotic behavior of $\mathrm{Brown}(y_0+\tilde{\sigma}_{s-t/2}+i\sigma_{t/2})$ with $t$ fixed and $s\to\infty$.
\subsubsection{Domain behavior}
The following theorem states that $\Omega_{s,t}$ has the shape of an ellipse in the limit with fixed $t$ as $s\to\infty$, except points close to the \emph{endpoints} of $\Omega_{s,t}\cap\mathbb R$. The limiting ellipse has a very short minor axis; it is a long and thin ellipse.
\begin{theorem}
Fix $t>0$. The following asymptotic behaviors of the graph of $\Omega_{s,t}$ hold.
\begin{enumerate}
\item Let $D_\nu = \sup\{\left\vert x-y\right\vert| x, y\in\mathrm{supp}\,\mu\}$. When $s\geq 4 D_\nu^2$, the function $b_{s,t}$ is unimodal. In particular, $\Omega_{s,t}\cap\mathbb R$ is an interval.
\item Given any $c>1$, we have
\[\left\vert
\sup \Omega_{s,t}\cap\mathbb R-\left(\tau(y_0)+2\sqrt{s}\right)\right\vert <\frac{c\left\vert \tau(y_0^2)-t\right\vert}{\sqrt{s}}\]
and
\[\left\vert
\inf \Omega_{s,t}\cap\mathbb R-\left(\tau(y_0)-2\sqrt{s}\right)\right\vert <\frac{c\left\vert \tau(y_0^2)-t\right\vert}{\sqrt{s}}\]
for all sufficiently large $s$. In particular,
\[ \Lambda_{y_0,s}\cap\mathbb R\subset \left(\tau(y_0)-2\sqrt{s}-\frac{c\left\vert \tau(y_0^2)-t\right\vert}{\sqrt{s}}, \tau(y_0)+\frac{c\left\vert \tau(y_0^2)-t\right\vert}{\sqrt{s}}\right)\]
for all large enough $s$.
\item Given any $\varphi_0\in(0,\pi/2)$, then for all large enough $s$, for all $\left\vert \cos\varphi\right\vert\leq \cos\varphi_0$, the unique $\alpha\in\mathbb R$ such that
\[H_{y_0,s}(\alpha+iv_{y_0,s}(\alpha)) = 2\sqrt{s}\cos\varphi.\]
satisfies
\[\left\vert U_{s,t}(\alpha+iv_{y_0,s}(\alpha))-\left[\frac{2s-t}{\sqrt{s}}\cos\varphi+i\frac{t}{\sqrt{s}}\sin\varphi\right]\right\vert<\frac{t}{(\sin\varphi_0)s^{3/2}}.\]
Furthermore, we have
\[\lim_{s\to \infty}\sup\{\left\vert \mathrm{Im}\,z\right\vert | z\in\Omega_{s,t}\} = 0.\]
\end{enumerate}
\end{theorem}
\begin{proof}
Point 1 follows directly from Theorem~\ref{thm:pushforward} and \cite[Theorem 3.2]{HasebeUeda2018} which states that $v_{y_0,s}$ is unimodal for $s\geq 4 D_\nu^2$, because, by Proposition~\ref{prop:alphast}, we have
\[b_{s,t} = \frac{t}{s}v_{y_0,s}.\]
Fix $t>0$. We now prove Point 2. Without loss of generality, we assume $\tau(y_0) = 0$. We first estimate $a_{1,r}(\alpha^*)$ where
\[\alpha^* = \sup\Lambda_{y_0/\sqrt{s},1}\cap\mathbb R.\]
We calculate
\begin{align*}
a_{1,r}(\alpha^*)-2 &= \alpha^*-2+(1-r)\sum_{n=0}^\infty\frac{\tau(y_0^n)}{s^{\frac{n}{2}}(\alpha^*)^{n+1}}\\
&=\alpha^*-1+\frac{1-\alpha^*}{\alpha^*}-\frac{t}{s\alpha^*}+\frac{\tau(y_0^2)}{s(\alpha^*)^3}+\sum_{n=3}^\infty\frac{\tau(y_0^n)}{s^{\frac{n}{2}}(\alpha^*)^{n+1}}\\
&=(\alpha^*-1)\frac{\alpha^*-1}{\alpha^*}+\frac{\tau(y_0^2)-t}{s(\alpha^*)^3}+\sum_{n=3}^\infty\frac{\tau(y_0^n)}{s^{\frac{n}{2}}(\alpha^*)^{n+1}}
\end{align*}
By Proposition~\ref{prop:vEndPoint} (with $s$ replaced by $s/\tau(y_0^2)$), given any $c>1$, for all large enough $s$, we have (by keeping only the order $1/s$ term)
\[\left\vert a_{1,r}(\alpha^*)-2\right\vert <\frac{c\left\vert \tau(y_0^2)-t\right\vert}{s}.\]
It follows that
\[\left\vert
\sup \Omega_{s,t}\cap\mathbb R-\left(\tau(y_0)+2\sqrt{s}\right)\right\vert <\frac{c\left\vert \tau(y_0^2)-t\right\vert}{\sqrt{s}}\]
for all sufficiently large $s$. The estimate for $\inf\Omega_{s,t}\cap\mathbb R$ is similar.
We now prove Point 3. By~\eqref{eq:EllipseImag},
\[\left\vert a+ib_{s,t}(a)-\sqrt{s}[(2-r)\cos\varphi+ir\sin\varphi]\right\vert = r\left\vert\alpha+iv_{y_0,s}(\alpha)-\sqrt{s}e^{i\varphi}\right\vert.
\]
Therefore, by Theorem~\ref{thm:vsAsymp}, for any $\varphi_0\in(0,\pi/2)$,
\begin{equation}
\label{eq:FixtEst}
\begin{split}
\left\vert a+ib_{s,t}(a)-\sqrt{s}[(2-r)\cos\varphi+ir\sin\varphi]\right\vert &= r\left\vert\alpha+iv_{y_0,s}(\alpha)-\sqrt{s}e^{i\varphi}\right\vert\\
&\leq \frac{t}{(\sin\varphi_0)s^{3/2}}
\end{split}
\end{equation}
for all sufficiently large $s$.
Let $\varphi_0=\frac{\pi}{6}$ so that $\sin\varphi>1/2$ for all $\varphi$ such that $\left\vert\cos\varphi\right\vert<\cos\varphi_0$. We label by $\alpha_\varphi$ the unique $\alpha\in\mathbb R$ such that
\[H_{y_0,s}(\alpha+iv_{y_0,s}(\alpha)) = 2\sqrt{s}\cos\varphi, \quad\left\vert\cos\varphi\right\vert\leq \cos\varphi_0.\]
By~\eqref{eq:FixtEst}, we have
\begin{equation*}
\sup \{b_{s,t}(a_{s,t}(\alpha))\vert \alpha_{\pi-\varphi_0}<\alpha<\alpha_{\varphi_0}\}>\frac{t}{\sqrt{s}}-\frac{2t}{s^{3/2}}.
\end{equation*}
Since
\[b_{s,t}(a_{s,t}(\alpha_{\varphi_0}))<\frac{t}{2\sqrt{s}}+\frac{2t}{s^{3/2}}\]
and, by Point 1, the function $b_{s,t}$ is unimodal,
\begin{equation}
\label{eq:EstOutside}
b_{s,t}(a_{s,t}(\alpha))<\frac{t}{2\sqrt{s}}+\frac{2t}{s^{3/2}},\quad \alpha\geq\alpha_{\varphi_0}\textrm{ or }\alpha\leq\alpha_{\pi-\varphi_0}.
\end{equation}
For all $\alpha_{\pi-\varphi_0}<\alpha<\alpha_{\varphi_0}$,
\begin{equation}
\label{eq:supabove}
\sup \{b_{s,t}(a_{s,t}(\alpha))\vert \alpha_{\pi-\varphi_0}<\alpha<\alpha_{\varphi_0}\}<\frac{t}{\sqrt{s}}+\frac{2t}{s^{3/2}}.
\end{equation}
Therefore, we conclude
\[\lim_{s\to \infty}\sup\{\left\vert \mathrm{Im}\,z\right\vert | z\in\Omega_{s,t}\} = 0\]
by~\eqref{eq:EstOutside} and~\eqref{eq:supabove}.
\end{proof}
\subsubsection{Density behavior}
If we consider the special case of $y_0 = 0$, $\mathrm{Brown}(y_0+\tilde\sigma_{s-t/2}+\sigma_{t/2})$ is just the elliptic law; as mentioned in ~\eqref{eq:ellipticLaw}, it has a constant density
\[\frac{1}{\pi}\frac{s}{(2s-t)t}.\]
If we fixed $t$ and let $s\to\infty$, this density converges to the constant $1/(2\pi t)$.
The following theorem states that if we consider an arbitrary self-adjoint initial condition $y_0$, the density of $\mathrm{Brown}(y_0+\tilde\sigma_{s-t/2}+i\sigma_{t/2})$ also converges to $1/(2\pi t)$; the convergence is uniform away the endpoints of $\Omega_{s,t}\cap\mathbb R$.
\begin{theorem}
Denote by $w_{y_0,s,t}$ the density of $\mathrm{Brown}(y_0+\tilde\sigma_{s-t/2}+i\sigma_{t/2})$. Then given any $c>1$ and $\varphi_0\in(0,\pi/2)$, we have
\[\left\vert w_{y_0,s,t}(a+ib) - \frac{1}{2\pi t}\right\vert <\frac{c}{4\pi s}, \quad\left\vert\psi_{y_0,s}(\alpha_{s,t}(a))\right\vert < 2\sqrt{s}\cos\varphi_0\]
for all sufficiently large $s$.
\end{theorem}
\begin{proof}
Let $c>1$ and $\varphi_0\in(0,\pi/2)$ be given. By Corollary~\ref{cor:stFormulas}, if we write $(a,b) = U_{s,t}(\alpha, \beta)$ for all $\alpha+i\beta\in\Lambda_{y_0,s}$. Then we have
\begin{equation}
\label{eq:Densityst}
w_{y_0,s,t}(a+ib) = \frac{1}{2\pi t}\frac{s \pi w_{y_0,s}(\alpha+i\beta)}{t/(2s)+(1-t/s)\pi s\cdot w_{y_0,s}(\alpha+i\beta)}
\end{equation}
for all $a+ib\in\Omega_{s,t}$.
By Theorem~\ref{thm:CircularDensity}, given any $1<c'<c$, we have
\[\left\vert \pi s\cdot w_{y_0,s}(\alpha+i\beta) - 1\right\vert < \frac{c'\tau(y_0^2)}{ s}\left(6+\frac{1}{\sin^3\varphi_0}\right),\quad \left\vert\psi_{y_0,s}(\alpha)\right\vert < 2\sqrt{s}\cos\varphi_0\]
for all large enough $s$. Then, we compute
\begin{align*}
\left\vert\frac{s \pi w_{y_0,s}(\alpha+i\beta)}{t/(2s)+(1-t/s)\pi s\cdot w_{y_0,s}(\alpha+i\beta)} - 1 \right\vert&=\frac{t}{s}\left\vert \frac{\pi s\cdot w_{y_0,s}(\alpha+i\beta)-1/2}{t/(2s)+(1-t/s)\pi s\cdot w_{y_0,s}(\alpha+i\beta)}\right\vert\\
&\leq \frac{c't}{s}\left[\frac{1}{2}+\frac{c'\tau(y_0^2)}{ s}\left(6+\frac{1}{2\sin^3\varphi_0}\right)\right]\\
&<\frac{ct}{2s}.
\end{align*}
for all large enough $s$, since $t/(2s)+(1-t/s)\pi s\cdot w_{y_0,s}(\alpha+i\beta)$ converges to $1$. Thus, using~\eqref{eq:Densityst}, we have the estimate
\[w_{y_0,s,t}(a+ib) - \frac{1}{2\pi t} = \frac{1}{2\pi t}\left\vert\frac{s \pi w_{y_0,s}(\alpha+i\beta)}{t/(2s)+(1-t/s)\pi s\cdot w_{y_0,s}(\alpha+i\beta)} - 1 \right\vert <\frac{c}{4\pi s}\]
for all sufficiently large $s$.
\end{proof}
\subsection{Set $s=t/2$ and let $s\to\infty$}
In this section, we investigate the asymptotic behavior of $\mathrm{Brown}(y_0+\sigma_{s-t/2}+i\tilde{\sigma}_{t/2})$ with $s=t/2$ and $s\to\infty$. Note that, when $s=t/2$, the random variable $y_0+\tilde{\sigma}_{s-t/2}+i\sigma_{t/2}$ is $y_0+i\sigma_{s}$.
\begin{theorem}
\label{thm:Skewasymptotic}
\begin{enumerate}
\item Let $D_\nu = \sup\{\left\vert x-y\right\vert| x, y\in\mathrm{supp}\,\mu\}$. When $s\geq 4 D_\nu^2$, the function $b_{s,t}$ is unimodal. In particular, $\Omega_{s,t}\cap\mathbb R$ is an interval.
\item We have
\[-\frac{4c\tau(y_0^2)}{\sqrt{s}}<\inf(\Omega_{s,t}\cap\mathbb R)-\tau(y_0)<0<\sup(\Omega_{s,t}\cap\mathbb R)-\tau(y_0)<\frac{4c\tau(y_0^2)}{\sqrt{s}}\]
for all $s$ large enough. In particular,
\[\Omega_{s,t}\cap \mathbb R\subset \left( \tau(y_0)- \frac{4c\tau(y_0^2)}{\sqrt{s}}, \tau(y_0)+ \frac{4c\tau(y_0^2)}{\sqrt{s}}\right)\]
for all $s$ large enough.
\item We also have
\[\left\vert\sup\{\left\vert\mathrm{Im}\,z\right\vert|z\in\Omega_{s,t}\} - 2\sqrt{s}\right\vert<\frac{2c}{\sqrt{s}}\]
for all large enough $s$.
\end{enumerate}
\end{theorem}
\begin{proof}
Point 1 follows directly from \cite[Theorem 3.2]{HasebeUeda2018} which states that $v_{y_0,s}$ is unimodal for $s\geq 4 D_\nu^2$, because, by Proposition~\ref{prop:alphast}, we have
\[b_{s,t} = 2v_{y_0,s}.\]
We now prove Point 2. Let $c>1$ be given. Without loss of generality, we assume $\tau(y_0) = 0$. Denote
\[M_s=\sup(\Lambda_s\cap\mathbb{R})\quad \textrm{and}\quad m_s=\inf(\Lambda_s\cap\mathbb{R}).\]
Then $\sup(\Omega_{y_0,s}\cap\mathbb R) = a_{y_0,s}(M_s)$ and $\inf(\Omega_{y_0,s}\cap\mathbb R) = a_{y_0,s}(m_s)$. First, $M_s > \sup(\mathrm{supp}\,\nu)$ by Point 1 of Theorem \ref{thm:vsAsymp}. Recall from Definition \ref{def:H} that (since $M_t$ is real)
\begin{equation}
\label{eq:asEst}
\begin{split}
a_{y_0, s}(M_s) &= H_{y_0,-s}(M_s)\\
&= M_s - s\int\frac{d\nu(x)}{M_s-x}\\
&= \frac{1}{M_s}(M_s^2-s)- \frac{s}{M_s^3}\sum_{n=2}^\infty\frac{\tau(y_0^n)}{M_s^{n-2}}.
\end{split}
\end{equation}
Now, by Theorem~\ref{thm:vsAsymp}, we have
\[\sqrt{s} - \frac{3c\tau(y_0^2)}{2\sqrt{s}}<M_s < \sqrt{s} + \frac{3c'\tau(y_0^2)}{2\sqrt{s}}\]
for all large enough $s$. Thus we can estimate $\left\vert a_{y_0, s}(M_t)\right\vert $ by \eqref{eq:asEst}
\begin{align*}
\left\vert a_{y_0,s}(M_s)\right\vert & = \left\vert(M_s-\sqrt{s})\left(1+\frac{\sqrt{s}}{M_s}\right)- \frac{s}{M_s^3}\sum_{n=2}^\infty\frac{\tau(y_0^n)}{M_s^{n-2}}\right\vert\\
&\leq \frac{3c\tau(y_0^2)}{\sqrt{s}}+\frac{c\tau(y_0^2)}{\sqrt{s}} = \frac{4c\tau(y_0^2)}{\sqrt{s}}.
\end{align*}
By that $\mathrm{Brown}(y_0+i\sigma_s)$ is symmetric about the real axis and the \emph{holomorphic} moments of $\mathrm{Brown}(y_0+i\sigma_s)$ agree with the holomorphic moments of $y_0+i\sigma_s$ \cite{Brown1986},
\begin{equation}
\label{eq:BrownInt}
\begin{split}
\int a \,d\mathrm{Brown}(y_0+i\sigma_s)(a+ib) &=\int (a+ib) \,d\mathrm{Brown}(y_0+i\sigma_s)(a+ib) \\
&= \tau(y_0+i\sigma_s) = 0.
\end{split}
\end{equation}
It is impossible that $a_{y_0,s}(M_s) \leq 0$; otherwise, since $\Omega_{y_0,s}$ is not a subset of the imaginary axis, the integral in \eqref{eq:BrownInt} is negative, contradicting that the integral is $0$.
The estimate for $a_{y_0,s}(m_s)$ is similar.
To prove Point 3, we let $\varphi_0\in(0,\pi/2)$ such that $1/(\sin\varphi_0)<c$. By Theorem~\ref{thm:vsAsymp}, if we write $
\alpha_\varphi$ the unique real number such that
\[H_{y_0,s}(\alpha_\varphi+iv_{y_0,s}(\alpha_\varphi)) = 2\sqrt{s}\cos\varphi,\quad\left\vert\cos\varphi\right\vert\leq\cos\varphi_0,\]
then
\[\left\vert\alpha_\varphi+iv_{y_0,s}(\alpha_\varphi) - \sqrt{s}e^{i\varphi}\right\vert<\frac{1}{(\sin\varphi_0)\sqrt{s}}.\]
Thus, we have
\[\sqrt{s}-\frac{1}{(\sin\varphi_0)\sqrt{s}}<\sup\{v_{y_0,s}(\alpha_\varphi)|\left\vert\cos\varphi\right\vert<\cos\varphi_0\}<\sqrt{s}+\frac{1}{(\sin\varphi_0)\sqrt{s}}.\]
Also, for all $\alpha\geq\alpha_{\varphi_0}$ or $\alpha\leq\alpha_{\pi-\varphi_0}$, we have, by unimodality of $v_{y_0,s}$,
\begin{align*}
v_{y_0,s}(\alpha)&<\sqrt{s}\sin\varphi_0+\frac{1}{(\sin\varphi_0)\sqrt{s}}\\
&<\sqrt{s}-\frac{1}{\sqrt{s}\sin\varphi_0}\\
&<\sup\{v_{y_0,s}(\alpha_\varphi)|\left\vert\cos\varphi\right\vert<\cos\varphi_0\}
\end{align*}
for all large enough $s$. It follows that
\[\left\vert\sup_{\alpha\in\mathbb R} v_{y_0, s}(\alpha) -\sqrt{s}\right\vert<\frac{1/(\sin\varphi_0)}{\sqrt{s}}<\frac{c}{\sqrt{s}}\]
for all sufficiently large $s$. Because $b_{s,t} = 2v_{y_0,s}$,
Point 3 of this theorem is established.
\end{proof}
\section{Acknowledgments}
The author would like to thank Brian Hall, Hari Bercovici, and Ping Zhong for useful conversations.
\bibliographystyle{acm}
| {
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{"url":"https:\/\/physics.stackexchange.com\/questions\/285815\/atmospheric-pressure-in-fbd-doubt","text":"# Atmospheric pressure In FBD doubt\n\nLet us say I have a beaker floating in a water tank. Suddenly I put some water in the beaker also. The figure here shows the exact situation\n\nNow I want to draw the FBD of the beaker. I am somehow confused with the role of the atmospheric pressure in drawing the FBD. I thought of two ways of drawing this and I am just unable to figure out which one is right.\n\nHere since the beaker is floating an $F_{buoyant}$ is there in the upward direction which the net mass (beaker+water in beaker) would put its weight downward.\n\nBut while solving one question I came to know that total force that a liquid exerts on the bottom of the container is $P_0+m_{liquid}g$ ($P_0$ is the atmospheric pressure). So should the downward force on the beaker be $(M_{beaker}+M_{water in beaker})g+P_0 S$ (S being the base area of the beaker) as in the second diagram down here.\n\nHere the $P_0S$ due to atmospher on the tube gets cancelled by the $P_0S$ exerted by the water in tank on the tube. But what about the $P_0S$ from the water present in the tube? Do the FBD's apply like the one in the new image?\n\nYou get the same answer either way. In the second case, you get$$(M_{beaker}+M_{water})g+P_0S=F_B+P_0S$$In the first case, you get $$(M_{beaker}+M_{water})g=F_B$$The key to this is that the buoyant force $F_B$ omits the effect of atmospheric pressure because it always cancels.\n\n\u2022 And what about the total force that the water in the beaker exerts on the beaker. Is it $P_0S+m_{water}g$ or is it just $m_{water}g$\n\u2013\u00a0user118752\nOct 12 '16 at 14:55\n\u2022 I have edited my question to clarify my doubt.\n\u2013\u00a0user118752\nOct 12 '16 at 16:32\n\u2022 What we are talking about here is the difference between using \"gauge pressure\" and using \"absolute pressure.\" For a problem like this, it doesn't matter whether you include the gas pressure or not. Both methods give exactly the same result, as you can see from the equations I wrote. The only time it is necessary to work with absolute pressure is when some of the gas is being compressed within the system. Google \"gauge pressure\" and see the discussion. Oct 12 '16 at 17:45","date":"2021-10-26 09:03:23","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\": 0.5023056864738464, \"perplexity\": 149.9958403899975}, \"config\": {\"markdown_headings\": true, \"markdown_code\": false, \"boilerplate_config\": {\"ratio_threshold\": 0.18, \"absolute_threshold\": 10, \"end_threshold\": 5, \"enable\": true}, \"remove_buttons\": true, \"remove_image_figures\": true, \"remove_link_clusters\": true, \"table_config\": {\"min_rows\": 2, \"min_cols\": 3, \"format\": \"plain\"}, \"remove_chinese\": true, \"remove_edit_buttons\": true, \"extract_latex\": true}, \"warc_path\": \"s3:\/\/commoncrawl\/crawl-data\/CC-MAIN-2021-43\/segments\/1634323587854.13\/warc\/CC-MAIN-20211026072759-20211026102759-00671.warc.gz\"}"} | null | null |
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} | 8,892 |
{"url":"http:\/\/gmatclub.com\/forum\/m15-q11-136279.html?fl=similar","text":"M15 Q11 : Retired Discussions [Locked]\nCheck GMAT Club App Tracker for the Latest School Decision Releases http:\/\/gmatclub.com\/AppTrack\n\n It is currently 09 Dec 2016, 07:22\n\n### GMAT Club Daily Prep\n\n#### Thank you for using the timer - this advanced tool can estimate your performance and suggest more practice questions. We have subscribed you to Daily Prep Questions via email.\n\nCustomized\nfor You\n\nwe will pick new questions that match your level based on your Timer History\n\nTrack\n\nevery week, we\u2019ll send you an estimated GMAT score based on your performance\n\nPractice\nPays\n\nwe will pick new questions that match your level based on your Timer History\n\n# Events & Promotions\n\n###### Events & Promotions in June\nOpen Detailed Calendar\n\n# M15 Q11\n\nAuthor Message\nManager\nJoined: 13 May 2010\nPosts: 124\nFollowers: 0\n\nKudos [?]: 12 [0], given: 4\n\n### Show Tags\n\n24 Jul 2012, 19:50\nIs $$|x - y| \\gt |x + y|$$ ?\n\n$$x^2 - y^2 = 9$$\n$$x - y = 2$$\n\nStatement (1) ALONE is sufficient, but Statement (2) ALONE is not sufficient\nStatement (2) ALONE is sufficient, but Statement (1) ALONE is not sufficient\nBOTH statements TOGETHER are sufficient, but NEITHER statement ALONE is sufficient\nEACH statement ALONE is sufficient\nStatements (1) and (2) TOGETHER are NOT sufficient\n\nStatement (1) by itself is insufficient. S1 gives us information about $$(x - y)(x + y)$$ but does not tell how $$(x - y)$$ and $$(x + y)$$ compare to each other.\n\nStatement (2) by itself is insufficient. S2 gives no information about $$(x + y)$$ .\n\nStatements (1) and (2) combined are sufficient. From S1 and S2 it follows that $$2(x + y) = 9$$ from where $$(x + y) = 4.5$$ . Now we can state that $$|x - y| = 2 \\lt |x + y| = 4.5$$ .\n\nMBA Section Director\nJoined: 19 Mar 2012\nPosts: 3545\nLocation: India\nGPA: 3.8\nWE: Marketing (Energy and Utilities)\nFollowers: 1477\n\nKudos [?]: 11495 [0], given: 1858\n\n### Show Tags\n\n24 Jul 2012, 20:27\nteal wrote:\nIs $$|x - y| \\gt |x + y|$$ ?\n\n$$x^2 - y^2 = 9$$\n$$x - y = 2$$\n\nStatement (1) ALONE is sufficient, but Statement (2) ALONE is not sufficient\nStatement (2) ALONE is sufficient, but Statement (1) ALONE is not sufficient\nBOTH statements TOGETHER are sufficient, but NEITHER statement ALONE is sufficient\nEACH statement ALONE is sufficient\nStatements (1) and (2) TOGETHER are NOT sufficient\n\nStatement (1) by itself is insufficient. S1 gives us information about $$(x - y)(x + y)$$ but does not tell how $$(x - y)$$ and $$(x + y)$$ compare to each other.\n\nStatement (2) by itself is insufficient. S2 gives no information about $$(x + y)$$ .\n\nStatements (1) and (2) combined are sufficient. From S1 and S2 it follows that $$2(x + y) = 9$$ from where $$(x + y) = 4.5$$ . Now we can state that $$|x - y| = 2 \\lt |x + y| = 4.5$$ .\n\nstatement 1\n$$x^2 - y^2 = 9$$\nor, (x+y)(x-y)=9\nClearly not sufficient (different combinations of x+y and x-y are possible)\n\nstatement 2\nx-y=2\nnot sufficient with no info on (x+y)\n\ncombining both together\nx+y=9\/2\nx-y=2\n\nso |x-y|<|x+y|\nSufficient\nHence C\n_________________\nManager\nJoined: 13 May 2010\nPosts: 124\nFollowers: 0\n\nKudos [?]: 12 [0], given: 4\n\n### Show Tags\n\n24 Jul 2012, 23:02\ncan you please suggest some numbers to prove statement 2 insuff??\nMBA Section Director\nJoined: 19 Mar 2012\nPosts: 3545\nLocation: India\nGPA: 3.8\nWE: Marketing (Energy and Utilities)\nFollowers: 1477\n\nKudos [?]: 11495 [0], given: 1858\n\n### Show Tags\n\n24 Jul 2012, 23:08\nconsider x=2 and y=4\nin this case |x+y| i.e 6>|x-y| i.e 2\nAgain consider x=2 and y=-4\nin this case |x+y| ie 2 < |x-y| i.e 6\nhope this helps.\n_________________\nMBA Section Director\nJoined: 19 Mar 2012\nPosts: 3545\nLocation: India\nGPA: 3.8\nWE: Marketing (Energy and Utilities)\nFollowers: 1477\n\nKudos [?]: 11495 [0], given: 1858\n\n### Show Tags\n\n24 Jul 2012, 23:10\nin general if you want to plug in numbers in questions like these, you need to consider positive, negative and fractional values of all the variables to eleminate\/consider one option\nCheers\n_________________\nIntern\nJoined: 24 Feb 2010\nPosts: 11\nFollowers: 0\n\nKudos [?]: 9 [0], given: 0\n\n### Show Tags\n\n25 Jul 2012, 00:59\nFor statement 2, use these values to prove that the statement alone is insufficient.\n\nx=2 and y=0\nx=3 and y=1\nx=-1 and y=-3\n\nAlways make a point to check for the inequality with 0 as a value.\n\nKind Regards,\nRavender\nMBA Section Director\nJoined: 19 Mar 2012\nPosts: 3545\nLocation: India\nGPA: 3.8\nWE: Marketing (Energy and Utilities)\nFollowers: 1477\n\nKudos [?]: 11495 [0], given: 1858\n\n### Show Tags\n\n25 Jul 2012, 01:07\npalsays wrote:\nFor statement 2, use these values to prove that the statement alone is insufficient.\n\nx=2 and y=0\nx=3 and y=1\nx=-1 and y=-3\n\nAlways make a point to check for the inequality with 0 as a value.\n\nKind Regards,\nRavender\n\n@palsays\nI dont think your values provide insufficiency\nfor x=2, y=0 |x+y|>|x-y|\nfor x=3, y=1 |x+y|>|x-y|\nfor x=-1, y=-3 |x+y|>|x-y|\n\nYou have to make one variable negative and one variable postive to show that |x+y|<|x-y|\nCheers\n_________________\nDirector\nJoined: 22 Mar 2011\nPosts: 612\nWE: Science (Education)\nFollowers: 98\n\nKudos [?]: 868 [0], given: 43\n\n### Show Tags\n\n25 Jul 2012, 01:10\nteal wrote:\nIs $$|x - y| \\gt |x + y|$$ ?\n\n$$x^2 - y^2 = 9$$\n$$x - y = 2$$\n\nStatement (1) ALONE is sufficient, but Statement (2) ALONE is not sufficient\nStatement (2) ALONE is sufficient, but Statement (1) ALONE is not sufficient\nBOTH statements TOGETHER are sufficient, but NEITHER statement ALONE is sufficient\nEACH statement ALONE is sufficient\nStatements (1) and (2) TOGETHER are NOT sufficient\n\nStatement (1) by itself is insufficient. S1 gives us information about $$(x - y)(x + y)$$ but does not tell how $$(x - y)$$ and $$(x + y)$$ compare to each other.\n\nStatement (2) by itself is insufficient. S2 gives no information about $$(x + y)$$ .\n\nStatements (1) and (2) combined are sufficient. From S1 and S2 it follows that $$2(x + y) = 9$$ from where $$(x + y) = 4.5$$ . Now we can state that $$|x - y| = 2 \\lt |x + y| = 4.5$$ .\n\nIn fact, the given inequality can be rewritten as $$(x-y)^2>(x+y)^2$$ - we can square both sides, as they are both positive. Rearranging the terms, the question becomes $$xy<0$$ (is the product xy negative)?\n\nThen, it is much easier to understand that neither (1), nor (2) alone is sufficient.\nTaking both statements, one can explicitly find the values of x and y (although not necessary), and check whether their product is negative.\nThat's why the correct answer should be C.\n_________________\n\nPhD in Applied Mathematics\nLove GMAT Quant questions and running.\n\nMBA Section Director\nJoined: 19 Mar 2012\nPosts: 3545\nLocation: India\nGPA: 3.8\nWE: Marketing (Energy and Utilities)\nFollowers: 1477\n\nKudos [?]: 11495 [0], given: 1858\n\n### Show Tags\n\n25 Jul 2012, 01:35\nyeah true that. Precisely my point.\n_________________\nRe: M15 Q11 \u00a0 [#permalink] 25 Jul 2012, 01:35\nSimilar topics Replies Last post\nSimilar\nTopics:\nPS: m 15 #31 3 20 Jun 2009, 11:05\nDS:m 15 #29 1 20 Jun 2009, 10:59\n4 m15,#10 18 28 Nov 2008, 01:10\n12 M15#18 22 19 Nov 2008, 18:24\n6 M15 #3 26 22 Oct 2008, 12:23\nDisplay posts from previous: Sort by\n\n# M15 Q11\n\nModerator: Bunuel\n\n Powered by phpBB \u00a9 phpBB Group and phpBB SEO Kindly note that the GMAT\u00ae test is a registered trademark of the Graduate Management Admission Council\u00ae, and this site has neither been reviewed nor endorsed by GMAC\u00ae.","date":"2016-12-09 15:22:54","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\": 0.32397979497909546, \"perplexity\": 6979.642120620309}, \"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-2016-50\/segments\/1480698542712.49\/warc\/CC-MAIN-20161202170902-00124-ip-10-31-129-80.ec2.internal.warc.gz\"}"} | null | null |
package com.baidu.oped.apm.common;
import static com.baidu.oped.apm.common.HistogramSchema.*;
import static com.baidu.oped.apm.common.ServiceTypeConstants.*;
import java.util.ArrayList;
import java.util.Collections;
import java.util.HashMap;
import java.util.List;
import java.util.Map;
import com.baidu.oped.apm.common.util.RpcCodeRange;
import com.baidu.oped.apm.common.util.apache.IntHashMap;
/**
* @author emeroad
* @author netspider
*/
public enum ServiceType {
// Undefined Service Code
UNDEFINED((short) -1, "UNDEFINED", TERMINAL, !RECORD_STATISTICS, !INCLUDE_DESTINATION, NORMAL_SCHEMA),
// Callee node that agent hasn't been installed
UNKNOWN((short) 1, "UNKNOWN", !TERMINAL, RECORD_STATISTICS, !INCLUDE_DESTINATION, NORMAL_SCHEMA),
// User
USER((short) 2, "USER", !TERMINAL, RECORD_STATISTICS, !INCLUDE_DESTINATION, NORMAL_SCHEMA),
// Group of UNKNOWN, used only for UI
UNKNOWN_GROUP((short) 3, "UNKNOWN_GROUP", !TERMINAL, RECORD_STATISTICS, !INCLUDE_DESTINATION, NORMAL_SCHEMA),
// Group of TEST, used for running tests
TEST((short) 5, "TEST", !TERMINAL, !RECORD_STATISTICS, !INCLUDE_DESTINATION, NORMAL_SCHEMA),
// Java applications, WAS
STAND_ALONE((short) 1000, "STAND_ALONE", !TERMINAL, RECORD_STATISTICS, !INCLUDE_DESTINATION, NORMAL_SCHEMA),
TEST_STAND_ALONE((short) 1005, "TEST_STAND_ALONE", !TERMINAL, RECORD_STATISTICS, !INCLUDE_DESTINATION, NORMAL_SCHEMA),
TOMCAT((short) 1010, "TOMCAT", !TERMINAL, RECORD_STATISTICS, !INCLUDE_DESTINATION, NORMAL_SCHEMA),
BLOC((short) 1020, "BLOC", !TERMINAL, RECORD_STATISTICS, !INCLUDE_DESTINATION, NORMAL_SCHEMA),
BLOC_INTERNAL_METHOD((short) 1021, "INTERNAL_METHOD", !TERMINAL, !RECORD_STATISTICS, !INCLUDE_DESTINATION, NORMAL_SCHEMA),
COLLECTOR((short) 1100, "COLLECTOR", !TERMINAL, !RECORD_STATISTICS, !INCLUDE_DESTINATION, NORMAL_SCHEMA),
/**
* Database
* shown only as xxx_EXECUTE_QUERY at the statistics info section in the server map
*/
// DB 2000
UNKNOWN_DB((short) 2050, "UNKNOWN_DB", TERMINAL, !RECORD_STATISTICS, INCLUDE_DESTINATION, NORMAL_SCHEMA),
UNKNOWN_DB_EXECUTE_QUERY((short) 2051, "UNKNOWN_DB", TERMINAL, RECORD_STATISTICS, INCLUDE_DESTINATION, NORMAL_SCHEMA),
MYSQL((short) 2100, "MYSQL", TERMINAL, !RECORD_STATISTICS, INCLUDE_DESTINATION, NORMAL_SCHEMA),
MYSQL_EXECUTE_QUERY((short) 2101, "MYSQL", TERMINAL, RECORD_STATISTICS, INCLUDE_DESTINATION, NORMAL_SCHEMA),
MSSQL((short) 2200, "MSSQLSERVER", TERMINAL, !RECORD_STATISTICS, INCLUDE_DESTINATION, NORMAL_SCHEMA),
MSSQL_EXECUTE_QUERY((short) 2201, "MSSQLSERVER", TERMINAL, RECORD_STATISTICS, INCLUDE_DESTINATION, NORMAL_SCHEMA),
ORACLE((short) 2300, "ORACLE", TERMINAL, !RECORD_STATISTICS, INCLUDE_DESTINATION, NORMAL_SCHEMA),
ORACLE_EXECUTE_QUERY((short) 2301, "ORACLE", TERMINAL, RECORD_STATISTICS, INCLUDE_DESTINATION, NORMAL_SCHEMA),
CUBRID((short) 2400, "CUBRID", TERMINAL, !RECORD_STATISTICS, INCLUDE_DESTINATION, NORMAL_SCHEMA),
CUBRID_EXECUTE_QUERY((short) 2401, "CUBRID", TERMINAL, RECORD_STATISTICS, true, NORMAL_SCHEMA),
// Internal method
// FIXME it's not clear to put internal method here. but do that for now.
INTERNAL_METHOD((short) 5000, "INTERNAL_METHOD", !TERMINAL, !RECORD_STATISTICS, !INCLUDE_DESTINATION, NORMAL_SCHEMA),
// Spring framework
SPRING((short) 5050, "SPRING", !TERMINAL, !RECORD_STATISTICS, !INCLUDE_DESTINATION, NORMAL_SCHEMA),
SPRING_MVC((short) 5051, "SPRING", !TERMINAL, !RECORD_STATISTICS, !INCLUDE_DESTINATION, NORMAL_SCHEMA),
// FIXME need to define how to handle spring related codes
SPRING_ORM_IBATIS((short) 5061, "SPRING", !TERMINAL, !RECORD_STATISTICS, !INCLUDE_DESTINATION, NORMAL_SCHEMA),
SPRING_BEAN((short) 5071, "SPRING_BEAN", !TERMINAL, !RECORD_STATISTICS, !INCLUDE_DESTINATION, NORMAL_SCHEMA),
// xBatis
IBATIS((short) 5500, "IBATIS", !TERMINAL, !RECORD_STATISTICS, !INCLUDE_DESTINATION, NORMAL_SCHEMA),
MYBATIS((short) 5510, "MYBATIS", !TERMINAL, !RECORD_STATISTICS, !INCLUDE_DESTINATION, NORMAL_SCHEMA),
// DBCP
DBCP((short) 6050, "DBCP", !TERMINAL, !RECORD_STATISTICS, !INCLUDE_DESTINATION, NORMAL_SCHEMA),
// Memory cache
MEMCACHED((short) 8050, "MEMCACHED", TERMINAL, RECORD_STATISTICS, !INCLUDE_DESTINATION, FAST_SCHEMA),
MEMCACHED_FUTURE_GET((short) 8051, "MEMCACHED", TERMINAL, !RECORD_STATISTICS, !INCLUDE_DESTINATION, FAST_SCHEMA),
ARCUS((short) 8100, "ARCUS", TERMINAL, RECORD_STATISTICS, INCLUDE_DESTINATION, FAST_SCHEMA),
ARCUS_FUTURE_GET((short) 8101, "ARCUS", TERMINAL, !RECORD_STATISTICS, INCLUDE_DESTINATION, FAST_SCHEMA),
ARCUS_EHCACHE_FUTURE_GET((short) 8102, "ARCUS-EHCACHE", TERMINAL, !RECORD_STATISTICS, INCLUDE_DESTINATION, FAST_SCHEMA),
// Redis & nBase-ARC
REDIS((short) 8200, "REDIS", TERMINAL, RECORD_STATISTICS, !INCLUDE_DESTINATION, FAST_SCHEMA),
NBASE_ARC((short) 8250, "NBASE_ARC", TERMINAL, RECORD_STATISTICS, INCLUDE_DESTINATION, FAST_SCHEMA),
// Connector, Client
HTTP_CLIENT((short) 9050, "HTTP_CLIENT", !TERMINAL, RECORD_STATISTICS, !INCLUDE_DESTINATION, NORMAL_SCHEMA),
HTTP_CLIENT_INTERNAL((short) 9051, "HTTP_CLIENT", !TERMINAL, !RECORD_STATISTICS, !INCLUDE_DESTINATION, NORMAL_SCHEMA),
JDK_HTTPURLCONNECTOR((short) 9055, "JDK_HTTPCONNECTOR", !TERMINAL, RECORD_STATISTICS, !INCLUDE_DESTINATION, NORMAL_SCHEMA),
NPC_CLIENT((short) 9060, "NPC_CLIENT", !TERMINAL, RECORD_STATISTICS, !INCLUDE_DESTINATION, NORMAL_SCHEMA),
NPC_CLIENT_INTERNAL((short) 9061, "NPC_CLIENT", !TERMINAL, !RECORD_STATISTICS, !INCLUDE_DESTINATION, NORMAL_SCHEMA),
NIMM_CLIENT((short) 9070, "NIMM_CLIENT", !TERMINAL, RECORD_STATISTICS, !INCLUDE_DESTINATION, NORMAL_SCHEMA);
public static final short WAS_START_INDEX = 1000;
public static final short WAS_END_INDEX = 2000;
private final short code;
private final String desc;
private final boolean terminal;
// FIXME record statistics of only rpc call currently. so is it all right to chane into isRecordRpc()
private final boolean recordStatistics;
// whether or not print out api including destinationId
private final boolean includeDestinationId;
private final HistogramSchema histogramSchema;
ServiceType(short code, String desc, boolean terminal, boolean recordStatistics, boolean includeDestinationId, HistogramSchema histogramSchema) {
this.code = code;
this.desc = desc;
this.terminal = terminal;
this.recordStatistics = recordStatistics;
this.includeDestinationId = includeDestinationId;
this.histogramSchema = histogramSchema;
}
// FIXME it may be not good to find serviceType by using this api
public static List<ServiceType> findDesc(String desc) {
if (desc == null) {
throw new NullPointerException("desc must not be null");
}
return STATISTICS_LOOKUP_TABLE.get(desc);
}
public boolean isInternalMethod() {
return this == INTERNAL_METHOD;
}
public boolean isRpcClient() {
return RpcCodeRange.isRpcRange(code);
}
public boolean isIndexable() {
return !terminal && !isRpcClient() && code > 1000;
}
// FIXME record statistics of only rpc call currently. so is it all right to chane into isRecordRpc()
public boolean isRecordStatistics() {
return recordStatistics;
}
public boolean isUnknown() {
return this == ServiceType.UNKNOWN; // || this == ServiceType.UNKNOWN_CLOUD;
}
// return true when the service type is USER or can not be identified
public boolean isUser() {
return this == ServiceType.USER;
}
public short getCode() {
return code;
}
public String getDesc() {
return desc;
}
public boolean isTerminal() {
return terminal;
}
public boolean isIncludeDestinationId() {
return includeDestinationId;
}
public HistogramSchema getHistogramSchema() {
return histogramSchema;
}
public boolean isWas() {
return isWas(this.code);
}
public static boolean isWas(final short code) {
return code >= WAS_START_INDEX && code < WAS_END_INDEX;
}
@Override
public String toString() {
return desc;
}
public static ServiceType findServiceType(short code) {
ServiceType serviceType = CODE_LOOKUP_TABLE.get(code);
if (serviceType == null) {
return UNDEFINED;
//return UNKNOWN;
}
return serviceType;
}
private static final IntHashMap<ServiceType> CODE_LOOKUP_TABLE = new IntHashMap<ServiceType>(256);
private static final Map<String, List<ServiceType>> STATISTICS_LOOKUP_TABLE = new HashMap<String, List<ServiceType>>(64);
static {
initializeLookupTable();
initializeStatisticsLookupTable();
}
private static void initializeStatisticsLookupTable() {
ServiceType[] values = ServiceType.values();
final Map<String, List<ServiceType>> temp = new HashMap<String, List<ServiceType>>();
for (ServiceType serviceType : values) {
if(serviceType.isRecordStatistics()) {
List<ServiceType> serviceTypeList = STATISTICS_LOOKUP_TABLE.get(serviceType.getDesc());
if (serviceTypeList == null) {
serviceTypeList = new ArrayList<ServiceType>();
temp.put(serviceType.getDesc(), serviceTypeList);
}
serviceTypeList.add(serviceType);
}
}
// Don't modify
for (Map.Entry<String, List<ServiceType>> entry : temp.entrySet()) {
List<ServiceType> serviceTypes = Collections.unmodifiableList(entry.getValue());
STATISTICS_LOOKUP_TABLE.put(entry.getKey(), serviceTypes);
}
}
private static void initializeLookupTable() {
ServiceType[] values = ServiceType.values();
for (ServiceType serviceType : values) {
ServiceType check = CODE_LOOKUP_TABLE.put(serviceType.code, serviceType);
if (check != null) {
throw new IllegalStateException("duplicated code found. code:" + serviceType.code);
}
}
}
}
| {
"redpajama_set_name": "RedPajamaGithub"
} | 6,888 |
name 'scala-jenkins-infra'
maintainer 'Lightbend, Inc.'
maintainer_email 'adriaan@lightbend.com'
license 'All rights reserved'
description 'Installs/Configures the Scala Jenkins infrastructure'
long_description IO.read(File.join(File.dirname(__FILE__), 'README.md'))
version '0.3.0'
# for chef_vault_item, which allows loading from plain databags when developing with vagrant
depends 'chef-vault'
depends 'magic_shell'
depends 'chef-client'
depends 'cron'
depends 'aws'
depends 'ebs'
depends 'windows'
depends 'java'
depends 'jenkins'
depends 'artifactory'
depends 'git'
depends 'git_user'
depends 'sbt-extras'
depends 'runit', '~> 1.5'
depends 'nodejs'
| {
"redpajama_set_name": "RedPajamaGithub"
} | 252 |
Q: Informix + Cobol in AIX I'm having troubles trying to compile a simple Cobol with Informix, it compiles, but it doesnt link and generate the executable
This is the message that I get:
ld: 0711-317 ERROR: Undefined symbol: .GETENV
ld: 0711-317 ERROR: Undefined symbol: .ECO_DB
ld: 0711-317 ERROR: Undefined symbol: .ECO_SQC
ld: 0711-317 ERROR: Undefined symbol: .ECO_XIM
ld: 0711-317 ERROR: Undefined symbol: .ECO_STM
ld: 0711-317 ERROR: Undefined symbol: .ECO_OC
ld: 0711-317 ERROR: Undefined symbol: .ECO_BOC
ld: 0711-317 ERROR: Undefined symbol: .ECO_SLC
ld: 0711-317 ERROR: Undefined symbol: .ECO_CMT
ld: 0711-317 ERROR: Undefined symbol: .ECO_CDB
ld: 0711-317 ERROR: Undefined symbol: .ECO_DCON
ld: 0711-317 ERROR: Undefined symbol: .ECO_CONN
ld: 0711-345 Use the -bloadmap or -bnoquiet option to obtain more information.
make: The error code from the last command is 8.
I though that is a was a problem with the libpath, but this is my .profile.
export PATH=$PATH:/usr/bin:/etc:/usr/sbin:/usr/ucb:$HOME/bin:/usr/bin/X11:/sbin:/usr/informix/lib:.
export INFORMIXDIR=/usr/informix
export PATH=$PATH:$INFORMIXDIR/bin
export INFORMIXCOB=cob2
export INFORMIXCOBDIR=/usr/lpp/cobol
export INFORMIXCOBTYPE=ibm
export INFORMIXCOBDIR=/usr/lpp/cobol
export COBDIR=/usr/lpp/cobol
export INFORMIXCOB=cob2
export LIBPATH=$LIBPATH:/usr/informix/lib:/usr/informix/lib/esql:/usr/lpp/cobol/lib
export LD_LIBRARY_PATH=$LD_LIBRARY_PATH:/usr/lpp/cobol/lib:/usr/informix/lib:/usr/informix/lib/esql
I have tried several thing but with no results..
this is my make, but I also try cob2 objet.o with the same result.
This is the makefile
this is my make, but I also try cob2 objet.o with the same result.
################################################################################
# Licensed Materials - Property of IBM
#
# ? Copyright IBM Corp. 2013
#
# The source code is provided "AS IS" without warranty of any kind. It is
# provided solely as a sample. It may be used, executed, copied, and modified
# for internal use only by Licensee without royalty.
################################################################################
################################################################################
#
# This makefile template can be used to build your AIX COBOL project. Comments
# are provided to help you understand how it works, and to help you edit it for
# your needs. There are 2 main sections:
# 1. Variables Section - defines targets to build, commands to use, and
# command options
# 2. Rules Section - defines how to build the targets. For each target, it
# provides the commands to run to build them.
#
################################################################################
# Add any other file extensions you will build or build from here
.SUFFIXES:
.SUFFIXES: .cbl .eco
################################################################################
# VARIABLES SECTION
#
# Define variables to control various targets, commands, comand options that are
# used in various places in the makefile
################################################################################
## BUILD TARGETS -- Define all targets to build. These are the final targets of
## the build, usually an exectable (linked objects), executables (batch programs),
## or a library (linked objects intended to be linked to by consuming projects).
## You may explicitly list your final target(s) by uncommenting the following
## lines, and commenting the automatic setting of TARGET below
#TARGETS = PROGRAM1 \
# PROGRAM2.exe \
# PROGRAM3
## If your project only contains batch and/or CICS programs and is simple
## enough, you can have the makefile determine the final targets automatically
## by uncommenting this line and changing "*.cbl *.sqb" to the source
## file extensions you use, as well as adding/removing the
## corresponding ' -e sed "s/\.cbl//" ' arguments to 'sed'
TARGETS = `ls -1 *.cbl *.eco 2> /dev/null | sed -e "s/\.cbl//" -e "s/\.eco//"`
## COMPILE COMMAND
## Single-threaded compile
COB = cob2
## COMPILE OPTIONS
## You will want to add your include paths here too (where copybooks are, if not
## in same directory as source), eg. -I/include/path1 -I/include/path2
COB_OPTS = -g
ESQLCOBOL = esqlcobol
## Link/bind options
LINK_OPTS=
## SQL COMPILE SETTINGS -- Uncomment and set the following variables if you use
## embedded SQL in your source
## DB2_DIR: DB2 install location
## DB_NAME: Database name
## DB_USER: DB2 userid
## DB_PASS: DB2 user password
#DB2_DIR = /opt/IBM/db2/V9.7
#DB_NAME =
#DB_USER =
#DB_PASS =
## Compile options for embedded SQL
#DB2_COB_OPTS = -qLIB -q"SQL('database $(DB_NAME) user $(DB_USER) using $(DB_PASS)')"
## DB2 Precompiler
DB2 = db2
DB2_PREP = $(DB2) prep
DB2_PREP_OPTS = target ibmcob
################################################################################
# RULES SECTION
#
# This section controls how targets are built from dependencies.
# The rule will get run if the dependencies change (become newer than the target)
# There are two types of rules:
# 1. Explicit rule. Builds one explicit target from an explicit list of
# dependencies
# <target>: <dependencies>
# <tab> <rules>
# Example:
# PROGRAM.exe: PROGRAM.cbl MYCOPY.cpy
# cob2 PROGRAM.cbl -o PROGRAM.exe -I./
# 2. Implicit rule. Builds targets with a specified file extension from a
# source file with the same basename and with a specified file extension. If
# only the target is specified then it builds a target with no file extension.
# .<target_extension>.<source_extension>:
# <tab> <rules>
# Example 1: For any .o targets that need to be built, if a .cbl file exists
# with the same name
# .cbl.o:
# cob2 $< -o $@
# Example 2: For any executable targets that need to be built with no file
# extension, if a .cbl file exists with the same name
# .cbl:
# cob2 $< -o $@
################################################################################
# 'all' is the default makefile target that is built for 'make' with no arguments
all: $(TARGETS)
## This rule builds batch COBOL source files to an exectuable program with the
## same basename as the source file with no file extension
.cbl:
$(COB) $< -o $@ $(COB_OPTS) $(DB2_COB_OPTS) $(LINK_OPTS)
## prar ficheros informix
.eco:
$(ESQLCOBOL) $< -o $@
## .SQB (EMBEDDED SQL) RULES FOR COMPILER VERSION 4.1.1.10 OR LATER
## Comment these .SQB rules if using compiler version 4.1.1.9 or earlier, and
## uncomment the .SQB rules following for compiler version 4.1.1.9 or earlier.
## This rule builds embedded SQL source in .sqb files.
.sqb:
$(COB) $< -o $@ $(COB_OPTS) $(DB2_COB_OPTS) $(LINK_OPTS)
## .SQB (EMBEDDED SQL) RULES FOR COMPILER VERSION 4.1.1.9 OR EARLIER
## Comment above .SQB rules if using compiler version 4.1.1.9 or earlier, and
## uncomment this rule.
## This rule preprocesses embedded SQL files, to produce .cbl files that will
## be compiled as usual
#.sqb.cbl:
# $(DB2) connect to $(DB_NAME) user $(DB_USER) using $(DB_PASS)
# $(DB2_PREP) $< $(DB2_PREP_OPTS)
## The clean rule gets run when you run 'make clean', which is the default
## Clean Build command for AIX COBOL projects in RD AIX & Linux. It should
## clean up all files created by the build. It is useful if you want to build
## from scratch (eg. if you change a copybook included by many source files).
clean:
rm -f $(TARGETS) core *.lst *.adt *.adt2 cmpout.xml
and this is the cobol source after preprocessing
0001 identification division.
0002 program-id. 'pruifx'.
0003 author. lmfa.
0004 date-written. Junio 2014.
0005
0006 ******************************************************************
0007 * prueba COBOL sql AIX (INFORMIX) *
0008 ******************************************************************
0009
0010 environment division.
0011 configuration section.
0012 special-names.
0013 decimal-point is comma.
0014
0015 input-output section.
0020 data division.
0025 working-storage section.
0026
0053 * exec sql include sqlca end-exec.
*************************************************************
* Title: sqlca.ibm
* Sccsid: @(#)sqlca.ibm 9.1 10/06/96 15:13:54
* Description:
* SQLCA include file for IBM COBOL Set for AIX 1.1
*************************************************************
77 SQLNOTFOUND PIC S9(10) VALUE 100.
01 SQLCA.
05 SQLCODE PIC S9(9) COMPUTATIONAL-5.
05 SQLERRM.
49 SQLERRML PIC S9(4) COMPUTATIONAL-5.
49 SQLERRMC PIC X(70).
05 SQLERRP PIC X(8).
05 SQLERRD OCCURS 6 TIMES
PIC S9(9) COMPUTATIONAL-5.
05 SQLWARN.
10 SQLWARN0 PIC X(1).
10 SQLWARN1 PIC X(1).
10 SQLWARN2 PIC X(1).
10 SQLWARN3 PIC X(1).
10 SQLWARN4 PIC X(1).
10 SQLWARN5 PIC X(1).
10 SQLWARN6 PIC X(1).
10 SQLWARN7 PIC X(1).
0053 * exec sql include tabdes end-exec.
*EXEC SQL BEGIN DECLARE SECTION END-EXEC.
01 TABDES.
02 DESCOD PICTURE X(3).
02 DESCLA PICTURE X(15).
02 DESDES PICTURE X(62).
*EXEC SQL END DECLARE SECTION END-EXEC.
0054
0055 ****** Definicion de variables para tablas *******************
0238 * exec sql begin declare section end-exec.
01 base-datos PICTURE x(32).
01 cnt-total PICTURE s9(4) value 200.
01 cnt-totals redefines cnt-total PICTURE s9(4).
01 cnt-pendientes PICTURE 9(4) value 0.
01 sql-area PICTURE x(6) value '119953'.
01 sql-fecha PICTURE x(10) value '23.05,201-'.
01 sql-gm PICTURE xx value 'GM'.
01 sql-rmp PICTURE xxx value 'RMP'.
01 sql-m1 PICTURE x(50) value '*'.
01 sql-m2 PICTURE xx value '*'.
01 sql-m3 PICTURE xxx value '*'.
0254 * exec sql end declare section end-exec.
0056
01 xor1 pic x(40) value '1234567890'.
01 xor2 pic x(40) value '0987654321'.
01 p1 pointer.
01 aa pic 9 value zeros.
01 cc pic 99 value 11.
01 campo pic x(64000).
01 P pointer.
01 ix-base pic x(13) value 'INFORMIX_BASE'.
01 ix-server pic x(14) value 'INFORMIXSERVER'.
*Beginning of ESQL/COBOL temporary variables.
77 SQLCODETMP PIC S9(9) COMP-5.
77 SQLWARNTMP PIC S9(9) COMP-5.
77 SQLSTATE PIC X(5).
77 SQLTYPE PIC S9(9) COMP-5.
77 SQLLEN PIC S9(9) COMP-5.
77 SQLINAME PIC S9(9) COMP-5.
77 SQLITYPE PIC S9(9) COMP-5.
77 SQLILEN PIC S9(9) COMP-5.
77 SQLTEXTLEN PIC S9(9) COMP-5.
77 SQL2TEXTLEN PIC S9(9) COMP-5.
77 SQL3TEXTLEN PIC S9(9) COMP-5.
77 SQLINCNT PIC S9(9) COMP-5.
77 SQLUSEFLAG PIC S9(9) COMP-5.
77 SQLOUTCNT PIC S9(9) COMP-5.
77 SQLDIRECTION PIC S9(9) COMP-5.
77 SQLVALUE PIC S9(9) COMP-5.
77 SQLSCRFLAG PIC S9(9) COMP-5.
77 SQLOBJLEN PIC S9(9) COMP-5.
77 SQLBINDTYPE PIC X(1).
77 SQLDUMLEN PIC S9(9) COMP-5 VALUE 1.
77 SQLOBJECT PIC X(18).
77 SQLTEXT PIC X(18).
77 SQL2TEXT PIC X(18).
77 SQL3TEXT PIC X(18).
77 SQLDUMMY PIC X(1) VALUE ' '.
77 ECO-DSH PIC X(7) VALUE 'eco_dsh'.
77 ECO-USH PIC X(7) VALUE 'eco_ush'.
77 ECO-GST PIC X(7) VALUE 'eco_gst'.
77 ECO-SQC PIC X(7) VALUE 'eco_sqc'.
77 ECO-LYR PIC X(7) VALUE 'eco_lyr'.
77 ECO-MSG PIC X(7) VALUE 'eco_msg'.
77 ECO-SQU PIC X(7) VALUE 'eco_squ'.
77 ECO-IQU PIC X(7) VALUE 'eco_iqu'.
77 ECO-SIG PIC X(7) VALUE 'eco_sig'.
77 ECO-SQS PIC X(7) VALUE 'eco_sqs'.
77 ECO-SQE PIC X(7) VALUE 'eco_sqe'.
77 ECO-SQB PIC X(7) VALUE 'eco_sqb'.
77 ECO-SQBCB PIC X(9) VALUE 'eco_sqbcb'.
77 ECO-SQD PIC X(7) VALUE 'eco_sqd'.
77 ECO-DTS PIC X(7) VALUE 'eco_dts'.
77 ECO-DAI PIC X(7) VALUE 'eco_dai'.
77 ECO-DSI PIC X(7) VALUE 'eco_dsi'.
77 ECO-DTC PIC X(7) VALUE 'eco_dtc'.
77 ECO-DTX PIC X(7) VALUE 'eco_dtx'.
77 ECO-INX PIC X(7) VALUE 'eco_inx'.
77 ECO-FIN PIC X(7) VALUE 'eco_fin'.
77 ECO-FFL PIC X(7) VALUE 'eco_ffl'.
77 ECO-DTCVASC PIC X(11) VALUE 'eco_dtcvasc'.
77 ECO-DTTOASC PIC X(11) VALUE 'eco_dttoasc'.
77 ECO-INCVASC PIC X(11) VALUE 'eco_incvasc'.
77 ECO-INTOASC PIC X(11) VALUE 'eco_intoasc'.
77 ECO-IMN PIC X(7) VALUE 'eco_imn'.
77 ECO-IDN PIC X(7) VALUE 'eco_idn'.
77 ECO-IDI PIC X(7) VALUE 'eco_idi'.
77 ECO-DAT PIC X(7) VALUE 'eco_dat'.
77 ECO-DAY PIC X(7) VALUE 'eco_day'.
77 ECO-DEF PIC X(7) VALUE 'eco_def'.
77 ECO-FMT PIC X(7) VALUE 'eco_fmt'.
77 ECO-JUL PIC X(7) VALUE 'eco_jul'.
77 ECO-MDY PIC X(7) VALUE 'eco_mdy'.
77 ECO-STR PIC X(7) VALUE 'eco_str'.
77 ECO-TDY PIC X(7) VALUE 'eco_tdy'.
*End of ESQL/COBOL temporary variables.
77 SQLTEXT01 PIC X(32) VALUE 'set isolation to committed re
- 'ad '.
77 SQLTEXTLEN01 PIC S9(9) COMP-5 VALUE 32.
77 SQLTEXT02 PIC X(44) VALUE 'select * from tabdes where de
- 'scod = ''UBC'' '.
77 SQLTEXTLEN02 PIC S9(9) COMP-5 VALUE 44.
77 SQLTEXT03 PIC X(44) VALUE 'select * from tabdes where de
- 'scod = ''UBC'' '.
77 SQLTEXTLEN03 PIC S9(9) COMP-5 VALUE 44.
local-storage section.
linkage section.
01 xor-campo pic x(64000).
0111
0231==*=================================================================
0232 procedure division.
0234
0235 Principal section.
Set P to address of ix-base
Call "getenv" using by value P returning base-datos
0497 * display 'INFORMIX_BASE' upon MY-ENV-NAME
0498 * accept base-datos from MY-ENV-VALUE
.
empezar.
move 'xxxxx@xxxxxx' to base-datos
0499 * exec sql database :base-datos end-exec
MOVE 0 TO SQLVALUE
MOVE 32 TO SQLOBJLEN
CALL 'eco_db' USING base-datos, SQLVALUE, SQLOBJLEN
CALL 'eco_sqc' USING SQLCA, SQLCODETMP, SQLWARNTMP, SQLSTATE
0500
0501 if not (sqlcode = zeros or = -377)
0502 display 'OpBD'
0503 go to sql-error
0504 end-if
0505
move 'set lock mode to wait 30' to sql-m1
* exec sql
* execute immediate :sql-m1
* end-exec
MOVE 50 TO SQLTEXTLEN
CALL 'eco_xim' USING sql-m1, SQLTEXTLEN
CALL 'eco_sqc' USING SQLCA, SQLCODETMP, SQLWARNTMP, SQLSTATE
display 'sqlcode->' sqlcode
0509
0511 *-----------------------------------------------------------------
0513 * exec sql
0514 * set isolation to committed read
0515 * end-exec
MOVE 0 TO SQLINCNT
CALL 'eco_stm' USING SQLTEXT01, SQLINCNT, SQLTEXTLEN01
CALL 'eco_sqc' USING SQLCA, SQLCODETMP, SQLWARNTMP, SQLSTATE
.
move spaces to tabdes
* exec sql
* select *
* into :tabdes
* from tabdes
* where descod = 'UBC'
* end-exec
MOVE 0 TO SQLINCNT
MOVE 3 TO SQLOUTCNT
CALL 'eco_oc' USING SQLOUTCNT
MOVE 233 TO SQLTYPE
MOVE 3 TO SQLLEN
CALL 'eco_boc' USING SQLTYPE, DESCOD OF tabdes, SQLLEN
MOVE 15 TO SQLLEN
CALL 'eco_boc' USING SQLTYPE, DESCLA OF tabdes, SQLLEN
MOVE 62 TO SQLLEN
CALL 'eco_boc' USING SQLTYPE, DESDES OF tabdes, SQLLEN
MOVE 0 TO SQLVALUE
CALL 'eco_slc' USING SQLTEXT02, SQLINCNT, SQLOUTCNT, SQLVALU
- E, SQLTEXTLEN02
CALL 'eco_sqc' USING SQLCA, SQLCODETMP, SQLWARNTMP, SQLSTATE
display 'TABDES:' tabdes
.
*---------------------------------------------------------------
* exec sql commit work end-exec
CALL 'eco_cmt'
CALL 'eco_sqc' USING SQLCA, SQLCODETMP, SQLWARNTMP, SQLSTATE
* exec sql close database end-exec
CALL 'eco_cdb'
CALL 'eco_sqc' USING SQLCA, SQLCODETMP, SQLWARNTMP, SQLSTATE
* exec sql disconnect all end-exec
MOVE 2 TO SQLVALUE
MOVE 0 TO SQL2TEXTLEN
CALL 'eco_dcon' USING SQLVALUE, SQLDUMMY, SQL2TEXTLEN
CALL 'eco_sqc' USING SQLCA, SQLCODETMP, SQLWARNTMP, SQLSTATE
* display 'INFORMIXSERVER' upon environment-name
* display 'predesa2' upon environment-value
move 'xxxxxx@xxxxxx' to base-datos
0871 * exec sql
0872 * connect to default with concurrent transaction
0873 * end-exec
MOVE 1 TO SQLVALUE
MOVE 0 TO SQLTEXTLEN
MOVE 0 TO SQL2TEXTLEN
MOVE 0 TO SQL3TEXTLEN
MOVE 0 TO SQLOBJLEN
MOVE 1 TO SQLUSEFLAG
CALL 'eco_conn' USING SQLVALUE, SQLDUMMY, SQLDUMMY, SQLDUMMY
, SQLDUMMY, SQLUSEFLAG, SQLTEXTLEN, SQL2TEXTLEN, SQL3TEXTLEN,
SQLOBJLEN
CALL 'eco_sqc' USING SQLCA, SQLCODETMP, SQLWARNTMP, SQLSTATE
0499 * exec sql database :base-datos end-exec
MOVE 0 TO SQLVALUE
MOVE 32 TO SQLOBJLEN
CALL 'eco_db' USING base-datos, SQLVALUE, SQLOBJLEN
CALL 'eco_sqc' USING SQLCA, SQLCODETMP, SQLWARNTMP, SQLSTATE
0500
0501 if not (sqlcode = zeros or = -377)
0502 display 'OpBD'
0503 go to sql-error
0504 end-if
0505
move 'set lock mode to wait 30' to sql-m1
* exec sql
* execute immediate :sql-m1
* end-exec
MOVE 50 TO SQLTEXTLEN
CALL 'eco_xim' USING sql-m1, SQLTEXTLEN
CALL 'eco_sqc' USING SQLCA, SQLCODETMP, SQLWARNTMP, SQLSTATE
display 'sqlcode->' sqlcode
move spaces to tabdes
* exec sql
* select *
* into :tabdes
* from tabdes
* where descod = 'UBC'
* end-exec
MOVE 0 TO SQLINCNT
MOVE 3 TO SQLOUTCNT
CALL 'eco_oc' USING SQLOUTCNT
MOVE 233 TO SQLTYPE
MOVE 3 TO SQLLEN
CALL 'eco_boc' USING SQLTYPE, DESCOD OF tabdes, SQLLEN
MOVE 15 TO SQLLEN
CALL 'eco_boc' USING SQLTYPE, DESCLA OF tabdes, SQLLEN
MOVE 62 TO SQLLEN
CALL 'eco_boc' USING SQLTYPE, DESDES OF tabdes, SQLLEN
MOVE 0 TO SQLVALUE
CALL 'eco_slc' USING SQLTEXT03, SQLINCNT, SQLOUTCNT, SQLVALU
- E, SQLTEXTLEN03
CALL 'eco_sqc' USING SQLCA, SQLCODETMP, SQLWARNTMP, SQLSTATE
display 'TABDES:' tabdes
* exec sql commit work end-exec
CALL 'eco_cmt'
CALL 'eco_sqc' USING SQLCA, SQLCODETMP, SQLWARNTMP, SQLSTATE
* exec sql close database end-exec
CALL 'eco_cdb'
CALL 'eco_sqc' USING SQLCA, SQLCODETMP, SQLWARNTMP, SQLSTATE
* exec sql disconnect all end-exec
MOVE 2 TO SQLVALUE
MOVE 0 TO SQL2TEXTLEN
CALL 'eco_dcon' USING SQLVALUE, SQLDUMMY, SQL2TEXTLEN
CALL 'eco_sqc' USING SQLCA, SQLCODETMP, SQLWARNTMP, SQLSTATE
* display 'INFORMIXSERVER' upon environment-name
* display 'desaobr2' upon environment-value
go to empezar.
.
sql-error.
display 'Sqlerror->' sqlcode
stop run
.
Any help would be appreciated
A: Because the COBOL for AIX compiler folds program-names to uppercase by default, if you have COBOL source that contains a call to a C function in mixed or lowercase characters, this function will be folded to uppercase characters. The linker will not find the program and will produce an error message that indicates an unresolved symbol.
You can use the PGMNAME compiler option to control how the COBOL for AIX compiler handles names. The default is PGMNAME(UPPER), but you can use PGMNAME(MIXED) to process the program-name as is, without truncation, translation or folding to uppercase. When you use PGMNAME(MIXED), remember to use the literal format of the program- name, that is, make the program-name a literal string, for example "programname" or you will see the following message:
IGYDS1046-E A user-defined word was found as a "PROGRAM-ID" name under the "PGMNAME(MIXED)" compiler option.
When "PGMNAME(MIXED)" is in effect, a literal is expected for the "PROGRAM-ID" name. The name is accepted in its uppercased format.
Also, when linking you need to specify the Informix library that contains the routines starting with "eco_". In the makefile that you specified I do not see any Informix libraries or object files being listed.
| {
"redpajama_set_name": "RedPajamaStackExchange"
} | 1,886 |
Q: Redis CRDB Eviction Policy I have read in the redis documentation that caching eviction policy for CRDB should be set to No Eviction .
"Note: Geo-Distributed CRDBs always operate in noeviction mode."
https://docs.redislabs.com/latest/rs/administering/database-operations/eviction-policy/
Reasoning for that is the garbage collection might cause inconsistencies as both the data center will have bidirectional synch.
I am not getting this point, can someone explain by giving a real world problem that might occur if suppose we have cache eviction policy LRU .
A: I got to know after doing some research that it is often a trouble to handle eviction when we have active replication. For example if one of the master runs out of memory and cache is trying to evict the keys to make some room for latest data, what might happen is - it will delete those keys from the other master even if there are no memory issues there. So until and unless there is really a good way to handle this ,eviction is not supported.
| {
"redpajama_set_name": "RedPajamaStackExchange"
} | 4,181 |
/***
* A control to add a Column Picker (right+click on any column header to reveal the column picker)
*
* USAGE:
*
* Add the slick.columnpicker.(js|css) files and register it with the grid.
*
* Available options, by defining a columnPicker object:
*
* var options = {
* enableCellNavigation: true,
* columnPicker: {
* columnTitle: "Columns", // default to empty string
*
* // the last 2 checkboxes titles
* hideForceFitButton: false, // show/hide checkbox near the end "Force Fit Columns" (default:false)
* hideSyncResizeButton: false, // show/hide checkbox near the end "Synchronous Resize" (default:false)
* forceFitTitle: "Force fit columns", // default to "Force fit columns"
* headerColumnValueExtractor: "Extract the column label" // default to column.name
* syncResizeTitle: "Synchronous resize", // default to "Synchronous resize"
* }
* };
*
* @class Slick.Controls.ColumnPicker
* @constructor
*/
(function ($) {
'use strict';
function SlickColumnPicker(columns, grid, options) {
var _grid = grid;
var _options = options;
var _gridUid = (grid && grid.getUID) ? grid.getUID() : '';
var $columnTitleElm;
var $list;
var $menu;
var columnCheckboxes;
var onColumnsChanged = new Slick.Event();
var defaults = {
fadeSpeed: 250,
// the last 2 checkboxes titles
hideForceFitButton: false,
hideSyncResizeButton: false,
forceFitTitle: "Force fit columns",
syncResizeTitle: "Synchronous resize",
headerColumnValueExtractor:
function (columnDef) {
return columnDef.name;
}
};
function init(grid) {
grid.onHeaderContextMenu.subscribe(handleHeaderContextMenu);
grid.onColumnsReordered.subscribe(updateColumnOrder);
_options = $.extend({}, defaults, options);
$menu = $("<div class='slick-columnpicker " + _gridUid + "' style='display:none' />").appendTo(document.body);
$("<button type='button' class='close' data-dismiss='slick-columnpicker' aria-label='Close'><span class='close' aria-hidden='true'>×</span></button>").appendTo($menu);
// user could pass a title on top of the columns list
if (_options.columnPickerTitle || (_options.columnPicker && _options.columnPicker.columnTitle)) {
var columnTitle = _options.columnPickerTitle || _options.columnPicker.columnTitle;
$columnTitleElm = $("<div class='title'/>").append(columnTitle);
$columnTitleElm.appendTo($menu);
}
$menu.on("click", updateColumn);
$list = $("<span class='slick-columnpicker-list' />");
// Hide the menu on outside click.
$(document.body).on("mousedown", handleBodyMouseDown);
// destroy the picker if user leaves the page
$(window).on("beforeunload", destroy);
}
function destroy() {
_grid.onHeaderContextMenu.unsubscribe(handleHeaderContextMenu);
_grid.onColumnsReordered.unsubscribe(updateColumnOrder);
if ($list) {
$list.remove();
}
if ($menu) {
$menu.off("click").remove();
}
$(document.body).off("mousedown", handleBodyMouseDown);
$(".slick-columnpicker." + _gridUid).hide(_options && _options.columnPicker && _options.columnPicker.fadeSpeed);
$columnTitleElm = null;
$list = null;
$menu = null;
$(window).off("beforeunload");
}
function handleBodyMouseDown(e) {
if (($menu && $menu[0] != e.target && !$.contains($menu[0], e.target)) || e.target.className == "close") {
$menu.hide(_options && _options.columnPicker && _options.columnPicker.fadeSpeed);
}
}
function handleHeaderContextMenu(e) {
e.preventDefault();
$list.empty();
updateColumnOrder();
columnCheckboxes = [];
var $li, $input, columnId;
var columnLabel, excludeCssClass;
for (var i = 0; i < columns.length; i++) {
columnId = columns[i].id;
excludeCssClass = columns[i].excludeFromColumnPicker ? "hidden" : "";
$li = $('<li class="' + excludeCssClass + '" />').appendTo($list);
$input = $("<input type='checkbox' id='" + _gridUid + "colpicker-" + columnId + "' />").data("column-id", columnId).appendTo($li);
columnCheckboxes.push($input);
if (_grid.getColumnIndex(columnId) != null) {
$input.attr("checked", "checked");
}
if (_options && _options.columnPicker && _options.columnPicker.headerColumnValueExtractor) {
columnLabel = _options.columnPicker.headerColumnValueExtractor(columns[i], _options);
} else {
columnLabel = defaults.headerColumnValueExtractor(columns[i], _options);
}
$("<label for='" + _gridUid + "colpicker-" + columnId + "' />")
.html(columnLabel)
.appendTo($li);
}
if (_options.columnPicker && (!_options.columnPicker.hideForceFitButton || !_options.columnPicker.hideSyncResizeButton)) {
$("<hr/>").appendTo($list);
}
if (!(_options.columnPicker && _options.columnPicker.hideForceFitButton)) {
var forceFitTitle = (_options.columnPicker && _options.columnPicker.forceFitTitle) || _options.forceFitTitle;
$li = $("<li />").appendTo($list);
$input = $("<input type='checkbox' id='" + _gridUid + "colpicker-forcefit' />").data("option", "autoresize").appendTo($li);
$("<label for='" + _gridUid + "colpicker-forcefit' />").text(forceFitTitle).appendTo($li);
if (_grid.getOptions().forceFitColumns) {
$input.attr("checked", "checked");
}
}
if (!(_options.columnPicker && _options.columnPicker.hideSyncResizeButton)) {
var syncResizeTitle = (_options.columnPicker && _options.columnPicker.syncResizeTitle) || _options.syncResizeTitle;
$li = $("<li />").appendTo($list);
$input = $("<input type='checkbox' id='" + _gridUid + "colpicker-syncresize' />").data("option", "syncresize").appendTo($li);
$("<label for='" + _gridUid + "colpicker-syncresize' />").text(syncResizeTitle).appendTo($li);
if (_grid.getOptions().syncColumnCellResize) {
$input.attr("checked", "checked");
}
}
$menu
.css("top", e.pageY - 10)
.css("left", e.pageX - 10)
.css("max-height", $(window).height() - e.pageY - 10)
.fadeIn(_options && _options.columnPicker && _options.columnPicker.fadeSpeed);
$list.appendTo($menu);
$li = null;
$input = null;
}
function updateColumnOrder() {
// Because columns can be reordered, we have to update the `columns`
// to reflect the new order, however we can't just take `grid.getColumns()`,
// as it does not include columns currently hidden by the picker.
// We create a new `columns` structure by leaving currently-hidden
// columns in their original ordinal position and interleaving the results
// of the current column sort.
var current = _grid.getColumns().slice(0);
var ordered = new Array(columns.length);
for (var i = 0; i < ordered.length; i++) {
if (_grid.getColumnIndex(columns[i].id) === undefined) {
// If the column doesn't return a value from getColumnIndex,
// it is hidden. Leave it in this position.
ordered[i] = columns[i];
} else {
// Otherwise, grab the next visible column.
ordered[i] = current.shift();
}
}
columns = ordered;
}
/** Update the Titles of each sections (command, customTitle, ...) */
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\section{Introduction}
The theory of Donaldson--Thomas invariants started around 2000 with the seminal work of R.\ Thomas \cite{Thomas1}. He associated integers to those moduli spaces of sheaves on a compact Calabi--Yau 3-fold which only contain stable sheaves. After some years, K.\ Behrend realized in \cite{Behrend} that these numbers, originally written as ``integrals'' over algebraic cycles or characteristic classes, can also be obtained by an integral over a constructible function, the so-called Behrend function, with respect to the measure given by the Euler characteristic. This new point of view did not only extend the theory to non-compact moduli spaces but revealed also the ``motivic nature'' of this new invariant. It has also been realized that quivers with potential provide another class of examples to which Donaldson--Thomas theory applies. Starting around 2006, D. Joyce \cite{JoyceI},\cite{JoyceCF},\cite{JoyceII},\cite{JoyceIII},\cite{JoyceMF},\cite{JoyceIV} and Y.\ Song \cite{JoyceDT} extended the theory using all kinds of ``motivic functions'' to produce (possibly rational) numbers even in the presence of semistable objects which is the generic situation when classifying objects in abelian categories. Around the same time, M.\ Kontsevich and Y.\ Soibelman \cite{KS1},\cite{KS2},\cite{KS3} independently proposed a theory producing even motives, some sort of refined ``numbers'', instead of simple numbers, also in the presence of semistable objects. The technical difficulties occurring in their approach disappear in the special situation of representations of quivers with potential. The case of zero potential has been intensively studied by M.\ Reineke in a series of papers \cite{Reineke2},\cite{Reineke3},\cite{Reineke4}.
Despite some computations of motivic or even numerical Donaldson--Thomas invariants for quivers with potential (see \cite{BBS},\cite{DaMe1},\cite{DaMe2},\cite{MMNS}), the true nature of Donaldson--Thomas invariants for quiver with potential remained mysterious for quite some time. A full understanding has been obtained recently and is the content of a series of papers \cite{DavisonMeinhardt4},\cite{DavisonMeinhardt3},\cite{MeinhardtReineke},\cite{Meinhardt4}. \\
The present text aims at giving a gentle introduction to Donaldson--Thomas theory in the case of quiver with potential. We have two reasons for our restriction to quivers. Firstly, so-called orientation data will not play any role, and secondly, we do not need to touch derived algebraic geometry. Apart from this, many important ideas and concepts are already visible in the case of quiver representations, and since the theory is fully understood, we belief that this is a good starting point for your journey towards an understanding of Donaldson--Thomas theory. There are more survey articles available focusing on different aspects of the theory. (see \cite{JoyceDT},\cite{KS3},\cite{Szendroi}) \\
Let us give a short outline of the paper. The next section starts very elementary by discussing the problem of classifying objects. The objects which are of interest to us form an abelian category although many ideas of section 2 also apply to ``non-linear'' moduli problems. We study in detail the difficulties arising from the construction of moduli spaces and develop slowly the concept of a (moduli) stack. Although the theory of stacks is rather rich and complicated, we can restrict ourselves to quotient stacks throughout this paper. Hence, a good understanding of a quotient stacks is inevitable. We try to illustrate this concept by giving important examples. We should mention that only very little knowledge of algebraic or complex geometry is needed. In many cases, you can easily replace ``schemes'' with ``varieties'' or ``complex manifolds''.\\
Section 3 provides the background on quivers and their representations. The point of view taken here is that quivers are the categorical (noncommutative) analogue of polynomial algebras in ordinary commutative algebra. In other words, they are a useful tool for practical computations when dealing with linear categories, but at the end of the day the result should only depend on the linear category and not its presentation as a quotient of the path category of a quiver by some ideal of relations. The relations important in this paper are given by noncommutative partial derivatives of a so-called potential. \\
The next two sections provide the language and the framework to formulate Donaldson--Thomas theory in section 6. We start in section 4 with the concept of ``motivic theories''. The best example the reader should have in mind are constructible functions. It should be clear that constructible functions can be pulled back and multiplied. Using fiberwise integrals with respect to the Euler characteristic, we can even push forward constructible functions. Moreover, every locally closed subscheme/subvariety/submanifold determines a constructible function, namely its characteristic function. In a nutshell, a motivic theory is just a generalization of this associating to every scheme $X$ an abelian group $R(X)$ of ``functions'' on $X$ which can be pulled back, pushed forward and multiplied. Moreover, to every locally closed subscheme in $X$ there is a ``characteristic function'' in $R(X)$ such that the characteristic function of a disjoint union is the sum of the characteristic function of its summands. Its is this property what makes a theory of generalized functions ``motivic''. As usual in algebraic geometry, the term ``function'' should be used with some care. Every function on say a complex variety $X$ determines a usual function from the set of points in $X$ to the coefficient ring $R(\mbox{point})$ of our theory, but this is not a one-one correspondence. \\
In section 5 we introduce vanishing cycles. We do not assume that the reader is familiar with any theory of vanishing cycles. As in the previous section, a vanishing cycle is just an additional structure on motivic theories formalizing the properties of ordinary classical vanishing cycles. The Behrend function mentioned at the beginning of this introduction provides a good example of a vanishing cycle on the theory of constructible functions. In fact, we will construct in a functorial way two vanishing cycles associated to a given motivic theory. The first construction is rather stupid, but the second one essentially covers all known nontrivial examples. At the end of sections 4 and 5 we extend motivic theories and vanishing cycles to quotient stacks as quotient stacks arise naturally in moduli problems. There is a way to circumvent stacks in Donaldson--Thomas theory by considering framed objects, but we belief that the usual approach of using stacks is more conceptual and should be known by anyone who wants to understand Donaldson--Thomas theory seriously.\\
In the last section 6 we finally introduce Donaldson--Thomas functions and invariants. After stating the main results, we consider many examples to illustrate the theory. Finally, we develop some tools used in Donaldson--Thomas theory such as Ringel--Hall algebras, an important integration map and the celebrated wall-crossing formula. \\
The reader will realize shortly that the text contains tons over exercises and examples. Most of the exercises are rather elementary and require some elementary computations and standard arguments. Nevertheless, we suggest to the reader to do them carefully in order to get your hands on the subject and to obtain a feeling about the objects involved. There is a lot of material in this text which is not part of the standard graduate courses at universities, and if you are not already familiar with the subject you certainly need some practice as we cannot provide a deep and lengthy discussion of the material presented here.\\
\textbf{Acknowledgments.} The paper is an expanded version of a couple of lectures the author has given in collaboration with Ben Davison at KIAS in February 2015. He is more than grateful to Michel van Garrel and Bumsig Kim for giving him the opportunity to visit this wonderful place. A lot of work on this paper has also been done at the University of Hong Kong, where the author gave another lecture series on Donaldson--Thomas theory. The author wants to use the opportunity to thank Matt Young for being such a wonderful host. He also wants to thank Jan Manschot for keeping up the pressure to finish this paper and for offering the opportunity to publish the paper. Finally, the author is very grateful to Markus Reineke for giving him as much support as possible.
\section{The problem of constructing a moduli space}
\subsection{Moduli spaces}
Let us start by recalling the general idea of a moduli space. Depending on the situation, mathematicians are trying to classify objects of various types. The general pattern is the following. There is some set (or class) of objects and isomorphisms between two objects. Such a structure is called a groupoid. A groupoid is a category with every morphism being an isomorphism. If the set of objects has cardinality one, a groupoid is just a group. The other extreme is a groupoid such that every morphism is the identity morphism of some object. Such groupoids are in one-to-one correspondence with ordinary sets. Hence, a groupoid interpolates between sets and groups. There are two main sources of groupoids.
\begin{example} \rm Let $X$ be a topological space. The fundamental groupoid $\pi_1(X)$ is the groupoid having the points of $X$ as objects, and given two points $x,y\in X$, the set of morphisms from $x$ to $y$ is the set of homotopy classes of paths from $x$ to $y$. Fixing a base point $x\in X$, the usual fundamental group $\pi_1(X,x)$ is just the automorphism group of $x$ considered as an object in the groupoid $\pi_1(X)$. Denote by $\pi_0(X)$ the set of path connected components, i.e.\ the set of objects in $\pi_1(X)$ up to isomorphism.
\end{example}
\begin{example} \rm
Given a category $\mathcal{C}$, one can consider the subcategory $\mathcal{I} so(\mathcal{C})$ of all isomorphisms in $\mathcal{C}$. Thus, $\mathcal{I} so (\mathcal{C})$ is a groupoid, and $\mathcal{C}/_\sim$ denotes the set of objects in $\mathcal{C}$ up to isomorphism.
\end{example}
These two examples are related as follows. To every (small) category one can construct a topological space $X_\mathcal{C}$ - the classifying space of $\mathcal{C}$ - such that $\pi_1(X_\mathcal{C})\cong \mathcal{I} so (\mathcal{C})$ and $\pi_0(X_\mathcal{C} )=\mathcal{C}/_\sim$. \\
Let us come back to the classification problem, say of objects in $\mathcal{C}$ up to isomorphism. The problem is to describe the set $\mathcal{C}/_\sim$. If it is discrete in a reasonable sense, one tries to find a parameterization by less complicated (discrete) objects. This applies for instance to the classification of semisimple algebraic groups or finite dimensional representations of the latter. In many other situations, $\mathcal{C}/_\sim$ is uncountable, and one wants to put a geometric structure on the set $\mathcal{C}/_\sim$ to obtain a ``moduli space''. However, if for instance $\mathcal{C}/_\sim$ has the cardinality of the field of complex numbers, one can always choose a bijection $\mathcal{C}/_\sim \cong M$ to the set of points of any complex variety or manifold $M$ of dimension greater than zero. Pulling back the geometric structure of $M$ along this bijection, we can equip $\mathcal{C}/_\sim$ in many different (non-isomorphic) ways with a structure of a complex manifold. Hence, we should ask:\\
\textbf{Question:} Is there a natural geometric structure on $\mathcal{C}/_\sim$? What does ``natural'' actually mean?\\
There is a very beautiful idea of what ``natural'' should mean, and which applies to many situations. Assume there is a notion of a family of objects in $\mathcal{C}$ over some ``base'' scheme/variety/(complex) manifold $S$, i.e.\ some object on $S$ which has ``fibers'' over $s\in S$, and these fibers should be objects in $\mathcal{C}$.
\begin{example} \rm
Given a $\mathbb{C}$-algebra $A$, a family of finite dimensional $A$-representations is a (holomorphic) vector bundle $V$ on $S$ and an $\mathbb{C}$-algebra homomorphisms $A\to \Gamma(S,\mathcal{E}nd(V))$ from $A$ into the algebra of sections of the endomorphism bundle of $V$.
\end{example}
\begin{example} \rm
Given a scheme/variety/manifold $X$ over $\mathbb{C}$ and some parameter space $S$, a family of coherent sheaves on $X$ parametrized by $S$ is just a coherent sheaf $E$ on $S\times X$ which is flat over $S$. The latter condition ensures that taking fibers and pull-backs of families behaves well. If $E$ is a family of zero dimensional sheaves on $X$, i.e.\ if the projection $p:\Supp(E)\to S$ has zero-dimensional fibers, flatness of $E$ over $S$ is equivalent to the requirement that $E$ is locally free over $S$. Using the coherence of $E$ once more, one can show that $p:\Supp(E)\to S$ is a finite morphism and if $X=\Spec A$ is affine, $E$ is completely determined by the vector bundle $V:=p_\ast E$ on $S$ together with a $\mathbb{C}$-algebra homomorphism $A\to \Gamma(S,\mathcal{E} nd(V))$. From that perspective, the previous example can be seen as a non-commutative version, namely families of zero dimensional sheaves on the non-commutative affine scheme $\Spec A$ for $A$ being a $\mathbb{C}$-algebra.
\end{example}
\begin{example} \rm A $G$-homogeneous space with respect to some (algebraic) group $G$ is a scheme $P$ with a right $G$-action such that $P\cong G$ as varieties with right $G$-action, where $G$ acts on $G$ by right multiplication. A (locally trivial) family of $G$-homogeneous spaces over $S$ is defined as a principal $G$-bundle on $S$.
\end{example}
Once a family is given, by taking the ``fiber'' over $s\in S$ we get an object in $\mathcal{C}$ and, hence, a point in $\mathcal{C}/_\sim$. Varying $s\in S$, we end up with a map $u:S\to \mathcal{C}/_\sim$. Moreover, we see that the pull-back of a family on $S$ along a morphism $f:S'\to S$ induces a morphism $u':S'\to \mathcal{C}/_\sim$ such that $u'=u\circ f$. Coming back to the question formulated above, we can now be more precise by asking: \\
\textbf{Question:} Is there a structure of a variety or scheme on $\mathcal{C}/_\sim$ such that for every family of objects over any $S$, the induced map $S\to \mathcal{C}/_\sim$ is a morphism of schemes? If so, is there any way to get back the family by knowing only the morphism $S\to \mathcal{C}/_\sim$?\\
If the first question has a positive answer, we call $\mathcal{M}=(\mathcal{C}/_\sim,\mbox{scheme structure}) $ a coarse moduli space for $\mathcal{C}$. If the second part of the question is also true, we should be able to (re)construct a ``universal'' family on $\mathcal{M}$ by considering the map $\id:\mathcal{M}\to \mathcal{M}$. Moreover, given a map $u':S'\to \mathcal{M}$ such that $u'=u\circ f$ for some map $f:S'\to S$, the family on $S'$ should be the pull-back of the family on $S$ associated to $u$ by uniqueness. As every morphism $u:S\to \mathcal{M}$ has an obvious factorization $S\xrightarrow{u} \mathcal{M}\xrightarrow{\id} \mathcal{M}$, we finally see that every family on $S$ must be the pull-back of the ``universal'' family on $\mathcal{M}$. In such case, we call $\mathcal{M}$ a fine moduli space.
\begin{example} \rm
Let $\mathcal{C}=\Vect_\mathbb{C}$ be the category of finite dimensional $\mathbb{C}$-vector spaces. A (locally trivial) family of finite dimensional vector spaces is just a vector bundle on some parameter space $S$. As a vector space is classified by its dimension, we can put the simplest scheme structure on $\Vect_\mathbb{C}/_\sim=\mathbb{N}$ by thinking of $\mathbb{N}$ as a disjoint union of countably many copies of $\Spec\mathbb{C}$. Given a vector bundle $V$, we obtain a well-defined morphism $S\to \mathbb{N}$ mapping $s\in S$ to the copy of $\Spec\mathbb{C}$ indexed by the dimension of the fiber $V_s$ of $V$ at $s$. The scheme $\mathbb{N}$ is a course moduli space, but apart from the zero dimensional case, it can never be a fine moduli space. Indeed, there is an obvious and essentially unique vector bundle on $\mathbb{N}$ inducing the identity map $\mathbb{N}\to \mathbb{N}$, but a vector bundle on $S$ can never be the pull-back of the one on $\mathbb{N}$ unless it is constant. Thus, $\mathbb{N}$ is not a fine moduli space.
\end{example}
These are also bad news for our previous examples concerning representations of an algebra $A$ or sheaves on a variety $X$. Indeed, for $A=\mathbb{C}$ or $X=\Spec \mathbb{C}$, we are back in the classification problem of finite dimensional $\mathbb{C}$-vector spaces.
\begin{example} \rm
Similar to the previous example, we see that the classification problem for homogeneous $G$-spaces has only a coarse moduli space given by $\Spec\mathbb{C}$.
\end{example}
There are several strategies to overcome the difficulty of constructing a fine moduli space.
\begin{example}[rigid families] \rm
One possibility is to rigidify families of objects. For example, instead of considering all vector bundles we could also restrict ourselves to constant vector bundles. In this particular case, $\mathbb{N}$ is even a fine moduli space. However, in many situations one wants to glue families together to form families of objects on bigger spaces. This is incompatible with the concept of rigidity, and we will not follow this path.
\end{example}
\begin{example}[weaker equivalence] \rm \label{weaker_equivalence}
Instead of classifying objects up to isomorphism, we could allow weaker equivalences. For example, we could identify to families $V^{(1)}$ and $V^{(2)}$ (over $S$) of vector spaces or representations of an algebra $A$ if there is a line bundle $L$ on $S$ such that $V^{(2)}=V^{(1)}\otimes_{\mathcal{O}_S} L$. By doing this, we can always replace a rank one bundle with the trivial rank one bundle. Hence, the moduli space $\Spec \mathbb{C}$ of one-dimensional vector spaces is a fine moduli space.
\end{example}
\begin{example}[projectivization] \rm \label{projectivization}
Similar to families of vector spaces of dimension $r$, one could look at locally trivial families $\mathcal{P}$ of projective spaces $\mathbb{P}^{r-1}$. The transition functions between local trivializations are regular functions with values in $\Aut(\mathbb{P}^{r-1})=\PGl(r)$. Every vector bundle $V$ of rank $r$ provides such a bundle by taking $\mathcal{P}:=\mathbb{P}(V)$, the bundle of lines or hyperplanes in $V$. Two vector bundles $V^{(1)}$, $V^{(2)}$ define isomorphic bundles $\mathbb{P}(V^{(1)})\cong \mathbb{P}(V^{(2)})$ if and only if $V^{(2)}=V^{(1)}\otimes_{\mathcal{O}_S} L$ for some line bundle $L$ on $S$, providing the bridge to the previous example. However, not every $\mathbb{P}^{r-1}$-bundle $\mathcal{P}$ can be realized as $\mathbb{P}(V)$ for some vector bundle $V$ on $S$. Given a $\mathbb{P}^{r-1}$-bundle $\mathcal{P}$, there is an associated locally trivial bundle $\mathcal{E}_{\mathcal{P}}$ of $\mathbb{C}$-algebras isomorphic to $\End_\mathbb{C}(\mathbb{C}^r)\cong\Mat_\mathbb{C}(r,r)$. Conversely, every locally trivial bundle $\mathcal{E}$ of $\mathbb{C}$-algebras isomorphic to $\Mat_\mathbb{C}(r,r)$ defines an associated $\mathbb{P}^{r-1}$-bundle $\mathcal{P}_{\mathcal{E}}$ as the transition functions of $\mathcal{E}$ must be in $\Aut(\Mat_\mathbb{C}(r,r))=\PGl(r)$. Thus, we have an equivalence of categories between locally trivial $\mathbb{P}^{r-1}$-bundles and locally trivial $\Mat_\mathbb{C}(r,r)$-bundles. If the $\mathbb{P}^{r-1}$-bundle $\mathcal{P}$ is given by $\mathbb{P}(V)$ for a vector bundle $V$ of rank $r$, then $\mathcal{E}_{\mathbb{P}(V)}=\mathcal{E} nd(V)$. Given a $\mathbb{C}$-algebra $A$, we can study families given by a locally free $\mathbb{P}^{r-1}$-bundle $\mathcal{P}$ or equivalently a locally free $\Mat_\mathbb{C}(r,r)$-bundle $\mathcal{E}$ and a homomorphism of algebras $A\to \Gamma(S,\mathcal{E})$. If $A=\mathbb{C}$, there is only a fine moduli space for $r=1$ as every $\mathbb{P}^0$-bundle must be constant. If the algebra $A$ is more complicated, there are also fine moduli spaces for $r>1$, but only for objects which are simple in a suitable sense. For $A=\mathbb{C}$ there are no simple vector spaces of dimension $r>1$.
\end{example}
As we have seen, the construction of fine moduli spaces can only be done in a few cases and severe restrictions. But even if we were only interested in coarse moduli spaces, a standard problem will occur as the following example shows.
\begin{example} \rm \label{S_equivalence} Instead of looking at representations of $A=\mathbb{C}$, we enter the next level of complexity by looking at finite dimensional representations of $A=\mathbb{C}[z]$. A one-dimensional representation $V$ is determined by the value of $z$ in $\End_\mathbb{C}(V)\cong \mathbb{C}$. In other words, a coarse moduli space is given by the complex affine line ${\mathbb{A}^1}$. Still, we have to face the problem discussed before that a line bundle on $S$ with $z$ acting by multiplication with a fixed number $c\in \mathbb{C}$ could almost never be the pull-back of a universal family under the constant map $S\to {\mathbb{A}^1}$ mapping $s\in S$ to $c\in {\mathbb{A}^1}$. Let us ignore the problem of finding a fine moduli space and continue with two-dimensional representations. Consider the trivial rank 2 bundle on $S={\mathbb{A}^1}$ with $z$ acting via the nilpotent matrix
\[ { 1 \;\; s \choose 0 \;\; 1 } \]
in the fiber over $s\in S={\mathbb{A}^1}$. The representations for $s\not=0$ are all isomorphic to each other, and our ``classifying map'' $u:S\to \mathcal{M}_2$ to a coarse moduli space $\mathcal{M}_2$ of rank 2 representations must be constant on $S\!\setminus\!\!\{0\}$. For $s=0$ we obtain a different representation and $u(0)$ must be another point in $\mathcal{M}_2$ if the latter parametrizes isomorphism types. However, such a discontinuous map $u:S\to \mathcal{M}_2$ cannot exist, and we have to abandon the idea of finding a coarse moduli space parameterizing isomorphism classes. One can show that a ``reasonable'' coarse moduli space is given by the GIT-quotient $\Mat_\mathbb{C}(2,2)/\!\!/\Gl(2)$ which is realized as $\Spec \mathbb{C}[\Mat_{\mathbb{C}}(2,2)]^{\Gl(2)}\cong \AA^2$ and similar for higher ranks. The classifying map $S\to \AA^2$ will map $s\in S$ to the unordered pair of eigenvalues of the $z$-action in the fiber over $s$. Such an unordered pair of eigenvalues is determined by the sum (corresponding to the trace) and the product (corresponding to the determinant) of the eigenvalues and similar for higher ranks. Therefore, $\mathcal{M}$ will parametrize unordered direct sums of one-dimensional representations. In other words, by passing from $\mathcal{C}/_\sim$ to $\mathcal{M}$, we identify each representation with the (unordered) direct sum of its simple Jordan--H\"older factors. Representations having the same Jordan--H\"older factors, i.e.\ corresponding to the same point in $\mathcal{M}$, are often called S-equivalent\footnote{The ``S'' in ``S-equivalent'' refers to semisimple, i.e.\ sums of simples, and should not be confused with our notation of a base of a family.}.
\end{example}
Let us summarize the lessons we have learned in the previous examples:
\begin{enumerate}
\item Constructing coarse moduli spaces has only a chance if we do not parametrize objects up to isomorphism but up to the weaker S-equivalence. In other words, classifying objects up to isomorphism is only possible for simple objects, i.e. objects without subobjects.
\item The construction of a universal family on the moduli space of simple objects might only work if we identify two families under a weaker equivalence (twist with a line bundle) or pass to some projectivization.
\end{enumerate}
We suggest to the reader to check these statements in the previous examples.
\subsection{Stability conditions}
Even though the set of objects in $\mathcal{C}$ up to isomorphism might be very large, the set of (isomorphism classes of) simple objects can be quite small, even finite. Thus, the ``coarse'' moduli space would not deliver much insight into the set of isomorphism types in $\mathcal{C}$. However, there is a simple but clever idea to overcome this problem. Instead of looking at $\mathcal{C}$, we should ``scan'' $\mathcal{C}$ by means of a collection $(\mathcal{C}_\mu)_{\mu\in T}$ of ``small'' full subcategories $\mathcal{C}_\mu\subseteq \mathcal{C}$. An object which might be far away from being simple or semisimple (direct sum of simples) can become semisimple or even simple in $\mathcal{C}_\mu$. By doing this, we can distinguish many S-equivalent objects either because they live in different subcategories or they live in the same subcategory $\mathcal{C}_\mu$ but have different Jordan--H\"older filtrations taken in $\mathcal{C}_\mu$. This brilliant idea is the essence of the concept of stability conditions. The following definition is due to Tom Bridgeland. However, there are more general definitions of stability conditions.
\begin{definition} \hfill
\begin{enumerate}
\item A central charge on a noetherian abelian category $\mathcal{C}$ is a function $Z$ on the set of objects in $\mathcal{C}$ with values in $\mathbb{H}_+:=\{r\exp(\sqrt{-1}\phi)\in \mathbb{C}\mid r\ge 0, \phi\in (0,\pi]\}$ such that $Z(E)=0$ implies $E=0$ and $Z(E)=Z(E')+Z(E'')$ for every short exact sequence $0\to E'\to E\to E''\to 0$. \\
\item Given a central charge $Z$, we call an object $E\in \mathcal{C}$ semistable if
\[ \arg Z(E'))\le \arg Z(E) \;\; \mbox{ for all subobjects } E'\subset E. \]
\item For $\mu\in (-\infty,+\infty]$ we denote with $\mathcal{C}_\mu$ the full subcategory of all semistable objects $E$ of slope $-\cot(\arg Z(E))=\mu$ and the zero object. It turns out that $\mathcal{C}_\mu$ is an abelian subcategory of $\mathcal{C}$ (cf.\ Exercise \ref{semistable_reps}).
\item A simple object in $\mathcal{C}_\mu$ is called stable. We assume that every semistable object of slope $\mu$ has a Jordan--H\"older filtration with stable subquotients of the same slope. Semisimple objects of $\mathcal{C}_\mu$, i.e.\ sums of stable objects of slope $\mu$, are called polystable.
\item Every object $E$ in $\mathcal{C}$ has a unique filtration $0 \subset E_1 \subset E_2 \subset \ldots \subset E_n=E$, the Harder--Narasimhan filtration, with semistable quotients $E_i/E_{i-1}$ of strictly decreasing slopes.
\end{enumerate}
\end{definition}
\begin{example}[The $r$-Kronecker quiver] \label{Kronecker} \rm
Let us illustrate this idea with a simple example. Consider the abelian category of $r$-tuples $\bar{x}$ of linear maps $x_i:V_1\to V_2 $ for $1\le i\le r$ between finite dimensional vector spaces $V_1,V_2$. Choosing two complex numbers $\zeta_1,\zeta_2 \in \mathbb{H}_+$, we get a central charge by putting $Z(\bar{x})=\zeta_1\dim V_1 + \zeta_2 \dim V_2$. Assume first that $\arg(\zeta_1)=\arg(\zeta_2)$. Then, all objects are semistable of the same slope $\mu=-\cot( \arg \zeta_1)$, and we have to face the old problems. Choose for instance $\dim V_1=\dim V_2=1$. The isomorphism type of such objects is determined by the choice of $r$ complex numbers $x_1,\ldots, x_r$ up to rescaling by $(g_1,g_2)\in \Gl(V_1)\times \Gl(V_2)=\mathbb{C}^\ast\times \mathbb{C}^\ast$ via $g_1x_ig_2^{-1}$. As the diagonal group $\{(g,g)\mid g\in\mathbb{C}^\ast\}$ acts trivially, we have the take the GIT quotient of $\AA^r$ by $\mathbb{C}^\ast$ which is just a point as $\Spec \mathbb{C}[x_1,\ldots,x_r]^{\mathbb{C}^\ast}=\Spec \mathbb{C}$. This corresponds to the fact that all objects have the same Jordan--H\"older factors $x_i=0:V_1 \to 0$ and $x_i=0:0 \to V_2$. Thus, all objects are S-equivalent to ``$V_1\oplus V_2$''$ = V_1\xrightarrow{0} V_2$. If $\arg \zeta_2 > \arg \zeta_1$, non of our objects with $\dim V_1=\dim V_2=1$ are semistable as the central charge $\zeta_2$ of the subobject $0:0 \to V_2$ has a bigger argument than the central charge $\zeta_1+\zeta_2$ of our given object. If, however, $\arg \zeta_2 < \arg \zeta_1$, all objects except for the semisimple $V_1\oplus V_2$ corresponding to $x_i=0$ for all $1\le i\le r$ are semistable of slope $\mu=- \Re e(\zeta_1+\zeta_2)/\Im m(\zeta_1+\zeta_2)$, and even stable. The moduli space $\mathcal{M}^{\zeta_1,\zeta_2}_{(1,1)}=\AA^r\!\setminus\!\!\{0\}/\mathbb{C}^\ast= \mathbb{P}^{r-1}$ parameterizing isomorphism classes of simple objects in $\mathcal{C}_\mu$ of dimension vector $(\dim V_1,\dim V_2)=(1,1)$ is even a fine moduli space if we identify two families of $r$-tuples of line bundle morphisms $x_i:V_1\to V_2$ on a parameter space $S$ as soon as they become isomorphic after twisting $V_1$ and $V_2$ with some line bundle $L$.
\end{example}
Note that coarse moduli spaces parameterizing S-equivalence classes of objects in $\mathcal{C}_\mu$ might not exist for all central charges, but one can show the existence for generic central charges and reasonable abelian categories. \\
We should also keep in mind that we paid a price for getting a refined version of S-equivalence, namely S-equivalence in subcategories. Indeed, coarse moduli spaces of (S-equivalence classes of) semistable objects can only ``see'' semistable objects but no objects with a non-trivial Harder--Narasimhan filtration. Hence, the construction of (coarse) moduli spaces remains unsatisfying.
\subsection{Moduli stacks}
There is, however, an alternative way to overcome all the problems seen in the previous examples. Following this approach, one can construct a fine moduli ``space'' with a universal family parameterizing all objects - not only simple or stable ones - up to isomorphism. According to the conservative law of mathematical difficulties, we also have to pay a price for getting such a beautiful solution of our moduli problem. It is hidden in the word ``space''. In fact, we have to leave our comfort zone of varieties or schemes and have to dive into the universe of more general spaces known as ``Artin stacks''.\\
Recall that a scheme $X$ is uniquely characterized by its set-valued functor $h_X:S\mapsto \Mor(S,X)$ of points. We have seen many set-valued functors before while studying moduli problems. The general pattern was the following. We considered set-valued contravariant functors $F:S\longmapsto \{ \mbox{families of objects in }\mathcal{C} \}/_\sim$ and a fine moduli space would be a scheme $\mathcal{M}$ such that $F\cong h_\mathcal{M}$, while a coarse moduli space is a scheme $\mathcal{M}$ together with a map $F\to h_\mathcal{M}$ which is universal with respect to all maps $F\to h_X$ of functors. One possibility of generalizing the concept of a space is to consider set-valued functors satisfying similar properties like the functor $h_X$. Note that if one has a collection of morphisms $U_i\to X$ defined on open subsets $U_i$ covering $S$ such that the maps agree on overlaps, one can glue the maps to form a global morphism $S\to X$. This sheaf property should also be satisfied by a general set-valued functor to be a reasonable generalization of a scheme. Such set-valued functors are also often called ``spaces''. A generalized space is called algebraic if it can be written as the ``quotient'' $X/_\sim$ of a scheme $X$ by an (\'{e}tale) equivalence relation. In other words, algebraic spaces are not to far away from schemes and many results for schemes can be generalized to algebraic spaces. In our situation of forming moduli spaces, this is still not the right approach to take, but shows already into the right direction. Indeed, the problems arising in the construction of universal families are related to the presence of (non-trivial) automorphisms. Thus, we should take automorphisms and isomorphisms more seriously into account. \\
Recall that a set with isomorphisms between points was just a groupoid studied at the beginning of this section. Hence, instead of looking at set-valued functors on the category of schemes, we should consider groupoid-valued contravariant functors. These functors should satisfy some gluing property which looks a bit more complicated than in the set-theoretic context. The best idea of remembering the gluing property is by looking at an example which is - as before - the baby example for all Artin stacks.
\begin{example} \rm \label{vector_bundle}
Consider the groupoid-valued functor $\Vect$ which maps any scheme $S$ to the groupoid of vector bundles (the objects) and isomorphisms between them (the morphisms). By pulling back vector bundles along morphisms $f:S'\to S$, we get indeed a contravariant functor.\footnote{Strictly speaking, we only get a pseudofunctor as $g^\ast\circ f^\ast$ is only equivalent to $(f\circ g)^\ast$, but we will ignore this technical problem as one can always resolve it.} Given two vector bundles $V, V'$ and an open cover $\cup_{i\in I}U_i=S$ of $S$ together with isomorphisms\footnote{We will always denote the pull-back along an inclusion $U\hookrightarrow S$ of an open subset by $|_U$.} $\phi_i:V|_{U_i} \to V'|_{U_i}$ on the open subsets $U_i$ such that they agree after restriction to the overlaps, i.e.\ $\phi_i|_{U_{ij}}=\phi_j|_{U_{ij}}$ with $U_{ij}=U_i\cap U_j$, we can
always find a unique global isomorphism $\phi:V\to V'$ such that $\phi_i=\phi|_{U_i}$. On the other hand, if we have vector bundles $V_i$ on $U_i$ and isomorphisms
$\phi_{ij}:V_i|_{U_{ij}} \to V_j|_{U_{ij}}$ such that the only possible composition $V_i|_{U_{ijk}} \to V_j|_{U_{ijk}} \to V_k|_{U_{ijk}} \to V_i|_{U_{ijk}}$ of their
restrictions to the triple overlaps $U_{ijk}=U_i\cap U_j\cap U_k$ is the identity (cocycle condition), one can use the transition isomorphisms $\phi_{ij}$ to glue the $V_i$ together, i.e.\ there is a vector bundle $V$ on $S$ and a family of isomorphisms $\phi_i:V|_{U_i}\to V_i$ such that the only possible composition $V|_{U_{ij}}\to V_i|_{U_{ij}} \to V_j|_{U_{ij}} \to V|_{U_{ij}}$ of their restrictions with $\phi_{ij}$ is the identity. This was the gluing property for isomorphisms and objects, and if we replace the word ``vector bundle'' with ``object'', we get the general form of the gluing property for a groupoid-valued functor.
\end{example}
\begin{definition}
A stack is a groupoid-valued contravariant functor\footnote{Again, we ignore the fact that $g^\ast\circ f^\ast$ might only be equivalent to $(f\circ g)^\ast$ for a pair $S''\xrightarrow{g} S' \xrightarrow{f} S$ of composable morphisms.} on the category of schemes satisfying the gluing property for isomorphisms and objects as seen in Example \ref{vector_bundle}
\end{definition}
In that perspective, a stack is like a (generalized) space with set-valued functors replaced with groupoid-valued functors.
\begin{exercise} Thinking of a set as a special groupoid with no nontrivial isomorphisms, show that every generalized space is a stack.
\end{exercise}
\begin{exercise} \label{moduli_algebra}
Fix a $\mathbb{C}$-algebra $A$. Show that the functor $A\rep$ associating to every scheme $S$ the groupoid of vector bundles $V$ with algebra homomorphisms $A\to \Gamma(S,\mathcal{E} nd(V))$ (the objects) and isomorphisms of vector bundles compatible with the algebra homomorphisms (the morphisms) is a stack. Prove the same for bundles $\mathcal{E}$ of matrix algebras and algebra homomorphisms $A\to \Gamma(S,\mathcal{E})$ as in Example \ref{projectivization}.
\end{exercise}
\begin{exercise}
Fix a scheme/variety/manifold $X$ over $\mathbb{C}$. Show that the functor $\Coh^X$ associating to every scheme $S$ the groupoid of coherent sheaves $E$ on $S\times X$ flat over $S$ (the objects) and isomorphisms between them (the morphisms) is a stack.
\end{exercise}
\begin{exercise}
Fix an algebraic group $G$. Show that the functor $\Spec \mathbb{C}/G$ associating to every scheme the groupoid of principal $G$-bundles (the objects) and isomorphisms between them (the morphisms) is a stack.
\end{exercise}
\begin{example} \rm
The following example is a generalization of the previous exercise. Fix an algebraic group $G$ and a scheme $X$ with a (right) $G$-action. There is a stack $X/G$ associating to every scheme $S$ the groupoid of pairs $(P\to S, m:P\to X)$, where $P\to S$ is a principal $G$-bundle and $m:P\to X$ is a $G$-equivariant map, with morphisms being given by $G$-bundle isomorphisms $u:P\to P'$ satisfying $m'\circ u =m$. The pull-back along a morphism $f:S'\to S$ is given by $(S'\times_S P \to S', m\circ \pr_P)$. The morphism $m:P\to X$ can also be interpreted as a section of the $X$-bundle $P\times_G X \to S$. The stack $X/G$ is called the quotient stack of $X$ with respect to the $G$-action.
\end{example}
When is comes to locally trivial families, there is some choice involved, namely the choice of the underlying (Grothendieck) topology. Intuitively, one would start with the Zariski topology, but the \'{e}tale or even the smooth topology have their advantages, too. In fact, the quotient stack $X/G$ defined above is usually taken with respect to the smooth or, equivalently, \'{e}tale topology. However, for so-called ``special'' groups $G$ like $\Gl(n)$ we could equivalently take the Zariski topology as every \'{e}tale locally trivial principal $G$-bundle is then already Zariski locally trivial. Notice that $\PGl(d)$ is not special and we should better take the \'{e}tale topology when it comes to principal $\PGl(d)$-bundles and quotient stacks $X/\PGl(d)$.
\begin{definition}
A 1-morphism (or morphism for short) from a stack $F$ to a stack $F'$ is a natural transformation $\eta:F\to F'$, i.e.\ a family of functors $\eta_S:F(S) \to F'(S)$ compatible with pull-backs along $f:S'\to S$ up to equivalence of functors. In other words, the functors $F'(f)\circ \eta_S$ and $\eta_{S'}\circ F(f)$ from $F(S)$ to $F'(S')$ are equivalent. A 2-morphism $\alpha:\eta\to \eta'$ between 1-morphisms is an invertible natural transformation $\alpha_S:\eta_S\to \eta'_S$ for every scheme $S$, compatible with pull-backs. In particular, given two stacks $F,F'$, we get a groupoid of morphisms $\MOR(F,F')$ with 1-morphisms being the objects and 2-morphisms being the morphisms. Hence, the category of stacks is a 2-category.
\end{definition}
Thinking of a set as being a groupoid having only identity morphisms, we can associate to every scheme $X$ a contravariant functor $h_X:S\mapsto \Mor(S,X)$. As we can glue morphisms, $h_X$ is indeed a stack. The following lemma is very important.
\begin{lemma}[Yoneda-Lemma] The covariant functor $h:X\mapsto h_X$ from schemes to stacks provides a full embedding of the category of schemes into the (2-)category of stacks. Moreover, there is an equivalence of groupoids $\MOR(h_X,F)\cong F(X)$ for every scheme $X$ and every stack $F$, natural in $X$ and $F$.
\end{lemma}
\begin{exercise}
Try to prove the Yoneda-Lemma.
\end{exercise}
The lemma is basically saying that the 2-category of stacks is an enlargement of the category of schemes, and we will drop the functor $h$ from notation. Though the definition of a stack looks very abstract, the reader should not think of a stack $F$ as a complicated functor, but rather as some object of a bigger 2-category containing the category of schemes. The groupoid-valued functor associated to $F$ can be (re)constructed by taking $X\mapsto \MOR(X,F)$. In other words, assume that you have a 2-category $\mathcal{C}$ with 2-morphisms being invertible, containing $\Sch_\CC$ as a full subcategory, and such that 1-morphisms starting at schemes and 2-morphisms between such 1-morphisms can be glued in a natural way. To every object $F\in \Obj(\mathcal{C})$ we can associate the groupoid-valued functor $\MOR(-,F)|_{\Sch_\CC}$ on the category of schemes. It satisfies the gluing axioms given above, and, hence, defines a stack. Thus, we get a covariant functor from $\mathcal{C}$ to the category of stacks showing that stacks form some sort of ``natural'' enlargement.
\begin{exercise} \label{stack_morphisms} \hfill
\begin{enumerate}
\item Let $X$ be a scheme with a right action of an algebraic group $G$. Consider the trivial principal $G$-bundle $\pr_X: X\times G\to X$ on $X$ and the $G$-equivariant map $m:X\times G\to X$ given by the group action. According to our definition of a quotient stack, the pair $(X\times G\to X,m)$ defines an element $\rho$ in $X/G(X)$. Check the Yoneda lemma by constructing a morphisms $\rho:X\to X/G$ which is called the ``standard atlas'' of $X/G$.
\item Given two schemes $X, Y$ and two algebraic groups $G, H$ acting on $X$ and $Y$ respectively. Assume that $\phi:G\to H$ is a group homomorphism and that $f:X\to Y$ is a morphism of schemes satisfying $f(xg)=f(x)\phi(g)$ for all $g\in G$ and $x\in X$. Construct a natural 1-morphism $\mathfrak{f}:X/G\to Y/H$ of stacks such that
\[ \xymatrix { X\ar[d]^{\rho_X} \ar[r]^f & Y \ar[d]^{\rho_Y} \\ X/G \ar[r]^{\mathfrak{f}} & Y/H} \]
commutes.\textbf{Warning:} Not every morphism $X/G\to Y/H$ is of this form.
\item Consider the special case $H=\{1\}$, and show that $\mathfrak{f}\mapsto f:=\mathfrak{f}\circ \rho_X$ defines an equivalence from $\MOR(X/G,Y)$ to the set $\Mor(X,Y)^G$ of $G$-invariant morphisms, thought as a groupoid.
\item More general, given a scheme $Y$ and a stack $F$, show that $\MOR(F,Y)$ is essentially a set, i.e.\ the only 2-morphisms are the identity morphisms.
\end{enumerate}
\end{exercise}
Let us come back to moduli spaces. The moduli problem of classifying $G$-homogeneous spaces $P$ together with $G$-equivariant maps $P\to X$ for some fixed scheme $X$ with an action of an algebraic group $G$, has a natural generalized ``moduli space'', namely the quotient stack $X/G$. This is not a deep insight, but just the definition of the associated moduli functor. Note that the isomorphism classes of the $\CC$-valued points of $X/G$, i.e.\ $X/G(\Spec\CC)/_\sim$, is the set of $G$-orbits in $X$ justifying the notation.\\
Quotient stacks are also very helpful when it comes to other moduli problems as the following example shows, and their usefulness cannot be overestimated.
\begin{example} \label{stack_example_2} \rm
Consider the stack of finite dimensional representations of a $\CC$-algebra $A$. Assume that $A$ is finitely presented, i.e.\ $A$ is generated by a set\footnote{The notation in this example has been chosen with an eye towards the next section.} $Q_1$ of finitely many elements $\alpha_1, \ldots, \alpha_n$ satisfying a finite set of relations $R=\{r_1,\ldots,r_m\}$. Fix a ``dimension'' $d\in \mathbb{N}$ and put $X_d =\Hom_\CC(\CC^d,\CC^d)^n=\prod_{\alpha\in Q_1}\Hom_\CC(\CC^d,\CC^d)$ and $X^R_d=\{(M_\alpha)_{\alpha\in Q_1}\mid r_j(M_{\alpha_1},\ldots,M_{\alpha_n})=0 \mbox{ for all }1\le j\le m\}$. We claim that the moduli stack $A\rep_d$ of $d$-dimensional representations of $A$ is equivalent to the quotient stack $X^R_d/\Gl(d)$ with $\Gl(d)$ acting by conjugation on $\Hom_\CC(\CC^d,\CC^d)$. Indeed, a family of $d$-dimensional representations on $S\in \Sch_\CC$ is uniquely determined by a vector bundle $V$ of rank $d$ on $S$ and vector bundle endomorphisms $\hat{\alpha}$ associated to $\alpha\in Q_1$ satisfying the relations $r_1,\ldots,r_m$. Consider the frame bundle $\Fr(V)=\{ (s,\tau) \mid s\in S, \tau\in \Hom_\CC(\CC^d,V_s)\mbox{ is invertible}\}$ of $V$ parameterizing all possible choices of a basis in all possible fibers of $V$. It comes with a projection to $S$ and a right action of $\Gl(d)$ by composition with $\tau\in \Hom_\CC(\CC^d,V_s)$.
\begin{exercise}
Show that $\Fr(V)$ is a principal $\Gl(d)$-bundle.
\end{exercise}
There is also a $\Gl(d)$-equivariant map $m(V,(\hat{\alpha})_{\alpha\in Q_1})$ from $\Fr(V)$ into $X^R_d$ mapping a pair $(s,\tau)$ to $(M_\alpha:=\tau^{-1}\circ \hat{\alpha}|_{V_s} \circ \tau)_{\alpha\in Q_1}$.
\begin{exercise}
Convince yourself that the map $(V,(\hat{\alpha})_{\alpha\in Q_1})\longmapsto \big(\Fr(V),m(V,(\hat{\alpha})_{\alpha\in Q_1})\big)$ extends to a functor from the groupoid of families of $d$-dimensional $A$-representations into the groupoid $X^R_d/\Gl(d)(S)$. Show furthermore that this functor is compatible with pull-backs, and, thus, defines a morphism $A\rep_d\to X^R_d/\Gl(d)$ of stacks.
\end{exercise}
Conversely, given a principal $\Gl(d)$-bundle $P$ on $S$ and a $\Gl(d)$-equivariant map $m:P\to X^R_d$, we can consider the trivial vector bundle $P\times \CC^d$ on $P$ which comes with a natural $\Gl(d)$-action compatible with the projection to $P$. Moreover, picking the component of $m$ associated to $\alpha\in Q_1$, we get an endomorphism $\hat{\alpha}$ of this trivial bundle. $\Gl(d)$-equivariance of $m$ ensures that $\hat{\alpha}$ commutes with the $\Gl(d)$-action on $P\times \CC^d$. By taking the $\Gl(d)$-quotient, we obtain a vector bundle $V=P\times_{\Gl(d)}\CC^d$ of rank $d$ on $S$ along with vector bundle endomorphisms $\hat{\alpha}$ satisfying the relations $r_1,\ldots,r_m$.
\begin{exercise}
Show that this construction extends to a functor between groupoids, compatible with pull-backs. Hence, we obtain a morphism from the quotient stack $X^R_d/\Gl(d)$ to $A\rep_d$. Prove that this morphism is an inverse (up to 2-isomorphism) of the morphism constructed above.
\end{exercise}
Thus, the claim is proven and the stack $A\rep$ of $A$-representations is isomorphic to $\sqcup_{d\in\mathbb{N}} X^R_d/\Gl(d)$.
\end{example}
\begin{exercise} \label{stack_projective_reps} Use a similar idea of frame bundles parameterizing tuples $(s\in S,\Mat_\mathbb{C}(r,r)\xrightarrow{\sim}\mathcal{E}_s)$ for a locally trivial family $\mathcal{E}$ on $S$ of $\mathbb{C}$-algebras isomorphic to $\Mat_\mathbb{C}(r,r)$ to show that the stack of projective $A$-representations is given by $\sqcup_{d\in\mathbb{N}} X^R_d/\PGl(d)$. As in Exercise \ref{stack_morphisms}(2), we obtain a morphism $X^R_d/\Gl(d)\longrightarrow X^R_d/\PGl(d)$ by means of the group homomorphism $\Gl(d)\to \PGl(d)$. Show that this morphism is mapping the $A$-representation on $V$ to the projective $A$-representation on $\mathbb{P}(V)$, in other words, forget $V$ and keep $\mathcal{E} nd(V)$ together with the algebra homomorphism $A\to \Gamma(S,\mathcal{E} nd(V))$.
\end{exercise}
\begin{example} \rm
The ``geometry'' of the moduli stack $\Coh^{X}$ of coherent sheaves on a smooth projective variety $X$ is more involved. First of all, it decomposes into components $\Coh^{X}_c$ indexed by numerical data like Chern classes similar to the dimension of a representation. Unfortunately, a component can not be written as a quotient stack. However, every component $\Coh^{X}_{c}$ is the nested union of ``open'' substacks $\Coh^X_{c,i}, i\in \mathbb{N},$ which can be written as a quotient stack $Y_{c,i}/\Gl(n_{c,i})$. Note that $n_{c,i}$ grows with $i$. More details can be found is section 9 of \cite{JoyceI}.
\end{example}
The following definition of a fiber product is very important.
\begin{definition}[fiber product]
Given two morphisms $\mathfrak{f}:\mathfrak{X}\to \mathfrak{Z}$ and $\mathfrak{g}:\mathfrak{Y}\to \mathfrak{Z}$ of groupoid-valued functors, we define the fiber product $\mathfrak{X}\times_\mathfrak{Z} \mathfrak{Y}$ as the groupoid-valued functor such that
\[ \Obj\mathfrak{X}\times_\mathfrak{Z} \mathfrak{Y}(S)=\{ (x,y,w)\mid x\in \Obj\mathfrak{X}(S), y\in \Obj\mathfrak{Y}(S), w\in \Mor_{\mathfrak{Z}(S)}(\mathfrak{f}_S(x),\mathfrak{g}_S(y)) \}, \] and
\begin{eqnarray*} \lefteqn{\Mor_{\mathfrak{X}\times_\mathfrak{Z}\mathfrak{Y}(S)}\Big((x,y,w),(x',y',w')\Big)} \\&=& \{ (u,v)\in \Mor_{\mathfrak{X}(S)}(x,x')\times \Mor_{\mathfrak{Y}(S)}(y,y') \mid \xymatrix @C=0.6cm @R=0.6cm{ \mathfrak{f}_S(x) \ar[r]^w \ar[d]_{\mathfrak{f}_S(u)} & \mathfrak{g}_S(y) \ar[d]^{\mathfrak{g}_S(v)} \\ \mathfrak{f}_S(x') \ar[r]^{w'} & \mathfrak{g}_S(y') } \mbox{commutes } \}
\end{eqnarray*}
for every $S\in \Sch_\CC$.
\end{definition}
\begin{exercise} Show the main properties of the fiber product.
\begin{enumerate}
\item Prove that $\mathfrak{X}\times_\mathfrak{Z} \mathfrak{Y}$ is a stack if $\mathfrak{X},\mathfrak{Y},\mathfrak{Z}$ were stacks.
\item Construct two morphisms $\pr_\mathfrak{X}: \mathfrak{X} \times_\mathfrak{Z} \mathfrak{Y} \longrightarrow \mathfrak{X}$ and $\pr_\mathfrak{Y}: \mathfrak{X} \times_\mathfrak{Z} \mathfrak{Y} \longrightarrow \mathfrak{Y}$ of groupoid-valued functors and a 2-morphism $\omega:\mathfrak{f}\circ\pr_\mathfrak{X}\to \mathfrak{g}\circ\pr_\mathfrak{Y}$. Show that the following universal property holds. Given a groupoid-valued functor $\mathfrak{T}$, two morphisms $\mathfrak{p}:\mathfrak{T}\to \mathfrak{X}$, $\mathfrak{q}:\mathfrak{T}\to \mathfrak{Y}$ and a 2-morphism $\eta:\mathfrak{f}\circ\mathfrak{p} \to \mathfrak{g}\circ\mathfrak{q}$, there is a unique morphism $\mathfrak{r}:\mathfrak{T}\to \mathfrak{X}\times_\mathfrak{Z} \mathfrak{Y}$ such that $\pr_\mathfrak{X}\circ\mathfrak{r}=\mathfrak{p}$ and $\pr_\mathfrak{Y}\circ\mathfrak{r}=\mathfrak{q}$.
\[ \xymatrix { \mathfrak{T} \ar@{.>}[dr]^{\mathfrak{r}} \ar@/^1pc/[drr]^{\mathfrak{p}} \ar@/_1pc/[ddr]_{\mathfrak{q}} & & \\ & \mathfrak{X}\times_\mathfrak{Z} \mathfrak{Y} \ar[r]^{\pr_\mathfrak{X}} \ar[d]_{\pr_\mathfrak{Y}} & \mathfrak{X} \ar[d]^\mathfrak{f} \ar@{=>}[dl]^\omega \\ & \mathfrak{Y} \ar[r]_\mathfrak{g} & \mathfrak{Z} } \]
\end{enumerate}
\end{exercise}
When it comes to quotient stacks, the following examples are very useful.
\begin{exercise} \hfill
\begin{enumerate}
\item Assume $\mathfrak{X}=X, \mathfrak{Y}=Y\in \Sch_\CC$ and $\mathfrak{Z}=Z/G$ for some algebraic group $G$ acting on a scheme $Z$. The morphisms $\mathfrak{f}:X\to Z/G$ and $\mathfrak{g}:Y \to Z/G$ are given by principal $G$-bundles $P\to X$ and $Q\to Y$ together with $G$-equivariant morphisms $f:P\to Z$ and $g:Q\to Z$ respectively. Show that the fiber product $X\times_{Z/G} Y$ is given by the scheme $Iso_{f,g}(P,Q)\subseteq Iso(P,Q)$ over $X\times Y$ given by $\{(x,y,w)\mid x\in X,y\in Y, w:P_x\xrightarrow{\sim}Q_y \;G\mbox{-equivariant such that }f|_{P_x}=g|_{Q_x}\circ w\}$.
\item Assume furthermore $X=Z$ and $P=X\times G\xrightarrow{\pr_X}X$ with $f:X\times G\to X$ being the group action. Hence, $\mathfrak{f}$ is the standard atlas $\rho:X\to X/G$. Show that $Iso_{f,g}(P,Q)$ is isomorphic to $Q$, and
\[ \xymatrix { Q \ar[r]^g \ar[d] & X \ar[d]^\rho \\ Y \ar[r]^{\mathfrak{g}} & X/G }\]
is the fiber product diagram, i.e.\ a cartesian square. Hence, $\rho:X\to X/G$ is the universal principal $G$-bundle.
\end{enumerate}
\end{exercise}
\begin{example} \label{example_fiber_product} \rm
Let $\phi:G\to K$ and $\psi:G\to K$ be homomorphisms between algebraic groups $G,H,K$ acting on $X,Y$ and $Z$ respectively. Moreover, let $f:X\to Z$ and $g:Y\to Z$ be two morphisms such that $f(xg)=f(x)\phi(g)$ and $g(yh)=g(y)\psi(h)$ for all $x\in X,y\in Y,g\in G,h\in H$. As we have seen in Exercise \ref{stack_morphisms}, this induces morphisms $\mathfrak{f}:X/G\to Z/K$ and $\mathfrak{g}:Y/H\to Z/K$. Then, $X/G\times_{Z/K} Y/H$ is the quotient stack $(X\times Y)\times_{(Z\times Z)} (Z\times K)/(G\times H)$ using the group actions $(x,y)(g,h)=(xg,yh)$, $(z,k)(z\phi(g),\phi(g)^{-1}k\psi(h))$, $(z_1,z_2)(g,h)=(z_1\phi(g),z_2\psi(h))$ of $G\times H$ on $X\times Y$, $Z\times K$, $Z\times Z$ and the $G\times H$-equivariant morphisms $X\times Y\ni (x,y)\mapsto (f(x),g(y))\in Z\times Z$, $Z\times K\ni(z,k) \mapsto (z,zk) \in Z\times Z$.
\end{example}
\begin{exercise} \label{fiber_projectivization}
Use the previous example to show that every fiber of the morphism $X^R_d/\Gl(d)\longrightarrow X^R_d/\PGl(d)$ constructed in Exercise \ref{stack_projective_reps} is isomorphic to $\Spec\mathbb{C}/\mathbb{G}_m$. Interpret this result in terms of (projective) $A$-representations. (cf.\ Example \ref{projectivization})
\end{exercise}
The following definition is slightly stronger than the one used in the literature as we do not have algebraic spaces at our disposal. However, it will be sufficient for our purposes.
\begin{definition} A morphism $\mathfrak{f}:\mathfrak{X}\to \mathfrak{Z}$ is called representable if for every morphisms $\mathfrak{g}:Y\to \mathfrak{Z}$ from a scheme $Y$ into $\mathfrak{Z}$, the fiber product $\mathfrak{X}\times_\mathfrak{Z} Y$ is (represented by) a scheme. In such a situation, we call $\mathfrak{f}$ smooth, surjective etc.\ if $\mathfrak{X}\times_\mathfrak{Z} Y \xrightarrow{\pr_Y} Y$ is smooth, surjective etc.
\end{definition}
\begin{exercise} \label{representable} Here are some examples of representable morphisms.
\begin{enumerate}
\item Show that every morphism between schemes is representable.
\item Prove that the standard atlas $\rho:X\to X/G$ is representable, smooth and surjective. Hint: Every algebraic group is a smooth scheme.
\item Use Example \ref{example_fiber_product} to show that $\mathfrak{f}:X/G\to Z/K$ is representable if $\phi:G\to K$ is injective. Give a counterexample for the converse statement.
\item Prove that the diagonal $\Delta_\mathfrak{Z}:\mathfrak{Z} \to \mathfrak{Z}\times_{\Spec\CC}\mathfrak{Z}$ is representable if and only if every morphism $\mathfrak{f}:X\to \mathfrak{Z}$ from a scheme $X$ is representable. Hint: $X\times_\mathfrak{Z} Y= (X\times Y)_{(\mathfrak{Z}\times \mathfrak{Z})} \mathfrak{Z}$.
\end{enumerate}
\end{exercise}
\begin{definition}
A stack $\mathfrak{X}$ is called algebraic or an Artin stack if
\begin{enumerate}
\item[(i)] $\Delta_\mathfrak{X}:\mathfrak{X}\to \mathfrak{X}\times \mathfrak{X}$ is representable (cf.\ Exercise \ref{representable}(4)) and
\item[(ii)] there is a smooth, surjective morphism $\rho:X\to \mathfrak{X}$ from a scheme $X$.
\end{enumerate}
In such a situation, we call $\rho:X\to \mathfrak{X}$ an atlas of $\mathfrak{X}$.
\end{definition}
In a suitable sense, the algebraic stack $\mathfrak{X}$ is a quotient of its atlas $X$ similar to the concept of an algebraic space. However, the quotient is taken in the category of groupoids and not in the category of sets as before. As we have seen in Exercise \ref{representable}, every quotient stack is an Artin stack with standard atlas $\rho:X\to X/G$. By taking $X^R=\sqcup_{d\in \mathbb{N}}X^R_d\to A\rep$, the moduli stack of finite dimensional representations of a finitely represented $\mathbb{C}$-algebra $A$ is also algebraic. Finally, using $Y=\sqcup_{c,i} Y_{c,i}\to \Coh^X$, we see that the moduli stack of coherent sheaves on a smooth projective variety $X$ is also an Artin stack.
\section{Quiver representations and their moduli}
\subsection{Quivers and $\CC$-linear categories}
Recall that a groupoid is a category generalizing groups and sets. Similarly, there is a categorical concept interpolating between $\CC$-algebras and sets.
These are the so-called $\CC$-linear categories. A category $\mathcal{A}$ is called $\CC$-linear if the morphism sets $\Mor_{\mathcal{A}}(x,y)$ have the structure of a $\CC$-vector space such that the composition of morphisms is $\CC$-bilinear.
As usual, we write $\Hom_{\mathcal{A}}(x,y)$ for the $\CC$-vector space of morphisms from $x$ to $y$ and $\End_{\mathcal{A}}(x)=\Hom_{\mathcal{A}}(x,x)$ for the $\CC$-algebra of endomorphisms of $x\in \Obj(\mathcal{A})$. A $\CC$-linear category with one object is just a $\CC$-algebra. On the other hand, $\CC$-linear categories with as less morphisms as possible are uniquely classified by their set of objects since any morphism must be zero or a multiple of the identity of some object.
Another standard example of a $\CC$-linear category is given by the category $\Vect_\CC$ of finite dimensional $\CC$-vector spaces. A finite dimensional representation of a $\CC$-linear category $\mathcal{A}$ is simply given by a functor $V:\mathcal{A}\to \Vect_\CC$. Indeed, if the category $\mathcal{A}$ has only one object $\star$, $V(\star)$ is just a finite dimensional representation of the endomorphism algebra $\End_\mathcal{A}(\star)$. As we have seen in the previous section, generators of algebras are very useful when it comes to the construction of moduli stacks. The analogue in the context of $\CC$-linear categories is called a quiver. A quiver consists of a set of objects $Q_0$ and a set of ``arrows'' $Q_1$ along with maps $s,t:Q_1 \to Q_0$ indicating the \underline{s}ource and the \underline{t}arget of an arrow. We do not require a composition law nor identity morphisms.
Given a $\CC$-linear category $\mathcal{A}$, a quiver in $\mathcal{A}$ satisfies $Q_0\subseteq\Obj(\mathcal{A}), Q_1\subseteq \Mor(\mathcal{A})$ and $s,t$ are given by restriction of the corresponding maps on $\Mor(\mathcal{A})$ to $Q_1$. We say that $\mathcal{A}$ is generated by a quiver $Q$, if the smallest $\CC$-linear category containing $Q$ is $\mathcal{A}$ which implies $Q_0=\Obj(\mathcal{A})$. There is a biggest $\CC$-linear category generated by a given quiver $Q$, the so-called path category $\CC Q$ of $Q$. A morphism of $\CC Q$ from $x\in Q_0$ to $y\in Q_0$ is a $\CC$-linear combination of chains $x=x_1\to x_2\to \ldots \to x_{n-1}\to x_n=y$ of composable arrows in $Q_1$. We also need to add an identity morphism and its $\CC$-linear multiples.
\begin{exercise} Construct a category of quivers such that $Q\mapsto \CC Q$ is a functor from this category to the category of (small) $\CC$-linear categories. Construct a right adjoint of this functor.
\end{exercise}
\begin{exercise} Show that there is a bijection between representations of $\CC Q$ and representations $V$ of $Q$ associating to every $i\in Q_0$ a vector space $V_i$ and to every arrow $\alpha:i\to j$ in $Q_1$ a $\CC$-linear map $V(\alpha):V_i\to V_j$.
\end{exercise}
Given a $\CC$-linear category $\mathcal{A}$ and a generating quiver $Q$ in $\mathcal{A}$, we get a full functor $\CC Q \twoheadrightarrow \mathcal{A}$ which is a bijection on the set of objects. The kernel is a $\CC$-linear subcategory $\mathcal{I}$ in $\CC Q$ which has the property $a\circ b\in \Mor(\mathcal{I})$ if $a\in \Mor(\mathcal{I})$ or $b\in \Mor(\mathcal{I})$ categorifying the concept of an ideal. A generating quiver for $\mathcal{I}$ is uniquely determined by its set of arrows
$R\subseteq \Mor(\CC Q)$ which are called relations. Conversely, every quiver $Q$ with relations give rise to a $\CC$-linear category $\CC Q/(R)$ uniquely defined up to isomorphism. Conversely, every $\CC$-linear category $\mathcal{A}$ can be written like this (up to isomorphism) in many ways.\\
Throughout this paper we will only consider finite quivers, i.e.\ $|Q_0|<\infty$ and $|Q_1|<\infty$ and similarly for the relations. Hence, the $\CC$-linear categories $\mathcal{A}$ which can be described by a finite quiver with finitely many relations are exactly the finitely presented $\CC$-linear categories.
\begin{exercise}
Show that the category of $\CC$-linear categories $\mathcal{A}$ with finite set of objects is equivalent to the category of $\CC$-algebras together with a distinguished finite set $\{e_i\}_{i\in I}$ of mutually orthogonal idempotent elements $e_i$ such that $1=\sum_{i\in I}e_i$. Hint: Put $A:=\oplus_{i,j\in \Obj(\mathcal{A})} \Hom_\mathcal{A}(i,j)$ and $e_i=\id_i$ for all $i\in \mathcal{A}$. Moreover, prove that the category of representations of such a $\CC$-linear category is isomorphic to the category of representations of the associated algebra.
\end{exercise}
Using the last exercise, we can also talk about the path $\CC$-algebra of a quiver with finite set $Q_0$ and its representations. Note that the path $\CC$-algebra has a distinguished family $(e_i)_{i\in Q_0}$ of mutually orthogonal idempotent elements summing up to $1$.
\subsection{Quiver moduli spaces and stacks}
Generalizing the moduli functor $A\rep$ of finite dimensional representations of a given $\CC$-algebra $A$ (see Example \ref{moduli_algebra}), we define the moduli functor $\mathcal{A}\rep$ of finite dimensional representations of a $\CC$-linear category $\mathcal{A}$ as follows. To every scheme $S$ over $\CC$ we associate the isomorphism groupoid $\mathcal{A}\rep(S)$ of the category of functors $\mathcal{A}\to \Vect_S$, where $\Vect_S$ is the category of vector bundles on $S$.
\begin{exercise}
Show that $\mathcal{A}\rep$ is a stack, i.e.\ satisfies the gluing axiom for groupoid-valued functors.
\end{exercise}
If $\mathcal{A}$ is represented by a quiver $Q$ with relations $R\subseteq \Mor(\CC Q)$, the category $\mathcal{A}\rep(S)$ is equivalent to the category of families $(V_i)_{i\in Q_0}$ of vector bundles on $S$ together with vector bundle morphisms $\hat{\alpha}=V(\alpha):V_i\to V_j$ such that $V(r)=r\big((\hat{\alpha})_{\alpha\in Q_1}\big)=0$ for all $r\in R$, where we extended $V$ from $Q_1$ to $\Mor(\CC Q_1)\supseteq Q_1$.
Let us assume that $Q_0,Q_1$ and $R$ are finite sets. Using Example \ref{stack_example_2}, it should not come as a surprise that $\mathcal{A}\rep$ is isomorphic to a disjoint union $\mathfrak{M}^R:=\sqcup_{d\in \mathbb{N}^{Q_0}} \mathfrak{M}^R_d$ of quotient stacks $\mathfrak{M}^R_d:=X^R_d/G_d$ with
\[X^R_d:=\{ (M_{\alpha})_{\alpha\in Q_1}\in X_d \mid r\big((M_\alpha)_{\alpha\in Q_1}\big)=0 \,\forall \, r\in R\} \subseteq X_d:= \prod_{Q_1\ni\alpha:i\to j} \Hom_\CC(\CC^{d_i},\CC^{d_j})\] and $G_d=\prod_{i\in Q_0}\Gl(d_i)$ acting on $X_d$ by simultaneous conjugation. The ``dimension vector'' $d\in \mathbb{N}^{Q_0}$ is fixing $\dim V=(\rk V_i)_{i\in Q_0}$. \\
Similarly, given a sequence of dimension vectors $d^{(1)},\ldots,d^{(r)}$ we denote with $X_{d^{(1)},\ldots,d^{(r)}}\subseteq X_{d^{(1)}+\ldots+d^{(r)}}$ the affine subvariety parameterizing linear maps preserving the standard flag $0 \subseteq \CC^{d^{(1)}_i} \subseteq \CC^{d^{(1)}_i}\oplus\CC^{d^{(2)}_i} \subseteq \ldots \subseteq \CC^{d^{(1)}_i}\oplus\ldots\oplus \CC^{d^{(r)}_i}$ for every $i\in Q_0$. The subgroup $G_{d^{(1)},\ldots,d^{(r)}}\subseteq G_{d^{(1)}+\ldots+d^{(r)}}$ is defined in the same way. Finally, we put $X^R_{d^{(1)},\ldots,d^{(r)}}:=X_{d^{(1)},\ldots,d^{(r)}}\cap X^R_{d^{(1)}+\ldots+d^{(r)}}$.
\begin{exercise}
Show that the stack of all successive extensions
\begin{eqnarray*}
&0\to V^{(1)}\to \hat{V}^{(2)} \to V^{(2)} \to 0, &\\
&0\to \hat{V}^{(2)}\to \hat{V}^{(3)} \to V^{(3)} \to 0, &\\
& \vdots & \\
&0\to \hat{V}^{(r-1)}\to \hat{V}^{(r)} \to V^{(r)} \to 0 &
\end{eqnarray*}
of quiver representations satisfying the relations $R$ and with $\dim V^{(j)}=d^{(j)}$ for all $1\le j\le r$ is given by the quotient stack $\mathfrak{M}^R_{d^{(1)},\ldots,d^{(r)}}=X^R_{d^{(1)},\ldots,d^{(r)}}/G_{d^{(1)},\ldots,d^{(r)}}$. Hint: The standard flag introduced above defines a standard successive extension of $Q_0$-graded vector spaces of prescribed dimension vectors. Given a family of successive extensions, consider the principal $G_{d^{(1)},\ldots,d^{(r)}}$-bundle parameterizing all isomorphism from the standard extension to the fibers of the family, and proceed as usual.
\end{exercise}
We are mainly interested in the following type of relations. A potential $W$ is an element of the vector space $\CC Q/[\CC Q,\CC Q]$, where $[\CC Q,\CC Q]$ denotes the $\CC$-linear span (and not the spanned ideal) of all commutators. Note that $\CC Q/[\CC Q,\CC Q]$ is the $0$-th Hochschild homology of the $\CC$-linear category $\CC Q$. Convince yourself that $W$ is essentially just a $\CC$-linear combination of equivalence classes of cycles in $Q$ with two cycles being equivalent if they can be transformed into each other by a cyclic permutation.
\begin{example} \rm
The three elements $[x,y]z=xyz-yxz$, $[z,x]y$ and $[y,z]x$ in $\CC Q^{(3)}$ of the 3-loop quiver $Q^{(3)}$
\[ \xymatrix { \bullet \ar@(u,r)^y \ar@(dr,dl)^z \ar@(l,u)^x }\]
define the same potential $W$.
\end{example}
For a fixed potential $W=\sum_{l=1}^L a_l \cdot[C_l]$ we define relations $\partial W/\partial \alpha\in \Hom_{\CC Q}(j,i)$ for every $\alpha:i\to j$ in $Q_1$ as follows.
\[ \frac{\partial W}{\partial \alpha}:=\sum_{l=1}^L a_l \cdot\sum_{C_l=u\alpha v} vu \]
with $a_l\in \CC$, where the second sum is over all occurrences of $\alpha$ in a fixed representative of an equivalence class $[C_l]$ of cycles in $Q$.
\begin{exercise}
Show that the definition of $\partial W/\partial \alpha$ is independent of the choice of the representative $C_l\in [C_l]$ for all $1\le l\le L$.
\end{exercise}
\begin{example} \rm
Using the potential $W=[x,y]z=xyz-yxz$ from the previous example, we compute
\begin{eqnarray*}
\frac{\partial W}{\partial x} & = & yz-zy\; =\;[y,z], \\
\frac{\partial W}{\partial y} & = & zx-xz\; =\;[z,x], \\
\frac{\partial W}{\partial z} & = & xy-yx\; =\;[x,y].
\end{eqnarray*}
Convince yourself that $W=[z,x]y$ and $W=[y,z]x$ provide the same relations.
\end{example}
Given a dimension vector $d\in \mathbb{N}^{Q_0}$ and a potential $W=\sum_{l=1}^L a_l \cdot[C_l]$ with $C_l=\alpha_l^{(1)}\circ \ldots \circ\alpha^{(n_l)}_l$, we define the following function
\[ \Tr(W)_d:X_d\ni (M_\alpha)_{\alpha\in Q_1} \longmapsto \sum_{l=1}^L a_l\cdot \Tr\big(M_{\alpha_l^{(1)}}\cdot \ldots \cdot M_{\alpha^{(n_l)}_l}\big) \in {\mathbb{A}^1} \]
which is independent of the choice of the representative $C_l\in [C_l]$ as the trace is invariant under cyclic permutation. By the same argument, $\Tr(W)_d$ is $G_d$-invariant, and induces a function $\mathfrak{Tr}(W)_d:\mathfrak{M}_d\to {\mathbb{A}^1}$ on the quotient stack.
\begin{exercise}
Let us take the relations $R=\{\partial W/\partial \alpha \mid \alpha\in Q_1\}$. Show that $X^R_d=\Crit(\Tr(W)_d)$ is the critical locus of $\Tr(W)_d$, and similarly $\mathfrak{M}^R_d=\Crit(\mathfrak{Tr}(W)_d)$.
\end{exercise}
Throughout the paper we will use the superscript $W$ instead of the superscript $R$ for $R=\{\partial W/\partial \alpha \mid \alpha\in Q_1\}$, and no superscript if $W=0$. We will also use the notation $\Jac(Q,W)$ for the so-called Jacobi algebra $\CC Q/(R)$.\\
The moduli stack $\mathfrak{M}^R_d$ has a coarse moduli space $\mathcal{M}^{W,ssimp}_d$ parameterizing semisimple (direct sums of simple) representations of dimension vector $d$. It is an affine scheme given by $\Spec \CC[X^R_d]^{G_d}$ with $\CC[X^R_d]^{G_d}$ denoting the $G_d$-invariant regular functions on the affine scheme $X^R_d$.
\begin{example} \rm
For the 3-loop quiver $Q^{(3)}$ with potential $W=[x,y]z$, the scheme $X^W_d$ parametrizes triples of commuting $d\times d$-matrices $M_x,M_y,M_z$. Hence, a simple representation of the Jacobi algebra $\Jac(Q^{(3)},W)=\CC[x,y,z]$ is one-dimensional and determined by $(M_x,M_y,M_z)\in \AA^3$. Therefore, $\mathcal{M}^W_d=\Sym^d(\AA^3)=(\AA^3)^d/S_d$.
\end{example}
Let us finally introduce a stability condition by choosing a tuple $\zeta\in \mathbb{H}_+^{Q_0}$ of complex numbers in the (extended) upper half plane $\mathbb{H}_+$ giving rise to the ``central charge'' $Z(V):=\zeta\cdot \dim V=\sum_{i\in Q_0}\zeta_i\dim V_i \in \mathbb{H}_+$ for every representation $V$ of $Q$.
\begin{definition} A representation $V\not=0$ of a quiver $Q$ (with relations) is called $\zeta$-semistable if
\[ \arg Z( V') \le \arg Z( V) \]
for all proper subrepresentations $V'\subset V$. If the inequality is strict, $V$ is called $\zeta$-stable. The real number $\mu(V):=-\cot(\arg Z(V))$ is called the slope of $V$. Hence, $V$ is semistable if and only if $\mu(V')\le \mu(V)$ for all proper subrepresentations $V'\subset V$.
\end{definition}
\begin{exercise} \label{semistable_reps} Let us show that semistable representations of the same slope $\mu$ form a nice full subcategory.
\begin{enumerate}
\item Consider a morphism $f:V^{(1)}\to V^{(2)}$ of semistable representations of slopes $\mu(V^{(1)})>\mu(V^{(2)})$. Show that $f=0$. Hint: Relate the slope of $V/\ker(f)=\im(f)$ to $\mu(V^{(1)})$ and to $\mu(V^{(2)})$ by drawing the central charges of all objects involved.
\item Using the notation of the first part, let us assume $\mu(V^{(1)})=\mu(V^{(2)})$ for the semistable representations $V^{(1)},V^{(2)}$. Show that $\ker(f)$ and $\coker(f)$ are also semistable of the same slope $\mu(V^{(1)})$. In particular, the semistable representations of a fixed slope $\mu$ form a full abelian subcategory.
\item Show that the stable objects of slope $\mu$ are the simple objects in the full abelian subcategory of semistable representations of slope $\mu$.
\item Prove that the extension of two semistable representations of slope $\mu$ is again semistable of the same slope.
\end{enumerate}
\end{exercise}
Every representation $V$ of a quiver (with relations) has a unique Harder--Narasimhan filtration, i.e.\ a finite filtration $0\subset V^{(1)} \subset \ldots\subset V^{(r)}=V$ such that the subquotients $V^{(i)}/V^{(i-1)}$ are semistable of slope $\mu^{(i)}$ satisfying $\mu^{(1)}>\ldots> \mu^{(r)}$.
\begin{exercise} Let us prove the last statement in three steps.
\begin{enumerate}
\item Show that $V$ has a maximal nonzero subrepresentation of maximal slope. Hint: Show that the set of slopes of subrepresentations of $V$ has a maximal element. Use Exercise \ref{semistable_reps}(4) to construct a maximal subrepresentation of maximal slope.
\item Use Exercise \ref{semistable_reps}(1) to construct a Harder--Narasimhan filtration. Hint: Let $V^{(1)}$ be the subrepresentation constructed in the first step, and let $V^{(2)}$ be the preimage of a maximal subrepresentation in $V/V^{(1)}$ of maximal slope. Proceed in this way, and use the previous exercise to estimate the slopes.
\item Prove the uniqueness of this filtration by applying Exercise \ref{semistable_reps}(1) once more.
\end{enumerate}
\end{exercise}
We denote by $X^{R,\zeta-ss}_d$ the subscheme of linear maps $(M_\alpha)_{\alpha\in Q_1}$ such that the induced quiver representation on $(\CC^{d_i})_{i\in Q_0}$ is $\zeta$-semistable. It is open and stable under the $G_d$-action. Hence, we can form the quotient stack $\mathfrak{M}^{R,\zeta-ss}_d=X^{R,\zeta-ss}_d/G_d$ of $\zeta$-semistable representations of dimension vector $d$. The open subscheme $X^{R,\zeta-st}_d\subseteq X^{R,\zeta-ss}_d$ and the open substack $\mathfrak{M}^{R,\zeta-st}_d\subset \mathfrak{M}^{R,\zeta-ss}_d$ of $\zeta$-stable representations are defined accordingly. \\
If $\zeta_i=-\theta_i+\sqrt{-1}$ with $\theta_i\in \mathbb{Z}$ for all $i\in Q_0$, one can linearize the $G_d$-action on the trivial line bundle over $X_d$ using the character \[G_d\ni (g_i)_{i\in Q_0} \longmapsto \prod_{i\in Q_0}\det(g_i)^{\theta\cdot d-|d|\theta_i} \in \mathbb{G}_m\]
with $\theta\cdot d=\sum_{i\in Q_0}\theta_id_i$ and $|d|:=\sum_{i\in Q_0}d_i$. A.\ King showed in \cite{King} that $X^{R,\zeta-ss}_d$ is the subscheme of semistable points in $X^R_d$ with respect to this linearization. Hence, a GIT-quotient $X^{R,\zeta-ss}_d/\!\!/G_d=\mathcal{M}^{R,\zeta-ss}_d$ with stable sublocus $X^{R,\zeta-st}_d/G_d=\mathcal{M}^{R,\zeta-st}_d$ exists. Using this, one can show that all moduli stacks $\mathfrak{M}^{R,\zeta-ss}_d$ have a coarse moduli space $\mathcal{M}^{R,\zeta-ss}_d$ parameterizing S-equivalence classes of $\zeta$-semistable objects, or, equivalently, isomorphisms classes of $\zeta$-polystable objects of dimension vector $d$ if $\zeta$ is in the complement of a countable union of real hypersurfaces in $\mathbb{H}_+^{Q_0}$. (see \cite{Meinhardt4} Example 3.32) A stability condition $\zeta$ having coarse moduli spaces $\mathcal{M}^{\zeta-ss}_d$ for all $d\in \mathbb{N}^{Q_0}$ is called geometric. In case $\zeta_i=\sqrt{-1}$ for all $i\in Q_0$, i.e.\ $\theta=0$, we write $\mathcal{M}^{R,ssimp}_d$ for $\mathcal{M}^{R,\zeta-ss}_d$ as its points correspond to isomorphism classes of semisimple $\mathbb{C} Q$-representations satisfying the relations $R$.
\begin{remark} \rm
Notice that $\mathbb{G}_m$, embedded into $G_d$ diagonally, acts trivially on $X_d$, and $G_d$ induces a $PG_d:=G_d/\mathbb{G}_m$-action on $X_d$. The character given above descends to a character on $PG_d$, and $X^{R,\zeta-ss}_d$ is also the semistable locus of $X^{R}_d$ with respect to the $PG_d$-linearization. Hence, $\mathcal{M}^{R,\zeta-ss}_d=X^{R,\zeta-ss}_d/\!\!/PG_d$ is also the coarse moduli space for $X^{R,\zeta-ss}_d/PG_d$, the stack of ``$d$-dimensional'' projective semistable quiver representations satisfying the relations $R$. It is not difficult to see that $\mathcal{M}^{R,\zeta-st}_d$ is in fact a fine moduli space for $X^{R,\zeta-st}_d/PG_d$, in other words, there is an isomorphism $X^{R,\zeta-st}_d/PG_d\cong \mathfrak{M}^{R,\zeta-st}_d$ of stacks. In particular, $\mathcal{M}^{R,\zeta-st}$ carries a universal family $\mathcal{P}$ of projective stable quiver representations satisfying our relations $R$. The reader should compare this with our final remarks in Example \ref{projectivization} and the two lessons we have mentioned after Example \ref{S_equivalence}. The morphism $X^{R,\zeta-st}_d/G_d\longrightarrow X^{R,\zeta-st}_d/PG_d\cong \mathcal{M}^{R,\zeta-st}_d$ is not an isomorphism. It is not hard to see that this map has a right inverse, i.e.\ a section, if $\gcd(d):=\gcd(d_i:i\in Q_0)=1$. (See \cite{Reineke5}, Section 5.4 for more details) Such a section is nothing else than a family $V=\bigoplus_{i\in Q_0}V_i$ of stable quiver representations on $\mathcal{M}^{R,\zeta-st}_d$ such that $\mathcal{P}=\mathbb{P}(V)$. The section and within the family is not unique. Any two sections corresponding to $V^{(1)}$ and $V^{(2)}$ differ (up to isomorphism) by a line bundle $L$ on $\mathcal{M}^{R,\zeta-st}$ with $V^{(2)}\cong V^{(1)}\otimes_{\mathcal{O}_{\mathcal{M}^{R,\zeta-st}_d}} L$ as in Example \ref{projectivization}. (See also Exercise \ref{fiber_projectivization}) Therefore, $V$ on $\mathcal{M}^{R,\zeta-ss}$ is only universal up to this weaker equivalence. It has been shown in \cite{Reineke6}, Thm.\ 3.4 that under some mild conditions on the pair $(d,\zeta)$ the moduli space $\mathcal{M}^{\zeta-st}_d$ has no ``universal family'' $V$, i.e.\ $X^{\zeta-st}_d/G_d\to \mathcal{M}^{\zeta-st}_d$ has no section, if $\gcd(d)>1$.
\end{remark}
\begin{definition} \label{generic_stability}
A stability condition $\zeta$ is called generic if $\langle d,d'\rangle=0$ for all $d,d'\in \Lambda^\zeta_\mu:=\{e\in \mathbb{N}^{Q_0}\mid e=0 \mbox{ or }e\mbox{ has slope }\mu\}$ and all $\mu\in \mathbb{R}$, where $\langle d,d' \rangle=(d,d')-(d',d)$ denotes the antisymmetrized Euler pairing
\[ (d,d')=\sum_{i\in Q_0}d_id'_i - \sum_{\alpha:i\to j}d_id'_j \]
satisfying $(\dim V,\dim V')=\dim \Hom_{\CC Q}(V,V')-\dim \Ext^1_{\CC Q}(V,V')$ for all $\CC Q$-representations $V,V'$.
\end{definition}
\section{From constructible functions to motivic theories}
\subsection{Constructible functions}
Let us start by recalling some facts about constructible functions. A constructible function is a function $a:X(\mathbb{C}) \to \mathbb{Z}$ on the set of (closed) points of a scheme/variety/manifold $X$ over $\mathbb{C}$ with only finitely many values on each connected component of $X$ and such that the level sets of $a$ are the (closed) points of locally closed subsets of $X$. We denote with $\Con(X)$ the group of constructible functions on $X$.
\begin{exercise}
Assume that $X$ is connected. Show that the map associating to every irreducible closed subset $V$ of $X$ its characteristic function extends to an isomorphism $\oplus_{x\in X}\mathbb{Z} x \cong \oplus_{V\subset X} \mathbb{Z} V\xrightarrow{\sim} \Con(X)$, where the first sum is over all not necessarily closed points $x\in X$, and the second sum is taken over all irreducible closed subsets $V\subset X$.
\end{exercise}
Apparently, we can pull back constructible functions and also multiply them pointwise. Contrary to the usual notation, we denote the pointwise product with $a\cap b$, i.e.\ $(a\cap b)(x)=a(x)b(x)$. The constant function $\mathbbm{1}_X(x)=1$ for all $x\in X(\mathbb{C})$ is the unit for the $\cap$-product. There is another product, the external product $a\boxtimes b =\pr_X^\ast(a)\cap\pr_Y^\ast(b)$ of two functions $a\in \Con(X)$ and $b\in \Con(Y)$ on $X\times Y$ such that $a\cap b=\Delta_X^\ast (a\boxtimes b)$ if $Y=X$. The unit for the $\boxtimes$-product is $1\in \mathbb{Z}=\Con(\Spec\mathbb{C})$. Moreover, we can define a push-forward of a constructible function
$a\in \Con(X)$ along a morphism $u:X\to Y$ of finite type by\footnote{For every scheme $X$ locally of finite type over $\mathbb{C}$, we denote with $X^{an}$ the ``analytification'' of $X$ which is an analytic space locally isomorphic to the vanishing locus of holomorphic functions on $\mathbb{C}^n$. If $X$ is smooth, $X^{an}$ is a complex manifold. In any case $X^{an}$ carries the analytic topology which is much finer than the Zariski topology on $X$.} $u_!(a)(y):=\int_{u^{-1}(y)^{an}} a\, d\chi_c:= \sum_{m\in \mathbb{Z}}m\chi_c\{x\in X\mid u(x)=y, a(x)=m\}^{an}$ for $y\in Y$. Here $\chi_c$ denotes the Euler characteristic with compact support, i.e.\ the alternating sum of the dimensions of the compactly supported cohomology. One can think of $\chi_c$ as a signed measure $\chi_c^X$ on $X$, even though it is only additive and not $\sigma$-additive. Given a constructible function $a$ on $X$, we get a new measure $a\cdot\chi_c^X$ on $X$ of density $a$ with respect to $\chi_c^X$. A push-forward of a measure is well-defined and $u_!(a\chi_c^X)$ has density $u_!(a)$ with respect to $\chi_c^Y$. Using the push-forward, one can define a third product for constructible functions on a monoidal scheme, i.e.\ a scheme $X$ with two maps $0:\Spec\CC\to X$ and $+:X\times X\to X$ of finite type satisfying an associativity and unit law. The convolution product is given by $a b=+_!(a\boxtimes b)$ and is commutative if $+$ is commutative. The unit is given by $0_!(1)$ with $1\in \Con(\Spec \CC)= \mathbb{Z}$ being the unit for the $\boxtimes$-product. If we had taken $X={\mathbb{A}^1}$, the convolution product is just the ``constructible version'' of the usual convolution product of integrable functions. The free commutative monoid generated by a scheme $X$ is given by $\Sym(X) = \sqcup_{n\in \mathbb{N}} \Sym^n(X)$ with $\Sym^n(X)=X^n/\!\!/S_n$, and $\oplus:\Sym(X) \times \Sym(X)\longrightarrow \Sym(X)$ is just the concatenation of unordered tuples of (geometric) points of $X$. The unit $0:\Spec\mathbb{C}=:\Sym^0(X) \hookrightarrow \Sym(X)$ is given by the ``empty tuple''. We can apply the definition of the convolution product $ab:=\oplus_!(a\boxtimes b)$ to $\Con(\Sym(X))$ making it into a commutative ring. This (convolution) ring has even more structure. Indeed, there is a family of maps $\sigma^n:\Con(X) \to \Con(\Sym^n(X))$ mapping the characteristic function of $V\subseteq X$ to the characteristic function of $\Sym^n(V)\subseteq \Sym^n(X)$.
\begin{example}\rm
Consider the example $X=\Spec \CC$. Then $\Sym(X)\cong\mathbb{N}$, and $\Con(X)\cong \mathbb{Z}[[t]]$ follows. The convolution product is just the ordinary product of power series and $\Sym^n(at)={ a+n-1 \choose n}t^n$. The pointwise $\cap$-product is known as the Hadamard product of two power series.
\end{example}
Let us collect the main properties of the structures described above.
\begin{proposition} \label{motivic_theories}
By taking pull-backs and push-forwards of constructible functions, we obtain a functor $\Con$ from the category $\Sch_\CC$ to the category of abelian groups which is both contravariant with respect to all morphisms and covariant with respect to all morphisms of finite type, i.e.\ for every morphism $u:X\to Y$
there is a group homomorphism $u^\ast:\Con(Y)\longrightarrow \Con(X)$, and if $u$ is of finite type, there is also a group homomorphism $u_!:\Con(X)\longrightarrow \Con(Y)$. Moreover, there is an ``exterior'' product
\[ \boxtimes: \Con(X)\otimes \Con(Y) \longrightarrow \Con(X\boxtimes Y) \]
defined for every pair $X,Y\in \Sch_\CC$ which is associative, symmetric\footnote{If $\tau:X\boxtimes Y \stackrel{\sim}{\to} Y\boxtimes X$ is the transposition, being symmetric means $\tau_!(a\boxtimes b )=\tau^\ast(a\boxtimes b) = b\boxtimes a$} and has a unit $1\in \Con(\Spec \CC )$. Finally, there are also operations
\[ \sigma^n: \Con(X) \longrightarrow \Con(\Sym^n( X)) \]
for $n\in \mathbb{N}$ such that $\sigma^n(1)=1$ holds for all $n\in \mathbb{N}$. Additionally, we have the following properties.
\begin{enumerate}
\item[(i)] Considered as a functor from $\Sch_\CC^{op}$ to abelian groups, $\Con$ commutes with all (not necessarily finite) products, i.e.\ the morphism
\[ \Con(X) \longrightarrow \prod_{X_i\in\pi_0(X)} \Con(X_i) \]
given by restriction to connected components is an isomorphism for all $X\in \Sch_\CC$.
\item[(ii)] ``Base change'' holds, i.e.\ for every cartesian diagram
\[ \xymatrix { X\times_Z Y \ar[r]^{\tilde{v}} \ar[d]_{\tilde{u}}& X \ar[d]^u \\ Y \ar[r]_v & Z } \]
with $u$ and, therefore, also $\tilde{u}$ of finite type, we have $\tilde{u}_!\circ \tilde{v}^\ast = v^\ast \circ u_!$.
\item[(iii)] The functor $\Con$ commutes with exterior products and $\sigma^n$, i.e.\ \\ $(u\boxtimes v)^\ast(a\boxtimes b)=u^\ast(a)\boxtimes v^\ast(b)$
for all $u:X\to X', v:Y\to Y', a\in \Con(X'), b\in \Con(Y')$. If $u, v$ are of finite type, then $(u\boxtimes v)_!(a\boxtimes b)=u_!(a)\boxtimes v_!(b)$ and $\Sym^n(u)_!(\sigma^n(a))=\sigma^n(u_!(a))$ for all $a\in \Con(X), b\in \Con(Y), n\in \mathbb{N}$.
\item[(iv)] Using the convolution product $ab=\oplus_!(a\boxtimes b)$ and thinking of $\Con(\Sym^n(X))$ as being a subgroup of $\Con(\Sym(X))$ by means of $\big(\Sym^n(X)\hookrightarrow\Sym(X)\big)_!$, we have
\[ \sigma^n(a+b)=\sum_{l=0}^n \sigma^l(a)\sigma^{n-l}(b) \]
with $\sigma^1(a)=a$ and $\sigma^0(a)=1\in \Con(\Spec\CC)\hookrightarrow \Con(\Sym(X))$ for all $a,b\in \Con(X)$.
\item[(v)] The ``motivic property'' holds, i.e.\ for every $X$ and every closed subscheme $Z\subseteq X$ giving rise to inclusions $i:Z \hookrightarrow X$ and $j:X\!\setminus\! Z \hookrightarrow X$, we have $i^\ast i_!=\id_{\Con(Z)}, j^\ast j_!=\id_{\Con(X\!\setminus\! Z)}, j^\ast i_!=i^\ast j_!=0$ and
\[ a=i_!i^\ast(a) + j_!j^\ast(a) \quad\forall a\in \Con(X).\]
\item[(vi)] The equation $\sigma^n(\mathbbm{1}_X)=\mathbbm{1}_{\Sym^n(X)}$ holds for all $X$ and all $n\in \mathbb{N}$ with $\mathbbm{1}_X=(X\to \Spec\mathbb{C})^\ast(1)$ and similarly for $\mathbbm{1}_{\Sym^n(X)}$.
\end{enumerate}
\end{proposition}
\begin{exercise}
Show that the projection formula $u_!(a\cap u^\ast(b))=u_!(a)\cap b$ holds for all $u:X\to Y$ of finite type and all $a\in R(X), b\in R(Y)$ by using the properties mentioned in Proposition \ref{motivic_theories} and $a\cap b=\Delta_X^\ast(a\boxtimes b)$. Hint: Consider the diagram
\[ \xymatrix { X \ar[d]_u \ar[r]^{\Delta_X} & X\times X \ar[r]^{\id_X \times u} & X\times Y \ar[d]^{u\times \id_Y} \\ Y \ar[rr]^{\Delta_Y} & & Y\times Y.} \]
\end{exercise}
\subsection{Motivic theories for schemes}
Generalizing constructible functions, we define a motivic theory\footnote{Motivic theories are special cases of reduced motivic $\lambda$-ring $(\Sch,ft)$-theories defined in \cite{DavisonMeinhardt3}. Every reduced motivic $\lambda$-ring $(\Sch,ft)$-theory is a motivic theory in our sense if $\sigma^n(\mathbbm{1}_X)=\mathbbm{1}_{\Sym^n(X)}$ holds for all $X$ and all $n\in \mathbb{N}$. (cf.\ Proposition \ref{motivic_theories}.(vi))} to be a rule associating to every scheme $X$ an abelian group $R(X)$, like $\Con(X)$, along with pull-backs $u^\ast:R(Y)\to R(X)$ for all morphisms $u:X\to Y$ and push-forwards $u_!:R(X)\to R(Y)$ if $u$ is of finite type. Moreover, there should be some associative, symmetric exterior product $\boxtimes:R(X)\times R(Y) \to R(X\times Y)$ with unit element $1\in R(\Spec\CC)$, and some operations $\sigma^n:R(X) \to R(\Sym^n(X))$ for all $n\in \mathbb{N}$, satisfying exactly the same properties as $\Con(X)$ given in Proposition \ref{motivic_theories}. Similar to the case $\Con$, we can construct a $\cap$-product $a\cap b=\Delta_X^\ast(a\boxtimes b)$ with unit $\mathbbm{1}_X=(X\to \Spec\mathbb{C})^\ast(1)$ and a convolution product $ab=+_!(a\boxtimes b)$ with unit $1_0=0_!(1)$ if, additionally, $(X,+,0)$ is a (commutative) monoid with $+$ being of finite type. Note that all these products coincide on $R(\Spec\CC)$.
\begin{exercise} \label{Euler_characteristics}
Given a motivic theory $R$ and a scheme $X$ with morphism $c:X\to \Spec\CC$, we define $[X]_R:=c_!c^\ast(1)\in R(\Spec\CC)$.
\begin{enumerate}
\item[(i)] Show that $[X]_R=[Z]_R+[X\!\setminus\! Z]_R$ (cut and paste relation) for every closed subscheme $Z\subseteq X$ and $[X\times Y]_R=[X]_R[Y]_R$ by applying the defining properties of Proposition \ref{motivic_theories}. In particular, $X\mapsto [X]_R\in R(\Spec\CC)$ is a generalization of the classical Euler characteristic $\chi_c:\Sch_\CC \to \mathbb{Z}=\Con(\Spec\CC)$.
\item[(ii)] Use {\rm (i)} to show $[\mathbb{P}^n]_R=\mathbb{L}_R^n+\ldots+\mathbb{L}_R+1=(\mathbb{L}^{n+1}_R-1)/(\mathbb{L}_R-1)$ with $\mathbb{L}_R:=[{\mathbb{A}^1}]_R.$
\end{enumerate}
\end{exercise}
\begin{exercise} \label{fiber_bundle} We use the notation introduced in the previous exercise.
\begin{enumerate}
\item[(i)] Assume $Y\to X$ is a Zariski locally trivial fiber bundle with fiber $F$. Use the cut and paste relation to prove $[Y]_R=[F]_R[X]_R$.
\item[(ii)] Use {\rm (i)} applied to the projection onto the first column and induction over $n\in \mathbb{N}$ to show $[\Gl(n)]_R=\prod_{i=0}^{n-1} (\mathbb{L}_R^n-\mathbb{L}_R^i)$.
\item[(iii)] Use {\rm (i)} to prove $[\Gr(k,n)]_R=[\Gl(n)]_R/[\Gl(k)]_R[\Gl(n-k)]_R$.
\end{enumerate}
\end{exercise}
A morphism $\eta:R\to R'$ between motivic theories is a collection of group homomorphisms $\eta_X:R(X)\to R'(X)$ commuting with pull-backs, push-forwards and exterior products. It is called a $\lambda$-morphism, if it additionally commutes with the $\sigma^n$-operations. Thus, we obtain a category of motivic theories containing the subcategory of motivic theories with $\lambda$-morphisms.\\
The rule $X\mapsto R(X)=0$ is the terminal object in the category of motivic theories. Moreover, the following holds.
\begin{lemma} \label{initial_object}
The category of motivic theories has an initial object given by the (completed) relative Grothendieck group $\underline{\Ka}_0(\Sch):X\mapsto \underline{\Ka}_0(\Sch_X)$ as constructed below. The unique morphisms starting at $\underline{\Ka}_0(\Sch)$ are even $\lambda$-morphisms.
\end{lemma}
Instead of giving an ad hoc definition of $\underline{\Ka}_0(\Sch)$, let us motivate the construction by looking at constructible functions. Starting with the constant function $1$ on $\Spec\CC$, we take the pull-back $c^\ast(1)=:\mathbbm{1}_X$ for the constant map $c:X \to \Spec\CC$ which is the constant function with value $1$ on $X$, but more importantly, it is the unit object for the $\cap$-product. For any morphism $v:V\to X$ of finite type, consider the function $v_!(\mathbbm{1}_V)\in \Con(X)$ and denote it by\footnote{We already introduced the shorthand $[V]_{\Con}$ for $[V\to \Spec\CC]_{\Con}$ in Exercise \ref{Euler_characteristics}.} $[V\xrightarrow{v} X]_{\Con}$. If $v:V\hookrightarrow X$ is the embedding of a locally closed subscheme, $[V\hookrightarrow X]_{\Con}$ is just the characteristic function of $V$. \\
Using Proposition \ref{motivic_theories}, we get
\begin{eqnarray}
&& \label{eq1} [V\to X]_{\Con}=[Z\to X]_{\Con} + [V\!\setminus\! Z \to X]_{\Con} \mbox{ for all closed }Z\subset V, \\
&& \label{eq2} 1=[\Spec\CC\xrightarrow{\id} \Spec \CC]_{\Con}, \\
&& u^\ast([W\to Y]_{\Con})=[X\times_Y W \to X]_{\Con} \mbox{ for all }u:X\to Y, \\
&& u_!([V\xrightarrow{v} X]_{\Con})=[V\xrightarrow{u\circ v} Y]_{\Con} \mbox{ if }u:X\to Y\mbox{ is of finite type}, \\
&& \label{eq5} [V\xrightarrow{v}X]_{\Con}\boxtimes[W\xrightarrow{w}Y]_{\Con}= [V\times W \xrightarrow{v\times w} X\times Y]_{\Con}, \\
&& \label{eq6} \sigma^n([V\to X]_{\Con})= [\Sym^n(V) \to \Sym^n(X)]_{\Con}.\\
&& \label{eq8} [V\xrightarrow{v}X]_{\Con}=[V'\xrightarrow{v'}X]_{\Con} \mbox{ if there is an isomorphism } \\
&& \nonumber v'':V\to V' \mbox { such that }v=v'\circ v'',
\end{eqnarray}
Obviously, the same must hold in every motivic category as they share the same properties, and so the same applies to the initial object if it exists. Moreover, for connected $X$ the group $\Con(X)$ is generated by all classes $[V\to X]_{\Con}$ satisfying relation (\ref{eq1}). The same must be true for the initial motivic theory since otherwise the subgroup spanned by the elements $[V\to X]_{init}$ for connected $X$ and extended by Property \ref{motivic_theories}(i) for non-connected $X$ is a proper subtheory of the initial theory which will lead to a contradiction. However, there are more relations in $\Con(X)$ as for example $[\mathbb{G}_m\xrightarrow{z^d}\mathbb{G}_m]=d[\mathbb{G}_m\xrightarrow{\id} \mathbb{G}_m]$ which might not hold in other motivic theories as for example the initial one. Dropping the subscript ``init'' we will, therefore, define our (hopefully) initial theory by associating to every connected scheme $X$ the group $\Ka_0(\Sch_X)$ generated by symbols $[V\xrightarrow{v}X]$ for every isomorphism class (due to equation (\ref{eq8})) of morphisms $v:V\to X$ of finite type, subject to the relation (\ref{eq1}). For non-connected $X$ we simply put
\[ \underline{\Ka}_0(\Sch_X):=\prod_{X_i\in \pi_0(X)} \Ka_0(\Sch_{X_i}). \]
To obtain a motivic theory $\underline{\Ka}_0(\Sch)$, we must define $1\in\Ka_0(\Sch_\CC), u^\ast, u_!,\boxtimes$ and $\sigma^n$ as in equations (\ref{eq2})--(\ref{eq6}), at least over connected components. It has been shown in \cite{GLMH1} that $\sigma^n$-operations satisfying these properties do indeed exist. Moreover, the authors prove that $\sigma^n(a\mathbb{L})=\sigma^n(a)\mathbb{L}^n$ holds for every $a\in \underline{\Ka}_0(X)$ and every $n\in \mathbb{N}$, were $\mathbb{L}=c_!c^\ast(1)=[\AA^1\xrightarrow{c} \Spec\mathbb{C}]\in \Ka_0(\Sch_\CC)$ is considered as an element of $\underline{\Ka}_0(\Sch_{\Sym(X)})$ via the embedding $0_!:\Ka_0(\Sch_\CC)\hookrightarrow \underline{\Ka}_0(\Sch_{\Sym(X)})$.
\begin{exercise} Use the properties of a morphism between motivic theories to show that $[V\to X] \mapsto [V\to X]_R$ defines a homomorphism $\eta_X:\Ka_0(\Sch_X)\to R(X)$ for connected $X$ which extends to a morphism $\eta:\underline{\Ka}_0(\Sch)\to R$ of motivic theories. Prove that this morphism is the only possible one. Hence, $\underline{\Ka}_0(\Sch)$ is the initial object in the category of motivic theories. Moreover, show that $\eta$ is a $\lambda$-morphism.
\end{exercise}
The initial property of $\underline{\Ka}_0(\Sch)$ is just a generalization of the well-known property that $X\to [X]\in \Ka_0(\Sch_\CC)$ is the universal Euler characteristics.\\
Let $R^{gm}(X)\subset R(X)$ be the subgroup generated by all elements $[V\to X]_R$ if $X$ is connected and $R^{gm}(X):=\prod_{X_i\in \pi_0(X)}R^{gm}(X_i)$ for general $X$. One should think of elements in $R^{gm}(X)$ as ``geometric'' since they are $\mathbb{Z}$-linear combinations of elements obtained by geometric constructions, namely pull-backs and push-forwards of the unit $1\in R(\Spec\mathbb{C})$.
\begin{exercise} \label{geometric_part} Show that $X\mapsto R^{gm}(X)$ defines a subtheory of $R$. By construction, it is the image of the $\lambda$-morphism $\eta:\underline{\Ka}_0(\Sch)\to R$ obtained in the previous exercise. Show that $\Con^{gm}=\Con$ and $\underline{\Ka}_0(\Sch)^{gm}=\underline{\Ka}_0(\Sch)$. Prove that $\sigma^n(a\mathbb{L}_R)=\sigma^n(a)\mathbb{L}_R^n$ holds for every $a\in R^{gm}(X)$ and every $n\in \mathbb{N}$.
\end{exercise}
\subsection{Motivic theories for quotient stacks}
In the previous section we generalized constructible functions and the classical Euler characteristic to more refined ``functions'' and invariants. When it comes to moduli problems, we should also be able to compute refined invariants of quotient stacks as they occur naturally in moduli problems. Hence, we need to extend motivic theories to disjoint unions of quotient stacks $X/G$ for schemes $X$ locally of finite type over $\mathbb{C}$ and linear algebraic groups $G$. This is the topic of this subsection.
\begin{exercise} \label{special}
Given a closed embedding $G\hookrightarrow \Gl(n)$ of a linear algebraic group $G$ and a $G$-action on a scheme $X$. Show that the morphism $X/G \to (X\times_G \Gl(n))/\Gl(n)$ induced by $X\ni x\mapsto (x,1)\in X\times_G\Gl(n)$ and $G\hookrightarrow \Gl(n)$ as in Exercise \ref{stack_morphisms} is in fact an isomorphism of quotient stacks. Hint: Given a principal $\Gl(n)$-bundle $P\to S$ and a $\Gl(n)$-equivariant morphism $\psi:P\to X\times_G\Gl(n)$, show that $\psi^{-1}(X)\to S$ for $X\hookrightarrow X\times_G\Gl(n)$ is a principal $G$-bundle over $S$. To prove local triviality of $\psi^{-1}(X)\to S$ one has to construct local sections of $\psi^{-1}(X)\to S$. For this, one can take a local section $\nu:U\to P$ of $P\to S$ and a lift $(f,g):\tilde{U} \to X\times \Gl(n)$ of $\psi\circ \nu:U\to X\times_G \Gl(n)$ on a possibly smaller \'{e}tale neighborhood $s\in \tilde{U}\subset U$ of $s\in S$. Then $\tilde{\nu} :\tilde{U}\ni t \longmapsto\to \nu(t)g(t)^{-1}\in \psi^{-1}(X)$ is a local section of $\psi^{-1}(X)\to S$. Note that if $G$ is special, these \'{e}tale neighborhoods $U$ and $\tilde{U}$ can even be replaced with Zariski neighborhoods.
\end{exercise}
\begin{definition} A stacky motivic theory $R$ is a rule associating to every disjoint union $\mathfrak{X}=\sqcup_{i\in I}X_i/G_i$ of quotient stacks with linear algebraic groups $G_i$ an abelian group $R(\mathfrak{X})$ along with pull-backs $u^\ast:R(\mathfrak{Y})\to R(\mathfrak{Y})$ for all (1-)morphisms $u:\mathfrak{X} \to \mathfrak{Y}$ and push-forwards $u_!:R(\mathfrak{X})\to R(\mathfrak{Y})$ if $u$ is of finite type. Moreover, there should be some associative, symmetric exterior product $\boxtimes:R(\mathfrak{X})\times R(\mathfrak{Y}) \to R(\mathfrak{X}\times \mathfrak{Y})$ with unit element $1\in R(\Spec\CC)$, and some operations $\sigma^n:R(X) \to R(\Sym^n(X))$ for all $n\in \mathbb{N}$ and all schemes $X$, satisfying the stacky analogue of the properties of $\Con(-)$ given in Proposition \ref{motivic_theories}.
\end{definition}
\begin{remark} \rm
There are two technical difficulties to overcome when we try to generalize Proposition \ref{motivic_theories}, which serves as our definition of a (stacky) motivic theory, to disjoint unions of quotient stacks. First of all, we need to explain what the correct generalization of a finite type morphism ought to be. For us, this is a (1-)morphism $u:\mathfrak{X}\to \mathfrak{Y}$ of algebraic stacks such the preimage of each ``connected component'' $Y/H$ (with connected $Y$) consists of only finitely many connected components $X_i/G_i$ of $\mathfrak{X}$. Secondly, we need to define $\Sym^n(X/G)$ for quotient stacks. There is an obvious candidate given by the quotient stack $X^n/(S_n\ltimes G^n)$. However, if $G=\{1\}$ is the trivial group, we get the quotient stack $X^n/S_n$ which is different from its coarse ``moduli space'' $\Sym^n(X)=X^n/\!\!/S_n$. To avoid these problems, we only require the existence of $\sigma^n$-operations for schemes $\mathfrak{X}=X$ and not for general disjoint unions of quotient stacks.
\end{remark}
\begin{example}\rm
There is no stacky motivic theory $R$ with $R|_{\Sch_\CC}=\Con$ such that the pull-back
$\rho^\ast:R(X/G) \to R(X)=\Con(X)$ is an embedding. Indeed, consider the case $X=\Spec\CC$ and $G=\mathbb{G}_m$. Then $X \xrightarrow{\rho} X/G \to \Spec\CC$ is the identity, and $\rho_!(\mathbbm{1}_X)$ cannot be zero. By assumption, $\rho^\ast\rho_!(\mathbbm{1}_X)$ is also nonzero. However, applying base change to the diagram
\[ \xymatrix { X\times G \ar[r]^m \ar[d]_{\pr_X} & X \ar[d]^\rho \\ X \ar[r]_\rho & X/G }\]
with $m:X\times G \to X$ denoting the (trivial) group action, $\rho^\ast\rho_!(\mathbbm{1}_X)=\pr_{X\, !}(\mathbbm{1}_{X\times G})=\chi_c(G)\mathbbm{1}_X =0$, a contradiction.
\end{example}
Applying the functoriality of the pull-back to the previous diagram, we obtain $\pr_X^{\ast}(b)=m^\ast(b)$ for $b=\rho^\ast(a)$. In other words, for every stacky motivic theory $R$, the image of $\rho^\ast$ is contained in the subgroup $R(X)^G:=\{a\in R(X)\mid \pr_X^{\ast}(b)=m^\ast(b)\}$ of ``$G$-invariant'' elements.
Despite the negative result given by the previous example, we will provide a functorial construction which associates to every motivic theory $R$ satisfying
\begin{equation} \label{eq7}
\sigma^n(a\mathbb{L}_R)=\sigma^n(a)\mathbb{L}_R^n \quad\forall \;a\in R(X)
\end{equation}
another motivic theory $R^{st}$ such that $R^{st}$ extends to a stacks motivic theory, also denoted with $R^{st}$, for which $\rho^\ast:R^{st}(X/G)\to R^{st}(X)^G$ is an isomorphism. Moreover, there is a morphism $R\to R^{st}|_{\Sch_\CC}$ of motivic theories satisfying the property that every morphism $R\to R'|_{\Sch_\CC}$ with $R'$ being a stacky motivic theory satisfying $\rho^\ast:R'(X/G) \xrightarrow{\sim} R'(X)^G$ must factorize though $R\to R^{st}$. In particular, the restriction functor from the category of stacky motivic theories $R'$ satisfying (\rm \ref{eq7}) and $\rho^\ast:R'(X/G) \xrightarrow{\sim} R'(X)^G$ has a left adjoint given by $R\to R^{st}$. As we will see, $\Con^{st}=0$.
Recall that a linear algebraic group $G$ was called special if every \'{e}tale locally trivial principal $G$-bundle is already Zariski locally trivial. In particular, given a closed embedding $G\hookrightarrow \Gl(n)$, the map $\Gl(n)\to \Gl(n)/G$ must be a Zariski locally trivial principal $G$-bundle. On can show that this property is already sufficient for being special. Hence, $\Gl(n)$ is special for every $n\in \mathbb{N}$. As a result of Exercise \ref{fiber_bundle} we get $[\Gl(n)]_R=[G]_R[\Gl(n)/G]_R$ in $R(\Spec\CC)$ for every motivic theory $R$. In particular, $[G]_R$ is invertible for every special group $G$ if and only if $[\Gl(n)]_R$ is invertible for every $n\in \mathbb{N}$.
\begin{definition}
Given a group $G$, a (1-)morphism $u:\mathfrak{P}\to \mathfrak{X}$ of stacks is called a principal $G$-bundle on $\mathfrak{X}$ if $u$ is representable and the pull-back $\tilde{u}:X\times_\mathfrak{X} \mathfrak{P} \longrightarrow X$ of $u$ along every morphism $X\to \mathfrak{X}$ with $X$ being a scheme is a principal $G$-bundle on $X$.
\end{definition}
\begin{exercise}
Given a stacky motivic theory $R$. We want to show in several steps that the condition $\rho^\ast: R(X/G)\to R(X)^G$ being an isomorphism for every special group $G$ is equivalent to the condition that $[\mathfrak{P}\xrightarrow{u} \mathfrak{X}]_R:=u_!(\mathbbm{1}_\mathfrak{P})=[G]\mathbbm{1}_\mathfrak{X}$ for every special group $G$ and every principal $G$-bundle $u:\mathfrak{P}\to \mathfrak{X}$ in the category of disjoint unions of quotient stacks.
\begin{enumerate}
\item Given a principal $G$-bundle $\mathfrak{P}\to \mathfrak{X}$ and assume for simplicity $\mathfrak{X}=X/\Gl(n)$. Consider the cartesian diagram
\[ \xymatrix {P \ar[r]^{\tilde{\rho}} \ar[d]^{\tilde{u}} & \mathfrak{P} \ar[d]^u \\ X \ar[r]^\rho & \mathfrak{X}.} \]
By assumption, $\tilde{u}:P\to X$ is a principal $G$-bundle. Use injectivity of $\rho^\ast$ to prove $u_!(\mathbbm{1}_\mathfrak{P})=[G]_R\mathbbm{1}_\mathfrak{X}$ (Hint: Use base change and Exercise \ref{fiber_bundle}.) Extend this result to arbitrary disjoint unions of quotient stacks.
\item Conversely, assume that $u_!(\mathbbm{1}_\mathfrak{P})=[G]_R\mathbbm{1}_\mathfrak{X}$ for every principal $G$-bundle in the category of quotient stacks.
Show first that $[G]_R$ is invertible in $R(\Spec\CC)$ with inverse $[\Spec\CC/G]_R$. (Hint: Consider the principal $G$-bundle $\Spec\CC \to \Spec\CC/G$.)
\item Secondly, prove that $\rho^\ast:R(X/G)\to R(X)^G$ is invertible by showing that $\rho_!(-)/[G]_R$ is an inverse. (see \cite{DavisonMeinhardt3}, Lemma 5.13 if you need help)
\end{enumerate}
\end{exercise}
For connected $X$ we define $R^{st}(X):=R(X)[ [\Gl(n)]^{-1}_R \mid n\in \mathbb{N}]$ using the $R(\Spec\CC)$-module structure of $R(X)$ by means of the $\boxtimes$-product. We extend it via $R^{st}(X)=\prod_{X_i\in \pi_0(X)}R^{st}(X_i)$ to non-connected $X$. The morphism $R(X)\to R^{st}(X)$ is obvious, and it is also easy to see how to extend $u_!, u^\ast$ for $u:X\to Y$ and the $\boxtimes$-product. The only nontrivial part is the extension of $\sigma^n$. For $X\in \Sch_\CC$ and $a\in R(X)$ define
\[ \sigma_t(a):=\sum_{n\in \mathbb{N}} \sigma^n(a)t^n \in R(\Sym(X)[[t]] \]
The Adams operations $\psi^n:R(X)\to R(\Sym^n(X))\subseteq R(\Sym(X))$ are defined by means of the series
\[ \psi_t(a):=\sum_{n\ge 1} \psi^n(a)t^{n} := \frac{d\log \sigma_t(a)}{d\log t}=t\sigma_t(-a)\frac{d\sigma_t(a)}{dt},\]
where the product is the convolution product in $R(\Sym(X))$. Using the properties of $\sigma^n$, we observe $\sigma_t(0)=1$ and $\sigma_t(a+b)=\sigma_t(a)\sigma_t(b)$. Thus, $\psi_t(0)=0$ as well as $\psi_t(a+b)=\psi_t(a)+\psi_t(b)$ follows. Property {\rm (\ref{eq7}) } implies $\psi^n(a P(\mathbb{L}_R))=\psi^n(a)P(\mathbb{L}_R^n)$ for every polynomial $P(x)\in \mathbb{Z}[x]$. Due to Exercise \ref{fiber_bundle}(ii), we have to extend $\psi^n$ to $R^{st}(X)$ by means of
\[ \psi^n\left( \frac{a}{\prod_{i\in I} [\Gl(m_i)]_R }\right):= \frac{\psi^n(a)}{\prod_{i\in I} P_{m_i}(\mathbb{L}^n_R)} \]
using the polynomial $P_m(x)=\prod_{j=0}^{m-1}(x^m-x^j)$ satisfying $[\Gl(m)]_R=P_m(\mathbb{L}_R)$. Having extended the Adams operations, we can also extend the $\sigma^n$-operations by putting
\[ \sigma_t(a)=\exp\left(\int \psi_t(a)\frac{dt}{t}\right). \]
Note that the last expression involves rational coefficient, but one can show that the rational coefficients disappear in the expression for
\[ \sigma_t\left(\frac{a}{\prod_{i\in I}[\Gl(m_i)]_R}\right)=\exp\left(\sum_{n\ge 1} \frac{\psi^n(a)t^n}{n\prod_{i\in I} P_{m_i}(\mathbb{L}_R^n)} \right) \]
if we express $\psi^n(a)$ in terms of $\sigma^m(a)$ for $1\le m\le n$. (See \cite{DavisonMeinhardt3}, Appendix B for more details.)
\begin{exercise}
Show that $\Con^{st}(X)=0$ for all $X$.
\end{exercise}
Now, as we have constructed $R^{st}$ on schemes, we will put $R^{st}(\mathfrak{X}):=R^{st}(X)^G$ for a quotient stack $\mathfrak{X}=X/G$ with special group $G$ and connected $X$. We have to show that this definition is independent of the presentation of the quotient stack. For this let $\mathfrak{X}\cong Y/H$ be another presentation with a special group $H$. Let us form the cartesian square
\[ \xymatrix @C=1.5cm @R=1.5cm { X\times_\mathfrak{X} Y\times G\times H \ar@/_1pc/[d] \ar@/^1pc/[d] \ar@/^1pc/[r] \ar@/_1pc/[r] & X\times_\mathfrak{X} Y \times H \ar@/^1pc/[d] \ar@/_1pc/[d] \ar[r]^{\rho''} & Y\times H \ar@/^1pc/[d]^{m_Y} \ar@/_1pc/[d]_{\pr_Y} \\
X \times_\mathfrak{X} Y \times G \ar[d]^{\tau''} \ar@/^1pc/[r] \ar@/_1pc/[r] & X\times_\mathfrak{X} Y \ar[d]^{\tau'} \ar[r]^{\rho'} & Y \ar[d]^\tau \\
X\times G \ar@/^1pc/[r]^{m_X} \ar@/_1pc/[r]_{\pr_X} & X \ar[r]^\rho & \mathfrak{X} }
\]
with $\rho',\rho''$ and $\tau',\tau''$ being $G$- respectively $H$-principal bundles. The other maps are either projections or actions of $G$ or $H$. Applying $R^{st}$, we get the following diagram with exact rows and columns by construction of $R^{st}$, where $K$ denotes the kernel of say $\pr_Y^\ast-m_Y^\ast$.
\[ \xymatrix @C=1.3cm { & 0 \ar[d] & 0 \ar[d] & 0 \ar[d] \\
& K \ar[r] \ar[d] & R^{st}(X) \ar[r]^{\pr_X^\ast-m_X^\ast} \ar[d]^{\tau'^\ast} & R^{st}(X\times G) \ar[d]^{\tau''^\ast} \\
0 \ar[r] & R^{st}(Y) \ar[r]^{\rho'^\ast} \ar[d]^{\pr_Y^\ast-m_Y^\ast} & R^{st}(X\times_\mathfrak{X} Y) \ar[r] \ar[d] & R^{st}(X\times_\mathfrak{X} Y \times G) \ar[d] \\
0 \ar[r] & R^{st}(Y\times H) \ar[r]_{\rho''^\ast} & R^{st}(X\times_\mathfrak{X} Y \times H) \ar[r] & R^{st}(X\times_\mathfrak{X} Y \times G\times H) }
\]
Hence, $R^{st}(X)^{G}\cong K\cong R^{st}(Y)^{H}$ showing that $R^{st}(\mathfrak{X})$ is independent of the choice of a presentation.
Given a morphism $u:X/G \to Y/H$ of quotient stacks, we form the cartesian diagram
\[ \xymatrix { X\times_{Y/H} Y \ar[d]_{\tilde{\tau}} \ar[r]^(0.4){\tilde{\rho}} & X/G\times_{Y/H} Y \ar[r]^(0.6){\tilde{u}} \ar[d] & Y \ar[d]^\tau \\
X \ar[r]^\rho & X/G \ar[r]^u & Y/H. } \]
For $a\in R^{st}(X/G)\cong R^{st}(X)^G$ and $b\in R^{st}(Y)^H$ we put
\begin{eqnarray*}
u_!(a)&:=&(\tilde{u}\circ \tilde{\rho})_!\tilde{\tau}^\ast(a)/[G]_R\in R^{st}(Y)^H\mbox{ and }\\
u^\ast(b)&:=&\tilde{\tau}_!(\tilde{u}\circ\tilde{\rho})^\ast(b)/[H]_R\in R^{st}(X)^G\\
a\boxtimes b&:=& a\boxtimes b\in R^{st}(X\times Y)^{G\times H}.
\end{eqnarray*}
For disjoint unions $\mathfrak{X}=\sqcup_{i\in I}X_i/G_i$ of connected quotient stacks, we can always assume that $G_i$ is special for all $i\in I$ due to Exercise \ref{special}. Then, we need to define $R^{st}(\mathfrak{X}):=\prod_{i\in I}R^{st}(X_i/G_i)$ according to Proposition \ref{motivic_theories}(i), and extend $u_!, u^\ast$ and $\boxtimes$ in the natural way.
Given a morphism $\eta:R\to R'|_{\Sch_\CC}$ with $\rho^\ast:R'(X/G)\to R'(X)^G$ being an isomorphism, we define $\eta_{X/G}:R^{st}(X/G)=R(X)^G \xrightarrow{\eta_X} R'(X)^G\xrightarrow{\rho^{\ast\, -1}} R'(X/G)$ with $\rho^{\ast\, -1}(a)=\rho_!(a)/[G]_R$ which is the only possible choice to extend $\eta$ to a morphism $R^{st}\to R'$ of stacky motivic theories. More details in a more general context are given in \cite{DavisonMeinhardt3}, Section 5.
\section{Vanishing cycles}
The aim of this section is to introduce the notion of a vanishing cycle taking values in a (stacky) motivic theory $R$. We start by considering vanishing cycles of morphisms $f:X\to {\mathbb{A}^1}$ defined on smooth schemes $X$.
\subsection{Vanishing cycles for schemes}
\begin{definition}
Given a motivic theory $R$, a vanishing cycle\footnote{The definition given here differs from the one given in \cite{DavisonMeinhardt3} for the sake of simplicity. We do not require the support property but a blow-up formula.} (with values in $R$) is a rule associating to every regular function $f:X\to {\mathbb{A}^1}$ on a smooth scheme/variety or complex manifold $X$ an element $\phi_f\in R(X)$ such that the following holds.
\begin{enumerate}
\item If $u:Y\to X$ is a smooth, then $\phi_{f\circ u}=f^\ast(\phi_f)$.
\item Let $X$ be a smooth variety containing a smooth closed subvariety $i:Y\hookrightarrow X$. Denote by $j:E\hookrightarrow \Bl_Y X$ the exceptional divisor of the blow-up $\pi:\Bl_Y X \to X$ of $X$ in $Y$. Then the formula \[ \pi_!\big( \phi_{f\circ \pi} - j_! \phi_{f\circ \pi\circ j}\big) = \phi_f - i_!\phi_{f\circ i}\] holds for every $f:X\to {\mathbb{A}^1}$.
\item Given two morphisms $f:X\to {\mathbb{A}^1}$ and $g:Y\to {\mathbb{A}^1}$ on smooth $X$ and $Y$, we introduce the notation $f\boxtimes g:X\times Y\xrightarrow{f\times g} {\mathbb{A}^1}\times {\mathbb{A}^1}\xrightarrow{+} {\mathbb{A}^1}$. Then $\phi_{f\boxtimes g}=\phi_f\boxtimes \phi_g$ in $R(X\times Y)$. Moreover, $\phi_{\Spec\CC\xrightarrow{0}{\mathbb{A}^1}}(1)=1$.
\end{enumerate}
\end{definition}
\begin{lemma} \label{vanishing_cycle_morphism}
A collection of elements $\phi_f\in R(X)$ for regular functions $f:X\to {\mathbb{A}^1}$ on smooth schemes $X$ satisfying the properties (1),(2) and (3) is equivalent to a collection of group homomorphisms\footnote{The collection of group homomorphisms $\phi_f$ is what is called a morphisms of (motivic) ring $(Sm,prop)$-theories over ${\mathbb{A}^1}$ in \cite{DavisonMeinhardt3}.} $\phi_f:\underline{\Ka}_0(\Sch_X)\to R(X)$ for all regular functions $f:X\to {\mathbb{A}^1}$ on arbitrary schemes $X$ such that the following diagrams commute
\[ \xymatrix @C=1.5cm{ \underline{\Ka}_0(\Sch_X) \ar[r]^{\phi_f} \ar[d]^{u^\ast} & R(X) \ar[d]^{u^\ast} \\ \underline{\Ka}_0(\Sch_Y) \ar[r]^{\phi_{f\circ u}} & R(Y) } \qquad\mbox{if }u:Y\to X\mbox{ is smooth,} \]
\[ \xymatrix @C=1.5cm{ \underline{\Ka}_0(\Sch_Y) \ar[r]^{\phi_{f\circ u}} \ar[d]^{u_!} & R(Y) \ar[d]^{u_!} \\ \underline{\Ka}_0(\Sch_X) \ar[r]^{\phi_{f}} & R(X) } \qquad\mbox{if }u:Y\to X\mbox{ is proper,} \]
\[ \xymatrix @C=2.5cm { \underline{\Ka}_0(\Sch_X)\otimes\underline{\Ka}_0(\Sch_Y) \ar[r]^{\phi_{f}\otimes \phi_g} \ar[d]^{\boxtimes} & R(X)\otimes R(Y) \ar[d]^{\boxtimes} \\ \underline{\Ka}_0(\Sch_{X\times Y}) \ar[r]^{\phi_{f\boxtimes g}} & R(X\times Y) } \]
and $\phi_{\Spec\mathbb{C}\xrightarrow{0}{\mathbb{A}^1}}(1)=1$.
\end{lemma}
\begin{exercise}
Proof the lemma using the following fact (see \cite{Bittner04}, Thm.\ 5.1). The group $\Ka_0(\Sch_Z)$ can also be written as the abelian group generated by symbols $[X\xrightarrow{p} Z]$ with smooth $X$ and proper $p$ subject the ``blow-up relation'': If $i:Y\hookrightarrow X$ is a smooth subvariety and $\pi:\Bl_Y X \to X$ the blow-up of $X$ in $Y$ with exceptional divisor $j:E\hookrightarrow \Bl_Y X$, then $[\Bl_Y X \xrightarrow{p\pi} Z] - [E \xrightarrow{p\pi j} Z] = [X\xrightarrow{p}Z]-[Y\xrightarrow{p i} Z]$. Hint: Given a function $f:Z\to {\mathbb{A}^1}$, try the Ansatz $\phi_f([X\xrightarrow{p} Z]):=p_!\phi_{f\circ p}$ for a proper morphism $X\xrightarrow{p}Z$ on a smooth scheme $X$, where $\phi_{f\circ p}\in R(X)$ on the right hand side is given by our family of elements. In particular, $\phi_f(\mathbbm{1}_X)=\phi_f\in R(X)$ for a regular function $f$ on a smooth scheme $X$.
\end{exercise}
We need to apologize for using the same symbol $\phi_f$ with two different meanings. However, with a bit of practice it should be clear from the context which interpretation is used.
\begin{example}\rm
For every motivic theory there is a canonical vanishing cycle such that $\phi^R_{can,f}=\mathbbm{1}_{X}\in R(X)$ for every $f:X\to {\mathbb{A}^1}$. Hence, it does not depend on $f$, and the map $\phi_f:\underline{\Ka}_0(\Sch_X)\to R(X)$ is just the morphism constructed in Lemma \ref{initial_object}.
\end{example}
Let us look at the following more interesting examples.
\begin{example} \rm
Let $R=\Con$. For $x\in X$ we fix a metric on a an analytic neighborhood of $x\in X^{an}$ for example by embedding such a neighborhood into $\mathbb{C}^n$. We form the so-called Milnor fiber $\MF_{f}(x):=f^{-1}(f(x)+\delta)\cap B_\varepsilon(x)$, where $B_\varepsilon(x)$ is a small open ball around $x\in X$ and $0<\delta \ll \varepsilon \ll 1$ are small real parameters. Notice, that the Milnor fiber depends on the choice of the metric and the choice of $\delta,\varepsilon$. However, its reduced cohomology and its Euler characteristic $\chi(\MF_f(x))$ are independent of the choices made. We finally define $\phi^{con}_f(x):=1-\chi(\MF_f(x))$ for sufficiently small $\delta,\varepsilon$. One can show that the properties listed above are satisfied. Moreover, $\phi^{con}_f$ agrees with the Behrend function of $\Crit(f)$ up to the sign $(-1)^{\dim X}$.
\end{example}
\begin{example} \rm
The previous example has a categorification. For a closed point $t\in {\mathbb{A}^1}(\mathbb{C})$ consider the cartesian diagram
\[ \xymatrix @C=1.5cm { X^{an} \ar[d]^{f} & X^{an}\times_\mathbb{C} \mathbb{C} \ar[d]^{\tilde{f}} \ar[l]^{q_t} \\ \mathbb{C} & \ar[l]^{t+\exp(-)} \mathbb{C}.}\]
Denoting the inclusion of the fiber $X_t=f^{-1}(t)$ into $X$ by $\iota_t$, the (classical) vanishing cycle $\phi^{perv}_f$ is defined via
\[ \oplus_{t\in {\mathbb{A}^1}(\mathbb{C})} \iota_{t\,!}\Cone(\mathbb{Q}_{X_t}\longrightarrow \iota_t^\ast q_{t\,\ast} q_t^\ast \mathbb{Q}_X), \]
in the derived category $D^b(X,\mathbb{Q})$ of sheaves of $\mathbb{Q}$-vector spaces on $X$. Here, $\mathbb{Q}_X$ respectively $\mathbb{Q}_{X_t}=\iota_t^\ast \mathbb{Q}_X$ denotes the locally constant sheaf on $X$ respectively $X_t$ with stalk $\mathbb{Q}$. The morphism is the restriction to $X_t$ of the adjunction $\mathbb{Q}_X \longrightarrow q_{t\,\ast} q_t^\ast \mathbb{Q}_X$. Spelling out the definition we see that the stalk of $\phi^{perv}_f$ at $x$ is given by the reduced cohomology of the Milnor fiber $\MF_f(x)$ shifted by $-1$. \\
Associating to every connected $X$ the Grouthendieck group $\Ka_0(D^b_{con}(X,\mathbb{Q}))=\Ka_0(\Perv(X))$ of the triangulated subcategory $D^b_{con}(X,\mathbb{Q})\subset D^b(X,\mathbb{Q})$ consisting of complexes of sheaves of $\mathbb{Q}$-vector spaces with constructible cohomology, we obtain a motivic theory $\underline{\Ka}_0(D^b_{con}(-,\mathbb{Q}))$ with $\underline{\Ka}_0(D^b_{con}(X,\mathbb{Q})):=\prod_{X_i\in \pi_0(X)} \Ka_0(D^b_{con}(X_i,\mathbb{Q}))$. Since $\phi_f^{perv}$ turns out to be a complex with constructible cohomology, we can take its class in $\underline{\Ka}_0(D^b_{con}(X,\mathbb{Q}))$ and get a vanishing cycle satisfying all required properties.
\end{example}
\begin{example} \rm
The previous example has a refinement $\phi_f^{mhm}\in D^b(\MHM(X)_{mon})$ involving (complexes of) ``monochromatic mixed Hodge modules with monodromy groups of the form $\mu_n$, the group of n-th roots of unity, for some $n\in \mathbb{N}$. Forgetting the Hodge and the monodromy structure, we get a functor $D^b(\MHM(X)_{mon})\longrightarrow D^b_{con}(X,\mathbb{Q})$ mapping $\phi_f^{mhm}$ to $\phi_f^{perv}$. By passing to Grothendieck groups, we get a vanishing cycle $\phi^{mhm}_f$ with values in $\underline{\Ka}_0(D^b(\MHM(X))_{mon})=\underline{\Ka}_0(\MHM(X)_{mon}):=\prod_{X_i\in \pi_0(X)} \Ka_0(\MHM(X_i)_{mon})$.
\end{example}
In the remaining part of this subsection we will construct vanishing cycles depending functorially on $R$. First of all we need to enlarge $R$ by defining a new motivic theory $R(-\times {\mathbb{A}^1})$ mapping $X$ to $R(X\times {\mathbb{A}^1})$ and using the exterior product
\[ R(X \times {\mathbb{A}^1})\otimes R(Y\times {\mathbb{A}^1}) \xrightarrow{\boxtimes} R( X\times Y \times \AA^2) \xrightarrow{(\id\times +)_!} R(X\times Y\times {\mathbb{A}^1}) \]
with unit $1':=0_!(1)\in R(\Spec\mathbb{C}\times{\mathbb{A}^1})=R({\mathbb{A}^1})$ and the $\sigma^n$-operations
\[ R(X\times {\mathbb{A}^1}) \xrightarrow{\sigma^n} R(\Sym^n(X\times {\mathbb{A}^1}))\longrightarrow R(\Sym^n(X)\times \Sym^n({\mathbb{A}^1})) \xrightarrow{(\id\times +)_!} R(\Sym^n(X)\times{\mathbb{A}^1}). \]
\begin{exercise}
Check the properties of a motivic theory given in Proposition \ref{motivic_theories}.
\end{exercise}
Given a scheme $X$, let $\mathbb{G}_m$ act on $X\times {\mathbb{A}^1}$ via $g(x,z)=(x,gz)$. For connected $X$ we denote with $R^{gm}_{\mathbb{G}_m}(X\times {\mathbb{A}^1})$ the subgroup of $R(X\times {\mathbb{A}^1})$ generated by elements $[Y\xrightarrow{f} X\times {\mathbb{A}^1}]_R$ such that $Y$ carries a good\footnote{An action of $\mathbb{G}_m$ on $Y$ is called good if every point $y\in Y$ has an affine $\mathbb{G}_m$-invariant neighborhood.} $\mathbb{G}_m$-action for which $f$ is homogeneous of some degree $d> 0$, i.e.\ $f(gy)=g^df(y)$. Notice, that such a $Y$ will carry many actions for which $f$ is homogeneous. Indeed, given $0\not=n\in \mathbb{N}$, let $\mathbb{G}_m$ act on $Y$ via $g\star y:=g^n y$ using the old action on the right hand side. Then, $f$ is homogeneous of degree $dn$ with respect to the new action. In particular, given finitely many generators $[Y_i\xrightarrow{f_i} X\times {\mathbb{A}^1}]$, we can always assume that the degrees of $f_i$ are equal. Finally, we put $R^{gm}_{\mathbb{G}_m}(X\times {\mathbb{A}^1}):=\prod_{X_i\in \pi_0(X)} R^{gm}_{\mathbb{G}_m}(X_i\times {\mathbb{A}^1})$ for non-connected $X$.
\begin{lemma} \label{monodromy_extension}
The subgroup $R^{gm}_{\mathbb{G}_m}(X\times {\mathbb{A}^1})\subset R(X\times {\mathbb{A}^1})$ is invariant under pull-backs, push-forwards, exterior products and the $\sigma^n$-operations. Moreover, $\pr_X^\ast$ maps $R^{gm}(X)$ onto a ``$\lambda$-ideal'' $I^{gm}_X\subseteq R^{gm}_{\mathbb{G}_m}(X\times {\mathbb{A}^1})$, i.e.\ $a\boxtimes b \in I^{gm}_{X\times Y}$ for $a\in I^{gm}_X, b\in R^{gm}_{\mathbb{G}_m}(Y\times {\mathbb{A}^1})$ and $\sigma^n(a)\in I^{gm}_{\Sym^n(X)}$ for $a\in I^{gm}_X$.
\end{lemma}
\begin{exercise}
Check the first sentence of the previous lemma.
\end{exercise}
\begin{proof}
To show that $I^{gm}_X$ is a $\lambda$-ideal, it suffices to look at generators $[V\times{\mathbb{A}^1}\xrightarrow{f\times \id_{{\mathbb{A}^1}}} X\times{\mathbb{A}^1}]$ and $[W \xrightarrow{(g,h)} Y\times {\mathbb{A}^1}]$ of $I^{gm}_X$ and $R^{gm}_{\mathbb{G}_m}(Y\times{\mathbb{A}^1})$ respectively with $[V\xrightarrow{f} X]\in R^{gm}(X)$. (cf.\ Exercise \ref{generators}) For the $\boxtimes$-product
\[ [V\times {\mathbb{A}^1}\to X\times {\mathbb{A}^1}] \boxtimes [W \to Y\times {\mathbb{A}^1}] = [V\times{\mathbb{A}^1}\times W\xrightarrow{u} X\times Y\times {\mathbb{A}^1}] \]
with $u(v,z,w)=(f(v),g(w),z+h(w))$ we use the isomorphism
\[ V\times {\mathbb{A}^1} \times W \ni(v,z,w)\longmapsto (v,w,z+h(w))\in V\times W\times {\mathbb{A}^1}\]
to show $[V\times{\mathbb{A}^1}\times W\xrightarrow{u} X\times Y\times {\mathbb{A}^1}]=\pr^\ast_{X\times Y}([V\times W\xrightarrow{f\times g} X\times Y])\in I^{gm}_{X\times Y}$.
We also have
\[ \sigma^n[V\times {\mathbb{A}^1}^1 \xrightarrow{f\times\id_{{\mathbb{A}^1}}} X\times {\mathbb{A}^1}]=[ \Sym^n(V\times {\mathbb{A}^1}) \xrightarrow{\tilde{p}} \Sym^n(X)\times {\mathbb{A}^1} ] \]
with $\tilde{p}$ being induced by the $S_n$-invariant morphism $p:(V\times {\mathbb{A}^1})^n\to \Sym^n(X)\times {\mathbb{A}^1}$ with $p\big((v_1,z_1),\ldots,(v_n,z_n)\big)=\big((f(v_1),\ldots,f(v_n)),z_1+\ldots + z_n\big)$. We define
\[ (V^n\times \AA^n)^0:=\{(v_1,\ldots,v_n,z_1,\ldots,z_n) \in V^n\times \AA^n \mid z_1 + \ldots z_n=0 \} \]
for all $n>0$. There is a $S_n$-equivariant isomorphism $\psi:(V^n\times \AA^n)^0\times {\mathbb{A}^1} \longrightarrow (V \times {\mathbb{A}^1})^n$ sending $((v_1,\ldots,v_n,z_1,\ldots,z_n),z)$ to $\big((v_1,z_1+z/n),\ldots,(v_n, z_n + z/n)\big)$. Then, $p\circ \psi=q\times \id_{\mathbb{A}^1}: (V^n\times \AA^n)^0\times {\mathbb{A}^1}\longrightarrow \Sym^n(X)\times {\mathbb{A}^1}$ for the $S_n$-invariant morphism $q:(V^n\times \AA^n)^0\twoheadrightarrow (V^n\times \AA^n)^0/\!\!/S_n\xrightarrow{\tilde{q}} \Sym^n(X)$ with $q(v_1,\ldots,v_n,z_1,\ldots,z_n)=(f(v_1),\ldots,f(v_n))$. Modding out the $S_n$-action, we see that
\[ [ \Sym^n(V\times {\mathbb{A}^1}) \xrightarrow{\tilde{p}} \Sym^n(X)\times {\mathbb{A}^1} ]=\pr^\ast_{\Sym^n}([(V^n\times {\mathbb{A}^1}^n)^0/\!\!/S_n \xrightarrow{\tilde{q}} \Sym^n(X)])\]
is indeed in $I^{gm}_{\Sym^n(X)}$.
\end{proof}
\begin{exercise} \label{generators} Convince yourself using the formula $\sigma^n(a+b)=\sum_{l=0}^n\pi^{(l)}_!(\sigma^l(a)\boxtimes \sigma^{n-l}(b))$ for the motivic theory $R(-\times {\mathbb{A}^1})$ with $\pi^{(l)}:\Sym^l(X)\times \Sym^{n-1}(X)\longrightarrow \Sym^n(X)$ being the natural map, that $a\in I^{gm}_X$ implies $\sigma^n(a)\in I^{gm}_{\Sym^n(X)}$ is indeed true if it already holds for generators $a=\pr^\ast_X([V\xrightarrow{f} X])$ of $I^{gm}_X$.
\end{exercise}
Due to Lemma \ref{monodromy_extension}, we can form the quotient $R^{gm}_{mon}(X)=R^{gm}_{\mathbb{G}_m}(X\times {\mathbb{A}^1})/\pr_X^\ast R^{gm}(X)$ and obtain a new motivic theory together with a morphism $R^{gm}\to R^{gm}_{mon}$ of motivic theories (cf.\ Exercise \ref{geometric_part}) given by $R^{gm}(X) \xrightarrow{(\id\times 0)_!} R^{gm}_{\mathbb{G}_m}(X\times {\mathbb{A}^1}) \twoheadrightarrow R^{gm}_{mon}(X)$.
\begin{exercise}
Fix a motivic theory $R$ and consider the map $R^{gm}_{\mathbb{G}_m}(X\times {\mathbb{A}^1}) \longrightarrow R^{gm}(X)$ given by $a\longmapsto (\id\times 0)^\ast(a)-(\id\times 1)^\ast(a)$, where $0,1:\Spec\CC \to {\mathbb{A}^1}$ are the obvious maps. As $\pr_X^\ast R^{gm}(X)$ is in the kernel, we obtain a well-defined group homomorphism $R^{gm}_{mon}(X)\to R^{gm}(X)$. Note that the composition $R^{gm}(X) \rightarrow R^{gm}_{mon}(X) \rightarrow R^{gm}(X)$ is the identity. Hence, $R^{gm}(X)$ is a direct summand of $R^{gm}_{mon}(X)$. However, show that the retraction $R^{gm}_{mon}(X)\to R^{gm}(X)$ is not a morphism of motivic theories and $R^{gm}$ is not a direct summand of the motivic theory $R^{gm}_{mon}$.
\end{exercise}
\begin{exercise}
Using the notation of the previous exercise, show that the kernel of $\Con^{gm}_{mon}(X) \longrightarrow \Con^{gm}(X)=\Con(X)$ is trivial, i.e.\ $\Con(X)=\Con^{gm}_{mon}(X)$. On the other hand, show that the kernel is nonzero for $X=\Spec\CC$ and $R=\underline{\Ka}_0(D^b(-,\mathbb{Q}))$, $R=\underline{\Ka}_0(\MHM(-)_{mon})$ and $R=\underline{\Ka}_0(\Sch)$.
\end{exercise}
If $R=R^{gm}$, we suppress the superscript ``gm'' from notation. This applies for instance to $\underline{\Ka}_0(\Sch)$ but also to $R^{gm}$ as $(R^{gm})^{gm}=R^{gm}$.
\begin{exercise}
Check that $R^{gm}_{mon}=(R^{gm})^{gm}_{mon}=(R^{gm})_{mon}$ using our convention for the last equation.
\end{exercise}
If $R^{gm}\subsetneq R$ is a proper subtheory, as for example for $\underline{\Ka}_0(D^b_{con}(-,\mathbb{Q}))$ or for $\underline{\Ka}_0(\MHM(-)_{mon})$, we can nevertheless define a theory $R_{mon}$ under the assumption that the formula
\[ \sigma^n(a\boxtimes b) =\sum_{\lambda \dashv n} \pi^{(n)}_!\Big(P^\lambda(\sigma^1(a),\ldots,\sigma^n(a))\boxtimes P^\lambda(\sigma^1(b),\ldots,\sigma^n(b))\Big) \]
holds in $R(\Sym^n(X\times Y))$ for all $0\not=n\in \mathbb{N}$, all $a\in R(X),b\in R(Y)$ and all $X,Y$, where the sum is taken over all partitions $\lambda=(\lambda_1\ge\ldots \ge \lambda_n\ge 0)$ of $n$ and
\[ P^\lambda(x_1,\ldots,x_n)=\det(x_{\lambda_i}+j-i)_{1\le i,j\le n}=\left|\begin{array}{cccc} x_{\lambda_1} & x_{\lambda_1+1} & \ldots & x_{\lambda_{1}+n-1} \\ x_{\lambda_2-1} & x_{\lambda_2} & \ldots & x_{\lambda_2+n-2} \\ \vdots & \vdots & \ddots & \vdots \\ x_{\lambda_n-n+1} & x_{\lambda_n-n+2} & \ldots & x_{\lambda_n} \end{array} \right| \]
is the polynomial from the Jacobi-Trudi formula with the convention $x_0=1$ and $x_m=0$ for $m<0$ or $m>n$. Here, $\pi^{(n)}:\Sym^n(X)\times \Sym^n(Y)\longrightarrow \Sym^n(X\times Y)$ is the obvious map. The expression $P^\lambda(\sigma^1(a),\ldots,\sigma^n(a))$ is computed in $R(\Sym(X))$ with respect to the convolution product, and is an element of $R(\Sym^n(X))$ since $\lambda_1 +\ldots + \lambda_n=n$. Similarly for $P^\lambda(\sigma^1(b),\ldots,\sigma^n(b))$. It can be show that this formula holds\footnote{The formula is a direct consequence of the assumption that $R(\Sym(X\times Y))$ is a special $\lambda$-ring which is true for any ``decategorification''.} whenever $R$ has a ``categorification'' as for example $\underline{\Ka}_0(D^b(\MHM_{mon}))$ or $\underline{\Ka}_0(D^b_{con}(-,\mathbb{Q}))$. In this case, we may replace $R^{gm}_{\mathbb{G}_m}(X\times {\mathbb{A}^1})$ with the $R(X)$-submodule $R_{\mathbb{G}_m}(X\times {\mathbb{A}^1})$ of $R(X\times {\mathbb{A}^1})$ generated by $R^{gm}_{\mathbb{G}_m}(X\times {\mathbb{A}^1})$. Note that $R(X\times {\mathbb{A}^1})$ is an $R(X)$-module using the convolution product and the embedding $R(X)\hookrightarrow R(X\times {\mathbb{A}^1})$ provided by the ``zero section'' $0_X=\id_X\times 0:X\hookrightarrow X\times {\mathbb{A}^1}$. The formula for $\sigma^n(a\boxtimes b)$ ensures that $R_{\mathbb{G}_m}(X\times {\mathbb{A}^1})$ is still closed under taking $\sigma^n:R(X\times {\mathbb{A}^1})\longrightarrow R(\Sym^n(X)\times {\mathbb{A}^1})$ and, thus, defines another motivic theory containing $R^{gm}_{\mathbb{G}_m}(-\times {\mathbb{A}^1})$ as a subtheory. Moreover, $\pr_X^\ast(R(X))=:I_X$ is a $\lambda$-ideal and the quotient $R_{mon}(X):=R_{\mathbb{G}_m}(X\times {\mathbb{A}^1})/I_X$ is a well-defined motivic theory which contains $R$ as a subtheory such that $R(X)\hookrightarrow R_{mon}(X)$ is a retract for every $X$. Moreover, the following diagram is cartesian
\[ \xymatrix { R^{gm} \ar@{^{(}->}[r] \ar@{^{(}->}[d] & R^{gm}_{mon} \ar@{^{(}->}[d] \\ R \ar@{^{(}->}[r] & R_{mon}.}\]
\begin{exercise} Show that $R^{gm}_{\mathbb{G}_m}(X\times {\mathbb{A}^1})$ is already an $R(X)$-module under the assumption $R=R^{gm}$. Also $I^{gm}_X=\pr^\ast_X (R(X))$ in this case. Hence, we do not get anything new by the previous construction whenever it applies, and putting $R_{mon}:=R^{gm}_{mon}$ for theories $R=R^{gm}$ will not cause any confusion.
\end{exercise}
\begin{example} \rm
There is a morphism $\underline{\Ka}_0(\MHM_{mon}) \longrightarrow \underline{\Ka}_0(\MHM)_{mon}$ of motivic theories. Roughly speaking, $\MHM_{mon}(X)$ is obtained by a categorification of the construction just described. One can show that for a regular function $f:X\to {\mathbb{A}^1}$ on a smooth scheme $X$ the image of $\phi^{mhm}_f\in \underline{\Ka}_0(\MHM_{mon}(X))$ under this morphism is already contained in the subgroup $\underline{\Ka}_0(\MHM(X))^{gm}_{mon}$. A similar statement holds for $\underline{\Ka}_0(D^b_{con}(-,\mathbb{Q}))$.
\end{example}
\begin{example} \rm
There is a morphism $\Ka^{\hat{\mu}}_0(\Sch_X)\to \underline{\Ka}_0(\Sch_X)_{mon}$ for every (connected) $X$ with $\Ka^{\hat{\mu}}_0(\Sch_X)$ being defined in \cite{DenefLoeser2}. In a nutshell, $\Ka^{\hat{\mu}}_0(\Sch_X)$ is constructed very similar to $\underline{\Ka}_0(\Sch_X)_{mon}$ by considering generators $[Y\xrightarrow{f} X\times {\mathbb{A}^1},\rho]$ with $Y$ carrying a $\mathbb{G}_m$-action $\rho:\mathbb{G}_m\times Y \to Y$ such that $f$ is homogeneous of degree $d> 0$. In contrast to our previous definition, the $\mathbb{G}_m$-action $\rho$ is part of the data, and $\Ka^{\hat{\mu}}_0(\Sch_X)\to \underline{\Ka}_0(\Sch_X)_{mon}$ forgets the $\mathbb{G}_m$-action $\rho$. In particular, given another homogeneous map $Y'\xrightarrow{f'} X\times {\mathbb{A}^1}$ with $Y'$ carrying a $\mathbb{G}_m$-action $\rho'$ and an isomorphism $\theta:Y'\xrightarrow{\sim} Y$ such that $f\theta=f'$ then $[Y\xrightarrow{f}X\times{\mathbb{A}^1}]=[Y'\xrightarrow{f'} X\times {\mathbb{A}^1}]$ in $\underline{\Ka}_0(\Sch_{X})_{mon}$, but the generators $[Y\xrightarrow{f}X\times {\mathbb{A}^1},\rho]$ and $[Y'\xrightarrow{f'}X\times {\mathbb{A}^1},\rho']$ of $\Ka^{\hat{\mu}}_0(\Sch_X)$ might be different unless $\theta$ is $\mathbb{G}_m$-equivariant. The relations in $\Ka^{\hat{\mu}}_0(\Sch_X)$ are the cut and paste relation for $\mathbb{G}_m$-invariant closed subschemes $Z\subset Y$ and $[Y\times {\mathbb{A}^1} \xrightarrow{u\times \id_{\mathbb{A}^1}} X\times {\mathbb{A}^1},\rho]=0$ for every $\mathbb{G}_m$-invariant morphism $u:Y\to X$ from a scheme $Y$ with a good $\mathbb{G}_m$-action. Here, $\rho$ is given by $g(y,z)=(gy,gz)$ using the $\mathbb{G}_m$-action on $Y$. There is a third relation dealing with linear actions of $\mu_d\subset \mathbb{G}_m$, the group of $d$-th roots of unity, which is also fulfilled in $\underline{\Ka}_0(\Sch_X)_{mon}$ as \'{e}tale locally trivial vector bundles are already Zariski locally trivial.
\end{example}
\begin{exercise} \label{functoriality}
Show that the construction $R\mapsto R^{gm}_{mon}$ is functorial in $R$.
\end{exercise}
The motivic theory $R^{gm}_{mon}$ will be the target of our vanishing cycle which we are going to construct now. For this let $f:X\to {\mathbb{A}^1}$ be a regular function on a smooth connected scheme and let $\mathcal{L}_n(X)$ be the scheme parameterizing all arcs of length $n$ in $X$, i.e.\ the scheme representing the set-valued functor $Y\mapsto \Mor(Y\times \Spec \CC[z]/(z^{n+1}), X)$. The standard action of $\mathbb{G}_m$ on ${\mathbb{A}^1}=\Spec \CC[z]$ given by $z\mapsto gz$ induces an action on $\Spec \CC[z]/(z^{n+1})$ and, hence, also on $\mathcal{L}_n(X)$. By functoriality applied to $f:X\to {\mathbb{A}^1}$, we also get a $\mathbb{G}_m$-equivariant morphism $\mathcal{L}_n(X)\xrightarrow{\mathcal{L}_n(f)} \mathcal{L}_n({\mathbb{A}^1})\cong \AA^{n+1}$, where $\mathbb{G}_m$ acts coordinatewise on the latter arc space with weights $0,1,\ldots,n$. Fix $t\in{\mathbb{A}^1}(\mathbb{C})$ and consider the map $f_n=\pr_{n+1}\circ \mathcal{L}_n(f):\mathcal{L}_n(X)|_{X_t} \longrightarrow {\mathbb{A}^1}$, a $\mathbb{G}_m$-equivariant map of degree $n$, and the projection $\pi_n:\mathcal{L}_n(X)\to X$ mapping an arc to its base point. By $\mathbb{G}_m$-equivariance, $[\mathcal{L}_n(X)|_{X_t} \xrightarrow{\pi_n\times f_n}X\times {\mathbb{A}^1}]$ is in $R^{gm}_{\mathbb{G}_m}(X_t\times {\mathbb{A}^1})$ and defines an element in $R^{gm}_{mon}(X_t)$. We form the generating series
\[ Z_{f,t}^R(T):=\sum_{n\ge 1} \mathbb{L}_R^{-n\dim X} [\mathcal{L}_n(X)|_{X_t} \xrightarrow{\pi_n\times f_n}X\times {\mathbb{A}^1}] \,T^n \mbox{ in }R^{gm}_{mon}(X_t)[[T]].\]
The following result is a consequence of Thm.\ 3.3.1 in the article \cite{DenefLoeser2} of Denef and Loeser or of Thm.\ 5.4 in Looijenga's paper \cite{Looijenga1}.
\begin{theorem}
The series $Z_{f,t}^R(T)$ is a Taylor expansion of a rational function in $T$. The latter has a regular value at $T=\infty$.
\end{theorem}
\begin{definition}
Using the same notation for the rational function, we define $\phi_{f,t}^R:=\mathbbm{1}_{X_t}+Z_{f,t}^R(\infty)\in R^{gm}_{mon}(X_t)$ and, finally, $\phi_f^R:=\sum_{t\in {\mathbb{A}^1}(\mathbb{C})} \iota_{t\,!}\phi_{f,t}^R \in R^{gm}_{mon}(X)$ to be the vanishing cycle of $f:X\to {\mathbb{A}^1}$. For non-connected $X$ we apply the definition to every connected component $X_i$ and define $\phi^R_f$ to be the family $(\phi^R_{f|_{X_i}})_{X_i\in \pi_{0}(X)}$ in $R^{gm}_{mon}(X)=\prod_{X_i\in \pi_0(X)}R^{gm}_{mon}(X_i)$.
\end{definition}
\begin{example} \rm One can show $\phi^{\Con}_f=\phi^{con}_f$ for all $f:X\to {\mathbb{A}^1}$ on smooth $X$.
\end{example}
\begin{example}\rm Let $f:X\to {\mathbb{A}^1}$ be a regular function on a smooth connected scheme $X$. Using the morphism $\Ka^{{\hat{\mu}}}_0(\Sch_X)\to \underline{\Ka}_0(\Sch_X)_{mon}$, the vanishing cycle $\phi^{mot}_f$ constructed by Denef and Loeser maps to $\phi^{\underline{\Ka}_0(\Sch)}_f$ up to the normalization factor $(-1)^{\dim X}$, and we will keep the shorter notation $\phi^{mot}_f$ for $\phi^{\underline{\Ka}_0(\Sch)}_f$.
\end{example}
\begin{example} \rm Let $f$ be a regular function on a smooth scheme as before. Using the map $\underline{\Ka}_0(\MHM(X)_{mon})\longrightarrow \underline{\Ka}_0(\MHM(X))_{mon}$ discussed earlier, the vanishing cycle $\phi^{mhm}_f$ maps to $\phi^{\underline{\Ka}_0(\MHM)}_f$, and we will keep the shorter notation $\phi^{mhm}_f$. Similarly for $\phi^{perv}_f$.
\end{example}
\begin{exercise} \label{functoriality2} Prove that $\phi^R_f$ is functorial in $R$. In particular, the diagram
\[ \xymatrix{ \underline{\Ka}_0(\Sch_X) \ar@{=}[r] \ar[d]^{\phi^{mot}} & \underline{\Ka}_0(\Sch_X) \ar[d]^{\phi^R_f} \\ \underline{\Ka}_0(\Sch_X)_{mon} \ar[r] & R^{gm}_{mon}(X) } \]
commutes where we used Lemma \ref{vanishing_cycle_morphism}, Lemma \ref{initial_object} and Exercise \ref{functoriality} to construct the corresponding morphisms. Conclude that $\phi^R_f$ is a vanishing cycle using the known fact that $\phi^{mot}$ is a vanishing cycle.
\end{exercise}
In order to compute the vanishing cycle in practice, we choose an embedded resolution of $X_t\subset X$, i.e.\ a smooth variety $Y$ together with a proper morphism $\pi:Y \rightarrow X$ such that $Y_t=(f\circ\pi)^{-1}(a)=\pi^{-1}(X_t)$ is a normal crossing divisor and $\pi: Y\!\setminus\! Y_t \xrightarrow{\sim} X\!\setminus\! X_t$. Denote the irreducible components of $Y_t$ by $E_i$ with $i\in J$ and let $m_i>0$ be the multiplicity of $f\circ \pi$ at $E_i$. Since $f\circ \pi$ is a section in $\mathcal{O}_Y(-\sum_{i\in J}m_iE_i)$, it induces a regular map to ${\mathbb{A}^1}$ from the total space of $\mathcal{O}_Y(\sum_{i\in I} m_iE_i)$ for any $\emptyset \neq I \subset J$. The latter space restricted to $E_I^\circ:=\cap_{i\in I}E_i \!\setminus\! \cup_{i\not\in I} E_i$ is just $\otimes_{i\in I} N_{E_i|Y}^{\otimes m_i}|_{E_I^\circ}$. By composition with the tensor product we get a regular map $f_I:N_I:=\prod_{i\in I}( N_{E_i|Y}\!\setminus\! E_i) |_{E_I^\circ} \longrightarrow {\mathbb{A}^1}$ which is obviously homogeneous of degree $m_i$ with respect to the $\mathbb{G}_m$-action on the factor $(N_{E_i|Y}\!\setminus\! E_i)|_{E_I^\circ}$ and homogeneous of degree $m_I:=\sum_{i\in I}m_i$ with respect to the diagonal $\mathbb{G}_m$-action. By composing with $\pi:Y \rightarrow X$, the projection $N_I \rightarrow E_I^\circ$ induces a map $\pi_I:N_I \rightarrow X_t$.
\begin{theorem}[\cite{DenefLoeser2} or \cite{Looijenga1}] \label{resolution}
Let $f:X\rightarrow \AA_k^1$ be a regular map and $\pi:Y \rightarrow X$ be an embedded resolution of $X_t$. In the notation just explained we have\footnote{This formula seems to differs from the one given in \cite{DenefLoeser2} or \cite{Looijenga1} by a sign. However, this is not true as the authors work with schemes over $X$ with good $\mu_d$-action. Given such a scheme $Y\xrightarrow{u} X$ with $\mu_d$-invariant $u$, the associated generator of $R^{gm}_{mon}(X)$ is $-[Y\times_{\mu_d} \mathbb{G}_m \ni (y,z)\mapsto (u(y),z^d)\in X\times {\mathbb{A}^1}]_R$. The sign here is chosen in such a way that if $Y$ carries a trivial $\mu_d$-action, then the generator is equivalent to $[Y\ni y \mapsto (u(y),0)\in X\times {\mathbb{A}^1}]$ in $R^{gm}(X)\subset R^{gm}_{mon}(X)$.}
\[ Z_{f,t}^R(\infty) \;= \sum_{\emptyset \neq I\subset J} (-1)^{|I|-1}[N_I\xrightarrow{\pi_I\times f_I} X_t\times {\mathbb{A}^1}] \,\in\, R^{gm}_{mon}(X_t). \]
\end{theorem}
\begin{corollary}[support property]\label{support_property}
Given a regular function $f:X\to {\mathbb{A}^1}$ on a smooth scheme $X$, the vanishing cycle $\phi^R_f$ is supported on $\Crit(f)$, i.e.\ $\phi^R_f$ is in the image of the embedding $ R(\Crit(f))^{gm}_{mon}\hookrightarrow R(X)^{gm}_{mon}$, where $\Crit(f)=\{df=0\}\subset X$ is the critical locus of $f$.
\end{corollary}
The following result might also be helpful when it comes to actual computations. Let $\int_X a \in R(\Spec\CC)$ be the short notation for $(X\to \Spec \CC)_!(a)$ for $a\in R(X)$.
\begin{theorem}[\cite{DaMe1}] \label{equivariant_case}
Let $X$ be a smooth variety with $\mathbb{G}_m$-action such that every point $x\in X$ has a neighborhood $U\subset X$ isomorphic to $\AA^{n(x)}\times U^{\mathbb{G}_m}$ with $\mathbb{G}_m$ acting by multiplication (with weight one) on $\AA^{n(x)}$. Let $f:X\to {\mathbb{A}^1}$ be a homogeneous function of degree $d> 0$. Then, $\int_X \phi^R_f=[X\xrightarrow{f}{\mathbb{A}^1}]$ in $R^{gm}_{mon}(\Spec\CC)$.
\end{theorem}
\begin{exercise} \label{square_root} \hfill
\begin{enumerate}
\item Prove that the scheme $\{(x,y)\in \AA^2\mid xy\not= 0\} \ni (x,y)\mapsto xy\in {\mathbb{A}^1}$ over ${\mathbb{A}^1}$ is isomorphic to the scheme $\mathbb{G}_m\times \mathbb{G}_m\ni (x,y) \to y \in {\mathbb{A}^1}$ over ${\mathbb{A}^1}$ and conclude $[ \{xy\not=0 \} \xrightarrow{xy} {\mathbb{A}^1}]_R=-[\mathbb{G}_m]_R=1-\mathbb{L}_R$ in $R^{gm}_{mon}(\Spec\mathbb{C})$.
\item Use this and any of the two previous theorems to show $\phi^R_{f,0}=\int_{\AA^2}\phi^R_f =\mathbb{L}_R$ for the function $f:\AA^2\ni (x,y)\mapsto xy\in {\mathbb{A}^1}$. Note that $\Crit(f)\cong\Spec\mathbb{C}$ is the origin $0\in \AA^2$. Hence, $\phi^R_f$ is located at the origin.
\item Use the result of the second part and the product formula for vanishing cycles to prove that $\phi_{z^2}^R$ for the function ${\mathbb{A}^1}\ni z\mapsto z^2\in {\mathbb{A}^1}$ is a square root $\mathbb{L}^{1/2}_R$ of $\mathbb{L}_R$.
\item Show that $\sigma^n(\mathbb{L}^{1/2}_R)=0$ for all $n\ge 2$. Hint: It is a well-known fact that $\AA^n\ni (z_1,\ldots,z_n)\longmapsto \big(\sum_{i=1}^n z_i^k\big)_{k=1}^n \in \AA^n$ has a factorization $\AA^n \twoheadrightarrow \Sym^n({\mathbb{A}^1}) \xrightarrow{\sim} \AA^n$ into the quotient map for the natural $S_n$-action on $\AA^n$ and an isomorphism. Remember that for every scheme $X$ the equation $[X\times {\mathbb{A}^1} \xrightarrow{\pr_{{\mathbb{A}^1}}} {\mathbb{A}^1}]=0$ holds in $R(\Spec\mathbb{C})_{mon}^{gm}$ by construction.
\end{enumerate}
\end{exercise}
\subsection{Vanishing cycles for quotient stacks}
The theory of vanishing cycles for regular functions $f:X\to {\mathbb{A}^1}$ on smooth schemes generalizes straight forward to functions $\mathfrak{f}:\mathfrak{X}\to {\mathbb{A}^1}$ on (disjoint unions of) smooth quotient stacks. Note that a quotient stack $X/G$ is called smooth if $X$ is smooth. A closed substack $\mathfrak{P}\subset \mathfrak{X}$ is given by $Y/G$ for a $G$-invariant closed subset $Y\subset X$. The blow-up of $\mathfrak{X}$ in $\mathfrak{P}$ is then simply given by the quotient stack $\Bl_\mathfrak{P} \mathfrak{X}=\Bl_Y X /G$ having exceptional divisor $\mathfrak{E}=E/G$. Given a quotient stack $X/G$ with smooth $X$ and a regular function $\mathfrak{f}:X/G\to {\mathbb{A}^1}$, we denote with $\Crit(\mathfrak{f})$ the quotient stack $\Crit(\mathfrak{f}\rho)/G$ with $\rho:X\to X/G$. The generalization to disjoint unions of quotient stacks is at hand.
\begin{definition}
Given a stacky motivic theory $R$, a stacky vanishing cycle (with values in $R$) is a rule associating to every regular function $\mathfrak{f}:\mathfrak{X}\to {\mathbb{A}^1}$ in a disjoint union of smooth quotient stacks an element $\phi_{\mathfrak{f}}\in R(\mathfrak{X})$ such that the following holds.
\begin{enumerate}
\item If $u:\mathfrak{P}\to \mathfrak{X}$ is a smooth, then $\phi_{\mathfrak{f}\circ u}=u^\ast(\phi_{\mathfrak{f}})$.
\item Let $\mathfrak{X}$ be a disjoint union of smooth quotients containing a smooth closed substack $i:\mathfrak{P}\hookrightarrow \mathfrak{X}$. Denote by $j:\mathfrak{E}\hookrightarrow \Bl_{\mathfrak{P}} \mathfrak{X}$ the exceptional divisor of the blow-up $\pi:\Bl_\mathfrak{P} \mathfrak{X} \to \mathfrak{X}$ of $\mathfrak{X}$ in $\mathfrak{P}$. Then the formula \[ \pi_!\big( \phi_{\mathfrak{f}\circ \pi} - j_! \phi_{\mathfrak{f}\circ \pi\circ j}\big) = \phi_{\mathfrak{f}} - i_!\phi_{\mathfrak{f}\circ i}\] holds for every $\mathfrak{f}:\mathfrak{X}\to {\mathbb{A}^1}$.
\item Given two morphisms $\mathfrak{f}:\mathfrak{X}\to {\mathbb{A}^1}$ and $\mathfrak{g}:\mathfrak{Y}\to {\mathbb{A}^1}$ with smooth $\mathfrak{X}$ and $\mathfrak{Y}$, we introduce the notation $\mathfrak{f}\boxtimes \mathfrak{g}:\mathfrak{X}\times \mathfrak{P}\xrightarrow{\mathfrak{f}\times \mathfrak{g}} {\mathbb{A}^1}\times {\mathbb{A}^1}\xrightarrow{+} {\mathbb{A}^1}$. Then $\phi_{\mathfrak{f}\boxtimes \mathfrak{g}}=\phi_{\mathfrak{f}}\boxtimes \phi_{\mathfrak{g}}$ in $R(\mathfrak{X}\times \mathfrak{P})$. Moreover, $\phi_{\Spec\CC\xrightarrow{0}{\mathbb{A}^1}}(1)=1$.
\end{enumerate}
\end{definition}
Recall, that we constructed a correspondence between motivic theories $R$ with $\mathbb{L}_R^{-1},(\mathbb{L}^n_R-1)^{-1}\in R(\Spec\CC)$ for all $0\not= n\in \mathbb{N}$ and stacky motivic theories satisfying $\rho^\ast:R(X/G)\xrightarrow{\sim} R(X)^G$ for every special group $G$.
\begin{lemma} \label{stacky_vanishing_cycles}
Let $R$ be a motivic theory such that $\mathbb{L}^{-1}_R, (\mathbb{L}^n_R-1)^{-1}\in R(\Spec\CC)$ for all $0\not= n\in \mathbb{N}$. The restriction to schemes provides a bijection between stacky vanishing cycles with values in $R^{st}$ and vanishing cycles with values in $R$.
\end{lemma}
\begin{proof}
By applying the first property of a stacky vanishing cycle to the smooth map $\rho:X\to X/G$, we see that $\phi_{\mathfrak{f}}$ is uniquely determined by $f=\mathfrak{f}\circ\rho:X\to {\mathbb{A}^1}$ as $\rho^\ast:R(X/G)\to R(X)$ is injective for special $G$.
Moreover, applying the first property once more to
\[ \xymatrix { X\times G \ar[r]^m \ar[d]^{\pr_X} & X \\ X} \]
we observe that the vanishing cycle $\phi_f$ of a $G$-invariant function $f:X\to {\mathbb{A}^1}$ is $G$-invariant. Hence, given a vanishing cycle with values in $R$, we can define $\phi^{st}_{\mathfrak{f}}$ to be the unique element in $R^{st}(X/G)$ mapping to $\phi_f=\phi_{\mathfrak{f}\circ \rho}$ under the isomorphism $\rho^\ast:R^{st}(X/G)\cong R(X)^G$. Alternatively, we can write $\phi^{st}_\mathfrak{f}=\rho_!(\phi_f)/[G]_R$ since $[G]_R^{-1}\rho_!$ is the inverse of $\rho^\ast$ on $R(X)^G$.
If $\mathfrak{f}:\mathfrak{X}\to {\mathbb{A}^1}$ is a function on a disjoint union of quotient stacks $\mathfrak{X}_i=X_i/G_i$, we may assume that $G_i$ is special for all $i$ (see Exercise \ref{special}) and define $\phi^{st}_\mathfrak{f}$ by means of the family $\phi^{st}_{\mathfrak{f}|_{\mathfrak{X}_i}}$ using the first property of a stacky motivic theory.
\end{proof}
\begin{exercise}
Complete the proof of the previous lemma by checking that $\phi^{st}_\mathfrak{f}$ on a quotient stack $\mathfrak{X}=X/G$ is independent of the choice of a presentation, i.e.\ if $X/G\cong Y/H$ for special groups $G$ and $H$, then $X$ is smooth if and only if $Y$ is smooth, and $\phi_{\mathfrak{f}\circ\rho_X}$ corresponds to $\phi_{\mathfrak{f}\circ\rho_Y}$ under the isomorphism $R(X)^G\cong R(Y)^H$ in this case. Moreover, show that $\phi^{st}$ satisfies the properties of a stacky vanishing cycle.
\end{exercise}
\begin{example} \rm
Given a motivic theory $R$ satisfying equation (\ref{eq7}) for all $a\in R(X), n\in \mathbb{N}$ and a vanishing cycle $\phi$ with values in $R$, we can adjoin inverses of $\mathbb{L}_R$ and $\mathbb{L}^n_R-1$ for all $n>0$. By applying the previous lemma to the new motivic theory and the induced vanishing cycle, we get a motivic theory $R^{st}$ and a vanishing cycle $\phi^{st}$ such that $\eta_X(\phi_f)=\phi^{st}_f$ for every $f:X\to {\mathbb{A}^1}$, where $\eta_X:R(X)\to R^{st}(X)$ is the adjunction morphism $R\to R^{st}|_{\Sch_\CC}$ from the previous section. In particular, we can apply this to $\phi^{R^{gm}}_{can}$ with values in $R^{gm}$ and also to $\phi^R$ with values in $R^{gm}_{mon}$ as equation (\ref{eq7}) holds in both cases (cf.\ Exercise \ref{geometric_part}). If $R$ satisfies equation (\ref{eq7}), we can also extend $\phi^R_{can}$ with values in $R$. Note that $\phi^{R,st}_{can, \mathfrak{f}} =\mathbbm{1}_{\mathfrak{X}}\in R^{st}(\mathfrak{X})$ for all $\mathfrak{f}:\mathfrak{X}\to {\mathbb{A}^1}$.
\end{example}
\section{Donaldson--Thomas theory}
After introducing a lot of technical notation, we are now in the position to provide the definition of Donaldson--Thomas functions and to state a couple of results in Donaldson--Thomas theory. We close this section by given a list of examples. There are basically three approaches to define Donaldson--Thomas functions (see \cite{DavisonMeinhardt3}). The one given here is due to Kontsevich and Soibelman.
\subsection{Definition and main results}
We start by fixing a stacky motivic theory $R$ satisfying $R(X/G)\cong R(X)^G$ for every quotient stack $X/G$ with special group $G$. Moreover, let $\phi$ be a stacky vanishing cycle with values in $R$ which is completely determined by its restriction to functions $f:X\to {\mathbb{A}^1}$ on smooth schemes $X$. (cf.\ Lemma \ref{stacky_vanishing_cycles}) As shown before, we could start with any vanishing cycle with values in a motivic theory and pass to the ``stackification''. Let us also assume that a square root $\mathbb{L}_R^{1/2}$ of $\mathbb{L}_R$ is contained in $R(\Spec\CC)$ such that $\sigma^n(\mathbb{L}_R^{1/2})=0$ for all $n\ge 2$.
\begin{example}\rm
Assume that $\sigma^n(a\mathbb{L}_R)=\sigma^n(a)\mathbb{L}^n_R$ holds for all $a\in R(X)$, all $n \in \mathbb{N}$ and all $X$. As shown in \cite{DavisonMeinhardt3}, Appendix B, one can extend the $\sigma^n$-operations to $R(X)[\mathbb{L}^{1/2}_R]$ such that $\sigma^n(-\mathbb{L}_R^{1/2})=(-\mathbb{L}_R^{1/2})^n$ for all $n\in \mathbb{N}$ or equivalently $\sigma^n(\mathbb{L}^{1/2}_R)=0$ for all $n\ge 2$. Thus, $R[\mathbb{L}^{1/2}_R]$ is a new motivic theory having the required square root of $\mathbb{L}_R$. We can apply the stackification to the canonical vanishing cycle $\phi^{R[\mathbb{L}_R^{1/2}]}_{can}$ of $R[\mathbb{L}_R^{1/2}]$ and obtain a pair $(R[\mathbb{L}^{1/2}_R]^{st}, \phi_{can}^{R[\mathbb{L}^{1/2}_R], st})$ satisfying our requirements. Note that the assumption on $R$ is always fulfilled if we replace $R$ with $R^{gm}$ (cf.\ Exercise \ref{geometric_part}).
\end{example}
\begin{example} \rm
For every motivic theory $R$ the element $[{\mathbb{A}^1}\ni z \mapsto z^2\in {\mathbb{A}^1}]_R\in R^{gm}_{mon}(\Spec \mathbb{C})$ is the required square root of $\mathbb{L}_{R^{gm}_{mon}}$ as shown in Exercise \ref{square_root}. The stackification of $\phi^R$ with values in $R^{gm}_{mon}$ will match our requirements. This applies in particular to $\phi^{mot}$ and $\phi^{mhm}$. Note that the stackification of $\Con=\Con^{gm}_{mon}$ and of $\underline{\Ka}_0(D^b_{con}(-,\mathbb{Q}))^{gm}_{mon}$ is zero as $[\Gl(n)]_R=0$ in both cases.
\end{example}
We fix a quiver $Q$ with potential $W$ and a geometric stability condition $\zeta$. Recall that $\mathfrak{M}^{\zeta-ss}$ was the stack of $\zeta$-semistable quiver representations with coarse moduli space $\mathcal{M}^{\zeta-ss}$ parameterizing polystable representations. Similarly $\mathfrak{M}$ was the stack of all quiver representations and $\mathcal{M}^{ssimp}$ its coarse moduli space parameterizing semisimple representations. There are various maps between these spaces as shown in the following diagram
\[ \xymatrix @C=2cm { \mathfrak{M}^{\zeta-ss} \ar@{^{(}->}[r] \ar[d]^{p^\zeta} & \mathfrak{M} \ar[d]^p & \\ \mathcal{M}^{\zeta-ss} \ar[r]^{q^\zeta} & \mathcal{M}^{ssimp} \ar[r]^{\dim\times \mathcal{T}r(W)} & \mathbb{N}^{Q_0}\times {\mathbb{A}^1}.}\]
Note that the maps in the lower horizontal row are homomorphisms of monoids with respect to (direct) sums. Moreover, $q^\zeta$ is proper. Denote the composition $\mathfrak{M}^{\zeta-ss} \hookrightarrow \mathfrak{M} \xrightarrow{\mathcal{T}r(W)\circ p} {\mathbb{A}^1}$ with $\mathfrak{Tr}(W)^\zeta$. For a fixed slope $\mu\in \mathbb{R}$, let us introduce the short hand $\phi_{\mathfrak{Tr}(W)^\zeta}(\mathcal{IC}_{\mathfrak{M}^{\zeta-ss}_\mu})$ for the object in $R(\mathfrak{M}^{\zeta-ss})\cong \prod_{d\in \mathbb{N}^{Q_0}} R(\mathfrak{M}^{\zeta-ss}_d)$ having components
\[ \mathbb{L}_R^{(d,d)/2}\phi_{\mathfrak{Tr}(W)^\zeta|_{\mathfrak{M}^{\zeta-ss}_d}}=\frac{\mathbb{L}_R^{(d,d)/2}}{[G_d]_R}\rho_{d!}\phi_{\Tr(W)_d|_{X_d^{\zeta-ss}}} \]
if $d$ has slope $\mu$ or $d=0$ and $0$ for the remaining dimension vectors $d$. The idea behind the notation is the following. The vanishing cycle $\phi$ defines a map $\underline{\Ka}_0(\Sch_{\mathfrak{M}^{\zeta-ss}})^{st} \longrightarrow R(\mathfrak{M}^{\zeta-ss})$ mapping $\mathbbm{1}_{\mathfrak{M}^{\zeta-ss}}=(\mathfrak{M}^{\zeta-ss}\to \Spec\CC)^\ast(1)$ to $\phi_{\mathfrak{Tr}(W)^\zeta}$. If we define $\mathcal{IC}_{\mathfrak{M}^{\zeta-ss}_\mu}$ to be the object in $\underline{\Ka}_0(\Sch_{\mathfrak{M}^{\zeta-ss}})[\mathbb{L}^{1/2}]^{st}$ whose restriction to $\mathfrak{M}^{\zeta-ss}_d$ is $\mathbb{L}^{(d,d)/2}\mathbbm{1}_{\mathfrak{M}^{\zeta-ss}_d}$ if $d$ has slope $\mu$ or $d=0$ and zero else, then $\phi_{\mathfrak{Tr}(W)^\zeta}(\mathcal{IC}_{\mathfrak{M}^{\zeta-ss}_\mu})$ is just the image of $\mathcal{IC}_{\mathfrak{M}^{\zeta-ss}_\mu}$ under the induced map $\underline{\Ka}_0(\Sch_{\mathfrak{M}^{\zeta-ss}})[\mathbb{L}^{1/2}]^{st} \longrightarrow R(\mathfrak{M}^{\zeta-ss})$. One should think of $\mathcal{IC}_{\mathfrak{M}^{\zeta-ss}_\mu}$ as (the class of) the motivic intersection complex of $\mathfrak{M}^{\zeta-ss}_\mu\subseteq \mathfrak{M}^{\zeta-ss}$.
Let us define the convolution product on $R(\mathcal{M}^{\zeta-ss})$ by means of
\[ R(\mathcal{M}^{\zeta-ss})\otimes R(\mathcal{M}^{\zeta-ss}) \xrightarrow{\boxtimes} R(\mathcal{M}^{\zeta-ss}\times \mathcal{M}^{\zeta-ss}) \xrightarrow{\oplus_!} R(\mathcal{M}^{\zeta-ss}) \]
and operations $\Sym^n:R(\mathcal{M}^{\zeta-ss}) \to R(\mathcal{M}^{\zeta-ss})$ for $n\in \mathbb{N}$ via
\begin{equation}\label{lambda_operations} R(\mathcal{M}^{\zeta-ss})\xrightarrow{\sigma^n} R(\Sym^n \mathcal{M}^{\zeta-ss}) \xrightarrow{\oplus_!} R(\mathcal{M}^{\zeta-ss}). \end{equation}
\begin{lemma}
For $a\in R(\mathcal{M}^{\zeta-ss})$ with $a_0:=a|_{\mathcal{M}^{\zeta-ss}_0}=0$ the infinite sum $\Sym(a):=\sum_{n\in \mathbb{N}} \Sym^n(a)$ has only finitely many nonzero summands after restriction to $\mathcal{M}^{\zeta-ss}_d$ and, hence, defines a well defined element in $R(\mathcal{M}^{\zeta-ss})\cong\prod_{d\in \mathbb{N}^{Q_0}} R(\mathcal{M}^{\zeta-ss}_d)$. Conversely, every element $b\in R(\mathcal{M}^{\zeta-ss})$ with $b_0=1\in R(\mathcal{M}^{\zeta-ss}_0)=R(\Spec\CC)$ can be written uniquely as $\Sym(a)$. The map $\Sym(-)$ is a group homomorphism form the additive group $\{a\in R(\mathcal{M}^{\zeta-ss})\mid a_0=0\}$ to the multiplicative group $\{b\in R(\mathcal{M}^{\zeta-ss})\mid b_0=1\}$. The same holds true if we replace $\mathcal{M}^{\zeta-ss}$ with $\mathcal{M}^{ssimp}$ or with $\mathbb{N}^{Q_0}$.
\end{lemma}
\begin{proof} Fix $d\not= 0$. Since $\oplus$ maps $\Sym^n \mathcal{M}^{\zeta-ss}_e$ to $\mathcal{M}^{\zeta-ss}_{ne}$, we get $\Sym^n(a)|_{\mathcal{M}^{\zeta-ss}_d}=0$ for all $n>|d|=\sum_{i\in Q_0}d_i$, and the infinite sum is finite after restriction to $\mathcal{M}^{\zeta-ss}_d$. Conversely, given $b$ we set $a_0=0$. Suppose $a_e\in R(\mathcal{M}^{\zeta-ss}_e)$ has been constructed for all
dimension vectors\footnote{We write $e<d$ if $d=e+e'$ with $0\not=e'\in \mathbb{N}^{Q_0}$.} $e<d$. We put
\[ a_d:=b_d- \sum_{{n_1e_1+\ldots n_re_r=d\atop 0\not=n_j\in \mathbb{N}, 0\not=e_i\not=e_j\in \mathbb{N}^{Q_0}}} \prod_{j=1}^r \Sym^{n_j}(a_{e_j}).\]
Then $b=\Sym(a)$ for $a\in R(\mathcal{M}^{\zeta-ss})$ with $a|_{\mathcal{M}^{\zeta-ss}_d}=a_d$. Using the properties of $\sigma^n:R(\mathcal{M}^{\zeta-ss})\longrightarrow R(\Sym^n\mathcal{M}^{\zeta-ss})$ we get $\Sym(0)=1$ and $\Sym(a+b)=\Sym(a)\Sym(b)$. In particular, $\{b\in R(\mathcal{M}^{\zeta-ss})\mid b_0=1\}\cong\{a\in R(\mathcal{M}^{\zeta-ss})\mid a_0=0\}$ as groups. The proof for $\mathcal{M}^{ssimp}$ and $\mathbb{N}^{Q_0}$ is similar.
\end{proof}
\begin{exercise} \label{master_equation}
Let $\iota:\mathcal{M}^{\zeta-st}\hookrightarrow \mathcal{M}^{\zeta-ss}$ be the inclusion of the moduli space of $\zeta$-stable representations. Show that $\mathbbm{1}_{\mathcal{M}^{\zeta-ss}}=\Sym\big(\iota_!\mathbbm{1}_{\mathcal{M}^{\zeta-st}}\big)$ holds in $\underline{\Ka}_0(\Sch_{\mathcal{M}^{\zeta-ss}})$. Hint: Proof that the strata of the Luna stratification of $\mathcal{M}^{\zeta-ss}$ and the strata of the natural stratification of $\sqcup_{{n_1, \ldots, n_r\atop d_i\not=d_j \forall i\not= j}}\prod_{i=1}^r(\mathcal{M}_{d_i}^{\zeta-ss})^{n_i}/\!\!/S_{n_i}$ given by the conjugacy types of the $S_{n_i}$-stabilizers coincide. In other words, the canonical map $\oplus:\Sym(\mathcal{M}^{\zeta-st}) \longrightarrow \mathcal{M}^{\zeta-ss}$ is a ``constructible'' isomorphism.
\end{exercise}
\begin{exercise}
Show that the restriction of $p^\zeta_!\big(\phi_{\mathfrak{Tr}(W)^\zeta}(\mathcal{IC}_{\mathfrak{M}^{\zeta-ss}_\mu})\big)$ to $\mathcal{M}^{\zeta-ss}_0$ is $1$.
\end{exercise}
Using the last exercise and the previous lemma, the following definition makes sense.
\begin{definition}
The Donaldson--Thomas function $\mathcal{DT}(Q,W)^\zeta\in R(\mathcal{M}^{\zeta-ss})$ is the unique element with $\mathcal{DT}(Q,W)^\zeta|_{\mathcal{M}^{\zeta-ss}_0}=0$ such that $\mathcal{DT}(Q,W)^\zeta_\mu:=\mathcal{DT}(Q,W)|_{\mathcal{M}^{\zeta-ss}_\mu}$ solves the equation
\[ p^\zeta_!\big(\phi_{\mathfrak{Tr}(W)^\zeta}(\mathcal{IC}_{\mathfrak{M}^{\zeta-ss}_\mu})\big)=\Sym\left( \frac{\mathcal{DT}(Q,W)^\zeta_\mu}{\mathbb{L}^{1/2}_R-\mathbb{L}^{-1/2}_R} \right) \]
in $R(\mathcal{M}^{\zeta-ss})$ for all $\mu\in (-\infty,+\infty]$. We also use the notation $\mathcal{DT}(Q,W)^\zeta_d:=\mathcal{DT}(Q,W)^\zeta|_{\mathcal{M}^{\zeta-ss}_d}$. The element $\int_{\mathcal{M}^{\zeta-ss}_d} \mathcal{DT}(Q,W)^\zeta_d=:\DT(Q,W)^\zeta_d\in R(\Spec \CC)$ is called the Donaldson--Thomas invariant of $(Q,W)$ with respect to $\zeta$ for dimension vector $d$. If $W=0$, we simply write $\mathcal{DT}(Q)^\zeta_d$ and $\DT(Q)^\zeta_d$.
\end{definition}
In view of Exercise \ref{master_equation}, one might hope that $\mathcal{DT}(Q,W)^\zeta_d/(\mathbb{L}_R^{1/2}-\mathbb{L}_R^{-1/2})$ is something like $p^\zeta_! \phi_{\mathfrak{Tr}(W)^\zeta}(j_!\mathcal{IC}_{\mathfrak{M}^{\zeta-st}_\mu})$, where $j:\mathfrak{M}^{\zeta-st}_\mu\hookrightarrow \mathfrak{M}^{\zeta-ss}_\mu$ denotes the inclusion, and $\mathcal{IC}_{\mathfrak{M}^{\zeta-st}_\mu}$ is defined similarly to $\mathcal{IC}_{\mathfrak{M}^{\zeta-ss}_\mu}$. Let us assume, we were allowed to commute $p^\zeta_!$ with $\phi_{\mathfrak{Tr}(W)^\zeta}$ which is a priori not clear as $p^\zeta$ is not proper. Then
\[ p^\zeta_! \phi_{\mathfrak{Tr}(W)^\zeta}(j_!\mathcal{IC}_{\mathfrak{M}^{\zeta-st}_\mu})= \phi_{\mathcal{T}r(W)\circ q^\zeta}\left(\frac{\iota_! \mathcal{IC}_{\mathcal{M}^{\zeta-ss}_\mu}}{\mathbb{L}^{1/2}_R-\mathbb{L}_R^{-1/2}}\right). \]
for $\mathcal{IC}_{\mathcal{M}^{\zeta-st}_d}=\mathbb{L}_R^{((d,d)-1)/2}\mathbbm{1}_{\mathcal{M}^{\zeta-st}_d}$, and $\mathcal{DT}(Q,W)^\zeta_d=\phi_{\mathcal{T}r(W)\circ q^\zeta_d} (\iota_!\mathcal{IC}_{\mathcal{M}^{\zeta-ss}_d})$ follows. This is not quite true. It turns out that the extension $\iota_!\mathcal{IC}_{\mathcal{M}^{\zeta-st}_\mu}$ of $\mathcal{IC}_{\mathcal{M}^{\zeta-st}_d}$ by zero has to be replaced with the ``correct'' extension $\mathcal{IC}_{\overline{\mathcal{M}^{\zeta-st}_d}}$ which restricts to $\mathcal{IC}_{\mathcal{M}^{\zeta-st}_d}$, but might also be nonzero on the boundary of $\mathcal{M}^{\zeta-st}_d$ inside $\mathcal{M}^{\zeta-ss}_d$. However, we have not defined $\mathcal{IC}_{\overline{\mathcal{M}^{\zeta-st}_d}}$ yet.
\begin{definition}
We denote with $\mathcal{IC}^{mot}_{\overline{\mathcal{M}^{\zeta-st}}}\in \underline{\Ka}_0(\Sch_{\mathcal{M}^{\zeta-ss}})[\mathbb{L}^{1/2}]^{st}$ the Donaldson--Thomas function $\mathcal{DT}(Q)^\zeta$ computed with respect to the stackification of the canonical vanishing cycle $\phi^{\underline{\Ka}_0(\Sch)[\mathbb{L}^{1/2}]}_{can}$. Note that $\overline{\mathcal{M}^{\zeta-st}_d}=\mathcal{M}^{\zeta-ss}_d$ if $\mathcal{M}^{\zeta-st}_d\not=\emptyset$ and $\overline{\mathcal{M}^{\zeta-st}_d}=\emptyset$ else.
\end{definition}
The following result justifies the definition.
\begin{theorem}[\cite{MeinhardtReineke}] \label{main_result_1}
If $\zeta$ is generic (see Definition \ref{generic_stability}), the element $\mathcal{IC}^{mot}_{\overline{\mathcal{M}^{\zeta-st}}}$ maps to the classical intersection complex\footnote{Strictly speaking one has to normalize the class of the classical (shifted) intersection complex of $\mathcal{M}^{\zeta-ss}_d$ by multiplication with $(-\mathbb{L}^{1/2})^{(d,d)-1}$ which does not change the underlying perverse sheaf.} $\mathcal{IC}_{\overline{\mathcal{M}^{\zeta-st}}}$ of the closure of $\mathfrak{M}^{\zeta-st}$ inside $\mathcal{M}^{\zeta-ss}$ under the map
$\underline{\Ka}_0(\Sch_{\mathcal{M}^{\zeta-ss}})[\mathbb{L}^{1/2}]^{st}\longrightarrow \underline{\Ka}_0(\MHM(\mathcal{M}^{\zeta-ss}))[\mathbb{L}^{1/2}]^{st}$ constructed in Lemma \ref{initial_object}.
\end{theorem}
\begin{example} \rm
Assuming equation (\ref{eq7}) for all $a\in R(X)$ and all $n\in \mathbb{N}$ so that $\phi^R_{can}$ has a stacky extension $\phi^{R[\mathbb{L}^{1/2}_R]}_{can}$. In this case, $\mathcal{DT}(Q,W)^\zeta=\mathcal{DT}(Q)^\zeta$ is just the image of $\mathcal{IC}^{mot}_{\overline{\mathcal{M}^{\zeta-st}}}$ under the canonical map $\underline{\Ka}_0(\Sch_{\mathcal{M}^{\zeta-ss}})[\mathbb{L}^{1/2}]^{st} \longrightarrow R(\mathcal{M}^{\zeta-ss})[\mathbb{L}^{1/2}]^{st}$ of Lemma \ref{initial_object}, and we may define $\mathcal{IC}_{\overline{\mathcal{M}^{\zeta-st}}}:=\mathcal{DT}(Q)^\zeta\in R(\mathcal{M}^{\zeta-ss})[\mathbb{L}^{1/2}]^{st}$. As for mixed Hodge modules one can show that $\mathcal{IC}_{\overline{\mathcal{M}^{\zeta-st}}}$ has a lift in $R(\mathcal{M}^{\zeta-ss})[\mathbb{L}^{-1/2}]$ if the canonical map $\underline{\Ka}_0(\Sch)\to R$ factorizes trough $\underline{\Ka}_0(\MHM)$ or if $R$ is a sheaf in the \'{e}tale topology with $[\mathbb{P}^r]$ acting as a nonzero divisor in each group $R(X)$ for all $r\in \mathbb{N}$. Note that we can replace $R$ with $R^{gm}$ to ensure equation (\ref{eq7}).
\end{example}
One can also prove the following result which should be seen as the analogue of Exercise \ref{master_equation}.
\begin{theorem}[\cite{DavisonMeinhardt3}] \label{main_result_2}
Recall that every stacky vanishing cycle with values in $R$ satisfying our assumptions defines a map $\phi_{\mathcal{T}r(W)\circ q^\zeta}:\underline{\Ka}_0(\Sch_{\mathcal{M}^{\zeta-ss}})^{st}\longrightarrow R(\mathcal{M}^{\zeta-ss})$. If this map commutes with the $\Sym^n$-operations of equation (\ref{lambda_operations}) for every $n\in \mathbb{N}$, and if $\zeta$ is generic, then $\mathcal{DT}(Q,W)^\zeta=\phi_{\mathcal{T}r(W)\circ q^\zeta}\big(\mathcal{IC}^{mot}_{\overline{\mathcal{M}^{\zeta-st}}}\big)$.
\end{theorem}
\begin{example} \rm The assumption on $\phi_{\mathcal{T}r(W) q^\zeta}$ is true for $\phi=\phi^{mhm}$. Hence, if $\zeta$ is generic, $\mathcal{DT}(Q,W)^\zeta_d=\phi^{mhm}_{\mathcal{T}r(W)\circ q^\zeta_d}\big(\mathcal{IC}_{\mathcal{M}^{\zeta-st}_d}\big)$ if $\mathcal{M}^{\zeta-st}_d\not=\emptyset$ and zero else. It is not known yet whether or not $\phi^{mot}_{\mathcal{T}r(W)\circ q^\zeta}$ commutes with the $\Sym^n$-operations, and we cannot apply the theorem. However, a counterexample is also not known, and conjecturally the theorem also holds for $\phi^{mot}$.
\end{example}
\begin{exercise} Use the support property of $\phi^R$ (see Corollary \ref{support_property}) to show that $\mathcal{DT}(Q,W)^\zeta_d$ is supported on $\mathcal{M}^{W,\zeta-ss}_d$, i.e.\ is an element in $R(\mathcal{M}^{W,\zeta-ss}_d)$, where $\mathcal{M}^{W,\zeta-ss}_d$ the moduli space parameterizing $\zeta$-polystable $\mathbb{C} Q$-representations $V$ of dimension $d$ such that $\partial W/\partial \alpha=0$ on $V$ for all $\alpha\in Q_1$.
\end{exercise}
\subsection{Examples}
The aim of this section is to provide some examples of (motivic) Donaldson--Thomas invariants.
\subsubsection{\rm \textbf{The m-loop quiver}}
Let us consider the quiver $Q^{(m)}$ with one vertex and $m$ loops.
The choice of a stability condition is irrelevant as $\mathfrak{M}^{\zeta-ss}=\mathfrak{M}$. We take the canonical vanishing cycle of $\underline{\Ka}_0(\Sch)[\mathbb{L}^{1/2}]$ and are only interested in Donaldson--Thomas invariants. As $\underline{\Ka}(\Sch_\mathbb{N})[\mathbb{L}^{1/2}]^{st}\cong \Ka(\Sch_\CC)[\mathbb{L}^{-1/2},(\mathbb{L}^n-1)^{-1}:n\in \mathbb{N}_\ast][[t]]$, we end up with the following power series
\[ A^{(m)}(t):=(\dim \circ p)_!\big(\phi_{\mathfrak{Tr}(W)}(\mathcal{IC}_{\mathfrak{M}})\big)=\sum_{d\ge 0}\frac{\mathbb{L}^{(m+1)d^2/2}}{[\Gl(d)]}t^d=\sum_{d\ge 0} \frac{\mathbb{L}^{(md^2+d)/2}}{\prod_{i=1}^d(\mathbb{L}^i-1)}t^d.\]
Note that the series is also well-defined for $m\in \mathbb{Z}$.
\begin{exercise}
Prove the identity $A^{(m)}(\mathbb{L} t)-A^{(m)}(t)=\mathbb{L}^{(m+1)/2}\,t\, A^{(m)}(\mathbb{L}^mt)$ for all $m\in \mathbb{Z}$.
\end{exercise}
For $m\in \mathbb{N}$ we introduce the series
\[ B^{(m)}(t):=A^{(m)}(\mathbb{L} t)/A^{(m)}(t)=\Sym\big( \sum_{d\ge 1} \mathbb{L}^{1/2}[\mathbb{P}^{d-1}]\DT(Q^{(m)})_d\,t^d\big), \]
where we used the properties of $\Sym$ and the fact that $\dim_!$ commutes with $\Sym$. Moreover, due to the previous exercise
\[ B^{(m)}(t)=1+\mathbb{L}^{(m+1)/2}t \prod_{i=0}^{m-1}B^{(m)}(\mathbb{L}^it). \]
For $m=0$ the empty product on the right hand side is $1$, and we obtain $B^{(0)}(t)=1+\mathbb{L}^{1/2}t$ as well as
\[ \DT(Q^{(0)})_d=\begin{cases} 1 & \mbox{for }d=1, \\ 0&\mbox{else.} \end{cases} \]
This is in fully agreement with Theorem \ref{main_result_1} as $\mathcal{M}^{simp}_d=\Spec\CC$ for $d=1$ and $\mathcal{M}^{simp}_d=\emptyset$ else.\\
For $m=1$, we get $B^{(1)}(t)=1+\mathbb{L} tB^{(1)}(t)$, and $B^{(1)}(t)=1/(1-\mathbb{L} t)=\sum_{d\in \mathbb{N}} \mathbb{L}^dt^d$ follows. Hence,
\[ \DT(Q^{(1)})_d=\begin{cases} \mathbb{L}^{1/2} & \mbox{for }d=1, \\ 0&\mbox{else.} \end{cases} \]
Again, this is in fully agreement with Theorem \ref{main_result_1} as $\mathcal{M}^{simp}_d={\mathbb{A}^1}$ for $d=1$ and $\mathcal{M}^{simp}_d=\emptyset$ else.\\
Solving the pseudo-algebraic equation for $m\ge 2$ is much more complicated, but the answer is given as follows. Note that $\mathbb{Z}/(d)=:C_d$ acts on the set $U_d:=\{ (a_1,\ldots,a_d)\in \mathbb{N}^d\mid a_1+\ldots+a_d=(m-1)d\}$ by cyclic permutation. We call $a=(a_1,\ldots,a_d)$ primitive if $\Stab_{C_d}(a)=\{0\}$, and almost primitive if $a$ is primitive or $m\equiv 0 (2), d\equiv 2 (4)$ and $a=(b_1,\ldots,b_{d/2},b_1,\ldots,b_{d/2})$ for some primitive $(b_1,\ldots,b_{d/2})$. The subset $U^{ap}_d:=\{ a\in U_d\mid a\mbox{ is almost primitive}\}$ is obviously stable under the $C_d$-action. Define $\deg(a)=\sum_{i=1}^d(d-i)a_i$ and $\deg(C_d\cdot a)=\min\{ \deg(a')\mid a'\in C_d\cdot a\}$.
\begin{theorem}[\cite{Reineke4}]
Let $d\ge 1$ and $m\ge 2$. Then $\dim(\mathcal{M}^{simp}_d)=(m-1)d^2+1$ and
\[\DT(Q^{(m)})_d=\mathbb{L}^{\frac{(m-1)d^2+1}{2}}\frac{1-\mathbb{L}^{-1}}{1-\mathbb{L}^{-d}}\sum_{C_d\cdot a\in U^{ap}_d/C_d} \mathbb{L}^{-\deg(C_d\cdot a)}. \]
In particular, $\chi_c(\DT(Q^{(m)})_d)=\frac{(-1)^{(m-1)d^2+1}}{d}|U^{as}_d/C_d|$.
\end{theorem}
\begin{exercise}
By taking the Euler characteristic of every coefficient of $B^{(m)}(t)$, we get a series $\beta^{(m)}(t)\in \mathbb{Z}[[t]]$ satisfying $\beta^{(m)}(t)=1+(-1)^{m+1}\beta^{(m)}(t)^m$, but also $\beta^{(m)}(t)=\prod_{d\ge 1}(1-t^d)^{d\Omega^{(m)}_d}$ using the shorthand $\Omega^{(m)}_d:=\chi_c(\DT(Q^{(m)})_d)$. Prove that
\[ \Omega^{(2)}_d=\chi_c(\DT(Q^{(2)})_d)=\frac{1}{2d^2}\sum_{n|d} (-1)^{n+1}\mu(d/n){2n \choose n} \]
for $d\ge 1$, where $\mu(-)$ denotes the M\"obius function. Hint: Solve the quadratic equation for $\beta^{(2)}(t)$ and use the logarithmic derivative to prove the formula
\[ \sum_{d\ge 1}2d^2\Omega^{(2)}_d \frac{t^d}{1-t^d}=\sum_{n\ge 1} \Big(\sum_{d|n} 2d^2\Omega^{(2)}_d\Big)t^n= 1-\frac{1}{\sqrt{1+4t}}. \]
The Taylor expansion of $1/\sqrt{1+4t}$ is $\sum_{n\in \mathbb{N}}(-1)^n{2n\choose n}t^n$. Finally, use the M\"obius function to solve for $\Omega^{(2)}_d$. \\
One can show $\Omega^{(2)}_d=F(d)$ for $F(d)$ being the coefficients introduced in \cite{JoyceMF}, equation (53).
\end{exercise}
\subsubsection{\rm\textbf{Dimension reduction}}
Let $Q$ be a quiver with potential $W$. Let $\zeta$ be a stability condition and $\mu\in [-\infty,\infty)$. Assume $\mathfrak{M}^{\zeta-ss}_d=\mathfrak{M}_d$ for all $d\in \Lambda_\mu$. As an example, we may take the King stability condition $\theta=0$ and $\mu=0$. Given a motivic theory $R$, the following arguments apply to $\phi^{R,st}$ with values in $R^{gm,st}_{mon}$. Without loss of generality, we consider the case $R=\underline{\Ka}_0(\Sch)$, i.e.\ $\phi=\phi^{mot,st}$. As in the previous example, we are only interested in Donaldson--Thomas invariants. A subset $C\subset Q_1$ such that every cycle in $W$ contains exactly one arrow in $C$ is called a cut of $W$. Let $\mathbb{G}_m$ act on $X_d=\prod_{\alpha:i\to j} \Hom(\CC^{d_i},\CC^{d_j})$ by multiplying a linear map corresponding to $\alpha\in C$ with $g\in \mathbb{G}_m$. By assumption, $\Tr(W)_d$ is homogeneous of degree one, and $\int_{X_d}\phi_{\Tr(W)_d}^{mot}$ is the residue class of $[X_d\xrightarrow{\Tr(W)_d}{\mathbb{A}^1}] $ in $\Ka_0(\Sch_\CC)_{mon}$ according to Theorem \ref{equivariant_case}. Consider the projection $\tau_d:X_d\to Y_d$ with $Y_d=\prod_{C\not\ni \alpha:i\to j}\Hom(\CC^{d_i},\CC^{d_j})$ which is a trivial vector bundle with fiber $F=\prod_{C\ni \alpha:i\to j}\Hom(\CC^{d_i},\CC^{d_j})$. As $\Tr(W)_d$ is linear along the fibers, we can think of it as being a section $\sigma^W_d$ of the dual bundle with fiber $F^\vee=\prod_{C\ni \alpha:i\to j}\Hom(\CC^{d_j},\CC^{d_i})$ using the trace pairing. Indeed, it maps a point $M=(M_\alpha)_{\alpha\not\in C}$ to $(\frac{\partial W}{\partial \alpha}(M))_{\alpha\in C}\in F^\vee$.
\begin{exercise}
Show that $[\tau_d^{-1}\{\sigma^W_d\not=0\} \xrightarrow{\Tr(W)_d}{\mathbb{A}^1}]$ is in $\pr_\CC^\ast \Ka_0(\Sch_\CC)\subset \Ka_{0,\mathbb{G}_m}(\Sch_{\mathbb{A}^1})$. Hint: Choose an open cover $\cup_{i\in I} U_i=\{\sigma^W_d\not= 0\}\subseteq Y_d$ such that $\tau_d: \tau_d^{-1}U_ i \to U_i$ splits into the kernel of $\Tr(W)_d$ and a complement of rank one.
\end{exercise}
Using the exercise, we obtain
\begin{eqnarray*} [X_d\xrightarrow{\Tr(W)_d}{\mathbb{A}^1}]&=&[\tau_d^{-1}\{\sigma^W_d=0\}\xrightarrow{0}{\mathbb{A}^1}]\;=\;[F][\{\sigma^W_d=0\}]\\ &=&\mathbb{L}^{\sum_{C\ni \alpha:i\to j}d_id_j}\left[\left\{\ M\in Y_d\mid \frac{\partial W}{\partial \alpha}(M)=0 \,\forall\, \alpha\in C\right\}\right]. \end{eqnarray*}
in $\Ka_0(\Sch_\mathbb{C})_{mon}$. Therefore,
\begin{eqnarray*} \lefteqn{ \Sym\left( \frac{\DT(Q,W)^\zeta_\mu}{\mathbb{L}^{1/2}-\mathbb{L}^{-1/2}} \right)\; =\; (\dim\circ p)_!\big(\phi_{\mathfrak{Tr}(W)}(\mathcal{IC}_{\mathfrak{M}_\mu})\big)} \\&=&\sum_{d\in \Lambda_\mu} \mathbb{L}^{(d,d)/2+\sum_{C\ni \alpha:i\to j}d_id_j}\frac{\left[\left\{M\in Y_d\mid \frac{\partial W}{\partial \alpha}(M)=0\, \forall \,\alpha\in C\right\}\right]}{[G_d]} \, t^d, \end{eqnarray*}
is actually in $\Ka_0(\Sch_\CC)[\mathbb{L}^{-1/2},(\mathbb{L}^n-1)^{-1}:n\in \mathbb{N}_\ast][[t_i\mid i\in Q_0]]$, where we used
\[\Ka_0(\Sch_\CC)[\mathbb{L}^{-1/2},(\mathbb{L}^n-1)^{-1}:n\in \mathbb{N}_\ast][[t_i: i\in Q_0]]=\underline{\Ka}(\Sch_{\mathbb{N}^{Q_0}})[\mathbb{L}^{1/2}]^{st} \subset \underline{\Ka}(\Sch_{\mathbb{N}^{Q_0}})^{st}_{mon}.\]
This reduction process is usually called dimension reduction, and \\
$\left\{M\in Y_d\mid \frac{\partial W}{\partial \alpha}(M)=0\, \forall \,\alpha\in C\right\}/G_d $
is the stack of $d$-dimensional representations of the algebra $\CC Q/(\alpha,\partial W/\partial \alpha\mid \alpha\in C)$.
\subsubsection{\rm\textbf{0-dimensional sheaves on a Calabi--Yau 3-fold}}
Let us illustrate the concept of dimension reduction using the quiver $Q^{(3)}$ with one vertex and three loops $x,y,z$. The choice of the stability condition does not matter. We take the potential $W=[x,y]z=xyz-yxz$, and $\CC Q/(\partial W/\partial\alpha\mid \alpha\in Q^{(3)}_1)=\CC[x,y,z]$ follows. Hence, representations of this algebra are just 0-dimensional sheaves of finite length $d$ on the Calabi--Yau 3-fold $\AA^3$. We can take $C=\{z\}$ and obtain $\CC Q/(z,\partial W/\partial z)=\CC[x,y]$, and representations of this algebra are 0-dimensional sheaves of finite length $d$ on the Calabi--Yau 2-fold $\AA^2$. Using $(d,d)^2=-2d^2$, we have to compute
\[ \sum_{d\in \mathbb{N}} \frac{\left[\left\{M\in Y_d\mid \frac{\partial W}{\partial \alpha}(M)=0\, \forall \,\alpha\in C\right\}\right]}{[\Gl(d)]}\,t^d \]
which has already been done by Feit and Fine half a century ago in \cite{FeitFine}. The answer is
\[ \Sym\Big( \frac{1}{\mathbb{L}-1}\sum_{d\ge 1} [\AA^2] t^d\Big), \]
and $DT(Q^{(3)},W)_d=\mathbb{L}^{3/2}=\int_{\AA^3} \mathcal{IC}^{mot}_{\AA^3}$ follows for all $d\ge 1$ if we define $\mathcal{IC}^{mot}_{X}=\mathbb{L}^{-\dim(X)/2}\mathbbm{1}_X\in \underline{\Ka}_0(\Sch_X)^{st}_{mon}$ for every smooth equidimensional variety $X$. This example has been generalized to arbitrary Calabi--Yau 3-folds by Behrend, Bryan, Szendr\H{o}i.
\begin{theorem}[\cite{BBS}] \label{skyscraper}
The motivic Donaldson--Thomas invariant for 0-dimensional sheaves of length $d\ge 1$ on a Calabi--Yau 3-fold $X$ is given by $\int_X \mathcal{IC}^{mot}_X=\mathbb{L}^{-3/2}[X]\in \Ka_0(\Sch_\CC)^{st}_{mon}$.
\end{theorem}
Note that the Donaldson--Thomas function $\mathcal{DT}(Q^{(3)},W)_d$ is supported on the moduli space $\Sym^d(\AA^3)$ of semisimple $\CC[x,y,z]$ representations but the sublocus of simple representations is empty for $d>1$. However, the space of simple $\CC Q^{(3)}$-representations is nonempty even for $d>1$ which is the reason why $\DT(Q^{(3)},W)_d\not=0$. It seems plausible that $\mathcal{DT}(Q^{(3)},W)_d$ is $\Delta_{d\,!}(\mathcal{IC}^{mot}_{\AA^3})$, where $\Delta_d:\AA^3\hookrightarrow \Sym^d(\AA^3)$ is the diagonal embedding.
\subsubsection{\rm\textbf{The 1-loop quiver with potential}}
Let us come back to the 1-loop quiver with $\CC Q^{(1)}=\CC[x]$ and choose an arbitrary potential $W\in \CC[x]$. Representations of the algebra $\CC Q/(dW/dx)=\CC[x]/(dW/dx)$ can be interpreted as (0-dimensional) sheaves of length $d$ on $\Crit(W)\subseteq {\mathbb{A}^1}$. Form the prime decomposition $dW/dx=c\prod_{i=1}^{r}(x-a_i)^{d_i-1}$ for some $1<d_i\in \mathbb{N}, c\in \CC^\times, a_i\in \CC$ satisfying $a_i\not=a_j$ for $i\not=j$. As before, the choice of a stability condition does not effect the Donaldson--Thomas function.
\begin{theorem}[\cite{DaMe1}] The Donaldson--Thomas function $\mathcal{DT}(Q^{(1)},W)_1$ computed using $\phi^{mot}$ is supported on $\Crit(W)$, i.e.\ is contained in $\underline{\Ka}_0(\Sch_{\Crit(W)})^{st}_{mon}\cong\prod_{i=1}^r \Ka_0(\Sch_\mathbb{C})^{st}_{mon}$, and its ``value'' at $a_i$ is given by $\mathbb{L}^{-1/2}[{\mathbb{A}^1}\xrightarrow{z^{d_i}} {\mathbb{A}^1}]\in \Ka_0(\Sch_\mathbb{C})^{st}_{mon}$. Moreover, $\mathcal{DT}(Q^{(1)},W)_d=0$ for $d\not=1$.
\end{theorem}
\begin{exercise}
Prove this result using Theorem \ref{main_result_2} and the explicit form of the vanishing cycle given by embedded resolutions as in Theorem \ref{resolution}.
\end{exercise}
The formula remains true if we replace $\phi^{mot}$ with $\phi^R$ due to Exercise \ref{functoriality2}.
\subsubsection{\rm\textbf{Sheaves on (-2)-curves}}
Consider the following quiver $Q$
\[ \xymatrix @C=2cm { \bullet \ar@(ul,dl)[]_X \ar@/^1pc/[r]_B \ar@/^2pc/[r]^A & \bullet \ar@(ur,dr)[]^Y \ar@/^1pc/[l]_C \ar@/^2pc/[l]^D }\]
with potential
\[ W_n=\frac{1}{n+1}\big(X^{n+1}-Y^{n+1}\big) - XCA+XDB+YAC-YBD \mbox{ for some }0<n\in \mathbb{N}. \]
The bounded derived category $D^b\Jac(Q,W_n)$ has also a geometric interpretation. For this consider the singular affine variety $X_n=\{x^2+y^2+(z+w^n)(z-w^n)=0\}\subset \AA^4$ which is a local model for a 3-fold with an $A_{n}$-singularity. By blowing-up $X_n$ in $\{x=z\pm w^d=0\}$ we get two minimal resolutions $Y_n^\pm$ with an exceptional locus $C\cong \mathbb{P}^1$. The normal bundle of $C$ inside $Y_n^\pm$ is $\mathcal{O}_{\mathbb{P}^1}(-1)\oplus \mathcal{O}_{\mathbb{P}^1}(-1)$ for $n=1$ and $\mathcal{O}_{\mathbb{P}^1}\oplus \mathcal{O}_{\mathbb{P}^1}(-2)$ for $n>1$. In particular, $Y_n^\pm$ is a Calabi--Yau 3-fold which has actually a locally trivial fibration over $C$ with fiber $\AA^2$. For $d=1$, $Y^\pm_1$ is isomorphic to the normal bundle $\mathcal{O}_{\mathbb{P}^1}(-1)\oplus \mathcal{O}_{\mathbb{P}^1}(-1)$ and known as the conifold resolution. For $d>1$ this fibration is not a vector bundle as the transition functions are not linear. The resolutions $Y_n^+$ and $Y_n^-$ are related via a flop over $X_n$, and also isomorphic to each other. Moreover,
\[ D^b\Jac(Q,W_n) \cong D^b\Coh(Y_n^\pm) \]
and (complexes of) nilpotent representations on the left hand side correspond to (complexes of) sheaves supported on $C\subseteq Y^\pm_n$. We are only interested in these objects and choose any stability condition $\zeta=(\zeta_1,\zeta_2)$ with $\zeta_1\nparallel \zeta_2$ in $\mathbb{R}^2\cong\mathbb{C}$.
\begin{theorem}[\cite{DaMe2}]
The Donaldson--Thomas invariant\\ $\DT(Q,W_n)^{nilp}_{(d_1,d_2)}\in \Ka_0(\Sch_\CC)^{st}_{mon}$ of nilpotent representations computed with respect to $\phi^{mot}$ is given by
\[ \DT(Q,W_n)^{\nilp}_{(d_1,d_2)}=\begin{cases} \mathbb{L}^{-3/2}[C]=\mathbb{L}^{-3/2}(\mathbb{L}+1) & \mbox{ if } 0\not= d_1=d_2, \\ \mathbb{L}^{-1/2}[{\mathbb{A}^1}\xrightarrow{z^{n+1}}{\mathbb{A}^1}] &\mbox{ if }|d_1-d_2|=1, \\ 0& \mbox{ else.} \end{cases} \]
\end{theorem}
Of course, the formula remains true if we replace $\phi^{mot}$ with $\phi^R$ due to Exercise \ref{functoriality2}. Note that $\DT(Q,W_n)^{nilp}_{(d_1,d_1)}$ is just ``counting'' 0-dimensional sheaves on $Y_n^\pm$ supported on $C$ which explains the answer in view of Theorem \ref{skyscraper}. For $|d_1-d_2|=1$, there is just one simple nilpotent $\Jac(Q,W_n)$-representation $V$ with $\Ext^1(V,V)$ being of dimension one. However, the obstruction of deforming $V$ as a representation of $\Jac(Q,W_n)$ is controlled by some potential of the form $z^{n+1}$ induced by $W_n$. Hence, we are back in the context of the previous example. The case of $n=1$ has been studied earlier by Morrison, Mozgovoy, Nagao, Szendr\H{o}i in \cite{MMNS}.
\subsection{The Ringel--Hall algebra}
In the previous section we have seen some examples of Donaldson--Thomas invariants and functions. In all of these cases the choice of the stability condition did not play a crucial role. However, for a generic quiver this is not the case and the Donaldson--Thomas functions and invariants change as we vary the stability condition. There is a wall and chamber structure on the moduli space of stability conditions and these changes will only happen if we jump over a wall into a different chamber. There is, however, a formula - the wall-crossing formula - relating the Donaldson--Thomas functions and invariants for various stability conditions. Before we state and prove the formula, let us introduce some fundamental objects in Donaldson--Thomas theory.
Fix two dimension vectors $d,d'$ and recall the following commutative diagram using the notation of section 2
\begin{equation*}
\xymatrix @C=2cm @R=1cm{
\mathfrak{M}_d \times \mathfrak{M}_{d'} \ar[d]_{p_d\times p_{d'}} & \mathfrak{M}_{d,d'} \ar[l]_{\pi_1\times \pi_3} \ar[r]^{\pi_2} & \mathfrak{M}_{d+d'} \ar[d]^{p_{d+d'}}\\
\mathcal{M}^{ssimp}_d\times \mathcal{M}^{ssimp}_{d'} \ar[rr]^(0.5)\oplus & & \mathcal{M}^{ssimp}_{d+d'} }
\end{equation*}
with $\mathfrak{M}_{d,d'}$ being the stack of short exact sequences $0\to V^{(1)} \to V^{(2)} \to V^{(3)} \to 0$ such that $\dim V^{(1)}=d, \dim V^{(3)}=d'$. The morphism $\pi_i$ maps a sequence to its i-th entry. By taking the disjoint union over all dimension vectors, we end up with
\[ \xymatrix @C=2.0cm{ \mathfrak{M}\times \mathfrak{M} & \mathfrak{Exact} \ar[r]^{\pi_2} \ar[l]_{\pi_1\times \pi_3} & \mathfrak{M}, }\]
where $\mathfrak{Exact}$ denotes the stack of all short exact sequences.
\begin{definition} For a given motivic theory $R$, we call the $R(\Spec\CC)$-module $R(\mathfrak{M})$ with the Ringel--Hall product
\[ \ast:R(\mathfrak{M})\otimes R(\mathfrak{M}) \xrightarrow{\boxtimes} R(\mathfrak{M}) \xrightarrow{(\pi_1\times\pi_3)^\ast} R(\mathfrak{Exact}) \xrightarrow{\pi_{2\,!}} R(\mathfrak{M}) \]
the Ringel--Hall algebra of the quiver $Q$ with respect to $R$.
\end{definition}
\begin{lemma} The Ringel--Hall algebra is an associative algebra with unit.
\end{lemma}
The proof is not very difficult but a nice exercise in dealing with successive extensions.
\begin{exercise} Let us fix three dimension vectors $d,d',d''$.
\begin{enumerate}
\item Consider the following diagram, where the maps are given by mapping (successive) extensions to its subquotients or intermediate extensions and also by the identity on the factors $\mathfrak{M}$ not being part of an extension.
\[ \xymatrix { \mathfrak{M}_d\times \mathfrak{M}_{d'}\times \mathfrak{M}_{d''} & \mathfrak{M}_d\times \mathfrak{M}_{d',d''} \ar[l] \ar[r] & \mathfrak{M}_d\times \mathfrak{M}_{d'+d''} \\
\mathfrak{M}_{d,d'}\times \mathfrak{M}_{d''} \ar[u] \ar[d] & \mathfrak{M}_{d,d',d''} \ar[l] \ar[r] \ar[d]\ar[u]& \mathfrak{M}_{d,d'+d''} \ar[d]\ar[u]\\
\mathfrak{M}_{d+d'}\times \mathfrak{M}_{d''} & \mathfrak{M}_{d+d',d''} \ar[r] \ar[l] & \mathfrak{M}_{d+d'+d''} } \]
Show that the diagram commutes and that every square is cartesian.
\item Use this diagram and the base change property of a motivic theory to prove associativity of the Ringel--Hall product.
\item The zero representation induces a map $\Spec\CC\xrightarrow{0}\mathfrak{M}$. Show that $1_0:=0_!(1)\in R(\mathfrak{M})$ is a unit for the Ringel--Hall product.
\end{enumerate}
\end{exercise}
Let us form the following cartesian product
\[ \xymatrix @C=1.5cm { \mathfrak{M}^{\zeta-ss}_{\mu,\mu'} \ar@{^{(}->}[r] \ar[d] & \mathfrak{Exact} \ar[r]^{\pi_2} \ar[d]^{\pi_1\times \pi_3} & \mathfrak{M} \\
\mathfrak{M}^{\zeta-ss}_\mu\times\mathfrak{M}^{\zeta-ss}_{\mu'} \ar@{^{(}->}[r] & \mathfrak{M}\times \mathfrak{M}. & }
\]
If $\mu > \mu'$, the composition $\mathfrak{M}^{\zeta-ss}_{\mu,\mu'} \to \mathfrak{M}$ is an isomorphism onto the image which is the substack of $\mathfrak{M}$ consisting of all representations whose Harder--Narasimhan filtration has only one subquotient in $\mathfrak{M}^{\zeta-ss}_\mu$ and another one in $\mathfrak{M}^{\zeta-ss}_{\mu'}$. Indeed, the functoriality of the Harder--Narasimhan filtration ensures that taking the Harder--Narasimhan filtration provides an inverse morphism to $\mathfrak{M}^{\zeta-ss}_{\mu,\mu'} \to \mathfrak{M}$. Fixing another slope $\mu''$ with $\mu'> \mu''$ we continue this way and take the fiber product
\[ \xymatrix @C=1.5cm { \mathfrak{M}^{\zeta-ss}_{\mu,\mu',\mu''} \ar@{^{(}->}[r] \ar[d] & \mathfrak{Exact} \ar[r]^{\pi_2} \ar[d]^{\pi_1\times \pi_3} & \mathfrak{M} \\
\mathfrak{M}^{\zeta-ss}_{\mu,\mu'}\times\mathfrak{M}^{\zeta-ss}_{\mu''} \ar@{^{(}->}[r] & \mathfrak{M}\times \mathfrak{M}. & }
\]
which can be identified with the substack of representations having a Harder--Narasimhan filtration with subquotients in $\mathfrak{M}^{\zeta-ss}_\mu,\mathfrak{M}^{\zeta-ss}_{\mu'},\mathfrak{M}^{\zeta-ss}_{\mu''}$. As every quiver representation has a unique Harder--Narasimhan filtration, we obtain a locally finite stratification of $\mathfrak{M}\!\setminus\!\{0\}$ with strata $\mathfrak{M}^{\zeta-ss}_{\mu_1,\ldots,\mu_r}\!\!\setminus\!\{0\} \hookrightarrow \mathfrak{M}\!\setminus\!\{0\}$, where $\mathfrak{M}^{\zeta-ss}_{\mu_1,\ldots,\mu_r}$ is defined as above by means of $r-1$ fiber products for every strictly decreasing sequence $\mu_1> \ldots > \mu_r$ in $(-\infty,+\infty]$ of length $r$. Using the notation
\[ \delta^\zeta_{\mu_1,\ldots,\mu_r}:= (\mathfrak{M}^{\zeta-ss}_{\mu_1,\ldots,\mu_r}\hookrightarrow \mathfrak{M})_!(\mathbbm{1}_{\mathfrak{M}^{\zeta-ss}_{\mu_1,\ldots,\mu_r}}) \in R(\mathfrak{M}), \]
this stratification can be written as
\[ \mathbbm{1}_{\mathfrak{M}}=1_0 + \sum_{{0<r\in \mathbb{N}\atop \mu_1> \ldots > \mu_r}} (\delta^\zeta_{\mu_1,\ldots,\mu_r}-1_0).\]
\begin{exercise} Prove the formula $\delta^\zeta_{\mu_1,\ldots,\mu_r}=\delta_{\mu_1}^\zeta\ast \ldots \ast \delta_{\mu_r}^\zeta$ for all sequences $\mu_1>\ldots>\mu_r$ of real numbers. \end{exercise}
Applying the formula proven in the exercise we can rewrite the infinite sum as an infinite product
\begin{equation} \label{wall_crossing_1} \mathbbm{1}_{\mathfrak{M}}= \prod_{\mu \searrow}^\ast \delta^\zeta_\mu \end{equation}
which is well-defined as for every dimension vector only finitely many factors contribute. Note that the (infinite) Ringel--Hall product has to be taken in decreasing order of the slopes. If $\zeta'$ is another stability condition, we conclude the formula
\begin{equation} \label{wall_crossing_2} \prod_{\mu \searrow}^\ast \delta^\zeta_\mu=\prod_{\mu \searrow}^\ast \delta^{\zeta'}_\mu \end{equation}
which relates elements in $R(\mathfrak{M})$ defined by means of two different stability conditions $\zeta$ and $\zeta'$. In order to obtain a similar formula for Donaldson--Thomas functions, we need to related the Ringel--Hall algebra with corresponding objects on the coarse moduli space which will be the topic of the next subsection.
\subsection{Integration map}
As the Donaldson--Thomas function was an object defined on $\mathcal{M}^{\zeta-ss}$, we cannot compare Donaldson--Thomas functions taken with respect to different stability conditions as $\mathcal{M}^{\zeta-ss}$ might change. To make them comparable, we need to push them down along $q^\zeta$ to $\mathcal{M}^{ssimp}$ which is the ``smallest'' of all moduli spaces. Fix a stacky vanishing cycle $\phi$, and use the fact that $q^\zeta_!$ commutes with $\Sym$, that the open embedding $j:\mathfrak{M}^{\zeta-ss}_\mu\hookrightarrow \mathfrak{M}$ is smooth, and the projection formula to conclude
\begin{eqnarray*} \Sym\left(\frac{q^\zeta_! \mathcal{DT}(Q,W)^\zeta_\mu}{\mathbb{L}^{1/2}-\mathbb{L}^{-1/2}}\right) &=& q^\zeta_! \Sym\left(\frac{\mathcal{DT}(Q,W)^\zeta_\mu}{\mathbb{L}^{1/2}-\mathbb{L}^{-1/2}}\right) \\
&=& (q^\zeta\circ p^\zeta)_!\big(\phi_{\mathfrak{Tr}(W)^\zeta}(\mathcal{IC}_{\mathfrak{M}^{\zeta-ss}_\mu})\big)\\
&=& (p\circ j)_!\big(\phi_{\mathfrak{Tr}(W)^\zeta}(j^\ast\mathcal{IC}_{\mathfrak{M}})\big)\\
&=& p_! j_!\big(j^\ast\phi_{\mathfrak{Tr}(W)}(\mathcal{IC}_{\mathfrak{M}})\big)\\
&=& p_!\big(\delta^\zeta_\mu\cap \phi_{\mathfrak{Tr}(W)}(\mathcal{IC}_{\mathfrak{M}})\big).
\end{eqnarray*}
\begin{definition}
The $R(\Spec\CC)$-linear map \[I^W:R(\mathfrak{M}) \ni a\longmapsto p_!\big(a\cap \phi_{\mathfrak{Tr}(W)}(\mathcal{IC}_\mathfrak{M})\big)\in R(\mathcal{M}^{ssimp})\] is called integration map.
\end{definition}
Hence, we have proven
\[ I^W(\delta^\zeta_\mu)=\Sym\left(\frac{q^\zeta_! \mathcal{DT}(Q,W)^\zeta_\mu}{\mathbb{L}^{1/2}-\mathbb{L}^{-1/2}}\right).\]
This formula can also been used to define Donaldson--Thomas functions $\underline{\mathcal{DT}}(Q,W)^\zeta_\mu \in R(\mathcal{M}^{ssimp})$ by means of
\[ I^W(\delta^\zeta_\mu)=\Sym\left(\frac{\underline{\mathcal{DT}}(Q,W)^\zeta_\mu}{\mathbb{L}^{1/2}-\mathbb{L}^{-1/2}}\right)\]
if $\zeta$ is not geometric, i.e.\ $\mathfrak{M}^{\zeta-ss}$ has no coarse moduli space. If $\zeta$ is geometric, then $\underline{\mathcal{DT}}(Q,W)^\zeta_\mu=q^\zeta_! \mathcal{DT}(Q,W)^\zeta_\mu$.
We wish to apply $I^W$ to equation (\ref{wall_crossing_1}) or (\ref{wall_crossing_2}) to obtain a wall-crossing formula for Donaldson--Thomas functions $\underline{\mathcal{DT}}(Q,W)^\zeta_\mu$. Unfortunately, $I^W$ will not be an $R(\Spec\CC)$-algebra homomorphism from the Ringel--Hall algebra $(R(\mathfrak{M}),\ast)$ to $R(\mathcal{M}^{ssimp})$ with the convolution product.
\begin{definition} Define the ``quantum'' or deformed convolution product on $R(\mathcal{M}^{ssimp})$ by means of
\[ (a_d)_{d\in \mathbb{N}^{Q_0}}\ast (b_{d'})_{d'\in \mathbb{N}^{Q_0}} := \left(\sum_{d+d'=d''} \mathbb{L}^{\langle d,d'\rangle/2} a_db_{d'}\right)_{d''\in \mathbb{N}^{Q_0}}\]
and similarly on $R(\mathbb{N}^{Q_0})$.
\end{definition}
As $\dim:\mathcal{M}^{ssimp}\to \mathbb{N}^{Q_0}$ is a monoid homomorphisms, it will preserve the convolution and, hence, also the deformed convolution product. The main result about the integration map was essentially proven by Reineke in \cite{Reineke_HN} for $W=0$ and by Kontsevich and Soibelman in \cite{KS1} for general potential. A rigorous proof can also be found in \cite{DavisonMeinhardt3}
\begin{theorem} \label{gmebra_homomorphism} The map $I^W:(R(\mathfrak{M}),\ast) \longrightarrow (R(\mathcal{M}^{ssimp}),\ast)$ is a homomorphism of $R(\Spec\CC)$-algebras.
\end{theorem}
\subsection{The wall-crossing identity}
Let us assume the conditions of Theorem \ref{gmebra_homomorphism}. By applying the integration map $I^W$ to the equations (\ref{wall_crossing_1}) and (\ref{wall_crossing_2}), we finally get the wall-crossing identity
\[ I^W(\mathbbm{1}_\mathfrak{M})= \prod_{\mu \searrow}^\ast \Sym\left(\frac{\underline{\mathcal{DT}}(Q,W)^\zeta_\mu}{\mathbb{L}^{1/2}-\mathbb{L}^{-1/2}}\right)=\prod_{\mu \searrow}^\ast \Sym\left(\frac{\underline{\mathcal{DT}}(Q,W)^{\zeta'}_\mu}{\mathbb{L}^{1/2}-\mathbb{L}^{-1/2}}\right) \]
relating the Donaldson--Thomas functions $\underline{\mathcal{DT}}(Q,W)^\zeta$ and $\underline{\mathcal{DT}}(Q,W)^{\zeta'}$ of two stability conditions $\zeta$ and $\zeta'$. Since $\dim_!$ commutes with the deformed convolution product, we obtain the same formula for the Donaldson--Thomas invariants
\[ \dim_!I^W(\mathbbm{1}_\mathfrak{M})= \prod_{\mu \searrow}^\ast \Sym\left(\frac{\DT(Q,W)^\zeta_\mu}{\mathbb{L}^{1/2}-\mathbb{L}^{-1/2}}\right)=\prod_{\mu \searrow}^\ast \Sym\left(\frac{\DT(Q,W)^{\zeta'}_\mu}{\mathbb{L}^{1/2}-\mathbb{L}^{-1/2}}\right). \]
Let us illustrate this formula with an example.
\begin{example}[cf.\ Example \ref{Kronecker}] \rm
Consider the $A_2$-quiver $Q: \bullet_1 \longrightarrow \bullet_2$ with potential $W=0$ and the canonical vanishing cycle of $\underline{\Ka}_0(\Sch)[\mathbb{L}^{1/2}]^{st}$.
\begin{exercise} Show that every representation $V_1\xrightarrow{M}V_2$ of $Q$ is a direct sum of copies of $S_1=(\CC\xrightarrow{0}0), S_2=(0\xrightarrow{0}\CC)$ and $S_{12}=(\CC\xrightarrow{\id}\CC)$. \end{exercise}
Fix a stability condition $\zeta=(\zeta_1,\zeta_2)$ satisfying $\arg(\zeta_1)<\arg(\zeta_2)$. Given a $\zeta$-semistable representation $V\cong S_1^{m_1}\oplus S_{12}^{m_{12}}\oplus S_2^{m_2}$ with $d_1=m_1+m_{12}$ and $d_2=m_2+m_{12}$, two of the multiplicities $m_1,m_{12},m_2$ must be zero. Moreover, $m_{12}$ must be zero, too, since $S_2\hookrightarrow S_{12}$ destabilizes $S_{12}$ and the latter cannot be (semi)stable. Thus, $V\cong S_1^{d_1}$ or $V\cong S_2^{d_2}$. In particular, the category of representations of dimension vector $(d_1,0)$ respectively $(0,d_2)$ is isomorphic to the category of representations of the quiver $Q^{(0)}$ with one vertex and no loop. Using $R(\mathbb{N}^{Q_0})\cong R[[t_1,t_2]]$ we, therefore, obtain
\[ \dim_!I^W(\mathbbm{1}_\mathfrak{M})= A^{(0)}(t_2)\ast A^{(0)}(t_1)=\Sym\left(\frac{t_2}{\mathbb{L}^{1/2}-\mathbb{L}^{-1/2}}\right)\ast \Sym\left(\frac{t_1}{\mathbb{L}^{1/2}-\mathbb{L}^{-1/2}}\right) .\]
On the other hand, if we assume $\arg(\zeta_1)>\arg(\zeta_2)$, the representation $S_{12}$ is $\zeta$-stable, and $V\cong S_{12}^{d_1}$ is another class of semistable objects. Thus,
\begin{eqnarray*} \lefteqn{\dim_!I^W(\mathbbm{1}_\mathfrak{M})\;=\; A^{(0)}(t_1)\ast A^{(0)}(t_1t_2)\ast A^{(0)}(t_2) }\\ &=&\Sym\left(\frac{t_1}{\mathbb{L}^{1/2}-\mathbb{L}^{-1/2}}\right)\ast\Sym\left(\frac{t_1t_2}{\mathbb{L}^{1/2}-\mathbb{L}^{-1/2}}\right)\ast \Sym\left(\frac{t_2}{\mathbb{L}^{1/2}-\mathbb{L}^{-1/2}}\right),\end{eqnarray*}
and the so-called quantum dilogarithm identity \[ A^{(0)}(t_2)\ast A^{(0)}(t_1)=A^{(0)}(t_1)\ast A^{(0)}(t_1t_2)\ast A^{(0)}(t_2)\] follows.
Let us also consider the case $\arg(\zeta_1)=\arg(\zeta_2)$. Then, all representations are semistable, and
\[ \dim_!I_W(\mathbbm{1}_\mathfrak{M})=\Sym\left(\frac{\sum_{(d_1,d_2)\not=(0,0)} \DT(Q)^\zeta_{(d_1,d_2)}\,t_1^{d_1}t_2^{d_2}}{\mathbb{L}^{1/2}-\mathbb{L}^{-1/2}}\right) \]
allows the computation of the Donaldson--Thomas invariants as the left hand side of the equation is already known by the previous two cases. For example, comparing the coefficients of $t_1t_2$ yields
\[ \frac{\mathbb{L}^{1/2}}{(\mathbb{L}^{1/2}-\mathbb{L}^{-1/2})^2}=\frac{\DT(Q)^\zeta_{(1,1)}}{\mathbb{L}^{1/2}-\mathbb{L}^{-1/2}}, \]
and $\DT(Q)^\zeta_{(1,1)}=\mathbb{L}/(\mathbb{L}-1)$ follows. Note that $\DT(Q)^\zeta_{(1,1)}\in \Ka_0(\Sch_\CC)[\mathbb{L}^{1/2}]^{st}$ cannot be lifted under the map $\Ka_0(\Sch_\CC)[\mathbb{L}^{-1/2}] \longrightarrow \Ka_0(\Sch_\CC)[\mathbb{L}^{1/2}]^{st}$ which does not contradict Theorem \ref{main_result_1} as $\zeta$ is not generic. In particular, the assumption of being generic cannot be dropped in Theorem \ref{main_result_1} and \ref{main_result_2}. \\
Let us finally say some words about the Donaldson--Thomas functions $\mathcal{DT}(Q)^\zeta_d$. For arbitrary stability condition $\zeta$ there is a unique polystable object of given dimension vector $d$ as the previous discussion shows. In particular, every stability condition is geometric with $\mathcal{M}^{\zeta-ss}_d=\mathcal{M}^{ssimp}_d=\Spec\CC$, and $\mathcal{DT}(Q)^\zeta_d=\DT(Q)^\zeta_d$ follows for all $d\in \mathbb{N}^{Q_0}$.
\end{example}
\bibliographystyle{plain}
| {
"redpajama_set_name": "RedPajamaArXiv"
} | 9,447 |
Q: Weyl group of $F_4$ and its degree $4$ representations The Weyl group of $ F_4 $
$$
W(F_4)
$$
is a solvable group of order 1152. $ W(F_4) $ can be realized as the symmetries of the 24-cell, which is a certain convex regular 4-polytope. Thus $ W(F_4) $ is a subgroup of $ O_4(\mathbb{R}) $.
Are there other degree 4 representations of $ W(F_4) $? For example is there a $ W(F_4) $ subgroup of $ SU_4 $? Is there a $ W(F_4) $ subgroup of $ PU_4 $ (i.e. a faithful projective degree 4 representation of $ W(F_4) $)?
A: It appears that while $ W(F_4) $ is a subgroup of $ O_4(\mathbb{R}) $ it is not a subgroup of $ SO_4(\mathbb{R}) $ or $ SU_4 $ or $ PU_4 $. That is, it has no faithful determinant $ 1 $, degree $ 4 $ irreps and also it has no degree $ 4
$ projective irreps.
The one primitive subgroup of $ SU_4 $ that has the right order to be a central extension of $ W(F_4) $ corresponding to a projective representation is group 21 from https://arxiv.org/abs/hep-th/9905212 but one can directly verify, using GAP say, that this group is not a central extension of $ W(F_4) $.
It is interesting to note that since $ W(F_4) \subseteq O_4(\mathbb{R}) \subset SO_6(\mathbb{R}) $ then we can lift $ W(F_4) $ through the double cover $ SU_4 \to SO_6(\mathbb{R}) $ to get a subgroup of $ SU_4 $ of order $ 2(1152)=4(576) $. This corresponds to a degree 4 projective irrep of the direct product $ S_4 \times S_4 $ (recall that $ W(F_4) $ is a central extension by cyclic 2 of $ S_4 \times S_4 $). This does not give an actual degree 4 projective irrep of $ W(F_4) $ but its basically the closest we can get.
As stated in the comments, $ W(F_4) $ does have four different faithful 4d irreps mapping into $ O_4(\mathbb{R}) $ (i.e. irreps of $ + $ type, meaning Frobenius-Schur indicator $ +1 $). These all correspond to nonconjugate $ W(F_4) $ subgroups of $ O_4(\mathbb{R}) $. Don't really know anything else about them though.
| {
"redpajama_set_name": "RedPajamaStackExchange"
} | 3,353 |
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